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233 changed files (+28428/-1282)
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@@ -0,0 +1,62 @@import data.nat.basic import data.nat.parity import tactic open nat /- These are pieces of data. -/ #check 2 + 2 def f (x : ℕ) := x + 3 #check f /- These are propositions, of type `Prop`. -/ #check 2 + 2 = 4 def fermat_last_theorem := ∀ x y z n : ℕ, n > 2 ∧ x * y * z ≠ 0 → x^n + y^n ≠ z^n #check fermat_last_theorem /- These are proofs of propositions. -/ theorem easy : 2 + 2 = 4 := rfl #check easy theorem hard : fermat_last_theorem := sorry #check hard /- Here are some proofs. -/ example : ∀ m n : nat, even n → even (m * n) := assume m n ⟨k, (hk : n = k + k)⟩, have hmn : m * n = m * k + m * k, by rw [hk, mul_add], show ∃ l, m * n = l + l, from ⟨_, hmn⟩ example : ∀ m n : nat, even n → even (m * n) := λ m n ⟨k, hk⟩, ⟨m * k, by rw [hk, mul_add]⟩ example : ∀ m n : nat, even n → even (m * n) := begin -- say m and n are natural numbers, and assume n=2*k rintros m n ⟨k, hk⟩, -- We need to prove m*n is twice a natural. Let's show it's twice m*k. use m * k, -- substitute in for n rw hk, -- and now it's obvious ring end example : ∀ m n : nat, even n → even (m * n) := by { rintros m n ⟨k, hk⟩, use m * k, rw hk, ring } example : ∀ m n : nat, even n → even (m * n) := by intros; simp * with parity_simps
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@@ -0,0 +1,6 @@import data.nat.basic import data.nat.parity import tactic open nat -- There are no exercises in this section.
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@@ -0,0 +1,186 @@import data.real.basic /- An example. -/ import data.real.basic example (a b c : ℝ) : (a * b) * c = b * (a * c) := begin rw mul_comm a b, rw mul_assoc b a c end /- Try these.-/ example (a b c : ℝ) : (c * b) * a = b * (a * c) := begin sorry end example (a b c : ℝ) : a * (b * c) = b * (a * c) := begin sorry end /- An example. -/ example (a b c : ℝ) : a * b * c = b * c * a := begin rw mul_assoc, rw mul_comm end /- Try doing the first of these without providing any arguments at all, and the second with only one argument. -/ example (a b c : ℝ) : a * (b * c) = b * (c * a) := begin sorry end example (a b c : ℝ) : a * (b * c) = b * (a * c) := begin sorry end /- Using facts from the local context. -/ example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := begin rw h', rw ←mul_assoc, rw h, rw mul_assoc end /- Try these. For the second one, use the theorem `sub_self`. -/ example (a b c d e f : ℝ) (h : b * c = e * f) : a * b * c * d = a * e * f * d := begin sorry end example (a b c d : ℝ) (hyp : c = b * a - d) (hyp' : d = a * b) : c = 0 := begin sorry end /- Examples. -/ example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by rw [h', ←mul_assoc, h, mul_assoc] section variables a b c d e f g : ℝ example (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by rw [h', ←mul_assoc, h, mul_assoc] end section variables a b c : ℝ #check a #check a + b #check (a : ℝ) #check mul_comm a b #check (mul_comm a b : a * b = b * a) #check mul_assoc c a b #check mul_comm a #check mul_comm #check @mul_comm end section variables a b : ℝ example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := begin rw [mul_add, add_mul, add_mul], rw [←add_assoc, add_assoc (a * a)], rw [mul_comm b a, ←two_mul] end example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := calc (a + b) * (a + b) = a * a + b * a + (a * b + b * b) : by rw [mul_add, add_mul, add_mul] ... = a * a + (b * a + a * b) + b * b : by rw [←add_assoc, add_assoc (a * a)] ... = a * a + 2 * (a * b) + b * b : by rw [mul_comm b a, ←two_mul] example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := calc (a + b) * (a + b) = a * a + b * a + (a * b + b * b) : begin sorry end ... = a * a + (b * a + a * b) + b * b : by sorry ... = a * a + 2 * (a * b) + b * b : by sorry end /- Try these. For the second, use the theorems listed underneath. -/ section variables a b c d : ℝ example : (a + b) * (c + d) = a * c + a * d + b * c + b * d := sorry example (a b : ℝ) : (a + b) * (a - b) = a^2 - b^2 := begin sorry end #check pow_two a #check mul_sub a b c #check add_mul a b c #check add_sub a b c #check sub_sub a b c #check add_zero a end /- Examples. -/ section variables a b c d : ℝ example (a b c d : ℝ) (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := begin rw hyp' at hyp, rw mul_comm d a at hyp, rw ← two_mul (a * d) at hyp, rw ← mul_assoc 2 a d at hyp, exact hyp end example : (c * b) * a = b * (a * c) := by ring example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by ring example : (a + b) * (a - b) = a^2 - b^2 := by ring example (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := begin rw [hyp, hyp'], ring end end
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@@ -0,0 +1,157 @@import algebra.ring import data.real.basic import tactic section variables (R : Type*) [ring R] #check (add_assoc : ∀ a b c : R, a + b + c = a + (b + c)) #check (add_comm : ∀ a b : R, a + b = b + a) #check (zero_add : ∀ a : R, 0 + a = a) #check (add_left_neg : ∀ a : R, -a + a = 0) #check (mul_assoc : ∀ a b c : R, a * b * c = a * (b * c)) #check (mul_one : ∀ a : R, a * 1 = a) #check (one_mul : ∀ a : R, 1 * a = a) #check (mul_add : ∀ a b c : R, a * (b + c) = a * b + a * c) #check (add_mul : ∀ a b c : R, (a + b) * c = a * c + b * c) end section variables (R : Type*) [comm_ring R] variables a b c d : R example : (c * b) * a = b * (a * c) := by ring example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by ring example : (a + b) * (a - b) = a^2 - b^2 := by ring example (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := begin rw [hyp, hyp'], ring end end namespace my_ring variables {R : Type*} [ring R] theorem add_zero (a : R) : a + 0 = a := by rw [add_comm, zero_add] theorem add_right_neg (a : R) : a + -a = 0 := by rw [add_comm, add_left_neg] #check @my_ring.add_zero #check @add_zero end my_ring namespace my_ring variables {R : Type*} [ring R] theorem neg_add_cancel_left (a b : R) : -a + (a + b) = b := by rw [←add_assoc, add_left_neg, zero_add] /- Prove these: -/ theorem add_neg_cancel_right (a b : R) : (a + b) + -b = a := sorry theorem add_left_cancel {a b c : R} (h : a + b = a + c) : b = c := sorry theorem add_right_cancel {a b c : R} (h : a + b = c + b) : a = c := sorry theorem mul_zero (a : R) : a * 0 = 0 := begin have h : a * 0 + a * 0 = a * 0 + 0, { rw [←mul_add, add_zero, add_zero] }, rw add_left_cancel h end theorem zero_mul (a : R) : 0 * a = 0 := sorry theorem neg_eq_of_add_eq_zero {a b : R} (h : a + b = 0) : -a = b := sorry theorem eq_neg_of_add_eq_zero {a b : R} (h : a + b = 0) : a = -b := sorry theorem neg_zero : (-0 : R) = 0 := begin apply neg_eq_of_add_eq_zero, rw add_zero end theorem neg_neg (a : R) : -(-a) = a := sorry end my_ring /- Examples. -/ section variables {R : Type*} [ring R] example (a b : R) : a - b = a + -b := sub_eq_add_neg a b end example (a b : ℝ) : a - b = a + -b := rfl example (a b : ℝ) : a - b = a + -b := by reflexivity namespace my_ring variables {R : Type*} [ring R] theorem self_sub (a : R) : a - a = 0 := sorry lemma one_add_one_eq_two : 1 + 1 = (2 : R) := by refl theorem two_mul (a : R) : 2 * a = a + a := sorry end my_ring section variables (A : Type*) [add_group A] #check (add_assoc : ∀ a b c : A, a + b + c = a + (b + c)) #check (zero_add : ∀ a : A, 0 + a = a) #check (add_left_neg : ∀ a : A, -a + a = 0) end section variables {G : Type*} [group G] #check (mul_assoc : ∀ a b c : G, a * b * c = a * (b * c)) #check (one_mul : ∀ a : G, 1 * a = a) #check (mul_left_inv : ∀ a : G, a⁻¹ * a = 1) namespace my_group theorem mul_right_inv (a : G) : a * a⁻¹ = 1 := sorry theorem mul_one (a : G) : a * 1 = a := sorry theorem mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a ⁻¹ := sorry end my_group end
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@@ -0,0 +1,159 @@import analysis.special_functions.log.basic variables a b c d e : ℝ open real #check (le_refl : ∀ a : ℝ, a ≤ a) #check (le_trans : a ≤ b → b ≤ c → a ≤ c) section variables (h : a ≤ b) (h' : b ≤ c) #check (le_refl : ∀ a : real, a ≤ a) #check (le_refl a : a ≤ a) #check (le_trans : a ≤ b → b ≤ c → a ≤ c) #check (le_trans h : b ≤ c → a ≤ c) #check (le_trans h h' : a ≤ c) end example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := begin apply le_trans, { apply h₀ }, apply h₁ end example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := begin apply le_trans h₀, apply h₁ end example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := by exact le_trans h₀ h₁ example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := le_trans h₀ h₁ example (x : ℝ) : x ≤ x := by apply le_refl example (x : ℝ) : x ≤ x := by exact le_refl x example (x : ℝ) : x ≤ x := le_refl x #check (le_refl : ∀ a, a ≤ a) #check (le_trans : a ≤ b → b ≤ c → a ≤ c) #check (lt_of_le_of_lt : a ≤ b → b < c → a < c) #check (lt_of_lt_of_le : a < b → b ≤ c → a < c) #check (lt_trans : a < b → b < c → a < c) /- Try this. -/ example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := sorry example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by linarith section example (h : 2 * a ≤ 3 * b) (h' : 1 ≤ a) (h'' : d = 2) : d + a ≤ 5 * b := by linarith end example (h : 1 ≤ a) (h' : b ≤ c) : 2 + a + exp b ≤ 3 * a + exp c := by linarith [exp_le_exp.mpr h'] #check (exp_le_exp : exp a ≤ exp b ↔ a ≤ b) #check (exp_lt_exp : exp a < exp b ↔ a < b) #check (log_le_log : 0 < a → 0 < b → (log a ≤ log b ↔ a ≤ b)) #check (log_lt_log : 0 < a → a < b → log a < log b) #check (add_le_add : a ≤ b → c ≤ d → a + c ≤ b + d) #check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b) #check (add_le_add_right : a ≤ b → ∀ c, a + c ≤ b + c) #check (add_lt_add_of_le_of_lt : a ≤ b → c < d → a + c < b + d) #check (add_lt_add_of_lt_of_le : a < b → c ≤ d → a + c < b + d) #check (add_lt_add_left : a < b → ∀ c, c + a < c + b) #check (add_lt_add_right : a < b → ∀ c, a + c < b + c) #check (add_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a + b) #check (add_pos : 0 < a → 0 < b → 0 < a + b) #check (add_pos_of_pos_of_nonneg : 0 < a → 0 ≤ b → 0 < a + b) #check (exp_pos : ∀ a, 0 < exp a) #check @add_le_add_left example (h : a ≤ b) : exp a ≤ exp b := begin rw exp_le_exp, exact h end example (h₀ : a ≤ b) (h₁ : c < d) : a + exp c + e < b + exp d + e := begin apply add_lt_add_of_lt_of_le, { apply add_lt_add_of_le_of_lt h₀, apply exp_lt_exp.mpr h₁ }, apply le_refl end example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := begin sorry end example : (0 : ℝ) < 1 := by norm_num example (h : a ≤ b) : log (1 + exp a) ≤ log (1 + exp b) := begin have h₀ : 0 < 1 + exp a, { sorry }, have h₁ : 0 < 1 + exp b, { sorry }, apply (log_le_log h₀ h₁).mpr, sorry end example : 0 ≤ a^2 := begin -- library_search, exact pow_two_nonneg a end example (h : a ≤ b) : c - exp b ≤ c - exp a := sorry example : 2*a*b ≤ a^2 + b^2 := begin have h : 0 ≤ a^2 - 2*a*b + b^2, calc a^2 - 2*a*b + b^2 = (a - b)^2 : by ring ... ≥ 0 : by apply pow_two_nonneg, calc 2*a*b = 2*a*b + 0 : by ring ... ≤ 2*a*b + (a^2 - 2*a*b + b^2) : add_le_add (le_refl _) h ... = a^2 + b^2 : by ring end example : 2*a*b ≤ a^2 + b^2 := begin have h : 0 ≤ a^2 - 2*a*b + b^2, calc a^2 - 2*a*b + b^2 = (a - b)^2 : by ring ... ≥ 0 : by apply pow_two_nonneg, linarith end example : abs (a*b) ≤ (a^2 + b^2) / 2 := sorry #check abs_le'.mpr
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@@ -0,0 +1,93 @@import data.real.basic section variables a b c d : ℝ #check (min_le_left a b : min a b ≤ a) #check (min_le_right a b : min a b ≤ b) #check (le_min : c ≤ a → c ≤ b → c ≤ min a b) example : min a b = min b a := begin apply le_antisymm, { show min a b ≤ min b a, apply le_min, { apply min_le_right }, apply min_le_left }, { show min b a ≤ min a b, apply le_min, { apply min_le_right }, apply min_le_left } end example : min a b = min b a := begin have h : ∀ x y, min x y ≤ min y x, { intros x y, apply le_min, apply min_le_right, apply min_le_left }, apply le_antisymm, apply h, apply h end example : min a b = min b a := begin apply le_antisymm, repeat { apply le_min, apply min_le_right, apply min_le_left } end example : max a b = max b a := sorry example : min (min a b) c = min a (min b c) := sorry lemma aux : min a b + c ≤ min (a + c) (b + c) := sorry example : min a b + c = min (a + c) (b + c) := sorry #check (abs_add : ∀ a b : ℝ, abs (a + b) ≤ abs a + abs b) example : abs a - abs b ≤ abs (a - b) := sorry end section variables w x y z : ℕ example (h₀ : x ∣ y) (h₁ : y ∣ z) : x ∣ z := dvd_trans h₀ h₁ example : x ∣ y * x * z := begin apply dvd_mul_of_dvd_left, apply dvd_mul_left end example : x ∣ x^2 := by apply dvd_mul_right example (h : x ∣ w) : x ∣ y * (x * z) + x^2 + w^2 := sorry end section variables m n : ℕ open nat #check (gcd_zero_right n : gcd n 0 = n) #check (gcd_zero_left n : gcd 0 n = n) #check (lcm_zero_right n : lcm n 0 = 0) #check (lcm_zero_left n : lcm 0 n = 0) example : gcd m n = gcd n m := sorry end
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@@ -0,0 +1,99 @@import topology.metric_space.basic section variables {α : Type*} [partial_order α] variables x y z : α #check x ≤ y #check (le_refl x : x ≤ x) #check (le_trans : x ≤ y → y ≤ z → x ≤ z) #check x < y #check (lt_irrefl x : ¬ x < x) #check (lt_trans : x < y → y < z → x < z) #check (lt_of_le_of_lt : x ≤ y → y < z → x < z) #check (lt_of_lt_of_le : x < y → y ≤ z → x < z) example : x < y ↔ x ≤ y ∧ x ≠ y := lt_iff_le_and_ne end section variables {α : Type*} [lattice α] variables x y z : α #check x ⊓ y #check (inf_le_left : x ⊓ y ≤ x) #check (inf_le_right : x ⊓ y ≤ y) #check (le_inf : z ≤ x → z ≤ y → z ≤ x ⊓ y) #check x ⊔ y #check (le_sup_left : x ≤ x ⊔ y) #check (le_sup_right: y ≤ x ⊔ y) #check (sup_le : x ≤ z → y ≤ z → x ⊔ y ≤ z) example : x ⊓ y = y ⊓ x := sorry example : x ⊓ y ⊓ z = x ⊓ (y ⊓ z) := sorry example : x ⊔ y = y ⊔ x := sorry example : x ⊔ y ⊔ z = x ⊔ (y ⊔ z) := sorry theorem absorb1 : x ⊓ (x ⊔ y) = x := sorry theorem absorb2 : x ⊔ (x ⊓ y) = x := sorry end section variables {α : Type*} [distrib_lattice α] variables x y z : α #check (inf_sup_left : x ⊓ (y ⊔ z) = (x ⊓ y) ⊔ (x ⊓ z)) #check (inf_sup_right : (x ⊔ y) ⊓ z = (x ⊓ z) ⊔ (y ⊓ z)) #check (sup_inf_left : x ⊔ (y ⊓ z) = (x ⊔ y) ⊓ (x ⊔ z)) #check (sup_inf_right : (x ⊓ y) ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) end section variables {α : Type*} [lattice α] variables a b c : α example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = (x ⊓ y) ⊔ (x ⊓ z)) : a ⊔ (b ⊓ c) = (a ⊔ b) ⊓ (a ⊔ c) := sorry example (h : ∀ x y z : α, x ⊔ (y ⊓ z) = (x ⊔ y) ⊓ (x ⊔ z)) : a ⊓ (b ⊔ c) = (a ⊓ b) ⊔ (a ⊓ c) := sorry end section variables {R : Type*} [ordered_ring R] variables a b c : R #check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b) #check (mul_pos : 0 < a → 0 < b → 0 < a * b) #check (mul_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a * b) example : a ≤ b → 0 ≤ b - a := sorry example : 0 ≤ b - a → a ≤ b := sorry example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := sorry end section variables {X : Type*} [metric_space X] variables x y z : X #check (dist_self x : dist x x = 0) #check (dist_comm x y : dist x y = dist y x) #check (dist_triangle x y z : dist x z ≤ dist x y + dist y z) example (x y : X) : 0 ≤ dist x y := sorry end
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@@ -0,0 +1,45 @@import data.real.basic example (a b c : ℝ) : (c * b) * a = b * (a * c) := begin rw mul_comm c b, rw mul_assoc b c a, rw mul_comm c a end example (a b c : ℝ) : a * (b * c) = b * (a * c) := begin rw ←mul_assoc a b c, rw mul_comm a b, rw mul_assoc b a c end example (a b c : ℝ) : a * (b * c) = b * (c * a) := begin rw mul_comm, rw mul_assoc end example (a b c : ℝ) : a * (b * c) = b * (a * c) := begin rw ←mul_assoc, rw mul_comm a, rw mul_assoc end example (a b c d e f : ℝ) (h : b * c = e * f) : a * b * c * d = a * e * f * d := begin rw mul_assoc a, rw h, rw ←mul_assoc end example (a b c d : ℝ) (hyp : c = b * a - d) (hyp' : d = a * b) : c = 0 := begin rw hyp, rw hyp', rw mul_comm, rw sub_self end
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@@ -0,0 +1,84 @@import algebra.ring import data.real.basic import tactic namespace my_ring variables {R : Type*} [ring R] theorem add_neg_cancel_right (a b : R) : (a + b) + -b = a := by rw [add_assoc, add_right_neg, add_zero] theorem add_left_cancel {a b c : R} (h : a + b = a + c) : b = c := by rw [←neg_add_cancel_left a b, h, neg_add_cancel_left] theorem add_right_cancel {a b c : R} (h : a + b = c + b) : a = c := by rw [←add_neg_cancel_right a b, h, add_neg_cancel_right] theorem zero_mul (a : R) : 0 * a = 0 := begin have h : 0 * a + 0 * a = 0 * a + 0, { rw [←add_mul, add_zero, add_zero] }, rw add_left_cancel h end theorem neg_eq_of_add_eq_zero {a b : R} (h : a + b = 0) : -a = b := by rw [←neg_add_cancel_left a b, h, add_zero] theorem eq_neg_of_add_eq_zero {a b : R} (h : a + b = 0) : a = -b := begin symmetry, apply neg_eq_of_add_eq_zero, rw [add_comm, h] end theorem neg_zero : (-0 : R) = 0 := begin apply neg_eq_of_add_eq_zero, rw add_zero end theorem neg_neg (a : R) : -(-a) = a := begin apply neg_eq_of_add_eq_zero, rw add_left_neg end end my_ring namespace my_ring variables {R : Type*} [ring R] theorem self_sub (a : R) : a - a = 0 := by rw [sub_eq_add_neg, add_right_neg] lemma one_add_one_eq_two : 1 + 1 = (2 : R) := by refl theorem two_mul (a : R) : 2 * a = a + a := by rw [←one_add_one_eq_two, add_mul, one_mul] end my_ring section variables {G : Type*} [group G] namespace my_group theorem mul_right_inv (a : G) : a * a⁻¹ = 1 := begin have h : (a * a⁻¹)⁻¹ * ((a * a⁻¹) * (a * a⁻¹)) = 1, { rw [mul_assoc, ←mul_assoc a⁻¹ a, mul_left_inv, one_mul, mul_left_inv] }, rw [←h, ←mul_assoc, mul_left_inv, one_mul] end theorem mul_one (a : G) : a * 1 = a := by rw [←mul_left_inv a, ←mul_assoc, mul_right_inv, one_mul] theorem mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a ⁻¹ := by rw [←one_mul (b⁻¹ * a⁻¹), ←mul_left_inv (a * b), mul_assoc, mul_assoc, ←mul_assoc b b⁻¹, mul_right_inv, one_mul, mul_right_inv, mul_one] end my_group end
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@@ -0,0 +1,79 @@import analysis.special_functions.log.basic variables a b c d e : ℝ open real example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := begin apply lt_of_le_of_lt h₀, apply lt_trans h₁, exact lt_of_le_of_lt h₂ h₃ end example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := begin apply add_le_add_left, rw exp_le_exp, apply add_le_add_left h₀ end -- an alterantive using `linarith`. example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := begin have : exp (a + d) ≤ exp (a + e), { rw exp_le_exp, linarith }, linarith [this] end example (h : a ≤ b) : log (1 + exp a) ≤ log (1 + exp b) := begin have h₀ : 0 < 1 + exp a, { linarith [exp_pos a]}, have h₁ : 0 < 1 + exp b, { linarith [exp_pos b] }, apply (log_le_log h₀ h₁).mpr, apply add_le_add_left (exp_le_exp.mpr h), end -- SOLUTION. example (h : a ≤ b) : c - exp b ≤ c - exp a := begin apply sub_le_sub_left, exact exp_le_exp.mpr h end -- alternatively: example (h : a ≤ b) : c - exp b ≤ c - exp a := by linarith [exp_le_exp.mpr h] theorem fact1 : a*b*2 ≤ a^2 + b^2 := begin have h : 0 ≤ a^2 - 2*a*b + b^2, calc a^2 - 2*a*b + b^2 = (a - b)^2 : by ring ... ≥ 0 : by apply pow_two_nonneg, linarith end theorem fact2 : -(a*b)*2 ≤ a^2 + b^2 := begin have h : 0 ≤ a^2 + 2*a*b + b^2, calc a^2 + 2*a*b + b^2 = (a + b)^2 : by ring ... ≥ 0 : by apply pow_two_nonneg, linarith end example : abs (a*b) ≤ (a^2 + b^2) / 2 := begin have h : (0 : ℝ) < 2, { norm_num }, apply abs_le'.mpr, split, { rw le_div_iff h, apply fact1 }, rw le_div_iff h, apply fact2, end
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@@ -0,0 +1,134 @@import data.real.basic section variables a b c d : ℝ #check (min_le_left a b : min a b ≤ a) #check (min_le_right a b : min a b ≤ b) #check (le_min : c ≤ a → c ≤ b → c ≤ min a b) example : max a b = max b a := begin apply le_antisymm, repeat { apply max_le, apply le_max_right, apply le_max_left } end example : min (min a b) c = min a (min b c) := begin apply le_antisymm, { apply le_min, { apply le_trans, apply min_le_left, apply min_le_left }, apply le_min, { apply le_trans, apply min_le_left, apply min_le_right }, apply min_le_right }, apply le_min, { apply le_min, { apply min_le_left }, apply le_trans, apply min_le_right, apply min_le_left }, apply le_trans, apply min_le_right, apply min_le_right end lemma aux : min a b + c ≤ min (a + c) (b + c) := begin apply le_min, { apply add_le_add_right, apply min_le_left }, apply add_le_add_right, apply min_le_right end example : min a b + c = min (a + c) (b + c) := begin apply le_antisymm, { apply aux }, have h : min (a + c) (b + c) = min (a + c) (b + c) - c + c, { rw sub_add_cancel }, rw h, apply add_le_add_right, rw sub_eq_add_neg, apply le_trans, apply aux, rw [add_neg_cancel_right, add_neg_cancel_right] end example : abs a - abs b ≤ abs (a - b) := calc abs a - abs b = abs (a - b + b) - abs b : by rw sub_add_cancel ... ≤ abs (a - b) + abs b - abs b : begin apply sub_le_sub_right, apply abs_add end ... ≤ abs (a - b) : by rw add_sub_cancel -- alternatively example : abs a - abs b ≤ abs (a - b) := begin have h := abs_add (a - b) b, rw sub_add_cancel at h, linarith end end section variables w x y z : ℕ example (h₀ : x ∣ y) (h₁ : y ∣ z) : x ∣ z := dvd_trans h₀ h₁ example : x ∣ y * x * z := begin apply dvd_mul_of_dvd_left, apply dvd_mul_left end example : x ∣ x^2 := by apply dvd_mul_right example (h : x ∣ w) : x ∣ y * (x * z) + x^2 + w^2 := begin apply dvd_add, { apply dvd_add, { apply dvd_mul_of_dvd_right, apply dvd_mul_right }, apply dvd_mul_right }, rw pow_two, apply dvd_mul_of_dvd_right, exact h end end section variables m n : ℕ open nat #check (gcd_zero_right n : gcd n 0 = n) #check (gcd_zero_left n : gcd 0 n = n) #check (lcm_zero_right n : lcm n 0 = 0) #check (lcm_zero_left n : lcm 0 n = 0) example : gcd m n = gcd n m := begin apply dvd_antisymm, repeat { apply dvd_gcd, apply gcd_dvd_right, apply gcd_dvd_left } end end
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@@ -0,0 +1,159 @@import topology.metric_space.basic section variables {α : Type*} [lattice α] variables x y z : α example : x ⊓ y = y ⊓ x := begin apply le_antisymm, repeat { apply le_inf, { apply inf_le_right }, apply inf_le_left } end example : x ⊓ y ⊓ z = x ⊓ (y ⊓ z) := begin apply le_antisymm, { apply le_inf, { apply le_trans, apply inf_le_left, apply inf_le_left }, apply le_inf, { apply le_trans, apply inf_le_left, apply inf_le_right }, apply inf_le_right }, apply le_inf, { apply le_inf, { apply inf_le_left }, apply le_trans, apply inf_le_right, apply inf_le_left }, apply le_trans, apply inf_le_right, apply inf_le_right end example : x ⊔ y = y ⊔ x := begin apply le_antisymm, repeat { apply sup_le, { apply le_sup_right }, apply le_sup_left } end example : x ⊔ y ⊔ z = x ⊔ (y ⊔ z) := begin apply le_antisymm, { apply sup_le, { apply sup_le, apply le_sup_left, { apply le_trans, apply @le_sup_left _ _ y z, apply le_sup_right } }, apply le_trans, apply @le_sup_right _ _ y z, apply le_sup_right }, apply sup_le, { apply le_trans, apply @le_sup_left _ _ x y, apply le_sup_left }, apply sup_le, { apply le_trans, apply @le_sup_right _ _ x y, apply le_sup_left }, apply le_sup_right end theorem absorb1 : x ⊓ (x ⊔ y) = x := begin apply le_antisymm, { apply inf_le_left }, apply le_inf, { apply le_refl }, apply le_sup_left end theorem absorb2 : x ⊔ (x ⊓ y) = x := begin apply le_antisymm, { apply sup_le, { apply le_refl }, apply inf_le_left }, apply le_sup_left end end section variables {α : Type*} [distrib_lattice α] variables x y z : α #check (inf_sup_left : x ⊓ (y ⊔ z) = (x ⊓ y) ⊔ (x ⊓ z)) #check (inf_sup_right : (x ⊔ y) ⊓ z = (x ⊓ z) ⊔ (y ⊓ z)) #check (sup_inf_left : x ⊔ (y ⊓ z) = (x ⊔ y) ⊓ (x ⊔ z)) #check (sup_inf_right : (x ⊓ y) ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) end section variables {α : Type*} [lattice α] variables a b c : α example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = (x ⊓ y) ⊔ (x ⊓ z)) : a ⊔ (b ⊓ c) = (a ⊔ b) ⊓ (a ⊔ c) := by rw [h, @inf_comm _ _ (a ⊔ b), absorb1, @inf_comm _ _ (a ⊔ b), h, ←sup_assoc, @inf_comm _ _ c a, absorb2, inf_comm] example (h : ∀ x y z : α, x ⊔ (y ⊓ z) = (x ⊔ y) ⊓ (x ⊔ z)) : a ⊓ (b ⊔ c) = (a ⊓ b) ⊔ (a ⊓ c) := by rw [h, @sup_comm _ _ (a ⊓ b), absorb2, @sup_comm _ _ (a ⊓ b), h, ←inf_assoc, @sup_comm _ _ c a, absorb1, sup_comm] end section variables {R : Type*} [ordered_ring R] variables a b c : R theorem aux1 : a ≤ b → 0 ≤ b - a := begin intro h, rw [←sub_self a, sub_eq_add_neg, sub_eq_add_neg, add_comm, add_comm b], apply add_le_add_left h end theorem aux2 : 0 ≤ b - a → a ≤ b := begin intro h, rw [←add_zero a, ←sub_add_cancel b a, add_comm (b - a)], apply add_le_add_left h end example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := begin have h1 : 0 ≤ (b - a) * c, { exact mul_nonneg (aux1 _ _ h) h' }, rw sub_mul at h1, exact aux2 _ _ h1 end end section variables {X : Type*} [metric_space X] variables x y z : X example (x y : X) : 0 ≤ dist x y := begin have : 0 ≤ dist x y + dist y x, { rw [←dist_self x], apply dist_triangle }, linarith [dist_comm x y] end end
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@@ -0,0 +1,188 @@import data.real.basic #check ∀ x : ℝ, 0 ≤ x → abs x = x #check ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε lemma my_lemma : ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := sorry section variables a b δ : ℝ variables (h₀ : 0 < δ) (h₁ : δ ≤ 1) variables (ha : abs a < δ) (hb : abs b < δ) #check my_lemma a b δ #check my_lemma a b δ h₀ h₁ #check my_lemma a b δ h₀ h₁ ha hb end lemma my_lemma2 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := sorry section variables a b δ : ℝ variables (h₀ : 0 < δ) (h₁ : δ ≤ 1) variables (ha : abs a < δ) (hb : abs b < δ) #check my_lemma2 h₀ h₁ ha hb end lemma my_lemma3 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := begin intros x y ε epos ele1 xlt ylt, sorry end lemma my_lemma4 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := begin intros x y ε epos ele1 xlt ylt, calc abs (x * y) = abs x * abs y : sorry ... ≤ abs x * ε : sorry ... < 1 * ε : sorry ... = ε : sorry end def fn_ub (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a def fn_lb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x section variables (f g : ℝ → ℝ) (a b : ℝ) example (hfa : fn_ub f a) (hgb : fn_ub g b) : fn_ub (λ x, f x + g x) (a + b) := begin intro x, dsimp, apply add_le_add, apply hfa, apply hgb end example (hfa : fn_lb f a) (hgb : fn_lb g b) : fn_lb (λ x, f x + g x) (a + b) := sorry example (nnf : fn_lb f 0) (nng : fn_lb g 0) : fn_lb (λ x, f x * g x) 0 := sorry example (hfa : fn_ub f a) (hfb : fn_ub g b) (nng : fn_lb g 0) (nna : 0 ≤ a) : fn_ub (λ x, f x * g x) (a * b) := sorry end section variables {α : Type*} {R : Type*} [ordered_cancel_add_comm_monoid R] #check @add_le_add def fn_ub' (f : α → R) (a : R) : Prop := ∀ x, f x ≤ a theorem fn_ub_add {f g : α → R} {a b : R} (hfa : fn_ub' f a) (hgb : fn_ub' g b) : fn_ub' (λ x, f x + g x) (a + b) := λ x, add_le_add (hfa x) (hgb x) end example (f : ℝ → ℝ) (h : monotone f) : ∀ {a b}, a ≤ b → f a ≤ f b := h section variables (f g : ℝ → ℝ) example (mf : monotone f) (mg : monotone g) : monotone (λ x, f x + g x) := begin intros a b aleb, apply add_le_add, apply mf aleb, apply mg aleb end example (mf : monotone f) (mg : monotone g) : monotone (λ x, f x + g x) := λ a b aleb, add_le_add (mf aleb) (mg aleb) example {c : ℝ} (mf : monotone f) (nnc : 0 ≤ c) : monotone (λ x, c * f x) := sorry example (mf : monotone f) (mg : monotone g) : monotone (λ x, f (g x)) := sorry def fn_even (f : ℝ → ℝ) : Prop := ∀ x, f x = f (-x) def fn_odd (f : ℝ → ℝ) : Prop := ∀ x, f x = - f (-x) example (ef : fn_even f) (eg : fn_even g) : fn_even (λ x, f x + g x) := begin intro x, calc (λ x, f x + g x) x = f x + g x : rfl ... = f (-x) + g (-x) : by rw [ef, eg] end example (of : fn_odd f) (og : fn_odd g) : fn_even (λ x, f x * g x) := sorry example (ef : fn_even f) (og : fn_odd g) : fn_odd (λ x, f x * g x) := sorry example (ef : fn_even f) (og : fn_odd g) : fn_even (λ x, f (g x)) := sorry end section variables {α : Type*} (r s t : set α) example : s ⊆ s := by { intros x xs, exact xs } theorem subset.refl : s ⊆ s := λ x xs, xs theorem subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := sorry end section variables {α : Type*} [partial_order α] variables (s : set α) (a b : α) def set_ub (s : set α) (a : α) := ∀ x, x ∈ s → x ≤ a example (h : set_ub s a) (h' : a ≤ b) : set_ub s b := sorry end section open function example (c : ℝ) : injective (λ x, x + c) := begin intros x₁ x₂ h', exact (add_left_inj c).mp h', end example {c : ℝ} (h : c ≠ 0) : injective (λ x, c * x) := sorry variables {α : Type*} {β : Type*} {γ : Type*} variables {g : β → γ} {f : α → β} example (injg : injective g) (injf : injective f) : injective (λ x, g (f x)) := sorry end
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@@ -0,0 +1,149 @@import data.real.basic example : ∃ x : ℝ, 2 < x ∧ x < 3 := begin use 5 / 2, norm_num end example : ∃ x : ℝ, 2 < x ∧ x < 3 := begin have h : 2 < (5 : ℝ) / 2 ∧ (5 : ℝ) / 2 < 3, by norm_num, exact ⟨5 / 2, h⟩ end example : ∃ x : ℝ, 2 < x ∧ x < 3 := ⟨5 / 2, by norm_num⟩ def fn_ub (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a def fn_lb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x def fn_has_ub (f : ℝ → ℝ) := ∃ a, fn_ub f a def fn_has_lb (f : ℝ → ℝ) := ∃ a, fn_lb f a theorem fn_ub_add {f g : ℝ → ℝ} {a b : ℝ} (hfa : fn_ub f a) (hgb : fn_ub g b) : fn_ub (λ x, f x + g x) (a + b) := λ x, add_le_add (hfa x) (hgb x) section variables {f g : ℝ → ℝ} example (ubf : fn_has_ub f) (ubg : fn_has_ub g) : fn_has_ub (λ x, f x + g x) := begin cases ubf with a ubfa, cases ubg with b ubfb, use a + b, apply fn_ub_add ubfa ubfb end example (lbf : fn_has_lb f) (lbg : fn_has_lb g) : fn_has_lb (λ x, f x + g x) := sorry example {c : ℝ} (ubf : fn_has_ub f) (h : c ≥ 0): fn_has_ub (λ x, c * f x) := sorry example (ubf : fn_has_ub f) (ubg : fn_has_ub g) : fn_has_ub (λ x, f x + g x) := begin rcases ubf with ⟨a, ubfa⟩, rcases ubg with ⟨b, ubfb⟩, exact ⟨a + b, fn_ub_add ubfa ubfb⟩ end example : fn_has_ub f → fn_has_ub g → fn_has_ub (λ x, f x + g x) := begin rintros ⟨a, ubfa⟩ ⟨b, ubfb⟩, exact ⟨a + b, fn_ub_add ubfa ubfb⟩ end example : fn_has_ub f → fn_has_ub g → fn_has_ub (λ x, f x + g x) := λ ⟨a, ubfa⟩ ⟨b, ubfb⟩, ⟨a + b, fn_ub_add ubfa ubfb⟩ end section variables {α : Type*} [comm_ring α] def sum_of_squares (x : α) := ∃ a b, x = a^2 + b^2 theorem sum_of_squares_mul {x y : α} (sosx : sum_of_squares x) (sosy : sum_of_squares y) : sum_of_squares (x * y) := begin rcases sosx with ⟨a, b, xeq⟩, rcases sosy with ⟨c, d, yeq⟩, rw [xeq, yeq], use [a*c - b*d, a*d + b*c], ring end theorem sum_of_squares_mul' {x y : α} (sosx : sum_of_squares x) (sosy : sum_of_squares y) : sum_of_squares (x * y) := begin rcases sosx with ⟨a, b, rfl⟩, rcases sosy with ⟨c, d, rfl⟩, use [a*c - b*d, a*d + b*c], ring end end section variables {a b c : ℕ} example (divab : a ∣ b) (divbc : b ∣ c) : a ∣ c := begin cases divab with d beq, cases divbc with e ceq, rw [ceq, beq], use (d * e), ring end example (divab : a ∣ b) (divac : a ∣ c) : a ∣ (b + c) := sorry end section open function example {c : ℝ} : surjective (λ x, x + c) := begin intro x, use x - c, dsimp, ring end example {c : ℝ} (h : c ≠ 0) : surjective (λ x, c * x) := sorry example (x y : ℝ) (h : x - y ≠ 0) : (x^2 - y^2) / (x - y) = x + y := by { field_simp [h], ring } example {f : ℝ → ℝ} (h : surjective f) : ∃ x, (f x)^2 = 4 := begin cases h 2 with x hx, use x, rw hx, norm_num end end section open function variables {α : Type*} {β : Type*} {γ : Type*} variables {g : β → γ} {f : α → β} example (surjg : surjective g) (surjf : surjective f) : surjective (λ x, g (f x)) := sorry end
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@@ -0,0 +1,159 @@import data.real.basic section variables a b : ℝ example (h : a < b) : ¬ b < a := begin intro h', have : a < a, from lt_trans h h', apply lt_irrefl a this end def fn_ub (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a def fn_lb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x def fn_has_ub (f : ℝ → ℝ) := ∃ a, fn_ub f a def fn_has_lb (f : ℝ → ℝ) := ∃ a, fn_lb f a variable f : ℝ → ℝ example (h : ∀ a, ∃ x, f x > a) : ¬ fn_has_ub f := begin intros fnub, cases fnub with a fnuba, cases h a with x hx, have : f x ≤ a, from fnuba x, linarith end example (h : ∀ a, ∃ x, f x < a) : ¬ fn_has_lb f := sorry example : ¬ fn_has_ub (λ x, x) := sorry #check (not_le_of_gt : a > b → ¬ a ≤ b) #check (not_lt_of_ge : a ≥ b → ¬ a < b) #check (lt_of_not_ge : ¬ a ≥ b → a < b) #check (le_of_not_gt : ¬ a > b → a ≤ b) example (h : monotone f) (h' : f a < f b) : a < b := sorry example (h : a ≤ b) (h' : f b < f a) : ¬ monotone f := sorry example : ¬ ∀ {f : ℝ → ℝ}, monotone f → ∀ {a b}, f a ≤ f b → a ≤ b := begin intro h, let f := λ x : ℝ, (0 : ℝ), have monof : monotone f, { sorry }, have h' : f 1 ≤ f 0, from le_refl _, sorry end example (x : ℝ) (h : ∀ ε > 0, x < ε) : x ≤ 0 := sorry end section variables {α : Type*} (P : α → Prop) (Q : Prop) example (h : ¬ ∃ x, P x) : ∀ x, ¬ P x := sorry example (h : ∀ x, ¬ P x) : ¬ ∃ x, P x := sorry example (h : ¬ ∀ x, P x) : ∃ x, ¬ P x := sorry example (h : ∃ x, ¬ P x) : ¬ ∀ x, P x := sorry open_locale classical example (h : ¬ ∀ x, P x) : ∃ x, ¬ P x := begin by_contradiction h', apply h, intro x, show P x, by_contradiction h'', exact h' ⟨x, h''⟩ end example (h : ¬ ¬ Q) : Q := sorry example (h : Q) : ¬ ¬ Q := sorry end open_locale classical section variable (f : ℝ → ℝ) example (h : ¬ fn_has_ub f) : ∀ a, ∃ x, f x > a := sorry example (h : ¬ ∀ a, ∃ x, f x > a) : fn_has_ub f := begin push_neg at h, exact h end example (h : ¬ fn_has_ub f) : ∀ a, ∃ x, f x > a := begin simp only [fn_has_ub, fn_ub] at h, push_neg at h, exact h end example (h : ¬ monotone f) : ∃ x y, x ≤ y ∧ f y < f x := sorry example (h : ¬ fn_has_ub f) : ∀ a, ∃ x, f x > a := begin contrapose! h, exact h end example (x : ℝ) (h : ∀ ε > 0, x ≤ ε) : x ≤ 0 := begin contrapose! h, use x / 2, split; linarith end end section variable a : ℕ example (h : 0 < 0) : a > 37 := begin exfalso, apply lt_irrefl 0 h end example (h : 0 < 0) : a > 37 := absurd h (lt_irrefl 0) example (h : 0 < 0) : a > 37 := begin have h' : ¬ 0 < 0, from lt_irrefl 0, contradiction end end
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@@ -0,0 +1,163 @@import data.real.basic import data.nat.prime example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬ y ≤ x) : x ≤ y ∧ x ≠ y := begin split, { assumption }, intro h, apply h₁, rw h end example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬ y ≤ x) : x ≤ y ∧ x ≠ y := ⟨h₀, λ h, h₁ (by rw h)⟩ example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬ y ≤ x) : x ≤ y ∧ x ≠ y := begin have h : x ≠ y, { contrapose! h₁, rw h₁ }, exact ⟨h₀, h⟩ end example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬ y ≤ x := begin cases h with h₀ h₁, contrapose! h₁, exact le_antisymm h₀ h₁ end example {x y : ℝ} : x ≤ y ∧ x ≠ y → ¬ y ≤ x := begin rintros ⟨h₀, h₁⟩ h', exact h₁ (le_antisymm h₀ h') end example {x y : ℝ} : x ≤ y ∧ x ≠ y → ¬ y ≤ x := λ ⟨h₀, h₁⟩ h', h₁ (le_antisymm h₀ h') example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬ y ≤ x := begin intro h', apply h.right, exact le_antisymm h.left h' end example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬ y ≤ x := λ h', h.right (le_antisymm h.left h') example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : m ∣ n ∧ ¬ n ∣ m := sorry example : ∃ x : ℝ, 2 < x ∧ x < 4 := ⟨5/2, by norm_num, by norm_num⟩ example (x y : ℝ) : (∃ z : ℝ, x < z ∧ z < y) → x < y := begin rintros ⟨z, xltz, zlty⟩, exact lt_trans xltz zlty end example (x y : ℝ) : (∃ z : ℝ, x < z ∧ z < y) → x < y := λ ⟨z, xltz, zlty⟩, lt_trans xltz zlty example : ∃ x : ℝ, 2 < x ∧ x < 4 := begin use 5 / 2, split; norm_num end example : ∃ m n : ℕ, 4 < m ∧ m < n ∧ n < 10 ∧ nat.prime m ∧ nat.prime n := begin use [5, 7], norm_num end example {x y : ℝ} : x ≤ y ∧ x ≠ y → x ≤ y ∧ ¬ y ≤ x := begin rintros ⟨h₀, h₁⟩, use [h₀, λ h', h₁ (le_antisymm h₀ h')] end example {x y : ℝ} (h : x ≤ y) : ¬ y ≤ x ↔ x ≠ y := begin split, { contrapose!, rintro rfl, reflexivity }, contrapose!, exact le_antisymm h end example {x y : ℝ} (h : x ≤ y) : ¬ y ≤ x ↔ x ≠ y := ⟨λ h₀ h₁, h₀ (by rw h₁), λ h₀ h₁, h₀ (le_antisymm h h₁)⟩ example {x y : ℝ} : x ≤ y ∧ ¬ y ≤ x ↔ x ≤ y ∧ x ≠ y := sorry theorem aux {x y : ℝ} (h : x^2 + y^2 = 0) : x = 0 := begin have h' : x^2 = 0, { sorry }, exact pow_eq_zero h' end example (x y : ℝ) : x^2 + y^2 = 0 ↔ x = 0 ∧ y = 0 := sorry section example (x y : ℝ) : abs (x + 3) < 5 → -8 < x ∧ x < 2 := begin rw abs_lt, intro h, split; linarith end example : 3 ∣ nat.gcd 6 15 := begin rw nat.dvd_gcd_iff, split; norm_num end end theorem not_monotone_iff {f : ℝ → ℝ}: ¬ monotone f ↔ ∃ x y, x ≤ y ∧ f x > f y := by { rw monotone, push_neg } example : ¬ monotone (λ x : ℝ, -x) := sorry section variables {α : Type*} [partial_order α] variables a b : α example : a < b ↔ a ≤ b ∧ a ≠ b := begin rw lt_iff_le_not_le, sorry end end section variables {α : Type*} [preorder α] variables a b c : α example : ¬ a < a := begin rw lt_iff_le_not_le, sorry end example : a < b → b < c → a < c := begin simp only [lt_iff_le_not_le], sorry end end
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@@ -0,0 +1,108 @@import data.real.basic section variables {x y : ℝ} example (h : y > x^2) : y > 0 ∨ y < -1 := by { left, linarith [pow_two_nonneg x] } example (h : -y > x^2 + 1) : y > 0 ∨ y < -1 := by { right, linarith [pow_two_nonneg x] } example (h : y > 0) : y > 0 ∨ y < -1 := or.inl h example (h : y < -1) : y > 0 ∨ y < -1 := or.inr h example : x < abs y → x < y ∨ x < -y := begin cases le_or_gt 0 y with h h, { rw abs_of_nonneg h, intro h, left, exact h }, rw abs_of_neg h, intro h, right, exact h end namespace my_abs theorem le_abs_self (x : ℝ) : x ≤ abs x := sorry theorem neg_le_abs_self (x : ℝ) : -x ≤ abs x := sorry theorem abs_add (x y : ℝ) : abs (x + y) ≤ abs x + abs y := sorry theorem lt_abs : x < abs y ↔ x < y ∨ x < -y := sorry theorem abs_lt : abs x < y ↔ - y < x ∧ x < y := sorry end my_abs end example {x : ℝ} (h : x ≠ 0) : x < 0 ∨ x > 0 := begin rcases lt_trichotomy x 0 with xlt | xeq | xgt, { left, exact xlt }, { contradiction }, right, exact xgt end example {m n k : ℕ} (h : m ∣ n ∨ m ∣ k) : m ∣ n * k := begin rcases h with ⟨a, rfl⟩ | ⟨b, rfl⟩, { rw [mul_assoc], apply dvd_mul_right }, rw [mul_comm, mul_assoc], apply dvd_mul_right end example {z : ℝ} (h : ∃ x y, z = x^2 + y^2 ∨ z = x^2 + y^2 + 1) : z ≥ 0 := sorry example {x : ℝ} (h : x^2 = 1) : x = 1 ∨ x = -1 := sorry example {x y : ℝ} (h : x^2 = y^2) : x = y ∨ x = -y := sorry section variables {R : Type*} [comm_ring R] [is_domain R] variables (x y : R) example (h : x^2 = 1) : x = 1 ∨ x = -1 := sorry example (h : x^2 = y^2) : x = y ∨ x = -y := sorry end example (P : Prop) : ¬ ¬ P → P := begin intro h, cases classical.em P, { assumption }, contradiction end section open_locale classical example (P : Prop) : ¬ ¬ P → P := begin intro h, by_cases h' : P, { assumption }, contradiction end example (P Q : Prop) : (P → Q) ↔ ¬ P ∨ Q := sorry end
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@@ -0,0 +1,119 @@import data.real.basic def converges_to (s : ℕ → ℝ) (a : ℝ) := ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε example : (λ x y : ℝ, (x + y)^2) = (λ x y : ℝ, x^2 + 2*x*y + y^2) := by { ext, ring } example (a b : ℝ) : abs a = abs (a - b + b) := by { congr, ring } example {a : ℝ} (h : 1 < a) : a < a * a := begin convert (mul_lt_mul_right _).2 h, { rw [one_mul] }, exact lt_trans zero_lt_one h end theorem converges_to_const (a : ℝ) : converges_to (λ x : ℕ, a) a := begin intros ε εpos, use 0, intros n nge, dsimp, rw [sub_self, abs_zero], apply εpos end theorem converges_to_add {s t : ℕ → ℝ} {a b : ℝ} (cs : converges_to s a) (ct : converges_to t b): converges_to (λ n, s n + t n) (a + b) := begin intros ε εpos, dsimp, have ε2pos : 0 < ε / 2, { linarith }, cases cs (ε / 2) ε2pos with Ns hs, cases ct (ε / 2) ε2pos with Nt ht, use max Ns Nt, sorry end theorem converges_to_mul_const {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : converges_to s a) : converges_to (λ n, c * s n) (c * a) := begin by_cases h : c = 0, { convert converges_to_const 0, { ext, rw [h, zero_mul] }, rw [h, zero_mul] }, have acpos : 0 < abs c, from abs_pos.mpr h, sorry end theorem exists_abs_le_of_converges_to {s : ℕ → ℝ} {a : ℝ} (cs : converges_to s a) : ∃ N b, ∀ n, N ≤ n → abs (s n) < b := begin cases cs 1 zero_lt_one with N h, use [N, abs a + 1], sorry end lemma aux {s t : ℕ → ℝ} {a : ℝ} (cs : converges_to s a) (ct : converges_to t 0) : converges_to (λ n, s n * t n) 0 := begin intros ε εpos, dsimp, rcases exists_abs_le_of_converges_to cs with ⟨N₀, B, h₀⟩, have Bpos : 0 < B, from lt_of_le_of_lt (abs_nonneg _) (h₀ N₀ (le_refl _)), have pos₀ : ε / B > 0, from div_pos εpos Bpos, cases ct _ pos₀ with N₁ h₁, sorry end theorem converges_to_mul {s t : ℕ → ℝ} {a b : ℝ} (cs : converges_to s a) (ct : converges_to t b): converges_to (λ n, s n * t n) (a * b) := begin have h₁ : converges_to (λ n, s n * (t n - b)) 0, { apply aux cs, convert converges_to_add ct (converges_to_const (-b)), ring }, convert (converges_to_add h₁ (converges_to_mul_const b cs)), { ext, ring }, ring end theorem converges_to_unique {s : ℕ → ℝ} {a b : ℝ} (sa : converges_to s a) (sb : converges_to s b) : a = b := begin by_contradiction abne, have : abs (a - b) > 0, { sorry }, let ε := abs (a - b) / 2, have εpos : ε > 0, { change abs (a - b) / 2 > 0, linarith }, cases sa ε εpos with Na hNa, cases sb ε εpos with Nb hNb, let N := max Na Nb, have absa : abs (s N - a) < ε, { sorry }, have absb : abs (s N - b) < ε, { sorry }, have : abs (a - b) < abs (a - b), { sorry }, exact lt_irrefl _ this end section variables {α : Type*} [linear_order α] def converges_to' (s : α → ℝ) (a : ℝ) := ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε end
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@@ -0,0 +1,151 @@import data.real.basic def fn_ub (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a def fn_lb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x section variables (f g : ℝ → ℝ) (a b : ℝ) example (hfa : fn_lb f a) (hgb : fn_lb g b) : fn_lb (λ x, f x + g x) (a + b) := begin intro x, apply add_le_add, apply hfa, apply hgb end example (nnf : fn_lb f 0) (nng : fn_lb g 0) : fn_lb (λ x, f x * g x) 0 := begin intro x, apply mul_nonneg, apply nnf, apply nng end example (hfa : fn_ub f a) (hfb : fn_ub g b) (nng : fn_lb g 0) (nna : 0 ≤ a) : fn_ub (λ x, f x * g x) (a * b) := begin intro x, apply mul_le_mul, apply hfa, apply hfb, apply nng, apply nna end end section variables (f g : ℝ → ℝ) example {c : ℝ} (mf : monotone f) (nnc : 0 ≤ c) : monotone (λ x, c * f x) := begin intros a b aleb, apply mul_le_mul_of_nonneg_left _ nnc, apply mf aleb end example {c : ℝ} (mf : monotone f) (nnc : 0 ≤ c) : monotone (λ x, c * f x) := λ a b aleb, mul_le_mul_of_nonneg_left (mf aleb) nnc example (mf : monotone f) (mg : monotone g) : monotone (λ x, f (g x)) := begin intros a b aleb, apply mf, apply mg, apply aleb end example (mf : monotone f) (mg : monotone g) : monotone (λ x, f (g x)) := λ a b aleb, mf (mg aleb) def fn_even (f : ℝ → ℝ) : Prop := ∀ x, f x = f (-x) def fn_odd (f : ℝ → ℝ) : Prop := ∀ x, f x = - f (-x) example (of : fn_odd f) (og : fn_odd g) : fn_even (λ x, f x * g x) := begin intro x, calc (λ x, f x * g x) x = f x * g x : rfl ... = f (- x) * g (- x) : by rw [of, og, neg_mul_neg] end example (ef : fn_even f) (og : fn_odd g) : fn_odd (λ x, f x * g x) := begin intro x, dsimp, rw [ef, og, neg_mul_eq_mul_neg] end example (ef : fn_even f) (og : fn_odd g) : fn_even (λ x, f (g x)) := begin intro x, dsimp, rw [og, ←ef] end end section variables {α : Type*} (r s t : set α) example : r ⊆ s → s ⊆ t → r ⊆ t := begin intros rsubs ssubt x xr, apply ssubt, apply rsubs, apply xr end theorem subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := λ rsubs ssubt x xr, ssubt (rsubs xr) end section variables {α : Type*} [partial_order α] variables (s : set α) (a b : α) def set_ub (s : set α) (a : α) := ∀ x, x ∈ s → x ≤ a example (h : set_ub s a) (h' : a ≤ b) : set_ub s b := begin intros x xs, apply le_trans (h x xs) h' end example (h : set_ub s a) (h' : a ≤ b) : set_ub s b := λ x xs, le_trans (h x xs) h' end section open function example {c : ℝ} (h : c ≠ 0) : injective (λ x, c * x) := begin intros x₁ x₂ h', apply (mul_right_inj' h).mp h' end variables {α : Type*} {β : Type*} {γ : Type*} variables {g : β → γ} {f : α → β} example (injg : injective g) (injf : injective f) : injective (λ x, g (f x)) := begin intros x₁ x₂ h, apply injf, apply injg, apply h end end
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@@ -0,0 +1,90 @@import data.real.basic def fn_ub (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a def fn_lb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x def fn_has_ub (f : ℝ → ℝ) := ∃ a, fn_ub f a def fn_has_lb (f : ℝ → ℝ) := ∃ a, fn_lb f a theorem fn_ub_add {f g : ℝ → ℝ} {a b : ℝ} (hfa : fn_ub f a) (hgb : fn_ub g b) : fn_ub (λ x, f x + g x) (a + b) := λ x, add_le_add (hfa x) (hgb x) section variables {f g : ℝ → ℝ} example (lbf : fn_has_lb f) (lbg : fn_has_lb g) : fn_has_lb (λ x, f x + g x) := begin cases lbf with a lbfa, cases lbg with b lbgb, use a + b, intro x, exact add_le_add (lbfa x) (lbgb x) end example {c : ℝ} (ubf : fn_has_ub f) (h : c ≥ 0): fn_has_ub (λ x, c * f x) := begin cases ubf with a lbfa, use c * a, intro x, exact mul_le_mul_of_nonneg_left (lbfa x) h end end section variables {a b c : ℕ} example (divab : a ∣ b) (divbc : b ∣ c) : a ∣ c := begin rcases divab with ⟨d, rfl⟩, rcases divbc with ⟨e, rfl⟩, use (d * e), ring end example (divab : a ∣ b) (divac : a ∣ c) : a ∣ (b + c) := begin rcases divab with ⟨d, rfl⟩, rcases divac with ⟨e, rfl⟩, use (d + e), ring end end section open function example {c : ℝ} (h : c ≠ 0) : surjective (λ x, c * x) := begin intro x, use x / c, dsimp, rw [mul_div_cancel' _ h] end example {c : ℝ} (h : c ≠ 0) : surjective (λ x, c * x) := begin intro x, use x / c, field_simp [h], ring end end section open function variables {α : Type*} {β : Type*} {γ : Type*} variables {g : β → γ} {f : α → β} example (surjg : surjective g) (surjf : surjective f) : surjective (λ x, g (f x)) := begin intro z, rcases surjg z with ⟨y, rfl⟩, rcases surjf y with ⟨x, rfl⟩, use [x, rfl] end end
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@@ -0,0 +1,132 @@import data.real.basic section variables a b : ℝ def fn_ub (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a def fn_lb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x def fn_has_ub (f : ℝ → ℝ) := ∃ a, fn_ub f a def fn_has_lb (f : ℝ → ℝ) := ∃ a, fn_lb f a variable f : ℝ → ℝ example (h : ∀ a, ∃ x, f x < a) : ¬ fn_has_lb f := begin rintros ⟨a, ha⟩, rcases h a with ⟨x, hx⟩, have := ha x, linarith end example : ¬ fn_has_ub (λ x, x) := begin rintros ⟨a, ha⟩, have : a + 1 ≤ a := ha (a + 1), linarith end example (h : monotone f) (h' : f a < f b) : a < b := begin apply lt_of_not_ge, intro h'', apply absurd h', apply not_lt_of_ge (h h'') end example (h : a ≤ b) (h' : f b < f a) : ¬ monotone f := begin intro h'', apply absurd h', apply not_lt_of_ge, apply h'' h end example : ¬ ∀ {f : ℝ → ℝ}, monotone f → ∀ {a b}, f a ≤ f b → a ≤ b := begin intro h, let f := λ x : ℝ, (0 : ℝ), have monof : monotone f, { intros a b leab, refl }, have h' : f 1 ≤ f 0, from le_refl _, have : (1 : ℝ) ≤ 0 := h monof h', linarith end example (x : ℝ) (h : ∀ ε > 0, x < ε) : x ≤ 0 := begin apply le_of_not_gt, intro h', linarith [h _ h'] end end section variables {α : Type*} (P : α → Prop) (Q : Prop) example (h : ¬ ∃ x, P x) : ∀ x, ¬ P x := begin intros x Px, apply h, use [x, Px] end example (h : ∀ x, ¬ P x) : ¬ ∃ x, P x := begin rintros ⟨x, Px⟩, exact h x Px end example (h : ∃ x, ¬ P x) : ¬ ∀ x, P x := begin intro h', rcases h with ⟨x, nPx⟩, apply nPx, apply h' end example (h : ¬ ¬ Q) : Q := begin by_contradiction h', exact h h' end example (h : Q) : ¬ ¬ Q := begin intro h', exact h' h end end open_locale classical section variable (f : ℝ → ℝ) example (h : ¬ fn_has_ub f) : ∀ a, ∃ x, f x > a := begin intro a, by_contradiction h', apply h, use a, intro x, apply le_of_not_gt, intro h'', apply h', use [x, h''] end example (h : ¬ monotone f) : ∃ x y, x ≤ y ∧ f y < f x := begin rw [monotone] at h, push_neg at h, exact h end end
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@@ -0,0 +1,108 @@import data.real.basic import data.nat.prime example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : m ∣ n ∧ ¬ n ∣ m := begin cases h with h0 h1, split, { exact h0 }, intro h2, apply h1, apply nat.dvd_antisymm h0 h2, end example {x y : ℝ} : x ≤ y ∧ ¬ y ≤ x ↔ x ≤ y ∧ x ≠ y := begin split, { rintros ⟨h0, h1⟩, split, { exact h0 }, intro h2, apply h1, rw h2 }, rintros ⟨h0, h1⟩, split, { exact h0 }, intro h2, apply h1, apply le_antisymm h0 h2 end theorem aux {x y : ℝ} (h : x^2 + y^2 = 0) : x = 0 := begin have h' : x^2 = 0, { linarith [pow_two_nonneg x, pow_two_nonneg y] }, exact pow_eq_zero h' end example (x y : ℝ) : x^2 + y^2 = 0 ↔ x = 0 ∧ y = 0 := begin split, { intro h, split, { exact aux h }, rw add_comm at h, exact aux h }, rintros ⟨rfl, rfl⟩, norm_num end theorem not_monotone_iff {f : ℝ → ℝ}: ¬ monotone f ↔ ∃ x y, x ≤ y ∧ f x > f y := by { rw monotone, push_neg } example : ¬ monotone (λ x : ℝ, -x) := begin rw not_monotone_iff, use [0, 1], norm_num end section variables {α : Type*} [partial_order α] variables a b : α example : a < b ↔ a ≤ b ∧ a ≠ b := begin rw lt_iff_le_not_le, split, { rintros ⟨h0, h1⟩, split, { exact h0 }, intro h2, apply h1, rw h2 }, rintros ⟨h0, h1⟩, split, { exact h0 }, intro h2, apply h1, apply le_antisymm h0 h2 end end section variables {α : Type*} [preorder α] variables a b c : α example : ¬ a < a := begin rw lt_iff_le_not_le, rintros ⟨h0, h1⟩, exact h1 h0 end example : a < b → b < c → a < c := begin simp only [lt_iff_le_not_le], rintros ⟨h0, h1⟩ ⟨h2, h3⟩, split, { apply le_trans h0 h2 }, intro h4, apply h1, apply le_trans h2 h4 end end
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@@ -0,0 +1,162 @@import data.real.basic section variables {x y : ℝ} namespace my_abs theorem le_abs_self (x : ℝ) : x ≤ abs x := begin cases le_or_gt 0 x with h h, { rw abs_of_nonneg h }, rw abs_of_neg h, linarith end theorem neg_le_abs_self (x : ℝ) : -x ≤ abs x := begin cases le_or_gt 0 x with h h, { rw abs_of_nonneg h, linarith }, rw abs_of_neg h end theorem abs_add (x y : ℝ) : abs (x + y) ≤ abs x + abs y := begin cases le_or_gt 0 (x + y) with h h, { rw abs_of_nonneg h, linarith [le_abs_self x, le_abs_self y] }, rw abs_of_neg h, linarith [neg_le_abs_self x, neg_le_abs_self y] end theorem lt_abs : x < abs y ↔ x < y ∨ x < -y := begin cases le_or_gt 0 y with h h, { rw abs_of_nonneg h, split, { intro h', left, exact h' }, intro h', cases h' with h' h', { exact h' }, linarith }, rw abs_of_neg h, split, { intro h', right, exact h' }, intro h', cases h' with h' h', { linarith }, exact h' end theorem abs_lt : abs x < y ↔ - y < x ∧ x < y := begin cases le_or_gt 0 x with h h, { rw abs_of_nonneg h, split, { intro h', split, { linarith }, exact h' }, intro h', cases h' with h1 h2, exact h2 }, rw abs_of_neg h, split, { intro h', split, { linarith }, linarith }, intro h', linarith end end my_abs end example {z : ℝ} (h : ∃ x y, z = x^2 + y^2 ∨ z = x^2 + y^2 + 1) : z ≥ 0 := by { rcases h with ⟨x, y, rfl | rfl⟩; linarith [sq_nonneg x, sq_nonneg y] } example {x : ℝ} (h : x^2 = 1) : x = 1 ∨ x = -1 := begin have h' : x^2 - 1 = 0, { rw [h, sub_self] }, have h'' : (x + 1) * (x - 1) = 0, { rw ← h', ring }, cases eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1, { right, exact eq_neg_iff_add_eq_zero.mpr h1 }, left, exact eq_of_sub_eq_zero h1 end example {x y : ℝ} (h : x^2 = y^2) : x = y ∨ x = -y := begin have h' : x^2 - y^2 = 0, { rw [h, sub_self] }, have h'' : (x + y) * (x - y) = 0, { rw ← h', ring }, cases eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1, { right, exact eq_neg_iff_add_eq_zero.mpr h1 }, left, exact eq_of_sub_eq_zero h1 end section variables {R : Type*} [comm_ring R] [is_domain R] variables (x y : R) example (h : x^2 = 1) : x = 1 ∨ x = -1 := begin have h' : x^2 - 1 = 0, { rw [h, sub_self] }, have h'' : (x + 1) * (x - 1) = 0, { rw ← h', ring }, cases eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1, { right, exact eq_neg_iff_add_eq_zero.mpr h1 }, left, exact eq_of_sub_eq_zero h1 end example (h : x^2 = y^2) : x = y ∨ x = -y := begin have h' : x^2 - y^2 = 0, { rw [h, sub_self] }, have h'' : (x + y) * (x - y) = 0, { rw ← h', ring }, cases eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1, { right, exact eq_neg_iff_add_eq_zero.mpr h1 }, left, exact eq_of_sub_eq_zero h1 end end section open_locale classical example (P Q : Prop) : (P → Q) ↔ ¬ P ∨ Q := begin split, { intro h, by_cases h' : P, { right, exact h h'}, left, exact h' }, rintros (h | h), { intro h', exact absurd h' h }, intro _, exact h end end
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@@ -0,0 +1,143 @@import data.real.basic def converges_to (s : ℕ → ℝ) (a : ℝ) := ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε theorem converges_to_const (a : ℝ) : converges_to (λ x : ℕ, a) a := begin intros ε εpos, use 0, intros n nge, dsimp, rw [sub_self, abs_zero], apply εpos end theorem converges_to_add {s t : ℕ → ℝ} {a b : ℝ} (cs : converges_to s a) (ct : converges_to t b): converges_to (λ n, s n + t n) (a + b) := begin intros ε εpos, dsimp, have ε2pos : 0 < ε / 2, { linarith }, cases cs (ε / 2) ε2pos with Ns hs, cases ct (ε / 2) ε2pos with Nt ht, use max Ns Nt, intros n hn, have ngeNs : n ≥ Ns := le_of_max_le_left hn, have ngeNt : n ≥ Nt := le_of_max_le_right hn, calc |s n + t n - (a + b)| = | s n - a + (t n - b) | : by { congr, ring } ... ≤ | s n - a | + | (t n - b) | : abs_add _ _ ... < ε / 2 + ε / 2 : add_lt_add (hs n ngeNs) (ht n ngeNt) ... = ε : by norm_num end theorem converges_to_mul_const {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : converges_to s a) : converges_to (λ n, c * s n) (c * a) := begin by_cases h : c = 0, { convert converges_to_const 0, { ext, rw [h, zero_mul] }, rw [h, zero_mul] }, have acpos : 0 < abs c, from abs_pos.mpr h, intros ε εpos, dsimp, have εcpos : 0 < ε / abs c, { apply div_pos εpos acpos }, cases cs (ε / abs c) εcpos with Ns hs, use Ns, intros n ngt, calc |c * s n - c * a| = |c| * |s n - a| : by { rw [←abs_mul, mul_sub] } ... < |c| * (ε / |c|) : mul_lt_mul_of_pos_left (hs n ngt) acpos ... = ε : mul_div_cancel' _ (ne_of_lt acpos).symm end theorem exists_abs_le_of_converges_to {s : ℕ → ℝ} {a : ℝ} (cs : converges_to s a) : ∃ N b, ∀ n, N ≤ n → abs (s n) < b := begin cases cs 1 zero_lt_one with N h, use [N, abs a + 1], intros n ngt, calc |s n| = |s n - a + a| : by { congr, abel } ... ≤ |s n - a| + |a| : abs_add _ _ ... < |a| + 1 : by linarith [h n ngt] end lemma aux {s t : ℕ → ℝ} {a : ℝ} (cs : converges_to s a) (ct : converges_to t 0) : converges_to (λ n, s n * t n) 0 := begin intros ε εpos, dsimp, rcases exists_abs_le_of_converges_to cs with ⟨N₀, B, h₀⟩, have Bpos : 0 < B, from lt_of_le_of_lt (abs_nonneg _) (h₀ N₀ (le_refl _)), have pos₀ : ε / B > 0, from div_pos εpos Bpos, cases ct _ pos₀ with N₁ h₁, use max N₀ N₁, intros n ngt, have ngeN₀ : n ≥ N₀ := le_of_max_le_left ngt, have ngeN₁ : n ≥ N₁ := le_of_max_le_right ngt, calc |s n * t n - 0| = |s n| * |t n - 0| : by rw [sub_zero, abs_mul, sub_zero] ... < B * (ε / B) : mul_lt_mul'' (h₀ n ngeN₀) (h₁ n ngeN₁) (abs_nonneg _) (abs_nonneg _) ... = ε : mul_div_cancel' _ (ne_of_lt Bpos).symm end theorem converges_to_muL {s t : ℕ → ℝ} {a b : ℝ} (cs : converges_to s a) (ct : converges_to t b): converges_to (λ n, s n * t n) (a * b) := begin have h₁ : converges_to (λ n, s n * (t n - b)) 0, { apply aux cs, convert converges_to_add ct (converges_to_const (-b)), ring }, convert (converges_to_add h₁ (converges_to_mul_const b cs)), { ext, ring }, ring end theorem converges_to_unique {s : ℕ → ℝ} {a b : ℝ} (sa : converges_to s a) (sb : converges_to s b) : a = b := begin by_contradiction abne, have : abs (a - b) > 0, { apply lt_of_le_of_ne, { apply abs_nonneg }, intro h'', apply abne, apply eq_of_abs_sub_eq_zero h''.symm, }, let ε := abs (a - b) / 2, have εpos : ε > 0, { change abs (a - b) / 2 > 0, linarith }, cases sa ε εpos with Na hNa, cases sb ε εpos with Nb hNb, let N := max Na Nb, have absa : abs (s N - a) < ε, { apply hNa, apply le_max_left }, have absb : abs (s N - b) < ε, { apply hNb, apply le_max_right }, have : abs (a - b) < abs (a - b), calc abs (a - b) = abs (- (s N - a) + (s N - b)) : by { congr, ring } ... ≤ abs (- (s N - a)) + abs (s N - b) : abs_add _ _ ... = abs (s N - a) + abs (s N - b) : by rw [abs_neg] ... < ε + ε : add_lt_add absa absb ... = abs (a - b) : by norm_num, exact lt_irrefl _ this end
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@@ -0,0 +1,273 @@import data.set.lattice import data.nat.parity import tactic section variable {α : Type*} variables (s t u : set α) open set example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := begin rw [subset_def, inter_def, inter_def], rw subset_def at h, dsimp, rintros x ⟨xs, xu⟩, exact ⟨h _ xs, xu⟩, end example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := begin simp only [subset_def, mem_inter_eq] at *, rintros x ⟨xs, xu⟩, exact ⟨h _ xs, xu⟩, end example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := begin intros x xsu, exact ⟨h xsu.1, xsu.2⟩ end theorem foo (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := λ x ⟨xs, xu⟩, ⟨h xs, xu⟩ example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by exact λ x ⟨xs, xu⟩, ⟨h xs, xu⟩ example : s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u) := begin intros x hx, have xs : x ∈ s := hx.1, have xtu : x ∈ t ∪ u := hx.2, cases xtu with xt xu, { left, show x ∈ s ∩ t, exact ⟨xs, xt⟩ }, right, show x ∈ s ∩ u, exact ⟨xs, xu⟩ end example : s ∩ (t ∪ u) ⊆ (s ∩ t) ∪ (s ∩ u) := begin rintros x ⟨xs, xt | xu⟩, { left, exact ⟨xs, xt⟩ }, right, exact ⟨xs, xu⟩ end example : (s ∩ t) ∪ (s ∩ u) ⊆ s ∩ (t ∪ u):= sorry example : s \ t \ u ⊆ s \ (t ∪ u) := begin intros x xstu, have xs : x ∈ s := xstu.1.1, have xnt : x ∉ t := xstu.1.2, have xnu : x ∉ u := xstu.2, split, { exact xs }, dsimp, intro xtu, -- x ∈ t ∨ x ∈ u cases xtu with xt xu, { show false, from xnt xt }, show false, from xnu xu end example : s \ t \ u ⊆ s \ (t ∪ u) := begin rintros x ⟨⟨xs, xnt⟩, xnu⟩, use xs, rintros (xt | xu); contradiction end example : s \ (t ∪ u) ⊆ s \ t \ u := sorry example : s ∩ t = t ∩ s := begin ext x, simp only [mem_inter_eq], split, { rintros ⟨xs, xt⟩, exact ⟨xt, xs⟩ }, rintros ⟨xt, xs⟩, exact ⟨xs, xt⟩ end example : s ∩ t = t ∩ s := set.ext $ λ x, ⟨λ ⟨xs, xt⟩, ⟨xt, xs⟩, λ ⟨xt, xs⟩, ⟨xs, xt⟩⟩ example : s ∩ t = t ∩ s := by ext x; simp [and.comm] example : s ∩ t = t ∩ s := begin apply subset.antisymm, { rintros x ⟨xs, xt⟩, exact ⟨xt, xs⟩ }, rintros x ⟨xt, xs⟩, exact ⟨xs, xt⟩ end example : s ∩ t = t ∩ s := subset.antisymm sorry sorry example : s ∩ (s ∪ t) = s := sorry example : s ∪ (s ∩ t) = s := sorry example : (s \ t) ∪ t = s ∪ t := sorry example : (s \ t) ∪ (t \ s) = (s ∪ t) \ (s ∩ t) := sorry def evens : set ℕ := {n | even n} def odds : set ℕ := {n | ¬ even n} example : evens ∪ odds = univ := begin rw [evens, odds], ext n, simp, apply classical.em end example (x : ℕ) (h : x ∈ (∅ : set ℕ)) : false := h example (x : ℕ) : x ∈ (univ : set ℕ) := trivial example : { n | nat.prime n } ∩ { n | n > 2} ⊆ { n | ¬ even n } := sorry #print prime #print nat.prime example (n : ℕ) : prime n ↔ nat.prime n := nat.prime_iff.symm example (n : ℕ) (h : prime n) : nat.prime n := by { rw nat.prime_iff, exact h } example (n : ℕ) (h : prime n) : nat.prime n := by rwa nat.prime_iff end section variables (s t : set ℕ) example (h₀ : ∀ x ∈ s, ¬ even x) (h₁ : ∀ x ∈ s, prime x) : ∀ x ∈ s, ¬ even x ∧ prime x := begin intros x xs, split, { apply h₀ x xs }, apply h₁ x xs end example (h : ∃ x ∈ s, ¬ even x ∧ prime x) : ∃ x ∈ s, prime x := begin rcases h with ⟨x, xs, _, prime_x⟩, use [x, xs, prime_x] end section variable (ssubt : s ⊆ t) include ssubt example (h₀ : ∀ x ∈ t, ¬ even x) (h₁ : ∀ x ∈ t, prime x) : ∀ x ∈ s, ¬ even x ∧ prime x := sorry example (h : ∃ x ∈ s, ¬ even x ∧ prime x) : ∃ x ∈ t, prime x := sorry end end section variables {α I : Type*} variables A B : I → set α variable s : set α open set example : s ∩ (⋃ i, A i) = ⋃ i, (A i ∩ s) := begin ext x, simp only [mem_inter_eq, mem_Union], split, { rintros ⟨xs, ⟨i, xAi⟩⟩, exact ⟨i, xAi, xs⟩ }, rintros ⟨i, xAi, xs⟩, exact ⟨xs, ⟨i, xAi⟩⟩ end example : (⋂ i, A i ∩ B i) = (⋂ i, A i) ∩ (⋂ i, B i) := begin ext x, simp only [mem_inter_eq, mem_Inter], split, { intro h, split, { intro i, exact (h i).1 }, intro i, exact (h i).2 }, rintros ⟨h1, h2⟩ i, split, { exact h1 i }, exact h2 i end open_locale classical example : s ∪ (⋂ i, A i) = ⋂ i, (A i ∪ s) := sorry def primes : set ℕ := {x | nat.prime x} example : (⋃ p ∈ primes, {x | p^2 ∣ x}) = {x | ∃ p ∈ primes, p^2 ∣ x} := by { ext, rw mem_Union₂, refl } example : (⋃ p ∈ primes, {x | p^2 ∣ x}) = {x | ∃ p ∈ primes, p^2 ∣ x} := by { ext, simp } example : (⋂ p ∈ primes, {x | ¬ p ∣ x}) ⊆ {x | x = 1} := begin intro x, contrapose!, simp, apply nat.exists_prime_and_dvd end example : (⋃ p ∈ primes, {x | x ≤ p}) = univ := sorry end section open set variables {α : Type*} (s : set (set α)) example : ⋃₀ s = ⋃ t ∈ s, t := begin ext x, rw mem_Union₂, refl end example : ⋂₀ s = ⋂ t ∈ s, t := begin ext x, rw mem_Inter₂, refl end end
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@@ -0,0 +1,216 @@import data.set.lattice import data.set.function import analysis.special_functions.log.basic section variables {α β : Type*} variable f : α → β variables s t : set α variables u v : set β open function open set example : f ⁻¹' (u ∩ v) = f ⁻¹' u ∩ f ⁻¹' v := by { ext, refl } example : f '' (s ∪ t) = f '' s ∪ f '' t := begin ext y, split, { rintros ⟨x, xs | xt, rfl⟩, { left, use [x, xs] }, right, use [x, xt] }, rintros (⟨x, xs, rfl⟩ | ⟨x, xt, rfl⟩), { use [x, or.inl xs] }, use [x, or.inr xt] end example : s ⊆ f ⁻¹' (f '' s) := begin intros x xs, show f x ∈ f '' s, use [x, xs] end example : f '' s ⊆ v ↔ s ⊆ f ⁻¹' v := sorry example (h : injective f) : f ⁻¹' (f '' s) ⊆ s := sorry example : f '' (f⁻¹' u) ⊆ u := sorry example (h : surjective f) : u ⊆ f '' (f⁻¹' u) := sorry example (h : s ⊆ t) : f '' s ⊆ f '' t := sorry example (h : u ⊆ v) : f ⁻¹' u ⊆ f ⁻¹' v := sorry example : f ⁻¹' (u ∪ v) = f ⁻¹' u ∪ f ⁻¹' v := sorry example : f '' (s ∩ t) ⊆ f '' s ∩ f '' t := sorry example (h : injective f) : f '' s ∩ f '' t ⊆ f '' (s ∩ t) := sorry example : f '' s \ f '' t ⊆ f '' (s \ t) := sorry example : f ⁻¹' u \ f ⁻¹' v ⊆ f ⁻¹' (u \ v) := sorry example : f '' s ∩ v = f '' (s ∩ f ⁻¹' v) := sorry example : f '' (s ∩ f ⁻¹' u) ⊆ f '' s ∪ u := sorry example : s ∩ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∩ u) := sorry example : s ∪ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∪ u) := sorry variables {I : Type*} (A : I → set α) (B : I → set β) example : f '' (⋃ i, A i) = ⋃ i, f '' A i := begin ext y, simp, split, { rintros ⟨x, ⟨i, xAi⟩, fxeq⟩, use [i, x, xAi, fxeq] }, rintros ⟨i, x, xAi, fxeq⟩, exact ⟨x, ⟨i, xAi⟩, fxeq⟩ end example : f '' (⋂ i, A i) ⊆ ⋂ i, f '' A i := begin intro y, simp, intros x h fxeq i, use [x, h i, fxeq], end example (i : I) (injf : injective f) : (⋂ i, f '' A i) ⊆ f '' (⋂ i, A i) := begin intro y, simp, intro h, rcases h i with ⟨x, xAi, fxeq⟩, use x, split, { intro i', rcases h i' with ⟨x', x'Ai, fx'eq⟩, have : f x = f x', by rw [fxeq, fx'eq], have : x = x', from injf this, rw this, exact x'Ai }, exact fxeq end example : f ⁻¹' (⋃ i, B i) = ⋃ i, f ⁻¹' (B i) := by { ext x, simp } example : f ⁻¹' (⋂ i, B i) = ⋂ i, f ⁻¹' (B i) := by { ext x, simp } end section open set real example : inj_on log { x | x > 0 } := begin intros x xpos y ypos, intro e, -- log x = log y calc x = exp (log x) : by rw exp_log xpos ... = exp (log y) : by rw e ... = y : by rw exp_log ypos end example : range exp = { y | y > 0 } := begin ext y, split, { rintros ⟨x, rfl⟩, apply exp_pos }, intro ypos, use log y, rw exp_log ypos end example : inj_on sqrt { x | x ≥ 0 } := sorry example : inj_on (λ x, x^2) { x : ℝ | x ≥ 0 } := sorry example : sqrt '' { x | x ≥ 0 } = {y | y ≥ 0} := sorry example : range (λ x, x^2) = {y : ℝ | y ≥ 0} := sorry end section variables {α β : Type*} [inhabited α] #check (default : α) variables (P : α → Prop) (h : ∃ x, P x) #check classical.some h example : P (classical.some h) := classical.some_spec h noncomputable theory open_locale classical def inverse (f : α → β) : β → α := λ y : β, if h : ∃ x, f x = y then classical.some h else default theorem inverse_spec {f : α → β} (y : β) (h : ∃ x, f x = y) : f (inverse f y) = y := begin rw inverse, dsimp, rw dif_pos h, exact classical.some_spec h end variable f : α → β open function example : injective f ↔ left_inverse (inverse f) f := sorry example : surjective f ↔ right_inverse (inverse f) f := sorry end section variable {α : Type*} open function theorem Cantor : ∀ f : α → set α, ¬ surjective f := begin intros f surjf, let S := { i | i ∉ f i}, rcases surjf S with ⟨j, h⟩, have h₁ : j ∉ f j, { intro h', have : j ∉ f j, { by rwa h at h' }, contradiction }, have h₂ : j ∈ S, sorry, have h₃ : j ∉ S, sorry, contradiction end end
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@@ -0,0 +1,108 @@import data.set.lattice import data.set.function import tactic open set open function noncomputable theory open_locale classical variables {α β : Type*} [nonempty β] section variables (f : α → β) (g : β → α) def sb_aux : ℕ → set α | 0 := univ \ (g '' univ) | (n + 1) := g '' (f '' sb_aux n) def sb_set := ⋃ n, sb_aux f g n def sb_fun (x : α) : β := if x ∈ sb_set f g then f x else inv_fun g x theorem sb_right_inv {x : α} (hx : x ∉ sb_set f g) : g (inv_fun g x) = x := begin have : x ∈ g '' univ, { contrapose! hx, rw [sb_set, mem_Union], use [0], rw [sb_aux, mem_diff], sorry }, have : ∃ y, g y = x, { sorry }, sorry end theorem sb_injective (hf: injective f) (hg : injective g) : injective (sb_fun f g) := begin set A := sb_set f g with A_def, set h := sb_fun f g with h_def, intros x₁ x₂, assume hxeq : h x₁ = h x₂, show x₁ = x₂, simp only [h_def, sb_fun, ←A_def] at hxeq, by_cases xA : x₁ ∈ A ∨ x₂ ∈ A, { wlog : x₁ ∈ A := xA using [x₁ x₂, x₂ x₁], have x₂A : x₂ ∈ A, { apply not_imp_self.mp, assume x₂nA : x₂ ∉ A, rw [if_pos xA, if_neg x₂nA] at hxeq, rw [A_def, sb_set, mem_Union] at xA, have x₂eq : x₂ = g (f x₁), { sorry }, rcases xA with ⟨n, hn⟩, rw [A_def, sb_set, mem_Union], use n + 1, simp [sb_aux], exact ⟨x₁, hn, x₂eq.symm⟩ }, sorry }, push_neg at xA, sorry end theorem sb_surjective (hf: injective f) (hg : injective g) : surjective (sb_fun f g) := begin set A := sb_set f g with A_def, set h := sb_fun f g with h_def, intro y, by_cases gyA : g y ∈ A, { rw [A_def, sb_set, mem_Union] at gyA, rcases gyA with ⟨n, hn⟩, cases n with n, { simp [sb_aux] at hn, contradiction }, simp [sb_aux] at hn, rcases hn with ⟨x, xmem, hx⟩, use x, have : x ∈ A, { rw [A_def, sb_set, mem_Union], exact ⟨n, xmem⟩ }, simp only [h_def, sb_fun, if_pos this], exact hg hx }, sorry end end theorem schroeder_bernstein {f : α → β} {g : β → α} (hf: injective f) (hg : injective g) : ∃ h : α → β, bijective h := ⟨sb_fun f g, sb_injective f g hf hg, sb_surjective f g hf hg⟩ /- Auxliary information -/ section variables (g : β → α) (x : α) #check (inv_fun g : α → β) #check (left_inverse_inv_fun : injective g → left_inverse (inv_fun g) g) #check (left_inverse_inv_fun : injective g → ∀ y, inv_fun g (g y) = y) #check (inv_fun_eq : (∃ y, g y = x) → g (inv_fun g x) = x) end
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@@ -0,0 +1,151 @@import data.set.lattice import data.nat.parity import tactic section variable {α : Type*} variables (s t u : set α) open set example : (s ∩ t) ∪ (s ∩ u) ⊆ s ∩ (t ∪ u):= begin rintros x (⟨xs, xt⟩ | ⟨xs, xu⟩), { use xs, left, exact xt }, use xs, right, exact xu end example : s \ (t ∪ u) ⊆ s \ t \ u := begin rintros x ⟨xs, xntu⟩, use xs, { intro xt, exact xntu (or.inl xt) }, intro xu, apply xntu (or.inr xu) end example : s ∩ t = t ∩ s := subset.antisymm (λ x ⟨xs, xt⟩, ⟨xt, xs⟩) (λ x ⟨xt, xs⟩, ⟨xs, xt⟩) example : s ∩ (s ∪ t) = s := begin ext x, split, { rintros ⟨xs, _⟩, exact xs }, intro xs, use xs, left, exact xs end example : s ∪ (s ∩ t) = s := begin ext x, split, { rintros (xs | ⟨xs, xt⟩); exact xs }, intro xs, left, exact xs end example : (s \ t) ∪ t = s ∪ t := begin ext x, split, { rintros (⟨xs, nxt⟩ | xt), { left, exact xs}, right, exact xt }, by_cases h : x ∈ t, { intro _, right, exact h }, rintros (xs | xt), { left, use [xs, h] }, right, use xt end example : (s \ t) ∪ (t \ s) = (s ∪ t) \ (s ∩ t) := begin ext x, split, { rintros (⟨xs, xnt⟩ | ⟨xt, xns⟩), { split, left, exact xs, rintros ⟨_, xt⟩, contradiction }, split , right, exact xt, rintros ⟨xs, _⟩, contradiction }, rintros ⟨xs | xt, nxst⟩, { left, use xs, intro xt, apply nxst, split; assumption }, right, use xt, intro xs, apply nxst, split; assumption end example : { n | nat.prime n } ∩ { n | n > 2} ⊆ { n | ¬ even n } := begin intro n, simp, intro nprime, cases nat.prime.eq_two_or_odd nprime with h h, { rw h, intro, linarith }, rw [nat.even_iff, h], norm_num end end section variables (s t : set ℕ) section variable (ssubt : s ⊆ t) include ssubt example (h₀ : ∀ x ∈ t, ¬ even x) (h₁ : ∀ x ∈ t, prime x) : ∀ x ∈ s, ¬ even x ∧ prime x := begin intros x xs, split, { apply h₀ x (ssubt xs) }, apply h₁ x (ssubt xs) end example (h : ∃ x ∈ s, ¬ even x ∧ prime x) : ∃ x ∈ t, prime x := begin rcases h with ⟨x, xs, _, px⟩, use [x, ssubt xs, px] end end end section variables {α I : Type*} variables A B : I → set α variable s : set α open set example : s ∪ (⋂ i, A i) = ⋂ i, (A i ∪ s) := begin ext x, simp only [mem_union, mem_Inter], split, { rintros (xs | xI), { intro i, right, exact xs }, intro i, left, exact xI i }, intro h, by_cases xs : x ∈ s, { left, exact xs }, right, intro i, cases h i, { assumption }, contradiction end def primes : set ℕ := {x | nat.prime x} example : (⋃ p ∈ primes, {x | x ≤ p}) = univ := begin apply eq_univ_of_forall, intro x, simp, rcases nat.exists_infinite_primes x with ⟨p, primep, pge⟩, use [p, pge, primep] end end
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@@ -0,0 +1,281 @@import data.set.lattice import data.set.function import analysis.special_functions.log.basic section variables {α β : Type*} variable f : α → β variables s t : set α variables u v : set β open function open set example : f '' s ⊆ v ↔ s ⊆ f ⁻¹' v := begin split, { intros h x xs, have : f x ∈ f '' s, from mem_image_of_mem _ xs, exact h this }, intros h y ymem, rcases ymem with ⟨x, xs, fxeq⟩, rw ← fxeq, apply h xs end example (h : injective f) : f ⁻¹' (f '' s) ⊆ s := begin rintros x ⟨y, ys, fxeq⟩, rw ← h fxeq, exact ys end example : f '' (f⁻¹' u) ⊆ u := begin rintros y ⟨x, xmem, rfl⟩, exact xmem end example (h : surjective f) : u ⊆ f '' (f⁻¹' u) := begin intros y yu, rcases h y with ⟨x, fxeq⟩, use x, split, { show f x ∈ u, rw fxeq, exact yu }, exact fxeq end example (h : s ⊆ t) : f '' s ⊆ f '' t := begin rintros y ⟨x, xs, fxeq⟩, use [x, h xs, fxeq] end example (h : u ⊆ v) : f ⁻¹' u ⊆ f ⁻¹' v := by intro x; apply h example : f ⁻¹' (u ∪ v) = f ⁻¹' u ∪ f ⁻¹' v := by ext x; refl example : f '' (s ∩ t) ⊆ f '' s ∩ f '' t := begin rintros y ⟨x, ⟨xs, xt⟩, rfl⟩, use [x, xs, rfl, x, xt, rfl] end example (h : injective f) : f '' s ∩ f '' t ⊆ f '' (s ∩ t) := begin rintros y ⟨⟨x₁, x₁s, rfl⟩, ⟨x₂, x₂t, fx₂eq⟩⟩, use [x₁, x₁s], rw ← h fx₂eq, exact x₂t end example : f '' s \ f '' t ⊆ f '' (s \ t) := begin rintros y ⟨⟨x₁, x₁s, rfl⟩, h⟩, use [x₁, x₁s], intro h', apply h, use [x₁, h', rfl] end example : f ⁻¹' u \ f ⁻¹' v ⊆ f ⁻¹' (u \ v) := λ x, id example : f '' s ∩ v = f '' (s ∩ f ⁻¹' v) := begin ext y, split, { rintros ⟨⟨x, xs, rfl⟩, fxv⟩, use [x, xs, fxv] }, rintros ⟨x, ⟨⟨xs, fxv⟩, rfl⟩⟩, use [x, xs, rfl, fxv], end example : f '' (s ∩ f ⁻¹' u) ⊆ f '' s ∩ u := begin rintros y ⟨x, ⟨xs, fxu⟩, rfl⟩, use [x, xs, rfl, fxu], end example : s ∩ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∩ u) := begin rintros x ⟨xs, fxu⟩, use [x, xs, rfl, fxu], end example : s ∪ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∪ u) := begin rintros x (xs | fxu), { left, use [x, xs, rfl] }, right, use fxu end variables {I : Type*} (A : I → set α) (B : I → set β) example : f '' (⋃ i, A i) = ⋃ i, f '' A i := begin ext y, simp, split, { rintros ⟨x, ⟨i, xAi⟩, fxeq⟩, use [i, x, xAi, fxeq] }, rintros ⟨i, x, xAi, fxeq⟩, exact ⟨x, ⟨i, xAi⟩, fxeq⟩ end example : f '' (⋂ i, A i) ⊆ ⋂ i, f '' A i := begin intro y, simp, intros x h fxeq i, use [x, h i, fxeq], end example (i : I) (injf : injective f) : (⋂ i, f '' A i) ⊆ f '' (⋂ i, A i) := begin intro y, simp, intro h, rcases h i with ⟨x, xAi, fxeq⟩, use x, split, { intro i', rcases h i' with ⟨x', x'Ai, fx'eq⟩, have : f x = f x', by rw [fxeq, fx'eq], have : x = x', from injf this, rw this, exact x'Ai }, exact fxeq end example : f ⁻¹' (⋃ i, B i) = ⋃ i, f ⁻¹' (B i) := by { ext x, simp } example : f ⁻¹' (⋂ i, B i) = ⋂ i, f ⁻¹' (B i) := by { ext x, simp } end section open set real example : inj_on sqrt { x | x ≥ 0 } := begin intros x xnonneg y ynonneg, intro e, calc x = (sqrt x)^2 : by rw sq_sqrt xnonneg ... = (sqrt y)^2 : by rw e ... = y : by rw sq_sqrt ynonneg end example : inj_on (λ x, x^2) { x : ℝ | x ≥ 0 } := begin intros x xnonneg y ynonneg, intro e, dsimp at *, calc x = sqrt (x^2) : by rw sqrt_sq xnonneg ... = sqrt (y^2) : by rw e ... = y : by rw sqrt_sq ynonneg, end example : sqrt '' { x | x ≥ 0 } = {y | y ≥ 0} := begin ext y, split, { rintros ⟨x, ⟨xnonneg, rfl⟩⟩, apply sqrt_nonneg }, intro ynonneg, use y^2, dsimp at *, split, apply pow_nonneg ynonneg, apply sqrt_sq, assumption, end example : range (λ x, x^2) = {y : ℝ | y ≥ 0} := begin ext y, split, { rintros ⟨x, rfl⟩, dsimp at *, apply pow_two_nonneg }, intro ynonneg, use sqrt y, exact sq_sqrt ynonneg, end end section variables {α β : Type*} [inhabited α] noncomputable theory open_locale classical def inverse (f : α → β) : β → α := λ y : β, if h : ∃ x, f x = y then classical.some h else default theorem inverse_spec {f : α → β} (y : β) (h : ∃ x, f x = y) : f (inverse f y) = y := begin rw inverse, dsimp, rw dif_pos h, exact classical.some_spec h end variable f : α → β open function example : injective f ↔ left_inverse (inverse f) f := begin split, { intros h y, apply h, apply inverse_spec, use y }, intros h x1 x2 e, rw [←h x1, ←h x2, e] end example : injective f ↔ left_inverse (inverse f) f := ⟨λ h y, h (inverse_spec _ ⟨y, rfl⟩), λ h x1 x2 e, by rw [←h x1, ←h x2, e]⟩ example : surjective f ↔ right_inverse (inverse f) f := begin split, { intros h y, apply inverse_spec, apply h }, intros h y, use (inverse f y), apply h end example : surjective f ↔ right_inverse (inverse f) f := ⟨λ h y, inverse_spec _ (h _), λ h y, ⟨inverse f y, h _⟩⟩ end section variable {α : Type*} open function theorem Cantor : ∀ f : α → set α, ¬ surjective f := begin intros f surjf, let S := { i | i ∉ f i}, rcases surjf S with ⟨j, h⟩, have h₁ : j ∉ f j, { intro h', have : j ∉ f j, by rwa h at h', contradiction }, have h₂ : j ∈ S, from h₁, have h₃ : j ∉ S, by rwa h at h₁, contradiction end end
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@@ -0,0 +1,94 @@import data.set.lattice import data.set.function import tactic open set open function noncomputable theory open_locale classical variables {α β : Type*} [nonempty β] section variables (f : α → β) (g : β → α) def sb_aux : ℕ → set α | 0 := univ \ (g '' univ) | (n + 1) := g '' (f '' sb_aux n) def sb_set := ⋃ n, sb_aux f g n def sb_fun (x : α) : β := if x ∈ sb_set f g then f x else inv_fun g x theorem sb_right_inv {x : α} (hx : x ∉ sb_set f g) : g (inv_fun g x) = x := begin have : x ∈ g '' univ, { contrapose! hx, rw [sb_set, mem_Union], use [0], rw [sb_aux, mem_diff], exact ⟨mem_univ _, hx⟩ }, have : ∃ y, g y = x, { simp at this, assumption }, exact inv_fun_eq this end theorem sb_injective (hf: injective f) (hg : injective g) : injective (sb_fun f g) := begin set A := sb_set f g with A_def, set h := sb_fun f g with h_def, intros x₁ x₂, assume hxeq : h x₁ = h x₂, show x₁ = x₂, simp only [h_def, sb_fun, ←A_def] at hxeq, by_cases xA : x₁ ∈ A ∨ x₂ ∈ A, { wlog : x₁ ∈ A := xA using [x₁ x₂, x₂ x₁], have x₂A : x₂ ∈ A, { apply not_imp_self.mp, assume x₂nA : x₂ ∉ A, rw [if_pos xA, if_neg x₂nA] at hxeq, rw [A_def, sb_set, mem_Union] at xA, have x₂eq : x₂ = g (f x₁), { rw [hxeq, sb_right_inv f g x₂nA] }, rcases xA with ⟨n, hn⟩, rw [A_def, sb_set, mem_Union], use n + 1, simp [sb_aux], exact ⟨x₁, hn, x₂eq.symm⟩ }, rw [if_pos xA, if_pos x₂A] at hxeq, exact hf hxeq }, push_neg at xA, rw [if_neg xA.1, if_neg xA.2] at hxeq, rw [←sb_right_inv f g xA.1, hxeq, sb_right_inv f g xA.2] end theorem sb_surjective (hf: injective f) (hg : injective g) : surjective (sb_fun f g) := begin set A := sb_set f g with A_def, set h := sb_fun f g with h_def, intro y, by_cases gyA : g y ∈ A, { rw [A_def, sb_set, mem_Union] at gyA, rcases gyA with ⟨n, hn⟩, cases n with n, { simp [sb_aux] at hn, contradiction }, simp [sb_aux] at hn, rcases hn with ⟨x, xmem, hx⟩, use x, have : x ∈ A, { rw [A_def, sb_set, mem_Union], exact ⟨n, xmem⟩ }, simp only [h_def, sb_fun, if_pos this], exact hg hx }, use g y, simp only [h_def, sb_fun, if_neg gyA], apply left_inverse_inv_fun hg end end
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@@ -0,0 +1,126 @@import data.nat.gcd import data.real.irrational #print nat.coprime example (m n : nat) (h : m.coprime n) : m.gcd n = 1 := h example (m n : nat) (h : m.coprime n) : m.gcd n = 1 := by { rw nat.coprime at h, exact h } example : nat.coprime 12 7 := by norm_num example : nat.gcd 12 8 = 4 := by norm_num #check @nat.prime_def_lt example (p : ℕ) (prime_p : nat.prime p) : 2 ≤ p ∧ ∀ (m : ℕ), m < p → m ∣ p → m = 1 := by rwa nat.prime_def_lt at prime_p #check nat.prime.eq_one_or_self_of_dvd example (p : ℕ) (prime_p : nat.prime p) : ∀ (m : ℕ), m ∣ p → m = 1 ∨ m = p := prime_p.eq_one_or_self_of_dvd example : nat.prime 17 := by norm_num -- commonly used example : nat.prime 2 := nat.prime_two example : nat.prime 3 := nat.prime_three #check @nat.prime.dvd_mul #check nat.prime.dvd_mul nat.prime_two #check nat.prime_two.dvd_mul lemma even_of_even_sqr {m : ℕ} (h : 2 ∣ m^2) : 2 ∣ m := begin rw [pow_two, nat.prime_two.dvd_mul] at h, cases h; assumption end example {m : ℕ} (h : 2 ∣ m^2) : 2 ∣ m := nat.prime.dvd_of_dvd_pow nat.prime_two h example (a b c : nat) (h : a * b = a * c) (h' : a ≠ 0) : b = c := begin -- library_search suggests the following: exact (mul_right_inj' h').mp h end example {m n : ℕ} (coprime_mn : m.coprime n) : m^2 ≠ 2 * n^2 := begin intro sqr_eq, have : 2 ∣ m, sorry, obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this, have : 2 * (2 * k^2) = 2 * n^2, { rw [←sqr_eq, meq], ring }, have : 2 * k^2 = n^2, sorry, have : 2 ∣ n, sorry, have : 2 ∣ m.gcd n, sorry, have : 2 ∣ 1, sorry, norm_num at this end example {m n p : ℕ} (coprime_mn : m.coprime n) (prime_p : p.prime) : m^2 ≠ p * n^2 := sorry #check nat.factors #check nat.prime_of_mem_factors #check nat.prod_factors #check nat.factors_unique theorem factorization_mul' {m n : ℕ} (mnez : m ≠ 0) (nnez : n ≠ 0) (p : ℕ) : (m * n).factorization p = m.factorization p + n.factorization p := by { rw nat.factorization_mul mnez nnez, refl } theorem factorization_pow' (n k p : ℕ) : (n^k).factorization p = k * n.factorization p := by { rw nat.factorization_pow, refl } theorem nat.prime.factorization' {p : ℕ} (prime_p : p.prime) : p.factorization p = 1 := by { rw prime_p.factorization, simp } example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.prime) : m^2 ≠ p * n^2 := begin intro sqr_eq, have nsqr_nez : n^2 ≠ 0, by simpa, have eq1 : nat.factorization (m^2) p = 2 * m.factorization p, sorry, have eq2 : (p * n^2).factorization p = 2 * n.factorization p + 1, sorry, have : (2 * m.factorization p) % 2 = (2 * n.factorization p + 1) % 2, { rw [←eq1, sqr_eq, eq2] }, rw [add_comm, nat.add_mul_mod_self_left, nat.mul_mod_right] at this, norm_num at this end example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m^k = r * n^k) {p : ℕ} (prime_p : p.prime) : k ∣ r.factorization p := begin cases r with r, { simp }, have npow_nz : n^k ≠ 0 := λ npowz, nnz (pow_eq_zero npowz), have eq1 : (m^k).factorization p = k * m.factorization p, sorry, have eq2 : (r.succ * n^k).factorization p = k * n.factorization p + r.succ.factorization p, sorry, have : r.succ.factorization p = k * m.factorization p - k * n.factorization p, { rw [←eq1, pow_eq, eq2, add_comm, nat.add_sub_cancel] }, rw this, sorry end #check multiplicity #check @irrational_nrt_of_n_not_dvd_multiplicity #check irrational_sqrt_two
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@@ -0,0 +1,151 @@import data.nat.prime import algebra.big_operators import tactic example (n : nat) : n.succ ≠ nat.zero := nat.succ_ne_zero n example (m n : nat) (h : m.succ = n.succ) : m = n := nat.succ.inj h def fac : ℕ → ℕ | 0 := 1 | (n + 1) := (n + 1) * fac n example : fac 0 = 1 := rfl example : fac 0 = 1 := by rw fac example : fac 0 = 1 := by simp [fac] example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := rfl example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := by rw fac example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := by simp [fac] theorem fac_pos (n : ℕ) : 0 < fac n := begin induction n with n ih, { rw fac, exact zero_lt_one }, rw fac, exact mul_pos n.succ_pos ih, end theorem dvd_fac {i n : ℕ} (ipos : 0 < i) (ile : i ≤ n) : i ∣ fac n := begin induction n with n ih, { exact absurd ipos (not_lt_of_ge ile) }, rw fac, cases nat.of_le_succ ile with h h, { apply dvd_mul_of_dvd_right (ih h) }, rw h, apply dvd_mul_right end theorem pow_two_le_fac (n : ℕ) : 2^(n-1) ≤ fac n := begin cases n with n, { simp [fac] }, sorry end section variables {α : Type*} (s : finset ℕ) (f : ℕ → ℕ) (n : ℕ) #check finset.sum s f #check finset.prod s f open_locale big_operators open finset example : s.sum f = ∑ x in s, f x := rfl example : s.prod f = ∏ x in s, f x := rfl example : (range n).sum f = ∑ x in range n, f x := rfl example : (range n).prod f = ∏ x in range n, f x := rfl example (f : ℕ → ℕ) : ∑ x in range 0, f x = 0 := finset.sum_range_zero f example (f : ℕ → ℕ) (n : ℕ): ∑ x in range n.succ, f x = (∑ x in range n, f x) + f n := finset.sum_range_succ f n example (f : ℕ → ℕ) : ∏ x in range 0, f x = 1 := finset.prod_range_zero f example (f : ℕ → ℕ) (n : ℕ): ∏ x in range n.succ, f x = (∏ x in range n, f x) * f n := finset.prod_range_succ f n example (n : ℕ) : fac n = ∏ i in range n, (i + 1) := begin induction n with n ih, { simp [fac] }, simp [fac, ih, prod_range_succ, mul_comm] end example (a b c d e f : ℕ) : a * ((b * c) * f * (d * e)) = d * (a * f * e) * (c * b) := by simp [mul_assoc, mul_comm, mul_left_comm] theorem sum_id (n : ℕ) : ∑ i in range (n + 1), i = n * (n + 1) / 2 := begin symmetry, apply nat.div_eq_of_eq_mul_right (by norm_num : 0 < 2), induction n with n ih, { simp }, rw [finset.sum_range_succ, mul_add 2, ←ih, nat.succ_eq_add_one], ring end theorem sum_sqr (n : ℕ) : ∑ i in range (n + 1), i^2 = n * (n + 1) * (2 *n + 1) / 6 := sorry end inductive my_nat | zero : my_nat | succ : my_nat → my_nat namespace my_nat def add : my_nat → my_nat → my_nat | x zero := x | x (succ y) := succ (add x y) def mul : my_nat → my_nat → my_nat | x zero := zero | x (succ y) := add (mul x y) x theorem zero_add (n : my_nat) : add zero n = n := begin induction n with n ih, { refl }, rw [add, ih] end theorem succ_add (m n : my_nat) : add (succ m) n = succ (add m n) := begin induction n with n ih, { refl }, rw [add, ih], refl end theorem add_comm (m n : my_nat) : add m n = add n m := begin induction n with n ih, { rw zero_add, refl }, rw [add, succ_add, ih] end theorem add_assoc (m n k : my_nat) : add (add m n) k = add m (add n k) := sorry theorem mul_add (m n k : my_nat) : mul m (add n k) = add (mul m n) (mul m k) := sorry theorem zero_mul (n : my_nat) : mul zero n = zero := sorry theorem succ_mul (m n : my_nat) : mul (succ m) n = add (mul m n) n := sorry theorem mul_comm (m n : my_nat) : mul m n = mul n m := sorry end my_nat
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@@ -0,0 +1,254 @@import data.nat.prime import algebra.big_operators import tactic open_locale big_operators theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := begin cases m, contradiction, cases m, contradiction, repeat { apply nat.succ_le_succ }, apply zero_le end example {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := begin by_contradiction h, push_neg at h, interval_cases m; contradiction end example {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := begin by_contradiction h, push_neg at h, revert m h h0 h1, dec_trivial end example {m : ℕ} (h : m < 2) : m = 0 ∨ m = 1 := by dec_trivial! example {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by omega theorem exists_prime_factor {n : nat} (h : 2 ≤ n) : ∃ p : nat, p.prime ∧ p ∣ n := begin by_cases np : n.prime, { use [n, np, dvd_rfl] }, induction n using nat.strong_induction_on with n ih, dsimp at ih, rw nat.prime_def_lt at np, push_neg at np, rcases np h with ⟨m, mltn, mdvdn, mne1⟩, have : m ≠ 0, { intro mz, rw [mz, zero_dvd_iff] at mdvdn, linarith }, have mgt2 : 2 ≤ m := two_le this mne1, by_cases mp : m.prime, { use [m, mp, mdvdn] }, rcases ih m mltn mgt2 mp with ⟨p, pp, pdvd⟩, use [p, pp, pdvd.trans mdvdn] end theorem primes_infinite : ∀ n, ∃ p > n, nat.prime p := begin intro n, have : 2 ≤ nat.factorial (n + 1) + 1, sorry, rcases exists_prime_factor this with ⟨p, pp, pdvd⟩, refine ⟨p, _, pp⟩, show p > n, by_contradiction ple, push_neg at ple, have : p ∣ nat.factorial (n + 1), sorry, have : p ∣ 1, sorry, show false, sorry end open finset section variables {α : Type*} [decidable_eq α] (r s t : finset α) example : r ∩ (s ∪ t) ⊆ (r ∩ s) ∪ (r ∩ t) := begin rw subset_iff, intro x, rw [mem_inter, mem_union, mem_union, mem_inter, mem_inter], tauto end example : r ∩ (s ∪ t) ⊆ (r ∩ s) ∪ (r ∩ t) := by { simp [subset_iff], intro x, tauto } example : (r ∩ s) ∪ (r ∩ t) ⊆ r ∩ (s ∪ t) := by { simp [subset_iff], intro x, tauto } example : (r ∩ s) ∪ (r ∩ t) = r ∩ (s ∪ t) := by { ext x, simp, tauto } end section variables {α : Type*} [decidable_eq α] (r s t : finset α) example : (r ∪ s) ∩ (r ∪ t) = r ∪ (s ∩ t) := sorry example : (r \ s \ t) = r \ (s ∪ t) := sorry end example (s : finset ℕ) (n : ℕ) (h : n ∈ s) : n ∣ (∏ i in s, i) := finset.dvd_prod_of_mem _ h theorem nat.prime.eq_of_dvd_of_prime {p q : ℕ} (prime_p : nat.prime p) (prime_q : nat.prime q) (h : p ∣ q) : p = q := sorry theorem mem_of_dvd_prod_primes {s : finset ℕ} {p : ℕ} (prime_p : p.prime) : (∀ n ∈ s, nat.prime n) → (p ∣ ∏ n in s, n) → p ∈ s := begin intros h₀ h₁, induction s using finset.induction_on with a s ans ih, { simp at h₁, linarith [prime_p.two_le] }, simp [finset.prod_insert ans, prime_p.dvd_mul] at h₀ h₁, rw mem_insert, sorry end example (s : finset ℕ) (x : ℕ) : x ∈ s.filter nat.prime ↔ x ∈ s ∧ x.prime := mem_filter theorem primes_infinite' : ∀ (s : finset nat), ∃ p, nat.prime p ∧ p ∉ s := begin intro s, by_contradiction h, push_neg at h, set s' := s.filter nat.prime with s'_def, have mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ n.prime, { intro n, simp [s'_def], apply h }, have : 2 ≤ (∏ i in s', i) + 1, sorry, rcases exists_prime_factor this with ⟨p, pp, pdvd⟩, have : p ∣ (∏ i in s', i), sorry, have : p ∣ 1, { convert nat.dvd_sub' pdvd this, simp }, show false, sorry end theorem bounded_of_ex_finset (Q : ℕ → Prop): (∃ s : finset ℕ, ∀ k, Q k → k ∈ s) → ∃ n, ∀ k, Q k → k < n := begin rintros ⟨s, hs⟩, use s.sup id + 1, intros k Qk, apply nat.lt_succ_of_le, show id k ≤ s.sup id, apply le_sup (hs k Qk) end theorem ex_finset_of_bounded (Q : ℕ → Prop) [decidable_pred Q] : (∃ n, ∀ k, Q k → k ≤ n) → (∃ s : finset ℕ, ∀ k, Q k ↔ k ∈ s) := begin rintros ⟨n, hn⟩, use (range (n + 1)).filter Q, intro k, simp [nat.lt_succ_iff], exact hn k end example : 27 % 4 = 3 := by norm_num example (n : ℕ) : (4 * n + 3) % 4 = 3 := by { rw [add_comm, nat.add_mul_mod_self_left], norm_num } theorem mod_4_eq_3_or_mod_4_eq_3 {m n : ℕ} (h : m * n % 4 = 3) : m % 4 = 3 ∨ n % 4 = 3 := begin revert h, rw [nat.mul_mod], have : m % 4 < 4 := nat.mod_lt m (by norm_num), interval_cases m % 4 with hm; simp [hm], have : n % 4 < 4 := nat.mod_lt n (by norm_num), interval_cases n % 4 with hn; simp [hn]; norm_num end theorem two_le_of_mod_4_eq_3 {n : ℕ} (h : n % 4 = 3) : 2 ≤ n := by apply two_le; { intro neq, rw neq at h, norm_num at h } theorem aux {m n : ℕ} (h₀ : m ∣ n) (h₁ : 2 ≤ m) (h₂ : m < n) : (n / m) ∣ n ∧ n / m < n := sorry theorem exists_prime_factor_mod_4_eq_3 {n : nat} (h : n % 4 = 3) : ∃ p : nat, p.prime ∧ p ∣ n ∧ p % 4 = 3 := begin by_cases np : n.prime, { use [n, np, dvd_rfl, h] }, induction n using nat.strong_induction_on with n ih, dsimp at ih, rw nat.prime_def_lt at np, push_neg at np, rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩, have mge2 : 2 ≤ m, { apply two_le _ mne1, intro mz, rw [mz, zero_dvd_iff] at mdvdn, linarith }, have neq : m * (n / m) = n := nat.mul_div_cancel' mdvdn, have : m % 4 = 3 ∨ (n / m) % 4 = 3, { apply mod_4_eq_3_or_mod_4_eq_3, rw [neq, h] }, cases this with h1 h1, { sorry }, sorry end example (m n : ℕ) (s : finset ℕ) (h : m ∈ erase s n) : m ≠ n ∧ m ∈ s := by rwa mem_erase at h example (m n : ℕ) (s : finset ℕ) (h : m ∈ erase s n) : m ≠ n ∧ m ∈ s := by { simp at h, assumption } theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, nat.prime p ∧ p % 4 = 3 := begin by_contradiction h, push_neg at h, cases h with n hn, have : ∃ s : finset nat, ∀ p : ℕ, p.prime ∧ p % 4 = 3 ↔ p ∈ s, { apply ex_finset_of_bounded, use n, contrapose! hn, rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩, exact ⟨p, pltn, pp, p4⟩ }, cases this with s hs, have h₀ : 2 ≤ 4 * (∏ i in erase s 3, i) + 3, sorry, have h₁ : (4 * (∏ i in erase s 3, i) + 3) % 4 = 3, sorry, rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩, have ps : p ∈ s, sorry, have pne3 : p ≠ 3, sorry, have : p ∣ 4 * (∏ i in erase s 3, i), sorry, have : p ∣ 3, sorry, have : p = 3, sorry, contradiction end
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@@ -0,0 +1,107 @@import data.nat.gcd import data.real.irrational lemma even_of_even_sqr {m : ℕ} (h : 2 ∣ m^2) : 2 ∣ m := begin rw [pow_two, nat.prime_two.dvd_mul] at h, cases h; assumption end example {m n : ℕ} (coprime_mn : m.coprime n) : m^2 ≠ 2 * n^2 := begin intro sqr_eq, have : 2 ∣ m, { apply even_of_even_sqr, rw sqr_eq, apply dvd_mul_right }, obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this, have : 2 * (2 * k^2) = 2 * n^2, { rw [←sqr_eq, meq], ring }, have : 2 * k^2 = n^2, from (mul_right_inj' (by norm_num)).mp this, have : 2 ∣ n, { apply even_of_even_sqr, rw ←this, apply dvd_mul_right }, have : 2 ∣ m.gcd n, by apply nat.dvd_gcd; assumption, have : 2 ∣ 1, { convert this, symmetry, exact coprime_mn }, norm_num at this end example {m n p : ℕ} (coprime_mn : m.coprime n) (prime_p : p.prime) : m^2 ≠ p * n^2 := begin intro sqr_eq, have : p ∣ m, { apply prime_p.dvd_of_dvd_pow, rw sqr_eq, apply dvd_mul_right }, obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this, have : p * (p * k^2) = p * n^2, { rw [←sqr_eq, meq], ring }, have : p * k^2 = n^2, { apply (mul_right_inj' _).mp this, exact prime_p.ne_zero }, have : p ∣ n, { apply prime_p.dvd_of_dvd_pow, rw ←this, apply dvd_mul_right }, have : p ∣ nat.gcd m n, { apply nat.dvd_gcd; assumption }, have : p ∣ 1, { convert this, symmetry, exact coprime_mn }, have : 2 ≤ 1, { apply prime_p.two_le.trans, exact nat.le_of_dvd zero_lt_one this }, norm_num at this end theorem factorization_mul' {m n : ℕ} (mnez : m ≠ 0) (nnez : n ≠ 0) (p : ℕ) : (m * n).factorization p = m.factorization p + n.factorization p := by { rw nat.factorization_mul mnez nnez, refl } theorem factorization_pow' (n k p : ℕ) : (n^k).factorization p = k * n.factorization p := by { rw nat.factorization_pow, refl } theorem nat.prime.factorization' {p : ℕ} (prime_p : p.prime) : p.factorization p = 1 := by { rw prime_p.factorization, simp } example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.prime) : m^2 ≠ p * n^2 := begin intro sqr_eq, have nsqr_nez : n^2 ≠ 0, by simpa, have eq1 : nat.factorization (m^2) p = 2 * m.factorization p, by { rw factorization_pow' }, have eq2 : (p * n^2).factorization p = 2 * n.factorization p + 1, { rw [factorization_mul' prime_p.ne_zero nsqr_nez, prime_p.factorization', factorization_pow', add_comm] }, have : (2 * m.factorization p) % 2 = (2 * n.factorization p + 1) % 2, { rw [←eq1, sqr_eq, eq2] }, rw [add_comm, nat.add_mul_mod_self_left, nat.mul_mod_right] at this, norm_num at this end example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m^k = r * n^k) {p : ℕ} (prime_p : p.prime) : k ∣ r.factorization p := begin cases r with r, { simp }, have npow_nz : n^k ≠ 0 := λ npowz, nnz (pow_eq_zero npowz), have eq1 : (m^k).factorization p = k * m.factorization p, by rw factorization_pow', have eq2 : (r.succ * n^k).factorization p = k * n.factorization p + r.succ.factorization p, { rw [factorization_mul' r.succ_ne_zero npow_nz, factorization_pow', add_comm] }, have : r.succ.factorization p = k * m.factorization p - k * n.factorization p, { rw [←eq1, pow_eq, eq2, add_comm, nat.add_sub_cancel] }, rw this, apply nat.dvd_sub'; apply nat.dvd_mul_right end
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@@ -0,0 +1,114 @@import data.nat.prime import algebra.big_operators import tactic def fac : ℕ → ℕ | 0 := 1 | (n + 1) := (n + 1) * fac n theorem pow_two_le_fac (n : ℕ) : 2^(n-1) ≤ fac n := begin cases n with n, { simp [fac] }, induction n with n ih, { simp [fac] }, simp at *, rw [pow_succ, fac], apply nat.mul_le_mul _ ih, repeat { apply nat.succ_le_succ }, apply zero_le end section variables {α : Type*} (s : finset ℕ) (f : ℕ → ℕ) (n : ℕ) open_locale big_operators open finset theorem sum_sqr (n : ℕ) : ∑ i in range (n + 1), i^2 = n * (n + 1) * (2 *n + 1) / 6 := begin symmetry, apply nat.div_eq_of_eq_mul_right (by norm_num : 0 < 6), induction n with n ih, { simp }, rw [finset.sum_range_succ, mul_add 6, ←ih, nat.succ_eq_add_one], ring end end inductive my_nat | zero : my_nat | succ : my_nat → my_nat namespace my_nat def add : my_nat → my_nat → my_nat | x zero := x | x (succ y) := succ (add x y) def mul : my_nat → my_nat → my_nat | x zero := zero | x (succ y) := add (mul x y) x theorem zero_add (n : my_nat) : add zero n = n := begin induction n with n ih, { refl }, rw [add, ih] end theorem succ_add (m n : my_nat) : add (succ m) n = succ (add m n) := begin induction n with n ih, { refl }, rw [add, ih], refl end theorem add_comm (m n : my_nat) : add m n = add n m := begin induction n with n ih, { rw zero_add, refl }, rw [add, succ_add, ih] end theorem add_assoc (m n k : my_nat) : add (add m n) k = add m (add n k) := begin induction k with k ih, { refl }, rw [add, ih], refl end theorem mul_add (m n k : my_nat) : mul m (add n k) = add (mul m n) (mul m k) := begin induction k with k ih, { refl }, rw [add, mul, mul, ih, add_assoc] end theorem zero_mul (n : my_nat) : mul zero n = zero := begin induction n with n ih, { refl }, rw [mul, ih], refl end theorem succ_mul (m n : my_nat) : mul (succ m) n = add (mul m n) n := begin induction n with n ih, { refl }, rw [mul, mul, ih, add_assoc, add_assoc, add_comm n, succ_add], refl end theorem mul_comm (m n : my_nat) : mul m n = mul n m := begin induction n with n ih, { rw [zero_mul], refl }, rw [mul, ih, succ_mul] end end my_nat
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@@ -0,0 +1,246 @@import data.nat.prime import algebra.big_operators import tactic open_locale big_operators theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := begin cases m, contradiction, cases m, contradiction, repeat { apply nat.succ_le_succ }, apply zero_le end theorem exists_prime_factor {n : nat} (h : 2 ≤ n) : ∃ p : nat, p.prime ∧ p ∣ n := begin by_cases np : n.prime, { use [n, np, dvd_rfl] }, induction n using nat.strong_induction_on with n ih, dsimp at ih, rw nat.prime_def_lt at np, push_neg at np, rcases np h with ⟨m, mltn, mdvdn, mne1⟩, have : m ≠ 0, { intro mz, rw [mz, zero_dvd_iff] at mdvdn, linarith }, have mgt2 : 2 ≤ m := two_le this mne1, by_cases mp : m.prime, { use [m, mp, mdvdn] }, rcases ih m mltn mgt2 mp with ⟨p, pp, pdvd⟩, use [p, pp, pdvd.trans mdvdn] end theorem primes_infinite : ∀ n, ∃ p > n, nat.prime p := begin intro n, have : 2 ≤ nat.factorial (n + 1) + 1, { apply nat.succ_le_succ, exact nat.succ_le_of_lt (nat.factorial_pos _) }, rcases exists_prime_factor this with ⟨p, pp, pdvd⟩, refine ⟨p, _, pp⟩, show p > n, by_contradiction ple, push_neg at ple, have : p ∣ nat.factorial (n + 1), { apply nat.dvd_factorial, apply pp.pos, linarith }, have : p ∣ 1, { convert nat.dvd_sub' pdvd this, simp }, show false, have := nat.le_of_dvd zero_lt_one this, linarith [pp.two_le] end open finset section variables {α : Type*} [decidable_eq α] (r s t : finset α) example : (r ∪ s) ∩ (r ∪ t) = r ∪ (s ∩ t) := begin ext x, rw [mem_inter, mem_union, mem_union, mem_union, mem_inter], tauto end example : (r ∪ s) ∩ (r ∪ t) = r ∪ (s ∩ t) := by { ext x, simp, tauto } example : (r \ s \ t) = r \ (s ∪ t) := begin ext x, rw [mem_sdiff, mem_sdiff, mem_sdiff, mem_union], tauto end example : (r \ s \ t) = r \ (s ∪ t) := by { ext x, simp, tauto } end theorem nat.prime.eq_of_dvd_of_prime {p q : ℕ} (prime_p : nat.prime p) (prime_q : nat.prime q) (h : p ∣ q) : p = q := begin cases prime_q.eq_one_or_self_of_dvd _ h, { linarith [prime_p.two_le] }, assumption end theorem mem_of_dvd_prod_primes {s : finset ℕ} {p : ℕ} (prime_p : p.prime) : (∀ n ∈ s, nat.prime n) → (p ∣ ∏ n in s, n) → p ∈ s := begin intros h₀ h₁, induction s using finset.induction_on with a s ans ih, { simp at h₁, linarith [prime_p.two_le] }, simp [finset.prod_insert ans, prime_p.dvd_mul] at h₀ h₁, rw mem_insert, cases h₁ with h₁ h₁, { left, exact prime_p.eq_of_dvd_of_prime h₀.1 h₁ }, right, exact ih h₀.2 h₁ end theorem primes_infinite' : ∀ (s : finset nat), ∃ p, nat.prime p ∧ p ∉ s := begin intro s, by_contradiction h, push_neg at h, set s' := s.filter nat.prime with s'_def, have mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ n.prime, { intro n, simp [s'_def], apply h }, have : 2 ≤ (∏ i in s', i) + 1, { apply nat.succ_le_succ, apply nat.succ_le_of_lt, apply finset.prod_pos, intros n ns', apply (mem_s'.mp ns').pos }, rcases exists_prime_factor this with ⟨p, pp, pdvd⟩, have : p ∣ (∏ i in s', i), { apply dvd_prod_of_mem, rw mem_s', apply pp }, have : p ∣ 1, { convert nat.dvd_sub' pdvd this, simp }, show false, have := nat.le_of_dvd zero_lt_one this, linarith [pp.two_le] end theorem bounded_of_ex_finset (Q : ℕ → Prop): (∃ s : finset ℕ, ∀ k, Q k → k ∈ s) → ∃ n, ∀ k, Q k → k < n := begin rintros ⟨s, hs⟩, use s.sup id + 1, intros k Qk, apply nat.lt_succ_of_le, show id k ≤ s.sup id, apply le_sup (hs k Qk) end theorem ex_finset_of_bounded (Q : ℕ → Prop) [decidable_pred Q] : (∃ n, ∀ k, Q k → k ≤ n) → (∃ s : finset ℕ, ∀ k, Q k ↔ k ∈ s) := begin rintros ⟨n, hn⟩, use (range (n + 1)).filter Q, intro k, simp [nat.lt_succ_iff], exact hn k end theorem mod_4_eq_3_or_mod_4_eq_3 {m n : ℕ} (h : m * n % 4 = 3) : m % 4 = 3 ∨ n % 4 = 3 := begin revert h, rw [nat.mul_mod], have : m % 4 < 4 := nat.mod_lt m (by norm_num), interval_cases m % 4 with hm; simp [hm], have : n % 4 < 4 := nat.mod_lt n (by norm_num), interval_cases n % 4 with hn; simp [hn]; norm_num end theorem two_le_of_mod_4_eq_3 {n : ℕ} (h : n % 4 = 3) : 2 ≤ n := by apply two_le; { intro neq, rw neq at h, norm_num at h } theorem aux {m n : ℕ} (h₀ : m ∣ n) (h₁ : 2 ≤ m) (h₂ : m < n) : (n / m) ∣ n ∧ n / m < n := begin split, { exact nat.div_dvd_of_dvd h₀ }, exact nat.div_lt_self (lt_of_le_of_lt (zero_le _) h₂) h₁ end theorem exists_prime_factor_mod_4_eq_3 {n : nat} (h : n % 4 = 3) : ∃ p : nat, p.prime ∧ p ∣ n ∧ p % 4 = 3 := begin by_cases np : n.prime, { use [n, np, dvd_rfl, h] }, induction n using nat.strong_induction_on with n ih, dsimp at ih, rw nat.prime_def_lt at np, push_neg at np, rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩, have mge2 : 2 ≤ m, { apply two_le _ mne1, intro mz, rw [mz, zero_dvd_iff] at mdvdn, linarith }, have neq : m * (n / m) = n := nat.mul_div_cancel' mdvdn, have : m % 4 = 3 ∨ (n / m) % 4 = 3, { apply mod_4_eq_3_or_mod_4_eq_3, rw [neq, h] }, cases this with h1 h1, { by_cases mp : m.prime, { use [m, mp, mdvdn, h1] }, rcases ih m mltn h1 mp with ⟨p, pp, pdvd, p4eq⟩, use [p, pp, pdvd.trans mdvdn, p4eq] }, obtain ⟨nmdvdn, nmltn⟩ := aux mdvdn mge2 mltn, by_cases nmp : (n / m).prime, { use [n / m, nmp, nmdvdn, h1] }, rcases ih (n / m) nmltn h1 nmp with ⟨p, pp, pdvd, p4eq⟩, use [p, pp, pdvd.trans nmdvdn, p4eq] end theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, nat.prime p ∧ p % 4 = 3 := begin by_contradiction h, push_neg at h, cases h with n hn, have : ∃ s : finset nat, ∀ p : ℕ, p.prime ∧ p % 4 = 3 ↔ p ∈ s, { apply ex_finset_of_bounded, use n, contrapose! hn, rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩, exact ⟨p, pltn, pp, p4⟩ }, cases this with s hs, have h₀ : 2 ≤ 4 * (∏ i in erase s 3, i) + 3, { apply le_add_left, norm_num }, have h₁ : (4 * (∏ i in erase s 3, i) + 3) % 4 = 3, { rw [add_comm, nat.add_mul_mod_self_left], norm_num }, rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩, have ps : p ∈ s, { rw ←hs p, exact ⟨pp, p4eq⟩ }, have pne3 : p ≠ 3, { intro peq, rw [peq, ←nat.dvd_add_iff_left (dvd_refl 3)] at pdvd, rw nat.prime_three.dvd_mul at pdvd, norm_num at pdvd, have : 3 ∈ s.erase 3, { apply mem_of_dvd_prod_primes nat.prime_three _ pdvd, intro n, simp [← hs n], tauto }, simp at this, exact this }, have : p ∣ 4 * (∏ i in erase s 3, i), { apply dvd_trans _ (dvd_mul_left _ _), apply dvd_prod_of_mem, simp, split; assumption }, have : p ∣ 3, { convert nat.dvd_sub' pdvd this, simp }, have : p = 3, { apply pp.eq_of_dvd_of_prime nat.prime_three this }, contradiction end
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@@ -0,0 +1,228 @@import algebra.big_operators.ring import data.real.basic @[ext] structure point := (x : ℝ) (y : ℝ) (z : ℝ) #check point.ext example (a b : point) (hx : a.x = b.x) (hy : a.y = b.y) (hz : a.z = b.z) : a = b := begin ext, repeat { assumption } end def my_point1 : point := { x := 2, y := -1, z := 4 } def my_point2 := { point . x := 2, y := -1, z := 4 } def my_point3 : point := ⟨2, -1, 4⟩ def my_point4 := point.mk 2 (-1) 4 structure point' := build :: (x : ℝ) (y : ℝ) (z : ℝ) #check point'.build 2 (-1) 4 namespace point def add (a b : point) : point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ def add' (a b : point) : point := { x := a.x + b.x, y := a.y + b.y, z := a.z + b.z } #check add my_point1 my_point2 #check my_point1.add my_point2 end point #check point.add my_point1 my_point2 #check my_point1.add my_point2 namespace point protected theorem add_comm (a b : point) : add a b = add b a := begin rw [add, add], ext; dsimp, repeat { apply add_comm } end example (a b : point) : add a b = add b a := by simp [add, add_comm] theorem add_x (a b : point) : (a.add b).x = a.x + b.x := rfl def add_alt : point → point → point | (point.mk x₁ y₁ z₁) (point.mk x₂ y₂ z₂) := ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ def add_alt' : point → point → point | ⟨x₁, y₁, z₁⟩ ⟨x₂, y₂, z₂⟩ := ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ theorem add_alt_x (a b : point) : (a.add_alt b).x = a.x + b.x := by { cases a, cases b, refl } theorem add_alt_comm (a b : point) : add_alt a b = add_alt b a := begin rcases a with ⟨xa, ya, za⟩, rcases b with ⟨xb, yb, zb⟩, rw [add_alt, add_alt], ext; dsimp, apply add_comm, repeat { apply add_comm }, end example (a b : point) : add_alt a b = add_alt b a := begin rcases a with ⟨xa, ya, za⟩, rcases b with ⟨xb, yb, zb⟩, simp [add_alt, add_comm] end example : ∀ a b : point, add_alt a b = add_alt b a := begin rintros ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩, simp [add_alt, add_comm] end example : ∀ a b : point, add a b = add b a := λ ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩, by simp [add, add_comm] protected theorem add_assoc (a b c : point) : (a.add b).add c = a.add (b.add c) := sorry def smul (r : ℝ) (a : point) : point := sorry theorem smul_distrib (r : ℝ) (a b : point) : (smul r a).add (smul r b) = smul r (a.add b) := sorry end point structure standard_two_simplex := (x : ℝ) (y : ℝ) (z : ℝ) (x_nonneg : 0 ≤ x) (y_nonneg : 0 ≤ y) (z_nonneg : 0 ≤ z) (sum_eq : x + y + z = 1) namespace standard_two_simplex def swap_xy (a : standard_two_simplex) : standard_two_simplex := { x := a.y, y := a.x, z := a.z, x_nonneg := a.y_nonneg, y_nonneg := a.x_nonneg, z_nonneg := a.z_nonneg, sum_eq := by rw [add_comm a.y a.x, a.sum_eq] } noncomputable theory def midpoint (a b : standard_two_simplex) : standard_two_simplex := { x := (a.x + b.x) / 2, y := (a.y + b.y) / 2, z := (a.z + b.z) / 2, x_nonneg := div_nonneg (add_nonneg a.x_nonneg b.x_nonneg) (by norm_num), y_nonneg := div_nonneg (add_nonneg a.y_nonneg b.y_nonneg) (by norm_num), z_nonneg := div_nonneg (add_nonneg a.z_nonneg b.z_nonneg) (by norm_num), sum_eq := by { field_simp, linarith [a.sum_eq, b.sum_eq]} } def weighted_average (lambda : real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) (a b : standard_two_simplex) : standard_two_simplex := sorry end standard_two_simplex open_locale big_operators structure standard_simplex (n : ℕ) := (v : fin n → ℝ) (nonneg : ∀ i : fin n, 0 ≤ v i) (sum_eq_one : ∑ i, v i = 1) namespace standard_simplex def midpoint (n : ℕ) (a b : standard_simplex n) : standard_simplex n := { v := λ i, (a.v i + b.v i) / 2, nonneg := begin intro i, apply div_nonneg, { linarith [a.nonneg i, b.nonneg i] }, norm_num end, sum_eq_one := begin simp [div_eq_mul_inv, ←finset.sum_mul, finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one], field_simp end } end standard_simplex structure is_linear (f : ℝ → ℝ) := (is_additive : ∀ x y, f (x + y) = f x + f y) (preserves_mul : ∀ x c, f (c * x) = c * f x) section variables (f : ℝ → ℝ) (linf : is_linear f) #check linf.is_additive #check linf.preserves_mul end def point'' := ℝ × ℝ × ℝ def is_linear' (f : ℝ → ℝ) := (∀ x y, f (x + y) = f x + f y) ∧ (∀ x c, f (c * x) = c * f x) def preal := { y : ℝ // 0 < y } section variable x : preal #check x.val #check x.property #check x.1 #check x.2 end def standard_two_simplex' := { p : ℝ × ℝ × ℝ // 0 ≤ p.1 ∧ 0 ≤ p.2.1 ∧ 0 ≤ p.2.2 ∧ p.1 + p.2.1 + p.2.2 = 1 } def standard_simplex' (n : ℕ) := { v : fin n → ℝ // (∀ i : fin n, 0 ≤ v i) ∧ (∑ i, v i = 1) } def std_simplex := Σ n : ℕ, standard_simplex n section variable s : std_simplex #check s.fst #check s.snd #check s.1 #check s.2 end
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@@ -0,0 +1,154 @@import data.real.basic structure group₁ (α : Type*) := (mul: α → α → α) (one: α) (inv: α → α) (mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z)) (mul_one: ∀ x : α, mul x one = x) (one_mul: ∀ x : α, mul x one = x) (mul_left_inv : ∀ x : α, mul (inv x) x = one) structure Group₁ := (α : Type*) (str : group₁ α) section variables (α β γ : Type*) variables (f : α ≃ β) (g : β ≃ γ) #check equiv α β #check (f.to_fun : α → β) #check (f.inv_fun : β → α) #check (f.right_inv: ∀ x : β, f (f.inv_fun x) = x) #check (f.left_inv: ∀ x : α, f.inv_fun (f x) = x) #check (equiv.refl α : α ≃ α) #check (f.symm : β ≃ α) #check (f.trans g : α ≃ γ) example (x : α) : (f.trans g).to_fun x = g.to_fun (f.to_fun x) := rfl example (x : α) : (f.trans g) x = g (f x) := rfl example : (f.trans g : α → γ) = g ∘ f := rfl end example (α : Type*) : equiv.perm α = (α ≃ α) := rfl def perm_group {α : Type*} : group₁ (equiv.perm α) := { mul := λ f g, equiv.trans g f, one := equiv.refl α, inv := equiv.symm, mul_assoc := λ f g h, (equiv.trans_assoc _ _ _).symm, one_mul := equiv.trans_refl, mul_one := equiv.refl_trans, mul_left_inv := equiv.self_trans_symm } structure add_group₁ (α : Type*) := (add : α → α → α) -- fill in the rest @[ext] structure point := (x : ℝ) (y : ℝ) (z : ℝ) namespace point def add (a b : point) : point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ def neg (a b : point) : point := sorry def zero : point := sorry def add_group_point : add_group point := sorry end point section variables {α : Type*} (f g : equiv.perm α) (n : ℕ) #check f * g #check mul_assoc f g g⁻¹ -- group power, defined for any group #check g^n example : f * g * (g⁻¹) = f := by { rw [mul_assoc, mul_right_inv, mul_one] } example : f * g * (g⁻¹) = f := mul_inv_cancel_right f g example {α : Type*} (f g : equiv.perm α) : g.symm.trans (g.trans f) = f := mul_inv_cancel_right f g end class group₂ (α : Type*) := (mul: α → α → α) (one: α) (inv: α → α) (mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z)) (mul_one: ∀ x : α, mul x one = x) (one_mul: ∀ x : α, mul x one = x) (mul_left_inv : ∀ x : α, mul (inv x) x = one) instance {α : Type*} : group₂ (equiv.perm α) := { mul := λ f g, equiv.trans g f, one := equiv.refl α, inv := equiv.symm, mul_assoc := λ f g h, (equiv.trans_assoc _ _ _).symm, one_mul := equiv.trans_refl, mul_one := equiv.refl_trans, mul_left_inv := equiv.self_trans_symm } #check @group₂.mul def my_square {α : Type*} [group₂ α] (x : α) := group₂.mul x x #check @my_square section variables {β : Type*} (f g : equiv.perm β) example : group₂.mul f g = g.trans f := rfl example : my_square f = f.trans f := rfl end instance : inhabited point := { default := ⟨0, 0, 0⟩ } #check (default : point) example : ([] : list point).head = default := rfl instance : has_add point := { add := point.add } section variables x y : point #check x + y example : x + y = point.add x y := rfl end instance has_mul_group₂ {α : Type*} [group₂ α] : has_mul α := ⟨group₂.mul⟩ instance has_one_group₂ {α : Type*} [group₂ α] : has_one α := ⟨group₂.one⟩ instance has_inv_group₂ {α : Type*} [group₂ α] : has_inv α := ⟨group₂.inv⟩ section variables {α : Type*} (f g : equiv.perm α) #check f * 1 * g⁻¹ def foo: f * 1 * g⁻¹ = g.symm.trans ((equiv.refl α).trans f) := rfl end class add_group₂ (α : Type*) := (add : α → α → α) -- fill in the rest
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@@ -0,0 +1,101 @@import algebra.big_operators.ring import data.real.basic @[ext] structure point := (x : ℝ) (y : ℝ) (z : ℝ) namespace point def add (a b : point) : point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ protected theorem add_assoc (a b c : point) : (a.add b).add c = a.add (b.add c) := by { simp [add, add_assoc] } def smul (r : ℝ) (a : point) : point := ⟨r * a.x, r * a.y, r * a.z⟩ theorem smul_distrib (r : ℝ) (a b : point) : (smul r a).add (smul r b) = smul r (a.add b) := by { simp [add, smul, mul_add] } end point structure standard_two_simplex := (x : ℝ) (y : ℝ) (z : ℝ) (x_nonneg : 0 ≤ x) (y_nonneg : 0 ≤ y) (z_nonneg : 0 ≤ z) (sum_eq : x + y + z = 1) namespace standard_two_simplex noncomputable theory def weighted_average (lambda : real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) (a b : standard_two_simplex) : standard_two_simplex := { x := lambda * a.x + (1 - lambda) * b.x, y := lambda * a.y + (1 - lambda) * b.y, z := lambda * a.z + (1 - lambda) * b.z, x_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.x_nonneg) (mul_nonneg (by linarith) b.x_nonneg), y_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.y_nonneg) (mul_nonneg (by linarith) b.y_nonneg), z_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.z_nonneg) (mul_nonneg (by linarith) b.z_nonneg), sum_eq := begin transitivity (a.x + a.y + a.z) * lambda + (b.x + b.y + b.z) * (1 - lambda), { ring }, simp [a.sum_eq, b.sum_eq] end } end standard_two_simplex open_locale big_operators structure standard_simplex (n : ℕ) := (v : fin n → ℝ) (nonneg : ∀ i : fin n, 0 ≤ v i) (sum_eq_one : ∑ i, v i = 1) namespace standard_simplex def midpoint (n : ℕ) (a b : standard_simplex n) : standard_simplex n := { v := λ i, (a.v i + b.v i) / 2, nonneg := begin intro i, apply div_nonneg, { linarith [a.nonneg i, b.nonneg i] }, norm_num end, sum_eq_one := begin simp [div_eq_mul_inv, ←finset.sum_mul, finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one], field_simp end } end standard_simplex namespace standard_simplex def weighted_average {n : ℕ} (lambda : real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) (a b : standard_simplex n) : standard_simplex n := { v := λ i, lambda * a.v i + (1 - lambda) * b.v i, nonneg := λ i, add_nonneg (mul_nonneg lambda_nonneg (a.nonneg i)) (mul_nonneg (by linarith) (b.nonneg i)), sum_eq_one := begin transitivity lambda * (∑ i, a.v i) + (1 - lambda) * (∑ i, b.v i), { rw [finset.sum_add_distrib, finset.mul_sum, finset.mul_sum] }, simp [a.sum_eq_one, b.sum_eq_one] end } end standard_simplex
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@@ -0,0 +1,65 @@import data.real.basic structure add_group₁ (α : Type*) := (add: α → α → α) (zero: α) (neg: α → α) (add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z)) (add_zero: ∀ x : α, add x zero = x) (zero_add: ∀ x : α, add x zero = x) (add_left_neg : ∀ x : α, add (neg x) x = zero) @[ext] structure point := (x : ℝ) (y : ℝ) (z : ℝ) namespace point def add (a b : point) : point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ def neg (a : point) : point := ⟨-a.x, -a.y, -a.z⟩ def zero : point := ⟨0, 0, 0⟩ def add_group_point : add_group₁ point := { add := point.add, zero := point.zero, neg := point.neg, add_assoc := by { simp [point.add, add_assoc] }, add_zero := by { simp [point.add, point.zero], intro, ext; refl }, zero_add := by { simp [point.add, point.zero], intro, ext; refl }, add_left_neg := by { simp [point.add, point.neg, point.zero] } } end point class add_group₂ (α : Type*) := (add: α → α → α) (zero: α) (neg: α → α) (add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z)) (add_zero: ∀ x : α, add x zero = x) (zero_add: ∀ x : α, add x zero = x) (add_left_neg : ∀ x : α, add (neg x) x = zero) instance has_add_add_group₂ {α : Type*} [add_group₂ α] : has_add α := ⟨add_group₂.add⟩ instance has_zero_add_group₂ {α : Type*} [add_group₂ α] : has_zero α := ⟨add_group₂.zero⟩ instance has_neg_add_group₂ {α : Type*} [add_group₂ α] : has_neg α := ⟨add_group₂.neg⟩ instance : add_group₂ point := { add := point.add, zero := point.zero, neg := point.neg, add_assoc := by { simp [point.add, add_assoc] }, add_zero := by { simp [point.add, point.zero], intro, ext; refl }, zero_add := by { simp [point.add, point.zero], intro, ext; refl }, add_left_neg := by { simp [point.add, point.neg, point.zero] } } section variables (x y : point) #check x + -y + 0 end
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@@ -0,0 +1,106 @@import topology.instances.real open set filter open_locale topological_space filter def principal {α : Type*} (s : set α) : filter α := { sets := {t | s ⊆ t}, univ_sets := sorry, sets_of_superset := sorry, inter_sets := sorry} example : filter ℕ := { sets := {s | ∃ a, ∀ b, a ≤ b → b ∈ s}, univ_sets := sorry, sets_of_superset := sorry, inter_sets := sorry } def tendsto₁ {X Y : Type*} (f : X → Y) (F : filter X) (G : filter Y) := ∀ V ∈ G, f ⁻¹' V ∈ F def tendsto₂ {X Y : Type*} (f : X → Y) (F : filter X) (G : filter Y) := map f F ≤ G example {X Y : Type*} (f : X → Y) (F : filter X) (G : filter Y) : tendsto₂ f F G ↔ tendsto₁ f F G := iff.rfl #check (@filter.map_mono : ∀ {α β} {m : α → β}, monotone (map m)) #check (@filter.map_map : ∀ {α β γ} {f : filter α} {m : α → β} {m' : β → γ}, map m' (map m f) = map (m' ∘ m) f) example {X Y Z : Type*} {F : filter X} {G : filter Y} {H : filter Z} {f : X → Y} {g : Y → Z} (hf : tendsto₁ f F G) (hg : tendsto₁ g G H) : tendsto₁ (g ∘ f) F H := sorry variables (f : ℝ → ℝ) (x₀ y₀ : ℝ) #check comap (coe : ℚ → ℝ) (𝓝 x₀) #check tendsto (f ∘ coe) (comap (coe : ℚ → ℝ) (𝓝 x₀)) (𝓝 y₀) section variables {α β γ : Type*} (F : filter α) {m : γ → β} {n : β → α} #check (comap_comap : comap m (comap n F) = comap (n ∘ m) F) end example : 𝓝 (x₀, y₀) = 𝓝 x₀ ×ᶠ 𝓝 y₀ := nhds_prod_eq #check le_inf_iff example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : tendsto f at_top (𝓝 (x₀, y₀)) ↔ tendsto (prod.fst ∘ f) at_top (𝓝 x₀) ∧ tendsto (prod.snd ∘ f) at_top (𝓝 y₀) := sorry example (x₀ : ℝ) : has_basis (𝓝 x₀) (λ ε : ℝ, 0 < ε) (λ ε, Ioo (x₀ - ε) (x₀ + ε)) := nhds_basis_Ioo_pos x₀ example (u : ℕ → ℝ) (x₀ : ℝ) : tendsto u at_top (𝓝 x₀) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, u n ∈ Ioo (x₀ - ε) (x₀ + ε) := begin have : at_top.has_basis (λ n : ℕ, true) Ici := at_top_basis, rw this.tendsto_iff (nhds_basis_Ioo_pos x₀), simp end example (P Q : ℕ → Prop) (hP : ∀ᶠ n in at_top, P n) (hQ : ∀ᶠ n in at_top, Q n) : ∀ᶠ n in at_top, P n ∧ Q n := hP.and hQ example (u v : ℕ → ℝ) (h : ∀ᶠ n in at_top, u n = v n) (x₀ : ℝ) : tendsto u at_top (𝓝 x₀) ↔ tendsto v at_top (𝓝 x₀) := tendsto_congr' h example (u v : ℕ → ℝ) (h : u =ᶠ[at_top] v) (x₀ : ℝ) : tendsto u at_top (𝓝 x₀) ↔ tendsto v at_top (𝓝 x₀) := tendsto_congr' h #check @eventually_of_forall #check @eventually.mono #check @eventually.and example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in at_top, P n) (hQ : ∀ᶠ n in at_top, Q n) (hR : ∀ᶠ n in at_top, P n ∧ Q n → R n) : ∀ᶠ n in at_top, R n := begin apply (hP.and (hQ.and hR)).mono, rintros n ⟨h, h', h''⟩, exact h'' ⟨h, h'⟩ end example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in at_top, P n) (hQ : ∀ᶠ n in at_top, Q n) (hR : ∀ᶠ n in at_top, P n ∧ Q n → R n) : ∀ᶠ n in at_top, R n := begin filter_upwards [hP, hQ, hR], intros n h h' h'', exact h'' ⟨h, h'⟩ end #check mem_closure_iff_cluster_pt #check le_principal_iff #check ne_bot_of_le example (u : ℕ → ℝ) (M : set ℝ) (x : ℝ) (hux : tendsto u at_top (𝓝 x)) (huM : ∀ᶠ n in at_top, u n ∈ M) : x ∈ closure M := sorry
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@@ -0,0 +1,282 @@import topology.instances.real import analysis.normed_space.banach_steinhaus open set filter open_locale topological_space filter variables {X : Type*} [metric_space X] (a b c : X) #check (dist a b : ℝ) #check (dist_nonneg : 0 ≤ dist a b) #check (dist_eq_zero : dist a b = 0 ↔ a = b) #check (dist_comm a b : dist a b = dist b a) #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) -- Note the next three lines are not quoted, their purpose is to make sure those things don't get renamed while we're looking elsewhere. #check emetric_space #check pseudo_metric_space #check pseudo_emetric_space example {u : ℕ → X} {a : X} : tendsto u at_top (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := metric.tendsto_at_top example {X Y : Type*} [metric_space X] [metric_space Y] {f : X → Y} : continuous f ↔ ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := metric.continuous_iff example {X Y : Type*} [metric_space X] [metric_space Y] {f : X → Y} (hf : continuous f) : continuous (λ p : X × X, dist (f p.1) (f p.2)) := by continuity example {X Y : Type*} [metric_space X] [metric_space Y] {f : X → Y} (hf : continuous f) : continuous (λ p : X × X, dist (f p.1) (f p.2)) := continuous_dist.comp ((hf.comp continuous_fst).prod_mk (hf.comp continuous_snd)) example {X Y : Type*} [metric_space X] [metric_space Y] {f : X → Y} (hf : continuous f) : continuous (λ p : X × X, dist (f p.1) (f p.2)) := begin apply continuous.dist, exact hf.comp continuous_fst, exact hf.comp continuous_snd end example {X Y : Type*} [metric_space X] [metric_space Y] {f : X → Y} (hf : continuous f) : continuous (λ p : X × X, dist (f p.1) (f p.2)) := (hf.comp continuous_fst).dist (hf.comp continuous_snd) example {X Y : Type*} [metric_space X] [metric_space Y] {f : X → Y} (hf : continuous f) : continuous (λ p : X × X, dist (f p.1) (f p.2)) := hf.fst'.dist hf.snd' example {f : ℝ → X} (hf : continuous f) : continuous (λ x : ℝ, f (x^2 + x)) := sorry example {X Y : Type*} [metric_space X] [metric_space Y] (f : X → Y) (a : X) : continuous_at f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x}, dist x a < δ → dist (f x) (f a) < ε := metric.continuous_at_iff variables r : ℝ example : metric.ball a r = {b | dist b a < r} := rfl example : metric.closed_ball a r = {b | dist b a ≤ r} := rfl example (hr : 0 < r) : a ∈ metric.ball a r := metric.mem_ball_self hr example (hr : 0 ≤ r) : a ∈ metric.closed_ball a r := metric.mem_closed_ball_self hr example (s : set X) : is_open s ↔ ∀ x ∈ s, ∃ ε > 0, metric.ball x ε ⊆ s := metric.is_open_iff example {s : set X} : is_closed s ↔ is_open sᶜ := is_open_compl_iff.symm example {s : set X} (hs : is_closed s) {u : ℕ → X} (hu : tendsto u at_top (𝓝 a)) (hus : ∀ n, u n ∈ s) : a ∈ s := hs.mem_of_tendsto hu (eventually_of_forall hus) example {s : set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ metric.ball b ε := metric.mem_closure_iff example {u : ℕ → X} (hu : tendsto u at_top (𝓝 a)) {s : set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := sorry example {x : X} {s : set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, metric.ball x ε ⊆ s := metric.nhds_basis_ball.mem_iff example {x : X} {s : set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, metric.closed_ball x ε ⊆ s := metric.nhds_basis_closed_ball.mem_iff example : is_compact (set.Icc 0 1 : set ℝ) := is_compact_Icc example {s : set X} (hs : is_compact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, strict_mono φ ∧ tendsto (u ∘ φ) at_top (𝓝 a) := hs.tendsto_subseq hu example {s : set X} (hs : is_compact s) (hs' : s.nonempty) {f : X → ℝ} (hfs : continuous_on f s) : ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := hs.exists_forall_le hs' hfs example {s : set X} (hs : is_compact s) (hs' : s.nonempty) {f : X → ℝ} (hfs : continuous_on f s) : ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := hs.exists_forall_ge hs' hfs example {s : set X} (hs : is_compact s) : is_closed s := hs.is_closed example {X : Type*} [metric_space X] [compact_space X] : is_compact (univ : set X) := compact_univ #check is_compact.is_closed example {X : Type*} [metric_space X] {Y : Type*} [metric_space Y] {f : X → Y} : uniform_continuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := metric.uniform_continuous_iff example {X : Type*} [metric_space X] [compact_space X] {Y : Type*} [metric_space Y] {f : X → Y} (hf : continuous f) : uniform_continuous f := sorry -- SOLUTIONs: example {X : Type*} [metric_space X] [compact_space X] {Y : Type*} [metric_space Y] {f : X → Y} (hf : continuous f) : uniform_continuous f := begin rw metric.uniform_continuous_iff, intros ε ε_pos, let φ : X × X → ℝ := λ p, dist (f p.1) (f p.2), have φ_cont : continuous φ := hf.fst'.dist hf.snd', let K := { p : X × X | ε ≤ φ p }, have K_closed : is_closed K := is_closed_le continuous_const φ_cont, have K_cpct : is_compact K := K_closed.is_compact, cases eq_empty_or_nonempty K with hK hK, { use [1, by norm_num], intros x y hxy, have : (x, y) ∉ K, by simp [hK], simpa [K] }, { rcases K_cpct.exists_forall_le hK continuous_dist.continuous_on with ⟨⟨x₀, x₁⟩, xx_in, H⟩, use dist x₀ x₁, split, { change _ < _, rw dist_pos, intro h, have : ε ≤ 0, by simpa [*] using xx_in, linarith }, { intros x x', contrapose!, intros hxx', exact H (x, x') hxx' } }, end example (u : ℕ → X) : cauchy_seq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := metric.cauchy_seq_iff example (u : ℕ → X) : cauchy_seq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := metric.cauchy_seq_iff' example [complete_space X] (u : ℕ → X) (hu : cauchy_seq u) : ∃ x, tendsto u at_top (𝓝 x) := cauchy_seq_tendsto_of_complete hu open_locale big_operators open finset lemma cauchy_seq_of_le_geometric_two' {u : ℕ → X} (hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : cauchy_seq u := begin rw metric.cauchy_seq_iff', intros ε ε_pos, obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε, { sorry }, use N, intros n hn, obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn, calc dist (u (N + k)) (u N) = dist (u (N+0)) (u (N + k)) : sorry ... ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) : sorry ... ≤ ∑ i in range k, (1/2 : ℝ)^(N+i) : sorry ... = 1/2^N*∑ i in range k, (1 / 2) ^ i : sorry ... ≤ 1/2^N*2 : sorry ... < ε : sorry end open metric example [complete_space X] (f : ℕ → set X) (ho : ∀ n, is_open (f n)) (hd : ∀ n, dense (f n)) : dense (⋂n, f n) := begin let B : ℕ → ℝ := λ n, (1/2)^n, have Bpos : ∀ n, 0 < B n, sorry, /- Translate the density assumption into two functions `center` and `radius` associating to any n, x, δ, δpos a center and a positive radius such that `closed_ball center radius` is included both in `f n` and in `closed_ball x δ`. We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ have : ∀ (n : ℕ) (x : X) (δ > 0), ∃ (y : X) (r > 0), r ≤ B (n+1) ∧ closed_ball y r ⊆ (closed_ball x δ) ∩ f n, { sorry }, choose! center radius Hpos HB Hball using this, intros x, rw mem_closure_iff_nhds_basis nhds_basis_closed_ball, intros ε εpos, /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball `closed_ball (c n) (r n)` is included in the previous ball and in `f n`, and such that `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs to all the `f n`. -/ let F : ℕ → (X × ℝ) := λn, nat.rec_on n (prod.mk x (min ε (B 0))) (λn p, prod.mk (center n p.1 p.2) (radius n p.1 p.2)), let c : ℕ → X := λ n, (F n).1, let r : ℕ → ℝ := λ n, (F n).2, have rpos : ∀ n, 0 < r n, { sorry }, have rB : ∀n, r n ≤ B n, { sorry }, have incl : ∀n, closed_ball (c (n+1)) (r (n+1)) ⊆ (closed_ball (c n) (r n)) ∩ (f n), { sorry }, have cdist : ∀ n, dist (c n) (c (n+1)) ≤ B n, { sorry }, have : cauchy_seq c, from cauchy_seq_of_le_geometric_two' cdist, -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. rcases cauchy_seq_tendsto_of_complete this with ⟨y, ylim⟩, -- this point `y` will be the desired point. We will check that it belongs to all -- `f n` and to `ball x ε`. use y, have I : ∀n, ∀ m ≥ n, closed_ball (c m) (r m) ⊆ closed_ball (c n) (r n), { sorry }, have yball : ∀n, y ∈ closed_ball (c n) (r n), { sorry }, sorry end
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@@ -0,0 +1,75 @@import topology.instances.real open set filter open_locale topological_space filter -- In the next example we could use `tauto` in each proof instead of knowing the lemmas example {α : Type*} (s : set α) : filter α := { sets := {t | s ⊆ t}, univ_sets := subset_univ s, sets_of_superset := λ U V hU hUV, subset.trans hU hUV, inter_sets := λ U V hU hV, subset_inter hU hV } example : filter ℕ := { sets := {s | ∃ a, ∀ b, a ≤ b → b ∈ s}, univ_sets := begin use 42, finish, end, sets_of_superset := begin rintros U V ⟨N, hN⟩ hUV, use N, tauto, end, inter_sets := begin rintros U V ⟨N, hN⟩ ⟨N', hN'⟩, use max N N', intros b hb, rw max_le_iff at hb, split ; tauto, end } def tendsto₁ {X Y : Type*} (f : X → Y) (F : filter X) (G : filter Y) := ∀ V ∈ G, f ⁻¹' V ∈ F example {X Y Z : Type*} {F : filter X} {G : filter Y} {H : filter Z} {f : X → Y} {g : Y → Z} (hf : tendsto₁ f F G) (hg : tendsto₁ g G H) : tendsto₁ (g ∘ f) F H := calc map (g ∘ f) F = map g (map f F) : by rw map_map ... ≤ map g G : map_mono hf ... ≤ H : hg example {X Y Z : Type*} {F : filter X} {G : filter Y} {H : filter Z} {f : X → Y} {g : Y → Z} (hf : tendsto₁ f F G) (hg : tendsto₁ g G H) : tendsto₁ (g ∘ f) F H := begin intros V hV, rw preimage_comp, apply hf, apply hg, exact hV end example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : tendsto f at_top (𝓝 (x₀, y₀)) ↔ tendsto (prod.fst ∘ f) at_top (𝓝 x₀) ∧ tendsto (prod.snd ∘ f) at_top (𝓝 y₀) := calc tendsto f at_top (𝓝 (x₀, y₀)) ↔ map f at_top ≤ 𝓝 (x₀, y₀) : iff.rfl ... ↔ map f at_top ≤ 𝓝 x₀ ×ᶠ 𝓝 y₀ : by rw nhds_prod_eq ... ↔ map f at_top ≤ (comap prod.fst (𝓝 x₀) ⊓ comap prod.snd (𝓝 y₀)) : iff.rfl ... ↔ map f at_top ≤ comap prod.fst (𝓝 x₀) ∧ map f at_top ≤ (comap prod.snd (𝓝 y₀)) : le_inf_iff ... ↔ map prod.fst (map f at_top) ≤ 𝓝 x₀ ∧ map prod.snd (map f at_top) ≤ 𝓝 y₀ : by rw [← map_le_iff_le_comap, ← map_le_iff_le_comap] ... ↔ map (prod.fst ∘ f) at_top ≤ 𝓝 x₀ ∧ map (prod.snd ∘ f) at_top ≤ 𝓝 y₀ : by rw [map_map, map_map] -- an alternative solution example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : tendsto f at_top (𝓝 (x₀, y₀)) ↔ tendsto (prod.fst ∘ f) at_top (𝓝 x₀) ∧ tendsto (prod.snd ∘ f) at_top (𝓝 y₀) := begin rw nhds_prod_eq, unfold tendsto filter.prod, rw [le_inf_iff, ← map_le_iff_le_comap, map_map, ← map_le_iff_le_comap, map_map] end example (u : ℕ → ℝ) (M : set ℝ) (x : ℝ) (hux : tendsto u at_top (𝓝 x)) (huM : ∀ᶠ n in at_top, u n ∈ M) : x ∈ closure M := mem_closure_iff_cluster_pt.mpr (ne_bot_of_le $ le_inf hux $ le_principal_iff.mpr huM)
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@@ -0,0 +1,135 @@import topology.instances.real import analysis.normed_space.banach_steinhaus open set filter open_locale topological_space filter example {f : ℝ → X} (hf : continuous f) : continuous (λ x : ℝ, f (x^2 + x)) := hf.comp $ (continuous_pow 2).add continuous_id example {u : ℕ → X} (hu : tendsto u at_top (𝓝 a)) {s : set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := begin rw metric.tendsto_at_top at hu, rw metric.mem_closure_iff, intros ε ε_pos, rcases hu ε ε_pos with ⟨N, hN⟩, refine ⟨u N, hs _, _⟩, rw dist_comm, exact hN N le_rfl end example {u : ℕ → X} (hu : ∀ (n : ℕ), dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : cauchy_seq u := begin rw metric.cauchy_seq_iff', intros ε ε_pos, obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε, { have : tendsto (λ N : ℕ, (1 / 2 ^ N * 2 : ℝ)) at_top (𝓝 0), { rw ← zero_mul (2 : ℝ), apply tendsto.mul, simp_rw ← one_div_pow (2 : ℝ), apply tendsto_pow_at_top_nhds_0_of_lt_1 ; linarith, exact tendsto_const_nhds }, rcases (at_top_basis.tendsto_iff (nhds_basis_Ioo_pos (0 : ℝ))).mp this ε ε_pos with ⟨N, H, hN⟩, exact ⟨N, by simpa using (hN N le_rfl).2⟩ }, use N, intros n hn, obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn, calc dist (u (N + k)) (u N) = dist (u (N+0)) (u (N + k)) : by rw [dist_comm, add_zero] ... ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) : dist_le_range_sum_dist (λ i, u (N+i)) k ... ≤ ∑ i in range k, (1/2 : ℝ) ^ (N+i) : sum_le_sum (λ i hi, hu $ N+i) ... = 1/2^N*∑ i in range k, (1 / 2)^i : by simp_rw [← one_div_pow, pow_add, ← mul_sum] ... ≤ 1/2^N*2 : mul_le_mul_of_nonneg_left (sum_geometric_two_le _) (one_div_nonneg.mpr (pow_nonneg (zero_le_two : (0 : ℝ) ≤ 2)_)) ... < ε : hN end example [complete_space X] (f : ℕ → set X) (ho : ∀ n, is_open (f n)) (hd : ∀ n, dense (f n)) : dense (⋂n, f n) := begin let B : ℕ → ℝ := λ n, (1/2)^n, have Bpos : ∀ n, 0 < B n, from λ n, (pow_pos sorry n), /- Translate the density assumption into two functions `center` and `radius` associating to any n, x, δ, δpos a center and a positive radius such that `closed_ball center radius` is included both in `f n` and in `closed_ball x δ`. We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ have : ∀ (n : ℕ) (x : X) (δ > 0), ∃ (y : X) (r > 0), r ≤ B (n+1) ∧ closed_ball y r ⊆ (closed_ball x δ) ∩ f n, { intros n x δ δpos, have : x ∈ closure (f n) := hd n x, rcases metric.mem_closure_iff.1 this (δ/2) (half_pos δpos) with ⟨y, ys, xy⟩, rw dist_comm at xy, obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closed_ball y r ⊆ f n := nhds_basis_closed_ball.mem_iff.1 (is_open_iff_mem_nhds.1 (ho n) y ys), refine ⟨y, min (min (δ/2) r) (B (n+1)), _, _, λz hz, ⟨_, _⟩⟩, show 0 < min (min (δ / 2) r) (B (n+1)), from lt_min (lt_min (half_pos δpos) rpos) (Bpos (n+1)), show min (min (δ / 2) r) (B (n+1)) ≤ B (n+1), from min_le_right _ _, show z ∈ closed_ball x δ, from calc dist z x ≤ dist z y + dist y x : dist_triangle _ _ _ ... ≤ (min (min (δ / 2) r) (B (n+1))) + (δ/2) : add_le_add hz xy.le ... ≤ δ/2 + δ/2 : add_le_add_right ((min_le_left _ _).trans (min_le_left _ _)) _ ... = δ : add_halves δ, show z ∈ f n, from hr (calc dist z y ≤ min (min (δ / 2) r) (B (n+1)) : hz ... ≤ r : (min_le_left _ _).trans (min_le_right _ _)) }, choose! center radius Hpos HB Hball using this, refine λ x, (mem_closure_iff_nhds_basis nhds_basis_closed_ball).2 (λ ε εpos, _), /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball `closed_ball (c n) (r n)` is included in the previous ball and in `f n`, and such that `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs to all the `f n`. -/ let F : ℕ → (X × ℝ) := λn, nat.rec_on n (prod.mk x (min ε (B 0))) (λn p, prod.mk (center n p.1 p.2) (radius n p.1 p.2)), let c : ℕ → X := λ n, (F n).1, let r : ℕ → ℝ := λ n, (F n).2, have rpos : ∀ n, 0 < r n, { assume n, induction n with n hn, exact lt_min εpos (Bpos 0), exact Hpos n (c n) (r n) hn }, have rB : ∀n, r n ≤ B n, { assume n, induction n with n hn, exact min_le_right _ _, exact HB n (c n) (r n) (rpos n) }, have incl : ∀n, closed_ball (c (n+1)) (r (n+1)) ⊆ (closed_ball (c n) (r n)) ∩ (f n) := λ n, Hball n (c n) (r n) (rpos n), have cdist : ∀ n, dist (c n) (c (n+1)) ≤ B n, { assume n, rw dist_comm, have A : c (n+1) ∈ closed_ball (c (n+1)) (r (n+1)) := mem_closed_ball_self (rpos $ n +1).le, have I := calc closed_ball (c (n+1)) (r (n+1)) ⊆ closed_ball (c n) (r n) : (incl n).trans (inter_subset_left _ _) ... ⊆ closed_ball (c n) (B n) : closed_ball_subset_closed_ball (rB n), exact I A }, have : cauchy_seq c, from cauchy_seq_of_le_geometric_two' cdist, -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. rcases cauchy_seq_tendsto_of_complete this with ⟨y, ylim⟩, -- this point `y` will be the desired point. We will check that it belongs to all -- `f n` and to `ball x ε`. use y, have I : ∀n, ∀ m ≥ n, closed_ball (c m) (r m) ⊆ closed_ball (c n) (r n), { assume n, refine nat.le_induction _ (λm hnm h, _), { exact subset.rfl }, { exact (incl m).trans ((set.inter_subset_left _ _).trans h) }}, have yball : ∀n, y ∈ closed_ball (c n) (r n), { assume n, refine is_closed_ball.mem_of_tendsto ylim _, refine (filter.eventually_ge_at_top n).mono (λ m hm, _), exact I n m hm (mem_closed_ball_self (rpos _).le) }, split, { suffices : ∀ n, y ∈ f n, by rwa set.mem_Inter, intro n, have : closed_ball (c (n+1)) (r (n+1)) ⊆ f n := subset.trans (incl n) (inter_subset_right _ _), exact this (yball (n+1)) }, calc dist y x ≤ r 0 : yball 0 ... ≤ ε : min_le_left _ _, end
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@@ -1,7 +1,7 @@import Mathlib.Data.Real.Basic -- An example. import Mathlib.Tactic import Mathlib.Data.Real.Basic -- An example. example (a b c : ℝ) : a * b * c = b * (a * c) := by rw [mul_comm a b] rw [mul_assoc b a c]
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@@ -46,7 +46,7 @@ example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c *section variable (a b c d e f g : ℝ) variable (a b c d e f : ℝ) example (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by rw [h', ← mul_assoc, h, mul_assoc]
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@@ -64,7 +64,6 @@ variable (a b c : ℝ)#check mul_assoc c a b #check mul_comm a #check mul_comm #check @mul_comm end
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@@ -40,8 +40,8 @@ theorem add_zero (a : R) : a + 0 = a := by rw [add_comm, zero_add]theorem add_right_neg (a : R) : a + -a = 0 := by rw [add_comm, add_left_neg] #check @MyRing.add_zero #check @add_zero #check MyRing.add_zero #check add_zero end MyRing
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@@ -74,7 +74,7 @@ example (h : 1 ≤ a) (h' : b ≤ c) : 2 + a + exp b ≤ 3 * a + exp c := by#check (add_pos : 0 < a → 0 < b → 0 < a + b) #check (add_pos_of_pos_of_nonneg : 0 < a → 0 ≤ b → 0 < a + b) #check (exp_pos : ∀ a, 0 < exp a) #check @add_le_add_left #check add_le_add_left example (h : a ≤ b) : exp a ≤ exp b := by rw [exp_le_exp]
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C02S04
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@@ -1,6 +1,3 @@import Mathlib.Data.Real.Basic import Mathlib.Data.Real.Basic example (a b c : ℝ) : c * b * a = b * (a * c) := by rw [mul_comm c b] rw [mul_assoc b c a]
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C02S04
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@@ -76,7 +77,7 @@ endsection variable {α : Type _} {R : Type _} [OrderedCancelAddCommMonoid R] #check @add_le_add #check add_le_add def FnUb' (f : α → R) (a : R) : Prop := ∀ x, f x ≤ a
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C03S02
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@@ -1,6 +1,6 @@import Mathlib.Tactic import Mathlib.Data.Real.Basic import Mathlib.Data.Nat.Prime import Mathlib.Tactic.NormNum namespace C03S04 example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := by
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C03S05
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@@ -1,6 +1,6 @@import Mathlib.Tactic import Mathlib.Data.Real.Basic import Mathlib.Data.Nat.Prime import Mathlib.Tactic.NormNum namespace C03S04 example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : m ∣ n ∧ ¬n ∣ m := by
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C03S05
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C03S06
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Data.Set.Lattice import Mathlib.Data.Set.Function import Mathlib.Analysis.SpecialFunctions.Log.Basic
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Data.Set.Lattice import Mathlib.Data.Set.Function import Mathlib.Analysis.SpecialFunctions.Log.Basic
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@@ -1,7 +1,6 @@import Mathlib.Tactic import Mathlib.Data.Nat.Factorization.Basic import Mathlib.Data.Nat.Prime import Mathlib.Tactic.NormNum.GCD import Mathlib.Tactic.NormNum.Prime #print Nat.coprime
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@@ -16,7 +15,7 @@ example : Nat.coprime 12 7 := by norm_numexample : Nat.gcd 12 8 = 4 := by norm_num #check @Nat.prime_def_lt #check Nat.prime_def_lt example (p : ℕ) (prime_p : Nat.Prime p) : 2 ≤ p ∧ ∀ m : ℕ, m < p → m ∣ p → m = 1 := by rwa [Nat.prime_def_lt] at prime_p
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@@ -35,7 +34,7 @@ example : Nat.Prime 2 :=example : Nat.Prime 3 := Nat.prime_three #check @Nat.Prime.dvd_mul #check Nat.Prime.dvd_mul #check Nat.Prime.dvd_mul Nat.prime_two #check Nat.prime_two.dvd_mul
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@@ -1,7 +1,6 @@import Mathlib.Data.Nat.Prime import Mathlib.Algebra.BigOperators.Order import Mathlib.Tactic import Mathlib.Tactic.IntervalCases open BigOperators
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@@ -1,7 +1,6 @@import Mathlib.Tactic import Mathlib.Data.Nat.Factorization.Basic import Mathlib.Data.Nat.Prime import Mathlib.Tactic.NormNum.GCD import Mathlib.Tactic.NormNum.Prime theorem even_of_even_sqr {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m := by rw [pow_two, Nat.prime_two.dvd_mul] at h
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@@ -1,7 +1,6 @@import Mathlib.Data.Nat.Prime import Mathlib.Algebra.BigOperators.Order import Mathlib.Tactic import Mathlib.Tactic.IntervalCases open BigOperators
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@@ -135,7 +135,7 @@ example (a b : ℤ) : a = b * (a / b) + a % b :=example (a b : ℤ) : b ≠ 0 → 0 ≤ a % b := Int.emod_nonneg a example (a b : ℤ) : b ≠ 0 → a % b < |b| := example (a b : ℤ) : b ≠ 0 → a % b < abs b := Int.emod_lt a namespace Int
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@@ -150,7 +150,7 @@ theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := byrw [div', mod'] linarith [Int.ediv_add_emod (a + b / 2) b] theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : |mod' a b| ≤ b / 2 := by theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by rw [mod', abs_le] constructor · linarith [Int.emod_nonneg (a + b / 2) h.ne']
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C06S02
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@@ -110,12 +111,12 @@ instance {α : Type _} : Group₂ (Equiv.Perm α) wheremul_one := Equiv.refl_trans mul_left_inv := Equiv.self_trans_symm #check @Group₂.mul #check Group₂.mul def mySquare {α : Type _} [Group₂ α] (x : α) := Group₂.mul x x #check @mySquare #check mySquare section variable {β : Type _} (f g : Equiv.Perm β)
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Data.Real.Basic namespace C06S02
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Topology.Instances.Real
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.GroupTheory.QuotientGroup
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Topology.Instances.Real
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.GroupTheory.QuotientGroup
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@@ -64,7 +64,7 @@ example (x₀ : ℝ) : HasBasis (𝓝 x₀) (fun ε : ℝ => 0 < ε) fun ε => Iexample (u : ℕ → ℝ) (x₀ : ℝ) : Tendsto u atTop (𝓝 x₀) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, u n ∈ Ioo (x₀ - ε) (x₀ + ε) := by have : atTop.HasBasis (fun n : ℕ => True) Ici := atTop_basis have : atTop.HasBasis (fun _ : ℕ => True) Ici := atTop_basis rw [this.tendsto_iff (nhds_basis_Ioo_pos x₀)] simp
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@@ -111,7 +111,7 @@ example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs :hs.exists_forall_ge hs' hfs example {s : Set X} (hs : IsCompact s) : IsClosed s := hs.IsClosed hs.isClosed example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := isCompact_univ
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@@ -165,19 +165,19 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :sorry /- Translate the density assumption into two functions `center` and `radius` associating to any n, x, δ, δpos a center and a positive radius such that `closed_ball center radius` is included both in `f n` and in `closed_ball x δ`. `closedBall center radius` is included both in `f n` and in `closedBall x δ`. We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ have : ∀ (n : ℕ) (x : X), ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closed_ball y r ⊆ closed_ball x δ ∩ f n := ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by sorry choose! center radius Hpos HB Hball using this intro x rw [mem_closure_iff_nhds_basis nhds_basis_closed_ball] rw [mem_closure_iff_nhds_basis nhds_basis_closedBall] intro ε εpos /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball `closed_ball (c n) (r n)` is included in the previous ball and in `f n`, and such that `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs to all the `f n`. -/ let F : ℕ → X × ℝ := fun n =>
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@@ -186,7 +186,7 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :let r : ℕ → ℝ := fun n => (F n).2 have rpos : ∀ n, 0 < r n := by sorry have rB : ∀ n, r n ≤ B n := by sorry have incl : ∀ n, closed_ball (c (n + 1)) (r (n + 1)) ⊆ closed_ball (c n) (r n) ∩ f n := by sorry have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by sorry have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`.
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@@ -194,7 +194,7 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :-- this point `y` will be the desired point. We will check that it belongs to all -- `f n` and to `ball x ε`. use y have I : ∀ n, ∀ m ≥ n, closed_ball (c m) (r m) ⊆ closed_ball (c n) (r n) := by sorry have yball : ∀ n, y ∈ closed_ball (c n) (r n) := by sorry have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry sorry
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@@ -69,4 +69,3 @@ example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) :example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x)) (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M := mem_closure_iff_clusterPt.mpr (neBot_of_le <| le_inf hux <| le_principal_iff.mpr huM)
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@@ -123,7 +123,7 @@ example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs :hs.exists_forall_ge hs' hfs example {s : Set X} (hs : IsCompact s) : IsClosed s := hs.IsClosed hs.isClosed example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := isCompact_univ
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@@ -146,13 +146,13 @@ example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpacehave φ_cont : Continuous φ := hf.fst'.dist hf.snd' let K := { p : X × X | ε ≤ φ p } have K_closed : IsClosed K := isClosed_le continuous_const φ_cont have K_cpct : IsCompact K := K_closed.is_compact have K_cpct : IsCompact K := K_closed.isCompact cases' eq_empty_or_nonempty K with hK hK · use 1, by norm_num intro x y hxy intro x y _ have : (x, y) ∉ K := by simp [hK] simpa [K] · rcases K_cpct.exists_forall_le hK continuous_dist.continuous_on with ⟨⟨x₀, x₁⟩, xx_in, H⟩ simpa using this · rcases K_cpct.exists_forall_le hK continuous_dist.continuousOn with ⟨⟨x₀, x₁⟩, xx_in, H⟩ use dist x₀ x₁ constructor · change _ < _
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@@ -205,8 +205,8 @@ example {u : ℕ → X} (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^simp_rw [← one_div_pow (2 : ℝ)] apply tendsto_pow_atTop_nhds_0_of_lt_1 <;> linarith exact tendsto_const_nhds rcases(atTop_basis.tendsto_iff (nhds_basis_Ioo_pos (0 : ℝ))).mp this ε ε_pos with ⟨N, H, hN⟩ exact ⟨N, by simpa using (hN N le_rfl).2⟩ rcases(atTop_basis.tendsto_iff (nhds_basis_Ioo_pos (0 : ℝ))).mp this ε ε_pos with ⟨N, _, hN⟩ exact ⟨N, by simpa using (hN N left_mem_Ici).2⟩ use N intro n hn obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn
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@@ -214,8 +214,8 @@ example {u : ℕ → X} (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := by rw [dist_comm, add_zero] _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := (dist_le_range_sum_dist (fun i => u (N + i)) k) _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := (sum_le_sum fun i hi => hu <| N + i) _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2) ^ i := by simp_rw [← one_div_pow, pow_add, ← mul_sum] _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := (sum_le_sum fun i _ => hu <| N + i) _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2 : ℝ) ^ i := by simp_rw [← one_div_pow, pow_add, ← mul_sum] _ ≤ 1 / 2 ^ N * 2 := (mul_le_mul_of_nonneg_left (sum_geometric_two_le _) (one_div_nonneg.mpr (pow_nonneg (zero_le_two : (0 : ℝ) ≤ 2) _)))
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@@ -231,19 +231,19 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :sorry /- Translate the density assumption into two functions `center` and `radius` associating to any n, x, δ, δpos a center and a positive radius such that `closed_ball center radius` is included both in `f n` and in `closed_ball x δ`. `closedBall center radius` is included both in `f n` and in `closedBall x δ`. We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ have : ∀ (n : ℕ) (x : X), ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closed_ball y r ⊆ closed_ball x δ ∩ f n := ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by sorry choose! center radius Hpos HB Hball using this intro x rw [mem_closure_iff_nhds_basis nhds_basis_closed_ball] rw [mem_closure_iff_nhds_basis nhds_basis_closedBall] intro ε εpos /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball `closed_ball (c n) (r n)` is included in the previous ball and in `f n`, and such that `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs to all the `f n`. -/ let F : ℕ → X × ℝ := fun n =>
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@@ -252,7 +252,7 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :let r : ℕ → ℝ := fun n => (F n).2 have rpos : ∀ n, 0 < r n := by sorry have rB : ∀ n, r n ≤ B n := by sorry have incl : ∀ n, closed_ball (c (n + 1)) (r (n + 1)) ⊆ closed_ball (c n) (r n) ∩ f n := by sorry have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by sorry have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`.
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@@ -260,8 +260,8 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :-- this point `y` will be the desired point. We will check that it belongs to all -- `f n` and to `ball x ε`. use y have I : ∀ n, ∀ m ≥ n, closed_ball (c m) (r m) ⊆ closed_ball (c n) (r n) := by sorry have yball : ∀ n, y ∈ closed_ball (c n) (r n) := by sorry have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry sorry example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) :
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@@ -270,23 +270,23 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :have Bpos : ∀ n, 0 < B n := fun n => pow_pos sorry n /- Translate the density assumption into two functions `center` and `radius` associating to any n, x, δ, δpos a center and a positive radius such that `closed_ball center radius` is included both in `f n` and in `closed_ball x δ`. `closedBall center radius` is included both in `f n` and in `closedBall x δ`. We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ have : ∀ (n : ℕ) (x : X), ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closed_ball y r ⊆ closed_ball x δ ∩ f n := by ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by intro n x δ δpos have : x ∈ closure (f n) := hd n x rcases Metric.mem_closure_iff.1 this (δ / 2) (half_pos δpos) with ⟨y, ys, xy⟩ rw [dist_comm] at xy obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closed_ball y r ⊆ f n := nhds_basis_closed_ball.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys) obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n := nhds_basis_closedBall.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys) refine' ⟨y, min (min (δ / 2) r) (B (n + 1)), _, _, fun z hz => ⟨_, _⟩⟩ show 0 < min (min (δ / 2) r) (B (n + 1)) exact lt_min (lt_min (half_pos δpos) rpos) (Bpos (n + 1)) show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1) exact min_le_right _ _ show z ∈ closed_ball x δ show z ∈ closedBall x δ exact calc dist z x ≤ dist z y + dist y x := dist_triangle _ _ _
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@@ -302,10 +302,10 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :_ ≤ r := (min_le_left _ _).trans (min_le_right _ _) ) choose! center radius Hpos HB Hball using this refine' fun x => (mem_closure_iff_nhds_basis nhds_basis_closed_ball).2 fun ε εpos => _ refine' fun x => (mem_closure_iff_nhds_basis nhds_basis_closedBall).2 fun ε εpos => _ /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball `closed_ball (c n) (r n)` is included in the previous ball and in `f n`, and such that `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs to all the `f n`. -/ let F : ℕ → X × ℝ := fun n =>
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@@ -322,18 +322,18 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :induction' n with n hn exact min_le_right _ _ exact HB n (c n) (r n) (rpos n) have incl : ∀ n, closed_ball (c (n + 1)) (r (n + 1)) ⊆ closed_ball (c n) (r n) ∩ f n := fun n => have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := fun n => Hball n (c n) (r n) (rpos n) have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by intro n rw [dist_comm] have A : c (n + 1) ∈ closed_ball (c (n + 1)) (r (n + 1)) := mem_closed_ball_self (rpos <| n + 1).le have A : c (n + 1) ∈ closedBall (c (n + 1)) (r (n + 1)) := mem_closedBall_self (rpos <| n + 1).le have I := calc closed_ball (c (n + 1)) (r (n + 1)) ⊆ closed_ball (c n) (r n) := closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) := (incl n).trans (inter_subset_left _ _) _ ⊆ closed_ball (c n) (B n) := closed_ball_subset_closed_ball (rB n) _ ⊆ closedBall (c n) (B n) := closedBall_subset_closedBall (rB n) exact I A have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist
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@@ -342,20 +342,20 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :-- this point `y` will be the desired point. We will check that it belongs to all -- `f n` and to `ball x ε`. use y have I : ∀ n, ∀ m ≥ n, closed_ball (c m) (r m) ⊆ closed_ball (c n) (r n) := by have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by intro n refine' Nat.le_induction _ fun m hnm h => _ · exact Subset.rfl · exact (incl m).trans ((Set.inter_subset_left _ _).trans h) have yball : ∀ n, y ∈ closed_ball (c n) (r n) := by have yball : ∀ n, y ∈ closedBall (c n) (r n) := by intro n refine' is_closed_ball.mem_of_tendsto ylim _ refine' isClosed_ball.mem_of_tendsto ylim _ refine' (Filter.eventually_ge_atTop n).mono fun m hm => _ exact I n m hm (mem_closed_ball_self (rpos _).le) exact I n m hm (mem_closedBall_self (rpos _).le) constructor · suffices ∀ n, y ∈ f n by rwa [Set.mem_iInter] intro n have : closed_ball (c (n + 1)) (r (n + 1)) ⊆ f n := have : closedBall (c (n + 1)) (r (n + 1)) ⊆ f n := Subset.trans (incl n) (inter_subset_right _ _) exact this (yball (n + 1)) calc
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@@ -135,7 +135,7 @@ example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X} (hA :· rw [continuous_iff_continuousAt] intro x suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 x by simp [ContinuousAt, (closed_nhds_basis _).tendsto_right_iff] simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff] intro V' V'_in V'_closed obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V' := aux (hφ x) V'_in suffices : ∀ y ∈ V, φ y ∈ V'
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@@ -147,7 +147,7 @@ example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X} (hA :exact mem_of_superset (preimage_mem_comap hVx) hV · intro a have lim : Tendsto f (𝓝 a) (𝓝 <| φ a) := by simpa [nhds_induced] using hφ a exact tendsto_nhds_unique limUnder f_cont.continuousAt exact tendsto_nhds_unique lim f_cont.continuousAt example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] {s : Set X} {a : X} : a ∈ closure s ↔ ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ Tendsto u atTop (𝓝 a) :=
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Topology.Instances.Real open Set Filter Topology
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@@ -79,9 +80,9 @@ example (u v : ℕ → ℝ) (h : u =ᶠ[atTop] v) (x₀ : ℝ) :Tendsto u atTop (𝓝 x₀) ↔ Tendsto v atTop (𝓝 x₀) := tendsto_congr' h #check @eventually_of_forall #check @Eventually.mono #check @Eventually.and #check eventually_of_forall #check Eventually.mono #check Eventually.and example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) (hR : ∀ᶠ n in atTop, P n ∧ Q n → R n) : ∀ᶠ n in atTop, R n := by
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus
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@@ -112,7 +113,7 @@ example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] {A : Set X}∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := sorry #check @HasBasis.tendsto_right_iff #check HasBasis.tendsto_right_iff example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] {s : Set X} {a : X} :
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Topology.Instances.Real open Set Filter Topology
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus
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@@ -126,7 +127,7 @@ example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] {A : Set X}∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := sorry #check @HasBasis.tendsto_right_iff #check HasBasis.tendsto_right_iff example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X} (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f)
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv import Mathlib.Analysis.Calculus.MeanValue
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@@ -27,6 +28,8 @@ example {f g : ℝ → ℝ} {x : ℝ} (hf : DifferentiableAt ℝ f x) (hg : Diffexample {f : ℝ → ℝ} {a : ℝ} (h : IsLocalMin f a) : deriv f a = 0 := h.deriv_eq_zero open Set example {f : ℝ → ℝ} {a b : ℝ} (hab : a < b) (hfc : ContinuousOn f (Icc a b)) (hfI : f a = f b) : ∃ c ∈ Ioo a b, deriv f c = 0 := exists_deriv_eq_zero f hab hfc hfI
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Analysis.NormedSpace.BanachSteinhaus import Mathlib.Analysis.NormedSpace.FiniteDimension import Mathlib.Analysis.Calculus.Inverse
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@@ -113,8 +114,6 @@ endopen Asymptotics open Asymptotics example {α : Type _} {E : Type _} [NormedGroup E] {F : Type _} [NormedGroup F] (c : ℝ) (l : Filter α) (f : α → E) (g : α → F) : IsBigOWith c l f g ↔ ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := isBigOWith_iff
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@@ -133,6 +132,8 @@ example {α : Type _} {E : Type _} [NormedAddCommGroup E] (l : Filter α) (f g :section open Topology variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
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@@ -171,7 +172,7 @@ example (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) :∀ᶠ x in 𝓝 (f a), f (hf.localInverse f f' a x) = x := hf.eventually_right_inverse example [CompleteSpace E] {f : E → F} {f' : E ≃L[𝕜] F} {a : E} example {f : E → F} {f' : E ≃L[𝕜] F} {a : E} (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) a) : HasStrictFDerivAt (HasStrictFDerivAt.localInverse f f' a hf) (f'.symm : F →L[𝕜] E) (f a) := HasStrictFDerivAt.to_localInverse hf
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv import Mathlib.Analysis.Calculus.MeanValue
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Analysis.NormedSpace.BanachSteinhaus import Mathlib.Analysis.NormedSpace.FiniteDimension import Mathlib.Analysis.Calculus.Inverse
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.MeasureTheory.Integral.IntervalIntegral import Mathlib.Analysis.SpecialFunctions.Integrals import Mathlib.Analysis.Convolution
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Analysis.NormedSpace.FiniteDimension import Mathlib.Analysis.Convolution import Mathlib.MeasureTheory.Function.Jacobian
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Analysis.NormedSpace.FiniteDimension import Mathlib.Analysis.Convolution import Mathlib.MeasureTheory.Function.Jacobian
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@@ -24,6 +25,8 @@ example {f g : α → E} (hf : Integrable f μ) (hg : Integrable g μ) :example {s : Set α} (c : E) : (∫ x in s, c ∂μ) = (μ s).toReal • c := set_integral_const c open Filter example {F : ℕ → α → E} {f : α → E} (bound : α → ℝ) (hmeas : ∀ n, AEStronglyMeasurable (F n) μ) (hint : Integrable bound μ) (hbound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a) (hlim : ∀ᵐ a ∂μ, Tendsto (fun n : ℕ ↦ F n a) atTop (𝓝 (f a))) :
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.MeasureTheory.Integral.IntervalIntegral import Mathlib.Analysis.SpecialFunctions.Integrals import Mathlib.Analysis.Convolution
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Analysis.NormedSpace.FiniteDimension import Mathlib.Analysis.Convolution import Mathlib.MeasureTheory.Function.Jacobian
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@@ -1,3 +1,4 @@import Mathlib.Tactic import Mathlib.Analysis.NormedSpace.FiniteDimension import Mathlib.Analysis.Convolution import Mathlib.MeasureTheory.Function.Jacobian
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@@ -0,0 +1,227 @@import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic noncomputable section @[ext] structure Point where x : ℝ y : ℝ z : ℝ #check Point.ext example (a b : Point) (hx : a.x = b.x) (hy : a.y = b.y) (hz : a.z = b.z) : a = b := by ext repeat' assumption def myPoint1 : Point where x := 2 y := -1 z := 4 def myPoint2 : Point := ⟨2, -1, 4⟩ def myPoint3 := Point.mk 2 (-1) 4 structure Point' where build :: x : ℝ y : ℝ z : ℝ #check Point'.build 2 (-1) 4 namespace Point def add (a b : Point) : Point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ def add' (a b : Point) : Point where x := a.x + b.x y := a.y + b.y z := a.z + b.z #check add myPoint1 myPoint2 #check myPoint1.add myPoint2 end Point #check Point.add myPoint1 myPoint2 #check myPoint1.add myPoint2 namespace Point protected theorem add_comm (a b : Point) : add a b = add b a := by rw [add, add] ext <;> dsimp repeat' apply add_comm example (a b : Point) : add a b = add b a := by simp [add, add_comm] theorem add_x (a b : Point) : (a.add b).x = a.x + b.x := rfl def addAlt : Point → Point → Point | Point.mk x₁ y₁ z₁, Point.mk x₂ y₂ z₂ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ def addAlt' : Point → Point → Point | ⟨x₁, y₁, z₁⟩, ⟨x₂, y₂, z₂⟩ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ theorem addAlt_x (a b : Point) : (a.addAlt b).x = a.x + b.x := by cases a cases b rfl theorem addAlt_comm (a b : Point) : addAlt a b = addAlt b a := by rcases a with ⟨xa, ya, za⟩ rcases b with ⟨xb, yb, zb⟩ rw [addAlt, addAlt] ext <;> dsimp apply add_comm repeat' apply add_comm example (a b : Point) : addAlt a b = addAlt b a := by rcases a with ⟨xa, ya, za⟩ rcases b with ⟨xb, yb, zb⟩ simp [addAlt, add_comm] example : ∀ a b : Point, addAlt a b = addAlt b a := by rintro ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ simp [addAlt, add_comm] example : ∀ a b : Point, add a b = add b a := fun ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ => by simp [add, add_comm] protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by sorry def smul (r : ℝ) (a : Point) : Point := sorry theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by sorry end Point structure StandardTwoSimplex where x : ℝ y : ℝ z : ℝ x_nonneg : 0 ≤ x y_nonneg : 0 ≤ y z_nonneg : 0 ≤ z sum_eq : x + y + z = 1 namespace StandardTwoSimplex def swapXy (a : StandardTwoSimplex) : StandardTwoSimplex where x := a.y y := a.x z := a.z x_nonneg := a.y_nonneg y_nonneg := a.x_nonneg z_nonneg := a.z_nonneg sum_eq := by rw [add_comm a.y a.x, a.sum_eq] noncomputable section def midpoint (a b : StandardTwoSimplex) : StandardTwoSimplex where x := (a.x + b.x) / 2 y := (a.y + b.y) / 2 z := (a.z + b.z) / 2 x_nonneg := div_nonneg (add_nonneg a.x_nonneg b.x_nonneg) (by norm_num) y_nonneg := div_nonneg (add_nonneg a.y_nonneg b.y_nonneg) (by norm_num) z_nonneg := div_nonneg (add_nonneg a.z_nonneg b.z_nonneg) (by norm_num) sum_eq := by field_simp; linarith [a.sum_eq, b.sum_eq] def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) (a b : StandardTwoSimplex) : StandardTwoSimplex where sorry end end StandardTwoSimplex open BigOperators structure StandardSimplex (n : ℕ) where V : Fin n → ℝ NonNeg : ∀ i : Fin n, 0 ≤ V i sum_eq_one : (∑ i, V i) = 1 namespace StandardSimplex def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n where V i := (a.V i + b.V i) / 2 NonNeg := by intro i apply div_nonneg · linarith [a.NonNeg i, b.NonNeg i] norm_num sum_eq_one := by simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] field_simp end StandardSimplex structure IsLinear (f : ℝ → ℝ) where is_additive : ∀ x y, f (x + y) = f x + f y preserves_mul : ∀ x c, f (c * x) = c * f x section variable (f : ℝ → ℝ) (linf : IsLinear f) #check linf.is_additive #check linf.preserves_mul end def Point'' := ℝ × ℝ × ℝ def IsLinear' (f : ℝ → ℝ) := (∀ x y, f (x + y) = f x + f y) ∧ ∀ x c, f (c * x) = c * f x def PReal := { y : ℝ // 0 < y } section variable (x : PReal) #check x.val #check x.property #check x.1 #check x.2 end def StandardTwoSimplex' := { p : ℝ × ℝ × ℝ // 0 ≤ p.1 ∧ 0 ≤ p.2.1 ∧ 0 ≤ p.2.2 ∧ p.1 + p.2.1 + p.2.2 = 1 } def StandardSimplex' (n : ℕ) := { v : Fin n → ℝ // (∀ i : Fin n, 0 ≤ v i) ∧ (∑ i, v i) = 1 } def StdSimplex := Σ n : ℕ, StandardSimplex n section variable (s : StdSimplex) #check s.fst #check s.snd #check s.1 #check s.2 end
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@@ -0,0 +1,172 @@import Mathlib.Data.Real.Basic structure Group₁ (α : Type _) where mul : α → α → α one : α inv : α → α mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) mul_one : ∀ x : α, mul x one = x one_mul : ∀ x : α, mul one x = x mul_left_inv : ∀ x : α, mul (inv x) x = one structure Group₁Cat where α : Type _ str : Group₁ α section variable (α β γ : Type _) variable (f : α ≃ β) (g : β ≃ γ) #check Equiv α β #check (f.toFun : α → β) #check (f.invFun : β → α) #check (f.right_inv : ∀ x : β, f (f.invFun x) = x) #check (f.left_inv : ∀ x : α, f.invFun (f x) = x) #check (Equiv.refl α : α ≃ α) #check (f.symm : β ≃ α) #check (f.trans g : α ≃ γ) example (x : α) : (f.trans g).toFun x = g.toFun (f.toFun x) := rfl example (x : α) : (f.trans g) x = g (f x) := rfl example : (f.trans g : α → γ) = g ∘ f := rfl end example (α : Type _) : Equiv.Perm α = (α ≃ α) := rfl def permGroup {α : Type _} : Group₁ (Equiv.Perm α) where mul f g := Equiv.trans g f one := Equiv.refl α inv := Equiv.symm mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm one_mul := Equiv.trans_refl mul_one := Equiv.refl_trans mul_left_inv := Equiv.self_trans_symm structure AddGroup₁ (α : Type _) where (add : α → α → α) -- fill in the rest @[ext] structure Point where x : ℝ y : ℝ z : ℝ namespace Point def add (a b : Point) : Point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ def neg (a : point) : point := sorry def zero : point := sorry def add_group_point : add_group₁ point := sorry end Point section variable {α : Type _} (f g : Equiv.Perm α) (n : ℕ) #check f * g #check mul_assoc f g g⁻¹ -- group power, defined for any group #check g ^ n example : f * g * g⁻¹ = f := by rw [mul_assoc, mul_right_inv, mul_one] example : f * g * g⁻¹ = f := mul_inv_cancel_right f g example {α : Type _} (f g : Equiv.Perm α) : g.symm.trans (g.trans f) = f := mul_inv_cancel_right f g end class Group₂ (α : Type _) where mul : α → α → α one : α inv : α → α mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) mul_one : ∀ x : α, mul x one = x one_mul : ∀ x : α, mul one x = x mul_left_inv : ∀ x : α, mul (inv x) x = one instance {α : Type _} : Group₂ (Equiv.Perm α) where mul f g := Equiv.trans g f one := Equiv.refl α inv := Equiv.symm mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm one_mul := Equiv.trans_refl mul_one := Equiv.refl_trans mul_left_inv := Equiv.self_trans_symm #check @Group₂.mul def mySquare {α : Type _} [Group₂ α] (x : α) := Group₂.mul x x #check @mySquare section variable {β : Type _} (f g : Equiv.Perm β) example : Group₂.mul f g = g.trans f := rfl example : mySquare f = f.trans f := rfl end instance : Inhabited Point where default := ⟨0, 0, 0⟩ #check (default : Point) example : ([] : List Point).headI = default := rfl instance : Add Point where add := Point.add section variable (x y : Point) #check x + y example : x + y = Point.add x y := rfl end instance hasMulGroup₂ {α : Type _} [Group₂ α] : Mul α := ⟨Group₂.mul⟩ instance hasOneGroup₂ {α : Type _} [Group₂ α] : One α := ⟨Group₂.one⟩ instance hasInvGroup₂ {α : Type _} [Group₂ α] : Inv α := ⟨Group₂.inv⟩ section variable {α : Type _} (f g : Equiv.Perm α) #check f * 1 * g⁻¹ def foo : f * 1 * g⁻¹ = g.symm.trans ((Equiv.refl α).trans f) := rfl end class AddGroup₂ (α : Type _) where add : α → α → α -- fill in the rest
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@@ -0,0 +1,271 @@import Mathlib.Data.Int.Basic import Mathlib.Algebra.EuclideanDomain.Basic import Mathlib.RingTheory.PrincipalIdealDomain import Mathlib.Tactic @[ext] structure gaussInt where re : ℤ im : ℤ namespace gaussInt instance : Zero gaussInt := ⟨⟨0, 0⟩⟩ instance : One gaussInt := ⟨⟨1, 0⟩⟩ instance : Add gaussInt := ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ instance : Neg gaussInt := ⟨fun x => ⟨-x.re, -x.im⟩⟩ instance : Mul gaussInt := ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := rfl theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := rfl theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := rfl theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := rfl theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := rfl @[simp] theorem zero_re : (0 : gaussInt).re = 0 := rfl @[simp] theorem zero_im : (0 : gaussInt).im = 0 := rfl @[simp] theorem one_re : (1 : gaussInt).re = 1 := rfl @[simp] theorem one_im : (1 : gaussInt).im = 0 := rfl @[simp] theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := rfl @[simp] theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := rfl @[simp] theorem neg_re (x : gaussInt) : (-x).re = -x.re := rfl @[simp] theorem neg_im (x : gaussInt) : (-x).im = -x.im := rfl @[simp] theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := rfl @[simp] theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := rfl instance instCommRing : CommRing gaussInt where zero := 0 one := 1 add := (· + ·) neg x := -x mul := (· * ·) add_assoc := by intros ext <;> simp <;> ring zero_add := by intros ext <;> simp add_zero := by intros ext <;> simp add_left_neg := by intros ext <;> simp add_comm := by intros ext <;> simp <;> ring mul_assoc := by intros ext <;> simp <;> ring one_mul := by intros ext <;> simp mul_one := by intros ext <;> simp left_distrib := by intros ext <;> simp <;> ring right_distrib := by intros ext <;> simp <;> ring mul_comm := by intros ext <;> simp <;> ring zero_mul := sorry mul_zero := sorry instance : Nontrivial gaussInt := by use 0, 1 rw [Ne, gaussInt.ext_iff] simp end gaussInt example (a b : ℤ) : a = b * (a / b) + a % b := Eq.symm <| Int.ediv_add_emod a b example (a b : ℤ) : b ≠ 0 → 0 ≤ a % b := Int.emod_nonneg a example (a b : ℤ) : b ≠ 0 → a % b < abs b := Int.emod_lt a namespace Int def div' (a b : ℤ) := (a + b / 2) / b def mod' (a b : ℤ) := (a + b / 2) % b - b / 2 theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by rw [div', mod'] linarith [Int.ediv_add_emod (a + b / 2) b] theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by rw [mod', abs_le] constructor · linarith [Int.emod_nonneg (a + b / 2) h.ne'] have := Int.emod_lt_of_pos (a + b / 2) h have := Int.ediv_add_emod b 2 have := Int.emod_lt_of_pos b zero_lt_two revert this; intro this -- FIXME, this should not be needed linarith theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] end Int theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by sorry namespace gaussInt def norm (x : gaussInt) := x.re ^ 2 + x.im ^ 2 @[simp] theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by sorry theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by sorry theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by sorry theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by sorry def conj (x : gaussInt) : gaussInt := ⟨x.re, -x.im⟩ @[simp] theorem conj_re (x : gaussInt) : (conj x).re = x.re := rfl @[simp] theorem conj_im (x : gaussInt) : (conj x).im = -x.im := rfl theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] instance : Div gaussInt := ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ instance : Mod gaussInt := ⟨fun x y => x - y * (x / y)⟩ theorem div_def (x y : gaussInt) : x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := rfl theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := rfl theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by have norm_y_pos : 0 < norm y := by rwa [norm_pos] have : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] ext <;> simp <;> ring have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by conv => lhs rw [← norm_conj y, ← norm_mul, this, norm] simp trans 2 * (y.norm / 2) ^ 2 · rw [two_mul] apply add_le_add <;> · rw [sq_le_sq] apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) apply le_abs_self rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] apply mul_le_mul_of_nonneg_left · apply Int.ediv_mul_le norm_num apply Int.ediv_nonneg (norm_nonneg y) norm_num have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos apply lt_of_le_of_lt this apply Int.ediv_lt_of_lt_mul · norm_num linarith theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := Int.natAbs_of_nonneg (norm_nonneg _) theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by apply Int.ofNat_lt.1 simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] apply norm_mod_lt x hy theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : ¬(norm (x * y)).natAbs < (norm x).natAbs := by apply not_lt_of_ge rw [norm_mul, Int.natAbs_mul] apply le_mul_of_one_le_right (Nat.zero_le _) apply Int.ofNat_le.1 rw [coe_natAbs_norm] exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) instance : EuclideanDomain gaussInt := { gaussInt.instCommRing with quotient := (· / ·) remainder := (· % ·) quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] quotient_zero := fun x => by simp [div_def, norm, Int.div'] rfl r := Measure (Int.natAbs ∘ norm) r_wellFounded := (measure (Int.natAbs ∘ norm)).2 remainder_lt := natAbs_norm_mod_lt mul_left_not_lt := not_norm_mul_left_lt_norm } example (x : gaussInt) : Irreducible x ↔ Prime x := PrincipalIdealRing.irreducible_iff_prime end gaussInt
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@@ -0,0 +1,96 @@import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic noncomputable section @[ext] structure Point where x : ℝ y : ℝ z : ℝ namespace Point def add (a b : Point) : Point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by simp [add, add_assoc] def smul (r : ℝ) (a : Point) : Point := ⟨r * a.x, r * a.y, r * a.z⟩ theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by simp [add, smul, mul_add] end Point structure StandardTwoSimplex where x : ℝ y : ℝ z : ℝ x_nonneg : 0 ≤ x y_nonneg : 0 ≤ y z_nonneg : 0 ≤ z sum_eq : x + y + z = 1 namespace StandardTwoSimplex noncomputable section def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) (a b : StandardTwoSimplex) : StandardTwoSimplex where x := lambda * a.x + (1 - lambda) * b.x y := lambda * a.y + (1 - lambda) * b.y z := lambda * a.z + (1 - lambda) * b.z x_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.x_nonneg) (mul_nonneg (by linarith) b.x_nonneg) y_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.y_nonneg) (mul_nonneg (by linarith) b.y_nonneg) z_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.z_nonneg) (mul_nonneg (by linarith) b.z_nonneg) sum_eq := by trans (a.x + a.y + a.z) * lambda + (b.x + b.y + b.z) * (1 - lambda) · ring simp [a.sum_eq, b.sum_eq] end end StandardTwoSimplex open BigOperators structure StandardSimplex (n : ℕ) where V : Fin n → ℝ NonNeg : ∀ i : Fin n, 0 ≤ V i sum_eq_one : (∑ i, V i) = 1 namespace StandardSimplex def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n where V i := (a.V i + b.V i) / 2 NonNeg := by intro i apply div_nonneg · linarith [a.NonNeg i, b.NonNeg i] norm_num sum_eq_one := by simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] field_simp end StandardSimplex namespace StandardSimplex def weightedAverage {n : ℕ} (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) (a b : StandardSimplex n) : StandardSimplex n where V i := lambda * a.V i + (1 - lambda) * b.V i NonNeg i := add_nonneg (mul_nonneg lambda_nonneg (a.NonNeg i)) (mul_nonneg (by linarith) (b.NonNeg i)) sum_eq_one := by trans (lambda * ∑ i, a.V i) + (1 - lambda) * ∑ i, b.V i · rw [Finset.sum_add_distrib, Finset.mul_sum, Finset.mul_sum] simp [a.sum_eq_one, b.sum_eq_one] end StandardSimplex
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@@ -0,0 +1,73 @@import Mathlib.Data.Real.Basic structure AddGroup₁ (α : Type _) where add : α → α → α zero : α neg : α → α add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) add_zero : ∀ x : α, add x zero = x zero_add : ∀ x : α, add x zero = x add_left_neg : ∀ x : α, add (neg x) x = zero @[ext] structure Point where x : ℝ y : ℝ z : ℝ namespace Point def add (a b : Point) : Point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ def neg (a : Point) : Point := ⟨-a.x, -a.y, -a.z⟩ def zero : Point := ⟨0, 0, 0⟩ def addGroupPoint : AddGroup₁ Point where add := Point.add zero := Point.zero neg := Point.neg add_assoc := by simp [Point.add, add_assoc] add_zero := by simp [Point.add, Point.zero] zero_add := by simp [Point.add, Point.zero] add_left_neg := by simp [Point.add, Point.neg, Point.zero] end Point class AddGroup₂ (α : Type _) where add : α → α → α zero : α neg : α → α add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) add_zero : ∀ x : α, add x zero = x zero_add : ∀ x : α, add x zero = x add_left_neg : ∀ x : α, add (neg x) x = zero instance hasAddAddGroup₂ {α : Type _} [AddGroup₂ α] : Add α := ⟨AddGroup₂.add⟩ instance hasZeroAddGroup₂ {α : Type _} [AddGroup₂ α] : Zero α := ⟨AddGroup₂.zero⟩ instance hasNegAddGroup₂ {α : Type _} [AddGroup₂ α] : Neg α := ⟨AddGroup₂.neg⟩ instance : AddGroup₂ Point where add := Point.add zero := Point.zero neg := Point.neg add_assoc := by simp [Point.add, add_assoc] add_zero := by simp [Point.add, Point.zero] zero_add := by simp [Point.add, Point.zero] add_left_neg := by simp [Point.add, Point.neg, Point.zero] section variable (x y : Point) #check x + -y + 0 end
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@@ -0,0 +1,285 @@import Mathlib.Data.Int.Basic import Mathlib.Algebra.EuclideanDomain.Basic import Mathlib.RingTheory.PrincipalIdealDomain import Mathlib.Tactic @[ext] structure gaussInt where re : ℤ im : ℤ namespace gaussInt instance : Zero gaussInt := ⟨⟨0, 0⟩⟩ instance : One gaussInt := ⟨⟨1, 0⟩⟩ instance : Add gaussInt := ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ instance : Neg gaussInt := ⟨fun x => ⟨-x.re, -x.im⟩⟩ instance : Mul gaussInt := ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := rfl theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := rfl theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := rfl theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := rfl theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := rfl @[simp] theorem zero_re : (0 : gaussInt).re = 0 := rfl @[simp] theorem zero_im : (0 : gaussInt).im = 0 := rfl @[simp] theorem one_re : (1 : gaussInt).re = 1 := rfl @[simp] theorem one_im : (1 : gaussInt).im = 0 := rfl @[simp] theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := rfl @[simp] theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := rfl @[simp] theorem neg_re (x : gaussInt) : (-x).re = -x.re := rfl @[simp] theorem neg_im (x : gaussInt) : (-x).im = -x.im := rfl @[simp] theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := rfl @[simp] theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := rfl instance instCommRing : CommRing gaussInt where zero := 0 one := 1 add := (· + ·) neg x := -x mul := (· * ·) add_assoc := by intros ext <;> simp <;> ring zero_add := by intros ext <;> simp add_zero := by intros ext <;> simp add_left_neg := by intros ext <;> simp add_comm := by intros ext <;> simp <;> ring mul_assoc := by intros ext <;> simp <;> ring one_mul := by intros ext <;> simp mul_one := by intros ext <;> simp left_distrib := by intros ext <;> simp <;> ring right_distrib := by intros ext <;> simp <;> ring mul_comm := by intros ext <;> simp <;> ring zero_mul := sorry mul_zero := sorry instance : Nontrivial gaussInt := by use 0, 1 rw [Ne, gaussInt.ext_iff] simp end gaussInt namespace Int def div' (a b : ℤ) := (a + b / 2) / b def mod' (a b : ℤ) := (a + b / 2) % b - b / 2 theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by rw [div', mod'] linarith [Int.ediv_add_emod (a + b / 2) b] theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by rw [mod', abs_le] constructor · linarith [Int.emod_nonneg (a + b / 2) h.ne'] have := Int.emod_lt_of_pos (a + b / 2) h have := Int.ediv_add_emod b 2 have := Int.emod_lt_of_pos b zero_lt_two revert this; intro this -- FIXME, this should not be needed linarith theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] end Int private theorem aux {α : Type _} [LinearOrderedRing α] {x y : α} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := haveI h' : x ^ 2 = 0 := by apply le_antisymm _ (sq_nonneg x) rw [← h] apply le_add_of_nonneg_right (sq_nonneg y) pow_eq_zero h' theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by constructor · intro h constructor · exact aux h rw [add_comm] at h exact aux h rintro ⟨rfl, rfl⟩ norm_num namespace gaussInt def norm (x : gaussInt) := x.re ^ 2 + x.im ^ 2 @[simp] theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by apply add_nonneg <;> apply sq_nonneg theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by rw [norm, sq_add_sq_eq_zero, gaussInt.ext_iff] rfl theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by rw [lt_iff_le_and_ne, ne_comm, Ne, norm_eq_zero] simp [norm_nonneg] theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by simp [norm] ring def conj (x : gaussInt) : gaussInt := ⟨x.re, -x.im⟩ @[simp] theorem conj_re (x : gaussInt) : (conj x).re = x.re := rfl @[simp] theorem conj_im (x : gaussInt) : (conj x).im = -x.im := rfl theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] instance : Div gaussInt := ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ instance : Mod gaussInt := ⟨fun x y => x - y * (x / y)⟩ theorem div_def (x y : gaussInt) : x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := rfl theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := rfl theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by have norm_y_pos : 0 < norm y := by rwa [norm_pos] have : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] ext <;> simp <;> ring have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by conv => lhs rw [← norm_conj y, ← norm_mul, this, norm] simp trans 2 * (y.norm / 2) ^ 2 · rw [two_mul] apply add_le_add <;> · rw [sq_le_sq] apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) apply le_abs_self rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] apply mul_le_mul_of_nonneg_left · apply Int.ediv_mul_le norm_num apply Int.ediv_nonneg (norm_nonneg y) norm_num have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos apply lt_of_le_of_lt this apply Int.ediv_lt_of_lt_mul · norm_num linarith theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := Int.natAbs_of_nonneg (norm_nonneg _) theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by apply Int.ofNat_lt.1 simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] apply norm_mod_lt x hy theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : ¬(norm (x * y)).natAbs < (norm x).natAbs := by apply not_lt_of_ge rw [norm_mul, Int.natAbs_mul] apply le_mul_of_one_le_right (Nat.zero_le _) apply Int.ofNat_le.1 rw [coe_natAbs_norm] exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) instance : EuclideanDomain gaussInt := { gaussInt.instCommRing with quotient := (· / ·) remainder := (· % ·) quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] quotient_zero := fun x => by simp [div_def, norm, Int.div'] rfl r := Measure (Int.natAbs ∘ norm) r_wellFounded := (measure (Int.natAbs ∘ norm)).2 remainder_lt := natAbs_norm_mod_lt mul_left_not_lt := not_norm_mul_left_lt_norm } example (x : gaussInt) : Irreducible x ↔ Prime x := PrincipalIdealRing.irreducible_iff_prime end gaussInt
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@@ -0,0 +1,164 @@import Mathlib.Data.Real.Basic import Mathlib.Data.Real.Basic -- An example. example (a b c : ℝ) : a * b * c = b * (a * c) := by rw [mul_comm a b] rw [mul_assoc b a c] -- Try these. example (a b c : ℝ) : c * b * a = b * (a * c) := by sorry example (a b c : ℝ) : a * (b * c) = b * (a * c) := by sorry -- An example. example (a b c : ℝ) : a * b * c = b * c * a := by rw [mul_assoc] rw [mul_comm] /- Try doing the first of these without providing any arguments at all, and the second with only one argument. -/ example (a b c : ℝ) : a * (b * c) = b * (c * a) := by sorry example (a b c : ℝ) : a * (b * c) = b * (a * c) := by sorry -- Using facts from the local context. example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by rw [h'] rw [← mul_assoc] rw [h] rw [mul_assoc] -- Try these. For the second one, use the theorem `sub_self`. example (a b c d e f : ℝ) (h : b * c = e * f) : a * b * c * d = a * e * f * d := by sorry example (a b c d : ℝ) (hyp : c = b * a - d) (hyp' : d = a * b) : c = 0 := by sorry -- Examples. example (a b c d e f : ℝ) (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by rw [h', ← mul_assoc, h, mul_assoc] section variable (a b c d e f g : ℝ) example (h : a * b = c * d) (h' : e = f) : a * (b * e) = c * (d * f) := by rw [h', ← mul_assoc, h, mul_assoc] end section variable (a b c : ℝ) #check a #check a + b #check (a : ℝ) #check mul_comm a b #check (mul_comm a b : a * b = b * a) #check mul_assoc c a b #check mul_comm a #check mul_comm #check @mul_comm end section variable (a b : ℝ) example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by rw [mul_add, add_mul, add_mul] rw [← add_assoc, add_assoc (a * a)] rw [mul_comm b a, ← two_mul] example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := calc (a + b) * (a + b) = a * a + b * a + (a * b + b * b) := by rw [mul_add, add_mul, add_mul] _ = a * a + (b * a + a * b) + b * b := by rw [← add_assoc, add_assoc (a * a)] _ = a * a + 2 * (a * b) + b * b := by rw [mul_comm b a, ← two_mul] example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := calc (a + b) * (a + b) = a * a + b * a + (a * b + b * b) := by sorry _ = a * a + (b * a + a * b) + b * b := by sorry _ = a * a + 2 * (a * b) + b * b := by sorry end -- Try these. For the second, use the theorems listed underneath. section variable (a b c d : ℝ) example : (a + b) * (c + d) = a * c + a * d + b * c + b * d := by sorry example (a b : ℝ) : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by sorry #check pow_two a #check mul_sub a b c #check add_mul a b c #check add_sub a b c #check sub_sub a b c #check add_zero a end -- Examples. section variable (a b c d : ℝ) example (a b c d : ℝ) (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by rw [hyp'] at hyp rw [mul_comm d a] at hyp rw [← two_mul (a * d)] at hyp rw [← mul_assoc 2 a d] at hyp exact hyp example : c * b * a = b * (a * c) := by ring example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by ring example : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by ring example (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by rw [hyp, hyp'] ring end example (a b c : ℕ) (h : a + b = c) : (a + b) * (a + b) = a * c + b * c := by nth_rw 2 [h] rw [add_mul]
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@@ -0,0 +1,168 @@import Mathlib.Algebra.Ring.Defs import Mathlib.Data.Real.Basic import Mathlib.Tactic section variable (R : Type _) [Ring R] #check (add_assoc : ∀ a b c : R, a + b + c = a + (b + c)) #check (add_comm : ∀ a b : R, a + b = b + a) #check (zero_add : ∀ a : R, 0 + a = a) #check (add_left_neg : ∀ a : R, -a + a = 0) #check (mul_assoc : ∀ a b c : R, a * b * c = a * (b * c)) #check (mul_one : ∀ a : R, a * 1 = a) #check (one_mul : ∀ a : R, 1 * a = a) #check (mul_add : ∀ a b c : R, a * (b + c) = a * b + a * c) #check (add_mul : ∀ a b c : R, (a + b) * c = a * c + b * c) end section variable (R : Type _) [CommRing R] variable (a b c d : R) example : c * b * a = b * (a * c) := by ring example : (a + b) * (a + b) = a * a + 2 * (a * b) + b * b := by ring example : (a + b) * (a - b) = a ^ 2 - b ^ 2 := by ring example (hyp : c = d * a + b) (hyp' : b = a * d) : c = 2 * a * d := by rw [hyp, hyp'] ring end namespace MyRing variable {R : Type _} [Ring R] theorem add_zero (a : R) : a + 0 = a := by rw [add_comm, zero_add] theorem add_right_neg (a : R) : a + -a = 0 := by rw [add_comm, add_left_neg] #check @MyRing.add_zero #check @add_zero end MyRing namespace MyRing variable {R : Type _} [Ring R] theorem neg_add_cancel_left (a b : R) : -a + (a + b) = b := by rw [← add_assoc, add_left_neg, zero_add] -- Prove these: theorem add_neg_cancel_right (a b : R) : a + b + -b = a := by sorry theorem add_left_cancel {a b c : R} (h : a + b = a + c) : b = c := by sorry theorem add_right_cancel {a b c : R} (h : a + b = c + b) : a = c := by sorry theorem mul_zero (a : R) : a * 0 = 0 := by have h : a * 0 + a * 0 = a * 0 + 0 := by rw [← mul_add, add_zero, add_zero] rw [add_left_cancel h] theorem zero_mul (a : R) : 0 * a = 0 := by sorry theorem neg_eq_of_add_eq_zero {a b : R} (h : a + b = 0) : -a = b := by sorry theorem eq_neg_of_add_eq_zero {a b : R} (h : a + b = 0) : a = -b := by sorry theorem neg_zero : (-0 : R) = 0 := by apply neg_eq_of_add_eq_zero rw [add_zero] theorem neg_neg (a : R) : - -a = a := by sorry end MyRing -- Examples. section variable {R : Type _} [Ring R] example (a b : R) : a - b = a + -b := sub_eq_add_neg a b end example (a b : ℝ) : a - b = a + -b := rfl example (a b : ℝ) : a - b = a + -b := by rfl namespace MyRing variable {R : Type _} [Ring R] theorem self_sub (a : R) : a - a = 0 := sorry theorem one_add_one_eq_two : 1 + 1 = (2 : R) := by norm_num theorem two_mul (a : R) : 2 * a = a + a := sorry end MyRing section variable (A : Type _) [AddGroup A] #check (add_assoc : ∀ a b c : A, a + b + c = a + (b + c)) #check (zero_add : ∀ a : A, 0 + a = a) #check (add_left_neg : ∀ a : A, -a + a = 0) end section variable {G : Type _} [Group G] #check (mul_assoc : ∀ a b c : G, a * b * c = a * (b * c)) #check (one_mul : ∀ a : G, 1 * a = a) #check (mul_left_inv : ∀ a : G, a⁻¹ * a = 1) namespace MyGroup theorem mul_right_inv (a : G) : a * a⁻¹ = 1 := by sorry theorem mul_one (a : G) : a * 1 = a := by sorry theorem mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a⁻¹ := by sorry end MyGroup end
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@@ -0,0 +1,156 @@import Mathlib.Analysis.SpecialFunctions.Log.Basic import Mathlib.Tactic variable (a b c d e : ℝ) open Real #check (le_refl : ∀ a : ℝ, a ≤ a) #check (le_trans : a ≤ b → b ≤ c → a ≤ c) section variable (h : a ≤ b) (h' : b ≤ c) #check (le_refl : ∀ a : Real, a ≤ a) #check (le_refl a : a ≤ a) #check (le_trans : a ≤ b → b ≤ c → a ≤ c) #check (le_trans h : b ≤ c → a ≤ c) #check (le_trans h h' : a ≤ c) end example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := by apply le_trans · apply h₀ . apply h₁ example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := by apply le_trans h₀ apply h₁ example (x y z : ℝ) (h₀ : x ≤ y) (h₁ : y ≤ z) : x ≤ z := le_trans h₀ h₁ example (x : ℝ) : x ≤ x := by apply le_refl example (x : ℝ) : x ≤ x := le_refl x #check (le_refl : ∀ a, a ≤ a) #check (le_trans : a ≤ b → b ≤ c → a ≤ c) #check (lt_of_le_of_lt : a ≤ b → b < c → a < c) #check (lt_of_lt_of_le : a < b → b ≤ c → a < c) #check (lt_trans : a < b → b < c → a < c) -- Try this. example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by sorry example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by linarith section example (h : 2 * a ≤ 3 * b) (h' : 1 ≤ a) (h'' : d = 2) : d + a ≤ 5 * b := by linarith end example (h : 1 ≤ a) (h' : b ≤ c) : 2 + a + exp b ≤ 3 * a + exp c := by linarith [exp_le_exp.mpr h'] #check (exp_le_exp : exp a ≤ exp b ↔ a ≤ b) #check (exp_lt_exp : exp a < exp b ↔ a < b) #check (log_le_log : 0 < a → 0 < b → (log a ≤ log b ↔ a ≤ b)) #check (log_lt_log : 0 < a → a < b → log a < log b) #check (add_le_add : a ≤ b → c ≤ d → a + c ≤ b + d) #check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b) #check (add_le_add_right : a ≤ b → ∀ c, a + c ≤ b + c) #check (add_lt_add_of_le_of_lt : a ≤ b → c < d → a + c < b + d) #check (add_lt_add_of_lt_of_le : a < b → c ≤ d → a + c < b + d) #check (add_lt_add_left : a < b → ∀ c, c + a < c + b) #check (add_lt_add_right : a < b → ∀ c, a + c < b + c) #check (add_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a + b) #check (add_pos : 0 < a → 0 < b → 0 < a + b) #check (add_pos_of_pos_of_nonneg : 0 < a → 0 ≤ b → 0 < a + b) #check (exp_pos : ∀ a, 0 < exp a) #check @add_le_add_left example (h : a ≤ b) : exp a ≤ exp b := by rw [exp_le_exp] exact h example (h₀ : a ≤ b) (h₁ : c < d) : a + exp c + e < b + exp d + e := by apply add_lt_add_of_lt_of_le · apply add_lt_add_of_le_of_lt h₀ apply exp_lt_exp.mpr h₁ apply le_refl example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := by sorry example : (0 : ℝ) < 1 := by norm_num example (h : a ≤ b) : log (1 + exp a) ≤ log (1 + exp b) := by have h₀ : 0 < 1 + exp a := by sorry have h₁ : 0 < 1 + exp b := by sorry apply (log_le_log h₀ h₁).mpr sorry example : 0 ≤ a ^ 2 := by -- library_search exact sq_nonneg a example (h : a ≤ b) : c - exp b ≤ c - exp a := by sorry example : 2 * a * b ≤ a ^ 2 + b ^ 2 := by have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 calc a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring _ ≥ 0 := by apply pow_two_nonneg calc 2 * a * b = 2 * a * b + 0 := by ring _ ≤ 2 * a * b + (a ^ 2 - 2 * a * b + b ^ 2) := add_le_add (le_refl _) h _ = a ^ 2 + b ^ 2 := by ring example : 2 * a * b ≤ a ^ 2 + b ^ 2 := by have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 calc a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring _ ≥ 0 := by apply pow_two_nonneg linarith example : abs (a * b) ≤ (a ^ 2 + b ^ 2) / 2 := by sorry #check abs_le'.mpr
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@@ -0,0 +1,98 @@import Mathlib.Data.Real.Basic section variable (a b c d : ℝ) #check (min_le_left a b : min a b ≤ a) #check (min_le_right a b : min a b ≤ b) #check (le_min : c ≤ a → c ≤ b → c ≤ min a b) example : min a b = min b a := by apply le_antisymm · show min a b ≤ min b a apply le_min · apply min_le_right apply min_le_left · show min b a ≤ min a b apply le_min · apply min_le_right apply min_le_left example : min a b = min b a := by have h : ∀ x y : ℝ, min x y ≤ min y x := by intro x y apply le_min apply min_le_right apply min_le_left apply le_antisymm apply h apply h example : min a b = min b a := by apply le_antisymm repeat apply le_min apply min_le_right apply min_le_left example : max a b = max b a := by sorry example : min (min a b) c = min a (min b c) := by sorry theorem aux : min a b + c ≤ min (a + c) (b + c) := by sorry example : min a b + c = min (a + c) (b + c) := by sorry #check (abs_add : ∀ a b : ℝ, abs (a + b) ≤ abs a + abs b) example : abs a - abs b ≤ abs (a - b) := sorry end section variable (w x y z : ℕ) example (h₀ : x ∣ y) (h₁ : y ∣ z) : x ∣ z := dvd_trans h₀ h₁ example : x ∣ y * x * z := by apply dvd_mul_of_dvd_left apply dvd_mul_left example : x ∣ x ^ 2 := by apply dvd_mul_left example (h : x ∣ w) : x ∣ y * (x * z) + x ^ 2 + w ^ 2 := by sorry end section variable (m n : ℕ) open Nat #check (gcd_zero_right n : gcd n 0 = n) #check (gcd_zero_left n : gcd 0 n = n) #check (lcm_zero_right n : lcm n 0 = 0) #check (lcm_zero_left n : lcm 0 n = 0) example : gcd m n = gcd n m := by sorry end
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@@ -0,0 +1,141 @@import Mathlib.Topology.MetricSpace.Basic section variable {α : Type _} [PartialOrder α] variable (x y z : α) #check x ≤ y #check (le_refl x : x ≤ x) #check (le_trans : x ≤ y → y ≤ z → x ≤ z) #check x < y #check (lt_irrefl x : ¬x < x) #check (lt_trans : x < y → y < z → x < z) #check (lt_of_le_of_lt : x ≤ y → y < z → x < z) #check (lt_of_lt_of_le : x < y → y ≤ z → x < z) example : x < y ↔ x ≤ y ∧ x ≠ y := lt_iff_le_and_ne end section variable {α : Type _} [Lattice α] variable (x y z : α) #check x ⊓ y #check (inf_le_left : x ⊓ y ≤ x) #check (inf_le_right : x ⊓ y ≤ y) #check (le_inf : z ≤ x → z ≤ y → z ≤ x ⊓ y) #check x ⊔ y #check (le_sup_left : x ≤ x ⊔ y) #check (le_sup_right : y ≤ x ⊔ y) #check (sup_le : x ≤ z → y ≤ z → x ⊔ y ≤ z) example : x ⊓ y = y ⊓ x := by sorry example : x ⊓ y ⊓ z = x ⊓ (y ⊓ z) := by sorry example : x ⊔ y = y ⊔ x := by sorry example : x ⊔ y ⊔ z = x ⊔ (y ⊔ z) := by sorry theorem absorb1 : x ⊓ (x ⊔ y) = x := by sorry theorem absorb2 : x ⊔ x ⊓ y = x := by sorry end section variable {α : Type _} [DistribLattice α] variable (x y z : α) #check (inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) #check (inf_sup_right : (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z) #check (sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) #check (sup_inf_right : x ⊓ y ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) end section variable {α : Type _} [Lattice α] variable (a b c : α) example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) : a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c) := by sorry example (h : ∀ x y z : α, x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) : a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c := by sorry end section variable {R : Type _} [StrictOrderedRing R] variable (a b c : R) #check (add_le_add_left : a ≤ b → ∀ c, c + a ≤ c + b) #check (mul_pos : 0 < a → 0 < b → 0 < a * b) #check (mul_nonneg : 0 ≤ a → 0 ≤ b → 0 ≤ a * b) example : a ≤ b → 0 ≤ b - a := by sorry example : 0 ≤ b - a → a ≤ b := by sorry example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := by sorry end section variable {X : Type _} [MetricSpace X] variable (x y z : X) #check (dist_self x : dist x x = 0) #check (dist_comm x y : dist x y = dist y x) #check (dist_triangle x y z : dist x z ≤ dist x y + dist y z) example (x y : X) : 0 ≤ dist x y := by sorry end
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@@ -0,0 +1,33 @@import Mathlib.Data.Real.Basic import Mathlib.Data.Real.Basic example (a b c : ℝ) : c * b * a = b * (a * c) := by rw [mul_comm c b] rw [mul_assoc b c a] rw [mul_comm c a] example (a b c : ℝ) : a * (b * c) = b * (a * c) := by rw [← mul_assoc a b c] rw [mul_comm a b] rw [mul_assoc b a c] example (a b c : ℝ) : a * (b * c) = b * (c * a) := by rw [mul_comm] rw [mul_assoc] example (a b c : ℝ) : a * (b * c) = b * (a * c) := by rw [← mul_assoc] rw [mul_comm a] rw [mul_assoc] example (a b c d e f : ℝ) (h : b * c = e * f) : a * b * c * d = a * e * f * d := by rw [mul_assoc a] rw [h] rw [← mul_assoc] example (a b c d : ℝ) (hyp : c = b * a - d) (hyp' : d = a * b) : c = 0 := by rw [hyp] rw [hyp'] rw [mul_comm] rw [sub_self]
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@@ -0,0 +1,76 @@import Mathlib.Algebra.Ring.Defs import Mathlib.Data.Real.Basic import Mathlib.Tactic namespace MyRing variable {R : Type _} [Ring R] theorem add_neg_cancel_right (a b : R) : a + b + -b = a := by rw [add_assoc, add_right_neg, add_zero] theorem add_left_cancel {a b c : R} (h : a + b = a + c) : b = c := by rw [← neg_add_cancel_left a b, h, neg_add_cancel_left] theorem add_right_cancel {a b c : R} (h : a + b = c + b) : a = c := by rw [← add_neg_cancel_right a b, h, add_neg_cancel_right] theorem zero_mul (a : R) : 0 * a = 0 := by have h : 0 * a + 0 * a = 0 * a + 0 := by rw [← add_mul, add_zero, add_zero] rw [add_left_cancel h] theorem neg_eq_of_add_eq_zero {a b : R} (h : a + b = 0) : -a = b := by rw [← neg_add_cancel_left a b, h, add_zero] theorem eq_neg_of_add_eq_zero {a b : R} (h : a + b = 0) : a = -b := by symm apply neg_eq_of_add_eq_zero rw [add_comm, h] theorem neg_zero : (-0 : R) = 0 := by apply neg_eq_of_add_eq_zero rw [add_zero] theorem neg_neg (a : R) : - -a = a := by apply neg_eq_of_add_eq_zero rw [add_left_neg] end MyRing namespace MyRing variable {R : Type _} [Ring R] theorem self_sub (a : R) : a - a = 0 := by rw [sub_eq_add_neg, add_right_neg] theorem one_add_one_eq_two : 1 + 1 = (2 : R) := by norm_num theorem two_mul (a : R) : 2 * a = a + a := by rw [← one_add_one_eq_two, add_mul, one_mul] end MyRing section variable {G : Type _} [Group G] namespace MyGroup theorem mul_right_inv (a : G) : a * a⁻¹ = 1 := by have h : (a * a⁻¹)⁻¹ * (a * a⁻¹ * (a * a⁻¹)) = 1 := by rw [mul_assoc, ← mul_assoc a⁻¹ a, mul_left_inv, one_mul, mul_left_inv] rw [← h, ← mul_assoc, mul_left_inv, one_mul] theorem mul_one (a : G) : a * 1 = a := by rw [← mul_left_inv a, ← mul_assoc, mul_right_inv, one_mul] theorem mul_inv_rev (a b : G) : (a * b)⁻¹ = b⁻¹ * a⁻¹ := by rw [← one_mul (b⁻¹ * a⁻¹), ← mul_left_inv (a * b), mul_assoc, mul_assoc, ← mul_assoc b b⁻¹, mul_right_inv, one_mul, mul_right_inv, mul_one] end MyGroup end
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@@ -0,0 +1,62 @@import Mathlib.Analysis.SpecialFunctions.Log.Basic import Mathlib.Tactic variable (a b c d e : ℝ) open Real example (h₀ : a ≤ b) (h₁ : b < c) (h₂ : c ≤ d) (h₃ : d < e) : a < e := by apply lt_of_le_of_lt h₀ apply lt_trans h₁ exact lt_of_le_of_lt h₂ h₃ example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := by apply add_le_add_left rw [exp_le_exp] apply add_le_add_left h₀ -- an alternative using `linarith`. example (h₀ : d ≤ e) : c + exp (a + d) ≤ c + exp (a + e) := by have : exp (a + d) ≤ exp (a + e) := by rw [exp_le_exp] linarith linarith [this] example (h : a ≤ b) : log (1 + exp a) ≤ log (1 + exp b) := by have h₀ : 0 < 1 + exp a := by linarith [exp_pos a] have h₁ : 0 < 1 + exp b := by linarith [exp_pos b] apply (log_le_log h₀ h₁).mpr apply add_le_add_left (exp_le_exp.mpr h) -- SOLUTION. example (h : a ≤ b) : c - exp b ≤ c - exp a := by apply sub_le_sub_left exact exp_le_exp.mpr h -- alternatively: example (h : a ≤ b) : c - exp b ≤ c - exp a := by linarith [exp_le_exp.mpr h] theorem fact1 : a * b * 2 ≤ a ^ 2 + b ^ 2 := by have h : 0 ≤ a ^ 2 - 2 * a * b + b ^ 2 calc a ^ 2 - 2 * a * b + b ^ 2 = (a - b) ^ 2 := by ring _ ≥ 0 := by apply pow_two_nonneg linarith theorem fact2 : -(a * b) * 2 ≤ a ^ 2 + b ^ 2 := by have h : 0 ≤ a ^ 2 + 2 * a * b + b ^ 2 calc a ^ 2 + 2 * a * b + b ^ 2 = (a + b) ^ 2 := by ring _ ≥ 0 := by apply pow_two_nonneg linarith example : abs (a * b) ≤ (a ^ 2 + b ^ 2) / 2 := by have h : (0 : ℝ) < 2 := by norm_num apply abs_le'.mpr constructor · rw [le_div_iff h] apply fact1 rw [le_div_iff h] apply fact2
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@@ -0,0 +1,124 @@import Mathlib.Data.Real.Basic section variable (a b c d : ℝ) #check (min_le_left a b : min a b ≤ a) #check (min_le_right a b : min a b ≤ b) #check (le_min : c ≤ a → c ≤ b → c ≤ min a b) example : max a b = max b a := by apply le_antisymm repeat' apply max_le apply le_max_right apply le_max_left example : min (min a b) c = min a (min b c) := by apply le_antisymm · apply le_min · apply le_trans apply min_le_left apply min_le_left apply le_min · apply le_trans apply min_le_left apply min_le_right apply min_le_right apply le_min · apply le_min · apply min_le_left apply le_trans apply min_le_right apply min_le_left apply le_trans apply min_le_right apply min_le_right theorem aux : min a b + c ≤ min (a + c) (b + c) := by apply le_min · apply add_le_add_right apply min_le_left apply add_le_add_right apply min_le_right example : min a b + c = min (a + c) (b + c) := by apply le_antisymm · apply aux have h : min (a + c) (b + c) = min (a + c) (b + c) - c + c := by rw [sub_add_cancel] rw [h] apply add_le_add_right rw [sub_eq_add_neg] apply le_trans apply aux rw [add_neg_cancel_right, add_neg_cancel_right] example : abs a - abs b ≤ abs (a - b) := calc abs a - abs b = abs (a - b + b) - abs b := by rw [sub_add_cancel] _ ≤ abs (a - b) + abs b - abs b := by apply sub_le_sub_right apply abs_add _ ≤ abs (a - b) := by rw [add_sub_cancel] -- alternatively example : abs a - abs b ≤ abs (a - b) := by have h := abs_add (a - b) b rw [sub_add_cancel] at h linarith end section variable (w x y z : ℕ) example (h₀ : x ∣ y) (h₁ : y ∣ z) : x ∣ z := dvd_trans h₀ h₁ example : x ∣ y * x * z := by apply dvd_mul_of_dvd_left apply dvd_mul_left example : x ∣ x ^ 2 := by apply dvd_mul_left example (h : x ∣ w) : x ∣ y * (x * z) + x ^ 2 + w ^ 2 := by apply dvd_add · apply dvd_add · apply dvd_mul_of_dvd_right apply dvd_mul_right apply dvd_mul_left rw [pow_two] apply dvd_mul_of_dvd_right exact h end section variable (m n : ℕ) open Nat #check (gcd_zero_right n : gcd n 0 = n) #check (gcd_zero_left n : gcd 0 n = n) #check (lcm_zero_right n : lcm n 0 = 0) #check (lcm_zero_left n : lcm 0 n = 0) example : gcd m n = gcd n m := by apply _root_.dvd_antisymm repeat' apply dvd_gcd apply gcd_dvd_right apply gcd_dvd_left end
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@@ -0,0 +1,149 @@import Mathlib.Topology.MetricSpace.Basic section variable {α : Type _} [Lattice α] variable (x y z : α) example : x ⊓ y = y ⊓ x := by apply le_antisymm repeat' apply le_inf · apply inf_le_right apply inf_le_left example : x ⊓ y ⊓ z = x ⊓ (y ⊓ z) := by apply le_antisymm · apply le_inf · apply le_trans apply inf_le_left apply inf_le_left apply le_inf · apply le_trans apply inf_le_left apply inf_le_right apply inf_le_right apply le_inf · apply le_inf · apply inf_le_left apply le_trans apply inf_le_right apply inf_le_left apply le_trans apply inf_le_right apply inf_le_right example : x ⊔ y = y ⊔ x := by apply le_antisymm repeat' apply sup_le · apply le_sup_right apply le_sup_left example : x ⊔ y ⊔ z = x ⊔ (y ⊔ z) := by apply le_antisymm · apply sup_le · apply sup_le apply le_sup_left · apply le_trans apply @le_sup_left _ _ y z apply le_sup_right apply le_trans apply @le_sup_right _ _ y z apply le_sup_right apply sup_le · apply le_trans apply @le_sup_left _ _ x y apply le_sup_left apply sup_le · apply le_trans apply @le_sup_right _ _ x y apply le_sup_left apply le_sup_right theorem absorb1 : x ⊓ (x ⊔ y) = x := by apply le_antisymm · apply inf_le_left apply le_inf · apply le_refl apply le_sup_left theorem absorb2 : x ⊔ x ⊓ y = x := by apply le_antisymm · apply sup_le · apply le_refl apply inf_le_left apply le_sup_left end section variable {α : Type _} [DistribLattice α] variable (x y z : α) #check (inf_sup_left : x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) #check (inf_sup_right : (x ⊔ y) ⊓ z = x ⊓ z ⊔ y ⊓ z) #check (sup_inf_left : x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) #check (sup_inf_right : x ⊓ y ⊔ z = (x ⊔ z) ⊓ (y ⊔ z)) end section variable {α : Type _} [Lattice α] variable (a b c : α) example (h : ∀ x y z : α, x ⊓ (y ⊔ z) = x ⊓ y ⊔ x ⊓ z) : a ⊔ b ⊓ c = (a ⊔ b) ⊓ (a ⊔ c) := by rw [h, @inf_comm _ _ (a ⊔ b), absorb1, @inf_comm _ _ (a ⊔ b), h, ← sup_assoc, @inf_comm _ _ c a, absorb2, inf_comm] example (h : ∀ x y z : α, x ⊔ y ⊓ z = (x ⊔ y) ⊓ (x ⊔ z)) : a ⊓ (b ⊔ c) = a ⊓ b ⊔ a ⊓ c := by rw [h, @sup_comm _ _ (a ⊓ b), absorb2, @sup_comm _ _ (a ⊓ b), h, ← inf_assoc, @sup_comm _ _ c a, absorb1, sup_comm] end section variable {R : Type _} [StrictOrderedRing R] variable (a b c : R) theorem aux1 : a ≤ b → 0 ≤ b - a := by intro h rw [← sub_self a, sub_eq_add_neg, sub_eq_add_neg, add_comm, add_comm b] apply add_le_add_left h theorem aux2 : 0 ≤ b - a → a ≤ b := by intro h rw [← add_zero a, ← sub_add_cancel b a, add_comm (b - a)] apply add_le_add_left h example (h : a ≤ b) (h' : 0 ≤ c) : a * c ≤ b * c := by have h1 : 0 ≤ (b - a) * c := mul_nonneg (aux1 _ _ h) h' rw [sub_mul] at h1 exact aux2 _ _ h1 end section variable {X : Type _} [MetricSpace X] variable (x y z : X) example (x y : X) : 0 ≤ dist x y :=by have : 0 ≤ dist x y + dist y x := by rw [← dist_self x] apply dist_triangle linarith [dist_comm x y] end
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@@ -0,0 +1,2 @@#eval "Hello, World!"
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@@ -0,0 +1,56 @@import Mathlib.Data.Nat.Basic import Mathlib.Data.Nat.Parity import Mathlib.Tactic open Nat -- These are pieces of data. #check 2 + 2 def f (x : ℕ) := x + 3 #check f -- These are propositions, of type `Prop`. #check 2 + 2 = 4 def FermatLastTheorem := ∀ x y z n : ℕ, n > 2 ∧ x * y * z ≠ 0 → x ^ n + y ^ n ≠ z ^ n #check FermatLastTheorem -- These are proofs of propositions. theorem easy : 2 + 2 = 4 := rfl #check easy theorem hard : FermatLastTheorem := sorry #check hard -- Here are some proofs. example : ∀ m n : Nat, Even n → Even (m * n) := fun m n ⟨k, (hk : n = k + k)⟩ => have hmn : m * n = m * k + m * k := by rw [hk, mul_add] show ∃ l, m * n = l + l from ⟨_, hmn⟩ example : ∀ m n : Nat, Even n → Even (m * n) := fun m n ⟨k, hk⟩ => ⟨m * k, by rw [hk, mul_add]⟩ example : ∀ m n : Nat, Even n → Even (m * n) := by -- say m and n are natural numbers, and assume n=2*k rintro m n ⟨k, hk⟩ -- We need to prove m*n is twice a natural number. Let's show it's twice m*k. use m * k -- substitute in for n rw [hk] -- and now it's obvious ring example : ∀ m n : Nat, Even n → Even (m * n) := by rintro m n ⟨k, hk⟩; use m * k; rw [hk]; ring example : ∀ m n : Nat, Even n → Even (m * n) := by intros; simp [*, parity_simps]
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@@ -0,0 +1,7 @@import Mathlib.Data.Nat.Basic import Mathlib.Data.Nat.Parity import Mathlib.Tactic open Nat -- There are no exercises in this section.
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@@ -0,0 +1,178 @@import Mathlib.Data.Real.Basic #check ∀ x : ℝ, 0 ≤ x → abs x = x #check ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε theorem my_lemma : ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := sorry section variable (a b δ : ℝ) variable (h₀ : 0 < δ) (h₁ : δ ≤ 1) variable (ha : abs a < δ) (hb : abs b < δ) #check my_lemma a b δ #check my_lemma a b δ h₀ h₁ #check my_lemma a b δ h₀ h₁ ha hb end theorem my_lemma2 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := sorry section variable (a b δ : ℝ) variable (h₀ : 0 < δ) (h₁ : δ ≤ 1) variable (ha : abs a < δ) (hb : abs b < δ) #check my_lemma2 h₀ h₁ ha hb end theorem my_lemma3 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := by intro x y ε epos ele1 xlt ylt sorry theorem my_lemma4 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := by intro x y ε epos ele1 xlt ylt calc abs (x * y) = abs x * abs y := sorry _ ≤ abs x * ε := sorry _ < 1 * ε := sorry _ = ε := sorry def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x section variable (f g : ℝ → ℝ) (a b : ℝ) example (hfa : FnUb f a) (hgb : FnUb g b) : FnUb (fun x ↦ f x + g x) (a + b) := by intro x dsimp apply add_le_add apply hfa apply hgb example (hfa : FnLb f a) (hgb : FnLb g b) : FnLb (fun x => f x + g x) (a + b) := sorry example (nnf : FnLb f 0) (nng : FnLb g 0) : FnLb (fun x => f x * g x) 0 := sorry example (hfa : FnUb f a) (hfb : FnUb g b) (nng : FnLb g 0) (nna : 0 ≤ a) : FnUb (fun x => f x * g x) (a * b) := sorry end section variable {α : Type _} {R : Type _} [OrderedCancelAddCommMonoid R] #check @add_le_add def FnUb' (f : α → R) (a : R) : Prop := ∀ x, f x ≤ a theorem fn_ub_add {f g : α → R} {a b : R} (hfa : FnUb' f a) (hgb : FnUb' g b) : FnUb' (fun x => f x + g x) (a + b) := fun x => add_le_add (hfa x) (hgb x) end example (f : ℝ → ℝ) (h : Monotone f) : ∀ {a b}, a ≤ b → f a ≤ f b := @h section variable (f g : ℝ → ℝ) example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f x + g x := by intro a b aleb apply add_le_add apply mf aleb apply mg aleb example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f x + g x := fun a b aleb => add_le_add (mf aleb) (mg aleb) example {c : ℝ} (mf : Monotone f) (nnc : 0 ≤ c) : Monotone fun x => c * f x := sorry example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f (g x) := sorry def FnEven (f : ℝ → ℝ) : Prop := ∀ x, f x = f (-x) def FnOdd (f : ℝ → ℝ) : Prop := ∀ x, f x = -f (-x) example (ef : FnEven f) (eg : FnEven g) : FnEven fun x => f x + g x := by intro x calc (fun x => f x + g x) x = f x + g x := rfl _ = f (-x) + g (-x) := by rw [ef, eg] example (of : FnOdd f) (og : FnOdd g) : FnEven fun x => f x * g x := by sorry example (ef : FnEven f) (og : FnOdd g) : FnOdd fun x => f x * g x := by sorry example (ef : FnEven f) (og : FnOdd g) : FnEven fun x => f (g x) := by sorry end section variable {α : Type _} (r s t : Set α) example : s ⊆ s := by intro x xs exact xs theorem Subset.refl : s ⊆ s := fun x xs => xs theorem Subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := by sorry end section variable {α : Type _} [PartialOrder α] variable (s : Set α) (a b : α) def SetUb (s : Set α) (a : α) := ∀ x, x ∈ s → x ≤ a example (h : SetUb s a) (h' : a ≤ b) : SetUb s b := sorry end section open Function example (c : ℝ) : Injective fun x => x + c := by intro x₁ x₂ h' exact (add_left_inj c).mp h' example {c : ℝ} (h : c ≠ 0) : Injective fun x => c * x := by sorry variable {α : Type _} {β : Type _} {γ : Type _} variable {g : β → γ} {f : α → β} example (injg : Injective g) (injf : Injective f) : Injective fun x => g (f x) := by sorry end
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@@ -0,0 +1,135 @@import Mathlib.Data.Real.Basic example : ∃ x : ℝ, 2 < x ∧ x < 3 := by use 5 / 2 norm_num example : ∃ x : ℝ, 2 < x ∧ x < 3 := have h : 2 < (5 : ℝ) / 2 ∧ (5 : ℝ) / 2 < 3 := by norm_num ⟨5 / 2, h⟩ example : ∃ x : ℝ, 2 < x ∧ x < 3 := ⟨5 / 2, by norm_num⟩ def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x def FnHasUb (f : ℝ → ℝ) := ∃ a, FnUb f a def FnHasLb (f : ℝ → ℝ) := ∃ a, FnLb f a theorem fnUb_add {f g : ℝ → ℝ} {a b : ℝ} (hfa : FnUb f a) (hgb : FnUb g b) : FnUb (fun x => f x + g x) (a + b) := fun x => add_le_add (hfa x) (hgb x) section variable {f g : ℝ → ℝ} example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x => f x + g x := by cases' ubf with a ubfa cases' ubg with b ubfb use a + b apply fnUb_add ubfa ubfb example (lbf : FnHasLb f) (lbg : FnHasLb g) : FnHasLb fun x => f x + g x := by sorry example {c : ℝ} (ubf : FnHasUb f) (h : c ≥ 0) : FnHasUb fun x => c * f x := by sorry example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x => f x + g x := by rcases ubf with ⟨a, ubfa⟩ rcases ubg with ⟨b, ubfb⟩ exact ⟨a + b, fnUb_add ubfa ubfb⟩ example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := by rintro ⟨a, ubfa⟩ ⟨b, ubfb⟩ exact ⟨a + b, fnUb_add ubfa ubfb⟩ example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := fun ⟨a, ubfa⟩ ⟨b, ubfb⟩ => ⟨a + b, fnUb_add ubfa ubfb⟩ end section variable {α : Type _} [CommRing α] def SumOfSquares (x : α) := ∃ a b, x = a ^ 2 + b ^ 2 theorem sumOfSquares_mul {x y : α} (sosx : SumOfSquares x) (sosy : SumOfSquares y) : SumOfSquares (x * y) := by rcases sosx with ⟨a, b, xeq⟩ rcases sosy with ⟨c, d, yeq⟩ rw [xeq, yeq] use a * c - b * d, a * d + b * c ring theorem sumOfSquares_mul' {x y : α} (sosx : SumOfSquares x) (sosy : SumOfSquares y) : SumOfSquares (x * y) := by rcases sosx with ⟨a, b, rfl⟩ rcases sosy with ⟨c, d, rfl⟩ use a * c - b * d, a * d + b * c ring end section variable {a b c : ℕ} example (divab : a ∣ b) (divbc : b ∣ c) : a ∣ c := by cases' divab with d beq cases' divbc with e ceq rw [ceq, beq] use d * e; ring example (divab : a ∣ b) (divac : a ∣ c) : a ∣ b + c := by sorry end section open Function example {c : ℝ} : Surjective fun x => x + c := by intro x use x - c dsimp; ring example {c : ℝ} (h : c ≠ 0) : Surjective fun x => c * x := by sorry example (x y : ℝ) (h : x - y ≠ 0) : (x ^ 2 - y ^ 2) / (x - y) = x + y := by field_simp [h] ring example {f : ℝ → ℝ} (h : Surjective f) : ∃ x, f x ^ 2 = 4 := by cases' h 2 with x hx use x rw [hx] norm_num end section open Function variable {α : Type _} {β : Type _} {γ : Type _} variable {g : β → γ} {f : α → β} example (surjg : Surjective g) (surjf : Surjective f) : Surjective fun x => g (f x) := by sorry end
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@@ -0,0 +1,143 @@import Mathlib.Data.Real.Basic section variable (a b : ℝ) example (h : a < b) : ¬b < a := by intro h' have : a < a := lt_trans h h' apply lt_irrefl a this def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x def FnHasUb (f : ℝ → ℝ) := ∃ a, FnUb f a def FnHasLb (f : ℝ → ℝ) := ∃ a, FnLb f a variable (f : ℝ → ℝ) example (h : ∀ a, ∃ x, f x > a) : ¬FnHasUb f := by intro fnub cases' fnub with a fnuba cases' h a with x hx have : f x ≤ a := fnuba x linarith example (h : ∀ a, ∃ x, f x < a) : ¬FnHasLb f := sorry example : ¬FnHasUb fun x => x := sorry #check (not_le_of_gt : a > b → ¬a ≤ b) #check (not_lt_of_ge : a ≥ b → ¬a < b) #check (lt_of_not_ge : ¬a ≥ b → a < b) #check (le_of_not_gt : ¬a > b → a ≤ b) example (h : Monotone f) (h' : f a < f b) : a < b := by sorry example (h : a ≤ b) (h' : f b < f a) : ¬Monotone f := by sorry example : ¬∀ {f : ℝ → ℝ}, Monotone f → ∀ {a b}, f a ≤ f b → a ≤ b := by intro h let f := fun x : ℝ => (0 : ℝ) have monof : Monotone f := by sorry have h' : f 1 ≤ f 0 := le_refl _ sorry example (x : ℝ) (h : ∀ ε > 0, x < ε) : x ≤ 0 := by sorry end section variable {α : Type _} (P : α → Prop) (Q : Prop) example (h : ¬∃ x, P x) : ∀ x, ¬P x := by sorry example (h : ∀ x, ¬P x) : ¬∃ x, P x := by sorry example (h : ¬∀ x, P x) : ∃ x, ¬P x := by sorry example (h : ∃ x, ¬P x) : ¬∀ x, P x := by sorry example (h : ¬∀ x, P x) : ∃ x, ¬P x := by by_contra h' apply h intro x show P x by_contra h'' exact h' ⟨x, h''⟩ example (h : ¬¬Q) : Q := by sorry example (h : Q) : ¬¬Q := by sorry end section variable (f : ℝ → ℝ) example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by sorry example (h : ¬∀ a, ∃ x, f x > a) : FnHasUb f := by push_neg at h exact h example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by simp only [FnHasUb, FnUb] at h push_neg at h exact h example (h : ¬Monotone f) : ∃ x y, x ≤ y ∧ f y < f x := by sorry example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by contrapose! h exact h example (x : ℝ) (h : ∀ ε > 0, x ≤ ε) : x ≤ 0 := by contrapose! h use x / 2 constructor <;> linarith end section variable (a : ℕ) example (h : 0 < 0) : a > 37 := by exfalso apply lt_irrefl 0 h example (h : 0 < 0) : a > 37 := absurd h (lt_irrefl 0) example (h : 0 < 0) : a > 37 := by have h' : ¬0 < 0 := lt_irrefl 0 contradiction end
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@@ -0,0 +1,138 @@import Mathlib.Data.Real.Basic import Mathlib.Data.Nat.Prime import Mathlib.Tactic.NormNum example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := by constructor · assumption intro h apply h₁ rw [h] example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := ⟨h₀, fun h => h₁ (by rw [h])⟩ example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := have h : x ≠ y := by contrapose! h₁ rw [h₁] ⟨h₀, h⟩ example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by cases' h with h₀ h₁ contrapose! h₁ exact le_antisymm h₀ h₁ example {x y : ℝ} : x ≤ y ∧ x ≠ y → ¬y ≤ x := by rintro ⟨h₀, h₁⟩ h' exact h₁ (le_antisymm h₀ h') example {x y : ℝ} : x ≤ y ∧ x ≠ y → ¬y ≤ x := fun ⟨h₀, h₁⟩ h' => h₁ (le_antisymm h₀ h') example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by intro h' apply h.right exact le_antisymm h.left h' example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := fun h' => h.right (le_antisymm h.left h') example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : m ∣ n ∧ ¬n ∣ m := sorry example : ∃ x : ℝ, 2 < x ∧ x < 4 := ⟨5 / 2, by norm_num, by norm_num⟩ example (x y : ℝ) : (∃ z : ℝ, x < z ∧ z < y) → x < y := by rintro ⟨z, xltz, zlty⟩ exact lt_trans xltz zlty example (x y : ℝ) : (∃ z : ℝ, x < z ∧ z < y) → x < y := fun ⟨z, xltz, zlty⟩ => lt_trans xltz zlty example : ∃ x : ℝ, 2 < x ∧ x < 4 := by use 5 / 2 constructor <;> norm_num example : ∃ m n : ℕ, 4 < m ∧ m < n ∧ n < 10 ∧ Nat.Prime m ∧ Nat.Prime n := by use 5 use 7 norm_num sorry example {x y : ℝ} : x ≤ y ∧ x ≠ y → x ≤ y ∧ ¬y ≤ x := by rintro ⟨h₀, h₁⟩ use h₀ exact fun h' => h₁ (le_antisymm h₀ h') example {x y : ℝ} (h : x ≤ y) : ¬y ≤ x ↔ x ≠ y := by constructor · contrapose! rintro rfl rfl contrapose! exact le_antisymm h example {x y : ℝ} (h : x ≤ y) : ¬y ≤ x ↔ x ≠ y := ⟨fun h₀ h₁ => h₀ (by rw [h₁]), fun h₀ h₁ => h₀ (le_antisymm h h₁)⟩ example {x y : ℝ} : x ≤ y ∧ ¬y ≤ x ↔ x ≤ y ∧ x ≠ y := sorry theorem aux {x y : ℝ} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := have h' : x ^ 2 = 0 := by sorry pow_eq_zero h' example (x y : ℝ) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := sorry section example (x : ℝ) : abs (x + 3) < 5 → -8 < x ∧ x < 2 := by rw [abs_lt] intro h constructor <;> linarith example : 3 ∣ Nat.gcd 6 15 := by rw [Nat.dvd_gcd_iff] constructor <;> norm_num end theorem not_monotone_iff {f : ℝ → ℝ} : ¬Monotone f ↔ ∃ x y, x ≤ y ∧ f x > f y := by rw [Monotone] push_neg rfl example : ¬Monotone fun x : ℝ => -x := by sorry section variable {α : Type _} [PartialOrder α] variable (a b : α) example : a < b ↔ a ≤ b ∧ a ≠ b := by rw [lt_iff_le_not_le] sorry end section variable {α : Type _} [Preorder α] variable (a b c : α) example : ¬a < a := by rw [lt_iff_le_not_le] sorry example : a < b → b < c → a < c := by simp only [lt_iff_le_not_le] sorry end
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@@ -0,0 +1,102 @@import Mathlib.Data.Real.Basic section variable {x y : ℝ} example (h : y > x ^ 2) : y > 0 ∨ y < -1 := by left linarith [pow_two_nonneg x] example (h : -y > x ^ 2 + 1) : y > 0 ∨ y < -1 := by right linarith [pow_two_nonneg x] example (h : y > 0) : y > 0 ∨ y < -1 := Or.inl h example (h : y < -1) : y > 0 ∨ y < -1 := Or.inr h example : x < abs y → x < y ∨ x < -y := by cases' le_or_gt 0 y with h h · rw [abs_of_nonneg h] intro h left exact h rw [abs_of_neg h] intro h; right; exact h namespace MyAbs theorem le_abs_self (x : ℝ) : x ≤ abs x := by sorry theorem neg_le_abs_self (x : ℝ) : -x ≤ abs x := by sorry theorem abs_add (x y : ℝ) : abs (x + y) ≤ abs x + abs y := by sorry theorem lt_abs : x < abs y ↔ x < y ∨ x < -y := by sorry theorem abs_lt : abs x < y ↔ -y < x ∧ x < y := by sorry end MyAbs end example {x : ℝ} (h : x ≠ 0) : x < 0 ∨ x > 0 := by rcases lt_trichotomy x 0 with (xlt | xeq | xgt) · left exact xlt · contradiction right; exact xgt example {m n k : ℕ} (h : m ∣ n ∨ m ∣ k) : m ∣ n * k := by rcases h with (⟨a, rfl⟩ | ⟨b, rfl⟩) · rw [mul_assoc] apply dvd_mul_right rw [mul_comm, mul_assoc] apply dvd_mul_right example {z : ℝ} (h : ∃ x y, z = x ^ 2 + y ^ 2 ∨ z = x ^ 2 + y ^ 2 + 1) : z ≥ 0 := by sorry example {x : ℝ} (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by sorry example {x y : ℝ} (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by sorry section variable {R : Type _} [CommRing R] [IsDomain R] variable (x y : R) example (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by sorry example (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by sorry end example (P : Prop) : ¬¬P → P := by intro h cases em P · assumption contradiction example (P : Prop) : ¬¬P → P := by intro h by_cases h' : P · assumption contradiction example (P Q : Prop) : P → Q ↔ ¬P ∨ Q := by sorry
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@@ -0,0 +1,96 @@import Mathlib.Data.Real.Basic def ConvergesTo (s : ℕ → ℝ) (a : ℝ) := ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε example : (fun x y : ℝ => (x + y) ^ 2) = fun x y : ℝ => x ^ 2 + 2 * x * y + y ^ 2 := by ext ring example (a b : ℝ) : abs a = abs (a - b + b) := by congr ring example {a : ℝ} (h : 1 < a) : a < a * a := by convert(mul_lt_mul_right _).2 h · rw [one_mul] exact lt_trans zero_lt_one h theorem convergesTo_const (a : ℝ) : ConvergesTo (fun x : ℕ => a) a := by intro ε εpos use 0 intro n nge; dsimp rw [sub_self, abs_zero] apply εpos theorem convergesTo_add {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : ConvergesTo (fun n => s n + t n) (a + b) := by intro ε εpos dsimp have ε2pos : 0 < ε / 2 := by linarith cases' cs (ε / 2) ε2pos with Ns hs cases' ct (ε / 2) ε2pos with Nt ht use max Ns Nt sorry theorem convergesTo_mul_const {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : ConvergesTo s a) : ConvergesTo (fun n => c * s n) (c * a) := by by_cases h : c = 0 · convert convergesTo_const 0 · rw [h, MulZeroClass.zero_mul] rw [h, MulZeroClass.zero_mul] have acpos : 0 < abs c := abs_pos.mpr h sorry theorem exists_abs_le_of_convergesTo {s : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) : ∃ N b, ∀ n, N ≤ n → abs (s n) < b := by cases' cs 1 zero_lt_one with N h use N, abs a + 1 sorry theorem aux {s t : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t 0) : ConvergesTo (fun n => s n * t n) 0 := by intro ε εpos dsimp rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩ have Bpos : 0 < B := lt_of_le_of_lt (abs_nonneg _) (h₀ N₀ (le_refl _)) have pos₀ : ε / B > 0 := div_pos εpos Bpos cases' ct _ pos₀ with N₁ h₁ sorry theorem convergesTo_mul {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : ConvergesTo (fun n => s n * t n) (a * b) := by have h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 := by apply aux cs convert convergesTo_add ct (convergesTo_const (-b)) ring have := convergesTo_add h₁ (convergesTo_mul_const b cs) convert convergesTo_add h₁ (convergesTo_mul_const b cs) using 1 · ext; ring ring theorem convergesTo_unique {s : ℕ → ℝ} {a b : ℝ} (sa : ConvergesTo s a) (sb : ConvergesTo s b) : a = b := by by_contra abne have : abs (a - b) > 0 := by sorry let ε := abs (a - b) / 2 have εpos : ε > 0 := by change abs (a - b) / 2 > 0 linarith cases' sa ε εpos with Na hNa cases' sb ε εpos with Nb hNb let N := max Na Nb have absa : abs (s N - a) < ε := by sorry have absb : abs (s N - b) < ε := by sorry have : abs (a - b) < abs (a - b) := by sorry exact lt_irrefl _ this section variable {α : Type _} [LinearOrder α] def ConvergesTo' (s : α → ℝ) (a : ℝ) := ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε end
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@@ -0,0 +1,131 @@import Mathlib.Data.Real.Basic def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x section variable (f g : ℝ → ℝ) (a b : ℝ) example (hfa : FnLb f a) (hgb : FnLb g b) : FnLb (fun x => f x + g x) (a + b) := by intro x apply add_le_add apply hfa apply hgb example (nnf : FnLb f 0) (nng : FnLb g 0) : FnLb (fun x => f x * g x) 0 := by intro x apply mul_nonneg apply nnf apply nng example (hfa : FnUb f a) (hfb : FnUb g b) (nng : FnLb g 0) (nna : 0 ≤ a) : FnUb (fun x => f x * g x) (a * b) := by intro x apply mul_le_mul apply hfa apply hfb apply nng apply nna end section variable (f g : ℝ → ℝ) example {c : ℝ} (mf : Monotone f) (nnc : 0 ≤ c) : Monotone fun x => c * f x := by intro a b aleb apply mul_le_mul_of_nonneg_left _ nnc apply mf aleb example {c : ℝ} (mf : Monotone f) (nnc : 0 ≤ c) : Monotone fun x => c * f x := fun a b aleb => mul_le_mul_of_nonneg_left (mf aleb) nnc example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f (g x) := by intro a b aleb apply mf apply mg apply aleb example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f (g x) := fun a b aleb => mf (mg aleb) def FnEven (f : ℝ → ℝ) : Prop := ∀ x, f x = f (-x) def FnOdd (f : ℝ → ℝ) : Prop := ∀ x, f x = -f (-x) example (of : FnOdd f) (og : FnOdd g) : FnEven fun x => f x * g x := by intro x calc (fun x => f x * g x) x = f x * g x := rfl _ = f (-x) * g (-x) := by rw [of, og, neg_mul_neg] example (ef : FnEven f) (og : FnOdd g) : FnOdd fun x => f x * g x := by intro x dsimp rw [ef, og, neg_mul_eq_mul_neg] example (ef : FnEven f) (og : FnOdd g) : FnEven fun x => f (g x) := by intro x dsimp rw [og, ← ef] end section variable {α : Type _} (r s t : Set α) example : r ⊆ s → s ⊆ t → r ⊆ t := by intro rsubs ssubt x xr apply ssubt apply rsubs apply xr theorem Subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := fun rsubs ssubt x xr => ssubt (rsubs xr) end section variable {α : Type _} [PartialOrder α] variable (s : Set α) (a b : α) def SetUb (s : Set α) (a : α) := ∀ x, x ∈ s → x ≤ a example (h : SetUb s a) (h' : a ≤ b) : SetUb s b := by intro x xs apply le_trans (h x xs) h' example (h : SetUb s a) (h' : a ≤ b) : SetUb s b := fun x xs => le_trans (h x xs) h' end section open Function example {c : ℝ} (h : c ≠ 0) : Injective fun x => c * x := by intro x₁ x₂ h' apply (mul_right_inj' h).mp h' variable {α : Type _} {β : Type _} {γ : Type _} variable {g : β → γ} {f : α → β} example (injg : Injective g) (injf : Injective f) : Injective fun x => g (f x) := by intro x₁ x₂ h apply injf apply injg apply h end
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@@ -0,0 +1,85 @@import Mathlib.Data.Real.Basic def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x def FnHasUb (f : ℝ → ℝ) := ∃ a, FnUb f a def FnHasLb (f : ℝ → ℝ) := ∃ a, FnLb f a theorem fnUb_add {f g : ℝ → ℝ} {a b : ℝ} (hfa : FnUb f a) (hgb : FnUb g b) : FnUb (fun x => f x + g x) (a + b) := fun x => add_le_add (hfa x) (hgb x) section variable {f g : ℝ → ℝ} example (lbf : FnHasLb f) (lbg : FnHasLb g) : FnHasLb fun x => f x + g x := by cases' lbf with a lbfa cases' lbg with b lbgb use a + b intro x exact add_le_add (lbfa x) (lbgb x) example {c : ℝ} (ubf : FnHasUb f) (h : c ≥ 0) : FnHasUb fun x => c * f x := by cases' ubf with a lbfa use c * a intro x exact mul_le_mul_of_nonneg_left (lbfa x) h end section variable {a b c : ℕ} example (divab : a ∣ b) (divbc : b ∣ c) : a ∣ c := by rcases divab with ⟨d, rfl⟩ rcases divbc with ⟨e, rfl⟩ use d * e; ring example (divab : a ∣ b) (divac : a ∣ c) : a ∣ b + c := by rcases divab with ⟨d, rfl⟩ rcases divac with ⟨e, rfl⟩ use d + e; ring end section open Function example {c : ℝ} (h : c ≠ 0) : Surjective fun x => c * x := by intro x use x / c dsimp; rw [mul_div_cancel' _ h] example {c : ℝ} (h : c ≠ 0) : Surjective fun x => c * x := by intro x use x / c field_simp [h] ; ring end section open Function variable {α : Type _} {β : Type _} {γ : Type _} variable {g : β → γ} {f : α → β} example (surjg : Surjective g) (surjf : Surjective f) : Surjective fun x => g (f x) := by intro z rcases surjg z with ⟨y, rfl⟩ rcases surjf y with ⟨x, rfl⟩ use x end
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@@ -0,0 +1,113 @@import Mathlib.Data.Real.Basic section variable (a b : ℝ) def FnUb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, f x ≤ a def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop := ∀ x, a ≤ f x def FnHasUb (f : ℝ → ℝ) := ∃ a, FnUb f a def FnHasLb (f : ℝ → ℝ) := ∃ a, FnLb f a variable (f : ℝ → ℝ) example (h : ∀ a, ∃ x, f x < a) : ¬FnHasLb f := by rintro ⟨a, ha⟩ rcases h a with ⟨x, hx⟩ have := ha x linarith example : ¬FnHasUb fun x => x := by rintro ⟨a, ha⟩ have : a + 1 ≤ a := ha (a + 1) linarith example (h : Monotone f) (h' : f a < f b) : a < b := by apply lt_of_not_ge intro h'' apply absurd h' apply not_lt_of_ge (h h'') example (h : a ≤ b) (h' : f b < f a) : ¬Monotone f := by intro h'' apply absurd h' apply not_lt_of_ge apply h'' h example : ¬∀ {f : ℝ → ℝ}, Monotone f → ∀ {a b}, f a ≤ f b → a ≤ b := by intro h let f := fun x : ℝ => (0 : ℝ) have monof : Monotone f := by intro a b leab rfl have h' : f 1 ≤ f 0 := le_refl _ have : (1 : ℝ) ≤ 0 := h monof h' linarith example (x : ℝ) (h : ∀ ε > 0, x < ε) : x ≤ 0 := by apply le_of_not_gt intro h' linarith [h _ h'] end section variable {α : Type _} (P : α → Prop) (Q : Prop) example (h : ¬∃ x, P x) : ∀ x, ¬P x := by intro x Px apply h use x exact Px example (h : ∀ x, ¬P x) : ¬∃ x, P x := by rintro ⟨x, Px⟩ exact h x Px example (h : ∃ x, ¬P x) : ¬∀ x, P x := by intro h' rcases h with ⟨x, nPx⟩ apply nPx apply h' example (h : ¬¬Q) : Q := by by_contra h' exact h h' example (h : Q) : ¬¬Q := by intro h' exact h' h end section variable (f : ℝ → ℝ) example (h : ¬FnHasUb f) : ∀ a, ∃ x, f x > a := by intro a by_contra h' apply h use a intro x apply le_of_not_gt intro h'' apply h' use x exact h'' example (h : ¬Monotone f) : ∃ x y, x ≤ y ∧ f y < f x := by rw [Monotone] at h push_neg at h exact h end
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@@ -0,0 +1,97 @@import Mathlib.Data.Real.Basic import Mathlib.Data.Nat.Prime import Mathlib.Tactic.NormNum example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : m ∣ n ∧ ¬n ∣ m := by cases' h with h0 h1 constructor · exact h0 intro h2 apply h1 apply Nat.dvd_antisymm h0 h2 example {x y : ℝ} : x ≤ y ∧ ¬y ≤ x ↔ x ≤ y ∧ x ≠ y := by constructor · rintro ⟨h0, h1⟩ constructor · exact h0 intro h2 apply h1 rw [h2] rintro ⟨h0, h1⟩ constructor · exact h0 intro h2 apply h1 apply le_antisymm h0 h2 theorem aux {x y : ℝ} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := have h' : x ^ 2 = 0 := by linarith [pow_two_nonneg x, pow_two_nonneg y] pow_eq_zero h' example (x y : ℝ) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by constructor · intro h constructor · exact aux h rw [add_comm] at h exact aux h rintro ⟨rfl, rfl⟩ norm_num theorem not_monotone_iff {f : ℝ → ℝ} : ¬Monotone f ↔ ∃ x y, x ≤ y ∧ f x > f y := by rw [Monotone] push_neg rfl example : ¬Monotone fun x : ℝ => -x := by rw [not_monotone_iff] use 0, 1 norm_num section variable {α : Type _} [PartialOrder α] variable (a b : α) example : a < b ↔ a ≤ b ∧ a ≠ b := by rw [lt_iff_le_not_le] constructor · rintro ⟨h0, h1⟩ constructor · exact h0 intro h2 apply h1 rw [h2] rintro ⟨h0, h1⟩ constructor · exact h0 intro h2 apply h1 apply le_antisymm h0 h2 end section variable {α : Type _} [Preorder α] variable (a b c : α) example : ¬a < a := by rw [lt_iff_le_not_le] rintro ⟨h0, h1⟩ exact h1 h0 example : a < b → b < c → a < c := by simp only [lt_iff_le_not_le] rintro ⟨h0, h1⟩ ⟨h2, h3⟩ constructor · apply le_trans h0 h2 intro h4 apply h1 apply le_trans h2 h4 end
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@@ -0,0 +1,142 @@import Mathlib.Data.Real.Basic section variable {x y : ℝ} namespace MyAbs theorem le_abs_self (x : ℝ) : x ≤ abs x := by cases' le_or_gt 0 x with h h · rw [abs_of_nonneg h] rw [abs_of_neg h] linarith theorem neg_le_abs_self (x : ℝ) : -x ≤ abs x := by cases' le_or_gt 0 x with h h · rw [abs_of_nonneg h] linarith rw [abs_of_neg h] theorem abs_add (x y : ℝ) : abs (x + y) ≤ abs x + abs y := by cases' le_or_gt 0 (x + y) with h h · rw [abs_of_nonneg h] linarith [le_abs_self x, le_abs_self y] rw [abs_of_neg h] linarith [neg_le_abs_self x, neg_le_abs_self y] theorem lt_abs : x < abs y ↔ x < y ∨ x < -y := by cases' le_or_gt 0 y with h h · rw [abs_of_nonneg h] constructor · intro h' left exact h' intro h' cases' h' with h' h' · exact h' linarith rw [abs_of_neg h] constructor · intro h' right exact h' intro h' cases' h' with h' h' · linarith exact h' theorem abs_lt : abs x < y ↔ -y < x ∧ x < y := by cases' le_or_gt 0 x with h h · rw [abs_of_nonneg h] constructor · intro h' constructor · linarith exact h' intro h' cases' h' with h1 h2 exact h2 rw [abs_of_neg h] constructor · intro h' constructor · linarith linarith intro h' linarith end MyAbs end example {z : ℝ} (h : ∃ x y, z = x ^ 2 + y ^ 2 ∨ z = x ^ 2 + y ^ 2 + 1) : z ≥ 0 := by rcases h with ⟨x, y, rfl | rfl⟩ <;> linarith [sq_nonneg x, sq_nonneg y] example {x : ℝ} (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by have h' : x ^ 2 - 1 = 0 := by rw [h, sub_self] have h'' : (x + 1) * (x - 1) = 0 := by rw [← h'] ring cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 · right exact eq_neg_iff_add_eq_zero.mpr h1 left exact eq_of_sub_eq_zero h1 example {x y : ℝ} (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by have h' : x ^ 2 - y ^ 2 = 0 := by rw [h, sub_self] have h'' : (x + y) * (x - y) = 0 := by rw [← h'] ring cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 · right exact eq_neg_iff_add_eq_zero.mpr h1 left exact eq_of_sub_eq_zero h1 section variable {R : Type _} [CommRing R] [IsDomain R] variable (x y : R) example (h : x ^ 2 = 1) : x = 1 ∨ x = -1 := by have h' : x ^ 2 - 1 = 0 := by rw [h, sub_self] have h'' : (x + 1) * (x - 1) = 0 := by rw [← h'] ring cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 · right exact eq_neg_iff_add_eq_zero.mpr h1 left exact eq_of_sub_eq_zero h1 example (h : x ^ 2 = y ^ 2) : x = y ∨ x = -y := by have h' : x ^ 2 - y ^ 2 = 0 := by rw [h, sub_self] have h'' : (x + y) * (x - y) = 0 := by rw [← h'] ring cases' eq_zero_or_eq_zero_of_mul_eq_zero h'' with h1 h1 · right exact eq_neg_iff_add_eq_zero.mpr h1 left exact eq_of_sub_eq_zero h1 end example (P Q : Prop) : P → Q ↔ ¬P ∨ Q := by constructor · intro h by_cases h' : P · right exact h h' left exact h' rintro (h | h) · intro h' exact absurd h' h intro exact h
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@@ -0,0 +1,126 @@import Mathlib.Data.Real.Basic def ConvergesTo (s : ℕ → ℝ) (a : ℝ) := ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε theorem convergesTo_const (a : ℝ) : ConvergesTo (fun x : ℕ => a) a := by intro ε εpos use 0 intro n nge; dsimp rw [sub_self, abs_zero] apply εpos theorem convergesTo_add {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : ConvergesTo (fun n => s n + t n) (a + b) := by intro ε εpos dsimp have ε2pos : 0 < ε / 2 := by linarith cases' cs (ε / 2) ε2pos with Ns hs cases' ct (ε / 2) ε2pos with Nt ht use max Ns Nt intro n hn have ngeNs : n ≥ Ns := le_of_max_le_left hn have ngeNt : n ≥ Nt := le_of_max_le_right hn calc |s n + t n - (a + b)| = |s n - a + (t n - b)| := by congr ring _ ≤ |s n - a| + |t n - b| := (abs_add _ _) _ < ε / 2 + ε / 2 := (add_lt_add (hs n ngeNs) (ht n ngeNt)) _ = ε := by norm_num theorem convergesTo_mul_const {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : ConvergesTo s a) : ConvergesTo (fun n => c * s n) (c * a) := by by_cases h : c = 0 · convert convergesTo_const 0 · rw [h, MulZeroClass.zero_mul] rw [h, MulZeroClass.zero_mul] have acpos : 0 < abs c := abs_pos.mpr h intro ε εpos dsimp have εcpos : 0 < ε / abs c := by apply div_pos εpos acpos cases' cs (ε / abs c) εcpos with Ns hs use Ns intro n ngt calc |c * s n - c * a| = |c| * |s n - a| := by rw [← abs_mul, mul_sub] _ < |c| * (ε / |c|) := (mul_lt_mul_of_pos_left (hs n ngt) acpos) _ = ε := mul_div_cancel' _ (ne_of_lt acpos).symm theorem exists_abs_le_of_converges_to {s : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) : ∃ N b, ∀ n, N ≤ n → abs (s n) < b := by cases' cs 1 zero_lt_one with N h use N, abs a + 1 intro n ngt calc |s n| = |s n - a + a| := by congr abel _ ≤ |s n - a| + |a| := (abs_add _ _) _ < |a| + 1 := by linarith [h n ngt] theorem aux {s t : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t 0) : ConvergesTo (fun n => s n * t n) 0 := by intro ε εpos dsimp rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩ have Bpos : 0 < B := lt_of_le_of_lt (abs_nonneg _) (h₀ N₀ (le_refl _)) have pos₀ : ε / B > 0 := div_pos εpos Bpos cases' ct _ pos₀ with N₁ h₁ use max N₀ N₁ intro n ngt have ngeN₀ : n ≥ N₀ := le_of_max_le_left ngt have ngeN₁ : n ≥ N₁ := le_of_max_le_right ngt calc |s n * t n - 0| = |s n| * |t n - 0| := by rw [sub_zero, abs_mul, sub_zero] _ < B * (ε / B) := (mul_lt_mul'' (h₀ n ngeN₀) (h₁ n ngeN₁) (abs_nonneg _) (abs_nonneg _)) _ = ε := mul_div_cancel' _ (ne_of_lt Bpos).symm theorem convergesTo_mul {s t : ℕ → ℝ} {a b : ℝ} (cs : ConvergesTo s a) (ct : ConvergesTo t b) : ConvergesTo (fun n => s n * t n) (a * b) := by have h₁ : ConvergesTo (fun n => s n * (t n + -b)) 0 := by apply aux cs convert convergesTo_add ct (convergesTo_const (-b)) ring have := convergesTo_add h₁ (convergesTo_mul_const b cs) convert convergesTo_add h₁ (convergesTo_mul_const b cs) using 1 · ext; ring ring theorem convergesTo_unique {s : ℕ → ℝ} {a b : ℝ} (sa : ConvergesTo s a) (sb : ConvergesTo s b) : a = b := by by_contra abne have : abs (a - b) > 0 := by apply lt_of_le_of_ne · apply abs_nonneg intro h'' apply abne apply eq_of_abs_sub_eq_zero h''.symm let ε := abs (a - b) / 2 have εpos : ε > 0 := by change abs (a - b) / 2 > 0 linarith cases' sa ε εpos with Na hNa cases' sb ε εpos with Nb hNb let N := max Na Nb have absa : abs (s N - a) < ε := by apply hNa apply le_max_left have absb : abs (s N - b) < ε := by apply hNb apply le_max_right have : abs (a - b) < abs (a - b) calc abs (a - b) = abs (-(s N - a) + (s N - b)) := by congr ring _ ≤ abs (-(s N - a)) + abs (s N - b) := (abs_add _ _) _ = abs (s N - a) + abs (s N - b) := by rw [abs_neg] _ < ε + ε := (add_lt_add absa absb) _ = abs (a - b) := by norm_num exact lt_irrefl _ this
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@@ -0,0 +1,118 @@import Mathlib.Data.Nat.Factorization.Basic import Mathlib.Data.Nat.Prime import Mathlib.Tactic.NormNum.GCD import Mathlib.Tactic.NormNum.Prime #print Nat.coprime example (m n : Nat) (h : m.coprime n) : m.gcd n = 1 := h example (m n : Nat) (h : m.coprime n) : m.gcd n = 1 := by rw [Nat.coprime] at h exact h example : Nat.coprime 12 7 := by norm_num example : Nat.gcd 12 8 = 4 := by norm_num #check @Nat.prime_def_lt example (p : ℕ) (prime_p : Nat.Prime p) : 2 ≤ p ∧ ∀ m : ℕ, m < p → m ∣ p → m = 1 := by rwa [Nat.prime_def_lt] at prime_p #check Nat.Prime.eq_one_or_self_of_dvd example (p : ℕ) (prime_p : Nat.Prime p) : ∀ m : ℕ, m ∣ p → m = 1 ∨ m = p := prime_p.eq_one_or_self_of_dvd example : Nat.Prime 17 := by norm_num -- commonly used example : Nat.Prime 2 := Nat.prime_two example : Nat.Prime 3 := Nat.prime_three #check @Nat.Prime.dvd_mul #check Nat.Prime.dvd_mul Nat.prime_two #check Nat.prime_two.dvd_mul theorem even_of_even_sqr {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m := by rw [pow_two, Nat.prime_two.dvd_mul] at h cases h <;> assumption example {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m := Nat.Prime.dvd_of_dvd_pow Nat.prime_two h example (a b c : Nat) (h : a * b = a * c) (h' : a ≠ 0) : b = c := -- library_search suggests the following: (mul_right_inj' h').mp h example {m n : ℕ} (coprime_mn : m.coprime n) : m ^ 2 ≠ 2 * n ^ 2 := by intro sqr_eq have : 2 ∣ m := by sorry, obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this have : 2 * (2 * k ^ 2) = 2 * n ^ 2 := by rw [← sqr_eq, meq] ring have : 2 * k ^ 2 = n ^ 2 := sorry, have : 2 ∣ n := by sorry, have : 2 ∣ m.gcd n := by sorry, have : 2 ∣ 1 := by sorry, norm_num at this example {m n p : ℕ} (coprime_mn : m.coprime n) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by sorry #check Nat.factors #check Nat.prime_of_mem_factors #check Nat.prod_factors #check Nat.factors_unique theorem factorization_mul' {m n : ℕ} (mnez : m ≠ 0) (nnez : n ≠ 0) (p : ℕ) : (m * n).factorization p = m.factorization p + n.factorization p := by rw [Nat.factorization_mul mnez nnez] rfl theorem factorization_pow' (n k p : ℕ) : (n ^ k).factorization p = k * n.factorization p := by rw [Nat.factorization_pow] rfl theorem Nat.Prime.factorization' {p : ℕ} (prime_p : p.Prime) : p.factorization p = 1 := by rw [prime_p.factorization] simp example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by intro sqr_eq have nsqr_nez : n ^ 2 ≠ 0 := by simpa have eq1 : Nat.factorization (m ^ 2) p = 2 * m.factorization p := by sorry, have eq2 : (p * n ^ 2).factorization p = 2 * n.factorization p + 1 := by sorry, have : 2 * m.factorization p % 2 = (2 * n.factorization p + 1) % 2 := by rw [← eq1, sqr_eq, eq2] rw [add_comm, Nat.add_mul_mod_self_left, Nat.mul_mod_right] at this norm_num at this example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m ^ k = r * n ^ k) {p : ℕ} (prime_p : p.Prime) : k ∣ r.factorization p := by cases' r with r · simp have npow_nz : n ^ k ≠ 0 := fun npowz => nnz (pow_eq_zero npowz) have eq1 : (m ^ k).factorization p = k * m.factorization p := by sorry, have eq2 : (r.succ * n ^ k).factorization p = k * n.factorization p + r.succ.factorization p := by sorry, have : r.succ.factorization p = k * m.factorization p - k * n.factorization p := by rw [← eq1, pow_eq, eq2, add_comm, Nat.add_sub_cancel] rw [this] sorry #check multiplicity
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@@ -0,0 +1,148 @@import Mathlib.Data.Nat.GCD.Basic import Mathlib.Algebra.BigOperators.Basic import Mathlib.Tactic example (n : Nat) : n.succ ≠ Nat.zero := Nat.succ_ne_zero n example (m n : Nat) (h : m.succ = n.succ) : m = n := Nat.succ.inj h def fac : ℕ → ℕ | 0 => 1 | n + 1 => (n + 1) * fac n example : fac 0 = 1 := rfl example : fac 0 = 1 := by rw [fac] example : fac 0 = 1 := by simp [fac] example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := rfl example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := by rw [fac] example (n : ℕ) : fac (n + 1) = (n + 1) * fac n := by simp [fac] theorem fac_pos (n : ℕ) : 0 < fac n := by induction' n with n ih · rw [fac] exact zero_lt_one rw [fac] exact mul_pos n.succ_pos ih theorem dvd_fac {i n : ℕ} (ipos : 0 < i) (ile : i ≤ n) : i ∣ fac n := by induction' n with n ih · exact absurd ipos (not_lt_of_ge ile) rw [fac] cases' Nat.of_le_succ ile with h h · apply dvd_mul_of_dvd_right (ih h) rw [h] apply dvd_mul_right theorem pow_two_le_fac (n : ℕ) : 2 ^ (n - 1) ≤ fac n := by cases' n with n · simp [fac] sorry section variable {α : Type _} (s : Finset ℕ) (f : ℕ → ℕ) (n : ℕ) #check Finset.sum s f #check Finset.prod s f open BigOperators open Finset example : s.sum f = ∑ x in s, f x := rfl example : s.prod f = ∏ x in s, f x := rfl example : (range n).sum f = ∑ x in range n, f x := rfl example : (range n).prod f = ∏ x in range n, f x := rfl example (f : ℕ → ℕ) : (∑ x in range 0, f x) = 0 := Finset.sum_range_zero f example (f : ℕ → ℕ) (n : ℕ) : (∑ x in range n.succ, f x) = (∑ x in range n, f x) + f n := Finset.sum_range_succ f n example (f : ℕ → ℕ) : (∏ x in range 0, f x) = 1 := Finset.prod_range_zero f example (f : ℕ → ℕ) (n : ℕ) : (∏ x in range n.succ, f x) = (∏ x in range n, f x) * f n := Finset.prod_range_succ f n example (n : ℕ) : fac n = ∏ i in range n, (i + 1) := by induction' n with n ih · rw [fac, prod_range_zero] rw [fac, ih, prod_range_succ, mul_comm] example (a b c d e f : ℕ) : a * (b * c * f * (d * e)) = d * (a * f * e) * (c * b) := by simp [mul_assoc, mul_comm, mul_left_comm] theorem sum_id (n : ℕ) : (∑ i in range (n + 1), i) = n * (n + 1) / 2 := by symm; apply Nat.div_eq_of_eq_mul_right (by norm_num : 0 < 2) induction' n with n ih · simp rw [Finset.sum_range_succ, mul_add 2, ← ih, Nat.succ_eq_add_one] ring theorem sum_sqr (n : ℕ) : (∑ i in range (n + 1), i ^ 2) = n * (n + 1) * (2 * n + 1) / 6 := by sorry end inductive MyNat | zero : MyNat | succ : MyNat → MyNat namespace MyNat def add : MyNat → MyNat → MyNat | x, zero => x | x, succ y => succ (add x y) def mul : MyNat → MyNat → MyNat | x, zero => zero | x, succ y => add (mul x y) x theorem zero_add (n : MyNat) : add zero n = n := by induction' n with n ih · rfl rw [add, ih] theorem succ_add (m n : MyNat) : add (succ m) n = succ (add m n) := by induction' n with n ih · rfl rw [add, ih] rfl theorem add_comm (m n : MyNat) : add m n = add n m := by induction' n with n ih · rw [zero_add] rfl rw [add, succ_add, ih] theorem add_assoc (m n k : MyNat) : add (add m n) k = add m (add n k) := by sorry theorem mul_add (m n k : MyNat) : mul m (add n k) = add (mul m n) (mul m k) := by sorry theorem zero_mul (n : MyNat) : mul zero n = zero := by sorry theorem succ_mul (m n : MyNat) : mul (succ m) n = add (mul m n) n := by sorry theorem mul_comm (m n : MyNat) : mul m n = mul n m := by sorry end MyNat
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@@ -0,0 +1,231 @@import Mathlib.Data.Nat.Prime import Mathlib.Algebra.BigOperators.Order import Mathlib.Tactic import Mathlib.Tactic.IntervalCases open BigOperators theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by cases m; contradiction case succ m => cases m; contradiction repeat' apply Nat.succ_le_succ apply zero_le example {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by by_contra h push_neg at h interval_cases m <;> contradiction example {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by by_contra h push_neg at h revert h0 h1 revert h m decide theorem exists_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ p : Nat, p.Prime ∧ p ∣ n := by by_cases np : n.Prime · use n, np induction' n using Nat.strong_induction_on with n ih dsimp at ih rw [Nat.prime_def_lt] at np push_neg at np rcases np h with ⟨m, mltn, mdvdn, mne1⟩ have : m ≠ 0 := by intro mz rw [mz, zero_dvd_iff] at mdvdn linarith have mgt2 : 2 ≤ m := two_le this mne1 by_cases mp : m.Prime · use m, mp exact mdvdn . rcases ih m mltn mgt2 mp with ⟨p, pp, pdvd⟩ use p, pp apply pdvd.trans mdvdn theorem primes_infinite : ∀ n, ∃ p > n, Nat.Prime p := by intro n have : 2 ≤ Nat.factorial (n + 1) + 1 := by sorry, rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ refine' ⟨p, _, pp⟩ show p > n by_contra ple push_neg at ple have : p ∣ Nat.factorial (n + 1) := by sorry, have : p ∣ 1 := by sorry, show False sorry open Finset section variable {α : Type _} [DecidableEq α] (r s t : Finset α) example : r ∩ (s ∪ t) ⊆ r ∩ s ∪ r ∩ t := by rw [subset_iff] intro x rw [mem_inter, mem_union, mem_union, mem_inter, mem_inter] tauto example : r ∩ (s ∪ t) ⊆ r ∩ s ∪ r ∩ t := by simp [subset_iff] intro x tauto example : r ∩ s ∪ r ∩ t ⊆ r ∩ (s ∪ t) := by simp [subset_iff] intro x tauto example : r ∩ s ∪ r ∩ t = r ∩ (s ∪ t) := by ext x simp tauto end section variable {α : Type _} [DecidableEq α] (r s t : Finset α) example : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by sorry example : (r \ s) \ t = r \ (s ∪ t) := by sorry end example (s : Finset ℕ) (n : ℕ) (h : n ∈ s) : n ∣ ∏ i in s, i := Finset.dvd_prod_of_mem _ h theorem Nat.Prime.eq_of_dvd_of_prime {p q : ℕ} (prime_p : Nat.Prime p) (prime_q : Nat.Prime q) (h : p ∣ q) : p = q := by sorry theorem mem_of_dvd_prod_primes {s : Finset ℕ} {p : ℕ} (prime_p : p.Prime) : (∀ n ∈ s, Nat.Prime n) → (p ∣ ∏ n in s, n) → p ∈ s := by intro h₀ h₁ induction' s using Finset.induction_on with a s ans ih · simp at h₁ linarith [prime_p.two_le] simp [Finset.prod_insert ans, prime_p.dvd_mul] at h₀ h₁ rw [mem_insert] sorry example (s : Finset ℕ) (x : ℕ) : x ∈ s.filter Nat.Prime ↔ x ∈ s ∧ x.Prime := mem_filter theorem primes_infinite' : ∀ s : Finset Nat, ∃ p, Nat.Prime p ∧ p ∉ s := by intro s by_contra h push_neg at h set s' := s.filter Nat.Prime with s'_def have mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ n.Prime := by intro n simp [s'_def] apply h have : 2 ≤ (∏ i in s', i) + 1 := by sorry, rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ have : p ∣ ∏ i in s', i := by sorry, have : p ∣ 1 := by convert Nat.dvd_sub' pdvd this simp show False sorry theorem bounded_of_ex_finset (Q : ℕ → Prop) : (∃ s : Finset ℕ, ∀ k, Q k → k ∈ s) → ∃ n, ∀ k, Q k → k < n := by rintro ⟨s, hs⟩ use s.sup id + 1 intro k Qk apply Nat.lt_succ_of_le show id k ≤ s.sup id apply le_sup (hs k Qk) theorem ex_finset_of_bounded (Q : ℕ → Prop) [DecidablePred Q] : (∃ n, ∀ k, Q k → k ≤ n) → ∃ s : Finset ℕ, ∀ k, Q k ↔ k ∈ s := by rintro ⟨n, hn⟩ use (range (n + 1)).filter Q intro k simp [Nat.lt_succ_iff] exact hn k example : 27 % 4 = 3 := by norm_num example (n : ℕ) : (4 * n + 3) % 4 = 3 := by rw [add_comm, Nat.add_mul_mod_self_left] norm_num theorem mod_4_eq_3_or_mod_4_eq_3 {m n : ℕ} (h : m * n % 4 = 3) : m % 4 = 3 ∨ n % 4 = 3 := by revert h rw [Nat.mul_mod] have : m % 4 < 4 := Nat.mod_lt m (by norm_num) interval_cases hm : m % 4 <;> simp [hm] have : n % 4 < 4 := Nat.mod_lt n (by norm_num) interval_cases hn : n % 4 <;> simp [hn] theorem two_le_of_mod_4_eq_3 {n : ℕ} (h : n % 4 = 3) : 2 ≤ n := by apply two_le <;> · intro neq rw [neq] at h norm_num at h theorem aux {m n : ℕ} (h₀ : m ∣ n) (h₁ : 2 ≤ m) (h₂ : m < n) : n / m ∣ n ∧ n / m < n := by sorry theorem exists_prime_factor_mod_4_eq_3 {n : Nat} (h : n % 4 = 3) : ∃ p : Nat, p.Prime ∧ p ∣ n ∧ p % 4 = 3 := by by_cases np : n.Prime · use n exact ⟨np, dvd_rfl, h⟩ induction' n using Nat.strong_induction_on with n ih dsimp at ih rw [Nat.prime_def_lt] at np push_neg at np rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩ have mge2 : 2 ≤ m := by apply two_le _ mne1 intro mz rw [mz, zero_dvd_iff] at mdvdn linarith have neq : m * (n / m) = n := Nat.mul_div_cancel' mdvdn have : m % 4 = 3 ∨ n / m % 4 = 3 := by apply mod_4_eq_3_or_mod_4_eq_3 rw [neq, h] cases' this with h1 h1 . sorry sorry example (m n : ℕ) (s : Finset ℕ) (h : m ∈ erase s n) : m ≠ n ∧ m ∈ s := by rwa [mem_erase] at h example (m n : ℕ) (s : Finset ℕ) (h : m ∈ erase s n) : m ≠ n ∧ m ∈ s := by simp at h assumption theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, Nat.Prime p ∧ p % 4 = 3 := by by_contra h push_neg at h cases' h with n hn have : ∃ s : Finset Nat, ∀ p : ℕ, p.Prime ∧ p % 4 = 3 ↔ p ∈ s := by apply ex_finset_of_bounded use n contrapose! hn rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩ exact ⟨p, pltn, pp, p4⟩ cases' this with s hs have h₁ : ((4 * ∏ i in erase s 3, i) + 3) % 4 = 3 := by sorry, rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩ have ps : p ∈ s := by sorry, have pne3 : p ≠ 3 := by sorry, have : p ∣ 4 * ∏ i in erase s 3, i := by sorry, have : p ∣ 3 := by sorry, have : p = 3 := by sorry, contradiction
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@@ -0,0 +1,101 @@import Mathlib.Data.Nat.Factorization.Basic import Mathlib.Data.Nat.Prime import Mathlib.Tactic.NormNum.GCD import Mathlib.Tactic.NormNum.Prime theorem even_of_even_sqr {m : ℕ} (h : 2 ∣ m ^ 2) : 2 ∣ m := by rw [pow_two, Nat.prime_two.dvd_mul] at h cases h <;> assumption example {m n : ℕ} (coprime_mn : m.coprime n) : m ^ 2 ≠ 2 * n ^ 2 := by intro sqr_eq have : 2 ∣ m := by apply even_of_even_sqr rw [sqr_eq] apply dvd_mul_right obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this have : 2 * (2 * k ^ 2) = 2 * n ^ 2 := by rw [← sqr_eq, meq] ring have : 2 * k ^ 2 = n ^ 2 := (mul_right_inj' (by norm_num)).mp this have : 2 ∣ n := by apply even_of_even_sqr rw [← this] apply dvd_mul_right have : 2 ∣ m.gcd n := by apply Nat.dvd_gcd <;> assumption have : 2 ∣ 1 := by convert this symm exact coprime_mn norm_num at this example {m n p : ℕ} (coprime_mn : m.coprime n) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by intro sqr_eq have : p ∣ m := by apply prime_p.dvd_of_dvd_pow rw [sqr_eq] apply dvd_mul_right obtain ⟨k, meq⟩ := dvd_iff_exists_eq_mul_left.mp this have : p * (p * k ^ 2) = p * n ^ 2 := by rw [← sqr_eq, meq] ring have : p * k ^ 2 = n ^ 2 := by apply (mul_right_inj' _).mp this exact prime_p.ne_zero have : p ∣ n := by apply prime_p.dvd_of_dvd_pow rw [← this] apply dvd_mul_right have : p ∣ Nat.gcd m n := by apply Nat.dvd_gcd <;> assumption have : p ∣ 1 := by convert this symm exact coprime_mn have : 2 ≤ 1 := by apply prime_p.two_le.trans exact Nat.le_of_dvd zero_lt_one this norm_num at this theorem factorization_mul' {m n : ℕ} (mnez : m ≠ 0) (nnez : n ≠ 0) (p : ℕ) : (m * n).factorization p = m.factorization p + n.factorization p := by rw [Nat.factorization_mul mnez nnez] rfl theorem factorization_pow' (n k p : ℕ) : (n ^ k).factorization p = k * n.factorization p := by rw [Nat.factorization_pow] rfl theorem Nat.Prime.factorization' {p : ℕ} (prime_p : p.Prime) : p.factorization p = 1 := by rw [prime_p.factorization] simp example {m n p : ℕ} (nnz : n ≠ 0) (prime_p : p.Prime) : m ^ 2 ≠ p * n ^ 2 := by intro sqr_eq have nsqr_nez : n ^ 2 ≠ 0 := by simpa have eq1 : Nat.factorization (m ^ 2) p = 2 * m.factorization p := by rw [factorization_pow'] have eq2 : (p * n ^ 2).factorization p = 2 * n.factorization p + 1 := by rw [factorization_mul' prime_p.ne_zero nsqr_nez, prime_p.factorization', factorization_pow', add_comm] have : 2 * m.factorization p % 2 = (2 * n.factorization p + 1) % 2 := by rw [← eq1, sqr_eq, eq2] rw [add_comm, Nat.add_mul_mod_self_left, Nat.mul_mod_right] at this norm_num at this example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m ^ k = r * n ^ k) {p : ℕ} (prime_p : p.Prime) : k ∣ r.factorization p := by cases' r with r · simp have npow_nz : n ^ k ≠ 0 := fun npowz => nnz (pow_eq_zero npowz) have eq1 : (m ^ k).factorization p = k * m.factorization p := by rw [factorization_pow'] have eq2 : (r.succ * n ^ k).factorization p = k * n.factorization p + r.succ.factorization p := by rw [factorization_mul' r.succ_ne_zero npow_nz, factorization_pow', add_comm] have : r.succ.factorization p = k * m.factorization p - k * n.factorization p := by rw [← eq1, pow_eq, eq2, add_comm, Nat.add_sub_cancel] rw [this] apply Nat.dvd_sub' <;> apply Nat.dvd_mul_right
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@@ -0,0 +1,98 @@import Mathlib.Data.Nat.GCD.Basic import Mathlib.Algebra.BigOperators.Basic import Mathlib.Tactic def fac : ℕ → ℕ | 0 => 1 | n + 1 => (n + 1) * fac n theorem pow_two_le_fac (n : ℕ) : 2 ^ (n - 1) ≤ fac n := by cases' n with n · simp [fac] induction' n with n ih · simp [fac] simp at * rw [pow_succ, fac] apply Nat.mul_le_mul _ ih repeat' apply Nat.succ_le_succ apply zero_le section variable {α : Type _} (s : Finset ℕ) (f : ℕ → ℕ) (n : ℕ) open BigOperators open Finset theorem sum_sqr (n : ℕ) : (∑ i in range (n + 1), i ^ 2) = n * (n + 1) * (2 * n + 1) / 6 := by symm; apply Nat.div_eq_of_eq_mul_right (by norm_num : 0 < 6) induction' n with n ih · simp rw [Finset.sum_range_succ, mul_add 6, ← ih, Nat.succ_eq_add_one] ring end inductive MyNat | zero : MyNat | succ : MyNat → MyNat namespace MyNat def add : MyNat → MyNat → MyNat | x, zero => x | x, succ y => succ (add x y) def mul : MyNat → MyNat → MyNat | x, zero => zero | x, succ y => add (mul x y) x theorem zero_add (n : MyNat) : add zero n = n := by induction' n with n ih · rfl rw [add, ih] theorem succ_add (m n : MyNat) : add (succ m) n = succ (add m n) := by induction' n with n ih · rfl rw [add, ih] rfl theorem add_comm (m n : MyNat) : add m n = add n m := by induction' n with n ih · rw [zero_add] rfl rw [add, succ_add, ih] theorem add_assoc (m n k : MyNat) : add (add m n) k = add m (add n k) := by induction' k with k ih · rfl rw [add, ih] rfl theorem mul_add (m n k : MyNat) : mul m (add n k) = add (mul m n) (mul m k) := by induction' k with k ih · rfl rw [add, mul, mul, ih, add_assoc] theorem zero_mul (n : MyNat) : mul zero n = zero := by induction' n with n ih · rfl rw [mul, ih] rfl theorem succ_mul (m n : MyNat) : mul (succ m) n = add (mul m n) n := by induction' n with n ih · rfl rw [mul, mul, ih, add_assoc, add_assoc, add_comm n, succ_add] rfl theorem mul_comm (m n : MyNat) : mul m n = mul n m := by induction' n with n ih · rw [zero_mul] rfl rw [mul, ih, succ_mul] end MyNat
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@@ -0,0 +1,241 @@import Mathlib.Data.Nat.Prime import Mathlib.Algebra.BigOperators.Order import Mathlib.Tactic import Mathlib.Tactic.IntervalCases open BigOperators theorem two_le {m : ℕ} (h0 : m ≠ 0) (h1 : m ≠ 1) : 2 ≤ m := by cases m; contradiction case succ m => cases m; contradiction repeat' apply Nat.succ_le_succ apply zero_le theorem exists_prime_factor {n : Nat} (h : 2 ≤ n) : ∃ p : Nat, p.Prime ∧ p ∣ n := by by_cases np : n.Prime · use n, np induction' n using Nat.strong_induction_on with n ih dsimp at ih rw [Nat.prime_def_lt] at np push_neg at np rcases np h with ⟨m, mltn, mdvdn, mne1⟩ have : m ≠ 0 := by intro mz rw [mz, zero_dvd_iff] at mdvdn linarith have mgt2 : 2 ≤ m := two_le this mne1 by_cases mp : m.Prime · use m, mp exact mdvdn . rcases ih m mltn mgt2 mp with ⟨p, pp, pdvd⟩ use p, pp apply pdvd.trans mdvdn theorem primes_infinite : ∀ n, ∃ p > n, Nat.Prime p := by intro n have : 2 ≤ Nat.factorial (n + 1) + 1 := by apply Nat.succ_le_succ exact Nat.succ_le_of_lt (Nat.factorial_pos _) rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ refine' ⟨p, _, pp⟩ show p > n by_contra ple push_neg at ple have : p ∣ Nat.factorial (n + 1) := by apply Nat.dvd_factorial apply pp.pos linarith have : p ∣ 1 := by convert Nat.dvd_sub' pdvd this simp show False have := Nat.le_of_dvd zero_lt_one this linarith [pp.two_le] open Finset section variable {α : Type _} [DecidableEq α] (r s t : Finset α) example : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by ext x rw [mem_inter, mem_union, mem_union, mem_union, mem_inter] tauto example : (r ∪ s) ∩ (r ∪ t) = r ∪ s ∩ t := by ext x simp tauto example : (r \ s) \ t = r \ (s ∪ t) := by ext x rw [mem_sdiff, mem_sdiff, mem_sdiff, mem_union] tauto example : (r \ s) \ t = r \ (s ∪ t) := by ext x simp tauto end theorem Nat.Prime.eq_of_dvd_of_prime {p q : ℕ} (prime_p : Nat.Prime p) (prime_q : Nat.Prime q) (h : p ∣ q) : p = q := by cases prime_q.eq_one_or_self_of_dvd _ h · linarith [prime_p.two_le] assumption theorem mem_of_dvd_prod_primes {s : Finset ℕ} {p : ℕ} (prime_p : p.Prime) : (∀ n ∈ s, Nat.Prime n) → (p ∣ ∏ n in s, n) → p ∈ s := by intro h₀ h₁ induction' s using Finset.induction_on with a s ans ih · simp at h₁ linarith [prime_p.two_le] simp [Finset.prod_insert ans, prime_p.dvd_mul] at h₀ h₁ rw [mem_insert] cases' h₁ with h₁ h₁ · left exact prime_p.eq_of_dvd_of_prime h₀.1 h₁ right exact ih h₀.2 h₁ theorem primes_infinite' : ∀ s : Finset Nat, ∃ p, Nat.Prime p ∧ p ∉ s := by intro s by_contra h push_neg at h set s' := s.filter Nat.Prime with s'_def have mem_s' : ∀ {n : ℕ}, n ∈ s' ↔ n.Prime := by intro n simp [s'_def] apply h have : 2 ≤ (∏ i in s', i) + 1 := by apply Nat.succ_le_succ apply Nat.succ_le_of_lt apply Finset.prod_pos intro n ns' apply (mem_s'.mp ns').pos rcases exists_prime_factor this with ⟨p, pp, pdvd⟩ have : p ∣ ∏ i in s', i := by apply dvd_prod_of_mem rw [mem_s'] apply pp have : p ∣ 1 := by convert Nat.dvd_sub' pdvd this simp show False have := Nat.le_of_dvd zero_lt_one this linarith [pp.two_le] theorem bounded_of_ex_finset (Q : ℕ → Prop) : (∃ s : Finset ℕ, ∀ k, Q k → k ∈ s) → ∃ n, ∀ k, Q k → k < n := by rintro ⟨s, hs⟩ use s.sup id + 1 intro k Qk apply Nat.lt_succ_of_le show id k ≤ s.sup id apply le_sup (hs k Qk) theorem ex_finset_of_bounded (Q : ℕ → Prop) [DecidablePred Q] : (∃ n, ∀ k, Q k → k ≤ n) → ∃ s : Finset ℕ, ∀ k, Q k ↔ k ∈ s := by rintro ⟨n, hn⟩ use (range (n + 1)).filter Q intro k simp [Nat.lt_succ_iff] exact hn k theorem mod_4_eq_3_or_mod_4_eq_3 {m n : ℕ} (h : m * n % 4 = 3) : m % 4 = 3 ∨ n % 4 = 3 := by revert h rw [Nat.mul_mod] have : m % 4 < 4 := Nat.mod_lt m (by norm_num) interval_cases hm : m % 4 <;> simp [hm] have : n % 4 < 4 := Nat.mod_lt n (by norm_num) interval_cases hn : n % 4 <;> simp [hn] theorem two_le_of_mod_4_eq_3 {n : ℕ} (h : n % 4 = 3) : 2 ≤ n := by apply two_le <;> · intro neq rw [neq] at h norm_num at h theorem aux {m n : ℕ} (h₀ : m ∣ n) (h₁ : 2 ≤ m) (h₂ : m < n) : n / m ∣ n ∧ n / m < n := by constructor · exact Nat.div_dvd_of_dvd h₀ exact Nat.div_lt_self (lt_of_le_of_lt (zero_le _) h₂) h₁ theorem exists_prime_factor_mod_4_eq_3 {n : Nat} (h : n % 4 = 3) : ∃ p : Nat, p.Prime ∧ p ∣ n ∧ p % 4 = 3 := by by_cases np : n.Prime · use n exact ⟨np, dvd_rfl, h⟩ induction' n using Nat.strong_induction_on with n ih dsimp at ih rw [Nat.prime_def_lt] at np push_neg at np rcases np (two_le_of_mod_4_eq_3 h) with ⟨m, mltn, mdvdn, mne1⟩ have mge2 : 2 ≤ m := by apply two_le _ mne1 intro mz rw [mz, zero_dvd_iff] at mdvdn linarith have neq : m * (n / m) = n := Nat.mul_div_cancel' mdvdn have : m % 4 = 3 ∨ n / m % 4 = 3 := by apply mod_4_eq_3_or_mod_4_eq_3 rw [neq, h] cases' this with h1 h1 · by_cases mp : m.Prime · use m exact ⟨mp, mdvdn, h1⟩ rcases ih m mltn h1 mp with ⟨p, pp, pdvd, p4eq⟩ use p exact ⟨pp, pdvd.trans mdvdn, p4eq⟩ obtain ⟨nmdvdn, nmltn⟩ := aux mdvdn mge2 mltn by_cases nmp : (n / m).Prime · use n / m exact ⟨nmp, nmdvdn, h1⟩ rcases ih (n / m) nmltn h1 nmp with ⟨p, pp, pdvd, p4eq⟩ use p exact ⟨pp, pdvd.trans nmdvdn, p4eq⟩ theorem primes_mod_4_eq_3_infinite : ∀ n, ∃ p > n, Nat.Prime p ∧ p % 4 = 3 := by by_contra h push_neg at h cases' h with n hn have : ∃ s : Finset Nat, ∀ p : ℕ, p.Prime ∧ p % 4 = 3 ↔ p ∈ s := by apply ex_finset_of_bounded use n contrapose! hn rcases hn with ⟨p, ⟨pp, p4⟩, pltn⟩ exact ⟨p, pltn, pp, p4⟩ cases' this with s hs have h₁ : ((4 * ∏ i in erase s 3, i) + 3) % 4 = 3 := by rw [add_comm, Nat.add_mul_mod_self_left] norm_num rcases exists_prime_factor_mod_4_eq_3 h₁ with ⟨p, pp, pdvd, p4eq⟩ have ps : p ∈ s := by rw [← hs p] exact ⟨pp, p4eq⟩ have pne3 : p ≠ 3 := by intro peq rw [peq, ← Nat.dvd_add_iff_left (dvd_refl 3)] at pdvd rw [Nat.prime_three.dvd_mul] at pdvd norm_num at pdvd have : 3 ∈ s.erase 3 := by apply mem_of_dvd_prod_primes Nat.prime_three _ pdvd intro n simp [← hs n] tauto simp at this have : p ∣ 4 * ∏ i in erase s 3, i := by apply dvd_trans _ (dvd_mul_left _ _) apply dvd_prod_of_mem simp constructor <;> assumption have : p ∣ 3 := by convert Nat.dvd_sub' pdvd this simp have : p = 3 := by apply pp.eq_of_dvd_of_prime Nat.prime_three this contradiction
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@@ -0,0 +1,254 @@import Mathlib.Data.Set.Lattice import Mathlib.Data.Nat.Prime import Mathlib.Data.Nat.Parity import Mathlib.Tactic section variable {α : Type _} variable (s t u : Set α) open Set example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by rw [subset_def, inter_def, inter_def] rw [subset_def] at h dsimp rintro x ⟨xs, xu⟩ exact ⟨h _ xs, xu⟩ example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by simp only [subset_def, mem_inter_iff] at * rintro x ⟨xs, xu⟩ exact ⟨h _ xs, xu⟩ example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by intro x xsu exact ⟨h xsu.1, xsu.2⟩ theorem foo (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := fun x ⟨xs, xu⟩ => ⟨h xs, xu⟩ example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := fun x ⟨xs, xu⟩ => ⟨h xs, xu⟩ example : s ∩ (t ∪ u) ⊆ s ∩ t ∪ s ∩ u := by intro x hx have xs : x ∈ s := hx.1 have xtu : x ∈ t ∪ u := hx.2 cases' xtu with xt xu · left show x ∈ s ∩ t exact ⟨xs, xt⟩ right show x ∈ s ∩ u exact ⟨xs, xu⟩ example : s ∩ (t ∪ u) ⊆ s ∩ t ∪ s ∩ u := by rintro x ⟨xs, xt | xu⟩ · left exact ⟨xs, xt⟩ right; exact ⟨xs, xu⟩ example : s ∩ t ∪ s ∩ u ⊆ s ∩ (t ∪ u) := by sorry example : (s \ t) \ u ⊆ s \ (t ∪ u) := by intro x xstu have xs : x ∈ s := xstu.1.1 have xnt : x ∉ t := xstu.1.2 have xnu : x ∉ u := xstu.2 constructor · exact xs intro xtu -- x ∈ t ∨ x ∈ u cases' xtu with xt xu · show False exact xnt xt show False; exact xnu xu example : (s \ t) \ u ⊆ s \ (t ∪ u) := by rintro x ⟨⟨xs, xnt⟩, xnu⟩ use xs rintro (xt | xu) <;> contradiction example : s \ (t ∪ u) ⊆ (s \ t) \ u := by sorry example : s ∩ t = t ∩ s := by ext x simp only [mem_inter_iff] constructor · rintro ⟨xs, xt⟩ exact ⟨xt, xs⟩ rintro ⟨xt, xs⟩; exact ⟨xs, xt⟩ example : s ∩ t = t ∩ s := Set.ext fun x => ⟨fun ⟨xs, xt⟩ => ⟨xt, xs⟩, fun ⟨xt, xs⟩ => ⟨xs, xt⟩⟩ example : s ∩ t = t ∩ s := by ext x; simp [and_comm] example : s ∩ t = t ∩ s := by apply Subset.antisymm · rintro x ⟨xs, xt⟩ exact ⟨xt, xs⟩ rintro x ⟨xt, xs⟩; exact ⟨xs, xt⟩ example : s ∩ t = t ∩ s := Subset.antisymm sorry sorry example : s ∩ (s ∪ t) = s := by sorry example : s ∪ s ∩ t = s := by sorry example : s \ t ∪ t = s ∪ t := by sorry example : s \ t ∪ t \ s = (s ∪ t) \ (s ∩ t) := by sorry def evens : Set ℕ := { n | Even n } def odds : Set ℕ := { n | ¬Even n } example : evens ∪ odds = univ := by rw [evens, odds] ext n simp apply Classical.em example (x : ℕ) (h : x ∈ (∅ : Set ℕ)) : False := h example (x : ℕ) : x ∈ (univ : Set ℕ) := trivial example : { n | Nat.Prime n } ∩ { n | n > 2 } ⊆ { n | ¬Even n } := by sorry #print Prime #print Nat.Prime example (n : ℕ) : Prime n ↔ Nat.Prime n := Nat.prime_iff.symm example (n : ℕ) (h : Prime n) : Nat.Prime n := by rw [Nat.prime_iff] exact h example (n : ℕ) (h : Prime n) : Nat.Prime n := by rwa [Nat.prime_iff] end section variable (s t : Set ℕ) example (h₀ : ∀ x ∈ s, ¬Even x) (h₁ : ∀ x ∈ s, Prime x) : ∀ x ∈ s, ¬Even x ∧ Prime x := by intro x xs constructor · apply h₀ x xs apply h₁ x xs example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ s, Prime x := by rcases h with ⟨x, xs, _, prime_x⟩ use x, xs exact prime_x section variable (ssubt : s ⊆ t) example (h₀ : ∀ x ∈ t, ¬Even x) (h₁ : ∀ x ∈ t, Prime x) : ∀ x ∈ s, ¬Even x ∧ Prime x := by sorry example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ t, Prime x := by sorry end end section variable {α I : Type _} variable (A B : I → Set α) variable (s : Set α) open Set example : (s ∩ ⋃ i, A i) = ⋃ i, A i ∩ s := by ext x simp only [mem_inter_iff, mem_iUnion] constructor · rintro ⟨xs, ⟨i, xAi⟩⟩ exact ⟨i, xAi, xs⟩ rintro ⟨i, xAi, xs⟩ exact ⟨xs, ⟨i, xAi⟩⟩ example : (⋂ i, A i ∩ B i) = (⋂ i, A i) ∩ ⋂ i, B i := by ext x simp only [mem_inter_iff, mem_iInter] constructor · intro h constructor · intro i exact (h i).1 intro i exact (h i).2 rintro ⟨h1, h2⟩ i constructor · exact h1 i exact h2 i example : (s ∪ ⋂ i, A i) = ⋂ i, A i ∪ s := by sorry def primes : Set ℕ := { x | Nat.Prime x } example : (⋃ p ∈ primes, { x | p ^ 2 ∣ x }) = { x | ∃ p ∈ primes, p ^ 2 ∣ x } :=by ext rw [mem_iUnion₂] simp example : (⋃ p ∈ primes, { x | p ^ 2 ∣ x }) = { x | ∃ p ∈ primes, p ^ 2 ∣ x } := by ext simp example : (⋂ p ∈ primes, { x | ¬p ∣ x }) ⊆ { x | x = 1 } := by intro x contrapose! simp apply Nat.exists_prime_and_dvd example : (⋃ p ∈ primes, { x | x ≤ p }) = univ := by sorry end section open Set variable {α : Type _} (s : Set (Set α)) example : ⋃₀ s = ⋃ t ∈ s, t := by ext x rw [mem_iUnion₂] simp example : ⋂₀ s = ⋂ t ∈ s, t := by ext x rw [mem_iInter₂] rfl end
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@@ -0,0 +1,217 @@import Mathlib.Data.Set.Lattice import Mathlib.Data.Set.Function import Mathlib.Analysis.SpecialFunctions.Log.Basic section variable {α β : Type _} variable (f : α → β) variable (s t : Set α) variable (u v : Set β) open Function open Set example : f ⁻¹' (u ∩ v) = f ⁻¹' u ∩ f ⁻¹' v := by ext rfl example : f '' (s ∪ t) = f '' s ∪ f '' t := by ext y; constructor · rintro ⟨x, xs | xt, rfl⟩ · left use x, xs right use x, xt rintro (⟨x, xs, rfl⟩ | ⟨x, xt, rfl⟩) · use x, Or.inl xs use x, Or.inr xt example : s ⊆ f ⁻¹' (f '' s) := by intro x xs show f x ∈ f '' s use x, xs example : f '' s ⊆ v ↔ s ⊆ f ⁻¹' v := by sorry example (h : Injective f) : f ⁻¹' (f '' s) ⊆ s := by sorry example : f '' (f ⁻¹' u) ⊆ u := by sorry example (h : Surjective f) : u ⊆ f '' (f ⁻¹' u) := by sorry example (h : s ⊆ t) : f '' s ⊆ f '' t := by sorry example (h : u ⊆ v) : f ⁻¹' u ⊆ f ⁻¹' v := by sorry example : f ⁻¹' (u ∪ v) = f ⁻¹' u ∪ f ⁻¹' v := by sorry example : f '' (s ∩ t) ⊆ f '' s ∩ f '' t := by sorry example (h : Injective f) : f '' s ∩ f '' t ⊆ f '' (s ∩ t) := by sorry example : f '' s \ f '' t ⊆ f '' (s \ t) := by sorry example : f ⁻¹' u \ f ⁻¹' v ⊆ f ⁻¹' (u \ v) := by sorry example : f '' s ∩ v = f '' (s ∩ f ⁻¹' v) := by sorry example : f '' (s ∩ f ⁻¹' u) ⊆ f '' s ∪ u := by sorry example : s ∩ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∩ u) := by sorry example : s ∪ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∪ u) := by sorry variable {I : Type _} (A : I → Set α) (B : I → Set β) example : (f '' ⋃ i, A i) = ⋃ i, f '' A i := by ext y; simp constructor · rintro ⟨x, ⟨i, xAi⟩, fxeq⟩ use i, x exact ⟨xAi, fxeq⟩ rintro ⟨i, x, xAi, fxeq⟩ exact ⟨x, ⟨i, xAi⟩, fxeq⟩ example : (f '' ⋂ i, A i) ⊆ ⋂ i, f '' A i := by intro y; simp intro x h fxeq i use x exact ⟨h i, fxeq⟩ example (i : I) (injf : Injective f) : (⋂ i, f '' A i) ⊆ f '' ⋂ i, A i := by intro y; simp intro h rcases h i with ⟨x, xAi, fxeq⟩ use x; constructor · intro i' rcases h i' with ⟨x', x'Ai, fx'eq⟩ have : f x = f x' := by rw [fxeq, fx'eq] have : x = x' := injf this rw [this] exact x'Ai exact fxeq example : (f ⁻¹' ⋃ i, B i) = ⋃ i, f ⁻¹' B i := by ext x simp example : (f ⁻¹' ⋂ i, B i) = ⋂ i, f ⁻¹' B i := by ext x simp example : InjOn f s ↔ ∀ x₁ ∈ s, ∀ x₂ ∈ s, f x₁ = f x₂ → x₁ = x₂ := Iff.refl _ end section open Set Real example : InjOn log { x | x > 0 } := by intro x xpos y ypos intro e -- log x = log y calc x = exp (log x) := by rw [exp_log xpos] _ = exp (log y) := by rw [e] _ = y := by rw [exp_log ypos] example : range exp = { y | y > 0 } := by ext y; constructor · rintro ⟨x, rfl⟩ apply exp_pos intro ypos use log y rw [exp_log ypos] example : InjOn sqrt { x | x ≥ 0 } := by sorry example : InjOn (fun x => x ^ 2) { x : ℝ | x ≥ 0 } := by sorry example : sqrt '' { x | x ≥ 0 } = { y | y ≥ 0 } := by sorry example : (range fun x => x ^ 2) = { y : ℝ | y ≥ 0 } := by sorry end section variable {α β : Type _} [Inhabited α] #check (default : α) variable (P : α → Prop) (h : ∃ x, P x) #check Classical.choose h example : P (Classical.choose h) := Classical.choose_spec h noncomputable section open Classical def inverse (f : α → β) : β → α := fun y : β => if h : ∃ x, f x = y then Classical.choose h else default theorem inverse_spec {f : α → β} (y : β) (h : ∃ x, f x = y) : f (inverse f y) = y := by rw [inverse]; dsimp; rw [dif_pos h] exact Classical.choose_spec h variable (f : α → β) open Function example : Injective f ↔ LeftInverse (inverse f) f := sorry example : Surjective f ↔ RightInverse (inverse f) f := sorry end section variable {α : Type _} open Function theorem Cantor : ∀ f : α → Set α, ¬Surjective f := by intro f surjf let S := { i | i ∉ f i } rcases surjf S with ⟨j, h⟩ have h₁ : j ∉ f j := by intro h' have : j ∉ f j := by rwa [h] at h' contradiction have h₂ : j ∈ S sorry have h₃ : j ∉ S sorry contradiction -- COMMENTS: TODO: improve this end
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@@ -0,0 +1,99 @@import Mathlib.Data.Set.Lattice import Mathlib.Data.Set.Function import Mathlib.Tactic open Set open Function noncomputable section open Classical variable {α β : Type _} [Nonempty β] section variable (f : α → β) (g : β → α) def sbAux : ℕ → Set α | 0 => univ \ g '' univ | n + 1 => g '' (f '' sbAux n) def sbSet := ⋃ n, sbAux f g n def sbFun (x : α) : β := if x ∈ sbSet f g then f x else invFun g x theorem sb_right_inv {x : α} (hx : x ∉ sbSet f g) : g (invFun g x) = x := by have : x ∈ g '' univ := by contrapose! hx rw [sbSet, mem_iUnion] use 0 rw [sbAux, mem_diff] sorry }, have : ∃ y, g y = x := by { sorry }, sorry theorem sb_injective (hf : Injective f) (hg : Injective g) : Injective (sbFun f g) := by set A := sbSet f g with A_def set h := sbFun f g with h_def intro x₁ x₂ intro (hxeq : h x₁ = h x₂) show x₁ = x₂ simp only [h_def, sbFun, ← A_def] at hxeq by_cases xA : x₁ ∈ A ∨ x₂ ∈ A · wlog x₁A : x₁ ∈ A generalizing x₁ x₂ hxeq xA · symm apply this hxeq.symm xA.symm (xA.resolve_left x₁A) have x₂A : x₂ ∈ A := by apply not_imp_self.mp intro (x₂nA : x₂ ∉ A) rw [if_pos x₁A, if_neg x₂nA] at hxeq rw [A_def, sbSet, mem_iUnion] at x₁A have x₂eq : x₂ = g (f x₁) := by . sorry rcases x₁A with ⟨n, hn⟩ rw [A_def, sbSet, mem_iUnion] use n + 1 simp [sbAux] exact ⟨x₁, hn, x₂eq.symm⟩ . sorry, push_neg at xA sorry theorem sb_surjective (hf : Injective f) (hg : Injective g) : Surjective (sbFun f g) := by set A := sbSet f g with A_def set h := sbFun f g with h_def intro y by_cases gyA : g y ∈ A · rw [A_def, sbSet, mem_iUnion] at gyA rcases gyA with ⟨n, hn⟩ cases' n with n · simp [sbAux] at hn simp [sbAux] at hn rcases hn with ⟨x, xmem, hx⟩ use x have : x ∈ A := by rw [A_def, sbSet, mem_iUnion] exact ⟨n, xmem⟩ simp only [h_def, sbFun, if_pos this] exact hg hx sorry end theorem schroeder_bernstein {f : α → β} {g : β → α} (hf : Injective f) (hg : Injective g) : ∃ h : α → β, Bijective h := ⟨sbFun f g, sb_injective f g hf hg, sb_surjective f g hf hg⟩ -- Auxiliary information section variable (g : β → α) (x : α) #check (invFun g : α → β) #check (leftInverse_invFun : Injective g → LeftInverse (invFun g) g) #check (leftInverse_invFun : Injective g → ∀ y, invFun g (g y) = y) #check (invFun_eq : (∃ y, g y = x) → g (invFun g x) = x) end
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@@ -0,0 +1,165 @@import Mathlib.Data.Set.Lattice import Mathlib.Data.Nat.Prime import Mathlib.Data.Nat.Parity import Mathlib.Tactic section variable {α : Type _} variable (s t u : Set α) open Set example : s ∩ t ∪ s ∩ u ⊆ s ∩ (t ∪ u) := by rintro x (⟨xs, xt⟩ | ⟨xs, xu⟩) · use xs left exact xt use xs; right; exact xu example : s \ (t ∪ u) ⊆ (s \ t) \ u := by rintro x ⟨xs, xntu⟩ constructor use xs · intro xt exact xntu (Or.inl xt) intro xu apply xntu (Or.inr xu) example : s ∩ t = t ∩ s := Subset.antisymm (fun x ⟨xs, xt⟩ => ⟨xt, xs⟩) fun x ⟨xt, xs⟩ => ⟨xs, xt⟩ example : s ∩ (s ∪ t) = s := by ext x; constructor · rintro ⟨xs, _⟩ exact xs intro xs use xs; left; exact xs example : s ∪ s ∩ t = s := by ext x; constructor · rintro (xs | ⟨xs, xt⟩) <;> exact xs intro xs; left; exact xs example : s \ t ∪ t = s ∪ t := by ext x; constructor · rintro (⟨xs, nxt⟩ | xt) · left exact xs right exact xt by_cases h : x ∈ t · intro right exact h rintro (xs | xt) · left use xs exact h right; exact xt example : s \ t ∪ t \ s = (s ∪ t) \ (s ∩ t) := by ext x; constructor · rintro (⟨xs, xnt⟩ | ⟨xt, xns⟩) · constructor left exact xs rintro ⟨_, xt⟩ contradiction constructor right exact xt rintro ⟨xs, _⟩ contradiction rintro ⟨xs | xt, nxst⟩ · left use xs intro xt apply nxst constructor <;> assumption right; use xt; intro xs apply nxst constructor <;> assumption example : { n | Nat.Prime n } ∩ { n | n > 2 } ⊆ { n | ¬Even n } := by intro n simp intro nprime cases' Nat.Prime.eq_two_or_odd nprime with h h · rw [h] intro linarith rw [Nat.even_iff, h] norm_num end section variable (s t : Set ℕ) section variable (ssubt : s ⊆ t) example (h₀ : ∀ x ∈ t, ¬Even x) (h₁ : ∀ x ∈ t, Prime x) : ∀ x ∈ s, ¬Even x ∧ Prime x := by intro x xs constructor · apply h₀ x (ssubt xs) apply h₁ x (ssubt xs) example (h : ∃ x ∈ s, ¬Even x ∧ Prime x) : ∃ x ∈ t, Prime x := by rcases h with ⟨x, xs, _, px⟩ use x, ssubt xs exact px end end section variable {α I : Type _} variable (A B : I → Set α) variable (s : Set α) open Set example : (s ∪ ⋂ i, A i) = ⋂ i, A i ∪ s := by ext x simp only [mem_union, mem_iInter] constructor · rintro (xs | xI) · intro i right exact xs intro i left exact xI i intro h by_cases xs : x ∈ s · left exact xs right intro i cases h i · assumption contradiction def primes : Set ℕ := { x | Nat.Prime x } example : (⋃ p ∈ primes, { x | x ≤ p }) = univ := by apply eq_univ_of_forall intro x simp rcases Nat.exists_infinite_primes x with ⟨p, primep, pge⟩ use p, pge exact primep end
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@@ -0,0 +1,256 @@import Mathlib.Data.Set.Lattice import Mathlib.Data.Set.Function import Mathlib.Analysis.SpecialFunctions.Log.Basic section variable {α β : Type _} variable (f : α → β) variable (s t : Set α) variable (u v : Set β) open Function open Set example : f '' s ⊆ v ↔ s ⊆ f ⁻¹' v := by constructor · intro h x xs have : f x ∈ f '' s := mem_image_of_mem _ xs exact h this intro h y ymem rcases ymem with ⟨x, xs, fxeq⟩ rw [← fxeq] apply h xs example (h : Injective f) : f ⁻¹' (f '' s) ⊆ s := by rintro x ⟨y, ys, fxeq⟩ rw [← h fxeq] exact ys example : f '' (f ⁻¹' u) ⊆ u := by rintro y ⟨x, xmem, rfl⟩ exact xmem example (h : Surjective f) : u ⊆ f '' (f ⁻¹' u) := by intro y yu rcases h y with ⟨x, fxeq⟩ use x constructor · show f x ∈ u rw [fxeq] exact yu exact fxeq example (h : s ⊆ t) : f '' s ⊆ f '' t := by rintro y ⟨x, xs, fxeq⟩ use x, h xs exact fxeq example (h : u ⊆ v) : f ⁻¹' u ⊆ f ⁻¹' v := by intro x; apply h example : f ⁻¹' (u ∪ v) = f ⁻¹' u ∪ f ⁻¹' v := by ext x; rfl example : f '' (s ∩ t) ⊆ f '' s ∩ f '' t := by rintro y ⟨x, ⟨xs, xt⟩, rfl⟩ constructor . use x, xs . use x, xt example (h : Injective f) : f '' s ∩ f '' t ⊆ f '' (s ∩ t) := by rintro y ⟨⟨x₁, x₁s, rfl⟩, ⟨x₂, x₂t, fx₂eq⟩⟩ use x₁ constructor . use x₁s rw [← h fx₂eq] exact x₂t . rfl example : f '' s \ f '' t ⊆ f '' (s \ t) := by rintro y ⟨⟨x₁, x₁s, rfl⟩, h⟩ use x₁ constructor . constructor . exact x₁s . intro h' apply h use x₁, h' . rfl example : f ⁻¹' u \ f ⁻¹' v ⊆ f ⁻¹' (u \ v) := fun x => id example : f '' s ∩ v = f '' (s ∩ f ⁻¹' v) := by ext y; constructor · rintro ⟨⟨x, xs, rfl⟩, fxv⟩ use x, ⟨xs, fxv⟩ rintro ⟨x, ⟨⟨xs, fxv⟩, rfl⟩⟩ exact ⟨⟨x, xs, rfl⟩, fxv⟩ example : f '' (s ∩ f ⁻¹' u) ⊆ f '' s ∩ u := by rintro y ⟨x, ⟨xs, fxu⟩, rfl⟩ exact ⟨⟨x, xs, rfl⟩, fxu⟩ example : s ∩ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∩ u) := by rintro x ⟨xs, fxu⟩ exact ⟨⟨x, xs, rfl⟩, fxu⟩ example : s ∪ f ⁻¹' u ⊆ f ⁻¹' (f '' s ∪ u) := by rintro x (xs | fxu) · left exact ⟨x, xs, rfl⟩ right; exact fxu variable {I : Type _} (A : I → Set α) (B : I → Set β) example : (f '' ⋃ i, A i) = ⋃ i, f '' A i := by ext y; simp constructor · rintro ⟨x, ⟨i, xAi⟩, fxeq⟩ use i, x exact ⟨xAi, fxeq⟩ rintro ⟨i, x, xAi, fxeq⟩ exact ⟨x, ⟨i, xAi⟩, fxeq⟩ example : (f '' ⋂ i, A i) ⊆ ⋂ i, f '' A i := by intro y; simp intro x h fxeq i use x exact ⟨h i, fxeq⟩ example (i : I) (injf : Injective f) : (⋂ i, f '' A i) ⊆ f '' ⋂ i, A i := by intro y; simp intro h rcases h i with ⟨x, xAi, fxeq⟩ use x; constructor · intro i' rcases h i' with ⟨x', x'Ai, fx'eq⟩ have : f x = f x' := by rw [fxeq, fx'eq] have : x = x' := injf this rw [this] exact x'Ai exact fxeq example : (f ⁻¹' ⋃ i, B i) = ⋃ i, f ⁻¹' B i := by ext x simp example : (f ⁻¹' ⋂ i, B i) = ⋂ i, f ⁻¹' B i := by ext x simp end section open Set Real example : InjOn sqrt { x | x ≥ 0 } := by intro x xnonneg y ynonneg intro e calc x = sqrt x ^ 2 := by rw [sq_sqrt xnonneg] _ = sqrt y ^ 2 := by rw [e] _ = y := by rw [sq_sqrt ynonneg] example : InjOn (fun x => x ^ 2) { x : ℝ | x ≥ 0 } := by intro x xnonneg y ynonneg intro e dsimp at * calc x = sqrt (x ^ 2) := by rw [sqrt_sq xnonneg] _ = sqrt (y ^ 2) := by rw [e] _ = y := by rw [sqrt_sq ynonneg] example : sqrt '' { x | x ≥ 0 } = { y | y ≥ 0 } := by ext y; constructor · rintro ⟨x, ⟨xnonneg, rfl⟩⟩ apply sqrt_nonneg intro ynonneg use y ^ 2 dsimp at * constructor apply pow_nonneg ynonneg apply sqrt_sq assumption example : (range fun x => x ^ 2) = { y : ℝ | y ≥ 0 } := by ext y constructor · rintro ⟨x, rfl⟩ dsimp at * apply pow_two_nonneg intro ynonneg use sqrt y exact sq_sqrt ynonneg end section variable {α β : Type _} [Inhabited α] noncomputable section open Classical def inverse (f : α → β) : β → α := fun y : β => if h : ∃ x, f x = y then Classical.choose h else default theorem inverse_spec {f : α → β} (y : β) (h : ∃ x, f x = y) : f (inverse f y) = y := by rw [inverse]; dsimp; rw [dif_pos h] exact Classical.choose_spec h variable (f : α → β) open Function example : Injective f ↔ LeftInverse (inverse f) f := by constructor · intro h y apply h apply inverse_spec use y intro h x1 x2 e rw [← h x1, ← h x2, e] example : Injective f ↔ LeftInverse (inverse f) f := ⟨fun h y => h (inverse_spec _ ⟨y, rfl⟩), fun h x1 x2 e => by rw [← h x1, ← h x2, e]⟩ example : Surjective f ↔ RightInverse (inverse f) f := by constructor · intro h y apply inverse_spec apply h intro h y use inverse f y apply h example : Surjective f ↔ RightInverse (inverse f) f := ⟨fun h y => inverse_spec _ (h _), fun h y => ⟨inverse f y, h _⟩⟩ end section variable {α : Type _} open Function theorem Cantor : ∀ f : α → Set α, ¬Surjective f := by intro f surjf let S := { i | i ∉ f i } rcases surjf S with ⟨j, h⟩ have h₁ : j ∉ f j := by intro h' have : j ∉ f j := by rwa [h] at h' contradiction have h₂ : j ∈ S := h₁ have h₃ : j ∉ S := by rwa [h] at h₁ contradiction end
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@@ -0,0 +1,92 @@import Mathlib.Data.Set.Lattice import Mathlib.Data.Set.Function import Mathlib.Tactic open Set open Function noncomputable section open Classical variable {α β : Type _} [Nonempty β] section variable (f : α → β) (g : β → α) def sbAux : ℕ → Set α | 0 => univ \ g '' univ | n + 1 => g '' (f '' sbAux n) def sbSet := ⋃ n, sbAux f g n def sbFun (x : α) : β := if x ∈ sbSet f g then f x else invFun g x theorem sb_right_inv {x : α} (hx : x ∉ sbSet f g) : g (invFun g x) = x := by have : x ∈ g '' univ := by contrapose! hx rw [sbSet, mem_iUnion] use 0 rw [sbAux, mem_diff] exact ⟨mem_univ _, hx⟩ have : ∃ y, g y = x := by simp at this assumption exact invFun_eq this theorem sb_injective (hf : Injective f) (hg : Injective g) : Injective (sbFun f g) := by set A := sbSet f g with A_def set h := sbFun f g with h_def intro x₁ x₂ intro (hxeq : h x₁ = h x₂) show x₁ = x₂ simp only [h_def, sbFun, ← A_def] at hxeq by_cases xA : x₁ ∈ A ∨ x₂ ∈ A · wlog x₁A : x₁ ∈ A generalizing x₁ x₂ hxeq xA · symm apply this hxeq.symm xA.symm (xA.resolve_left x₁A) have x₂A : x₂ ∈ A := by apply not_imp_self.mp intro (x₂nA : x₂ ∉ A) rw [if_pos x₁A, if_neg x₂nA] at hxeq rw [A_def, sbSet, mem_iUnion] at x₁A have x₂eq : x₂ = g (f x₁) := by rw [hxeq, sb_right_inv f g x₂nA] rcases x₁A with ⟨n, hn⟩ rw [A_def, sbSet, mem_iUnion] use n + 1 simp [sbAux] exact ⟨x₁, hn, x₂eq.symm⟩ rw [if_pos x₁A, if_pos x₂A] at hxeq exact hf hxeq push_neg at xA rw [if_neg xA.1, if_neg xA.2] at hxeq rw [← sb_right_inv f g xA.1, hxeq, sb_right_inv f g xA.2] theorem sb_surjective (hf : Injective f) (hg : Injective g) : Surjective (sbFun f g) := by set A := sbSet f g with A_def set h := sbFun f g with h_def intro y by_cases gyA : g y ∈ A · rw [A_def, sbSet, mem_iUnion] at gyA rcases gyA with ⟨n, hn⟩ cases' n with n · simp [sbAux] at hn simp [sbAux] at hn rcases hn with ⟨x, xmem, hx⟩ use x have : x ∈ A := by rw [A_def, sbSet, mem_iUnion] exact ⟨n, xmem⟩ simp only [h_def, sbFun, if_pos this] exact hg hx use g y simp only [h_def, sbFun, if_neg gyA] apply leftInverse_invFun hg end
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@@ -0,0 +1,227 @@import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic noncomputable section @[ext] structure Point where x : ℝ y : ℝ z : ℝ #check Point.ext example (a b : Point) (hx : a.x = b.x) (hy : a.y = b.y) (hz : a.z = b.z) : a = b := by ext repeat' assumption def myPoint1 : Point where x := 2 y := -1 z := 4 def myPoint2 : Point := ⟨2, -1, 4⟩ def myPoint3 := Point.mk 2 (-1) 4 structure Point' where build :: x : ℝ y : ℝ z : ℝ #check Point'.build 2 (-1) 4 namespace Point def add (a b : Point) : Point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ def add' (a b : Point) : Point where x := a.x + b.x y := a.y + b.y z := a.z + b.z #check add myPoint1 myPoint2 #check myPoint1.add myPoint2 end Point #check Point.add myPoint1 myPoint2 #check myPoint1.add myPoint2 namespace Point protected theorem add_comm (a b : Point) : add a b = add b a := by rw [add, add] ext <;> dsimp repeat' apply add_comm example (a b : Point) : add a b = add b a := by simp [add, add_comm] theorem add_x (a b : Point) : (a.add b).x = a.x + b.x := rfl def addAlt : Point → Point → Point | Point.mk x₁ y₁ z₁, Point.mk x₂ y₂ z₂ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ def addAlt' : Point → Point → Point | ⟨x₁, y₁, z₁⟩, ⟨x₂, y₂, z₂⟩ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ theorem addAlt_x (a b : Point) : (a.addAlt b).x = a.x + b.x := by cases a cases b rfl theorem addAlt_comm (a b : Point) : addAlt a b = addAlt b a := by rcases a with ⟨xa, ya, za⟩ rcases b with ⟨xb, yb, zb⟩ rw [addAlt, addAlt] ext <;> dsimp apply add_comm repeat' apply add_comm example (a b : Point) : addAlt a b = addAlt b a := by rcases a with ⟨xa, ya, za⟩ rcases b with ⟨xb, yb, zb⟩ simp [addAlt, add_comm] example : ∀ a b : Point, addAlt a b = addAlt b a := by rintro ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ simp [addAlt, add_comm] example : ∀ a b : Point, add a b = add b a := fun ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ => by simp [add, add_comm] protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by sorry def smul (r : ℝ) (a : Point) : Point := sorry theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by sorry end Point structure StandardTwoSimplex where x : ℝ y : ℝ z : ℝ x_nonneg : 0 ≤ x y_nonneg : 0 ≤ y z_nonneg : 0 ≤ z sum_eq : x + y + z = 1 namespace StandardTwoSimplex def swapXy (a : StandardTwoSimplex) : StandardTwoSimplex where x := a.y y := a.x z := a.z x_nonneg := a.y_nonneg y_nonneg := a.x_nonneg z_nonneg := a.z_nonneg sum_eq := by rw [add_comm a.y a.x, a.sum_eq] noncomputable section def midpoint (a b : StandardTwoSimplex) : StandardTwoSimplex where x := (a.x + b.x) / 2 y := (a.y + b.y) / 2 z := (a.z + b.z) / 2 x_nonneg := div_nonneg (add_nonneg a.x_nonneg b.x_nonneg) (by norm_num) y_nonneg := div_nonneg (add_nonneg a.y_nonneg b.y_nonneg) (by norm_num) z_nonneg := div_nonneg (add_nonneg a.z_nonneg b.z_nonneg) (by norm_num) sum_eq := by field_simp; linarith [a.sum_eq, b.sum_eq] def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) (a b : StandardTwoSimplex) : StandardTwoSimplex where sorry end end StandardTwoSimplex open BigOperators structure StandardSimplex (n : ℕ) where V : Fin n → ℝ NonNeg : ∀ i : Fin n, 0 ≤ V i sum_eq_one : (∑ i, V i) = 1 namespace StandardSimplex def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n where V i := (a.V i + b.V i) / 2 NonNeg := by intro i apply div_nonneg · linarith [a.NonNeg i, b.NonNeg i] norm_num sum_eq_one := by simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] field_simp end StandardSimplex structure IsLinear (f : ℝ → ℝ) where is_additive : ∀ x y, f (x + y) = f x + f y preserves_mul : ∀ x c, f (c * x) = c * f x section variable (f : ℝ → ℝ) (linf : IsLinear f) #check linf.is_additive #check linf.preserves_mul end def Point'' := ℝ × ℝ × ℝ def IsLinear' (f : ℝ → ℝ) := (∀ x y, f (x + y) = f x + f y) ∧ ∀ x c, f (c * x) = c * f x def PReal := { y : ℝ // 0 < y } section variable (x : PReal) #check x.val #check x.property #check x.1 #check x.2 end def StandardTwoSimplex' := { p : ℝ × ℝ × ℝ // 0 ≤ p.1 ∧ 0 ≤ p.2.1 ∧ 0 ≤ p.2.2 ∧ p.1 + p.2.1 + p.2.2 = 1 } def StandardSimplex' (n : ℕ) := { v : Fin n → ℝ // (∀ i : Fin n, 0 ≤ v i) ∧ (∑ i, v i) = 1 } def StdSimplex := Σ n : ℕ, StandardSimplex n section variable (s : StdSimplex) #check s.fst #check s.snd #check s.1 #check s.2 end
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@@ -0,0 +1,172 @@import Mathlib.Data.Real.Basic structure Group₁ (α : Type _) where mul : α → α → α one : α inv : α → α mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) mul_one : ∀ x : α, mul x one = x one_mul : ∀ x : α, mul one x = x mul_left_inv : ∀ x : α, mul (inv x) x = one structure Group₁Cat where α : Type _ str : Group₁ α section variable (α β γ : Type _) variable (f : α ≃ β) (g : β ≃ γ) #check Equiv α β #check (f.toFun : α → β) #check (f.invFun : β → α) #check (f.right_inv : ∀ x : β, f (f.invFun x) = x) #check (f.left_inv : ∀ x : α, f.invFun (f x) = x) #check (Equiv.refl α : α ≃ α) #check (f.symm : β ≃ α) #check (f.trans g : α ≃ γ) example (x : α) : (f.trans g).toFun x = g.toFun (f.toFun x) := rfl example (x : α) : (f.trans g) x = g (f x) := rfl example : (f.trans g : α → γ) = g ∘ f := rfl end example (α : Type _) : Equiv.Perm α = (α ≃ α) := rfl def permGroup {α : Type _} : Group₁ (Equiv.Perm α) where mul f g := Equiv.trans g f one := Equiv.refl α inv := Equiv.symm mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm one_mul := Equiv.trans_refl mul_one := Equiv.refl_trans mul_left_inv := Equiv.self_trans_symm structure AddGroup₁ (α : Type _) where (add : α → α → α) -- fill in the rest @[ext] structure Point where x : ℝ y : ℝ z : ℝ namespace Point def add (a b : Point) : Point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ def neg (a : point) : point := sorry def zero : point := sorry def add_group_point : add_group₁ point := sorry end Point section variable {α : Type _} (f g : Equiv.Perm α) (n : ℕ) #check f * g #check mul_assoc f g g⁻¹ -- group power, defined for any group #check g ^ n example : f * g * g⁻¹ = f := by rw [mul_assoc, mul_right_inv, mul_one] example : f * g * g⁻¹ = f := mul_inv_cancel_right f g example {α : Type _} (f g : Equiv.Perm α) : g.symm.trans (g.trans f) = f := mul_inv_cancel_right f g end class Group₂ (α : Type _) where mul : α → α → α one : α inv : α → α mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) mul_one : ∀ x : α, mul x one = x one_mul : ∀ x : α, mul one x = x mul_left_inv : ∀ x : α, mul (inv x) x = one instance {α : Type _} : Group₂ (Equiv.Perm α) where mul f g := Equiv.trans g f one := Equiv.refl α inv := Equiv.symm mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm one_mul := Equiv.trans_refl mul_one := Equiv.refl_trans mul_left_inv := Equiv.self_trans_symm #check @Group₂.mul def mySquare {α : Type _} [Group₂ α] (x : α) := Group₂.mul x x #check @mySquare section variable {β : Type _} (f g : Equiv.Perm β) example : Group₂.mul f g = g.trans f := rfl example : mySquare f = f.trans f := rfl end instance : Inhabited Point where default := ⟨0, 0, 0⟩ #check (default : Point) example : ([] : List Point).headI = default := rfl instance : Add Point where add := Point.add section variable (x y : Point) #check x + y example : x + y = Point.add x y := rfl end instance hasMulGroup₂ {α : Type _} [Group₂ α] : Mul α := ⟨Group₂.mul⟩ instance hasOneGroup₂ {α : Type _} [Group₂ α] : One α := ⟨Group₂.one⟩ instance hasInvGroup₂ {α : Type _} [Group₂ α] : Inv α := ⟨Group₂.inv⟩ section variable {α : Type _} (f g : Equiv.Perm α) #check f * 1 * g⁻¹ def foo : f * 1 * g⁻¹ = g.symm.trans ((Equiv.refl α).trans f) := rfl end class AddGroup₂ (α : Type _) where add : α → α → α -- fill in the rest
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@@ -0,0 +1,271 @@import Mathlib.Data.Int.Basic import Mathlib.Algebra.EuclideanDomain.Basic import Mathlib.RingTheory.PrincipalIdealDomain import Mathlib.Tactic @[ext] structure gaussInt where re : ℤ im : ℤ namespace gaussInt instance : Zero gaussInt := ⟨⟨0, 0⟩⟩ instance : One gaussInt := ⟨⟨1, 0⟩⟩ instance : Add gaussInt := ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ instance : Neg gaussInt := ⟨fun x => ⟨-x.re, -x.im⟩⟩ instance : Mul gaussInt := ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := rfl theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := rfl theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := rfl theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := rfl theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := rfl @[simp] theorem zero_re : (0 : gaussInt).re = 0 := rfl @[simp] theorem zero_im : (0 : gaussInt).im = 0 := rfl @[simp] theorem one_re : (1 : gaussInt).re = 1 := rfl @[simp] theorem one_im : (1 : gaussInt).im = 0 := rfl @[simp] theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := rfl @[simp] theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := rfl @[simp] theorem neg_re (x : gaussInt) : (-x).re = -x.re := rfl @[simp] theorem neg_im (x : gaussInt) : (-x).im = -x.im := rfl @[simp] theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := rfl @[simp] theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := rfl instance instCommRing : CommRing gaussInt where zero := 0 one := 1 add := (· + ·) neg x := -x mul := (· * ·) add_assoc := by intros ext <;> simp <;> ring zero_add := by intros ext <;> simp add_zero := by intros ext <;> simp add_left_neg := by intros ext <;> simp add_comm := by intros ext <;> simp <;> ring mul_assoc := by intros ext <;> simp <;> ring one_mul := by intros ext <;> simp mul_one := by intros ext <;> simp left_distrib := by intros ext <;> simp <;> ring right_distrib := by intros ext <;> simp <;> ring mul_comm := by intros ext <;> simp <;> ring zero_mul := sorry mul_zero := sorry instance : Nontrivial gaussInt := by use 0, 1 rw [Ne, gaussInt.ext_iff] simp end gaussInt example (a b : ℤ) : a = b * (a / b) + a % b := Eq.symm <| Int.ediv_add_emod a b example (a b : ℤ) : b ≠ 0 → 0 ≤ a % b := Int.emod_nonneg a example (a b : ℤ) : b ≠ 0 → a % b < abs b := Int.emod_lt a namespace Int def div' (a b : ℤ) := (a + b / 2) / b def mod' (a b : ℤ) := (a + b / 2) % b - b / 2 theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by rw [div', mod'] linarith [Int.ediv_add_emod (a + b / 2) b] theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by rw [mod', abs_le] constructor · linarith [Int.emod_nonneg (a + b / 2) h.ne'] have := Int.emod_lt_of_pos (a + b / 2) h have := Int.ediv_add_emod b 2 have := Int.emod_lt_of_pos b zero_lt_two revert this; intro this -- FIXME, this should not be needed linarith theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] end Int theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by sorry namespace gaussInt def norm (x : gaussInt) := x.re ^ 2 + x.im ^ 2 @[simp] theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by sorry theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by sorry theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by sorry theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by sorry def conj (x : gaussInt) : gaussInt := ⟨x.re, -x.im⟩ @[simp] theorem conj_re (x : gaussInt) : (conj x).re = x.re := rfl @[simp] theorem conj_im (x : gaussInt) : (conj x).im = -x.im := rfl theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] instance : Div gaussInt := ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ instance : Mod gaussInt := ⟨fun x y => x - y * (x / y)⟩ theorem div_def (x y : gaussInt) : x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := rfl theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := rfl theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by have norm_y_pos : 0 < norm y := by rwa [norm_pos] have : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] ext <;> simp <;> ring have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by conv => lhs rw [← norm_conj y, ← norm_mul, this, norm] simp trans 2 * (y.norm / 2) ^ 2 · rw [two_mul] apply add_le_add <;> · rw [sq_le_sq] apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) apply le_abs_self rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] apply mul_le_mul_of_nonneg_left · apply Int.ediv_mul_le norm_num apply Int.ediv_nonneg (norm_nonneg y) norm_num have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos apply lt_of_le_of_lt this apply Int.ediv_lt_of_lt_mul · norm_num linarith theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := Int.natAbs_of_nonneg (norm_nonneg _) theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by apply Int.ofNat_lt.1 simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] apply norm_mod_lt x hy theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : ¬(norm (x * y)).natAbs < (norm x).natAbs := by apply not_lt_of_ge rw [norm_mul, Int.natAbs_mul] apply le_mul_of_one_le_right (Nat.zero_le _) apply Int.ofNat_le.1 rw [coe_natAbs_norm] exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) instance : EuclideanDomain gaussInt := { gaussInt.instCommRing with quotient := (· / ·) remainder := (· % ·) quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] quotient_zero := fun x => by simp [div_def, norm, Int.div'] rfl r := Measure (Int.natAbs ∘ norm) r_wellFounded := (measure (Int.natAbs ∘ norm)).2 remainder_lt := natAbs_norm_mod_lt mul_left_not_lt := not_norm_mul_left_lt_norm } example (x : gaussInt) : Irreducible x ↔ Prime x := PrincipalIdealRing.irreducible_iff_prime end gaussInt
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@@ -0,0 +1,96 @@import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic noncomputable section @[ext] structure Point where x : ℝ y : ℝ z : ℝ namespace Point def add (a b : Point) : Point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by simp [add, add_assoc] def smul (r : ℝ) (a : Point) : Point := ⟨r * a.x, r * a.y, r * a.z⟩ theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by simp [add, smul, mul_add] end Point structure StandardTwoSimplex where x : ℝ y : ℝ z : ℝ x_nonneg : 0 ≤ x y_nonneg : 0 ≤ y z_nonneg : 0 ≤ z sum_eq : x + y + z = 1 namespace StandardTwoSimplex noncomputable section def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) (a b : StandardTwoSimplex) : StandardTwoSimplex where x := lambda * a.x + (1 - lambda) * b.x y := lambda * a.y + (1 - lambda) * b.y z := lambda * a.z + (1 - lambda) * b.z x_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.x_nonneg) (mul_nonneg (by linarith) b.x_nonneg) y_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.y_nonneg) (mul_nonneg (by linarith) b.y_nonneg) z_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.z_nonneg) (mul_nonneg (by linarith) b.z_nonneg) sum_eq := by trans (a.x + a.y + a.z) * lambda + (b.x + b.y + b.z) * (1 - lambda) · ring simp [a.sum_eq, b.sum_eq] end end StandardTwoSimplex open BigOperators structure StandardSimplex (n : ℕ) where V : Fin n → ℝ NonNeg : ∀ i : Fin n, 0 ≤ V i sum_eq_one : (∑ i, V i) = 1 namespace StandardSimplex def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n where V i := (a.V i + b.V i) / 2 NonNeg := by intro i apply div_nonneg · linarith [a.NonNeg i, b.NonNeg i] norm_num sum_eq_one := by simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] field_simp end StandardSimplex namespace StandardSimplex def weightedAverage {n : ℕ} (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) (a b : StandardSimplex n) : StandardSimplex n where V i := lambda * a.V i + (1 - lambda) * b.V i NonNeg i := add_nonneg (mul_nonneg lambda_nonneg (a.NonNeg i)) (mul_nonneg (by linarith) (b.NonNeg i)) sum_eq_one := by trans (lambda * ∑ i, a.V i) + (1 - lambda) * ∑ i, b.V i · rw [Finset.sum_add_distrib, Finset.mul_sum, Finset.mul_sum] simp [a.sum_eq_one, b.sum_eq_one] end StandardSimplex
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@@ -0,0 +1,73 @@import Mathlib.Data.Real.Basic structure AddGroup₁ (α : Type _) where add : α → α → α zero : α neg : α → α add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) add_zero : ∀ x : α, add x zero = x zero_add : ∀ x : α, add x zero = x add_left_neg : ∀ x : α, add (neg x) x = zero @[ext] structure Point where x : ℝ y : ℝ z : ℝ namespace Point def add (a b : Point) : Point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ def neg (a : Point) : Point := ⟨-a.x, -a.y, -a.z⟩ def zero : Point := ⟨0, 0, 0⟩ def addGroupPoint : AddGroup₁ Point where add := Point.add zero := Point.zero neg := Point.neg add_assoc := by simp [Point.add, add_assoc] add_zero := by simp [Point.add, Point.zero] zero_add := by simp [Point.add, Point.zero] add_left_neg := by simp [Point.add, Point.neg, Point.zero] end Point class AddGroup₂ (α : Type _) where add : α → α → α zero : α neg : α → α add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) add_zero : ∀ x : α, add x zero = x zero_add : ∀ x : α, add x zero = x add_left_neg : ∀ x : α, add (neg x) x = zero instance hasAddAddGroup₂ {α : Type _} [AddGroup₂ α] : Add α := ⟨AddGroup₂.add⟩ instance hasZeroAddGroup₂ {α : Type _} [AddGroup₂ α] : Zero α := ⟨AddGroup₂.zero⟩ instance hasNegAddGroup₂ {α : Type _} [AddGroup₂ α] : Neg α := ⟨AddGroup₂.neg⟩ instance : AddGroup₂ Point where add := Point.add zero := Point.zero neg := Point.neg add_assoc := by simp [Point.add, add_assoc] add_zero := by simp [Point.add, Point.zero] zero_add := by simp [Point.add, Point.zero] add_left_neg := by simp [Point.add, Point.neg, Point.zero] section variable (x y : Point) #check x + -y + 0 end
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@@ -0,0 +1,285 @@import Mathlib.Data.Int.Basic import Mathlib.Algebra.EuclideanDomain.Basic import Mathlib.RingTheory.PrincipalIdealDomain import Mathlib.Tactic @[ext] structure gaussInt where re : ℤ im : ℤ namespace gaussInt instance : Zero gaussInt := ⟨⟨0, 0⟩⟩ instance : One gaussInt := ⟨⟨1, 0⟩⟩ instance : Add gaussInt := ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ instance : Neg gaussInt := ⟨fun x => ⟨-x.re, -x.im⟩⟩ instance : Mul gaussInt := ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := rfl theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := rfl theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := rfl theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := rfl theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := rfl @[simp] theorem zero_re : (0 : gaussInt).re = 0 := rfl @[simp] theorem zero_im : (0 : gaussInt).im = 0 := rfl @[simp] theorem one_re : (1 : gaussInt).re = 1 := rfl @[simp] theorem one_im : (1 : gaussInt).im = 0 := rfl @[simp] theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := rfl @[simp] theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := rfl @[simp] theorem neg_re (x : gaussInt) : (-x).re = -x.re := rfl @[simp] theorem neg_im (x : gaussInt) : (-x).im = -x.im := rfl @[simp] theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := rfl @[simp] theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := rfl instance instCommRing : CommRing gaussInt where zero := 0 one := 1 add := (· + ·) neg x := -x mul := (· * ·) add_assoc := by intros ext <;> simp <;> ring zero_add := by intros ext <;> simp add_zero := by intros ext <;> simp add_left_neg := by intros ext <;> simp add_comm := by intros ext <;> simp <;> ring mul_assoc := by intros ext <;> simp <;> ring one_mul := by intros ext <;> simp mul_one := by intros ext <;> simp left_distrib := by intros ext <;> simp <;> ring right_distrib := by intros ext <;> simp <;> ring mul_comm := by intros ext <;> simp <;> ring zero_mul := sorry mul_zero := sorry instance : Nontrivial gaussInt := by use 0, 1 rw [Ne, gaussInt.ext_iff] simp end gaussInt namespace Int def div' (a b : ℤ) := (a + b / 2) / b def mod' (a b : ℤ) := (a + b / 2) % b - b / 2 theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by rw [div', mod'] linarith [Int.ediv_add_emod (a + b / 2) b] theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by rw [mod', abs_le] constructor · linarith [Int.emod_nonneg (a + b / 2) h.ne'] have := Int.emod_lt_of_pos (a + b / 2) h have := Int.ediv_add_emod b 2 have := Int.emod_lt_of_pos b zero_lt_two revert this; intro this -- FIXME, this should not be needed linarith theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] end Int private theorem aux {α : Type _} [LinearOrderedRing α] {x y : α} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := haveI h' : x ^ 2 = 0 := by apply le_antisymm _ (sq_nonneg x) rw [← h] apply le_add_of_nonneg_right (sq_nonneg y) pow_eq_zero h' theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by constructor · intro h constructor · exact aux h rw [add_comm] at h exact aux h rintro ⟨rfl, rfl⟩ norm_num namespace gaussInt def norm (x : gaussInt) := x.re ^ 2 + x.im ^ 2 @[simp] theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by apply add_nonneg <;> apply sq_nonneg theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by rw [norm, sq_add_sq_eq_zero, gaussInt.ext_iff] rfl theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by rw [lt_iff_le_and_ne, ne_comm, Ne, norm_eq_zero] simp [norm_nonneg] theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by simp [norm] ring def conj (x : gaussInt) : gaussInt := ⟨x.re, -x.im⟩ @[simp] theorem conj_re (x : gaussInt) : (conj x).re = x.re := rfl @[simp] theorem conj_im (x : gaussInt) : (conj x).im = -x.im := rfl theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] instance : Div gaussInt := ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ instance : Mod gaussInt := ⟨fun x y => x - y * (x / y)⟩ theorem div_def (x y : gaussInt) : x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := rfl theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := rfl theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by have norm_y_pos : 0 < norm y := by rwa [norm_pos] have : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] ext <;> simp <;> ring have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by conv => lhs rw [← norm_conj y, ← norm_mul, this, norm] simp trans 2 * (y.norm / 2) ^ 2 · rw [two_mul] apply add_le_add <;> · rw [sq_le_sq] apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) apply le_abs_self rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] apply mul_le_mul_of_nonneg_left · apply Int.ediv_mul_le norm_num apply Int.ediv_nonneg (norm_nonneg y) norm_num have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos apply lt_of_le_of_lt this apply Int.ediv_lt_of_lt_mul · norm_num linarith theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := Int.natAbs_of_nonneg (norm_nonneg _) theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by apply Int.ofNat_lt.1 simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] apply norm_mod_lt x hy theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : ¬(norm (x * y)).natAbs < (norm x).natAbs := by apply not_lt_of_ge rw [norm_mul, Int.natAbs_mul] apply le_mul_of_one_le_right (Nat.zero_le _) apply Int.ofNat_le.1 rw [coe_natAbs_norm] exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) instance : EuclideanDomain gaussInt := { gaussInt.instCommRing with quotient := (· / ·) remainder := (· % ·) quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] quotient_zero := fun x => by simp [div_def, norm, Int.div'] rfl r := Measure (Int.natAbs ∘ norm) r_wellFounded := (measure (Int.natAbs ∘ norm)).2 remainder_lt := natAbs_norm_mod_lt mul_left_not_lt := not_norm_mul_left_lt_norm } example (x : gaussInt) : Irreducible x ↔ Prime x := PrincipalIdealRing.irreducible_iff_prime end gaussInt
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@@ -0,0 +1,110 @@import Mathlib.Topology.Instances.Real open Set Filter Topology def principal {α : Type _} (s : Set α) : Filter α where sets := { t | s ⊆ t } univ_sets := sorry sets_of_superset := sorry inter_sets := sorry example : Filter ℕ := { sets := { s | ∃ a, ∀ b, a ≤ b → b ∈ s } univ_sets := sorry sets_of_superset := sorry inter_sets := sorry } def Tendsto₁ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := ∀ V ∈ G, f ⁻¹' V ∈ F def Tendsto₂ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := map f F ≤ G example {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) : Tendsto₂ f F G ↔ Tendsto₁ f F G := Iff.rfl #check (@Filter.map_mono : ∀ {α β} {m : α → β}, Monotone (map m)) #check (@Filter.map_map : ∀ {α β γ} {f : Filter α} {m : α → β} {m' : β → γ}, map m' (map m f) = map (m' ∘ m) f) example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := sorry variable (f : ℝ → ℝ) (x₀ y₀ : ℝ) #check comap ((↑) : ℚ → ℝ) (𝓝 x₀) #check Tendsto (f ∘ (↑)) (comap ((↑) : ℚ → ℝ) (𝓝 x₀)) (𝓝 y₀) section variable {α β γ : Type _} (F : Filter α) {m : γ → β} {n : β → α} #check (comap_comap : comap m (comap n F) = comap (n ∘ m) F) end example : 𝓝 (x₀, y₀) = 𝓝 x₀ ×ᶠ 𝓝 y₀ := nhds_prod_eq #check le_inf_iff example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : Tendsto f atTop (𝓝 (x₀, y₀)) ↔ Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := sorry example (x₀ : ℝ) : HasBasis (𝓝 x₀) (fun ε : ℝ => 0 < ε) fun ε => Ioo (x₀ - ε) (x₀ + ε) := nhds_basis_Ioo_pos x₀ example (u : ℕ → ℝ) (x₀ : ℝ) : Tendsto u atTop (𝓝 x₀) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, u n ∈ Ioo (x₀ - ε) (x₀ + ε) := by have : atTop.HasBasis (fun _ : ℕ => True) Ici := atTop_basis rw [this.tendsto_iff (nhds_basis_Ioo_pos x₀)] simp example (P Q : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) : ∀ᶠ n in atTop, P n ∧ Q n := hP.and hQ example (u v : ℕ → ℝ) (h : ∀ᶠ n in atTop, u n = v n) (x₀ : ℝ) : Tendsto u atTop (𝓝 x₀) ↔ Tendsto v atTop (𝓝 x₀) := tendsto_congr' h example (u v : ℕ → ℝ) (h : u =ᶠ[atTop] v) (x₀ : ℝ) : Tendsto u atTop (𝓝 x₀) ↔ Tendsto v atTop (𝓝 x₀) := tendsto_congr' h #check @eventually_of_forall #check @Eventually.mono #check @Eventually.and example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) (hR : ∀ᶠ n in atTop, P n ∧ Q n → R n) : ∀ᶠ n in atTop, R n := by apply (hP.and (hQ.and hR)).mono rintro n ⟨h, h', h''⟩ exact h'' ⟨h, h'⟩ example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in atTop, Q n) (hR : ∀ᶠ n in atTop, P n ∧ Q n → R n) : ∀ᶠ n in atTop, R n := by filter_upwards [hP, hQ, hR] intro n h h' h'' exact h'' ⟨h, h'⟩ #check mem_closure_iff_clusterPt #check le_principal_iff #check neBot_of_le example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x)) (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M := sorry
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@@ -0,0 +1,200 @@import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus open Set Filter open Topology Filter variable {X : Type _} [MetricSpace X] (a b c : X) #check (dist a b : ℝ) #check (dist_nonneg : 0 ≤ dist a b) #check (dist_eq_zero : dist a b = 0 ↔ a = b) #check (dist_comm a b : dist a b = dist b a) #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) -- Note the next three lines are not quoted, their purpose is to make sure those things don't get renamed while we're looking elsewhere. #check EMetricSpace #check PseudoMetricSpace #check PseudoEMetricSpace example {u : ℕ → X} {a : X} : Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := Metric.tendsto_atTop example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} : Continuous f ↔ ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := Metric.continuous_iff example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := by continuity example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := continuous_dist.comp ((hf.comp continuous_fst).prod_mk (hf.comp continuous_snd)) example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := by apply Continuous.dist exact hf.comp continuous_fst exact hf.comp continuous_snd example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := (hf.comp continuous_fst).dist (hf.comp continuous_snd) example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := hf.fst'.dist hf.snd' example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := sorry example {X Y : Type _} [MetricSpace X] [MetricSpace Y] (f : X → Y) (a : X) : ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x}, dist x a < δ → dist (f x) (f a) < ε := Metric.continuousAt_iff variable (r : ℝ) example : Metric.ball a r = { b | dist b a < r } := rfl example : Metric.closedBall a r = { b | dist b a ≤ r } := rfl example (hr : 0 < r) : a ∈ Metric.ball a r := Metric.mem_ball_self hr example (hr : 0 ≤ r) : a ∈ Metric.closedBall a r := Metric.mem_closedBall_self hr example (s : Set X) : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s := Metric.isOpen_iff example {s : Set X} : IsClosed s ↔ IsOpen (sᶜ) := isOpen_compl_iff.symm example {s : Set X} (hs : IsClosed s) {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) (hus : ∀ n, u n ∈ s) : a ∈ s := hs.mem_of_tendsto hu (eventually_of_forall hus) example {s : Set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ Metric.ball b ε := Metric.mem_closure_iff example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := sorry example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.ball x ε ⊆ s := Metric.nhds_basis_ball.mem_iff example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.closedBall x ε ⊆ s := Metric.nhds_basis_closedBall.mem_iff example : IsCompact (Set.Icc 0 1 : Set ℝ) := isCompact_Icc example {s : Set X} (hs : IsCompact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := hs.tendsto_subseq hu example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := hs.exists_forall_le hs' hfs example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := hs.exists_forall_ge hs' hfs example {s : Set X} (hs : IsCompact s) : IsClosed s := hs.isClosed example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := isCompact_univ #check IsCompact.isClosed example {X : Type _} [MetricSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} : UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := Metric.uniformContinuous_iff example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} (hf : Continuous f) : UniformContinuous f := sorry example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := Metric.cauchySeq_iff example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := Metric.cauchySeq_iff' example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : ∃ x, Tendsto u atTop (𝓝 x) := cauchySeq_tendsto_of_complete hu open BigOperators open Finset theorem cauchySeq_of_le_geometric_two' {u : ℕ → X} (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by rw [Metric.cauchySeq_iff'] intro ε ε_pos obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry use N intro n hn obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn calc dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := sorry _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2) ^ i := sorry _ ≤ 1 / 2 ^ N * 2 := sorry _ < ε := sorry open Metric example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : Dense (⋂ n, f n) := by let B : ℕ → ℝ := fun n => (1 / 2) ^ n have Bpos : ∀ n, 0 < B n sorry /- Translate the density assumption into two functions `center` and `radius` associating to any n, x, δ, δpos a center and a positive radius such that `closedBall center radius` is included both in `f n` and in `closedBall x δ`. We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ have : ∀ (n : ℕ) (x : X), ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by sorry choose! center radius Hpos HB Hball using this intro x rw [mem_closure_iff_nhds_basis nhds_basis_closedBall] intro ε εpos /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs to all the `f n`. -/ let F : ℕ → X × ℝ := fun n => Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) let c : ℕ → X := fun n => (F n).1 let r : ℕ → ℝ := fun n => (F n).2 have rpos : ∀ n, 0 < r n := by sorry have rB : ∀ n, r n ≤ B n := by sorry have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by sorry have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ -- this point `y` will be the desired point. We will check that it belongs to all -- `f n` and to `ball x ε`. use y have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry sorry
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@@ -0,0 +1,151 @@import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus open Set Filter open Topology Filter section variable {X : Type _} [TopologicalSpace X] example : IsOpen (univ : Set X) := isOpen_univ example : IsOpen (∅ : Set X) := isOpen_empty example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋃ i, s i) := isOpen_iUnion hs example {ι : Type _} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋂ i, s i) := isOpen_iInter hs variable {Y : Type _} [TopologicalSpace Y] example {f : X → Y} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) := continuous_def example {f : X → Y} {x : X} : ContinuousAt f x ↔ map f (𝓝 x) ≤ 𝓝 (f x) := Iff.rfl example {f : X → Y} {x : X} : ContinuousAt f x ↔ ∀ U ∈ 𝓝 (f x), ∀ᶠ x in 𝓝 x, f x ∈ U := Iff.rfl example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := mem_nhds_iff example (x : X) : pure x ≤ 𝓝 x := pure_le_nhds x example (x : X) (P : X → Prop) (h : ∀ᶠ y in 𝓝 x, P y) : P x := pure_le_nhds x h example {P : X → Prop} {x : X} (h : ∀ᶠ y in 𝓝 x, P y) : ∀ᶠ y in 𝓝 x, ∀ᶠ z in 𝓝 y, P z := eventually_eventually_nhds.mpr h #check TopologicalSpace.mkOfNhds #check TopologicalSpace.nhds_mkOfNhds example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := sorry end -- BOTH. variable {X Y : Type _} example (f : X → Y) : TopologicalSpace X → TopologicalSpace Y := TopologicalSpace.coinduced f example (f : X → Y) : TopologicalSpace Y → TopologicalSpace X := TopologicalSpace.induced f example (f : X → Y) (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) : TopologicalSpace.coinduced f T_X ≤ T_Y ↔ T_X ≤ TopologicalSpace.induced f T_Y := coinduced_le_iff_le_induced #check coinduced_compose #check induced_compose example {T T' : TopologicalSpace X} : T ≤ T' ↔ ∀ s, T'.IsOpen s → T.IsOpen s := Iff.rfl example (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) (f : X → Y) : Continuous f ↔ TopologicalSpace.coinduced f T_X ≤ T_Y := continuous_iff_coinduced_le example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) (g : Y → Z) : @Continuous Y Z (TopologicalSpace.coinduced f T_X) T_Z g ↔ @Continuous X Z T_X T_Z (g ∘ f) := by rw [continuous_iff_coinduced_le, coinduced_compose, continuous_iff_coinduced_le] example (ι : Type _) (X : ι → Type _) (T_X : ∀ i, TopologicalSpace <| X i) : (Pi.topologicalSpace : TopologicalSpace (∀ i, X i)) = ⨅ i, TopologicalSpace.induced (fun x => x i) (T_X i) := rfl example [TopologicalSpace X] [T2Space X] {u : ℕ → X} {a b : X} (ha : Tendsto u atTop (𝓝 a)) (hb : Tendsto u atTop (𝓝 b)) : a = b := tendsto_nhds_unique ha hb example [TopologicalSpace X] [RegularSpace X] (a : X) : (𝓝 a).HasBasis (fun s : Set X => s ∈ 𝓝 a ∧ IsClosed s) id := closed_nhds_basis a example [TopologicalSpace X] {x : X} : (𝓝 x).HasBasis (fun t : Set X => t ∈ 𝓝 x ∧ IsOpen t) id := nhds_basis_opens' x theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y} (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := sorry example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] {A : Set X} (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f) (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) <| 𝓝 x) <| 𝓝 c) : ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := sorry #check @HasBasis.tendsto_right_iff example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] {s : Set X} {a : X} : a ∈ closure s ↔ ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ Tendsto u atTop (𝓝 a) := mem_closure_iff_seq_limit variable [TopologicalSpace X] example {F : Filter X} {x : X} : ClusterPt x F ↔ NeBot (𝓝 x ⊓ F) := Iff.rfl example {s : Set X} : IsCompact s ↔ ∀ (F : Filter X) [NeBot F], F ≤ 𝓟 s → ∃ a ∈ s, ClusterPt a F := Iff.rfl example [TopologicalSpace.FirstCountableTopology X] {s : Set X} {u : ℕ → X} (hs : IsCompact s) (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := hs.tendsto_subseq hu variable [TopologicalSpace Y] example {x : X} {F : Filter X} {G : Filter Y} (H : ClusterPt x F) {f : X → Y} (hfx : ContinuousAt f x) (hf : Tendsto f F G) : ClusterPt (f x) G := ClusterPt.map H hfx hf example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : IsCompact (f '' s) := by intro F F_ne F_le have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by sorry have Hne : (𝓟 s ⊓ comap f F).NeBot := by sorry have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left sorry example {ι : Type _} {s : Set X} (hs : IsCompact s) (U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) (hsU : s ⊆ ⋃ i, U i) : ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i := hs.elim_finite_subcover U hUo hsU example [CompactSpace X] : IsCompact (univ : Set X) := isCompact_univ
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@@ -0,0 +1,71 @@import Mathlib.Topology.Instances.Real open Set Filter Topology -- In the next example we could use `tauto` in each proof instead of knowing the lemmas example {α : Type _} (s : Set α) : Filter α := { sets := { t | s ⊆ t } univ_sets := subset_univ s sets_of_superset := fun hU hUV => Subset.trans hU hUV inter_sets := fun hU hV => subset_inter hU hV } example : Filter ℕ := { sets := { s | ∃ a, ∀ b, a ≤ b → b ∈ s } univ_sets := by use 42 simp sets_of_superset := by rintro U V ⟨N, hN⟩ hUV use N tauto inter_sets := by rintro U V ⟨N, hN⟩ ⟨N', hN'⟩ use max N N' intro b hb rw [max_le_iff] at hb constructor <;> tauto } def Tendsto₁ {X Y : Type _} (f : X → Y) (F : Filter X) (G : Filter Y) := ∀ V ∈ G, f ⁻¹' V ∈ F example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := calc map (g ∘ f) F = map g (map f F) := by rw [map_map] _ ≤ map g G := (map_mono hf) _ ≤ H := hg example {X Y Z : Type _} {F : Filter X} {G : Filter Y} {H : Filter Z} {f : X → Y} {g : Y → Z} (hf : Tendsto₁ f F G) (hg : Tendsto₁ g G H) : Tendsto₁ (g ∘ f) F H := by intro V hV rw [preimage_comp] apply hf apply hg exact hV example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : Tendsto f atTop (𝓝 (x₀, y₀)) ↔ Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := calc Tendsto f atTop (𝓝 (x₀, y₀)) ↔ map f atTop ≤ 𝓝 (x₀, y₀) := Iff.rfl _ ↔ map f atTop ≤ 𝓝 x₀ ×ᶠ 𝓝 y₀ := by rw [nhds_prod_eq] _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ⊓ comap Prod.snd (𝓝 y₀) := Iff.rfl _ ↔ map f atTop ≤ comap Prod.fst (𝓝 x₀) ∧ map f atTop ≤ comap Prod.snd (𝓝 y₀) := le_inf_iff _ ↔ map Prod.fst (map f atTop) ≤ 𝓝 x₀ ∧ map Prod.snd (map f atTop) ≤ 𝓝 y₀ := by rw [← map_le_iff_le_comap, ← map_le_iff_le_comap] _ ↔ map (Prod.fst ∘ f) atTop ≤ 𝓝 x₀ ∧ map (Prod.snd ∘ f) atTop ≤ 𝓝 y₀ := by rw [map_map, map_map] -- an alternative solution example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) : Tendsto f atTop (𝓝 (x₀, y₀)) ↔ Tendsto (Prod.fst ∘ f) atTop (𝓝 x₀) ∧ Tendsto (Prod.snd ∘ f) atTop (𝓝 y₀) := by rw [nhds_prod_eq] unfold Tendsto Filter.prod rw [le_inf_iff, ← map_le_iff_le_comap, map_map, ← map_le_iff_le_comap, map_map] example (u : ℕ → ℝ) (M : Set ℝ) (x : ℝ) (hux : Tendsto u atTop (𝓝 x)) (huM : ∀ᶠ n in atTop, u n ∈ M) : x ∈ closure M := mem_closure_iff_clusterPt.mpr (neBot_of_le <| le_inf hux <| le_principal_iff.mpr huM)
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@@ -0,0 +1,365 @@import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus open Set Filter open Topology Filter variable {X : Type _} [MetricSpace X] (a b c : X) #check (dist a b : ℝ) #check (dist_nonneg : 0 ≤ dist a b) #check (dist_eq_zero : dist a b = 0 ↔ a = b) #check (dist_comm a b : dist a b = dist b a) #check (dist_triangle a b c : dist a c ≤ dist a b + dist b c) -- Note the next three lines are not quoted, their purpose is to make sure those things don't get renamed while we're looking elsewhere. #check EMetricSpace #check PseudoMetricSpace #check PseudoEMetricSpace example {u : ℕ → X} {a : X} : Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε := Metric.tendsto_atTop example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} : Continuous f ↔ ∀ x : X, ∀ ε > 0, ∃ δ > 0, ∀ x', dist x' x < δ → dist (f x') (f x) < ε := Metric.continuous_iff example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := by continuity example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := continuous_dist.comp ((hf.comp continuous_fst).prod_mk (hf.comp continuous_snd)) example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := by apply Continuous.dist exact hf.comp continuous_fst exact hf.comp continuous_snd example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := (hf.comp continuous_fst).dist (hf.comp continuous_snd) example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} (hf : Continuous f) : Continuous fun p : X × X => dist (f p.1) (f p.2) := hf.fst'.dist hf.snd' example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := sorry example {f : ℝ → X} (hf : Continuous f) : Continuous fun x : ℝ => f (x ^ 2 + x) := hf.comp <| (continuous_pow 2).add continuous_id example {X Y : Type _} [MetricSpace X] [MetricSpace Y] (f : X → Y) (a : X) : ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ {x}, dist x a < δ → dist (f x) (f a) < ε := Metric.continuousAt_iff variable (r : ℝ) example : Metric.ball a r = { b | dist b a < r } := rfl example : Metric.closedBall a r = { b | dist b a ≤ r } := rfl example (hr : 0 < r) : a ∈ Metric.ball a r := Metric.mem_ball_self hr example (hr : 0 ≤ r) : a ∈ Metric.closedBall a r := Metric.mem_closedBall_self hr example (s : Set X) : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, Metric.ball x ε ⊆ s := Metric.isOpen_iff example {s : Set X} : IsClosed s ↔ IsOpen (sᶜ) := isOpen_compl_iff.symm example {s : Set X} (hs : IsClosed s) {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) (hus : ∀ n, u n ∈ s) : a ∈ s := hs.mem_of_tendsto hu (eventually_of_forall hus) example {s : Set X} : a ∈ closure s ↔ ∀ ε > 0, ∃ b ∈ s, a ∈ Metric.ball b ε := Metric.mem_closure_iff example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := sorry example {u : ℕ → X} (hu : Tendsto u atTop (𝓝 a)) {s : Set X} (hs : ∀ n, u n ∈ s) : a ∈ closure s := by rw [Metric.tendsto_atTop] at hu rw [Metric.mem_closure_iff] intro ε ε_pos rcases hu ε ε_pos with ⟨N, hN⟩ refine' ⟨u N, hs _, _⟩ rw [dist_comm] exact hN N le_rfl example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.ball x ε ⊆ s := Metric.nhds_basis_ball.mem_iff example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ ε > 0, Metric.closedBall x ε ⊆ s := Metric.nhds_basis_closedBall.mem_iff example : IsCompact (Set.Icc 0 1 : Set ℝ) := isCompact_Icc example {s : Set X} (hs : IsCompact s) {u : ℕ → X} (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := hs.tendsto_subseq hu example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : ∃ x ∈ s, ∀ y ∈ s, f x ≤ f y := hs.exists_forall_le hs' hfs example {s : Set X} (hs : IsCompact s) (hs' : s.Nonempty) {f : X → ℝ} (hfs : ContinuousOn f s) : ∃ x ∈ s, ∀ y ∈ s, f y ≤ f x := hs.exists_forall_ge hs' hfs example {s : Set X} (hs : IsCompact s) : IsClosed s := hs.isClosed example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := isCompact_univ #check IsCompact.isClosed example {X : Type _} [MetricSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} : UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ {a b : X}, dist a b < δ → dist (f a) (f b) < ε := Metric.uniformContinuous_iff example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} (hf : Continuous f) : UniformContinuous f := sorry example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} (hf : Continuous f) : UniformContinuous f := by rw [Metric.uniformContinuous_iff] intro ε ε_pos let φ : X × X → ℝ := fun p => dist (f p.1) (f p.2) have φ_cont : Continuous φ := hf.fst'.dist hf.snd' let K := { p : X × X | ε ≤ φ p } have K_closed : IsClosed K := isClosed_le continuous_const φ_cont have K_cpct : IsCompact K := K_closed.isCompact cases' eq_empty_or_nonempty K with hK hK · use 1, by norm_num intro x y _ have : (x, y) ∉ K := by simp [hK] simpa using this · rcases K_cpct.exists_forall_le hK continuous_dist.continuousOn with ⟨⟨x₀, x₁⟩, xx_in, H⟩ use dist x₀ x₁ constructor · change _ < _ rw [dist_pos] intro h have : ε ≤ 0 := by simpa [*] using xx_in linarith · intro x x' contrapose! intro hxx' exact H (x, x') hxx' example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε := Metric.cauchySeq_iff example (u : ℕ → X) : CauchySeq u ↔ ∀ ε > 0, ∃ N : ℕ, ∀ n ≥ N, dist (u n) (u N) < ε := Metric.cauchySeq_iff' example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : ∃ x, Tendsto u atTop (𝓝 x) := cauchySeq_tendsto_of_complete hu open BigOperators open Finset theorem cauchySeq_of_le_geometric_two' {u : ℕ → X} (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by rw [Metric.cauchySeq_iff'] intro ε ε_pos obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by sorry use N intro n hn obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn calc dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := sorry _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := sorry _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := sorry _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2) ^ i := sorry _ ≤ 1 / 2 ^ N * 2 := sorry _ < ε := sorry example {u : ℕ → X} (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^ n) : CauchySeq u := by rw [Metric.cauchySeq_iff'] intro ε ε_pos obtain ⟨N, hN⟩ : ∃ N : ℕ, 1 / 2 ^ N * 2 < ε := by have : Tendsto (fun N : ℕ => (1 / 2 ^ N * 2 : ℝ)) atTop (𝓝 0) := by rw [← MulZeroClass.zero_mul (2 : ℝ)] apply Tendsto.mul simp_rw [← one_div_pow (2 : ℝ)] apply tendsto_pow_atTop_nhds_0_of_lt_1 <;> linarith exact tendsto_const_nhds rcases(atTop_basis.tendsto_iff (nhds_basis_Ioo_pos (0 : ℝ))).mp this ε ε_pos with ⟨N, _, hN⟩ exact ⟨N, by simpa using (hN N left_mem_Ici).2⟩ use N intro n hn obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn calc dist (u (N + k)) (u N) = dist (u (N + 0)) (u (N + k)) := by rw [dist_comm, add_zero] _ ≤ ∑ i in range k, dist (u (N + i)) (u (N + (i + 1))) := (dist_le_range_sum_dist (fun i => u (N + i)) k) _ ≤ ∑ i in range k, (1 / 2 : ℝ) ^ (N + i) := (sum_le_sum fun i _ => hu <| N + i) _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2 : ℝ) ^ i := by simp_rw [← one_div_pow, pow_add, ← mul_sum] _ ≤ 1 / 2 ^ N * 2 := (mul_le_mul_of_nonneg_left (sum_geometric_two_le _) (one_div_nonneg.mpr (pow_nonneg (zero_le_two : (0 : ℝ) ≤ 2) _))) _ < ε := hN open Metric example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : Dense (⋂ n, f n) := by let B : ℕ → ℝ := fun n => (1 / 2) ^ n have Bpos : ∀ n, 0 < B n sorry /- Translate the density assumption into two functions `center` and `radius` associating to any n, x, δ, δpos a center and a positive radius such that `closedBall center radius` is included both in `f n` and in `closedBall x δ`. We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ have : ∀ (n : ℕ) (x : X), ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by sorry choose! center radius Hpos HB Hball using this intro x rw [mem_closure_iff_nhds_basis nhds_basis_closedBall] intro ε εpos /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs to all the `f n`. -/ let F : ℕ → X × ℝ := fun n => Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) let c : ℕ → X := fun n => (F n).1 let r : ℕ → ℝ := fun n => (F n).2 have rpos : ∀ n, 0 < r n := by sorry have rB : ∀ n, r n ≤ B n := by sorry have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := by sorry have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by sorry have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ -- this point `y` will be the desired point. We will check that it belongs to all -- `f n` and to `ball x ε`. use y have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry sorry example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) : Dense (⋂ n, f n) := by let B : ℕ → ℝ := fun n => (1 / 2) ^ n have Bpos : ∀ n, 0 < B n := fun n => pow_pos sorry n /- Translate the density assumption into two functions `center` and `radius` associating to any n, x, δ, δpos a center and a positive radius such that `closedBall center radius` is included both in `f n` and in `closedBall x δ`. We can also require `radius ≤ (1/2)^(n+1)`, to ensure we get a Cauchy sequence later. -/ have : ∀ (n : ℕ) (x : X), ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by intro n x δ δpos have : x ∈ closure (f n) := hd n x rcases Metric.mem_closure_iff.1 this (δ / 2) (half_pos δpos) with ⟨y, ys, xy⟩ rw [dist_comm] at xy obtain ⟨r, rpos, hr⟩ : ∃ r > 0, closedBall y r ⊆ f n := nhds_basis_closedBall.mem_iff.1 (isOpen_iff_mem_nhds.1 (ho n) y ys) refine' ⟨y, min (min (δ / 2) r) (B (n + 1)), _, _, fun z hz => ⟨_, _⟩⟩ show 0 < min (min (δ / 2) r) (B (n + 1)) exact lt_min (lt_min (half_pos δpos) rpos) (Bpos (n + 1)) show min (min (δ / 2) r) (B (n + 1)) ≤ B (n + 1) exact min_le_right _ _ show z ∈ closedBall x δ exact calc dist z x ≤ dist z y + dist y x := dist_triangle _ _ _ _ ≤ min (min (δ / 2) r) (B (n + 1)) + δ / 2 := (add_le_add hz xy.le) _ ≤ δ / 2 + δ / 2 := (add_le_add_right ((min_le_left _ _).trans (min_le_left _ _)) _) _ = δ := add_halves δ show z ∈ f n exact hr (calc dist z y ≤ min (min (δ / 2) r) (B (n + 1)) := hz _ ≤ r := (min_le_left _ _).trans (min_le_right _ _) ) choose! center radius Hpos HB Hball using this refine' fun x => (mem_closure_iff_nhds_basis nhds_basis_closedBall).2 fun ε εpos => _ /- `ε` is positive. We have to find a point in the ball of radius `ε` around `x` belonging to all `f n`. For this, we construct inductively a sequence `F n = (c n, r n)` such that the closed ball `closedBall (c n) (r n)` is included in the previous ball and in `f n`, and such that `r n` is small enough to ensure that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs to all the `f n`. -/ let F : ℕ → X × ℝ := fun n => Nat.recOn n (Prod.mk x (min ε (B 0))) fun n p => Prod.mk (center n p.1 p.2) (radius n p.1 p.2) let c : ℕ → X := fun n => (F n).1 let r : ℕ → ℝ := fun n => (F n).2 have rpos : ∀ n, 0 < r n := by intro n induction' n with n hn exact lt_min εpos (Bpos 0) exact Hpos n (c n) (r n) hn have rB : ∀ n, r n ≤ B n := by intro n induction' n with n hn exact min_le_right _ _ exact HB n (c n) (r n) (rpos n) have incl : ∀ n, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := fun n => Hball n (c n) (r n) (rpos n) have cdist : ∀ n, dist (c n) (c (n + 1)) ≤ B n := by intro n rw [dist_comm] have A : c (n + 1) ∈ closedBall (c (n + 1)) (r (n + 1)) := mem_closedBall_self (rpos <| n + 1).le have I := calc closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) := (incl n).trans (inter_subset_left _ _) _ ⊆ closedBall (c n) (B n) := closedBall_subset_closedBall (rB n) exact I A have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist -- as the sequence `c n` is Cauchy in a complete space, it converges to a limit `y`. rcases cauchySeq_tendsto_of_complete this with ⟨y, ylim⟩ -- this point `y` will be the desired point. We will check that it belongs to all -- `f n` and to `ball x ε`. use y have I : ∀ n, ∀ m ≥ n, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by intro n refine' Nat.le_induction _ fun m hnm h => _ · exact Subset.rfl · exact (incl m).trans ((Set.inter_subset_left _ _).trans h) have yball : ∀ n, y ∈ closedBall (c n) (r n) := by intro n refine' isClosed_ball.mem_of_tendsto ylim _ refine' (Filter.eventually_ge_atTop n).mono fun m hm => _ exact I n m hm (mem_closedBall_self (rpos _).le) constructor · suffices ∀ n, y ∈ f n by rwa [Set.mem_iInter] intro n have : closedBall (c (n + 1)) (r (n + 1)) ⊆ f n := Subset.trans (incl n) (inter_subset_right _ _) exact this (yball (n + 1)) calc dist y x ≤ r 0 := yball 0 _ ≤ ε := min_le_left _ _
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@@ -0,0 +1,202 @@import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus open Set Filter open Topology Filter section variable {X : Type _} [TopologicalSpace X] example : IsOpen (univ : Set X) := isOpen_univ example : IsOpen (∅ : Set X) := isOpen_empty example {ι : Type _} {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋃ i, s i) := isOpen_iUnion hs example {ι : Type _} [Fintype ι] {s : ι → Set X} (hs : ∀ i, IsOpen <| s i) : IsOpen (⋂ i, s i) := isOpen_iInter hs variable {Y : Type _} [TopologicalSpace Y] example {f : X → Y} : Continuous f ↔ ∀ s, IsOpen s → IsOpen (f ⁻¹' s) := continuous_def example {f : X → Y} {x : X} : ContinuousAt f x ↔ map f (𝓝 x) ≤ 𝓝 (f x) := Iff.rfl example {f : X → Y} {x : X} : ContinuousAt f x ↔ ∀ U ∈ 𝓝 (f x), ∀ᶠ x in 𝓝 x, f x ∈ U := Iff.rfl example {x : X} {s : Set X} : s ∈ 𝓝 x ↔ ∃ t, t ⊆ s ∧ IsOpen t ∧ x ∈ t := mem_nhds_iff example (x : X) : pure x ≤ 𝓝 x := pure_le_nhds x example (x : X) (P : X → Prop) (h : ∀ᶠ y in 𝓝 x, P y) : P x := pure_le_nhds x h example {P : X → Prop} {x : X} (h : ∀ᶠ y in 𝓝 x, P y) : ∀ᶠ y in 𝓝 x, ∀ᶠ z in 𝓝 y, P z := eventually_eventually_nhds.mpr h #check TopologicalSpace.mkOfNhds #check TopologicalSpace.nhds_mkOfNhds example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := sorry example {α : Type _} (n : α → Filter α) (H₀ : ∀ a, pure a ≤ n a) (H : ∀ a : α, ∀ p : α → Prop, (∀ᶠ x in n a, p x) → ∀ᶠ y in n a, ∀ᶠ x in n y, p x) : ∀ a, ∀ s ∈ n a, ∃ t ∈ n a, t ⊆ s ∧ ∀ a' ∈ t, s ∈ n a' := by intro a s s_in refine' ⟨{ y | s ∈ n y }, H a (fun x => x ∈ s) s_in, _, by tauto⟩ rintro y (hy : s ∈ n y) exact H₀ y hy end -- BOTH. variable {X Y : Type _} example (f : X → Y) : TopologicalSpace X → TopologicalSpace Y := TopologicalSpace.coinduced f example (f : X → Y) : TopologicalSpace Y → TopologicalSpace X := TopologicalSpace.induced f example (f : X → Y) (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) : TopologicalSpace.coinduced f T_X ≤ T_Y ↔ T_X ≤ TopologicalSpace.induced f T_Y := coinduced_le_iff_le_induced #check coinduced_compose #check induced_compose example {T T' : TopologicalSpace X} : T ≤ T' ↔ ∀ s, T'.IsOpen s → T.IsOpen s := Iff.rfl example (T_X : TopologicalSpace X) (T_Y : TopologicalSpace Y) (f : X → Y) : Continuous f ↔ TopologicalSpace.coinduced f T_X ≤ T_Y := continuous_iff_coinduced_le example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : TopologicalSpace Z) (g : Y → Z) : @Continuous Y Z (TopologicalSpace.coinduced f T_X) T_Z g ↔ @Continuous X Z T_X T_Z (g ∘ f) := by rw [continuous_iff_coinduced_le, coinduced_compose, continuous_iff_coinduced_le] example (ι : Type _) (X : ι → Type _) (T_X : ∀ i, TopologicalSpace <| X i) : (Pi.topologicalSpace : TopologicalSpace (∀ i, X i)) = ⨅ i, TopologicalSpace.induced (fun x => x i) (T_X i) := rfl example [TopologicalSpace X] [T2Space X] {u : ℕ → X} {a b : X} (ha : Tendsto u atTop (𝓝 a)) (hb : Tendsto u atTop (𝓝 b)) : a = b := tendsto_nhds_unique ha hb example [TopologicalSpace X] [RegularSpace X] (a : X) : (𝓝 a).HasBasis (fun s : Set X => s ∈ 𝓝 a ∧ IsClosed s) id := closed_nhds_basis a example [TopologicalSpace X] {x : X} : (𝓝 x).HasBasis (fun t : Set X => t ∈ 𝓝 x ∧ IsOpen t) id := nhds_basis_opens' x theorem aux {X Y A : Type _} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y} (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := sorry example {X Y A : Type _} [TopologicalSpace X] {c : A → X} {f : A → Y} {x : X} {F : Filter Y} (h : Tendsto f (comap c (𝓝 x)) F) {V' : Set Y} (V'_in : V' ∈ F) : ∃ V ∈ 𝓝 x, IsOpen V ∧ c ⁻¹' V ⊆ f ⁻¹' V' := by simpa [and_assoc] using ((nhds_basis_opens' x).comap c).tendsto_left_iff.mp h V' V'_in example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] {A : Set X} (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f) (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) <| 𝓝 x) <| 𝓝 c) : ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := sorry #check @HasBasis.tendsto_right_iff example [TopologicalSpace X] [TopologicalSpace Y] [T3Space Y] {A : Set X} (hA : ∀ x, x ∈ closure A) {f : A → Y} (f_cont : Continuous f) (hf : ∀ x : X, ∃ c : Y, Tendsto f (comap (↑) <| 𝓝 x) <| 𝓝 c) : ∃ φ : X → Y, Continuous φ ∧ ∀ a : A, φ a = f a := by choose φ hφ using hf use φ constructor · rw [continuous_iff_continuousAt] intro x suffices ∀ V' ∈ 𝓝 (φ x), IsClosed V' → φ ⁻¹' V' ∈ 𝓝 x by simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff] intro V' V'_in V'_closed obtain ⟨V, V_in, V_op, hV⟩ : ∃ V ∈ 𝓝 x, IsOpen V ∧ (↑) ⁻¹' V ⊆ f ⁻¹' V' := aux (hφ x) V'_in suffices : ∀ y ∈ V, φ y ∈ V' exact mem_of_superset V_in this intro y y_in have hVx : V ∈ 𝓝 y := V_op.mem_nhds y_in haveI : (comap ((↑) : A → X) (𝓝 y)).NeBot := by simpa [mem_closure_iff_comap_neBot] using hA y apply V'_closed.mem_of_tendsto (hφ y) exact mem_of_superset (preimage_mem_comap hVx) hV · intro a have lim : Tendsto f (𝓝 a) (𝓝 <| φ a) := by simpa [nhds_induced] using hφ a exact tendsto_nhds_unique lim f_cont.continuousAt example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] {s : Set X} {a : X} : a ∈ closure s ↔ ∃ u : ℕ → X, (∀ n, u n ∈ s) ∧ Tendsto u atTop (𝓝 a) := mem_closure_iff_seq_limit variable [TopologicalSpace X] example {F : Filter X} {x : X} : ClusterPt x F ↔ NeBot (𝓝 x ⊓ F) := Iff.rfl example {s : Set X} : IsCompact s ↔ ∀ (F : Filter X) [NeBot F], F ≤ 𝓟 s → ∃ a ∈ s, ClusterPt a F := Iff.rfl example [TopologicalSpace.FirstCountableTopology X] {s : Set X} {u : ℕ → X} (hs : IsCompact s) (hu : ∀ n, u n ∈ s) : ∃ a ∈ s, ∃ φ : ℕ → ℕ, StrictMono φ ∧ Tendsto (u ∘ φ) atTop (𝓝 a) := hs.tendsto_subseq hu variable [TopologicalSpace Y] example {x : X} {F : Filter X} {G : Filter Y} (H : ClusterPt x F) {f : X → Y} (hfx : ContinuousAt f x) (hf : Tendsto f F G) : ClusterPt (f x) G := ClusterPt.map H hfx hf example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : IsCompact (f '' s) := by intro F F_ne F_le have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by sorry have Hne : (𝓟 s ⊓ comap f F).NeBot := by sorry have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left sorry example [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) {s : Set X} (hs : IsCompact s) : IsCompact (f '' s) := by intro F F_ne F_le have map_eq : map f (𝓟 s ⊓ comap f F) = 𝓟 (f '' s) ⊓ F := by rw [Filter.push_pull, map_principal] have Hne : (𝓟 s ⊓ comap f F).NeBot := by apply NeBot.of_map rwa [map_eq, inf_of_le_right F_le] have Hle : 𝓟 s ⊓ comap f F ≤ 𝓟 s := inf_le_left rcases hs Hle with ⟨x, x_in, hx⟩ refine' ⟨f x, mem_image_of_mem f x_in, _⟩ apply hx.map hf.continuousAt rw [Tendsto, map_eq] exact inf_le_right example {ι : Type _} {s : Set X} (hs : IsCompact s) (U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) (hsU : s ⊆ ⋃ i, U i) : ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i := hs.elim_finite_subcover U hUo hsU example [CompactSpace X] : IsCompact (univ : Set X) := isCompact_univ
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@@ -30,15 +28,11 @@<nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > <a href="index.html" class="icon icon-home"> Mathematics in Lean <a href="index.html" class="icon icon-home"> Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="text" name="q" placeholder="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
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@@ -77,8 +71,8 @@<div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> <li class="breadcrumb-item active"><span class="section-number">1. </span>Introduction</li> <li><a href="index.html" class="icon icon-home"></a> »</li> <li><span class="section-number">1. </span>Introduction</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C01_Introduction.rst.txt" rel="nofollow"> View page source</a> </li>
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@@ -88,10 +82,10 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="introduction"> <span id="id1"></span><h1><span class="section-number">1. </span>Introduction<a class="headerlink" href="#introduction" title="Permalink to this heading"></a></h1> <section id="getting-started"> <h2><span class="section-number">1.1. </span>Getting Started<a class="headerlink" href="#getting-started" title="Permalink to this heading"></a></h2> <div class="section" id="introduction"> <span id="id1"></span><h1><span class="section-number">1. </span>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline"></a></h1> <div class="section" id="getting-started"> <h2><span class="section-number">1.1. </span>Getting Started<a class="headerlink" href="#getting-started" title="Permalink to this headline"></a></h2> <p>The goal of this book is to teach you to formalize mathematics using the Lean 4 interactive proof assistant. It assumes that you know some mathematics, but it does not require much.
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@@ -163,9 +157,9 @@ Lean is about <em>doing</em> mathematics interactively, not just reading about iWorking through the exercises is central to the experience. You can always compare your solutions to the ones in the <code class="docutils literal notranslate"><span class="pre">solutions</span></code> folder associated with each section.</p> </section> <section id="overview"> <h2><span class="section-number">1.2. </span>Overview<a class="headerlink" href="#overview" title="Permalink to this heading"></a></h2> </div> <div class="section" id="overview"> <h2><span class="section-number">1.2. </span>Overview<a class="headerlink" href="#overview" title="Permalink to this headline"></a></h2> <p>Put simply, Lean is a tool for building complex expressions in a formal language known as <em>dependent type theory</em>.</p> <p id="index-0">Every expression has a <em>type</em>, and you can use the <cite>#check</cite> command to
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@@ -333,8 +327,8 @@ Giovanni Mascellani, Hunter Monroe, Pietro Monticone, Bartosz Piotrowski,and Guilherme Silva. Our work has been partially supported by the Hoskinson Center for Formal Mathematics.</p> </section> </section> </div> </div> </div>
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@@ -31,15 +29,11 @@<nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > <a href="index.html" class="icon icon-home"> Mathematics in Lean <a href="index.html" class="icon icon-home"> Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="text" name="q" placeholder="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
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@@ -81,8 +75,8 @@<div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> <li class="breadcrumb-item active"><span class="section-number">2. </span>Basics</li> <li><a href="index.html" class="icon icon-home"></a> »</li> <li><span class="section-number">2. </span>Basics</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C02_Basics.rst.txt" rel="nofollow"> View page source</a> </li>
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@@ -92,14 +86,14 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="basics"> <span id="id1"></span><h1><span class="section-number">2. </span>Basics<a class="headerlink" href="#basics" title="Permalink to this heading"></a></h1> <div class="section" id="basics"> <span id="id1"></span><h1><span class="section-number">2. </span>Basics<a class="headerlink" href="#basics" title="Permalink to this headline"></a></h1> <p>This chapter is designed to introduce you to the nuts and bolts of mathematical reasoning in Lean: calculating, applying lemmas and theorems, and reasoning about generic structures.</p> <section id="calculating"> <h2><span class="section-number">2.1. </span>Calculating<a class="headerlink" href="#calculating" title="Permalink to this heading"></a></h2> <div class="section" id="calculating"> <h2><span class="section-number">2.1. </span>Calculating<a class="headerlink" href="#calculating" title="Permalink to this headline"></a></h2> <p>We generally learn to carry out mathematical calculations without thinking of them as proofs. But when we justify each step in a calculation,
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@@ -121,7 +115,10 @@ so the left-hand side of <code class="docutils literal notranslate"><span class=However, it is generally good style to be mindful of Lean’s notational conventions and leave out parentheses when Lean does as well.</p> <p>Let’s try out <code class="docutils literal notranslate"><span class="pre">rw</span></code>.</p> <div class="highlight-lean notranslate" id="index-1"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">b</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">c</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <div class="highlight-lean notranslate" id="index-1"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="n">Mathlib.Tactic</span> <span class="kn">import</span> <span class="n">Mathlib.Data.Real.Basic</span> <span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">*</span> <span class="n">c</span> <span class="bp">=</span> <span class="n">b</span> <span class="bp">*</span> <span class="o">(</span><span class="n">a</span> <span class="bp">*</span> <span class="n">c</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mul_comm</span> <span class="n">a</span> <span class="n">b</span><span class="o">]</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mul_assoc</span> <span class="n">b</span> <span class="n">a</span> <span class="n">c</span><span class="o">]</span> </pre></div>
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@@ -235,7 +232,7 @@ by listing the relevant identities separated by commas inside the square bracketa comma in any list of rewrites.</p> <p>Another trick is that we can declare variables once and for all outside an example or theorem. Lean then includes them automatically.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="n">d</span> <span class="n">e</span> <span class="n">f</span> <span class="n">g</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">c</span> <span class="n">d</span> <span class="n">e</span> <span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">d</span><span class="o">)</span> <span class="o">(</span><span class="n">h'</span> <span class="o">:</span> <span class="n">e</span> <span class="bp">=</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">*</span> <span class="o">(</span><span class="n">b</span> <span class="bp">*</span> <span class="n">e</span><span class="o">)</span> <span class="bp">=</span> <span class="n">c</span> <span class="bp">*</span> <span class="o">(</span><span class="n">d</span> <span class="bp">*</span> <span class="n">f</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">h'</span><span class="o">,</span> <span class="bp">←</span> <span class="n">mul_assoc</span><span class="o">,</span> <span class="n">h</span><span class="o">,</span> <span class="n">mul_assoc</span><span class="o">]</span>
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@@ -258,7 +255,6 @@ command to determine the type of an expression:</p><span class="k">#check</span> <span class="n">mul_assoc</span> <span class="n">c</span> <span class="n">a</span> <span class="n">b</span> <span class="k">#check</span> <span class="n">mul_comm</span> <span class="n">a</span> <span class="k">#check</span> <span class="n">mul_comm</span> <span class="k">#check</span> <span class="bp">@</span><span class="n">mul_comm</span> <span class="kd">end</span> </pre></div>
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@@ -390,9 +386,9 @@ occurrence of <code class="docutils literal notranslate"><span class="pre">a</sp<span class="n">rw</span> <span class="o">[</span><span class="n">add_mul</span><span class="o">]</span> </pre></div> </div> </section> <section id="proving-identities-in-algebraic-structures"> <span id="id2"></span><h2><span class="section-number">2.2. </span>Proving Identities in Algebraic Structures<a class="headerlink" href="#proving-identities-in-algebraic-structures" title="Permalink to this heading"></a></h2> </div> <div class="section" id="proving-identities-in-algebraic-structures"> <span id="id2"></span><h2><span class="section-number">2.2. </span>Proving Identities in Algebraic Structures<a class="headerlink" href="#proving-identities-in-algebraic-structures" title="Permalink to this headline"></a></h2> <p id="index-7">Mathematically, a ring consists of a collection of objects, <span class="math notranslate nohighlight">\(R\)</span>, operations <span class="math notranslate nohighlight">\(+\)</span> <span class="math notranslate nohighlight">\(\times\)</span>, and constants <span class="math notranslate nohighlight">\(0\)</span> and <span class="math notranslate nohighlight">\(1\)</span>, and an operation <span class="math notranslate nohighlight">\(x \mapsto -x\)</span> such that:</p>
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@@ -498,8 +494,8 @@ as ring axioms, because they follow from the other axioms.</p><span class="kd">theorem</span> <span class="n">add_right_neg</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">+</span> <span class="bp">-</span><span class="n">a</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">add_comm</span><span class="o">,</span> <span class="n">add_left_neg</span><span class="o">]</span> <span class="k">#check</span> <span class="bp">@</span><span class="n">MyRing.add_zero</span> <span class="k">#check</span> <span class="bp">@</span><span class="n">add_zero</span> <span class="k">#check</span> <span class="n">MyRing.add_zero</span> <span class="k">#check</span> <span class="n">add_zero</span> <span class="kd">end</span> <span class="n">MyRing</span> </pre></div>
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@@ -711,9 +707,9 @@ It may seem odd that the algebraic structures are called<cite>noncomm_ring</cite> and <cite>ring</cite>. This is partly for historical reasons, but also for the convenience of using a shorter name for the tactic that deals with commutative rings, since it is used more often.</p> </section> <section id="using-theorems-and-lemmas"> <span id="id3"></span><h2><span class="section-number">2.3. </span>Using Theorems and Lemmas<a class="headerlink" href="#using-theorems-and-lemmas" title="Permalink to this heading"></a></h2> </div> <div class="section" id="using-theorems-and-lemmas"> <span id="id3"></span><h2><span class="section-number">2.3. </span>Using Theorems and Lemmas<a class="headerlink" href="#using-theorems-and-lemmas" title="Permalink to this headline"></a></h2> <p id="index-15">Rewriting is great for proving equations, but what about other sorts of theorems? For example, how can we prove an inequality,
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@@ -837,7 +833,7 @@ inequalities on the real numbers.</p><span class="k">#check</span> <span class="o">(</span><span class="n">add_pos</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">a</span> <span class="bp">→</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">b</span> <span class="bp">→</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">add_pos_of_pos_of_nonneg</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">a</span> <span class="bp">→</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">b</span> <span class="bp">→</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">)</span> <span class="k">#check</span> <span class="o">(</span><span class="n">exp_pos</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">a</span><span class="o">,</span> <span class="mi">0</span> <span class="bp"><</span> <span class="n">exp</span> <span class="n">a</span><span class="o">)</span> <span class="k">#check</span> <span class="bp">@</span><span class="n">add_le_add_left</span> <span class="k">#check</span> <span class="n">add_le_add_left</span> </pre></div> </div> <p>Some of the theorems, <code class="docutils literal notranslate"><span class="pre">exp_le_exp</span></code>, <code class="docutils literal notranslate"><span class="pre">exp_lt_exp</span></code>, and <code class="docutils literal notranslate"><span class="pre">log_le_log</span></code>
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@@ -978,9 +974,9 @@ following theorem. You can use the theorem <code class="docutils literal notrans</div> <p>If you managed to solve this, congratulations! You are well on your way to becoming a master formalizer.</p> </section> <section id="more-examples-using-apply-and-rw"> <span id="more-on-order-and-divisibility"></span><h2><span class="section-number">2.4. </span>More examples using apply and rw<a class="headerlink" href="#more-examples-using-apply-and-rw" title="Permalink to this heading"></a></h2> </div> <div class="section" id="more-examples-using-apply-and-rw"> <span id="more-on-order-and-divisibility"></span><h2><span class="section-number">2.4. </span>More examples using apply and rw<a class="headerlink" href="#more-examples-using-apply-and-rw" title="Permalink to this headline"></a></h2> <p id="index-20">The <code class="docutils literal notranslate"><span class="pre">min</span></code> function on the real numbers is uniquely characterized by the following three facts:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="o">(</span><span class="n">min_le_left</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">min</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">≤</span> <span class="n">a</span><span class="o">)</span>
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@@ -1187,9 +1183,9 @@ theorem and the version <code class="docutils literal notranslate"><span class="the one specifically for the natural numbers. You can use <code class="docutils literal notranslate"><span class="pre">_root_.dvd_antisymm</span></code> to specify the generic one; either one will work.</p> </section> <section id="proving-facts-about-algebraic-structures"> <span id="id4"></span><h2><span class="section-number">2.5. </span>Proving Facts about Algebraic Structures<a class="headerlink" href="#proving-facts-about-algebraic-structures" title="Permalink to this heading"></a></h2> </div> <div class="section" id="proving-facts-about-algebraic-structures"> <span id="id4"></span><h2><span class="section-number">2.5. </span>Proving Facts about Algebraic Structures<a class="headerlink" href="#proving-facts-about-algebraic-structures" title="Permalink to this headline"></a></h2> <p id="index-26">In <a class="reference internal" href="#proving-identities-in-algebraic-structures"><span class="std std-numref">Section 2.2</span></a>, we saw that many common identities governing the real numbers hold in more general classes of algebraic structures,
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@@ -1402,8 +1398,8 @@ always nonnegative:</p></div> <p>We recommend making use of the theorem <code class="docutils literal notranslate"><span class="pre">nonneg_of_mul_nonneg_left</span></code>. As you may have guessed, this theorem is called <code class="docutils literal notranslate"><span class="pre">dist_nonneg</span></code> in mathlib.</p> </section> </section> </div> </div> </div>
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@@ -13,11 +12,10 @@<script src="_static/js/html5shiv.min.js"></script> <![endif]--> <script src="_static/jquery.js"></script> <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/jquery.js"></script> <script src="_static/underscore.js"></script> <script src="_static/doctools.js"></script> <script src="_static/sphinx_highlight.js"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" />
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@@ -31,15 +29,11 @@<nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > <a href="index.html" class="icon icon-home"> Mathematics in Lean <a href="index.html" class="icon icon-home"> Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="text" name="q" placeholder="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
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@@ -82,8 +76,8 @@<div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> <li class="breadcrumb-item active"><span class="section-number">3. </span>Logic</li> <li><a href="index.html" class="icon icon-home"></a> »</li> <li><span class="section-number">3. </span>Logic</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C03_Logic.rst.txt" rel="nofollow"> View page source</a> </li>
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@@ -93,8 +87,8 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="logic"> <span id="id1"></span><h1><span class="section-number">3. </span>Logic<a class="headerlink" href="#logic" title="Permalink to this heading"></a></h1> <div class="section" id="logic"> <span id="id1"></span><h1><span class="section-number">3. </span>Logic<a class="headerlink" href="#logic" title="Permalink to this headline"></a></h1> <p>In the last chapter, we dealt with equations, inequalities, and basic mathematical statements like “<span class="math notranslate nohighlight">\(x\)</span> divides <span class="math notranslate nohighlight">\(y\)</span>.”
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@@ -104,8 +98,8 @@ using logical terms like “and,” “or,” “not,”“if … then,” “every,” and “some.” In this chapter, we show you how to work with statements that are built up in this way.</p> <section id="implication-and-the-universal-quantifier"> <span id="id2"></span><h2><span class="section-number">3.1. </span>Implication and the Universal Quantifier<a class="headerlink" href="#implication-and-the-universal-quantifier" title="Permalink to this heading"></a></h2> <div class="section" id="implication-and-the-universal-quantifier"> <span id="id2"></span><h2><span class="section-number">3.1. </span>Implication and the Universal Quantifier<a class="headerlink" href="#implication-and-the-universal-quantifier" title="Permalink to this headline"></a></h2> <p>Consider the statement after the <code class="docutils literal notranslate"><span class="pre">#check</span></code>:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="bp">∀</span> <span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">,</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">x</span> <span class="bp">→</span> <span class="bp">|</span><span class="n">x</span><span class="bp">|</span> <span class="bp">=</span> <span class="n">x</span> </pre></div>
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@@ -287,7 +281,7 @@ So if we prove the theorem <code class="docutils literal notranslate"><span clasit will apply in all these instances.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">{</span><span class="n">R</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">OrderedCancelAddCommMonoid</span> <span class="n">R</span><span class="o">]</span> <span class="k">#check</span> <span class="bp">@</span><span class="n">add_le_add</span> <span class="k">#check</span> <span class="n">add_le_add</span> <span class="kd">def</span> <span class="n">FnUb'</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">R</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">R</span><span class="o">)</span> <span class="o">:</span> <span class="kt">Prop</span> <span class="o">:=</span> <span class="bp">∀</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">≤</span> <span class="n">a</span>
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@@ -493,9 +487,9 @@ a lemma name.</p><span class="gr">sorry</span> </pre></div> </div> </section> <section id="the-existential-quantifier"> <span id="id3"></span><h2><span class="section-number">3.2. </span>The Existential Quantifier<a class="headerlink" href="#the-existential-quantifier" title="Permalink to this heading"></a></h2> </div> <div class="section" id="the-existential-quantifier"> <span id="id3"></span><h2><span class="section-number">3.2. </span>The Existential Quantifier<a class="headerlink" href="#the-existential-quantifier" title="Permalink to this headline"></a></h2> <p>The existential quantifier, which can be entered as <code class="docutils literal notranslate"><span class="pre">\ex</span></code> in VS Code, is used to represent the phrase “there exists.” The formal expression <code class="docutils literal notranslate"><span class="pre">∃</span> <span class="pre">x</span> <span class="pre">:</span> <span class="pre">ℝ,</span> <span class="pre">2</span> <span class="pre"><</span> <span class="pre">x</span> <span class="pre">∧</span> <span class="pre">x</span> <span class="pre"><</span> <span class="pre">3</span></code> in Lean says
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@@ -763,9 +757,9 @@ the composition of surjective functions is surjective.</p><span class="gr">sorry</span> </pre></div> </div> </section> <section id="negation"> <span id="id4"></span><h2><span class="section-number">3.3. </span>Negation<a class="headerlink" href="#negation" title="Permalink to this heading"></a></h2> </div> <div class="section" id="negation"> <span id="id4"></span><h2><span class="section-number">3.3. </span>Negation<a class="headerlink" href="#negation" title="Permalink to this headline"></a></h2> <p>The symbol <code class="docutils literal notranslate"><span class="pre">¬</span></code> is meant to express negation, so <code class="docutils literal notranslate"><span class="pre">¬</span> <span class="pre">x</span> <span class="pre"><</span> <span class="pre">y</span></code> says that <code class="docutils literal notranslate"><span class="pre">x</span></code> is not less than <code class="docutils literal notranslate"><span class="pre">y</span></code>, <code class="docutils literal notranslate"><span class="pre">¬</span> <span class="pre">x</span> <span class="pre">=</span> <span class="pre">y</span></code> (or, equivalently, <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">≠</span> <span class="pre">y</span></code>) says that
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@@ -1030,9 +1024,9 @@ Finally, the <code class="docutils literal notranslate"><span class="pre">contraby finding a contradiction in the hypotheses, such as a pair of the form <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">:</span> <span class="pre">P</span></code> and <code class="docutils literal notranslate"><span class="pre">h'</span> <span class="pre">:</span> <span class="pre">¬</span> <span class="pre">P</span></code>. Of course, in this example, <code class="docutils literal notranslate"><span class="pre">linarith</span></code> also works.</p> </section> <section id="conjunction-and-bi-implication"> <span id="conjunction-and-biimplication"></span><h2><span class="section-number">3.4. </span>Conjunction and Bi-implication<a class="headerlink" href="#conjunction-and-bi-implication" title="Permalink to this heading"></a></h2> </div> <div class="section" id="conjunction-and-bi-implication"> <span id="conjunction-and-biimplication"></span><h2><span class="section-number">3.4. </span>Conjunction and Bi-implication<a class="headerlink" href="#conjunction-and-bi-implication" title="Permalink to this headline"></a></h2> <p id="index-18">You have already seen that the conjunction symbol, <code class="docutils literal notranslate"><span class="pre">∧</span></code>, is used to express “and.” The <code class="docutils literal notranslate"><span class="pre">constructor</span></code> tactic allows you to prove a statement of
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@@ -1262,9 +1256,9 @@ to be instantiated to different values.</p><span class="gr">sorry</span> </pre></div> </div> </section> <section id="disjunction"> <span id="id5"></span><h2><span class="section-number">3.5. </span>Disjunction<a class="headerlink" href="#disjunction" title="Permalink to this heading"></a></h2> </div> <div class="section" id="disjunction"> <span id="id5"></span><h2><span class="section-number">3.5. </span>Disjunction<a class="headerlink" href="#disjunction" title="Permalink to this headline"></a></h2> <p id="index-21">The canonical way to prove a disjunction <code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">∨</span> <span class="pre">B</span></code> is to prove <code class="docutils literal notranslate"><span class="pre">A</span></code> or to prove <code class="docutils literal notranslate"><span class="pre">B</span></code>. The <code class="docutils literal notranslate"><span class="pre">left</span></code> tactic chooses <code class="docutils literal notranslate"><span class="pre">A</span></code>,
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@@ -1457,9 +1451,9 @@ using <code class="docutils literal notranslate"><span class="pre">by_cases</spa<span class="gr">sorry</span> </pre></div> </div> </section> <section id="sequences-and-convergence"> <span id="id6"></span><h2><span class="section-number">3.6. </span>Sequences and Convergence<a class="headerlink" href="#sequences-and-convergence" title="Permalink to this heading"></a></h2> </div> <div class="section" id="sequences-and-convergence"> <span id="id6"></span><h2><span class="section-number">3.6. </span>Sequences and Convergence<a class="headerlink" href="#sequences-and-convergence" title="Permalink to this headline"></a></h2> <p>We now have enough skills at our disposal to do some real mathematics. In Lean, we can represent a sequence <span class="math notranslate nohighlight">\(s_0, s_1, s_2, \ldots\)</span> of real numbers as a function <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">:</span> <span class="pre">ℕ</span> <span class="pre">→</span> <span class="pre">ℝ</span></code>.
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@@ -1694,8 +1688,8 @@ for dealing with convergence in vastly more general terms,not only abstracting away particular features of the domain and codomain, but also abstracting over different types of convergence.</p> </section> </section> </div> </div> </div>
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@@ -31,15 +29,11 @@<nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > <a href="index.html" class="icon icon-home"> Mathematics in Lean <a href="index.html" class="icon icon-home"> Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="text" name="q" placeholder="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
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@@ -79,8 +73,8 @@<div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> <li class="breadcrumb-item active"><span class="section-number">4. </span>Sets and Functions</li> <li><a href="index.html" class="icon icon-home"></a> »</li> <li><span class="section-number">4. </span>Sets and Functions</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C04_Sets_and_Functions.rst.txt" rel="nofollow"> View page source</a> </li>
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@@ -90,8 +84,8 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="sets-and-functions"> <span id="id1"></span><h1><span class="section-number">4. </span>Sets and Functions<a class="headerlink" href="#sets-and-functions" title="Permalink to this heading"></a></h1> <div class="section" id="sets-and-functions"> <span id="id1"></span><h1><span class="section-number">4. </span>Sets and Functions<a class="headerlink" href="#sets-and-functions" title="Permalink to this headline"></a></h1> <p>The vocabulary of sets, relations, and functions provides a uniform language for carrying out constructions in all the branches of mathematics.
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@@ -123,8 +117,8 @@ such as a set natural numbers or a set of functionsfrom real numbers to real numbers. The distinction between types and set takes some getting used to, but this chapter will take you through the essentials.</p> <section id="sets"> <span id="id2"></span><h2><span class="section-number">4.1. </span>Sets<a class="headerlink" href="#sets" title="Permalink to this heading"></a></h2> <div class="section" id="sets"> <span id="id2"></span><h2><span class="section-number">4.1. </span>Sets<a class="headerlink" href="#sets" title="Permalink to this headline"></a></h2> <p id="index-0">If <code class="docutils literal notranslate"><span class="pre">α</span></code> is any type, the type <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">α</span></code> consists of sets of elements of <code class="docutils literal notranslate"><span class="pre">α</span></code>. This type supports the usual set-theoretic operations and relations.
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@@ -565,9 +559,9 @@ and intersection.</p></div> <p>In the library, these identities are called <code class="docutils literal notranslate"><span class="pre">sUnion_eq_biUnion</span></code> and <code class="docutils literal notranslate"><span class="pre">sInter_eq_biInter</span></code>.</p> </section> <section id="functions"> <span id="id3"></span><h2><span class="section-number">4.2. </span>Functions<a class="headerlink" href="#functions" title="Permalink to this heading"></a></h2> </div> <div class="section" id="functions"> <span id="id3"></span><h2><span class="section-number">4.2. </span>Functions<a class="headerlink" href="#functions" title="Permalink to this headline"></a></h2> <p>If <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">α</span> <span class="pre">→</span> <span class="pre">β</span></code> is a function and <code class="docutils literal notranslate"><span class="pre">p</span></code> is a set of elements of type <code class="docutils literal notranslate"><span class="pre">β</span></code>, the library defines <code class="docutils literal notranslate"><span class="pre">preimage</span> <span class="pre">f</span> <span class="pre">p</span></code>, written <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">⁻¹'</span> <span class="pre">p</span></code>,
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@@ -892,9 +886,9 @@ and then fill in the two lines that are missing.</p><span class="n">contradiction</span> </pre></div> </div> </section> <section id="the-schroder-bernstein-theorem"> <span id="the-schroeder-bernstein-theorem"></span><h2><span class="section-number">4.3. </span>The Schröder-Bernstein Theorem<a class="headerlink" href="#the-schroder-bernstein-theorem" title="Permalink to this heading"></a></h2> </div> <div class="section" id="the-schroder-bernstein-theorem"> <span id="the-schroeder-bernstein-theorem"></span><h2><span class="section-number">4.3. </span>The Schröder-Bernstein Theorem<a class="headerlink" href="#the-schroder-bernstein-theorem" title="Permalink to this headline"></a></h2> <p>We close this chapter with an elementary but nontrivial theorem of set theory. Let <span class="math notranslate nohighlight">\(\alpha\)</span> and <span class="math notranslate nohighlight">\(\beta\)</span> be sets. (In our formalization, they will actually be types.)
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@@ -1153,8 +1147,8 @@ and the proof uses the fact that <code class="docutils literal notranslate"><spa<span class="o">⟨</span><span class="n">sbFun</span> <span class="n">f</span> <span class="n">g</span><span class="o">,</span> <span class="n">sb_injective</span> <span class="n">f</span> <span class="n">g</span> <span class="n">hf</span> <span class="n">hg</span><span class="o">,</span> <span class="n">sb_surjective</span> <span class="n">f</span> <span class="n">g</span> <span class="n">hf</span> <span class="n">hg</span><span class="o">⟩</span> </pre></div> </div> </section> </section> </div> </div> </div>
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@@ -13,11 +12,10 @@<script src="_static/js/html5shiv.min.js"></script> <![endif]--> <script src="_static/jquery.js"></script> <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/jquery.js"></script> <script src="_static/underscore.js"></script> <script src="_static/doctools.js"></script> <script src="_static/sphinx_highlight.js"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" />
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@@ -31,15 +29,11 @@<nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > <a href="index.html" class="icon icon-home"> Mathematics in Lean <a href="index.html" class="icon icon-home"> Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="text" name="q" placeholder="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
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@@ -79,8 +73,8 @@<div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> <li class="breadcrumb-item active"><span class="section-number">5. </span>Elementary Number Theory</li> <li><a href="index.html" class="icon icon-home"></a> »</li> <li><span class="section-number">5. </span>Elementary Number Theory</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C05_Number_Theory.rst.txt" rel="nofollow"> View page source</a> </li>
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@@ -90,15 +84,15 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="elementary-number-theory"> <span id="number-theory"></span><h1><span class="section-number">5. </span>Elementary Number Theory<a class="headerlink" href="#elementary-number-theory" title="Permalink to this heading"></a></h1> <div class="section" id="elementary-number-theory"> <span id="number-theory"></span><h1><span class="section-number">5. </span>Elementary Number Theory<a class="headerlink" href="#elementary-number-theory" title="Permalink to this headline"></a></h1> <p>In this chapter, we show you how to formalize some elementary results in number theory. As we deal with more substantive mathematical content, the proofs will get longer and more involved, building on the skills you have already mastered.</p> <section id="irrational-roots"> <span id="section-irrational-roots"></span><h2><span class="section-number">5.1. </span>Irrational Roots<a class="headerlink" href="#irrational-roots" title="Permalink to this heading"></a></h2> <div class="section" id="irrational-roots"> <span id="section-irrational-roots"></span><h2><span class="section-number">5.1. </span>Irrational Roots<a class="headerlink" href="#irrational-roots" title="Permalink to this headline"></a></h2> <p>Let’s start with a fact known to the ancient greeks, namely, that the square root of 2 is irrational. If we suppose otherwise,
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@@ -151,7 +145,7 @@ to the natural numbers.</p><p>We also need the notion of a prime number, <code class="docutils literal notranslate"><span class="pre">Nat.Prime</span></code>. The theorem <code class="docutils literal notranslate"><span class="pre">Nat.prime_def_lt</span></code> provides one familiar characterization, and <code class="docutils literal notranslate"><span class="pre">Nat.Prime.eq_one_or_self_of_dvd</span></code> provides another.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="bp">@</span><span class="n">Nat.prime_def_lt</span> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="n">Nat.prime_def_lt</span> <span class="kd">example</span> <span class="o">(</span><span class="n">p</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">)</span> <span class="o">(</span><span class="n">prime_p</span> <span class="o">:</span> <span class="n">Nat.Prime</span> <span class="n">p</span><span class="o">)</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">≤</span> <span class="n">p</span> <span class="bp">∧</span> <span class="bp">∀</span> <span class="n">m</span> <span class="o">:</span> <span class="n">ℕ</span><span class="o">,</span> <span class="n">m</span> <span class="bp"><</span> <span class="n">p</span> <span class="bp">→</span> <span class="n">m</span> <span class="bp">∣</span> <span class="n">p</span> <span class="bp">→</span> <span class="n">m</span> <span class="bp">=</span> <span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rwa</span> <span class="o">[</span><span class="n">Nat.prime_def_lt</span><span class="o">]</span> <span class="n">at</span> <span class="n">prime_p</span>
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@@ -186,7 +180,7 @@ if the square of a number is even, then that number is even as well.Mathlib defines the predicate <code class="docutils literal notranslate"><span class="pre">Even</span></code> in <code class="docutils literal notranslate"><span class="pre">Data.Nat.Parity</span></code>, but for reasons that will become clear below, we will simply use <code class="docutils literal notranslate"><span class="pre">2</span> <span class="pre">∣</span> <span class="pre">m</span></code> to express that <code class="docutils literal notranslate"><span class="pre">m</span></code> is even.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="bp">@</span><span class="n">Nat.Prime.dvd_mul</span> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="n">Nat.Prime.dvd_mul</span> <span class="k">#check</span> <span class="n">Nat.Prime.dvd_mul</span> <span class="n">Nat.prime_two</span> <span class="k">#check</span> <span class="n">Nat.prime_two.dvd_mul</span>
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@@ -397,9 +391,9 @@ and that it takes values in the extended natural numbers <code class="docutils lwhich adds the value infinity to the natural numbers. In the next chapter, we will begin to develop the means to appreciate the way that Lean supports this sort of generality.</p> </section> <section id="induction-and-recursion"> <span id="section-induction-and-recursion"></span><h2><span class="section-number">5.2. </span>Induction and Recursion<a class="headerlink" href="#induction-and-recursion" title="Permalink to this heading"></a></h2> </div> <div class="section" id="induction-and-recursion"> <span id="section-induction-and-recursion"></span><h2><span class="section-number">5.2. </span>Induction and Recursion<a class="headerlink" href="#induction-and-recursion" title="Permalink to this headline"></a></h2> <p>The set of natural numbers <span class="math notranslate nohighlight">\(\mathbb{N} = \{ 0, 1, 2, \ldots \}\)</span> is not only fundamentally important in its own right, but also a plays a central role in the construction of new mathematical objects.
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@@ -702,9 +696,9 @@ The function <code class="docutils literal notranslate"><span class="pre">pred</<span class="kd">end</span> <span class="n">MyNat</span> </pre></div> </div> </section> <section id="infinitely-many-primes"> <span id="section-infinitely-many-primes"></span><h2><span class="section-number">5.3. </span>Infinitely Many Primes<a class="headerlink" href="#infinitely-many-primes" title="Permalink to this heading"></a></h2> </div> <div class="section" id="infinitely-many-primes"> <span id="section-infinitely-many-primes"></span><h2><span class="section-number">5.3. </span>Infinitely Many Primes<a class="headerlink" href="#infinitely-many-primes" title="Permalink to this headline"></a></h2> <p>Let us continue our exploration of induction and recursion with another mathematical standard: a proof that there are infinitely many primes. One way to formulate this is as the statement that
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@@ -1144,8 +1138,8 @@ along the way.</p></div> <p>If you managed to complete the proof, congratulations! This has been a serious feat of formalization.</p> </section> </section> </div> </div> </div>
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@@ -31,15 +29,11 @@<nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > <a href="index.html" class="icon icon-home"> Mathematics in Lean <a href="index.html" class="icon icon-home"> Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="text" name="q" placeholder="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
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@@ -90,8 +84,8 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="structures"> <span id="id1"></span><h1><span class="section-number">6. </span>Structures<a class="headerlink" href="#structures" title="Permalink to this heading"></a></h1> <div class="section" id="structures"> <span id="id1"></span><h1><span class="section-number">6. </span>Structures<a class="headerlink" href="#structures" title="Permalink to this headline"></a></h1> <p>Modern mathematics makes essential use of algebraic structures, which encapsulate patterns that can be instantiated in
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@@ -110,8 +104,8 @@ It will also show you how to define and usealgebraic structures on your own.</p> <p>For more technical detail, you can consult <a class="reference external" href="https://leanprover.github.io/theorem_proving_in_lean/">Theorem Proving in Lean</a>, and a paper by Anne Baanen, <a class="reference external" href="https://arxiv.org/abs/2202.01629">Use and abuse of instance parameters in the Lean mathematical library</a>.</p> <section id="defining-structures"> <span id="section-structures"></span><h2><span class="section-number">6.1. </span>Defining structures<a class="headerlink" href="#defining-structures" title="Permalink to this heading"></a></h2> <div class="section" id="defining-structures"> <span id="section-structures"></span><h2><span class="section-number">6.1. </span>Defining structures<a class="headerlink" href="#defining-structures" title="Permalink to this headline"></a></h2> <p>In the broadest sense of the term, a <em>structure</em> is a specification of a collection of data, possibly with constraints that the data is required to satisfy.
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@@ -488,9 +482,9 @@ as long as we redefine the old accessors in terms of the new definition.Moreover, as we are about to see, Lean provides support for weaving structures together into a rich, interconnected hierarchy, and for managing the interactions between them.</p> </section> <section id="algebraic-structures"> <span id="section-algebraic-structures"></span><h2><span class="section-number">6.2. </span>Algebraic Structures<a class="headerlink" href="#algebraic-structures" title="Permalink to this heading"></a></h2> </div> <div class="section" id="algebraic-structures"> <span id="section-algebraic-structures"></span><h2><span class="section-number">6.2. </span>Algebraic Structures<a class="headerlink" href="#algebraic-structures" title="Permalink to this headline"></a></h2> <p>To clarify what we mean by the phrase <em>algebraic structure</em>, it will help to consider some examples.</p> <ol class="arabic simple">
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@@ -885,12 +879,12 @@ without troubling us with the details.</p></pre></div> </div> <p>The following illustrates their use.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="bp">@</span><span class="n">Group₂.mul</span> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="n">Group₂.mul</span> <span class="kd">def</span> <span class="n">mySquare</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">Group₂</span> <span class="n">α</span><span class="o">]</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">α</span><span class="o">)</span> <span class="o">:=</span> <span class="n">Group₂.mul</span> <span class="n">x</span> <span class="n">x</span> <span class="k">#check</span> <span class="bp">@</span><span class="n">mySquare</span> <span class="k">#check</span> <span class="n">mySquare</span> <span class="kn">section</span> <span class="kd">variable</span> <span class="o">{</span><span class="n">β</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">(</span><span class="n">f</span> <span class="n">g</span> <span class="o">:</span> <span class="n">Equiv.Perm</span> <span class="n">β</span><span class="o">)</span>
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@@ -1037,9 +1031,9 @@ because it configures automation that invisibly governs the interpretation ofthe expressions we type. When used wisely, however, class inference is a powerful tool. It is what makes algebraic reasoning possible in Lean.</p> </section> <section id="building-the-gaussian-integers"> <span id="section-building-the-gaussian-integers"></span><h2><span class="section-number">6.3. </span>Building the Gaussian Integers<a class="headerlink" href="#building-the-gaussian-integers" title="Permalink to this heading"></a></h2> </div> <div class="section" id="building-the-gaussian-integers"> <span id="section-building-the-gaussian-integers"></span><h2><span class="section-number">6.3. </span>Building the Gaussian Integers<a class="headerlink" href="#building-the-gaussian-integers" title="Permalink to this headline"></a></h2> <p>We will now illustrate the use of the algebraic hierarchy in Lean by building an important mathematical object, the <em>Gaussian integers</em>, and showing that it is a Euclidean domain. In other words, according to
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@@ -1533,8 +1527,8 @@ the notions of being prime and being irreducible coincide.</p><span class="n">PrincipalIdealRing.irreducible_iff_prime</span> </pre></div> </div> </section> </section> </div> </div> </div>
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@@ -30,15 +28,11 @@<nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > <a href="index.html" class="icon icon-home"> Mathematics in Lean <a href="index.html" class="icon icon-home"> Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="text" name="q" placeholder="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
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@@ -78,8 +72,8 @@<div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> <li class="breadcrumb-item active"><span class="section-number">7. </span>Hierarchies</li> <li><a href="index.html" class="icon icon-home"></a> »</li> <li><span class="section-number">7. </span>Hierarchies</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C07_Hierarchies.rst.txt" rel="nofollow"> View page source</a> </li>
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@@ -89,8 +83,8 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="hierarchies"> <span id="id1"></span><h1><span class="section-number">7. </span>Hierarchies<a class="headerlink" href="#hierarchies" title="Permalink to this heading"></a></h1> <div class="section" id="hierarchies"> <span id="id1"></span><h1><span class="section-number">7. </span>Hierarchies<a class="headerlink" href="#hierarchies" title="Permalink to this headline"></a></h1> <p>We have seen in <a class="reference internal" href="C06_Structures.html#structures"><span class="std std-numref">Chapter 6</span></a> how to define the class of groups and build instances of this class, and then how to build an instance of the commutative ring class. But of course there is a hierarchy here: a
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@@ -106,8 +100,8 @@ the following chapters and come back here for a second reading.</p>so we will used indices to distinguish our version. For instance we will have <code class="docutils literal notranslate"><span class="pre">Ring₁</span></code> as our version of <code class="docutils literal notranslate"><span class="pre">Ring</span></code>. Since we will gradually explain more powerful ways of formalizing structures, those indices will sometimes grow beyond one.</p> <section id="basics"> <span id="section-hierarchies-basics"></span><h2><span class="section-number">7.1. </span>Basics<a class="headerlink" href="#basics" title="Permalink to this heading"></a></h2> <div class="section" id="basics"> <span id="section-hierarchies-basics"></span><h2><span class="section-number">7.1. </span>Basics<a class="headerlink" href="#basics" title="Permalink to this headline"></a></h2> <p>At the very bottom of all hierarchies in Lean, we find data-carrying classes. The following class records that the given type <code class="docutils literal notranslate"><span class="pre">α</span></code> is endowed with a distinguished element called <code class="docutils literal notranslate"><span class="pre">one</span></code>. At this stage, it has no property at all.</p>
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@@ -490,7 +484,7 @@ type <code class="docutils literal notranslate"><span class="pre">β</span><</pre></div> </div> <p>There is something interesting going on here. While it isn’t too surprising that the ring structure on <code class="docutils literal notranslate"><span class="pre">R</span></code> is a parameter in this definition, you probably expected <code class="docutils literal notranslate"><span class="pre">AddCommGroup3</span> <span class="pre">M</span></code> ring structure on <code class="docutils literal notranslate"><span class="pre">R</span></code> is a parameter in this definition, you probably expected <code class="docutils literal notranslate"><span class="pre">AddCommGroup₃</span> <span class="pre">M</span></code> to be part of the <code class="docutils literal notranslate"><span class="pre">extends</span></code> clause just as <code class="docutils literal notranslate"><span class="pre">SMul₃</span> <span class="pre">R</span> <span class="pre">M</span></code> is. Trying to do that would lead to a mysterious sounding error message: <code class="docutils literal notranslate"><span class="pre">cannot</span> <span class="pre">find</span> <span class="pre">synthesization</span> <span class="pre">order</span> <span class="pre">for</span> <span class="pre">instance</span> <span class="pre">Module₁.toAddCommGroup₃</span> <span class="pre">with</span> <span class="pre">type</span> <span class="pre">(R</span> <span class="pre">:</span> <span class="pre">Type)</span> <span class="pre">→</span> <span class="pre">[inst</span> <span class="pre">:</span> <span class="pre">Ring₃</span> <span class="pre">R]</span> <span class="pre">→</span> <span class="pre">{M</span> <span class="pre">:</span> <span class="pre">Type}</span> <span class="pre">→</span> <span class="pre">[self</span> <span class="pre">:</span> <span class="pre">Module₁</span> <span class="pre">R</span> <span class="pre">M]</span> <span class="pre">→</span> <span class="pre">AddCommGroup₃</span> <span class="pre">M</span>
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@@ -632,9 +626,9 @@ to incorporate a type class <code class="docutils literal notranslate"><span clathat every preorder comes with a <code class="docutils literal notranslate"><span class="pre"><₁</span></code> which has a default value built from <code class="docutils literal notranslate"><span class="pre">≤₁</span></code> and a <code class="docutils literal notranslate"><span class="pre">Prop</span></code>-valued field asserting the natural relation between those two comparison operators. -/</p> </section> <section id="morphisms"> <span id="section-hierarchies-morphisms"></span><h2><span class="section-number">7.2. </span>Morphisms<a class="headerlink" href="#morphisms" title="Permalink to this heading"></a></h2> </div> <div class="section" id="morphisms"> <span id="section-hierarchies-morphisms"></span><h2><span class="section-number">7.2. </span>Morphisms<a class="headerlink" href="#morphisms" title="Permalink to this headline"></a></h2> <p>So far in this chapter, we discussed how to create a hierarchy of mathematical structures. But defining structures is not really completed until we have morphisms. There are two main approaches here. The most obvious one is to define a predicate on functions.</p>
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@@ -836,9 +830,9 @@ definitions below.</p><span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> </section> <section id="sub-objects"> <span id="section-hierarchies-subobjects"></span><h2><span class="section-number">7.3. </span>Sub-objects<a class="headerlink" href="#sub-objects" title="Permalink to this heading"></a></h2> </div> <div class="section" id="sub-objects"> <span id="section-hierarchies-subobjects"></span><h2><span class="section-number">7.3. </span>Sub-objects<a class="headerlink" href="#sub-objects" title="Permalink to this headline"></a></h2> <p>After defining some algebraic structure and its morphisms, the next step is to consider sets that inherit this algebraic structure, for instance subgroups or subrings. This largely overlaps our previous topic. Indeed a set in <code class="docutils literal notranslate"><span class="pre">X</span></code> is implemented as a function from
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@@ -944,7 +938,9 @@ convenient notation are built and code duplication is avoided in Mathlib. Here tis the <code class="docutils literal notranslate"><span class="pre">HasQuotient</span></code> class which allows notations like <code class="docutils literal notranslate"><span class="pre">M</span> <span class="pre">⧸</span> <span class="pre">N</span></code>. Beware the quotient symbol <code class="docutils literal notranslate"><span class="pre">⧸</span></code> is a special unicode character, not a regular ASCII division symbol.</p> <p>As an example, we will build the quotient of a commutative monoid by a submonoid, leave proofs to you.</p> to you. In the last example, you can use <code class="docutils literal notranslate"><span class="pre">Setoid.refl</span></code> but it won’t automatically pick up the relevant <code class="docutils literal notranslate"><span class="pre">Setoid</span></code> structure. You can fix this issue by providing all arguments using the <code class="docutils literal notranslate"><span class="pre">@</span></code> syntax, as in <code class="docutils literal notranslate"><span class="pre">@Setoid.refl</span> <span class="pre">M</span> <span class="pre">N.Setoid</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">Submonoid.Setoid</span> <span class="o">[</span><span class="n">CommMonoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">(</span><span class="n">N</span> <span class="o">:</span> <span class="n">Submonoid</span> <span class="n">M</span><span class="o">)</span> <span class="o">:</span> <span class="n">Setoid</span> <span class="n">M</span> <span class="n">where</span> <span class="n">r</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">↦</span> <span class="bp">∃</span> <span class="n">w</span> <span class="bp">∈</span> <span class="n">N</span><span class="o">,</span> <span class="bp">∃</span> <span class="n">z</span> <span class="bp">∈</span> <span class="n">N</span><span class="o">,</span> <span class="n">x</span><span class="bp">*</span><span class="n">w</span> <span class="bp">=</span> <span class="n">y</span><span class="bp">*</span><span class="n">z</span> <span class="n">iseqv</span> <span class="o">:=</span> <span class="o">{</span>
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@@ -972,8 +968,8 @@ to you.</p><span class="gr">sorry</span> </pre></div> </div> </section> </section> </div> </div> </div>
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@@ -13,11 +12,10 @@<script src="_static/js/html5shiv.min.js"></script> <![endif]--> <script src="_static/jquery.js"></script> <script src="_static/_sphinx_javascript_frameworks_compat.js"></script> <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/jquery.js"></script> <script src="_static/underscore.js"></script> <script src="_static/doctools.js"></script> <script src="_static/sphinx_highlight.js"></script> <script async="async" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" />
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@@ -31,15 +29,11 @@<nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > <a href="index.html" class="icon icon-home"> Mathematics in Lean <a href="index.html" class="icon icon-home"> Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="text" name="q" placeholder="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
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@@ -91,8 +85,8 @@<div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> <li class="breadcrumb-item active"><span class="section-number">8. </span>Topology</li> <li><a href="index.html" class="icon icon-home"></a> »</li> <li><span class="section-number">8. </span>Topology</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C08_Topology.rst.txt" rel="nofollow"> View page source</a> </li>
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@@ -102,8 +96,8 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <span class="target" id="topology"></span><section id="index-0"> <span id="id1"></span><h1><span class="section-number">8. </span>Topology<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> <span class="target" id="topology"></span><div class="section" id="index-0"> <span id="id1"></span><h1><span class="section-number">8. </span>Topology<a class="headerlink" href="#index-0" title="Permalink to this headline"></a></h1> <p>Calculus is based on the concept of a function, which is used to model quantities that depend on one another. For example, it is common to study quantities that change over time.
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@@ -170,8 +164,8 @@ and simply note that the rest can be proved “in the same way.”Formalizing mathematics requires making the relevant notion of “sameness” fully explicit, and that is exactly what Bourbaki’s theory of filters manages to do.</p> <section id="filters"> <span id="index-1"></span><span id="id2"></span><h2><span class="section-number">8.1. </span>Filters<a class="headerlink" href="#filters" title="Permalink to this heading"></a></h2> <div class="section" id="filters"> <span id="index-1"></span><span id="id2"></span><h2><span class="section-number">8.1. </span>Filters<a class="headerlink" href="#filters" title="Permalink to this headline"></a></h2> <p>A <em>filter</em> on a type <code class="docutils literal notranslate"><span class="pre">X</span></code> is a collection of sets of <code class="docutils literal notranslate"><span class="pre">X</span></code> that satisfies three conditions that we will spell out below. The notion supports two related ideas:</p>
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@@ -471,9 +465,9 @@ Given <code class="docutils literal notranslate"><span class="pre">F</span> <spa<li><p>the condition <code class="docutils literal notranslate"><span class="pre">U</span> <span class="pre">∈</span> <span class="pre">F</span> <span class="pre">→</span> <span class="pre">U</span> <span class="pre">⊆</span> <span class="pre">V</span> <span class="pre">→</span> <span class="pre">V</span> <span class="pre">∈</span> <span class="pre">F</span></code> ensures <code class="docutils literal notranslate"><span class="pre">(∀ᶠ</span> <span class="pre">x</span> <span class="pre">in</span> <span class="pre">F,</span> <span class="pre">P</span> <span class="pre">x)</span> <span class="pre">→</span> <span class="pre">(∀</span> <span class="pre">x,</span> <span class="pre">P</span> <span class="pre">x</span> <span class="pre">→</span> <span class="pre">Q</span> <span class="pre">x)</span> <span class="pre">→</span> <span class="pre">∀ᶠ</span> <span class="pre">x</span> <span class="pre">in</span> <span class="pre">F,</span> <span class="pre">Q</span> <span class="pre">x</span></code>, and</p></li> <li><p>the condition <code class="docutils literal notranslate"><span class="pre">U</span> <span class="pre">∈</span> <span class="pre">F</span> <span class="pre">→</span> <span class="pre">V</span> <span class="pre">∈</span> <span class="pre">F</span> <span class="pre">→</span> <span class="pre">U</span> <span class="pre">∩</span> <span class="pre">V</span> <span class="pre">∈</span> <span class="pre">F</span></code> ensures <code class="docutils literal notranslate"><span class="pre">(∀ᶠ</span> <span class="pre">x</span> <span class="pre">in</span> <span class="pre">F,</span> <span class="pre">P</span> <span class="pre">x)</span> <span class="pre">→</span> <span class="pre">(∀ᶠ</span> <span class="pre">x</span> <span class="pre">in</span> <span class="pre">F,</span> <span class="pre">Q</span> <span class="pre">x)</span> <span class="pre">→</span> <span class="pre">∀ᶠ</span> <span class="pre">x</span> <span class="pre">in</span> <span class="pre">F,</span> <span class="pre">P</span> <span class="pre">x</span> <span class="pre">∧</span> <span class="pre">Q</span> <span class="pre">x</span></code>.</p></li> </ul> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="bp">@</span><span class="n">eventually_of_forall</span> <span class="k">#check</span> <span class="bp">@</span><span class="n">Eventually.mono</span> <span class="k">#check</span> <span class="bp">@</span><span class="n">Eventually.and</span> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="n">eventually_of_forall</span> <span class="k">#check</span> <span class="n">Eventually.mono</span> <span class="k">#check</span> <span class="n">Eventually.and</span> </pre></div> </div> <p>The second item, corresponding to <code class="docutils literal notranslate"><span class="pre">eventually.mono</span></code>, supports nice ways
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@@ -524,9 +518,9 @@ by definition, the assumption <code class="docutils literal notranslate"><span c<span class="gr">sorry</span> </pre></div> </div> </section> <section id="metric-spaces"> <span id="index-2"></span><span id="id3"></span><h2><span class="section-number">8.2. </span>Metric spaces<a class="headerlink" href="#metric-spaces" title="Permalink to this heading"></a></h2> </div> <div class="section" id="metric-spaces"> <span id="index-2"></span><span id="id3"></span><h2><span class="section-number">8.2. </span>Metric spaces<a class="headerlink" href="#metric-spaces" title="Permalink to this headline"></a></h2> <p>Examples in the previous section focus on sequences of real numbers. In this section we will go up a bit in generality and focus on metric spaces. A metric space is a type <code class="docutils literal notranslate"><span class="pre">X</span></code> equipped with a distance function <code class="docutils literal notranslate"><span class="pre">dist</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">ℝ</span></code> which is a generalization of the function <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">x</span> <span class="pre">y</span> <span class="pre">↦</span> <span class="pre">|x</span> <span class="pre">-</span> <span class="pre">y|</span></code> from the case where <code class="docutils literal notranslate"><span class="pre">X</span> <span class="pre">=</span> <span class="pre">ℝ</span></code>.</p>
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@@ -544,8 +538,8 @@ the function <code class="docutils literal notranslate"><span class="pre">fun</sThey are called <code class="docutils literal notranslate"><span class="pre">EMetricSpace</span></code>, <code class="docutils literal notranslate"><span class="pre">PseudoMetricSpace</span></code> and <code class="docutils literal notranslate"><span class="pre">PseudoEMetricSpace</span></code> respectively (here “e” stands for “extended”).</p> <p>Note that our journey from <code class="docutils literal notranslate"><span class="pre">ℝ</span></code> to metric spaces jumped over the special case of normed spaces that also require linear algebra and will be explained as part of the calculus chapter.</p> <section id="convergence-and-continuity"> <h3><span class="section-number">8.2.1. </span>Convergence and continuity<a class="headerlink" href="#convergence-and-continuity" title="Permalink to this heading"></a></h3> <div class="section" id="convergence-and-continuity"> <h3><span class="section-number">8.2.1. </span>Convergence and continuity<a class="headerlink" href="#convergence-and-continuity" title="Permalink to this headline"></a></h3> <p>Using distance functions, we can already define convergent sequences and continuous functions between metric spaces. They are actually defined in a more general setting covered in the next section, but we have lemmas recasting the definition is terms of distances.</p>
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@@ -632,9 +626,9 @@ and get our final proof, now bordering obfuscation.</p><span class="n">Metric.continuousAt_iff</span> </pre></div> </div> </section> <section id="balls-open-sets-and-closed-sets"> <h3><span class="section-number">8.2.2. </span>Balls, open sets and closed sets<a class="headerlink" href="#balls-open-sets-and-closed-sets" title="Permalink to this heading"></a></h3> </div> <div class="section" id="balls-open-sets-and-closed-sets"> <h3><span class="section-number">8.2.2. </span>Balls, open sets and closed sets<a class="headerlink" href="#balls-open-sets-and-closed-sets" title="Permalink to this headline"></a></h3> <p>Once we have a distance function, the most important geometric definitions are (open) balls and closed balls.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">(</span><span class="n">r</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span>
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@@ -689,9 +683,9 @@ argument so we can invoke <code class="docutils literal notranslate"><span class<span class="n">Metric.nhds_basis_closedBall.mem_iff</span> </pre></div> </div> </section> <section id="compactness"> <h3><span class="section-number">8.2.3. </span>Compactness<a class="headerlink" href="#compactness" title="Permalink to this heading"></a></h3> </div> <div class="section" id="compactness"> <h3><span class="section-number">8.2.3. </span>Compactness<a class="headerlink" href="#compactness" title="Permalink to this headline"></a></h3> <p>Compactness is an important topological notion. It distinguishes subsets of a metric space that enjoy the same kind of properties as segments in reals compared to other intervals:</p> <ul class="simple">
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@@ -732,9 +726,9 @@ are deduced from more general versions, some of which will be discussed in later</pre></div> </div> <p>In a compact metric space any closed set is compact, this is <code class="docutils literal notranslate"><span class="pre">IsCompact.isClosed</span></code>.</p> </section> <section id="uniformly-continuous-functions"> <h3><span class="section-number">8.2.4. </span>Uniformly continuous functions<a class="headerlink" href="#uniformly-continuous-functions" title="Permalink to this heading"></a></h3> </div> <div class="section" id="uniformly-continuous-functions"> <h3><span class="section-number">8.2.4. </span>Uniformly continuous functions<a class="headerlink" href="#uniformly-continuous-functions" title="Permalink to this headline"></a></h3> <p>We now turn to uniformity notions on metric spaces : uniformly continuous functions, Cauchy sequences and completeness. Again those are defined in a more general context but we have lemmas in the metric name space to access their elementary definitions. We start with uniform continuity.</p>
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@@ -763,9 +757,9 @@ of the distance function on <code class="docutils literal notranslate"><span cla<span class="gr">sorry</span> </pre></div> </div> </section> <section id="completeness"> <h3><span class="section-number">8.2.5. </span>Completeness<a class="headerlink" href="#completeness" title="Permalink to this heading"></a></h3> </div> <div class="section" id="completeness"> <h3><span class="section-number">8.2.5. </span>Completeness<a class="headerlink" href="#completeness" title="Permalink to this headline"></a></h3> <p>A Cauchy sequence in a metric space is a sequence whose terms get closer and closer to each other. There are a couple of equivalent ways to state that idea. In particular converging sequences are Cauchy. The converse is true only in so-called <em>complete</em>
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@@ -854,12 +848,12 @@ define something inductively in the middle of a proof using <code class="docutil<span class="gr">sorry</span> </pre></div> </div> </section> </section> <section id="topological-spaces"> <span id="index-4"></span><span id="id4"></span><h2><span class="section-number">8.3. </span>Topological spaces<a class="headerlink" href="#topological-spaces" title="Permalink to this heading"></a></h2> <section id="fundamentals"> <h3><span class="section-number">8.3.1. </span>Fundamentals<a class="headerlink" href="#fundamentals" title="Permalink to this heading"></a></h3> </div> </div> <div class="section" id="topological-spaces"> <span id="index-4"></span><span id="id4"></span><h2><span class="section-number">8.3. </span>Topological spaces<a class="headerlink" href="#topological-spaces" title="Permalink to this headline"></a></h2> <div class="section" id="fundamentals"> <h3><span class="section-number">8.3.1. </span>Fundamentals<a class="headerlink" href="#fundamentals" title="Permalink to this headline"></a></h3> <p>We now go up in generality and introduce topological spaces. We will review the two main ways to define topological spaces and then explain how the category of topological spaces is much better behaved than the category of metric spaces. Note that we won’t be using mathlib category theory here, only having
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@@ -1046,9 +1040,9 @@ Let us explore that constraint “on paper” using notation <span class</div> <p>This ends our tour of how mathlib thinks that topological spaces fix defects of the theory of metric spaces by being a more functorial theory and having a complete lattice structure for any fixed type.</p> </section> <section id="separation-and-countability"> <h3><span class="section-number">8.3.2. </span>Separation and countability<a class="headerlink" href="#separation-and-countability" title="Permalink to this heading"></a></h3> </div> <div class="section" id="separation-and-countability"> <h3><span class="section-number">8.3.2. </span>Separation and countability<a class="headerlink" href="#separation-and-countability" title="Permalink to this headline"></a></h3> <p>We saw that the category of topological spaces have very nice properties. The price to pay for this is existence of rather pathological topological spaces. There are a number of assumptions you can make on a topological space to ensure its behavior
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@@ -1123,7 +1117,7 @@ together with the fact that <code class="docutils literal notranslate"><span cla<span class="bp">∃</span> <span class="n">φ</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">→</span> <span class="n">Y</span><span class="o">,</span> <span class="n">Continuous</span> <span class="n">φ</span> <span class="bp">∧</span> <span class="bp">∀</span> <span class="n">a</span> <span class="o">:</span> <span class="n">A</span><span class="o">,</span> <span class="n">φ</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">f</span> <span class="n">a</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="k">#check</span> <span class="bp">@</span><span class="n">HasBasis.tendsto_right_iff</span> <span class="k">#check</span> <span class="n">HasBasis.tendsto_right_iff</span> </pre></div> </div> <p>In addition to separation property, the main kind of assumption you can make on a topological
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@@ -1136,9 +1130,9 @@ of sets can be understood using sequences.</p><span class="n">mem_closure_iff_seq_limit</span> </pre></div> </div> </section> <section id="id5"> <h3><span class="section-number">8.3.3. </span>Compactness<a class="headerlink" href="#id5" title="Permalink to this heading"></a></h3> </div> <div class="section" id="id5"> <h3><span class="section-number">8.3.3. </span>Compactness<a class="headerlink" href="#id5" title="Permalink to this headline"></a></h3> <p>Let us now discuss how compactness is defined for topological spaces. As usual there are several ways to think about it and mathlib goes for the filter version.</p> <p>We first need to define cluster points of filters. Given a filter <code class="docutils literal notranslate"><span class="pre">F</span></code> on a topological space <code class="docutils literal notranslate"><span class="pre">X</span></code>,
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@@ -1199,9 +1193,9 @@ cover <code class="docutils literal notranslate"><span class="pre">s</span></cod<span class="n">isCompact_univ</span> </pre></div> </div> </section> </section> </section> </div> </div> </div> </div>
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@@ -31,15 +29,11 @@<nav data-toggle="wy-nav-shift" class="wy-nav-side"> <div class="wy-side-scroll"> <div class="wy-side-nav-search" > <a href="index.html" class="icon icon-home"> Mathematics in Lean <a href="index.html" class="icon icon-home"> Mathematics in Lean </a> <div role="search"> <form id="rtd-search-form" class="wy-form" action="search.html" method="get"> <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="text" name="q" placeholder="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
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@@ -95,8 +89,8 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <span class="target" id="differential-calculus"></span><section id="index-0"> <span id="id1"></span><h1><span class="section-number">9. </span>Differential Calculus<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> <span class="target" id="differential-calculus"></span><div class="section" id="index-0"> <span id="id1"></span><h1><span class="section-number">9. </span>Differential Calculus<a class="headerlink" href="#index-0" title="Permalink to this headline"></a></h1> <p>We now consider the formalization of notions from <em>analysis</em>, starting with differentiation in this chapter and turning integration and measure theory in the next.
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@@ -105,8 +99,8 @@ setting of functions from the real numbers to the real numbers,which is familiar from any introductory calculus class. In <a class="reference internal" href="#normed-spaces"><span class="std std-numref">Section 9.2</span></a>, we then consider the notion of a derivative in a much broader setting.</p> <section id="elementary-differential-calculus"> <span id="index-1"></span><span id="id2"></span><h2><span class="section-number">9.1. </span>Elementary Differential Calculus<a class="headerlink" href="#elementary-differential-calculus" title="Permalink to this heading"></a></h2> <div class="section" id="elementary-differential-calculus"> <span id="index-1"></span><span id="id2"></span><h2><span class="section-number">9.1. </span>Elementary Differential Calculus<a class="headerlink" href="#elementary-differential-calculus" title="Permalink to this headline"></a></h2> <p>Let <code class="docutils literal notranslate"><span class="pre">f</span></code> be a function from the reals to the reals. There is a difference between talking about the derivative of <code class="docutils literal notranslate"><span class="pre">f</span></code> at a single point and talking about the derivative function.
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@@ -160,7 +154,9 @@ definition of <code class="docutils literal notranslate"><span class="pre">deriv</div> <p>We can eve state Rolle’s theorem without any differentiability assumptions, which seems even weirder.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">hab</span> <span class="o">:</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">b</span><span class="o">)</span> <span class="o">(</span><span class="n">hfc</span> <span class="o">:</span> <span class="n">ContinuousOn</span> <span class="n">f</span> <span class="o">(</span><span class="n">Icc</span> <span class="n">a</span> <span class="n">b</span><span class="o">))</span> <span class="o">(</span><span class="n">hfI</span> <span class="o">:</span> <span class="n">f</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">f</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">open</span> <span class="n">Set</span> <span class="kd">example</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">ℝ</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">}</span> <span class="o">(</span><span class="n">hab</span> <span class="o">:</span> <span class="n">a</span> <span class="bp"><</span> <span class="n">b</span><span class="o">)</span> <span class="o">(</span><span class="n">hfc</span> <span class="o">:</span> <span class="n">ContinuousOn</span> <span class="n">f</span> <span class="o">(</span><span class="n">Icc</span> <span class="n">a</span> <span class="n">b</span><span class="o">))</span> <span class="o">(</span><span class="n">hfI</span> <span class="o">:</span> <span class="n">f</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">f</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="bp">∃</span> <span class="n">c</span> <span class="bp">∈</span> <span class="n">Ioo</span> <span class="n">a</span> <span class="n">b</span><span class="o">,</span> <span class="n">deriv</span> <span class="n">f</span> <span class="n">c</span> <span class="bp">=</span> <span class="mi">0</span> <span class="o">:=</span> <span class="n">exists_deriv_eq_zero</span> <span class="n">f</span> <span class="n">hab</span> <span class="n">hfc</span> <span class="n">hfI</span> </pre></div>
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@@ -177,11 +173,11 @@ seems even weirder.</p><span class="kd">example</span> <span class="o">:</span> <span class="n">deriv</span> <span class="n">sin</span> <span class="n">π</span> <span class="bp">=</span> <span class="bp">-</span><span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> </pre></div> </div> </section> <section id="differential-calculus-in-normed-spaces"> <span id="normed-spaces"></span><span id="index-2"></span><h2><span class="section-number">9.2. </span>Differential Calculus in Normed Spaces<a class="headerlink" href="#differential-calculus-in-normed-spaces" title="Permalink to this heading"></a></h2> <section id="id3"> <h3><span class="section-number">9.2.1. </span>Normed spaces<a class="headerlink" href="#id3" title="Permalink to this heading"></a></h3> </div> <div class="section" id="differential-calculus-in-normed-spaces"> <span id="normed-spaces"></span><span id="index-2"></span><h2><span class="section-number">9.2. </span>Differential Calculus in Normed Spaces<a class="headerlink" href="#differential-calculus-in-normed-spaces" title="Permalink to this headline"></a></h2> <div class="section" id="id3"> <h3><span class="section-number">9.2.1. </span>Normed spaces<a class="headerlink" href="#id3" title="Permalink to this headline"></a></h3> <p>Differentiation can be generalized beyond <code class="docutils literal notranslate"><span class="pre">ℝ</span></code> using the notion of a <em>normed vector space</em>, which encapsulates both direction and distance. We start with the notion of a <em>normed group</em>, which as an additive commutative
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@@ -244,9 +240,9 @@ complete as long as the field itself is complete.</p><span class="n">FiniteDimensional.complete</span> <span class="bp">𝕜</span> <span class="n">E</span> </pre></div> </div> </section> <section id="continuous-linear-maps"> <h3><span class="section-number">9.2.2. </span>Continuous linear maps<a class="headerlink" href="#continuous-linear-maps" title="Permalink to this heading"></a></h3> </div> <div class="section" id="continuous-linear-maps"> <h3><span class="section-number">9.2.2. </span>Continuous linear maps<a class="headerlink" href="#continuous-linear-maps" title="Permalink to this headline"></a></h3> <p>We now turn to the morphisms in the category of normed spaces, namely, continuous linear maps. In mathlib, the type of <code class="docutils literal notranslate"><span class="pre">𝕜</span></code>-linear continuous maps between normed spaces
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@@ -326,9 +322,9 @@ Minor ingredients include <code class="docutils literal notranslate"><span class<span class="gr">sorry</span> </pre></div> </div> </section> <section id="asymptotic-comparisons"> <h3><span class="section-number">9.2.3. </span>Asymptotic comparisons<a class="headerlink" href="#asymptotic-comparisons" title="Permalink to this heading"></a></h3> </div> <div class="section" id="asymptotic-comparisons"> <h3><span class="section-number">9.2.3. </span>Asymptotic comparisons<a class="headerlink" href="#asymptotic-comparisons" title="Permalink to this headline"></a></h3> <p>Defining differentiability also requires asymptotic comparisons. Mathlib has an extensive library covering the big O and little o relations, whose definitions are shown below.
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@@ -337,8 +333,6 @@ notation.Here we will only use little o to define differentiability.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">open</span> <span class="n">Asymptotics</span> <span class="kn">open</span> <span class="n">Asymptotics</span> <span class="kd">example</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">{</span><span class="n">E</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedGroup</span> <span class="n">E</span><span class="o">]</span> <span class="o">{</span><span class="n">F</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedGroup</span> <span class="n">F</span><span class="o">]</span> <span class="o">(</span><span class="n">c</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">l</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">α</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">E</span><span class="o">)</span> <span class="o">(</span><span class="n">g</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">F</span><span class="o">)</span> <span class="o">:</span> <span class="n">IsBigOWith</span> <span class="n">c</span> <span class="n">l</span> <span class="n">f</span> <span class="n">g</span> <span class="bp">↔</span> <span class="bp">∀ᶠ</span> <span class="n">x</span> <span class="k">in</span> <span class="n">l</span><span class="o">,</span> <span class="bp">‖</span><span class="n">f</span> <span class="n">x</span><span class="bp">‖</span> <span class="bp">≤</span> <span class="n">c</span> <span class="bp">*</span> <span class="bp">‖</span><span class="n">g</span> <span class="n">x</span><span class="bp">‖</span> <span class="o">:=</span> <span class="n">isBigOWith_iff</span>
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@@ -356,15 +350,17 @@ Here we will only use little o to define differentiability.</p><span class="n">Iff.rfl</span> </pre></div> </div> </section> <section id="differentiability"> <h3><span class="section-number">9.2.4. </span>Differentiability<a class="headerlink" href="#differentiability" title="Permalink to this heading"></a></h3> </div> <div class="section" id="differentiability"> <h3><span class="section-number">9.2.4. </span>Differentiability<a class="headerlink" href="#differentiability" title="Permalink to this headline"></a></h3> <p>We are now ready to discuss differentiable functions between normed spaces. In analogy the elementary one-dimensional, mathlib defines a predicate <code class="docutils literal notranslate"><span class="pre">has_fderiv_at</span></code> and a function <code class="docutils literal notranslate"><span class="pre">fderiv</span></code>. Here the letter “f” stands for <em>Fréchet</em>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="bp">𝕜</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NontriviallyNormedField</span> <span class="bp">𝕜</span><span class="o">]</span> <span class="o">{</span><span class="n">E</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedAddCommGroup</span> <span class="n">E</span><span class="o">]</span> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">open</span> <span class="n">Topology</span> <span class="kd">variable</span> <span class="o">{</span><span class="bp">𝕜</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NontriviallyNormedField</span> <span class="bp">𝕜</span><span class="o">]</span> <span class="o">{</span><span class="n">E</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedAddCommGroup</span> <span class="n">E</span><span class="o">]</span> <span class="o">[</span><span class="n">NormedSpace</span> <span class="bp">𝕜</span> <span class="n">E</span><span class="o">]</span> <span class="o">{</span><span class="n">F</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">NormedAddCommGroup</span> <span class="n">F</span><span class="o">]</span> <span class="o">[</span><span class="n">NormedSpace</span> <span class="bp">𝕜</span> <span class="n">F</span><span class="o">]</span> <span class="kd">example</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span> <span class="n">F</span><span class="o">)</span> <span class="o">(</span><span class="n">f'</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">F</span><span class="o">)</span> <span class="o">(</span><span class="n">x₀</span> <span class="o">:</span> <span class="n">E</span><span class="o">)</span> <span class="o">:</span>
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@@ -425,7 +421,7 @@ from the left and from the right, and that it is strictly differentiable.</p><span class="bp">∀ᶠ</span> <span class="n">x</span> <span class="k">in</span> <span class="bp">𝓝</span> <span class="o">(</span><span class="n">f</span> <span class="n">a</span><span class="o">),</span> <span class="n">f</span> <span class="o">(</span><span class="n">hf.localInverse</span> <span class="n">f</span> <span class="n">f'</span> <span class="n">a</span> <span class="n">x</span><span class="o">)</span> <span class="bp">=</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">hf.eventually_right_inverse</span> <span class="kd">example</span> <span class="o">[</span><span class="n">CompleteSpace</span> <span class="n">E</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span> <span class="n">F</span><span class="o">}</span> <span class="o">{</span><span class="n">f'</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">≃</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">F</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="o">:</span> <span class="n">E</span><span class="o">}</span> <span class="kd">example</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span> <span class="n">F</span><span class="o">}</span> <span class="o">{</span><span class="n">f'</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">≃</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">F</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="o">:</span> <span class="n">E</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">HasStrictFDerivAt</span> <span class="n">f</span> <span class="o">(</span><span class="n">f'</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">→</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">F</span><span class="o">)</span> <span class="n">a</span><span class="o">)</span> <span class="o">:</span> <span class="n">HasStrictFDerivAt</span> <span class="o">(</span><span class="n">HasStrictFDerivAt.localInverse</span> <span class="n">f</span> <span class="n">f'</span> <span class="n">a</span> <span class="n">hf</span><span class="o">)</span> <span class="o">(</span><span class="n">f'.symm</span> <span class="o">:</span> <span class="n">F</span> <span class="bp">→</span><span class="n">L</span><span class="o">[</span><span class="bp">𝕜</span><span class="o">]</span> <span class="n">E</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">a</span><span class="o">)</span> <span class="o">:=</span> <span class="n">HasStrictFDerivAt.to_localInverse</span> <span class="n">hf</span>
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@@ -439,9 +435,9 @@ For example, you may want to use one-sided derivatives in theone-dimensional setting. The means to do so are found in mathlib in a more general context; see <code class="docutils literal notranslate"><span class="pre">HasFDerivWithinAt</span></code> or the even more general <code class="docutils literal notranslate"><span class="pre">HasFDerivAtFilter</span></code>.</p> </section> </section> </section> </div> </div> </div> </div>
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@@ -90,10 +84,10 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <span class="target" id="integration-and-measure-theory"></span><section id="index-0"> <span id="id1"></span><h1><span class="section-number">10. </span>Integration and Measure Theory<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> <section id="elementary-integration"> <span id="index-1"></span><span id="id2"></span><h2><span class="section-number">10.1. </span>Elementary Integration<a class="headerlink" href="#elementary-integration" title="Permalink to this heading"></a></h2> <span class="target" id="integration-and-measure-theory"></span><div class="section" id="index-0"> <span id="id1"></span><h1><span class="section-number">10. </span>Integration and Measure Theory<a class="headerlink" href="#index-0" title="Permalink to this headline"></a></h1> <div class="section" id="elementary-integration"> <span id="index-1"></span><span id="id2"></span><h2><span class="section-number">10.1. </span>Elementary Integration<a class="headerlink" href="#elementary-integration" title="Permalink to this headline"></a></h2> <p>We first focus on integration of functions on finite intervals in <code class="docutils literal notranslate"><span class="pre">ℝ</span></code>. We can integrate elementary functions.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">open</span> <span class="n">MeasureTheory</span> <span class="n">intervalIntegral</span>
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@@ -130,9 +124,9 @@ which are not shown here, are not equivalent.)</p><span class="n">rfl</span> </pre></div> </div> </section> <section id="measure-theory"> <span id="index-2"></span><span id="id3"></span><h2><span class="section-number">10.2. </span>Measure Theory<a class="headerlink" href="#measure-theory" title="Permalink to this heading"></a></h2> </div> <div class="section" id="measure-theory"> <span id="index-2"></span><span id="id3"></span><h2><span class="section-number">10.2. </span>Measure Theory<a class="headerlink" href="#measure-theory" title="Permalink to this headline"></a></h2> <p>The general context for integration in mathlib is measure theory. Even the elementary integrals of the previous section are in fact Bochner integrals. Bochner integration is a generalization of Lebesgue integration where the target space can be any Banach space,
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@@ -140,14 +134,14 @@ not necessarily finite dimensional.</p><p>The first component in the development of measure theory is the notion of a <span class="math notranslate nohighlight">\(\sigma\)</span>-algebra of sets, which are called the <em>measurable</em> sets. The type class <code class="docutils literal notranslate"><span class="pre">measurable_space</span></code> serves to equip a type with such a structure. The type class <code class="docutils literal notranslate"><span class="pre">MeasurableSpace</span></code> serves to equip a type with such a structure. The sets <code class="docutils literal notranslate"><span class="pre">empty</span></code> and <code class="docutils literal notranslate"><span class="pre">univ</span></code> are measurable, the complement of a measurable set is measurable, and a countable union or intersection of measurable sets is measurable. Note that these axioms are redundant; if you <code class="docutils literal notranslate"><span class="pre">#print</span> <span class="pre">measurable_space</span></code>, Note that these axioms are redundant; if you <code class="docutils literal notranslate"><span class="pre">#print</span> <span class="pre">MeasurableSpace</span></code>, you will see the ones that mathlib uses. As the examples below show, countability assumptions can be expressed using the <code class="docutils literal notranslate"><span class="pre">encodable</span></code> type class.</p> <code class="docutils literal notranslate"><span class="pre">Encodable</span></code> type class.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">α</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="o">[</span><span class="n">MeasurableSpace</span> <span class="n">α</span><span class="o">]</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">MeasurableSet</span> <span class="o">(</span><span class="bp">∅</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">α</span><span class="o">)</span> <span class="o">:=</span>
-
@@ -207,9 +201,9 @@ almost everywhere.</p><span class="n">Iff.rfl</span> </pre></div> </div> </section> <section id="integration"> <span id="id4"></span><h2><span class="section-number">10.3. </span>Integration<a class="headerlink" href="#integration" title="Permalink to this heading"></a></h2> </div> <div class="section" id="integration"> <span id="id4"></span><h2><span class="section-number">10.3. </span>Integration<a class="headerlink" href="#integration" title="Permalink to this headline"></a></h2> <p>Now that we have measurable spaces and measures we can consider integrals. As explained above, mathlib uses a very general notion of integration that allows any Banach space as the target.
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@@ -240,7 +234,9 @@ So in all cases we have the following lemma.</p><p>We now quickly explain how to access the most important theorems in integration theory, starting with the dominated convergence theorem. There are several versions in mathlib, and here we only show the most basic one.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">F</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">E</span><span class="o">}</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">E</span><span class="o">}</span> <span class="o">(</span><span class="n">bound</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">hmeas</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="n">AEStronglyMeasurable</span> <span class="o">(</span><span class="n">F</span> <span class="n">n</span><span class="o">)</span> <span class="n">μ</span><span class="o">)</span> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">open</span> <span class="n">Filter</span> <span class="kd">example</span> <span class="o">{</span><span class="n">F</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">→</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">E</span><span class="o">}</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">E</span><span class="o">}</span> <span class="o">(</span><span class="n">bound</span> <span class="o">:</span> <span class="n">α</span> <span class="bp">→</span> <span class="n">ℝ</span><span class="o">)</span> <span class="o">(</span><span class="n">hmeas</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="n">AEStronglyMeasurable</span> <span class="o">(</span><span class="n">F</span> <span class="n">n</span><span class="o">)</span> <span class="n">μ</span><span class="o">)</span> <span class="o">(</span><span class="n">hint</span> <span class="o">:</span> <span class="n">Integrable</span> <span class="n">bound</span> <span class="n">μ</span><span class="o">)</span> <span class="o">(</span><span class="n">hbound</span> <span class="o">:</span> <span class="bp">∀</span> <span class="n">n</span><span class="o">,</span> <span class="bp">∀ᵐ</span> <span class="n">a</span> <span class="bp">∂</span><span class="n">μ</span><span class="o">,</span> <span class="bp">‖</span><span class="n">F</span> <span class="n">n</span> <span class="n">a</span><span class="bp">‖</span> <span class="bp">≤</span> <span class="n">bound</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">hlim</span> <span class="o">:</span> <span class="bp">∀ᵐ</span> <span class="n">a</span> <span class="bp">∂</span><span class="n">μ</span><span class="o">,</span> <span class="n">Tendsto</span> <span class="o">(</span><span class="k">fun</span> <span class="n">n</span> <span class="o">:</span> <span class="n">ℕ</span> <span class="bp">↦</span> <span class="n">F</span> <span class="n">n</span> <span class="n">a</span><span class="o">)</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="o">(</span><span class="n">f</span> <span class="n">a</span><span class="o">)))</span> <span class="o">:</span> <span class="n">Tendsto</span> <span class="o">(</span><span class="k">fun</span> <span class="n">n</span> <span class="bp">↦</span> <span class="bp">∫</span> <span class="n">a</span><span class="o">,</span> <span class="n">F</span> <span class="n">n</span> <span class="n">a</span> <span class="bp">∂</span><span class="n">μ</span><span class="o">)</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">𝓝</span> <span class="o">(</span><span class="bp">∫</span> <span class="n">a</span><span class="o">,</span> <span class="n">f</span> <span class="n">a</span> <span class="bp">∂</span><span class="n">μ</span><span class="o">))</span> <span class="o">:=</span>
-
@@ -282,8 +278,8 @@ gives finite mass to compact sets, and give positive mass to open sets.</p><span class="n">integral_image_eq_integral_abs_det_fderiv_smul</span> <span class="n">μ</span> <span class="n">hs</span> <span class="n">hf</span> <span class="n">h_inj</span> <span class="n">g</span> </pre></div> </div> </section> </section> </div> </div> </div>
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@@ -1,123 +0,0 @@/* Compatability shim for jQuery and underscores.js. * * Copyright Sphinx contributors * Released under the two clause BSD licence */ /** * small helper function to urldecode strings * * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/decodeURIComponent#Decoding_query_parameters_from_a_URL */ jQuery.urldecode = function(x) { if (!x) { return x } return decodeURIComponent(x.replace(/\+/g, ' ')); }; /** * small helper function to urlencode strings */ jQuery.urlencode = encodeURIComponent; /** * This function returns the parsed url parameters of the * current request. Multiple values per key are supported, * it will always return arrays of strings for the value parts. */ jQuery.getQueryParameters = function(s) { if (typeof s === 'undefined') s = document.location.search; var parts = s.substr(s.indexOf('?') + 1).split('&'); var result = {}; for (var i = 0; i < parts.length; i++) { var tmp = parts[i].split('=', 2); var key = jQuery.urldecode(tmp[0]); var value = jQuery.urldecode(tmp[1]); if (key in result) result[key].push(value); else result[key] = [value]; } return result; }; /** * highlight a given string on a jquery object by wrapping it in * span elements with the given class name. */ jQuery.fn.highlightText = function(text, className) { function highlight(node, addItems) { if (node.nodeType === 3) { var val = node.nodeValue; var pos = val.toLowerCase().indexOf(text); if (pos >= 0 && !jQuery(node.parentNode).hasClass(className) && !jQuery(node.parentNode).hasClass("nohighlight")) { var span; var isInSVG = jQuery(node).closest("body, svg, foreignObject").is("svg"); if (isInSVG) { span = document.createElementNS("http://www.w3.org/2000/svg", "tspan"); } else { span = document.createElement("span"); span.className = className; } span.appendChild(document.createTextNode(val.substr(pos, text.length))); node.parentNode.insertBefore(span, node.parentNode.insertBefore( document.createTextNode(val.substr(pos + text.length)), node.nextSibling)); node.nodeValue = val.substr(0, pos); if (isInSVG) { var rect = document.createElementNS("http://www.w3.org/2000/svg", "rect"); var bbox = node.parentElement.getBBox(); rect.x.baseVal.value = bbox.x; rect.y.baseVal.value = bbox.y; rect.width.baseVal.value = bbox.width; rect.height.baseVal.value = bbox.height; rect.setAttribute('class', className); addItems.push({ "parent": node.parentNode, "target": rect}); } } } else if (!jQuery(node).is("button, select, textarea")) { jQuery.each(node.childNodes, function() { highlight(this, addItems); }); } } var addItems = []; var result = this.each(function() { highlight(this, addItems); }); for (var i = 0; i < addItems.length; ++i) { jQuery(addItems[i].parent).before(addItems[i].target); } return result; }; /* * backward compatibility for jQuery.browser * This will be supported until firefox bug is fixed. */ if (!jQuery.browser) { jQuery.uaMatch = function(ua) { ua = ua.toLowerCase(); var match = /(chrome)[ \/]([\w.]+)/.exec(ua) || /(webkit)[ \/]([\w.]+)/.exec(ua) || /(opera)(?:.*version|)[ \/]([\w.]+)/.exec(ua) || /(msie) ([\w.]+)/.exec(ua) || ua.indexOf("compatible") < 0 && /(mozilla)(?:.*? rv:([\w.]+)|)/.exec(ua) || []; return { browser: match[ 1 ] || "", version: match[ 2 ] || "0" }; }; jQuery.browser = {}; jQuery.browser[jQuery.uaMatch(navigator.userAgent).browser] = true; }
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@@ -4,7 +4,7 @@* * Sphinx stylesheet -- basic theme. * * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS. * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */
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@@ -222,7 +222,7 @@ table.modindextable td {/* -- general body styles --------------------------------------------------- */ div.body { min-width: 360px; min-width: 450px; max-width: 800px; }
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@@ -237,6 +237,16 @@ a.headerlink {visibility: hidden; } a.brackets:before, span.brackets > a:before{ content: "["; } a.brackets:after, span.brackets > a:after { content: "]"; } h1:hover > a.headerlink, h2:hover > a.headerlink, h3:hover > a.headerlink,
-
@@ -325,16 +335,12 @@ p.sidebar-title {font-weight: bold; } nav.contents, aside.topic, div.admonition, div.topic, blockquote { clear: left; } /* -- topics ---------------------------------------------------------------- */ nav.contents, aside.topic, div.topic { border: 1px solid #ccc; padding: 7px;
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@@ -373,8 +379,6 @@ div.body p.centered {div.sidebar > :last-child, aside.sidebar > :last-child, nav.contents > :last-child, aside.topic > :last-child, div.topic > :last-child, div.admonition > :last-child { margin-bottom: 0;
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@@ -382,8 +386,6 @@ div.admonition > :last-child {div.sidebar::after, aside.sidebar::after, nav.contents::after, aside.topic::after, div.topic::after, div.admonition::after, blockquote::after {
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@@ -426,6 +428,10 @@ table.docutils td, table.docutils th {border-bottom: 1px solid #aaa; } table.footnote td, table.footnote th { border: 0 !important; } th { text-align: left; padding-right: 5px;
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@@ -609,26 +615,19 @@ ul.simple p {margin-bottom: 0; } aside.footnote > span, div.citation > span { dl.footnote > dt, dl.citation > dt { float: left; margin-right: 0.5em; } aside.footnote > span:last-of-type, div.citation > span:last-of-type { padding-right: 0.5em; } aside.footnote > p { margin-left: 2em; } div.citation > p { margin-left: 4em; } aside.footnote > p:last-of-type, div.citation > p:last-of-type { dl.footnote > dd, dl.citation > dd { margin-bottom: 0em; } aside.footnote > p:last-of-type:after, div.citation > p:last-of-type:after { dl.footnote > dd:after, dl.citation > dd:after { content: ""; clear: both; }
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@@ -645,6 +644,10 @@ dl.field-list > dt {padding-right: 5px; } dl.field-list > dt:after { content: ":"; } dl.field-list > dd { padding-left: 0.5em; margin-top: 0em;
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@@ -2,155 +2,325 @@* doctools.js * ~~~~~~~~~~~ * * Base JavaScript utilities for all Sphinx HTML documentation. * Sphinx JavaScript utilities for all documentation. * * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS. * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */ "use strict"; const BLACKLISTED_KEY_CONTROL_ELEMENTS = new Set([ "TEXTAREA", "INPUT", "SELECT", "BUTTON", ]); const _ready = (callback) => { if (document.readyState !== "loading") { callback(); } else { document.addEventListener("DOMContentLoaded", callback); /** * select a different prefix for underscore */ $u = _.noConflict(); /** * make the code below compatible with browsers without * an installed firebug like debugger if (!window.console || !console.firebug) { var names = ["log", "debug", "info", "warn", "error", "assert", "dir", "dirxml", "group", "groupEnd", "time", "timeEnd", "count", "trace", "profile", "profileEnd"]; window.console = {}; for (var i = 0; i < names.length; ++i) window.console[names[i]] = function() {}; } */ /** * small helper function to urldecode strings * * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/decodeURIComponent#Decoding_query_parameters_from_a_URL */ jQuery.urldecode = function(x) { if (!x) { return x } return decodeURIComponent(x.replace(/\+/g, ' ')); }; /** * small helper function to urlencode strings */ jQuery.urlencode = encodeURIComponent; /** * This function returns the parsed url parameters of the * current request. Multiple values per key are supported, * it will always return arrays of strings for the value parts. */ jQuery.getQueryParameters = function(s) { if (typeof s === 'undefined') s = document.location.search; var parts = s.substr(s.indexOf('?') + 1).split('&'); var result = {}; for (var i = 0; i < parts.length; i++) { var tmp = parts[i].split('=', 2); var key = jQuery.urldecode(tmp[0]); var value = jQuery.urldecode(tmp[1]); if (key in result) result[key].push(value); else result[key] = [value]; } return result; }; /** * highlight a given string on a jquery object by wrapping it in * span elements with the given class name. */ jQuery.fn.highlightText = function(text, className) { function highlight(node, addItems) { if (node.nodeType === 3) { var val = node.nodeValue; var pos = val.toLowerCase().indexOf(text); if (pos >= 0 && !jQuery(node.parentNode).hasClass(className) && !jQuery(node.parentNode).hasClass("nohighlight")) { var span; var isInSVG = jQuery(node).closest("body, svg, foreignObject").is("svg"); if (isInSVG) { span = document.createElementNS("http://www.w3.org/2000/svg", "tspan"); } else { span = document.createElement("span"); span.className = className; } span.appendChild(document.createTextNode(val.substr(pos, text.length))); node.parentNode.insertBefore(span, node.parentNode.insertBefore( document.createTextNode(val.substr(pos + text.length)), node.nextSibling)); node.nodeValue = val.substr(0, pos); if (isInSVG) { var rect = document.createElementNS("http://www.w3.org/2000/svg", "rect"); var bbox = node.parentElement.getBBox(); rect.x.baseVal.value = bbox.x; rect.y.baseVal.value = bbox.y; rect.width.baseVal.value = bbox.width; rect.height.baseVal.value = bbox.height; rect.setAttribute('class', className); addItems.push({ "parent": node.parentNode, "target": rect}); } } } else if (!jQuery(node).is("button, select, textarea")) { jQuery.each(node.childNodes, function() { highlight(this, addItems); }); } } var addItems = []; var result = this.each(function() { highlight(this, addItems); }); for (var i = 0; i < addItems.length; ++i) { jQuery(addItems[i].parent).before(addItems[i].target); } return result; }; /* * backward compatibility for jQuery.browser * This will be supported until firefox bug is fixed. */ if (!jQuery.browser) { jQuery.uaMatch = function(ua) { ua = ua.toLowerCase(); var match = /(chrome)[ \/]([\w.]+)/.exec(ua) || /(webkit)[ \/]([\w.]+)/.exec(ua) || /(opera)(?:.*version|)[ \/]([\w.]+)/.exec(ua) || /(msie) ([\w.]+)/.exec(ua) || ua.indexOf("compatible") < 0 && /(mozilla)(?:.*? rv:([\w.]+)|)/.exec(ua) || []; return { browser: match[ 1 ] || "", version: match[ 2 ] || "0" }; }; jQuery.browser = {}; jQuery.browser[jQuery.uaMatch(navigator.userAgent).browser] = true; } /** * Small JavaScript module for the documentation. */ const Documentation = { init: () => { Documentation.initDomainIndexTable(); Documentation.initOnKeyListeners(); var Documentation = { init : function() { this.fixFirefoxAnchorBug(); this.highlightSearchWords(); this.initIndexTable(); if (DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS) { this.initOnKeyListeners(); } }, /** * i18n support */ TRANSLATIONS: {}, PLURAL_EXPR: (n) => (n === 1 ? 0 : 1), LOCALE: "unknown", TRANSLATIONS : {}, PLURAL_EXPR : function(n) { return n === 1 ? 0 : 1; }, LOCALE : 'unknown', // gettext and ngettext don't access this so that the functions // can safely bound to a different name (_ = Documentation.gettext) gettext: (string) => { const translated = Documentation.TRANSLATIONS[string]; switch (typeof translated) { case "undefined": return string; // no translation case "string": return translated; // translation exists default: return translated[0]; // (singular, plural) translation tuple exists } gettext : function(string) { var translated = Documentation.TRANSLATIONS[string]; if (typeof translated === 'undefined') return string; return (typeof translated === 'string') ? translated : translated[0]; }, ngettext: (singular, plural, n) => { const translated = Documentation.TRANSLATIONS[singular]; if (typeof translated !== "undefined") return translated[Documentation.PLURAL_EXPR(n)]; return n === 1 ? singular : plural; ngettext : function(singular, plural, n) { var translated = Documentation.TRANSLATIONS[singular]; if (typeof translated === 'undefined') return (n == 1) ? singular : plural; return translated[Documentation.PLURALEXPR(n)]; }, addTranslations: (catalog) => { Object.assign(Documentation.TRANSLATIONS, catalog.messages); Documentation.PLURAL_EXPR = new Function( "n", `return (${catalog.plural_expr})` ); Documentation.LOCALE = catalog.locale; addTranslations : function(catalog) { for (var key in catalog.messages) this.TRANSLATIONS[key] = catalog.messages[key]; this.PLURAL_EXPR = new Function('n', 'return +(' + catalog.plural_expr + ')'); this.LOCALE = catalog.locale; }, /** * helper function to focus on search bar * add context elements like header anchor links */ focusSearchBar: () => { document.querySelectorAll("input[name=q]")[0]?.focus(); addContextElements : function() { $('div[id] > :header:first').each(function() { $('<a class="headerlink">\u00B6</a>'). attr('href', '#' + this.id). attr('title', _('Permalink to this headline')). appendTo(this); }); $('dt[id]').each(function() { $('<a class="headerlink">\u00B6</a>'). attr('href', '#' + this.id). attr('title', _('Permalink to this definition')). appendTo(this); }); }, /** * Initialise the domain index toggle buttons * workaround a firefox stupidity * see: https://bugzilla.mozilla.org/show_bug.cgi?id=645075 */ initDomainIndexTable: () => { const toggler = (el) => { const idNumber = el.id.substr(7); const toggledRows = document.querySelectorAll(`tr.cg-${idNumber}`); if (el.src.substr(-9) === "minus.png") { el.src = `${el.src.substr(0, el.src.length - 9)}plus.png`; toggledRows.forEach((el) => (el.style.display = "none")); } else { el.src = `${el.src.substr(0, el.src.length - 8)}minus.png`; toggledRows.forEach((el) => (el.style.display = "")); fixFirefoxAnchorBug : function() { if (document.location.hash && $.browser.mozilla) window.setTimeout(function() { document.location.href += ''; }, 10); }, /** * highlight the search words provided in the url in the text */ highlightSearchWords : function() { var params = $.getQueryParameters(); var terms = (params.highlight) ? params.highlight[0].split(/\s+/) : []; if (terms.length) { var body = $('div.body'); if (!body.length) { body = $('body'); } }; window.setTimeout(function() { $.each(terms, function() { body.highlightText(this.toLowerCase(), 'highlighted'); }); }, 10); $('<p class="highlight-link"><a href="javascript:Documentation.' + 'hideSearchWords()">' + _('Hide Search Matches') + '</a></p>') .appendTo($('#searchbox')); } }, /** * init the domain index toggle buttons */ initIndexTable : function() { var togglers = $('img.toggler').click(function() { var src = $(this).attr('src'); var idnum = $(this).attr('id').substr(7); $('tr.cg-' + idnum).toggle(); if (src.substr(-9) === 'minus.png') $(this).attr('src', src.substr(0, src.length-9) + 'plus.png'); else $(this).attr('src', src.substr(0, src.length-8) + 'minus.png'); }).css('display', ''); if (DOCUMENTATION_OPTIONS.COLLAPSE_INDEX) { togglers.click(); } }, const togglerElements = document.querySelectorAll("img.toggler"); togglerElements.forEach((el) => el.addEventListener("click", (event) => toggler(event.currentTarget)) ); togglerElements.forEach((el) => (el.style.display = "")); if (DOCUMENTATION_OPTIONS.COLLAPSE_INDEX) togglerElements.forEach(toggler); /** * helper function to hide the search marks again */ hideSearchWords : function() { $('#searchbox .highlight-link').fadeOut(300); $('span.highlighted').removeClass('highlighted'); var url = new URL(window.location); url.searchParams.delete('highlight'); window.history.replaceState({}, '', url); }, initOnKeyListeners: () => { // only install a listener if it is really needed if ( !DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS && !DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS ) return; document.addEventListener("keydown", (event) => { // bail for input elements if (BLACKLISTED_KEY_CONTROL_ELEMENTS.has(document.activeElement.tagName)) return; // bail with special keys if (event.altKey || event.ctrlKey || event.metaKey) return; if (!event.shiftKey) { switch (event.key) { case "ArrowLeft": if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS) break; const prevLink = document.querySelector('link[rel="prev"]'); if (prevLink && prevLink.href) { window.location.href = prevLink.href; event.preventDefault(); /** * make the url absolute */ makeURL : function(relativeURL) { return DOCUMENTATION_OPTIONS.URL_ROOT + '/' + relativeURL; }, /** * get the current relative url */ getCurrentURL : function() { var path = document.location.pathname; var parts = path.split(/\//); $.each(DOCUMENTATION_OPTIONS.URL_ROOT.split(/\//), function() { if (this === '..') parts.pop(); }); var url = parts.join('/'); return path.substring(url.lastIndexOf('/') + 1, path.length - 1); }, initOnKeyListeners: function() { $(document).keydown(function(event) { var activeElementType = document.activeElement.tagName; // don't navigate when in search box, textarea, dropdown or button if (activeElementType !== 'TEXTAREA' && activeElementType !== 'INPUT' && activeElementType !== 'SELECT' && activeElementType !== 'BUTTON' && !event.altKey && !event.ctrlKey && !event.metaKey && !event.shiftKey) { switch (event.keyCode) { case 37: // left var prevHref = $('link[rel="prev"]').prop('href'); if (prevHref) { window.location.href = prevHref; return false; } break; case "ArrowRight": if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS) break; const nextLink = document.querySelector('link[rel="next"]'); if (nextLink && nextLink.href) { window.location.href = nextLink.href; event.preventDefault(); case 39: // right var nextHref = $('link[rel="next"]').prop('href'); if (nextHref) { window.location.href = nextHref; return false; } break; } } // some keyboard layouts may need Shift to get / switch (event.key) { case "/": if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) break; Documentation.focusSearchBar(); event.preventDefault(); } }); }, } }; // quick alias for translations const _ = Documentation.gettext; _ = Documentation.gettext; _ready(Documentation.init); $(document).ready(function() { Documentation.init(); });
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@@ -1,14 +1,12 @@var DOCUMENTATION_OPTIONS = { URL_ROOT: document.getElementById("documentation_options").getAttribute('data-url_root'), VERSION: '0.1', LANGUAGE: 'en', LANGUAGE: 'None', COLLAPSE_INDEX: false, BUILDER: 'html', FILE_SUFFIX: '.html', LINK_SUFFIX: '.html', HAS_SOURCE: true, SOURCELINK_SUFFIX: '.txt', NAVIGATION_WITH_KEYS: false, SHOW_SEARCH_SUMMARY: true, ENABLE_SEARCH_SHORTCUTS: true, NAVIGATION_WITH_KEYS: false };
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@@ -0,0 +1,10872 @@/*! * jQuery JavaScript Library v3.5.1 * https://jquery.com/ * * Includes Sizzle.js * https://sizzlejs.com/ * * Copyright JS Foundation and other contributors * Released under the MIT license * https://jquery.org/license * * Date: 2020-05-04T22:49Z */ ( function( global, factory ) { "use strict"; if ( typeof module === "object" && typeof module.exports === "object" ) { // For CommonJS and CommonJS-like environments where a proper `window` // is present, execute the factory and get jQuery. // For environments that do not have a `window` with a `document` // (such as Node.js), expose a factory as module.exports. // This accentuates the need for the creation of a real `window`. // e.g. var jQuery = require("jquery")(window); // See ticket #14549 for more info. module.exports = global.document ? factory( global, true ) : function( w ) { if ( !w.document ) { throw new Error( "jQuery requires a window with a document" ); } return factory( w ); }; } else { factory( global ); } // Pass this if window is not defined yet } )( typeof window !== "undefined" ? window : this, function( window, noGlobal ) { // Edge <= 12 - 13+, Firefox <=18 - 45+, IE 10 - 11, Safari 5.1 - 9+, iOS 6 - 9.1 // throw exceptions when non-strict code (e.g., ASP.NET 4.5) accesses strict mode // arguments.callee.caller (trac-13335). But as of jQuery 3.0 (2016), strict mode should be common // enough that all such attempts are guarded in a try block. "use strict"; var arr = []; var getProto = Object.getPrototypeOf; var slice = arr.slice; var flat = arr.flat ? function( array ) { return arr.flat.call( array ); } : function( array ) { return arr.concat.apply( [], array ); }; var push = arr.push; var indexOf = arr.indexOf; var class2type = {}; var toString = class2type.toString; var hasOwn = class2type.hasOwnProperty; var fnToString = hasOwn.toString; var ObjectFunctionString = fnToString.call( Object ); var support = {}; var isFunction = function isFunction( obj ) { // Support: Chrome <=57, Firefox <=52 // In some browsers, typeof returns "function" for HTML <object> elements // (i.e., `typeof document.createElement( "object" ) === "function"`). // We don't want to classify *any* DOM node as a function. return typeof obj === "function" && typeof obj.nodeType !== "number"; }; var isWindow = function isWindow( obj ) { return obj != null && obj === obj.window; }; var document = window.document; var preservedScriptAttributes = { type: true, src: true, nonce: true, noModule: true }; function DOMEval( code, node, doc ) { doc = doc || document; var i, val, script = doc.createElement( "script" ); script.text = code; if ( node ) { for ( i in preservedScriptAttributes ) { // Support: Firefox 64+, Edge 18+ // Some browsers don't support the "nonce" property on scripts. // On the other hand, just using `getAttribute` is not enough as // the `nonce` attribute is reset to an empty string whenever it // becomes browsing-context connected. // See https://github.com/whatwg/html/issues/2369 // See https://html.spec.whatwg.org/#nonce-attributes // The `node.getAttribute` check was added for the sake of // `jQuery.globalEval` so that it can fake a nonce-containing node // via an object. val = node[ i ] || node.getAttribute && node.getAttribute( i ); if ( val ) { script.setAttribute( i, val ); } } } doc.head.appendChild( script ).parentNode.removeChild( script ); } function toType( obj ) { if ( obj == null ) { return obj + ""; } // Support: Android <=2.3 only (functionish RegExp) return typeof obj === "object" || typeof obj === "function" ? class2type[ toString.call( obj ) ] || "object" : typeof obj; } /* global Symbol */ // Defining this global in .eslintrc.json would create a danger of using the global // unguarded in another place, it seems safer to define global only for this module var version = "3.5.1", // Define a local copy of jQuery jQuery = function( selector, context ) { // The jQuery object is actually just the init constructor 'enhanced' // Need init if jQuery is called (just allow error to be thrown if not included) return new jQuery.fn.init( selector, context ); }; jQuery.fn = jQuery.prototype = { // The current version of jQuery being used jquery: version, constructor: jQuery, // The default length of a jQuery object is 0 length: 0, toArray: function() { return slice.call( this ); }, // Get the Nth element in the matched element set OR // Get the whole matched element set as a clean array get: function( num ) { // Return all the elements in a clean array if ( num == null ) { return slice.call( this ); } // Return just the one element from the set return num < 0 ? this[ num + this.length ] : this[ num ]; }, // Take an array of elements and push it onto the stack // (returning the new matched element set) pushStack: function( elems ) { // Build a new jQuery matched element set var ret = jQuery.merge( this.constructor(), elems ); // Add the old object onto the stack (as a reference) ret.prevObject = this; // Return the newly-formed element set return ret; }, // Execute a callback for every element in the matched set. each: function( callback ) { return jQuery.each( this, callback ); }, map: function( callback ) { return this.pushStack( jQuery.map( this, function( elem, i ) { return callback.call( elem, i, elem ); } ) ); }, slice: function() { return this.pushStack( slice.apply( this, arguments ) ); }, first: function() { return this.eq( 0 ); }, last: function() { return this.eq( -1 ); }, even: function() { return this.pushStack( jQuery.grep( this, function( _elem, i ) { return ( i + 1 ) % 2; } ) ); }, odd: function() { return this.pushStack( jQuery.grep( this, function( _elem, i ) { return i % 2; } ) ); }, eq: function( i ) { var len = this.length, j = +i + ( i < 0 ? len : 0 ); return this.pushStack( j >= 0 && j < len ? [ this[ j ] ] : [] ); }, end: function() { return this.prevObject || this.constructor(); }, // For internal use only. // Behaves like an Array's method, not like a jQuery method. push: push, sort: arr.sort, splice: arr.splice }; jQuery.extend = jQuery.fn.extend = function() { var options, name, src, copy, copyIsArray, clone, target = arguments[ 0 ] || {}, i = 1, length = arguments.length, deep = false; // Handle a deep copy situation if ( typeof target === "boolean" ) { deep = target; // Skip the boolean and the target target = arguments[ i ] || {}; i++; } // Handle case when target is a string or something (possible in deep copy) if ( typeof target !== "object" && !isFunction( target ) ) { target = {}; } // Extend jQuery itself if only one argument is passed if ( i === length ) { target = this; i--; } for ( ; i < length; i++ ) { // Only deal with non-null/undefined values if ( ( options = arguments[ i ] ) != null ) { // Extend the base object for ( name in options ) { copy = options[ name ]; // Prevent Object.prototype pollution // Prevent never-ending loop if ( name === "__proto__" || target === copy ) { continue; } // Recurse if we're merging plain objects or arrays if ( deep && copy && ( jQuery.isPlainObject( copy ) || ( copyIsArray = Array.isArray( copy ) ) ) ) { src = target[ name ]; // Ensure proper type for the source value if ( copyIsArray && !Array.isArray( src ) ) { clone = []; } else if ( !copyIsArray && !jQuery.isPlainObject( src ) ) { clone = {}; } else { clone = src; } copyIsArray = false; // Never move original objects, clone them target[ name ] = jQuery.extend( deep, clone, copy ); // Don't bring in undefined values } else if ( copy !== undefined ) { target[ name ] = copy; } } } } // Return the modified object return target; }; jQuery.extend( { // Unique for each copy of jQuery on the page expando: "jQuery" + ( version + Math.random() ).replace( /\D/g, "" ), // Assume jQuery is ready without the ready module isReady: true, error: function( msg ) { throw new Error( msg ); }, noop: function() {}, isPlainObject: function( obj ) { var proto, Ctor; // Detect obvious negatives // Use toString instead of jQuery.type to catch host objects if ( !obj || toString.call( obj ) !== "[object Object]" ) { return false; } proto = getProto( obj ); // Objects with no prototype (e.g., `Object.create( null )`) are plain if ( !proto ) { return true; } // Objects with prototype are plain iff they were constructed by a global Object function Ctor = hasOwn.call( proto, "constructor" ) && proto.constructor; return typeof Ctor === "function" && fnToString.call( Ctor ) === ObjectFunctionString; }, isEmptyObject: function( obj ) { var name; for ( name in obj ) { return false; } return true; }, // Evaluates a script in a provided context; falls back to the global one // if not specified. globalEval: function( code, options, doc ) { DOMEval( code, { nonce: options && options.nonce }, doc ); }, each: function( obj, callback ) { var length, i = 0; if ( isArrayLike( obj ) ) { length = obj.length; for ( ; i < length; i++ ) { if ( callback.call( obj[ i ], i, obj[ i ] ) === false ) { break; } } } else { for ( i in obj ) { if ( callback.call( obj[ i ], i, obj[ i ] ) === false ) { break; } } } return obj; }, // results is for internal usage only makeArray: function( arr, results ) { var ret = results || []; if ( arr != null ) { if ( isArrayLike( Object( arr ) ) ) { jQuery.merge( ret, typeof arr === "string" ? [ arr ] : arr ); } else { push.call( ret, arr ); } } return ret; }, inArray: function( elem, arr, i ) { return arr == null ? -1 : indexOf.call( arr, elem, i ); }, // Support: Android <=4.0 only, PhantomJS 1 only // push.apply(_, arraylike) throws on ancient WebKit merge: function( first, second ) { var len = +second.length, j = 0, i = first.length; for ( ; j < len; j++ ) { first[ i++ ] = second[ j ]; } first.length = i; return first; }, grep: function( elems, callback, invert ) { var callbackInverse, matches = [], i = 0, length = elems.length, callbackExpect = !invert; // Go through the array, only saving the items // that pass the validator function for ( ; i < length; i++ ) { callbackInverse = !callback( elems[ i ], i ); if ( callbackInverse !== callbackExpect ) { matches.push( elems[ i ] ); } } return matches; }, // arg is for internal usage only map: function( elems, callback, arg ) { var length, value, i = 0, ret = []; // Go through the array, translating each of the items to their new values if ( isArrayLike( elems ) ) { length = elems.length; for ( ; i < length; i++ ) { value = callback( elems[ i ], i, arg ); if ( value != null ) { ret.push( value ); } } // Go through every key on the object, } else { for ( i in elems ) { value = callback( elems[ i ], i, arg ); if ( value != null ) { ret.push( value ); } } } // Flatten any nested arrays return flat( ret ); }, // A global GUID counter for objects guid: 1, // jQuery.support is not used in Core but other projects attach their // properties to it so it needs to exist. support: support } ); if ( typeof Symbol === "function" ) { jQuery.fn[ Symbol.iterator ] = arr[ Symbol.iterator ]; } // Populate the class2type map jQuery.each( "Boolean Number String Function Array Date RegExp Object Error Symbol".split( " " ), function( _i, name ) { class2type[ "[object " + name + "]" ] = name.toLowerCase(); } ); function isArrayLike( obj ) { // Support: real iOS 8.2 only (not reproducible in simulator) // `in` check used to prevent JIT error (gh-2145) // hasOwn isn't used here due to false negatives // regarding Nodelist length in IE var length = !!obj && "length" in obj && obj.length, type = toType( obj ); if ( isFunction( obj ) || isWindow( obj ) ) { return false; } return type === "array" || length === 0 || typeof length === "number" && length > 0 && ( length - 1 ) in obj; } var Sizzle = /*! * Sizzle CSS Selector Engine v2.3.5 * https://sizzlejs.com/ * * Copyright JS Foundation and other contributors * Released under the MIT license * https://js.foundation/ * * Date: 2020-03-14 */ ( function( window ) { var i, support, Expr, getText, isXML, tokenize, compile, select, outermostContext, sortInput, hasDuplicate, // Local document vars setDocument, document, docElem, documentIsHTML, rbuggyQSA, rbuggyMatches, matches, contains, // Instance-specific data expando = "sizzle" + 1 * new Date(), preferredDoc = window.document, dirruns = 0, done = 0, classCache = createCache(), tokenCache = createCache(), compilerCache = createCache(), nonnativeSelectorCache = createCache(), sortOrder = function( a, b ) { if ( a === b ) { hasDuplicate = true; } return 0; }, // Instance methods hasOwn = ( {} ).hasOwnProperty, arr = [], pop = arr.pop, pushNative = arr.push, push = arr.push, slice = arr.slice, // Use a stripped-down indexOf as it's faster than native // https://jsperf.com/thor-indexof-vs-for/5 indexOf = function( list, elem ) { var i = 0, len = list.length; for ( ; i < len; i++ ) { if ( list[ i ] === elem ) { return i; } } return -1; }, booleans = "checked|selected|async|autofocus|autoplay|controls|defer|disabled|hidden|" + "ismap|loop|multiple|open|readonly|required|scoped", // Regular expressions // http://www.w3.org/TR/css3-selectors/#whitespace whitespace = "[\\x20\\t\\r\\n\\f]", // https://www.w3.org/TR/css-syntax-3/#ident-token-diagram identifier = "(?:\\\\[\\da-fA-F]{1,6}" + whitespace + "?|\\\\[^\\r\\n\\f]|[\\w-]|[^\0-\\x7f])+", // Attribute selectors: http://www.w3.org/TR/selectors/#attribute-selectors attributes = "\\[" + whitespace + "*(" + identifier + ")(?:" + whitespace + // Operator (capture 2) "*([*^$|!~]?=)" + whitespace + // "Attribute values must be CSS identifiers [capture 5] // or strings [capture 3 or capture 4]" "*(?:'((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\"|(" + identifier + "))|)" + whitespace + "*\\]", pseudos = ":(" + identifier + ")(?:\\((" + // To reduce the number of selectors needing tokenize in the preFilter, prefer arguments: // 1. quoted (capture 3; capture 4 or capture 5) "('((?:\\\\.|[^\\\\'])*)'|\"((?:\\\\.|[^\\\\\"])*)\")|" + // 2. simple (capture 6) "((?:\\\\.|[^\\\\()[\\]]|" + attributes + ")*)|" + // 3. anything else (capture 2) ".*" + ")\\)|)", // Leading and non-escaped trailing whitespace, capturing some non-whitespace characters preceding the latter rwhitespace = new RegExp( whitespace + "+", "g" ), rtrim = new RegExp( "^" + whitespace + "+|((?:^|[^\\\\])(?:\\\\.)*)" + whitespace + "+$", "g" ), rcomma = new RegExp( "^" + whitespace + "*," + whitespace + "*" ), rcombinators = new RegExp( "^" + whitespace + "*([>+~]|" + whitespace + ")" + whitespace + "*" ), rdescend = new RegExp( whitespace + "|>" ), rpseudo = new RegExp( pseudos ), ridentifier = new RegExp( "^" + identifier + "$" ), matchExpr = { "ID": new RegExp( "^#(" + identifier + ")" ), "CLASS": new RegExp( "^\\.(" + identifier + ")" ), "TAG": new RegExp( "^(" + identifier + "|[*])" ), "ATTR": new RegExp( "^" + attributes ), "PSEUDO": new RegExp( "^" + pseudos ), "CHILD": new RegExp( "^:(only|first|last|nth|nth-last)-(child|of-type)(?:\\(" + whitespace + "*(even|odd|(([+-]|)(\\d*)n|)" + whitespace + "*(?:([+-]|)" + whitespace + "*(\\d+)|))" + whitespace + "*\\)|)", "i" ), "bool": new RegExp( "^(?:" + booleans + ")$", "i" ), // For use in libraries implementing .is() // We use this for POS matching in `select` "needsContext": new RegExp( "^" + whitespace + "*[>+~]|:(even|odd|eq|gt|lt|nth|first|last)(?:\\(" + whitespace + "*((?:-\\d)?\\d*)" + whitespace + "*\\)|)(?=[^-]|$)", "i" ) }, rhtml = /HTML$/i, rinputs = /^(?:input|select|textarea|button)$/i, rheader = /^h\d$/i, rnative = /^[^{]+\{\s*\[native \w/, // Easily-parseable/retrievable ID or TAG or CLASS selectors rquickExpr = /^(?:#([\w-]+)|(\w+)|\.([\w-]+))$/, rsibling = /[+~]/, // CSS escapes // http://www.w3.org/TR/CSS21/syndata.html#escaped-characters runescape = new RegExp( "\\\\[\\da-fA-F]{1,6}" + whitespace + "?|\\\\([^\\r\\n\\f])", "g" ), funescape = function( escape, nonHex ) { var high = "0x" + escape.slice( 1 ) - 0x10000; return nonHex ? // Strip the backslash prefix from a non-hex escape sequence nonHex : // Replace a hexadecimal escape sequence with the encoded Unicode code point // Support: IE <=11+ // For values outside the Basic Multilingual Plane (BMP), manually construct a // surrogate pair high < 0 ? String.fromCharCode( high + 0x10000 ) : String.fromCharCode( high >> 10 | 0xD800, high & 0x3FF | 0xDC00 ); }, // CSS string/identifier serialization // https://drafts.csswg.org/cssom/#common-serializing-idioms rcssescape = /([\0-\x1f\x7f]|^-?\d)|^-$|[^\0-\x1f\x7f-\uFFFF\w-]/g, fcssescape = function( ch, asCodePoint ) { if ( asCodePoint ) { // U+0000 NULL becomes U+FFFD REPLACEMENT CHARACTER if ( ch === "\0" ) { return "\uFFFD"; } // Control characters and (dependent upon position) numbers get escaped as code points return ch.slice( 0, -1 ) + "\\" + ch.charCodeAt( ch.length - 1 ).toString( 16 ) + " "; } // Other potentially-special ASCII characters get backslash-escaped return "\\" + ch; }, // Used for iframes // See setDocument() // Removing the function wrapper causes a "Permission Denied" // error in IE unloadHandler = function() { setDocument(); }, inDisabledFieldset = addCombinator( function( elem ) { return elem.disabled === true && elem.nodeName.toLowerCase() === "fieldset"; }, { dir: "parentNode", next: "legend" } ); // Optimize for push.apply( _, NodeList ) try { push.apply( ( arr = slice.call( preferredDoc.childNodes ) ), preferredDoc.childNodes ); // Support: Android<4.0 // Detect silently failing push.apply // eslint-disable-next-line no-unused-expressions arr[ preferredDoc.childNodes.length ].nodeType; } catch ( e ) { push = { apply: arr.length ? // Leverage slice if possible function( target, els ) { pushNative.apply( target, slice.call( els ) ); } : // Support: IE<9 // Otherwise append directly function( target, els ) { var j = target.length, i = 0; // Can't trust NodeList.length while ( ( target[ j++ ] = els[ i++ ] ) ) {} target.length = j - 1; } }; } function Sizzle( selector, context, results, seed ) { var m, i, elem, nid, match, groups, newSelector, newContext = context && context.ownerDocument, // nodeType defaults to 9, since context defaults to document nodeType = context ? context.nodeType : 9; results = results || []; // Return early from calls with invalid selector or context if ( typeof selector !== "string" || !selector || nodeType !== 1 && nodeType !== 9 && nodeType !== 11 ) { return results; } // Try to shortcut find operations (as opposed to filters) in HTML documents if ( !seed ) { setDocument( context ); context = context || document; if ( documentIsHTML ) { // If the selector is sufficiently simple, try using a "get*By*" DOM method // (excepting DocumentFragment context, where the methods don't exist) if ( nodeType !== 11 && ( match = rquickExpr.exec( selector ) ) ) { // ID selector if ( ( m = match[ 1 ] ) ) { // Document context if ( nodeType === 9 ) { if ( ( elem = context.getElementById( m ) ) ) { // Support: IE, Opera, Webkit // TODO: identify versions // getElementById can match elements by name instead of ID if ( elem.id === m ) { results.push( elem ); return results; } } else { return results; } // Element context } else { // Support: IE, Opera, Webkit // TODO: identify versions // getElementById can match elements by name instead of ID if ( newContext && ( elem = newContext.getElementById( m ) ) && contains( context, elem ) && elem.id === m ) { results.push( elem ); return results; } } // Type selector } else if ( match[ 2 ] ) { push.apply( results, context.getElementsByTagName( selector ) ); return results; // Class selector } else if ( ( m = match[ 3 ] ) && support.getElementsByClassName && context.getElementsByClassName ) { push.apply( results, context.getElementsByClassName( m ) ); return results; } } // Take advantage of querySelectorAll if ( support.qsa && !nonnativeSelectorCache[ selector + " " ] && ( !rbuggyQSA || !rbuggyQSA.test( selector ) ) && // Support: IE 8 only // Exclude object elements ( nodeType !== 1 || context.nodeName.toLowerCase() !== "object" ) ) { newSelector = selector; newContext = context; // qSA considers elements outside a scoping root when evaluating child or // descendant combinators, which is not what we want. // In such cases, we work around the behavior by prefixing every selector in the // list with an ID selector referencing the scope context. // The technique has to be used as well when a leading combinator is used // as such selectors are not recognized by querySelectorAll. // Thanks to Andrew Dupont for this technique. if ( nodeType === 1 && ( rdescend.test( selector ) || rcombinators.test( selector ) ) ) { // Expand context for sibling selectors newContext = rsibling.test( selector ) && testContext( context.parentNode ) || context; // We can use :scope instead of the ID hack if the browser // supports it & if we're not changing the context. if ( newContext !== context || !support.scope ) { // Capture the context ID, setting it first if necessary if ( ( nid = context.getAttribute( "id" ) ) ) { nid = nid.replace( rcssescape, fcssescape ); } else { context.setAttribute( "id", ( nid = expando ) ); } } // Prefix every selector in the list groups = tokenize( selector ); i = groups.length; while ( i-- ) { groups[ i ] = ( nid ? "#" + nid : ":scope" ) + " " + toSelector( groups[ i ] ); } newSelector = groups.join( "," ); } try { push.apply( results, newContext.querySelectorAll( newSelector ) ); return results; } catch ( qsaError ) { nonnativeSelectorCache( selector, true ); } finally { if ( nid === expando ) { context.removeAttribute( "id" ); } } } } } // All others return select( selector.replace( rtrim, "$1" ), context, results, seed ); } /** * Create key-value caches of limited size * @returns {function(string, object)} Returns the Object data after storing it on itself with * property name the (space-suffixed) string and (if the cache is larger than Expr.cacheLength) * deleting the oldest entry */ function createCache() { var keys = []; function cache( key, value ) { // Use (key + " ") to avoid collision with native prototype properties (see Issue #157) if ( keys.push( key + " " ) > Expr.cacheLength ) { // Only keep the most recent entries delete cache[ keys.shift() ]; } return ( cache[ key + " " ] = value ); } return cache; } /** * Mark a function for special use by Sizzle * @param {Function} fn The function to mark */ function markFunction( fn ) { fn[ expando ] = true; return fn; } /** * Support testing using an element * @param {Function} fn Passed the created element and returns a boolean result */ function assert( fn ) { var el = document.createElement( "fieldset" ); try { return !!fn( el ); } catch ( e ) { return false; } finally { // Remove from its parent by default if ( el.parentNode ) { el.parentNode.removeChild( el ); } // release memory in IE el = null; } } /** * Adds the same handler for all of the specified attrs * @param {String} attrs Pipe-separated list of attributes * @param {Function} handler The method that will be applied */ function addHandle( attrs, handler ) { var arr = attrs.split( "|" ), i = arr.length; while ( i-- ) { Expr.attrHandle[ arr[ i ] ] = handler; } } /** * Checks document order of two siblings * @param {Element} a * @param {Element} b * @returns {Number} Returns less than 0 if a precedes b, greater than 0 if a follows b */ function siblingCheck( a, b ) { var cur = b && a, diff = cur && a.nodeType === 1 && b.nodeType === 1 && a.sourceIndex - b.sourceIndex; // Use IE sourceIndex if available on both nodes if ( diff ) { return diff; } // Check if b follows a if ( cur ) { while ( ( cur = cur.nextSibling ) ) { if ( cur === b ) { return -1; } } } return a ? 1 : -1; } /** * Returns a function to use in pseudos for input types * @param {String} type */ function createInputPseudo( type ) { return function( elem ) { var name = elem.nodeName.toLowerCase(); return name === "input" && elem.type === type; }; } /** * Returns a function to use in pseudos for buttons * @param {String} type */ function createButtonPseudo( type ) { return function( elem ) { var name = elem.nodeName.toLowerCase(); return ( name === "input" || name === "button" ) && elem.type === type; }; } /** * Returns a function to use in pseudos for :enabled/:disabled * @param {Boolean} disabled true for :disabled; false for :enabled */ function createDisabledPseudo( disabled ) { // Known :disabled false positives: fieldset[disabled] > legend:nth-of-type(n+2) :can-disable return function( elem ) { // Only certain elements can match :enabled or :disabled // https://html.spec.whatwg.org/multipage/scripting.html#selector-enabled // https://html.spec.whatwg.org/multipage/scripting.html#selector-disabled if ( "form" in elem ) { // Check for inherited disabledness on relevant non-disabled elements: // * listed form-associated elements in a disabled fieldset // https://html.spec.whatwg.org/multipage/forms.html#category-listed // https://html.spec.whatwg.org/multipage/forms.html#concept-fe-disabled // * option elements in a disabled optgroup // https://html.spec.whatwg.org/multipage/forms.html#concept-option-disabled // All such elements have a "form" property. if ( elem.parentNode && elem.disabled === false ) { // Option elements defer to a parent optgroup if present if ( "label" in elem ) { if ( "label" in elem.parentNode ) { return elem.parentNode.disabled === disabled; } else { return elem.disabled === disabled; } } // Support: IE 6 - 11 // Use the isDisabled shortcut property to check for disabled fieldset ancestors return elem.isDisabled === disabled || // Where there is no isDisabled, check manually /* jshint -W018 */ elem.isDisabled !== !disabled && inDisabledFieldset( elem ) === disabled; } return elem.disabled === disabled; // Try to winnow out elements that can't be disabled before trusting the disabled property. // Some victims get caught in our net (label, legend, menu, track), but it shouldn't // even exist on them, let alone have a boolean value. } else if ( "label" in elem ) { return elem.disabled === disabled; } // Remaining elements are neither :enabled nor :disabled return false; }; } /** * Returns a function to use in pseudos for positionals * @param {Function} fn */ function createPositionalPseudo( fn ) { return markFunction( function( argument ) { argument = +argument; return markFunction( function( seed, matches ) { var j, matchIndexes = fn( [], seed.length, argument ), i = matchIndexes.length; // Match elements found at the specified indexes while ( i-- ) { if ( seed[ ( j = matchIndexes[ i ] ) ] ) { seed[ j ] = !( matches[ j ] = seed[ j ] ); } } } ); } ); } /** * Checks a node for validity as a Sizzle context * @param {Element|Object=} context * @returns {Element|Object|Boolean} The input node if acceptable, otherwise a falsy value */ function testContext( context ) { return context && typeof context.getElementsByTagName !== "undefined" && context; } // Expose support vars for convenience support = Sizzle.support = {}; /** * Detects XML nodes * @param {Element|Object} elem An element or a document * @returns {Boolean} True iff elem is a non-HTML XML node */ isXML = Sizzle.isXML = function( elem ) { var namespace = elem.namespaceURI, docElem = ( elem.ownerDocument || elem ).documentElement; // Support: IE <=8 // Assume HTML when documentElement doesn't yet exist, such as inside loading iframes // https://bugs.jquery.com/ticket/4833 return !rhtml.test( namespace || docElem && docElem.nodeName || "HTML" ); }; /** * Sets document-related variables once based on the current document * @param {Element|Object} [doc] An element or document object to use to set the document * @returns {Object} Returns the current document */ setDocument = Sizzle.setDocument = function( node ) { var hasCompare, subWindow, doc = node ? node.ownerDocument || node : preferredDoc; // Return early if doc is invalid or already selected // Support: IE 11+, Edge 17 - 18+ // IE/Edge sometimes throw a "Permission denied" error when strict-comparing // two documents; shallow comparisons work. // eslint-disable-next-line eqeqeq if ( doc == document || doc.nodeType !== 9 || !doc.documentElement ) { return document; } // Update global variables document = doc; docElem = document.documentElement; documentIsHTML = !isXML( document ); // Support: IE 9 - 11+, Edge 12 - 18+ // Accessing iframe documents after unload throws "permission denied" errors (jQuery #13936) // Support: IE 11+, Edge 17 - 18+ // IE/Edge sometimes throw a "Permission denied" error when strict-comparing // two documents; shallow comparisons work. // eslint-disable-next-line eqeqeq if ( preferredDoc != document && ( subWindow = document.defaultView ) && subWindow.top !== subWindow ) { // Support: IE 11, Edge if ( subWindow.addEventListener ) { subWindow.addEventListener( "unload", unloadHandler, false ); // Support: IE 9 - 10 only } else if ( subWindow.attachEvent ) { subWindow.attachEvent( "onunload", unloadHandler ); } } // Support: IE 8 - 11+, Edge 12 - 18+, Chrome <=16 - 25 only, Firefox <=3.6 - 31 only, // Safari 4 - 5 only, Opera <=11.6 - 12.x only // IE/Edge & older browsers don't support the :scope pseudo-class. // Support: Safari 6.0 only // Safari 6.0 supports :scope but it's an alias of :root there. support.scope = assert( function( el ) { docElem.appendChild( el ).appendChild( document.createElement( "div" ) ); return typeof el.querySelectorAll !== "undefined" && !el.querySelectorAll( ":scope fieldset div" ).length; } ); /* Attributes ---------------------------------------------------------------------- */ // Support: IE<8 // Verify that getAttribute really returns attributes and not properties // (excepting IE8 booleans) support.attributes = assert( function( el ) { el.className = "i"; return !el.getAttribute( "className" ); } ); /* getElement(s)By* ---------------------------------------------------------------------- */ // Check if getElementsByTagName("*") returns only elements support.getElementsByTagName = assert( function( el ) { el.appendChild( document.createComment( "" ) ); return !el.getElementsByTagName( "*" ).length; } ); // Support: IE<9 support.getElementsByClassName = rnative.test( document.getElementsByClassName ); // Support: IE<10 // Check if getElementById returns elements by name // The broken getElementById methods don't pick up programmatically-set names, // so use a roundabout getElementsByName test support.getById = assert( function( el ) { docElem.appendChild( el ).id = expando; return !document.getElementsByName || !document.getElementsByName( expando ).length; } ); // ID filter and find if ( support.getById ) { Expr.filter[ "ID" ] = function( id ) { var attrId = id.replace( runescape, funescape ); return function( elem ) { return elem.getAttribute( "id" ) === attrId; }; }; Expr.find[ "ID" ] = function( id, context ) { if ( typeof context.getElementById !== "undefined" && documentIsHTML ) { var elem = context.getElementById( id ); return elem ? [ elem ] : []; } }; } else { Expr.filter[ "ID" ] = function( id ) { var attrId = id.replace( runescape, funescape ); return function( elem ) { var node = typeof elem.getAttributeNode !== "undefined" && elem.getAttributeNode( "id" ); return node && node.value === attrId; }; }; // Support: IE 6 - 7 only // getElementById is not reliable as a find shortcut Expr.find[ "ID" ] = function( id, context ) { if ( typeof context.getElementById !== "undefined" && documentIsHTML ) { var node, i, elems, elem = context.getElementById( id ); if ( elem ) { // Verify the id attribute node = elem.getAttributeNode( "id" ); if ( node && node.value === id ) { return [ elem ]; } // Fall back on getElementsByName elems = context.getElementsByName( id ); i = 0; while ( ( elem = elems[ i++ ] ) ) { node = elem.getAttributeNode( "id" ); if ( node && node.value === id ) { return [ elem ]; } } } return []; } }; } // Tag Expr.find[ "TAG" ] = support.getElementsByTagName ? function( tag, context ) { if ( typeof context.getElementsByTagName !== "undefined" ) { return context.getElementsByTagName( tag ); // DocumentFragment nodes don't have gEBTN } else if ( support.qsa ) { return context.querySelectorAll( tag ); } } : function( tag, context ) { var elem, tmp = [], i = 0, // By happy coincidence, a (broken) gEBTN appears on DocumentFragment nodes too results = context.getElementsByTagName( tag ); // Filter out possible comments if ( tag === "*" ) { while ( ( elem = results[ i++ ] ) ) { if ( elem.nodeType === 1 ) { tmp.push( elem ); } } return tmp; } return results; }; // Class Expr.find[ "CLASS" ] = support.getElementsByClassName && function( className, context ) { if ( typeof context.getElementsByClassName !== "undefined" && documentIsHTML ) { return context.getElementsByClassName( className ); } }; /* QSA/matchesSelector ---------------------------------------------------------------------- */ // QSA and matchesSelector support // matchesSelector(:active) reports false when true (IE9/Opera 11.5) rbuggyMatches = []; // qSa(:focus) reports false when true (Chrome 21) // We allow this because of a bug in IE8/9 that throws an error // whenever `document.activeElement` is accessed on an iframe // So, we allow :focus to pass through QSA all the time to avoid the IE error // See https://bugs.jquery.com/ticket/13378 rbuggyQSA = []; if ( ( support.qsa = rnative.test( document.querySelectorAll ) ) ) { // Build QSA regex // Regex strategy adopted from Diego Perini assert( function( el ) { var input; // Select is set to empty string on purpose // This is to test IE's treatment of not explicitly // setting a boolean content attribute, // since its presence should be enough // https://bugs.jquery.com/ticket/12359 docElem.appendChild( el ).innerHTML = "<a id='" + expando + "'></a>" + "<select id='" + expando + "-\r\\' msallowcapture=''>" + "<option selected=''></option></select>"; // Support: IE8, Opera 11-12.16 // Nothing should be selected when empty strings follow ^= or $= or *= // The test attribute must be unknown in Opera but "safe" for WinRT // https://msdn.microsoft.com/en-us/library/ie/hh465388.aspx#attribute_section if ( el.querySelectorAll( "[msallowcapture^='']" ).length ) { rbuggyQSA.push( "[*^$]=" + whitespace + "*(?:''|\"\")" ); } // Support: IE8 // Boolean attributes and "value" are not treated correctly if ( !el.querySelectorAll( "[selected]" ).length ) { rbuggyQSA.push( "\\[" + whitespace + "*(?:value|" + booleans + ")" ); } // Support: Chrome<29, Android<4.4, Safari<7.0+, iOS<7.0+, PhantomJS<1.9.8+ if ( !el.querySelectorAll( "[id~=" + expando + "-]" ).length ) { rbuggyQSA.push( "~=" ); } // Support: IE 11+, Edge 15 - 18+ // IE 11/Edge don't find elements on a `[name='']` query in some cases. // Adding a temporary attribute to the document before the selection works // around the issue. // Interestingly, IE 10 & older don't seem to have the issue. input = document.createElement( "input" ); input.setAttribute( "name", "" ); el.appendChild( input ); if ( !el.querySelectorAll( "[name='']" ).length ) { rbuggyQSA.push( "\\[" + whitespace + "*name" + whitespace + "*=" + whitespace + "*(?:''|\"\")" ); } // Webkit/Opera - :checked should return selected option elements // http://www.w3.org/TR/2011/REC-css3-selectors-20110929/#checked // IE8 throws error here and will not see later tests if ( !el.querySelectorAll( ":checked" ).length ) { rbuggyQSA.push( ":checked" ); } // Support: Safari 8+, iOS 8+ // https://bugs.webkit.org/show_bug.cgi?id=136851 // In-page `selector#id sibling-combinator selector` fails if ( !el.querySelectorAll( "a#" + expando + "+*" ).length ) { rbuggyQSA.push( ".#.+[+~]" ); } // Support: Firefox <=3.6 - 5 only // Old Firefox doesn't throw on a badly-escaped identifier. el.querySelectorAll( "\\\f" ); rbuggyQSA.push( "[\\r\\n\\f]" ); } ); assert( function( el ) { el.innerHTML = "<a href='' disabled='disabled'></a>" + "<select disabled='disabled'><option/></select>"; // Support: Windows 8 Native Apps // The type and name attributes are restricted during .innerHTML assignment var input = document.createElement( "input" ); input.setAttribute( "type", "hidden" ); el.appendChild( input ).setAttribute( "name", "D" ); // Support: IE8 // Enforce case-sensitivity of name attribute if ( el.querySelectorAll( "[name=d]" ).length ) { rbuggyQSA.push( "name" + whitespace + "*[*^$|!~]?=" ); } // FF 3.5 - :enabled/:disabled and hidden elements (hidden elements are still enabled) // IE8 throws error here and will not see later tests if ( el.querySelectorAll( ":enabled" ).length !== 2 ) { rbuggyQSA.push( ":enabled", ":disabled" ); } // Support: IE9-11+ // IE's :disabled selector does not pick up the children of disabled fieldsets docElem.appendChild( el ).disabled = true; if ( el.querySelectorAll( ":disabled" ).length !== 2 ) { rbuggyQSA.push( ":enabled", ":disabled" ); } // Support: Opera 10 - 11 only // Opera 10-11 does not throw on post-comma invalid pseudos el.querySelectorAll( "*,:x" ); rbuggyQSA.push( ",.*:" ); } ); } if ( ( support.matchesSelector = rnative.test( ( matches = docElem.matches || docElem.webkitMatchesSelector || docElem.mozMatchesSelector || docElem.oMatchesSelector || docElem.msMatchesSelector ) ) ) ) { assert( function( el ) { // Check to see if it's possible to do matchesSelector // on a disconnected node (IE 9) support.disconnectedMatch = matches.call( el, "*" ); // This should fail with an exception // Gecko does not error, returns false instead matches.call( el, "[s!='']:x" ); rbuggyMatches.push( "!=", pseudos ); } ); } rbuggyQSA = rbuggyQSA.length && new RegExp( rbuggyQSA.join( "|" ) ); rbuggyMatches = rbuggyMatches.length && new RegExp( rbuggyMatches.join( "|" ) ); /* Contains ---------------------------------------------------------------------- */ hasCompare = rnative.test( docElem.compareDocumentPosition ); // Element contains another // Purposefully self-exclusive // As in, an element does not contain itself contains = hasCompare || rnative.test( docElem.contains ) ? function( a, b ) { var adown = a.nodeType === 9 ? a.documentElement : a, bup = b && b.parentNode; return a === bup || !!( bup && bup.nodeType === 1 && ( adown.contains ? adown.contains( bup ) : a.compareDocumentPosition && a.compareDocumentPosition( bup ) & 16 ) ); } : function( a, b ) { if ( b ) { while ( ( b = b.parentNode ) ) { if ( b === a ) { return true; } } } return false; }; /* Sorting ---------------------------------------------------------------------- */ // Document order sorting sortOrder = hasCompare ? function( a, b ) { // Flag for duplicate removal if ( a === b ) { hasDuplicate = true; return 0; } // Sort on method existence if only one input has compareDocumentPosition var compare = !a.compareDocumentPosition - !b.compareDocumentPosition; if ( compare ) { return compare; } // Calculate position if both inputs belong to the same document // Support: IE 11+, Edge 17 - 18+ // IE/Edge sometimes throw a "Permission denied" error when strict-comparing // two documents; shallow comparisons work. // eslint-disable-next-line eqeqeq compare = ( a.ownerDocument || a ) == ( b.ownerDocument || b ) ? a.compareDocumentPosition( b ) : // Otherwise we know they are disconnected 1; // Disconnected nodes if ( compare & 1 || ( !support.sortDetached && b.compareDocumentPosition( a ) === compare ) ) { // Choose the first element that is related to our preferred document // Support: IE 11+, Edge 17 - 18+ // IE/Edge sometimes throw a "Permission denied" error when strict-comparing // two documents; shallow comparisons work. // eslint-disable-next-line eqeqeq if ( a == document || a.ownerDocument == preferredDoc && contains( preferredDoc, a ) ) { return -1; } // Support: IE 11+, Edge 17 - 18+ // IE/Edge sometimes throw a "Permission denied" error when strict-comparing // two documents; shallow comparisons work. // eslint-disable-next-line eqeqeq if ( b == document || b.ownerDocument == preferredDoc && contains( preferredDoc, b ) ) { return 1; } // Maintain original order return sortInput ? ( indexOf( sortInput, a ) - indexOf( sortInput, b ) ) : 0; } return compare & 4 ? -1 : 1; } : function( a, b ) { // Exit early if the nodes are identical if ( a === b ) { hasDuplicate = true; return 0; } var cur, i = 0, aup = a.parentNode, bup = b.parentNode, ap = [ a ], bp = [ b ]; // Parentless nodes are either documents or disconnected if ( !aup || !bup ) { // Support: IE 11+, Edge 17 - 18+ // IE/Edge sometimes throw a "Permission denied" error when strict-comparing // two documents; shallow comparisons work. /* eslint-disable eqeqeq */ return a == document ? -1 : b == document ? 1 : /* eslint-enable eqeqeq */ aup ? -1 : bup ? 1 : sortInput ? ( indexOf( sortInput, a ) - indexOf( sortInput, b ) ) : 0; // If the nodes are siblings, we can do a quick check } else if ( aup === bup ) { return siblingCheck( a, b ); } // Otherwise we need full lists of their ancestors for comparison cur = a; while ( ( cur = cur.parentNode ) ) { ap.unshift( cur ); } cur = b; while ( ( cur = cur.parentNode ) ) { bp.unshift( cur ); } // Walk down the tree looking for a discrepancy while ( ap[ i ] === bp[ i ] ) { i++; } return i ? // Do a sibling check if the nodes have a common ancestor siblingCheck( ap[ i ], bp[ i ] ) : // Otherwise nodes in our document sort first // Support: IE 11+, Edge 17 - 18+ // IE/Edge sometimes throw a "Permission denied" error when strict-comparing // two documents; shallow comparisons work. /* eslint-disable eqeqeq */ ap[ i ] == preferredDoc ? -1 : bp[ i ] == preferredDoc ? 1 : /* eslint-enable eqeqeq */ 0; }; return document; }; Sizzle.matches = function( expr, elements ) { return Sizzle( expr, null, null, elements ); }; Sizzle.matchesSelector = function( elem, expr ) { setDocument( elem ); if ( support.matchesSelector && documentIsHTML && !nonnativeSelectorCache[ expr + " " ] && ( !rbuggyMatches || !rbuggyMatches.test( expr ) ) && ( !rbuggyQSA || !rbuggyQSA.test( expr ) ) ) { try { var ret = matches.call( elem, expr ); // IE 9's matchesSelector returns false on disconnected nodes if ( ret || support.disconnectedMatch || // As well, disconnected nodes are said to be in a document // fragment in IE 9 elem.document && elem.document.nodeType !== 11 ) { return ret; } } catch ( e ) { nonnativeSelectorCache( expr, true ); } } return Sizzle( expr, document, null, [ elem ] ).length > 0; }; Sizzle.contains = function( context, elem ) { // Set document vars if needed // Support: IE 11+, Edge 17 - 18+ // IE/Edge sometimes throw a "Permission denied" error when strict-comparing // two documents; shallow comparisons work. // eslint-disable-next-line eqeqeq if ( ( context.ownerDocument || context ) != document ) { setDocument( context ); } return contains( context, elem ); }; Sizzle.attr = function( elem, name ) { // Set document vars if needed // Support: IE 11+, Edge 17 - 18+ // IE/Edge sometimes throw a "Permission denied" error when strict-comparing // two documents; shallow comparisons work. // eslint-disable-next-line eqeqeq if ( ( elem.ownerDocument || elem ) != document ) { setDocument( elem ); } var fn = Expr.attrHandle[ name.toLowerCase() ], // Don't get fooled by Object.prototype properties (jQuery #13807) val = fn && hasOwn.call( Expr.attrHandle, name.toLowerCase() ) ? fn( elem, name, !documentIsHTML ) : undefined; return val !== undefined ? val : support.attributes || !documentIsHTML ? elem.getAttribute( name ) : ( val = elem.getAttributeNode( name ) ) && val.specified ? val.value : null; }; Sizzle.escape = function( sel ) { return ( sel + "" ).replace( rcssescape, fcssescape ); }; Sizzle.error = function( msg ) { throw new Error( "Syntax error, unrecognized expression: " + msg ); }; /** * Document sorting and removing duplicates * @param {ArrayLike} results */ Sizzle.uniqueSort = function( results ) { var elem, duplicates = [], j = 0, i = 0; // Unless we *know* we can detect duplicates, assume their presence hasDuplicate = !support.detectDuplicates; sortInput = !support.sortStable && results.slice( 0 ); results.sort( sortOrder ); if ( hasDuplicate ) { while ( ( elem = results[ i++ ] ) ) { if ( elem === results[ i ] ) { j = duplicates.push( i ); } } while ( j-- ) { results.splice( duplicates[ j ], 1 ); } } // Clear input after sorting to release objects // See https://github.com/jquery/sizzle/pull/225 sortInput = null; return results; }; /** * Utility function for retrieving the text value of an array of DOM nodes * @param {Array|Element} elem */ getText = Sizzle.getText = function( elem ) { var node, ret = "", i = 0, nodeType = elem.nodeType; if ( !nodeType ) { // If no nodeType, this is expected to be an array while ( ( node = elem[ i++ ] ) ) { // Do not traverse comment nodes ret += getText( node ); } } else if ( nodeType === 1 || nodeType === 9 || nodeType === 11 ) { // Use textContent for elements // innerText usage removed for consistency of new lines (jQuery #11153) if ( typeof elem.textContent === "string" ) { return elem.textContent; } else { // Traverse its children for ( elem = elem.firstChild; elem; elem = elem.nextSibling ) { ret += getText( elem ); } } } else if ( nodeType === 3 || nodeType === 4 ) { return elem.nodeValue; } // Do not include comment or processing instruction nodes return ret; }; Expr = Sizzle.selectors = { // Can be adjusted by the user cacheLength: 50, createPseudo: markFunction, match: matchExpr, attrHandle: {}, find: {}, relative: { ">": { dir: "parentNode", first: true }, " ": { dir: "parentNode" }, "+": { dir: "previousSibling", first: true }, "~": { dir: "previousSibling" } }, preFilter: { "ATTR": function( match ) { match[ 1 ] = match[ 1 ].replace( runescape, funescape ); // Move the given value to match[3] whether quoted or unquoted match[ 3 ] = ( match[ 3 ] || match[ 4 ] || match[ 5 ] || "" ).replace( runescape, funescape ); if ( match[ 2 ] === "~=" ) { match[ 3 ] = " " + match[ 3 ] + " "; } return match.slice( 0, 4 ); }, "CHILD": function( match ) { /* matches from matchExpr["CHILD"] 1 type (only|nth|...) 2 what (child|of-type) 3 argument (even|odd|\d*|\d*n([+-]\d+)?|...) 4 xn-component of xn+y argument ([+-]?\d*n|) 5 sign of xn-component 6 x of xn-component 7 sign of y-component 8 y of y-component */ match[ 1 ] = match[ 1 ].toLowerCase(); if ( match[ 1 ].slice( 0, 3 ) === "nth" ) { // nth-* requires argument if ( !match[ 3 ] ) { Sizzle.error( match[ 0 ] ); } // numeric x and y parameters for Expr.filter.CHILD // remember that false/true cast respectively to 0/1 match[ 4 ] = +( match[ 4 ] ? match[ 5 ] + ( match[ 6 ] || 1 ) : 2 * ( match[ 3 ] === "even" || match[ 3 ] === "odd" ) ); match[ 5 ] = +( ( match[ 7 ] + match[ 8 ] ) || match[ 3 ] === "odd" ); // other types prohibit arguments } else if ( match[ 3 ] ) { Sizzle.error( match[ 0 ] ); } return match; }, "PSEUDO": function( match ) { var excess, unquoted = !match[ 6 ] && match[ 2 ]; if ( matchExpr[ "CHILD" ].test( match[ 0 ] ) ) { return null; } // Accept quoted arguments as-is if ( match[ 3 ] ) { match[ 2 ] = match[ 4 ] || match[ 5 ] || ""; // Strip excess characters from unquoted arguments } else if ( unquoted && rpseudo.test( unquoted ) && // Get excess from tokenize (recursively) ( excess = tokenize( unquoted, true ) ) && // advance to the next closing parenthesis ( excess = unquoted.indexOf( ")", unquoted.length - excess ) - unquoted.length ) ) { // excess is a negative index match[ 0 ] = match[ 0 ].slice( 0, excess ); match[ 2 ] = unquoted.slice( 0, excess ); } // Return only captures needed by the pseudo filter method (type and argument) return match.slice( 0, 3 ); } }, filter: { "TAG": function( nodeNameSelector ) { var nodeName = nodeNameSelector.replace( runescape, funescape ).toLowerCase(); return nodeNameSelector === "*" ? function() { return true; } : function( elem ) { return elem.nodeName && elem.nodeName.toLowerCase() === nodeName; }; }, "CLASS": function( className ) { var pattern = classCache[ className + " " ]; return pattern || ( pattern = new RegExp( "(^|" + whitespace + ")" + className + "(" + whitespace + "|$)" ) ) && classCache( className, function( elem ) { return pattern.test( typeof elem.className === "string" && elem.className || typeof elem.getAttribute !== "undefined" && elem.getAttribute( "class" ) || "" ); } ); }, "ATTR": function( name, operator, check ) { return function( elem ) { var result = Sizzle.attr( elem, name ); if ( result == null ) { return operator === "!="; } if ( !operator ) { return true; } result += ""; /* eslint-disable max-len */ return operator === "=" ? result === check : operator === "!=" ? result !== check : operator === "^=" ? check && result.indexOf( check ) === 0 : operator === "*=" ? check && result.indexOf( check ) > -1 : operator === "$=" ? check && result.slice( -check.length ) === check : operator === "~=" ? ( " " + result.replace( rwhitespace, " " ) + " " ).indexOf( check ) > -1 : operator === "|=" ? result === check || result.slice( 0, check.length + 1 ) === check + "-" : false; /* eslint-enable max-len */ }; }, "CHILD": function( type, what, _argument, first, last ) { var simple = type.slice( 0, 3 ) !== "nth", forward = type.slice( -4 ) !== "last", ofType = what === "of-type"; return first === 1 && last === 0 ? // Shortcut for :nth-*(n) function( elem ) { return !!elem.parentNode; } : function( elem, _context, xml ) { var cache, uniqueCache, outerCache, node, nodeIndex, start, dir = simple !== forward ? "nextSibling" : "previousSibling", parent = elem.parentNode, name = ofType && elem.nodeName.toLowerCase(), useCache = !xml && !ofType, diff = false; if ( parent ) { // :(first|last|only)-(child|of-type) if ( simple ) { while ( dir ) { node = elem; while ( ( node = node[ dir ] ) ) { if ( ofType ? node.nodeName.toLowerCase() === name : node.nodeType === 1 ) { return false; } } // Reverse direction for :only-* (if we haven't yet done so) start = dir = type === "only" && !start && "nextSibling"; } return true; } start = [ forward ? parent.firstChild : parent.lastChild ]; // non-xml :nth-child(...) stores cache data on `parent` if ( forward && useCache ) { // Seek `elem` from a previously-cached index // ...in a gzip-friendly way node = parent; outerCache = node[ expando ] || ( node[ expando ] = {} ); // Support: IE <9 only // Defend against cloned attroperties (jQuery gh-1709) uniqueCache = outerCache[ node.uniqueID ] || ( outerCache[ node.uniqueID ] = {} ); cache = uniqueCache[ type ] || []; nodeIndex = cache[ 0 ] === dirruns && cache[ 1 ]; diff = nodeIndex && cache[ 2 ]; node = nodeIndex && parent.childNodes[ nodeIndex ]; while ( ( node = ++nodeIndex && node && node[ dir ] || // Fallback to seeking `elem` from the start ( diff = nodeIndex = 0 ) || start.pop() ) ) { // When found, cache indexes on `parent` and break if ( node.nodeType === 1 && ++diff && node === elem ) { uniqueCache[ type ] = [ dirruns, nodeIndex, diff ]; break; } } } else { // Use previously-cached element index if available if ( useCache ) { // ...in a gzip-friendly way node = elem; outerCache = node[ expando ] || ( node[ expando ] = {} ); // Support: IE <9 only // Defend against cloned attroperties (jQuery gh-1709) uniqueCache = outerCache[ node.uniqueID ] || ( outerCache[ node.uniqueID ] = {} ); cache = uniqueCache[ type ] || []; nodeIndex = cache[ 0 ] === dirruns && cache[ 1 ]; diff = nodeIndex; } // xml :nth-child(...) // or :nth-last-child(...) or :nth(-last)?-of-type(...) if ( diff === false ) { // Use the same loop as above to seek `elem` from the start while ( ( node = ++nodeIndex && node && node[ dir ] || ( diff = nodeIndex = 0 ) || start.pop() ) ) { if ( ( ofType ? node.nodeName.toLowerCase() === name : node.nodeType === 1 ) && ++diff ) { // Cache the index of each encountered element if ( useCache ) { outerCache = node[ expando ] || ( node[ expando ] = {} ); // Support: IE <9 only // Defend against cloned attroperties (jQuery gh-1709) uniqueCache = outerCache[ node.uniqueID ] || ( outerCache[ node.uniqueID ] = {} ); uniqueCache[ type ] = [ dirruns, diff ]; } if ( node === elem ) { break; } } } } } // Incorporate the offset, then check against cycle size diff -= last; return diff === first || ( diff % first === 0 && diff / first >= 0 ); } }; }, "PSEUDO": function( pseudo, argument ) { // pseudo-class names are case-insensitive // http://www.w3.org/TR/selectors/#pseudo-classes // Prioritize by case sensitivity in case custom pseudos are added with uppercase letters // Remember that setFilters inherits from pseudos var args, fn = Expr.pseudos[ pseudo ] || Expr.setFilters[ pseudo.toLowerCase() ] || Sizzle.error( "unsupported pseudo: " + pseudo ); // The user may use createPseudo to indicate that // arguments are needed to create the filter function // just as Sizzle does if ( fn[ expando ] ) { return fn( argument ); } // But maintain support for old signatures if ( fn.length > 1 ) { args = [ pseudo, pseudo, "", argument ]; return Expr.setFilters.hasOwnProperty( pseudo.toLowerCase() ) ? markFunction( function( seed, matches ) { var idx, matched = fn( seed, argument ), i = matched.length; while ( i-- ) { idx = indexOf( seed, matched[ i ] ); seed[ idx ] = !( matches[ idx ] = matched[ i ] ); } } ) : function( elem ) { return fn( elem, 0, args ); }; } return fn; } }, pseudos: { // Potentially complex pseudos "not": markFunction( function( selector ) { // Trim the selector passed to compile // to avoid treating leading and trailing // spaces as combinators var input = [], results = [], matcher = compile( selector.replace( rtrim, "$1" ) ); return matcher[ expando ] ? markFunction( function( seed, matches, _context, xml ) { var elem, unmatched = matcher( seed, null, xml, [] ), i = seed.length; // Match elements unmatched by `matcher` while ( i-- ) { if ( ( elem = unmatched[ i ] ) ) { seed[ i ] = !( matches[ i ] = elem ); } } } ) : function( elem, _context, xml ) { input[ 0 ] = elem; matcher( input, null, xml, results ); // Don't keep the element (issue #299) input[ 0 ] = null; return !results.pop(); }; } ), "has": markFunction( function( selector ) { return function( elem ) { return Sizzle( selector, elem ).length > 0; }; } ), "contains": markFunction( function( text ) { text = text.replace( runescape, funescape ); return function( elem ) { return ( elem.textContent || getText( elem ) ).indexOf( text ) > -1; }; } ), // "Whether an element is represented by a :lang() selector // is based solely on the element's language value // being equal to the identifier C, // or beginning with the identifier C immediately followed by "-". // The matching of C against the element's language value is performed case-insensitively. // The identifier C does not have to be a valid language name." // http://www.w3.org/TR/selectors/#lang-pseudo "lang": markFunction( function( lang ) { // lang value must be a valid identifier if ( !ridentifier.test( lang || "" ) ) { Sizzle.error( "unsupported lang: " + lang ); } lang = lang.replace( runescape, funescape ).toLowerCase(); return function( elem ) { var elemLang; do { if ( ( elemLang = documentIsHTML ? elem.lang : elem.getAttribute( "xml:lang" ) || elem.getAttribute( "lang" ) ) ) { elemLang = elemLang.toLowerCase(); return elemLang === lang || elemLang.indexOf( lang + "-" ) === 0; } } while ( ( elem = elem.parentNode ) && elem.nodeType === 1 ); return false; }; } ), // Miscellaneous "target": function( elem ) { var hash = window.location && window.location.hash; return hash && hash.slice( 1 ) === elem.id; }, "root": function( elem ) { return elem === docElem; }, "focus": function( elem ) { return elem === document.activeElement && ( !document.hasFocus || document.hasFocus() ) && !!( elem.type || elem.href || ~elem.tabIndex ); }, // Boolean properties "enabled": createDisabledPseudo( false ), "disabled": createDisabledPseudo( true ), "checked": function( elem ) { // In CSS3, :checked should return both checked and selected elements // http://www.w3.org/TR/2011/REC-css3-selectors-20110929/#checked var nodeName = elem.nodeName.toLowerCase(); return ( nodeName === "input" && !!elem.checked ) || ( nodeName === "option" && !!elem.selected ); }, "selected": function( elem ) { // Accessing this property makes selected-by-default // options in Safari work properly if ( elem.parentNode ) { // eslint-disable-next-line no-unused-expressions elem.parentNode.selectedIndex; } return elem.selected === true; }, // Contents "empty": function( elem ) { // http://www.w3.org/TR/selectors/#empty-pseudo // :empty is negated by element (1) or content nodes (text: 3; cdata: 4; entity ref: 5), // but not by others (comment: 8; processing instruction: 7; etc.) // nodeType < 6 works because attributes (2) do not appear as children for ( elem = elem.firstChild; elem; elem = elem.nextSibling ) { if ( elem.nodeType < 6 ) { return false; } } return true; }, "parent": function( elem ) { return !Expr.pseudos[ "empty" ]( elem ); }, // Element/input types "header": function( elem ) { return rheader.test( elem.nodeName ); }, "input": function( elem ) { return rinputs.test( elem.nodeName ); }, "button": function( elem ) { var name = elem.nodeName.toLowerCase(); return name === "input" && elem.type === "button" || name === "button"; }, "text": function( elem ) { var attr; return elem.nodeName.toLowerCase() === "input" && elem.type === "text" && // Support: IE<8 // New HTML5 attribute values (e.g., "search") appear with elem.type === "text" ( ( attr = elem.getAttribute( "type" ) ) == null || attr.toLowerCase() === "text" ); }, // Position-in-collection "first": createPositionalPseudo( function() { return [ 0 ]; } ), "last": createPositionalPseudo( function( _matchIndexes, length ) { return [ length - 1 ]; } ), "eq": createPositionalPseudo( function( _matchIndexes, length, argument ) { return [ argument < 0 ? argument + length : argument ]; } ), "even": createPositionalPseudo( function( matchIndexes, length ) { var i = 0; for ( ; i < length; i += 2 ) { matchIndexes.push( i ); } return matchIndexes; } ), "odd": createPositionalPseudo( function( matchIndexes, length ) { var i = 1; for ( ; i < length; i += 2 ) { matchIndexes.push( i ); } return matchIndexes; } ), "lt": createPositionalPseudo( function( matchIndexes, length, argument ) { var i = argument < 0 ? argument + length : argument > length ? length : argument; for ( ; --i >= 0; ) { matchIndexes.push( i ); } return matchIndexes; } ), "gt": createPositionalPseudo( function( matchIndexes, length, argument ) { var i = argument < 0 ? argument + length : argument; for ( ; ++i < length; ) { matchIndexes.push( i ); } return matchIndexes; } ) } }; Expr.pseudos[ "nth" ] = Expr.pseudos[ "eq" ]; // Add button/input type pseudos for ( i in { radio: true, checkbox: true, file: true, password: true, image: true } ) { Expr.pseudos[ i ] = createInputPseudo( i ); } for ( i in { submit: true, reset: true } ) { Expr.pseudos[ i ] = createButtonPseudo( i ); } // Easy API for creating new setFilters function setFilters() {} setFilters.prototype = Expr.filters = Expr.pseudos; Expr.setFilters = new setFilters(); tokenize = Sizzle.tokenize = function( selector, parseOnly ) { var matched, match, tokens, type, soFar, groups, preFilters, cached = tokenCache[ selector + " " ]; if ( cached ) { return parseOnly ? 0 : cached.slice( 0 ); } soFar = selector; groups = []; preFilters = Expr.preFilter; while ( soFar ) { // Comma and first run if ( !matched || ( match = rcomma.exec( soFar ) ) ) { if ( match ) { // Don't consume trailing commas as valid soFar = soFar.slice( match[ 0 ].length ) || soFar; } groups.push( ( tokens = [] ) ); } matched = false; // Combinators if ( ( match = rcombinators.exec( soFar ) ) ) { matched = match.shift(); tokens.push( { value: matched, // Cast descendant combinators to space type: match[ 0 ].replace( rtrim, " " ) } ); soFar = soFar.slice( matched.length ); } // Filters for ( type in Expr.filter ) { if ( ( match = matchExpr[ type ].exec( soFar ) ) && ( !preFilters[ type ] || ( match = preFilters[ type ]( match ) ) ) ) { matched = match.shift(); tokens.push( { value: matched, type: type, matches: match } ); soFar = soFar.slice( matched.length ); } } if ( !matched ) { break; } } // Return the length of the invalid excess // if we're just parsing // Otherwise, throw an error or return tokens return parseOnly ? soFar.length : soFar ? Sizzle.error( selector ) : // Cache the tokens tokenCache( selector, groups ).slice( 0 ); }; function toSelector( tokens ) { var i = 0, len = tokens.length, selector = ""; for ( ; i < len; i++ ) { selector += tokens[ i ].value; } return selector; } function addCombinator( matcher, combinator, base ) { var dir = combinator.dir, skip = combinator.next, key = skip || dir, checkNonElements = base && key === "parentNode", doneName = done++; return combinator.first ? // Check against closest ancestor/preceding element function( elem, context, xml ) { while ( ( elem = elem[ dir ] ) ) { if ( elem.nodeType === 1 || checkNonElements ) { return matcher( elem, context, xml ); } } return false; } : // Check against all ancestor/preceding elements function( elem, context, xml ) { var oldCache, uniqueCache, outerCache, newCache = [ dirruns, doneName ]; // We can't set arbitrary data on XML nodes, so they don't benefit from combinator caching if ( xml ) { while ( ( elem = elem[ dir ] ) ) { if ( elem.nodeType === 1 || checkNonElements ) { if ( matcher( elem, context, xml ) ) { return true; } } } } else { while ( ( elem = elem[ dir ] ) ) { if ( elem.nodeType === 1 || checkNonElements ) { outerCache = elem[ expando ] || ( elem[ expando ] = {} ); // Support: IE <9 only // Defend against cloned attroperties (jQuery gh-1709) uniqueCache = outerCache[ elem.uniqueID ] || ( outerCache[ elem.uniqueID ] = {} ); if ( skip && skip === elem.nodeName.toLowerCase() ) { elem = elem[ dir ] || elem; } else if ( ( oldCache = uniqueCache[ key ] ) && oldCache[ 0 ] === dirruns && oldCache[ 1 ] === doneName ) { // Assign to newCache so results back-propagate to previous elements return ( newCache[ 2 ] = oldCache[ 2 ] ); } else { // Reuse newcache so results back-propagate to previous elements uniqueCache[ key ] = newCache; // A match means we're done; a fail means we have to keep checking if ( ( newCache[ 2 ] = matcher( elem, context, xml ) ) ) { return true; } } } } } return false; }; } function elementMatcher( matchers ) { return matchers.length > 1 ? function( elem, context, xml ) { var i = matchers.length; while ( i-- ) { if ( !matchers[ i ]( elem, context, xml ) ) { return false; } } return true; } : matchers[ 0 ]; } function multipleContexts( selector, contexts, results ) { var i = 0, len = contexts.length; for ( ; i < len; i++ ) { Sizzle( selector, contexts[ i ], results ); } return results; } function condense( unmatched, map, filter, context, xml ) { var elem, newUnmatched = [], i = 0, len = unmatched.length, mapped = map != null; for ( ; i < len; i++ ) { if ( ( elem = unmatched[ i ] ) ) { if ( !filter || filter( elem, context, xml ) ) { newUnmatched.push( elem ); if ( mapped ) { map.push( i ); } } } } return newUnmatched; } function setMatcher( preFilter, selector, matcher, postFilter, postFinder, postSelector ) { if ( postFilter && !postFilter[ expando ] ) { postFilter = setMatcher( postFilter ); } if ( postFinder && !postFinder[ expando ] ) { postFinder = setMatcher( postFinder, postSelector ); } return markFunction( function( seed, results, context, xml ) { var temp, i, elem, preMap = [], postMap = [], preexisting = results.length, // Get initial elements from seed or context elems = seed || multipleContexts( selector || "*", context.nodeType ? [ context ] : context, [] ), // Prefilter to get matcher input, preserving a map for seed-results synchronization matcherIn = preFilter && ( seed || !selector ) ? condense( elems, preMap, preFilter, context, xml ) : elems, matcherOut = matcher ? // If we have a postFinder, or filtered seed, or non-seed postFilter or preexisting results, postFinder || ( seed ? preFilter : preexisting || postFilter ) ? // ...intermediate processing is necessary [] : // ...otherwise use results directly results : matcherIn; // Find primary matches if ( matcher ) { matcher( matcherIn, matcherOut, context, xml ); } // Apply postFilter if ( postFilter ) { temp = condense( matcherOut, postMap ); postFilter( temp, [], context, xml ); // Un-match failing elements by moving them back to matcherIn i = temp.length; while ( i-- ) { if ( ( elem = temp[ i ] ) ) { matcherOut[ postMap[ i ] ] = !( matcherIn[ postMap[ i ] ] = elem ); } } } if ( seed ) { if ( postFinder || preFilter ) { if ( postFinder ) { // Get the final matcherOut by condensing this intermediate into postFinder contexts temp = []; i = matcherOut.length; while ( i-- ) { if ( ( elem = matcherOut[ i ] ) ) { // Restore matcherIn since elem is not yet a final match temp.push( ( matcherIn[ i ] = elem ) ); } } postFinder( null, ( matcherOut = [] ), temp, xml ); } // Move matched elements from seed to results to keep them synchronized i = matcherOut.length; while ( i-- ) { if ( ( elem = matcherOut[ i ] ) && ( temp = postFinder ? indexOf( seed, elem ) : preMap[ i ] ) > -1 ) { seed[ temp ] = !( results[ temp ] = elem ); } } } // Add elements to results, through postFinder if defined } else { matcherOut = condense( matcherOut === results ? matcherOut.splice( preexisting, matcherOut.length ) : matcherOut ); if ( postFinder ) { postFinder( null, results, matcherOut, xml ); } else { push.apply( results, matcherOut ); } } } ); } function matcherFromTokens( tokens ) { var checkContext, matcher, j, len = tokens.length, leadingRelative = Expr.relative[ tokens[ 0 ].type ], implicitRelative = leadingRelative || Expr.relative[ " " ], i = leadingRelative ? 1 : 0, // The foundational matcher ensures that elements are reachable from top-level context(s) matchContext = addCombinator( function( elem ) { return elem === checkContext; }, implicitRelative, true ), matchAnyContext = addCombinator( function( elem ) { return indexOf( checkContext, elem ) > -1; }, implicitRelative, true ), matchers = [ function( elem, context, xml ) { var ret = ( !leadingRelative && ( xml || context !== outermostContext ) ) || ( ( checkContext = context ).nodeType ? matchContext( elem, context, xml ) : matchAnyContext( elem, context, xml ) ); // Avoid hanging onto element (issue #299) checkContext = null; return ret; } ]; for ( ; i < len; i++ ) { if ( ( matcher = Expr.relative[ tokens[ i ].type ] ) ) { matchers = [ addCombinator( elementMatcher( matchers ), matcher ) ]; } else { matcher = Expr.filter[ tokens[ i ].type ].apply( null, tokens[ i ].matches ); // Return special upon seeing a positional matcher if ( matcher[ expando ] ) { // Find the next relative operator (if any) for proper handling j = ++i; for ( ; j < len; j++ ) { if ( Expr.relative[ tokens[ j ].type ] ) { break; } } return setMatcher( i > 1 && elementMatcher( matchers ), i > 1 && toSelector( // If the preceding token was a descendant combinator, insert an implicit any-element `*` tokens .slice( 0, i - 1 ) .concat( { value: tokens[ i - 2 ].type === " " ? "*" : "" } ) ).replace( rtrim, "$1" ), matcher, i < j && matcherFromTokens( tokens.slice( i, j ) ), j < len && matcherFromTokens( ( tokens = tokens.slice( j ) ) ), j < len && toSelector( tokens ) ); } matchers.push( matcher ); } } return elementMatcher( matchers ); } function matcherFromGroupMatchers( elementMatchers, setMatchers ) { var bySet = setMatchers.length > 0, byElement = elementMatchers.length > 0, superMatcher = function( seed, context, xml, results, outermost ) { var elem, j, matcher, matchedCount = 0, i = "0", unmatched = seed && [], setMatched = [], contextBackup = outermostContext, // We must always have either seed elements or outermost context elems = seed || byElement && Expr.find[ "TAG" ]( "*", outermost ), // Use integer dirruns iff this is the outermost matcher dirrunsUnique = ( dirruns += contextBackup == null ? 1 : Math.random() || 0.1 ), len = elems.length; if ( outermost ) { // Support: IE 11+, Edge 17 - 18+ // IE/Edge sometimes throw a "Permission denied" error when strict-comparing // two documents; shallow comparisons work. // eslint-disable-next-line eqeqeq outermostContext = context == document || context || outermost; } // Add elements passing elementMatchers directly to results // Support: IE<9, Safari // Tolerate NodeList properties (IE: "length"; Safari: <number>) matching elements by id for ( ; i !== len && ( elem = elems[ i ] ) != null; i++ ) { if ( byElement && elem ) { j = 0; // Support: IE 11+, Edge 17 - 18+ // IE/Edge sometimes throw a "Permission denied" error when strict-comparing // two documents; shallow comparisons work. // eslint-disable-next-line eqeqeq if ( !context && elem.ownerDocument != document ) { setDocument( elem ); xml = !documentIsHTML; } while ( ( matcher = elementMatchers[ j++ ] ) ) { if ( matcher( elem, context || document, xml ) ) { results.push( elem ); break; } } if ( outermost ) { dirruns = dirrunsUnique; } } // Track unmatched elements for set filters if ( bySet ) { // They will have gone through all possible matchers if ( ( elem = !matcher && elem ) ) { matchedCount--; } // Lengthen the array for every element, matched or not if ( seed ) { unmatched.push( elem ); } } } // `i` is now the count of elements visited above, and adding it to `matchedCount` // makes the latter nonnegative. matchedCount += i; // Apply set filters to unmatched elements // NOTE: This can be skipped if there are no unmatched elements (i.e., `matchedCount` // equals `i`), unless we didn't visit _any_ elements in the above loop because we have // no element matchers and no seed. // Incrementing an initially-string "0" `i` allows `i` to remain a string only in that // case, which will result in a "00" `matchedCount` that differs from `i` but is also // numerically zero. if ( bySet && i !== matchedCount ) { j = 0; while ( ( matcher = setMatchers[ j++ ] ) ) { matcher( unmatched, setMatched, context, xml ); } if ( seed ) { // Reintegrate element matches to eliminate the need for sorting if ( matchedCount > 0 ) { while ( i-- ) { if ( !( unmatched[ i ] || setMatched[ i ] ) ) { setMatched[ i ] = pop.call( results ); } } } // Discard index placeholder values to get only actual matches setMatched = condense( setMatched ); } // Add matches to results push.apply( results, setMatched ); // Seedless set matches succeeding multiple successful matchers stipulate sorting if ( outermost && !seed && setMatched.length > 0 && ( matchedCount + setMatchers.length ) > 1 ) { Sizzle.uniqueSort( results ); } } // Override manipulation of globals by nested matchers if ( outermost ) { dirruns = dirrunsUnique; outermostContext = contextBackup; } return unmatched; }; return bySet ? markFunction( superMatcher ) : superMatcher; } compile = Sizzle.compile = function( selector, match /* Internal Use Only */ ) { var i, setMatchers = [], elementMatchers = [], cached = compilerCache[ selector + " " ]; if ( !cached ) { // Generate a function of recursive functions that can be used to check each element if ( !match ) { match = tokenize( selector ); } i = match.length; while ( i-- ) { cached = matcherFromTokens( match[ i ] ); if ( cached[ expando ] ) { setMatchers.push( cached ); } else { elementMatchers.push( cached ); } } // Cache the compiled function cached = compilerCache( selector, matcherFromGroupMatchers( elementMatchers, setMatchers ) ); // Save selector and tokenization cached.selector = selector; } return cached; }; /** * A low-level selection function that works with Sizzle's compiled * selector functions * @param {String|Function} selector A selector or a pre-compiled * selector function built with Sizzle.compile * @param {Element} context * @param {Array} [results] * @param {Array} [seed] A set of elements to match against */ select = Sizzle.select = function( selector, context, results, seed ) { var i, tokens, token, type, find, compiled = typeof selector === "function" && selector, match = !seed && tokenize( ( selector = compiled.selector || selector ) ); results = results || []; // Try to minimize operations if there is only one selector in the list and no seed // (the latter of which guarantees us context) if ( match.length === 1 ) { // Reduce context if the leading compound selector is an ID tokens = match[ 0 ] = match[ 0 ].slice( 0 ); if ( tokens.length > 2 && ( token = tokens[ 0 ] ).type === "ID" && context.nodeType === 9 && documentIsHTML && Expr.relative[ tokens[ 1 ].type ] ) { context = ( Expr.find[ "ID" ]( token.matches[ 0 ] .replace( runescape, funescape ), context ) || [] )[ 0 ]; if ( !context ) { return results; // Precompiled matchers will still verify ancestry, so step up a level } else if ( compiled ) { context = context.parentNode; } selector = selector.slice( tokens.shift().value.length ); } // Fetch a seed set for right-to-left matching i = matchExpr[ "needsContext" ].test( selector ) ? 0 : tokens.length; while ( i-- ) { token = tokens[ i ]; // Abort if we hit a combinator if ( Expr.relative[ ( type = token.type ) ] ) { break; } if ( ( find = Expr.find[ type ] ) ) { // Search, expanding context for leading sibling combinators if ( ( seed = find( token.matches[ 0 ].replace( runescape, funescape ), rsibling.test( tokens[ 0 ].type ) && testContext( context.parentNode ) || context ) ) ) { // If seed is empty or no tokens remain, we can return early tokens.splice( i, 1 ); selector = seed.length && toSelector( tokens ); if ( !selector ) { push.apply( results, seed ); return results; } break; } } } } // Compile and execute a filtering function if one is not provided // Provide `match` to avoid retokenization if we modified the selector above ( compiled || compile( selector, match ) )( seed, context, !documentIsHTML, results, !context || rsibling.test( selector ) && testContext( context.parentNode ) || context ); return results; }; // One-time assignments // Sort stability support.sortStable = expando.split( "" ).sort( sortOrder ).join( "" ) === expando; // Support: Chrome 14-35+ // Always assume duplicates if they aren't passed to the comparison function support.detectDuplicates = !!hasDuplicate; // Initialize against the default document setDocument(); // Support: Webkit<537.32 - Safari 6.0.3/Chrome 25 (fixed in Chrome 27) // Detached nodes confoundingly follow *each other* support.sortDetached = assert( function( el ) { // Should return 1, but returns 4 (following) return el.compareDocumentPosition( document.createElement( "fieldset" ) ) & 1; } ); // Support: IE<8 // Prevent attribute/property "interpolation" // https://msdn.microsoft.com/en-us/library/ms536429%28VS.85%29.aspx if ( !assert( function( el ) { el.innerHTML = "<a href='#'></a>"; return el.firstChild.getAttribute( "href" ) === "#"; } ) ) { addHandle( "type|href|height|width", function( elem, name, isXML ) { if ( !isXML ) { return elem.getAttribute( name, name.toLowerCase() === "type" ? 1 : 2 ); } } ); } // Support: IE<9 // Use defaultValue in place of getAttribute("value") if ( !support.attributes || !assert( function( el ) { el.innerHTML = "<input/>"; el.firstChild.setAttribute( "value", "" ); return el.firstChild.getAttribute( "value" ) === ""; } ) ) { addHandle( "value", function( elem, _name, isXML ) { if ( !isXML && elem.nodeName.toLowerCase() === "input" ) { return elem.defaultValue; } } ); } // Support: IE<9 // Use getAttributeNode to fetch booleans when getAttribute lies if ( !assert( function( el ) { return el.getAttribute( "disabled" ) == null; } ) ) { addHandle( booleans, function( elem, name, isXML ) { var val; if ( !isXML ) { return elem[ name ] === true ? name.toLowerCase() : ( val = elem.getAttributeNode( name ) ) && val.specified ? val.value : null; } } ); } return Sizzle; } )( window ); jQuery.find = Sizzle; jQuery.expr = Sizzle.selectors; // Deprecated jQuery.expr[ ":" ] = jQuery.expr.pseudos; jQuery.uniqueSort = jQuery.unique = Sizzle.uniqueSort; jQuery.text = Sizzle.getText; jQuery.isXMLDoc = Sizzle.isXML; jQuery.contains = Sizzle.contains; jQuery.escapeSelector = Sizzle.escape; var dir = function( elem, dir, until ) { var matched = [], truncate = until !== undefined; while ( ( elem = elem[ dir ] ) && elem.nodeType !== 9 ) { if ( elem.nodeType === 1 ) { if ( truncate && jQuery( elem ).is( until ) ) { break; } matched.push( elem ); } } return matched; }; var siblings = function( n, elem ) { var matched = []; for ( ; n; n = n.nextSibling ) { if ( n.nodeType === 1 && n !== elem ) { matched.push( n ); } } return matched; }; var rneedsContext = jQuery.expr.match.needsContext; function nodeName( elem, name ) { return elem.nodeName && elem.nodeName.toLowerCase() === name.toLowerCase(); }; var rsingleTag = ( /^<([a-z][^\/\0>:\x20\t\r\n\f]*)[\x20\t\r\n\f]*\/?>(?:<\/\1>|)$/i ); // Implement the identical functionality for filter and not function winnow( elements, qualifier, not ) { if ( isFunction( qualifier ) ) { return jQuery.grep( elements, function( elem, i ) { return !!qualifier.call( elem, i, elem ) !== not; } ); } // Single element if ( qualifier.nodeType ) { return jQuery.grep( elements, function( elem ) { return ( elem === qualifier ) !== not; } ); } // Arraylike of elements (jQuery, arguments, Array) if ( typeof qualifier !== "string" ) { return jQuery.grep( elements, function( elem ) { return ( indexOf.call( qualifier, elem ) > -1 ) !== not; } ); } // Filtered directly for both simple and complex selectors return jQuery.filter( qualifier, elements, not ); } jQuery.filter = function( expr, elems, not ) { var elem = elems[ 0 ]; if ( not ) { expr = ":not(" + expr + ")"; } if ( elems.length === 1 && elem.nodeType === 1 ) { return jQuery.find.matchesSelector( elem, expr ) ? [ elem ] : []; } return jQuery.find.matches( expr, jQuery.grep( elems, function( elem ) { return elem.nodeType === 1; } ) ); }; jQuery.fn.extend( { find: function( selector ) { var i, ret, len = this.length, self = this; if ( typeof selector !== "string" ) { return this.pushStack( jQuery( selector ).filter( function() { for ( i = 0; i < len; i++ ) { if ( jQuery.contains( self[ i ], this ) ) { return true; } } } ) ); } ret = this.pushStack( [] ); for ( i = 0; i < len; i++ ) { jQuery.find( selector, self[ i ], ret ); } return len > 1 ? jQuery.uniqueSort( ret ) : ret; }, filter: function( selector ) { return this.pushStack( winnow( this, selector || [], false ) ); }, not: function( selector ) { return this.pushStack( winnow( this, selector || [], true ) ); }, is: function( selector ) { return !!winnow( this, // If this is a positional/relative selector, check membership in the returned set // so $("p:first").is("p:last") won't return true for a doc with two "p". typeof selector === "string" && rneedsContext.test( selector ) ? jQuery( selector ) : selector || [], false ).length; } } ); // Initialize a jQuery object // A central reference to the root jQuery(document) var rootjQuery, // A simple way to check for HTML strings // Prioritize #id over <tag> to avoid XSS via location.hash (#9521) // Strict HTML recognition (#11290: must start with <) // Shortcut simple #id case for speed rquickExpr = /^(?:\s*(<[\w\W]+>)[^>]*|#([\w-]+))$/, init = jQuery.fn.init = function( selector, context, root ) { var match, elem; // HANDLE: $(""), $(null), $(undefined), $(false) if ( !selector ) { return this; } // Method init() accepts an alternate rootjQuery // so migrate can support jQuery.sub (gh-2101) root = root || rootjQuery; // Handle HTML strings if ( typeof selector === "string" ) { if ( selector[ 0 ] === "<" && selector[ selector.length - 1 ] === ">" && selector.length >= 3 ) { // Assume that strings that start and end with <> are HTML and skip the regex check match = [ null, selector, null ]; } else { match = rquickExpr.exec( selector ); } // Match html or make sure no context is specified for #id if ( match && ( match[ 1 ] || !context ) ) { // HANDLE: $(html) -> $(array) if ( match[ 1 ] ) { context = context instanceof jQuery ? context[ 0 ] : context; // Option to run scripts is true for back-compat // Intentionally let the error be thrown if parseHTML is not present jQuery.merge( this, jQuery.parseHTML( match[ 1 ], context && context.nodeType ? context.ownerDocument || context : document, true ) ); // HANDLE: $(html, props) if ( rsingleTag.test( match[ 1 ] ) && jQuery.isPlainObject( context ) ) { for ( match in context ) { // Properties of context are called as methods if possible if ( isFunction( this[ match ] ) ) { this[ match ]( context[ match ] ); // ...and otherwise set as attributes } else { this.attr( match, context[ match ] ); } } } return this; // HANDLE: $(#id) } else { elem = document.getElementById( match[ 2 ] ); if ( elem ) { // Inject the element directly into the jQuery object this[ 0 ] = elem; this.length = 1; } return this; } // HANDLE: $(expr, $(...)) } else if ( !context || context.jquery ) { return ( context || root ).find( selector ); // HANDLE: $(expr, context) // (which is just equivalent to: $(context).find(expr) } else { return this.constructor( context ).find( selector ); } // HANDLE: $(DOMElement) } else if ( selector.nodeType ) { this[ 0 ] = selector; this.length = 1; return this; // HANDLE: $(function) // Shortcut for document ready } else if ( isFunction( selector ) ) { return root.ready !== undefined ? root.ready( selector ) : // Execute immediately if ready is not present selector( jQuery ); } return jQuery.makeArray( selector, this ); }; // Give the init function the jQuery prototype for later instantiation init.prototype = jQuery.fn; // Initialize central reference rootjQuery = jQuery( document ); var rparentsprev = /^(?:parents|prev(?:Until|All))/, // Methods guaranteed to produce a unique set when starting from a unique set guaranteedUnique = { children: true, contents: true, next: true, prev: true }; jQuery.fn.extend( { has: function( target ) { var targets = jQuery( target, this ), l = targets.length; return this.filter( function() { var i = 0; for ( ; i < l; i++ ) { if ( jQuery.contains( this, targets[ i ] ) ) { return true; } } } ); }, closest: function( selectors, context ) { var cur, i = 0, l = this.length, matched = [], targets = typeof selectors !== "string" && jQuery( selectors ); // Positional selectors never match, since there's no _selection_ context if ( !rneedsContext.test( selectors ) ) { for ( ; i < l; i++ ) { for ( cur = this[ i ]; cur && cur !== context; cur = cur.parentNode ) { // Always skip document fragments if ( cur.nodeType < 11 && ( targets ? targets.index( cur ) > -1 : // Don't pass non-elements to Sizzle cur.nodeType === 1 && jQuery.find.matchesSelector( cur, selectors ) ) ) { matched.push( cur ); break; } } } } return this.pushStack( matched.length > 1 ? jQuery.uniqueSort( matched ) : matched ); }, // Determine the position of an element within the set index: function( elem ) { // No argument, return index in parent if ( !elem ) { return ( this[ 0 ] && this[ 0 ].parentNode ) ? this.first().prevAll().length : -1; } // Index in selector if ( typeof elem === "string" ) { return indexOf.call( jQuery( elem ), this[ 0 ] ); } // Locate the position of the desired element return indexOf.call( this, // If it receives a jQuery object, the first element is used elem.jquery ? elem[ 0 ] : elem ); }, add: function( selector, context ) { return this.pushStack( jQuery.uniqueSort( jQuery.merge( this.get(), jQuery( selector, context ) ) ) ); }, addBack: function( selector ) { return this.add( selector == null ? this.prevObject : this.prevObject.filter( selector ) ); } } ); function sibling( cur, dir ) { while ( ( cur = cur[ dir ] ) && cur.nodeType !== 1 ) {} return cur; } jQuery.each( { parent: function( elem ) { var parent = elem.parentNode; return parent && parent.nodeType !== 11 ? parent : null; }, parents: function( elem ) { return dir( elem, "parentNode" ); }, parentsUntil: function( elem, _i, until ) { return dir( elem, "parentNode", until ); }, next: function( elem ) { return sibling( elem, "nextSibling" ); }, prev: function( elem ) { return sibling( elem, "previousSibling" ); }, nextAll: function( elem ) { return dir( elem, "nextSibling" ); }, prevAll: function( elem ) { return dir( elem, "previousSibling" ); }, nextUntil: function( elem, _i, until ) { return dir( elem, "nextSibling", until ); }, prevUntil: function( elem, _i, until ) { return dir( elem, "previousSibling", until ); }, siblings: function( elem ) { return siblings( ( elem.parentNode || {} ).firstChild, elem ); }, children: function( elem ) { return siblings( elem.firstChild ); }, contents: function( elem ) { if ( elem.contentDocument != null && // Support: IE 11+ // <object> elements with no `data` attribute has an object // `contentDocument` with a `null` prototype. getProto( elem.contentDocument ) ) { return elem.contentDocument; } // Support: IE 9 - 11 only, iOS 7 only, Android Browser <=4.3 only // Treat the template element as a regular one in browsers that // don't support it. if ( nodeName( elem, "template" ) ) { elem = elem.content || elem; } return jQuery.merge( [], elem.childNodes ); } }, function( name, fn ) { jQuery.fn[ name ] = function( until, selector ) { var matched = jQuery.map( this, fn, until ); if ( name.slice( -5 ) !== "Until" ) { selector = until; } if ( selector && typeof selector === "string" ) { matched = jQuery.filter( selector, matched ); } if ( this.length > 1 ) { // Remove duplicates if ( !guaranteedUnique[ name ] ) { jQuery.uniqueSort( matched ); } // Reverse order for parents* and prev-derivatives if ( rparentsprev.test( name ) ) { matched.reverse(); } } return this.pushStack( matched ); }; } ); var rnothtmlwhite = ( /[^\x20\t\r\n\f]+/g ); // Convert String-formatted options into Object-formatted ones function createOptions( options ) { var object = {}; jQuery.each( options.match( rnothtmlwhite ) || [], function( _, flag ) { object[ flag ] = true; } ); return object; } /* * Create a callback list using the following parameters: * * options: an optional list of space-separated options that will change how * the callback list behaves or a more traditional option object * * By default a callback list will act like an event callback list and can be * "fired" multiple times. * * Possible options: * * once: will ensure the callback list can only be fired once (like a Deferred) * * memory: will keep track of previous values and will call any callback added * after the list has been fired right away with the latest "memorized" * values (like a Deferred) * * unique: will ensure a callback can only be added once (no duplicate in the list) * * stopOnFalse: interrupt callings when a callback returns false * */ jQuery.Callbacks = function( options ) { // Convert options from String-formatted to Object-formatted if needed // (we check in cache first) options = typeof options === "string" ? createOptions( options ) : jQuery.extend( {}, options ); var // Flag to know if list is currently firing firing, // Last fire value for non-forgettable lists memory, // Flag to know if list was already fired fired, // Flag to prevent firing locked, // Actual callback list list = [], // Queue of execution data for repeatable lists queue = [], // Index of currently firing callback (modified by add/remove as needed) firingIndex = -1, // Fire callbacks fire = function() { // Enforce single-firing locked = locked || options.once; // Execute callbacks for all pending executions, // respecting firingIndex overrides and runtime changes fired = firing = true; for ( ; queue.length; firingIndex = -1 ) { memory = queue.shift(); while ( ++firingIndex < list.length ) { // Run callback and check for early termination if ( list[ firingIndex ].apply( memory[ 0 ], memory[ 1 ] ) === false && options.stopOnFalse ) { // Jump to end and forget the data so .add doesn't re-fire firingIndex = list.length; memory = false; } } } // Forget the data if we're done with it if ( !options.memory ) { memory = false; } firing = false; // Clean up if we're done firing for good if ( locked ) { // Keep an empty list if we have data for future add calls if ( memory ) { list = []; // Otherwise, this object is spent } else { list = ""; } } }, // Actual Callbacks object self = { // Add a callback or a collection of callbacks to the list add: function() { if ( list ) { // If we have memory from a past run, we should fire after adding if ( memory && !firing ) { firingIndex = list.length - 1; queue.push( memory ); } ( function add( args ) { jQuery.each( args, function( _, arg ) { if ( isFunction( arg ) ) { if ( !options.unique || !self.has( arg ) ) { list.push( arg ); } } else if ( arg && arg.length && toType( arg ) !== "string" ) { // Inspect recursively add( arg ); } } ); } )( arguments ); if ( memory && !firing ) { fire(); } } return this; }, // Remove a callback from the list remove: function() { jQuery.each( arguments, function( _, arg ) { var index; while ( ( index = jQuery.inArray( arg, list, index ) ) > -1 ) { list.splice( index, 1 ); // Handle firing indexes if ( index <= firingIndex ) { firingIndex--; } } } ); return this; }, // Check if a given callback is in the list. // If no argument is given, return whether or not list has callbacks attached. has: function( fn ) { return fn ? jQuery.inArray( fn, list ) > -1 : list.length > 0; }, // Remove all callbacks from the list empty: function() { if ( list ) { list = []; } return this; }, // Disable .fire and .add // Abort any current/pending executions // Clear all callbacks and values disable: function() { locked = queue = []; list = memory = ""; return this; }, disabled: function() { return !list; }, // Disable .fire // Also disable .add unless we have memory (since it would have no effect) // Abort any pending executions lock: function() { locked = queue = []; if ( !memory && !firing ) { list = memory = ""; } return this; }, locked: function() { return !!locked; }, // Call all callbacks with the given context and arguments fireWith: function( context, args ) { if ( !locked ) { args = args || []; args = [ context, args.slice ? args.slice() : args ]; queue.push( args ); if ( !firing ) { fire(); } } return this; }, // Call all the callbacks with the given arguments fire: function() { self.fireWith( this, arguments ); return this; }, // To know if the callbacks have already been called at least once fired: function() { return !!fired; } }; return self; }; function Identity( v ) { return v; } function Thrower( ex ) { throw ex; } function adoptValue( value, resolve, reject, noValue ) { var method; try { // Check for promise aspect first to privilege synchronous behavior if ( value && isFunction( ( method = value.promise ) ) ) { method.call( value ).done( resolve ).fail( reject ); // Other thenables } else if ( value && isFunction( ( method = value.then ) ) ) { method.call( value, resolve, reject ); // Other non-thenables } else { // Control `resolve` arguments by letting Array#slice cast boolean `noValue` to integer: // * false: [ value ].slice( 0 ) => resolve( value ) // * true: [ value ].slice( 1 ) => resolve() resolve.apply( undefined, [ value ].slice( noValue ) ); } // For Promises/A+, convert exceptions into rejections // Since jQuery.when doesn't unwrap thenables, we can skip the extra checks appearing in // Deferred#then to conditionally suppress rejection. } catch ( value ) { // Support: Android 4.0 only // Strict mode functions invoked without .call/.apply get global-object context reject.apply( undefined, [ value ] ); } } jQuery.extend( { Deferred: function( func ) { var tuples = [ // action, add listener, callbacks, // ... .then handlers, argument index, [final state] [ "notify", "progress", jQuery.Callbacks( "memory" ), jQuery.Callbacks( "memory" ), 2 ], [ "resolve", "done", jQuery.Callbacks( "once memory" ), jQuery.Callbacks( "once memory" ), 0, "resolved" ], [ "reject", "fail", jQuery.Callbacks( "once memory" ), jQuery.Callbacks( "once memory" ), 1, "rejected" ] ], state = "pending", promise = { state: function() { return state; }, always: function() { deferred.done( arguments ).fail( arguments ); return this; }, "catch": function( fn ) { return promise.then( null, fn ); }, // Keep pipe for back-compat pipe: function( /* fnDone, fnFail, fnProgress */ ) { var fns = arguments; return jQuery.Deferred( function( newDefer ) { jQuery.each( tuples, function( _i, tuple ) { // Map tuples (progress, done, fail) to arguments (done, fail, progress) var fn = isFunction( fns[ tuple[ 4 ] ] ) && fns[ tuple[ 4 ] ]; // deferred.progress(function() { bind to newDefer or newDefer.notify }) // deferred.done(function() { bind to newDefer or newDefer.resolve }) // deferred.fail(function() { bind to newDefer or newDefer.reject }) deferred[ tuple[ 1 ] ]( function() { var returned = fn && fn.apply( this, arguments ); if ( returned && isFunction( returned.promise ) ) { returned.promise() .progress( newDefer.notify ) .done( newDefer.resolve ) .fail( newDefer.reject ); } else { newDefer[ tuple[ 0 ] + "With" ]( this, fn ? [ returned ] : arguments ); } } ); } ); fns = null; } ).promise(); }, then: function( onFulfilled, onRejected, onProgress ) { var maxDepth = 0; function resolve( depth, deferred, handler, special ) { return function() { var that = this, args = arguments, mightThrow = function() { var returned, then; // Support: Promises/A+ section 2.3.3.3.3 // https://promisesaplus.com/#point-59 // Ignore double-resolution attempts if ( depth < maxDepth ) { return; } returned = handler.apply( that, args ); // Support: Promises/A+ section 2.3.1 // https://promisesaplus.com/#point-48 if ( returned === deferred.promise() ) { throw new TypeError( "Thenable self-resolution" ); } // Support: Promises/A+ sections 2.3.3.1, 3.5 // https://promisesaplus.com/#point-54 // https://promisesaplus.com/#point-75 // Retrieve `then` only once then = returned && // Support: Promises/A+ section 2.3.4 // https://promisesaplus.com/#point-64 // Only check objects and functions for thenability ( typeof returned === "object" || typeof returned === "function" ) && returned.then; // Handle a returned thenable if ( isFunction( then ) ) { // Special processors (notify) just wait for resolution if ( special ) { then.call( returned, resolve( maxDepth, deferred, Identity, special ), resolve( maxDepth, deferred, Thrower, special ) ); // Normal processors (resolve) also hook into progress } else { // ...and disregard older resolution values maxDepth++; then.call( returned, resolve( maxDepth, deferred, Identity, special ), resolve( maxDepth, deferred, Thrower, special ), resolve( maxDepth, deferred, Identity, deferred.notifyWith ) ); } // Handle all other returned values } else { // Only substitute handlers pass on context // and multiple values (non-spec behavior) if ( handler !== Identity ) { that = undefined; args = [ returned ]; } // Process the value(s) // Default process is resolve ( special || deferred.resolveWith )( that, args ); } }, // Only normal processors (resolve) catch and reject exceptions process = special ? mightThrow : function() { try { mightThrow(); } catch ( e ) { if ( jQuery.Deferred.exceptionHook ) { jQuery.Deferred.exceptionHook( e, process.stackTrace ); } // Support: Promises/A+ section 2.3.3.3.4.1 // https://promisesaplus.com/#point-61 // Ignore post-resolution exceptions if ( depth + 1 >= maxDepth ) { // Only substitute handlers pass on context // and multiple values (non-spec behavior) if ( handler !== Thrower ) { that = undefined; args = [ e ]; } deferred.rejectWith( that, args ); } } }; // Support: Promises/A+ section 2.3.3.3.1 // https://promisesaplus.com/#point-57 // Re-resolve promises immediately to dodge false rejection from // subsequent errors if ( depth ) { process(); } else { // Call an optional hook to record the stack, in case of exception // since it's otherwise lost when execution goes async if ( jQuery.Deferred.getStackHook ) { process.stackTrace = jQuery.Deferred.getStackHook(); } window.setTimeout( process ); } }; } return jQuery.Deferred( function( newDefer ) { // progress_handlers.add( ... ) tuples[ 0 ][ 3 ].add( resolve( 0, newDefer, isFunction( onProgress ) ? onProgress : Identity, newDefer.notifyWith ) ); // fulfilled_handlers.add( ... ) tuples[ 1 ][ 3 ].add( resolve( 0, newDefer, isFunction( onFulfilled ) ? onFulfilled : Identity ) ); // rejected_handlers.add( ... ) tuples[ 2 ][ 3 ].add( resolve( 0, newDefer, isFunction( onRejected ) ? onRejected : Thrower ) ); } ).promise(); }, // Get a promise for this deferred // If obj is provided, the promise aspect is added to the object promise: function( obj ) { return obj != null ? jQuery.extend( obj, promise ) : promise; } }, deferred = {}; // Add list-specific methods jQuery.each( tuples, function( i, tuple ) { var list = tuple[ 2 ], stateString = tuple[ 5 ]; // promise.progress = list.add // promise.done = list.add // promise.fail = list.add promise[ tuple[ 1 ] ] = list.add; // Handle state if ( stateString ) { list.add( function() { // state = "resolved" (i.e., fulfilled) // state = "rejected" state = stateString; }, // rejected_callbacks.disable // fulfilled_callbacks.disable tuples[ 3 - i ][ 2 ].disable, // rejected_handlers.disable // fulfilled_handlers.disable tuples[ 3 - i ][ 3 ].disable, // progress_callbacks.lock tuples[ 0 ][ 2 ].lock, // progress_handlers.lock tuples[ 0 ][ 3 ].lock ); } // progress_handlers.fire // fulfilled_handlers.fire // rejected_handlers.fire list.add( tuple[ 3 ].fire ); // deferred.notify = function() { deferred.notifyWith(...) } // deferred.resolve = function() { deferred.resolveWith(...) } // deferred.reject = function() { deferred.rejectWith(...) } deferred[ tuple[ 0 ] ] = function() { deferred[ tuple[ 0 ] + "With" ]( this === deferred ? undefined : this, arguments ); return this; }; // deferred.notifyWith = list.fireWith // deferred.resolveWith = list.fireWith // deferred.rejectWith = list.fireWith deferred[ tuple[ 0 ] + "With" ] = list.fireWith; } ); // Make the deferred a promise promise.promise( deferred ); // Call given func if any if ( func ) { func.call( deferred, deferred ); } // All done! return deferred; }, // Deferred helper when: function( singleValue ) { var // count of uncompleted subordinates remaining = arguments.length, // count of unprocessed arguments i = remaining, // subordinate fulfillment data resolveContexts = Array( i ), resolveValues = slice.call( arguments ), // the master Deferred master = jQuery.Deferred(), // subordinate callback factory updateFunc = function( i ) { return function( value ) { resolveContexts[ i ] = this; resolveValues[ i ] = arguments.length > 1 ? slice.call( arguments ) : value; if ( !( --remaining ) ) { master.resolveWith( resolveContexts, resolveValues ); } }; }; // Single- and empty arguments are adopted like Promise.resolve if ( remaining <= 1 ) { adoptValue( singleValue, master.done( updateFunc( i ) ).resolve, master.reject, !remaining ); // Use .then() to unwrap secondary thenables (cf. gh-3000) if ( master.state() === "pending" || isFunction( resolveValues[ i ] && resolveValues[ i ].then ) ) { return master.then(); } } // Multiple arguments are aggregated like Promise.all array elements while ( i-- ) { adoptValue( resolveValues[ i ], updateFunc( i ), master.reject ); } return master.promise(); } } ); // These usually indicate a programmer mistake during development, // warn about them ASAP rather than swallowing them by default. var rerrorNames = /^(Eval|Internal|Range|Reference|Syntax|Type|URI)Error$/; jQuery.Deferred.exceptionHook = function( error, stack ) { // Support: IE 8 - 9 only // Console exists when dev tools are open, which can happen at any time if ( window.console && window.console.warn && error && rerrorNames.test( error.name ) ) { window.console.warn( "jQuery.Deferred exception: " + error.message, error.stack, stack ); } }; jQuery.readyException = function( error ) { window.setTimeout( function() { throw error; } ); }; // The deferred used on DOM ready var readyList = jQuery.Deferred(); jQuery.fn.ready = function( fn ) { readyList .then( fn ) // Wrap jQuery.readyException in a function so that the lookup // happens at the time of error handling instead of callback // registration. .catch( function( error ) { jQuery.readyException( error ); } ); return this; }; jQuery.extend( { // Is the DOM ready to be used? Set to true once it occurs. isReady: false, // A counter to track how many items to wait for before // the ready event fires. See #6781 readyWait: 1, // Handle when the DOM is ready ready: function( wait ) { // Abort if there are pending holds or we're already ready if ( wait === true ? --jQuery.readyWait : jQuery.isReady ) { return; } // Remember that the DOM is ready jQuery.isReady = true; // If a normal DOM Ready event fired, decrement, and wait if need be if ( wait !== true && --jQuery.readyWait > 0 ) { return; } // If there are functions bound, to execute readyList.resolveWith( document, [ jQuery ] ); } } ); jQuery.ready.then = readyList.then; // The ready event handler and self cleanup method function completed() { document.removeEventListener( "DOMContentLoaded", completed ); window.removeEventListener( "load", completed ); jQuery.ready(); } // Catch cases where $(document).ready() is called // after the browser event has already occurred. // Support: IE <=9 - 10 only // Older IE sometimes signals "interactive" too soon if ( document.readyState === "complete" || ( document.readyState !== "loading" && !document.documentElement.doScroll ) ) { // Handle it asynchronously to allow scripts the opportunity to delay ready window.setTimeout( jQuery.ready ); } else { // Use the handy event callback document.addEventListener( "DOMContentLoaded", completed ); // A fallback to window.onload, that will always work window.addEventListener( "load", completed ); } // Multifunctional method to get and set values of a collection // The value/s can optionally be executed if it's a function var access = function( elems, fn, key, value, chainable, emptyGet, raw ) { var i = 0, len = elems.length, bulk = key == null; // Sets many values if ( toType( key ) === "object" ) { chainable = true; for ( i in key ) { access( elems, fn, i, key[ i ], true, emptyGet, raw ); } // Sets one value } else if ( value !== undefined ) { chainable = true; if ( !isFunction( value ) ) { raw = true; } if ( bulk ) { // Bulk operations run against the entire set if ( raw ) { fn.call( elems, value ); fn = null; // ...except when executing function values } else { bulk = fn; fn = function( elem, _key, value ) { return bulk.call( jQuery( elem ), value ); }; } } if ( fn ) { for ( ; i < len; i++ ) { fn( elems[ i ], key, raw ? value : value.call( elems[ i ], i, fn( elems[ i ], key ) ) ); } } } if ( chainable ) { return elems; } // Gets if ( bulk ) { return fn.call( elems ); } return len ? fn( elems[ 0 ], key ) : emptyGet; }; // Matches dashed string for camelizing var rmsPrefix = /^-ms-/, rdashAlpha = /-([a-z])/g; // Used by camelCase as callback to replace() function fcamelCase( _all, letter ) { return letter.toUpperCase(); } // Convert dashed to camelCase; used by the css and data modules // Support: IE <=9 - 11, Edge 12 - 15 // Microsoft forgot to hump their vendor prefix (#9572) function camelCase( string ) { return string.replace( rmsPrefix, "ms-" ).replace( rdashAlpha, fcamelCase ); } var acceptData = function( owner ) { // Accepts only: // - Node // - Node.ELEMENT_NODE // - Node.DOCUMENT_NODE // - Object // - Any return owner.nodeType === 1 || owner.nodeType === 9 || !( +owner.nodeType ); }; function Data() { this.expando = jQuery.expando + Data.uid++; } Data.uid = 1; Data.prototype = { cache: function( owner ) { // Check if the owner object already has a cache var value = owner[ this.expando ]; // If not, create one if ( !value ) { value = {}; // We can accept data for non-element nodes in modern browsers, // but we should not, see #8335. // Always return an empty object. if ( acceptData( owner ) ) { // If it is a node unlikely to be stringify-ed or looped over // use plain assignment if ( owner.nodeType ) { owner[ this.expando ] = value; // Otherwise secure it in a non-enumerable property // configurable must be true to allow the property to be // deleted when data is removed } else { Object.defineProperty( owner, this.expando, { value: value, configurable: true } ); } } } return value; }, set: function( owner, data, value ) { var prop, cache = this.cache( owner ); // Handle: [ owner, key, value ] args // Always use camelCase key (gh-2257) if ( typeof data === "string" ) { cache[ camelCase( data ) ] = value; // Handle: [ owner, { properties } ] args } else { // Copy the properties one-by-one to the cache object for ( prop in data ) { cache[ camelCase( prop ) ] = data[ prop ]; } } return cache; }, get: function( owner, key ) { return key === undefined ? this.cache( owner ) : // Always use camelCase key (gh-2257) owner[ this.expando ] && owner[ this.expando ][ camelCase( key ) ]; }, access: function( owner, key, value ) { // In cases where either: // // 1. No key was specified // 2. A string key was specified, but no value provided // // Take the "read" path and allow the get method to determine // which value to return, respectively either: // // 1. The entire cache object // 2. The data stored at the key // if ( key === undefined || ( ( key && typeof key === "string" ) && value === undefined ) ) { return this.get( owner, key ); } // When the key is not a string, or both a key and value // are specified, set or extend (existing objects) with either: // // 1. An object of properties // 2. A key and value // this.set( owner, key, value ); // Since the "set" path can have two possible entry points // return the expected data based on which path was taken[*] return value !== undefined ? value : key; }, remove: function( owner, key ) { var i, cache = owner[ this.expando ]; if ( cache === undefined ) { return; } if ( key !== undefined ) { // Support array or space separated string of keys if ( Array.isArray( key ) ) { // If key is an array of keys... // We always set camelCase keys, so remove that. key = key.map( camelCase ); } else { key = camelCase( key ); // If a key with the spaces exists, use it. // Otherwise, create an array by matching non-whitespace key = key in cache ? [ key ] : ( key.match( rnothtmlwhite ) || [] ); } i = key.length; while ( i-- ) { delete cache[ key[ i ] ]; } } // Remove the expando if there's no more data if ( key === undefined || jQuery.isEmptyObject( cache ) ) { // Support: Chrome <=35 - 45 // Webkit & Blink performance suffers when deleting properties // from DOM nodes, so set to undefined instead // https://bugs.chromium.org/p/chromium/issues/detail?id=378607 (bug restricted) if ( owner.nodeType ) { owner[ this.expando ] = undefined; } else { delete owner[ this.expando ]; } } }, hasData: function( owner ) { var cache = owner[ this.expando ]; return cache !== undefined && !jQuery.isEmptyObject( cache ); } }; var dataPriv = new Data(); var dataUser = new Data(); // Implementation Summary // // 1. Enforce API surface and semantic compatibility with 1.9.x branch // 2. Improve the module's maintainability by reducing the storage // paths to a single mechanism. // 3. Use the same single mechanism to support "private" and "user" data. // 4. _Never_ expose "private" data to user code (TODO: Drop _data, _removeData) // 5. Avoid exposing implementation details on user objects (eg. expando properties) // 6. Provide a clear path for implementation upgrade to WeakMap in 2014 var rbrace = /^(?:\{[\w\W]*\}|\[[\w\W]*\])$/, rmultiDash = /[A-Z]/g; function getData( data ) { if ( data === "true" ) { return true; } if ( data === "false" ) { return false; } if ( data === "null" ) { return null; } // Only convert to a number if it doesn't change the string if ( data === +data + "" ) { return +data; } if ( rbrace.test( data ) ) { return JSON.parse( data ); } return data; } function dataAttr( elem, key, data ) { var name; // If nothing was found internally, try to fetch any // data from the HTML5 data-* attribute if ( data === undefined && elem.nodeType === 1 ) { name = "data-" + key.replace( rmultiDash, "-$&" ).toLowerCase(); data = elem.getAttribute( name ); if ( typeof data === "string" ) { try { data = getData( data ); } catch ( e ) {} // Make sure we set the data so it isn't changed later dataUser.set( elem, key, data ); } else { data = undefined; } } return data; } jQuery.extend( { hasData: function( elem ) { return dataUser.hasData( elem ) || dataPriv.hasData( elem ); }, data: function( elem, name, data ) { return dataUser.access( elem, name, data ); }, removeData: function( elem, name ) { dataUser.remove( elem, name ); }, // TODO: Now that all calls to _data and _removeData have been replaced // with direct calls to dataPriv methods, these can be deprecated. _data: function( elem, name, data ) { return dataPriv.access( elem, name, data ); }, _removeData: function( elem, name ) { dataPriv.remove( elem, name ); } } ); jQuery.fn.extend( { data: function( key, value ) { var i, name, data, elem = this[ 0 ], attrs = elem && elem.attributes; // Gets all values if ( key === undefined ) { if ( this.length ) { data = dataUser.get( elem ); if ( elem.nodeType === 1 && !dataPriv.get( elem, "hasDataAttrs" ) ) { i = attrs.length; while ( i-- ) { // Support: IE 11 only // The attrs elements can be null (#14894) if ( attrs[ i ] ) { name = attrs[ i ].name; if ( name.indexOf( "data-" ) === 0 ) { name = camelCase( name.slice( 5 ) ); dataAttr( elem, name, data[ name ] ); } } } dataPriv.set( elem, "hasDataAttrs", true ); } } return data; } // Sets multiple values if ( typeof key === "object" ) { return this.each( function() { dataUser.set( this, key ); } ); } return access( this, function( value ) { var data; // The calling jQuery object (element matches) is not empty // (and therefore has an element appears at this[ 0 ]) and the // `value` parameter was not undefined. An empty jQuery object // will result in `undefined` for elem = this[ 0 ] which will // throw an exception if an attempt to read a data cache is made. if ( elem && value === undefined ) { // Attempt to get data from the cache // The key will always be camelCased in Data data = dataUser.get( elem, key ); if ( data !== undefined ) { return data; } // Attempt to "discover" the data in // HTML5 custom data-* attrs data = dataAttr( elem, key ); if ( data !== undefined ) { return data; } // We tried really hard, but the data doesn't exist. return; } // Set the data... this.each( function() { // We always store the camelCased key dataUser.set( this, key, value ); } ); }, null, value, arguments.length > 1, null, true ); }, removeData: function( key ) { return this.each( function() { dataUser.remove( this, key ); } ); } } ); jQuery.extend( { queue: function( elem, type, data ) { var queue; if ( elem ) { type = ( type || "fx" ) + "queue"; queue = dataPriv.get( elem, type ); // Speed up dequeue by getting out quickly if this is just a lookup if ( data ) { if ( !queue || Array.isArray( data ) ) { queue = dataPriv.access( elem, type, jQuery.makeArray( data ) ); } else { queue.push( data ); } } return queue || []; } }, dequeue: function( elem, type ) { type = type || "fx"; var queue = jQuery.queue( elem, type ), startLength = queue.length, fn = queue.shift(), hooks = jQuery._queueHooks( elem, type ), next = function() { jQuery.dequeue( elem, type ); }; // If the fx queue is dequeued, always remove the progress sentinel if ( fn === "inprogress" ) { fn = queue.shift(); startLength--; } if ( fn ) { // Add a progress sentinel to prevent the fx queue from being // automatically dequeued if ( type === "fx" ) { queue.unshift( "inprogress" ); } // Clear up the last queue stop function delete hooks.stop; fn.call( elem, next, hooks ); } if ( !startLength && hooks ) { hooks.empty.fire(); } }, // Not public - generate a queueHooks object, or return the current one _queueHooks: function( elem, type ) { var key = type + "queueHooks"; return dataPriv.get( elem, key ) || dataPriv.access( elem, key, { empty: jQuery.Callbacks( "once memory" ).add( function() { dataPriv.remove( elem, [ type + "queue", key ] ); } ) } ); } } ); jQuery.fn.extend( { queue: function( type, data ) { var setter = 2; if ( typeof type !== "string" ) { data = type; type = "fx"; setter--; } if ( arguments.length < setter ) { return jQuery.queue( this[ 0 ], type ); } return data === undefined ? this : this.each( function() { var queue = jQuery.queue( this, type, data ); // Ensure a hooks for this queue jQuery._queueHooks( this, type ); if ( type === "fx" && queue[ 0 ] !== "inprogress" ) { jQuery.dequeue( this, type ); } } ); }, dequeue: function( type ) { return this.each( function() { jQuery.dequeue( this, type ); } ); }, clearQueue: function( type ) { return this.queue( type || "fx", [] ); }, // Get a promise resolved when queues of a certain type // are emptied (fx is the type by default) promise: function( type, obj ) { var tmp, count = 1, defer = jQuery.Deferred(), elements = this, i = this.length, resolve = function() { if ( !( --count ) ) { defer.resolveWith( elements, [ elements ] ); } }; if ( typeof type !== "string" ) { obj = type; type = undefined; } type = type || "fx"; while ( i-- ) { tmp = dataPriv.get( elements[ i ], type + "queueHooks" ); if ( tmp && tmp.empty ) { count++; tmp.empty.add( resolve ); } } resolve(); return defer.promise( obj ); } } ); var pnum = ( /[+-]?(?:\d*\.|)\d+(?:[eE][+-]?\d+|)/ ).source; var rcssNum = new RegExp( "^(?:([+-])=|)(" + pnum + ")([a-z%]*)$", "i" ); var cssExpand = [ "Top", "Right", "Bottom", "Left" ]; var documentElement = document.documentElement; var isAttached = function( elem ) { return jQuery.contains( elem.ownerDocument, elem ); }, composed = { composed: true }; // Support: IE 9 - 11+, Edge 12 - 18+, iOS 10.0 - 10.2 only // Check attachment across shadow DOM boundaries when possible (gh-3504) // Support: iOS 10.0-10.2 only // Early iOS 10 versions support `attachShadow` but not `getRootNode`, // leading to errors. We need to check for `getRootNode`. if ( documentElement.getRootNode ) { isAttached = function( elem ) { return jQuery.contains( elem.ownerDocument, elem ) || elem.getRootNode( composed ) === elem.ownerDocument; }; } var isHiddenWithinTree = function( elem, el ) { // isHiddenWithinTree might be called from jQuery#filter function; // in that case, element will be second argument elem = el || elem; // Inline style trumps all return elem.style.display === "none" || elem.style.display === "" && // Otherwise, check computed style // Support: Firefox <=43 - 45 // Disconnected elements can have computed display: none, so first confirm that elem is // in the document. isAttached( elem ) && jQuery.css( elem, "display" ) === "none"; }; function adjustCSS( elem, prop, valueParts, tween ) { var adjusted, scale, maxIterations = 20, currentValue = tween ? function() { return tween.cur(); } : function() { return jQuery.css( elem, prop, "" ); }, initial = currentValue(), unit = valueParts && valueParts[ 3 ] || ( jQuery.cssNumber[ prop ] ? "" : "px" ), // Starting value computation is required for potential unit mismatches initialInUnit = elem.nodeType && ( jQuery.cssNumber[ prop ] || unit !== "px" && +initial ) && rcssNum.exec( jQuery.css( elem, prop ) ); if ( initialInUnit && initialInUnit[ 3 ] !== unit ) { // Support: Firefox <=54 // Halve the iteration target value to prevent interference from CSS upper bounds (gh-2144) initial = initial / 2; // Trust units reported by jQuery.css unit = unit || initialInUnit[ 3 ]; // Iteratively approximate from a nonzero starting point initialInUnit = +initial || 1; while ( maxIterations-- ) { // Evaluate and update our best guess (doubling guesses that zero out). // Finish if the scale equals or crosses 1 (making the old*new product non-positive). jQuery.style( elem, prop, initialInUnit + unit ); if ( ( 1 - scale ) * ( 1 - ( scale = currentValue() / initial || 0.5 ) ) <= 0 ) { maxIterations = 0; } initialInUnit = initialInUnit / scale; } initialInUnit = initialInUnit * 2; jQuery.style( elem, prop, initialInUnit + unit ); // Make sure we update the tween properties later on valueParts = valueParts || []; } if ( valueParts ) { initialInUnit = +initialInUnit || +initial || 0; // Apply relative offset (+=/-=) if specified adjusted = valueParts[ 1 ] ? initialInUnit + ( valueParts[ 1 ] + 1 ) * valueParts[ 2 ] : +valueParts[ 2 ]; if ( tween ) { tween.unit = unit; tween.start = initialInUnit; tween.end = adjusted; } } return adjusted; } var defaultDisplayMap = {}; function getDefaultDisplay( elem ) { var temp, doc = elem.ownerDocument, nodeName = elem.nodeName, display = defaultDisplayMap[ nodeName ]; if ( display ) { return display; } temp = doc.body.appendChild( doc.createElement( nodeName ) ); display = jQuery.css( temp, "display" ); temp.parentNode.removeChild( temp ); if ( display === "none" ) { display = "block"; } defaultDisplayMap[ nodeName ] = display; return display; } function showHide( elements, show ) { var display, elem, values = [], index = 0, length = elements.length; // Determine new display value for elements that need to change for ( ; index < length; index++ ) { elem = elements[ index ]; if ( !elem.style ) { continue; } display = elem.style.display; if ( show ) { // Since we force visibility upon cascade-hidden elements, an immediate (and slow) // check is required in this first loop unless we have a nonempty display value (either // inline or about-to-be-restored) if ( display === "none" ) { values[ index ] = dataPriv.get( elem, "display" ) || null; if ( !values[ index ] ) { elem.style.display = ""; } } if ( elem.style.display === "" && isHiddenWithinTree( elem ) ) { values[ index ] = getDefaultDisplay( elem ); } } else { if ( display !== "none" ) { values[ index ] = "none"; // Remember what we're overwriting dataPriv.set( elem, "display", display ); } } } // Set the display of the elements in a second loop to avoid constant reflow for ( index = 0; index < length; index++ ) { if ( values[ index ] != null ) { elements[ index ].style.display = values[ index ]; } } return elements; } jQuery.fn.extend( { show: function() { return showHide( this, true ); }, hide: function() { return showHide( this ); }, toggle: function( state ) { if ( typeof state === "boolean" ) { return state ? this.show() : this.hide(); } return this.each( function() { if ( isHiddenWithinTree( this ) ) { jQuery( this ).show(); } else { jQuery( this ).hide(); } } ); } } ); var rcheckableType = ( /^(?:checkbox|radio)$/i ); var rtagName = ( /<([a-z][^\/\0>\x20\t\r\n\f]*)/i ); var rscriptType = ( /^$|^module$|\/(?:java|ecma)script/i ); ( function() { var fragment = document.createDocumentFragment(), div = fragment.appendChild( document.createElement( "div" ) ), input = document.createElement( "input" ); // Support: Android 4.0 - 4.3 only // Check state lost if the name is set (#11217) // Support: Windows Web Apps (WWA) // `name` and `type` must use .setAttribute for WWA (#14901) input.setAttribute( "type", "radio" ); input.setAttribute( "checked", "checked" ); input.setAttribute( "name", "t" ); div.appendChild( input ); // Support: Android <=4.1 only // Older WebKit doesn't clone checked state correctly in fragments support.checkClone = div.cloneNode( true ).cloneNode( true ).lastChild.checked; // Support: IE <=11 only // Make sure textarea (and checkbox) defaultValue is properly cloned div.innerHTML = "<textarea>x</textarea>"; support.noCloneChecked = !!div.cloneNode( true ).lastChild.defaultValue; // Support: IE <=9 only // IE <=9 replaces <option> tags with their contents when inserted outside of // the select element. div.innerHTML = "<option></option>"; support.option = !!div.lastChild; } )(); // We have to close these tags to support XHTML (#13200) var wrapMap = { // XHTML parsers do not magically insert elements in the // same way that tag soup parsers do. So we cannot shorten // this by omitting <tbody> or other required elements. thead: [ 1, "<table>", "</table>" ], col: [ 2, "<table><colgroup>", "</colgroup></table>" ], tr: [ 2, "<table><tbody>", "</tbody></table>" ], td: [ 3, "<table><tbody><tr>", "</tr></tbody></table>" ], _default: [ 0, "", "" ] }; wrapMap.tbody = wrapMap.tfoot = wrapMap.colgroup = wrapMap.caption = wrapMap.thead; wrapMap.th = wrapMap.td; // Support: IE <=9 only if ( !support.option ) { wrapMap.optgroup = wrapMap.option = [ 1, "<select multiple='multiple'>", "</select>" ]; } function getAll( context, tag ) { // Support: IE <=9 - 11 only // Use typeof to avoid zero-argument method invocation on host objects (#15151) var ret; if ( typeof context.getElementsByTagName !== "undefined" ) { ret = context.getElementsByTagName( tag || "*" ); } else if ( typeof context.querySelectorAll !== "undefined" ) { ret = context.querySelectorAll( tag || "*" ); } else { ret = []; } if ( tag === undefined || tag && nodeName( context, tag ) ) { return jQuery.merge( [ context ], ret ); } return ret; } // Mark scripts as having already been evaluated function setGlobalEval( elems, refElements ) { var i = 0, l = elems.length; for ( ; i < l; i++ ) { dataPriv.set( elems[ i ], "globalEval", !refElements || dataPriv.get( refElements[ i ], "globalEval" ) ); } } var rhtml = /<|&#?\w+;/; function buildFragment( elems, context, scripts, selection, ignored ) { var elem, tmp, tag, wrap, attached, j, fragment = context.createDocumentFragment(), nodes = [], i = 0, l = elems.length; for ( ; i < l; i++ ) { elem = elems[ i ]; if ( elem || elem === 0 ) { // Add nodes directly if ( toType( elem ) === "object" ) { // Support: Android <=4.0 only, PhantomJS 1 only // push.apply(_, arraylike) throws on ancient WebKit jQuery.merge( nodes, elem.nodeType ? [ elem ] : elem ); // Convert non-html into a text node } else if ( !rhtml.test( elem ) ) { nodes.push( context.createTextNode( elem ) ); // Convert html into DOM nodes } else { tmp = tmp || fragment.appendChild( context.createElement( "div" ) ); // Deserialize a standard representation tag = ( rtagName.exec( elem ) || [ "", "" ] )[ 1 ].toLowerCase(); wrap = wrapMap[ tag ] || wrapMap._default; tmp.innerHTML = wrap[ 1 ] + jQuery.htmlPrefilter( elem ) + wrap[ 2 ]; // Descend through wrappers to the right content j = wrap[ 0 ]; while ( j-- ) { tmp = tmp.lastChild; } // Support: Android <=4.0 only, PhantomJS 1 only // push.apply(_, arraylike) throws on ancient WebKit jQuery.merge( nodes, tmp.childNodes ); // Remember the top-level container tmp = fragment.firstChild; // Ensure the created nodes are orphaned (#12392) tmp.textContent = ""; } } } // Remove wrapper from fragment fragment.textContent = ""; i = 0; while ( ( elem = nodes[ i++ ] ) ) { // Skip elements already in the context collection (trac-4087) if ( selection && jQuery.inArray( elem, selection ) > -1 ) { if ( ignored ) { ignored.push( elem ); } continue; } attached = isAttached( elem ); // Append to fragment tmp = getAll( fragment.appendChild( elem ), "script" ); // Preserve script evaluation history if ( attached ) { setGlobalEval( tmp ); } // Capture executables if ( scripts ) { j = 0; while ( ( elem = tmp[ j++ ] ) ) { if ( rscriptType.test( elem.type || "" ) ) { scripts.push( elem ); } } } } return fragment; } var rkeyEvent = /^key/, rmouseEvent = /^(?:mouse|pointer|contextmenu|drag|drop)|click/, rtypenamespace = /^([^.]*)(?:\.(.+)|)/; function returnTrue() { return true; } function returnFalse() { return false; } // Support: IE <=9 - 11+ // focus() and blur() are asynchronous, except when they are no-op. // So expect focus to be synchronous when the element is already active, // and blur to be synchronous when the element is not already active. // (focus and blur are always synchronous in other supported browsers, // this just defines when we can count on it). function expectSync( elem, type ) { return ( elem === safeActiveElement() ) === ( type === "focus" ); } // Support: IE <=9 only // Accessing document.activeElement can throw unexpectedly // https://bugs.jquery.com/ticket/13393 function safeActiveElement() { try { return document.activeElement; } catch ( err ) { } } function on( elem, types, selector, data, fn, one ) { var origFn, type; // Types can be a map of types/handlers if ( typeof types === "object" ) { // ( types-Object, selector, data ) if ( typeof selector !== "string" ) { // ( types-Object, data ) data = data || selector; selector = undefined; } for ( type in types ) { on( elem, type, selector, data, types[ type ], one ); } return elem; } if ( data == null && fn == null ) { // ( types, fn ) fn = selector; data = selector = undefined; } else if ( fn == null ) { if ( typeof selector === "string" ) { // ( types, selector, fn ) fn = data; data = undefined; } else { // ( types, data, fn ) fn = data; data = selector; selector = undefined; } } if ( fn === false ) { fn = returnFalse; } else if ( !fn ) { return elem; } if ( one === 1 ) { origFn = fn; fn = function( event ) { // Can use an empty set, since event contains the info jQuery().off( event ); return origFn.apply( this, arguments ); }; // Use same guid so caller can remove using origFn fn.guid = origFn.guid || ( origFn.guid = jQuery.guid++ ); } return elem.each( function() { jQuery.event.add( this, types, fn, data, selector ); } ); } /* * Helper functions for managing events -- not part of the public interface. * Props to Dean Edwards' addEvent library for many of the ideas. */ jQuery.event = { global: {}, add: function( elem, types, handler, data, selector ) { var handleObjIn, eventHandle, tmp, events, t, handleObj, special, handlers, type, namespaces, origType, elemData = dataPriv.get( elem ); // Only attach events to objects that accept data if ( !acceptData( elem ) ) { return; } // Caller can pass in an object of custom data in lieu of the handler if ( handler.handler ) { handleObjIn = handler; handler = handleObjIn.handler; selector = handleObjIn.selector; } // Ensure that invalid selectors throw exceptions at attach time // Evaluate against documentElement in case elem is a non-element node (e.g., document) if ( selector ) { jQuery.find.matchesSelector( documentElement, selector ); } // Make sure that the handler has a unique ID, used to find/remove it later if ( !handler.guid ) { handler.guid = jQuery.guid++; } // Init the element's event structure and main handler, if this is the first if ( !( events = elemData.events ) ) { events = elemData.events = Object.create( null ); } if ( !( eventHandle = elemData.handle ) ) { eventHandle = elemData.handle = function( e ) { // Discard the second event of a jQuery.event.trigger() and // when an event is called after a page has unloaded return typeof jQuery !== "undefined" && jQuery.event.triggered !== e.type ? jQuery.event.dispatch.apply( elem, arguments ) : undefined; }; } // Handle multiple events separated by a space types = ( types || "" ).match( rnothtmlwhite ) || [ "" ]; t = types.length; while ( t-- ) { tmp = rtypenamespace.exec( types[ t ] ) || []; type = origType = tmp[ 1 ]; namespaces = ( tmp[ 2 ] || "" ).split( "." ).sort(); // There *must* be a type, no attaching namespace-only handlers if ( !type ) { continue; } // If event changes its type, use the special event handlers for the changed type special = jQuery.event.special[ type ] || {}; // If selector defined, determine special event api type, otherwise given type type = ( selector ? special.delegateType : special.bindType ) || type; // Update special based on newly reset type special = jQuery.event.special[ type ] || {}; // handleObj is passed to all event handlers handleObj = jQuery.extend( { type: type, origType: origType, data: data, handler: handler, guid: handler.guid, selector: selector, needsContext: selector && jQuery.expr.match.needsContext.test( selector ), namespace: namespaces.join( "." ) }, handleObjIn ); // Init the event handler queue if we're the first if ( !( handlers = events[ type ] ) ) { handlers = events[ type ] = []; handlers.delegateCount = 0; // Only use addEventListener if the special events handler returns false if ( !special.setup || special.setup.call( elem, data, namespaces, eventHandle ) === false ) { if ( elem.addEventListener ) { elem.addEventListener( type, eventHandle ); } } } if ( special.add ) { special.add.call( elem, handleObj ); if ( !handleObj.handler.guid ) { handleObj.handler.guid = handler.guid; } } // Add to the element's handler list, delegates in front if ( selector ) { handlers.splice( handlers.delegateCount++, 0, handleObj ); } else { handlers.push( handleObj ); } // Keep track of which events have ever been used, for event optimization jQuery.event.global[ type ] = true; } }, // Detach an event or set of events from an element remove: function( elem, types, handler, selector, mappedTypes ) { var j, origCount, tmp, events, t, handleObj, special, handlers, type, namespaces, origType, elemData = dataPriv.hasData( elem ) && dataPriv.get( elem ); if ( !elemData || !( events = elemData.events ) ) { return; } // Once for each type.namespace in types; type may be omitted types = ( types || "" ).match( rnothtmlwhite ) || [ "" ]; t = types.length; while ( t-- ) { tmp = rtypenamespace.exec( types[ t ] ) || []; type = origType = tmp[ 1 ]; namespaces = ( tmp[ 2 ] || "" ).split( "." ).sort(); // Unbind all events (on this namespace, if provided) for the element if ( !type ) { for ( type in events ) { jQuery.event.remove( elem, type + types[ t ], handler, selector, true ); } continue; } special = jQuery.event.special[ type ] || {}; type = ( selector ? special.delegateType : special.bindType ) || type; handlers = events[ type ] || []; tmp = tmp[ 2 ] && new RegExp( "(^|\\.)" + namespaces.join( "\\.(?:.*\\.|)" ) + "(\\.|$)" ); // Remove matching events origCount = j = handlers.length; while ( j-- ) { handleObj = handlers[ j ]; if ( ( mappedTypes || origType === handleObj.origType ) && ( !handler || handler.guid === handleObj.guid ) && ( !tmp || tmp.test( handleObj.namespace ) ) && ( !selector || selector === handleObj.selector || selector === "**" && handleObj.selector ) ) { handlers.splice( j, 1 ); if ( handleObj.selector ) { handlers.delegateCount--; } if ( special.remove ) { special.remove.call( elem, handleObj ); } } } // Remove generic event handler if we removed something and no more handlers exist // (avoids potential for endless recursion during removal of special event handlers) if ( origCount && !handlers.length ) { if ( !special.teardown || special.teardown.call( elem, namespaces, elemData.handle ) === false ) { jQuery.removeEvent( elem, type, elemData.handle ); } delete events[ type ]; } } // Remove data and the expando if it's no longer used if ( jQuery.isEmptyObject( events ) ) { dataPriv.remove( elem, "handle events" ); } }, dispatch: function( nativeEvent ) { var i, j, ret, matched, handleObj, handlerQueue, args = new Array( arguments.length ), // Make a writable jQuery.Event from the native event object event = jQuery.event.fix( nativeEvent ), handlers = ( dataPriv.get( this, "events" ) || Object.create( null ) )[ event.type ] || [], special = jQuery.event.special[ event.type ] || {}; // Use the fix-ed jQuery.Event rather than the (read-only) native event args[ 0 ] = event; for ( i = 1; i < arguments.length; i++ ) { args[ i ] = arguments[ i ]; } event.delegateTarget = this; // Call the preDispatch hook for the mapped type, and let it bail if desired if ( special.preDispatch && special.preDispatch.call( this, event ) === false ) { return; } // Determine handlers handlerQueue = jQuery.event.handlers.call( this, event, handlers ); // Run delegates first; they may want to stop propagation beneath us i = 0; while ( ( matched = handlerQueue[ i++ ] ) && !event.isPropagationStopped() ) { event.currentTarget = matched.elem; j = 0; while ( ( handleObj = matched.handlers[ j++ ] ) && !event.isImmediatePropagationStopped() ) { // If the event is namespaced, then each handler is only invoked if it is // specially universal or its namespaces are a superset of the event's. if ( !event.rnamespace || handleObj.namespace === false || event.rnamespace.test( handleObj.namespace ) ) { event.handleObj = handleObj; event.data = handleObj.data; ret = ( ( jQuery.event.special[ handleObj.origType ] || {} ).handle || handleObj.handler ).apply( matched.elem, args ); if ( ret !== undefined ) { if ( ( event.result = ret ) === false ) { event.preventDefault(); event.stopPropagation(); } } } } } // Call the postDispatch hook for the mapped type if ( special.postDispatch ) { special.postDispatch.call( this, event ); } return event.result; }, handlers: function( event, handlers ) { var i, handleObj, sel, matchedHandlers, matchedSelectors, handlerQueue = [], delegateCount = handlers.delegateCount, cur = event.target; // Find delegate handlers if ( delegateCount && // Support: IE <=9 // Black-hole SVG <use> instance trees (trac-13180) cur.nodeType && // Support: Firefox <=42 // Suppress spec-violating clicks indicating a non-primary pointer button (trac-3861) // https://www.w3.org/TR/DOM-Level-3-Events/#event-type-click // Support: IE 11 only // ...but not arrow key "clicks" of radio inputs, which can have `button` -1 (gh-2343) !( event.type === "click" && event.button >= 1 ) ) { for ( ; cur !== this; cur = cur.parentNode || this ) { // Don't check non-elements (#13208) // Don't process clicks on disabled elements (#6911, #8165, #11382, #11764) if ( cur.nodeType === 1 && !( event.type === "click" && cur.disabled === true ) ) { matchedHandlers = []; matchedSelectors = {}; for ( i = 0; i < delegateCount; i++ ) { handleObj = handlers[ i ]; // Don't conflict with Object.prototype properties (#13203) sel = handleObj.selector + " "; if ( matchedSelectors[ sel ] === undefined ) { matchedSelectors[ sel ] = handleObj.needsContext ? jQuery( sel, this ).index( cur ) > -1 : jQuery.find( sel, this, null, [ cur ] ).length; } if ( matchedSelectors[ sel ] ) { matchedHandlers.push( handleObj ); } } if ( matchedHandlers.length ) { handlerQueue.push( { elem: cur, handlers: matchedHandlers } ); } } } } // Add the remaining (directly-bound) handlers cur = this; if ( delegateCount < handlers.length ) { handlerQueue.push( { elem: cur, handlers: handlers.slice( delegateCount ) } ); } return handlerQueue; }, addProp: function( name, hook ) { Object.defineProperty( jQuery.Event.prototype, name, { enumerable: true, configurable: true, get: isFunction( hook ) ? function() { if ( this.originalEvent ) { return hook( this.originalEvent ); } } : function() { if ( this.originalEvent ) { return this.originalEvent[ name ]; } }, set: function( value ) { Object.defineProperty( this, name, { enumerable: true, configurable: true, writable: true, value: value } ); } } ); }, fix: function( originalEvent ) { return originalEvent[ jQuery.expando ] ? originalEvent : new jQuery.Event( originalEvent ); }, special: { load: { // Prevent triggered image.load events from bubbling to window.load noBubble: true }, click: { // Utilize native event to ensure correct state for checkable inputs setup: function( data ) { // For mutual compressibility with _default, replace `this` access with a local var. // `|| data` is dead code meant only to preserve the variable through minification. var el = this || data; // Claim the first handler if ( rcheckableType.test( el.type ) && el.click && nodeName( el, "input" ) ) { // dataPriv.set( el, "click", ... ) leverageNative( el, "click", returnTrue ); } // Return false to allow normal processing in the caller return false; }, trigger: function( data ) { // For mutual compressibility with _default, replace `this` access with a local var. // `|| data` is dead code meant only to preserve the variable through minification. var el = this || data; // Force setup before triggering a click if ( rcheckableType.test( el.type ) && el.click && nodeName( el, "input" ) ) { leverageNative( el, "click" ); } // Return non-false to allow normal event-path propagation return true; }, // For cross-browser consistency, suppress native .click() on links // Also prevent it if we're currently inside a leveraged native-event stack _default: function( event ) { var target = event.target; return rcheckableType.test( target.type ) && target.click && nodeName( target, "input" ) && dataPriv.get( target, "click" ) || nodeName( target, "a" ); } }, beforeunload: { postDispatch: function( event ) { // Support: Firefox 20+ // Firefox doesn't alert if the returnValue field is not set. if ( event.result !== undefined && event.originalEvent ) { event.originalEvent.returnValue = event.result; } } } } }; // Ensure the presence of an event listener that handles manually-triggered // synthetic events by interrupting progress until reinvoked in response to // *native* events that it fires directly, ensuring that state changes have // already occurred before other listeners are invoked. function leverageNative( el, type, expectSync ) { // Missing expectSync indicates a trigger call, which must force setup through jQuery.event.add if ( !expectSync ) { if ( dataPriv.get( el, type ) === undefined ) { jQuery.event.add( el, type, returnTrue ); } return; } // Register the controller as a special universal handler for all event namespaces dataPriv.set( el, type, false ); jQuery.event.add( el, type, { namespace: false, handler: function( event ) { var notAsync, result, saved = dataPriv.get( this, type ); if ( ( event.isTrigger & 1 ) && this[ type ] ) { // Interrupt processing of the outer synthetic .trigger()ed event // Saved data should be false in such cases, but might be a leftover capture object // from an async native handler (gh-4350) if ( !saved.length ) { // Store arguments for use when handling the inner native event // There will always be at least one argument (an event object), so this array // will not be confused with a leftover capture object. saved = slice.call( arguments ); dataPriv.set( this, type, saved ); // Trigger the native event and capture its result // Support: IE <=9 - 11+ // focus() and blur() are asynchronous notAsync = expectSync( this, type ); this[ type ](); result = dataPriv.get( this, type ); if ( saved !== result || notAsync ) { dataPriv.set( this, type, false ); } else { result = {}; } if ( saved !== result ) { // Cancel the outer synthetic event event.stopImmediatePropagation(); event.preventDefault(); return result.value; } // If this is an inner synthetic event for an event with a bubbling surrogate // (focus or blur), assume that the surrogate already propagated from triggering the // native event and prevent that from happening again here. // This technically gets the ordering wrong w.r.t. to `.trigger()` (in which the // bubbling surrogate propagates *after* the non-bubbling base), but that seems // less bad than duplication. } else if ( ( jQuery.event.special[ type ] || {} ).delegateType ) { event.stopPropagation(); } // If this is a native event triggered above, everything is now in order // Fire an inner synthetic event with the original arguments } else if ( saved.length ) { // ...and capture the result dataPriv.set( this, type, { value: jQuery.event.trigger( // Support: IE <=9 - 11+ // Extend with the prototype to reset the above stopImmediatePropagation() jQuery.extend( saved[ 0 ], jQuery.Event.prototype ), saved.slice( 1 ), this ) } ); // Abort handling of the native event event.stopImmediatePropagation(); } } } ); } jQuery.removeEvent = function( elem, type, handle ) { // This "if" is needed for plain objects if ( elem.removeEventListener ) { elem.removeEventListener( type, handle ); } }; jQuery.Event = function( src, props ) { // Allow instantiation without the 'new' keyword if ( !( this instanceof jQuery.Event ) ) { return new jQuery.Event( src, props ); } // Event object if ( src && src.type ) { this.originalEvent = src; this.type = src.type; // Events bubbling up the document may have been marked as prevented // by a handler lower down the tree; reflect the correct value. this.isDefaultPrevented = src.defaultPrevented || src.defaultPrevented === undefined && // Support: Android <=2.3 only src.returnValue === false ? returnTrue : returnFalse; // Create target properties // Support: Safari <=6 - 7 only // Target should not be a text node (#504, #13143) this.target = ( src.target && src.target.nodeType === 3 ) ? src.target.parentNode : src.target; this.currentTarget = src.currentTarget; this.relatedTarget = src.relatedTarget; // Event type } else { this.type = src; } // Put explicitly provided properties onto the event object if ( props ) { jQuery.extend( this, props ); } // Create a timestamp if incoming event doesn't have one this.timeStamp = src && src.timeStamp || Date.now(); // Mark it as fixed this[ jQuery.expando ] = true; }; // jQuery.Event is based on DOM3 Events as specified by the ECMAScript Language Binding // https://www.w3.org/TR/2003/WD-DOM-Level-3-Events-20030331/ecma-script-binding.html jQuery.Event.prototype = { constructor: jQuery.Event, isDefaultPrevented: returnFalse, isPropagationStopped: returnFalse, isImmediatePropagationStopped: returnFalse, isSimulated: false, preventDefault: function() { var e = this.originalEvent; this.isDefaultPrevented = returnTrue; if ( e && !this.isSimulated ) { e.preventDefault(); } }, stopPropagation: function() { var e = this.originalEvent; this.isPropagationStopped = returnTrue; if ( e && !this.isSimulated ) { e.stopPropagation(); } }, stopImmediatePropagation: function() { var e = this.originalEvent; this.isImmediatePropagationStopped = returnTrue; if ( e && !this.isSimulated ) { e.stopImmediatePropagation(); } this.stopPropagation(); } }; // Includes all common event props including KeyEvent and MouseEvent specific props jQuery.each( { altKey: true, bubbles: true, cancelable: true, changedTouches: true, ctrlKey: true, detail: true, eventPhase: true, metaKey: true, pageX: true, pageY: true, shiftKey: true, view: true, "char": true, code: true, charCode: true, key: true, keyCode: true, button: true, buttons: true, clientX: true, clientY: true, offsetX: true, offsetY: true, pointerId: true, pointerType: true, screenX: true, screenY: true, targetTouches: true, toElement: true, touches: true, which: function( event ) { var button = event.button; // Add which for key events if ( event.which == null && rkeyEvent.test( event.type ) ) { return event.charCode != null ? event.charCode : event.keyCode; } // Add which for click: 1 === left; 2 === middle; 3 === right if ( !event.which && button !== undefined && rmouseEvent.test( event.type ) ) { if ( button & 1 ) { return 1; } if ( button & 2 ) { return 3; } if ( button & 4 ) { return 2; } return 0; } return event.which; } }, jQuery.event.addProp ); jQuery.each( { focus: "focusin", blur: "focusout" }, function( type, delegateType ) { jQuery.event.special[ type ] = { // Utilize native event if possible so blur/focus sequence is correct setup: function() { // Claim the first handler // dataPriv.set( this, "focus", ... ) // dataPriv.set( this, "blur", ... ) leverageNative( this, type, expectSync ); // Return false to allow normal processing in the caller return false; }, trigger: function() { // Force setup before trigger leverageNative( this, type ); // Return non-false to allow normal event-path propagation return true; }, delegateType: delegateType }; } ); // Create mouseenter/leave events using mouseover/out and event-time checks // so that event delegation works in jQuery. // Do the same for pointerenter/pointerleave and pointerover/pointerout // // Support: Safari 7 only // Safari sends mouseenter too often; see: // https://bugs.chromium.org/p/chromium/issues/detail?id=470258 // for the description of the bug (it existed in older Chrome versions as well). jQuery.each( { mouseenter: "mouseover", mouseleave: "mouseout", pointerenter: "pointerover", pointerleave: "pointerout" }, function( orig, fix ) { jQuery.event.special[ orig ] = { delegateType: fix, bindType: fix, handle: function( event ) { var ret, target = this, related = event.relatedTarget, handleObj = event.handleObj; // For mouseenter/leave call the handler if related is outside the target. // NB: No relatedTarget if the mouse left/entered the browser window if ( !related || ( related !== target && !jQuery.contains( target, related ) ) ) { event.type = handleObj.origType; ret = handleObj.handler.apply( this, arguments ); event.type = fix; } return ret; } }; } ); jQuery.fn.extend( { on: function( types, selector, data, fn ) { return on( this, types, selector, data, fn ); }, one: function( types, selector, data, fn ) { return on( this, types, selector, data, fn, 1 ); }, off: function( types, selector, fn ) { var handleObj, type; if ( types && types.preventDefault && types.handleObj ) { // ( event ) dispatched jQuery.Event handleObj = types.handleObj; jQuery( types.delegateTarget ).off( handleObj.namespace ? handleObj.origType + "." + handleObj.namespace : handleObj.origType, handleObj.selector, handleObj.handler ); return this; } if ( typeof types === "object" ) { // ( types-object [, selector] ) for ( type in types ) { this.off( type, selector, types[ type ] ); } return this; } if ( selector === false || typeof selector === "function" ) { // ( types [, fn] ) fn = selector; selector = undefined; } if ( fn === false ) { fn = returnFalse; } return this.each( function() { jQuery.event.remove( this, types, fn, selector ); } ); } } ); var // Support: IE <=10 - 11, Edge 12 - 13 only // In IE/Edge using regex groups here causes severe slowdowns. // See https://connect.microsoft.com/IE/feedback/details/1736512/ rnoInnerhtml = /<script|<style|<link/i, // checked="checked" or checked rchecked = /checked\s*(?:[^=]|=\s*.checked.)/i, rcleanScript = /^\s*<!(?:\[CDATA\[|--)|(?:\]\]|--)>\s*$/g; // Prefer a tbody over its parent table for containing new rows function manipulationTarget( elem, content ) { if ( nodeName( elem, "table" ) && nodeName( content.nodeType !== 11 ? content : content.firstChild, "tr" ) ) { return jQuery( elem ).children( "tbody" )[ 0 ] || elem; } return elem; } // Replace/restore the type attribute of script elements for safe DOM manipulation function disableScript( elem ) { elem.type = ( elem.getAttribute( "type" ) !== null ) + "/" + elem.type; return elem; } function restoreScript( elem ) { if ( ( elem.type || "" ).slice( 0, 5 ) === "true/" ) { elem.type = elem.type.slice( 5 ); } else { elem.removeAttribute( "type" ); } return elem; } function cloneCopyEvent( src, dest ) { var i, l, type, pdataOld, udataOld, udataCur, events; if ( dest.nodeType !== 1 ) { return; } // 1. Copy private data: events, handlers, etc. if ( dataPriv.hasData( src ) ) { pdataOld = dataPriv.get( src ); events = pdataOld.events; if ( events ) { dataPriv.remove( dest, "handle events" ); for ( type in events ) { for ( i = 0, l = events[ type ].length; i < l; i++ ) { jQuery.event.add( dest, type, events[ type ][ i ] ); } } } } // 2. Copy user data if ( dataUser.hasData( src ) ) { udataOld = dataUser.access( src ); udataCur = jQuery.extend( {}, udataOld ); dataUser.set( dest, udataCur ); } } // Fix IE bugs, see support tests function fixInput( src, dest ) { var nodeName = dest.nodeName.toLowerCase(); // Fails to persist the checked state of a cloned checkbox or radio button. if ( nodeName === "input" && rcheckableType.test( src.type ) ) { dest.checked = src.checked; // Fails to return the selected option to the default selected state when cloning options } else if ( nodeName === "input" || nodeName === "textarea" ) { dest.defaultValue = src.defaultValue; } } function domManip( collection, args, callback, ignored ) { // Flatten any nested arrays args = flat( args ); var fragment, first, scripts, hasScripts, node, doc, i = 0, l = collection.length, iNoClone = l - 1, value = args[ 0 ], valueIsFunction = isFunction( value ); // We can't cloneNode fragments that contain checked, in WebKit if ( valueIsFunction || ( l > 1 && typeof value === "string" && !support.checkClone && rchecked.test( value ) ) ) { return collection.each( function( index ) { var self = collection.eq( index ); if ( valueIsFunction ) { args[ 0 ] = value.call( this, index, self.html() ); } domManip( self, args, callback, ignored ); } ); } if ( l ) { fragment = buildFragment( args, collection[ 0 ].ownerDocument, false, collection, ignored ); first = fragment.firstChild; if ( fragment.childNodes.length === 1 ) { fragment = first; } // Require either new content or an interest in ignored elements to invoke the callback if ( first || ignored ) { scripts = jQuery.map( getAll( fragment, "script" ), disableScript ); hasScripts = scripts.length; // Use the original fragment for the last item // instead of the first because it can end up // being emptied incorrectly in certain situations (#8070). for ( ; i < l; i++ ) { node = fragment; if ( i !== iNoClone ) { node = jQuery.clone( node, true, true ); // Keep references to cloned scripts for later restoration if ( hasScripts ) { // Support: Android <=4.0 only, PhantomJS 1 only // push.apply(_, arraylike) throws on ancient WebKit jQuery.merge( scripts, getAll( node, "script" ) ); } } callback.call( collection[ i ], node, i ); } if ( hasScripts ) { doc = scripts[ scripts.length - 1 ].ownerDocument; // Reenable scripts jQuery.map( scripts, restoreScript ); // Evaluate executable scripts on first document insertion for ( i = 0; i < hasScripts; i++ ) { node = scripts[ i ]; if ( rscriptType.test( node.type || "" ) && !dataPriv.access( node, "globalEval" ) && jQuery.contains( doc, node ) ) { if ( node.src && ( node.type || "" ).toLowerCase() !== "module" ) { // Optional AJAX dependency, but won't run scripts if not present if ( jQuery._evalUrl && !node.noModule ) { jQuery._evalUrl( node.src, { nonce: node.nonce || node.getAttribute( "nonce" ) }, doc ); } } else { DOMEval( node.textContent.replace( rcleanScript, "" ), node, doc ); } } } } } } return collection; } function remove( elem, selector, keepData ) { var node, nodes = selector ? jQuery.filter( selector, elem ) : elem, i = 0; for ( ; ( node = nodes[ i ] ) != null; i++ ) { if ( !keepData && node.nodeType === 1 ) { jQuery.cleanData( getAll( node ) ); } if ( node.parentNode ) { if ( keepData && isAttached( node ) ) { setGlobalEval( getAll( node, "script" ) ); } node.parentNode.removeChild( node ); } } return elem; } jQuery.extend( { htmlPrefilter: function( html ) { return html; }, clone: function( elem, dataAndEvents, deepDataAndEvents ) { var i, l, srcElements, destElements, clone = elem.cloneNode( true ), inPage = isAttached( elem ); // Fix IE cloning issues if ( !support.noCloneChecked && ( elem.nodeType === 1 || elem.nodeType === 11 ) && !jQuery.isXMLDoc( elem ) ) { // We eschew Sizzle here for performance reasons: https://jsperf.com/getall-vs-sizzle/2 destElements = getAll( clone ); srcElements = getAll( elem ); for ( i = 0, l = srcElements.length; i < l; i++ ) { fixInput( srcElements[ i ], destElements[ i ] ); } } // Copy the events from the original to the clone if ( dataAndEvents ) { if ( deepDataAndEvents ) { srcElements = srcElements || getAll( elem ); destElements = destElements || getAll( clone ); for ( i = 0, l = srcElements.length; i < l; i++ ) { cloneCopyEvent( srcElements[ i ], destElements[ i ] ); } } else { cloneCopyEvent( elem, clone ); } } // Preserve script evaluation history destElements = getAll( clone, "script" ); if ( destElements.length > 0 ) { setGlobalEval( destElements, !inPage && getAll( elem, "script" ) ); } // Return the cloned set return clone; }, cleanData: function( elems ) { var data, elem, type, special = jQuery.event.special, i = 0; for ( ; ( elem = elems[ i ] ) !== undefined; i++ ) { if ( acceptData( elem ) ) { if ( ( data = elem[ dataPriv.expando ] ) ) { if ( data.events ) { for ( type in data.events ) { if ( special[ type ] ) { jQuery.event.remove( elem, type ); // This is a shortcut to avoid jQuery.event.remove's overhead } else { jQuery.removeEvent( elem, type, data.handle ); } } } // Support: Chrome <=35 - 45+ // Assign undefined instead of using delete, see Data#remove elem[ dataPriv.expando ] = undefined; } if ( elem[ dataUser.expando ] ) { // Support: Chrome <=35 - 45+ // Assign undefined instead of using delete, see Data#remove elem[ dataUser.expando ] = undefined; } } } } } ); jQuery.fn.extend( { detach: function( selector ) { return remove( this, selector, true ); }, remove: function( selector ) { return remove( this, selector ); }, text: function( value ) { return access( this, function( value ) { return value === undefined ? jQuery.text( this ) : this.empty().each( function() { if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) { this.textContent = value; } } ); }, null, value, arguments.length ); }, append: function() { return domManip( this, arguments, function( elem ) { if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) { var target = manipulationTarget( this, elem ); target.appendChild( elem ); } } ); }, prepend: function() { return domManip( this, arguments, function( elem ) { if ( this.nodeType === 1 || this.nodeType === 11 || this.nodeType === 9 ) { var target = manipulationTarget( this, elem ); target.insertBefore( elem, target.firstChild ); } } ); }, before: function() { return domManip( this, arguments, function( elem ) { if ( this.parentNode ) { this.parentNode.insertBefore( elem, this ); } } ); }, after: function() { return domManip( this, arguments, function( elem ) { if ( this.parentNode ) { this.parentNode.insertBefore( elem, this.nextSibling ); } } ); }, empty: function() { var elem, i = 0; for ( ; ( elem = this[ i ] ) != null; i++ ) { if ( elem.nodeType === 1 ) { // Prevent memory leaks jQuery.cleanData( getAll( elem, false ) ); // Remove any remaining nodes elem.textContent = ""; } } return this; }, clone: function( dataAndEvents, deepDataAndEvents ) { dataAndEvents = dataAndEvents == null ? false : dataAndEvents; deepDataAndEvents = deepDataAndEvents == null ? dataAndEvents : deepDataAndEvents; return this.map( function() { return jQuery.clone( this, dataAndEvents, deepDataAndEvents ); } ); }, html: function( value ) { return access( this, function( value ) { var elem = this[ 0 ] || {}, i = 0, l = this.length; if ( value === undefined && elem.nodeType === 1 ) { return elem.innerHTML; } // See if we can take a shortcut and just use innerHTML if ( typeof value === "string" && !rnoInnerhtml.test( value ) && !wrapMap[ ( rtagName.exec( value ) || [ "", "" ] )[ 1 ].toLowerCase() ] ) { value = jQuery.htmlPrefilter( value ); try { for ( ; i < l; i++ ) { elem = this[ i ] || {}; // Remove element nodes and prevent memory leaks if ( elem.nodeType === 1 ) { jQuery.cleanData( getAll( elem, false ) ); elem.innerHTML = value; } } elem = 0; // If using innerHTML throws an exception, use the fallback method } catch ( e ) {} } if ( elem ) { this.empty().append( value ); } }, null, value, arguments.length ); }, replaceWith: function() { var ignored = []; // Make the changes, replacing each non-ignored context element with the new content return domManip( this, arguments, function( elem ) { var parent = this.parentNode; if ( jQuery.inArray( this, ignored ) < 0 ) { jQuery.cleanData( getAll( this ) ); if ( parent ) { parent.replaceChild( elem, this ); } } // Force callback invocation }, ignored ); } } ); jQuery.each( { appendTo: "append", prependTo: "prepend", insertBefore: "before", insertAfter: "after", replaceAll: "replaceWith" }, function( name, original ) { jQuery.fn[ name ] = function( selector ) { var elems, ret = [], insert = jQuery( selector ), last = insert.length - 1, i = 0; for ( ; i <= last; i++ ) { elems = i === last ? this : this.clone( true ); jQuery( insert[ i ] )[ original ]( elems ); // Support: Android <=4.0 only, PhantomJS 1 only // .get() because push.apply(_, arraylike) throws on ancient WebKit push.apply( ret, elems.get() ); } return this.pushStack( ret ); }; } ); var rnumnonpx = new RegExp( "^(" + pnum + ")(?!px)[a-z%]+$", "i" ); var getStyles = function( elem ) { // Support: IE <=11 only, Firefox <=30 (#15098, #14150) // IE throws on elements created in popups // FF meanwhile throws on frame elements through "defaultView.getComputedStyle" var view = elem.ownerDocument.defaultView; if ( !view || !view.opener ) { view = window; } return view.getComputedStyle( elem ); }; var swap = function( elem, options, callback ) { var ret, name, old = {}; // Remember the old values, and insert the new ones for ( name in options ) { old[ name ] = elem.style[ name ]; elem.style[ name ] = options[ name ]; } ret = callback.call( elem ); // Revert the old values for ( name in options ) { elem.style[ name ] = old[ name ]; } return ret; }; var rboxStyle = new RegExp( cssExpand.join( "|" ), "i" ); ( function() { // Executing both pixelPosition & boxSizingReliable tests require only one layout // so they're executed at the same time to save the second computation. function computeStyleTests() { // This is a singleton, we need to execute it only once if ( !div ) { return; } container.style.cssText = "position:absolute;left:-11111px;width:60px;" + "margin-top:1px;padding:0;border:0"; div.style.cssText = "position:relative;display:block;box-sizing:border-box;overflow:scroll;" + "margin:auto;border:1px;padding:1px;" + "width:60%;top:1%"; documentElement.appendChild( container ).appendChild( div ); var divStyle = window.getComputedStyle( div ); pixelPositionVal = divStyle.top !== "1%"; // Support: Android 4.0 - 4.3 only, Firefox <=3 - 44 reliableMarginLeftVal = roundPixelMeasures( divStyle.marginLeft ) === 12; // Support: Android 4.0 - 4.3 only, Safari <=9.1 - 10.1, iOS <=7.0 - 9.3 // Some styles come back with percentage values, even though they shouldn't div.style.right = "60%"; pixelBoxStylesVal = roundPixelMeasures( divStyle.right ) === 36; // Support: IE 9 - 11 only // Detect misreporting of content dimensions for box-sizing:border-box elements boxSizingReliableVal = roundPixelMeasures( divStyle.width ) === 36; // Support: IE 9 only // Detect overflow:scroll screwiness (gh-3699) // Support: Chrome <=64 // Don't get tricked when zoom affects offsetWidth (gh-4029) div.style.position = "absolute"; scrollboxSizeVal = roundPixelMeasures( div.offsetWidth / 3 ) === 12; documentElement.removeChild( container ); // Nullify the div so it wouldn't be stored in the memory and // it will also be a sign that checks already performed div = null; } function roundPixelMeasures( measure ) { return Math.round( parseFloat( measure ) ); } var pixelPositionVal, boxSizingReliableVal, scrollboxSizeVal, pixelBoxStylesVal, reliableTrDimensionsVal, reliableMarginLeftVal, container = document.createElement( "div" ), div = document.createElement( "div" ); // Finish early in limited (non-browser) environments if ( !div.style ) { return; } // Support: IE <=9 - 11 only // Style of cloned element affects source element cloned (#8908) div.style.backgroundClip = "content-box"; div.cloneNode( true ).style.backgroundClip = ""; support.clearCloneStyle = div.style.backgroundClip === "content-box"; jQuery.extend( support, { boxSizingReliable: function() { computeStyleTests(); return boxSizingReliableVal; }, pixelBoxStyles: function() { computeStyleTests(); return pixelBoxStylesVal; }, pixelPosition: function() { computeStyleTests(); return pixelPositionVal; }, reliableMarginLeft: function() { computeStyleTests(); return reliableMarginLeftVal; }, scrollboxSize: function() { computeStyleTests(); return scrollboxSizeVal; }, // Support: IE 9 - 11+, Edge 15 - 18+ // IE/Edge misreport `getComputedStyle` of table rows with width/height // set in CSS while `offset*` properties report correct values. // Behavior in IE 9 is more subtle than in newer versions & it passes // some versions of this test; make sure not to make it pass there! reliableTrDimensions: function() { var table, tr, trChild, trStyle; if ( reliableTrDimensionsVal == null ) { table = document.createElement( "table" ); tr = document.createElement( "tr" ); trChild = document.createElement( "div" ); table.style.cssText = "position:absolute;left:-11111px"; tr.style.height = "1px"; trChild.style.height = "9px"; documentElement .appendChild( table ) .appendChild( tr ) .appendChild( trChild ); trStyle = window.getComputedStyle( tr ); reliableTrDimensionsVal = parseInt( trStyle.height ) > 3; documentElement.removeChild( table ); } return reliableTrDimensionsVal; } } ); } )(); function curCSS( elem, name, computed ) { var width, minWidth, maxWidth, ret, // Support: Firefox 51+ // Retrieving style before computed somehow // fixes an issue with getting wrong values // on detached elements style = elem.style; computed = computed || getStyles( elem ); // getPropertyValue is needed for: // .css('filter') (IE 9 only, #12537) // .css('--customProperty) (#3144) if ( computed ) { ret = computed.getPropertyValue( name ) || computed[ name ]; if ( ret === "" && !isAttached( elem ) ) { ret = jQuery.style( elem, name ); } // A tribute to the "awesome hack by Dean Edwards" // Android Browser returns percentage for some values, // but width seems to be reliably pixels. // This is against the CSSOM draft spec: // https://drafts.csswg.org/cssom/#resolved-values if ( !support.pixelBoxStyles() && rnumnonpx.test( ret ) && rboxStyle.test( name ) ) { // Remember the original values width = style.width; minWidth = style.minWidth; maxWidth = style.maxWidth; // Put in the new values to get a computed value out style.minWidth = style.maxWidth = style.width = ret; ret = computed.width; // Revert the changed values style.width = width; style.minWidth = minWidth; style.maxWidth = maxWidth; } } return ret !== undefined ? // Support: IE <=9 - 11 only // IE returns zIndex value as an integer. ret + "" : ret; } function addGetHookIf( conditionFn, hookFn ) { // Define the hook, we'll check on the first run if it's really needed. return { get: function() { if ( conditionFn() ) { // Hook not needed (or it's not possible to use it due // to missing dependency), remove it. delete this.get; return; } // Hook needed; redefine it so that the support test is not executed again. return ( this.get = hookFn ).apply( this, arguments ); } }; } var cssPrefixes = [ "Webkit", "Moz", "ms" ], emptyStyle = document.createElement( "div" ).style, vendorProps = {}; // Return a vendor-prefixed property or undefined function vendorPropName( name ) { // Check for vendor prefixed names var capName = name[ 0 ].toUpperCase() + name.slice( 1 ), i = cssPrefixes.length; while ( i-- ) { name = cssPrefixes[ i ] + capName; if ( name in emptyStyle ) { return name; } } } // Return a potentially-mapped jQuery.cssProps or vendor prefixed property function finalPropName( name ) { var final = jQuery.cssProps[ name ] || vendorProps[ name ]; if ( final ) { return final; } if ( name in emptyStyle ) { return name; } return vendorProps[ name ] = vendorPropName( name ) || name; } var // Swappable if display is none or starts with table // except "table", "table-cell", or "table-caption" // See here for display values: https://developer.mozilla.org/en-US/docs/CSS/display rdisplayswap = /^(none|table(?!-c[ea]).+)/, rcustomProp = /^--/, cssShow = { position: "absolute", visibility: "hidden", display: "block" }, cssNormalTransform = { letterSpacing: "0", fontWeight: "400" }; function setPositiveNumber( _elem, value, subtract ) { // Any relative (+/-) values have already been // normalized at this point var matches = rcssNum.exec( value ); return matches ? // Guard against undefined "subtract", e.g., when used as in cssHooks Math.max( 0, matches[ 2 ] - ( subtract || 0 ) ) + ( matches[ 3 ] || "px" ) : value; } function boxModelAdjustment( elem, dimension, box, isBorderBox, styles, computedVal ) { var i = dimension === "width" ? 1 : 0, extra = 0, delta = 0; // Adjustment may not be necessary if ( box === ( isBorderBox ? "border" : "content" ) ) { return 0; } for ( ; i < 4; i += 2 ) { // Both box models exclude margin if ( box === "margin" ) { delta += jQuery.css( elem, box + cssExpand[ i ], true, styles ); } // If we get here with a content-box, we're seeking "padding" or "border" or "margin" if ( !isBorderBox ) { // Add padding delta += jQuery.css( elem, "padding" + cssExpand[ i ], true, styles ); // For "border" or "margin", add border if ( box !== "padding" ) { delta += jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles ); // But still keep track of it otherwise } else { extra += jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles ); } // If we get here with a border-box (content + padding + border), we're seeking "content" or // "padding" or "margin" } else { // For "content", subtract padding if ( box === "content" ) { delta -= jQuery.css( elem, "padding" + cssExpand[ i ], true, styles ); } // For "content" or "padding", subtract border if ( box !== "margin" ) { delta -= jQuery.css( elem, "border" + cssExpand[ i ] + "Width", true, styles ); } } } // Account for positive content-box scroll gutter when requested by providing computedVal if ( !isBorderBox && computedVal >= 0 ) { // offsetWidth/offsetHeight is a rounded sum of content, padding, scroll gutter, and border // Assuming integer scroll gutter, subtract the rest and round down delta += Math.max( 0, Math.ceil( elem[ "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ) ] - computedVal - delta - extra - 0.5 // If offsetWidth/offsetHeight is unknown, then we can't determine content-box scroll gutter // Use an explicit zero to avoid NaN (gh-3964) ) ) || 0; } return delta; } function getWidthOrHeight( elem, dimension, extra ) { // Start with computed style var styles = getStyles( elem ), // To avoid forcing a reflow, only fetch boxSizing if we need it (gh-4322). // Fake content-box until we know it's needed to know the true value. boxSizingNeeded = !support.boxSizingReliable() || extra, isBorderBox = boxSizingNeeded && jQuery.css( elem, "boxSizing", false, styles ) === "border-box", valueIsBorderBox = isBorderBox, val = curCSS( elem, dimension, styles ), offsetProp = "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ); // Support: Firefox <=54 // Return a confounding non-pixel value or feign ignorance, as appropriate. if ( rnumnonpx.test( val ) ) { if ( !extra ) { return val; } val = "auto"; } // Support: IE 9 - 11 only // Use offsetWidth/offsetHeight for when box sizing is unreliable. // In those cases, the computed value can be trusted to be border-box. if ( ( !support.boxSizingReliable() && isBorderBox || // Support: IE 10 - 11+, Edge 15 - 18+ // IE/Edge misreport `getComputedStyle` of table rows with width/height // set in CSS while `offset*` properties report correct values. // Interestingly, in some cases IE 9 doesn't suffer from this issue. !support.reliableTrDimensions() && nodeName( elem, "tr" ) || // Fall back to offsetWidth/offsetHeight when value is "auto" // This happens for inline elements with no explicit setting (gh-3571) val === "auto" || // Support: Android <=4.1 - 4.3 only // Also use offsetWidth/offsetHeight for misreported inline dimensions (gh-3602) !parseFloat( val ) && jQuery.css( elem, "display", false, styles ) === "inline" ) && // Make sure the element is visible & connected elem.getClientRects().length ) { isBorderBox = jQuery.css( elem, "boxSizing", false, styles ) === "border-box"; // Where available, offsetWidth/offsetHeight approximate border box dimensions. // Where not available (e.g., SVG), assume unreliable box-sizing and interpret the // retrieved value as a content box dimension. valueIsBorderBox = offsetProp in elem; if ( valueIsBorderBox ) { val = elem[ offsetProp ]; } } // Normalize "" and auto val = parseFloat( val ) || 0; // Adjust for the element's box model return ( val + boxModelAdjustment( elem, dimension, extra || ( isBorderBox ? "border" : "content" ), valueIsBorderBox, styles, // Provide the current computed size to request scroll gutter calculation (gh-3589) val ) ) + "px"; } jQuery.extend( { // Add in style property hooks for overriding the default // behavior of getting and setting a style property cssHooks: { opacity: { get: function( elem, computed ) { if ( computed ) { // We should always get a number back from opacity var ret = curCSS( elem, "opacity" ); return ret === "" ? "1" : ret; } } } }, // Don't automatically add "px" to these possibly-unitless properties cssNumber: { "animationIterationCount": true, "columnCount": true, "fillOpacity": true, "flexGrow": true, "flexShrink": true, "fontWeight": true, "gridArea": true, "gridColumn": true, "gridColumnEnd": true, "gridColumnStart": true, "gridRow": true, "gridRowEnd": true, "gridRowStart": true, "lineHeight": true, "opacity": true, "order": true, "orphans": true, "widows": true, "zIndex": true, "zoom": true }, // Add in properties whose names you wish to fix before // setting or getting the value cssProps: {}, // Get and set the style property on a DOM Node style: function( elem, name, value, extra ) { // Don't set styles on text and comment nodes if ( !elem || elem.nodeType === 3 || elem.nodeType === 8 || !elem.style ) { return; } // Make sure that we're working with the right name var ret, type, hooks, origName = camelCase( name ), isCustomProp = rcustomProp.test( name ), style = elem.style; // Make sure that we're working with the right name. We don't // want to query the value if it is a CSS custom property // since they are user-defined. if ( !isCustomProp ) { name = finalPropName( origName ); } // Gets hook for the prefixed version, then unprefixed version hooks = jQuery.cssHooks[ name ] || jQuery.cssHooks[ origName ]; // Check if we're setting a value if ( value !== undefined ) { type = typeof value; // Convert "+=" or "-=" to relative numbers (#7345) if ( type === "string" && ( ret = rcssNum.exec( value ) ) && ret[ 1 ] ) { value = adjustCSS( elem, name, ret ); // Fixes bug #9237 type = "number"; } // Make sure that null and NaN values aren't set (#7116) if ( value == null || value !== value ) { return; } // If a number was passed in, add the unit (except for certain CSS properties) // The isCustomProp check can be removed in jQuery 4.0 when we only auto-append // "px" to a few hardcoded values. if ( type === "number" && !isCustomProp ) { value += ret && ret[ 3 ] || ( jQuery.cssNumber[ origName ] ? "" : "px" ); } // background-* props affect original clone's values if ( !support.clearCloneStyle && value === "" && name.indexOf( "background" ) === 0 ) { style[ name ] = "inherit"; } // If a hook was provided, use that value, otherwise just set the specified value if ( !hooks || !( "set" in hooks ) || ( value = hooks.set( elem, value, extra ) ) !== undefined ) { if ( isCustomProp ) { style.setProperty( name, value ); } else { style[ name ] = value; } } } else { // If a hook was provided get the non-computed value from there if ( hooks && "get" in hooks && ( ret = hooks.get( elem, false, extra ) ) !== undefined ) { return ret; } // Otherwise just get the value from the style object return style[ name ]; } }, css: function( elem, name, extra, styles ) { var val, num, hooks, origName = camelCase( name ), isCustomProp = rcustomProp.test( name ); // Make sure that we're working with the right name. We don't // want to modify the value if it is a CSS custom property // since they are user-defined. if ( !isCustomProp ) { name = finalPropName( origName ); } // Try prefixed name followed by the unprefixed name hooks = jQuery.cssHooks[ name ] || jQuery.cssHooks[ origName ]; // If a hook was provided get the computed value from there if ( hooks && "get" in hooks ) { val = hooks.get( elem, true, extra ); } // Otherwise, if a way to get the computed value exists, use that if ( val === undefined ) { val = curCSS( elem, name, styles ); } // Convert "normal" to computed value if ( val === "normal" && name in cssNormalTransform ) { val = cssNormalTransform[ name ]; } // Make numeric if forced or a qualifier was provided and val looks numeric if ( extra === "" || extra ) { num = parseFloat( val ); return extra === true || isFinite( num ) ? num || 0 : val; } return val; } } ); jQuery.each( [ "height", "width" ], function( _i, dimension ) { jQuery.cssHooks[ dimension ] = { get: function( elem, computed, extra ) { if ( computed ) { // Certain elements can have dimension info if we invisibly show them // but it must have a current display style that would benefit return rdisplayswap.test( jQuery.css( elem, "display" ) ) && // Support: Safari 8+ // Table columns in Safari have non-zero offsetWidth & zero // getBoundingClientRect().width unless display is changed. // Support: IE <=11 only // Running getBoundingClientRect on a disconnected node // in IE throws an error. ( !elem.getClientRects().length || !elem.getBoundingClientRect().width ) ? swap( elem, cssShow, function() { return getWidthOrHeight( elem, dimension, extra ); } ) : getWidthOrHeight( elem, dimension, extra ); } }, set: function( elem, value, extra ) { var matches, styles = getStyles( elem ), // Only read styles.position if the test has a chance to fail // to avoid forcing a reflow. scrollboxSizeBuggy = !support.scrollboxSize() && styles.position === "absolute", // To avoid forcing a reflow, only fetch boxSizing if we need it (gh-3991) boxSizingNeeded = scrollboxSizeBuggy || extra, isBorderBox = boxSizingNeeded && jQuery.css( elem, "boxSizing", false, styles ) === "border-box", subtract = extra ? boxModelAdjustment( elem, dimension, extra, isBorderBox, styles ) : 0; // Account for unreliable border-box dimensions by comparing offset* to computed and // faking a content-box to get border and padding (gh-3699) if ( isBorderBox && scrollboxSizeBuggy ) { subtract -= Math.ceil( elem[ "offset" + dimension[ 0 ].toUpperCase() + dimension.slice( 1 ) ] - parseFloat( styles[ dimension ] ) - boxModelAdjustment( elem, dimension, "border", false, styles ) - 0.5 ); } // Convert to pixels if value adjustment is needed if ( subtract && ( matches = rcssNum.exec( value ) ) && ( matches[ 3 ] || "px" ) !== "px" ) { elem.style[ dimension ] = value; value = jQuery.css( elem, dimension ); } return setPositiveNumber( elem, value, subtract ); } }; } ); jQuery.cssHooks.marginLeft = addGetHookIf( support.reliableMarginLeft, function( elem, computed ) { if ( computed ) { return ( parseFloat( curCSS( elem, "marginLeft" ) ) || elem.getBoundingClientRect().left - swap( elem, { marginLeft: 0 }, function() { return elem.getBoundingClientRect().left; } ) ) + "px"; } } ); // These hooks are used by animate to expand properties jQuery.each( { margin: "", padding: "", border: "Width" }, function( prefix, suffix ) { jQuery.cssHooks[ prefix + suffix ] = { expand: function( value ) { var i = 0, expanded = {}, // Assumes a single number if not a string parts = typeof value === "string" ? value.split( " " ) : [ value ]; for ( ; i < 4; i++ ) { expanded[ prefix + cssExpand[ i ] + suffix ] = parts[ i ] || parts[ i - 2 ] || parts[ 0 ]; } return expanded; } }; if ( prefix !== "margin" ) { jQuery.cssHooks[ prefix + suffix ].set = setPositiveNumber; } } ); jQuery.fn.extend( { css: function( name, value ) { return access( this, function( elem, name, value ) { var styles, len, map = {}, i = 0; if ( Array.isArray( name ) ) { styles = getStyles( elem ); len = name.length; for ( ; i < len; i++ ) { map[ name[ i ] ] = jQuery.css( elem, name[ i ], false, styles ); } return map; } return value !== undefined ? jQuery.style( elem, name, value ) : jQuery.css( elem, name ); }, name, value, arguments.length > 1 ); } } ); function Tween( elem, options, prop, end, easing ) { return new Tween.prototype.init( elem, options, prop, end, easing ); } jQuery.Tween = Tween; Tween.prototype = { constructor: Tween, init: function( elem, options, prop, end, easing, unit ) { this.elem = elem; this.prop = prop; this.easing = easing || jQuery.easing._default; this.options = options; this.start = this.now = this.cur(); this.end = end; this.unit = unit || ( jQuery.cssNumber[ prop ] ? "" : "px" ); }, cur: function() { var hooks = Tween.propHooks[ this.prop ]; return hooks && hooks.get ? hooks.get( this ) : Tween.propHooks._default.get( this ); }, run: function( percent ) { var eased, hooks = Tween.propHooks[ this.prop ]; if ( this.options.duration ) { this.pos = eased = jQuery.easing[ this.easing ]( percent, this.options.duration * percent, 0, 1, this.options.duration ); } else { this.pos = eased = percent; } this.now = ( this.end - this.start ) * eased + this.start; if ( this.options.step ) { this.options.step.call( this.elem, this.now, this ); } if ( hooks && hooks.set ) { hooks.set( this ); } else { Tween.propHooks._default.set( this ); } return this; } }; Tween.prototype.init.prototype = Tween.prototype; Tween.propHooks = { _default: { get: function( tween ) { var result; // Use a property on the element directly when it is not a DOM element, // or when there is no matching style property that exists. if ( tween.elem.nodeType !== 1 || tween.elem[ tween.prop ] != null && tween.elem.style[ tween.prop ] == null ) { return tween.elem[ tween.prop ]; } // Passing an empty string as a 3rd parameter to .css will automatically // attempt a parseFloat and fallback to a string if the parse fails. // Simple values such as "10px" are parsed to Float; // complex values such as "rotate(1rad)" are returned as-is. result = jQuery.css( tween.elem, tween.prop, "" ); // Empty strings, null, undefined and "auto" are converted to 0. return !result || result === "auto" ? 0 : result; }, set: function( tween ) { // Use step hook for back compat. // Use cssHook if its there. // Use .style if available and use plain properties where available. if ( jQuery.fx.step[ tween.prop ] ) { jQuery.fx.step[ tween.prop ]( tween ); } else if ( tween.elem.nodeType === 1 && ( jQuery.cssHooks[ tween.prop ] || tween.elem.style[ finalPropName( tween.prop ) ] != null ) ) { jQuery.style( tween.elem, tween.prop, tween.now + tween.unit ); } else { tween.elem[ tween.prop ] = tween.now; } } } }; // Support: IE <=9 only // Panic based approach to setting things on disconnected nodes Tween.propHooks.scrollTop = Tween.propHooks.scrollLeft = { set: function( tween ) { if ( tween.elem.nodeType && tween.elem.parentNode ) { tween.elem[ tween.prop ] = tween.now; } } }; jQuery.easing = { linear: function( p ) { return p; }, swing: function( p ) { return 0.5 - Math.cos( p * Math.PI ) / 2; }, _default: "swing" }; jQuery.fx = Tween.prototype.init; // Back compat <1.8 extension point jQuery.fx.step = {}; var fxNow, inProgress, rfxtypes = /^(?:toggle|show|hide)$/, rrun = /queueHooks$/; function schedule() { if ( inProgress ) { if ( document.hidden === false && window.requestAnimationFrame ) { window.requestAnimationFrame( schedule ); } else { window.setTimeout( schedule, jQuery.fx.interval ); } jQuery.fx.tick(); } } // Animations created synchronously will run synchronously function createFxNow() { window.setTimeout( function() { fxNow = undefined; } ); return ( fxNow = Date.now() ); } // Generate parameters to create a standard animation function genFx( type, includeWidth ) { var which, i = 0, attrs = { height: type }; // If we include width, step value is 1 to do all cssExpand values, // otherwise step value is 2 to skip over Left and Right includeWidth = includeWidth ? 1 : 0; for ( ; i < 4; i += 2 - includeWidth ) { which = cssExpand[ i ]; attrs[ "margin" + which ] = attrs[ "padding" + which ] = type; } if ( includeWidth ) { attrs.opacity = attrs.width = type; } return attrs; } function createTween( value, prop, animation ) { var tween, collection = ( Animation.tweeners[ prop ] || [] ).concat( Animation.tweeners[ "*" ] ), index = 0, length = collection.length; for ( ; index < length; index++ ) { if ( ( tween = collection[ index ].call( animation, prop, value ) ) ) { // We're done with this property return tween; } } } function defaultPrefilter( elem, props, opts ) { var prop, value, toggle, hooks, oldfire, propTween, restoreDisplay, display, isBox = "width" in props || "height" in props, anim = this, orig = {}, style = elem.style, hidden = elem.nodeType && isHiddenWithinTree( elem ), dataShow = dataPriv.get( elem, "fxshow" ); // Queue-skipping animations hijack the fx hooks if ( !opts.queue ) { hooks = jQuery._queueHooks( elem, "fx" ); if ( hooks.unqueued == null ) { hooks.unqueued = 0; oldfire = hooks.empty.fire; hooks.empty.fire = function() { if ( !hooks.unqueued ) { oldfire(); } }; } hooks.unqueued++; anim.always( function() { // Ensure the complete handler is called before this completes anim.always( function() { hooks.unqueued--; if ( !jQuery.queue( elem, "fx" ).length ) { hooks.empty.fire(); } } ); } ); } // Detect show/hide animations for ( prop in props ) { value = props[ prop ]; if ( rfxtypes.test( value ) ) { delete props[ prop ]; toggle = toggle || value === "toggle"; if ( value === ( hidden ? "hide" : "show" ) ) { // Pretend to be hidden if this is a "show" and // there is still data from a stopped show/hide if ( value === "show" && dataShow && dataShow[ prop ] !== undefined ) { hidden = true; // Ignore all other no-op show/hide data } else { continue; } } orig[ prop ] = dataShow && dataShow[ prop ] || jQuery.style( elem, prop ); } } // Bail out if this is a no-op like .hide().hide() propTween = !jQuery.isEmptyObject( props ); if ( !propTween && jQuery.isEmptyObject( orig ) ) { return; } // Restrict "overflow" and "display" styles during box animations if ( isBox && elem.nodeType === 1 ) { // Support: IE <=9 - 11, Edge 12 - 15 // Record all 3 overflow attributes because IE does not infer the shorthand // from identically-valued overflowX and overflowY and Edge just mirrors // the overflowX value there. opts.overflow = [ style.overflow, style.overflowX, style.overflowY ]; // Identify a display type, preferring old show/hide data over the CSS cascade restoreDisplay = dataShow && dataShow.display; if ( restoreDisplay == null ) { restoreDisplay = dataPriv.get( elem, "display" ); } display = jQuery.css( elem, "display" ); if ( display === "none" ) { if ( restoreDisplay ) { display = restoreDisplay; } else { // Get nonempty value(s) by temporarily forcing visibility showHide( [ elem ], true ); restoreDisplay = elem.style.display || restoreDisplay; display = jQuery.css( elem, "display" ); showHide( [ elem ] ); } } // Animate inline elements as inline-block if ( display === "inline" || display === "inline-block" && restoreDisplay != null ) { if ( jQuery.css( elem, "float" ) === "none" ) { // Restore the original display value at the end of pure show/hide animations if ( !propTween ) { anim.done( function() { style.display = restoreDisplay; } ); if ( restoreDisplay == null ) { display = style.display; restoreDisplay = display === "none" ? "" : display; } } style.display = "inline-block"; } } } if ( opts.overflow ) { style.overflow = "hidden"; anim.always( function() { style.overflow = opts.overflow[ 0 ]; style.overflowX = opts.overflow[ 1 ]; style.overflowY = opts.overflow[ 2 ]; } ); } // Implement show/hide animations propTween = false; for ( prop in orig ) { // General show/hide setup for this element animation if ( !propTween ) { if ( dataShow ) { if ( "hidden" in dataShow ) { hidden = dataShow.hidden; } } else { dataShow = dataPriv.access( elem, "fxshow", { display: restoreDisplay } ); } // Store hidden/visible for toggle so `.stop().toggle()` "reverses" if ( toggle ) { dataShow.hidden = !hidden; } // Show elements before animating them if ( hidden ) { showHide( [ elem ], true ); } /* eslint-disable no-loop-func */ anim.done( function() { /* eslint-enable no-loop-func */ // The final step of a "hide" animation is actually hiding the element if ( !hidden ) { showHide( [ elem ] ); } dataPriv.remove( elem, "fxshow" ); for ( prop in orig ) { jQuery.style( elem, prop, orig[ prop ] ); } } ); } // Per-property setup propTween = createTween( hidden ? dataShow[ prop ] : 0, prop, anim ); if ( !( prop in dataShow ) ) { dataShow[ prop ] = propTween.start; if ( hidden ) { propTween.end = propTween.start; propTween.start = 0; } } } } function propFilter( props, specialEasing ) { var index, name, easing, value, hooks; // camelCase, specialEasing and expand cssHook pass for ( index in props ) { name = camelCase( index ); easing = specialEasing[ name ]; value = props[ index ]; if ( Array.isArray( value ) ) { easing = value[ 1 ]; value = props[ index ] = value[ 0 ]; } if ( index !== name ) { props[ name ] = value; delete props[ index ]; } hooks = jQuery.cssHooks[ name ]; if ( hooks && "expand" in hooks ) { value = hooks.expand( value ); delete props[ name ]; // Not quite $.extend, this won't overwrite existing keys. // Reusing 'index' because we have the correct "name" for ( index in value ) { if ( !( index in props ) ) { props[ index ] = value[ index ]; specialEasing[ index ] = easing; } } } else { specialEasing[ name ] = easing; } } } function Animation( elem, properties, options ) { var result, stopped, index = 0, length = Animation.prefilters.length, deferred = jQuery.Deferred().always( function() { // Don't match elem in the :animated selector delete tick.elem; } ), tick = function() { if ( stopped ) { return false; } var currentTime = fxNow || createFxNow(), remaining = Math.max( 0, animation.startTime + animation.duration - currentTime ), // Support: Android 2.3 only // Archaic crash bug won't allow us to use `1 - ( 0.5 || 0 )` (#12497) temp = remaining / animation.duration || 0, percent = 1 - temp, index = 0, length = animation.tweens.length; for ( ; index < length; index++ ) { animation.tweens[ index ].run( percent ); } deferred.notifyWith( elem, [ animation, percent, remaining ] ); // If there's more to do, yield if ( percent < 1 && length ) { return remaining; } // If this was an empty animation, synthesize a final progress notification if ( !length ) { deferred.notifyWith( elem, [ animation, 1, 0 ] ); } // Resolve the animation and report its conclusion deferred.resolveWith( elem, [ animation ] ); return false; }, animation = deferred.promise( { elem: elem, props: jQuery.extend( {}, properties ), opts: jQuery.extend( true, { specialEasing: {}, easing: jQuery.easing._default }, options ), originalProperties: properties, originalOptions: options, startTime: fxNow || createFxNow(), duration: options.duration, tweens: [], createTween: function( prop, end ) { var tween = jQuery.Tween( elem, animation.opts, prop, end, animation.opts.specialEasing[ prop ] || animation.opts.easing ); animation.tweens.push( tween ); return tween; }, stop: function( gotoEnd ) { var index = 0, // If we are going to the end, we want to run all the tweens // otherwise we skip this part length = gotoEnd ? animation.tweens.length : 0; if ( stopped ) { return this; } stopped = true; for ( ; index < length; index++ ) { animation.tweens[ index ].run( 1 ); } // Resolve when we played the last frame; otherwise, reject if ( gotoEnd ) { deferred.notifyWith( elem, [ animation, 1, 0 ] ); deferred.resolveWith( elem, [ animation, gotoEnd ] ); } else { deferred.rejectWith( elem, [ animation, gotoEnd ] ); } return this; } } ), props = animation.props; propFilter( props, animation.opts.specialEasing ); for ( ; index < length; index++ ) { result = Animation.prefilters[ index ].call( animation, elem, props, animation.opts ); if ( result ) { if ( isFunction( result.stop ) ) { jQuery._queueHooks( animation.elem, animation.opts.queue ).stop = result.stop.bind( result ); } return result; } } jQuery.map( props, createTween, animation ); if ( isFunction( animation.opts.start ) ) { animation.opts.start.call( elem, animation ); } // Attach callbacks from options animation .progress( animation.opts.progress ) .done( animation.opts.done, animation.opts.complete ) .fail( animation.opts.fail ) .always( animation.opts.always ); jQuery.fx.timer( jQuery.extend( tick, { elem: elem, anim: animation, queue: animation.opts.queue } ) ); return animation; } jQuery.Animation = jQuery.extend( Animation, { tweeners: { "*": [ function( prop, value ) { var tween = this.createTween( prop, value ); adjustCSS( tween.elem, prop, rcssNum.exec( value ), tween ); return tween; } ] }, tweener: function( props, callback ) { if ( isFunction( props ) ) { callback = props; props = [ "*" ]; } else { props = props.match( rnothtmlwhite ); } var prop, index = 0, length = props.length; for ( ; index < length; index++ ) { prop = props[ index ]; Animation.tweeners[ prop ] = Animation.tweeners[ prop ] || []; Animation.tweeners[ prop ].unshift( callback ); } }, prefilters: [ defaultPrefilter ], prefilter: function( callback, prepend ) { if ( prepend ) { Animation.prefilters.unshift( callback ); } else { Animation.prefilters.push( callback ); } } } ); jQuery.speed = function( speed, easing, fn ) { var opt = speed && typeof speed === "object" ? jQuery.extend( {}, speed ) : { complete: fn || !fn && easing || isFunction( speed ) && speed, duration: speed, easing: fn && easing || easing && !isFunction( easing ) && easing }; // Go to the end state if fx are off if ( jQuery.fx.off ) { opt.duration = 0; } else { if ( typeof opt.duration !== "number" ) { if ( opt.duration in jQuery.fx.speeds ) { opt.duration = jQuery.fx.speeds[ opt.duration ]; } else { opt.duration = jQuery.fx.speeds._default; } } } // Normalize opt.queue - true/undefined/null -> "fx" if ( opt.queue == null || opt.queue === true ) { opt.queue = "fx"; } // Queueing opt.old = opt.complete; opt.complete = function() { if ( isFunction( opt.old ) ) { opt.old.call( this ); } if ( opt.queue ) { jQuery.dequeue( this, opt.queue ); } }; return opt; }; jQuery.fn.extend( { fadeTo: function( speed, to, easing, callback ) { // Show any hidden elements after setting opacity to 0 return this.filter( isHiddenWithinTree ).css( "opacity", 0 ).show() // Animate to the value specified .end().animate( { opacity: to }, speed, easing, callback ); }, animate: function( prop, speed, easing, callback ) { var empty = jQuery.isEmptyObject( prop ), optall = jQuery.speed( speed, easing, callback ), doAnimation = function() { // Operate on a copy of prop so per-property easing won't be lost var anim = Animation( this, jQuery.extend( {}, prop ), optall ); // Empty animations, or finishing resolves immediately if ( empty || dataPriv.get( this, "finish" ) ) { anim.stop( true ); } }; doAnimation.finish = doAnimation; return empty || optall.queue === false ? this.each( doAnimation ) : this.queue( optall.queue, doAnimation ); }, stop: function( type, clearQueue, gotoEnd ) { var stopQueue = function( hooks ) { var stop = hooks.stop; delete hooks.stop; stop( gotoEnd ); }; if ( typeof type !== "string" ) { gotoEnd = clearQueue; clearQueue = type; type = undefined; } if ( clearQueue ) { this.queue( type || "fx", [] ); } return this.each( function() { var dequeue = true, index = type != null && type + "queueHooks", timers = jQuery.timers, data = dataPriv.get( this ); if ( index ) { if ( data[ index ] && data[ index ].stop ) { stopQueue( data[ index ] ); } } else { for ( index in data ) { if ( data[ index ] && data[ index ].stop && rrun.test( index ) ) { stopQueue( data[ index ] ); } } } for ( index = timers.length; index--; ) { if ( timers[ index ].elem === this && ( type == null || timers[ index ].queue === type ) ) { timers[ index ].anim.stop( gotoEnd ); dequeue = false; timers.splice( index, 1 ); } } // Start the next in the queue if the last step wasn't forced. // Timers currently will call their complete callbacks, which // will dequeue but only if they were gotoEnd. if ( dequeue || !gotoEnd ) { jQuery.dequeue( this, type ); } } ); }, finish: function( type ) { if ( type !== false ) { type = type || "fx"; } return this.each( function() { var index, data = dataPriv.get( this ), queue = data[ type + "queue" ], hooks = data[ type + "queueHooks" ], timers = jQuery.timers, length = queue ? queue.length : 0; // Enable finishing flag on private data data.finish = true; // Empty the queue first jQuery.queue( this, type, [] ); if ( hooks && hooks.stop ) { hooks.stop.call( this, true ); } // Look for any active animations, and finish them for ( index = timers.length; index--; ) { if ( timers[ index ].elem === this && timers[ index ].queue === type ) { timers[ index ].anim.stop( true ); timers.splice( index, 1 ); } } // Look for any animations in the old queue and finish them for ( index = 0; index < length; index++ ) { if ( queue[ index ] && queue[ index ].finish ) { queue[ index ].finish.call( this ); } } // Turn off finishing flag delete data.finish; } ); } } ); jQuery.each( [ "toggle", "show", "hide" ], function( _i, name ) { var cssFn = jQuery.fn[ name ]; jQuery.fn[ name ] = function( speed, easing, callback ) { return speed == null || typeof speed === "boolean" ? cssFn.apply( this, arguments ) : this.animate( genFx( name, true ), speed, easing, callback ); }; } ); // Generate shortcuts for custom animations jQuery.each( { slideDown: genFx( "show" ), slideUp: genFx( "hide" ), slideToggle: genFx( "toggle" ), fadeIn: { opacity: "show" }, fadeOut: { opacity: "hide" }, fadeToggle: { opacity: "toggle" } }, function( name, props ) { jQuery.fn[ name ] = function( speed, easing, callback ) { return this.animate( props, speed, easing, callback ); }; } ); jQuery.timers = []; jQuery.fx.tick = function() { var timer, i = 0, timers = jQuery.timers; fxNow = Date.now(); for ( ; i < timers.length; i++ ) { timer = timers[ i ]; // Run the timer and safely remove it when done (allowing for external removal) if ( !timer() && timers[ i ] === timer ) { timers.splice( i--, 1 ); } } if ( !timers.length ) { jQuery.fx.stop(); } fxNow = undefined; }; jQuery.fx.timer = function( timer ) { jQuery.timers.push( timer ); jQuery.fx.start(); }; jQuery.fx.interval = 13; jQuery.fx.start = function() { if ( inProgress ) { return; } inProgress = true; schedule(); }; jQuery.fx.stop = function() { inProgress = null; }; jQuery.fx.speeds = { slow: 600, fast: 200, // Default speed _default: 400 }; // Based off of the plugin by Clint Helfers, with permission. // https://web.archive.org/web/20100324014747/http://blindsignals.com/index.php/2009/07/jquery-delay/ jQuery.fn.delay = function( time, type ) { time = jQuery.fx ? jQuery.fx.speeds[ time ] || time : time; type = type || "fx"; return this.queue( type, function( next, hooks ) { var timeout = window.setTimeout( next, time ); hooks.stop = function() { window.clearTimeout( timeout ); }; } ); }; ( function() { var input = document.createElement( "input" ), select = document.createElement( "select" ), opt = select.appendChild( document.createElement( "option" ) ); input.type = "checkbox"; // Support: Android <=4.3 only // Default value for a checkbox should be "on" support.checkOn = input.value !== ""; // Support: IE <=11 only // Must access selectedIndex to make default options select support.optSelected = opt.selected; // Support: IE <=11 only // An input loses its value after becoming a radio input = document.createElement( "input" ); input.value = "t"; input.type = "radio"; support.radioValue = input.value === "t"; } )(); var boolHook, attrHandle = jQuery.expr.attrHandle; jQuery.fn.extend( { attr: function( name, value ) { return access( this, jQuery.attr, name, value, arguments.length > 1 ); }, removeAttr: function( name ) { return this.each( function() { jQuery.removeAttr( this, name ); } ); } } ); jQuery.extend( { attr: function( elem, name, value ) { var ret, hooks, nType = elem.nodeType; // Don't get/set attributes on text, comment and attribute nodes if ( nType === 3 || nType === 8 || nType === 2 ) { return; } // Fallback to prop when attributes are not supported if ( typeof elem.getAttribute === "undefined" ) { return jQuery.prop( elem, name, value ); } // Attribute hooks are determined by the lowercase version // Grab necessary hook if one is defined if ( nType !== 1 || !jQuery.isXMLDoc( elem ) ) { hooks = jQuery.attrHooks[ name.toLowerCase() ] || ( jQuery.expr.match.bool.test( name ) ? boolHook : undefined ); } if ( value !== undefined ) { if ( value === null ) { jQuery.removeAttr( elem, name ); return; } if ( hooks && "set" in hooks && ( ret = hooks.set( elem, value, name ) ) !== undefined ) { return ret; } elem.setAttribute( name, value + "" ); return value; } if ( hooks && "get" in hooks && ( ret = hooks.get( elem, name ) ) !== null ) { return ret; } ret = jQuery.find.attr( elem, name ); // Non-existent attributes return null, we normalize to undefined return ret == null ? undefined : ret; }, attrHooks: { type: { set: function( elem, value ) { if ( !support.radioValue && value === "radio" && nodeName( elem, "input" ) ) { var val = elem.value; elem.setAttribute( "type", value ); if ( val ) { elem.value = val; } return value; } } } }, removeAttr: function( elem, value ) { var name, i = 0, // Attribute names can contain non-HTML whitespace characters // https://html.spec.whatwg.org/multipage/syntax.html#attributes-2 attrNames = value && value.match( rnothtmlwhite ); if ( attrNames && elem.nodeType === 1 ) { while ( ( name = attrNames[ i++ ] ) ) { elem.removeAttribute( name ); } } } } ); // Hooks for boolean attributes boolHook = { set: function( elem, value, name ) { if ( value === false ) { // Remove boolean attributes when set to false jQuery.removeAttr( elem, name ); } else { elem.setAttribute( name, name ); } return name; } }; jQuery.each( jQuery.expr.match.bool.source.match( /\w+/g ), function( _i, name ) { var getter = attrHandle[ name ] || jQuery.find.attr; attrHandle[ name ] = function( elem, name, isXML ) { var ret, handle, lowercaseName = name.toLowerCase(); if ( !isXML ) { // Avoid an infinite loop by temporarily removing this function from the getter handle = attrHandle[ lowercaseName ]; attrHandle[ lowercaseName ] = ret; ret = getter( elem, name, isXML ) != null ? lowercaseName : null; attrHandle[ lowercaseName ] = handle; } return ret; }; } ); var rfocusable = /^(?:input|select|textarea|button)$/i, rclickable = /^(?:a|area)$/i; jQuery.fn.extend( { prop: function( name, value ) { return access( this, jQuery.prop, name, value, arguments.length > 1 ); }, removeProp: function( name ) { return this.each( function() { delete this[ jQuery.propFix[ name ] || name ]; } ); } } ); jQuery.extend( { prop: function( elem, name, value ) { var ret, hooks, nType = elem.nodeType; // Don't get/set properties on text, comment and attribute nodes if ( nType === 3 || nType === 8 || nType === 2 ) { return; } if ( nType !== 1 || !jQuery.isXMLDoc( elem ) ) { // Fix name and attach hooks name = jQuery.propFix[ name ] || name; hooks = jQuery.propHooks[ name ]; } if ( value !== undefined ) { if ( hooks && "set" in hooks && ( ret = hooks.set( elem, value, name ) ) !== undefined ) { return ret; } return ( elem[ name ] = value ); } if ( hooks && "get" in hooks && ( ret = hooks.get( elem, name ) ) !== null ) { return ret; } return elem[ name ]; }, propHooks: { tabIndex: { get: function( elem ) { // Support: IE <=9 - 11 only // elem.tabIndex doesn't always return the // correct value when it hasn't been explicitly set // https://web.archive.org/web/20141116233347/http://fluidproject.org/blog/2008/01/09/getting-setting-and-removing-tabindex-values-with-javascript/ // Use proper attribute retrieval(#12072) var tabindex = jQuery.find.attr( elem, "tabindex" ); if ( tabindex ) { return parseInt( tabindex, 10 ); } if ( rfocusable.test( elem.nodeName ) || rclickable.test( elem.nodeName ) && elem.href ) { return 0; } return -1; } } }, propFix: { "for": "htmlFor", "class": "className" } } ); // Support: IE <=11 only // Accessing the selectedIndex property // forces the browser to respect setting selected // on the option // The getter ensures a default option is selected // when in an optgroup // eslint rule "no-unused-expressions" is disabled for this code // since it considers such accessions noop if ( !support.optSelected ) { jQuery.propHooks.selected = { get: function( elem ) { /* eslint no-unused-expressions: "off" */ var parent = elem.parentNode; if ( parent && parent.parentNode ) { parent.parentNode.selectedIndex; } return null; }, set: function( elem ) { /* eslint no-unused-expressions: "off" */ var parent = elem.parentNode; if ( parent ) { parent.selectedIndex; if ( parent.parentNode ) { parent.parentNode.selectedIndex; } } } }; } jQuery.each( [ "tabIndex", "readOnly", "maxLength", "cellSpacing", "cellPadding", "rowSpan", "colSpan", "useMap", "frameBorder", "contentEditable" ], function() { jQuery.propFix[ this.toLowerCase() ] = this; } ); // Strip and collapse whitespace according to HTML spec // https://infra.spec.whatwg.org/#strip-and-collapse-ascii-whitespace function stripAndCollapse( value ) { var tokens = value.match( rnothtmlwhite ) || []; return tokens.join( " " ); } function getClass( elem ) { return elem.getAttribute && elem.getAttribute( "class" ) || ""; } function classesToArray( value ) { if ( Array.isArray( value ) ) { return value; } if ( typeof value === "string" ) { return value.match( rnothtmlwhite ) || []; } return []; } jQuery.fn.extend( { addClass: function( value ) { var classes, elem, cur, curValue, clazz, j, finalValue, i = 0; if ( isFunction( value ) ) { return this.each( function( j ) { jQuery( this ).addClass( value.call( this, j, getClass( this ) ) ); } ); } classes = classesToArray( value ); if ( classes.length ) { while ( ( elem = this[ i++ ] ) ) { curValue = getClass( elem ); cur = elem.nodeType === 1 && ( " " + stripAndCollapse( curValue ) + " " ); if ( cur ) { j = 0; while ( ( clazz = classes[ j++ ] ) ) { if ( cur.indexOf( " " + clazz + " " ) < 0 ) { cur += clazz + " "; } } // Only assign if different to avoid unneeded rendering. finalValue = stripAndCollapse( cur ); if ( curValue !== finalValue ) { elem.setAttribute( "class", finalValue ); } } } } return this; }, removeClass: function( value ) { var classes, elem, cur, curValue, clazz, j, finalValue, i = 0; if ( isFunction( value ) ) { return this.each( function( j ) { jQuery( this ).removeClass( value.call( this, j, getClass( this ) ) ); } ); } if ( !arguments.length ) { return this.attr( "class", "" ); } classes = classesToArray( value ); if ( classes.length ) { while ( ( elem = this[ i++ ] ) ) { curValue = getClass( elem ); // This expression is here for better compressibility (see addClass) cur = elem.nodeType === 1 && ( " " + stripAndCollapse( curValue ) + " " ); if ( cur ) { j = 0; while ( ( clazz = classes[ j++ ] ) ) { // Remove *all* instances while ( cur.indexOf( " " + clazz + " " ) > -1 ) { cur = cur.replace( " " + clazz + " ", " " ); } } // Only assign if different to avoid unneeded rendering. finalValue = stripAndCollapse( cur ); if ( curValue !== finalValue ) { elem.setAttribute( "class", finalValue ); } } } } return this; }, toggleClass: function( value, stateVal ) { var type = typeof value, isValidValue = type === "string" || Array.isArray( value ); if ( typeof stateVal === "boolean" && isValidValue ) { return stateVal ? this.addClass( value ) : this.removeClass( value ); } if ( isFunction( value ) ) { return this.each( function( i ) { jQuery( this ).toggleClass( value.call( this, i, getClass( this ), stateVal ), stateVal ); } ); } return this.each( function() { var className, i, self, classNames; if ( isValidValue ) { // Toggle individual class names i = 0; self = jQuery( this ); classNames = classesToArray( value ); while ( ( className = classNames[ i++ ] ) ) { // Check each className given, space separated list if ( self.hasClass( className ) ) { self.removeClass( className ); } else { self.addClass( className ); } } // Toggle whole class name } else if ( value === undefined || type === "boolean" ) { className = getClass( this ); if ( className ) { // Store className if set dataPriv.set( this, "__className__", className ); } // If the element has a class name or if we're passed `false`, // then remove the whole classname (if there was one, the above saved it). // Otherwise bring back whatever was previously saved (if anything), // falling back to the empty string if nothing was stored. if ( this.setAttribute ) { this.setAttribute( "class", className || value === false ? "" : dataPriv.get( this, "__className__" ) || "" ); } } } ); }, hasClass: function( selector ) { var className, elem, i = 0; className = " " + selector + " "; while ( ( elem = this[ i++ ] ) ) { if ( elem.nodeType === 1 && ( " " + stripAndCollapse( getClass( elem ) ) + " " ).indexOf( className ) > -1 ) { return true; } } return false; } } ); var rreturn = /\r/g; jQuery.fn.extend( { val: function( value ) { var hooks, ret, valueIsFunction, elem = this[ 0 ]; if ( !arguments.length ) { if ( elem ) { hooks = jQuery.valHooks[ elem.type ] || jQuery.valHooks[ elem.nodeName.toLowerCase() ]; if ( hooks && "get" in hooks && ( ret = hooks.get( elem, "value" ) ) !== undefined ) { return ret; } ret = elem.value; // Handle most common string cases if ( typeof ret === "string" ) { return ret.replace( rreturn, "" ); } // Handle cases where value is null/undef or number return ret == null ? "" : ret; } return; } valueIsFunction = isFunction( value ); return this.each( function( i ) { var val; if ( this.nodeType !== 1 ) { return; } if ( valueIsFunction ) { val = value.call( this, i, jQuery( this ).val() ); } else { val = value; } // Treat null/undefined as ""; convert numbers to string if ( val == null ) { val = ""; } else if ( typeof val === "number" ) { val += ""; } else if ( Array.isArray( val ) ) { val = jQuery.map( val, function( value ) { return value == null ? "" : value + ""; } ); } hooks = jQuery.valHooks[ this.type ] || jQuery.valHooks[ this.nodeName.toLowerCase() ]; // If set returns undefined, fall back to normal setting if ( !hooks || !( "set" in hooks ) || hooks.set( this, val, "value" ) === undefined ) { this.value = val; } } ); } } ); jQuery.extend( { valHooks: { option: { get: function( elem ) { var val = jQuery.find.attr( elem, "value" ); return val != null ? val : // Support: IE <=10 - 11 only // option.text throws exceptions (#14686, #14858) // Strip and collapse whitespace // https://html.spec.whatwg.org/#strip-and-collapse-whitespace stripAndCollapse( jQuery.text( elem ) ); } }, select: { get: function( elem ) { var value, option, i, options = elem.options, index = elem.selectedIndex, one = elem.type === "select-one", values = one ? null : [], max = one ? index + 1 : options.length; if ( index < 0 ) { i = max; } else { i = one ? index : 0; } // Loop through all the selected options for ( ; i < max; i++ ) { option = options[ i ]; // Support: IE <=9 only // IE8-9 doesn't update selected after form reset (#2551) if ( ( option.selected || i === index ) && // Don't return options that are disabled or in a disabled optgroup !option.disabled && ( !option.parentNode.disabled || !nodeName( option.parentNode, "optgroup" ) ) ) { // Get the specific value for the option value = jQuery( option ).val(); // We don't need an array for one selects if ( one ) { return value; } // Multi-Selects return an array values.push( value ); } } return values; }, set: function( elem, value ) { var optionSet, option, options = elem.options, values = jQuery.makeArray( value ), i = options.length; while ( i-- ) { option = options[ i ]; /* eslint-disable no-cond-assign */ if ( option.selected = jQuery.inArray( jQuery.valHooks.option.get( option ), values ) > -1 ) { optionSet = true; } /* eslint-enable no-cond-assign */ } // Force browsers to behave consistently when non-matching value is set if ( !optionSet ) { elem.selectedIndex = -1; } return values; } } } } ); // Radios and checkboxes getter/setter jQuery.each( [ "radio", "checkbox" ], function() { jQuery.valHooks[ this ] = { set: function( elem, value ) { if ( Array.isArray( value ) ) { return ( elem.checked = jQuery.inArray( jQuery( elem ).val(), value ) > -1 ); } } }; if ( !support.checkOn ) { jQuery.valHooks[ this ].get = function( elem ) { return elem.getAttribute( "value" ) === null ? "on" : elem.value; }; } } ); // Return jQuery for attributes-only inclusion support.focusin = "onfocusin" in window; var rfocusMorph = /^(?:focusinfocus|focusoutblur)$/, stopPropagationCallback = function( e ) { e.stopPropagation(); }; jQuery.extend( jQuery.event, { trigger: function( event, data, elem, onlyHandlers ) { var i, cur, tmp, bubbleType, ontype, handle, special, lastElement, eventPath = [ elem || document ], type = hasOwn.call( event, "type" ) ? event.type : event, namespaces = hasOwn.call( event, "namespace" ) ? event.namespace.split( "." ) : []; cur = lastElement = tmp = elem = elem || document; // Don't do events on text and comment nodes if ( elem.nodeType === 3 || elem.nodeType === 8 ) { return; } // focus/blur morphs to focusin/out; ensure we're not firing them right now if ( rfocusMorph.test( type + jQuery.event.triggered ) ) { return; } if ( type.indexOf( "." ) > -1 ) { // Namespaced trigger; create a regexp to match event type in handle() namespaces = type.split( "." ); type = namespaces.shift(); namespaces.sort(); } ontype = type.indexOf( ":" ) < 0 && "on" + type; // Caller can pass in a jQuery.Event object, Object, or just an event type string event = event[ jQuery.expando ] ? event : new jQuery.Event( type, typeof event === "object" && event ); // Trigger bitmask: & 1 for native handlers; & 2 for jQuery (always true) event.isTrigger = onlyHandlers ? 2 : 3; event.namespace = namespaces.join( "." ); event.rnamespace = event.namespace ? new RegExp( "(^|\\.)" + namespaces.join( "\\.(?:.*\\.|)" ) + "(\\.|$)" ) : null; // Clean up the event in case it is being reused event.result = undefined; if ( !event.target ) { event.target = elem; } // Clone any incoming data and prepend the event, creating the handler arg list data = data == null ? [ event ] : jQuery.makeArray( data, [ event ] ); // Allow special events to draw outside the lines special = jQuery.event.special[ type ] || {}; if ( !onlyHandlers && special.trigger && special.trigger.apply( elem, data ) === false ) { return; } // Determine event propagation path in advance, per W3C events spec (#9951) // Bubble up to document, then to window; watch for a global ownerDocument var (#9724) if ( !onlyHandlers && !special.noBubble && !isWindow( elem ) ) { bubbleType = special.delegateType || type; if ( !rfocusMorph.test( bubbleType + type ) ) { cur = cur.parentNode; } for ( ; cur; cur = cur.parentNode ) { eventPath.push( cur ); tmp = cur; } // Only add window if we got to document (e.g., not plain obj or detached DOM) if ( tmp === ( elem.ownerDocument || document ) ) { eventPath.push( tmp.defaultView || tmp.parentWindow || window ); } } // Fire handlers on the event path i = 0; while ( ( cur = eventPath[ i++ ] ) && !event.isPropagationStopped() ) { lastElement = cur; event.type = i > 1 ? bubbleType : special.bindType || type; // jQuery handler handle = ( dataPriv.get( cur, "events" ) || Object.create( null ) )[ event.type ] && dataPriv.get( cur, "handle" ); if ( handle ) { handle.apply( cur, data ); } // Native handler handle = ontype && cur[ ontype ]; if ( handle && handle.apply && acceptData( cur ) ) { event.result = handle.apply( cur, data ); if ( event.result === false ) { event.preventDefault(); } } } event.type = type; // If nobody prevented the default action, do it now if ( !onlyHandlers && !event.isDefaultPrevented() ) { if ( ( !special._default || special._default.apply( eventPath.pop(), data ) === false ) && acceptData( elem ) ) { // Call a native DOM method on the target with the same name as the event. // Don't do default actions on window, that's where global variables be (#6170) if ( ontype && isFunction( elem[ type ] ) && !isWindow( elem ) ) { // Don't re-trigger an onFOO event when we call its FOO() method tmp = elem[ ontype ]; if ( tmp ) { elem[ ontype ] = null; } // Prevent re-triggering of the same event, since we already bubbled it above jQuery.event.triggered = type; if ( event.isPropagationStopped() ) { lastElement.addEventListener( type, stopPropagationCallback ); } elem[ type ](); if ( event.isPropagationStopped() ) { lastElement.removeEventListener( type, stopPropagationCallback ); } jQuery.event.triggered = undefined; if ( tmp ) { elem[ ontype ] = tmp; } } } } return event.result; }, // Piggyback on a donor event to simulate a different one // Used only for `focus(in | out)` events simulate: function( type, elem, event ) { var e = jQuery.extend( new jQuery.Event(), event, { type: type, isSimulated: true } ); jQuery.event.trigger( e, null, elem ); } } ); jQuery.fn.extend( { trigger: function( type, data ) { return this.each( function() { jQuery.event.trigger( type, data, this ); } ); }, triggerHandler: function( type, data ) { var elem = this[ 0 ]; if ( elem ) { return jQuery.event.trigger( type, data, elem, true ); } } } ); // Support: Firefox <=44 // Firefox doesn't have focus(in | out) events // Related ticket - https://bugzilla.mozilla.org/show_bug.cgi?id=687787 // // Support: Chrome <=48 - 49, Safari <=9.0 - 9.1 // focus(in | out) events fire after focus & blur events, // which is spec violation - http://www.w3.org/TR/DOM-Level-3-Events/#events-focusevent-event-order // Related ticket - https://bugs.chromium.org/p/chromium/issues/detail?id=449857 if ( !support.focusin ) { jQuery.each( { focus: "focusin", blur: "focusout" }, function( orig, fix ) { // Attach a single capturing handler on the document while someone wants focusin/focusout var handler = function( event ) { jQuery.event.simulate( fix, event.target, jQuery.event.fix( event ) ); }; jQuery.event.special[ fix ] = { setup: function() { // Handle: regular nodes (via `this.ownerDocument`), window // (via `this.document`) & document (via `this`). var doc = this.ownerDocument || this.document || this, attaches = dataPriv.access( doc, fix ); if ( !attaches ) { doc.addEventListener( orig, handler, true ); } dataPriv.access( doc, fix, ( attaches || 0 ) + 1 ); }, teardown: function() { var doc = this.ownerDocument || this.document || this, attaches = dataPriv.access( doc, fix ) - 1; if ( !attaches ) { doc.removeEventListener( orig, handler, true ); dataPriv.remove( doc, fix ); } else { dataPriv.access( doc, fix, attaches ); } } }; } ); } var location = window.location; var nonce = { guid: Date.now() }; var rquery = ( /\?/ ); // Cross-browser xml parsing jQuery.parseXML = function( data ) { var xml; if ( !data || typeof data !== "string" ) { return null; } // Support: IE 9 - 11 only // IE throws on parseFromString with invalid input. try { xml = ( new window.DOMParser() ).parseFromString( data, "text/xml" ); } catch ( e ) { xml = undefined; } if ( !xml || xml.getElementsByTagName( "parsererror" ).length ) { jQuery.error( "Invalid XML: " + data ); } return xml; }; var rbracket = /\[\]$/, rCRLF = /\r?\n/g, rsubmitterTypes = /^(?:submit|button|image|reset|file)$/i, rsubmittable = /^(?:input|select|textarea|keygen)/i; function buildParams( prefix, obj, traditional, add ) { var name; if ( Array.isArray( obj ) ) { // Serialize array item. jQuery.each( obj, function( i, v ) { if ( traditional || rbracket.test( prefix ) ) { // Treat each array item as a scalar. add( prefix, v ); } else { // Item is non-scalar (array or object), encode its numeric index. buildParams( prefix + "[" + ( typeof v === "object" && v != null ? i : "" ) + "]", v, traditional, add ); } } ); } else if ( !traditional && toType( obj ) === "object" ) { // Serialize object item. for ( name in obj ) { buildParams( prefix + "[" + name + "]", obj[ name ], traditional, add ); } } else { // Serialize scalar item. add( prefix, obj ); } } // Serialize an array of form elements or a set of // key/values into a query string jQuery.param = function( a, traditional ) { var prefix, s = [], add = function( key, valueOrFunction ) { // If value is a function, invoke it and use its return value var value = isFunction( valueOrFunction ) ? valueOrFunction() : valueOrFunction; s[ s.length ] = encodeURIComponent( key ) + "=" + encodeURIComponent( value == null ? "" : value ); }; if ( a == null ) { return ""; } // If an array was passed in, assume that it is an array of form elements. if ( Array.isArray( a ) || ( a.jquery && !jQuery.isPlainObject( a ) ) ) { // Serialize the form elements jQuery.each( a, function() { add( this.name, this.value ); } ); } else { // If traditional, encode the "old" way (the way 1.3.2 or older // did it), otherwise encode params recursively. for ( prefix in a ) { buildParams( prefix, a[ prefix ], traditional, add ); } } // Return the resulting serialization return s.join( "&" ); }; jQuery.fn.extend( { serialize: function() { return jQuery.param( this.serializeArray() ); }, serializeArray: function() { return this.map( function() { // Can add propHook for "elements" to filter or add form elements var elements = jQuery.prop( this, "elements" ); return elements ? jQuery.makeArray( elements ) : this; } ) .filter( function() { var type = this.type; // Use .is( ":disabled" ) so that fieldset[disabled] works return this.name && !jQuery( this ).is( ":disabled" ) && rsubmittable.test( this.nodeName ) && !rsubmitterTypes.test( type ) && ( this.checked || !rcheckableType.test( type ) ); } ) .map( function( _i, elem ) { var val = jQuery( this ).val(); if ( val == null ) { return null; } if ( Array.isArray( val ) ) { return jQuery.map( val, function( val ) { return { name: elem.name, value: val.replace( rCRLF, "\r\n" ) }; } ); } return { name: elem.name, value: val.replace( rCRLF, "\r\n" ) }; } ).get(); } } ); var r20 = /%20/g, rhash = /#.*$/, rantiCache = /([?&])_=[^&]*/, rheaders = /^(.*?):[ \t]*([^\r\n]*)$/mg, // #7653, #8125, #8152: local protocol detection rlocalProtocol = /^(?:about|app|app-storage|.+-extension|file|res|widget):$/, rnoContent = /^(?:GET|HEAD)$/, rprotocol = /^\/\//, /* Prefilters * 1) They are useful to introduce custom dataTypes (see ajax/jsonp.js for an example) * 2) These are called: * - BEFORE asking for a transport * - AFTER param serialization (s.data is a string if s.processData is true) * 3) key is the dataType * 4) the catchall symbol "*" can be used * 5) execution will start with transport dataType and THEN continue down to "*" if needed */ prefilters = {}, /* Transports bindings * 1) key is the dataType * 2) the catchall symbol "*" can be used * 3) selection will start with transport dataType and THEN go to "*" if needed */ transports = {}, // Avoid comment-prolog char sequence (#10098); must appease lint and evade compression allTypes = "*/".concat( "*" ), // Anchor tag for parsing the document origin originAnchor = document.createElement( "a" ); originAnchor.href = location.href; // Base "constructor" for jQuery.ajaxPrefilter and jQuery.ajaxTransport function addToPrefiltersOrTransports( structure ) { // dataTypeExpression is optional and defaults to "*" return function( dataTypeExpression, func ) { if ( typeof dataTypeExpression !== "string" ) { func = dataTypeExpression; dataTypeExpression = "*"; } var dataType, i = 0, dataTypes = dataTypeExpression.toLowerCase().match( rnothtmlwhite ) || []; if ( isFunction( func ) ) { // For each dataType in the dataTypeExpression while ( ( dataType = dataTypes[ i++ ] ) ) { // Prepend if requested if ( dataType[ 0 ] === "+" ) { dataType = dataType.slice( 1 ) || "*"; ( structure[ dataType ] = structure[ dataType ] || [] ).unshift( func ); // Otherwise append } else { ( structure[ dataType ] = structure[ dataType ] || [] ).push( func ); } } } }; } // Base inspection function for prefilters and transports function inspectPrefiltersOrTransports( structure, options, originalOptions, jqXHR ) { var inspected = {}, seekingTransport = ( structure === transports ); function inspect( dataType ) { var selected; inspected[ dataType ] = true; jQuery.each( structure[ dataType ] || [], function( _, prefilterOrFactory ) { var dataTypeOrTransport = prefilterOrFactory( options, originalOptions, jqXHR ); if ( typeof dataTypeOrTransport === "string" && !seekingTransport && !inspected[ dataTypeOrTransport ] ) { options.dataTypes.unshift( dataTypeOrTransport ); inspect( dataTypeOrTransport ); return false; } else if ( seekingTransport ) { return !( selected = dataTypeOrTransport ); } } ); return selected; } return inspect( options.dataTypes[ 0 ] ) || !inspected[ "*" ] && inspect( "*" ); } // A special extend for ajax options // that takes "flat" options (not to be deep extended) // Fixes #9887 function ajaxExtend( target, src ) { var key, deep, flatOptions = jQuery.ajaxSettings.flatOptions || {}; for ( key in src ) { if ( src[ key ] !== undefined ) { ( flatOptions[ key ] ? target : ( deep || ( deep = {} ) ) )[ key ] = src[ key ]; } } if ( deep ) { jQuery.extend( true, target, deep ); } return target; } /* Handles responses to an ajax request: * - finds the right dataType (mediates between content-type and expected dataType) * - returns the corresponding response */ function ajaxHandleResponses( s, jqXHR, responses ) { var ct, type, finalDataType, firstDataType, contents = s.contents, dataTypes = s.dataTypes; // Remove auto dataType and get content-type in the process while ( dataTypes[ 0 ] === "*" ) { dataTypes.shift(); if ( ct === undefined ) { ct = s.mimeType || jqXHR.getResponseHeader( "Content-Type" ); } } // Check if we're dealing with a known content-type if ( ct ) { for ( type in contents ) { if ( contents[ type ] && contents[ type ].test( ct ) ) { dataTypes.unshift( type ); break; } } } // Check to see if we have a response for the expected dataType if ( dataTypes[ 0 ] in responses ) { finalDataType = dataTypes[ 0 ]; } else { // Try convertible dataTypes for ( type in responses ) { if ( !dataTypes[ 0 ] || s.converters[ type + " " + dataTypes[ 0 ] ] ) { finalDataType = type; break; } if ( !firstDataType ) { firstDataType = type; } } // Or just use first one finalDataType = finalDataType || firstDataType; } // If we found a dataType // We add the dataType to the list if needed // and return the corresponding response if ( finalDataType ) { if ( finalDataType !== dataTypes[ 0 ] ) { dataTypes.unshift( finalDataType ); } return responses[ finalDataType ]; } } /* Chain conversions given the request and the original response * Also sets the responseXXX fields on the jqXHR instance */ function ajaxConvert( s, response, jqXHR, isSuccess ) { var conv2, current, conv, tmp, prev, converters = {}, // Work with a copy of dataTypes in case we need to modify it for conversion dataTypes = s.dataTypes.slice(); // Create converters map with lowercased keys if ( dataTypes[ 1 ] ) { for ( conv in s.converters ) { converters[ conv.toLowerCase() ] = s.converters[ conv ]; } } current = dataTypes.shift(); // Convert to each sequential dataType while ( current ) { if ( s.responseFields[ current ] ) { jqXHR[ s.responseFields[ current ] ] = response; } // Apply the dataFilter if provided if ( !prev && isSuccess && s.dataFilter ) { response = s.dataFilter( response, s.dataType ); } prev = current; current = dataTypes.shift(); if ( current ) { // There's only work to do if current dataType is non-auto if ( current === "*" ) { current = prev; // Convert response if prev dataType is non-auto and differs from current } else if ( prev !== "*" && prev !== current ) { // Seek a direct converter conv = converters[ prev + " " + current ] || converters[ "* " + current ]; // If none found, seek a pair if ( !conv ) { for ( conv2 in converters ) { // If conv2 outputs current tmp = conv2.split( " " ); if ( tmp[ 1 ] === current ) { // If prev can be converted to accepted input conv = converters[ prev + " " + tmp[ 0 ] ] || converters[ "* " + tmp[ 0 ] ]; if ( conv ) { // Condense equivalence converters if ( conv === true ) { conv = converters[ conv2 ]; // Otherwise, insert the intermediate dataType } else if ( converters[ conv2 ] !== true ) { current = tmp[ 0 ]; dataTypes.unshift( tmp[ 1 ] ); } break; } } } } // Apply converter (if not an equivalence) if ( conv !== true ) { // Unless errors are allowed to bubble, catch and return them if ( conv && s.throws ) { response = conv( response ); } else { try { response = conv( response ); } catch ( e ) { return { state: "parsererror", error: conv ? e : "No conversion from " + prev + " to " + current }; } } } } } } return { state: "success", data: response }; } jQuery.extend( { // Counter for holding the number of active queries active: 0, // Last-Modified header cache for next request lastModified: {}, etag: {}, ajaxSettings: { url: location.href, type: "GET", isLocal: rlocalProtocol.test( location.protocol ), global: true, processData: true, async: true, contentType: "application/x-www-form-urlencoded; charset=UTF-8", /* timeout: 0, data: null, dataType: null, username: null, password: null, cache: null, throws: false, traditional: false, headers: {}, */ accepts: { "*": allTypes, text: "text/plain", html: "text/html", xml: "application/xml, text/xml", json: "application/json, text/javascript" }, contents: { xml: /\bxml\b/, html: /\bhtml/, json: /\bjson\b/ }, responseFields: { xml: "responseXML", text: "responseText", json: "responseJSON" }, // Data converters // Keys separate source (or catchall "*") and destination types with a single space converters: { // Convert anything to text "* text": String, // Text to html (true = no transformation) "text html": true, // Evaluate text as a json expression "text json": JSON.parse, // Parse text as xml "text xml": jQuery.parseXML }, // For options that shouldn't be deep extended: // you can add your own custom options here if // and when you create one that shouldn't be // deep extended (see ajaxExtend) flatOptions: { url: true, context: true } }, // Creates a full fledged settings object into target // with both ajaxSettings and settings fields. // If target is omitted, writes into ajaxSettings. ajaxSetup: function( target, settings ) { return settings ? // Building a settings object ajaxExtend( ajaxExtend( target, jQuery.ajaxSettings ), settings ) : // Extending ajaxSettings ajaxExtend( jQuery.ajaxSettings, target ); }, ajaxPrefilter: addToPrefiltersOrTransports( prefilters ), ajaxTransport: addToPrefiltersOrTransports( transports ), // Main method ajax: function( url, options ) { // If url is an object, simulate pre-1.5 signature if ( typeof url === "object" ) { options = url; url = undefined; } // Force options to be an object options = options || {}; var transport, // URL without anti-cache param cacheURL, // Response headers responseHeadersString, responseHeaders, // timeout handle timeoutTimer, // Url cleanup var urlAnchor, // Request state (becomes false upon send and true upon completion) completed, // To know if global events are to be dispatched fireGlobals, // Loop variable i, // uncached part of the url uncached, // Create the final options object s = jQuery.ajaxSetup( {}, options ), // Callbacks context callbackContext = s.context || s, // Context for global events is callbackContext if it is a DOM node or jQuery collection globalEventContext = s.context && ( callbackContext.nodeType || callbackContext.jquery ) ? jQuery( callbackContext ) : jQuery.event, // Deferreds deferred = jQuery.Deferred(), completeDeferred = jQuery.Callbacks( "once memory" ), // Status-dependent callbacks statusCode = s.statusCode || {}, // Headers (they are sent all at once) requestHeaders = {}, requestHeadersNames = {}, // Default abort message strAbort = "canceled", // Fake xhr jqXHR = { readyState: 0, // Builds headers hashtable if needed getResponseHeader: function( key ) { var match; if ( completed ) { if ( !responseHeaders ) { responseHeaders = {}; while ( ( match = rheaders.exec( responseHeadersString ) ) ) { responseHeaders[ match[ 1 ].toLowerCase() + " " ] = ( responseHeaders[ match[ 1 ].toLowerCase() + " " ] || [] ) .concat( match[ 2 ] ); } } match = responseHeaders[ key.toLowerCase() + " " ]; } return match == null ? null : match.join( ", " ); }, // Raw string getAllResponseHeaders: function() { return completed ? responseHeadersString : null; }, // Caches the header setRequestHeader: function( name, value ) { if ( completed == null ) { name = requestHeadersNames[ name.toLowerCase() ] = requestHeadersNames[ name.toLowerCase() ] || name; requestHeaders[ name ] = value; } return this; }, // Overrides response content-type header overrideMimeType: function( type ) { if ( completed == null ) { s.mimeType = type; } return this; }, // Status-dependent callbacks statusCode: function( map ) { var code; if ( map ) { if ( completed ) { // Execute the appropriate callbacks jqXHR.always( map[ jqXHR.status ] ); } else { // Lazy-add the new callbacks in a way that preserves old ones for ( code in map ) { statusCode[ code ] = [ statusCode[ code ], map[ code ] ]; } } } return this; }, // Cancel the request abort: function( statusText ) { var finalText = statusText || strAbort; if ( transport ) { transport.abort( finalText ); } done( 0, finalText ); return this; } }; // Attach deferreds deferred.promise( jqXHR ); // Add protocol if not provided (prefilters might expect it) // Handle falsy url in the settings object (#10093: consistency with old signature) // We also use the url parameter if available s.url = ( ( url || s.url || location.href ) + "" ) .replace( rprotocol, location.protocol + "//" ); // Alias method option to type as per ticket #12004 s.type = options.method || options.type || s.method || s.type; // Extract dataTypes list s.dataTypes = ( s.dataType || "*" ).toLowerCase().match( rnothtmlwhite ) || [ "" ]; // A cross-domain request is in order when the origin doesn't match the current origin. if ( s.crossDomain == null ) { urlAnchor = document.createElement( "a" ); // Support: IE <=8 - 11, Edge 12 - 15 // IE throws exception on accessing the href property if url is malformed, // e.g. http://example.com:80x/ try { urlAnchor.href = s.url; // Support: IE <=8 - 11 only // Anchor's host property isn't correctly set when s.url is relative urlAnchor.href = urlAnchor.href; s.crossDomain = originAnchor.protocol + "//" + originAnchor.host !== urlAnchor.protocol + "//" + urlAnchor.host; } catch ( e ) { // If there is an error parsing the URL, assume it is crossDomain, // it can be rejected by the transport if it is invalid s.crossDomain = true; } } // Convert data if not already a string if ( s.data && s.processData && typeof s.data !== "string" ) { s.data = jQuery.param( s.data, s.traditional ); } // Apply prefilters inspectPrefiltersOrTransports( prefilters, s, options, jqXHR ); // If request was aborted inside a prefilter, stop there if ( completed ) { return jqXHR; } // We can fire global events as of now if asked to // Don't fire events if jQuery.event is undefined in an AMD-usage scenario (#15118) fireGlobals = jQuery.event && s.global; // Watch for a new set of requests if ( fireGlobals && jQuery.active++ === 0 ) { jQuery.event.trigger( "ajaxStart" ); } // Uppercase the type s.type = s.type.toUpperCase(); // Determine if request has content s.hasContent = !rnoContent.test( s.type ); // Save the URL in case we're toying with the If-Modified-Since // and/or If-None-Match header later on // Remove hash to simplify url manipulation cacheURL = s.url.replace( rhash, "" ); // More options handling for requests with no content if ( !s.hasContent ) { // Remember the hash so we can put it back uncached = s.url.slice( cacheURL.length ); // If data is available and should be processed, append data to url if ( s.data && ( s.processData || typeof s.data === "string" ) ) { cacheURL += ( rquery.test( cacheURL ) ? "&" : "?" ) + s.data; // #9682: remove data so that it's not used in an eventual retry delete s.data; } // Add or update anti-cache param if needed if ( s.cache === false ) { cacheURL = cacheURL.replace( rantiCache, "$1" ); uncached = ( rquery.test( cacheURL ) ? "&" : "?" ) + "_=" + ( nonce.guid++ ) + uncached; } // Put hash and anti-cache on the URL that will be requested (gh-1732) s.url = cacheURL + uncached; // Change '%20' to '+' if this is encoded form body content (gh-2658) } else if ( s.data && s.processData && ( s.contentType || "" ).indexOf( "application/x-www-form-urlencoded" ) === 0 ) { s.data = s.data.replace( r20, "+" ); } // Set the If-Modified-Since and/or If-None-Match header, if in ifModified mode. if ( s.ifModified ) { if ( jQuery.lastModified[ cacheURL ] ) { jqXHR.setRequestHeader( "If-Modified-Since", jQuery.lastModified[ cacheURL ] ); } if ( jQuery.etag[ cacheURL ] ) { jqXHR.setRequestHeader( "If-None-Match", jQuery.etag[ cacheURL ] ); } } // Set the correct header, if data is being sent if ( s.data && s.hasContent && s.contentType !== false || options.contentType ) { jqXHR.setRequestHeader( "Content-Type", s.contentType ); } // Set the Accepts header for the server, depending on the dataType jqXHR.setRequestHeader( "Accept", s.dataTypes[ 0 ] && s.accepts[ s.dataTypes[ 0 ] ] ? s.accepts[ s.dataTypes[ 0 ] ] + ( s.dataTypes[ 0 ] !== "*" ? ", " + allTypes + "; q=0.01" : "" ) : s.accepts[ "*" ] ); // Check for headers option for ( i in s.headers ) { jqXHR.setRequestHeader( i, s.headers[ i ] ); } // Allow custom headers/mimetypes and early abort if ( s.beforeSend && ( s.beforeSend.call( callbackContext, jqXHR, s ) === false || completed ) ) { // Abort if not done already and return return jqXHR.abort(); } // Aborting is no longer a cancellation strAbort = "abort"; // Install callbacks on deferreds completeDeferred.add( s.complete ); jqXHR.done( s.success ); jqXHR.fail( s.error ); // Get transport transport = inspectPrefiltersOrTransports( transports, s, options, jqXHR ); // If no transport, we auto-abort if ( !transport ) { done( -1, "No Transport" ); } else { jqXHR.readyState = 1; // Send global event if ( fireGlobals ) { globalEventContext.trigger( "ajaxSend", [ jqXHR, s ] ); } // If request was aborted inside ajaxSend, stop there if ( completed ) { return jqXHR; } // Timeout if ( s.async && s.timeout > 0 ) { timeoutTimer = window.setTimeout( function() { jqXHR.abort( "timeout" ); }, s.timeout ); } try { completed = false; transport.send( requestHeaders, done ); } catch ( e ) { // Rethrow post-completion exceptions if ( completed ) { throw e; } // Propagate others as results done( -1, e ); } } // Callback for when everything is done function done( status, nativeStatusText, responses, headers ) { var isSuccess, success, error, response, modified, statusText = nativeStatusText; // Ignore repeat invocations if ( completed ) { return; } completed = true; // Clear timeout if it exists if ( timeoutTimer ) { window.clearTimeout( timeoutTimer ); } // Dereference transport for early garbage collection // (no matter how long the jqXHR object will be used) transport = undefined; // Cache response headers responseHeadersString = headers || ""; // Set readyState jqXHR.readyState = status > 0 ? 4 : 0; // Determine if successful isSuccess = status >= 200 && status < 300 || status === 304; // Get response data if ( responses ) { response = ajaxHandleResponses( s, jqXHR, responses ); } // Use a noop converter for missing script if ( !isSuccess && jQuery.inArray( "script", s.dataTypes ) > -1 ) { s.converters[ "text script" ] = function() {}; } // Convert no matter what (that way responseXXX fields are always set) response = ajaxConvert( s, response, jqXHR, isSuccess ); // If successful, handle type chaining if ( isSuccess ) { // Set the If-Modified-Since and/or If-None-Match header, if in ifModified mode. if ( s.ifModified ) { modified = jqXHR.getResponseHeader( "Last-Modified" ); if ( modified ) { jQuery.lastModified[ cacheURL ] = modified; } modified = jqXHR.getResponseHeader( "etag" ); if ( modified ) { jQuery.etag[ cacheURL ] = modified; } } // if no content if ( status === 204 || s.type === "HEAD" ) { statusText = "nocontent"; // if not modified } else if ( status === 304 ) { statusText = "notmodified"; // If we have data, let's convert it } else { statusText = response.state; success = response.data; error = response.error; isSuccess = !error; } } else { // Extract error from statusText and normalize for non-aborts error = statusText; if ( status || !statusText ) { statusText = "error"; if ( status < 0 ) { status = 0; } } } // Set data for the fake xhr object jqXHR.status = status; jqXHR.statusText = ( nativeStatusText || statusText ) + ""; // Success/Error if ( isSuccess ) { deferred.resolveWith( callbackContext, [ success, statusText, jqXHR ] ); } else { deferred.rejectWith( callbackContext, [ jqXHR, statusText, error ] ); } // Status-dependent callbacks jqXHR.statusCode( statusCode ); statusCode = undefined; if ( fireGlobals ) { globalEventContext.trigger( isSuccess ? "ajaxSuccess" : "ajaxError", [ jqXHR, s, isSuccess ? success : error ] ); } // Complete completeDeferred.fireWith( callbackContext, [ jqXHR, statusText ] ); if ( fireGlobals ) { globalEventContext.trigger( "ajaxComplete", [ jqXHR, s ] ); // Handle the global AJAX counter if ( !( --jQuery.active ) ) { jQuery.event.trigger( "ajaxStop" ); } } } return jqXHR; }, getJSON: function( url, data, callback ) { return jQuery.get( url, data, callback, "json" ); }, getScript: function( url, callback ) { return jQuery.get( url, undefined, callback, "script" ); } } ); jQuery.each( [ "get", "post" ], function( _i, method ) { jQuery[ method ] = function( url, data, callback, type ) { // Shift arguments if data argument was omitted if ( isFunction( data ) ) { type = type || callback; callback = data; data = undefined; } // The url can be an options object (which then must have .url) return jQuery.ajax( jQuery.extend( { url: url, type: method, dataType: type, data: data, success: callback }, jQuery.isPlainObject( url ) && url ) ); }; } ); jQuery.ajaxPrefilter( function( s ) { var i; for ( i in s.headers ) { if ( i.toLowerCase() === "content-type" ) { s.contentType = s.headers[ i ] || ""; } } } ); jQuery._evalUrl = function( url, options, doc ) { return jQuery.ajax( { url: url, // Make this explicit, since user can override this through ajaxSetup (#11264) type: "GET", dataType: "script", cache: true, async: false, global: false, // Only evaluate the response if it is successful (gh-4126) // dataFilter is not invoked for failure responses, so using it instead // of the default converter is kludgy but it works. converters: { "text script": function() {} }, dataFilter: function( response ) { jQuery.globalEval( response, options, doc ); } } ); }; jQuery.fn.extend( { wrapAll: function( html ) { var wrap; if ( this[ 0 ] ) { if ( isFunction( html ) ) { html = html.call( this[ 0 ] ); } // The elements to wrap the target around wrap = jQuery( html, this[ 0 ].ownerDocument ).eq( 0 ).clone( true ); if ( this[ 0 ].parentNode ) { wrap.insertBefore( this[ 0 ] ); } wrap.map( function() { var elem = this; while ( elem.firstElementChild ) { elem = elem.firstElementChild; } return elem; } ).append( this ); } return this; }, wrapInner: function( html ) { if ( isFunction( html ) ) { return this.each( function( i ) { jQuery( this ).wrapInner( html.call( this, i ) ); } ); } return this.each( function() { var self = jQuery( this ), contents = self.contents(); if ( contents.length ) { contents.wrapAll( html ); } else { self.append( html ); } } ); }, wrap: function( html ) { var htmlIsFunction = isFunction( html ); return this.each( function( i ) { jQuery( this ).wrapAll( htmlIsFunction ? html.call( this, i ) : html ); } ); }, unwrap: function( selector ) { this.parent( selector ).not( "body" ).each( function() { jQuery( this ).replaceWith( this.childNodes ); } ); return this; } } ); jQuery.expr.pseudos.hidden = function( elem ) { return !jQuery.expr.pseudos.visible( elem ); }; jQuery.expr.pseudos.visible = function( elem ) { return !!( elem.offsetWidth || elem.offsetHeight || elem.getClientRects().length ); }; jQuery.ajaxSettings.xhr = function() { try { return new window.XMLHttpRequest(); } catch ( e ) {} }; var xhrSuccessStatus = { // File protocol always yields status code 0, assume 200 0: 200, // Support: IE <=9 only // #1450: sometimes IE returns 1223 when it should be 204 1223: 204 }, xhrSupported = jQuery.ajaxSettings.xhr(); support.cors = !!xhrSupported && ( "withCredentials" in xhrSupported ); support.ajax = xhrSupported = !!xhrSupported; jQuery.ajaxTransport( function( options ) { var callback, errorCallback; // Cross domain only allowed if supported through XMLHttpRequest if ( support.cors || xhrSupported && !options.crossDomain ) { return { send: function( headers, complete ) { var i, xhr = options.xhr(); xhr.open( options.type, options.url, options.async, options.username, options.password ); // Apply custom fields if provided if ( options.xhrFields ) { for ( i in options.xhrFields ) { xhr[ i ] = options.xhrFields[ i ]; } } // Override mime type if needed if ( options.mimeType && xhr.overrideMimeType ) { xhr.overrideMimeType( options.mimeType ); } // X-Requested-With header // For cross-domain requests, seeing as conditions for a preflight are // akin to a jigsaw puzzle, we simply never set it to be sure. // (it can always be set on a per-request basis or even using ajaxSetup) // For same-domain requests, won't change header if already provided. if ( !options.crossDomain && !headers[ "X-Requested-With" ] ) { headers[ "X-Requested-With" ] = "XMLHttpRequest"; } // Set headers for ( i in headers ) { xhr.setRequestHeader( i, headers[ i ] ); } // Callback callback = function( type ) { return function() { if ( callback ) { callback = errorCallback = xhr.onload = xhr.onerror = xhr.onabort = xhr.ontimeout = xhr.onreadystatechange = null; if ( type === "abort" ) { xhr.abort(); } else if ( type === "error" ) { // Support: IE <=9 only // On a manual native abort, IE9 throws // errors on any property access that is not readyState if ( typeof xhr.status !== "number" ) { complete( 0, "error" ); } else { complete( // File: protocol always yields status 0; see #8605, #14207 xhr.status, xhr.statusText ); } } else { complete( xhrSuccessStatus[ xhr.status ] || xhr.status, xhr.statusText, // Support: IE <=9 only // IE9 has no XHR2 but throws on binary (trac-11426) // For XHR2 non-text, let the caller handle it (gh-2498) ( xhr.responseType || "text" ) !== "text" || typeof xhr.responseText !== "string" ? { binary: xhr.response } : { text: xhr.responseText }, xhr.getAllResponseHeaders() ); } } }; }; // Listen to events xhr.onload = callback(); errorCallback = xhr.onerror = xhr.ontimeout = callback( "error" ); // Support: IE 9 only // Use onreadystatechange to replace onabort // to handle uncaught aborts if ( xhr.onabort !== undefined ) { xhr.onabort = errorCallback; } else { xhr.onreadystatechange = function() { // Check readyState before timeout as it changes if ( xhr.readyState === 4 ) { // Allow onerror to be called first, // but that will not handle a native abort // Also, save errorCallback to a variable // as xhr.onerror cannot be accessed window.setTimeout( function() { if ( callback ) { errorCallback(); } } ); } }; } // Create the abort callback callback = callback( "abort" ); try { // Do send the request (this may raise an exception) xhr.send( options.hasContent && options.data || null ); } catch ( e ) { // #14683: Only rethrow if this hasn't been notified as an error yet if ( callback ) { throw e; } } }, abort: function() { if ( callback ) { callback(); } } }; } } ); // Prevent auto-execution of scripts when no explicit dataType was provided (See gh-2432) jQuery.ajaxPrefilter( function( s ) { if ( s.crossDomain ) { s.contents.script = false; } } ); // Install script dataType jQuery.ajaxSetup( { accepts: { script: "text/javascript, application/javascript, " + "application/ecmascript, application/x-ecmascript" }, contents: { script: /\b(?:java|ecma)script\b/ }, converters: { "text script": function( text ) { jQuery.globalEval( text ); return text; } } } ); // Handle cache's special case and crossDomain jQuery.ajaxPrefilter( "script", function( s ) { if ( s.cache === undefined ) { s.cache = false; } if ( s.crossDomain ) { s.type = "GET"; } } ); // Bind script tag hack transport jQuery.ajaxTransport( "script", function( s ) { // This transport only deals with cross domain or forced-by-attrs requests if ( s.crossDomain || s.scriptAttrs ) { var script, callback; return { send: function( _, complete ) { script = jQuery( "<script>" ) .attr( s.scriptAttrs || {} ) .prop( { charset: s.scriptCharset, src: s.url } ) .on( "load error", callback = function( evt ) { script.remove(); callback = null; if ( evt ) { complete( evt.type === "error" ? 404 : 200, evt.type ); } } ); // Use native DOM manipulation to avoid our domManip AJAX trickery document.head.appendChild( script[ 0 ] ); }, abort: function() { if ( callback ) { callback(); } } }; } } ); var oldCallbacks = [], rjsonp = /(=)\?(?=&|$)|\?\?/; // Default jsonp settings jQuery.ajaxSetup( { jsonp: "callback", jsonpCallback: function() { var callback = oldCallbacks.pop() || ( jQuery.expando + "_" + ( nonce.guid++ ) ); this[ callback ] = true; return callback; } } ); // Detect, normalize options and install callbacks for jsonp requests jQuery.ajaxPrefilter( "json jsonp", function( s, originalSettings, jqXHR ) { var callbackName, overwritten, responseContainer, jsonProp = s.jsonp !== false && ( rjsonp.test( s.url ) ? "url" : typeof s.data === "string" && ( s.contentType || "" ) .indexOf( "application/x-www-form-urlencoded" ) === 0 && rjsonp.test( s.data ) && "data" ); // Handle iff the expected data type is "jsonp" or we have a parameter to set if ( jsonProp || s.dataTypes[ 0 ] === "jsonp" ) { // Get callback name, remembering preexisting value associated with it callbackName = s.jsonpCallback = isFunction( s.jsonpCallback ) ? s.jsonpCallback() : s.jsonpCallback; // Insert callback into url or form data if ( jsonProp ) { s[ jsonProp ] = s[ jsonProp ].replace( rjsonp, "$1" + callbackName ); } else if ( s.jsonp !== false ) { s.url += ( rquery.test( s.url ) ? "&" : "?" ) + s.jsonp + "=" + callbackName; } // Use data converter to retrieve json after script execution s.converters[ "script json" ] = function() { if ( !responseContainer ) { jQuery.error( callbackName + " was not called" ); } return responseContainer[ 0 ]; }; // Force json dataType s.dataTypes[ 0 ] = "json"; // Install callback overwritten = window[ callbackName ]; window[ callbackName ] = function() { responseContainer = arguments; }; // Clean-up function (fires after converters) jqXHR.always( function() { // If previous value didn't exist - remove it if ( overwritten === undefined ) { jQuery( window ).removeProp( callbackName ); // Otherwise restore preexisting value } else { window[ callbackName ] = overwritten; } // Save back as free if ( s[ callbackName ] ) { // Make sure that re-using the options doesn't screw things around s.jsonpCallback = originalSettings.jsonpCallback; // Save the callback name for future use oldCallbacks.push( callbackName ); } // Call if it was a function and we have a response if ( responseContainer && isFunction( overwritten ) ) { overwritten( responseContainer[ 0 ] ); } responseContainer = overwritten = undefined; } ); // Delegate to script return "script"; } } ); // Support: Safari 8 only // In Safari 8 documents created via document.implementation.createHTMLDocument // collapse sibling forms: the second one becomes a child of the first one. // Because of that, this security measure has to be disabled in Safari 8. // https://bugs.webkit.org/show_bug.cgi?id=137337 support.createHTMLDocument = ( function() { var body = document.implementation.createHTMLDocument( "" ).body; body.innerHTML = "<form></form><form></form>"; return body.childNodes.length === 2; } )(); // Argument "data" should be string of html // context (optional): If specified, the fragment will be created in this context, // defaults to document // keepScripts (optional): If true, will include scripts passed in the html string jQuery.parseHTML = function( data, context, keepScripts ) { if ( typeof data !== "string" ) { return []; } if ( typeof context === "boolean" ) { keepScripts = context; context = false; } var base, parsed, scripts; if ( !context ) { // Stop scripts or inline event handlers from being executed immediately // by using document.implementation if ( support.createHTMLDocument ) { context = document.implementation.createHTMLDocument( "" ); // Set the base href for the created document // so any parsed elements with URLs // are based on the document's URL (gh-2965) base = context.createElement( "base" ); base.href = document.location.href; context.head.appendChild( base ); } else { context = document; } } parsed = rsingleTag.exec( data ); scripts = !keepScripts && []; // Single tag if ( parsed ) { return [ context.createElement( parsed[ 1 ] ) ]; } parsed = buildFragment( [ data ], context, scripts ); if ( scripts && scripts.length ) { jQuery( scripts ).remove(); } return jQuery.merge( [], parsed.childNodes ); }; /** * Load a url into a page */ jQuery.fn.load = function( url, params, callback ) { var selector, type, response, self = this, off = url.indexOf( " " ); if ( off > -1 ) { selector = stripAndCollapse( url.slice( off ) ); url = url.slice( 0, off ); } // If it's a function if ( isFunction( params ) ) { // We assume that it's the callback callback = params; params = undefined; // Otherwise, build a param string } else if ( params && typeof params === "object" ) { type = "POST"; } // If we have elements to modify, make the request if ( self.length > 0 ) { jQuery.ajax( { url: url, // If "type" variable is undefined, then "GET" method will be used. // Make value of this field explicit since // user can override it through ajaxSetup method type: type || "GET", dataType: "html", data: params } ).done( function( responseText ) { // Save response for use in complete callback response = arguments; self.html( selector ? // If a selector was specified, locate the right elements in a dummy div // Exclude scripts to avoid IE 'Permission Denied' errors jQuery( "<div>" ).append( jQuery.parseHTML( responseText ) ).find( selector ) : // Otherwise use the full result responseText ); // If the request succeeds, this function gets "data", "status", "jqXHR" // but they are ignored because response was set above. // If it fails, this function gets "jqXHR", "status", "error" } ).always( callback && function( jqXHR, status ) { self.each( function() { callback.apply( this, response || [ jqXHR.responseText, status, jqXHR ] ); } ); } ); } return this; }; jQuery.expr.pseudos.animated = function( elem ) { return jQuery.grep( jQuery.timers, function( fn ) { return elem === fn.elem; } ).length; }; jQuery.offset = { setOffset: function( elem, options, i ) { var curPosition, curLeft, curCSSTop, curTop, curOffset, curCSSLeft, calculatePosition, position = jQuery.css( elem, "position" ), curElem = jQuery( elem ), props = {}; // Set position first, in-case top/left are set even on static elem if ( position === "static" ) { elem.style.position = "relative"; } curOffset = curElem.offset(); curCSSTop = jQuery.css( elem, "top" ); curCSSLeft = jQuery.css( elem, "left" ); calculatePosition = ( position === "absolute" || position === "fixed" ) && ( curCSSTop + curCSSLeft ).indexOf( "auto" ) > -1; // Need to be able to calculate position if either // top or left is auto and position is either absolute or fixed if ( calculatePosition ) { curPosition = curElem.position(); curTop = curPosition.top; curLeft = curPosition.left; } else { curTop = parseFloat( curCSSTop ) || 0; curLeft = parseFloat( curCSSLeft ) || 0; } if ( isFunction( options ) ) { // Use jQuery.extend here to allow modification of coordinates argument (gh-1848) options = options.call( elem, i, jQuery.extend( {}, curOffset ) ); } if ( options.top != null ) { props.top = ( options.top - curOffset.top ) + curTop; } if ( options.left != null ) { props.left = ( options.left - curOffset.left ) + curLeft; } if ( "using" in options ) { options.using.call( elem, props ); } else { if ( typeof props.top === "number" ) { props.top += "px"; } if ( typeof props.left === "number" ) { props.left += "px"; } curElem.css( props ); } } }; jQuery.fn.extend( { // offset() relates an element's border box to the document origin offset: function( options ) { // Preserve chaining for setter if ( arguments.length ) { return options === undefined ? 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"" : ( text + "" ).replace( rtrim, "" ); }; // Register as a named AMD module, since jQuery can be concatenated with other // files that may use define, but not via a proper concatenation script that // understands anonymous AMD modules. A named AMD is safest and most robust // way to register. Lowercase jquery is used because AMD module names are // derived from file names, and jQuery is normally delivered in a lowercase // file name. Do this after creating the global so that if an AMD module wants // to call noConflict to hide this version of jQuery, it will work. // Note that for maximum portability, libraries that are not jQuery should // declare themselves as anonymous modules, and avoid setting a global if an // AMD loader is present. jQuery is a special case. 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@@ -5,12 +5,12 @@* This script contains the language-specific data used by searchtools.js, * namely the list of stopwords, stemmer, scorer and splitter. * * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS. * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */ var stopwords = ["a", "and", "are", "as", "at", "be", "but", "by", "for", "if", "in", "into", "is", "it", "near", "no", "not", "of", "on", "or", "such", "that", "the", "their", "then", "there", "these", "they", "this", "to", "was", "will", "with"]; var stopwords = ["a","and","are","as","at","be","but","by","for","if","in","into","is","it","near","no","not","of","on","or","such","that","the","their","then","there","these","they","this","to","was","will","with"]; /* Non-minified version is copied as a separate JS file, is available */
-
@@ -197,3 +197,101 @@ var Stemmer = function() {} } var splitChars = (function() { var result = {}; var singles = [96, 180, 187, 191, 215, 247, 749, 885, 903, 907, 909, 930, 1014, 1648, 1748, 1809, 2416, 2473, 2481, 2526, 2601, 2609, 2612, 2615, 2653, 2702, 2706, 2729, 2737, 2740, 2857, 2865, 2868, 2910, 2928, 2948, 2961, 2971, 2973, 3085, 3089, 3113, 3124, 3213, 3217, 3241, 3252, 3295, 3341, 3345, 3369, 3506, 3516, 3633, 3715, 3721, 3736, 3744, 3748, 3750, 3756, 3761, 3781, 3912, 4239, 4347, 4681, 4695, 4697, 4745, 4785, 4799, 4801, 4823, 4881, 5760, 5901, 5997, 6313, 7405, 8024, 8026, 8028, 8030, 8117, 8125, 8133, 8181, 8468, 8485, 8487, 8489, 8494, 8527, 11311, 11359, 11687, 11695, 11703, 11711, 11719, 11727, 11735, 12448, 12539, 43010, 43014, 43019, 43587, 43696, 43713, 64286, 64297, 64311, 64317, 64319, 64322, 64325, 65141]; var i, j, start, end; for (i = 0; i < singles.length; i++) { result[singles[i]] = true; } var ranges = [[0, 47], [58, 64], [91, 94], [123, 169], [171, 177], [182, 184], [706, 709], [722, 735], [741, 747], [751, 879], [888, 889], [894, 901], [1154, 1161], [1318, 1328], [1367, 1368], [1370, 1376], [1416, 1487], [1515, 1519], [1523, 1568], [1611, 1631], [1642, 1645], [1750, 1764], [1767, 1773], [1789, 1790], [1792, 1807], [1840, 1868], [1958, 1968], [1970, 1983], [2027, 2035], [2038, 2041], [2043, 2047], [2070, 2073], [2075, 2083], [2085, 2087], [2089, 2307], [2362, 2364], [2366, 2383], [2385, 2391], [2402, 2405], [2419, 2424], [2432, 2436], [2445, 2446], [2449, 2450], [2483, 2485], [2490, 2492], [2494, 2509], [2511, 2523], [2530, 2533], [2546, 2547], [2554, 2564], [2571, 2574], [2577, 2578], [2618, 2648], [2655, 2661], [2672, 2673], [2677, 2692], [2746, 2748], [2750, 2767], [2769, 2783], [2786, 2789], [2800, 2820], [2829, 2830], [2833, 2834], [2874, 2876], [2878, 2907], [2914, 2917], [2930, 2946], [2955, 2957], [2966, 2968], [2976, 2978], [2981, 2983], [2987, 2989], [3002, 3023], [3025, 3045], [3059, 3076], [3130, 3132], [3134, 3159], [3162, 3167], [3170, 3173], [3184, 3191], [3199, 3204], [3258, 3260], [3262, 3293], [3298, 3301], [3312, 3332], [3386, 3388], [3390, 3423], [3426, 3429], [3446, 3449], [3456, 3460], [3479, 3481], [3518, 3519], [3527, 3584], [3636, 3647], [3655, 3663], [3674, 3712], [3717, 3718], [3723, 3724], [3726, 3731], [3752, 3753], [3764, 3772], [3774, 3775], [3783, 3791], [3802, 3803], [3806, 3839], [3841, 3871], [3892, 3903], [3949, 3975], [3980, 4095], [4139, 4158], [4170, 4175], [4182, 4185], [4190, 4192], [4194, 4196], [4199, 4205], [4209, 4212], [4226, 4237], [4250, 4255], [4294, 4303], [4349, 4351], [4686, 4687], [4702, 4703], [4750, 4751], [4790, 4791], [4806, 4807], [4886, 4887], [4955, 4968], [4989, 4991], [5008, 5023], [5109, 5120], [5741, 5742], [5787, 5791], [5867, 5869], [5873, 5887], [5906, 5919], [5938, 5951], [5970, 5983], [6001, 6015], [6068, 6102], [6104, 6107], [6109, 6111], [6122, 6127], [6138, 6159], [6170, 6175], [6264, 6271], [6315, 6319], [6390, 6399], [6429, 6469], [6510, 6511], [6517, 6527], [6572, 6592], [6600, 6607], [6619, 6655], [6679, 6687], [6741, 6783], [6794, 6799], [6810, 6822], [6824, 6916], [6964, 6980], [6988, 6991], [7002, 7042], [7073, 7085], [7098, 7167], [7204, 7231], [7242, 7244], [7294, 7400], [7410, 7423], [7616, 7679], [7958, 7959], [7966, 7967], [8006, 8007], [8014, 8015], [8062, 8063], [8127, 8129], [8141, 8143], [8148, 8149], [8156, 8159], [8173, 8177], [8189, 8303], [8306, 8307], [8314, 8318], [8330, 8335], [8341, 8449], [8451, 8454], [8456, 8457], [8470, 8472], [8478, 8483], [8506, 8507], [8512, 8516], [8522, 8525], [8586, 9311], [9372, 9449], [9472, 10101], [10132, 11263], [11493, 11498], [11503, 11516], [11518, 11519], [11558, 11567], [11622, 11630], [11632, 11647], [11671, 11679], [11743, 11822], [11824, 12292], [12296, 12320], [12330, 12336], [12342, 12343], [12349, 12352], [12439, 12444], [12544, 12548], [12590, 12592], [12687, 12689], [12694, 12703], [12728, 12783], [12800, 12831], [12842, 12880], [12896, 12927], [12938, 12976], [12992, 13311], [19894, 19967], [40908, 40959], [42125, 42191], [42238, 42239], [42509, 42511], [42540, 42559], [42592, 42593], [42607, 42622], [42648, 42655], [42736, 42774], [42784, 42785], [42889, 42890], [42893, 43002], [43043, 43055], [43062, 43071], [43124, 43137], [43188, 43215], [43226, 43249], [43256, 43258], [43260, 43263], [43302, 43311], [43335, 43359], [43389, 43395], [43443, 43470], [43482, 43519], [43561, 43583], [43596, 43599], [43610, 43615], [43639, 43641], [43643, 43647], [43698, 43700], [43703, 43704], [43710, 43711], [43715, 43738], [43742, 43967], [44003, 44015], [44026, 44031], [55204, 55215], [55239, 55242], [55292, 55295], [57344, 63743], [64046, 64047], [64110, 64111], [64218, 64255], [64263, 64274], [64280, 64284], [64434, 64466], [64830, 64847], [64912, 64913], [64968, 65007], [65020, 65135], [65277, 65295], [65306, 65312], [65339, 65344], [65371, 65381], [65471, 65473], [65480, 65481], [65488, 65489], [65496, 65497]]; for (i = 0; i < ranges.length; i++) { start = ranges[i][0]; end = ranges[i][1]; for (j = start; j <= end; j++) { result[j] = true; } } return result; })(); function splitQuery(query) { var result = []; var start = -1; for (var i = 0; i < query.length; i++) { if (splitChars[query.charCodeAt(i)]) { if (start !== -1) { result.push(query.slice(start, i)); start = -1; } } else if (start === -1) { start = i; } } if (start !== -1) { result.push(query.slice(start)); } return result; }
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@@ -4,24 +4,22 @@* * Sphinx JavaScript utilities for the full-text search. * * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS. * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */ "use strict"; /** * Simple result scoring code. */ if (typeof Scorer === "undefined") { if (!Scorer) { /** * Simple result scoring code. */ var Scorer = { // Implement the following function to further tweak the score for each result // The function takes a result array [docname, title, anchor, descr, score, filename] // The function takes a result array [filename, title, anchor, descr, score] // and returns the new score. /* score: result => { const [docname, title, anchor, descr, score, filename] = result return score score: function(result) { return result[4]; }, */
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@@ -30,11 +28,9 @@ if (typeof Scorer === "undefined") {// or matches in the last dotted part of the object name objPartialMatch: 6, // Additive scores depending on the priority of the object objPrio: { 0: 15, // used to be importantResults 1: 5, // used to be objectResults 2: -5, // used to be unimportantResults }, objPrio: {0: 15, // used to be importantResults 1: 5, // used to be objectResults 2: -5}, // used to be unimportantResults // Used when the priority is not in the mapping. objPrioDefault: 0,
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@@ -43,495 +39,456 @@ if (typeof Scorer === "undefined") {partialTitle: 7, // query found in terms term: 5, partialTerm: 2, partialTerm: 2 }; } const _removeChildren = (element) => { while (element && element.lastChild) element.removeChild(element.lastChild); }; /** * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping */ const _escapeRegExp = (string) => string.replace(/[.*+\-?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string const _displayItem = (item, searchTerms) => { const docBuilder = DOCUMENTATION_OPTIONS.BUILDER; const docUrlRoot = DOCUMENTATION_OPTIONS.URL_ROOT; const docFileSuffix = DOCUMENTATION_OPTIONS.FILE_SUFFIX; const docLinkSuffix = DOCUMENTATION_OPTIONS.LINK_SUFFIX; const showSearchSummary = DOCUMENTATION_OPTIONS.SHOW_SEARCH_SUMMARY; const [docName, title, anchor, descr, score, _filename] = item; let listItem = document.createElement("li"); let requestUrl; let linkUrl; if (docBuilder === "dirhtml") { // dirhtml builder let dirname = docName + "/"; if (dirname.match(/\/index\/$/)) dirname = dirname.substring(0, dirname.length - 6); else if (dirname === "index/") dirname = ""; requestUrl = docUrlRoot + dirname; linkUrl = requestUrl; } else { // normal html builders requestUrl = docUrlRoot + docName + docFileSuffix; linkUrl = docName + docLinkSuffix; if (!splitQuery) { function splitQuery(query) { return query.split(/\s+/); } let linkEl = listItem.appendChild(document.createElement("a")); linkEl.href = linkUrl + anchor; linkEl.dataset.score = score; linkEl.innerHTML = title; if (descr) listItem.appendChild(document.createElement("span")).innerHTML = " (" + descr + ")"; else if (showSearchSummary) fetch(requestUrl) .then((responseData) => responseData.text()) .then((data) => { if (data) listItem.appendChild( Search.makeSearchSummary(data, searchTerms) ); }); Search.output.appendChild(listItem); }; const _finishSearch = (resultCount) => { Search.stopPulse(); Search.title.innerText = _("Search Results"); if (!resultCount) Search.status.innerText = Documentation.gettext( "Your search did not match any documents. Please make sure that all words are spelled correctly and that you've selected enough categories." ); else Search.status.innerText = _( `Search finished, found ${resultCount} page(s) matching the search query.` ); }; const _displayNextItem = ( results, resultCount, searchTerms ) => { // results left, load the summary and display it // this is intended to be dynamic (don't sub resultsCount) if (results.length) { _displayItem(results.pop(), searchTerms); setTimeout( () => _displayNextItem(results, resultCount, searchTerms), 5 ); } // search finished, update title and status message else _finishSearch(resultCount); }; /** * Default splitQuery function. Can be overridden in ``sphinx.search`` with a * custom function per language. * * The regular expression works by splitting the string on consecutive characters * that are not Unicode letters, numbers, underscores, or emoji characters. * This is the same as ``\W+`` in Python, preserving the surrogate pair area. */ if (typeof splitQuery === "undefined") { var splitQuery = (query) => query .split(/[^\p{Letter}\p{Number}_\p{Emoji_Presentation}]+/gu) .filter(term => term) // remove remaining empty strings } /** * Search Module */ const Search = { _index: null, _queued_query: null, _pulse_status: -1, htmlToText: (htmlString) => { const htmlElement = new DOMParser().parseFromString(htmlString, 'text/html'); htmlElement.querySelectorAll(".headerlink").forEach((el) => { el.remove() }); const docContent = htmlElement.querySelector('[role="main"]'); if (docContent !== undefined) return docContent.textContent; console.warn( "Content block not found. Sphinx search tries to obtain it via '[role=main]'. Could you check your theme or template." ); return ""; var Search = { _index : null, _queued_query : null, _pulse_status : -1, htmlToText : function(htmlString) { var virtualDocument = document.implementation.createHTMLDocument('virtual'); var htmlElement = $(htmlString, virtualDocument); htmlElement.find('.headerlink').remove(); docContent = htmlElement.find('[role=main]')[0]; if(docContent === undefined) { console.warn("Content block not found. Sphinx search tries to obtain it " + "via '[role=main]'. Could you check your theme or template."); return ""; } return docContent.textContent || docContent.innerText; }, init: () => { const query = new URLSearchParams(window.location.search).get("q"); document .querySelectorAll('input[name="q"]') .forEach((el) => (el.value = query)); if (query) Search.performSearch(query); init : function() { var params = $.getQueryParameters(); if (params.q) { var query = params.q[0]; $('input[name="q"]')[0].value = query; this.performSearch(query); } }, loadIndex: (url) => (document.body.appendChild(document.createElement("script")).src = url), loadIndex : function(url) { $.ajax({type: "GET", url: url, data: null, dataType: "script", cache: true, complete: function(jqxhr, textstatus) { if (textstatus != "success") { document.getElementById("searchindexloader").src = url; } }}); }, setIndex: (index) => { Search._index = index; if (Search._queued_query !== null) { const query = Search._queued_query; Search._queued_query = null; Search.query(query); setIndex : function(index) { var q; this._index = index; if ((q = this._queued_query) !== null) { this._queued_query = null; Search.query(q); } }, hasIndex: () => Search._index !== null, deferQuery: (query) => (Search._queued_query = query), hasIndex : function() { return this._index !== null; }, stopPulse: () => (Search._pulse_status = -1), deferQuery : function(query) { this._queued_query = query; }, startPulse: () => { if (Search._pulse_status >= 0) return; stopPulse : function() { this._pulse_status = 0; }, const pulse = () => { startPulse : function() { if (this._pulse_status >= 0) return; function pulse() { var i; Search._pulse_status = (Search._pulse_status + 1) % 4; Search.dots.innerText = ".".repeat(Search._pulse_status); if (Search._pulse_status >= 0) window.setTimeout(pulse, 500); }; var dotString = ''; for (i = 0; i < Search._pulse_status; i++) dotString += '.'; Search.dots.text(dotString); if (Search._pulse_status > -1) window.setTimeout(pulse, 500); } pulse(); }, /** * perform a search for something (or wait until index is loaded) */ performSearch: (query) => { performSearch : function(query) { // create the required interface elements const searchText = document.createElement("h2"); searchText.textContent = _("Searching"); const searchSummary = document.createElement("p"); searchSummary.classList.add("search-summary"); searchSummary.innerText = ""; const searchList = document.createElement("ul"); searchList.classList.add("search"); const out = document.getElementById("search-results"); Search.title = out.appendChild(searchText); Search.dots = Search.title.appendChild(document.createElement("span")); Search.status = out.appendChild(searchSummary); Search.output = out.appendChild(searchList); const searchProgress = document.getElementById("search-progress"); // Some themes don't use the search progress node if (searchProgress) { searchProgress.innerText = _("Preparing search..."); } Search.startPulse(); this.out = $('#search-results'); this.title = $('<h2>' + _('Searching') + '</h2>').appendTo(this.out); this.dots = $('<span></span>').appendTo(this.title); this.status = $('<p class="search-summary"> </p>').appendTo(this.out); this.output = $('<ul class="search"/>').appendTo(this.out); $('#search-progress').text(_('Preparing search...')); this.startPulse(); // index already loaded, the browser was quick! if (Search.hasIndex()) Search.query(query); else Search.deferQuery(query); if (this.hasIndex()) this.query(query); else this.deferQuery(query); }, /** * execute search (requires search index to be loaded) */ query: (query) => { const filenames = Search._index.filenames; const docNames = Search._index.docnames; const titles = Search._index.titles; const allTitles = Search._index.alltitles; const indexEntries = Search._index.indexentries; // stem the search terms and add them to the correct list const stemmer = new Stemmer(); const searchTerms = new Set(); const excludedTerms = new Set(); const highlightTerms = new Set(); const objectTerms = new Set(splitQuery(query.toLowerCase().trim())); splitQuery(query.trim()).forEach((queryTerm) => { const queryTermLower = queryTerm.toLowerCase(); // maybe skip this "word" // stopwords array is from language_data.js if ( stopwords.indexOf(queryTermLower) !== -1 || queryTerm.match(/^\d+$/) ) return; query : function(query) { var i; // stem the searchterms and add them to the correct list var stemmer = new Stemmer(); var searchterms = []; var excluded = []; var hlterms = []; var tmp = splitQuery(query); var objectterms = []; for (i = 0; i < tmp.length; i++) { if (tmp[i] !== "") { objectterms.push(tmp[i].toLowerCase()); } if ($u.indexOf(stopwords, tmp[i].toLowerCase()) != -1 || tmp[i] === "") { // skip this "word" continue; } // stem the word let word = stemmer.stemWord(queryTermLower); var word = stemmer.stemWord(tmp[i].toLowerCase()); // prevent stemmer from cutting word smaller than two chars if(word.length < 3 && tmp[i].length >= 3) { word = tmp[i]; } var toAppend; // select the correct list if (word[0] === "-") excludedTerms.add(word.substr(1)); if (word[0] == '-') { toAppend = excluded; word = word.substr(1); } else { searchTerms.add(word); highlightTerms.add(queryTermLower); toAppend = searchterms; hlterms.push(tmp[i].toLowerCase()); } }); if (SPHINX_HIGHLIGHT_ENABLED) { // set in sphinx_highlight.js localStorage.setItem("sphinx_highlight_terms", [...highlightTerms].join(" ")) // only add if not already in the list if (!$u.contains(toAppend, word)) toAppend.push(word); } var highlightstring = '?highlight=' + $.urlencode(hlterms.join(" ")); // console.debug("SEARCH: searching for:"); // console.info("required: ", [...searchTerms]); // console.info("excluded: ", [...excludedTerms]); // array of [docname, title, anchor, descr, score, filename] let results = []; _removeChildren(document.getElementById("search-progress")); const queryLower = query.toLowerCase(); for (const [title, foundTitles] of Object.entries(allTitles)) { if (title.toLowerCase().includes(queryLower) && (queryLower.length >= title.length/2)) { for (const [file, id] of foundTitles) { let score = Math.round(100 * queryLower.length / title.length) results.push([ docNames[file], titles[file] !== title ? `${titles[file]} > ${title}` : title, id !== null ? "#" + id : "", null, score, filenames[file], ]); } } } // console.debug('SEARCH: searching for:'); // console.info('required: ', searchterms); // console.info('excluded: ', excluded); // search for explicit entries in index directives for (const [entry, foundEntries] of Object.entries(indexEntries)) { if (entry.includes(queryLower) && (queryLower.length >= entry.length/2)) { for (const [file, id] of foundEntries) { let score = Math.round(100 * queryLower.length / entry.length) results.push([ docNames[file], titles[file], id ? "#" + id : "", null, score, filenames[file], ]); } } } // prepare search var terms = this._index.terms; var titleterms = this._index.titleterms; // array of [filename, title, anchor, descr, score] var results = []; $('#search-progress').empty(); // lookup as object objectTerms.forEach((term) => results.push(...Search.performObjectSearch(term, objectTerms)) ); for (i = 0; i < objectterms.length; i++) { var others = [].concat(objectterms.slice(0, i), objectterms.slice(i+1, objectterms.length)); results = results.concat(this.performObjectSearch(objectterms[i], others)); } // lookup as search terms in fulltext results.push(...Search.performTermsSearch(searchTerms, excludedTerms)); results = results.concat(this.performTermsSearch(searchterms, excluded, terms, titleterms)); // let the scorer override scores with a custom scoring function if (Scorer.score) results.forEach((item) => (item[4] = Scorer.score(item))); if (Scorer.score) { for (i = 0; i < results.length; i++) results[i][4] = Scorer.score(results[i]); } // now sort the results by score (in opposite order of appearance, since the // display function below uses pop() to retrieve items) and then // alphabetically results.sort((a, b) => { const leftScore = a[4]; const rightScore = b[4]; if (leftScore === rightScore) { results.sort(function(a, b) { var left = a[4]; var right = b[4]; if (left > right) { return 1; } else if (left < right) { return -1; } else { // same score: sort alphabetically const leftTitle = a[1].toLowerCase(); const rightTitle = b[1].toLowerCase(); if (leftTitle === rightTitle) return 0; return leftTitle > rightTitle ? -1 : 1; // inverted is intentional left = a[1].toLowerCase(); right = b[1].toLowerCase(); return (left > right) ? -1 : ((left < right) ? 1 : 0); } return leftScore > rightScore ? 1 : -1; }); // remove duplicate search results // note the reversing of results, so that in the case of duplicates, the highest-scoring entry is kept let seen = new Set(); results = results.reverse().reduce((acc, result) => { let resultStr = result.slice(0, 4).concat([result[5]]).map(v => String(v)).join(','); if (!seen.has(resultStr)) { acc.push(result); seen.add(resultStr); } return acc; }, []); results = results.reverse(); // for debugging //Search.lastresults = results.slice(); // a copy // console.info("search results:", Search.lastresults); //console.info('search results:', Search.lastresults); // print the results _displayNextItem(results, results.length, searchTerms); var resultCount = results.length; function displayNextItem() { // results left, load the summary and display it if (results.length) { var item = results.pop(); var listItem = $('<li></li>'); var requestUrl = ""; var linkUrl = ""; if (DOCUMENTATION_OPTIONS.BUILDER === 'dirhtml') { // dirhtml builder var dirname = item[0] + '/'; if (dirname.match(/\/index\/$/)) { dirname = dirname.substring(0, dirname.length-6); } else if (dirname == 'index/') { dirname = ''; } requestUrl = DOCUMENTATION_OPTIONS.URL_ROOT + dirname; linkUrl = requestUrl; } else { // normal html builders requestUrl = DOCUMENTATION_OPTIONS.URL_ROOT + item[0] + DOCUMENTATION_OPTIONS.FILE_SUFFIX; linkUrl = item[0] + DOCUMENTATION_OPTIONS.LINK_SUFFIX; } listItem.append($('<a/>').attr('href', linkUrl + highlightstring + item[2]).html(item[1])); if (item[3]) { listItem.append($('<span> (' + item[3] + ')</span>')); Search.output.append(listItem); setTimeout(function() { displayNextItem(); }, 5); } else if (DOCUMENTATION_OPTIONS.HAS_SOURCE) { $.ajax({url: requestUrl, dataType: "text", complete: function(jqxhr, textstatus) { var data = jqxhr.responseText; if (data !== '' && data !== undefined) { var summary = Search.makeSearchSummary(data, searchterms, hlterms); if (summary) { listItem.append(summary); } } Search.output.append(listItem); setTimeout(function() { displayNextItem(); }, 5); }}); } else { // no source available, just display title Search.output.append(listItem); setTimeout(function() { displayNextItem(); }, 5); } } // search finished, update title and status message else { Search.stopPulse(); Search.title.text(_('Search Results')); if (!resultCount) Search.status.text(_('Your search did not match any documents. Please make sure that all words are spelled correctly and that you\'ve selected enough categories.')); else Search.status.text(_('Search finished, found %s page(s) matching the search query.').replace('%s', resultCount)); Search.status.fadeIn(500); } } displayNextItem(); }, /** * search for object names */ performObjectSearch: (object, objectTerms) => { const filenames = Search._index.filenames; const docNames = Search._index.docnames; const objects = Search._index.objects; const objNames = Search._index.objnames; const titles = Search._index.titles; const results = []; const objectSearchCallback = (prefix, match) => { const name = match[4] const fullname = (prefix ? prefix + "." : "") + name; const fullnameLower = fullname.toLowerCase(); if (fullnameLower.indexOf(object) < 0) return; let score = 0; const parts = fullnameLower.split("."); // check for different match types: exact matches of full name or // "last name" (i.e. last dotted part) if (fullnameLower === object || parts.slice(-1)[0] === object) score += Scorer.objNameMatch; else if (parts.slice(-1)[0].indexOf(object) > -1) score += Scorer.objPartialMatch; // matches in last name const objName = objNames[match[1]][2]; const title = titles[match[0]]; // If more than one term searched for, we require other words to be // found in the name/title/description const otherTerms = new Set(objectTerms); otherTerms.delete(object); if (otherTerms.size > 0) { const haystack = `${prefix} ${name} ${objName} ${title}`.toLowerCase(); if ( [...otherTerms].some((otherTerm) => haystack.indexOf(otherTerm) < 0) ) return; performObjectSearch : function(object, otherterms) { var filenames = this._index.filenames; var docnames = this._index.docnames; var objects = this._index.objects; var objnames = this._index.objnames; var titles = this._index.titles; var i; var results = []; for (var prefix in objects) { for (var iMatch = 0; iMatch != objects[prefix].length; ++iMatch) { var match = objects[prefix][iMatch]; var name = match[4]; var fullname = (prefix ? prefix + '.' : '') + name; var fullnameLower = fullname.toLowerCase() if (fullnameLower.indexOf(object) > -1) { var score = 0; var parts = fullnameLower.split('.'); // check for different match types: exact matches of full name or // "last name" (i.e. last dotted part) if (fullnameLower == object || parts[parts.length - 1] == object) { score += Scorer.objNameMatch; // matches in last name } else if (parts[parts.length - 1].indexOf(object) > -1) { score += Scorer.objPartialMatch; } var objname = objnames[match[1]][2]; var title = titles[match[0]]; // If more than one term searched for, we require other words to be // found in the name/title/description if (otherterms.length > 0) { var haystack = (prefix + ' ' + name + ' ' + objname + ' ' + title).toLowerCase(); var allfound = true; for (i = 0; i < otherterms.length; i++) { if (haystack.indexOf(otherterms[i]) == -1) { allfound = false; break; } } if (!allfound) { continue; } } var descr = objname + _(', in ') + title; var anchor = match[3]; if (anchor === '') anchor = fullname; else if (anchor == '-') anchor = objnames[match[1]][1] + '-' + fullname; // add custom score for some objects according to scorer if (Scorer.objPrio.hasOwnProperty(match[2])) { score += Scorer.objPrio[match[2]]; } else { score += Scorer.objPrioDefault; } results.push([docnames[match[0]], fullname, '#'+anchor, descr, score, filenames[match[0]]]); } } } let anchor = match[3]; if (anchor === "") anchor = fullname; else if (anchor === "-") anchor = objNames[match[1]][1] + "-" + fullname; const descr = objName + _(", in ") + title; // add custom score for some objects according to scorer if (Scorer.objPrio.hasOwnProperty(match[2])) score += Scorer.objPrio[match[2]]; else score += Scorer.objPrioDefault; results.push([ docNames[match[0]], fullname, "#" + anchor, descr, score, filenames[match[0]], ]); }; Object.keys(objects).forEach((prefix) => objects[prefix].forEach((array) => objectSearchCallback(prefix, array) ) ); return results; }, /** * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions */ escapeRegExp : function(string) { return string.replace(/[.*+\-?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string }, /** * search for full-text terms in the index */ performTermsSearch: (searchTerms, excludedTerms) => { // prepare search const terms = Search._index.terms; const titleTerms = Search._index.titleterms; const filenames = Search._index.filenames; const docNames = Search._index.docnames; const titles = Search._index.titles; performTermsSearch : function(searchterms, excluded, terms, titleterms) { var docnames = this._index.docnames; var filenames = this._index.filenames; var titles = this._index.titles; const scoreMap = new Map(); const fileMap = new Map(); var i, j, file; var fileMap = {}; var scoreMap = {}; var results = []; // perform the search on the required terms searchTerms.forEach((word) => { const files = []; const arr = [ { files: terms[word], score: Scorer.term }, { files: titleTerms[word], score: Scorer.title }, for (i = 0; i < searchterms.length; i++) { var word = searchterms[i]; var files = []; var _o = [ {files: terms[word], score: Scorer.term}, {files: titleterms[word], score: Scorer.title} ]; // add support for partial matches if (word.length > 2) { const escapedWord = _escapeRegExp(word); Object.keys(terms).forEach((term) => { if (term.match(escapedWord) && !terms[word]) arr.push({ files: terms[term], score: Scorer.partialTerm }); }); Object.keys(titleTerms).forEach((term) => { if (term.match(escapedWord) && !titleTerms[word]) arr.push({ files: titleTerms[word], score: Scorer.partialTitle }); }); var word_regex = this.escapeRegExp(word); for (var w in terms) { if (w.match(word_regex) && !terms[word]) { _o.push({files: terms[w], score: Scorer.partialTerm}) } } for (var w in titleterms) { if (w.match(word_regex) && !titleterms[word]) { _o.push({files: titleterms[w], score: Scorer.partialTitle}) } } } // no match but word was a required one if (arr.every((record) => record.files === undefined)) return; if ($u.every(_o, function(o){return o.files === undefined;})) { break; } // found search word in contents arr.forEach((record) => { if (record.files === undefined) return; let recordFiles = record.files; if (recordFiles.length === undefined) recordFiles = [recordFiles]; files.push(...recordFiles); // set score for the word in each file recordFiles.forEach((file) => { if (!scoreMap.has(file)) scoreMap.set(file, {}); scoreMap.get(file)[word] = record.score; }); $u.each(_o, function(o) { var _files = o.files; if (_files === undefined) return if (_files.length === undefined) _files = [_files]; files = files.concat(_files); // set score for the word in each file to Scorer.term for (j = 0; j < _files.length; j++) { file = _files[j]; if (!(file in scoreMap)) scoreMap[file] = {}; scoreMap[file][word] = o.score; } }); // create the mapping files.forEach((file) => { if (fileMap.has(file) && fileMap.get(file).indexOf(word) === -1) fileMap.get(file).push(word); else fileMap.set(file, [word]); }); }); for (j = 0; j < files.length; j++) { file = files[j]; if (file in fileMap && fileMap[file].indexOf(word) === -1) fileMap[file].push(word); else fileMap[file] = [word]; } } // now check if the files don't contain excluded terms const results = []; for (const [file, wordList] of fileMap) { // check if all requirements are matched for (file in fileMap) { var valid = true; // as search terms with length < 3 are discarded const filteredTermCount = [...searchTerms].filter( (term) => term.length > 2 ).length; // check if all requirements are matched var filteredTermCount = // as search terms with length < 3 are discarded: ignore searchterms.filter(function(term){return term.length > 2}).length if ( wordList.length !== searchTerms.size && wordList.length !== filteredTermCount ) continue; fileMap[file].length != searchterms.length && fileMap[file].length != filteredTermCount ) continue; // ensure that none of the excluded terms is in the search result if ( [...excludedTerms].some( (term) => terms[term] === file || titleTerms[term] === file || (terms[term] || []).includes(file) || (titleTerms[term] || []).includes(file) ) ) break; for (i = 0; i < excluded.length; i++) { if (terms[excluded[i]] == file || titleterms[excluded[i]] == file || $u.contains(terms[excluded[i]] || [], file) || $u.contains(titleterms[excluded[i]] || [], file)) { valid = false; break; } } // select one (max) score for the file. const score = Math.max(...wordList.map((w) => scoreMap.get(file)[w])); // add result to the result list results.push([ docNames[file], titles[file], "", null, score, filenames[file], ]); // if we have still a valid result we can add it to the result list if (valid) { // select one (max) score for the file. // for better ranking, we should calculate ranking by using words statistics like basic tf-idf... var score = $u.max($u.map(fileMap[file], function(w){return scoreMap[file][w]})); results.push([docnames[file], titles[file], '', null, score, filenames[file]]); } } return results; },
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@@ -539,28 +496,34 @@ const Search = {/** * helper function to return a node containing the * search summary for a given text. keywords is a list * of stemmed words. * of stemmed words, hlwords is the list of normal, unstemmed * words. the first one is used to find the occurrence, the * latter for highlighting it. */ makeSearchSummary: (htmlText, keywords) => { const text = Search.htmlToText(htmlText); if (text === "") return null; const textLower = text.toLowerCase(); const actualStartPosition = [...keywords] .map((k) => textLower.indexOf(k.toLowerCase())) .filter((i) => i > -1) .slice(-1)[0]; const startWithContext = Math.max(actualStartPosition - 120, 0); const top = startWithContext === 0 ? "" : "..."; const tail = startWithContext + 240 < text.length ? "..." : ""; let summary = document.createElement("p"); summary.classList.add("context"); summary.textContent = top + text.substr(startWithContext, 240).trim() + tail; return summary; }, makeSearchSummary : function(htmlText, keywords, hlwords) { var text = Search.htmlToText(htmlText); if (text == "") { return null; } var textLower = text.toLowerCase(); var start = 0; $.each(keywords, function() { var i = textLower.indexOf(this.toLowerCase()); if (i > -1) start = i; }); start = Math.max(start - 120, 0); var excerpt = ((start > 0) ? '...' : '') + $.trim(text.substr(start, 240)) + ((start + 240 - text.length) ? '...' : ''); var rv = $('<p class="context"></p>').text(excerpt); $.each(hlwords, function() { rv = rv.highlightText(this, 'highlighted'); }); return rv; } }; _ready(Search.init); $(document).ready(function() { Search.init(); });
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html/_static/sphinx_highlight.js (deleted)
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@@ -1,144 +0,0 @@/* Highlighting utilities for Sphinx HTML documentation. */ "use strict"; const SPHINX_HIGHLIGHT_ENABLED = true /** * highlight a given string on a node by wrapping it in * span elements with the given class name. */ const _highlight = (node, addItems, text, className) => { if (node.nodeType === Node.TEXT_NODE) { const val = node.nodeValue; const parent = node.parentNode; const pos = val.toLowerCase().indexOf(text); if ( pos >= 0 && !parent.classList.contains(className) && !parent.classList.contains("nohighlight") ) { let span; const closestNode = parent.closest("body, svg, foreignObject"); const isInSVG = closestNode && closestNode.matches("svg"); if (isInSVG) { span = document.createElementNS("http://www.w3.org/2000/svg", "tspan"); } else { span = document.createElement("span"); span.classList.add(className); } span.appendChild(document.createTextNode(val.substr(pos, text.length))); parent.insertBefore( span, parent.insertBefore( document.createTextNode(val.substr(pos + text.length)), node.nextSibling ) ); node.nodeValue = val.substr(0, pos); if (isInSVG) { const rect = document.createElementNS( "http://www.w3.org/2000/svg", "rect" ); const bbox = parent.getBBox(); rect.x.baseVal.value = bbox.x; rect.y.baseVal.value = bbox.y; rect.width.baseVal.value = bbox.width; rect.height.baseVal.value = bbox.height; rect.setAttribute("class", className); addItems.push({ parent: parent, target: rect }); } } } else if (node.matches && !node.matches("button, select, textarea")) { node.childNodes.forEach((el) => _highlight(el, addItems, text, className)); } }; const _highlightText = (thisNode, text, className) => { let addItems = []; _highlight(thisNode, addItems, text, className); addItems.forEach((obj) => obj.parent.insertAdjacentElement("beforebegin", obj.target) ); }; /** * Small JavaScript module for the documentation. */ const SphinxHighlight = { /** * highlight the search words provided in localstorage in the text */ highlightSearchWords: () => { if (!SPHINX_HIGHLIGHT_ENABLED) return; // bail if no highlight // get and clear terms from localstorage const url = new URL(window.location); const highlight = localStorage.getItem("sphinx_highlight_terms") || url.searchParams.get("highlight") || ""; localStorage.removeItem("sphinx_highlight_terms") url.searchParams.delete("highlight"); window.history.replaceState({}, "", url); // get individual terms from highlight string const terms = highlight.toLowerCase().split(/\s+/).filter(x => x); if (terms.length === 0) return; // nothing to do // There should never be more than one element matching "div.body" const divBody = document.querySelectorAll("div.body"); const body = divBody.length ? divBody[0] : document.querySelector("body"); window.setTimeout(() => { terms.forEach((term) => _highlightText(body, term, "highlighted")); }, 10); const searchBox = document.getElementById("searchbox"); if (searchBox === null) return; searchBox.appendChild( document .createRange() .createContextualFragment( '<p class="highlight-link">' + '<a href="javascript:SphinxHighlight.hideSearchWords()">' + _("Hide Search Matches") + "</a></p>" ) ); }, /** * helper function to hide the search marks again */ hideSearchWords: () => { document .querySelectorAll("#searchbox .highlight-link") .forEach((el) => el.remove()); document .querySelectorAll("span.highlighted") .forEach((el) => el.classList.remove("highlighted")); localStorage.removeItem("sphinx_highlight_terms") }, initEscapeListener: () => { // only install a listener if it is really needed if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) return; document.addEventListener("keydown", (event) => { // bail for input elements if (BLACKLISTED_KEY_CONTROL_ELEMENTS.has(document.activeElement.tagName)) return; // bail with special keys if (event.shiftKey || event.altKey || event.ctrlKey || event.metaKey) return; if (DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS && (event.key === "Escape")) { SphinxHighlight.hideSearchWords(); event.preventDefault(); } }); }, }; _ready(SphinxHighlight.highlightSearchWords); _ready(SphinxHighlight.initEscapeListener);
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-
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@@ -0,0 +1,2042 @@(function (global, factory) { typeof exports === 'object' && typeof module !== 'undefined' ? module.exports = factory() : typeof define === 'function' && define.amd ? define('underscore', factory) : (global = typeof globalThis !== 'undefined' ? globalThis : global || self, (function () { var current = global._; var exports = global._ = factory(); exports.noConflict = function () { global._ = current; return exports; }; }())); }(this, (function () { // Underscore.js 1.13.1 // https://underscorejs.org // (c) 2009-2021 Jeremy Ashkenas, Julian Gonggrijp, and DocumentCloud and Investigative Reporters & Editors // Underscore may be freely distributed under the MIT license. // Current version. var VERSION = '1.13.1'; // Establish the root object, `window` (`self`) in the browser, `global` // on the server, or `this` in some virtual machines. We use `self` // instead of `window` for `WebWorker` support. var root = typeof self == 'object' && self.self === self && self || typeof global == 'object' && global.global === global && global || Function('return this')() || {}; // Save bytes in the minified (but not gzipped) version: var ArrayProto = Array.prototype, ObjProto = Object.prototype; var SymbolProto = typeof Symbol !== 'undefined' ? Symbol.prototype : null; // Create quick reference variables for speed access to core prototypes. var push = ArrayProto.push, slice = ArrayProto.slice, toString = ObjProto.toString, hasOwnProperty = ObjProto.hasOwnProperty; // Modern feature detection. var supportsArrayBuffer = typeof ArrayBuffer !== 'undefined', supportsDataView = typeof DataView !== 'undefined'; // All **ECMAScript 5+** native function implementations that we hope to use // are declared here. var nativeIsArray = Array.isArray, nativeKeys = Object.keys, nativeCreate = Object.create, nativeIsView = supportsArrayBuffer && ArrayBuffer.isView; // Create references to these builtin functions because we override them. var _isNaN = isNaN, _isFinite = isFinite; // Keys in IE < 9 that won't be iterated by `for key in ...` and thus missed. var hasEnumBug = !{toString: null}.propertyIsEnumerable('toString'); var nonEnumerableProps = ['valueOf', 'isPrototypeOf', 'toString', 'propertyIsEnumerable', 'hasOwnProperty', 'toLocaleString']; // The largest integer that can be represented exactly. var MAX_ARRAY_INDEX = Math.pow(2, 53) - 1; // Some functions take a variable number of arguments, or a few expected // arguments at the beginning and then a variable number of values to operate // on. This helper accumulates all remaining arguments past the function’s // argument length (or an explicit `startIndex`), into an array that becomes // the last argument. Similar to ES6’s "rest parameter". function restArguments(func, startIndex) { startIndex = startIndex == null ? func.length - 1 : +startIndex; return function() { var length = Math.max(arguments.length - startIndex, 0), rest = Array(length), index = 0; for (; index < length; index++) { rest[index] = arguments[index + startIndex]; } switch (startIndex) { case 0: return func.call(this, rest); case 1: return func.call(this, arguments[0], rest); case 2: return func.call(this, arguments[0], arguments[1], rest); } var args = Array(startIndex + 1); for (index = 0; index < startIndex; index++) { args[index] = arguments[index]; } args[startIndex] = rest; return func.apply(this, args); }; } // Is a given variable an object? function isObject(obj) { var type = typeof obj; return type === 'function' || type === 'object' && !!obj; } // Is a given value equal to null? function isNull(obj) { return obj === null; } // Is a given variable undefined? function isUndefined(obj) { return obj === void 0; } // Is a given value a boolean? function isBoolean(obj) { return obj === true || obj === false || toString.call(obj) === '[object Boolean]'; } // Is a given value a DOM element? function isElement(obj) { return !!(obj && obj.nodeType === 1); } // Internal function for creating a `toString`-based type tester. function tagTester(name) { var tag = '[object ' + name + ']'; return function(obj) { return toString.call(obj) === tag; }; } var isString = tagTester('String'); var isNumber = tagTester('Number'); var isDate = tagTester('Date'); var isRegExp = tagTester('RegExp'); var isError = tagTester('Error'); var isSymbol = tagTester('Symbol'); var isArrayBuffer = tagTester('ArrayBuffer'); var isFunction = tagTester('Function'); // Optimize `isFunction` if appropriate. Work around some `typeof` bugs in old // v8, IE 11 (#1621), Safari 8 (#1929), and PhantomJS (#2236). var nodelist = root.document && root.document.childNodes; if (typeof /./ != 'function' && typeof Int8Array != 'object' && typeof nodelist != 'function') { isFunction = function(obj) { return typeof obj == 'function' || false; }; } var isFunction$1 = isFunction; var hasObjectTag = tagTester('Object'); // In IE 10 - Edge 13, `DataView` has string tag `'[object Object]'`. // In IE 11, the most common among them, this problem also applies to // `Map`, `WeakMap` and `Set`. var hasStringTagBug = ( supportsDataView && hasObjectTag(new DataView(new ArrayBuffer(8))) ), isIE11 = (typeof Map !== 'undefined' && hasObjectTag(new Map)); var isDataView = tagTester('DataView'); // In IE 10 - Edge 13, we need a different heuristic // to determine whether an object is a `DataView`. function ie10IsDataView(obj) { return obj != null && isFunction$1(obj.getInt8) && isArrayBuffer(obj.buffer); } var isDataView$1 = (hasStringTagBug ? ie10IsDataView : isDataView); // Is a given value an array? // Delegates to ECMA5's native `Array.isArray`. var isArray = nativeIsArray || tagTester('Array'); // Internal function to check whether `key` is an own property name of `obj`. function has$1(obj, key) { return obj != null && hasOwnProperty.call(obj, key); } var isArguments = tagTester('Arguments'); // Define a fallback version of the method in browsers (ahem, IE < 9), where // there isn't any inspectable "Arguments" type. (function() { if (!isArguments(arguments)) { isArguments = function(obj) { return has$1(obj, 'callee'); }; } }()); var isArguments$1 = isArguments; // Is a given object a finite number? function isFinite$1(obj) { return !isSymbol(obj) && _isFinite(obj) && !isNaN(parseFloat(obj)); } // Is the given value `NaN`? function isNaN$1(obj) { return isNumber(obj) && _isNaN(obj); } // Predicate-generating function. Often useful outside of Underscore. function constant(value) { return function() { return value; }; } // Common internal logic for `isArrayLike` and `isBufferLike`. function createSizePropertyCheck(getSizeProperty) { return function(collection) { var sizeProperty = getSizeProperty(collection); return typeof sizeProperty == 'number' && sizeProperty >= 0 && sizeProperty <= MAX_ARRAY_INDEX; } } // Internal helper to generate a function to obtain property `key` from `obj`. function shallowProperty(key) { return function(obj) { return obj == null ? void 0 : obj[key]; }; } // Internal helper to obtain the `byteLength` property of an object. var getByteLength = shallowProperty('byteLength'); // Internal helper to determine whether we should spend extensive checks against // `ArrayBuffer` et al. var isBufferLike = createSizePropertyCheck(getByteLength); // Is a given value a typed array? var typedArrayPattern = /\[object ((I|Ui)nt(8|16|32)|Float(32|64)|Uint8Clamped|Big(I|Ui)nt64)Array\]/; function isTypedArray(obj) { // `ArrayBuffer.isView` is the most future-proof, so use it when available. // Otherwise, fall back on the above regular expression. return nativeIsView ? (nativeIsView(obj) && !isDataView$1(obj)) : isBufferLike(obj) && typedArrayPattern.test(toString.call(obj)); } var isTypedArray$1 = supportsArrayBuffer ? isTypedArray : constant(false); // Internal helper to obtain the `length` property of an object. var getLength = shallowProperty('length'); // Internal helper to create a simple lookup structure. // `collectNonEnumProps` used to depend on `_.contains`, but this led to // circular imports. `emulatedSet` is a one-off solution that only works for // arrays of strings. function emulatedSet(keys) { var hash = {}; for (var l = keys.length, i = 0; i < l; ++i) hash[keys[i]] = true; return { contains: function(key) { return hash[key]; }, push: function(key) { hash[key] = true; return keys.push(key); } }; } // Internal helper. Checks `keys` for the presence of keys in IE < 9 that won't // be iterated by `for key in ...` and thus missed. Extends `keys` in place if // needed. function collectNonEnumProps(obj, keys) { keys = emulatedSet(keys); var nonEnumIdx = nonEnumerableProps.length; var constructor = obj.constructor; var proto = isFunction$1(constructor) && constructor.prototype || ObjProto; // Constructor is a special case. var prop = 'constructor'; if (has$1(obj, prop) && !keys.contains(prop)) keys.push(prop); while (nonEnumIdx--) { prop = nonEnumerableProps[nonEnumIdx]; if (prop in obj && obj[prop] !== proto[prop] && !keys.contains(prop)) { keys.push(prop); } } } // Retrieve the names of an object's own properties. // Delegates to **ECMAScript 5**'s native `Object.keys`. function keys(obj) { if (!isObject(obj)) return []; if (nativeKeys) return nativeKeys(obj); var keys = []; for (var key in obj) if (has$1(obj, key)) keys.push(key); // Ahem, IE < 9. if (hasEnumBug) collectNonEnumProps(obj, keys); return keys; } // Is a given array, string, or object empty? // An "empty" object has no enumerable own-properties. function isEmpty(obj) { if (obj == null) return true; // Skip the more expensive `toString`-based type checks if `obj` has no // `.length`. var length = getLength(obj); if (typeof length == 'number' && ( isArray(obj) || isString(obj) || isArguments$1(obj) )) return length === 0; return getLength(keys(obj)) === 0; } // Returns whether an object has a given set of `key:value` pairs. function isMatch(object, attrs) { var _keys = keys(attrs), length = _keys.length; if (object == null) return !length; var obj = Object(object); for (var i = 0; i < length; i++) { var key = _keys[i]; if (attrs[key] !== obj[key] || !(key in obj)) return false; } return true; } // If Underscore is called as a function, it returns a wrapped object that can // be used OO-style. This wrapper holds altered versions of all functions added // through `_.mixin`. Wrapped objects may be chained. function _$1(obj) { if (obj instanceof _$1) return obj; if (!(this instanceof _$1)) return new _$1(obj); this._wrapped = obj; } _$1.VERSION = VERSION; // Extracts the result from a wrapped and chained object. _$1.prototype.value = function() { return this._wrapped; }; // Provide unwrapping proxies for some methods used in engine operations // such as arithmetic and JSON stringification. _$1.prototype.valueOf = _$1.prototype.toJSON = _$1.prototype.value; _$1.prototype.toString = function() { return String(this._wrapped); }; // Internal function to wrap or shallow-copy an ArrayBuffer, // typed array or DataView to a new view, reusing the buffer. function toBufferView(bufferSource) { return new Uint8Array( bufferSource.buffer || bufferSource, bufferSource.byteOffset || 0, getByteLength(bufferSource) ); } // We use this string twice, so give it a name for minification. var tagDataView = '[object DataView]'; // Internal recursive comparison function for `_.isEqual`. function eq(a, b, aStack, bStack) { // Identical objects are equal. `0 === -0`, but they aren't identical. // See the [Harmony `egal` proposal](https://wiki.ecmascript.org/doku.php?id=harmony:egal). if (a === b) return a !== 0 || 1 / a === 1 / b; // `null` or `undefined` only equal to itself (strict comparison). if (a == null || b == null) return false; // `NaN`s are equivalent, but non-reflexive. if (a !== a) return b !== b; // Exhaust primitive checks var type = typeof a; if (type !== 'function' && type !== 'object' && typeof b != 'object') return false; return deepEq(a, b, aStack, bStack); } // Internal recursive comparison function for `_.isEqual`. function deepEq(a, b, aStack, bStack) { // Unwrap any wrapped objects. if (a instanceof _$1) a = a._wrapped; if (b instanceof _$1) b = b._wrapped; // Compare `[[Class]]` names. var className = toString.call(a); if (className !== toString.call(b)) return false; // Work around a bug in IE 10 - Edge 13. if (hasStringTagBug && className == '[object Object]' && isDataView$1(a)) { if (!isDataView$1(b)) return false; className = tagDataView; } switch (className) { // These types are compared by value. case '[object RegExp]': // RegExps are coerced to strings for comparison (Note: '' + /a/i === '/a/i') case '[object String]': // Primitives and their corresponding object wrappers are equivalent; thus, `"5"` is // equivalent to `new String("5")`. return '' + a === '' + b; case '[object Number]': // `NaN`s are equivalent, but non-reflexive. // Object(NaN) is equivalent to NaN. if (+a !== +a) return +b !== +b; // An `egal` comparison is performed for other numeric values. return +a === 0 ? 1 / +a === 1 / b : +a === +b; case '[object Date]': case '[object Boolean]': // Coerce dates and booleans to numeric primitive values. Dates are compared by their // millisecond representations. Note that invalid dates with millisecond representations // of `NaN` are not equivalent. return +a === +b; case '[object Symbol]': return SymbolProto.valueOf.call(a) === SymbolProto.valueOf.call(b); case '[object ArrayBuffer]': case tagDataView: // Coerce to typed array so we can fall through. return deepEq(toBufferView(a), toBufferView(b), aStack, bStack); } var areArrays = className === '[object Array]'; if (!areArrays && isTypedArray$1(a)) { var byteLength = getByteLength(a); if (byteLength !== getByteLength(b)) return false; if (a.buffer === b.buffer && a.byteOffset === b.byteOffset) return true; areArrays = true; } if (!areArrays) { if (typeof a != 'object' || typeof b != 'object') return false; // Objects with different constructors are not equivalent, but `Object`s or `Array`s // from different frames are. var aCtor = a.constructor, bCtor = b.constructor; if (aCtor !== bCtor && !(isFunction$1(aCtor) && aCtor instanceof aCtor && isFunction$1(bCtor) && bCtor instanceof bCtor) && ('constructor' in a && 'constructor' in b)) { return false; } } // Assume equality for cyclic structures. The algorithm for detecting cyclic // structures is adapted from ES 5.1 section 15.12.3, abstract operation `JO`. // Initializing stack of traversed objects. // It's done here since we only need them for objects and arrays comparison. aStack = aStack || []; bStack = bStack || []; var length = aStack.length; while (length--) { // Linear search. Performance is inversely proportional to the number of // unique nested structures. if (aStack[length] === a) return bStack[length] === b; } // Add the first object to the stack of traversed objects. aStack.push(a); bStack.push(b); // Recursively compare objects and arrays. if (areArrays) { // Compare array lengths to determine if a deep comparison is necessary. length = a.length; if (length !== b.length) return false; // Deep compare the contents, ignoring non-numeric properties. while (length--) { if (!eq(a[length], b[length], aStack, bStack)) return false; } } else { // Deep compare objects. var _keys = keys(a), key; length = _keys.length; // Ensure that both objects contain the same number of properties before comparing deep equality. if (keys(b).length !== length) return false; while (length--) { // Deep compare each member key = _keys[length]; if (!(has$1(b, key) && eq(a[key], b[key], aStack, bStack))) return false; } } // Remove the first object from the stack of traversed objects. aStack.pop(); bStack.pop(); return true; } // Perform a deep comparison to check if two objects are equal. function isEqual(a, b) { return eq(a, b); } // Retrieve all the enumerable property names of an object. function allKeys(obj) { if (!isObject(obj)) return []; var keys = []; for (var key in obj) keys.push(key); // Ahem, IE < 9. if (hasEnumBug) collectNonEnumProps(obj, keys); return keys; } // Since the regular `Object.prototype.toString` type tests don't work for // some types in IE 11, we use a fingerprinting heuristic instead, based // on the methods. It's not great, but it's the best we got. // The fingerprint method lists are defined below. function ie11fingerprint(methods) { var length = getLength(methods); return function(obj) { if (obj == null) return false; // `Map`, `WeakMap` and `Set` have no enumerable keys. var keys = allKeys(obj); if (getLength(keys)) return false; for (var i = 0; i < length; i++) { if (!isFunction$1(obj[methods[i]])) return false; } // If we are testing against `WeakMap`, we need to ensure that // `obj` doesn't have a `forEach` method in order to distinguish // it from a regular `Map`. return methods !== weakMapMethods || !isFunction$1(obj[forEachName]); }; } // In the interest of compact minification, we write // each string in the fingerprints only once. var forEachName = 'forEach', hasName = 'has', commonInit = ['clear', 'delete'], mapTail = ['get', hasName, 'set']; // `Map`, `WeakMap` and `Set` each have slightly different // combinations of the above sublists. var mapMethods = commonInit.concat(forEachName, mapTail), weakMapMethods = commonInit.concat(mapTail), setMethods = ['add'].concat(commonInit, forEachName, hasName); var isMap = isIE11 ? ie11fingerprint(mapMethods) : tagTester('Map'); var isWeakMap = isIE11 ? ie11fingerprint(weakMapMethods) : tagTester('WeakMap'); var isSet = isIE11 ? ie11fingerprint(setMethods) : tagTester('Set'); var isWeakSet = tagTester('WeakSet'); // Retrieve the values of an object's properties. function values(obj) { var _keys = keys(obj); var length = _keys.length; var values = Array(length); for (var i = 0; i < length; i++) { values[i] = obj[_keys[i]]; } return values; } // Convert an object into a list of `[key, value]` pairs. // The opposite of `_.object` with one argument. function pairs(obj) { var _keys = keys(obj); var length = _keys.length; var pairs = Array(length); for (var i = 0; i < length; i++) { pairs[i] = [_keys[i], obj[_keys[i]]]; } return pairs; } // Invert the keys and values of an object. The values must be serializable. function invert(obj) { var result = {}; var _keys = keys(obj); for (var i = 0, length = _keys.length; i < length; i++) { result[obj[_keys[i]]] = _keys[i]; } return result; } // Return a sorted list of the function names available on the object. function functions(obj) { var names = []; for (var key in obj) { if (isFunction$1(obj[key])) names.push(key); } return names.sort(); } // An internal function for creating assigner functions. function createAssigner(keysFunc, defaults) { return function(obj) { var length = arguments.length; if (defaults) obj = Object(obj); if (length < 2 || obj == null) return obj; for (var index = 1; index < length; index++) { var source = arguments[index], keys = keysFunc(source), l = keys.length; for (var i = 0; i < l; i++) { var key = keys[i]; if (!defaults || obj[key] === void 0) obj[key] = source[key]; } } return obj; }; } // Extend a given object with all the properties in passed-in object(s). var extend = createAssigner(allKeys); // Assigns a given object with all the own properties in the passed-in // object(s). // (https://developer.mozilla.org/docs/Web/JavaScript/Reference/Global_Objects/Object/assign) var extendOwn = createAssigner(keys); // Fill in a given object with default properties. var defaults = createAssigner(allKeys, true); // Create a naked function reference for surrogate-prototype-swapping. function ctor() { return function(){}; } // An internal function for creating a new object that inherits from another. function baseCreate(prototype) { if (!isObject(prototype)) return {}; if (nativeCreate) return nativeCreate(prototype); var Ctor = ctor(); Ctor.prototype = prototype; var result = new Ctor; Ctor.prototype = null; return result; } // Creates an object that inherits from the given prototype object. // If additional properties are provided then they will be added to the // created object. function create(prototype, props) { var result = baseCreate(prototype); if (props) extendOwn(result, props); return result; } // Create a (shallow-cloned) duplicate of an object. function clone(obj) { if (!isObject(obj)) return obj; return isArray(obj) ? obj.slice() : extend({}, obj); } // Invokes `interceptor` with the `obj` and then returns `obj`. // The primary purpose of this method is to "tap into" a method chain, in // order to perform operations on intermediate results within the chain. function tap(obj, interceptor) { interceptor(obj); return obj; } // Normalize a (deep) property `path` to array. // Like `_.iteratee`, this function can be customized. function toPath$1(path) { return isArray(path) ? path : [path]; } _$1.toPath = toPath$1; // Internal wrapper for `_.toPath` to enable minification. // Similar to `cb` for `_.iteratee`. function toPath(path) { return _$1.toPath(path); } // Internal function to obtain a nested property in `obj` along `path`. function deepGet(obj, path) { var length = path.length; for (var i = 0; i < length; i++) { if (obj == null) return void 0; obj = obj[path[i]]; } return length ? obj : void 0; } // Get the value of the (deep) property on `path` from `object`. // If any property in `path` does not exist or if the value is // `undefined`, return `defaultValue` instead. // The `path` is normalized through `_.toPath`. function get(object, path, defaultValue) { var value = deepGet(object, toPath(path)); return isUndefined(value) ? defaultValue : value; } // Shortcut function for checking if an object has a given property directly on // itself (in other words, not on a prototype). Unlike the internal `has` // function, this public version can also traverse nested properties. function has(obj, path) { path = toPath(path); var length = path.length; for (var i = 0; i < length; i++) { var key = path[i]; if (!has$1(obj, key)) return false; obj = obj[key]; } return !!length; } // Keep the identity function around for default iteratees. function identity(value) { return value; } // Returns a predicate for checking whether an object has a given set of // `key:value` pairs. function matcher(attrs) { attrs = extendOwn({}, attrs); return function(obj) { return isMatch(obj, attrs); }; } // Creates a function that, when passed an object, will traverse that object’s // properties down the given `path`, specified as an array of keys or indices. function property(path) { path = toPath(path); return function(obj) { return deepGet(obj, path); }; } // Internal function that returns an efficient (for current engines) version // of the passed-in callback, to be repeatedly applied in other Underscore // functions. function optimizeCb(func, context, argCount) { if (context === void 0) return func; switch (argCount == null ? 3 : argCount) { case 1: return function(value) { return func.call(context, value); }; // The 2-argument case is omitted because we’re not using it. case 3: return function(value, index, collection) { return func.call(context, value, index, collection); }; case 4: return function(accumulator, value, index, collection) { return func.call(context, accumulator, value, index, collection); }; } return function() { return func.apply(context, arguments); }; } // An internal function to generate callbacks that can be applied to each // element in a collection, returning the desired result — either `_.identity`, // an arbitrary callback, a property matcher, or a property accessor. function baseIteratee(value, context, argCount) { if (value == null) return identity; if (isFunction$1(value)) return optimizeCb(value, context, argCount); if (isObject(value) && !isArray(value)) return matcher(value); return property(value); } // External wrapper for our callback generator. Users may customize // `_.iteratee` if they want additional predicate/iteratee shorthand styles. // This abstraction hides the internal-only `argCount` argument. function iteratee(value, context) { return baseIteratee(value, context, Infinity); } _$1.iteratee = iteratee; // The function we call internally to generate a callback. It invokes // `_.iteratee` if overridden, otherwise `baseIteratee`. function cb(value, context, argCount) { if (_$1.iteratee !== iteratee) return _$1.iteratee(value, context); return baseIteratee(value, context, argCount); } // Returns the results of applying the `iteratee` to each element of `obj`. // In contrast to `_.map` it returns an object. function mapObject(obj, iteratee, context) { iteratee = cb(iteratee, context); var _keys = keys(obj), length = _keys.length, results = {}; for (var index = 0; index < length; index++) { var currentKey = _keys[index]; results[currentKey] = iteratee(obj[currentKey], currentKey, obj); } return results; } // Predicate-generating function. Often useful outside of Underscore. function noop(){} // Generates a function for a given object that returns a given property. function propertyOf(obj) { if (obj == null) return noop; return function(path) { return get(obj, path); }; } // Run a function **n** times. function times(n, iteratee, context) { var accum = Array(Math.max(0, n)); iteratee = optimizeCb(iteratee, context, 1); for (var i = 0; i < n; i++) accum[i] = iteratee(i); return accum; } // Return a random integer between `min` and `max` (inclusive). function random(min, max) { if (max == null) { max = min; min = 0; } return min + Math.floor(Math.random() * (max - min + 1)); } // A (possibly faster) way to get the current timestamp as an integer. var now = Date.now || function() { return new Date().getTime(); }; // Internal helper to generate functions for escaping and unescaping strings // to/from HTML interpolation. function createEscaper(map) { var escaper = function(match) { return map[match]; }; // Regexes for identifying a key that needs to be escaped. var source = '(?:' + keys(map).join('|') + ')'; var testRegexp = RegExp(source); var replaceRegexp = RegExp(source, 'g'); return function(string) { string = string == null ? '' : '' + string; return testRegexp.test(string) ? string.replace(replaceRegexp, escaper) : string; }; } // Internal list of HTML entities for escaping. var escapeMap = { '&': '&', '<': '<', '>': '>', '"': '"', "'": ''', '`': '`' }; // Function for escaping strings to HTML interpolation. var _escape = createEscaper(escapeMap); // Internal list of HTML entities for unescaping. var unescapeMap = invert(escapeMap); // Function for unescaping strings from HTML interpolation. var _unescape = createEscaper(unescapeMap); // By default, Underscore uses ERB-style template delimiters. Change the // following template settings to use alternative delimiters. var templateSettings = _$1.templateSettings = { evaluate: /<%([\s\S]+?)%>/g, interpolate: /<%=([\s\S]+?)%>/g, escape: /<%-([\s\S]+?)%>/g }; // When customizing `_.templateSettings`, if you don't want to define an // interpolation, evaluation or escaping regex, we need one that is // guaranteed not to match. var noMatch = /(.)^/; // Certain characters need to be escaped so that they can be put into a // string literal. var escapes = { "'": "'", '\\': '\\', '\r': 'r', '\n': 'n', '\u2028': 'u2028', '\u2029': 'u2029' }; var escapeRegExp = /\\|'|\r|\n|\u2028|\u2029/g; function escapeChar(match) { return '\\' + escapes[match]; } // In order to prevent third-party code injection through // `_.templateSettings.variable`, we test it against the following regular // expression. It is intentionally a bit more liberal than just matching valid // identifiers, but still prevents possible loopholes through defaults or // destructuring assignment. var bareIdentifier = /^\s*(\w|\$)+\s*$/; // JavaScript micro-templating, similar to John Resig's implementation. // Underscore templating handles arbitrary delimiters, preserves whitespace, // and correctly escapes quotes within interpolated code. // NB: `oldSettings` only exists for backwards compatibility. function template(text, settings, oldSettings) { if (!settings && oldSettings) settings = oldSettings; settings = defaults({}, settings, _$1.templateSettings); // Combine delimiters into one regular expression via alternation. var matcher = RegExp([ (settings.escape || noMatch).source, (settings.interpolate || noMatch).source, (settings.evaluate || noMatch).source ].join('|') + '|$', 'g'); // Compile the template source, escaping string literals appropriately. var index = 0; var source = "__p+='"; text.replace(matcher, function(match, escape, interpolate, evaluate, offset) { source += text.slice(index, offset).replace(escapeRegExp, escapeChar); index = offset + match.length; if (escape) { source += "'+\n((__t=(" + escape + "))==null?'':_.escape(__t))+\n'"; } else if (interpolate) { source += "'+\n((__t=(" + interpolate + "))==null?'':__t)+\n'"; } else if (evaluate) { source += "';\n" + evaluate + "\n__p+='"; } // Adobe VMs need the match returned to produce the correct offset. return match; }); source += "';\n"; var argument = settings.variable; if (argument) { // Insure against third-party code injection. (CVE-2021-23358) if (!bareIdentifier.test(argument)) throw new Error( 'variable is not a bare identifier: ' + argument ); } else { // If a variable is not specified, place data values in local scope. source = 'with(obj||{}){\n' + source + '}\n'; argument = 'obj'; } source = "var __t,__p='',__j=Array.prototype.join," + "print=function(){__p+=__j.call(arguments,'');};\n" + source + 'return __p;\n'; var render; try { render = new Function(argument, '_', source); } catch (e) { e.source = source; throw e; } var template = function(data) { return render.call(this, data, _$1); }; // Provide the compiled source as a convenience for precompilation. template.source = 'function(' + argument + '){\n' + source + '}'; return template; } // Traverses the children of `obj` along `path`. If a child is a function, it // is invoked with its parent as context. Returns the value of the final // child, or `fallback` if any child is undefined. function result(obj, path, fallback) { path = toPath(path); var length = path.length; if (!length) { return isFunction$1(fallback) ? fallback.call(obj) : fallback; } for (var i = 0; i < length; i++) { var prop = obj == null ? void 0 : obj[path[i]]; if (prop === void 0) { prop = fallback; i = length; // Ensure we don't continue iterating. } obj = isFunction$1(prop) ? prop.call(obj) : prop; } return obj; } // Generate a unique integer id (unique within the entire client session). // Useful for temporary DOM ids. var idCounter = 0; function uniqueId(prefix) { var id = ++idCounter + ''; return prefix ? prefix + id : id; } // Start chaining a wrapped Underscore object. function chain(obj) { var instance = _$1(obj); instance._chain = true; return instance; } // Internal function to execute `sourceFunc` bound to `context` with optional // `args`. Determines whether to execute a function as a constructor or as a // normal function. function executeBound(sourceFunc, boundFunc, context, callingContext, args) { if (!(callingContext instanceof boundFunc)) return sourceFunc.apply(context, args); var self = baseCreate(sourceFunc.prototype); var result = sourceFunc.apply(self, args); if (isObject(result)) return result; return self; } // Partially apply a function by creating a version that has had some of its // arguments pre-filled, without changing its dynamic `this` context. `_` acts // as a placeholder by default, allowing any combination of arguments to be // pre-filled. Set `_.partial.placeholder` for a custom placeholder argument. var partial = restArguments(function(func, boundArgs) { var placeholder = partial.placeholder; var bound = function() { var position = 0, length = boundArgs.length; var args = Array(length); for (var i = 0; i < length; i++) { args[i] = boundArgs[i] === placeholder ? arguments[position++] : boundArgs[i]; } while (position < arguments.length) args.push(arguments[position++]); return executeBound(func, bound, this, this, args); }; return bound; }); partial.placeholder = _$1; // Create a function bound to a given object (assigning `this`, and arguments, // optionally). var bind = restArguments(function(func, context, args) { if (!isFunction$1(func)) throw new TypeError('Bind must be called on a function'); var bound = restArguments(function(callArgs) { return executeBound(func, bound, context, this, args.concat(callArgs)); }); return bound; }); // Internal helper for collection methods to determine whether a collection // should be iterated as an array or as an object. // Related: https://people.mozilla.org/~jorendorff/es6-draft.html#sec-tolength // Avoids a very nasty iOS 8 JIT bug on ARM-64. #2094 var isArrayLike = createSizePropertyCheck(getLength); // Internal implementation of a recursive `flatten` function. function flatten$1(input, depth, strict, output) { output = output || []; if (!depth && depth !== 0) { depth = Infinity; } else if (depth <= 0) { return output.concat(input); } var idx = output.length; for (var i = 0, length = getLength(input); i < length; i++) { var value = input[i]; if (isArrayLike(value) && (isArray(value) || isArguments$1(value))) { // Flatten current level of array or arguments object. if (depth > 1) { flatten$1(value, depth - 1, strict, output); idx = output.length; } else { var j = 0, len = value.length; while (j < len) output[idx++] = value[j++]; } } else if (!strict) { output[idx++] = value; } } return output; } // Bind a number of an object's methods to that object. Remaining arguments // are the method names to be bound. Useful for ensuring that all callbacks // defined on an object belong to it. var bindAll = restArguments(function(obj, keys) { keys = flatten$1(keys, false, false); var index = keys.length; if (index < 1) throw new Error('bindAll must be passed function names'); while (index--) { var key = keys[index]; obj[key] = bind(obj[key], obj); } return obj; }); // Memoize an expensive function by storing its results. function memoize(func, hasher) { var memoize = function(key) { var cache = memoize.cache; var address = '' + (hasher ? hasher.apply(this, arguments) : key); if (!has$1(cache, address)) cache[address] = func.apply(this, arguments); return cache[address]; }; memoize.cache = {}; return memoize; } // Delays a function for the given number of milliseconds, and then calls // it with the arguments supplied. var delay = restArguments(function(func, wait, args) { return setTimeout(function() { return func.apply(null, args); }, wait); }); // Defers a function, scheduling it to run after the current call stack has // cleared. var defer = partial(delay, _$1, 1); // Returns a function, that, when invoked, will only be triggered at most once // during a given window of time. Normally, the throttled function will run // as much as it can, without ever going more than once per `wait` duration; // but if you'd like to disable the execution on the leading edge, pass // `{leading: false}`. To disable execution on the trailing edge, ditto. function throttle(func, wait, options) { var timeout, context, args, result; var previous = 0; if (!options) options = {}; var later = function() { previous = options.leading === false ? 0 : now(); timeout = null; result = func.apply(context, args); if (!timeout) context = args = null; }; var throttled = function() { var _now = now(); if (!previous && options.leading === false) previous = _now; var remaining = wait - (_now - previous); context = this; args = arguments; if (remaining <= 0 || remaining > wait) { if (timeout) { clearTimeout(timeout); timeout = null; } previous = _now; result = func.apply(context, args); if (!timeout) context = args = null; } else if (!timeout && options.trailing !== false) { timeout = setTimeout(later, remaining); } return result; }; throttled.cancel = function() { clearTimeout(timeout); previous = 0; timeout = context = args = null; }; return throttled; } // When a sequence of calls of the returned function ends, the argument // function is triggered. The end of a sequence is defined by the `wait` // parameter. If `immediate` is passed, the argument function will be // triggered at the beginning of the sequence instead of at the end. function debounce(func, wait, immediate) { var timeout, previous, args, result, context; var later = function() { var passed = now() - previous; if (wait > passed) { timeout = setTimeout(later, wait - passed); } else { timeout = null; if (!immediate) result = func.apply(context, args); // This check is needed because `func` can recursively invoke `debounced`. if (!timeout) args = context = null; } }; var debounced = restArguments(function(_args) { context = this; args = _args; previous = now(); if (!timeout) { timeout = setTimeout(later, wait); if (immediate) result = func.apply(context, args); } return result; }); debounced.cancel = function() { clearTimeout(timeout); timeout = args = context = null; }; return debounced; } // Returns the first function passed as an argument to the second, // allowing you to adjust arguments, run code before and after, and // conditionally execute the original function. function wrap(func, wrapper) { return partial(wrapper, func); } // Returns a negated version of the passed-in predicate. function negate(predicate) { return function() { return !predicate.apply(this, arguments); }; } // Returns a function that is the composition of a list of functions, each // consuming the return value of the function that follows. function compose() { var args = arguments; var start = args.length - 1; return function() { var i = start; var result = args[start].apply(this, arguments); while (i--) result = args[i].call(this, result); return result; }; } // Returns a function that will only be executed on and after the Nth call. function after(times, func) { return function() { if (--times < 1) { return func.apply(this, arguments); } }; } // Returns a function that will only be executed up to (but not including) the // Nth call. function before(times, func) { var memo; return function() { if (--times > 0) { memo = func.apply(this, arguments); } if (times <= 1) func = null; return memo; }; } // Returns a function that will be executed at most one time, no matter how // often you call it. Useful for lazy initialization. var once = partial(before, 2); // Returns the first key on an object that passes a truth test. function findKey(obj, predicate, context) { predicate = cb(predicate, context); var _keys = keys(obj), key; for (var i = 0, length = _keys.length; i < length; i++) { key = _keys[i]; if (predicate(obj[key], key, obj)) return key; } } // Internal function to generate `_.findIndex` and `_.findLastIndex`. function createPredicateIndexFinder(dir) { return function(array, predicate, context) { predicate = cb(predicate, context); var length = getLength(array); var index = dir > 0 ? 0 : length - 1; for (; index >= 0 && index < length; index += dir) { if (predicate(array[index], index, array)) return index; } return -1; }; } // Returns the first index on an array-like that passes a truth test. var findIndex = createPredicateIndexFinder(1); // Returns the last index on an array-like that passes a truth test. var findLastIndex = createPredicateIndexFinder(-1); // Use a comparator function to figure out the smallest index at which // an object should be inserted so as to maintain order. Uses binary search. function sortedIndex(array, obj, iteratee, context) { iteratee = cb(iteratee, context, 1); var value = iteratee(obj); var low = 0, high = getLength(array); while (low < high) { var mid = Math.floor((low + high) / 2); if (iteratee(array[mid]) < value) low = mid + 1; else high = mid; } return low; } // Internal function to generate the `_.indexOf` and `_.lastIndexOf` functions. function createIndexFinder(dir, predicateFind, sortedIndex) { return function(array, item, idx) { var i = 0, length = getLength(array); if (typeof idx == 'number') { if (dir > 0) { i = idx >= 0 ? idx : Math.max(idx + length, i); } else { length = idx >= 0 ? Math.min(idx + 1, length) : idx + length + 1; } } else if (sortedIndex && idx && length) { idx = sortedIndex(array, item); return array[idx] === item ? idx : -1; } if (item !== item) { idx = predicateFind(slice.call(array, i, length), isNaN$1); return idx >= 0 ? idx + i : -1; } for (idx = dir > 0 ? i : length - 1; idx >= 0 && idx < length; idx += dir) { if (array[idx] === item) return idx; } return -1; }; } // Return the position of the first occurrence of an item in an array, // or -1 if the item is not included in the array. // If the array is large and already in sort order, pass `true` // for **isSorted** to use binary search. var indexOf = createIndexFinder(1, findIndex, sortedIndex); // Return the position of the last occurrence of an item in an array, // or -1 if the item is not included in the array. var lastIndexOf = createIndexFinder(-1, findLastIndex); // Return the first value which passes a truth test. function find(obj, predicate, context) { var keyFinder = isArrayLike(obj) ? findIndex : findKey; var key = keyFinder(obj, predicate, context); if (key !== void 0 && key !== -1) return obj[key]; } // Convenience version of a common use case of `_.find`: getting the first // object containing specific `key:value` pairs. function findWhere(obj, attrs) { return find(obj, matcher(attrs)); } // The cornerstone for collection functions, an `each` // implementation, aka `forEach`. // Handles raw objects in addition to array-likes. Treats all // sparse array-likes as if they were dense. function each(obj, iteratee, context) { iteratee = optimizeCb(iteratee, context); var i, length; if (isArrayLike(obj)) { for (i = 0, length = obj.length; i < length; i++) { iteratee(obj[i], i, obj); } } else { var _keys = keys(obj); for (i = 0, length = _keys.length; i < length; i++) { iteratee(obj[_keys[i]], _keys[i], obj); } } return obj; } // Return the results of applying the iteratee to each element. function map(obj, iteratee, context) { iteratee = cb(iteratee, context); var _keys = !isArrayLike(obj) && keys(obj), length = (_keys || obj).length, results = Array(length); for (var index = 0; index < length; index++) { var currentKey = _keys ? _keys[index] : index; results[index] = iteratee(obj[currentKey], currentKey, obj); } return results; } // Internal helper to create a reducing function, iterating left or right. function createReduce(dir) { // Wrap code that reassigns argument variables in a separate function than // the one that accesses `arguments.length` to avoid a perf hit. (#1991) var reducer = function(obj, iteratee, memo, initial) { var _keys = !isArrayLike(obj) && keys(obj), length = (_keys || obj).length, index = dir > 0 ? 0 : length - 1; if (!initial) { memo = obj[_keys ? _keys[index] : index]; index += dir; } for (; index >= 0 && index < length; index += dir) { var currentKey = _keys ? _keys[index] : index; memo = iteratee(memo, obj[currentKey], currentKey, obj); } return memo; }; return function(obj, iteratee, memo, context) { var initial = arguments.length >= 3; return reducer(obj, optimizeCb(iteratee, context, 4), memo, initial); }; } // **Reduce** builds up a single result from a list of values, aka `inject`, // or `foldl`. var reduce = createReduce(1); // The right-associative version of reduce, also known as `foldr`. var reduceRight = createReduce(-1); // Return all the elements that pass a truth test. function filter(obj, predicate, context) { var results = []; predicate = cb(predicate, context); each(obj, function(value, index, list) { if (predicate(value, index, list)) results.push(value); }); return results; } // Return all the elements for which a truth test fails. function reject(obj, predicate, context) { return filter(obj, negate(cb(predicate)), context); } // Determine whether all of the elements pass a truth test. function every(obj, predicate, context) { predicate = cb(predicate, context); var _keys = !isArrayLike(obj) && keys(obj), length = (_keys || obj).length; for (var index = 0; index < length; index++) { var currentKey = _keys ? _keys[index] : index; if (!predicate(obj[currentKey], currentKey, obj)) return false; } return true; } // Determine if at least one element in the object passes a truth test. function some(obj, predicate, context) { predicate = cb(predicate, context); var _keys = !isArrayLike(obj) && keys(obj), length = (_keys || obj).length; for (var index = 0; index < length; index++) { var currentKey = _keys ? _keys[index] : index; if (predicate(obj[currentKey], currentKey, obj)) return true; } return false; } // Determine if the array or object contains a given item (using `===`). function contains(obj, item, fromIndex, guard) { if (!isArrayLike(obj)) obj = values(obj); if (typeof fromIndex != 'number' || guard) fromIndex = 0; return indexOf(obj, item, fromIndex) >= 0; } // Invoke a method (with arguments) on every item in a collection. var invoke = restArguments(function(obj, path, args) { var contextPath, func; if (isFunction$1(path)) { func = path; } else { path = toPath(path); contextPath = path.slice(0, -1); path = path[path.length - 1]; } return map(obj, function(context) { var method = func; if (!method) { if (contextPath && contextPath.length) { context = deepGet(context, contextPath); } if (context == null) return void 0; method = context[path]; } return method == null ? method : method.apply(context, args); }); }); // Convenience version of a common use case of `_.map`: fetching a property. function pluck(obj, key) { return map(obj, property(key)); } // Convenience version of a common use case of `_.filter`: selecting only // objects containing specific `key:value` pairs. function where(obj, attrs) { return filter(obj, matcher(attrs)); } // Return the maximum element (or element-based computation). function max(obj, iteratee, context) { var result = -Infinity, lastComputed = -Infinity, value, computed; if (iteratee == null || typeof iteratee == 'number' && typeof obj[0] != 'object' && obj != null) { obj = isArrayLike(obj) ? obj : values(obj); for (var i = 0, length = obj.length; i < length; i++) { value = obj[i]; if (value != null && value > result) { result = value; } } } else { iteratee = cb(iteratee, context); each(obj, function(v, index, list) { computed = iteratee(v, index, list); if (computed > lastComputed || computed === -Infinity && result === -Infinity) { result = v; lastComputed = computed; } }); } return result; } // Return the minimum element (or element-based computation). function min(obj, iteratee, context) { var result = Infinity, lastComputed = Infinity, value, computed; if (iteratee == null || typeof iteratee == 'number' && typeof obj[0] != 'object' && obj != null) { obj = isArrayLike(obj) ? obj : values(obj); for (var i = 0, length = obj.length; i < length; i++) { value = obj[i]; if (value != null && value < result) { result = value; } } } else { iteratee = cb(iteratee, context); each(obj, function(v, index, list) { computed = iteratee(v, index, list); if (computed < lastComputed || computed === Infinity && result === Infinity) { result = v; lastComputed = computed; } }); } return result; } // Sample **n** random values from a collection using the modern version of the // [Fisher-Yates shuffle](https://en.wikipedia.org/wiki/Fisher–Yates_shuffle). // If **n** is not specified, returns a single random element. // The internal `guard` argument allows it to work with `_.map`. function sample(obj, n, guard) { if (n == null || guard) { if (!isArrayLike(obj)) obj = values(obj); return obj[random(obj.length - 1)]; } var sample = isArrayLike(obj) ? clone(obj) : values(obj); var length = getLength(sample); n = Math.max(Math.min(n, length), 0); var last = length - 1; for (var index = 0; index < n; index++) { var rand = random(index, last); var temp = sample[index]; sample[index] = sample[rand]; sample[rand] = temp; } return sample.slice(0, n); } // Shuffle a collection. function shuffle(obj) { return sample(obj, Infinity); } // Sort the object's values by a criterion produced by an iteratee. function sortBy(obj, iteratee, context) { var index = 0; iteratee = cb(iteratee, context); return pluck(map(obj, function(value, key, list) { return { value: value, index: index++, criteria: iteratee(value, key, list) }; }).sort(function(left, right) { var a = left.criteria; var b = right.criteria; if (a !== b) { if (a > b || a === void 0) return 1; if (a < b || b === void 0) return -1; } return left.index - right.index; }), 'value'); } // An internal function used for aggregate "group by" operations. function group(behavior, partition) { return function(obj, iteratee, context) { var result = partition ? [[], []] : {}; iteratee = cb(iteratee, context); each(obj, function(value, index) { var key = iteratee(value, index, obj); behavior(result, value, key); }); return result; }; } // Groups the object's values by a criterion. Pass either a string attribute // to group by, or a function that returns the criterion. var groupBy = group(function(result, value, key) { if (has$1(result, key)) result[key].push(value); else result[key] = [value]; }); // Indexes the object's values by a criterion, similar to `_.groupBy`, but for // when you know that your index values will be unique. var indexBy = group(function(result, value, key) { result[key] = value; }); // Counts instances of an object that group by a certain criterion. Pass // either a string attribute to count by, or a function that returns the // criterion. var countBy = group(function(result, value, key) { if (has$1(result, key)) result[key]++; else result[key] = 1; }); // Split a collection into two arrays: one whose elements all pass the given // truth test, and one whose elements all do not pass the truth test. var partition = group(function(result, value, pass) { result[pass ? 0 : 1].push(value); }, true); // Safely create a real, live array from anything iterable. var reStrSymbol = /[^\ud800-\udfff]|[\ud800-\udbff][\udc00-\udfff]|[\ud800-\udfff]/g; function toArray(obj) { if (!obj) return []; if (isArray(obj)) return slice.call(obj); if (isString(obj)) { // Keep surrogate pair characters together. return obj.match(reStrSymbol); } if (isArrayLike(obj)) return map(obj, identity); return values(obj); } // Return the number of elements in a collection. function size(obj) { if (obj == null) return 0; return isArrayLike(obj) ? obj.length : keys(obj).length; } // Internal `_.pick` helper function to determine whether `key` is an enumerable // property name of `obj`. function keyInObj(value, key, obj) { return key in obj; } // Return a copy of the object only containing the allowed properties. var pick = restArguments(function(obj, keys) { var result = {}, iteratee = keys[0]; if (obj == null) return result; if (isFunction$1(iteratee)) { if (keys.length > 1) iteratee = optimizeCb(iteratee, keys[1]); keys = allKeys(obj); } else { iteratee = keyInObj; keys = flatten$1(keys, false, false); obj = Object(obj); } for (var i = 0, length = keys.length; i < length; i++) { var key = keys[i]; var value = obj[key]; if (iteratee(value, key, obj)) result[key] = value; } return result; }); // Return a copy of the object without the disallowed properties. var omit = restArguments(function(obj, keys) { var iteratee = keys[0], context; if (isFunction$1(iteratee)) { iteratee = negate(iteratee); if (keys.length > 1) context = keys[1]; } else { keys = map(flatten$1(keys, false, false), String); iteratee = function(value, key) { return !contains(keys, key); }; } return pick(obj, iteratee, context); }); // Returns everything but the last entry of the array. Especially useful on // the arguments object. Passing **n** will return all the values in // the array, excluding the last N. function initial(array, n, guard) { return slice.call(array, 0, Math.max(0, array.length - (n == null || guard ? 1 : n))); } // Get the first element of an array. Passing **n** will return the first N // values in the array. The **guard** check allows it to work with `_.map`. function first(array, n, guard) { if (array == null || array.length < 1) return n == null || guard ? void 0 : []; if (n == null || guard) return array[0]; return initial(array, array.length - n); } // Returns everything but the first entry of the `array`. Especially useful on // the `arguments` object. Passing an **n** will return the rest N values in the // `array`. function rest(array, n, guard) { return slice.call(array, n == null || guard ? 1 : n); } // Get the last element of an array. Passing **n** will return the last N // values in the array. function last(array, n, guard) { if (array == null || array.length < 1) return n == null || guard ? void 0 : []; if (n == null || guard) return array[array.length - 1]; return rest(array, Math.max(0, array.length - n)); } // Trim out all falsy values from an array. function compact(array) { return filter(array, Boolean); } // Flatten out an array, either recursively (by default), or up to `depth`. // Passing `true` or `false` as `depth` means `1` or `Infinity`, respectively. function flatten(array, depth) { return flatten$1(array, depth, false); } // Take the difference between one array and a number of other arrays. // Only the elements present in just the first array will remain. var difference = restArguments(function(array, rest) { rest = flatten$1(rest, true, true); return filter(array, function(value){ return !contains(rest, value); }); }); // Return a version of the array that does not contain the specified value(s). var without = restArguments(function(array, otherArrays) { return difference(array, otherArrays); }); // Produce a duplicate-free version of the array. If the array has already // been sorted, you have the option of using a faster algorithm. // The faster algorithm will not work with an iteratee if the iteratee // is not a one-to-one function, so providing an iteratee will disable // the faster algorithm. function uniq(array, isSorted, iteratee, context) { if (!isBoolean(isSorted)) { context = iteratee; iteratee = isSorted; isSorted = false; } if (iteratee != null) iteratee = cb(iteratee, context); var result = []; var seen = []; for (var i = 0, length = getLength(array); i < length; i++) { var value = array[i], computed = iteratee ? iteratee(value, i, array) : value; if (isSorted && !iteratee) { if (!i || seen !== computed) result.push(value); seen = computed; } else if (iteratee) { if (!contains(seen, computed)) { seen.push(computed); result.push(value); } } else if (!contains(result, value)) { result.push(value); } } return result; } // Produce an array that contains the union: each distinct element from all of // the passed-in arrays. var union = restArguments(function(arrays) { return uniq(flatten$1(arrays, true, true)); }); // Produce an array that contains every item shared between all the // passed-in arrays. function intersection(array) { var result = []; var argsLength = arguments.length; for (var i = 0, length = getLength(array); i < length; i++) { var item = array[i]; if (contains(result, item)) continue; var j; for (j = 1; j < argsLength; j++) { if (!contains(arguments[j], item)) break; } if (j === argsLength) result.push(item); } return result; } // Complement of zip. Unzip accepts an array of arrays and groups // each array's elements on shared indices. function unzip(array) { var length = array && max(array, getLength).length || 0; var result = Array(length); for (var index = 0; index < length; index++) { result[index] = pluck(array, index); } return result; } // Zip together multiple lists into a single array -- elements that share // an index go together. var zip = restArguments(unzip); // Converts lists into objects. Pass either a single array of `[key, value]` // pairs, or two parallel arrays of the same length -- one of keys, and one of // the corresponding values. Passing by pairs is the reverse of `_.pairs`. function object(list, values) { var result = {}; for (var i = 0, length = getLength(list); i < length; i++) { if (values) { result[list[i]] = values[i]; } else { result[list[i][0]] = list[i][1]; } } return result; } // Generate an integer Array containing an arithmetic progression. A port of // the native Python `range()` function. 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@@ -83,8 +77,8 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="mathematics-in-lean"> <h1>Mathematics in Lean<a class="headerlink" href="#mathematics-in-lean" title="Permalink to this heading"></a></h1> <div class="section" id="mathematics-in-lean"> <h1>Mathematics in Lean<a class="headerlink" href="#mathematics-in-lean" title="Permalink to this headline"></a></h1> <div class="toctree-wrapper compound"> <ul> <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a><ul>
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@@ -1,1 +1,1 @@Search.setIndex({"docnames": ["C01_Introduction", "C02_Basics", "C03_Logic", "C04_Sets_and_Functions", "C05_Number_Theory", "C06_Structures", "C07_Hierarchies", "C08_Topology", "C09_Differential_Calculus", "C10_Integration_and_Measure_Theory", "genindex", "index"], "filenames": ["C01_Introduction.rst", "C02_Basics.rst", "C03_Logic.rst", "C04_Sets_and_Functions.rst", "C05_Number_Theory.rst", "C06_Structures.rst", "C07_Hierarchies.rst", "C08_Topology.rst", "C09_Differential_Calculus.rst", "C10_Integration_and_Measure_Theory.rst", "genindex.rst", "index.rst"], "titles": ["<span class=\"section-number\">1. </span>Introduction", "<span class=\"section-number\">2. </span>Basics", "<span class=\"section-number\">3. </span>Logic", "<span class=\"section-number\">4. </span>Sets and Functions", "<span class=\"section-number\">5. </span>Elementary Number Theory", "<span class=\"section-number\">6. </span>Structures", "<span class=\"section-number\">7. </span>Hierarchies", "<span class=\"section-number\">8. </span>Topology", "<span class=\"section-number\">9. </span>Differential Calculus", "<span class=\"section-number\">10. </span>Integration and Measure Theory", "Index", "Mathematics in Lean"], "terms": {"The": [0, 1, 4, 5, 6, 7, 8, 9, 11], "goal": [0, 1, 2, 3, 4, 5, 7], "thi": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], "book": [0, 2, 7], "i": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], "teach": 0, "you": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], "formal": [0, 1, 2, 3, 4, 5, 6, 7, 8], "mathemat": [0, 1, 2, 3, 4, 5, 6, 7], "us": [0, 2, 3, 4, 5, 6, 7, 8, 9, 11], "lean": [0, 1, 2, 3, 4, 5, 6, 7, 8], "4": [0, 1, 2, 4, 5, 7, 8], "interact": [0, 5, 6, 9], "proof": [0, 1, 2, 3, 4, 5, 6, 7, 8], "assist": [0, 2, 5], "It": [0, 1, 2, 3, 4, 5, 6, 7, 8], "assum": [0, 1, 2, 3, 4, 6, 7], "know": [0, 1, 2, 3, 4, 5, 6, 7, 8], "some": [0, 1, 2, 3, 4, 5, 6, 7, 8], "doe": [0, 1, 2, 3, 4, 5, 6, 7, 8], "requir": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], "much": [0, 2, 4, 6, 7, 8], "although": [0, 3, 4, 5, 6, 7], "we": [0, 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:6,test:4,text:[0,3,5,7],th:4,than:[0,1,2,3,4,5,6,7,8],thank:[6,7],thei:[0,1,2,3,4,5,6,7,8],them:[0,1,2,3,4,5,6,7],theme:3,themselv:[0,4,5],theorem:[0,2,4,5,7,8,9,11],theoret:[2,3,7],theori:[0,1,2,3,6,7,8,11],therefor:[1,4,5,7],thereof:0,thi:[0,1,2,3,4,5,6,7,8,9],thing:[0,1,2,3,4,5,6,7],think:[0,1,2,3,4,5,6,7,8],third:[1,2,3,5,7],tho:1,thorough:0,those:[0,1,2,3,5,6,7,8],though:[2,3,4,5],three:[1,2,4,5,7],through:[0,1,3,4,5,6,7],thu:[1,3,4,5,7],thumb:4,tighter:[1,3],tightli:5,time:[1,4,5,6,7,8,9],timesav:1,tiresom:7,to_addit:6,to_localinvers:8,to_real:9,tofun:[5,6],togeth:[1,2,3,4,5,7],tomuloneclass:6,too:[0,1,3,5,6,7],tool:[0,1,2,5,6],top:[0,6,7,8],topic:[4,6],topolog:[1,5,6,8,11],topological_spac:7,topological_structur:7,topologicalspac:[7,8],toreal:9,total:1,tour:[7,8],trace:6,tradeoff:3,train:[1,6],tran:[2,4,5,6],trans_assoc:5,trans_refl:5,transform:[2,3],transit:[1,2,5,7],translat:[4,6,7],trap:6,treat:[2,3,5,6,7,8],tri:[1,2,4,6],triangl:[1,2,5],trick:[1,2,4,6,8],tricki:[2,3,4,6],trigger:6,tripl:3,trivial:[3,4,7],troubl:[2,3,5],truncat:4,truth:[1,4],tupl:5,turn:[2,3,4,5,6,7,8],tutori:0,twice:[0,1,7],two:[0,1,2,3,4,5,6,7,9],two_l:4,two_le_of_mod_4_eq_3:4,two_mul:1,ty:0,type:[0,1,2,3,4,5,6,7,8,9],typeclass:6,typic:1,u:[2,3,5,7,9],ubf:2,ubfa:2,ubg:2,ubgb:2,ultim:0,un:3,unbundl:6,unchang:[1,2],uncom:1,uncount:7,uncurri:7,under:[5,6,7],underli:[0,1,5,6],underneath:[1,3],underscor:[1,2,3],understand:[0,2,3,4,5,6,7],understood:[4,7],unfold:[1,2,3,4,5,7],unicod:[1,3,6],unif:6,uniform:[3,7,8],uniformcontinu:7,uniformcontinuous_iff:7,uniformli:8,union:[1,3,4,5,6,8,9],union_def:3,uniqu:[1,2,3,4,5,7],unit:[5,6,7],univ:[3,7,8,9],univ_set:7,univers:[1,3,5,7,11],unknown:6,unless:[1,2,5],unlik:[3,4],unnecessari:3,unpack:[2,3],unpleas:[6,7],unrel:6,unshad:3,unspecifi:6,unsurprisingli:3,until:[1,2,4,5,6,7],up:[0,2,3,4,5,6,7],updat:0,upper:[1,2],us:[0,2,3,4,5,6,7,8,9,11],usabl:6,usag:1,user:[0,2,6],usual:[1,2,3,4,5,7,9],v:[2,3,5,7],val:5,valid:2,valu:[1,2,3,4,5,6,7,8,9],varepsilon:[2,7],variabl:[1,2,3,4,5,6,7,8,9],variant:[1,2,7],variat:[1,3,4,5,7,8],varieti:7,variou:[2,3,5,6,9],vastli:2,ve:[6,7],vector:[1,3,6,8],verbos:3,veri:[0,2,4,6,7,9],verifi:2,versa:[1,3],version:[0,1,2,3,4,5,6,7,8,9],vertic:[2,5],vi:5,via:7,vice:[1,3],view:[5,6,7],visibl:1,vocabulari:3,volum:9,vs:[0,1,2,3,5],w:[1,6],wa:[1,3],wai:[0,1,2,3,4,5,6,7,9],want:[0,1,2,3,4,5,6,7,8,9],wasn:7,we:[0,1,2,3,4,5,6,7,8,9],weak:6,weaker:[1,2],weav:5,web:[0,1,4],weight:5,weightedaverag:5,weird:[6,7],weirder:8,welcom:[0,1],well:[0,1,2,3,4,5,6,7],were:[2,4,5,7],what:[0,1,2,3,4,5,6,7],whatev:[2,4],whatsnew:6,when:[1,2,3,4,5,6,7,8],whenev:[1,2,4,5],where:[0,1,2,3,4,5,6,7,8,9],wherea:[0,1,3,5,7],wherebi:[5,7],whether:[1,2,3,4,6,7],which:[0,1,2,3,4,5,6,7,8,9],who:[0,7],whole:5,whose:[1,2,4,5,6,7,8],why:[2,4,6,7],wide:2,window:[0,1,2,5],wise:5,wish:7,with_top:8,within:[1,2,7],without:[1,2,4,5,6,7,8],withtop:8,wlog:3,won:[2,4,6,7],word:[1,2,3,4,5,7],work:[0,1,2,3,4,5,6,7,8],world:0,worri:[1,3,4,5],worth:[1,2,5],would:[1,2,3,4,5,6,7,8],wouldn:6,wrap:[6,7],write:[0,1,2,4,5,6,7,8,9],written:[1,2,3,4,5,6,7,8],wrong:3,wrote:4,x:[0,1,2,3,4,5,6,7,8,9],x_0:7,x_1:[2,3],x_2:[2,3],x_i:7,x_nonneg:5,xa:[3,5],xai:3,xb:5,xeq:[2,3],xgt:2,xlt:2,xltz:2,xmem:3,xnt:3,xnu:3,xpo:3,xs:[2,3],xstu:3,xsu:3,xt:3,xtu:3,xu:3,xy:[2,5],y:[0,1,2,3,4,5,6,7,8,9],y_nonneg:5,ya:5,yb:5,yball:7,yeq:2,yet:[2,3,4,5,6,7],yield:[1,2,3,4,5,7],ylim:7,ylt:2,you:[0,1,2,3,4,5,6,7,8,9],your:[0,1,2,3,5,6,7],yourself:[2,5],ypo:3,z:[0,1,2,5,6,7,8,9],z_nonneg:5,za:5,zb:5,zermelo:2,zero:[2,3,4,5,6,7,8,9],zero_add:[1,4,5,6],zero_def:5,zero_dvd_iff:4,zero_im:5,zero_l:[4,5],zero_lt_on:[2,4],zero_lt_two:5,zero_mul:[1,4,5,6],zero_r:5,zero_smul:6,zf:2,zlty:2,zsmul:6,zulip:[0,4]},titles:["<span class=\"section-number\">1. </span>Introduction","<span class=\"section-number\">2. </span>Basics","<span class=\"section-number\">3. </span>Logic","<span class=\"section-number\">4. </span>Sets and Functions","<span class=\"section-number\">5. </span>Elementary Number Theory","<span class=\"section-number\">6. </span>Structures","<span class=\"section-number\">7. </span>Hierarchies","<span class=\"section-number\">8. </span>Topology","<span class=\"section-number\">9. </span>Differential Calculus","<span class=\"section-number\">10. </span>Integration and Measure Theory","Index","Mathematics in Lean"],titleterms:{"function":[3,7],"schr\u00f6der":3,The:[2,3],about:1,algebra:[1,5],appli:1,asymptot:8,ball:7,basic:[1,6],bernstein:3,bi:2,build:5,calcul:1,calculu:8,close:7,compact:7,comparison:8,complet:7,conjunct:2,continu:[7,8],converg:[2,7],countabl:7,defin:5,differenti:8,disjunct:2,elementari:[4,8,9],exampl:1,existenti:2,fact:1,filter:7,fundament:7,gaussian:5,get:0,hierarchi:6,ident:1,implic:2,index:10,induct:4,infinit:4,integ:5,integr:9,introduct:0,irrat:4,lean:11,lemma:1,linear:8,logic:2,mani:4,map:8,mathemat:11,measur:9,metric:7,more:1,morphism:6,negat:2,norm:8,number:4,object:6,open:7,overview:0,prime:4,prove:1,quantifi:2,recurs:4,root:4,rw:1,separ:7,sequenc:2,set:[3,7],space:[7,8],start:0,structur:[1,5],sub:6,theorem:[1,3],theori:[4,9],topolog:7,uniformli:7,univers:2,us:1}})
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