Update mer. 28 juin 2023 16:38:39 CEST

Changes

80 changed files (+1077/-13924)

• MIL/C01_Introduction/S02_Overview.lean
  • @@ -32,12 +32,12 @@ theorem hard : FermatLastTheorem :=
    #check hard -- Here are some proofs. - example : ∀ m n : Nat, Even n → Even (m * n) := fun m n ⟨k, (hk : n = k + k)⟩ => + 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]⟩ + 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
• MIL/C02_Basics/S03_Using_Theorems_and_Lemmas.lean
  • @@ -97,7 +97,7 @@ example (h : a ≤ b) : log (1 + exp a) ≤ log (1 + exp b) := by
    sorry example : 0 ≤ a ^ 2 := by - -- library_search + -- apply? exact sq_nonneg a example (h : a ≤ b) : c - exp b ≤ c - exp a := by
  • @@ -123,7 +123,7 @@ example : 2 * a * b ≤ a ^ 2 + b ^ 2 := by
    _ ≥ 0 := by apply pow_two_nonneg linarith - example : abs (a * b) ≤ (a ^ 2 + b ^ 2) / 2 := by + example : |a * b| ≤ (a ^ 2 + b ^ 2) / 2 := by sorry #check abs_le'.mpr
• MIL/C02_Basics/S04_More_on_Order_and_Divisibility.lean
  • @@ -48,9 +48,9 @@ theorem aux : min a b + c ≤ min (a + c) (b + c) := by
    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) + #check (abs_add : ∀ a b : ℝ, |a + b| ≤ |a| + |b|) - example : abs a - abs b ≤ abs (a - b) := + example : |a| - |b| ≤ |a - b| := sorry end
• MIL/C02_Basics/solutions/Solutions_S03_Using_Theorems_and_Lemmas.lean
  • @@ -50,7 +50,7 @@ theorem fact2 : -(a * b) * 2 ≤ a ^ 2 + b ^ 2 := by
    _ ≥ 0 := by apply pow_two_nonneg linarith - example : abs (a * b) ≤ (a ^ 2 + b ^ 2) / 2 := by + example : |a * b| ≤ (a ^ 2 + b ^ 2) / 2 := by have h : (0 : ℝ) < 2 := by norm_num apply abs_le'.mpr constructor
• MIL/C02_Basics/solutions/Solutions_S04_More_on_Order_and_Divisibility.lean
  • @@ -54,17 +54,17 @@ example : min a b + c = min (a + c) (b + c) := by
    apply aux rw [add_neg_cancel_right, add_neg_cancel_right] - example : abs a - abs b ≤ abs (a - b) := + example : |a| - |b| ≤ |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 + |a| - |b| = |a - b + b| - |b| := by rw [sub_add_cancel] + _ ≤ |a - b| + |b| - |b| := by apply sub_le_sub_right apply abs_add - _ ≤ abs (a - b) := by rw [add_sub_cancel] + _ ≤ |a - b| := by rw [add_sub_cancel] -- alternatively - example : abs a - abs b ≤ abs (a - b) := by + example : |a| - |b| ≤ |a - b| := by have h := abs_add (a - b) b rw [sub_add_cancel] at h linarith
• MIL/C03_Logic/S01_Implication_and_the_Universal_Quantifier.lean
  • @@ -1,17 +1,17 @@
    import Mathlib.Data.Real.Basic namespace C03S01 - #check ∀ x : ℝ, 0 ≤ x → abs x = x + #check ∀ x : ℝ, 0 ≤ x → |x| = x - #check ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε + #check ∀ x y ε : ℝ, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |x * y| < ε - theorem my_lemma : ∀ 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 → |x| < ε → |y| < ε → |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) + variable (ha : |a| < δ) (hb : |b| < δ) #check my_lemma a b δ #check my_lemma a b δ h₀ h₁
  • @@ -19,29 +19,29 @@ variable (ha : abs a < δ) (hb : abs b < δ)
    end - theorem my_lemma2 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → abs x < ε → abs y < ε → abs (x * y) < ε := + theorem my_lemma2 : ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |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) + variable (ha : |a| < δ) (hb : |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 + ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |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 + ∀ {x y ε : ℝ}, 0 < ε → ε ≤ 1 → |x| < ε → |y| < ε → |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 + |x * y| = |x| * |y| := sorry + _ ≤ |x| * ε := sorry _ < 1 * ε := sorry _ = ε := sorry
  • @@ -61,14 +61,14 @@ example (hfa : FnUb f a) (hgb : FnUb g b) : FnUb (fun x ↦ f x + g x) (a + b) :
    apply hfa apply hgb - example (hfa : FnLb f a) (hgb : FnLb g b) : FnLb (fun x => f x + g x) (a + b) := + 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 := + 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) := + FnUb (fun x ↦ f x * g x) (a * b) := sorry end
  • @@ -82,7 +82,7 @@ def FnUb' (f : α → R) (a : R) : Prop :=
    ∀ x, f x ≤ a theorem fnUb_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) + FnUb' (fun x ↦ f x + g x) (a + b) := fun x ↦ add_le_add (hfa x) (hgb x) end
  • @@ -92,19 +92,19 @@ example (f : ℝ → ℝ) (h : Monotone f) : ∀ {a b}, a ≤ b → f a ≤ f b
    section variable (f g : ℝ → ℝ) - example (mf : Monotone f) (mg : Monotone g) : Monotone fun x => f x + g x := by + 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 (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 := + 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) := + example (mf : Monotone f) (mg : Monotone g) : Monotone fun x ↦ f (g x) := sorry def FnEven (f : ℝ → ℝ) : Prop :=
  • @@ -113,20 +113,20 @@ def FnEven (f : ℝ → ℝ) : Prop :=
    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 + 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 + (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 + 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 + 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 + example (ef : FnEven f) (og : FnOdd g) : FnEven fun x ↦ f (g x) := by sorry end
  • @@ -139,7 +139,7 @@ example : s ⊆ s := by
    intro x xs exact xs - theorem Subset.refl : s ⊆ s := fun x xs => xs + theorem Subset.refl : s ⊆ s := fun x xs ↦ xs theorem Subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := by sorry
  • @@ -162,17 +162,17 @@ section
    open Function - example (c : ℝ) : Injective fun x => x + c := by + 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 + 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 + example (injg : Injective g) (injf : Injective f) : Injective fun x ↦ g (f x) := by sorry end
• MIL/C03_Logic/S02_The_Existential_Quantifier.lean
  • @@ -25,36 +25,36 @@ 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) + 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 + example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x := by cases' ubf with a ubfa cases' ubg with b ubgb use a + b apply fnUb_add ubfa ubgb - example (lbf : FnHasLb f) (lbg : FnHasLb g) : FnHasLb fun x => f x + g x := by + 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 + 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 + example (ubf : FnHasUb f) (ubg : FnHasUb g) : FnHasUb fun x ↦ f x + g x := by rcases ubf with ⟨a, ubfa⟩ rcases ubg with ⟨b, ubgb⟩ exact ⟨a + b, fnUb_add ubfa ubgb⟩ - example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := by + example : FnHasUb f → FnHasUb g → FnHasUb fun x ↦ f x + g x := by rintro ⟨a, ubfa⟩ ⟨b, ubgb⟩ exact ⟨a + b, fnUb_add ubfa ubgb⟩ - example : FnHasUb f → FnHasUb g → FnHasUb fun x => f x + g x := - fun ⟨a, ubfa⟩ ⟨b, ubgb⟩ => ⟨a + b, fnUb_add ubfa ubgb⟩ + example : FnHasUb f → FnHasUb g → FnHasUb fun x ↦ f x + g x := + fun ⟨a, ubfa⟩ ⟨b, ubgb⟩ ↦ ⟨a + b, fnUb_add ubfa ubgb⟩ end
  • @@ -100,12 +100,12 @@ section
    open Function - example {c : ℝ} : Surjective fun x => x + c := by + 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 + 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
  • @@ -125,7 +125,7 @@ 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 + example (surjg : Surjective g) (surjf : Surjective f) : Surjective fun x ↦ g (f x) := by sorry end
• MIL/C03_Logic/S03_Negation.lean
  • @@ -33,7 +33,7 @@ example (h : ∀ a, ∃ x, f x > a) : ¬FnHasUb f := by
    example (h : ∀ a, ∃ x, f x < a) : ¬FnHasLb f := sorry - example : ¬FnHasUb fun x => x := + example : ¬FnHasUb fun x ↦ x := sorry #check (not_le_of_gt : a > b → ¬a ≤ b)
  • @@ -49,7 +49,7 @@ example (h : a ≤ b) (h' : f b < f a) : ¬Monotone f := by
    example : ¬∀ {f : ℝ → ℝ}, Monotone f → ∀ {a b}, f a ≤ f b → a ≤ b := by intro h - let f := fun x : ℝ => (0 : ℝ) + let f := fun x : ℝ ↦ (0 : ℝ) have monof : Monotone f := by sorry have h' : f 1 ≤ f 0 := le_refl _ sorry
• MIL/C03_Logic/S04_Conjunction_and_Bi-implication.lean
  • @@ -11,7 +11,7 @@ example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y :=
    rw [h] example {x y : ℝ} (h₀ : x ≤ y) (h₁ : ¬y ≤ x) : x ≤ y ∧ x ≠ y := - ⟨h₀, fun h => h₁ (by rw [h])⟩ + ⟨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
  • @@ -29,7 +29,7 @@ example {x y : ℝ} : x ≤ y ∧ x ≠ y → ¬y ≤ x := by
    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') + fun ⟨h₀, h₁⟩ h' ↦ h₁ (le_antisymm h₀ h') example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by intro h'
  • @@ -37,7 +37,7 @@ example {x y : ℝ} (h : x ≤ y ∧ x ≠ y) : ¬y ≤ x := by
    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') + fun h' ↦ h.right (le_antisymm h.left h') example {m n : ℕ} (h : m ∣ n ∧ m ≠ n) : m ∣ n ∧ ¬n ∣ m := sorry
  • @@ -50,7 +50,7 @@ example (x y : ℝ) : (∃ z : ℝ, x < z ∧ z < y) → x < y := by
    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 + fun ⟨z, xltz, zlty⟩ ↦ lt_trans xltz zlty example : ∃ x : ℝ, 2 < x ∧ x < 4 := by use 5 / 2
  • @@ -65,7 +65,7 @@ example : ∃ m n : ℕ, 4 < m ∧ m < n ∧ n < 10 ∧ Nat.Prime m ∧ Nat.Prim
    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') + exact fun h' ↦ h₁ (le_antisymm h₀ h') example {x y : ℝ} (h : x ≤ y) : ¬y ≤ x ↔ x ≠ y := by constructor
  • @@ -76,7 +76,7 @@ example {x y : ℝ} (h : x ≤ y) : ¬y ≤ x ↔ x ≠ y := by
    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)⟩ + ⟨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
  • @@ -90,7 +90,7 @@ example (x y : ℝ) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 :=
    section - example (x : ℝ) : abs (x + 3) < 5 → -8 < x ∧ x < 2 := by + example (x : ℝ) : |x + 3| < 5 → -8 < x ∧ x < 2 := by rw [abs_lt] intro h constructor <;> linarith
  • @@ -106,7 +106,7 @@ theorem not_monotone_iff {f : ℝ → ℝ} : ¬Monotone f ↔ ∃ x y, x ≤ y
    push_neg rfl - example : ¬Monotone fun x : ℝ => -x := by + example : ¬Monotone fun x : ℝ ↦ -x := by sorry section
  • @@ -132,4 +132,3 @@ example : a < b → b < c → a < c := by
    sorry end -
• MIL/C03_Logic/S05_Disjunction.lean
  • @@ -19,7 +19,7 @@ example (h : y > 0) : y > 0 ∨ y < -1 :=
    example (h : y < -1) : y > 0 ∨ y < -1 := Or.inr h - example : x < abs y → x < y ∨ x < -y := by + example : x < |y| → x < y ∨ x < -y := by cases' le_or_gt 0 y with h h · rw [abs_of_nonneg h] intro h
  • @@ -30,19 +30,19 @@ example : x < abs y → x < y ∨ x < -y := by
    namespace MyAbs - theorem le_abs_self (x : ℝ) : x ≤ abs x := by + theorem le_abs_self (x : ℝ) : x ≤ |x| := by sorry - theorem neg_le_abs_self (x : ℝ) : -x ≤ abs x := by + theorem neg_le_abs_self (x : ℝ) : -x ≤ |x| := by sorry - theorem abs_add (x y : ℝ) : abs (x + y) ≤ abs x + abs y := by + theorem abs_add (x y : ℝ) : |x + y| ≤ |x| + |y| := by sorry - theorem lt_abs : x < abs y ↔ x < y ∨ x < -y := by + theorem lt_abs : x < |y| ↔ x < y ∨ x < -y := by sorry - theorem abs_lt : abs x < y ↔ -y < x ∧ x < y := by + theorem abs_lt : |x| < y ↔ -y < x ∧ x < y := by sorry end MyAbs
• MIL/C03_Logic/S06_Sequences_and_Convergence.lean
  • @@ -2,13 +2,13 @@ import Mathlib.Data.Real.Basic
    namespace C03S06 def ConvergesTo (s : ℕ → ℝ) (a : ℝ) := - ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε + ∀ ε > 0, ∃ N, ∀ n ≥ N, |s n - a| < ε - example : (fun x y : ℝ => (x + y) ^ 2) = fun x y : ℝ => x ^ 2 + 2 * x * y + y ^ 2 := by + 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 + example (a b : ℝ) : |a| = |a - b + b| := by congr ring
  • @@ -17,18 +17,18 @@ example {a : ℝ} (h : 1 < a) : a < a * a := by
    · rw [one_mul] exact lt_trans zero_lt_one h - theorem convergesTo_const (a : ℝ) : ConvergesTo (fun x : ℕ => a) a := by + theorem convergesTo_const (a : ℝ) : ConvergesTo (fun x : ℕ ↦ a) a := by intro ε εpos use 0 - intro n nge; dsimp + intro n nge 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 + ConvergesTo (fun n ↦ s n + t n) (a + b) := by intro ε εpos - dsimp + dsimp -- this line is not needed but cleans up the goal a bit. have ε2pos : 0 < ε / 2 := by linarith cases' cs (ε / 2) ε2pos with Ns hs cases' ct (ε / 2) ε2pos with Nt ht
  • @@ -36,22 +36,24 @@ theorem convergesTo_add {s t : ℕ → ℝ} {a b : ℝ}
    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 + 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 + · rw [h] + ring + rw [h] + ring + have acpos : 0 < |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 + ∃ N b, ∀ n, N ≤ n → |s n| < b := by cases' cs 1 zero_lt_one with N h - use N, abs a + 1 + use N, |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 + ConvergesTo (fun n ↦ s n * t n) 0 := by intro ε εpos dsimp rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩
  • @@ -62,8 +64,8 @@ theorem aux {s t : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) (ct : Converges
    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 + 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
  • @@ -76,24 +78,24 @@ 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 : |a - b| > 0 := by sorry + let ε := |a - b| / 2 have εpos : ε > 0 := by - change abs (a - b) / 2 > 0 + change |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 + have absa : |s N - a| < ε := by sorry + have absb : |s N - b| < ε := by sorry + have : |a - b| < |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 + ∀ ε > 0, ∃ N, ∀ n ≥ N, |s n - a| < ε end
• MIL/C03_Logic/solutions/Solutions_S01_Implication_and_the_Universal_Quantifier.lean
  • @@ -10,20 +10,20 @@ def FnLb (f : ℝ → ℝ) (a : ℝ) : Prop :=
    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 + 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 + 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 + FnUb (fun x ↦ f x * g x) (a * b) := by intro x apply mul_le_mul apply hfa
  • @@ -36,22 +36,22 @@ end
    section variable (f g : ℝ → ℝ) - example {c : ℝ} (mf : Monotone f) (nnc : 0 ≤ c) : Monotone fun x => c * f x := by + 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 {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 + 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) + 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)
  • @@ -59,19 +59,19 @@ def FnEven (f : ℝ → ℝ) : Prop :=
    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 + 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 + (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 + 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 + example (ef : FnEven f) (og : FnOdd g) : FnEven fun x ↦ f (g x) := by intro x dsimp rw [og, ← ef]
  • @@ -89,7 +89,7 @@ example : r ⊆ s → s ⊆ t → r ⊆ t := by
    apply xr theorem Subset.trans : r ⊆ s → s ⊆ t → r ⊆ t := - fun rsubs ssubt x xr => ssubt (rsubs xr) + fun rsubs ssubt x xr ↦ ssubt (rsubs xr) end
  • @@ -105,7 +105,7 @@ example (h : SetUb s a) (h' : a ≤ b) : SetUb s b := by
    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' + fun x xs ↦ le_trans (h x xs) h' end
  • @@ -113,14 +113,14 @@ section
    open Function - example {c : ℝ} (h : c ≠ 0) : Injective fun x => c * x := by + 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 + example (injg : Injective g) (injf : Injective f) : Injective fun x ↦ g (f x) := by intro x₁ x₂ h apply injf apply injg
• MIL/C03_Logic/solutions/Solutions_S02_The_Existential_Quantifier.lean
  • @@ -14,21 +14,21 @@ 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) + 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 + 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 + 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
  • @@ -55,12 +55,12 @@ section
    open Function - example {c : ℝ} (h : c ≠ 0) : Surjective fun x => c * x := by + 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 + example {c : ℝ} (h : c ≠ 0) : Surjective fun x ↦ c * x := by intro x use x / c field_simp [h] ; ring
  • @@ -72,7 +72,7 @@ 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 + 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⟩
• MIL/C03_Logic/solutions/Solutions_S03_Negation.lean
  • @@ -24,7 +24,7 @@ example (h : ∀ a, ∃ x, f x < a) : ¬FnHasLb f := by
    have := ha x linarith - example : ¬FnHasUb fun x => x := by + example : ¬FnHasUb fun x ↦ x := by rintro ⟨a, ha⟩ have : a + 1 ≤ a := ha (a + 1) linarith
  • @@ -43,7 +43,7 @@ example (h : a ≤ b) (h' : f b < f a) : ¬Monotone f := by
    example : ¬∀ {f : ℝ → ℝ}, Monotone f → ∀ {a b}, f a ≤ f b → a ≤ b := by intro h - let f := fun x : ℝ => (0 : ℝ) + let f := fun x : ℝ ↦ (0 : ℝ) have monof : Monotone f := by intro a b leab rfl
• MIL/C03_Logic/solutions/Solutions_S04_Conjunction_and_Bi-implication.lean
  • @@ -45,7 +45,7 @@ theorem not_monotone_iff {f : ℝ → ℝ} : ¬Monotone f ↔ ∃ x y, x ≤ y
    push_neg rfl - example : ¬Monotone fun x : ℝ => -x := by + example : ¬Monotone fun x : ℝ ↦ -x := by rw [not_monotone_iff] use 0, 1 norm_num
  • @@ -91,4 +91,3 @@ example : a < b → b < c → a < c := by
    apply le_trans h2 h4 end -
• MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean
  • @@ -7,26 +7,26 @@ variable {x y : ℝ}
    namespace MyAbs - theorem le_abs_self (x : ℝ) : x ≤ abs x := by + theorem le_abs_self (x : ℝ) : x ≤ |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 + theorem neg_le_abs_self (x : ℝ) : -x ≤ |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 + theorem abs_add (x y : ℝ) : |x + y| ≤ |x| + |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 + theorem lt_abs : x < |y| ↔ x < y ∨ x < -y := by cases' le_or_gt 0 y with h h · rw [abs_of_nonneg h] constructor
  • @@ -47,7 +47,7 @@ theorem lt_abs : x < abs y ↔ x < y ∨ x < -y := by
    · linarith exact h' - theorem abs_lt : abs x < y ↔ -y < x ∧ x < y := by + theorem abs_lt : |x| < y ↔ -y < x ∧ x < y := by cases' le_or_gt 0 x with h h · rw [abs_of_nonneg h] constructor
  • @@ -137,5 +137,3 @@ example (P Q : Prop) : P → Q ↔ ¬P ∨ Q := by
    exact absurd h' h intro exact h - -
• MIL/C03_Logic/solutions/Solutions_S06_Sequences_and_Convergence.lean
  • @@ -2,18 +2,18 @@ import Mathlib.Data.Real.Basic
    namespace C03S06 def ConvergesTo (s : ℕ → ℝ) (a : ℝ) := - ∀ ε > 0, ∃ N, ∀ n ≥ N, abs (s n - a) < ε + ∀ ε > 0, ∃ N, ∀ n ≥ N, |s n - a| < ε - theorem convergesTo_const (a : ℝ) : ConvergesTo (fun x : ℕ => a) a := by + theorem convergesTo_const (a : ℝ) : ConvergesTo (fun x : ℕ ↦ a) a := by intro ε εpos use 0 - intro n nge; dsimp + intro n nge 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 + ConvergesTo (fun n ↦ s n + t n) (a + b) := by intro ε εpos dsimp have ε2pos : 0 < ε / 2 := by linarith
  • @@ -32,16 +32,18 @@ theorem convergesTo_add {s t : ℕ → ℝ} {a b : ℝ}
    _ = 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 + 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 + · rw [h] + ring + rw [h] + ring + have acpos : 0 < |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 + have εcpos : 0 < ε / |c| := by apply div_pos εpos acpos + cases' cs (ε / |c|) εcpos with Ns hs use Ns intro n ngt calc
  • @@ -50,9 +52,9 @@ theorem convergesTo_mul_const {s : ℕ → ℝ} {a : ℝ} (c : ℝ) (cs : Conver
    _ = ε := mul_div_cancel' _ (ne_of_lt acpos).symm 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 + ∃ N b, ∀ n, N ≤ n → |s n| < b := by cases' cs 1 zero_lt_one with N h - use N, abs a + 1 + use N, |a| + 1 intro n ngt calc |s n| = |s n - a + a| := by
  • @@ -62,7 +64,7 @@ theorem exists_abs_le_of_convergesTo {s : ℕ → ℝ} {a : ℝ} (cs : Converges
    _ < |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 + ConvergesTo (fun n ↦ s n * t n) 0 := by intro ε εpos dsimp rcases exists_abs_le_of_convergesTo cs with ⟨N₀, B, h₀⟩
  • @@ -80,8 +82,8 @@ theorem aux {s t : ℕ → ℝ} {a : ℝ} (cs : ConvergesTo s a) (ct : Converges
    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 + 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
  • @@ -94,34 +96,34 @@ 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 + have : |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 + let ε := |a - b| / 2 have εpos : ε > 0 := by - change abs (a - b) / 2 > 0 + change |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 + have absa : |s N - a| < ε := by apply hNa apply le_max_left - have absb : abs (s N - b) < ε := by + have absb : |s N - b| < ε := by apply hNb apply le_max_right - have : abs (a - b) < abs (a - b) + have : |a - b| < |a - b| calc - abs (a - b) = abs (-(s N - a) + (s N - b)) := by + |a - b| = |(-(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] + _ ≤ |(-(s N - a))| + |s N - b| := (abs_add _ _) + _ = |s N - a| + |s N - b| := by rw [abs_neg] _ < ε + ε := (add_lt_add absa absb) - _ = abs (a - b) := by norm_num + _ = |a - b| := by norm_num exact lt_irrefl _ this
• MIL/C04_Sets_and_Functions/S01_Sets.lean
  • @@ -11,7 +11,7 @@ open Set
    example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by rw [subset_def, inter_def, inter_def] rw [subset_def] at h - dsimp + simp only [mem_setOf] rintro x ⟨xs, xu⟩ exact ⟨h _ xs, xu⟩
  • @@ -25,10 +25,10 @@ example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := by
    exact ⟨h xsu.1, xsu.2⟩ theorem foo (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := - fun x ⟨xs, xu⟩ => ⟨h xs, xu⟩ + fun x ⟨xs, xu⟩ ↦ ⟨h xs, xu⟩ example (h : s ⊆ t) : s ∩ u ⊆ t ∩ u := - fun x ⟨xs, xu⟩ => ⟨h xs, xu⟩ + fun x ⟨xs, xu⟩ ↦ ⟨h xs, xu⟩ example : s ∩ (t ∪ u) ⊆ s ∩ t ∪ s ∩ u := by intro x hx
  • @@ -80,7 +80,7 @@ example : s ∩ t = t ∩ s := by
    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⟩⟩ + 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]
• MIL/C04_Sets_and_Functions/S02_Functions.lean
  • @@ -145,13 +145,13 @@ example : range exp = { y | y > 0 } := by
    example : InjOn sqrt { x | x ≥ 0 } := by sorry - example : InjOn (fun x => x ^ 2) { x : ℝ | x ≥ 0 } := by + 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 + example : (range fun x ↦ x ^ 2) = { y : ℝ | y ≥ 0 } := by sorry end
  • @@ -172,7 +172,7 @@ noncomputable section
    open Classical - def inverse (f : α → β) : β → α := fun y : β => + 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
• MIL/C04_Sets_and_Functions/solutions/Solutions_S01_Sets.lean
  • @@ -26,7 +26,7 @@ example : s \ (t ∪ u) ⊆ (s \ t) \ u := by
    example : s ∩ t = t ∩ s := Subset.antisymm - (fun x ⟨xs, xt⟩ => ⟨xt, xs⟩) fun x ⟨xt, xs⟩ => ⟨xs, xt⟩ + (fun x ⟨xs, xt⟩ ↦ ⟨xt, xs⟩) fun x ⟨xt, xs⟩ ↦ ⟨xs, xt⟩ example : s ∩ (s ∪ t) = s := by ext x; constructor
• MIL/C04_Sets_and_Functions/solutions/Solutions_S02_Functions.lean
  • @@ -79,7 +79,7 @@ example : f '' s \ f '' t ⊆ f '' (s \ t) := by
    . rfl example : f ⁻¹' u \ f ⁻¹' v ⊆ f ⁻¹' (u \ v) := - fun x => id + fun x ↦ id example : f '' s ∩ v = f '' (s ∩ f ⁻¹' v) := by ext y; constructor
  • @@ -155,7 +155,7 @@ example : InjOn sqrt { x | x ≥ 0 } := by
    _ = y := by rw [sq_sqrt ynonneg] - example : InjOn (fun x => x ^ 2) { x : ℝ | x ≥ 0 } := by + example : InjOn (fun x ↦ x ^ 2) { x : ℝ | x ≥ 0 } := by intro x xnonneg y ynonneg intro e dsimp at *
  • @@ -177,7 +177,7 @@ example : sqrt '' { x | x ≥ 0 } = { y | y ≥ 0 } := by
    apply sqrt_sq assumption - example : (range fun x => x ^ 2) = { y : ℝ | y ≥ 0 } := by + example : (range fun x ↦ x ^ 2) = { y : ℝ | y ≥ 0 } := by ext y constructor · rintro ⟨x, rfl⟩
  • @@ -196,7 +196,7 @@ noncomputable section
    open Classical - def inverse (f : α → β) : β → α := fun y : β => + 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
  • @@ -217,7 +217,7 @@ example : Injective f ↔ LeftInverse (inverse f) f := by
    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]⟩ + ⟨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
  • @@ -229,7 +229,7 @@ example : Surjective f ↔ RightInverse (inverse f) f := by
    apply h example : Surjective f ↔ RightInverse (inverse f) f := - ⟨fun h y => inverse_spec _ (h _), fun h y => ⟨inverse f y, h _⟩⟩ + ⟨fun h y ↦ inverse_spec _ (h _), fun h y ↦ ⟨inverse f y, h _⟩⟩ end
• MIL/C05_Number_Theory/S01_Irrational_Roots.lean
  • @@ -47,7 +47,7 @@ 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: + -- apply? suggests the following: (mul_right_inj' h').mp h example {m n : ℕ} (coprime_mn : m.coprime n) : m ^ 2 ≠ 2 * n ^ 2 := by
  • @@ -106,7 +106,7 @@ example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m ^ k = r * n ^ k) {p : ℕ} (
    k ∣ r.factorization p := by cases' r with r · simp - have npow_nz : n ^ k ≠ 0 := fun npowz => nnz (pow_eq_zero npowz) + 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 =
• MIL/C05_Number_Theory/solutions/Solutions_S01_Irrational_Roots.lean
  • @@ -91,7 +91,7 @@ example {m n k r : ℕ} (nnz : n ≠ 0) (pow_eq : m ^ k = r * n ^ k) {p : ℕ} (
    k ∣ r.factorization p := by cases' r with r · simp - have npow_nz : n ^ k ≠ 0 := fun npowz => nnz (pow_eq_zero npowz) + 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 =
• MIL/C06_Abstract_Algebra/S01_Structures.lean
  • @@ -44,11 +44,13 @@ def add' (a b : Point) : Point where
    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
  • @@ -96,14 +98,10 @@ example : ∀ a b : Point, add a b = add b a := fun ⟨xa, ya, za⟩ ⟨xb, yb,
    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 + 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
  • @@ -140,9 +138,9 @@ def midpoint (a b : StandardTwoSimplex) : StandardTwoSimplex
    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 := + (a b : StandardTwoSimplex) : StandardTwoSimplex + where sorry - end end StandardTwoSimplex
  • @@ -165,8 +163,7 @@ def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n
    · 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] + simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] field_simp end StandardSimplex
  • @@ -176,9 +173,11 @@ structure IsLinear (f : ℝ → ℝ) where
    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
  • @@ -193,11 +192,15 @@ def PReal :=
    { y : ℝ // 0 < y } section + variable (x : PReal) #check x.val + #check x.property + #check x.1 + #check x.2 end
  • @@ -211,6 +214,7 @@ def StandardSimplex' (n : ℕ) :=
    def StdSimplex := Σ n : ℕ, StandardSimplex n section + variable (s : StdSimplex) #check s.fst
• MIL/C06_Abstract_Algebra/S02_Algebraic_Structures.lean
  • @@ -14,6 +14,7 @@ structure Group₁Cat where
    str : Group₁ α section + variable (α β γ : Type _) variable (f : α ≃ β) (g : β ≃ γ)
  • @@ -64,15 +65,15 @@ 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 neg (a : point) : point := sorry - def add_group_point : AddGroup₁ 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
  • @@ -156,6 +157,7 @@ instance hasInvGroup₂ {α : Type _} [Group₂ α] : Inv α :=
    ⟨Group₂.inv⟩ section + variable {α : Type _} (f g : Equiv.Perm α) #check f * 1 * g⁻¹
• MIL/C06_Abstract_Algebra/S03_Building_the_Gaussian_Integers.lean
  • @@ -37,8 +37,7 @@ theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ :=
    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⟩ := + 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]
  • @@ -91,13 +90,13 @@ instance instCommRing : CommRing gaussInt where
    intros ext <;> simp <;> ring zero_add := by - intro + intros ext <;> simp add_zero := by - intro + intros ext <;> simp add_left_neg := by - intro + intros ext <;> simp add_comm := by intros
  • @@ -106,10 +105,10 @@ instance instCommRing : CommRing gaussInt where
    intros ext <;> simp <;> ring one_mul := by - intro + intros ext <;> simp mul_one := by - intro + intros ext <;> simp left_distrib := by intros
  • @@ -136,7 +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 < abs b := + example (a b : ℤ) : b ≠ 0 → a % b < |b| := Int.emod_lt a namespace Int
  • @@ -151,7 +150,7 @@ 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 + theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : |mod' a b| ≤ b / 2 := by rw [mod', abs_le] constructor · linarith [Int.emod_nonneg (a + b / 2) h.ne']
  • @@ -208,11 +207,9 @@ theorem div_def (x y : gaussInt) :
    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 + 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 + 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
  • @@ -241,8 +238,7 @@ theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) :
    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 + 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
  • @@ -260,8 +256,7 @@ 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_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
• MIL/C06_Abstract_Algebra/solutions/Solutions_S01_Structures.lean
  • @@ -20,8 +20,7 @@ protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c)
    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 + 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
  • @@ -40,8 +39,8 @@ namespace StandardTwoSimplex
    noncomputable section def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) - (a b : StandardTwoSimplex) : StandardTwoSimplex - where + (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
  • @@ -75,8 +74,7 @@ def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n
    · 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] + simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] field_simp end StandardSimplex
• MIL/C06_Abstract_Algebra/solutions/Solutions_S03_Building_the_Gaussian_Integers.lean
  • @@ -37,8 +37,7 @@ theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ :=
    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⟩ := + 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]
  • @@ -91,13 +90,13 @@ instance instCommRing : CommRing gaussInt where
    intros ext <;> simp <;> ring zero_add := by - intro + intros ext <;> simp add_zero := by - intro + intros ext <;> simp add_left_neg := by - intro + intros ext <;> simp add_comm := by intros
  • @@ -106,10 +105,10 @@ instance instCommRing : CommRing gaussInt where
    intros ext <;> simp <;> ring one_mul := by - intro + intros ext <;> simp mul_one := by - intro + intros ext <;> simp left_distrib := by intros
  • @@ -222,11 +221,9 @@ theorem div_def (x y : gaussInt) :
    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 + 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 + 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
  • @@ -255,8 +252,7 @@ theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) :
    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 + 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
  • @@ -274,8 +270,7 @@ 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_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
• MIL/C06_Structures/S01_Structures.lean
  • @@ -92,7 +92,7 @@ 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 + 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
• MIL/C06_Structures/S03_Building_the_Gaussian_Integers.lean
  • @@ -17,13 +17,13 @@ instance : One gaussInt :=
    ⟨⟨1, 0⟩⟩ instance : Add gaussInt := - ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ + ⟨fun x y ↦ ⟨x.re + y.re, x.im + y.im⟩⟩ instance : Neg gaussInt := - ⟨fun x => ⟨-x.re, -x.im⟩⟩ + ⟨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⟩⟩ + ⟨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
  • @@ -144,7 +144,7 @@ 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 < abs b := + example (a b : ℤ) : b ≠ 0 → a % b < |b| := Int.emod_lt a namespace Int
  • @@ -159,7 +159,7 @@ 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 + theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : |mod' a b| ≤ b / 2 := by rw [mod', abs_le] constructor · linarith [Int.emod_nonneg (a + b / 2) h.ne']
  • @@ -204,10 +204,10 @@ theorem conj_im (x : gaussInt) : (conj x).im = -x.im :=
    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)⟩⟩ + ⟨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)⟩ + ⟨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)⟩ :=
  • @@ -224,8 +224,8 @@ theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) :
    have H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y · calc norm (x % y) * norm y = norm (x % y * conj y) := by simp only [norm_mul, norm_conj] - _ = abs (Int.mod' (x.re * y.re + x.im * y.im) (norm y)) ^ 2 - + abs (Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)) ^ 2 := by simp [H1, norm, sq_abs] + _ = |Int.mod' (x.re * y.re + x.im * y.im) (norm y)| ^ 2 + + |Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)| ^ 2 := by simp [H1, norm, sq_abs] _ ≤ (y.norm / 2) ^ 2 + (y.norm / 2) ^ 2 := by gcongr <;> apply Int.abs_mod'_le _ _ norm_y_pos _ = norm y / 2 * (norm y / 2 * 2) := by ring _ ≤ norm y / 2 * norm y := by gcongr; apply Int.ediv_mul_le; norm_num
  • @@ -258,8 +258,8 @@ instance : EuclideanDomain gaussInt :=
    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 + 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)
• MIL/C06_Structures/solutions/Solutions_S03_Building_the_Gaussian_Integers.lean
  • @@ -17,13 +17,13 @@ instance : One gaussInt :=
    ⟨⟨1, 0⟩⟩ instance : Add gaussInt := - ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ + ⟨fun x y ↦ ⟨x.re + y.re, x.im + y.im⟩⟩ instance : Neg gaussInt := - ⟨fun x => ⟨-x.re, -x.im⟩⟩ + ⟨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⟩⟩ + ⟨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
  • @@ -150,7 +150,7 @@ 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 + theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : |mod' a b| ≤ b / 2 := by rw [mod', abs_le] constructor · linarith [Int.emod_nonneg (a + b / 2) h.ne']
  • @@ -218,10 +218,10 @@ theorem conj_im (x : gaussInt) : (conj x).im = -x.im :=
    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)⟩⟩ + ⟨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)⟩ + ⟨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)⟩ :=
  • @@ -238,8 +238,8 @@ theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) :
    have H2 : norm (x % y) * norm y ≤ norm y / 2 * norm y · calc norm (x % y) * norm y = norm (x % y * conj y) := by simp only [norm_mul, norm_conj] - _ = abs (Int.mod' (x.re * y.re + x.im * y.im) (norm y)) ^ 2 - + abs (Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)) ^ 2 := by simp [H1, norm, sq_abs] + _ = |Int.mod' (x.re * y.re + x.im * y.im) (norm y)| ^ 2 + + |Int.mod' (-(x.re * y.im) + x.im * y.re) (norm y)| ^ 2 := by simp [H1, norm, sq_abs] _ ≤ (y.norm / 2) ^ 2 + (y.norm / 2) ^ 2 := by gcongr <;> apply Int.abs_mod'_le _ _ norm_y_pos _ = norm y / 2 * (norm y / 2 * 2) := by ring _ ≤ norm y / 2 * norm y := by gcongr; apply Int.ediv_mul_le; norm_num
  • @@ -272,8 +272,8 @@ instance : EuclideanDomain gaussInt :=
    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 + 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)
• MIL/C07_Hierarchies/S03_Subobjects.lean
  • @@ -52,7 +52,7 @@ instance [Monoid M] : SubmonoidClass₁ (Submonoid₁ M) M where
    instance [Monoid M] : Inf (Submonoid₁ M) := - ⟨fun S₁ S₂ => + ⟨fun S₁ S₂ ↦ { carrier := S₁ ∩ S₂ one_mem := ⟨S₁.one_mem, S₂.one_mem⟩ mul_mem := fun ⟨hx, hx'⟩ ⟨hy, hy'⟩ ↦ ⟨S₁.mul_mem hx hy, S₂.mul_mem hx' hy'⟩ }⟩
• MIL/C07_Hierarchies/solutions/Solutions_S01_Basics.lean
  • @@ -344,5 +344,5 @@ class LT₁ (α : Type) where
    class PreOrder₂ (α : Type) extends LE₁ α, LT₁ α where le_refl : ∀ a : α, a ≤₁ a le_trans : ∀ a b c : α, a ≤₁ b → b ≤₁ c → a ≤₁ c - lt := fun a b => a ≤₁ b ∧ ¬b ≤₁ a + lt := fun a b ↦ a ≤₁ b ∧ ¬b ≤₁ a lt_iff_le_not_le : ∀ a b : α, a <₁ b ↔ a ≤₁ b ∧ ¬b ≤₁ a := by intros; rfl
• MIL/C07_Hierarchies/solutions/Solutions_S03_Subobjects.lean
  • @@ -80,7 +80,7 @@ instance [Group G] : SubgroupClass₁ (Subgroup₁ G) G :=
    instance [Monoid M] : Inf (Submonoid₁ M) := - ⟨fun S₁ S₂ => + ⟨fun S₁ S₂ ↦ { carrier := S₁ ∩ S₂ one_mem := ⟨S₁.one_mem, S₂.one_mem⟩ mul_mem := fun ⟨hx, hx'⟩ ⟨hy, hy'⟩ ↦ ⟨S₁.mul_mem hx hy, S₂.mul_mem hx' hy'⟩ }⟩
• MIL/C07_Topology/S01_Filters.lean
  • @@ -42,6 +42,7 @@ variable (f : ℝ → ℝ) (x₀ y₀ : ℝ)
    #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)
  • @@ -63,7 +64,7 @@ example (x₀ : ℝ) : HasBasis (𝓝 x₀) (fun ε : ℝ => 0 < ε) fun ε => I
    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 + have : atTop.HasBasis (fun n : ℕ => True) Ici := atTop_basis rw [this.tendsto_iff (nhds_basis_Ioo_pos x₀)] simp
  • @@ -80,7 +81,9 @@ example (u v : ℕ → ℝ) (h : u =ᶠ[atTop] v) (x₀ : ℝ) :
    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)
  • @@ -96,7 +99,9 @@ example (P Q R : ℕ → Prop) (hP : ∀ᶠ n in atTop, P n) (hQ : ∀ᶠ n in a
    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))
• MIL/C07_Topology/S02_Metric_Spaces.lean
  • @@ -2,28 +2,33 @@ 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 := + 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 := + 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) :
  • @@ -81,8 +86,7 @@ example {s : Set X} (hs : IsClosed s) {u : ℕ → X} (hu : Tendsto u atTop (
    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 := + 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 :=
  • @@ -98,18 +102,16 @@ 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) : + 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) : + 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 + hs.IsClosed example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := isCompact_univ
  • @@ -117,25 +119,20 @@ example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X)
    #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 := + 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} + 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 := + 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 := + 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) := + example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : ∃ x, Tendsto u atTop (𝓝 x) := cauchySeq_tendsto_of_complete hu open BigOperators
  • @@ -168,31 +165,28 @@ 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 - `closedBall center radius` is included both in `f n` and in `closedBall x δ`. + `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) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := + ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closed_ball y r ⊆ closed_ball 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] + rw [mem_closure_iff_nhds_basis nhds_basis_closed_ball] 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`. -/ + /- `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 := 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) + 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 incl : ∀ n, closed_ball (c (n + 1)) (r (n + 1)) ⊆ closed_ball (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`.
  • @@ -200,7 +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, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry - have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry + 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 sorry
• MIL/C07_Topology/S03_Topological_Spaces.lean
  • @@ -1,9 +1,12 @@
    import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus - open Set Filter Topology + open Set Filter + + open Topology Filter section + variable {X : Type _} [TopologicalSpace X] example : IsOpen (univ : Set X) :=
  • @@ -15,8 +18,7 @@ example : IsOpen (∅ : Set X) :=
    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) := + 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]
  • @@ -77,10 +79,8 @@ 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 + 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) :
  • @@ -96,13 +96,11 @@ 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 := + 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) : + 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
  • @@ -114,8 +112,7 @@ example [TopologicalSpace X] [TopologicalSpace Y] [RegularSpace Y] {A : Set X}
    #check @HasBasis.tendsto_right_iff - example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] - {s : Set X} {a : X} : + 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
  • @@ -124,8 +121,7 @@ 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 := + 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)
• MIL/C07_Topology/solutions/Solutions_S01_Filters.lean
  • @@ -69,3 +69,4 @@ 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) +
• MIL/C07_Topology/solutions/Solutions_S02_Metric_Spaces.lean
  • @@ -2,28 +2,33 @@ 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 := + 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 := + 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) :
  • @@ -84,8 +89,7 @@ example {s : Set X} (hs : IsClosed s) {u : ℕ → X} (hu : Tendsto u atTop (
    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 := + 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
  • @@ -110,18 +114,16 @@ 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) : + 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) : + 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 + hs.IsClosed example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X) := isCompact_univ
  • @@ -129,12 +131,10 @@ example {X : Type _} [MetricSpace X] [CompactSpace X] : IsCompact (univ : Set X)
    #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 := + 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} + example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace Y] {f : X → Y} (hf : Continuous f) : UniformContinuous f := sorry
  • @@ -146,13 +146,13 @@ example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace
    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 + have K_cpct : IsCompact K := K_closed.is_compact cases' eq_empty_or_nonempty K with hK hK · use 1, by norm_num - intro x y _ + intro x y hxy 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⟩ + simpa [K] + · rcases K_cpct.exists_forall_le hK continuous_dist.continuous_on with ⟨⟨x₀, x₁⟩, xx_in, H⟩ use dist x₀ x₁ constructor · change _ < _
  • @@ -165,16 +165,13 @@ example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace
    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 := + 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 := + 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) := + example [CompleteSpace X] (u : ℕ → X) (hu : CauchySeq u) : ∃ x, Tendsto u atTop (𝓝 x) := cauchySeq_tendsto_of_complete hu open BigOperators
  • @@ -208,8 +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, _, hN⟩ - exact ⟨N, by simpa using (hN N left_mem_Ici).2⟩ + 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⟩ use N intro n hn obtain ⟨k, rfl : n = N + k⟩ := le_iff_exists_add.mp hn
  • @@ -217,8 +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 _ => hu <| N + i) - _ = 1 / 2 ^ N * ∑ i in range k, (1 / 2 : ℝ) ^ 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 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) _)))
  • @@ -234,31 +231,28 @@ 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 - `closedBall center radius` is included both in `f n` and in `closedBall x δ`. + `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) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := + ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closed_ball y r ⊆ closed_ball 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] + rw [mem_closure_iff_nhds_basis nhds_basis_closed_ball] 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`. -/ + /- `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 := 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) + 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 incl : ∀ n, closed_ball (c (n + 1)) (r (n + 1)) ⊆ closed_ball (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`.
  • @@ -266,8 +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, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by sorry - have yball : ∀ n, y ∈ closedBall (c n) (r n) := by sorry + 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 sorry example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd : ∀ n, Dense (f n)) :
  • @@ -276,23 +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 - `closedBall center radius` is included both in `f n` and in `closedBall x δ`. + `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) ∧ closedBall y r ⊆ closedBall x δ ∩ f n := by + ∀ δ > 0, ∃ y : X, ∃ r > 0, r ≤ B (n + 1) ∧ closed_ball y r ⊆ closed_ball 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) + 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) 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 + show z ∈ closed_ball x δ exact calc dist z x ≤ dist z y + dist y x := dist_triangle _ _ _
  • @@ -308,10 +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_closedBall).2 fun ε εpos => _ + refine' fun x => (mem_closure_iff_nhds_basis nhds_basis_closed_ball).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 + `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 := fun n =>
  • @@ -328,18 +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, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) ∩ f n := fun n => + have incl : ∀ n, closed_ball (c (n + 1)) (r (n + 1)) ⊆ closed_ball (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 A : c (n + 1) ∈ closed_ball (c (n + 1)) (r (n + 1)) := + mem_closed_ball_self (rpos <| n + 1).le have I := calc - closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (c n) (r n) := + closed_ball (c (n + 1)) (r (n + 1)) ⊆ closed_ball (c n) (r n) := (incl n).trans (inter_subset_left _ _) - _ ⊆ closedBall (c n) (B n) := closedBall_subset_closedBall (rB n) + _ ⊆ closed_ball (c n) (B n) := closed_ball_subset_closed_ball (rB n) exact I A have : CauchySeq c := cauchySeq_of_le_geometric_two' cdist
  • @@ -348,20 +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, closedBall (c m) (r m) ⊆ closedBall (c n) (r n) := by + have I : ∀ n, ∀ m ≥ n, closed_ball (c m) (r m) ⊆ closed_ball (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 + have yball : ∀ n, y ∈ closed_ball (c n) (r n) := by intro n - refine' isClosed_ball.mem_of_tendsto ylim _ + refine' is_closed_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) + exact I n m hm (mem_closed_ball_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 := + have : closed_ball (c (n + 1)) (r (n + 1)) ⊆ f n := Subset.trans (incl n) (inter_subset_right _ _) exact this (yball (n + 1)) calc
• MIL/C07_Topology/solutions/Solutions_S03_Topological_Spaces.lean
  • @@ -1,9 +1,12 @@
    import Mathlib.Topology.Instances.Real import Mathlib.Analysis.NormedSpace.BanachSteinhaus - open Set Filter Topology + open Set Filter + + open Topology Filter section + variable {X : Type _} [TopologicalSpace X] example : IsOpen (univ : Set X) :=
  • @@ -15,8 +18,7 @@ example : IsOpen (∅ : Set X) :=
    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) := + 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]
  • @@ -85,10 +87,8 @@ 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 + 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) :
  • @@ -104,19 +104,16 @@ 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 := + 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) : + 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) : + 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
  • @@ -138,7 +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 - simpa [ContinuousAt, (closed_nhds_basis (φ x)).tendsto_right_iff] + simp [ContinuousAt, (closed_nhds_basis _).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'
  • @@ -150,10 +147,9 @@ 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 lim f_cont.continuousAt + exact tendsto_nhds_unique limUnder f_cont.continuousAt - example [TopologicalSpace X] [TopologicalSpace.FirstCountableTopology X] - {s : Set X} {a : X} : + 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
  • @@ -162,8 +158,7 @@ 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 := + 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)
• MIL/C08_Topology/S01_Filters.lean
  • @@ -58,12 +58,12 @@ example (f : ℕ → ℝ × ℝ) (x₀ y₀ : ℝ) :
    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) := + 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 + have : atTop.HasBasis (fun _ : ℕ ↦ True) Ici := atTop_basis rw [this.tendsto_iff (nhds_basis_Ioo_pos x₀)] simp
• MIL/C08_Topology/S02_Metric_Spaces.lean
  • @@ -27,27 +27,27 @@ example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} :
    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 + 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 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 + 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) := + 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) := + 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) := + 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) :
  • @@ -163,7 +163,7 @@ 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 + 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
  • @@ -184,11 +184,11 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :
    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 => + 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 + 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
• MIL/C08_Topology/S03_Topological_Spaces.lean
  • @@ -85,7 +85,7 @@ example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : Topological
    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) := + ⨅ 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))
  • @@ -93,11 +93,11 @@ example [TopologicalSpace X] [T2Space X] {u : ℕ → X} {a b : X} (ha : Tendsto
    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 := + (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 := + (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}
• MIL/C08_Topology/solutions/Solutions_S01_Filters.lean
  • @@ -6,8 +6,8 @@ open Set Filter Topology
    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 } + 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 }
• MIL/C08_Topology/solutions/Solutions_S02_Metric_Spaces.lean
  • @@ -27,30 +27,30 @@ example {X Y : Type _} [MetricSpace X] [MetricSpace Y] {f : X → Y} :
    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 + 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 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 + 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) := + 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) := + 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) := + 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) := + 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) :
  • @@ -142,7 +142,7 @@ example {X : Type _} [MetricSpace X] [CompactSpace X] {Y : Type _} [MetricSpace
    (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) + 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
  • @@ -202,8 +202,8 @@ example {u : ℕ → X} (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^
    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)] + have : Tendsto (fun N : ℕ ↦ (1 / 2 ^ N * 2 : ℝ)) atTop (𝓝 0) := by + rw [← zero_mul (2 : ℝ)] apply Tendsto.mul simp_rw [← one_div_pow (2 : ℝ)] apply tendsto_pow_atTop_nhds_0_of_lt_1 <;> linarith
  • @@ -216,8 +216,8 @@ example {u : ℕ → X} (hu : ∀ n : ℕ, dist (u n) (u (n + 1)) ≤ (1 / 2) ^
    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) + (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 _)
  • @@ -229,7 +229,7 @@ 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 + 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
  • @@ -250,11 +250,11 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :
    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 => + 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 + 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
  • @@ -272,8 +272,8 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :
    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 + 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`.
  • @@ -287,7 +287,7 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :
    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 => ⟨_, _⟩⟩ + 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)
  • @@ -308,16 +308,16 @@ 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_closedBall).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 `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 + 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
  • @@ -328,7 +328,7 @@ 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, closedBall (c (n + 1)) (r (n + 1)) ⊆ closedBall (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
  • @@ -350,13 +350,13 @@ example [CompleteSpace X] (f : ℕ → Set X) (ho : ∀ n, IsOpen (f n)) (hd :
    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 => _ + 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 => _ + 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]
• MIL/C08_Topology/solutions/Solutions_S03_Topological_Spaces.lean
  • @@ -55,7 +55,7 @@ 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⟩ + 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
  • @@ -93,7 +93,7 @@ example {Z : Type _} (f : X → Y) (T_X : TopologicalSpace X) (T_Z : Topological
    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) := + ⨅ 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))
  • @@ -101,11 +101,11 @@ example [TopologicalSpace X] [T2Space X] {u : ℕ → X} {a b : X} (ha : Tendsto
    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 := + (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 := + (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}
• MIL/C09_Differential_Calculus/S02_Differential_Calculus_in_Normed_Spaces.lean
  • @@ -27,7 +27,7 @@ example (x y : E) : ‖x + y‖ ≤ ‖x‖ + ‖y‖ :=
    example : MetricSpace E := by infer_instance example {X : Type _} [TopologicalSpace X] {f : X → E} (hf : Continuous f) : - Continuous fun x => ‖f x‖ := + Continuous fun x ↦ ‖f x‖ := hf.norm variable [NormedSpace ℝ E]
  • @@ -89,7 +89,7 @@ open Metric
    example {ι : Type _} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : ∃ C', ∀ i, ‖g i‖ ≤ C' := by -- sequence of subsets consisting of those `x : E` with norms `‖g i x‖` bounded by `n` - let e : ℕ → Set E := fun n => ⋂ i : ι, { x : E | ‖g i x‖ ≤ n } + let e : ℕ → Set E := fun n ↦ ⋂ i : ι, { x : E | ‖g i x‖ ≤ n } -- each of these sets is closed have hc : ∀ n : ℕ, IsClosed (e n) sorry
  • @@ -105,7 +105,7 @@ example {ι : Type _} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x,
    have real_norm_le : ∀ z ∈ ball x ε, ∀ (i : ι), ‖g i z‖ ≤ m sorry have εk_pos : 0 < ε / ‖k‖ := sorry - refine' ⟨(m + m : ℕ) / (ε / ‖k‖), fun i => ContinuousLinearMap.op_norm_le_of_shell ε_pos _ hk _⟩ + refine' ⟨(m + m : ℕ) / (ε / ‖k‖), fun i ↦ ContinuousLinearMap.op_norm_le_of_shell ε_pos _ hk _⟩ sorry sorry
  • @@ -137,7 +137,7 @@ variable {𝕜 : Type _} [NontriviallyNormedField 𝕜] {E : Type _} [NormedAddC
    [NormedSpace 𝕜 E] {F : Type _} [NormedAddCommGroup F] [NormedSpace 𝕜 F] example (f : E → F) (f' : E →L[𝕜] F) (x₀ : E) : - HasFDerivAt f f' x₀ ↔ (fun x => f x - f x₀ - f' (x - x₀)) =o[𝓝 x₀] fun x => x - x₀ := + HasFDerivAt f f' x₀ ↔ (fun x ↦ f x - f x₀ - f' (x - x₀)) =o[𝓝 x₀] fun x ↦ x - x₀ := Iff.rfl example (f : E → F) (f' : E →L[𝕜] F) (x₀ : E) (hff' : HasFDerivAt f f' x₀) : fderiv 𝕜 f x₀ = f' :=
  • @@ -148,8 +148,8 @@ example (n : ℕ) (f : E → F) : E → E[×n]→L[𝕜] F :=
    example (n : WithTop ℕ) {f : E → F} : ContDiff 𝕜 n f ↔ - (∀ m : ℕ, (m : WithTop ℕ) ≤ n → Continuous fun x => iteratedFDeriv 𝕜 m f x) ∧ - ∀ m : ℕ, (m : WithTop ℕ) < n → Differentiable 𝕜 fun x => iteratedFDeriv 𝕜 m f x := + (∀ m : ℕ, (m : WithTop ℕ) ≤ n → Continuous fun x ↦ iteratedFDeriv 𝕜 m f x) ∧ + ∀ m : ℕ, (m : WithTop ℕ) < n → Differentiable 𝕜 fun x ↦ iteratedFDeriv 𝕜 m f x := contDiff_iff_continuous_differentiable example {𝕂 : Type _} [IsROrC 𝕂] {E : Type _} [NormedAddCommGroup E] [NormedSpace 𝕂 E] {F : Type _}
• MIL/C09_Differential_Calculus/solutions/Solutions_S02_Differential_Calculus_in_Normed_Spaces.lean
  • @@ -21,16 +21,16 @@ open Metric
    example {ι : Type _} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x, ∃ C, ∀ i, ‖g i x‖ ≤ C) : ∃ C', ∀ i, ‖g i‖ ≤ C' := by -- sequence of subsets consisting of those `x : E` with norms `‖g i x‖` bounded by `n` - let e : ℕ → Set E := fun n => ⋂ i : ι, { x : E | ‖g i x‖ ≤ n } + let e : ℕ → Set E := fun n ↦ ⋂ i : ι, { x : E | ‖g i x‖ ≤ n } -- each of these sets is closed - have hc : ∀ n : ℕ, IsClosed (e n) := fun i => - isClosed_iInter fun i => isClosed_le (g i).cont.norm continuous_const + have hc : ∀ n : ℕ, IsClosed (e n) := fun i ↦ + isClosed_iInter fun i ↦ isClosed_le (g i).cont.norm continuous_const -- the union is the entire space; this is where we use `h` have hU : (⋃ n : ℕ, e n) = univ := by - refine' eq_univ_of_forall fun x => _ + refine' eq_univ_of_forall fun x ↦ _ cases' h x with C hC obtain ⟨m, hm⟩ := exists_nat_ge C - exact ⟨e m, mem_range_self m, mem_iInter.mpr fun i => le_trans (hC i) hm⟩ + exact ⟨e m, mem_range_self m, mem_iInter.mpr fun i ↦ le_trans (hC i) hm⟩ /- apply the Baire category theorem to conclude that for some `m : ℕ`, `e m` contains some `x` -/ obtain ⟨m : ℕ, x : E, hx : x ∈ interior (e m)⟩ := nonempty_interior_of_iUnion_of_closed hc hU
  • @@ -42,7 +42,7 @@ example {ι : Type _} [CompleteSpace E] {g : ι → E →L[𝕜] F} (h : ∀ x,
    replace hz := mem_iInter.mp (interior_iInter_subset _ (hε hz)) i apply interior_subset hz have εk_pos : 0 < ε / ‖k‖ := div_pos ε_pos (zero_lt_one.trans hk) - refine' ⟨(m + m : ℕ) / (ε / ‖k‖), fun i => ContinuousLinearMap.op_norm_le_of_shell ε_pos _ hk _⟩ + refine' ⟨(m + m : ℕ) / (ε / ‖k‖), fun i ↦ ContinuousLinearMap.op_norm_le_of_shell ε_pos _ hk _⟩ · exact div_nonneg (Nat.cast_nonneg _) εk_pos.le intro y le_y y_lt calc
• MIL/C10_Integration_and_Measure_Theory/S01_Elementary_Integration.lean
  • @@ -21,7 +21,7 @@ example (a b : ℝ) : (∫ x in a..b, x) = (b ^ 2 - a ^ 2) / 2 :=
    example {a b : ℝ} (h : (0 : ℝ) ∉ [[a, b]]) : (∫ x in a..b, 1 / x) = Real.log (b / a) := integral_one_div h - example (f : ℝ → ℝ) (hf : Continuous f) (a b : ℝ) : deriv (fun u => ∫ x : ℝ in a..u, f x) b = f b := + example (f : ℝ → ℝ) (hf : Continuous f) (a b : ℝ) : deriv (fun u ↦ ∫ x : ℝ in a..u, f x) b = f b := (integral_hasStrictDerivAt_right (hf.intervalIntegrable _ _) (hf.stronglyMeasurableAtFilter _ _) hf.continuousAt).hasDerivAt.deriv
  • @@ -31,5 +31,5 @@ example {f : ℝ → ℝ} {a b : ℝ} {f' : ℝ → ℝ} (h : ∀ x ∈ [[a, b]]
    open Convolution - example (f : ℝ → ℝ) (g : ℝ → ℝ) : f ⋆ g = fun x => ∫ t, f t * g (x - t) := + example (f : ℝ → ℝ) (g : ℝ → ℝ) : f ⋆ g = fun x ↦ ∫ t, f t * g (x - t) := rfl
• MIL/C10_Integration_and_Measure_Theory/S03_Integration.lean
  • @@ -26,8 +26,8 @@ example {s : Set α} (c : E) : (∫ x in s, c ∂μ) = (μ s).toReal • c :=
    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))) : - Tendsto (fun n => ∫ a, F n a ∂μ) atTop (𝓝 (∫ a, f a ∂μ)) := + (hlim : ∀ᵐ a ∂μ, Tendsto (fun n : ℕ ↦ F n a) atTop (𝓝 (f a))) : + Tendsto (fun n ↦ ∫ a, F n a ∂μ) atTop (𝓝 (∫ a, f a ∂μ)) := tendsto_integral_of_dominated_convergence bound hmeas hint hbound hlim example {α : Type _} [MeasurableSpace α] {μ : Measure α} [SigmaFinite μ] {β : Type _}
  • @@ -47,7 +47,7 @@ variable {𝕜 : Type _} {G : Type _} {E : Type _} {E' : Type _} {F : Type _} [N
    [Sub G] example (f : G → E) (g : G → E') (L : E →L[𝕜] E' →L[𝕜] F) (μ : Measure G) : - f ⋆[L, μ] g = fun x => ∫ t, L (f t) (g (x - t)) ∂μ := + f ⋆[L, μ] g = fun x ↦ ∫ t, L (f t) (g (x - t)) ∂μ := rfl end
• html/.buildinfo
  • @@ -1,4 +1,4 @@
    # Sphinx build info version 1 # This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done. - config: 88166e69df2da4dbd33869bb72fe3995 + config: 798c3985ff3075064f96be164dceb162 tags: 645f666f9bcd5a90fca523b33c5a78b7
• html/C01_Introduction.html
  • @@ -1,7 +1,7 @@
    <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>1. Introduction &mdash; Mathematics in Lean 0.1 documentation</title>
  • @@ -13,11 +13,11 @@
    <script src="_static/js/html5shiv.min.js"></script> <![endif]--> - <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/_sphinx_javascript_frameworks_compat.js"></script> + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.js"></script> <script src="_static/doctools.js"></script> + <script src="_static/sphinx_highlight.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" /> <link rel="search" title="Search" href="search.html" />
  • @@ -30,11 +30,15 @@
    <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" /> + <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
  • @@ -49,7 +53,7 @@
    <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> - <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Elementary Number Theory</a></li> <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li>
  • @@ -73,8 +77,8 @@
    <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> - <li><a href="index.html" class="icon icon-home"></a> &raquo;</li> - <li><span class="section-number">1. </span>Introduction</li> + <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 class="wy-breadcrumbs-aside"> <a href="_sources/C01_Introduction.rst.txt" rel="nofollow"> View page source</a> </li>
  • @@ -130,9 +134,7 @@ version of the library, <code class="docutils literal notranslate"><span class="
    choose <code class="docutils literal notranslate"><span class="pre">Open</span> <span class="pre">Folder</span></code> from the <code class="docutils literal notranslate"><span class="pre">File</span></code> menu. Be sure to open the folder <code class="docutils literal notranslate"><span class="pre">mathematics_in_lean</span></code>, not any other folder.</p></li> </ol> - <p>Opening any Lean file will simultaneously open this - book in a VS Code window. You can update to a newer version by tying - <code class="docutils literal notranslate"><span class="pre">git</span> <span class="pre">pull</span></code> followed by <code class="docutils literal notranslate"><span class="pre">lake</span> <span class="pre">exe</span> <span class="pre">cache</span> <span class="pre">get</span></code> inside + <p>You can update to a newer version by tying <code class="docutils literal notranslate"><span class="pre">git</span> <span class="pre">pull</span></code> followed by <code class="docutils literal notranslate"><span class="pre">lake</span> <span class="pre">exe</span> <span class="pre">cache</span> <span class="pre">get</span></code> inside the <code class="docutils literal notranslate"><span class="pre">mathematics_in_lean</span></code> folder.</p> <p>Alternatively, you can run Lean and VS Code in the cloud, using <a class="reference external" href="https://gitpod.io/">Gitpod</a>.
  • @@ -227,17 +229,18 @@ we can write down the expressions themselves
    or we can provide Lean with <em>instructions</em> as to how to construct them. For example, the following expression represents a proof of the fact that if <code class="docutils literal notranslate"><span class="pre">n</span></code> is even then so is <code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">*</span> <span class="pre">n</span></code>:</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">Nat</span><span class="o">,</span> <span class="n">Even</span> <span class="n">n</span> <span class="bp">&#8594;</span> <span class="n">Even</span> <span class="o">(</span><span class="n">m</span> <span class="bp">*</span> <span class="n">n</span><span class="o">)</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">m</span> <span class="n">n</span> <span class="o">&#10216;</span><span class="n">k</span><span class="o">,</span> <span class="o">(</span><span class="n">hk</span> <span class="o">:</span> <span class="n">n</span> <span class="bp">=</span> <span class="n">k</span> <span class="bp">+</span> <span class="n">k</span><span class="o">)&#10217;</span> <span class="bp">=&gt;</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">Nat</span><span class="o">,</span> <span class="n">Even</span> <span class="n">n</span> <span class="bp">&#8594;</span> <span class="n">Even</span> <span class="o">(</span><span class="n">m</span> <span class="bp">*</span> <span class="n">n</span><span class="o">)</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">m</span> <span class="n">n</span> <span class="o">&#10216;</span><span class="n">k</span><span class="o">,</span> <span class="o">(</span><span class="n">hk</span> <span class="o">:</span> <span class="n">n</span> <span class="bp">=</span> <span class="n">k</span> <span class="bp">+</span> <span class="n">k</span><span class="o">)&#10217;</span> <span class="bp">&#8614;</span> <span class="k">have</span> <span class="n">hmn</span> <span class="o">:</span> <span class="n">m</span> <span class="bp">*</span> <span class="n">n</span> <span class="bp">=</span> <span class="n">m</span> <span class="bp">*</span> <span class="n">k</span> <span class="bp">+</span> <span class="n">m</span> <span class="bp">*</span> <span class="n">k</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">hk</span><span class="o">,</span> <span class="n">mul_add</span><span class="o">]</span> <span class="k">show</span> <span class="bp">&#8707;</span> <span class="n">l</span><span class="o">,</span> <span class="n">m</span> <span class="bp">*</span> <span class="n">n</span> <span class="bp">=</span> <span class="n">l</span> <span class="bp">+</span> <span class="n">l</span> <span class="k">from</span> <span class="o">&#10216;</span><span class="n">_</span><span class="o">,</span> <span class="n">hmn</span><span class="o">&#10217;</span> </pre></div> </div> <p>The <em>proof term</em> can be compressed to a single line:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">Nat</span><span class="o">,</span> <span class="n">Even</span> <span class="n">n</span> <span class="bp">&#8594;</span> <span class="n">Even</span> <span class="o">(</span><span class="n">m</span> <span class="bp">*</span> <span class="n">n</span><span class="o">)</span> <span class="o">:=</span> - <span class="k">fun</span> <span class="n">m</span> <span class="n">n</span> <span class="o">&#10216;</span><span class="n">k</span><span class="o">,</span> <span class="n">hk</span><span class="o">&#10217;</span> <span class="bp">=&gt;</span> <span class="o">&#10216;</span><span class="n">m</span> <span class="bp">*</span> <span class="n">k</span><span class="o">,</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">hk</span><span class="o">,</span> <span class="n">mul_add</span><span class="o">]&#10217;</span> + <span class="k">fun</span> <span class="n">m</span> <span class="n">n</span> <span class="o">&#10216;</span><span class="n">k</span><span class="o">,</span> <span class="n">hk</span><span class="o">&#10217;</span> <span class="bp">&#8614;</span> <span class="o">&#10216;</span><span class="n">m</span> <span class="bp">*</span> <span class="n">k</span><span class="o">,</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">hk</span><span class="o">,</span> <span class="n">mul_add</span><span class="o">]&#10217;</span> </pre></div> </div> - <p>The following is, instead, a <em>tactic-style</em> proof of the same theorem:</p> + <p>The following is, instead, a <em>tactic-style</em> proof of the same theorem, where lines + starting with <code class="docutils literal notranslate"><span class="pre">--</span></code> are comments, hence ignored by Lean:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">m</span> <span class="n">n</span> <span class="o">:</span> <span class="n">Nat</span><span class="o">,</span> <span class="n">Even</span> <span class="n">n</span> <span class="bp">&#8594;</span> <span class="n">Even</span> <span class="o">(</span><span class="n">m</span> <span class="bp">*</span> <span class="n">n</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="c1">-- say m and n are natural numbers, and assume n=2*k</span> <span class="n">rintro</span> <span class="n">m</span> <span class="n">n</span> <span class="o">&#10216;</span><span class="n">k</span><span class="o">,</span> <span class="n">hk</span><span class="o">&#10217;</span>
  • @@ -299,11 +302,11 @@ powerful and ever-growing library, <em>mathlib</em>.
    As a result, we can show you how to use some of the mathematical objects and theorems in the library, and some of the very useful tactics. - This book is not meant to be used as an overview of the library; + This book is not meant to be used as an complete overview of the library; the <a class="reference external" href="https://leanprover-community.github.io/">community</a> web pages contain extensive documentation. Rather, our goal is to introduce you to the style of thinking that - underlies that formalization, + underlies that formalization, and point out basic entry points so that you are comfortable browsing the library and finding things on your own.</p> <p>Interactive theorem proving can be frustrating,
• html/C02_Basics.html
  • @@ -1,7 +1,7 @@
    <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>2. Basics &mdash; Mathematics in Lean 0.1 documentation</title>
  • @@ -13,11 +13,11 @@
    <script src="_static/js/html5shiv.min.js"></script> <![endif]--> - <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/_sphinx_javascript_frameworks_compat.js"></script> + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.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" />
  • @@ -31,11 +31,15 @@
    <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" /> + <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
  • @@ -47,13 +51,13 @@
    <li class="toctree-l2"><a class="reference internal" href="#calculating">2.1. Calculating</a></li> <li class="toctree-l2"><a class="reference internal" href="#proving-identities-in-algebraic-structures">2.2. Proving Identities in Algebraic Structures</a></li> <li class="toctree-l2"><a class="reference internal" href="#using-theorems-and-lemmas">2.3. Using Theorems and Lemmas</a></li> - <li class="toctree-l2"><a class="reference internal" href="#more-on-order-and-divisibility">2.4. More on Order and Divisibility</a></li> + <li class="toctree-l2"><a class="reference internal" href="#more-examples-using-apply-and-rw">2.4. More examples using apply and rw</a></li> <li class="toctree-l2"><a class="reference internal" href="#proving-facts-about-algebraic-structures">2.5. Proving Facts about Algebraic Structures</a></li> </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> - <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Elementary Number Theory</a></li> <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li>
  • @@ -77,8 +81,8 @@
    <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> - <li><a href="index.html" class="icon icon-home"></a> &raquo;</li> - <li><span class="section-number">2. </span>Basics</li> + <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 class="wy-breadcrumbs-aside"> <a href="_sources/C02_Basics.rst.txt" rel="nofollow"> View page source</a> </li>
  • @@ -576,8 +580,16 @@ After that, we are back to proving the original goal,
    except a new hypothesis <code class="docutils literal notranslate"><span class="pre">h</span></code> has been added: having proved it, we are now free to use it. At this point, the goal is exactly the result of <code class="docutils literal notranslate"><span class="pre">add_left_cancel</span> <span class="pre">h</span></code>. + .. index:: apply, tactics ; apply, exact, tactics ; exact We could equally well have closed the proof with - <code class="docutils literal notranslate"><span class="pre">apply</span> <span class="pre">add_left_cancel</span> <span class="pre">h</span></code> or <code class="docutils literal notranslate"><span class="pre">exact</span> <span class="pre">add_left_cancel</span> <span class="pre">h</span></code>.</p> + <code class="docutils literal notranslate"><span class="pre">apply</span> <span class="pre">add_left_cancel</span> <span class="pre">h</span></code> or <code class="docutils literal notranslate"><span class="pre">exact</span> <span class="pre">add_left_cancel</span> <span class="pre">h</span></code>. + The <code class="docutils literal notranslate"><span class="pre">exact</span></code> tactic takes as argument a proof term which completely proves the + current goal, without creating any new goal. The <code class="docutils literal notranslate"><span class="pre">apply</span></code> tactic is a variant + whose argument is not necessarily a complete proof. The missing pieces are either + inferred automatically by Lean or become new goals to prove. + While the <code class="docutils literal notranslate"><span class="pre">exact</span></code> tactic is technically redundant since it is strictly less powerful + than <code class="docutils literal notranslate"><span class="pre">apply</span></code>, it makes proof scripts slightly clearer tho + human readers and easier to maintain when the library evolves.</p> <p>Remember that multiplication is not assumed to be commutative, so the following theorem also requires some work.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">zero_mul</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="mi">0</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>
  • @@ -620,11 +632,10 @@ addition of the additive inverse.</p>
    <span class="n">rfl</span> </pre></div> </div> - <p id="index-12">The proof term <code class="docutils literal notranslate"><span class="pre">rfl</span></code> is short for <code class="docutils literal notranslate"><span class="pre">reflexivity</span></code>. + <p id="index-12">The proof term <code class="docutils literal notranslate"><span class="pre">rfl</span></code> is short for &#8220;reflexivity&#8221;. Presenting it as a proof of <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">-</span> <span class="pre">b</span> <span class="pre">=</span> <span class="pre">a</span> <span class="pre">+</span> <span class="pre">-b</span></code> forces Lean to unfold the definition and recognize both sides as being the same. - The <code class="docutils literal notranslate"><span class="pre">reflexivity</span></code> tactic, which can be abbreviated as <code class="docutils literal notranslate"><span class="pre">rfl</span></code>, - does the same. + The <code class="docutils literal notranslate"><span class="pre">rfl</span></code> tactic does the same. This is an instance of what is known as a <em>definitional equality</em> in Lean&#8217;s underlying logic. This means that not only can one rewrite with <code class="docutils literal notranslate"><span class="pre">sub_eq_add_neg</span></code>
  • @@ -718,7 +729,7 @@ In this section, we will make good use of these tools.</p>
    <p>As we explain in more detail in <a class="reference internal" href="C03_Logic.html#implication-and-the-universal-quantifier"><span class="std std-numref">Section 3.1</span></a>, the implicit parentheses in the statement of <code class="docutils literal notranslate"><span class="pre">le_trans</span></code> associate to the right, so it should be interpreted as <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">&#8804;</span> <span class="pre">b</span> <span class="pre">&#8594;</span> <span class="pre">(b</span> <span class="pre">&#8804;</span> <span class="pre">c</span> <span class="pre">&#8594;</span> <span class="pre">a</span> <span class="pre">&#8804;</span> <span class="pre">c)</span></code>. - The library designers have set the arguments to <code class="docutils literal notranslate"><span class="pre">le_trans</span></code> implicit, + The library designers have set the arguments <code class="docutils literal notranslate"><span class="pre">a</span></code>, <code class="docutils literal notranslate"><span class="pre">b</span></code> and <code class="docutils literal notranslate"><span class="pre">c</span></code> to <code class="docutils literal notranslate"><span class="pre">le_trans</span></code> implicit, so that Lean will <em>not</em> let you provide them explicitly (unless you really insist, as we will discuss later). Rather, it expects to infer them from the context in which they are used.
  • @@ -881,31 +892,33 @@ part of formalization.
    There are a number of strategies you can use:</p> <ul class="simple"> <li><p>You can browse mathlib in its - <a class="reference external" href="https://github.com/leanprover-community/mathlib">GitHub repository</a>.</p></li> + <a class="reference external" href="https://github.com/leanprover-community/mathlib4">GitHub repository</a>.</p></li> <li><p>You can use the API documentation on the mathlib - <a class="reference external" href="https://leanprover-community.github.io/mathlib_docs/">web pages</a>.</p></li> - <li><p>You can rely on mathlib naming conventions and tab completion in - the editor to guess a theorem name. + <a class="reference external" href="https://leanprover-community.github.io/mathlib4_docs/">web pages</a>.</p></li> + <li><p>You can rely on mathlib naming conventions and Ctrl-space completion in + the editor to guess a theorem name (or Cmd-space on a Mac keyboard). In Lean, a theorem named <code class="docutils literal notranslate"><span class="pre">A_of_B_of_C</span></code> establishes something of the form <code class="docutils literal notranslate"><span class="pre">A</span></code> from hypotheses of the form <code class="docutils literal notranslate"><span class="pre">B</span></code> and <code class="docutils literal notranslate"><span class="pre">C</span></code>, where <code class="docutils literal notranslate"><span class="pre">A</span></code>, <code class="docutils literal notranslate"><span class="pre">B</span></code>, and <code class="docutils literal notranslate"><span class="pre">C</span></code> approximate the way we might read the goals out loud. So a theorem establishing something like <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">+</span> <span class="pre">y</span> <span class="pre">&#8804;</span> <span class="pre">...</span></code> will probably start with <code class="docutils literal notranslate"><span class="pre">add_le</span></code>. - Typing <code class="docutils literal notranslate"><span class="pre">add_le</span></code> and hitting tab will give you some helpful choices.</p></li> + Typing <code class="docutils literal notranslate"><span class="pre">add_le</span></code> and hitting Ctrl-space will give you some helpful choices. + Note that hitting Ctrl-space twice displays more information about the available + completions.</p></li> <li><p>If you right-click on an existing theorem name in VS Code, the editor will show a menu with the option to jump to the file where the theorem is defined, and you can find similar theorems nearby.</p></li> - <li><p>You can use the <code class="docutils literal notranslate"><span class="pre">library_search</span></code> tactic, + <li><p>You can use the <code class="docutils literal notranslate"><span class="pre">apply?</span></code> tactic, which tries to find the relevant theorem in the library.</p></li> </ul> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">&#8804;</span> <span class="n">a</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> - <span class="c1">-- library_search</span> + <span class="c1">-- apply?</span> <span class="n">exact</span> <span class="n">sq_nonneg</span> <span class="n">a</span> </pre></div> </div> - <p>To try out <code class="docutils literal notranslate"><span class="pre">library_search</span></code> in this example, + <p>To try out <code class="docutils literal notranslate"><span class="pre">apply?</span></code> in this example, delete the <code class="docutils literal notranslate"><span class="pre">exact</span></code> command and uncomment the previous line. Using these tricks, see if you can find what you need to do the
  • @@ -914,7 +927,7 @@ next example:</p>
    <span class="gr">sorry</span> </pre></div> </div> - <p>Using the same tricks, confirm that <code class="docutils literal notranslate"><span class="pre">linarith</span></code> instead of <code class="docutils literal notranslate"><span class="pre">library_search</span></code> + <p>Using the same tricks, confirm that <code class="docutils literal notranslate"><span class="pre">linarith</span></code> instead of <code class="docutils literal notranslate"><span class="pre">apply?</span></code> can also finish the job.</p> <p>Here is another example of an inequality:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">a</span> <span class="bp">*</span> <span class="n">b</span> <span class="bp">&#8804;</span> <span class="n">a</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span>
  • @@ -957,7 +970,7 @@ linear arithmetic, and <code class="docutils literal notranslate"><span class="p
    </div> <p>How nice! We challenge you to use these ideas to prove the following theorem. You can use the theorem <code class="docutils literal notranslate"><span class="pre">abs_le'.mpr</span></code>.</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">abs</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="bp">&#8804;</span> <span class="o">(</span><span class="n">a</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">/</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="bp">|</span><span class="n">a</span> <span class="bp">*</span> <span class="n">b</span><span class="bp">|</span> <span class="bp">&#8804;</span> <span class="o">(</span><span class="n">a</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">b</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">/</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">#check</span> <span class="n">abs_le&#39;.mpr</span>
  • @@ -966,8 +979,8 @@ following theorem. You can use the theorem <code class="docutils literal notrans
    <p>If you managed to solve this, congratulations! You are well on your way to becoming a master formalizer.</p> </section> - <section id="more-on-order-and-divisibility"> - <span id="id4"></span><h2><span class="section-number">2.4. </span>More on Order and Divisibility<a class="headerlink" href="#more-on-order-and-divisibility" title="Permalink to this heading">&#61633;</a></h2> + <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">&#61633;</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">&#8804;</span> <span class="n">a</span><span class="o">)</span>
  • @@ -1102,11 +1115,11 @@ As a hint, you can use the theorem <code class="docutils literal notranslate"><s
    and the <code class="docutils literal notranslate"><span class="pre">linarith</span></code> tactic.</p> <p id="index-23">Lean&#8217;s naming convention is made manifest in the library&#8217;s name for the triangle inequality:</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="o">(</span><span class="n">abs_add</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">,</span> <span class="n">abs</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="bp">&#8804;</span> <span class="n">abs</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">abs</span> <span class="n">b</span><span class="o">)</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="o">(</span><span class="n">abs_add</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">,</span> <span class="bp">|</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="bp">|</span> <span class="bp">&#8804;</span> <span class="bp">|</span><span class="n">a</span><span class="bp">|</span> <span class="bp">+</span> <span class="bp">|</span><span class="n">b</span><span class="bp">|</span><span class="o">)</span> </pre></div> </div> <p>Use it to prove the following variant:</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">abs</span> <span class="n">a</span> <span class="bp">-</span> <span class="n">abs</span> <span class="n">b</span> <span class="bp">&#8804;</span> <span class="n">abs</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> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="bp">|</span><span class="n">a</span><span class="bp">|</span> <span class="bp">-</span> <span class="bp">|</span><span class="n">b</span><span class="bp">|</span> <span class="bp">&#8804;</span> <span class="bp">|</span><span class="n">a</span> <span class="bp">-</span> <span class="n">b</span><span class="bp">|</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div>
  • @@ -1176,7 +1189,7 @@ You can use <code class="docutils literal notranslate"><span class="pre">_root_.
    either one will work.</p> </section> <section id="proving-facts-about-algebraic-structures"> - <span id="id5"></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">&#61633;</a></h2> + <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">&#61633;</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,
• html/C03_Logic.html
  • @@ -1,7 +1,7 @@
    <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>3. Logic &mdash; Mathematics in Lean 0.1 documentation</title>
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  • @@ -31,11 +31,15 @@
    <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" /> + <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
  • @@ -54,7 +58,7 @@
    </ul> </li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> - <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Elementary Number Theory</a></li> <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li>
  • @@ -78,8 +82,8 @@
    <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> - <li><a href="index.html" class="icon icon-home"></a> &raquo;</li> - <li><span class="section-number">3. </span>Logic</li> + <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 class="wy-breadcrumbs-aside"> <a href="_sources/C03_Logic.rst.txt" rel="nofollow"> View page source</a> </li>
  • @@ -103,13 +107,13 @@ 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">&#61633;</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">&#8704;</span> <span class="n">x</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">,</span> <span class="mi">0</span> <span class="bp">&#8804;</span> <span class="n">x</span> <span class="bp">&#8594;</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">x</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="bp">&#8704;</span> <span class="n">x</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">,</span> <span class="mi">0</span> <span class="bp">&#8804;</span> <span class="n">x</span> <span class="bp">&#8594;</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> </div> <p>In words, we would say &#8220;for every real number <code class="docutils literal notranslate"><span class="pre">x</span></code>, if <code class="docutils literal notranslate"><span class="pre">0</span> <span class="pre">&#8804;</span> <span class="pre">x</span></code> then the absolute value of <code class="docutils literal notranslate"><span class="pre">x</span></code> equals <code class="docutils literal notranslate"><span class="pre">x</span></code>&#8221;. We can also have more complicated statements like:</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="bp">&#8704;</span> <span class="n">x</span> <span class="n">y</span> <span class="n">&#949;</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">,</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">&#949;</span> <span class="bp">&#8804;</span> <span class="mi">1</span> <span class="bp">&#8594;</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">abs</span> <span class="n">y</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">abs</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="bp">&lt;</span> <span class="n">&#949;</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="bp">&#8704;</span> <span class="n">x</span> <span class="n">y</span> <span class="n">&#949;</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">,</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">&#949;</span> <span class="bp">&#8804;</span> <span class="mi">1</span> <span class="bp">&#8594;</span> <span class="bp">|</span><span class="n">x</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="bp">|</span><span class="n">y</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="bp">|</span><span class="n">x</span> <span class="bp">*</span> <span class="n">y</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> </pre></div> </div> <p>In words, we would say &#8220;for every <code class="docutils literal notranslate"><span class="pre">x</span></code>, <code class="docutils literal notranslate"><span class="pre">y</span></code>, and <code class="docutils literal notranslate"><span class="pre">&#949;</span></code>,
  • @@ -119,7 +123,7 @@ then the absolute value of <code class="docutils literal notranslate"><span clas
    In Lean, in a sequence of implications there are implicit parentheses grouped to the right. So the expression above means - &#8220;if <code class="docutils literal notranslate"><span class="pre">0</span> <span class="pre">&lt;</span> <span class="pre">&#949;</span></code> then if <code class="docutils literal notranslate"><span class="pre">&#949;</span> <span class="pre">&#8804;</span> <span class="pre">1</span></code> then if <code class="docutils literal notranslate"><span class="pre">abs</span> <span class="pre">x</span> <span class="pre">&lt;</span> <span class="pre">&#949;</span></code> &#8230;&#8221; + &#8220;if <code class="docutils literal notranslate"><span class="pre">0</span> <span class="pre">&lt;</span> <span class="pre">&#949;</span></code> then if <code class="docutils literal notranslate"><span class="pre">&#949;</span> <span class="pre">&#8804;</span> <span class="pre">1</span></code> then if <code class="docutils literal notranslate"><span class="pre">|x|</span> <span class="pre">&lt;</span> <span class="pre">&#949;</span></code> &#8230;&#8221; As a result, the expression says that all the assumptions together imply the conclusion.</p> <p>You have already seen that even though the universal quantifier
  • @@ -127,14 +131,16 @@ in this statement
    ranges over objects and the implication arrows introduce hypotheses, Lean treats the two in very similar ways. In particular, if you have proved a theorem of that form, - you can apply it to objects and hypotheses in the same way:</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">my_lemma</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">x</span> <span class="n">y</span> <span class="n">&#949;</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">,</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">&#949;</span> <span class="bp">&#8804;</span> <span class="mi">1</span> <span class="bp">&#8594;</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">abs</span> <span class="n">y</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">abs</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="bp">&lt;</span> <span class="n">&#949;</span> <span class="o">:=</span> + you can apply it to objects and hypotheses in the same way. + We will use as an example the following statement that we will help you to prove a + bit later:</p> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">my_lemma</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">x</span> <span class="n">y</span> <span class="n">&#949;</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">,</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">&#949;</span> <span class="bp">&#8804;</span> <span class="mi">1</span> <span class="bp">&#8594;</span> <span class="bp">|</span><span class="n">x</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="bp">|</span><span class="n">y</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="bp">|</span><span class="n">x</span> <span class="bp">*</span> <span class="n">y</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="kn">section</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">&#948;</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">h&#8320;</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">&#948;</span><span class="o">)</span> <span class="o">(</span><span class="n">h&#8321;</span> <span class="o">:</span> <span class="n">&#948;</span> <span class="bp">&#8804;</span> <span class="mi">1</span><span class="o">)</span> - <span class="kd">variable</span> <span class="o">(</span><span class="n">ha</span> <span class="o">:</span> <span class="n">abs</span> <span class="n">a</span> <span class="bp">&lt;</span> <span class="n">&#948;</span><span class="o">)</span> <span class="o">(</span><span class="n">hb</span> <span class="o">:</span> <span class="n">abs</span> <span class="n">b</span> <span class="bp">&lt;</span> <span class="n">&#948;</span><span class="o">)</span> + <span class="kd">variable</span> <span class="o">(</span><span class="n">ha</span> <span class="o">:</span> <span class="bp">|</span><span class="n">a</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#948;</span><span class="o">)</span> <span class="o">(</span><span class="n">hb</span> <span class="o">:</span> <span class="bp">|</span><span class="n">b</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#948;</span><span class="o">)</span> <span class="k">#check</span> <span class="n">my_lemma</span> <span class="n">a</span> <span class="n">b</span> <span class="n">&#948;</span> <span class="k">#check</span> <span class="n">my_lemma</span> <span class="n">a</span> <span class="n">b</span> <span class="n">&#948;</span> <span class="n">h&#8320;</span> <span class="n">h&#8321;</span>
  • @@ -148,13 +154,13 @@ to use curly brackets to make quantified variables implicit
    when they can be inferred from subsequent hypotheses. When we do that, we can just apply a lemma to the hypotheses without mentioning the objects.</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">my_lemma2</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="n">&#949;</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">},</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">&#949;</span> <span class="bp">&#8804;</span> <span class="mi">1</span> <span class="bp">&#8594;</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">abs</span> <span class="n">y</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">abs</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="bp">&lt;</span> <span class="n">&#949;</span> <span class="o">:=</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">my_lemma2</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="n">&#949;</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">},</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">&#949;</span> <span class="bp">&#8804;</span> <span class="mi">1</span> <span class="bp">&#8594;</span> <span class="bp">|</span><span class="n">x</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="bp">|</span><span class="n">y</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="bp">|</span><span class="n">x</span> <span class="bp">*</span> <span class="n">y</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="kn">section</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="n">&#948;</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="kd">variable</span> <span class="o">(</span><span class="n">h&#8320;</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">&#948;</span><span class="o">)</span> <span class="o">(</span><span class="n">h&#8321;</span> <span class="o">:</span> <span class="n">&#948;</span> <span class="bp">&#8804;</span> <span class="mi">1</span><span class="o">)</span> - <span class="kd">variable</span> <span class="o">(</span><span class="n">ha</span> <span class="o">:</span> <span class="n">abs</span> <span class="n">a</span> <span class="bp">&lt;</span> <span class="n">&#948;</span><span class="o">)</span> <span class="o">(</span><span class="n">hb</span> <span class="o">:</span> <span class="n">abs</span> <span class="n">b</span> <span class="bp">&lt;</span> <span class="n">&#948;</span><span class="o">)</span> + <span class="kd">variable</span> <span class="o">(</span><span class="n">ha</span> <span class="o">:</span> <span class="bp">|</span><span class="n">a</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#948;</span><span class="o">)</span> <span class="o">(</span><span class="n">hb</span> <span class="o">:</span> <span class="bp">|</span><span class="n">b</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#948;</span><span class="o">)</span> <span class="k">#check</span> <span class="n">my_lemma2</span> <span class="n">h&#8320;</span> <span class="n">h&#8321;</span> <span class="n">ha</span> <span class="n">hb</span>
  • @@ -163,13 +169,13 @@ mentioning the objects.</p>
    </div> <p>At this stage, you also know that if you use the <code class="docutils literal notranslate"><span class="pre">apply</span></code> tactic to apply <code class="docutils literal notranslate"><span class="pre">my_lemma</span></code> - to a goal of the form <code class="docutils literal notranslate"><span class="pre">abs</span> <span class="pre">(a</span> <span class="pre">*</span> <span class="pre">b)</span> <span class="pre">&lt;</span> <span class="pre">&#948;</span></code>, + to a goal of the form <code class="docutils literal notranslate"><span class="pre">|a</span> <span class="pre">*</span> <span class="pre">b|</span> <span class="pre">&lt;</span> <span class="pre">&#948;</span></code>, you are left with new goals that require you to prove each of the hypotheses.</p> <p id="index-0">To prove a statement like this, use the <code class="docutils literal notranslate"><span class="pre">intro</span></code> tactic. Take a look at what it does in this example:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">my_lemma3</span> <span class="o">:</span> - <span class="bp">&#8704;</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="n">&#949;</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">},</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">&#949;</span> <span class="bp">&#8804;</span> <span class="mi">1</span> <span class="bp">&#8594;</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">abs</span> <span class="n">y</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">abs</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="bp">&lt;</span> <span class="n">&#949;</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="bp">&#8704;</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="n">&#949;</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">},</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">&#949;</span> <span class="bp">&#8804;</span> <span class="mi">1</span> <span class="bp">&#8594;</span> <span class="bp">|</span><span class="n">x</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="bp">|</span><span class="n">y</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="bp">|</span><span class="n">x</span> <span class="bp">*</span> <span class="n">y</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">y</span> <span class="n">&#949;</span> <span class="n">epos</span> <span class="n">ele1</span> <span class="n">xlt</span> <span class="n">ylt</span> <span class="gr">sorry</span> </pre></div>
  • @@ -190,11 +196,11 @@ In a moment, we will see why it is sometimes necessary to
    introduce variables and hypotheses after the proof begins.</p> <p>To help you prove the lemma, we will start you off:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">my_lemma4</span> <span class="o">:</span> - <span class="bp">&#8704;</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="n">&#949;</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">},</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">&#949;</span> <span class="bp">&#8804;</span> <span class="mi">1</span> <span class="bp">&#8594;</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">abs</span> <span class="n">y</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">abs</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="bp">&lt;</span> <span class="n">&#949;</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="bp">&#8704;</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="n">&#949;</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">},</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="n">&#949;</span> <span class="bp">&#8804;</span> <span class="mi">1</span> <span class="bp">&#8594;</span> <span class="bp">|</span><span class="n">x</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="bp">|</span><span class="n">y</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">&#8594;</span> <span class="bp">|</span><span class="n">x</span> <span class="bp">*</span> <span class="n">y</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">y</span> <span class="n">&#949;</span> <span class="n">epos</span> <span class="n">ele1</span> <span class="n">xlt</span> <span class="n">ylt</span> <span class="k">calc</span> - <span class="n">abs</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="bp">=</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">abs</span> <span class="n">y</span> <span class="o">:=</span> <span class="gr">sorry</span> - <span class="n">_</span> <span class="bp">&#8804;</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">&#949;</span> <span class="o">:=</span> <span class="gr">sorry</span> + <span class="bp">|</span><span class="n">x</span> <span class="bp">*</span> <span class="n">y</span><span class="bp">|</span> <span class="bp">=</span> <span class="bp">|</span><span class="n">x</span><span class="bp">|</span> <span class="bp">*</span> <span class="bp">|</span><span class="n">y</span><span class="bp">|</span> <span class="o">:=</span> <span class="gr">sorry</span> + <span class="n">_</span> <span class="bp">&#8804;</span> <span class="bp">|</span><span class="n">x</span><span class="bp">|</span> <span class="bp">*</span> <span class="n">&#949;</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">_</span> <span class="bp">&lt;</span> <span class="mi">1</span> <span class="bp">*</span> <span class="n">&#949;</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="n">_</span> <span class="bp">=</span> <span class="n">&#949;</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div>
  • @@ -203,7 +209,7 @@ introduce variables and hypotheses after the proof begins.</p>
    <code class="docutils literal notranslate"><span class="pre">abs_mul</span></code>, <code class="docutils literal notranslate"><span class="pre">mul_le_mul</span></code>, <code class="docutils literal notranslate"><span class="pre">abs_nonneg</span></code>, <code class="docutils literal notranslate"><span class="pre">mul_lt_mul_right</span></code>, and <code class="docutils literal notranslate"><span class="pre">one_mul</span></code>. Remember that you can find theorems like these using - tab completion. + Ctrl-space completion (or Cmd-space completion on a Mac). Remember also that you can use <code class="docutils literal notranslate"><span class="pre">.mp</span></code> and <code class="docutils literal notranslate"><span class="pre">.mpr</span></code> or <code class="docutils literal notranslate"><span class="pre">.1</span></code> and <code class="docutils literal notranslate"><span class="pre">.2</span></code> to extract the two directions of an if-and-only-if statement.</p>
  • @@ -224,8 +230,9 @@ on the values of <code class="docutils literal notranslate"><span class="pre">f<
    <span class="bp">&#8704;</span> <span class="n">x</span><span class="o">,</span> <span class="n">a</span> <span class="bp">&#8804;</span> <span class="n">f</span> <span class="n">x</span> </pre></div> </div> - <p id="index-1">In the next example, <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">x</span> <span class="pre">&#8614;</span> <span class="pre">f</span> <span class="pre">x</span> <span class="pre">+</span> <span class="pre">g</span> <span class="pre">x</span></code> is a name for the - function that maps <code class="docutils literal notranslate"><span class="pre">x</span></code> to <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span> <span class="pre">+</span> <span class="pre">g</span> <span class="pre">x</span></code>.</p> + <p id="index-1">In the next example, <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">x</span> <span class="pre">&#8614;</span> <span class="pre">f</span> <span class="pre">x</span> <span class="pre">+</span> <span class="pre">g</span> <span class="pre">x</span></code> is the + function that maps <code class="docutils literal notranslate"><span class="pre">x</span></code> to <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span> <span class="pre">+</span> <span class="pre">g</span> <span class="pre">x</span></code>. Going from the expression <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span> <span class="pre">+</span> <span class="pre">g</span> <span class="pre">x</span></code> + to this function is called a lambda abstraction in type theory.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">hfa</span> <span class="o">:</span> <span class="n">FnUb</span> <span class="n">f</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">hgb</span> <span class="o">:</span> <span class="n">FnUb</span> <span class="n">g</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnUb</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</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="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">dsimp</span>
  • @@ -254,14 +261,14 @@ and gives you more control over how the goal is transformed.</p>
    The last two <code class="docutils literal notranslate"><span class="pre">apply</span></code> commands force Lean to unfold the definitions of <code class="docutils literal notranslate"><span class="pre">FnUb</span></code> in the hypotheses. Try carrying out similar proofs of these:</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">hfa</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="n">f</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">hgb</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="n">g</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</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="o">)</span> <span class="o">:=</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">hfa</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="n">f</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">hgb</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="n">g</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</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="o">)</span> <span class="o">:=</span> <span class="gr">sorry</span> - <span class="kd">example</span> <span class="o">(</span><span class="n">nnf</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="n">f</span> <span class="mi">0</span><span class="o">)</span> <span class="o">(</span><span class="n">nng</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="n">g</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">g</span> <span class="n">x</span><span class="o">)</span> <span class="mi">0</span> <span class="o">:=</span> + <span class="kd">example</span> <span class="o">(</span><span class="n">nnf</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="n">f</span> <span class="mi">0</span><span class="o">)</span> <span class="o">(</span><span class="n">nng</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="n">g</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">g</span> <span class="n">x</span><span class="o">)</span> <span class="mi">0</span> <span class="o">:=</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">(</span><span class="n">hfa</span> <span class="o">:</span> <span class="n">FnUb</span> <span class="n">f</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">hfb</span> <span class="o">:</span> <span class="n">FnUb</span> <span class="n">g</span> <span class="n">b</span><span class="o">)</span> <span class="o">(</span><span class="n">nng</span> <span class="o">:</span> <span class="n">FnLb</span> <span class="n">g</span> <span class="mi">0</span><span class="o">)</span> <span class="o">(</span><span class="n">nna</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">&#8804;</span> <span class="n">a</span><span class="o">)</span> <span class="o">:</span> - <span class="n">FnUb</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">g</span> <span class="n">x</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="o">)</span> <span class="o">:=</span> + <span class="n">FnUb</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">g</span> <span class="n">x</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="o">)</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div>
  • @@ -286,7 +293,7 @@ it will apply in all these instances.</p>
    <span class="bp">&#8704;</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">&#8804;</span> <span class="n">a</span> <span class="kd">theorem</span> <span class="n">fnUb_add</span> <span class="o">{</span><span class="n">f</span> <span class="n">g</span> <span class="o">:</span> <span class="n">&#945;</span> <span class="bp">&#8594;</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">hfa</span> <span class="o">:</span> <span class="n">FnUb&#39;</span> <span class="n">f</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">hgb</span> <span class="o">:</span> <span class="n">FnUb&#39;</span> <span class="n">g</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> - <span class="n">FnUb&#39;</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</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="o">)</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">add_le_add</span> <span class="o">(</span><span class="n">hfa</span> <span class="n">x</span><span class="o">)</span> <span class="o">(</span><span class="n">hgb</span> <span class="n">x</span><span class="o">)</span> + <span class="n">FnUb&#39;</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</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="o">)</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">add_le_add</span> <span class="o">(</span><span class="n">hfa</span> <span class="n">x</span><span class="o">)</span> <span class="o">(</span><span class="n">hgb</span> <span class="n">x</span><span class="o">)</span> </pre></div> </div> <p>You have already seen square brackets like these in
  • @@ -316,7 +323,7 @@ and then apply the resulting expression to the goal.
    Or you can apply it to the goal and let Lean help you work backwards by displaying the remaining hypotheses as new subgoals.</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">mf</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">mg</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">mf</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">mg</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">a</span> <span class="n">b</span> <span class="n">aleb</span> <span class="n">apply</span> <span class="n">add_le_add</span> <span class="n">apply</span> <span class="n">mf</span> <span class="n">aleb</span>
  • @@ -327,21 +334,21 @@ as new subgoals.</p>
    to give a proof term instead. To describe a proof that temporarily introduces objects <code class="docutils literal notranslate"><span class="pre">a</span></code> and <code class="docutils literal notranslate"><span class="pre">b</span></code> and a hypothesis <code class="docutils literal notranslate"><span class="pre">aleb</span></code>, - Lean uses the notation <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">a</span> <span class="pre">b</span> <span class="pre">aleb</span> <span class="pre">=&gt;</span> <span class="pre">...</span></code>. + Lean uses the notation <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">a</span> <span class="pre">b</span> <span class="pre">aleb</span> <span class="pre">&#8614;</span> <span class="pre">...</span></code>. This is analogous to the way that an expression - like <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">x</span> <span class="pre">=&gt;</span> <span class="pre">x^2</span></code> describes a function + like <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">x</span> <span class="pre">&#8614;</span> <span class="pre">x^2</span></code> describes a function by temporarily naming an object, <code class="docutils literal notranslate"><span class="pre">x</span></code>, and then using it to describe a value. So the <code class="docutils literal notranslate"><span class="pre">intro</span></code> command in the previous proof corresponds to the lambda abstraction in the next proof term. The <code class="docutils literal notranslate"><span class="pre">apply</span></code> commands then correspond to building the application of the theorem to its arguments.</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">mf</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">mg</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> - <span class="k">fun</span> <span class="n">a</span> <span class="n">b</span> <span class="n">aleb</span> <span class="bp">=&gt;</span> <span class="n">add_le_add</span> <span class="o">(</span><span class="n">mf</span> <span class="n">aleb</span><span class="o">)</span> <span class="o">(</span><span class="n">mg</span> <span class="n">aleb</span><span class="o">)</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">mf</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">mg</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> + <span class="k">fun</span> <span class="n">a</span> <span class="n">b</span> <span class="n">aleb</span> <span class="bp">&#8614;</span> <span class="n">add_le_add</span> <span class="o">(</span><span class="n">mf</span> <span class="n">aleb</span><span class="o">)</span> <span class="o">(</span><span class="n">mg</span> <span class="n">aleb</span><span class="o">)</span> </pre></div> </div> <p>Here is a useful trick: if you start writing - the proof term <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">a</span> <span class="pre">b</span> <span class="pre">aleb</span> <span class="pre">=&gt;</span> <span class="pre">_</span></code> using + the proof term <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">a</span> <span class="pre">b</span> <span class="pre">aleb</span> <span class="pre">&#8614;</span> <span class="pre">_</span></code> using an underscore where the rest of the expression should go, Lean will flag an error,
  • @@ -351,10 +358,10 @@ hover over the squiggly error marker,
    Lean will show you the goal that the remaining expression has to solve.</p> <p>Try proving these, with either tactics or proof terms:</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">c</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">(</span><span class="n">mf</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">nnc</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">&#8804;</span> <span class="n">c</span><span class="o">)</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">f</span> <span class="n">x</span> <span class="o">:=</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">c</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">(</span><span class="n">mf</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">nnc</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">&#8804;</span> <span class="n">c</span><span class="o">)</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">f</span> <span class="n">x</span> <span class="o">:=</span> <span class="gr">sorry</span> - <span class="kd">example</span> <span class="o">(</span><span class="n">mf</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">mg</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">f</span> <span class="o">(</span><span class="n">g</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> + <span class="kd">example</span> <span class="o">(</span><span class="n">mf</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">mg</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="o">(</span><span class="n">g</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div>
  • @@ -372,27 +379,27 @@ You can complete the proofs of the others.</p>
    <span class="kd">def</span> <span class="n">FnOdd</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="kt">Prop</span> <span class="o">:=</span> <span class="bp">&#8704;</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="bp">-</span><span class="n">f</span> <span class="o">(</span><span class="bp">-</span><span class="n">x</span><span class="o">)</span> - <span class="kd">example</span> <span class="o">(</span><span class="n">ef</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">eg</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="kd">example</span> <span class="o">(</span><span class="n">ef</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">eg</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x</span> <span class="k">calc</span> - <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span><span class="o">)</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">rfl</span> + <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span><span class="o">)</span> <span class="n">x</span> <span class="bp">=</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">rfl</span> <span class="n">_</span> <span class="bp">=</span> <span class="n">f</span> <span class="o">(</span><span class="bp">-</span><span class="n">x</span><span class="o">)</span> <span class="bp">+</span> <span class="n">g</span> <span class="o">(</span><span class="bp">-</span><span class="n">x</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">ef</span><span class="o">,</span> <span class="n">eg</span><span class="o">]</span> - <span class="kd">example</span> <span class="o">(</span><span class="n">of</span> <span class="o">:</span> <span class="n">FnOdd</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">og</span> <span class="o">:</span> <span class="n">FnOdd</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="kd">example</span> <span class="o">(</span><span class="n">of</span> <span class="o">:</span> <span class="n">FnOdd</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">og</span> <span class="o">:</span> <span class="n">FnOdd</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> - <span class="kd">example</span> <span class="o">(</span><span class="n">ef</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">og</span> <span class="o">:</span> <span class="n">FnOdd</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnOdd</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="kd">example</span> <span class="o">(</span><span class="n">ef</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">og</span> <span class="o">:</span> <span class="n">FnOdd</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnOdd</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> - <span class="kd">example</span> <span class="o">(</span><span class="n">ef</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">og</span> <span class="o">:</span> <span class="n">FnOdd</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">f</span> <span class="o">(</span><span class="n">g</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="kd">example</span> <span class="o">(</span><span class="n">ef</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">og</span> <span class="o">:</span> <span class="n">FnOdd</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnEven</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="o">(</span><span class="n">g</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p id="index-4">The first proof can be shortened using <code class="docutils literal notranslate"><span class="pre">dsimp</span></code> or <code class="docutils literal notranslate"><span class="pre">change</span></code> - to get rid of the lambda. + to get rid of the lambda abstraction. But you can check that the subsequent <code class="docutils literal notranslate"><span class="pre">rw</span></code> won&#8217;t work - unless we get rid of the lambda explicitly, + unless we get rid of the lambda abstraction explicitly, because otherwise it cannot find the patterns <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">x</span></code> and <code class="docutils literal notranslate"><span class="pre">g</span> <span class="pre">x</span></code> in the expression. Contrary to some other tactics, <code class="docutils literal notranslate"><span class="pre">rw</span></code> operates on the syntactic level,
  • @@ -400,14 +407,24 @@ it won&#8217;t unfold definitions or apply reductions for you
    (it has a variant called <code class="docutils literal notranslate"><span class="pre">erw</span></code> that tries a little harder in this direction, but not much harder).</p> <p>You can find implicit universal quantifiers all over the place, - once you know how to spot them. - Mathlib includes a good library for rudimentary set theory. - Lean&#8217;s logical foundation imposes the restriction that when - we talk about sets, we are always talking about sets of - elements of some type. If <code class="docutils literal notranslate"><span class="pre">x</span></code> has type <code class="docutils literal notranslate"><span class="pre">&#945;</span></code> and <code class="docutils literal notranslate"><span class="pre">s</span></code> has - type <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">&#945;</span></code>, then <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">&#8712;</span> <span class="pre">s</span></code> is a proposition that - asserts that <code class="docutils literal notranslate"><span class="pre">x</span></code> is an element of <code class="docutils literal notranslate"><span class="pre">s</span></code>. - If <code class="docutils literal notranslate"><span class="pre">s</span></code> and <code class="docutils literal notranslate"><span class="pre">t</span></code> are of type <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">&#945;</span></code>, + once you know how to spot them.</p> + <p>Mathlib includes a good library for manipulating sets. Recall that Lean does not + use foundations based on set theory, so here the word set has its mundane meaning + of a collection of mathematical objets of some given type <code class="docutils literal notranslate"><span class="pre">&#945;</span></code>. + If <code class="docutils literal notranslate"><span class="pre">x</span></code> has type <code class="docutils literal notranslate"><span class="pre">&#945;</span></code> and <code class="docutils literal notranslate"><span class="pre">s</span></code> has type <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">&#945;</span></code>, then <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">&#8712;</span> <span class="pre">s</span></code> is a proposition + that asserts that <code class="docutils literal notranslate"><span class="pre">x</span></code> is an element of <code class="docutils literal notranslate"><span class="pre">s</span></code>. If <code class="docutils literal notranslate"><span class="pre">y</span></code> has some different type <code class="docutils literal notranslate"><span class="pre">&#946;</span></code> then the + expression <code class="docutils literal notranslate"><span class="pre">y</span> <span class="pre">&#8712;</span> <span class="pre">s</span></code> makes no sense. Here &#8220;makes no sense&#8221; means &#8220;has no type hence Lean does not + accept it as a well-formed statement&#8221;. This contrasts with Zermelo-Fraenkel set theory for instance + where <code class="docutils literal notranslate"><span class="pre">a</span> <span class="pre">&#8712;</span> <span class="pre">b</span></code> is a well-formed statement for every mathematical objects <code class="docutils literal notranslate"><span class="pre">a</span></code> and <code class="docutils literal notranslate"><span class="pre">b</span></code>. + For instance <code class="docutils literal notranslate"><span class="pre">sin</span> <span class="pre">&#8712;</span> <span class="pre">cos</span></code> is a well-formed statement in ZF. This defect of set theoretic + foundations is an important motivation for not using it in a proof assistant which is meant to assist + us by detecting meaningless expressions. In Lean <code class="docutils literal notranslate"><span class="pre">sin</span></code> has type <code class="docutils literal notranslate"><span class="pre">&#8477;</span> <span class="pre">&#8594;</span> <span class="pre">&#8477;</span></code> and <code class="docutils literal notranslate"><span class="pre">cos</span></code> has type + <code class="docutils literal notranslate"><span class="pre">&#8477;</span> <span class="pre">&#8594;</span> <span class="pre">&#8477;</span></code> which is not equal to <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">(&#8477;</span> <span class="pre">&#8594;</span> <span class="pre">&#8477;)</span></code>, even after unfolding definitions, so the statement + <code class="docutils literal notranslate"><span class="pre">sin</span> <span class="pre">&#8712;</span> <span class="pre">cos</span></code> makes no sense. + One can also use Lean to work on set theory itself. For instance the independence of the continuum + hypothesis from the axioms of Zermelo-Fraenkel has been formalized in Lean. But such a meta-theory + of set theory is completely beyond the scope of this book.</p> + <p>If <code class="docutils literal notranslate"><span class="pre">s</span></code> and <code class="docutils literal notranslate"><span class="pre">t</span></code> are of type <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">&#945;</span></code>, then the subset relation <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">&#8838;</span> <span class="pre">t</span></code> is defined to mean <code class="docutils literal notranslate"><span class="pre">&#8704;</span> <span class="pre">{x</span> <span class="pre">:</span> <span class="pre">&#945;},</span> <span class="pre">x</span> <span class="pre">&#8712;</span> <span class="pre">s</span> <span class="pre">&#8594;</span> <span class="pre">x</span> <span class="pre">&#8712;</span> <span class="pre">t</span></code>. The variable in the quantifier is marked implicit so that
  • @@ -422,7 +439,7 @@ and asks you to do the same for transitivity.</p>
    <span class="n">intro</span> <span class="n">x</span> <span class="n">xs</span> <span class="n">exact</span> <span class="n">xs</span> - <span class="kd">theorem</span> <span class="n">Subset.refl</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">&#8838;</span> <span class="n">s</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="n">xs</span> <span class="bp">=&gt;</span> <span class="n">xs</span> + <span class="kd">theorem</span> <span class="n">Subset.refl</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">&#8838;</span> <span class="n">s</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="n">xs</span> <span class="bp">&#8614;</span> <span class="n">xs</span> <span class="kd">theorem</span> <span class="n">Subset.trans</span> <span class="o">:</span> <span class="n">r</span> <span class="bp">&#8838;</span> <span class="n">s</span> <span class="bp">&#8594;</span> <span class="n">s</span> <span class="bp">&#8838;</span> <span class="n">t</span> <span class="bp">&#8594;</span> <span class="n">r</span> <span class="bp">&#8838;</span> <span class="n">t</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span>
  • @@ -455,14 +472,16 @@ Mathlib defines <code class="docutils literal notranslate"><span class="pre">Fun
    The next example shows that, on the real numbers, any function that adds a constant is injective. We then ask you to show that multiplication by a nonzero - constant is also injective.</p> + constant is also injective, using the lemma name in the example as a source + of inspiration. Recall you should use Ctrl-space completion after guessing the beginning of + a lemma name.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">open</span> <span class="n">Function</span> - <span class="kd">example</span> <span class="o">(</span><span class="n">c</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="n">Injective</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="kd">example</span> <span class="o">(</span><span class="n">c</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="n">Injective</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x&#8321;</span> <span class="n">x&#8322;</span> <span class="n">h&#39;</span> <span class="n">exact</span> <span class="o">(</span><span class="n">add_left_inj</span> <span class="n">c</span><span class="o">)</span><span class="bp">.</span><span class="n">mp</span> <span class="n">h&#39;</span> - <span class="kd">example</span> <span class="o">{</span><span class="n">c</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">&#8800;</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">Injective</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="kd">example</span> <span class="o">{</span><span class="n">c</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">&#8800;</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">Injective</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div>
  • @@ -470,7 +489,7 @@ constant is also injective.</p>
    <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">&#945;</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">&#946;</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">&#947;</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="kd">variable</span> <span class="o">{</span><span class="n">g</span> <span class="o">:</span> <span class="n">&#946;</span> <span class="bp">&#8594;</span> <span class="n">&#947;</span><span class="o">}</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">&#945;</span> <span class="bp">&#8594;</span> <span class="n">&#946;</span><span class="o">}</span> - <span class="kd">example</span> <span class="o">(</span><span class="n">injg</span> <span class="o">:</span> <span class="n">Injective</span> <span class="n">g</span><span class="o">)</span> <span class="o">(</span><span class="n">injf</span> <span class="o">:</span> <span class="n">Injective</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">Injective</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">g</span> <span class="o">(</span><span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="kd">example</span> <span class="o">(</span><span class="n">injg</span> <span class="o">:</span> <span class="n">Injective</span> <span class="n">g</span><span class="o">)</span> <span class="o">(</span><span class="n">injf</span> <span class="o">:</span> <span class="n">Injective</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">Injective</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">g</span> <span class="o">(</span><span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div>
  • @@ -540,10 +559,10 @@ without specifying the bound:</p>
    </div> <p>We can use the theorem <code class="docutils literal notranslate"><span class="pre">FnUb_add</span></code> from the last section to prove that if <code class="docutils literal notranslate"><span class="pre">f</span></code> and <code class="docutils literal notranslate"><span class="pre">g</span></code> have upper bounds, - then so does <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">x</span> <span class="pre">=&gt;</span> <span class="pre">f</span> <span class="pre">x</span> <span class="pre">+</span> <span class="pre">g</span> <span class="pre">x</span></code>.</p> + then so does <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">x</span> <span class="pre">&#8614;</span> <span class="pre">f</span> <span class="pre">x</span> <span class="pre">+</span> <span class="pre">g</span> <span class="pre">x</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">f</span> <span class="n">g</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span><span class="o">}</span> - <span class="kd">example</span> <span class="o">(</span><span class="n">ubf</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">ubg</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="kd">example</span> <span class="o">(</span><span class="n">ubf</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">ubg</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">cases&#39;</span> <span class="n">ubf</span> <span class="k">with</span> <span class="n">a</span> <span class="n">ubfa</span> <span class="n">cases&#39;</span> <span class="n">ubg</span> <span class="k">with</span> <span class="n">b</span> <span class="n">ubgb</span> <span class="n">use</span> <span class="n">a</span> <span class="bp">+</span> <span class="n">b</span>
  • @@ -573,10 +592,10 @@ from the last section into named theorems,
    as we did with <code class="docutils literal notranslate"><span class="pre">fn_ub_add</span></code>, or you can insert the arguments directly into the proofs.</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">lbf</span> <span class="o">:</span> <span class="n">FnHasLb</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">lbg</span> <span class="o">:</span> <span class="n">FnHasLb</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnHasLb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">lbf</span> <span class="o">:</span> <span class="n">FnHasLb</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">lbg</span> <span class="o">:</span> <span class="n">FnHasLb</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnHasLb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> - <span class="kd">example</span> <span class="o">{</span><span class="n">c</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">(</span><span class="n">ubf</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">&#8805;</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">f</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="kd">example</span> <span class="o">{</span><span class="n">c</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">(</span><span class="n">ubf</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">&#8805;</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">f</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div>
  • @@ -592,20 +611,20 @@ them as a pattern <code class="docutils literal notranslate"><span class="pre">&
    The <code class="docutils literal notranslate"><span class="pre">rintro</span></code> tactic is a combination of <code class="docutils literal notranslate"><span class="pre">intro</span></code> and <code class="docutils literal notranslate"><span class="pre">rcases</span></code>. These examples illustrate their use:</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">ubf</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">ubg</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">ubf</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">f</span><span class="o">)</span> <span class="o">(</span><span class="n">ubg</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">g</span><span class="o">)</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rcases</span> <span class="n">ubf</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">a</span><span class="o">,</span> <span class="n">ubfa</span><span class="o">&#10217;</span> <span class="n">rcases</span> <span class="n">ubg</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">b</span><span class="o">,</span> <span class="n">ubgb</span><span class="o">&#10217;</span> <span class="n">exact</span> <span class="o">&#10216;</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">,</span> <span class="n">fnUb_add</span> <span class="n">ubfa</span> <span class="n">ubgb</span><span class="o">&#10217;</span> - <span class="kd">example</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">f</span> <span class="bp">&#8594;</span> <span class="n">FnHasUb</span> <span class="n">g</span> <span class="bp">&#8594;</span> <span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="kd">example</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">f</span> <span class="bp">&#8594;</span> <span class="n">FnHasUb</span> <span class="n">g</span> <span class="bp">&#8594;</span> <span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rintro</span> <span class="o">&#10216;</span><span class="n">a</span><span class="o">,</span> <span class="n">ubfa</span><span class="o">&#10217;</span> <span class="o">&#10216;</span><span class="n">b</span><span class="o">,</span> <span class="n">ubgb</span><span class="o">&#10217;</span> <span class="n">exact</span> <span class="o">&#10216;</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">,</span> <span class="n">fnUb_add</span> <span class="n">ubfa</span> <span class="n">ubgb</span><span class="o">&#10217;</span> </pre></div> </div> - <p>In fact, Lean also supports a pattern-matching lambda + <p>In fact, Lean also supports a pattern-matching fun in expressions and proof terms:</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">f</span> <span class="bp">&#8594;</span> <span class="n">FnHasUb</span> <span class="n">g</span> <span class="bp">&#8594;</span> <span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> - <span class="k">fun</span> <span class="o">&#10216;</span><span class="n">a</span><span class="o">,</span> <span class="n">ubfa</span><span class="o">&#10217;</span> <span class="o">&#10216;</span><span class="n">b</span><span class="o">,</span> <span class="n">ubgb</span><span class="o">&#10217;</span> <span class="bp">=&gt;</span> <span class="o">&#10216;</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">,</span> <span class="n">fnUb_add</span> <span class="n">ubfa</span> <span class="n">ubgb</span><span class="o">&#10217;</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">FnHasUb</span> <span class="n">f</span> <span class="bp">&#8594;</span> <span class="n">FnHasUb</span> <span class="n">g</span> <span class="bp">&#8594;</span> <span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">g</span> <span class="n">x</span> <span class="o">:=</span> + <span class="k">fun</span> <span class="o">&#10216;</span><span class="n">a</span><span class="o">,</span> <span class="n">ubfa</span><span class="o">&#10217;</span> <span class="o">&#10216;</span><span class="n">b</span><span class="o">,</span> <span class="n">ubgb</span><span class="o">&#10217;</span> <span class="bp">&#8614;</span> <span class="o">&#10216;</span><span class="n">a</span> <span class="bp">+</span> <span class="n">b</span><span class="o">,</span> <span class="n">fnUb_add</span> <span class="n">ubfa</span> <span class="n">ubgb</span><span class="o">&#10217;</span> </pre></div> </div> <p>These are power-user moves, and there is no harm
  • @@ -658,7 +677,7 @@ then <span class="math notranslate nohighlight">\(xy\)</span> is the norm of <sp
    Our cryptic proof illustrates the fact that the proof that is easiest to formalize isn&#8217;t always the most perspicuous one. - In the chapters to come, + In <a class="reference internal" href="C06_Structures.html#section-building-the-gaussian-integers"><span class="std std-numref">Section 6.3</span></a>, we will provide you with the means to define the Gaussian integers and use them to provide an alternative proof.</p> <p>The pattern of unpacking an equation inside an existential quantifier
  • @@ -705,14 +724,14 @@ such that <span class="math notranslate nohighlight">\(f(x) = y\)</span>.
    Notice that this statement includes both a universal and an existential quantifier, which explains why the next example makes use of both <code class="docutils literal notranslate"><span class="pre">intro</span></code> and <code class="docutils literal notranslate"><span class="pre">use</span></code>.</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">c</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">:</span> <span class="n">Surjective</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">c</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">:</span> <span class="n">Surjective</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">c</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">x</span> <span class="n">use</span> <span class="n">x</span> <span class="bp">-</span> <span class="n">c</span> <span class="n">dsimp</span><span class="bp">;</span> <span class="n">ring</span> </pre></div> </div> - <p>Try this example yourself:</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">c</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">&#8800;</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">Surjective</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> + <p>Try this example yourself using the theorem <code class="docutils literal notranslate"><span class="pre">mul_div_cancel'</span></code>.:</p> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">c</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">&#8800;</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">Surjective</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div>
  • @@ -724,8 +743,7 @@ It can be used in conjunction with the <code class="docutils literal notranslate
    <span class="n">ring</span> </pre></div> </div> - <p>You can use the theorem <code class="docutils literal notranslate"><span class="pre">div_mul_cancel</span></code>. - The next example uses a surjectivity hypothesis + <p>The next example uses a surjectivity hypothesis by applying it to a suitable value. Note that you can use <code class="docutils literal notranslate"><span class="pre">cases'</span></code> with any expression, not just a hypothesis.</p>
  • @@ -741,7 +759,7 @@ the composition of surjective functions is surjective.</p>
    <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">&#945;</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">&#946;</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">&#947;</span> <span class="o">:</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">}</span> <span class="kd">variable</span> <span class="o">{</span><span class="n">g</span> <span class="o">:</span> <span class="n">&#946;</span> <span class="bp">&#8594;</span> <span class="n">&#947;</span><span class="o">}</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">&#945;</span> <span class="bp">&#8594;</span> <span class="n">&#946;</span><span class="o">}</span> - <span class="kd">example</span> <span class="o">(</span><span class="n">surjg</span> <span class="o">:</span> <span class="n">Surjective</span> <span class="n">g</span><span class="o">)</span> <span class="o">(</span><span class="n">surjf</span> <span class="o">:</span> <span class="n">Surjective</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">Surjective</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">g</span> <span class="o">(</span><span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="kd">example</span> <span class="o">(</span><span class="n">surjg</span> <span class="o">:</span> <span class="n">Surjective</span> <span class="n">g</span><span class="o">)</span> <span class="o">(</span><span class="n">surjf</span> <span class="o">:</span> <span class="n">Surjective</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">Surjective</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">g</span> <span class="o">(</span><span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div>
  • @@ -798,11 +816,14 @@ which says that a function has an upper bound.</p>
    <span class="n">linarith</span> </pre></div> </div> + <p>Remember that it is often convenient to use <code class="docutils literal notranslate"><span class="pre">linarith</span></code> + when a goal follows from linear equations and + inequalities that in the context.</p> <p>See if you can prove these in a similar way:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">a</span><span class="o">,</span> <span class="bp">&#8707;</span> <span class="n">x</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">a</span><span class="o">)</span> <span class="o">:</span> <span class="bp">&#172;</span><span class="n">FnHasLb</span> <span class="n">f</span> <span class="o">:=</span> <span class="gr">sorry</span> - <span class="kd">example</span> <span class="o">:</span> <span class="bp">&#172;</span><span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">x</span> <span class="o">:=</span> + <span class="kd">example</span> <span class="o">:</span> <span class="bp">&#172;</span><span class="n">FnHasUb</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">x</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div>
  • @@ -824,9 +845,6 @@ Use some of the theorems just enumerated to prove the following:</p>
    <span class="gr">sorry</span> </pre></div> </div> - <p>Remember that it is often convenient to use <code class="docutils literal notranslate"><span class="pre">linarith</span></code> - when a goal follows from linear equations and - inequalities that in the context.</p> <p>We can show that the first example in the last snippet cannot be proved if we replace <code class="docutils literal notranslate"><span class="pre">&lt;</span></code> by <code class="docutils literal notranslate"><span class="pre">&#8804;</span></code>. Notice that we can prove the negation of a universally
  • @@ -834,7 +852,7 @@ quantified statement by giving a counterexample.
    Complete the proof.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="bp">&#172;&#8704;</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span><span class="o">},</span> <span class="n">Monotone</span> <span class="n">f</span> <span class="bp">&#8594;</span> <span class="bp">&#8704;</span> <span class="o">{</span><span class="n">a</span> <span class="n">b</span><span class="o">},</span> <span class="n">f</span> <span class="n">a</span> <span class="bp">&#8804;</span> <span class="n">f</span> <span class="n">b</span> <span class="bp">&#8594;</span> <span class="n">a</span> <span class="bp">&#8804;</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">h</span> - <span class="k">let</span> <span class="n">f</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">=&gt;</span> <span class="o">(</span><span class="mi">0</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> + <span class="k">let</span> <span class="n">f</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">&#8614;</span> <span class="o">(</span><span class="mi">0</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="k">have</span> <span class="n">monof</span> <span class="o">:</span> <span class="n">Monotone</span> <span class="n">f</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="n">h&#39;</span> <span class="o">:</span> <span class="n">f</span> <span class="mi">1</span> <span class="bp">&#8804;</span> <span class="n">f</span> <span class="mi">0</span> <span class="o">:=</span> <span class="n">le_refl</span> <span class="n">_</span> <span class="gr">sorry</span>
  • @@ -844,7 +862,7 @@ Complete the proof.</p>
    which adds a <em>local definition</em> to the context. If you put the cursor after the <code class="docutils literal notranslate"><span class="pre">let</span></code> command, in the goal window you will see that the definition - <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">&#8477;</span> <span class="pre">&#8594;</span> <span class="pre">&#8477;</span> <span class="pre">:=</span> <span class="pre">fun</span> <span class="pre">x</span> <span class="pre">=&gt;</span> <span class="pre">0</span></code> has been added to the context. + <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">&#8477;</span> <span class="pre">&#8594;</span> <span class="pre">&#8477;</span> <span class="pre">:=</span> <span class="pre">fun</span> <span class="pre">x</span> <span class="pre">&#8614;</span> <span class="pre">0</span></code> has been added to the context. Lean will unfold the definition of <code class="docutils literal notranslate"><span class="pre">f</span></code> when it has to. In particular, when we prove <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">1</span> <span class="pre">&#8804;</span> <span class="pre">f</span> <span class="pre">0</span></code> with <code class="docutils literal notranslate"><span class="pre">le_refl</span></code>, Lean reduces <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">1</span></code> and <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">0</span></code> to <code class="docutils literal notranslate"><span class="pre">0</span></code>.</p>
  • @@ -1039,7 +1057,7 @@ The first is a slick proof-term version of the
    previous proof, which drops into tactic mode at the keyword <code class="docutils literal notranslate"><span class="pre">by</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">(</span><span class="n">h&#8320;</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&#8804;</span> <span class="n">y</span><span class="o">)</span> <span class="o">(</span><span class="n">h&#8321;</span> <span class="o">:</span> <span class="bp">&#172;</span><span class="n">y</span> <span class="bp">&#8804;</span> <span class="n">x</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&#8804;</span> <span class="n">y</span> <span class="bp">&#8743;</span> <span class="n">x</span> <span class="bp">&#8800;</span> <span class="n">y</span> <span class="o">:=</span> - <span class="o">&#10216;</span><span class="n">h&#8320;</span><span class="o">,</span> <span class="k">fun</span> <span class="n">h</span> <span class="bp">=&gt;</span> <span class="n">h&#8321;</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">])&#10217;</span> + <span class="o">&#10216;</span><span class="n">h&#8320;</span><span class="o">,</span> <span class="k">fun</span> <span class="n">h</span> <span class="bp">&#8614;</span> <span class="n">h&#8321;</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">])&#10217;</span> <span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">(</span><span class="n">h&#8320;</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&#8804;</span> <span class="n">y</span><span class="o">)</span> <span class="o">(</span><span class="n">h&#8321;</span> <span class="o">:</span> <span class="bp">&#172;</span><span class="n">y</span> <span class="bp">&#8804;</span> <span class="n">x</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&#8804;</span> <span class="n">y</span> <span class="bp">&#8743;</span> <span class="n">x</span> <span class="bp">&#8800;</span> <span class="n">y</span> <span class="o">:=</span> <span class="k">have</span> <span class="n">h</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&#8800;</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span>
  • @@ -1051,7 +1069,7 @@ which drops into tactic mode at the keyword <code class="docutils literal notran
    <p><em>Using</em> a conjunction instead of proving one involves unpacking the proofs of the two parts. You can use the <code class="docutils literal notranslate"><span class="pre">cases</span></code> tactic for that, - as well as <code class="docutils literal notranslate"><span class="pre">rcases</span></code>, <code class="docutils literal notranslate"><span class="pre">rintro</span></code>, or a pattern-matching lambda, + as well as <code class="docutils literal notranslate"><span class="pre">rcases</span></code>, <code class="docutils literal notranslate"><span class="pre">rintro</span></code>, or a pattern-matching <code class="docutils literal notranslate"><span class="pre">fun</span></code>, all in a manner similar to the way they are used with the existential quantifier.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&#8804;</span> <span class="n">y</span> <span class="bp">&#8743;</span> <span class="n">x</span> <span class="bp">&#8800;</span> <span class="n">y</span><span class="o">)</span> <span class="o">:</span> <span class="bp">&#172;</span><span class="n">y</span> <span class="bp">&#8804;</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span>
  • @@ -1064,7 +1082,7 @@ the existential quantifier.</p>
    <span class="n">exact</span> <span class="n">h&#8321;</span> <span class="o">(</span><span class="n">le_antisymm</span> <span class="n">h&#8320;</span> <span class="n">h&#39;</span><span class="o">)</span> <span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&#8804;</span> <span class="n">y</span> <span class="bp">&#8743;</span> <span class="n">x</span> <span class="bp">&#8800;</span> <span class="n">y</span> <span class="bp">&#8594;</span> <span class="bp">&#172;</span><span class="n">y</span> <span class="bp">&#8804;</span> <span class="n">x</span> <span class="o">:=</span> - <span class="k">fun</span> <span class="o">&#10216;</span><span class="n">h&#8320;</span><span class="o">,</span> <span class="n">h&#8321;</span><span class="o">&#10217;</span> <span class="n">h&#39;</span> <span class="bp">=&gt;</span> <span class="n">h&#8321;</span> <span class="o">(</span><span class="n">le_antisymm</span> <span class="n">h&#8320;</span> <span class="n">h&#39;</span><span class="o">)</span> + <span class="k">fun</span> <span class="o">&#10216;</span><span class="n">h&#8320;</span><span class="o">,</span> <span class="n">h&#8321;</span><span class="o">&#10217;</span> <span class="n">h&#39;</span> <span class="bp">&#8614;</span> <span class="n">h&#8321;</span> <span class="o">(</span><span class="n">le_antisymm</span> <span class="n">h&#8320;</span> <span class="n">h&#39;</span><span class="o">)</span> </pre></div> </div> <p>In contrast to using an existential quantifier,
  • @@ -1078,7 +1096,7 @@ or, equivalently, <code class="docutils literal notranslate"><span class="pre">h
    <span class="n">exact</span> <span class="n">le_antisymm</span> <span class="n">h.left</span> <span class="n">h&#39;</span> <span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&#8804;</span> <span class="n">y</span> <span class="bp">&#8743;</span> <span class="n">x</span> <span class="bp">&#8800;</span> <span class="n">y</span><span class="o">)</span> <span class="o">:</span> <span class="bp">&#172;</span><span class="n">y</span> <span class="bp">&#8804;</span> <span class="n">x</span> <span class="o">:=</span> - <span class="k">fun</span> <span class="n">h&#39;</span> <span class="bp">=&gt;</span> <span class="n">h.right</span> <span class="o">(</span><span class="n">le_antisymm</span> <span class="n">h.left</span> <span class="n">h&#39;</span><span class="o">)</span> + <span class="k">fun</span> <span class="n">h&#39;</span> <span class="bp">&#8614;</span> <span class="n">h.right</span> <span class="o">(</span><span class="n">le_antisymm</span> <span class="n">h.left</span> <span class="n">h&#39;</span><span class="o">)</span> </pre></div> </div> <p>Try using these techniques to come up with various ways of proving of the following:</p>
  • @@ -1096,7 +1114,7 @@ with anonymous constructors, <code class="docutils literal notranslate"><span cl
    <span class="n">exact</span> <span class="n">lt_trans</span> <span class="n">xltz</span> <span class="n">zlty</span> <span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="bp">&#8707;</span> <span class="n">z</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">,</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">z</span> <span class="bp">&#8743;</span> <span class="n">z</span> <span class="bp">&lt;</span> <span class="n">y</span><span class="o">)</span> <span class="bp">&#8594;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">y</span> <span class="o">:=</span> - <span class="k">fun</span> <span class="o">&#10216;</span><span class="n">z</span><span class="o">,</span> <span class="n">xltz</span><span class="o">,</span> <span class="n">zlty</span><span class="o">&#10217;</span> <span class="bp">=&gt;</span> <span class="n">lt_trans</span> <span class="n">xltz</span> <span class="n">zlty</span> + <span class="k">fun</span> <span class="o">&#10216;</span><span class="n">z</span><span class="o">,</span> <span class="n">xltz</span><span class="o">,</span> <span class="n">zlty</span><span class="o">&#10217;</span> <span class="bp">&#8614;</span> <span class="n">lt_trans</span> <span class="n">xltz</span> <span class="n">zlty</span> </pre></div> </div> <p>You can also use the <code class="docutils literal notranslate"><span class="pre">use</span></code> tactic:</p>
  • @@ -1113,7 +1131,7 @@ with anonymous constructors, <code class="docutils literal notranslate"><span cl
    <span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&#8804;</span> <span class="n">y</span> <span class="bp">&#8743;</span> <span class="n">x</span> <span class="bp">&#8800;</span> <span class="n">y</span> <span class="bp">&#8594;</span> <span class="n">x</span> <span class="bp">&#8804;</span> <span class="n">y</span> <span class="bp">&#8743;</span> <span class="bp">&#172;</span><span class="n">y</span> <span class="bp">&#8804;</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rintro</span> <span class="o">&#10216;</span><span class="n">h&#8320;</span><span class="o">,</span> <span class="n">h&#8321;</span><span class="o">&#10217;</span> <span class="n">use</span> <span class="n">h&#8320;</span> - <span class="n">exact</span> <span class="k">fun</span> <span class="n">h&#39;</span> <span class="bp">=&gt;</span> <span class="n">h&#8321;</span> <span class="o">(</span><span class="n">le_antisymm</span> <span class="n">h&#8320;</span> <span class="n">h&#39;</span><span class="o">)</span> + <span class="n">exact</span> <span class="k">fun</span> <span class="n">h&#39;</span> <span class="bp">&#8614;</span> <span class="n">h&#8321;</span> <span class="o">(</span><span class="n">le_antisymm</span> <span class="n">h&#8320;</span> <span class="n">h&#39;</span><span class="o">)</span> </pre></div> </div> <p>In the first example, the semicolon after the <code class="docutils literal notranslate"><span class="pre">constructor</span></code> command tells Lean to use the
  • @@ -1136,7 +1154,7 @@ just as you would if you were proving a conjunction.</p>
    <span class="n">exact</span> <span class="n">le_antisymm</span> <span class="n">h</span> <span class="kd">example</span> <span class="o">{</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&#8804;</span> <span class="n">y</span><span class="o">)</span> <span class="o">:</span> <span class="bp">&#172;</span><span class="n">y</span> <span class="bp">&#8804;</span> <span class="n">x</span> <span class="bp">&#8596;</span> <span class="n">x</span> <span class="bp">&#8800;</span> <span class="n">y</span> <span class="o">:=</span> - <span class="o">&#10216;</span><span class="k">fun</span> <span class="n">h&#8320;</span> <span class="n">h&#8321;</span> <span class="bp">=&gt;</span> <span class="n">h&#8320;</span> <span class="o">(</span><span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">h&#8321;</span><span class="o">]),</span> <span class="k">fun</span> <span class="n">h&#8320;</span> <span class="n">h&#8321;</span> <span class="bp">=&gt;</span> <span class="n">h&#8320;</span> <span class="o">(</span><span class="n">le_antisymm</span> <span class="n">h</span> <span class="n">h&#8321;</span><span class="o">)&#10217;</span> + <span class="o">&#10216;</span><span class="k">fun</span> <span class="n">h&#8320;</span> <span class="n">h&#8321;</span> <span class="bp">&#8614;</span> <span class="n">h&#8320;</span> <span class="o">(</span><span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">h&#8321;</span><span class="o">]),</span> <span class="k">fun</span> <span class="n">h&#8320;</span> <span class="n">h&#8321;</span> <span class="bp">&#8614;</span> <span class="n">h&#8320;</span> <span class="o">(</span><span class="n">le_antisymm</span> <span class="n">h</span> <span class="n">h&#8321;</span><span class="o">)&#10217;</span> </pre></div> </div> <p>The last proof term is inscrutable. Remember that you can
  • @@ -1170,11 +1188,11 @@ and you can also use it with <code class="docutils literal notranslate"><span cl
    It is often convenient to rewrite a statement to an equivalent one. In the next example, we use <code class="docutils literal notranslate"><span class="pre">abs_lt</span></code> to - replace an expression of the form <code class="docutils literal notranslate"><span class="pre">abs</span> <span class="pre">x</span> <span class="pre">&lt;</span> <span class="pre">y</span></code> + replace an expression of the form <code class="docutils literal notranslate"><span class="pre">|x|</span> <span class="pre">&lt;</span> <span class="pre">y</span></code> by the equivalent expression <code class="docutils literal notranslate"><span class="pre">-</span> <span class="pre">y</span> <span class="pre">&lt;</span> <span class="pre">x</span> <span class="pre">&#8743;</span> <span class="pre">x</span> <span class="pre">&lt;</span> <span class="pre">y</span></code>, and in the one after that we use <code class="docutils literal notranslate"><span class="pre">Nat.dvd_gcd_iff</span></code> to replace an expression of the form <code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">&#8739;</span> <span class="pre">Nat.gcd</span> <span class="pre">n</span> <span class="pre">k</span></code> by the equivalent expression <code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">&#8739;</span> <span class="pre">n</span> <span class="pre">&#8743;</span> <span class="pre">m</span> <span class="pre">&#8739;</span> <span class="pre">k</span></code>.</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="n">abs</span> <span class="o">(</span><span class="n">x</span> <span class="bp">+</span> <span class="mi">3</span><span class="o">)</span> <span class="bp">&lt;</span> <span class="mi">5</span> <span class="bp">&#8594;</span> <span class="bp">-</span><span class="mi">8</span> <span class="bp">&lt;</span> <span class="n">x</span> <span class="bp">&#8743;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="bp">|</span><span class="n">x</span> <span class="bp">+</span> <span class="mi">3</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="mi">5</span> <span class="bp">&#8594;</span> <span class="bp">-</span><span class="mi">8</span> <span class="bp">&lt;</span> <span class="n">x</span> <span class="bp">&#8743;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">abs_lt</span><span class="o">]</span> <span class="n">intro</span> <span class="n">h</span> <span class="n">constructor</span> <span class="bp">&lt;;&gt;</span> <span class="n">linarith</span>
  • @@ -1194,7 +1212,7 @@ the proof of the theorem is needed.)</p>
    <span class="n">push_neg</span> <span class="n">rfl</span> - <span class="kd">example</span> <span class="o">:</span> <span class="bp">&#172;</span><span class="n">Monotone</span> <span class="k">fun</span> <span class="n">x</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">=&gt;</span> <span class="bp">-</span><span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="kd">example</span> <span class="o">:</span> <span class="bp">&#172;</span><span class="n">Monotone</span> <span class="k">fun</span> <span class="n">x</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">&#8614;</span> <span class="bp">-</span><span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div>
  • @@ -1295,7 +1313,7 @@ the <code class="docutils literal notranslate"><span class="pre">cases</span></c
    As usual, we can tell Lean what names to use for the hypotheses. In the next example, we tell Lean to use the name <code class="docutils literal notranslate"><span class="pre">h</span></code> on each branch.</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">abs</span> <span class="n">y</span> <span class="bp">&#8594;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">y</span> <span class="bp">&#8744;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="bp">-</span><span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="bp">|</span><span class="n">y</span><span class="bp">|</span> <span class="bp">&#8594;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">y</span> <span class="bp">&#8744;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="bp">-</span><span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">cases&#39;</span> <span class="n">le_or_gt</span> <span class="mi">0</span> <span class="n">y</span> <span class="k">with</span> <span class="n">h</span> <span class="n">h</span> <span class="bp">&#183;</span> <span class="n">rw</span> <span class="o">[</span><span class="n">abs_of_nonneg</span> <span class="n">h</span><span class="o">]</span> <span class="n">intro</span> <span class="n">h</span>
  • @@ -1307,9 +1325,9 @@ to use the name <code class="docutils literal notranslate"><span class="pre">h</
    </div> <p>The absolute value function is defined in such a way that we can immediately prove that - <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">&#8805;</span> <span class="pre">0</span></code> implies <code class="docutils literal notranslate"><span class="pre">abs</span> <span class="pre">x</span> <span class="pre">=</span> <span class="pre">x</span></code> + <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">&#8805;</span> <span class="pre">0</span></code> implies <code class="docutils literal notranslate"><span class="pre">|x|</span> <span class="pre">=</span> <span class="pre">x</span></code> (this is the theorem <code class="docutils literal notranslate"><span class="pre">abs_of_nonneg</span></code>) - and <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">&lt;</span> <span class="pre">0</span></code> implies <code class="docutils literal notranslate"><span class="pre">abs</span> <span class="pre">x</span> <span class="pre">=</span> <span class="pre">-x</span></code> (this is <code class="docutils literal notranslate"><span class="pre">abs_of_neg</span></code>). + and <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">&lt;</span> <span class="pre">0</span></code> implies <code class="docutils literal notranslate"><span class="pre">|x|</span> <span class="pre">=</span> <span class="pre">-x</span></code> (this is <code class="docutils literal notranslate"><span class="pre">abs_of_neg</span></code>). The expression <code class="docutils literal notranslate"><span class="pre">le_or_gt</span> <span class="pre">0</span> <span class="pre">x</span></code> establishes <code class="docutils literal notranslate"><span class="pre">0</span> <span class="pre">&#8804;</span> <span class="pre">x</span> <span class="pre">&#8744;</span> <span class="pre">x</span> <span class="pre">&lt;</span> <span class="pre">0</span></code>, allowing us to split on those two cases. Try proving the triangle inequality using the two
  • @@ -1317,23 +1335,23 @@ first two theorems in the next snippet.
    They are given the same names they have in mathlib.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">namespace</span> <span class="n">MyAbs</span> - <span class="kd">theorem</span> <span class="n">le_abs_self</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&#8804;</span> <span class="n">abs</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="kd">theorem</span> <span class="n">le_abs_self</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&#8804;</span> <span class="bp">|</span><span class="n">x</span><span class="bp">|</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> - <span class="kd">theorem</span> <span class="n">neg_le_abs_self</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="bp">-</span><span class="n">x</span> <span class="bp">&#8804;</span> <span class="n">abs</span> <span class="n">x</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="kd">theorem</span> <span class="n">neg_le_abs_self</span> <span class="o">(</span><span class="n">x</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="bp">-</span><span class="n">x</span> <span class="bp">&#8804;</span> <span class="bp">|</span><span class="n">x</span><span class="bp">|</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> - <span class="kd">theorem</span> <span class="n">abs_add</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="n">abs</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="bp">&#8804;</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp">+</span> <span class="n">abs</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="kd">theorem</span> <span class="n">abs_add</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="bp">|</span><span class="n">x</span> <span class="bp">+</span> <span class="n">y</span><span class="bp">|</span> <span class="bp">&#8804;</span> <span class="bp">|</span><span class="n">x</span><span class="bp">|</span> <span class="bp">+</span> <span class="bp">|</span><span class="n">y</span><span class="bp">|</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div> <p>In case you enjoyed these (pun intended) and you want more practice with disjunction, try these.</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">lt_abs</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">abs</span> <span class="n">y</span> <span class="bp">&#8596;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">y</span> <span class="bp">&#8744;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="bp">-</span><span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">lt_abs</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="bp">|</span><span class="n">y</span><span class="bp">|</span> <span class="bp">&#8596;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">y</span> <span class="bp">&#8744;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="bp">-</span><span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> - <span class="kd">theorem</span> <span class="n">abs_lt</span> <span class="o">:</span> <span class="n">abs</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">y</span> <span class="bp">&#8596;</span> <span class="bp">-</span><span class="n">y</span> <span class="bp">&lt;</span> <span class="n">x</span> <span class="bp">&#8743;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="kd">theorem</span> <span class="n">abs_lt</span> <span class="o">:</span> <span class="bp">|</span><span class="n">x</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">y</span> <span class="bp">&#8596;</span> <span class="bp">-</span><span class="n">y</span> <span class="bp">&lt;</span> <span class="n">x</span> <span class="bp">&#8743;</span> <span class="n">x</span> <span class="bp">&lt;</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div>
  • @@ -1452,7 +1470,7 @@ that is, there is a number <span class="math notranslate nohighlight">\(N\)</spa
    <span class="math notranslate nohighlight">\(n \ge N\)</span>, <span class="math notranslate nohighlight">\(| s_n - a | &lt; \varepsilon\)</span>. In Lean, we can render this as follows:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">ConvergesTo</span> <span class="o">(</span><span class="n">s</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:=</span> - <span class="bp">&#8704;</span> <span class="n">&#949;</span> <span class="bp">&gt;</span> <span class="mi">0</span><span class="o">,</span> <span class="bp">&#8707;</span> <span class="n">N</span><span class="o">,</span> <span class="bp">&#8704;</span> <span class="n">n</span> <span class="bp">&#8805;</span> <span class="n">N</span><span class="o">,</span> <span class="n">abs</span> <span class="o">(</span><span class="n">s</span> <span class="n">n</span> <span class="bp">-</span> <span class="n">a</span><span class="o">)</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> + <span class="bp">&#8704;</span> <span class="n">&#949;</span> <span class="bp">&gt;</span> <span class="mi">0</span><span class="o">,</span> <span class="bp">&#8707;</span> <span class="n">N</span><span class="o">,</span> <span class="bp">&#8704;</span> <span class="n">n</span> <span class="bp">&#8805;</span> <span class="n">N</span><span class="o">,</span> <span class="bp">|</span><span class="n">s</span> <span class="n">n</span> <span class="bp">-</span> <span class="n">a</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> </pre></div> </div> <p>The notation <code class="docutils literal notranslate"><span class="pre">&#8704;</span> <span class="pre">&#949;</span> <span class="pre">&gt;</span> <span class="pre">0,</span> <span class="pre">...</span></code> is a convenient abbreviation
  • @@ -1472,7 +1490,7 @@ value for every <span class="math notranslate nohighlight">\(x\)</span>.
    The <code class="docutils literal notranslate"><span class="pre">ext</span></code> tactic enables us to prove an equation between functions by proving that their values are the same at all the values of their arguments.</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">=&gt;</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="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">=</span> <span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">=&gt;</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">y</span> <span class="bp">+</span> <span class="n">y</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">&#8614;</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="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">=</span> <span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">&#8614;</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">x</span> <span class="bp">*</span> <span class="n">y</span> <span class="bp">+</span> <span class="n">y</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ext</span> <span class="n">ring</span> </pre></div>
  • @@ -1484,7 +1502,7 @@ above proof.
    The second tactic, the <code class="docutils literal notranslate"><span class="pre">congr</span></code> tactic, allows us to prove an equation between two expressions by reconciling the parts that are different:</p> - <div class="highlight-lean notranslate"><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="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="n">abs</span> <span class="n">a</span> <span class="bp">=</span> <span class="n">abs</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">b</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> + <div class="highlight-lean notranslate"><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="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="bp">|</span><span class="n">a</span><span class="bp">|</span> <span class="bp">=</span> <span class="bp">|</span><span class="n">a</span> <span class="bp">-</span> <span class="n">b</span> <span class="bp">+</span> <span class="n">b</span><span class="bp">|</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">congr</span> <span class="n">ring</span> </pre></div>
  • @@ -1514,10 +1532,10 @@ and Lean can&#8217;t fill it in for us automatically,
    it simply leaves it for us as another goal.</p> <p>The following shows that any constant sequence <span class="math notranslate nohighlight">\(a, a, a, \ldots\)</span> converges.</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">convergesTo_const</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">=&gt;</span> <span class="n">a</span><span class="o">)</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">convergesTo_const</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8614;</span> <span class="n">a</span><span class="o">)</span> <span class="n">a</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">&#949;</span> <span class="n">&#949;pos</span> <span class="n">use</span> <span class="mi">0</span> - <span class="n">intro</span> <span class="n">n</span> <span class="n">nge</span><span class="bp">;</span> <span class="n">dsimp</span> + <span class="n">intro</span> <span class="n">n</span> <span class="n">nge</span> <span class="n">rw</span> <span class="o">[</span><span class="n">sub_self</span><span class="o">,</span> <span class="n">abs_zero</span><span class="o">]</span> <span class="n">apply</span> <span class="n">&#949;pos</span> </pre></div>
  • @@ -1545,9 +1563,9 @@ The following example begins to implement this strategy.
    See if you can finish it off.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">convergesTo_add</span> <span class="o">{</span><span class="n">s</span> <span class="n">t</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</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">&#8477;</span><span class="o">}</span> <span class="o">(</span><span class="n">cs</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">s</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">ct</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">t</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> - <span class="n">ConvergesTo</span> <span class="o">(</span><span class="k">fun</span> <span class="n">n</span> <span class="bp">=&gt;</span> <span class="n">s</span> <span class="n">n</span> <span class="bp">+</span> <span class="n">t</span> <span class="n">n</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="o">)</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="n">ConvergesTo</span> <span class="o">(</span><span class="k">fun</span> <span class="n">n</span> <span class="bp">&#8614;</span> <span class="n">s</span> <span class="n">n</span> <span class="bp">+</span> <span class="n">t</span> <span class="n">n</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="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">&#949;</span> <span class="n">&#949;pos</span> - <span class="n">dsimp</span> + <span class="n">dsimp</span> <span class="c1">-- this line is not needed but cleans up the goal a bit.</span> <span class="k">have</span> <span class="n">&#949;2pos</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">/</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">linarith</span> <span class="n">cases&#39;</span> <span class="n">cs</span> <span class="o">(</span><span class="n">&#949;</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="n">&#949;2pos</span> <span class="k">with</span> <span class="n">Ns</span> <span class="n">hs</span> <span class="n">cases&#39;</span> <span class="n">ct</span> <span class="o">(</span><span class="n">&#949;</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="n">&#949;2pos</span> <span class="k">with</span> <span class="n">Nt</span> <span class="n">ht</span>
  • @@ -1558,8 +1576,8 @@ See if you can finish it off.</p>
    <p>As hints, you can use <code class="docutils literal notranslate"><span class="pre">le_of_max_le_left</span></code> and <code class="docutils literal notranslate"><span class="pre">le_of_max_le_right</span></code>, and <code class="docutils literal notranslate"><span class="pre">norm_num</span></code> can prove <code class="docutils literal notranslate"><span class="pre">&#949;</span> <span class="pre">/</span> <span class="pre">2</span> <span class="pre">+</span> <span class="pre">&#949;</span> <span class="pre">/</span> <span class="pre">2</span> <span class="pre">=</span> <span class="pre">&#949;</span></code>. Also, it is helpful to use the <code class="docutils literal notranslate"><span class="pre">congr</span></code> tactic to - show that <code class="docutils literal notranslate"><span class="pre">abs</span> <span class="pre">(s</span> <span class="pre">n</span> <span class="pre">+</span> <span class="pre">t</span> <span class="pre">n</span> <span class="pre">-</span> <span class="pre">(a</span> <span class="pre">+</span> <span class="pre">b))</span></code> is equal to - <code class="docutils literal notranslate"><span class="pre">abs</span> <span class="pre">((s</span> <span class="pre">n</span> <span class="pre">-</span> <span class="pre">a)</span> <span class="pre">+</span> <span class="pre">(t</span> <span class="pre">n</span> <span class="pre">-</span> <span class="pre">b)),</span></code> + show that <code class="docutils literal notranslate"><span class="pre">|s</span> <span class="pre">n</span> <span class="pre">+</span> <span class="pre">t</span> <span class="pre">n</span> <span class="pre">-</span> <span class="pre">(a</span> <span class="pre">+</span> <span class="pre">b)|</span></code> is equal to + <code class="docutils literal notranslate"><span class="pre">|(s</span> <span class="pre">n</span> <span class="pre">-</span> <span class="pre">a)</span> <span class="pre">+</span> <span class="pre">(t</span> <span class="pre">n</span> <span class="pre">-</span> <span class="pre">b)|,</span></code> since then you can use the triangle inequality. Notice that we marked all the variables <code class="docutils literal notranslate"><span class="pre">s</span></code>, <code class="docutils literal notranslate"><span class="pre">t</span></code>, <code class="docutils literal notranslate"><span class="pre">a</span></code>, and <code class="docutils literal notranslate"><span class="pre">b</span></code> implicit because they can be inferred from the hypotheses.</p>
  • @@ -1568,19 +1586,21 @@ of addition is tricky.
    We will get there by proving some auxiliary statements first. See if you can also finish off the next proof, which shows that if <code class="docutils literal notranslate"><span class="pre">s</span></code> converges to <code class="docutils literal notranslate"><span class="pre">a</span></code>, - then <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">n</span> <span class="pre">=&gt;</span> <span class="pre">c</span> <span class="pre">*</span> <span class="pre">s</span> <span class="pre">n</span></code> converges to <code class="docutils literal notranslate"><span class="pre">c</span> <span class="pre">*</span> <span class="pre">a</span></code>. + then <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">n</span> <span class="pre">&#8614;</span> <span class="pre">c</span> <span class="pre">*</span> <span class="pre">s</span> <span class="pre">n</span></code> converges to <code class="docutils literal notranslate"><span class="pre">c</span> <span class="pre">*</span> <span class="pre">a</span></code>. It is helpful to split into cases depending on whether <code class="docutils literal notranslate"><span class="pre">c</span></code> is equal to zero or not. We have taken care of the zero case, and we have left you to prove the result with the extra assumption that <code class="docutils literal notranslate"><span class="pre">c</span></code> is nonzero.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">convergesTo_mul_const</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">(</span><span class="n">c</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">(</span><span class="n">cs</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">s</span> <span class="n">a</span><span class="o">)</span> <span class="o">:</span> - <span class="n">ConvergesTo</span> <span class="o">(</span><span class="k">fun</span> <span class="n">n</span> <span class="bp">=&gt;</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">s</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">c</span> <span class="bp">*</span> <span class="n">a</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="n">ConvergesTo</span> <span class="o">(</span><span class="k">fun</span> <span class="n">n</span> <span class="bp">&#8614;</span> <span class="n">c</span> <span class="bp">*</span> <span class="n">s</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">c</span> <span class="bp">*</span> <span class="n">a</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">by_cases</span> <span class="n">h</span> <span class="o">:</span> <span class="n">c</span> <span class="bp">=</span> <span class="mi">0</span> <span class="bp">&#183;</span> <span class="n">convert</span> <span class="n">convergesTo_const</span> <span class="mi">0</span> - <span class="bp">&#183;</span> <span class="n">rw</span> <span class="o">[</span><span class="n">h</span><span class="o">,</span> <span class="n">MulZeroClass.zero_mul</span><span class="o">]</span> - <span class="n">rw</span> <span class="o">[</span><span class="n">h</span><span class="o">,</span> <span class="n">MulZeroClass.zero_mul</span><span class="o">]</span> - <span class="k">have</span> <span class="n">acpos</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">abs</span> <span class="n">c</span> <span class="o">:=</span> <span class="n">abs_pos.mpr</span> <span class="n">h</span> + <span class="bp">&#183;</span> <span class="n">rw</span> <span class="o">[</span><span class="n">h</span><span class="o">]</span> + <span class="n">ring</span> + <span class="n">rw</span> <span class="o">[</span><span class="n">h</span><span class="o">]</span> + <span class="n">ring</span> + <span class="k">have</span> <span class="n">acpos</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="bp">|</span><span class="n">c</span><span class="bp">|</span> <span class="o">:=</span> <span class="n">abs_pos.mpr</span> <span class="n">h</span> <span class="gr">sorry</span> </pre></div> </div>
  • @@ -1589,9 +1609,9 @@ it shows that a convergent sequence is eventually bounded
    in absolute value. We have started you off; see if you can finish it.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">exists_abs_le_of_convergesTo</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">(</span><span class="n">cs</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">s</span> <span class="n">a</span><span class="o">)</span> <span class="o">:</span> - <span class="bp">&#8707;</span> <span class="n">N</span> <span class="n">b</span><span class="o">,</span> <span class="bp">&#8704;</span> <span class="n">n</span><span class="o">,</span> <span class="n">N</span> <span class="bp">&#8804;</span> <span class="n">n</span> <span class="bp">&#8594;</span> <span class="n">abs</span> <span class="o">(</span><span class="n">s</span> <span class="n">n</span><span class="o">)</span> <span class="bp">&lt;</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="bp">&#8707;</span> <span class="n">N</span> <span class="n">b</span><span class="o">,</span> <span class="bp">&#8704;</span> <span class="n">n</span><span class="o">,</span> <span class="n">N</span> <span class="bp">&#8804;</span> <span class="n">n</span> <span class="bp">&#8594;</span> <span class="bp">|</span><span class="n">s</span> <span class="n">n</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">b</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">cases&#39;</span> <span class="n">cs</span> <span class="mi">1</span> <span class="n">zero_lt_one</span> <span class="k">with</span> <span class="n">N</span> <span class="n">h</span> - <span class="n">use</span> <span class="n">N</span><span class="o">,</span> <span class="n">abs</span> <span class="n">a</span> <span class="bp">+</span> <span class="mi">1</span> + <span class="n">use</span> <span class="n">N</span><span class="o">,</span> <span class="bp">|</span><span class="n">a</span><span class="bp">|</span> <span class="bp">+</span> <span class="mi">1</span> <span class="gr">sorry</span> </pre></div> </div>
  • @@ -1602,13 +1622,13 @@ and we will see at the end of this section that it
    holds more generally.</p> <p>The next lemma is auxiliary: we prove that if <code class="docutils literal notranslate"><span class="pre">s</span></code> converges to <code class="docutils literal notranslate"><span class="pre">a</span></code> and <code class="docutils literal notranslate"><span class="pre">t</span></code> converges to <code class="docutils literal notranslate"><span class="pre">0</span></code>, - then <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">n</span> <span class="pre">=&gt;</span> <span class="pre">s</span> <span class="pre">n</span> <span class="pre">*</span> <span class="pre">t</span> <span class="pre">n</span></code> converges to <code class="docutils literal notranslate"><span class="pre">0</span></code>. + then <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">n</span> <span class="pre">&#8614;</span> <span class="pre">s</span> <span class="pre">n</span> <span class="pre">*</span> <span class="pre">t</span> <span class="pre">n</span></code> converges to <code class="docutils literal notranslate"><span class="pre">0</span></code>. To do so, we use the previous theorem to find a <code class="docutils literal notranslate"><span class="pre">B</span></code> that bounds <code class="docutils literal notranslate"><span class="pre">s</span></code> beyond some point <code class="docutils literal notranslate"><span class="pre">N&#8320;</span></code>. See if you can understand the strategy we have outlined and finish the proof.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">aux</span> <span class="o">{</span><span class="n">s</span> <span class="n">t</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">{</span><span class="n">a</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">}</span> <span class="o">(</span><span class="n">cs</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">s</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">ct</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">t</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> - <span class="n">ConvergesTo</span> <span class="o">(</span><span class="k">fun</span> <span class="n">n</span> <span class="bp">=&gt;</span> <span class="n">s</span> <span class="n">n</span> <span class="bp">*</span> <span class="n">t</span> <span class="n">n</span><span class="o">)</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="n">ConvergesTo</span> <span class="o">(</span><span class="k">fun</span> <span class="n">n</span> <span class="bp">&#8614;</span> <span class="n">s</span> <span class="n">n</span> <span class="bp">*</span> <span class="n">t</span> <span class="n">n</span><span class="o">)</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">&#949;</span> <span class="n">&#949;pos</span> <span class="n">dsimp</span> <span class="n">rcases</span> <span class="n">exists_abs_le_of_convergesTo</span> <span class="n">cs</span> <span class="k">with</span> <span class="o">&#10216;</span><span class="n">N&#8320;</span><span class="o">,</span> <span class="n">B</span><span class="o">,</span> <span class="n">h&#8320;</span><span class="o">&#10217;</span>
  • @@ -1623,8 +1643,8 @@ We are now within striking distance of our theorem.
    The following proof finishes it off.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">convergesTo_mul</span> <span class="o">{</span><span class="n">s</span> <span class="n">t</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</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">&#8477;</span><span class="o">}</span> <span class="o">(</span><span class="n">cs</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">s</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">ct</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">t</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> - <span class="n">ConvergesTo</span> <span class="o">(</span><span class="k">fun</span> <span class="n">n</span> <span class="bp">=&gt;</span> <span class="n">s</span> <span class="n">n</span> <span class="bp">*</span> <span class="n">t</span> <span class="n">n</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="o">)</span> <span class="o">:=</span> <span class="kd">by</span> - <span class="k">have</span> <span class="n">h&#8321;</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="o">(</span><span class="k">fun</span> <span class="n">n</span> <span class="bp">=&gt;</span> <span class="n">s</span> <span class="n">n</span> <span class="bp">*</span> <span class="o">(</span><span class="n">t</span> <span class="n">n</span> <span class="bp">+</span> <span class="bp">-</span><span class="n">b</span><span class="o">))</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="n">ConvergesTo</span> <span class="o">(</span><span class="k">fun</span> <span class="n">n</span> <span class="bp">&#8614;</span> <span class="n">s</span> <span class="n">n</span> <span class="bp">*</span> <span class="n">t</span> <span class="n">n</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="o">)</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="k">have</span> <span class="n">h&#8321;</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="o">(</span><span class="k">fun</span> <span class="n">n</span> <span class="bp">&#8614;</span> <span class="n">s</span> <span class="n">n</span> <span class="bp">*</span> <span class="o">(</span><span class="n">t</span> <span class="n">n</span> <span class="bp">+</span> <span class="bp">-</span><span class="n">b</span><span class="o">))</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">aux</span> <span class="n">cs</span> <span class="n">convert</span> <span class="n">convergesTo_add</span> <span class="n">ct</span> <span class="o">(</span><span class="n">convergesTo_const</span> <span class="o">(</span><span class="bp">-</span><span class="n">b</span><span class="o">))</span> <span class="n">ring</span>
  • @@ -1643,17 +1663,17 @@ you can delete the proof sketch and try proving it from scratch.)</p>
    <span class="o">(</span><span class="n">sa</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">s</span> <span class="n">a</span><span class="o">)</span> <span class="o">(</span><span class="n">sb</span> <span class="o">:</span> <span class="n">ConvergesTo</span> <span class="n">s</span> <span class="n">b</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="o">:=</span> <span class="kd">by</span> <span class="n">by_contra</span> <span class="n">abne</span> - <span class="k">have</span> <span class="o">:</span> <span class="n">abs</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="bp">&gt;</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> - <span class="k">let</span> <span class="n">&#949;</span> <span class="o">:=</span> <span class="n">abs</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="bp">/</span> <span class="mi">2</span> + <span class="k">have</span> <span class="o">:</span> <span class="bp">|</span><span class="n">a</span> <span class="bp">-</span> <span class="n">b</span><span class="bp">|</span> <span class="bp">&gt;</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> + <span class="k">let</span> <span class="n">&#949;</span> <span class="o">:=</span> <span class="bp">|</span><span class="n">a</span> <span class="bp">-</span> <span class="n">b</span><span class="bp">|</span> <span class="bp">/</span> <span class="mi">2</span> <span class="k">have</span> <span class="n">&#949;pos</span> <span class="o">:</span> <span class="n">&#949;</span> <span class="bp">&gt;</span> <span class="mi">0</span> <span class="o">:=</span> <span class="kd">by</span> - <span class="n">change</span> <span class="n">abs</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="bp">/</span> <span class="mi">2</span> <span class="bp">&gt;</span> <span class="mi">0</span> + <span class="n">change</span> <span class="bp">|</span><span class="n">a</span> <span class="bp">-</span> <span class="n">b</span><span class="bp">|</span> <span class="bp">/</span> <span class="mi">2</span> <span class="bp">&gt;</span> <span class="mi">0</span> <span class="n">linarith</span> <span class="n">cases&#39;</span> <span class="n">sa</span> <span class="n">&#949;</span> <span class="n">&#949;pos</span> <span class="k">with</span> <span class="n">Na</span> <span class="n">hNa</span> <span class="n">cases&#39;</span> <span class="n">sb</span> <span class="n">&#949;</span> <span class="n">&#949;pos</span> <span class="k">with</span> <span class="n">Nb</span> <span class="n">hNb</span> <span class="k">let</span> <span class="n">N</span> <span class="o">:=</span> <span class="n">max</span> <span class="n">Na</span> <span class="n">Nb</span> - <span class="k">have</span> <span class="n">absa</span> <span class="o">:</span> <span class="n">abs</span> <span class="o">(</span><span class="n">s</span> <span class="n">N</span> <span class="bp">-</span> <span class="n">a</span><span class="o">)</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> - <span class="k">have</span> <span class="n">absb</span> <span class="o">:</span> <span class="n">abs</span> <span class="o">(</span><span class="n">s</span> <span class="n">N</span> <span class="bp">-</span> <span class="n">b</span><span class="o">)</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> - <span class="k">have</span> <span class="o">:</span> <span class="n">abs</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="bp">&lt;</span> <span class="n">abs</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="kd">by</span> <span class="gr">sorry</span> + <span class="k">have</span> <span class="n">absa</span> <span class="o">:</span> <span class="bp">|</span><span class="n">s</span> <span class="n">N</span> <span class="bp">-</span> <span class="n">a</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> + <span class="k">have</span> <span class="n">absb</span> <span class="o">:</span> <span class="bp">|</span><span class="n">s</span> <span class="n">N</span> <span class="bp">-</span> <span class="n">b</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> + <span class="k">have</span> <span class="o">:</span> <span class="bp">|</span><span class="n">a</span> <span class="bp">-</span> <span class="n">b</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="bp">|</span><span class="n">a</span> <span class="bp">-</span> <span class="n">b</span><span class="bp">|</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="n">exact</span> <span class="n">lt_irrefl</span> <span class="n">_</span> <span class="n">this</span> </pre></div> </div>
  • @@ -1666,7 +1686,7 @@ everywhere by any linear order <code class="docutils literal notranslate"><span
    <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">{</span><span class="n">&#945;</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">LinearOrder</span> <span class="n">&#945;</span><span class="o">]</span> <span class="kd">def</span> <span class="n">ConvergesTo&#39;</span> <span class="o">(</span><span class="n">s</span> <span class="o">:</span> <span class="n">&#945;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:=</span> - <span class="bp">&#8704;</span> <span class="n">&#949;</span> <span class="bp">&gt;</span> <span class="mi">0</span><span class="o">,</span> <span class="bp">&#8707;</span> <span class="n">N</span><span class="o">,</span> <span class="bp">&#8704;</span> <span class="n">n</span> <span class="bp">&#8805;</span> <span class="n">N</span><span class="o">,</span> <span class="n">abs</span> <span class="o">(</span><span class="n">s</span> <span class="n">n</span> <span class="bp">-</span> <span class="n">a</span><span class="o">)</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> + <span class="bp">&#8704;</span> <span class="n">&#949;</span> <span class="bp">&gt;</span> <span class="mi">0</span><span class="o">,</span> <span class="bp">&#8707;</span> <span class="n">N</span><span class="o">,</span> <span class="bp">&#8704;</span> <span class="n">n</span> <span class="bp">&#8805;</span> <span class="n">N</span><span class="o">,</span> <span class="bp">|</span><span class="n">s</span> <span class="n">n</span> <span class="bp">-</span> <span class="n">a</span><span class="bp">|</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> </pre></div> </div> <p>In <a class="reference internal" href="C08_Topology.html#filters"><span class="std std-numref">Section 8.1</span></a>, we will see that mathlib has mechanisms
• html/C04_Sets_and_Functions.html
  • @@ -1,7 +1,7 @@
    <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>4. Sets and Functions &mdash; Mathematics in Lean 0.1 documentation</title>
  • @@ -13,16 +13,16 @@
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  • @@ -31,11 +31,15 @@
    <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" /> + <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
  • @@ -51,7 +55,7 @@
    <li class="toctree-l2"><a class="reference internal" href="#the-schroder-bernstein-theorem">4.3. The Schr&#246;der-Bernstein Theorem</a></li> </ul> </li> - <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Elementary Number Theory</a></li> <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li>
  • @@ -75,8 +79,8 @@
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  • @@ -156,7 +160,7 @@ you can see the effects of these commands.</p>
    <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">&#8838;</span> <span class="n">t</span><span class="o">)</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">&#8745;</span> <span class="n">u</span> <span class="bp">&#8838;</span> <span class="n">t</span> <span class="bp">&#8745;</span> <span class="n">u</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">subset_def</span><span class="o">,</span> <span class="n">inter_def</span><span class="o">,</span> <span class="n">inter_def</span><span class="o">]</span> <span class="n">rw</span> <span class="o">[</span><span class="n">subset_def</span><span class="o">]</span> <span class="n">at</span> <span class="n">h</span> - <span class="n">dsimp</span> + <span class="n">simp</span> <span class="n">only</span> <span class="o">[</span><span class="n">mem_setOf</span><span class="o">]</span> <span class="n">rintro</span> <span class="n">x</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">&#10217;</span> <span class="n">exact</span> <span class="o">&#10216;</span><span class="n">h</span> <span class="n">_</span> <span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">&#10217;</span>
  • @@ -180,10 +184,10 @@ to make sense of the <code class="docutils literal notranslate"><span class="pre
    Lean is forced to expand the definitions. The following examples also illustrate the phenomenon:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">theorem</span> <span class="n">foo</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">&#8838;</span> <span class="n">t</span><span class="o">)</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">&#8745;</span> <span class="n">u</span> <span class="bp">&#8838;</span> <span class="n">t</span> <span class="bp">&#8745;</span> <span class="n">u</span> <span class="o">:=</span> - <span class="k">fun</span> <span class="n">x</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">&#10217;</span> <span class="bp">=&gt;</span> <span class="o">&#10216;</span><span class="n">h</span> <span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">&#10217;</span> + <span class="k">fun</span> <span class="n">x</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">&#10217;</span> <span class="bp">&#8614;</span> <span class="o">&#10216;</span><span class="n">h</span> <span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">&#10217;</span> <span class="kd">example</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">&#8838;</span> <span class="n">t</span><span class="o">)</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">&#8745;</span> <span class="n">u</span> <span class="bp">&#8838;</span> <span class="n">t</span> <span class="bp">&#8745;</span> <span class="n">u</span> <span class="o">:=</span> - <span class="k">fun</span> <span class="n">x</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">&#10217;</span> <span class="bp">=&gt;</span> <span class="o">&#10216;</span><span class="n">h</span> <span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">&#10217;</span> + <span class="k">fun</span> <span class="n">x</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">&#10217;</span> <span class="bp">&#8614;</span> <span class="o">&#10216;</span><span class="n">h</span> <span class="n">xs</span><span class="o">,</span> <span class="n">xu</span><span class="o">&#10217;</span> </pre></div> </div> <p>Due to a quirk of how Lean processes its input,
  • @@ -281,15 +285,10 @@ does not harm the proof.
    In fact, if you like inscrutable proof terms, the following one-line proof is for you:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">&#8745;</span> <span class="n">t</span> <span class="bp">=</span> <span class="n">t</span> <span class="bp">&#8745;</span> <span class="n">s</span> <span class="o">:=</span> - <span class="n">Set.ext</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="o">&#10216;</span><span class="k">fun</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">&#10217;</span> <span class="bp">=&gt;</span> <span class="o">&#10216;</span><span class="n">xt</span><span class="o">,</span> <span class="n">xs</span><span class="o">&#10217;,</span> <span class="k">fun</span> <span class="o">&#10216;</span><span class="n">xt</span><span class="o">,</span> <span class="n">xs</span><span class="o">&#10217;</span> <span class="bp">=&gt;</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">&#10217;&#10217;</span> + <span class="n">Set.ext</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="o">&#10216;</span><span class="k">fun</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">&#10217;</span> <span class="bp">&#8614;</span> <span class="o">&#10216;</span><span class="n">xt</span><span class="o">,</span> <span class="n">xs</span><span class="o">&#10217;,</span> <span class="k">fun</span> <span class="o">&#10216;</span><span class="n">xt</span><span class="o">,</span> <span class="n">xs</span><span class="o">&#10217;</span> <span class="bp">&#8614;</span> <span class="o">&#10216;</span><span class="n">xs</span><span class="o">,</span> <span class="n">xt</span><span class="o">&#10217;&#10217;</span> </pre></div> </div> - <p>The dollar sign is a useful syntax: - writing <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">$</span> <span class="pre">...</span></code> - is essentially the same as writing <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">(...)</span></code>, - but it saves us the trouble of having to close - a set of parentheses at the end of a long expression. - Here is an even shorter proof, + <p>Here is an even shorter proof, using the simplifier:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">s</span> <span class="bp">&#8745;</span> <span class="n">t</span> <span class="bp">=</span> <span class="n">t</span> <span class="bp">&#8745;</span> <span class="n">s</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ext</span> <span class="n">x</span><span class="bp">;</span> <span class="n">simp</span> <span class="o">[</span><span class="n">and_comm</span><span class="o">]</span> </pre></div>
  • @@ -511,8 +510,8 @@ at an appropriate point in the proof.</p>
    </div> <p>Mathlib also has bounded unions and intersections, which are analogous to the bounded quantifiers. - You can unpack their meaning with <code class="docutils literal notranslate"><span class="pre">mem_Union&#8322;</span></code> - and <code class="docutils literal notranslate"><span class="pre">mem_Inter&#8322;</span></code>. + You can unpack their meaning with <code class="docutils literal notranslate"><span class="pre">mem_iUnion&#8322;</span></code> + and <code class="docutils literal notranslate"><span class="pre">mem_iInter&#8322;</span></code>. As the following examples show, Lean&#8217;s simplifier carries out these replacements as well.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">def</span> <span class="n">primes</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">&#8469;</span> <span class="o">:=</span>
  • @@ -777,13 +776,13 @@ the statements to a subset of the domain type.</p>
    <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">InjOn</span> <span class="n">sqrt</span> <span class="o">{</span> <span class="n">x</span> <span class="bp">|</span> <span class="n">x</span> <span class="bp">&#8805;</span> <span class="mi">0</span> <span class="o">}</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> - <span class="kd">example</span> <span class="o">:</span> <span class="n">InjOn</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="o">{</span> <span class="n">x</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">|</span> <span class="n">x</span> <span class="bp">&#8805;</span> <span class="mi">0</span> <span class="o">}</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="kd">example</span> <span class="o">:</span> <span class="n">InjOn</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="o">{</span> <span class="n">x</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">|</span> <span class="n">x</span> <span class="bp">&#8805;</span> <span class="mi">0</span> <span class="o">}</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="kd">example</span> <span class="o">:</span> <span class="n">sqrt</span> <span class="bp">&#39;&#39;</span> <span class="o">{</span> <span class="n">x</span> <span class="bp">|</span> <span class="n">x</span> <span class="bp">&#8805;</span> <span class="mi">0</span> <span class="o">}</span> <span class="bp">=</span> <span class="o">{</span> <span class="n">y</span> <span class="bp">|</span> <span class="n">y</span> <span class="bp">&#8805;</span> <span class="mi">0</span> <span class="o">}</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> - <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">range</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">=</span> <span class="o">{</span> <span class="n">y</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">|</span> <span class="n">y</span> <span class="bp">&#8805;</span> <span class="mi">0</span> <span class="o">}</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="kd">example</span> <span class="o">:</span> <span class="o">(</span><span class="n">range</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">=</span> <span class="o">{</span> <span class="n">y</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">|</span> <span class="n">y</span> <span class="bp">&#8805;</span> <span class="mi">0</span> <span class="o">}</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> </pre></div> </div>
  • @@ -826,7 +825,7 @@ as follows:</p>
    <span class="kn">open</span> <span class="n">Classical</span> - <span class="kd">def</span> <span class="n">inverse</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">&#945;</span> <span class="bp">&#8594;</span> <span class="n">&#946;</span><span class="o">)</span> <span class="o">:</span> <span class="n">&#946;</span> <span class="bp">&#8594;</span> <span class="n">&#945;</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">y</span> <span class="o">:</span> <span class="n">&#946;</span> <span class="bp">=&gt;</span> + <span class="kd">def</span> <span class="n">inverse</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">&#945;</span> <span class="bp">&#8594;</span> <span class="n">&#946;</span><span class="o">)</span> <span class="o">:</span> <span class="n">&#946;</span> <span class="bp">&#8594;</span> <span class="n">&#945;</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">y</span> <span class="o">:</span> <span class="n">&#946;</span> <span class="bp">&#8614;</span> <span class="k">if</span> <span class="n">h</span> <span class="o">:</span> <span class="bp">&#8707;</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">y</span> <span class="k">then</span> <span class="n">Classical.choose</span> <span class="n">h</span> <span class="k">else</span> <span class="n">default</span> <span class="kd">theorem</span> <span class="n">inverse_spec</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">&#945;</span> <span class="bp">&#8594;</span> <span class="n">&#946;</span><span class="o">}</span> <span class="o">(</span><span class="n">y</span> <span class="o">:</span> <span class="n">&#946;</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">&#8707;</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">y</span><span class="o">)</span> <span class="o">:</span> <span class="n">f</span> <span class="o">(</span><span class="n">inverse</span> <span class="n">f</span> <span class="n">y</span><span class="o">)</span> <span class="bp">=</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span>
  • @@ -1162,7 +1161,7 @@ and the proof uses the fact that <code class="docutils literal notranslate"><spa
    </div> <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> <a href="C03_Logic.html" class="btn btn-neutral float-left" title="3. Logic" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> - <a href="C05_Number_Theory.html" class="btn btn-neutral float-right" title="5. Number Theory" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right" aria-hidden="true"></span></a> + <a href="C05_Number_Theory.html" class="btn btn-neutral float-right" title="5. Elementary Number Theory" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right" aria-hidden="true"></span></a> </div> <hr/>
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  • @@ -1,10 +1,10 @@
    <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> - <title>5. Number Theory &mdash; Mathematics in Lean 0.1 documentation</title> + <title>5. Elementary Number Theory &mdash; Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" /> <link rel="stylesheet" href="_static/css/theme.css" type="text/css" /> <link rel="stylesheet" href="_static/css/custom.css" type="text/css" />
  • @@ -13,11 +13,11 @@
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  • @@ -31,11 +31,15 @@
    <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" /> + <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
  • @@ -46,7 +50,7 @@
    <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> - <li class="toctree-l1 current"><a class="current reference internal" href="#">5. Number Theory</a><ul> + <li class="toctree-l1 current"><a class="current reference internal" href="#">5. Elementary Number Theory</a><ul> <li class="toctree-l2"><a class="reference internal" href="#irrational-roots">5.1. Irrational Roots</a></li> <li class="toctree-l2"><a class="reference internal" href="#induction-and-recursion">5.2. Induction and Recursion</a></li> <li class="toctree-l2"><a class="reference internal" href="#infinitely-many-primes">5.3. Infinitely Many Primes</a></li>
  • @@ -75,8 +79,8 @@
    <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> - <li><a href="index.html" class="icon icon-home"></a> &raquo;</li> - <li><span class="section-number">5. </span>Number Theory</li> + <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 class="wy-breadcrumbs-aside"> <a href="_sources/C05_Number_Theory.rst.txt" rel="nofollow"> View page source</a> </li>
  • @@ -86,8 +90,8 @@
    <div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> - <section id="number-theory"> - <span id="id1"></span><h1><span class="section-number">5. </span>Number Theory<a class="headerlink" href="#number-theory" title="Permalink to this heading">&#61633;</a></h1> + <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">&#61633;</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,
  • @@ -209,7 +213,7 @@ and if all else fails,
    don&#8217;t hesitate to ask on <a class="reference external" href="https://leanprover.zulipchat.com/">Zulip</a>.</p> <div class="highlight-lean notranslate"><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">Nat</span><span class="o">)</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">a</span> <span class="bp">*</span> <span class="n">c</span><span class="o">)</span> <span class="o">(</span><span class="n">h&#39;</span> <span class="o">:</span> <span class="n">a</span> <span class="bp">&#8800;</span> <span class="mi">0</span><span class="o">)</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">c</span> <span class="o">:=</span> - <span class="c1">-- library_search suggests the following:</span> + <span class="c1">-- apply? suggests the following:</span> <span class="o">(</span><span class="n">mul_right_inj&#39;</span> <span class="n">h&#39;</span><span class="o">)</span><span class="bp">.</span><span class="n">mp</span> <span class="n">h</span> </pre></div> </div>
  • @@ -344,7 +348,7 @@ to finish it off.</p>
    <span class="n">k</span> <span class="bp">&#8739;</span> <span class="n">r.factorization</span> <span class="n">p</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">cases&#39;</span> <span class="n">r</span> <span class="k">with</span> <span class="n">r</span> <span class="bp">&#183;</span> <span class="n">simp</span> - <span class="k">have</span> <span class="n">npow_nz</span> <span class="o">:</span> <span class="n">n</span> <span class="bp">^</span> <span class="n">k</span> <span class="bp">&#8800;</span> <span class="mi">0</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">npowz</span> <span class="bp">=&gt;</span> <span class="n">nnz</span> <span class="o">(</span><span class="n">pow_eq_zero</span> <span class="n">npowz</span><span class="o">)</span> + <span class="k">have</span> <span class="n">npow_nz</span> <span class="o">:</span> <span class="n">n</span> <span class="bp">^</span> <span class="n">k</span> <span class="bp">&#8800;</span> <span class="mi">0</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">npowz</span> <span class="bp">&#8614;</span> <span class="n">nnz</span> <span class="o">(</span><span class="n">pow_eq_zero</span> <span class="n">npowz</span><span class="o">)</span> <span class="k">have</span> <span class="n">eq1</span> <span class="o">:</span> <span class="o">(</span><span class="n">m</span> <span class="bp">^</span> <span class="n">k</span><span class="o">)</span><span class="bp">.</span><span class="n">factorization</span> <span class="n">p</span> <span class="bp">=</span> <span class="n">k</span> <span class="bp">*</span> <span class="n">m.factorization</span> <span class="n">p</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="n">eq2</span> <span class="o">:</span> <span class="o">(</span><span class="n">r.succ</span> <span class="bp">*</span> <span class="n">n</span> <span class="bp">^</span> <span class="n">k</span><span class="o">)</span><span class="bp">.</span><span class="n">factorization</span> <span class="n">p</span> <span class="bp">=</span>
• html/C06_Structures.html
  • @@ -1,7 +1,7 @@
    <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>6. Structures &mdash; Mathematics in Lean 0.1 documentation</title>
  • @@ -13,17 +13,17 @@
    <script src="_static/js/html5shiv.min.js"></script> <![endif]--> - <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/_sphinx_javascript_frameworks_compat.js"></script> + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.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" /> <link rel="search" title="Search" href="search.html" /> <link rel="next" title="7. Hierarchies" href="C07_Hierarchies.html" /> - <link rel="prev" title="5. Number Theory" href="C05_Number_Theory.html" /> + <link rel="prev" title="5. Elementary Number Theory" href="C05_Number_Theory.html" /> </head> <body class="wy-body-for-nav">
  • @@ -31,11 +31,15 @@
    <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" /> + <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
  • @@ -46,7 +50,7 @@
    <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> - <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Elementary Number Theory</a></li> <li class="toctree-l1 current"><a class="current reference internal" href="#">6. Structures</a><ul> <li class="toctree-l2"><a class="reference internal" href="#defining-structures">6.1. Defining structures</a></li> <li class="toctree-l2"><a class="reference internal" href="#algebraic-structures">6.2. Algebraic Structures</a></li>
  • @@ -75,8 +79,8 @@
    <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> - <li><a href="index.html" class="icon icon-home"></a> &raquo;</li> - <li><span class="section-number">6. </span>Structures</li> + <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> + <li class="breadcrumb-item active"><span class="section-number">6. </span>Structures</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C06_Structures.rst.txt" rel="nofollow"> View page source</a> </li>
  • @@ -257,7 +261,7 @@ components, which we can do with <code class="docutils literal notranslate"><spa
    <span class="n">rintro</span> <span class="o">&#10216;</span><span class="n">xa</span><span class="o">,</span> <span class="n">ya</span><span class="o">,</span> <span class="n">za</span><span class="o">&#10217;</span> <span class="o">&#10216;</span><span class="n">xb</span><span class="o">,</span> <span class="n">yb</span><span class="o">,</span> <span class="n">zb</span><span class="o">&#10217;</span> <span class="n">simp</span> <span class="o">[</span><span class="n">addAlt</span><span class="o">,</span> <span class="n">add_comm</span><span class="o">]</span> - <span class="kd">example</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">,</span> <span class="n">add</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">add</span> <span class="n">b</span> <span class="n">a</span> <span class="o">:=</span> <span class="k">fun</span> <span class="o">&#10216;</span><span class="n">xa</span><span class="o">,</span> <span class="n">ya</span><span class="o">,</span> <span class="n">za</span><span class="o">&#10217;</span> <span class="o">&#10216;</span><span class="n">xb</span><span class="o">,</span> <span class="n">yb</span><span class="o">,</span> <span class="n">zb</span><span class="o">&#10217;</span> <span class="bp">=&gt;</span> <span class="kd">by</span> + <span class="kd">example</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">Point</span><span class="o">,</span> <span class="n">add</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">add</span> <span class="n">b</span> <span class="n">a</span> <span class="o">:=</span> <span class="k">fun</span> <span class="o">&#10216;</span><span class="n">xa</span><span class="o">,</span> <span class="n">ya</span><span class="o">,</span> <span class="n">za</span><span class="o">&#10217;</span> <span class="o">&#10216;</span><span class="n">xb</span><span class="o">,</span> <span class="n">yb</span><span class="o">,</span> <span class="n">zb</span><span class="o">&#10217;</span> <span class="bp">&#8614;</span> <span class="kd">by</span> <span class="n">simp</span> <span class="o">[</span><span class="n">add</span><span class="o">,</span> <span class="n">add_comm</span><span class="o">]</span> </pre></div> </div>
  • @@ -923,7 +927,7 @@ Here is one way it comes up.
    In Lean&#8217;s foundation, a data type <code class="docutils literal notranslate"><span class="pre">&#945;</span></code> may be empty. In a number of applications, however, it is useful to know that a type has at least one element. - For example, the function <code class="docutils literal notranslate"><span class="pre">List.head</span></code>, which returns the first + For example, the function <code class="docutils literal notranslate"><span class="pre">List.headI</span></code>, which returns the first element of a list, can return the default value when the list is empty. To make that work, the Lean library defines a class <code class="docutils literal notranslate"><span class="pre">Inhabited</span> <span class="pre">&#945;</span></code>, which does nothing more than store a default value.
  • @@ -1069,13 +1073,13 @@ be a square root of <span class="math notranslate nohighlight">\(-1\)</span>. Th
    <span class="o">&#10216;&#10216;</span><span class="mi">1</span><span class="o">,</span> <span class="mi">0</span><span class="o">&#10217;&#10217;</span> <span class="kd">instance</span> <span class="o">:</span> <span class="n">Add</span> <span class="n">gaussInt</span> <span class="o">:=</span> - <span class="o">&#10216;</span><span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">=&gt;</span> <span class="o">&#10216;</span><span class="n">x.re</span> <span class="bp">+</span> <span class="n">y.re</span><span class="o">,</span> <span class="n">x.im</span> <span class="bp">+</span> <span class="n">y.im</span><span class="o">&#10217;&#10217;</span> + <span class="o">&#10216;</span><span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">&#8614;</span> <span class="o">&#10216;</span><span class="n">x.re</span> <span class="bp">+</span> <span class="n">y.re</span><span class="o">,</span> <span class="n">x.im</span> <span class="bp">+</span> <span class="n">y.im</span><span class="o">&#10217;&#10217;</span> <span class="kd">instance</span> <span class="o">:</span> <span class="n">Neg</span> <span class="n">gaussInt</span> <span class="o">:=</span> - <span class="o">&#10216;</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="o">&#10216;</span><span class="bp">-</span><span class="n">x.re</span><span class="o">,</span> <span class="bp">-</span><span class="n">x.im</span><span class="o">&#10217;&#10217;</span> + <span class="o">&#10216;</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="o">&#10216;</span><span class="bp">-</span><span class="n">x.re</span><span class="o">,</span> <span class="bp">-</span><span class="n">x.im</span><span class="o">&#10217;&#10217;</span> <span class="kd">instance</span> <span class="o">:</span> <span class="n">Mul</span> <span class="n">gaussInt</span> <span class="o">:=</span> - <span class="o">&#10216;</span><span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">=&gt;</span> <span class="o">&#10216;</span><span class="n">x.re</span> <span class="bp">*</span> <span class="n">y.re</span> <span class="bp">-</span> <span class="n">x.im</span> <span class="bp">*</span> <span class="n">y.im</span><span class="o">,</span> <span class="n">x.re</span> <span class="bp">*</span> <span class="n">y.im</span> <span class="bp">+</span> <span class="n">x.im</span> <span class="bp">*</span> <span class="n">y.re</span><span class="o">&#10217;&#10217;</span> + <span class="o">&#10216;</span><span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">&#8614;</span> <span class="o">&#10216;</span><span class="n">x.re</span> <span class="bp">*</span> <span class="n">y.re</span> <span class="bp">-</span> <span class="n">x.im</span> <span class="bp">*</span> <span class="n">y.im</span><span class="o">,</span> <span class="n">x.re</span> <span class="bp">*</span> <span class="n">y.im</span> <span class="bp">+</span> <span class="n">x.im</span> <span class="bp">*</span> <span class="n">y.re</span><span class="o">&#10217;&#10217;</span> </pre></div> </div> <p>As noted in <a class="reference internal" href="#section-structures"><span class="std std-numref">Section 6.1</span></a>, it is a good idea to put all the definitions
  • @@ -1238,7 +1242,7 @@ satisfy the following:</p>
    <span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">&#8484;</span><span class="o">)</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">&#8800;</span> <span class="mi">0</span> <span class="bp">&#8594;</span> <span class="mi">0</span> <span class="bp">&#8804;</span> <span class="n">a</span> <span class="bp">%</span> <span class="n">b</span> <span class="o">:=</span> <span class="n">Int.emod_nonneg</span> <span class="n">a</span> - <span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">&#8484;</span><span class="o">)</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">&#8800;</span> <span class="mi">0</span> <span class="bp">&#8594;</span> <span class="n">a</span> <span class="bp">%</span> <span class="n">b</span> <span class="bp">&lt;</span> <span class="n">abs</span> <span class="n">b</span> <span class="o">:=</span> + <span class="kd">example</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">&#8484;</span><span class="o">)</span> <span class="o">:</span> <span class="n">b</span> <span class="bp">&#8800;</span> <span class="mi">0</span> <span class="bp">&#8594;</span> <span class="n">a</span> <span class="bp">%</span> <span class="n">b</span> <span class="bp">&lt;</span> <span class="bp">|</span><span class="n">b</span><span class="bp">|</span> <span class="o">:=</span> <span class="n">Int.emod_lt</span> <span class="n">a</span> </pre></div> </div>
  • @@ -1337,7 +1341,7 @@ from the remainder.</p>
    <span class="n">rw</span> <span class="o">[</span><span class="n">div&#39;</span><span class="o">,</span> <span class="n">mod&#39;</span><span class="o">]</span> <span class="n">linarith</span> <span class="o">[</span><span class="n">Int.ediv_add_emod</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="mi">2</span><span class="o">)</span> <span class="n">b</span><span class="o">]</span> - <span class="kd">theorem</span> <span class="n">abs_mod&#39;_le</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">&#8484;</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="n">abs</span> <span class="o">(</span><span class="n">mod&#39;</span> <span class="n">a</span> <span class="n">b</span><span class="o">)</span> <span class="bp">&#8804;</span> <span class="n">b</span> <span class="bp">/</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="kd">theorem</span> <span class="n">abs_mod&#39;_le</span> <span class="o">(</span><span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">&#8484;</span><span class="o">)</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">b</span><span class="o">)</span> <span class="o">:</span> <span class="bp">|</span><span class="n">mod&#39;</span> <span class="n">a</span> <span class="n">b</span><span class="bp">|</span> <span class="bp">&#8804;</span> <span class="n">b</span> <span class="bp">/</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mod&#39;</span><span class="o">,</span> <span class="n">abs_le</span><span class="o">]</span> <span class="n">constructor</span> <span class="bp">&#183;</span> <span class="n">linarith</span> <span class="o">[</span><span class="n">Int.emod_nonneg</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="mi">2</span><span class="o">)</span> <span class="n">h.ne&#39;</span><span class="o">]</span>
  • @@ -1406,7 +1410,7 @@ then the real and imaginary parts of <code class="docutils literal notranslate">
    <span class="math notranslate nohighlight">\((a + bi) (c - di)\)</span>, and the denominators are both equal to the norm of <span class="math notranslate nohighlight">\(c + di\)</span>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="o">:</span> <span class="n">Div</span> <span class="n">gaussInt</span> <span class="o">:=</span> - <span class="o">&#10216;</span><span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">=&gt;</span> <span class="o">&#10216;</span><span class="n">Int.div&#39;</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">conj</span> <span class="n">y</span><span class="o">)</span><span class="bp">.</span><span class="n">re</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span><span class="o">),</span> <span class="n">Int.div&#39;</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">conj</span> <span class="n">y</span><span class="o">)</span><span class="bp">.</span><span class="n">im</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span><span class="o">)&#10217;&#10217;</span> + <span class="o">&#10216;</span><span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">&#8614;</span> <span class="o">&#10216;</span><span class="n">Int.div&#39;</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">conj</span> <span class="n">y</span><span class="o">)</span><span class="bp">.</span><span class="n">re</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span><span class="o">),</span> <span class="n">Int.div&#39;</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">conj</span> <span class="n">y</span><span class="o">)</span><span class="bp">.</span><span class="n">im</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span><span class="o">)&#10217;&#10217;</span> </pre></div> </div> <p>Having defined <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">/</span> <span class="pre">y</span></code>, We define <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">%</span> <span class="pre">y</span></code> to be the remainder,
  • @@ -1414,7 +1418,7 @@ of <span class="math notranslate nohighlight">\(c + di\)</span>.</p>
    theorems <code class="docutils literal notranslate"><span class="pre">div_def</span></code> and <code class="docutils literal notranslate"><span class="pre">mod_def</span></code> so that we can use them with <code class="docutils literal notranslate"><span class="pre">simp</span></code> and <code class="docutils literal notranslate"><span class="pre">rewrite</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="o">:</span> <span class="n">Mod</span> <span class="n">gaussInt</span> <span class="o">:=</span> - <span class="o">&#10216;</span><span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">=&gt;</span> <span class="n">x</span> <span class="bp">-</span> <span class="n">y</span> <span class="bp">*</span> <span class="o">(</span><span class="n">x</span> <span class="bp">/</span> <span class="n">y</span><span class="o">)&#10217;</span> + <span class="o">&#10216;</span><span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">&#8614;</span> <span class="n">x</span> <span class="bp">-</span> <span class="n">y</span> <span class="bp">*</span> <span class="o">(</span><span class="n">x</span> <span class="bp">/</span> <span class="n">y</span><span class="o">)&#10217;</span> <span class="kd">theorem</span> <span class="n">div_def</span> <span class="o">(</span><span class="n">x</span> <span class="n">y</span> <span class="o">:</span> <span class="n">gaussInt</span><span class="o">)</span> <span class="o">:</span> <span class="n">x</span> <span class="bp">/</span> <span class="n">y</span> <span class="bp">=</span> <span class="o">&#10216;</span><span class="n">Int.div&#39;</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">conj</span> <span class="n">y</span><span class="o">)</span><span class="bp">.</span><span class="n">re</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span><span class="o">),</span> <span class="n">Int.div&#39;</span> <span class="o">(</span><span class="n">x</span> <span class="bp">*</span> <span class="n">conj</span> <span class="n">y</span><span class="o">)</span><span class="bp">.</span><span class="n">im</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span><span class="o">)&#10217;</span> <span class="o">:=</span>
  • @@ -1457,8 +1461,8 @@ to step through the details and see if you can find a nicer argument.</p>
    <span class="k">have</span> <span class="n">H2</span> <span class="o">:</span> <span class="n">norm</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="bp">*</span> <span class="n">norm</span> <span class="n">y</span> <span class="bp">&#8804;</span> <span class="n">norm</span> <span class="n">y</span> <span class="bp">/</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">norm</span> <span class="n">y</span> <span class="bp">&#183;</span> <span class="k">calc</span> <span class="n">norm</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="bp">*</span> <span class="n">norm</span> <span class="n">y</span> <span class="bp">=</span> <span class="n">norm</span> <span class="o">(</span><span class="n">x</span> <span class="bp">%</span> <span class="n">y</span> <span class="bp">*</span> <span class="n">conj</span> <span class="n">y</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="n">only</span> <span class="o">[</span><span class="n">norm_mul</span><span class="o">,</span> <span class="n">norm_conj</span><span class="o">]</span> - <span class="n">_</span> <span class="bp">=</span> <span class="n">abs</span> <span class="o">(</span><span class="n">Int.mod&#39;</span> <span class="o">(</span><span class="n">x.re</span> <span class="bp">*</span> <span class="n">y.re</span> <span class="bp">+</span> <span class="n">x.im</span> <span class="bp">*</span> <span class="n">y.im</span><span class="o">)</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span><span class="o">))</span> <span class="bp">^</span> <span class="mi">2</span> - <span class="bp">+</span> <span class="n">abs</span> <span class="o">(</span><span class="n">Int.mod&#39;</span> <span class="o">(</span><span class="bp">-</span><span class="o">(</span><span class="n">x.re</span> <span class="bp">*</span> <span class="n">y.im</span><span class="o">)</span> <span class="bp">+</span> <span class="n">x.im</span> <span class="bp">*</span> <span class="n">y.re</span><span class="o">)</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span><span class="o">))</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="o">[</span><span class="n">H1</span><span class="o">,</span> <span class="n">norm</span><span class="o">,</span> <span class="n">sq_abs</span><span class="o">]</span> + <span class="n">_</span> <span class="bp">=</span> <span class="bp">|</span><span class="n">Int.mod&#39;</span> <span class="o">(</span><span class="n">x.re</span> <span class="bp">*</span> <span class="n">y.re</span> <span class="bp">+</span> <span class="n">x.im</span> <span class="bp">*</span> <span class="n">y.im</span><span class="o">)</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span><span class="o">)</span><span class="bp">|</span> <span class="bp">^</span> <span class="mi">2</span> + <span class="bp">+</span> <span class="bp">|</span><span class="n">Int.mod&#39;</span> <span class="o">(</span><span class="bp">-</span><span class="o">(</span><span class="n">x.re</span> <span class="bp">*</span> <span class="n">y.im</span><span class="o">)</span> <span class="bp">+</span> <span class="n">x.im</span> <span class="bp">*</span> <span class="n">y.re</span><span class="o">)</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span><span class="o">)</span><span class="bp">|</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> <span class="o">[</span><span class="n">H1</span><span class="o">,</span> <span class="n">norm</span><span class="o">,</span> <span class="n">sq_abs</span><span class="o">]</span> <span class="n">_</span> <span class="bp">&#8804;</span> <span class="o">(</span><span class="n">y.norm</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="o">(</span><span class="n">y.norm</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">^</span> <span class="mi">2</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">gcongr</span> <span class="bp">&lt;;&gt;</span> <span class="n">apply</span> <span class="n">Int.abs_mod&#39;_le</span> <span class="n">_</span> <span class="n">_</span> <span class="n">norm_y_pos</span> <span class="n">_</span> <span class="bp">=</span> <span class="n">norm</span> <span class="n">y</span> <span class="bp">/</span> <span class="mi">2</span> <span class="bp">*</span> <span class="o">(</span><span class="n">norm</span> <span class="n">y</span> <span class="bp">/</span> <span class="mi">2</span> <span class="bp">*</span> <span class="mi">2</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">ring</span> <span class="n">_</span> <span class="bp">&#8804;</span> <span class="n">norm</span> <span class="n">y</span> <span class="bp">/</span> <span class="mi">2</span> <span class="bp">*</span> <span class="n">norm</span> <span class="n">y</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">gcongr</span><span class="bp">;</span> <span class="n">apply</span> <span class="n">Int.ediv_mul_le</span><span class="bp">;</span> <span class="n">norm_num</span>
  • @@ -1513,8 +1517,8 @@ and in that case, the required properties are the theorems
    <span class="n">quotient</span> <span class="o">:=</span> <span class="o">(</span><span class="bp">&#183;</span> <span class="bp">/</span> <span class="bp">&#183;</span><span class="o">)</span> <span class="n">remainder</span> <span class="o">:=</span> <span class="o">(</span><span class="bp">&#183;</span> <span class="bp">%</span> <span class="bp">&#183;</span><span class="o">)</span> <span class="n">quotient_mul_add_remainder_eq</span> <span class="o">:=</span> - <span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">=&gt;</span> <span class="kd">by</span> <span class="n">simp</span> <span class="n">only</span><span class="bp">;</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mod_def</span><span class="o">,</span> <span class="n">add_comm</span><span class="o">,</span> <span class="n">sub_add_cancel</span><span class="o">]</span> - <span class="n">quotient_zero</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="kd">by</span> + <span class="k">fun</span> <span class="n">x</span> <span class="n">y</span> <span class="bp">&#8614;</span> <span class="kd">by</span> <span class="n">simp</span> <span class="n">only</span><span class="bp">;</span> <span class="n">rw</span> <span class="o">[</span><span class="n">mod_def</span><span class="o">,</span> <span class="n">add_comm</span><span class="o">,</span> <span class="n">sub_add_cancel</span><span class="o">]</span> + <span class="n">quotient_zero</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="kd">by</span> <span class="n">simp</span> <span class="o">[</span><span class="n">div_def</span><span class="o">,</span> <span class="n">norm</span><span class="o">,</span> <span class="n">Int.div&#39;</span><span class="o">]</span> <span class="n">rfl</span> <span class="n">r</span> <span class="o">:=</span> <span class="n">Measure</span> <span class="o">(</span><span class="n">Int.natAbs</span> <span class="bp">&#8728;</span> <span class="n">norm</span><span class="o">)</span>
  • @@ -1536,7 +1540,7 @@ the notions of being prime and being irreducible coincide.</p>
    </div> </div> <footer><div class="rst-footer-buttons" role="navigation" aria-label="Footer"> - <a href="C05_Number_Theory.html" class="btn btn-neutral float-left" title="5. Number Theory" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> + <a href="C05_Number_Theory.html" class="btn btn-neutral float-left" title="5. Elementary Number Theory" accesskey="p" rel="prev"><span class="fa fa-arrow-circle-left" aria-hidden="true"></span> Previous</a> <a href="C07_Hierarchies.html" class="btn btn-neutral float-right" title="7. Hierarchies" accesskey="n" rel="next">Next <span class="fa fa-arrow-circle-right" aria-hidden="true"></span></a> </div>
• html/C07_Hierarchies.html
  • @@ -1,7 +1,7 @@
    <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>7. Hierarchies &mdash; Mathematics in Lean 0.1 documentation</title>
  • @@ -13,11 +13,11 @@
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  • @@ -30,11 +30,15 @@
    <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" /> + <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
  • @@ -45,7 +49,7 @@
    <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> - <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Elementary Number Theory</a></li> <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> <li class="toctree-l1 current"><a class="current reference internal" href="#">7. Hierarchies</a><ul> <li class="toctree-l2"><a class="reference internal" href="#basics">7.1. Basics</a></li>
  • @@ -74,8 +78,8 @@
    <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> - <li><a href="index.html" class="icon icon-home"></a> &raquo;</li> - <li><span class="section-number">7. </span>Hierarchies</li> + <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 class="wy-breadcrumbs-aside"> <a href="_sources/C07_Hierarchies.rst.txt" rel="nofollow"> View page source</a> </li>
  • @@ -914,7 +918,7 @@ always form a complete lattice, and this structure is used a lot. For instance y
    the lemma saying that an intersection of submonoids is a submonoid. But this won&#8217;t be a lemma, this will be an infimum construction. Let us do the case of two submonoids.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">instance</span> <span class="o">[</span><span class="n">Monoid</span> <span class="n">M</span><span class="o">]</span> <span class="o">:</span> <span class="n">Inf</span> <span class="o">(</span><span class="n">Submonoid&#8321;</span> <span class="n">M</span><span class="o">)</span> <span class="o">:=</span> - <span class="o">&#10216;</span><span class="k">fun</span> <span class="n">S&#8321;</span> <span class="n">S&#8322;</span> <span class="bp">=&gt;</span> + <span class="o">&#10216;</span><span class="k">fun</span> <span class="n">S&#8321;</span> <span class="n">S&#8322;</span> <span class="bp">&#8614;</span> <span class="o">{</span> <span class="n">carrier</span> <span class="o">:=</span> <span class="n">S&#8321;</span> <span class="bp">&#8745;</span> <span class="n">S&#8322;</span> <span class="n">one_mem</span> <span class="o">:=</span> <span class="o">&#10216;</span><span class="n">S&#8321;.one_mem</span><span class="o">,</span> <span class="n">S&#8322;.one_mem</span><span class="o">&#10217;</span> <span class="n">mul_mem</span> <span class="o">:=</span> <span class="k">fun</span> <span class="o">&#10216;</span><span class="n">hx</span><span class="o">,</span> <span class="n">hx&#39;</span><span class="o">&#10217;</span> <span class="o">&#10216;</span><span class="n">hy</span><span class="o">,</span> <span class="n">hy&#39;</span><span class="o">&#10217;</span> <span class="bp">&#8614;</span> <span class="o">&#10216;</span><span class="n">S&#8321;.mul_mem</span> <span class="n">hx</span> <span class="n">hy</span><span class="o">,</span> <span class="n">S&#8322;.mul_mem</span> <span class="n">hx&#39;</span> <span class="n">hy&#39;</span><span class="o">&#10217;</span> <span class="o">}&#10217;</span>
• html/C08_Topology.html
  • @@ -1,7 +1,7 @@
    <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>8. Topology &mdash; Mathematics in Lean 0.1 documentation</title>
  • @@ -13,11 +13,11 @@
    <script src="_static/js/html5shiv.min.js"></script> <![endif]--> - <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/_sphinx_javascript_frameworks_compat.js"></script> + <script data-url_root="./" id="documentation_options" src="_static/documentation_options.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" />
  • @@ -31,11 +31,15 @@
    <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" /> + <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
  • @@ -46,7 +50,7 @@
    <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> - <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Elementary Number Theory</a></li> <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1 current"><a class="current reference internal" href="#">8. Topology</a><ul>
  • @@ -87,8 +91,8 @@
    <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> - <li><a href="index.html" class="icon icon-home"></a> &raquo;</li> - <li><span class="section-number">8. </span>Topology</li> + <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 class="wy-breadcrumbs-aside"> <a href="_sources/C08_Topology.rst.txt" rel="nofollow"> View page source</a> </li>
  • @@ -177,7 +181,7 @@ supports two related ideas:</p>
    </ul> <p>The filters that correspond to these descriptions will be defined later in this section, but we can already name them:</p> <ul class="simple"> - <li><p><code class="docutils literal notranslate"><span class="pre">(at_top</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">&#8469;)</span></code>, made of sets of <code class="docutils literal notranslate"><span class="pre">&#8469;</span></code> containing <code class="docutils literal notranslate"><span class="pre">{n</span> <span class="pre">|</span> <span class="pre">n</span> <span class="pre">&#8805;</span> <span class="pre">N}</span></code> for some <code class="docutils literal notranslate"><span class="pre">N</span></code></p></li> + <li><p><code class="docutils literal notranslate"><span class="pre">(atTop</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">&#8469;)</span></code>, made of sets of <code class="docutils literal notranslate"><span class="pre">&#8469;</span></code> containing <code class="docutils literal notranslate"><span class="pre">{n</span> <span class="pre">|</span> <span class="pre">n</span> <span class="pre">&#8805;</span> <span class="pre">N}</span></code> for some <code class="docutils literal notranslate"><span class="pre">N</span></code></p></li> <li><p><code class="docutils literal notranslate"><span class="pre">&#120029;</span> <span class="pre">x</span></code>, made of neighborhoods of <code class="docutils literal notranslate"><span class="pre">x</span></code> in a topological space</p></li> <li><p><code class="docutils literal notranslate"><span class="pre">&#120036;</span> <span class="pre">X</span></code>, made of entourages of a uniform space (uniform spaces generalize metric spaces and topological groups)</p></li> <li><p><code class="docutils literal notranslate"><span class="pre">&#956;.a_e</span></code> , made of sets whose complement has zero measure with respect to a measure <code class="docutils literal notranslate"><span class="pre">&#956;</span></code>.</p></li>
  • @@ -204,7 +208,7 @@ condition then says that <code class="docutils literal notranslate"><span class=
    large set is sufficiently large and the third one says that the intersection of two sufficiently large sets is sufficiently large.</p> <p>It may be even more useful to think of a filter on a type <code class="docutils literal notranslate"><span class="pre">X</span></code> - as a generalized element of <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">X</span></code>. For instance, <code class="docutils literal notranslate"><span class="pre">at_top</span></code> is the + as a generalized element of <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">X</span></code>. For instance, <code class="docutils literal notranslate"><span class="pre">atTop</span></code> is the &#8220;set of very large numbers&#8221; and <code class="docutils literal notranslate"><span class="pre">&#120029;</span> <span class="pre">x&#8320;</span></code> is the &#8220;set of points very close to <code class="docutils literal notranslate"><span class="pre">x&#8320;</span></code>.&#8221; One manifestation of this view is that we can associate to any <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">:</span> <span class="pre">Set</span> <span class="pre">X</span></code> the so-called <em>principal filter</em> consisting of all sets that contain <code class="docutils literal notranslate"><span class="pre">s</span></code>.
  • @@ -218,7 +222,7 @@ For the purpose of demonstration, we ask you to take this opportunity to work ou
    <span class="n">inter_sets</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> - <p>For our second example, we ask you to define the filter <code class="docutils literal notranslate"><span class="pre">at_top</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">&#8469;</span></code>. + <p>For our second example, we ask you to define the filter <code class="docutils literal notranslate"><span class="pre">atTop</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">&#8469;</span></code>. (We could use any type with a preorder instead of <code class="docutils literal notranslate"><span class="pre">&#8469;</span></code>.)</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">Filter</span> <span class="n">&#8469;</span> <span class="o">:=</span> <span class="o">{</span> <span class="n">sets</span> <span class="o">:=</span> <span class="o">{</span> <span class="n">s</span> <span class="bp">|</span> <span class="bp">&#8707;</span> <span class="n">a</span><span class="o">,</span> <span class="bp">&#8704;</span> <span class="n">b</span><span class="o">,</span> <span class="n">a</span> <span class="bp">&#8804;</span> <span class="n">b</span> <span class="bp">&#8594;</span> <span class="n">b</span> <span class="bp">&#8712;</span> <span class="n">s</span> <span class="o">}</span>
  • @@ -239,7 +243,7 @@ as follows:</p>
    <span class="bp">&#8704;</span> <span class="n">V</span> <span class="bp">&#8712;</span> <span class="n">G</span><span class="o">,</span> <span class="n">f</span> <span class="bp">&#8315;&#185;&#39;</span> <span class="n">V</span> <span class="bp">&#8712;</span> <span class="n">F</span> </pre></div> </div> - <p>When <code class="docutils literal notranslate"><span class="pre">X</span></code> is <code class="docutils literal notranslate"><span class="pre">&#8469;</span></code> and <code class="docutils literal notranslate"><span class="pre">Y</span></code> is <code class="docutils literal notranslate"><span class="pre">&#8477;</span></code>, <code class="docutils literal notranslate"><span class="pre">Tendsto&#8321;</span> <span class="pre">u</span> <span class="pre">at_top</span> <span class="pre">(&#120029;</span> <span class="pre">x)</span></code> is equivalent to saying that the sequence <code class="docutils literal notranslate"><span class="pre">u</span> <span class="pre">:</span> <span class="pre">&#8469;</span> <span class="pre">&#8594;</span> <span class="pre">&#8477;</span></code> + <p>When <code class="docutils literal notranslate"><span class="pre">X</span></code> is <code class="docutils literal notranslate"><span class="pre">&#8469;</span></code> and <code class="docutils literal notranslate"><span class="pre">Y</span></code> is <code class="docutils literal notranslate"><span class="pre">&#8477;</span></code>, <code class="docutils literal notranslate"><span class="pre">Tendsto&#8321;</span> <span class="pre">u</span> <span class="pre">atTop</span> <span class="pre">(&#120029;</span> <span class="pre">x)</span></code> is equivalent to saying that the sequence <code class="docutils literal notranslate"><span class="pre">u</span> <span class="pre">:</span> <span class="pre">&#8469;</span> <span class="pre">&#8594;</span> <span class="pre">&#8477;</span></code> converges to the real number <code class="docutils literal notranslate"><span class="pre">x</span></code>. When both <code class="docutils literal notranslate"><span class="pre">X</span></code> and <code class="docutils literal notranslate"><span class="pre">Y</span></code> are <code class="docutils literal notranslate"><span class="pre">&#8477;</span></code>, <code class="docutils literal notranslate"><span class="pre">Tendsto</span> <span class="pre">f</span> <span class="pre">(&#120029;</span> <span class="pre">x&#8320;)</span> <span class="pre">(&#120029;</span> <span class="pre">y&#8320;)</span></code> is equivalent to the familiar notion <span class="math notranslate nohighlight">\(\lim_{x \to x&#8320;} f(x) = y&#8320;\)</span>. All of the other kinds of limits mentioned in the introduction are
  • @@ -270,7 +274,7 @@ This inclusion is order preserving, so the order relation on <code class="docuti
    between generalized sets. In this analogy, pushforward is analogous to the direct image. And, indeed, <code class="docutils literal notranslate"><span class="pre">map</span> <span class="pre">f</span> <span class="pre">(&#120031;</span> <span class="pre">s)</span> <span class="pre">=</span> <span class="pre">&#120031;</span> <span class="pre">(f</span> <span class="pre">''</span> <span class="pre">s)</span></code>.</p> <p>We can now understand intuitively why a sequence <code class="docutils literal notranslate"><span class="pre">u</span> <span class="pre">:</span> <span class="pre">&#8469;</span> <span class="pre">&#8594;</span> <span class="pre">&#8477;</span></code> converges to - a point <code class="docutils literal notranslate"><span class="pre">x&#8320;</span></code> if and only if we have <code class="docutils literal notranslate"><span class="pre">map</span> <span class="pre">u</span> <span class="pre">at_top</span> <span class="pre">&#8804;</span> <span class="pre">&#120029;</span> <span class="pre">x&#8320;</span></code>. + a point <code class="docutils literal notranslate"><span class="pre">x&#8320;</span></code> if and only if we have <code class="docutils literal notranslate"><span class="pre">map</span> <span class="pre">u</span> <span class="pre">atTop</span> <span class="pre">&#8804;</span> <span class="pre">&#120029;</span> <span class="pre">x&#8320;</span></code>. The inequality means the &#8220;direct image under <code class="docutils literal notranslate"><span class="pre">u</span></code>&#8221; of &#8220;the set of very big natural numbers&#8221; is &#8220;included&#8221; in &#8220;the set of points very close to <code class="docutils literal notranslate"><span class="pre">x&#8320;</span></code>.&#8221;</p> <p>As promised, the definition of <code class="docutils literal notranslate"><span class="pre">Tendsto&#8322;</span></code> does not exhibit any quantifiers or sets.
  • @@ -337,7 +341,7 @@ which is to say, it reverses the order of the arguments.</p>
    </div> <p>The product operation is defined in terms of the pullback operation and the <code class="docutils literal notranslate"><span class="pre">inf</span></code> operation:</p> <blockquote> - <div><p><code class="docutils literal notranslate"><span class="pre">F</span> <span class="pre">&#215;&#738;</span> <span class="pre">G</span> <span class="pre">=</span> <span class="pre">(comap</span> <span class="pre">prod.fst</span> <span class="pre">F)</span> <span class="pre">&#8851;</span> <span class="pre">(comap</span> <span class="pre">prod.snd</span> <span class="pre">G)</span></code>.</p> + <div><p><code class="docutils literal notranslate"><span class="pre">F</span> <span class="pre">&#215;&#738;</span> <span class="pre">G</span> <span class="pre">=</span> <span class="pre">(comap</span> <span class="pre">Prod.fst</span> <span class="pre">F)</span> <span class="pre">&#8851;</span> <span class="pre">(comap</span> <span class="pre">Prod.snd</span> <span class="pre">G)</span></code>.</p> </div></blockquote> <p>Here the <code class="docutils literal notranslate"><span class="pre">inf</span></code> operation refers to the lattice structure on <code class="docutils literal notranslate"><span class="pre">filter</span> <span class="pre">X</span></code> for any type <code class="docutils literal notranslate"><span class="pre">X</span></code>, whereby <code class="docutils literal notranslate"><span class="pre">F</span> <span class="pre">&#8851;</span> <span class="pre">G</span></code> is the greatest filter that is smaller than both <code class="docutils literal notranslate"><span class="pre">F</span></code> and <code class="docutils literal notranslate"><span class="pre">G</span></code>.
  • @@ -383,15 +387,15 @@ given <code class="docutils literal notranslate"><span class="pre">x&#8320;</spa
    from the subtype corresponding to <code class="docutils literal notranslate"><span class="pre">s</span></code> is nontrivial if and only if <code class="docutils literal notranslate"><span class="pre">x&#8320;</span></code> belongs to the closure of <code class="docutils literal notranslate"><span class="pre">s</span></code>.</p> <p>In order to manage lemmas that do need to assume some filter is nontrivial, mathlib has - a type class <code class="docutils literal notranslate"><span class="pre">Filter.ne_bot</span></code>, and the library has lemmas that assume - <code class="docutils literal notranslate"><span class="pre">(F</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">X)</span> <span class="pre">[F.ne_bot]</span></code>. The instance database knows, for example, that <code class="docutils literal notranslate"><span class="pre">(at_top</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">&#8469;).ne_bot</span></code>, + a type class <code class="docutils literal notranslate"><span class="pre">Filter.NeBot</span></code>, and the library has lemmas that assume + <code class="docutils literal notranslate"><span class="pre">(F</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">X)</span> <span class="pre">[F.NeBot]</span></code>. The instance database knows, for example, that <code class="docutils literal notranslate"><span class="pre">(atTop</span> <span class="pre">:</span> <span class="pre">Filter</span> <span class="pre">&#8469;).NeBot</span></code>, and it knows that pushing forward a nontrivial filter gives a nontrivial filter. - As a result, a lemma assuming <code class="docutils literal notranslate"><span class="pre">[F.ne_bot]</span></code> will automatically apply to <code class="docutils literal notranslate"><span class="pre">map</span> <span class="pre">u</span> <span class="pre">at_top</span></code> for any sequence <code class="docutils literal notranslate"><span class="pre">u</span></code>.</p> + As a result, a lemma assuming <code class="docutils literal notranslate"><span class="pre">[F.NeBot]</span></code> will automatically apply to <code class="docutils literal notranslate"><span class="pre">map</span> <span class="pre">u</span> <span class="pre">atTop</span></code> for any sequence <code class="docutils literal notranslate"><span class="pre">u</span></code>.</p> <p>Our tour of the algebraic properties of filters and their relation to limits is essentially done, but we have not yet justified our claim to have recaptured the usual limit notions. - Superficially, it may seem that <code class="docutils literal notranslate"><span class="pre">Tendsto</span> <span class="pre">u</span> <span class="pre">at_top</span> <span class="pre">(&#120029;</span> <span class="pre">x&#8320;)</span></code> + Superficially, it may seem that <code class="docutils literal notranslate"><span class="pre">Tendsto</span> <span class="pre">u</span> <span class="pre">atTop</span> <span class="pre">(&#120029;</span> <span class="pre">x&#8320;)</span></code> is stronger than the notion of convergence defined in <a class="reference internal" href="C03_Logic.html#sequences-and-convergence"><span class="std std-numref">Section 3.6</span></a> because we ask that <em>every</em> neighborhood of <code class="docutils literal notranslate"><span class="pre">x&#8320;</span></code> - has a preimage belonging to <code class="docutils literal notranslate"><span class="pre">at_top</span></code>, whereas the usual definition only requires + has a preimage belonging to <code class="docutils literal notranslate"><span class="pre">atTop</span></code>, whereas the usual definition only requires this for the standard neighborhoods <code class="docutils literal notranslate"><span class="pre">Ioo</span> <span class="pre">(x&#8320;</span> <span class="pre">-</span> <span class="pre">&#949;)</span> <span class="pre">(x&#8320;</span> <span class="pre">+</span> <span class="pre">&#949;)</span></code>. The key is that, by definition, every neighborhood contains such a standard one. This observation leads to the notion of a <em>filter basis</em>.</p>
  • @@ -404,11 +408,11 @@ a predicate on <code class="docutils literal notranslate"><span class="pre">&#95
    In the case of <code class="docutils literal notranslate"><span class="pre">&#120029;</span> <span class="pre">x&#8320;</span></code>, we want <code class="docutils literal notranslate"><span class="pre">&#953;</span></code> to be <code class="docutils literal notranslate"><span class="pre">&#8477;</span></code>, we write <code class="docutils literal notranslate"><span class="pre">&#949;</span></code> for <code class="docutils literal notranslate"><span class="pre">i</span></code>, and the predicate should select the positive values of <code class="docutils literal notranslate"><span class="pre">&#949;</span></code>. So the fact that the sets <code class="docutils literal notranslate"><span class="pre">Ioo</span>&#160; <span class="pre">(x&#8320;</span> <span class="pre">-</span> <span class="pre">&#949;)</span> <span class="pre">(x&#8320;</span> <span class="pre">+</span> <span class="pre">&#949;)</span></code> form a basis for the neighborhood topology on <code class="docutils literal notranslate"><span class="pre">&#8477;</span></code> is stated as follows:</p> - <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">x&#8320;</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="n">HasBasis</span> <span class="o">(</span><span class="bp">&#120029;</span> <span class="n">x&#8320;</span><span class="o">)</span> <span class="o">(</span><span class="k">fun</span> <span class="n">&#949;</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">=&gt;</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">&#949;</span><span class="o">)</span> <span class="k">fun</span> <span class="n">&#949;</span> <span class="bp">=&gt;</span> <span class="n">Ioo</span> <span class="o">(</span><span class="n">x&#8320;</span> <span class="bp">-</span> <span class="n">&#949;</span><span class="o">)</span> <span class="o">(</span><span class="n">x&#8320;</span> <span class="bp">+</span> <span class="n">&#949;</span><span class="o">)</span> <span class="o">:=</span> + <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">x&#8320;</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="n">HasBasis</span> <span class="o">(</span><span class="bp">&#120029;</span> <span class="n">x&#8320;</span><span class="o">)</span> <span class="o">(</span><span class="k">fun</span> <span class="n">&#949;</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">&#8614;</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">&#949;</span><span class="o">)</span> <span class="k">fun</span> <span class="n">&#949;</span> <span class="bp">&#8614;</span> <span class="n">Ioo</span> <span class="o">(</span><span class="n">x&#8320;</span> <span class="bp">-</span> <span class="n">&#949;</span><span class="o">)</span> <span class="o">(</span><span class="n">x&#8320;</span> <span class="bp">+</span> <span class="n">&#949;</span><span class="o">)</span> <span class="o">:=</span> <span class="n">nhds_basis_Ioo_pos</span> <span class="n">x&#8320;</span> </pre></div> </div> - <p>There is also a nice basis for the filter <code class="docutils literal notranslate"><span class="pre">at_top</span></code>. The lemma + <p>There is also a nice basis for the filter <code class="docutils literal notranslate"><span class="pre">atTop</span></code>. The lemma <code class="docutils literal notranslate"><span class="pre">Filter.has_basis.tendsto_iff</span></code> allows us to reformulate a statement of the form <code class="docutils literal notranslate"><span class="pre">Tendsto</span> <span class="pre">f</span> <span class="pre">F</span> <span class="pre">G</span></code> given bases for <code class="docutils literal notranslate"><span class="pre">F</span></code> and <code class="docutils literal notranslate"><span class="pre">G</span></code>.
  • @@ -416,7 +420,7 @@ Putting these pieces together gives us essentially the notion of convergence
    that we used in <a class="reference internal" href="C03_Logic.html#sequences-and-convergence"><span class="std std-numref">Section 3.6</span></a>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">u</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">(</span><span class="n">x&#8320;</span> <span class="o">:</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="n">Tendsto</span> <span class="n">u</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">&#120029;</span> <span class="n">x&#8320;</span><span class="o">)</span> <span class="bp">&#8596;</span> <span class="bp">&#8704;</span> <span class="n">&#949;</span> <span class="bp">&gt;</span> <span class="mi">0</span><span class="o">,</span> <span class="bp">&#8707;</span> <span class="n">N</span><span class="o">,</span> <span class="bp">&#8704;</span> <span class="n">n</span> <span class="bp">&#8805;</span> <span class="n">N</span><span class="o">,</span> <span class="n">u</span> <span class="n">n</span> <span class="bp">&#8712;</span> <span class="n">Ioo</span> <span class="o">(</span><span class="n">x&#8320;</span> <span class="bp">-</span> <span class="n">&#949;</span><span class="o">)</span> <span class="o">(</span><span class="n">x&#8320;</span> <span class="bp">+</span> <span class="n">&#949;</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> - <span class="k">have</span> <span class="o">:</span> <span class="n">atTop.HasBasis</span> <span class="o">(</span><span class="k">fun</span> <span class="n">_</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">=&gt;</span> <span class="n">True</span><span class="o">)</span> <span class="n">Ici</span> <span class="o">:=</span> <span class="n">atTop_basis</span> + <span class="k">have</span> <span class="o">:</span> <span class="n">atTop.HasBasis</span> <span class="o">(</span><span class="k">fun</span> <span class="n">_</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8614;</span> <span class="n">True</span><span class="o">)</span> <span class="n">Ici</span> <span class="o">:=</span> <span class="n">atTop_basis</span> <span class="n">rw</span> <span class="o">[</span><span class="n">this.tendsto_iff</span> <span class="o">(</span><span class="n">nhds_basis_Ioo_pos</span> <span class="n">x&#8320;</span><span class="o">)]</span> <span class="n">simp</span> </pre></div>
  • @@ -430,11 +434,11 @@ Using <code class="docutils literal notranslate"><span class="pre">cases</span><
    eventually prove <code class="docutils literal notranslate"><span class="pre">&#8704;</span> <span class="pre">n</span> <span class="pre">&#8805;</span> <span class="pre">N,</span> <span class="pre">P</span> <span class="pre">n</span> <span class="pre">&#8743;</span> <span class="pre">Q</span> <span class="pre">n</span></code>. Doing this repeatedly becomes tiresome.</p> <p>We can do better by noting that the statement &#8220;<code class="docutils literal notranslate"><span class="pre">P</span> <span class="pre">n</span></code> and <code class="docutils literal notranslate"><span class="pre">Q</span> <span class="pre">n</span></code> hold for large enough <code class="docutils literal notranslate"><span class="pre">n</span></code>&#8221; means - that we have <code class="docutils literal notranslate"><span class="pre">{n</span> <span class="pre">|</span> <span class="pre">P</span> <span class="pre">n}</span> <span class="pre">&#8712;</span> <span class="pre">at_top</span></code> and <code class="docutils literal notranslate"><span class="pre">{n</span> <span class="pre">|</span> <span class="pre">Q</span> <span class="pre">n}</span> <span class="pre">&#8712;</span> <span class="pre">at_top</span></code>. - The fact that <code class="docutils literal notranslate"><span class="pre">at_top</span></code> is a filter implies that the intersection of two elements of <code class="docutils literal notranslate"><span class="pre">at_top</span></code> - is again in <code class="docutils literal notranslate"><span class="pre">at_top</span></code>, so we have <code class="docutils literal notranslate"><span class="pre">{n</span> <span class="pre">|</span> <span class="pre">P</span> <span class="pre">n</span> <span class="pre">&#8743;</span> <span class="pre">Q</span> <span class="pre">n}</span> <span class="pre">&#8712;</span> <span class="pre">at_top</span></code>. - Writing <code class="docutils literal notranslate"><span class="pre">{n</span> <span class="pre">|</span> <span class="pre">P</span> <span class="pre">n}</span> <span class="pre">&#8712;</span> <span class="pre">at_top</span></code> is unpleasant, - but we can use the more suggestive notation <code class="docutils literal notranslate"><span class="pre">&#8704;&#7584;</span> <span class="pre">n</span> <span class="pre">in</span> <span class="pre">at_top,</span> <span class="pre">P</span> <span class="pre">n</span></code>. + that we have <code class="docutils literal notranslate"><span class="pre">{n</span> <span class="pre">|</span> <span class="pre">P</span> <span class="pre">n}</span> <span class="pre">&#8712;</span> <span class="pre">atTop</span></code> and <code class="docutils literal notranslate"><span class="pre">{n</span> <span class="pre">|</span> <span class="pre">Q</span> <span class="pre">n}</span> <span class="pre">&#8712;</span> <span class="pre">atTop</span></code>. + The fact that <code class="docutils literal notranslate"><span class="pre">atTop</span></code> is a filter implies that the intersection of two elements of <code class="docutils literal notranslate"><span class="pre">atTop</span></code> + is again in <code class="docutils literal notranslate"><span class="pre">atTop</span></code>, so we have <code class="docutils literal notranslate"><span class="pre">{n</span> <span class="pre">|</span> <span class="pre">P</span> <span class="pre">n</span> <span class="pre">&#8743;</span> <span class="pre">Q</span> <span class="pre">n}</span> <span class="pre">&#8712;</span> <span class="pre">atTop</span></code>. + Writing <code class="docutils literal notranslate"><span class="pre">{n</span> <span class="pre">|</span> <span class="pre">P</span> <span class="pre">n}</span> <span class="pre">&#8712;</span> <span class="pre">atTop</span></code> is unpleasant, + but we can use the more suggestive notation <code class="docutils literal notranslate"><span class="pre">&#8704;&#7584;</span> <span class="pre">n</span> <span class="pre">in</span> <span class="pre">atTop,</span> <span class="pre">P</span> <span class="pre">n</span></code>. Here the superscripted <code class="docutils literal notranslate"><span class="pre">f</span></code> stands for &#8220;Filter.&#8221; You can think of the notation as saying that for all <code class="docutils literal notranslate"><span class="pre">n</span></code> in the &#8220;set of very large numbers,&#8221; <code class="docutils literal notranslate"><span class="pre">P</span> <span class="pre">n</span></code> holds. The <code class="docutils literal notranslate"><span class="pre">&#8704;&#7584;</span></code> notation stands for <code class="docutils literal notranslate"><span class="pre">Filter.Eventually</span></code>, and the lemma <code class="docutils literal notranslate"><span class="pre">Filter.Eventually.and</span></code> uses the intersection property of filters to do what we just described:</p>
  • @@ -494,7 +498,7 @@ Compare:</p>
    used with <code class="docutils literal notranslate"><span class="pre">eventually</span></code> to say that a property holds for almost every point.</p> <p>There is a dual version of <code class="docutils literal notranslate"><span class="pre">&#8704;&#7584;</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></code>, which is occasionally useful: <code class="docutils literal notranslate"><span class="pre">&#8707;&#7584;</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></code> means - <code class="docutils literal notranslate"><span class="pre">{x</span> <span class="pre">|</span> <span class="pre">&#172;P</span> <span class="pre">x}</span> <span class="pre">&#8713;</span> <span class="pre">F</span></code>. For example, <code class="docutils literal notranslate"><span class="pre">&#8707;&#7584;</span> <span class="pre">n</span> <span class="pre">in</span> <span class="pre">at_top,</span> <span class="pre">P</span> <span class="pre">n</span></code> means there are arbitrarily large <code class="docutils literal notranslate"><span class="pre">n</span></code> such that <code class="docutils literal notranslate"><span class="pre">P</span> <span class="pre">n</span></code> holds. + <code class="docutils literal notranslate"><span class="pre">{x</span> <span class="pre">|</span> <span class="pre">&#172;P</span> <span class="pre">x}</span> <span class="pre">&#8713;</span> <span class="pre">F</span></code>. For example, <code class="docutils literal notranslate"><span class="pre">&#8707;&#7584;</span> <span class="pre">n</span> <span class="pre">in</span> <span class="pre">atTop,</span> <span class="pre">P</span> <span class="pre">n</span></code> means there are arbitrarily large <code class="docutils literal notranslate"><span class="pre">n</span></code> such that <code class="docutils literal notranslate"><span class="pre">P</span> <span class="pre">n</span></code> holds. The <code class="docutils literal notranslate"><span class="pre">&#8707;&#7584;</span></code> notation stands for <code class="docutils literal notranslate"><span class="pre">Filter.frequently</span></code>.</p> <p>For a more sophisticated example, consider the following statement about a sequence <code class="docutils literal notranslate"><span class="pre">u</span></code>, a set <code class="docutils literal notranslate"><span class="pre">M</span></code>, and a value <code class="docutils literal notranslate"><span class="pre">x</span></code>:</p>
  • @@ -504,12 +508,13 @@ sufficiently large <code class="docutils literal notranslate"><span class="pre">
    </div></blockquote> <p>This can be formalized as follows:</p> <blockquote> - <div><p><code class="docutils literal notranslate"><span class="pre">Tendsto</span> <span class="pre">u</span> <span class="pre">at_top</span> <span class="pre">(&#120029;</span> <span class="pre">x)</span> <span class="pre">&#8594;</span> <span class="pre">(&#8704;&#7584;</span> <span class="pre">n</span> <span class="pre">in</span> <span class="pre">at_top,</span> <span class="pre">u</span> <span class="pre">n</span> <span class="pre">&#8712;</span> <span class="pre">M)</span> <span class="pre">&#8594;</span> <span class="pre">x</span> <span class="pre">&#8712;</span> <span class="pre">closure</span> <span class="pre">M</span></code>.</p> + <div><p><code class="docutils literal notranslate"><span class="pre">Tendsto</span> <span class="pre">u</span> <span class="pre">atTop</span> <span class="pre">(&#120029;</span> <span class="pre">x)</span> <span class="pre">&#8594;</span> <span class="pre">(&#8704;&#7584;</span> <span class="pre">n</span> <span class="pre">in</span> <span class="pre">atTop,</span> <span class="pre">u</span> <span class="pre">n</span> <span class="pre">&#8712;</span> <span class="pre">M)</span> <span class="pre">&#8594;</span> <span class="pre">x</span> <span class="pre">&#8712;</span> <span class="pre">closure</span> <span class="pre">M</span></code>.</p> </div></blockquote> <p>This is a special case of the theorem <code class="docutils literal notranslate"><span class="pre">mem_closure_of_tendsto</span></code> from the topology library. See if you can prove it using the quoted lemmas, - using the fact that <code class="docutils literal notranslate"><span class="pre">cluster_pt</span> <span class="pre">x</span> <span class="pre">F</span></code> means <code class="docutils literal notranslate"><span class="pre">(&#120029;</span> <span class="pre">x</span> <span class="pre">&#8851;</span> <span class="pre">F).ne_bot</span></code>.</p> + using the fact that <code class="docutils literal notranslate"><span class="pre">ClusterPt</span> <span class="pre">x</span> <span class="pre">F</span></code> means <code class="docutils literal notranslate"><span class="pre">(&#120029;</span> <span class="pre">x</span> <span class="pre">&#8851;</span> <span class="pre">F).NeBot</span></code> and that, + by definition, the assumption <code class="docutils literal notranslate"><span class="pre">&#8704;&#7584;</span> <span class="pre">n</span> <span class="pre">in</span> <span class="pre">atTop,</span> <span class="pre">u</span> <span class="pre">n</span> <span class="pre">&#8712;</span> <span class="pre">M</span></code> means <code class="docutils literal notranslate"><span class="pre">M</span> <span class="pre">&#8712;</span> <span class="pre">map</span> <span class="pre">u</span> <span class="pre">atTop</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="n">mem_closure_iff_clusterPt</span> <span class="k">#check</span> <span class="n">le_principal_iff</span> <span class="k">#check</span> <span class="n">neBot_of_le</span>
  • @@ -560,7 +565,7 @@ in an exercise below. Notice that Lean knows how to treat a product of two metri
    it makes sense to consider continuous functions from <code class="docutils literal notranslate"><span class="pre">X</span> <span class="pre">&#215;</span> <span class="pre">X</span></code> to <code class="docutils literal notranslate"><span class="pre">&#8477;</span></code>. In particular the (uncurried version of the) distance function is such a function.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">X</span> <span class="n">Y</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">MetricSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">&#8594;</span> <span class="n">Y</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> - <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">p</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">&#215;</span> <span class="n">X</span> <span class="bp">=&gt;</span> <span class="n">dist</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.1</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.2</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">continuity</span> + <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">p</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">&#215;</span> <span class="n">X</span> <span class="bp">&#8614;</span> <span class="n">dist</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.1</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.2</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">continuity</span> </pre></div> </div> <p>This tactic is a bit slow, so it is also useful to know
  • @@ -575,7 +580,7 @@ those two continuities using <code class="docutils literal notranslate"><span cl
    <code class="docutils literal notranslate"><span class="pre">(hf.comp</span> <span class="pre">continuous_fst).prod_mk</span> <span class="pre">(hf.comp</span> <span class="pre">continuous_snd)</span> <span class="pre">:</span> <span class="pre">Continuous</span> <span class="pre">(fun</span> <span class="pre">p</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">&#215;</span> <span class="pre">X</span> <span class="pre">&#8614;</span> <span class="pre">(f</span> <span class="pre">p.1,</span> <span class="pre">f</span> <span class="pre">p.2))</span></code> and compose once more to get our full proof.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">X</span> <span class="n">Y</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">MetricSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">&#8594;</span> <span class="n">Y</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> - <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">p</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">&#215;</span> <span class="n">X</span> <span class="bp">=&gt;</span> <span class="n">dist</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.1</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.2</span><span class="o">)</span> <span class="o">:=</span> + <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">p</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">&#215;</span> <span class="n">X</span> <span class="bp">&#8614;</span> <span class="n">dist</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.1</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.2</span><span class="o">)</span> <span class="o">:=</span> <span class="n">continuous_dist.comp</span> <span class="o">((</span><span class="n">hf.comp</span> <span class="n">continuous_fst</span><span class="o">)</span><span class="bp">.</span><span class="n">prod_mk</span> <span class="o">(</span><span class="n">hf.comp</span> <span class="n">continuous_snd</span><span class="o">))</span> </pre></div> </div>
  • @@ -593,13 +598,13 @@ composition and refuse to apply this lemma. It is especially bad at this when pr
    which is nicer to Lean&#8217;s elaborator and also provides a shorter proof when directly providing a full proof term, as can be seen from the following two new proofs of the above statement:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">X</span> <span class="n">Y</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">MetricSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">&#8594;</span> <span class="n">Y</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> - <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">p</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">&#215;</span> <span class="n">X</span> <span class="bp">=&gt;</span> <span class="n">dist</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.1</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.2</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> + <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">p</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">&#215;</span> <span class="n">X</span> <span class="bp">&#8614;</span> <span class="n">dist</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.1</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.2</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">apply</span> <span class="n">Continuous.dist</span> <span class="n">exact</span> <span class="n">hf.comp</span> <span class="n">continuous_fst</span> <span class="n">exact</span> <span class="n">hf.comp</span> <span class="n">continuous_snd</span> <span class="kd">example</span> <span class="o">{</span><span class="n">X</span> <span class="n">Y</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">MetricSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">&#8594;</span> <span class="n">Y</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> - <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">p</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">&#215;</span> <span class="n">X</span> <span class="bp">=&gt;</span> <span class="n">dist</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.1</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.2</span><span class="o">)</span> <span class="o">:=</span> + <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">p</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">&#215;</span> <span class="n">X</span> <span class="bp">&#8614;</span> <span class="n">dist</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.1</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.2</span><span class="o">)</span> <span class="o">:=</span> <span class="o">(</span><span class="n">hf.comp</span> <span class="n">continuous_fst</span><span class="o">)</span><span class="bp">.</span><span class="n">dist</span> <span class="o">(</span><span class="n">hf.comp</span> <span class="n">continuous_snd</span><span class="o">)</span> </pre></div> </div>
  • @@ -611,13 +616,13 @@ to type, let us wrap this discussion with a last bit of compression offered
    by <code class="docutils literal notranslate"><span class="pre">Continuous.fst'</span></code> which allows to compress <code class="docutils literal notranslate"><span class="pre">hf.comp</span> <span class="pre">continuous_fst</span></code> to <code class="docutils literal notranslate"><span class="pre">hf.fst'</span></code> (and the same with <code class="docutils literal notranslate"><span class="pre">snd</span></code>) and get our final proof, now bordering obfuscation.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">{</span><span class="n">X</span> <span class="n">Y</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">MetricSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">[</span><span class="n">MetricSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">&#8594;</span> <span class="n">Y</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> - <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">p</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">&#215;</span> <span class="n">X</span> <span class="bp">=&gt;</span> <span class="n">dist</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.1</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.2</span><span class="o">)</span> <span class="o">:=</span> + <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">p</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">&#215;</span> <span class="n">X</span> <span class="bp">&#8614;</span> <span class="n">dist</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.1</span><span class="o">)</span> <span class="o">(</span><span class="n">f</span> <span class="n">p.2</span><span class="o">)</span> <span class="o">:=</span> <span class="n">hf.fst&#39;.dist</span> <span class="n">hf.snd&#39;</span> </pre></div> </div> <p>It&#8217;s your turn now to prove some continuity lemma. After trying the continuity tactic, you will need <code class="docutils literal notranslate"><span class="pre">Continuous.add</span></code>, <code class="docutils literal notranslate"><span class="pre">continuous_pow</span></code> and <code class="docutils literal notranslate"><span class="pre">continuous_id</span></code> to do it by hand.</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">&#8477;</span> <span class="bp">&#8594;</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">x</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">=&gt;</span> <span class="n">f</span> <span class="o">(</span><span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> + <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">&#8477;</span> <span class="bp">&#8594;</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">x</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="o">(</span><span class="n">x</span> <span class="bp">^</span> <span class="mi">2</span> <span class="bp">+</span> <span class="n">x</span><span class="o">)</span> <span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div>
  • @@ -807,7 +812,7 @@ define something inductively in the middle of a proof using <code class="docutil
    <span class="kd">example</span> <span class="o">[</span><span class="n">CompleteSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">Set</span> <span class="n">X</span><span class="o">)</span> <span class="o">(</span><span class="n">ho</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">n</span><span class="o">,</span> <span class="n">IsOpen</span> <span class="o">(</span><span class="n">f</span> <span class="n">n</span><span class="o">))</span> <span class="o">(</span><span class="n">hd</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">n</span><span class="o">,</span> <span class="n">Dense</span> <span class="o">(</span><span class="n">f</span> <span class="n">n</span><span class="o">))</span> <span class="o">:</span> <span class="n">Dense</span> <span class="o">(</span><span class="bp">&#8898;</span> <span class="n">n</span><span class="o">,</span> <span class="n">f</span> <span class="n">n</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> - <span class="k">let</span> <span class="n">B</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">n</span> <span class="bp">=&gt;</span> <span class="o">(</span><span class="mi">1</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">^</span> <span class="n">n</span> + <span class="k">let</span> <span class="n">B</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">n</span> <span class="bp">&#8614;</span> <span class="o">(</span><span class="mi">1</span> <span class="bp">/</span> <span class="mi">2</span><span class="o">)</span> <span class="bp">^</span> <span class="n">n</span> <span class="k">have</span> <span class="n">Bpos</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">n</span><span class="o">,</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">B</span> <span class="n">n</span> <span class="gr">sorry</span> <span class="c">/-</span><span class="cm"> Translate the density assumption into two functions `center` and `radius` associating</span>
  • @@ -828,11 +833,11 @@ define something inductively in the middle of a proof using <code class="docutil
    <span class="cm"> in the previous ball and in `f n`, and such that `r n` is small enough to ensure</span> <span class="cm"> that `c n` is a Cauchy sequence. Then `c n` converges to a limit which belongs</span> <span class="cm"> to all the `f n`. -/</span> - <span class="k">let</span> <span class="n">F</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">X</span> <span class="bp">&#215;</span> <span class="n">&#8477;</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">n</span> <span class="bp">=&gt;</span> + <span class="k">let</span> <span class="n">F</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">X</span> <span class="bp">&#215;</span> <span class="n">&#8477;</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">n</span> <span class="bp">&#8614;</span> <span class="n">Nat.recOn</span> <span class="n">n</span> <span class="o">(</span><span class="n">Prod.mk</span> <span class="n">x</span> <span class="o">(</span><span class="n">min</span> <span class="n">&#949;</span> <span class="o">(</span><span class="n">B</span> <span class="mi">0</span><span class="o">)))</span> - <span class="k">fun</span> <span class="n">n</span> <span class="n">p</span> <span class="bp">=&gt;</span> <span class="n">Prod.mk</span> <span class="o">(</span><span class="n">center</span> <span class="n">n</span> <span class="n">p.1</span> <span class="n">p.2</span><span class="o">)</span> <span class="o">(</span><span class="n">radius</span> <span class="n">n</span> <span class="n">p.1</span> <span class="n">p.2</span><span class="o">)</span> - <span class="k">let</span> <span class="n">c</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">X</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">n</span> <span class="bp">=&gt;</span> <span class="o">(</span><span class="n">F</span> <span class="n">n</span><span class="o">)</span><span class="bp">.</span><span class="mi">1</span> - <span class="k">let</span> <span class="n">r</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">n</span> <span class="bp">=&gt;</span> <span class="o">(</span><span class="n">F</span> <span class="n">n</span><span class="o">)</span><span class="bp">.</span><span class="mi">2</span> + <span class="k">fun</span> <span class="n">n</span> <span class="n">p</span> <span class="bp">&#8614;</span> <span class="n">Prod.mk</span> <span class="o">(</span><span class="n">center</span> <span class="n">n</span> <span class="n">p.1</span> <span class="n">p.2</span><span class="o">)</span> <span class="o">(</span><span class="n">radius</span> <span class="n">n</span> <span class="n">p.1</span> <span class="n">p.2</span><span class="o">)</span> + <span class="k">let</span> <span class="n">c</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">X</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">n</span> <span class="bp">&#8614;</span> <span class="o">(</span><span class="n">F</span> <span class="n">n</span><span class="o">)</span><span class="bp">.</span><span class="mi">1</span> + <span class="k">let</span> <span class="n">r</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">n</span> <span class="bp">&#8614;</span> <span class="o">(</span><span class="n">F</span> <span class="n">n</span><span class="o">)</span><span class="bp">.</span><span class="mi">2</span> <span class="k">have</span> <span class="n">rpos</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">n</span><span class="o">,</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">r</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="n">rB</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">n</span><span class="o">,</span> <span class="n">r</span> <span class="n">n</span> <span class="bp">&#8804;</span> <span class="n">B</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="n">incl</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">n</span><span class="o">,</span> <span class="n">closedBall</span> <span class="o">(</span><span class="n">c</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">))</span> <span class="o">(</span><span class="n">r</span> <span class="o">(</span><span class="n">n</span> <span class="bp">+</span> <span class="mi">1</span><span class="o">))</span> <span class="bp">&#8838;</span> <span class="n">closedBall</span> <span class="o">(</span><span class="n">c</span> <span class="n">n</span><span class="o">)</span> <span class="o">(</span><span class="n">r</span> <span class="n">n</span><span class="o">)</span> <span class="bp">&#8745;</span> <span class="n">f</span> <span class="n">n</span> <span class="o">:=</span> <span class="kd">by</span>
  • @@ -1035,7 +1040,7 @@ Let us explore that constraint &#8220;on paper&#8221; using notation <span class
    <p>So we see that what is the topology we want on <code class="docutils literal notranslate"><span class="pre">&#928;</span> <span class="pre">i,</span> <span class="pre">X</span> <span class="pre">i</span></code>:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">(</span><span class="n">&#953;</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">X</span> <span class="o">:</span> <span class="n">&#953;</span> <span class="bp">&#8594;</span> <span class="kt">Type</span> <span class="n">_</span><span class="o">)</span> <span class="o">(</span><span class="n">T_X</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">i</span><span class="o">,</span> <span class="n">TopologicalSpace</span> <span class="bp">&lt;|</span> <span class="n">X</span> <span class="n">i</span><span class="o">)</span> <span class="o">:</span> <span class="o">(</span><span class="k">Pi</span><span class="bp">.</span><span class="n">topologicalSpace</span> <span class="o">:</span> <span class="n">TopologicalSpace</span> <span class="o">(</span><span class="bp">&#8704;</span> <span class="n">i</span><span class="o">,</span> <span class="n">X</span> <span class="n">i</span><span class="o">))</span> <span class="bp">=</span> - <span class="bp">&#10757;</span> <span class="n">i</span><span class="o">,</span> <span class="n">TopologicalSpace.induced</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">x</span> <span class="n">i</span><span class="o">)</span> <span class="o">(</span><span class="n">T_X</span> <span class="n">i</span><span class="o">)</span> <span class="o">:=</span> + <span class="bp">&#10757;</span> <span class="n">i</span><span class="o">,</span> <span class="n">TopologicalSpace.induced</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">x</span> <span class="n">i</span><span class="o">)</span> <span class="o">(</span><span class="n">T_X</span> <span class="n">i</span><span class="o">)</span> <span class="o">:=</span> <span class="n">rfl</span> </pre></div> </div>
  • @@ -1056,13 +1061,13 @@ neighborhood.</p>
    <span class="n">tendsto_nhds_unique</span> <span class="n">ha</span> <span class="n">hb</span> <span class="kd">example</span> <span class="o">[</span><span class="n">TopologicalSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">[</span><span class="n">RegularSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">(</span><span class="n">a</span> <span class="o">:</span> <span class="n">X</span><span class="o">)</span> <span class="o">:</span> - <span class="o">(</span><span class="bp">&#120029;</span> <span class="n">a</span><span class="o">)</span><span class="bp">.</span><span class="n">HasBasis</span> <span class="o">(</span><span class="k">fun</span> <span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span> <span class="bp">=&gt;</span> <span class="n">s</span> <span class="bp">&#8712;</span> <span class="bp">&#120029;</span> <span class="n">a</span> <span class="bp">&#8743;</span> <span class="n">IsClosed</span> <span class="n">s</span><span class="o">)</span> <span class="n">id</span> <span class="o">:=</span> + <span class="o">(</span><span class="bp">&#120029;</span> <span class="n">a</span><span class="o">)</span><span class="bp">.</span><span class="n">HasBasis</span> <span class="o">(</span><span class="k">fun</span> <span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span> <span class="bp">&#8614;</span> <span class="n">s</span> <span class="bp">&#8712;</span> <span class="bp">&#120029;</span> <span class="n">a</span> <span class="bp">&#8743;</span> <span class="n">IsClosed</span> <span class="n">s</span><span class="o">)</span> <span class="n">id</span> <span class="o">:=</span> <span class="n">closed_nhds_basis</span> <span class="n">a</span> </pre></div> </div> <p>Note that, in every topological space, each point has a basis of open neighborhood, by definition.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">[</span><span class="n">TopologicalSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">{</span><span class="n">x</span> <span class="o">:</span> <span class="n">X</span><span class="o">}</span> <span class="o">:</span> - <span class="o">(</span><span class="bp">&#120029;</span> <span class="n">x</span><span class="o">)</span><span class="bp">.</span><span class="n">HasBasis</span> <span class="o">(</span><span class="k">fun</span> <span class="n">t</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span> <span class="bp">=&gt;</span> <span class="n">t</span> <span class="bp">&#8712;</span> <span class="bp">&#120029;</span> <span class="n">x</span> <span class="bp">&#8743;</span> <span class="n">IsOpen</span> <span class="n">t</span><span class="o">)</span> <span class="n">id</span> <span class="o">:=</span> + <span class="o">(</span><span class="bp">&#120029;</span> <span class="n">x</span><span class="o">)</span><span class="bp">.</span><span class="n">HasBasis</span> <span class="o">(</span><span class="k">fun</span> <span class="n">t</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span> <span class="bp">&#8614;</span> <span class="n">t</span> <span class="bp">&#8712;</span> <span class="bp">&#120029;</span> <span class="n">x</span> <span class="bp">&#8743;</span> <span class="n">IsOpen</span> <span class="n">t</span><span class="o">)</span> <span class="n">id</span> <span class="o">:=</span> <span class="n">nhds_basis_opens&#39;</span> <span class="n">x</span> </pre></div> </div>
  • @@ -1172,7 +1177,7 @@ looks like in metric spaces.</p>
    </div> <p>As an exercise, we will prove that the image of a compact set under a continuous map is compact. In addition to what we saw already, you should use <code class="docutils literal notranslate"><span class="pre">Filter.push_pull</span></code> and - <code class="docutils literal notranslate"><span class="pre">ne_bot.of_map</span></code>.</p> + <code class="docutils literal notranslate"><span class="pre">NeBot.of_map</span></code>.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">[</span><span class="n">TopologicalSpace</span> <span class="n">Y</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">&#8594;</span> <span class="n">Y</span><span class="o">}</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</span><span class="o">)</span> <span class="o">{</span><span class="n">s</span> <span class="o">:</span> <span class="n">Set</span> <span class="n">X</span><span class="o">}</span> <span class="o">(</span><span class="n">hs</span> <span class="o">:</span> <span class="n">IsCompact</span> <span class="n">s</span><span class="o">)</span> <span class="o">:</span> <span class="n">IsCompact</span> <span class="o">(</span><span class="n">f</span> <span class="bp">&#39;&#39;</span> <span class="n">s</span><span class="o">)</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">intro</span> <span class="n">F</span> <span class="n">F_ne</span> <span class="n">F_le</span>
• html/C09_Differential_Calculus.html
  • @@ -1,7 +1,7 @@
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  • @@ -13,11 +13,11 @@
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  • @@ -31,11 +31,15 @@
    <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" /> + <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
  • @@ -46,7 +50,7 @@
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  • @@ -201,7 +205,7 @@ Lean and mathlib know this.</p>
    <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">example</span> <span class="o">:</span> <span class="n">MetricSpace</span> <span class="n">E</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">infer_instance</span> <span class="kd">example</span> <span class="o">{</span><span class="n">X</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">TopologicalSpace</span> <span class="n">X</span><span class="o">]</span> <span class="o">{</span><span class="n">f</span> <span class="o">:</span> <span class="n">X</span> <span class="bp">&#8594;</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">Continuous</span> <span class="n">f</span><span class="o">)</span> <span class="o">:</span> - <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="bp">&#8214;</span><span class="n">f</span> <span class="n">x</span><span class="bp">&#8214;</span> <span class="o">:=</span> + <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="bp">&#8214;</span><span class="n">f</span> <span class="n">x</span><span class="bp">&#8214;</span> <span class="o">:=</span> <span class="n">hf.norm</span> </pre></div> </div>
  • @@ -301,7 +305,7 @@ Minor ingredients include <code class="docutils literal notranslate"><span class
    <span class="kd">example</span> <span class="o">{</span><span class="n">&#953;</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">CompleteSpace</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">&#953;</span> <span class="bp">&#8594;</span> <span class="n">E</span> <span class="bp">&#8594;</span><span class="n">L</span><span class="o">[</span><span class="bp">&#120156;</span><span class="o">]</span> <span class="n">F</span><span class="o">}</span> <span class="o">(</span><span class="n">h</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">x</span><span class="o">,</span> <span class="bp">&#8707;</span> <span class="n">C</span><span class="o">,</span> <span class="bp">&#8704;</span> <span class="n">i</span><span class="o">,</span> <span class="bp">&#8214;</span><span class="n">g</span> <span class="n">i</span> <span class="n">x</span><span class="bp">&#8214;</span> <span class="bp">&#8804;</span> <span class="n">C</span><span class="o">)</span> <span class="o">:</span> <span class="bp">&#8707;</span> <span class="n">C&#39;</span><span class="o">,</span> <span class="bp">&#8704;</span> <span class="n">i</span><span class="o">,</span> <span class="bp">&#8214;</span><span class="n">g</span> <span class="n">i</span><span class="bp">&#8214;</span> <span class="bp">&#8804;</span> <span class="n">C&#39;</span> <span class="o">:=</span> <span class="kd">by</span> <span class="c1">-- sequence of subsets consisting of those `x : E` with norms `&#8214;g i x&#8214;` bounded by `n`</span> - <span class="k">let</span> <span class="n">e</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">Set</span> <span class="n">E</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">n</span> <span class="bp">=&gt;</span> <span class="bp">&#8898;</span> <span class="n">i</span> <span class="o">:</span> <span class="n">&#953;</span><span class="o">,</span> <span class="o">{</span> <span class="n">x</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">|</span> <span class="bp">&#8214;</span><span class="n">g</span> <span class="n">i</span> <span class="n">x</span><span class="bp">&#8214;</span> <span class="bp">&#8804;</span> <span class="n">n</span> <span class="o">}</span> + <span class="k">let</span> <span class="n">e</span> <span class="o">:</span> <span class="n">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">Set</span> <span class="n">E</span> <span class="o">:=</span> <span class="k">fun</span> <span class="n">n</span> <span class="bp">&#8614;</span> <span class="bp">&#8898;</span> <span class="n">i</span> <span class="o">:</span> <span class="n">&#953;</span><span class="o">,</span> <span class="o">{</span> <span class="n">x</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">|</span> <span class="bp">&#8214;</span><span class="n">g</span> <span class="n">i</span> <span class="n">x</span><span class="bp">&#8214;</span> <span class="bp">&#8804;</span> <span class="n">n</span> <span class="o">}</span> <span class="c1">-- each of these sets is closed</span> <span class="k">have</span> <span class="n">hc</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">n</span> <span class="o">:</span> <span class="n">&#8469;</span><span class="o">,</span> <span class="n">IsClosed</span> <span class="o">(</span><span class="n">e</span> <span class="n">n</span><span class="o">)</span> <span class="gr">sorry</span>
  • @@ -317,7 +321,7 @@ Minor ingredients include <code class="docutils literal notranslate"><span class
    <span class="k">have</span> <span class="n">real_norm_le</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">z</span> <span class="bp">&#8712;</span> <span class="n">ball</span> <span class="n">x</span> <span class="n">&#949;</span><span class="o">,</span> <span class="bp">&#8704;</span> <span class="o">(</span><span class="n">i</span> <span class="o">:</span> <span class="n">&#953;</span><span class="o">),</span> <span class="bp">&#8214;</span><span class="n">g</span> <span class="n">i</span> <span class="n">z</span><span class="bp">&#8214;</span> <span class="bp">&#8804;</span> <span class="n">m</span> <span class="gr">sorry</span> <span class="k">have</span> <span class="n">&#949;k_pos</span> <span class="o">:</span> <span class="mi">0</span> <span class="bp">&lt;</span> <span class="n">&#949;</span> <span class="bp">/</span> <span class="bp">&#8214;</span><span class="n">k</span><span class="bp">&#8214;</span> <span class="o">:=</span> <span class="gr">sorry</span> - <span class="n">refine&#39;</span> <span class="o">&#10216;(</span><span class="n">m</span> <span class="bp">+</span> <span class="n">m</span> <span class="o">:</span> <span class="n">&#8469;</span><span class="o">)</span> <span class="bp">/</span> <span class="o">(</span><span class="n">&#949;</span> <span class="bp">/</span> <span class="bp">&#8214;</span><span class="n">k</span><span class="bp">&#8214;</span><span class="o">),</span> <span class="k">fun</span> <span class="n">i</span> <span class="bp">=&gt;</span> <span class="n">ContinuousLinearMap.op_norm_le_of_shell</span> <span class="n">&#949;_pos</span> <span class="n">_</span> <span class="n">hk</span> <span class="n">_</span><span class="o">&#10217;</span> + <span class="n">refine&#39;</span> <span class="o">&#10216;(</span><span class="n">m</span> <span class="bp">+</span> <span class="n">m</span> <span class="o">:</span> <span class="n">&#8469;</span><span class="o">)</span> <span class="bp">/</span> <span class="o">(</span><span class="n">&#949;</span> <span class="bp">/</span> <span class="bp">&#8214;</span><span class="n">k</span><span class="bp">&#8214;</span><span class="o">),</span> <span class="k">fun</span> <span class="n">i</span> <span class="bp">&#8614;</span> <span class="n">ContinuousLinearMap.op_norm_le_of_shell</span> <span class="n">&#949;_pos</span> <span class="n">_</span> <span class="n">hk</span> <span class="n">_</span><span class="o">&#10217;</span> <span class="gr">sorry</span> <span class="gr">sorry</span> </pre></div>
  • @@ -364,7 +368,7 @@ Here the letter
    <span class="o">[</span><span class="n">NormedSpace</span> <span class="bp">&#120156;</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">&#120156;</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">&#8594;</span> <span class="n">F</span><span class="o">)</span> <span class="o">(</span><span class="n">f&#39;</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">&#8594;</span><span class="n">L</span><span class="o">[</span><span class="bp">&#120156;</span><span class="o">]</span> <span class="n">F</span><span class="o">)</span> <span class="o">(</span><span class="n">x&#8320;</span> <span class="o">:</span> <span class="n">E</span><span class="o">)</span> <span class="o">:</span> - <span class="n">HasFDerivAt</span> <span class="n">f</span> <span class="n">f&#39;</span> <span class="n">x&#8320;</span> <span class="bp">&#8596;</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">-</span> <span class="n">f</span> <span class="n">x&#8320;</span> <span class="bp">-</span> <span class="n">f&#39;</span> <span class="o">(</span><span class="n">x</span> <span class="bp">-</span> <span class="n">x&#8320;</span><span class="o">))</span> <span class="bp">=</span><span class="n">o</span><span class="o">[</span><span class="bp">&#120029;</span> <span class="n">x&#8320;</span><span class="o">]</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">x</span> <span class="bp">-</span> <span class="n">x&#8320;</span> <span class="o">:=</span> + <span class="n">HasFDerivAt</span> <span class="n">f</span> <span class="n">f&#39;</span> <span class="n">x&#8320;</span> <span class="bp">&#8596;</span> <span class="o">(</span><span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">f</span> <span class="n">x</span> <span class="bp">-</span> <span class="n">f</span> <span class="n">x&#8320;</span> <span class="bp">-</span> <span class="n">f&#39;</span> <span class="o">(</span><span class="n">x</span> <span class="bp">-</span> <span class="n">x&#8320;</span><span class="o">))</span> <span class="bp">=</span><span class="n">o</span><span class="o">[</span><span class="bp">&#120029;</span> <span class="n">x&#8320;</span><span class="o">]</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">x</span> <span class="bp">-</span> <span class="n">x&#8320;</span> <span class="o">:=</span> <span class="n">Iff.rfl</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">&#8594;</span> <span class="n">F</span><span class="o">)</span> <span class="o">(</span><span class="n">f&#39;</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">&#8594;</span><span class="n">L</span><span class="o">[</span><span class="bp">&#120156;</span><span class="o">]</span> <span class="n">F</span><span class="o">)</span> <span class="o">(</span><span class="n">x&#8320;</span> <span class="o">:</span> <span class="n">E</span><span class="o">)</span> <span class="o">(</span><span class="n">hff&#39;</span> <span class="o">:</span> <span class="n">HasFDerivAt</span> <span class="n">f</span> <span class="n">f&#39;</span> <span class="n">x&#8320;</span><span class="o">)</span> <span class="o">:</span> <span class="n">fderiv</span> <span class="bp">&#120156;</span> <span class="n">f</span> <span class="n">x&#8320;</span> <span class="bp">=</span> <span class="n">f&#39;</span> <span class="o">:=</span>
  • @@ -383,8 +387,8 @@ So <span class="math notranslate nohighlight">\(\mathcal{C}^\infty\)</span> func
    <span class="kd">example</span> <span class="o">(</span><span class="n">n</span> <span class="o">:</span> <span class="n">WithTop</span> <span class="n">&#8469;</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">&#8594;</span> <span class="n">F</span><span class="o">}</span> <span class="o">:</span> <span class="n">ContDiff</span> <span class="bp">&#120156;</span> <span class="n">n</span> <span class="n">f</span> <span class="bp">&#8596;</span> - <span class="o">(</span><span class="bp">&#8704;</span> <span class="n">m</span> <span class="o">:</span> <span class="n">&#8469;</span><span class="o">,</span> <span class="o">(</span><span class="n">m</span> <span class="o">:</span> <span class="n">WithTop</span> <span class="n">&#8469;</span><span class="o">)</span> <span class="bp">&#8804;</span> <span class="n">n</span> <span class="bp">&#8594;</span> <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">iteratedFDeriv</span> <span class="bp">&#120156;</span> <span class="n">m</span> <span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="bp">&#8743;</span> - <span class="bp">&#8704;</span> <span class="n">m</span> <span class="o">:</span> <span class="n">&#8469;</span><span class="o">,</span> <span class="o">(</span><span class="n">m</span> <span class="o">:</span> <span class="n">WithTop</span> <span class="n">&#8469;</span><span class="o">)</span> <span class="bp">&lt;</span> <span class="n">n</span> <span class="bp">&#8594;</span> <span class="n">Differentiable</span> <span class="bp">&#120156;</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="n">iteratedFDeriv</span> <span class="bp">&#120156;</span> <span class="n">m</span> <span class="n">f</span> <span class="n">x</span> <span class="o">:=</span> + <span class="o">(</span><span class="bp">&#8704;</span> <span class="n">m</span> <span class="o">:</span> <span class="n">&#8469;</span><span class="o">,</span> <span class="o">(</span><span class="n">m</span> <span class="o">:</span> <span class="n">WithTop</span> <span class="n">&#8469;</span><span class="o">)</span> <span class="bp">&#8804;</span> <span class="n">n</span> <span class="bp">&#8594;</span> <span class="n">Continuous</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">iteratedFDeriv</span> <span class="bp">&#120156;</span> <span class="n">m</span> <span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="bp">&#8743;</span> + <span class="bp">&#8704;</span> <span class="n">m</span> <span class="o">:</span> <span class="n">&#8469;</span><span class="o">,</span> <span class="o">(</span><span class="n">m</span> <span class="o">:</span> <span class="n">WithTop</span> <span class="n">&#8469;</span><span class="o">)</span> <span class="bp">&lt;</span> <span class="n">n</span> <span class="bp">&#8594;</span> <span class="n">Differentiable</span> <span class="bp">&#120156;</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="n">iteratedFDeriv</span> <span class="bp">&#120156;</span> <span class="n">m</span> <span class="n">f</span> <span class="n">x</span> <span class="o">:=</span> <span class="n">contDiff_iff_continuous_differentiable</span> </pre></div> </div>
• html/C10_Integration_and_Measure_Theory.html
  • @@ -1,7 +1,7 @@
    <!DOCTYPE html> <html class="writer-html5" lang="en" > <head> - <meta charset="utf-8" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" /> + <meta charset="utf-8" /><meta name="generator" content="Docutils 0.18.1: http://docutils.sourceforge.net/" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <title>10. Integration and Measure Theory &mdash; Mathematics in Lean 0.1 documentation</title>
  • @@ -13,11 +13,11 @@
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  • @@ -31,11 +31,15 @@
    <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" /> + <input type="text" name="q" placeholder="Search docs" aria-label="Search docs" /> <input type="hidden" name="check_keywords" value="yes" /> <input type="hidden" name="area" value="default" /> </form>
  • @@ -46,7 +50,7 @@
    <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> - <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Elementary Number Theory</a></li> <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li>
  • @@ -75,8 +79,8 @@
    <div class="rst-content"> <div role="navigation" aria-label="Page navigation"> <ul class="wy-breadcrumbs"> - <li><a href="index.html" class="icon icon-home"></a> &raquo;</li> - <li><span class="section-number">10. </span>Integration and Measure Theory</li> + <li><a href="index.html" class="icon icon-home" aria-label="Home"></a></li> + <li class="breadcrumb-item active"><span class="section-number">10. </span>Integration and Measure Theory</li> <li class="wy-breadcrumbs-aside"> <a href="_sources/C10_Integration_and_Measure_Theory.rst.txt" rel="nofollow"> View page source</a> </li>
  • @@ -110,7 +114,7 @@ says that integration provides an inverse to differentiation and the second one
    specifies how to compute integrals of derivatives. (These two parts are very closely related, but their optimal versions, which are not shown here, are not equivalent.)</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">&#8477;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</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">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="n">deriv</span> <span class="o">(</span><span class="k">fun</span> <span class="n">u</span> <span class="bp">=&gt;</span> <span class="bp">&#8747;</span> <span class="n">x</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="k">in</span> <span class="n">a..u</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="n">b</span> <span class="bp">=</span> <span class="n">f</span> <span class="n">b</span> <span class="o">:=</span> + <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">&#8477;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">(</span><span class="n">hf</span> <span class="o">:</span> <span class="n">Continuous</span> <span class="n">f</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">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="n">deriv</span> <span class="o">(</span><span class="k">fun</span> <span class="n">u</span> <span class="bp">&#8614;</span> <span class="bp">&#8747;</span> <span class="n">x</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="k">in</span> <span class="n">a..u</span><span class="o">,</span> <span class="n">f</span> <span class="n">x</span><span class="o">)</span> <span class="n">b</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="n">integral_hasStrictDerivAt_right</span> <span class="o">(</span><span class="n">hf.intervalIntegrable</span> <span class="n">_</span> <span class="n">_</span><span class="o">)</span> <span class="o">(</span><span class="n">hf.stronglyMeasurableAtFilter</span> <span class="n">_</span> <span class="n">_</span><span class="o">)</span> <span class="n">hf.continuousAt</span><span class="o">)</span><span class="bp">.</span><span class="n">hasDerivAt.deriv</span>
  • @@ -122,7 +126,7 @@ which are not shown here, are not equivalent.)</p>
    <p>Convolution is also defined in mathlib and its basic properties are proved.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">open</span> <span class="n">Convolution</span> - <span class="kd">example</span> <span class="o">(</span><span class="n">f</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">(</span><span class="n">g</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">&#8902;</span> <span class="n">g</span> <span class="bp">=</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="bp">&#8747;</span> <span class="n">t</span><span class="o">,</span> <span class="n">f</span> <span class="n">t</span> <span class="bp">*</span> <span class="n">g</span> <span class="o">(</span><span class="n">x</span> <span class="bp">-</span> <span class="n">t</span><span class="o">)</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">&#8477;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">(</span><span class="n">g</span> <span class="o">:</span> <span class="n">&#8477;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">:</span> <span class="n">f</span> <span class="bp">&#8902;</span> <span class="n">g</span> <span class="bp">=</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="bp">&#8747;</span> <span class="n">t</span><span class="o">,</span> <span class="n">f</span> <span class="n">t</span> <span class="bp">*</span> <span class="n">g</span> <span class="o">(</span><span class="n">x</span> <span class="bp">-</span> <span class="n">t</span><span class="o">)</span> <span class="o">:=</span> <span class="n">rfl</span> </pre></div> </div>
  • @@ -238,8 +242,8 @@ 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">&#8469;</span> <span class="bp">&#8594;</span> <span class="n">&#945;</span> <span class="bp">&#8594;</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">&#945;</span> <span class="bp">&#8594;</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">&#945;</span> <span class="bp">&#8594;</span> <span class="n">&#8477;</span><span class="o">)</span> <span class="o">(</span><span class="n">hmeas</span> <span class="o">:</span> <span class="bp">&#8704;</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">&#956;</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">&#956;</span><span class="o">)</span> <span class="o">(</span><span class="n">hbound</span> <span class="o">:</span> <span class="bp">&#8704;</span> <span class="n">n</span><span class="o">,</span> <span class="bp">&#8704;&#7504;</span> <span class="n">a</span> <span class="bp">&#8706;</span><span class="n">&#956;</span><span class="o">,</span> <span class="bp">&#8214;</span><span class="n">F</span> <span class="n">n</span> <span class="n">a</span><span class="bp">&#8214;</span> <span class="bp">&#8804;</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">&#8704;&#7504;</span> <span class="n">a</span> <span class="bp">&#8706;</span><span class="n">&#956;</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">&#8469;</span> <span class="bp">=&gt;</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">&#120029;</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">=&gt;</span> <span class="bp">&#8747;</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">&#8706;</span><span class="n">&#956;</span><span class="o">)</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">&#120029;</span> <span class="o">(</span><span class="bp">&#8747;</span> <span class="n">a</span><span class="o">,</span> <span class="n">f</span> <span class="n">a</span> <span class="bp">&#8706;</span><span class="n">&#956;</span><span class="o">))</span> <span class="o">:=</span> + <span class="o">(</span><span class="n">hlim</span> <span class="o">:</span> <span class="bp">&#8704;&#7504;</span> <span class="n">a</span> <span class="bp">&#8706;</span><span class="n">&#956;</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">&#8469;</span> <span class="bp">&#8614;</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">&#120029;</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">&#8614;</span> <span class="bp">&#8747;</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">&#8706;</span><span class="n">&#956;</span><span class="o">)</span> <span class="n">atTop</span> <span class="o">(</span><span class="bp">&#120029;</span> <span class="o">(</span><span class="bp">&#8747;</span> <span class="n">a</span><span class="o">,</span> <span class="n">f</span> <span class="n">a</span> <span class="bp">&#8706;</span><span class="n">&#956;</span><span class="o">))</span> <span class="o">:=</span> <span class="n">tendsto_integral_of_dominated_convergence</span> <span class="n">bound</span> <span class="n">hmeas</span> <span class="n">hint</span> <span class="n">hbound</span> <span class="n">hlim</span> </pre></div> </div>
  • @@ -260,7 +264,7 @@ continuous bilinear form.</p>
    <span class="o">[</span><span class="n">Sub</span> <span class="n">G</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">G</span> <span class="bp">&#8594;</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">G</span> <span class="bp">&#8594;</span> <span class="n">E&#39;</span><span class="o">)</span> <span class="o">(</span><span class="n">L</span> <span class="o">:</span> <span class="n">E</span> <span class="bp">&#8594;</span><span class="n">L</span><span class="o">[</span><span class="bp">&#120156;</span><span class="o">]</span> <span class="n">E&#39;</span> <span class="bp">&#8594;</span><span class="n">L</span><span class="o">[</span><span class="bp">&#120156;</span><span class="o">]</span> <span class="n">F</span><span class="o">)</span> <span class="o">(</span><span class="n">&#956;</span> <span class="o">:</span> <span class="n">Measure</span> <span class="n">G</span><span class="o">)</span> <span class="o">:</span> - <span class="n">f</span> <span class="bp">&#8902;</span><span class="o">[</span><span class="n">L</span><span class="o">,</span> <span class="n">&#956;</span><span class="o">]</span> <span class="n">g</span> <span class="bp">=</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">=&gt;</span> <span class="bp">&#8747;</span> <span class="n">t</span><span class="o">,</span> <span class="n">L</span> <span class="o">(</span><span class="n">f</span> <span class="n">t</span><span class="o">)</span> <span class="o">(</span><span class="n">g</span> <span class="o">(</span><span class="n">x</span> <span class="bp">-</span> <span class="n">t</span><span class="o">))</span> <span class="bp">&#8706;</span><span class="n">&#956;</span> <span class="o">:=</span> + <span class="n">f</span> <span class="bp">&#8902;</span><span class="o">[</span><span class="n">L</span><span class="o">,</span> <span class="n">&#956;</span><span class="o">]</span> <span class="n">g</span> <span class="bp">=</span> <span class="k">fun</span> <span class="n">x</span> <span class="bp">&#8614;</span> <span class="bp">&#8747;</span> <span class="n">t</span><span class="o">,</span> <span class="n">L</span> <span class="o">(</span><span class="n">f</span> <span class="n">t</span><span class="o">)</span> <span class="o">(</span><span class="n">g</span> <span class="o">(</span><span class="n">x</span> <span class="bp">-</span> <span class="n">t</span><span class="o">))</span> <span class="bp">&#8706;</span><span class="n">&#956;</span> <span class="o">:=</span> <span class="n">rfl</span> </pre></div> </div>
• html/_sources/C05_Number_Theory.rst.txt
  • @@ -1,7 +1,7 @@
    .. _number_theory: - Number Theory - ============= + Elementary Number Theory + ======================== In this chapter, we show you how to formalize some elementary results in number theory.
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  • @@ -1,20 +1,9 @@
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• html/_static/basic.css
  • @@ -4,7 +4,7 @@
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  • @@ -237,16 +237,6 @@ a.headerlink {
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  • @@ -335,13 +325,17 @@ p.sidebar-title {
    font-weight: bold; } - div.admonition, div.topic, aside.topic, blockquote { + nav.contents, + aside.topic, + div.admonition, div.topic, blockquote { clear: left; } /* -- topics ---------------------------------------------------------------- */ - div.topic, aside.topic { + nav.contents, + aside.topic, + div.topic { border: 1px solid #ccc; padding: 7px; margin: 10px 0 10px 0;
  • @@ -379,16 +373,18 @@ div.body p.centered {
    div.sidebar > :last-child, aside.sidebar > :last-child, - div.topic > :last-child, + nav.contents > :last-child, aside.topic > :last-child, + div.topic > :last-child, div.admonition > :last-child { margin-bottom: 0; } div.sidebar::after, aside.sidebar::after, - div.topic::after, + nav.contents::after, aside.topic::after, + div.topic::after, div.admonition::after, blockquote::after { display: block;
  • @@ -613,25 +609,6 @@ ul.simple p {
    margin-bottom: 0; } - /* Docutils 0.17 and older (footnotes & citations) */ - dl.footnote > dt, - dl.citation > dt { - float: left; - margin-right: 0.5em; - } - - dl.footnote > dd, - dl.citation > dd { - margin-bottom: 0em; - } - - dl.footnote > dd:after, - dl.citation > dd:after { - content: ""; - clear: both; - } - - /* Docutils 0.18+ (footnotes & citations) */ aside.footnote > span, div.citation > span { float: left;
  • @@ -656,8 +633,6 @@ div.citation > p:last-of-type:after {
    clear: both; } - /* Footnotes & citations ends */ - dl.field-list { display: grid; grid-template-columns: fit-content(30%) auto;
  • @@ -670,10 +645,6 @@ dl.field-list > dt {
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• html/_static/css/theme.css
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.rst-content aside.footnote p,html.writer-html5 .rst-content div.citation p{font-size:.9rem;line-height:1.2rem;margin-bottom:12px}html.writer-html5 .rst-content aside.citation span.backrefs,html.writer-html5 .rst-content aside.footnote span.backrefs,html.writer-html5 .rst-content div.citation span.backrefs{text-align:left;font-style:italic;margin-left:.65rem;word-break:break-word;word-spacing:-.1rem;max-width:5rem}html.writer-html5 .rst-content aside.citation span.backrefs>a,html.writer-html5 .rst-content aside.footnote span.backrefs>a,html.writer-html5 .rst-content div.citation span.backrefs>a{word-break:keep-all}html.writer-html5 .rst-content aside.citation span.backrefs>a:not(:first-child):before,html.writer-html5 .rst-content aside.footnote span.backrefs>a:not(:first-child):before,html.writer-html5 .rst-content div.citation span.backrefs>a:not(:first-child):before{content:" "}html.writer-html5 .rst-content aside.citation span.label,html.writer-html5 .rst-content aside.footnote span.label,html.writer-html5 .rst-content div.citation span.label{line-height:1.2rem}html.writer-html5 .rst-content aside.citation-list,html.writer-html5 .rst-content aside.footnote-list,html.writer-html5 .rst-content div.citation-list{margin-bottom:24px}html.writer-html5 .rst-content dl.option-list kbd{font-size:.9rem}.rst-content table.docutils.footnote,html.writer-html4 .rst-content table.docutils.citation,html.writer-html5 .rst-content aside.footnote,html.writer-html5 .rst-content aside.footnote-list aside.footnote,html.writer-html5 .rst-content div.citation-list>div.citation,html.writer-html5 .rst-content dl.citation,html.writer-html5 .rst-content dl.footnote{color:grey}.rst-content table.docutils.footnote code,.rst-content table.docutils.footnote tt,html.writer-html4 .rst-content table.docutils.citation code,html.writer-html4 .rst-content table.docutils.citation tt,html.writer-html5 .rst-content aside.footnote-list aside.footnote code,html.writer-html5 .rst-content aside.footnote-list aside.footnote tt,html.writer-html5 .rst-content aside.footnote code,html.writer-html5 .rst-content aside.footnote tt,html.writer-html5 .rst-content div.citation-list>div.citation code,html.writer-html5 .rst-content div.citation-list>div.citation tt,html.writer-html5 .rst-content dl.citation code,html.writer-html5 .rst-content dl.citation tt,html.writer-html5 .rst-content dl.footnote code,html.writer-html5 .rst-content dl.footnote tt{color:#555}.rst-content .wy-table-responsive.citation,.rst-content .wy-table-responsive.footnote{margin-bottom:0}.rst-content .wy-table-responsive.citation+:not(.citation),.rst-content .wy-table-responsive.footnote+:not(.footnote){margin-top:24px}.rst-content .wy-table-responsive.citation:last-child,.rst-content .wy-table-responsive.footnote:last-child{margin-bottom:24px}.rst-content table.docutils th{border-color:#e1e4e5}html.writer-html5 .rst-content table.docutils th{border:1px solid #e1e4e5}html.writer-html5 .rst-content 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.rst-content tt{font-weight:700;color:#404040;overflow-wrap:normal}.rst-content kbd,.rst-content pre,.rst-content samp{font-family:SFMono-Regular,Menlo,Monaco,Consolas,Liberation Mono,Courier New,Courier,monospace}.rst-content a code,.rst-content a tt{color:#2980b9}.rst-content dl{margin-bottom:24px}.rst-content dl dt{font-weight:700;margin-bottom:12px}.rst-content dl ol,.rst-content dl p,.rst-content dl table,.rst-content dl ul{margin-bottom:12px}.rst-content dl dd{margin:0 0 12px 24px;line-height:24px}.rst-content dl dd>ol:last-child,.rst-content dl dd>p:last-child,.rst-content dl dd>table:last-child,.rst-content dl dd>ul:last-child{margin-bottom:0}html.writer-html4 .rst-content dl:not(.docutils),html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.citation):not(.glossary):not(.simple){margin-bottom:24px}html.writer-html4 .rst-content dl:not(.docutils)>dt,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.citation):not(.glossary):not(.simple)>dt{display:table;margin:6px 0;font-size:90%;line-height:normal;background:#e7f2fa;color:#2980b9;border-top:3px solid #6ab0de;padding:6px;position:relative}html.writer-html4 .rst-content dl:not(.docutils)>dt:before,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.citation):not(.glossary):not(.simple)>dt:before{color:#6ab0de}html.writer-html4 .rst-content dl:not(.docutils)>dt .headerlink,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.citation):not(.glossary):not(.simple)>dt .headerlink{color:#404040;font-size:100%!important}html.writer-html4 .rst-content dl:not(.docutils) dl:not(.option-list):not(.field-list):not(.footnote):not(.citation):not(.glossary):not(.simple)>dt,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.citation):not(.glossary):not(.simple) dl:not(.option-list):not(.field-list):not(.footnote):not(.citation):not(.glossary):not(.simple)>dt{margin-bottom:6px;border:none;border-left:3px solid #ccc;background:#f0f0f0;color:#555}html.writer-html4 .rst-content dl:not(.docutils) dl:not(.option-list):not(.field-list):not(.footnote):not(.citation):not(.glossary):not(.simple)>dt .headerlink,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.citation):not(.glossary):not(.simple) dl:not(.option-list):not(.field-list):not(.footnote):not(.citation):not(.glossary):not(.simple)>dt .headerlink{color:#404040;font-size:100%!important}html.writer-html4 .rst-content dl:not(.docutils)>dt:first-child,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.citation):not(.glossary):not(.simple)>dt:first-child{margin-top:0}html.writer-html4 .rst-content dl:not(.docutils) code.descclassname,html.writer-html4 .rst-content dl:not(.docutils) code.descname,html.writer-html4 .rst-content dl:not(.docutils) tt.descclassname,html.writer-html4 .rst-content dl:not(.docutils) tt.descname,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.citation):not(.glossary):not(.simple) code.descclassname,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.citation):not(.glossary):not(.simple) code.descname,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.citation):not(.glossary):not(.simple) tt.descclassname,html.writer-html5 .rst-content dl[class]:not(.option-list):not(.field-list):not(.footnote):not(.citation):not(.glossary):not(.simple) tt.descname{background-color:transparent;border:none;padding:0;font-size:100%!important}html.writer-html4 .rst-content dl:not(.docutils) code.descname,html.writer-html4 .rst-content 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.versionmodified{font-style:italic}@media screen and (max-width:480px){.rst-content .sidebar{width:100%}}span[id*=MathJax-Span]{color:#404040}.math{text-align:center}@font-face{font-family:Lato;src:url(fonts/lato-normal.woff2?bd03a2cc277bbbc338d464e679fe9942) format("woff2"),url(fonts/lato-normal.woff?27bd77b9162d388cb8d4c4217c7c5e2a) format("woff");font-weight:400;font-style:normal;font-display:block}@font-face{font-family:Lato;src:url(fonts/lato-bold.woff2?cccb897485813c7c256901dbca54ecf2) format("woff2"),url(fonts/lato-bold.woff?d878b6c29b10beca227e9eef4246111b) format("woff");font-weight:700;font-style:normal;font-display:block}@font-face{font-family:Lato;src:url(fonts/lato-bold-italic.woff2?0b6bb6725576b072c5d0b02ecdd1900d) format("woff2"),url(fonts/lato-bold-italic.woff?9c7e4e9eb485b4a121c760e61bc3707c) format("woff");font-weight:700;font-style:italic;font-display:block}@font-face{font-family:Lato;src:url(fonts/lato-normal-italic.woff2?4eb103b4d12be57cb1d040ed5e162e9d) 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• html/_static/doctools.js
  • @@ -4,12 +4,19 @@
    * * Base JavaScript utilities for all Sphinx HTML documentation. * - * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS. + * :copyright: Copyright 2007-2023 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();
  • @@ -18,73 +25,11 @@ const _ready = (callback) => {
    } }; - /** - * 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 Documentation = { init: () => { - Documentation.highlightSearchWords(); Documentation.initDomainIndexTable(); Documentation.initOnKeyListeners(); },
  • @@ -126,51 +71,6 @@ const Documentation = {
    Documentation.LOCALE = catalog.locale; }, - /** - * highlight the search words provided in the url in the text - */ - highlightSearchWords: () => { - const highlight = - new URLSearchParams(window.location.search).get("highlight") || ""; - 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:Documentation.hideSearchWords()">' + - Documentation.gettext("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")); - const url = new URL(window.location); - url.searchParams.delete("highlight"); - window.history.replaceState({}, "", url); - }, - /** * helper function to focus on search bar */
  • @@ -210,15 +110,11 @@ const Documentation = {
    ) return; - const blacklistedElements = new Set([ - "TEXTAREA", - "INPUT", - "SELECT", - "BUTTON", - ]); document.addEventListener("keydown", (event) => { - if (blacklistedElements.has(document.activeElement.tagName)) return; // bail for input elements - if (event.altKey || event.ctrlKey || event.metaKey) return; // bail with special keys + // 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) {
  • @@ -240,10 +136,6 @@ const Documentation = {
    event.preventDefault(); } break; - case "Escape": - if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) break; - Documentation.hideSearchWords(); - event.preventDefault(); } }
• html/_static/documentation_options.js
  • @@ -10,5 +10,5 @@ var DOCUMENTATION_OPTIONS = {
    SOURCELINK_SUFFIX: '.txt', NAVIGATION_WITH_KEYS: false, SHOW_SEARCH_SUMMARY: true, - ENABLE_SEARCH_SHORTCUTS: false, + ENABLE_SEARCH_SHORTCUTS: true, };
• html/_static/jquery-3.6.0.js (deleted)
  • @@ -1,10881 +0,0 @@
    - /*! - * jQuery JavaScript Library v3.6.0 - * https://jquery.com/ - * - * Includes Sizzle.js - * https://sizzlejs.com/ - * - * Copyright OpenJS Foundation and other contributors - * Released under the MIT license - * https://jquery.org/license - * - * Date: 2021-03-02T17:08Z - */ - ( 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. - // Support: QtWeb <=3.8.5, WebKit <=534.34, wkhtmltopdf tool <=0.12.5 - // Plus for old WebKit, typeof returns "function" for HTML collections - // (e.g., `typeof document.getElementsByTagName("div") === "function"`). (gh-4756) - return typeof obj === "function" && typeof obj.nodeType !== "number" && - typeof obj.item !== "function"; - }; - - - 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 = "3.6.0", - - // 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 v2.3.6 - * https://sizzlejs.com/ - * - * Copyright JS Foundation and other contributors - * Released under the MIT license - * https://js.foundation/ - * - * Date: 2021-02-16 - */ - ( 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 && elem.namespaceURI, - docElem = elem && ( 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 primary Deferred - primary = 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 ) ) { - primary.resolveWith( resolveContexts, resolveValues ); - } - }; - }; - - // Single- and empty arguments are adopted like Promise.resolve - if ( remaining <= 1 ) { - adoptValue( singleValue, primary.done( updateFunc( i ) ).resolve, primary.reject, - !remaining ); - - // Use .then() to unwrap secondary thenables (cf. gh-3000) - if ( primary.state() === "pending" || - isFunction( resolveValues[ i ] && resolveValues[ i ].then ) ) { - - return primary.then(); - } - } - - // Multiple arguments are aggregated like Promise.all array elements - while ( i-- ) { - adoptValue( resolveValues[ i ], updateFunc( i ), primary.reject ); - } - - return primary.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 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(); - - // Support: Chrome 86+ - // In Chrome, if an element having a focusout handler is blurred by - // clicking outside of it, it invokes the handler synchronously. If - // that handler calls `.remove()` on the element, the data is cleared, - // leaving `result` undefined. We need to guard against this. - return result && 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: true - }, 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; - }, - - // Suppress native focus or blur as it's already being fired - // in leverageNative. - _default: function() { - 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! - // - // Support: Firefox 70+ - // Only Firefox includes border widths - // in computed dimensions. (gh-4529) - 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;border-collapse:separate"; - tr.style.cssText = "border:1px solid"; - - // Support: Chrome 86+ - // Height set through cssText does not get applied. - // Computed height then comes back as 0. - tr.style.height = "1px"; - trChild.style.height = "9px"; - - // Support: Android 8 Chrome 86+ - // In our bodyBackground.html iframe, - // display for all div elements is set to "inline", - // which causes a problem only in Android 8 Chrome 86. - // Ensuring the div is display: block - // gets around this issue. - trChild.style.display = "block"; - - documentElement - .appendChild( table ) - .appendChild( tr ) - .appendChild( trChild ); - - trStyle = window.getComputedStyle( tr ); - reliableTrDimensionsVal = ( parseInt( trStyle.height, 10 ) + - parseInt( trStyle.borderTopWidth, 10 ) + - parseInt( trStyle.borderBottomWidth, 10 ) ) === tr.offsetHeight; - - 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, parserErrorElem; - 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 ) {} - - parserErrorElem = xml && xml.getElementsByTagName( "parsererror" )[ 0 ]; - if ( !xml || parserErrorElem ) { - jQuery.error( "Invalid XML: " + ( - parserErrorElem ? - jQuery.map( parserErrorElem.childNodes, function( el ) { - return el.textContent; - } ).join( "\n" ) : - 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 but not if jsonp - if ( !isSuccess && - jQuery.inArray( "script", s.dataTypes ) > -1 && - jQuery.inArray( "json", s.dataTypes ) < 0 ) { - 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 { - 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 ? - this : - this.each( function( i ) { - jQuery.offset.setOffset( this, options, i ); - } ); - } - - var rect, win, - elem = this[ 0 ]; - - if ( !elem ) { - return; - } - - // Return zeros for disconnected and hidden (display: none) elements (gh-2310) - // Support: IE <=11 only - // Running getBoundingClientRect on a - // disconnected node in IE throws an error - if ( !elem.getClientRects().length ) { - return { top: 0, left: 0 }; - } - - // Get document-relative position by adding viewport scroll to viewport-relative gBCR - rect = elem.getBoundingClientRect(); - win = elem.ownerDocument.defaultView; - return { - top: rect.top + win.pageYOffset, - left: rect.left + win.pageXOffset - }; - }, - - // position() relates an element's margin box to its offset parent's padding box - // This corresponds to the behavior of CSS absolute positioning - position: function() { - if ( !this[ 0 ] ) { - return; - } - - var offsetParent, offset, doc, - elem = this[ 0 ], - parentOffset = { top: 0, left: 0 }; - - // position:fixed elements are offset from the viewport, which itself always has zero offset - if ( jQuery.css( elem, "position" ) === "fixed" ) { - - // Assume position:fixed implies availability of getBoundingClientRect - offset = elem.getBoundingClientRect(); - - } else { - offset = this.offset(); - - // Account for the *real* offset parent, which can be the document or its root element - // when a statically positioned element is identified - doc = elem.ownerDocument; - offsetParent = elem.offsetParent || doc.documentElement; - while ( offsetParent && - ( offsetParent === doc.body || offsetParent === doc.documentElement ) && - jQuery.css( offsetParent, "position" ) === "static" ) { - - offsetParent = offsetParent.parentNode; - } - if ( offsetParent && offsetParent !== elem && offsetParent.nodeType === 1 ) { - - // Incorporate borders into its offset, since they are outside its content origin - parentOffset = jQuery( offsetParent ).offset(); - parentOffset.top += jQuery.css( offsetParent, "borderTopWidth", true ); - parentOffset.left += jQuery.css( offsetParent, "borderLeftWidth", true ); - } - } - - // Subtract parent offsets and element margins - return { - top: offset.top - parentOffset.top - jQuery.css( elem, "marginTop", true ), - left: offset.left - parentOffset.left - jQuery.css( elem, "marginLeft", true ) - }; - }, - - // This method will return documentElement in the following cases: - // 1) For the element inside the iframe without offsetParent, this method will return - // documentElement of the parent window - // 2) For the hidden or detached element - // 3) For body or html element, i.e. in case of the html node - it will return itself - // - // but those exceptions were never presented as a real life use-cases - // and might be considered as more preferable results. - // - // This logic, however, is not guaranteed and can change at any point in the future - offsetParent: function() { - return this.map( function() { - var offsetParent = this.offsetParent; - - while ( offsetParent && jQuery.css( offsetParent, "position" ) === "static" ) { - offsetParent = offsetParent.offsetParent; - } - - return offsetParent || documentElement; - } ); - } - } ); - - // Create scrollLeft and scrollTop methods - jQuery.each( { scrollLeft: "pageXOffset", scrollTop: "pageYOffset" }, function( method, prop ) { - var top = "pageYOffset" === prop; - - jQuery.fn[ method ] = function( val ) { - return access( this, function( elem, method, val ) { - - // Coalesce documents and windows - var win; - if ( isWindow( elem ) ) { - win = elem; - } else if ( elem.nodeType === 9 ) { - win = elem.defaultView; - } - - if ( val === undefined ) { - return win ? win[ prop ] : elem[ method ]; - } - - if ( win ) { - win.scrollTo( - !top ? val : win.pageXOffset, - top ? val : win.pageYOffset - ); - - } else { - elem[ method ] = val; - } - }, method, val, arguments.length ); - }; - } ); - - // Support: Safari <=7 - 9.1, Chrome <=37 - 49 - // Add the top/left cssHooks using jQuery.fn.position - // Webkit bug: https://bugs.webkit.org/show_bug.cgi?id=29084 - // Blink bug: https://bugs.chromium.org/p/chromium/issues/detail?id=589347 - // getComputedStyle returns percent when specified for top/left/bottom/right; - // rather than make the css module depend on the offset module, just check for it here - jQuery.each( [ "top", "left" ], function( _i, prop ) { - jQuery.cssHooks[ prop ] = addGetHookIf( support.pixelPosition, - function( elem, computed ) { - if ( computed ) { - computed = curCSS( elem, prop ); - - // If curCSS returns percentage, fallback to offset - return rnumnonpx.test( computed ) ? - jQuery( elem ).position()[ prop ] + "px" : - computed; - } - } - ); - } ); - - - // Create innerHeight, innerWidth, height, width, outerHeight and outerWidth methods - jQuery.each( { Height: "height", Width: "width" }, function( name, type ) { - jQuery.each( { - padding: "inner" + name, - content: type, - "": "outer" + name - }, function( defaultExtra, funcName ) { - - // Margin is only for outerHeight, outerWidth - jQuery.fn[ funcName ] = function( margin, value ) { - var chainable = arguments.length && ( defaultExtra || typeof margin !== "boolean" ), - extra = defaultExtra || ( margin === true || value === true ? "margin" : "border" ); - - return access( this, function( elem, type, value ) { - var doc; - - if ( isWindow( elem ) ) { - - // $( window ).outerWidth/Height return w/h including scrollbars (gh-1729) - return funcName.indexOf( "outer" ) === 0 ? - elem[ "inner" + name ] : - elem.document.documentElement[ "client" + name ]; - } - - // Get document width or height - if ( elem.nodeType === 9 ) { - doc = elem.documentElement; - - // Either scroll[Width/Height] or offset[Width/Height] or client[Width/Height], - // whichever is greatest - return Math.max( - elem.body[ "scroll" + name ], doc[ "scroll" + name ], - elem.body[ "offset" + name ], doc[ "offset" + name ], - doc[ "client" + name ] - ); - } - - return value === undefined ? - - // Get width or height on the element, requesting but not forcing parseFloat - jQuery.css( elem, type, extra ) : - - // Set width or height on the element - jQuery.style( elem, type, value, extra ); - }, type, chainable ? margin : undefined, chainable ); - }; - } ); - } ); - - - jQuery.each( [ - "ajaxStart", - "ajaxStop", - "ajaxComplete", - "ajaxError", - "ajaxSuccess", - "ajaxSend" - ], function( _i, type ) { - jQuery.fn[ type ] = function( fn ) { - return this.on( type, fn ); - }; - } ); - - - - - jQuery.fn.extend( { - - bind: function( types, data, fn ) { - return this.on( types, null, data, fn ); - }, - unbind: function( types, fn ) { - return this.off( types, null, fn ); - }, - - delegate: function( selector, types, data, fn ) { - return this.on( types, selector, data, fn ); - }, - undelegate: function( selector, types, fn ) { - - // ( namespace ) or ( selector, types [, fn] ) - return arguments.length === 1 ? - this.off( selector, "**" ) : - this.off( types, selector || "**", fn ); - }, - - hover: function( fnOver, fnOut ) { - return this.mouseenter( fnOver ).mouseleave( fnOut || fnOver ); - } - } ); - - jQuery.each( - ( "blur focus focusin focusout resize scroll click dblclick " + - "mousedown mouseup mousemove mouseover mouseout mouseenter mouseleave " + - "change select submit keydown keypress keyup contextmenu" ).split( " " ), - function( _i, name ) { - - // Handle event binding - jQuery.fn[ name ] = function( data, fn ) { - return arguments.length > 0 ? - this.on( name, null, data, fn ) : - this.trigger( name ); - }; - } - ); - - - - - // Support: Android <=4.0 only - // Make sure we trim BOM and NBSP - var rtrim = /^[\s\uFEFF\xA0]+|[\s\uFEFF\xA0]+$/g; - - // Bind a function to a context, optionally partially applying any - // arguments. - // jQuery.proxy is deprecated to promote standards (specifically Function#bind) - // However, it is not slated for removal any time soon - jQuery.proxy = function( fn, context ) { - var tmp, args, proxy; - - if ( typeof context === "string" ) { - tmp = fn[ context ]; - context = fn; - fn = tmp; - } - - // Quick check to determine if target is callable, in the spec - // this throws a TypeError, but we will just return undefined. - if ( !isFunction( fn ) ) { - return undefined; - } - - // Simulated bind - args = slice.call( arguments, 2 ); - proxy = function() { - return fn.apply( context || this, args.concat( slice.call( arguments ) ) ); - }; - - // Set the guid of unique handler to the same of original handler, so it can be removed - proxy.guid = fn.guid = fn.guid || jQuery.guid++; - - return proxy; - }; - - jQuery.holdReady = function( hold ) { - if ( hold ) { - jQuery.readyWait++; - } else { - jQuery.ready( true ); - } - }; - jQuery.isArray = Array.isArray; - jQuery.parseJSON = JSON.parse; - jQuery.nodeName = nodeName; - jQuery.isFunction = isFunction; - jQuery.isWindow = isWindow; - jQuery.camelCase = camelCase; - jQuery.type = toType; - - jQuery.now = Date.now; - - jQuery.isNumeric = function( obj ) { - - // As of jQuery 3.0, isNumeric is limited to - // strings and numbers (primitives or objects) - // that can be coerced to finite numbers (gh-2662) - var type = jQuery.type( obj ); - return ( type === "number" || type === "string" ) && - - // parseFloat NaNs numeric-cast false positives ("") - // ...but misinterprets leading-number strings, particularly hex literals ("0x...") - // subtraction forces infinities to NaN - !isNaN( obj - parseFloat( obj ) ); - }; - - jQuery.trim = function( text ) { - return text == null ? - "" : - ( 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. For more information, see - // https://github.com/jrburke/requirejs/wiki/Updating-existing-libraries#wiki-anon - - if ( typeof define === "function" && define.amd ) { - define( "jquery", [], function() { - return jQuery; - } ); - } - - - - - var - - // Map over jQuery in case of overwrite - _jQuery = window.jQuery, - - // Map over the $ in case of overwrite - _$ = window.$; - - jQuery.noConflict = function( deep ) { - if ( window.$ === jQuery ) { - window.$ = _$; - } - - if ( deep && window.jQuery === jQuery ) { - window.jQuery = _jQuery; - } - - return jQuery; - }; - - // Expose jQuery and $ identifiers, even in AMD - // (#7102#comment:10, https://github.com/jquery/jquery/pull/557) - // and CommonJS for browser emulators (#13566) - if ( typeof noGlobal === "undefined" ) { - window.jQuery = window.$ = jQuery; - } - - - - - return jQuery; - } );
• html/_static/language_data.js
  • @@ -5,7 +5,7 @@
    * This script contains the language-specific data used by searchtools.js, * namely the list of stopwords, stemmer, scorer and splitter. * - * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS. + * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */
• html/_static/searchtools.js
  • @@ -4,7 +4,7 @@
    * * Sphinx JavaScript utilities for the full-text search. * - * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS. + * :copyright: Copyright 2007-2023 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */
  • @@ -57,14 +57,14 @@ const _removeChildren = (element) => {
    const _escapeRegExp = (string) => string.replace(/[.*+\-?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string - const _displayItem = (item, highlightTerms, searchTerms) => { + 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] = item; + const [docName, title, anchor, descr, score, _filename] = item; let listItem = document.createElement("li"); let requestUrl;
  • @@ -82,13 +82,12 @@ const _displayItem = (item, highlightTerms, searchTerms) => {
    requestUrl = docUrlRoot + docName + docFileSuffix; linkUrl = docName + docLinkSuffix; } - const params = new URLSearchParams(); - params.set("highlight", [...highlightTerms].join(" ")); let linkEl = listItem.appendChild(document.createElement("a")); - linkEl.href = linkUrl + "?" + params.toString() + anchor; + linkEl.href = linkUrl + anchor; + linkEl.dataset.score = score; linkEl.innerHTML = title; if (descr) - listItem.appendChild(document.createElement("span")).innerText = + listItem.appendChild(document.createElement("span")).innerHTML = " (" + descr + ")"; else if (showSearchSummary) fetch(requestUrl)
  • @@ -96,7 +95,7 @@ const _displayItem = (item, highlightTerms, searchTerms) => {
    .then((data) => { if (data) listItem.appendChild( - Search.makeSearchSummary(data, searchTerms, highlightTerms) + Search.makeSearchSummary(data, searchTerms) ); }); Search.output.appendChild(listItem);
  • @@ -116,15 +115,14 @@ const _finishSearch = (resultCount) => {
    const _displayNextItem = ( results, resultCount, - highlightTerms, 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(), highlightTerms, searchTerms); + _displayItem(results.pop(), searchTerms); setTimeout( - () => _displayNextItem(results, resultCount, highlightTerms, searchTerms), + () => _displayNextItem(results, resultCount, searchTerms), 5 ); }
  • @@ -155,10 +153,8 @@ const Search = {
    _pulse_status: -1, htmlToText: (htmlString) => { - const htmlElement = document - .createRange() - .createContextualFragment(htmlString); - _removeChildren(htmlElement.querySelectorAll(".headerlink")); + 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(
  • @@ -239,6 +235,12 @@ const Search = {
    * 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();
  • @@ -266,6 +268,10 @@ const Search = {
    } }); + if (SPHINX_HIGHLIGHT_ENABLED) { // set in sphinx_highlight.js + localStorage.setItem("sphinx_highlight_terms", [...highlightTerms].join(" ")) + } + // console.debug("SEARCH: searching for:"); // console.info("required: ", [...searchTerms]); // console.info("excluded: ", [...excludedTerms]);
  • @@ -274,6 +280,40 @@ const Search = {
    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], + ]); + } + } + } + + // 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], + ]); + } + } + } + // lookup as object objectTerms.forEach((term) => results.push(...Search.performObjectSearch(term, objectTerms))
  • @@ -320,7 +360,7 @@ const Search = {
    // console.info("search results:", Search.lastresults); // print the results - _displayNextItem(results, results.length, highlightTerms, searchTerms); + _displayNextItem(results, results.length, searchTerms); }, /**
  • @@ -401,8 +441,8 @@ const Search = {
    // prepare search const terms = Search._index.terms; const titleTerms = Search._index.titleterms; - const docNames = Search._index.docnames; const filenames = Search._index.filenames; + const docNames = Search._index.docnames; const titles = Search._index.titles; const scoreMap = new Map();
  • @@ -499,16 +539,15 @@ const Search = {
    /** * helper function to return a node containing the * search summary for a given text. keywords is a list - * of stemmed words, highlightWords is the list of normal, unstemmed - * words. the first one is used to find the occurrence, the - * latter for highlighting it. + * of stemmed words. */ - makeSearchSummary: (htmlText, keywords, highlightWords) => { - const text = Search.htmlToText(htmlText).toLowerCase(); + makeSearchSummary: (htmlText, keywords) => { + const text = Search.htmlToText(htmlText); if (text === "") return null; + const textLower = text.toLowerCase(); const actualStartPosition = [...keywords] - .map((k) => text.indexOf(k.toLowerCase())) + .map((k) => textLower.indexOf(k.toLowerCase())) .filter((i) => i > -1) .slice(-1)[0]; const startWithContext = Math.max(actualStartPosition - 120, 0);
  • @@ -516,13 +555,9 @@ const Search = {
    const top = startWithContext === 0 ? "" : "..."; const tail = startWithContext + 240 < text.length ? "..." : ""; - let summary = document.createElement("div"); + let summary = document.createElement("p"); summary.classList.add("context"); - summary.innerText = top + text.substr(startWithContext, 240).trim() + tail; - - highlightWords.forEach((highlightWord) => - _highlightText(summary, highlightWord, "highlighted") - ); + summary.textContent = top + text.substr(startWithContext, 240).trim() + tail; return summary; },
• html/_static/sphinx_highlight.js (new)
  • @@ -0,0 +1,144 @@
    + /* 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);
• html/_static/underscore-1.13.1.js (deleted)
  • @@ -1,2042 +0,0 @@
    - (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 = { - '&': '&amp;', - '<': '&lt;', - '>': '&gt;', - '"': '&quot;', - "'": '&#x27;', - '`': '&#x60;' - }; - - // 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. 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    <li class="toctree-l1"><a class="reference internal" href="C02_Basics.html">2. Basics</a></li> <li class="toctree-l1"><a class="reference internal" href="C03_Logic.html">3. Logic</a></li> <li class="toctree-l1"><a class="reference internal" href="C04_Sets_and_Functions.html">4. Sets and Functions</a></li> - <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a></li> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Elementary Number Theory</a></li> <li class="toctree-l1"><a class="reference internal" href="C06_Structures.html">6. Structures</a></li> <li class="toctree-l1"><a class="reference internal" href="C07_Hierarchies.html">7. Hierarchies</a></li> <li class="toctree-l1"><a class="reference internal" href="C08_Topology.html">8. Topology</a></li>
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    <li class="toctree-l2"><a class="reference internal" href="C04_Sets_and_Functions.html#the-schroder-bernstein-theorem">4.3. The Schr&#246;der-Bernstein Theorem</a></li> </ul> </li> - <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Number Theory</a><ul> + <li class="toctree-l1"><a class="reference internal" href="C05_Number_Theory.html">5. Elementary Number Theory</a><ul> <li class="toctree-l2"><a class="reference internal" href="C05_Number_Theory.html#irrational-roots">5.1. Irrational Roots</a></li> <li class="toctree-l2"><a class="reference internal" href="C05_Number_Theory.html#induction-and-recursion">5.2. Induction and Recursion</a></li> <li class="toctree-l2"><a class="reference internal" href="C05_Number_Theory.html#infinitely-many-primes">5.3. Infinitely Many Primes</a></li>
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