Changes
30 changed files (+1050/-2217)
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@@ -1,223 +0,0 @@import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic noncomputable section @[ext] structure Point where x : ℝ y : ℝ z : ℝ #check Point.ext example (a b : Point) (hx : a.x = b.x) (hy : a.y = b.y) (hz : a.z = b.z) : a = b := by ext repeat' assumption def myPoint1 : Point where x := 2 y := -1 z := 4 def myPoint2 : Point := ⟨2, -1, 4⟩ def myPoint3 := Point.mk 2 (-1) 4 structure Point' where build :: x : ℝ y : ℝ z : ℝ #check Point'.build 2 (-1) 4 namespace Point def add (a b : Point) : Point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ def add' (a b : Point) : Point where x := a.x + b.x y := a.y + b.y z := a.z + b.z #check add myPoint1 myPoint2 #check myPoint1.add myPoint2 end Point #check Point.add myPoint1 myPoint2 #check myPoint1.add myPoint2 namespace Point protected theorem add_comm (a b : Point) : add a b = add b a := by rw [add, add] ext <;> dsimp repeat' apply add_comm example (a b : Point) : add a b = add b a := by simp [add, add_comm] theorem add_x (a b : Point) : (a.add b).x = a.x + b.x := rfl def addAlt : Point → Point → Point | Point.mk x₁ y₁ z₁, Point.mk x₂ y₂ z₂ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ def addAlt' : Point → Point → Point | ⟨x₁, y₁, z₁⟩, ⟨x₂, y₂, z₂⟩ => ⟨x₁ + x₂, y₁ + y₂, z₁ + z₂⟩ theorem addAlt_x (a b : Point) : (a.addAlt b).x = a.x + b.x := by cases a cases b rfl theorem addAlt_comm (a b : Point) : addAlt a b = addAlt b a := by rcases a with ⟨xa, ya, za⟩ rcases b with ⟨xb, yb, zb⟩ rw [addAlt, addAlt] ext <;> dsimp apply add_comm repeat' apply add_comm example (a b : Point) : addAlt a b = addAlt b a := by rcases a with ⟨xa, ya, za⟩ rcases b with ⟨xb, yb, zb⟩ simp [addAlt, add_comm] example : ∀ a b : Point, addAlt a b = addAlt b a := by rintro ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ simp [addAlt, add_comm] example : ∀ a b : Point, add a b = add b a := fun ⟨xa, ya, za⟩ ⟨xb, yb, zb⟩ => by simp [add, add_comm] protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by sorry def smul (r : ℝ) (a : Point) : Point := sorry theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by sorry end Point structure StandardTwoSimplex where x : ℝ y : ℝ z : ℝ x_nonneg : 0 ≤ x y_nonneg : 0 ≤ y z_nonneg : 0 ≤ z sum_eq : x + y + z = 1 namespace StandardTwoSimplex def swapXy (a : StandardTwoSimplex) : StandardTwoSimplex where x := a.y y := a.x z := a.z x_nonneg := a.y_nonneg y_nonneg := a.x_nonneg z_nonneg := a.z_nonneg sum_eq := by rw [add_comm a.y a.x, a.sum_eq] noncomputable section def midpoint (a b : StandardTwoSimplex) : StandardTwoSimplex where x := (a.x + b.x) / 2 y := (a.y + b.y) / 2 z := (a.z + b.z) / 2 x_nonneg := div_nonneg (add_nonneg a.x_nonneg b.x_nonneg) (by norm_num) y_nonneg := div_nonneg (add_nonneg a.y_nonneg b.y_nonneg) (by norm_num) z_nonneg := div_nonneg (add_nonneg a.z_nonneg b.z_nonneg) (by norm_num) sum_eq := by field_simp; linarith [a.sum_eq, b.sum_eq] def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) (a b : StandardTwoSimplex) : StandardTwoSimplex := sorry end end StandardTwoSimplex open BigOperators structure StandardSimplex (n : ℕ) where V : Fin n → ℝ NonNeg : ∀ i : Fin n, 0 ≤ V i sum_eq_one : (∑ i, V i) = 1 namespace StandardSimplex def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n where V i := (a.V i + b.V i) / 2 NonNeg := by intro i apply div_nonneg · linarith [a.NonNeg i, b.NonNeg i] norm_num sum_eq_one := by simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] field_simp end StandardSimplex structure IsLinear (f : ℝ → ℝ) where is_additive : ∀ x y, f (x + y) = f x + f y preserves_mul : ∀ x c, f (c * x) = c * f x section variable (f : ℝ → ℝ) (linf : IsLinear f) #check linf.is_additive #check linf.preserves_mul end def Point'' := ℝ × ℝ × ℝ def IsLinear' (f : ℝ → ℝ) := (∀ x y, f (x + y) = f x + f y) ∧ ∀ x c, f (c * x) = c * f x def PReal := { y : ℝ // 0 < y } section variable (x : PReal) #check x.val #check x.property #check x.1 #check x.2 end def StandardTwoSimplex' := { p : ℝ × ℝ × ℝ // 0 ≤ p.1 ∧ 0 ≤ p.2.1 ∧ 0 ≤ p.2.2 ∧ p.1 + p.2.1 + p.2.2 = 1 } def StandardSimplex' (n : ℕ) := { v : Fin n → ℝ // (∀ i : Fin n, 0 ≤ v i) ∧ (∑ i, v i) = 1 } def StdSimplex := Σ n : ℕ, StandardSimplex n section variable (s : StdSimplex) #check s.fst #check s.snd #check s.1 #check s.2 end
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@@ -1,170 +0,0 @@import Mathlib.Data.Real.Basic structure Group₁ (α : Type _) where mul : α → α → α one : α inv : α → α mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) mul_one : ∀ x : α, mul x one = x one_mul : ∀ x : α, mul one x = x mul_left_inv : ∀ x : α, mul (inv x) x = one structure Group₁Cat where α : Type _ str : Group₁ α section variable (α β γ : Type _) variable (f : α ≃ β) (g : β ≃ γ) #check Equiv α β #check (f.toFun : α → β) #check (f.invFun : β → α) #check (f.right_inv : ∀ x : β, f (f.invFun x) = x) #check (f.left_inv : ∀ x : α, f.invFun (f x) = x) #check (Equiv.refl α : α ≃ α) #check (f.symm : β ≃ α) #check (f.trans g : α ≃ γ) example (x : α) : (f.trans g).toFun x = g.toFun (f.toFun x) := rfl example (x : α) : (f.trans g) x = g (f x) := rfl example : (f.trans g : α → γ) = g ∘ f := rfl end example (α : Type _) : Equiv.Perm α = (α ≃ α) := rfl def permGroup {α : Type _} : Group₁ (Equiv.Perm α) where mul f g := Equiv.trans g f one := Equiv.refl α inv := Equiv.symm mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm one_mul := Equiv.trans_refl mul_one := Equiv.refl_trans mul_left_inv := Equiv.self_trans_symm structure AddGroup₁ (α : Type _) where (add : α → α → α) -- fill in the rest @[ext] structure Point where x : ℝ y : ℝ z : ℝ namespace Point def add (a b : Point) : Point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ def neg (a : Point) : Point := sorry def zero : Point := sorry def add_group_point : AddGroup₁ point := sorry end Point section variable {α : Type _} (f g : Equiv.Perm α) (n : ℕ) #check f * g #check mul_assoc f g g⁻¹ -- group power, defined for any group #check g ^ n example : f * g * g⁻¹ = f := by rw [mul_assoc, mul_right_inv, mul_one] example : f * g * g⁻¹ = f := mul_inv_cancel_right f g example {α : Type _} (f g : Equiv.Perm α) : g.symm.trans (g.trans f) = f := mul_inv_cancel_right f g end class Group₂ (α : Type _) where mul : α → α → α one : α inv : α → α mul_assoc : ∀ x y z : α, mul (mul x y) z = mul x (mul y z) mul_one : ∀ x : α, mul x one = x one_mul : ∀ x : α, mul one x = x mul_left_inv : ∀ x : α, mul (inv x) x = one instance {α : Type _} : Group₂ (Equiv.Perm α) where mul f g := Equiv.trans g f one := Equiv.refl α inv := Equiv.symm mul_assoc f g h := (Equiv.trans_assoc _ _ _).symm one_mul := Equiv.trans_refl mul_one := Equiv.refl_trans mul_left_inv := Equiv.self_trans_symm #check @Group₂.mul def mySquare {α : Type _} [Group₂ α] (x : α) := Group₂.mul x x #check @mySquare section variable {β : Type _} (f g : Equiv.Perm β) example : Group₂.mul f g = g.trans f := rfl example : mySquare f = f.trans f := rfl end instance : Inhabited Point where default := ⟨0, 0, 0⟩ #check (default : Point) example : ([] : List Point).headI = default := rfl instance : Add Point where add := Point.add section variable (x y : Point) #check x + y example : x + y = Point.add x y := rfl end instance hasMulGroup₂ {α : Type _} [Group₂ α] : Mul α := ⟨Group₂.mul⟩ instance hasOneGroup₂ {α : Type _} [Group₂ α] : One α := ⟨Group₂.one⟩ instance hasInvGroup₂ {α : Type _} [Group₂ α] : Inv α := ⟨Group₂.inv⟩ section variable {α : Type _} (f g : Equiv.Perm α) #check f * 1 * g⁻¹ def foo : f * 1 * g⁻¹ = g.symm.trans ((Equiv.refl α).trans f) := rfl end class AddGroup₂ (α : Type _) where add : α → α → α -- fill in the rest
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@@ -1,276 +0,0 @@import Mathlib.Data.Int.Basic import Mathlib.Algebra.EuclideanDomain.Basic import Mathlib.RingTheory.PrincipalIdealDomain import Mathlib.Tactic @[ext] structure gaussInt where re : ℤ im : ℤ namespace gaussInt instance : Zero gaussInt := ⟨⟨0, 0⟩⟩ instance : One gaussInt := ⟨⟨1, 0⟩⟩ instance : Add gaussInt := ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ instance : Neg gaussInt := ⟨fun x => ⟨-x.re, -x.im⟩⟩ instance : Mul gaussInt := ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := rfl theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := rfl theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := rfl theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := rfl theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := rfl @[simp] theorem zero_re : (0 : gaussInt).re = 0 := rfl @[simp] theorem zero_im : (0 : gaussInt).im = 0 := rfl @[simp] theorem one_re : (1 : gaussInt).re = 1 := rfl @[simp] theorem one_im : (1 : gaussInt).im = 0 := rfl @[simp] theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := rfl @[simp] theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := rfl @[simp] theorem neg_re (x : gaussInt) : (-x).re = -x.re := rfl @[simp] theorem neg_im (x : gaussInt) : (-x).im = -x.im := rfl @[simp] theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := rfl @[simp] theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := rfl instance instCommRing : CommRing gaussInt where zero := 0 one := 1 add := (· + ·) neg x := -x mul := (· * ·) add_assoc := by intros ext <;> simp <;> ring zero_add := by intro ext <;> simp add_zero := by intro ext <;> simp add_left_neg := by intro ext <;> simp add_comm := by intros ext <;> simp <;> ring mul_assoc := by intros ext <;> simp <;> ring one_mul := by intro ext <;> simp mul_one := by intro ext <;> simp left_distrib := by intros ext <;> simp <;> ring right_distrib := by intros ext <;> simp <;> ring mul_comm := by intros ext <;> simp <;> ring zero_mul := sorry mul_zero := sorry instance : Nontrivial gaussInt := by use 0, 1 rw [Ne, gaussInt.ext_iff] simp end gaussInt example (a b : ℤ) : a = b * (a / b) + a % b := Eq.symm <| Int.ediv_add_emod a b example (a b : ℤ) : b ≠ 0 → 0 ≤ a % b := Int.emod_nonneg a example (a b : ℤ) : b ≠ 0 → a % b < abs b := Int.emod_lt a namespace Int def div' (a b : ℤ) := (a + b / 2) / b def mod' (a b : ℤ) := (a + b / 2) % b - b / 2 theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by rw [div', mod'] linarith [Int.ediv_add_emod (a + b / 2) b] theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by rw [mod', abs_le] constructor · linarith [Int.emod_nonneg (a + b / 2) h.ne'] have := Int.emod_lt_of_pos (a + b / 2) h have := Int.ediv_add_emod b 2 have := Int.emod_lt_of_pos b zero_lt_two revert this; intro this -- FIXME, this should not be needed linarith theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] end Int theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by sorry namespace gaussInt def norm (x : gaussInt) := x.re ^ 2 + x.im ^ 2 @[simp] theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by sorry theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by sorry theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by sorry theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by sorry def conj (x : gaussInt) : gaussInt := ⟨x.re, -x.im⟩ @[simp] theorem conj_re (x : gaussInt) : (conj x).re = x.re := rfl @[simp] theorem conj_im (x : gaussInt) : (conj x).im = -x.im := rfl theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] instance : Div gaussInt := ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ instance : Mod gaussInt := ⟨fun x y => x - y * (x / y)⟩ theorem div_def (x y : gaussInt) : x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := rfl theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := rfl theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by have norm_y_pos : 0 < norm y := by rwa [norm_pos] have : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] ext <;> simp <;> ring have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by conv => lhs rw [← norm_conj y, ← norm_mul, this, norm] simp trans 2 * (y.norm / 2) ^ 2 · rw [two_mul] apply add_le_add <;> · rw [sq_le_sq] apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) apply le_abs_self rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] apply mul_le_mul_of_nonneg_left · apply Int.ediv_mul_le norm_num apply Int.ediv_nonneg (norm_nonneg y) norm_num have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos apply lt_of_le_of_lt this apply Int.ediv_lt_of_lt_mul · norm_num linarith theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := Int.natAbs_of_nonneg (norm_nonneg _) theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by apply Int.ofNat_lt.1 simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] apply norm_mod_lt x hy theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : ¬(norm (x * y)).natAbs < (norm x).natAbs := by apply not_lt_of_ge rw [norm_mul, Int.natAbs_mul] apply le_mul_of_one_le_right (Nat.zero_le _) apply Int.ofNat_le.1 rw [coe_natAbs_norm] exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) instance : EuclideanDomain gaussInt := { gaussInt.instCommRing with quotient := (· / ·) remainder := (· % ·) quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] quotient_zero := fun x => by simp [div_def, norm, Int.div'] rfl r := Measure (Int.natAbs ∘ norm) r_wellFounded := (measure (Int.natAbs ∘ norm)).2 remainder_lt := natAbs_norm_mod_lt mul_left_not_lt := not_norm_mul_left_lt_norm } example (x : gaussInt) : Irreducible x ↔ Prime x := PrincipalIdealRing.irreducible_iff_prime end gaussInt
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@@ -1,98 +0,0 @@import Mathlib.Algebra.BigOperators.Ring import Mathlib.Data.Real.Basic noncomputable section @[ext] structure Point where x : ℝ y : ℝ z : ℝ namespace Point def add (a b : Point) : Point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ protected theorem add_assoc (a b c : Point) : (a.add b).add c = a.add (b.add c) := by simp [add, add_assoc] def smul (r : ℝ) (a : Point) : Point := ⟨r * a.x, r * a.y, r * a.z⟩ theorem smul_distrib (r : ℝ) (a b : Point) : (smul r a).add (smul r b) = smul r (a.add b) := by simp [add, smul, mul_add] end Point structure StandardTwoSimplex where x : ℝ y : ℝ z : ℝ x_nonneg : 0 ≤ x y_nonneg : 0 ≤ y z_nonneg : 0 ≤ z sum_eq : x + y + z = 1 namespace StandardTwoSimplex noncomputable section def weightedAverage (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) (a b : StandardTwoSimplex) : StandardTwoSimplex where x := lambda * a.x + (1 - lambda) * b.x y := lambda * a.y + (1 - lambda) * b.y z := lambda * a.z + (1 - lambda) * b.z x_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.x_nonneg) (mul_nonneg (by linarith) b.x_nonneg) y_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.y_nonneg) (mul_nonneg (by linarith) b.y_nonneg) z_nonneg := add_nonneg (mul_nonneg lambda_nonneg a.z_nonneg) (mul_nonneg (by linarith) b.z_nonneg) sum_eq := by trans (a.x + a.y + a.z) * lambda + (b.x + b.y + b.z) * (1 - lambda) · ring simp [a.sum_eq, b.sum_eq] end end StandardTwoSimplex open BigOperators structure StandardSimplex (n : ℕ) where V : Fin n → ℝ NonNeg : ∀ i : Fin n, 0 ≤ V i sum_eq_one : (∑ i, V i) = 1 namespace StandardSimplex def midpoint (n : ℕ) (a b : StandardSimplex n) : StandardSimplex n where V i := (a.V i + b.V i) / 2 NonNeg := by intro i apply div_nonneg · linarith [a.NonNeg i, b.NonNeg i] norm_num sum_eq_one := by simp [div_eq_mul_inv, ← Finset.sum_mul, Finset.sum_add_distrib, a.sum_eq_one, b.sum_eq_one] field_simp end StandardSimplex namespace StandardSimplex def weightedAverage {n : ℕ} (lambda : Real) (lambda_nonneg : 0 ≤ lambda) (lambda_le : lambda ≤ 1) (a b : StandardSimplex n) : StandardSimplex n where V i := lambda * a.V i + (1 - lambda) * b.V i NonNeg i := add_nonneg (mul_nonneg lambda_nonneg (a.NonNeg i)) (mul_nonneg (by linarith) (b.NonNeg i)) sum_eq_one := by trans (lambda * ∑ i, a.V i) + (1 - lambda) * ∑ i, b.V i · rw [Finset.sum_add_distrib, Finset.mul_sum, Finset.mul_sum] simp [a.sum_eq_one, b.sum_eq_one] end StandardSimplex
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@@ -1,73 +0,0 @@import Mathlib.Data.Real.Basic structure AddGroup₁ (α : Type _) where add : α → α → α zero : α neg : α → α add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) add_zero : ∀ x : α, add x zero = x zero_add : ∀ x : α, add x zero = x add_left_neg : ∀ x : α, add (neg x) x = zero @[ext] structure Point where x : ℝ y : ℝ z : ℝ namespace Point def add (a b : Point) : Point := ⟨a.x + b.x, a.y + b.y, a.z + b.z⟩ def neg (a : Point) : Point := ⟨-a.x, -a.y, -a.z⟩ def zero : Point := ⟨0, 0, 0⟩ def addGroupPoint : AddGroup₁ Point where add := Point.add zero := Point.zero neg := Point.neg add_assoc := by simp [Point.add, add_assoc] add_zero := by simp [Point.add, Point.zero] zero_add := by simp [Point.add, Point.zero] add_left_neg := by simp [Point.add, Point.neg, Point.zero] end Point class AddGroup₂ (α : Type _) where add : α → α → α zero : α neg : α → α add_assoc : ∀ x y z : α, add (add x y) z = add x (add y z) add_zero : ∀ x : α, add x zero = x zero_add : ∀ x : α, add x zero = x add_left_neg : ∀ x : α, add (neg x) x = zero instance hasAddAddGroup₂ {α : Type _} [AddGroup₂ α] : Add α := ⟨AddGroup₂.add⟩ instance hasZeroAddGroup₂ {α : Type _} [AddGroup₂ α] : Zero α := ⟨AddGroup₂.zero⟩ instance hasNegAddGroup₂ {α : Type _} [AddGroup₂ α] : Neg α := ⟨AddGroup₂.neg⟩ instance : AddGroup₂ Point where add := Point.add zero := Point.zero neg := Point.neg add_assoc := by simp [Point.add, add_assoc] add_zero := by simp [Point.add, Point.zero] zero_add := by simp [Point.add, Point.zero] add_left_neg := by simp [Point.add, Point.neg, Point.zero] section variable (x y : Point) #check x + -y + 0 end
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@@ -1,290 +0,0 @@import Mathlib.Data.Int.Basic import Mathlib.Algebra.EuclideanDomain.Basic import Mathlib.RingTheory.PrincipalIdealDomain import Mathlib.Tactic @[ext] structure gaussInt where re : ℤ im : ℤ namespace gaussInt instance : Zero gaussInt := ⟨⟨0, 0⟩⟩ instance : One gaussInt := ⟨⟨1, 0⟩⟩ instance : Add gaussInt := ⟨fun x y => ⟨x.re + y.re, x.im + y.im⟩⟩ instance : Neg gaussInt := ⟨fun x => ⟨-x.re, -x.im⟩⟩ instance : Mul gaussInt := ⟨fun x y => ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩⟩ theorem zero_def : (0 : gaussInt) = ⟨0, 0⟩ := rfl theorem one_def : (1 : gaussInt) = ⟨1, 0⟩ := rfl theorem add_def (x y : gaussInt) : x + y = ⟨x.re + y.re, x.im + y.im⟩ := rfl theorem neg_def (x : gaussInt) : -x = ⟨-x.re, -x.im⟩ := rfl theorem mul_def (x y : gaussInt) : x * y = ⟨x.re * y.re - x.im * y.im, x.re * y.im + x.im * y.re⟩ := rfl @[simp] theorem zero_re : (0 : gaussInt).re = 0 := rfl @[simp] theorem zero_im : (0 : gaussInt).im = 0 := rfl @[simp] theorem one_re : (1 : gaussInt).re = 1 := rfl @[simp] theorem one_im : (1 : gaussInt).im = 0 := rfl @[simp] theorem add_re (x y : gaussInt) : (x + y).re = x.re + y.re := rfl @[simp] theorem add_im (x y : gaussInt) : (x + y).im = x.im + y.im := rfl @[simp] theorem neg_re (x : gaussInt) : (-x).re = -x.re := rfl @[simp] theorem neg_im (x : gaussInt) : (-x).im = -x.im := rfl @[simp] theorem mul_re (x y : gaussInt) : (x * y).re = x.re * y.re - x.im * y.im := rfl @[simp] theorem mul_im (x y : gaussInt) : (x * y).im = x.re * y.im + x.im * y.re := rfl instance instCommRing : CommRing gaussInt where zero := 0 one := 1 add := (· + ·) neg x := -x mul := (· * ·) add_assoc := by intros ext <;> simp <;> ring zero_add := by intro ext <;> simp add_zero := by intro ext <;> simp add_left_neg := by intro ext <;> simp add_comm := by intros ext <;> simp <;> ring mul_assoc := by intros ext <;> simp <;> ring one_mul := by intro ext <;> simp mul_one := by intro ext <;> simp left_distrib := by intros ext <;> simp <;> ring right_distrib := by intros ext <;> simp <;> ring mul_comm := by intros ext <;> simp <;> ring zero_mul := sorry mul_zero := sorry instance : Nontrivial gaussInt := by use 0, 1 rw [Ne, gaussInt.ext_iff] simp end gaussInt namespace Int def div' (a b : ℤ) := (a + b / 2) / b def mod' (a b : ℤ) := (a + b / 2) % b - b / 2 theorem div'_add_mod' (a b : ℤ) : b * div' a b + mod' a b = a := by rw [div', mod'] linarith [Int.ediv_add_emod (a + b / 2) b] theorem abs_mod'_le (a b : ℤ) (h : 0 < b) : abs (mod' a b) ≤ b / 2 := by rw [mod', abs_le] constructor · linarith [Int.emod_nonneg (a + b / 2) h.ne'] have := Int.emod_lt_of_pos (a + b / 2) h have := Int.ediv_add_emod b 2 have := Int.emod_lt_of_pos b zero_lt_two revert this; intro this -- FIXME, this should not be needed linarith theorem mod'_eq (a b : ℤ) : mod' a b = a - b * div' a b := by linarith [div'_add_mod' a b] end Int private theorem aux {α : Type _} [LinearOrderedRing α] {x y : α} (h : x ^ 2 + y ^ 2 = 0) : x = 0 := haveI h' : x ^ 2 = 0 := by apply le_antisymm _ (sq_nonneg x) rw [← h] apply le_add_of_nonneg_right (sq_nonneg y) pow_eq_zero h' theorem sq_add_sq_eq_zero {α : Type _} [LinearOrderedRing α] (x y : α) : x ^ 2 + y ^ 2 = 0 ↔ x = 0 ∧ y = 0 := by constructor · intro h constructor · exact aux h rw [add_comm] at h exact aux h rintro ⟨rfl, rfl⟩ norm_num namespace gaussInt def norm (x : gaussInt) := x.re ^ 2 + x.im ^ 2 @[simp] theorem norm_nonneg (x : gaussInt) : 0 ≤ norm x := by apply add_nonneg <;> apply sq_nonneg theorem norm_eq_zero (x : gaussInt) : norm x = 0 ↔ x = 0 := by rw [norm, sq_add_sq_eq_zero, gaussInt.ext_iff] rfl theorem norm_pos (x : gaussInt) : 0 < norm x ↔ x ≠ 0 := by rw [lt_iff_le_and_ne, ne_comm, Ne, norm_eq_zero] simp [norm_nonneg] theorem norm_mul (x y : gaussInt) : norm (x * y) = norm x * norm y := by simp [norm] ring def conj (x : gaussInt) : gaussInt := ⟨x.re, -x.im⟩ @[simp] theorem conj_re (x : gaussInt) : (conj x).re = x.re := rfl @[simp] theorem conj_im (x : gaussInt) : (conj x).im = -x.im := rfl theorem norm_conj (x : gaussInt) : norm (conj x) = norm x := by simp [norm] instance : Div gaussInt := ⟨fun x y => ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩⟩ instance : Mod gaussInt := ⟨fun x y => x - y * (x / y)⟩ theorem div_def (x y : gaussInt) : x / y = ⟨Int.div' (x * conj y).re (norm y), Int.div' (x * conj y).im (norm y)⟩ := rfl theorem mod_def (x y : gaussInt) : x % y = x - y * (x / y) := rfl theorem norm_mod_lt (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : (x % y).norm < y.norm := by have norm_y_pos : 0 < norm y := by rwa [norm_pos] have : x % y * conj y = ⟨Int.mod' (x * conj y).re (norm y), Int.mod' (x * conj y).im (norm y)⟩ := by rw [mod_def, sub_mul, Int.mod'_eq, Int.mod'_eq, sub_eq_add_neg, div_def, norm] ext <;> simp <;> ring have : norm (x % y) * norm y ≤ norm y / 2 * norm y := by conv => lhs rw [← norm_conj y, ← norm_mul, this, norm] simp trans 2 * (y.norm / 2) ^ 2 · rw [two_mul] apply add_le_add <;> · rw [sq_le_sq] apply le_trans (Int.abs_mod'_le _ _ norm_y_pos) apply le_abs_self rw [pow_two, ← mul_assoc, mul_comm, mul_comm (2 : ℤ)] apply mul_le_mul_of_nonneg_left · apply Int.ediv_mul_le norm_num apply Int.ediv_nonneg (norm_nonneg y) norm_num have : norm (x % y) ≤ norm y / 2 := le_of_mul_le_mul_right this norm_y_pos apply lt_of_le_of_lt this apply Int.ediv_lt_of_lt_mul · norm_num linarith theorem coe_natAbs_norm (x : gaussInt) : (x.norm.natAbs : ℤ) = x.norm := Int.natAbs_of_nonneg (norm_nonneg _) theorem natAbs_norm_mod_lt (x y : gaussInt) (hy : y ≠ 0) : (x % y).norm.natAbs < y.norm.natAbs := by apply Int.ofNat_lt.1 simp only [Int.coe_natAbs, abs_of_nonneg, norm_nonneg] apply norm_mod_lt x hy theorem not_norm_mul_left_lt_norm (x : gaussInt) {y : gaussInt} (hy : y ≠ 0) : ¬(norm (x * y)).natAbs < (norm x).natAbs := by apply not_lt_of_ge rw [norm_mul, Int.natAbs_mul] apply le_mul_of_one_le_right (Nat.zero_le _) apply Int.ofNat_le.1 rw [coe_natAbs_norm] exact Int.add_one_le_of_lt ((norm_pos _).mpr hy) instance : EuclideanDomain gaussInt := { gaussInt.instCommRing with quotient := (· / ·) remainder := (· % ·) quotient_mul_add_remainder_eq := fun x y => by simp only; rw [mod_def, add_comm, sub_add_cancel] quotient_zero := fun x => by simp [div_def, norm, Int.div'] rfl r := Measure (Int.natAbs ∘ norm) r_wellFounded := (measure (Int.natAbs ∘ norm)).2 remainder_lt := natAbs_norm_mod_lt mul_left_not_lt := not_norm_mul_left_lt_norm } example (x : gaussInt) : Irreducible x ↔ Prime x := PrincipalIdealRing.irreducible_iff_prime end gaussInt
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@@ -1,8 +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="viewport" content="width=device-width, initial-scale=1.0" /> <title>1. Introduction — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" />
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@@ -16,7 +15,6 @@<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 src="_static/doctools.js"></script> <script src="_static/js/theme.js"></script> <link rel="index" title="Index" href="genindex.html" />
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@@ -84,10 +82,10 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="introduction"> <span id="id1"></span><h1><span class="section-number">1. </span>Introduction<a class="headerlink" href="#introduction" title="Permalink to this heading"></a></h1> <section id="getting-started"> <h2><span class="section-number">1.1. </span>Getting Started<a class="headerlink" href="#getting-started" title="Permalink to this heading"></a></h2> <div class="section" id="introduction"> <span id="id1"></span><h1><span class="section-number">1. </span>Introduction<a class="headerlink" href="#introduction" title="Permalink to this headline"></a></h1> <div class="section" id="getting-started"> <h2><span class="section-number">1.1. </span>Getting Started<a class="headerlink" href="#getting-started" title="Permalink to this headline"></a></h2> <p>The goal of this book is to teach you to formalize mathematics using the Lean 4 interactive proof assistant. It assumes that you know some mathematics, but it does not require much.
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@@ -124,7 +122,7 @@ these <a class="reference external" href="https://leanprover-community.github.ioIt isn’t enough just to install the Lean 4 extension for VSCode; make sure you complete the steps so that Lean 4 and elan are installed.</p></li> <li><p>Make sure you have <a class="reference external" href="https://git-scm.com/">git</a> installed. In a terminal, navigate to the folder where you want to put a copy of the repository, and type <code class="docutils literal notranslate"><span class="pre">git</span> <span class="pre">clone</span> <span class="pre">git@github.com:leanprover-community/mathematics_in_lean.git</span></code> repository, and type <code class="docutils literal notranslate"><span class="pre">git</span> <span class="pre">clone</span> <span class="pre">https://github.com/leanprover-community/mathematics_in_lean.git</span></code> to fetch it from github.</p></li> <li><p>Navigate to <code class="docutils literal notranslate"><span class="pre">mathematics_in_lean</span></code>, and execute <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> to fetch a compiled version of the library, <code class="docutils literal notranslate"><span class="pre">Mathlib</span></code>.</p></li>
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@@ -179,9 +177,9 @@ You don’t have to do all of them; when you feel comfortable that you havethe relevant skills, feel free to move on. You can always compare your solutions to the ones in the <code class="docutils literal notranslate"><span class="pre">solutions</span></code> folder associated with each section.</p> </section> <section id="overview"> <h2><span class="section-number">1.2. </span>Overview<a class="headerlink" href="#overview" title="Permalink to this heading"></a></h2> </div> <div class="section" id="overview"> <h2><span class="section-number">1.2. </span>Overview<a class="headerlink" href="#overview" title="Permalink to this headline"></a></h2> <p>Put simply, Lean is a tool for building complex expressions in a formal language known as <em>dependent type theory</em>.</p> <p id="index-0">Every expression has a <em>type</em>, and you can use the <cite>#check</cite> command to
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@@ -350,8 +348,8 @@ Giovanni Mascellani, Hunter Monroe, Pietro Monticone, Oliver Nash,Bartosz Piotrowski, and Guilherme Silva. Our work has been partially supported by the Hoskinson Center for Formal Mathematics.</p> </section> </section> </div> </div> </div>
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@@ -16,7 +15,6 @@<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 src="_static/doctools.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>
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@@ -88,14 +86,14 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="basics"> <span id="id1"></span><h1><span class="section-number">2. </span>Basics<a class="headerlink" href="#basics" title="Permalink to this heading"></a></h1> <div class="section" id="basics"> <span id="id1"></span><h1><span class="section-number">2. </span>Basics<a class="headerlink" href="#basics" title="Permalink to this headline"></a></h1> <p>This chapter is designed to introduce you to the nuts and bolts of mathematical reasoning in Lean: calculating, applying lemmas and theorems, and reasoning about generic structures.</p> <section id="calculating"> <h2><span class="section-number">2.1. </span>Calculating<a class="headerlink" href="#calculating" title="Permalink to this heading"></a></h2> <div class="section" id="calculating"> <h2><span class="section-number">2.1. </span>Calculating<a class="headerlink" href="#calculating" title="Permalink to this headline"></a></h2> <p>We generally learn to carry out mathematical calculations without thinking of them as proofs. But when we justify each step in a calculation,
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@@ -388,9 +386,9 @@ occurrence of <code class="docutils literal notranslate"><span class="pre">a</sp<span class="n">rw</span> <span class="o">[</span><span class="n">add_mul</span><span class="o">]</span> </pre></div> </div> </section> <section id="proving-identities-in-algebraic-structures"> <span id="id2"></span><h2><span class="section-number">2.2. </span>Proving Identities in Algebraic Structures<a class="headerlink" href="#proving-identities-in-algebraic-structures" title="Permalink to this heading"></a></h2> </div> <div class="section" id="proving-identities-in-algebraic-structures"> <span id="id2"></span><h2><span class="section-number">2.2. </span>Proving Identities in Algebraic Structures<a class="headerlink" href="#proving-identities-in-algebraic-structures" title="Permalink to this headline"></a></h2> <p id="index-7">Mathematically, a ring consists of a collection of objects, <span class="math notranslate nohighlight">\(R\)</span>, operations <span class="math notranslate nohighlight">\(+\)</span> <span class="math notranslate nohighlight">\(\times\)</span>, and constants <span class="math notranslate nohighlight">\(0\)</span> and <span class="math notranslate nohighlight">\(1\)</span>, and an operation <span class="math notranslate nohighlight">\(x \mapsto -x\)</span> such that:</p>
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@@ -708,9 +706,9 @@ It may seem odd that the algebraic structures are called<cite>noncomm_ring</cite> and <cite>ring</cite>. This is partly for historical reasons, but also for the convenience of using a shorter name for the tactic that deals with commutative rings, since it is used more often.</p> </section> <section id="using-theorems-and-lemmas"> <span id="id3"></span><h2><span class="section-number">2.3. </span>Using Theorems and Lemmas<a class="headerlink" href="#using-theorems-and-lemmas" title="Permalink to this heading"></a></h2> </div> <div class="section" id="using-theorems-and-lemmas"> <span id="id3"></span><h2><span class="section-number">2.3. </span>Using Theorems and Lemmas<a class="headerlink" href="#using-theorems-and-lemmas" title="Permalink to this headline"></a></h2> <p id="index-16">Rewriting is great for proving equations, but what about other sorts of theorems? For example, how can we prove an inequality,
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@@ -975,9 +973,9 @@ following theorem. You can use the theorem <code class="docutils literal notrans</div> <p>If you managed to solve this, congratulations! You are well on your way to becoming a master formalizer.</p> </section> <section id="more-examples-using-apply-and-rw"> <span id="more-on-order-and-divisibility"></span><h2><span class="section-number">2.4. </span>More examples using apply and rw<a class="headerlink" href="#more-examples-using-apply-and-rw" title="Permalink to this heading"></a></h2> </div> <div class="section" id="more-examples-using-apply-and-rw"> <span id="more-on-order-and-divisibility"></span><h2><span class="section-number">2.4. </span>More examples using apply and rw<a class="headerlink" href="#more-examples-using-apply-and-rw" title="Permalink to this headline"></a></h2> <p id="index-21">The <code class="docutils literal notranslate"><span class="pre">min</span></code> function on the real numbers is uniquely characterized by the following three facts:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="o">(</span><span class="n">min_le_left</span> <span class="n">a</span> <span class="n">b</span> <span class="o">:</span> <span class="n">min</span> <span class="n">a</span> <span class="n">b</span> <span class="bp">≤</span> <span class="n">a</span><span class="o">)</span>
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@@ -1186,9 +1184,9 @@ prove the following:</p><dt>either one will work.</dt><dd><p>either one will work.</p> </dd> </dl> </section> <section id="proving-facts-about-algebraic-structures"> <span id="id4"></span><h2><span class="section-number">2.5. </span>Proving Facts about Algebraic Structures<a class="headerlink" href="#proving-facts-about-algebraic-structures" title="Permalink to this heading"></a></h2> </div> <div class="section" id="proving-facts-about-algebraic-structures"> <span id="id4"></span><h2><span class="section-number">2.5. </span>Proving Facts about Algebraic Structures<a class="headerlink" href="#proving-facts-about-algebraic-structures" title="Permalink to this headline"></a></h2> <p id="index-27">In <a class="reference internal" href="#proving-identities-in-algebraic-structures"><span class="std std-numref">Section 2.2</span></a>, we saw that many common identities governing the real numbers hold in more general classes of algebraic structures,
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@@ -1401,8 +1399,8 @@ always nonnegative:</p></div> <p>We recommend making use of the theorem <code class="docutils literal notranslate"><span class="pre">nonneg_of_mul_nonneg_left</span></code>. As you may have guessed, this theorem is called <code class="docutils literal notranslate"><span class="pre">dist_nonneg</span></code> in mathlib.</p> </section> </section> </div> </div> </div>
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@@ -16,7 +15,6 @@<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 src="_static/doctools.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>
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@@ -89,8 +87,8 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="logic"> <span id="id1"></span><h1><span class="section-number">3. </span>Logic<a class="headerlink" href="#logic" title="Permalink to this heading"></a></h1> <div class="section" id="logic"> <span id="id1"></span><h1><span class="section-number">3. </span>Logic<a class="headerlink" href="#logic" title="Permalink to this headline"></a></h1> <p>In the last chapter, we dealt with equations, inequalities, and basic mathematical statements like “<span class="math notranslate nohighlight">\(x\)</span> divides <span class="math notranslate nohighlight">\(y\)</span>.”
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@@ -100,8 +98,8 @@ using logical terms like “and,” “or,” “not,”“if … then,” “every,” and “some.” In this chapter, we show you how to work with statements that are built up in this way.</p> <section id="implication-and-the-universal-quantifier"> <span id="id2"></span><h2><span class="section-number">3.1. </span>Implication and the Universal Quantifier<a class="headerlink" href="#implication-and-the-universal-quantifier" title="Permalink to this heading"></a></h2> <div class="section" id="implication-and-the-universal-quantifier"> <span id="id2"></span><h2><span class="section-number">3.1. </span>Implication and the Universal Quantifier<a class="headerlink" href="#implication-and-the-universal-quantifier" title="Permalink to this headline"></a></h2> <p>Consider the statement after the <code class="docutils literal notranslate"><span class="pre">#check</span></code>:</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="k">#check</span> <span class="bp">∀</span> <span class="n">x</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">,</span> <span class="mi">0</span> <span class="bp">≤</span> <span class="n">x</span> <span class="bp">→</span> <span class="bp">|</span><span class="n">x</span><span class="bp">|</span> <span class="bp">=</span> <span class="n">x</span> </pre></div>
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@@ -489,9 +487,9 @@ a lemma name.</p><span class="gr">sorry</span> </pre></div> </div> </section> <section id="the-existential-quantifier"> <span id="id3"></span><h2><span class="section-number">3.2. </span>The Existential Quantifier<a class="headerlink" href="#the-existential-quantifier" title="Permalink to this heading"></a></h2> </div> <div class="section" id="the-existential-quantifier"> <span id="id3"></span><h2><span class="section-number">3.2. </span>The Existential Quantifier<a class="headerlink" href="#the-existential-quantifier" title="Permalink to this headline"></a></h2> <p>The existential quantifier, which can be entered as <code class="docutils literal notranslate"><span class="pre">\ex</span></code> in VS Code, is used to represent the phrase “there exists.” The formal expression <code class="docutils literal notranslate"><span class="pre">∃</span> <span class="pre">x</span> <span class="pre">:</span> <span class="pre">ℝ,</span> <span class="pre">2</span> <span class="pre"><</span> <span class="pre">x</span> <span class="pre">∧</span> <span class="pre">x</span> <span class="pre"><</span> <span class="pre">3</span></code> in Lean says
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@@ -796,9 +794,9 @@ the composition of surjective functions is surjective.</p><span class="gr">sorry</span> </pre></div> </div> </section> <section id="negation"> <span id="id4"></span><h2><span class="section-number">3.3. </span>Negation<a class="headerlink" href="#negation" title="Permalink to this heading"></a></h2> </div> <div class="section" id="negation"> <span id="id4"></span><h2><span class="section-number">3.3. </span>Negation<a class="headerlink" href="#negation" title="Permalink to this headline"></a></h2> <p>The symbol <code class="docutils literal notranslate"><span class="pre">¬</span></code> is meant to express negation, so <code class="docutils literal notranslate"><span class="pre">¬</span> <span class="pre">x</span> <span class="pre"><</span> <span class="pre">y</span></code> says that <code class="docutils literal notranslate"><span class="pre">x</span></code> is not less than <code class="docutils literal notranslate"><span class="pre">y</span></code>, <code class="docutils literal notranslate"><span class="pre">¬</span> <span class="pre">x</span> <span class="pre">=</span> <span class="pre">y</span></code> (or, equivalently, <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">≠</span> <span class="pre">y</span></code>) says that
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@@ -1063,9 +1061,9 @@ Finally, the <code class="docutils literal notranslate"><span class="pre">contraby finding a contradiction in the hypotheses, such as a pair of the form <code class="docutils literal notranslate"><span class="pre">h</span> <span class="pre">:</span> <span class="pre">P</span></code> and <code class="docutils literal notranslate"><span class="pre">h'</span> <span class="pre">:</span> <span class="pre">¬</span> <span class="pre">P</span></code>. Of course, in this example, <code class="docutils literal notranslate"><span class="pre">linarith</span></code> also works.</p> </section> <section id="conjunction-and-bi-implication"> <span id="conjunction-and-biimplication"></span><h2><span class="section-number">3.4. </span>Conjunction and Bi-implication<a class="headerlink" href="#conjunction-and-bi-implication" title="Permalink to this heading"></a></h2> </div> <div class="section" id="conjunction-and-bi-implication"> <span id="conjunction-and-biimplication"></span><h2><span class="section-number">3.4. </span>Conjunction and Bi-implication<a class="headerlink" href="#conjunction-and-bi-implication" title="Permalink to this headline"></a></h2> <p id="index-18">You have already seen that the conjunction symbol, <code class="docutils literal notranslate"><span class="pre">∧</span></code>, is used to express “and.” The <code class="docutils literal notranslate"><span class="pre">constructor</span></code> tactic allows you to prove a statement of
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@@ -1324,9 +1322,9 @@ to be instantiated to different values.</p><span class="gr">sorry</span> </pre></div> </div> </section> <section id="disjunction"> <span id="id5"></span><h2><span class="section-number">3.5. </span>Disjunction<a class="headerlink" href="#disjunction" title="Permalink to this heading"></a></h2> </div> <div class="section" id="disjunction"> <span id="id5"></span><h2><span class="section-number">3.5. </span>Disjunction<a class="headerlink" href="#disjunction" title="Permalink to this headline"></a></h2> <p id="index-21">The canonical way to prove a disjunction <code class="docutils literal notranslate"><span class="pre">A</span> <span class="pre">∨</span> <span class="pre">B</span></code> is to prove <code class="docutils literal notranslate"><span class="pre">A</span></code> or to prove <code class="docutils literal notranslate"><span class="pre">B</span></code>. The <code class="docutils literal notranslate"><span class="pre">left</span></code> tactic chooses <code class="docutils literal notranslate"><span class="pre">A</span></code>,
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@@ -1567,9 +1565,9 @@ using <code class="docutils literal notranslate"><span class="pre">by_cases</spa<span class="gr">sorry</span> </pre></div> </div> </section> <section id="sequences-and-convergence"> <span id="id6"></span><h2><span class="section-number">3.6. </span>Sequences and Convergence<a class="headerlink" href="#sequences-and-convergence" title="Permalink to this heading"></a></h2> </div> <div class="section" id="sequences-and-convergence"> <span id="id6"></span><h2><span class="section-number">3.6. </span>Sequences and Convergence<a class="headerlink" href="#sequences-and-convergence" title="Permalink to this headline"></a></h2> <p>We now have enough skills at our disposal to do some real mathematics. In Lean, we can represent a sequence <span class="math notranslate nohighlight">\(s_0, s_1, s_2, \ldots\)</span> of real numbers as a function <code class="docutils literal notranslate"><span class="pre">s</span> <span class="pre">:</span> <span class="pre">ℕ</span> <span class="pre">→</span> <span class="pre">ℝ</span></code>.
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@@ -1804,8 +1802,8 @@ for dealing with convergence in vastly more general terms,not only abstracting away particular features of the domain and codomain, but also abstracting over different types of convergence.</p> </section> </section> </div> </div> </div>
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@@ -86,8 +84,8 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="sets-and-functions"> <span id="id1"></span><h1><span class="section-number">4. </span>Sets and Functions<a class="headerlink" href="#sets-and-functions" title="Permalink to this heading"></a></h1> <div class="section" id="sets-and-functions"> <span id="id1"></span><h1><span class="section-number">4. </span>Sets and Functions<a class="headerlink" href="#sets-and-functions" title="Permalink to this headline"></a></h1> <p>The vocabulary of sets, relations, and functions provides a uniform language for carrying out constructions in all the branches of mathematics.
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@@ -119,8 +117,8 @@ such as a set natural numbers or a set of functionsfrom real numbers to real numbers. The distinction between types and set takes some getting used to, but this chapter will take you through the essentials.</p> <section id="sets"> <span id="id2"></span><h2><span class="section-number">4.1. </span>Sets<a class="headerlink" href="#sets" title="Permalink to this heading"></a></h2> <div class="section" id="sets"> <span id="id2"></span><h2><span class="section-number">4.1. </span>Sets<a class="headerlink" href="#sets" title="Permalink to this headline"></a></h2> <p id="index-0">If <code class="docutils literal notranslate"><span class="pre">α</span></code> is any type, the type <code class="docutils literal notranslate"><span class="pre">Set</span> <span class="pre">α</span></code> consists of sets of elements of <code class="docutils literal notranslate"><span class="pre">α</span></code>. This type supports the usual set-theoretic operations and relations.
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@@ -557,9 +555,9 @@ and intersection.</p></div> <p>In the library, these identities are called <code class="docutils literal notranslate"><span class="pre">sUnion_eq_biUnion</span></code> and <code class="docutils literal notranslate"><span class="pre">sInter_eq_biInter</span></code>.</p> </section> <section id="functions"> <span id="id3"></span><h2><span class="section-number">4.2. </span>Functions<a class="headerlink" href="#functions" title="Permalink to this heading"></a></h2> </div> <div class="section" id="functions"> <span id="id3"></span><h2><span class="section-number">4.2. </span>Functions<a class="headerlink" href="#functions" title="Permalink to this headline"></a></h2> <p>If <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">:</span> <span class="pre">α</span> <span class="pre">→</span> <span class="pre">β</span></code> is a function and <code class="docutils literal notranslate"><span class="pre">p</span></code> is a set of elements of type <code class="docutils literal notranslate"><span class="pre">β</span></code>, the library defines <code class="docutils literal notranslate"><span class="pre">preimage</span> <span class="pre">f</span> <span class="pre">p</span></code>, written <code class="docutils literal notranslate"><span class="pre">f</span> <span class="pre">⁻¹'</span> <span class="pre">p</span></code>,
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@@ -884,9 +882,9 @@ and then fill in the two lines that are missing.</p><span class="n">contradiction</span> </pre></div> </div> </section> <section id="the-schroder-bernstein-theorem"> <span id="the-schroeder-bernstein-theorem"></span><h2><span class="section-number">4.3. </span>The Schröder-Bernstein Theorem<a class="headerlink" href="#the-schroder-bernstein-theorem" title="Permalink to this heading"></a></h2> </div> <div class="section" id="the-schroder-bernstein-theorem"> <span id="the-schroeder-bernstein-theorem"></span><h2><span class="section-number">4.3. </span>The Schröder-Bernstein Theorem<a class="headerlink" href="#the-schroder-bernstein-theorem" title="Permalink to this headline"></a></h2> <p>We close this chapter with an elementary but nontrivial theorem of set theory. Let <span class="math notranslate nohighlight">\(\alpha\)</span> and <span class="math notranslate nohighlight">\(\beta\)</span> be sets. (In our formalization, they will actually be types.)
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@@ -1145,8 +1143,8 @@ and the proof uses the fact that <code class="docutils literal notranslate"><spa<span class="o">⟨</span><span class="n">sbFun</span> <span class="n">f</span> <span class="n">g</span><span class="o">,</span> <span class="n">sb_injective</span> <span class="n">f</span> <span class="n">g</span> <span class="n">hf</span> <span class="n">hg</span><span class="o">,</span> <span class="n">sb_surjective</span> <span class="n">f</span> <span class="n">g</span> <span class="n">hf</span> <span class="n">hg</span><span class="o">⟩</span> </pre></div> </div> </section> </section> </div> </div> </div>
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@@ -86,15 +84,15 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="elementary-number-theory"> <span id="number-theory"></span><h1><span class="section-number">5. </span>Elementary Number Theory<a class="headerlink" href="#elementary-number-theory" title="Permalink to this heading"></a></h1> <div class="section" id="elementary-number-theory"> <span id="number-theory"></span><h1><span class="section-number">5. </span>Elementary Number Theory<a class="headerlink" href="#elementary-number-theory" title="Permalink to this headline"></a></h1> <p>In this chapter, we show you how to formalize some elementary results in number theory. As we deal with more substantive mathematical content, the proofs will get longer and more involved, building on the skills you have already mastered.</p> <section id="irrational-roots"> <span id="section-irrational-roots"></span><h2><span class="section-number">5.1. </span>Irrational Roots<a class="headerlink" href="#irrational-roots" title="Permalink to this heading"></a></h2> <div class="section" id="irrational-roots"> <span id="section-irrational-roots"></span><h2><span class="section-number">5.1. </span>Irrational Roots<a class="headerlink" href="#irrational-roots" title="Permalink to this headline"></a></h2> <p>Let’s start with a fact known to the ancient greeks, namely, that the square root of 2 is irrational. If we suppose otherwise,
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@@ -393,9 +391,9 @@ and that it takes values in the extended natural numbers <code class="docutils lwhich adds the value infinity to the natural numbers. In the next chapter, we will begin to develop the means to appreciate the way that Lean supports this sort of generality.</p> </section> <section id="induction-and-recursion"> <span id="section-induction-and-recursion"></span><h2><span class="section-number">5.2. </span>Induction and Recursion<a class="headerlink" href="#induction-and-recursion" title="Permalink to this heading"></a></h2> </div> <div class="section" id="induction-and-recursion"> <span id="section-induction-and-recursion"></span><h2><span class="section-number">5.2. </span>Induction and Recursion<a class="headerlink" href="#induction-and-recursion" title="Permalink to this headline"></a></h2> <p>The set of natural numbers <span class="math notranslate nohighlight">\(\mathbb{N} = \{ 0, 1, 2, \ldots \}\)</span> is not only fundamentally important in its own right, but also a plays a central role in the construction of new mathematical objects.
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@@ -698,9 +696,9 @@ The function <code class="docutils literal notranslate"><span class="pre">pred</<span class="kd">end</span> <span class="n">MyNat</span> </pre></div> </div> </section> <section id="infinitely-many-primes"> <span id="section-infinitely-many-primes"></span><h2><span class="section-number">5.3. </span>Infinitely Many Primes<a class="headerlink" href="#infinitely-many-primes" title="Permalink to this heading"></a></h2> </div> <div class="section" id="infinitely-many-primes"> <span id="section-infinitely-many-primes"></span><h2><span class="section-number">5.3. </span>Infinitely Many Primes<a class="headerlink" href="#infinitely-many-primes" title="Permalink to this headline"></a></h2> <p>Let us continue our exploration of induction and recursion with another mathematical standard: a proof that there are infinitely many primes. One way to formulate this is as the statement that
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@@ -1140,8 +1138,8 @@ along the way.</p></div> <p>If you managed to complete the proof, congratulations! This has been a serious feat of formalization.</p> </section> </section> </div> </div> </div>
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@@ -86,8 +84,8 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="structures"> <span id="id1"></span><h1><span class="section-number">6. </span>Structures<a class="headerlink" href="#structures" title="Permalink to this heading"></a></h1> <div class="section" id="structures"> <span id="id1"></span><h1><span class="section-number">6. </span>Structures<a class="headerlink" href="#structures" title="Permalink to this headline"></a></h1> <p>Modern mathematics makes essential use of algebraic structures, which encapsulate patterns that can be instantiated in
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@@ -106,8 +104,8 @@ It will also show you how to define and usealgebraic structures on your own.</p> <p>For more technical detail, you can consult <a class="reference external" href="https://leanprover.github.io/theorem_proving_in_lean/">Theorem Proving in Lean</a>, and a paper by Anne Baanen, <a class="reference external" href="https://arxiv.org/abs/2202.01629">Use and abuse of instance parameters in the Lean mathematical library</a>.</p> <section id="defining-structures"> <span id="section-structures"></span><h2><span class="section-number">6.1. </span>Defining structures<a class="headerlink" href="#defining-structures" title="Permalink to this heading"></a></h2> <div class="section" id="defining-structures"> <span id="section-structures"></span><h2><span class="section-number">6.1. </span>Defining structures<a class="headerlink" href="#defining-structures" title="Permalink to this headline"></a></h2> <p>In the broadest sense of the term, a <em>structure</em> is a specification of a collection of data, possibly with constraints that the data is required to satisfy.
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@@ -484,9 +482,9 @@ as long as we redefine the old accessors in terms of the new definition.Moreover, as we are about to see, Lean provides support for weaving structures together into a rich, interconnected hierarchy, and for managing the interactions between them.</p> </section> <section id="algebraic-structures"> <span id="section-algebraic-structures"></span><h2><span class="section-number">6.2. </span>Algebraic Structures<a class="headerlink" href="#algebraic-structures" title="Permalink to this heading"></a></h2> </div> <div class="section" id="algebraic-structures"> <span id="section-algebraic-structures"></span><h2><span class="section-number">6.2. </span>Algebraic Structures<a class="headerlink" href="#algebraic-structures" title="Permalink to this headline"></a></h2> <p>To clarify what we mean by the phrase <em>algebraic structure</em>, it will help to consider some examples.</p> <ol class="arabic simple">
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@@ -1033,9 +1031,9 @@ because it configures automation that invisibly governs the interpretation ofthe expressions we type. When used wisely, however, class inference is a powerful tool. It is what makes algebraic reasoning possible in Lean.</p> </section> <section id="building-the-gaussian-integers"> <span id="section-building-the-gaussian-integers"></span><h2><span class="section-number">6.3. </span>Building the Gaussian Integers<a class="headerlink" href="#building-the-gaussian-integers" title="Permalink to this heading"></a></h2> </div> <div class="section" id="building-the-gaussian-integers"> <span id="section-building-the-gaussian-integers"></span><h2><span class="section-number">6.3. </span>Building the Gaussian Integers<a class="headerlink" href="#building-the-gaussian-integers" title="Permalink to this headline"></a></h2> <p>We will now illustrate the use of the algebraic hierarchy in Lean by building an important mathematical object, the <em>Gaussian integers</em>, and showing that it is a Euclidean domain. In other words, according to
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@@ -1529,8 +1527,8 @@ the notions of being prime and being irreducible coincide.</p><span class="n">PrincipalIdealRing.irreducible_iff_prime</span> </pre></div> </div> </section> </section> </div> </div> </div>
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@@ -85,8 +83,8 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="hierarchies"> <span id="id1"></span><h1><span class="section-number">7. </span>Hierarchies<a class="headerlink" href="#hierarchies" title="Permalink to this heading"></a></h1> <div class="section" id="hierarchies"> <span id="id1"></span><h1><span class="section-number">7. </span>Hierarchies<a class="headerlink" href="#hierarchies" title="Permalink to this headline"></a></h1> <p>We have seen in <a class="reference internal" href="C06_Structures.html#structures"><span class="std std-numref">Chapter 6</span></a> how to define the class of groups and build instances of this class, and then how to build an instance of the commutative ring class. But of course there is a hierarchy here: a
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@@ -102,8 +100,8 @@ the following chapters and come back here for a second reading.</p>so we will used indices to distinguish our version. For instance we will have <code class="docutils literal notranslate"><span class="pre">Ring₁</span></code> as our version of <code class="docutils literal notranslate"><span class="pre">Ring</span></code>. Since we will gradually explain more powerful ways of formalizing structures, those indices will sometimes grow beyond one.</p> <section id="basics"> <span id="section-hierarchies-basics"></span><h2><span class="section-number">7.1. </span>Basics<a class="headerlink" href="#basics" title="Permalink to this heading"></a></h2> <div class="section" id="basics"> <span id="section-hierarchies-basics"></span><h2><span class="section-number">7.1. </span>Basics<a class="headerlink" href="#basics" title="Permalink to this headline"></a></h2> <p>At the very bottom of all hierarchies in Lean, we find data-carrying classes. The following class records that the given type <code class="docutils literal notranslate"><span class="pre">α</span></code> is endowed with a distinguished element called <code class="docutils literal notranslate"><span class="pre">one</span></code>. At this stage, it has no property at all.</p>
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@@ -628,9 +626,9 @@ to incorporate a type class <code class="docutils literal notranslate"><span clathat every preorder comes with a <code class="docutils literal notranslate"><span class="pre"><₁</span></code> which has a default value built from <code class="docutils literal notranslate"><span class="pre">≤₁</span></code> and a <code class="docutils literal notranslate"><span class="pre">Prop</span></code>-valued field asserting the natural relation between those two comparison operators. -/</p> </section> <section id="morphisms"> <span id="section-hierarchies-morphisms"></span><h2><span class="section-number">7.2. </span>Morphisms<a class="headerlink" href="#morphisms" title="Permalink to this heading"></a></h2> </div> <div class="section" id="morphisms"> <span id="section-hierarchies-morphisms"></span><h2><span class="section-number">7.2. </span>Morphisms<a class="headerlink" href="#morphisms" title="Permalink to this headline"></a></h2> <p>So far in this chapter, we discussed how to create a hierarchy of mathematical structures. But defining structures is not really completed until we have morphisms. There are two main approaches here. The most obvious one is to define a predicate on functions.</p>
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@@ -832,9 +830,9 @@ definitions below.</p><span class="o">:=</span> <span class="gr">sorry</span> </pre></div> </div> </section> <section id="sub-objects"> <span id="section-hierarchies-subobjects"></span><h2><span class="section-number">7.3. </span>Sub-objects<a class="headerlink" href="#sub-objects" title="Permalink to this heading"></a></h2> </div> <div class="section" id="sub-objects"> <span id="section-hierarchies-subobjects"></span><h2><span class="section-number">7.3. </span>Sub-objects<a class="headerlink" href="#sub-objects" title="Permalink to this headline"></a></h2> <p>After defining some algebraic structure and its morphisms, the next step is to consider sets that inherit this algebraic structure, for instance subgroups or subrings. This largely overlaps our previous topic. Indeed a set in <code class="docutils literal notranslate"><span class="pre">X</span></code> is implemented as a function from
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@@ -970,8 +968,8 @@ the <code class="docutils literal notranslate"><span class="pre">@</span></c<span class="gr">sorry</span> </pre></div> </div> </section> </section> </div> </div> </div>
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@@ -16,7 +15,6 @@<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 src="_static/doctools.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>
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@@ -98,8 +96,8 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <span class="target" id="topology"></span><section id="index-0"> <span id="id1"></span><h1><span class="section-number">8. </span>Topology<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> <span class="target" id="topology"></span><div class="section" id="index-0"> <span id="id1"></span><h1><span class="section-number">8. </span>Topology<a class="headerlink" href="#index-0" title="Permalink to this headline"></a></h1> <p>Calculus is based on the concept of a function, which is used to model quantities that depend on one another. For example, it is common to study quantities that change over time.
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@@ -166,8 +164,8 @@ and simply note that the rest can be proved “in the same way.”Formalizing mathematics requires making the relevant notion of “sameness” fully explicit, and that is exactly what Bourbaki’s theory of filters manages to do.</p> <section id="filters"> <span id="index-1"></span><span id="id2"></span><h2><span class="section-number">8.1. </span>Filters<a class="headerlink" href="#filters" title="Permalink to this heading"></a></h2> <div class="section" id="filters"> <span id="index-1"></span><span id="id2"></span><h2><span class="section-number">8.1. </span>Filters<a class="headerlink" href="#filters" title="Permalink to this headline"></a></h2> <p>A <em>filter</em> on a type <code class="docutils literal notranslate"><span class="pre">X</span></code> is a collection of sets of <code class="docutils literal notranslate"><span class="pre">X</span></code> that satisfies three conditions that we will spell out below. The notion supports two related ideas:</p>
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@@ -520,9 +518,9 @@ by definition, the assumption <code class="docutils literal notranslate"><span c<span class="gr">sorry</span> </pre></div> </div> </section> <section id="metric-spaces"> <span id="index-2"></span><span id="id3"></span><h2><span class="section-number">8.2. </span>Metric spaces<a class="headerlink" href="#metric-spaces" title="Permalink to this heading"></a></h2> </div> <div class="section" id="metric-spaces"> <span id="index-2"></span><span id="id3"></span><h2><span class="section-number">8.2. </span>Metric spaces<a class="headerlink" href="#metric-spaces" title="Permalink to this headline"></a></h2> <p>Examples in the previous section focus on sequences of real numbers. In this section we will go up a bit in generality and focus on metric spaces. A metric space is a type <code class="docutils literal notranslate"><span class="pre">X</span></code> equipped with a distance function <code class="docutils literal notranslate"><span class="pre">dist</span> <span class="pre">:</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">X</span> <span class="pre">→</span> <span class="pre">ℝ</span></code> which is a generalization of the function <code class="docutils literal notranslate"><span class="pre">fun</span> <span class="pre">x</span> <span class="pre">y</span> <span class="pre">↦</span> <span class="pre">|x</span> <span class="pre">-</span> <span class="pre">y|</span></code> from the case where <code class="docutils literal notranslate"><span class="pre">X</span> <span class="pre">=</span> <span class="pre">ℝ</span></code>.</p>
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@@ -540,8 +538,8 @@ the function <code class="docutils literal notranslate"><span class="pre">fun</sThey are called <code class="docutils literal notranslate"><span class="pre">EMetricSpace</span></code>, <code class="docutils literal notranslate"><span class="pre">PseudoMetricSpace</span></code> and <code class="docutils literal notranslate"><span class="pre">PseudoEMetricSpace</span></code> respectively (here “e” stands for “extended”).</p> <p>Note that our journey from <code class="docutils literal notranslate"><span class="pre">ℝ</span></code> to metric spaces jumped over the special case of normed spaces that also require linear algebra and will be explained as part of the calculus chapter.</p> <section id="convergence-and-continuity"> <h3><span class="section-number">8.2.1. </span>Convergence and continuity<a class="headerlink" href="#convergence-and-continuity" title="Permalink to this heading"></a></h3> <div class="section" id="convergence-and-continuity"> <h3><span class="section-number">8.2.1. </span>Convergence and continuity<a class="headerlink" href="#convergence-and-continuity" title="Permalink to this headline"></a></h3> <p>Using distance functions, we can already define convergent sequences and continuous functions between metric spaces. They are actually defined in a more general setting covered in the next section, but we have lemmas recasting the definition is terms of distances.</p>
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@@ -628,9 +626,9 @@ and get our final proof, now bordering obfuscation.</p><span class="n">Metric.continuousAt_iff</span> </pre></div> </div> </section> <section id="balls-open-sets-and-closed-sets"> <h3><span class="section-number">8.2.2. </span>Balls, open sets and closed sets<a class="headerlink" href="#balls-open-sets-and-closed-sets" title="Permalink to this heading"></a></h3> </div> <div class="section" id="balls-open-sets-and-closed-sets"> <h3><span class="section-number">8.2.2. </span>Balls, open sets and closed sets<a class="headerlink" href="#balls-open-sets-and-closed-sets" title="Permalink to this headline"></a></h3> <p>Once we have a distance function, the most important geometric definitions are (open) balls and closed balls.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kd">variable</span> <span class="o">(</span><span class="n">r</span> <span class="o">:</span> <span class="n">ℝ</span><span class="o">)</span>
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@@ -685,9 +683,9 @@ argument so we can invoke <code class="docutils literal notranslate"><span class<span class="n">Metric.nhds_basis_closedBall.mem_iff</span> </pre></div> </div> </section> <section id="compactness"> <h3><span class="section-number">8.2.3. </span>Compactness<a class="headerlink" href="#compactness" title="Permalink to this heading"></a></h3> </div> <div class="section" id="compactness"> <h3><span class="section-number">8.2.3. </span>Compactness<a class="headerlink" href="#compactness" title="Permalink to this headline"></a></h3> <p>Compactness is an important topological notion. It distinguishes subsets of a metric space that enjoy the same kind of properties as segments in reals compared to other intervals:</p> <ul class="simple">
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@@ -728,9 +726,9 @@ are deduced from more general versions, some of which will be discussed in later</pre></div> </div> <p>In a compact metric space any closed set is compact, this is <code class="docutils literal notranslate"><span class="pre">IsCompact.isClosed</span></code>.</p> </section> <section id="uniformly-continuous-functions"> <h3><span class="section-number">8.2.4. </span>Uniformly continuous functions<a class="headerlink" href="#uniformly-continuous-functions" title="Permalink to this heading"></a></h3> </div> <div class="section" id="uniformly-continuous-functions"> <h3><span class="section-number">8.2.4. </span>Uniformly continuous functions<a class="headerlink" href="#uniformly-continuous-functions" title="Permalink to this headline"></a></h3> <p>We now turn to uniformity notions on metric spaces : uniformly continuous functions, Cauchy sequences and completeness. Again those are defined in a more general context but we have lemmas in the metric name space to access their elementary definitions. We start with uniform continuity.</p>
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@@ -759,9 +757,9 @@ of the distance function on <code class="docutils literal notranslate"><span cla<span class="gr">sorry</span> </pre></div> </div> </section> <section id="completeness"> <h3><span class="section-number">8.2.5. </span>Completeness<a class="headerlink" href="#completeness" title="Permalink to this heading"></a></h3> </div> <div class="section" id="completeness"> <h3><span class="section-number">8.2.5. </span>Completeness<a class="headerlink" href="#completeness" title="Permalink to this headline"></a></h3> <p>A Cauchy sequence in a metric space is a sequence whose terms get closer and closer to each other. There are a couple of equivalent ways to state that idea. In particular converging sequences are Cauchy. The converse is true only in so-called <em>complete</em>
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@@ -850,12 +848,12 @@ define something inductively in the middle of a proof using <code class="docutil<span class="gr">sorry</span> </pre></div> </div> </section> </section> <section id="topological-spaces"> <span id="index-4"></span><span id="id4"></span><h2><span class="section-number">8.3. </span>Topological spaces<a class="headerlink" href="#topological-spaces" title="Permalink to this heading"></a></h2> <section id="fundamentals"> <h3><span class="section-number">8.3.1. </span>Fundamentals<a class="headerlink" href="#fundamentals" title="Permalink to this heading"></a></h3> </div> </div> <div class="section" id="topological-spaces"> <span id="index-4"></span><span id="id4"></span><h2><span class="section-number">8.3. </span>Topological spaces<a class="headerlink" href="#topological-spaces" title="Permalink to this headline"></a></h2> <div class="section" id="fundamentals"> <h3><span class="section-number">8.3.1. </span>Fundamentals<a class="headerlink" href="#fundamentals" title="Permalink to this headline"></a></h3> <p>We now go up in generality and introduce topological spaces. We will review the two main ways to define topological spaces and then explain how the category of topological spaces is much better behaved than the category of metric spaces. Note that we won’t be using mathlib category theory here, only having
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@@ -1042,9 +1040,9 @@ Let us explore that constraint “on paper” using notation <span class</div> <p>This ends our tour of how mathlib thinks that topological spaces fix defects of the theory of metric spaces by being a more functorial theory and having a complete lattice structure for any fixed type.</p> </section> <section id="separation-and-countability"> <h3><span class="section-number">8.3.2. </span>Separation and countability<a class="headerlink" href="#separation-and-countability" title="Permalink to this heading"></a></h3> </div> <div class="section" id="separation-and-countability"> <h3><span class="section-number">8.3.2. </span>Separation and countability<a class="headerlink" href="#separation-and-countability" title="Permalink to this headline"></a></h3> <p>We saw that the category of topological spaces have very nice properties. The price to pay for this is existence of rather pathological topological spaces. There are a number of assumptions you can make on a topological space to ensure its behavior
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@@ -1132,9 +1130,9 @@ of sets can be understood using sequences.</p><span class="n">mem_closure_iff_seq_limit</span> </pre></div> </div> </section> <section id="id5"> <h3><span class="section-number">8.3.3. </span>Compactness<a class="headerlink" href="#id5" title="Permalink to this heading"></a></h3> </div> <div class="section" id="id5"> <h3><span class="section-number">8.3.3. </span>Compactness<a class="headerlink" href="#id5" title="Permalink to this headline"></a></h3> <p>Let us now discuss how compactness is defined for topological spaces. As usual there are several ways to think about it and mathlib goes for the filter version.</p> <p>We first need to define cluster points of filters. Given a filter <code class="docutils literal notranslate"><span class="pre">F</span></code> on a topological space <code class="docutils literal notranslate"><span class="pre">X</span></code>,
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@@ -1195,9 +1193,9 @@ cover <code class="docutils literal notranslate"><span class="pre">s</span></cod<span class="n">isCompact_univ</span> </pre></div> </div> </section> </section> </section> </div> </div> </div> </div>
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@@ -91,8 +89,8 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <span class="target" id="differential-calculus"></span><section id="index-0"> <span id="id1"></span><h1><span class="section-number">9. </span>Differential Calculus<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> <span class="target" id="differential-calculus"></span><div class="section" id="index-0"> <span id="id1"></span><h1><span class="section-number">9. </span>Differential Calculus<a class="headerlink" href="#index-0" title="Permalink to this headline"></a></h1> <p>We now consider the formalization of notions from <em>analysis</em>, starting with differentiation in this chapter and turning integration and measure theory in the next.
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@@ -101,8 +99,8 @@ setting of functions from the real numbers to the real numbers,which is familiar from any introductory calculus class. In <a class="reference internal" href="#normed-spaces"><span class="std std-numref">Section 9.2</span></a>, we then consider the notion of a derivative in a much broader setting.</p> <section id="elementary-differential-calculus"> <span id="index-1"></span><span id="id2"></span><h2><span class="section-number">9.1. </span>Elementary Differential Calculus<a class="headerlink" href="#elementary-differential-calculus" title="Permalink to this heading"></a></h2> <div class="section" id="elementary-differential-calculus"> <span id="index-1"></span><span id="id2"></span><h2><span class="section-number">9.1. </span>Elementary Differential Calculus<a class="headerlink" href="#elementary-differential-calculus" title="Permalink to this headline"></a></h2> <p>Let <code class="docutils literal notranslate"><span class="pre">f</span></code> be a function from the reals to the reals. There is a difference between talking about the derivative of <code class="docutils literal notranslate"><span class="pre">f</span></code> at a single point and talking about the derivative function.
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@@ -175,11 +173,11 @@ seems even weirder.</p><span class="kd">example</span> <span class="o">:</span> <span class="n">deriv</span> <span class="n">sin</span> <span class="n">π</span> <span class="bp">=</span> <span class="bp">-</span><span class="mi">1</span> <span class="o">:=</span> <span class="kd">by</span> <span class="n">simp</span> </pre></div> </div> </section> <section id="differential-calculus-in-normed-spaces"> <span id="normed-spaces"></span><span id="index-2"></span><h2><span class="section-number">9.2. </span>Differential Calculus in Normed Spaces<a class="headerlink" href="#differential-calculus-in-normed-spaces" title="Permalink to this heading"></a></h2> <section id="id3"> <h3><span class="section-number">9.2.1. </span>Normed spaces<a class="headerlink" href="#id3" title="Permalink to this heading"></a></h3> </div> <div class="section" id="differential-calculus-in-normed-spaces"> <span id="normed-spaces"></span><span id="index-2"></span><h2><span class="section-number">9.2. </span>Differential Calculus in Normed Spaces<a class="headerlink" href="#differential-calculus-in-normed-spaces" title="Permalink to this headline"></a></h2> <div class="section" id="id3"> <h3><span class="section-number">9.2.1. </span>Normed spaces<a class="headerlink" href="#id3" title="Permalink to this headline"></a></h3> <p>Differentiation can be generalized beyond <code class="docutils literal notranslate"><span class="pre">ℝ</span></code> using the notion of a <em>normed vector space</em>, which encapsulates both direction and distance. We start with the notion of a <em>normed group</em>, which as an additive commutative
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@@ -242,9 +240,9 @@ complete as long as the field itself is complete.</p><span class="n">FiniteDimensional.complete</span> <span class="bp">𝕜</span> <span class="n">E</span> </pre></div> </div> </section> <section id="continuous-linear-maps"> <h3><span class="section-number">9.2.2. </span>Continuous linear maps<a class="headerlink" href="#continuous-linear-maps" title="Permalink to this heading"></a></h3> </div> <div class="section" id="continuous-linear-maps"> <h3><span class="section-number">9.2.2. </span>Continuous linear maps<a class="headerlink" href="#continuous-linear-maps" title="Permalink to this headline"></a></h3> <p>We now turn to the morphisms in the category of normed spaces, namely, continuous linear maps. In mathlib, the type of <code class="docutils literal notranslate"><span class="pre">𝕜</span></code>-linear continuous maps between normed spaces
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@@ -324,9 +322,9 @@ Minor ingredients include <code class="docutils literal notranslate"><span class<span class="gr">sorry</span> </pre></div> </div> </section> <section id="asymptotic-comparisons"> <h3><span class="section-number">9.2.3. </span>Asymptotic comparisons<a class="headerlink" href="#asymptotic-comparisons" title="Permalink to this heading"></a></h3> </div> <div class="section" id="asymptotic-comparisons"> <h3><span class="section-number">9.2.3. </span>Asymptotic comparisons<a class="headerlink" href="#asymptotic-comparisons" title="Permalink to this headline"></a></h3> <p>Defining differentiability also requires asymptotic comparisons. Mathlib has an extensive library covering the big O and little o relations, whose definitions are shown below.
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@@ -352,9 +350,9 @@ Here we will only use little o to define differentiability.</p><span class="n">Iff.rfl</span> </pre></div> </div> </section> <section id="differentiability"> <h3><span class="section-number">9.2.4. </span>Differentiability<a class="headerlink" href="#differentiability" title="Permalink to this heading"></a></h3> </div> <div class="section" id="differentiability"> <h3><span class="section-number">9.2.4. </span>Differentiability<a class="headerlink" href="#differentiability" title="Permalink to this headline"></a></h3> <p>We are now ready to discuss differentiable functions between normed spaces. In analogy the elementary one-dimensional, mathlib defines a predicate <code class="docutils literal notranslate"><span class="pre">HasFDerivAt</span></code> and a function <code class="docutils literal notranslate"><span class="pre">fderiv</span></code>.
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@@ -437,9 +435,9 @@ For example, you may want to use one-sided derivatives in theone-dimensional setting. The means to do so are found in mathlib in a more general context; see <code class="docutils literal notranslate"><span class="pre">HasFDerivWithinAt</span></code> or the even more general <code class="docutils literal notranslate"><span class="pre">HasFDerivAtFilter</span></code>.</p> </section> </section> </section> </div> </div> </div> </div>
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@@ -86,10 +84,10 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <span class="target" id="integration-and-measure-theory"></span><section id="index-0"> <span id="id1"></span><h1><span class="section-number">10. </span>Integration and Measure Theory<a class="headerlink" href="#index-0" title="Permalink to this heading"></a></h1> <section id="elementary-integration"> <span id="index-1"></span><span id="id2"></span><h2><span class="section-number">10.1. </span>Elementary Integration<a class="headerlink" href="#elementary-integration" title="Permalink to this heading"></a></h2> <span class="target" id="integration-and-measure-theory"></span><div class="section" id="index-0"> <span id="id1"></span><h1><span class="section-number">10. </span>Integration and Measure Theory<a class="headerlink" href="#index-0" title="Permalink to this headline"></a></h1> <div class="section" id="elementary-integration"> <span id="index-1"></span><span id="id2"></span><h2><span class="section-number">10.1. </span>Elementary Integration<a class="headerlink" href="#elementary-integration" title="Permalink to this headline"></a></h2> <p>We first focus on integration of functions on finite intervals in <code class="docutils literal notranslate"><span class="pre">ℝ</span></code>. We can integrate elementary functions.</p> <div class="highlight-lean notranslate"><div class="highlight"><pre><span></span><span class="kn">open</span> <span class="n">MeasureTheory</span> <span class="n">intervalIntegral</span>
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@@ -126,9 +124,9 @@ which are not shown here, are not equivalent.)</p><span class="n">rfl</span> </pre></div> </div> </section> <section id="measure-theory"> <span id="index-2"></span><span id="id3"></span><h2><span class="section-number">10.2. </span>Measure Theory<a class="headerlink" href="#measure-theory" title="Permalink to this heading"></a></h2> </div> <div class="section" id="measure-theory"> <span id="index-2"></span><span id="id3"></span><h2><span class="section-number">10.2. </span>Measure Theory<a class="headerlink" href="#measure-theory" title="Permalink to this headline"></a></h2> <p>The general context for integration in mathlib is measure theory. Even the elementary integrals of the previous section are in fact Bochner integrals. Bochner integration is a generalization of Lebesgue integration where the target space can be any Banach space,
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@@ -203,9 +201,9 @@ almost everywhere.</p><span class="n">Iff.rfl</span> </pre></div> </div> </section> <section id="integration"> <span id="id4"></span><h2><span class="section-number">10.3. </span>Integration<a class="headerlink" href="#integration" title="Permalink to this heading"></a></h2> </div> <div class="section" id="integration"> <span id="id4"></span><h2><span class="section-number">10.3. </span>Integration<a class="headerlink" href="#integration" title="Permalink to this headline"></a></h2> <p>Now that we have measurable spaces and measures we can consider integrals. As explained above, mathlib uses a very general notion of integration that allows any Banach space as the target.
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@@ -280,8 +278,8 @@ gives finite mass to compact sets, and give positive mass to open sets.</p><span class="n">integral_image_eq_integral_abs_det_fderiv_smul</span> <span class="n">μ</span> <span class="n">hs</span> <span class="n">hf</span> <span class="n">h_inj</span> <span class="n">g</span> </pre></div> </div> </section> </section> </div> </div> </div>
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@@ -1,134 +0,0 @@/* * _sphinx_javascript_frameworks_compat.js * ~~~~~~~~~~ * * Compatability shim for jQuery and underscores.js. * * WILL BE REMOVED IN Sphinx 6.0 * xref RemovedInSphinx60Warning * */ /** * select a different prefix for underscore */ $u = _.noConflict(); /** * small helper function to urldecode strings * * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/decodeURIComponent#Decoding_query_parameters_from_a_URL */ jQuery.urldecode = function(x) { if (!x) { return x } return decodeURIComponent(x.replace(/\+/g, ' ')); }; /** * small helper function to urlencode strings */ jQuery.urlencode = encodeURIComponent; /** * This function returns the parsed url parameters of the * current request. Multiple values per key are supported, * it will always return arrays of strings for the value parts. */ jQuery.getQueryParameters = function(s) { if (typeof s === 'undefined') s = document.location.search; var parts = s.substr(s.indexOf('?') + 1).split('&'); var result = {}; for (var i = 0; i < parts.length; i++) { var tmp = parts[i].split('=', 2); var key = jQuery.urldecode(tmp[0]); var value = jQuery.urldecode(tmp[1]); if (key in result) result[key].push(value); else result[key] = [value]; } return result; }; /** * highlight a given string on a jquery object by wrapping it in * span elements with the given class name. */ jQuery.fn.highlightText = function(text, className) { function highlight(node, addItems) { if (node.nodeType === 3) { var val = node.nodeValue; var pos = val.toLowerCase().indexOf(text); if (pos >= 0 && !jQuery(node.parentNode).hasClass(className) && !jQuery(node.parentNode).hasClass("nohighlight")) { var span; var isInSVG = jQuery(node).closest("body, svg, foreignObject").is("svg"); if (isInSVG) { span = document.createElementNS("http://www.w3.org/2000/svg", "tspan"); } else { span = document.createElement("span"); span.className = className; } span.appendChild(document.createTextNode(val.substr(pos, text.length))); node.parentNode.insertBefore(span, node.parentNode.insertBefore( document.createTextNode(val.substr(pos + text.length)), node.nextSibling)); node.nodeValue = val.substr(0, pos); if (isInSVG) { var rect = document.createElementNS("http://www.w3.org/2000/svg", "rect"); var bbox = node.parentElement.getBBox(); rect.x.baseVal.value = bbox.x; rect.y.baseVal.value = bbox.y; rect.width.baseVal.value = bbox.width; rect.height.baseVal.value = bbox.height; rect.setAttribute('class', className); addItems.push({ "parent": node.parentNode, "target": rect}); } } } else if (!jQuery(node).is("button, select, textarea")) { jQuery.each(node.childNodes, function() { highlight(this, addItems); }); } } var addItems = []; var result = this.each(function() { highlight(this, addItems); }); for (var i = 0; i < addItems.length; ++i) { jQuery(addItems[i].parent).before(addItems[i].target); } return result; }; /* * backward compatibility for jQuery.browser * This will be supported until firefox bug is fixed. */ if (!jQuery.browser) { jQuery.uaMatch = function(ua) { ua = ua.toLowerCase(); var match = /(chrome)[ \/]([\w.]+)/.exec(ua) || /(webkit)[ \/]([\w.]+)/.exec(ua) || /(opera)(?:.*version|)[ \/]([\w.]+)/.exec(ua) || /(msie) ([\w.]+)/.exec(ua) || ua.indexOf("compatible") < 0 && /(mozilla)(?:.*? rv:([\w.]+)|)/.exec(ua) || []; return { browser: match[ 1 ] || "", version: match[ 2 ] || "0" }; }; jQuery.browser = {}; jQuery.browser[jQuery.uaMatch(navigator.userAgent).browser] = true; }
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@@ -222,7 +222,7 @@ table.modindextable td {/* -- general body styles --------------------------------------------------- */ div.body { min-width: 360px; min-width: 450px; max-width: 800px; }
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@@ -335,13 +335,13 @@ p.sidebar-title {font-weight: bold; } div.admonition, div.topic, aside.topic, blockquote { div.admonition, div.topic, blockquote { clear: left; } /* -- topics ---------------------------------------------------------------- */ div.topic, aside.topic { div.topic { border: 1px solid #ccc; padding: 7px; margin: 10px 0 10px 0;
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@@ -380,7 +380,6 @@ div.body p.centered {div.sidebar > :last-child, aside.sidebar > :last-child, div.topic > :last-child, aside.topic > :last-child, div.admonition > :last-child { margin-bottom: 0; }
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@@ -388,7 +387,6 @@ div.admonition > :last-child {div.sidebar::after, aside.sidebar::after, div.topic::after, aside.topic::after, div.admonition::after, blockquote::after { display: block;
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@@ -430,6 +428,10 @@ table.docutils td, table.docutils th {border-bottom: 1px solid #aaa; } table.footnote td, table.footnote th { border: 0 !important; } th { text-align: left; padding-right: 5px;
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@@ -613,7 +615,6 @@ ul.simple p {margin-bottom: 0; } /* Docutils 0.17 and older (footnotes & citations) */ dl.footnote > dt, dl.citation > dt { float: left;
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@@ -631,33 +632,6 @@ dl.citation > dd:after {clear: both; } /* Docutils 0.18+ (footnotes & citations) */ aside.footnote > span, div.citation > span { float: left; } aside.footnote > span:last-of-type, div.citation > span:last-of-type { padding-right: 0.5em; } aside.footnote > p { margin-left: 2em; } div.citation > p { margin-left: 4em; } aside.footnote > p:last-of-type, div.citation > p:last-of-type { margin-bottom: 0em; } aside.footnote > p:last-of-type:after, div.citation > p:last-of-type:after { content: ""; clear: both; } /* Footnotes & citations ends */ dl.field-list { display: grid; grid-template-columns: fit-content(30%) auto;
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@@ -2,263 +2,325 @@* doctools.js * ~~~~~~~~~~~ * * Base JavaScript utilities for all Sphinx HTML documentation. * Sphinx JavaScript utilities for all documentation. * * :copyright: Copyright 2007-2022 by the Sphinx team, see AUTHORS. * :license: BSD, see LICENSE for details. * */ "use strict"; const _ready = (callback) => { if (document.readyState !== "loading") { callback(); } else { document.addEventListener("DOMContentLoaded", callback); /** * select a different prefix for underscore */ $u = _.noConflict(); /** * make the code below compatible with browsers without * an installed firebug like debugger if (!window.console || !console.firebug) { var names = ["log", "debug", "info", "warn", "error", "assert", "dir", "dirxml", "group", "groupEnd", "time", "timeEnd", "count", "trace", "profile", "profileEnd"]; window.console = {}; for (var i = 0; i < names.length; ++i) window.console[names[i]] = function() {}; } */ /** * small helper function to urldecode strings * * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/decodeURIComponent#Decoding_query_parameters_from_a_URL */ jQuery.urldecode = function(x) { if (!x) { return x } return decodeURIComponent(x.replace(/\+/g, ' ')); }; /** * highlight a given string on a node by wrapping it in * span elements with the given class name. * small helper function to urlencode strings */ 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; jQuery.urlencode = encodeURIComponent; 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); } /** * This function returns the parsed url parameters of the * current request. Multiple values per key are supported, * it will always return arrays of strings for the value parts. */ jQuery.getQueryParameters = function(s) { if (typeof s === 'undefined') s = document.location.search; var parts = s.substr(s.indexOf('?') + 1).split('&'); var result = {}; for (var i = 0; i < parts.length; i++) { var tmp = parts[i].split('=', 2); var key = jQuery.urldecode(tmp[0]); var value = jQuery.urldecode(tmp[1]); if (key in result) result[key].push(value); else result[key] = [value]; } return result; }; span.appendChild(document.createTextNode(val.substr(pos, text.length))); parent.insertBefore( span, parent.insertBefore( /** * highlight a given string on a jquery object by wrapping it in * span elements with the given class name. */ jQuery.fn.highlightText = function(text, className) { function highlight(node, addItems) { if (node.nodeType === 3) { var val = node.nodeValue; var pos = val.toLowerCase().indexOf(text); if (pos >= 0 && !jQuery(node.parentNode).hasClass(className) && !jQuery(node.parentNode).hasClass("nohighlight")) { var span; var isInSVG = jQuery(node).closest("body, svg, foreignObject").is("svg"); if (isInSVG) { span = document.createElementNS("http://www.w3.org/2000/svg", "tspan"); } else { span = document.createElement("span"); span.className = className; } span.appendChild(document.createTextNode(val.substr(pos, text.length))); node.parentNode.insertBefore(span, node.parentNode.insertBefore( document.createTextNode(val.substr(pos + text.length)), node.nextSibling ) ); node.nodeValue = val.substr(0, pos); if (isInSVG) { 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 }); node.nextSibling)); node.nodeValue = val.substr(0, pos); if (isInSVG) { var rect = document.createElementNS("http://www.w3.org/2000/svg", "rect"); var bbox = node.parentElement.getBBox(); rect.x.baseVal.value = bbox.x; rect.y.baseVal.value = bbox.y; rect.width.baseVal.value = bbox.width; rect.height.baseVal.value = bbox.height; rect.setAttribute('class', className); addItems.push({ "parent": node.parentNode, "target": rect}); } } } } else if (node.matches && !node.matches("button, select, textarea")) { node.childNodes.forEach((el) => _highlight(el, addItems, text, className)); else if (!jQuery(node).is("button, select, textarea")) { jQuery.each(node.childNodes, function() { highlight(this, addItems); }); } } }; const _highlightText = (thisNode, text, className) => { let addItems = []; _highlight(thisNode, addItems, text, className); addItems.forEach((obj) => obj.parent.insertAdjacentElement("beforebegin", obj.target) ); var addItems = []; var result = this.each(function() { highlight(this, addItems); }); for (var i = 0; i < addItems.length; ++i) { jQuery(addItems[i].parent).before(addItems[i].target); } return result; }; /* * backward compatibility for jQuery.browser * This will be supported until firefox bug is fixed. */ if (!jQuery.browser) { jQuery.uaMatch = function(ua) { ua = ua.toLowerCase(); var match = /(chrome)[ \/]([\w.]+)/.exec(ua) || /(webkit)[ \/]([\w.]+)/.exec(ua) || /(opera)(?:.*version|)[ \/]([\w.]+)/.exec(ua) || /(msie) ([\w.]+)/.exec(ua) || ua.indexOf("compatible") < 0 && /(mozilla)(?:.*? rv:([\w.]+)|)/.exec(ua) || []; return { browser: match[ 1 ] || "", version: match[ 2 ] || "0" }; }; jQuery.browser = {}; jQuery.browser[jQuery.uaMatch(navigator.userAgent).browser] = true; } /** * Small JavaScript module for the documentation. */ const Documentation = { init: () => { Documentation.highlightSearchWords(); Documentation.initDomainIndexTable(); Documentation.initOnKeyListeners(); var Documentation = { init : function() { this.fixFirefoxAnchorBug(); this.highlightSearchWords(); this.initIndexTable(); if (DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS) { this.initOnKeyListeners(); } }, /** * i18n support */ TRANSLATIONS: {}, PLURAL_EXPR: (n) => (n === 1 ? 0 : 1), LOCALE: "unknown", TRANSLATIONS : {}, PLURAL_EXPR : function(n) { return n === 1 ? 0 : 1; }, LOCALE : 'unknown', // gettext and ngettext don't access this so that the functions // can safely bound to a different name (_ = Documentation.gettext) gettext: (string) => { const translated = Documentation.TRANSLATIONS[string]; switch (typeof translated) { case "undefined": return string; // no translation case "string": return translated; // translation exists default: return translated[0]; // (singular, plural) translation tuple exists } gettext : function(string) { var translated = Documentation.TRANSLATIONS[string]; if (typeof translated === 'undefined') return string; return (typeof translated === 'string') ? translated : translated[0]; }, ngettext: (singular, plural, n) => { const translated = Documentation.TRANSLATIONS[singular]; if (typeof translated !== "undefined") return translated[Documentation.PLURAL_EXPR(n)]; return n === 1 ? singular : plural; ngettext : function(singular, plural, n) { var translated = Documentation.TRANSLATIONS[singular]; if (typeof translated === 'undefined') return (n == 1) ? singular : plural; return translated[Documentation.PLURALEXPR(n)]; }, addTranslations: (catalog) => { Object.assign(Documentation.TRANSLATIONS, catalog.messages); Documentation.PLURAL_EXPR = new Function( "n", `return (${catalog.plural_expr})` ); Documentation.LOCALE = catalog.locale; addTranslations : function(catalog) { for (var key in catalog.messages) this.TRANSLATIONS[key] = catalog.messages[key]; this.PLURAL_EXPR = new Function('n', 'return +(' + catalog.plural_expr + ')'); this.LOCALE = catalog.locale; }, /** * highlight the search words provided in the url in the text * add context elements like header anchor links */ 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>" ) ); addContextElements : function() { $('div[id] > :header:first').each(function() { $('<a class="headerlink">\u00B6</a>'). attr('href', '#' + this.id). attr('title', _('Permalink to this headline')). appendTo(this); }); $('dt[id]').each(function() { $('<a class="headerlink">\u00B6</a>'). attr('href', '#' + this.id). attr('title', _('Permalink to this definition')). appendTo(this); }); }, /** * helper function to hide the search marks again * workaround a firefox stupidity * see: https://bugzilla.mozilla.org/show_bug.cgi?id=645075 */ 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); fixFirefoxAnchorBug : function() { if (document.location.hash && $.browser.mozilla) window.setTimeout(function() { document.location.href += ''; }, 10); }, /** * helper function to focus on search bar * highlight the search words provided in the url in the text */ focusSearchBar: () => { document.querySelectorAll("input[name=q]")[0]?.focus(); highlightSearchWords : function() { var params = $.getQueryParameters(); var terms = (params.highlight) ? params.highlight[0].split(/\s+/) : []; if (terms.length) { var body = $('div.body'); if (!body.length) { body = $('body'); } window.setTimeout(function() { $.each(terms, function() { body.highlightText(this.toLowerCase(), 'highlighted'); }); }, 10); $('<p class="highlight-link"><a href="javascript:Documentation.' + 'hideSearchWords()">' + _('Hide Search Matches') + '</a></p>') .appendTo($('#searchbox')); } }, /** * Initialise the domain index toggle buttons * init the domain index toggle buttons */ initDomainIndexTable: () => { const toggler = (el) => { const idNumber = el.id.substr(7); const toggledRows = document.querySelectorAll(`tr.cg-${idNumber}`); if (el.src.substr(-9) === "minus.png") { el.src = `${el.src.substr(0, el.src.length - 9)}plus.png`; toggledRows.forEach((el) => (el.style.display = "none")); } else { el.src = `${el.src.substr(0, el.src.length - 8)}minus.png`; toggledRows.forEach((el) => (el.style.display = "")); } }; const togglerElements = document.querySelectorAll("img.toggler"); togglerElements.forEach((el) => el.addEventListener("click", (event) => toggler(event.currentTarget)) ); togglerElements.forEach((el) => (el.style.display = "")); if (DOCUMENTATION_OPTIONS.COLLAPSE_INDEX) togglerElements.forEach(toggler); initIndexTable : function() { var togglers = $('img.toggler').click(function() { var src = $(this).attr('src'); var idnum = $(this).attr('id').substr(7); $('tr.cg-' + idnum).toggle(); if (src.substr(-9) === 'minus.png') $(this).attr('src', src.substr(0, src.length-9) + 'plus.png'); else $(this).attr('src', src.substr(0, src.length-8) + 'minus.png'); }).css('display', ''); if (DOCUMENTATION_OPTIONS.COLLAPSE_INDEX) { togglers.click(); } }, initOnKeyListeners: () => { // only install a listener if it is really needed if ( !DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS && !DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS ) return; /** * helper function to hide the search marks again */ hideSearchWords : function() { $('#searchbox .highlight-link').fadeOut(300); $('span.highlighted').removeClass('highlighted'); var url = new URL(window.location); url.searchParams.delete('highlight'); window.history.replaceState({}, '', url); }, 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 /** * make the url absolute */ makeURL : function(relativeURL) { return DOCUMENTATION_OPTIONS.URL_ROOT + '/' + relativeURL; }, if (!event.shiftKey) { switch (event.key) { case "ArrowLeft": if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS) break; /** * get the current relative url */ getCurrentURL : function() { var path = document.location.pathname; var parts = path.split(/\//); $.each(DOCUMENTATION_OPTIONS.URL_ROOT.split(/\//), function() { if (this === '..') parts.pop(); }); var url = parts.join('/'); return path.substring(url.lastIndexOf('/') + 1, path.length - 1); }, const prevLink = document.querySelector('link[rel="prev"]'); if (prevLink && prevLink.href) { window.location.href = prevLink.href; event.preventDefault(); initOnKeyListeners: function() { $(document).keydown(function(event) { var activeElementType = document.activeElement.tagName; // don't navigate when in search box, textarea, dropdown or button if (activeElementType !== 'TEXTAREA' && activeElementType !== 'INPUT' && activeElementType !== 'SELECT' && activeElementType !== 'BUTTON' && !event.altKey && !event.ctrlKey && !event.metaKey && !event.shiftKey) { switch (event.keyCode) { case 37: // left var prevHref = $('link[rel="prev"]').prop('href'); if (prevHref) { window.location.href = prevHref; return false; } break; case "ArrowRight": if (!DOCUMENTATION_OPTIONS.NAVIGATION_WITH_KEYS) break; const nextLink = document.querySelector('link[rel="next"]'); if (nextLink && nextLink.href) { window.location.href = nextLink.href; event.preventDefault(); case 39: // right var nextHref = $('link[rel="next"]').prop('href'); if (nextHref) { window.location.href = nextHref; return false; } break; case "Escape": if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) break; Documentation.hideSearchWords(); event.preventDefault(); } } // some keyboard layouts may need Shift to get / switch (event.key) { case "/": if (!DOCUMENTATION_OPTIONS.ENABLE_SEARCH_SHORTCUTS) break; Documentation.focusSearchBar(); event.preventDefault(); } }); }, } }; // quick alias for translations const _ = Documentation.gettext; _ = Documentation.gettext; _ready(Documentation.init); $(document).ready(function() { Documentation.init(); });
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@@ -1,14 +1,12 @@var DOCUMENTATION_OPTIONS = { URL_ROOT: document.getElementById("documentation_options").getAttribute('data-url_root'), VERSION: '0.1', LANGUAGE: 'en', LANGUAGE: 'None', COLLAPSE_INDEX: false, BUILDER: 'html', FILE_SUFFIX: '.html', LINK_SUFFIX: '.html', HAS_SOURCE: true, SOURCELINK_SUFFIX: '.txt', NAVIGATION_WITH_KEYS: false, SHOW_SEARCH_SUMMARY: true, ENABLE_SEARCH_SHORTCUTS: false, NAVIGATION_WITH_KEYS: false };
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@@ -1,15 +1,15 @@/*! * jQuery JavaScript Library v3.6.0 * jQuery JavaScript Library v3.5.1 * https://jquery.com/ * * Includes Sizzle.js * https://sizzlejs.com/ * * Copyright OpenJS Foundation and other contributors * Copyright JS Foundation and other contributors * Released under the MIT license * https://jquery.org/license * * Date: 2021-03-02T17:08Z * Date: 2020-05-04T22:49Z */ ( function( global, factory ) {
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@@ -76,16 +76,12 @@ 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"; }; // Support: Chrome <=57, Firefox <=52 // In some browsers, typeof returns "function" for HTML <object> elements // (i.e., `typeof document.createElement( "object" ) === "function"`). // We don't want to classify *any* DOM node as a function. return typeof obj === "function" && typeof obj.nodeType !== "number"; }; var isWindow = function isWindow( obj ) {
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@@ -151,7 +147,7 @@ function toType( obj ) {var version = "3.6.0", version = "3.5.1", // Define a local copy of jQuery jQuery = function( selector, context ) {
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@@ -405,7 +401,7 @@ jQuery.extend( {if ( isArrayLike( Object( arr ) ) ) { jQuery.merge( ret, typeof arr === "string" ? [ arr ] : arr [ arr ] : arr ); } else { push.call( ret, arr );
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@@ -500,9 +496,9 @@ if ( typeof Symbol === "function" ) {// 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( _i, name ) { class2type[ "[object " + name + "]" ] = name.toLowerCase(); } ); function isArrayLike( obj ) {
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@@ -522,14 +518,14 @@ function isArrayLike( obj ) {} var Sizzle = /*! * Sizzle CSS Selector Engine v2.3.6 * Sizzle CSS Selector Engine v2.3.5 * https://sizzlejs.com/ * * Copyright JS Foundation and other contributors * Released under the MIT license * https://js.foundation/ * * Date: 2021-02-16 * Date: 2020-03-14 */ ( function( window ) { var i,
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@@ -1112,8 +1108,8 @@ support = Sizzle.support = {};* @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; var namespace = elem.namespaceURI, docElem = ( elem.ownerDocument || elem ).documentElement; // Support: IE <=8 // Assume HTML when documentElement doesn't yet exist, such as inside loading iframes
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@@ -3028,9 +3024,9 @@ var rneedsContext = jQuery.expr.match.needsContext;function nodeName( elem, name ) { return elem.nodeName && elem.nodeName.toLowerCase() === name.toLowerCase(); 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 );
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@@ -4001,8 +3997,8 @@ jQuery.extend( {resolveContexts = Array( i ), resolveValues = slice.call( arguments ), // the primary Deferred primary = jQuery.Deferred(), // the master Deferred master = jQuery.Deferred(), // subordinate callback factory updateFunc = function( i ) {
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@@ -4010,30 +4006,30 @@ jQuery.extend( {resolveContexts[ i ] = this; resolveValues[ i ] = arguments.length > 1 ? slice.call( arguments ) : value; if ( !( --remaining ) ) { primary.resolveWith( resolveContexts, resolveValues ); master.resolveWith( resolveContexts, resolveValues ); } }; }; // Single- and empty arguments are adopted like Promise.resolve if ( remaining <= 1 ) { adoptValue( singleValue, primary.done( updateFunc( i ) ).resolve, primary.reject, adoptValue( singleValue, master.done( updateFunc( i ) ).resolve, master.reject, !remaining ); // Use .then() to unwrap secondary thenables (cf. gh-3000) if ( primary.state() === "pending" || if ( master.state() === "pending" || isFunction( resolveValues[ i ] && resolveValues[ i ].then ) ) { return primary.then(); return master.then(); } } // Multiple arguments are aggregated like Promise.all array elements while ( i-- ) { adoptValue( resolveValues[ i ], updateFunc( i ), primary.reject ); adoptValue( resolveValues[ i ], updateFunc( i ), master.reject ); } return primary.promise(); return master.promise(); } } );
-
@@ -4184,8 +4180,8 @@ var access = function( elems, fn, key, value, chainable, emptyGet, raw ) {for ( ; i < len; i++ ) { fn( elems[ i ], key, raw ? value : value.call( elems[ i ], i, fn( elems[ i ], key ) ) value : value.call( elems[ i ], i, fn( elems[ i ], key ) ) ); } }
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@@ -5093,7 +5089,10 @@ function buildFragment( elems, context, scripts, selection, ignored ) {} var rtypenamespace = /^([^.]*)(?:\.(.+)|)/; var rkeyEvent = /^key/, rmouseEvent = /^(?:mouse|pointer|contextmenu|drag|drop)|click/, rtypenamespace = /^([^.]*)(?:\.(.+)|)/; function returnTrue() { return true;
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@@ -5388,8 +5387,8 @@ jQuery.event = {event = jQuery.event.fix( nativeEvent ), handlers = ( dataPriv.get( this, "events" ) || Object.create( null ) )[ event.type ] || [], 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
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@@ -5513,12 +5512,12 @@ jQuery.event = {get: isFunction( hook ) ? function() { if ( this.originalEvent ) { return hook( this.originalEvent ); return hook( this.originalEvent ); } } : function() { if ( this.originalEvent ) { return this.originalEvent[ name ]; return this.originalEvent[ name ]; } },
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@@ -5657,13 +5656,7 @@ function leverageNative( el, type, expectSync ) {// 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; return result.value; } // If this is an inner synthetic event for an event with a bubbling surrogate
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@@ -5828,7 +5821,34 @@ jQuery.each( {targetTouches: true, toElement: true, touches: true, which: true which: function( event ) { var button = event.button; // Add which for key events if ( event.which == null && rkeyEvent.test( event.type ) ) { return event.charCode != null ? event.charCode : event.keyCode; } // Add which for click: 1 === left; 2 === middle; 3 === right if ( !event.which && button !== undefined && rmouseEvent.test( event.type ) ) { if ( button & 1 ) { return 1; } if ( button & 2 ) { return 3; } if ( button & 4 ) { return 2; } return 0; } return event.which; } }, jQuery.event.addProp ); jQuery.each( { focus: "focusin", blur: "focusout" }, function( type, delegateType ) {
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@@ -5854,12 +5874,6 @@ jQuery.each( { focus: "focusin", blur: "focusout" }, function( type, delegateTypreturn true; }, // Suppress native focus or blur as it's already being fired // in leverageNative. _default: function() { return true; }, delegateType: delegateType }; } );
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@@ -6527,10 +6541,6 @@ var rboxStyle = new RegExp( cssExpand.join( "|" ), "i" );// 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 ) {
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@@ -6538,32 +6548,17 @@ var rboxStyle = new RegExp( cssExpand.join( "|" ), "i" );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. table.style.cssText = "position:absolute;left:-11111px"; 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; reliableTrDimensionsVal = parseInt( trStyle.height ) > 3; documentElement.removeChild( table ); }
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@@ -7027,10 +7022,10 @@ jQuery.each( [ "height", "width" ], function( _i, dimension ) {// 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 ); swap( elem, cssShow, function() { return getWidthOrHeight( elem, dimension, extra ); } ) : getWidthOrHeight( elem, dimension, extra ); } },
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@@ -7089,7 +7084,7 @@ jQuery.cssHooks.marginLeft = addGetHookIf( support.reliableMarginLeft,swap( elem, { marginLeft: 0 }, function() { return elem.getBoundingClientRect().left; } ) ) + "px"; ) + "px"; } } );
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@@ -7228,7 +7223,7 @@ Tween.propHooks = {if ( jQuery.fx.step[ tween.prop ] ) { jQuery.fx.step[ tween.prop ]( tween ); } else if ( tween.elem.nodeType === 1 && ( jQuery.cssHooks[ tween.prop ] || jQuery.cssHooks[ tween.prop ] || tween.elem.style[ finalPropName( tween.prop ) ] != null ) ) { jQuery.style( tween.elem, tween.prop, tween.now + tween.unit ); } else {
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@@ -7473,7 +7468,7 @@ function defaultPrefilter( elem, props, opts ) {anim.done( function() { /* eslint-enable no-loop-func */ /* eslint-enable no-loop-func */ // The final step of a "hide" animation is actually hiding the element if ( !hidden ) {
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@@ -7593,7 +7588,7 @@ function Animation( elem, properties, options ) {tweens: [], createTween: function( prop, end ) { var tween = jQuery.Tween( elem, animation.opts, prop, end, animation.opts.specialEasing[ prop ] || animation.opts.easing ); animation.opts.specialEasing[ prop ] || animation.opts.easing ); animation.tweens.push( tween ); return tween; },
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@@ -7766,8 +7761,7 @@ jQuery.fn.extend( {anim.stop( true ); } }; doAnimation.finish = doAnimation; doAnimation.finish = doAnimation; return empty || optall.queue === false ? this.each( doAnimation ) :
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@@ -8407,8 +8401,8 @@ jQuery.fn.extend( {if ( this.setAttribute ) { this.setAttribute( "class", className || value === false ? "" : dataPriv.get( this, "__className__" ) || "" "" : dataPriv.get( this, "__className__" ) || "" ); } }
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@@ -8423,7 +8417,7 @@ jQuery.fn.extend( {while ( ( elem = this[ i++ ] ) ) { if ( elem.nodeType === 1 && ( " " + stripAndCollapse( getClass( elem ) ) + " " ).indexOf( className ) > -1 ) { return true; return true; } }
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@@ -8713,7 +8707,9 @@ jQuery.extend( jQuery.event, {special.bindType || type; // jQuery handler handle = ( dataPriv.get( cur, "events" ) || Object.create( null ) )[ event.type ] && handle = ( dataPriv.get( cur, "events" ) || Object.create( null ) )[ event.type ] && dataPriv.get( cur, "handle" ); if ( handle ) { handle.apply( cur, data );
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@@ -8860,7 +8856,7 @@ var rquery = ( /\?/ );// Cross-browser xml parsing jQuery.parseXML = function( data ) { var xml, parserErrorElem; var xml; if ( !data || typeof data !== "string" ) { return null; }
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@@ -8869,17 +8865,12 @@ jQuery.parseXML = function( data ) {// IE throws on parseFromString with invalid input. try { xml = ( new window.DOMParser() ).parseFromString( data, "text/xml" ); } catch ( e ) {} } catch ( e ) { xml = undefined; } 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 ) ); if ( !xml || xml.getElementsByTagName( "parsererror" ).length ) { jQuery.error( "Invalid XML: " + data ); } return xml; };
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@@ -8980,14 +8971,16 @@ jQuery.fn.extend( {// 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() { } ) .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 ) { } ) .map( function( _i, elem ) { var val = jQuery( this ).val(); if ( val == null ) {
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@@ -9040,8 +9033,7 @@ var// Anchor tag for parsing the document origin originAnchor = document.createElement( "a" ); originAnchor.href = location.href; originAnchor.href = location.href; // Base "constructor" for jQuery.ajaxPrefilter and jQuery.ajaxTransport function addToPrefiltersOrTransports( structure ) {
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@@ -9422,8 +9414,8 @@ jQuery.extend( {// 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, jQuery( callbackContext ) : jQuery.event, // Deferreds deferred = jQuery.Deferred(),
-
@@ -9735,10 +9727,8 @@ jQuery.extend( {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 ) { // Use a noop converter for missing script if ( !isSuccess && jQuery.inArray( "script", s.dataTypes ) > -1 ) { s.converters[ "text script" ] = function() {}; }
-
@@ -10476,6 +10466,12 @@ jQuery.offset = {options.using.call( elem, props ); } else { if ( typeof props.top === "number" ) { props.top += "px"; } if ( typeof props.left === "number" ) { props.left += "px"; } curElem.css( props ); } }
-
@@ -10644,11 +10640,8 @@ jQuery.each( [ "top", "left" ], function( _i, prop ) {// 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 ) { jQuery.each( { padding: "inner" + name, content: type, "": "outer" + name }, function( defaultExtra, funcName ) { // Margin is only for outerHeight, outerWidth jQuery.fn[ funcName ] = function( margin, value ) {
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@@ -10733,8 +10726,7 @@ jQuery.fn.extend( {} } ); jQuery.each( ( "blur focus focusin focusout resize scroll click dblclick " + 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 ) {
-
@@ -10745,8 +10737,7 @@ jQuery.each(this.on( name, null, data, fn ) : this.trigger( name ); }; } ); } );
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-
-
-
@@ -10,7 +10,7 @@* */ var stopwords = ["a", "and", "are", "as", "at", "be", "but", "by", "for", "if", "in", "into", "is", "it", "near", "no", "not", "of", "on", "or", "such", "that", "the", "their", "then", "there", "these", "they", "this", "to", "was", "will", "with"]; var stopwords = ["a","and","are","as","at","be","but","by","for","if","in","into","is","it","near","no","not","of","on","or","such","that","the","their","then","there","these","they","this","to","was","will","with"]; /* Non-minified version is copied as a separate JS file, is available */
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@@ -197,3 +197,101 @@ var Stemmer = function() {} } var splitChars = (function() { var result = {}; var singles = [96, 180, 187, 191, 215, 247, 749, 885, 903, 907, 909, 930, 1014, 1648, 1748, 1809, 2416, 2473, 2481, 2526, 2601, 2609, 2612, 2615, 2653, 2702, 2706, 2729, 2737, 2740, 2857, 2865, 2868, 2910, 2928, 2948, 2961, 2971, 2973, 3085, 3089, 3113, 3124, 3213, 3217, 3241, 3252, 3295, 3341, 3345, 3369, 3506, 3516, 3633, 3715, 3721, 3736, 3744, 3748, 3750, 3756, 3761, 3781, 3912, 4239, 4347, 4681, 4695, 4697, 4745, 4785, 4799, 4801, 4823, 4881, 5760, 5901, 5997, 6313, 7405, 8024, 8026, 8028, 8030, 8117, 8125, 8133, 8181, 8468, 8485, 8487, 8489, 8494, 8527, 11311, 11359, 11687, 11695, 11703, 11711, 11719, 11727, 11735, 12448, 12539, 43010, 43014, 43019, 43587, 43696, 43713, 64286, 64297, 64311, 64317, 64319, 64322, 64325, 65141]; var i, j, start, end; for (i = 0; i < singles.length; i++) { result[singles[i]] = true; } var ranges = [[0, 47], [58, 64], [91, 94], [123, 169], [171, 177], [182, 184], [706, 709], [722, 735], [741, 747], [751, 879], [888, 889], [894, 901], [1154, 1161], [1318, 1328], [1367, 1368], [1370, 1376], [1416, 1487], [1515, 1519], [1523, 1568], [1611, 1631], [1642, 1645], [1750, 1764], [1767, 1773], [1789, 1790], [1792, 1807], [1840, 1868], [1958, 1968], [1970, 1983], [2027, 2035], [2038, 2041], [2043, 2047], [2070, 2073], [2075, 2083], [2085, 2087], [2089, 2307], [2362, 2364], [2366, 2383], [2385, 2391], [2402, 2405], [2419, 2424], [2432, 2436], [2445, 2446], [2449, 2450], [2483, 2485], [2490, 2492], [2494, 2509], [2511, 2523], [2530, 2533], [2546, 2547], [2554, 2564], [2571, 2574], [2577, 2578], [2618, 2648], [2655, 2661], [2672, 2673], [2677, 2692], [2746, 2748], [2750, 2767], [2769, 2783], [2786, 2789], [2800, 2820], [2829, 2830], [2833, 2834], [2874, 2876], [2878, 2907], [2914, 2917], [2930, 2946], [2955, 2957], [2966, 2968], [2976, 2978], [2981, 2983], [2987, 2989], [3002, 3023], [3025, 3045], [3059, 3076], [3130, 3132], [3134, 3159], [3162, 3167], [3170, 3173], [3184, 3191], [3199, 3204], [3258, 3260], [3262, 3293], [3298, 3301], [3312, 3332], [3386, 3388], [3390, 3423], [3426, 3429], [3446, 3449], [3456, 3460], [3479, 3481], [3518, 3519], [3527, 3584], [3636, 3647], [3655, 3663], [3674, 3712], [3717, 3718], [3723, 3724], [3726, 3731], [3752, 3753], [3764, 3772], [3774, 3775], [3783, 3791], [3802, 3803], [3806, 3839], [3841, 3871], [3892, 3903], [3949, 3975], [3980, 4095], [4139, 4158], [4170, 4175], [4182, 4185], [4190, 4192], [4194, 4196], [4199, 4205], [4209, 4212], [4226, 4237], [4250, 4255], [4294, 4303], [4349, 4351], [4686, 4687], [4702, 4703], [4750, 4751], [4790, 4791], [4806, 4807], [4886, 4887], [4955, 4968], [4989, 4991], [5008, 5023], [5109, 5120], [5741, 5742], [5787, 5791], [5867, 5869], [5873, 5887], [5906, 5919], [5938, 5951], [5970, 5983], [6001, 6015], [6068, 6102], [6104, 6107], [6109, 6111], [6122, 6127], [6138, 6159], [6170, 6175], [6264, 6271], [6315, 6319], [6390, 6399], [6429, 6469], [6510, 6511], [6517, 6527], [6572, 6592], [6600, 6607], [6619, 6655], [6679, 6687], [6741, 6783], [6794, 6799], [6810, 6822], [6824, 6916], [6964, 6980], [6988, 6991], [7002, 7042], [7073, 7085], [7098, 7167], [7204, 7231], [7242, 7244], [7294, 7400], [7410, 7423], [7616, 7679], [7958, 7959], [7966, 7967], [8006, 8007], [8014, 8015], [8062, 8063], [8127, 8129], [8141, 8143], [8148, 8149], [8156, 8159], [8173, 8177], [8189, 8303], [8306, 8307], [8314, 8318], [8330, 8335], [8341, 8449], [8451, 8454], [8456, 8457], [8470, 8472], [8478, 8483], [8506, 8507], [8512, 8516], [8522, 8525], [8586, 9311], [9372, 9449], [9472, 10101], [10132, 11263], [11493, 11498], [11503, 11516], [11518, 11519], [11558, 11567], [11622, 11630], [11632, 11647], [11671, 11679], [11743, 11822], [11824, 12292], [12296, 12320], [12330, 12336], [12342, 12343], [12349, 12352], [12439, 12444], [12544, 12548], [12590, 12592], [12687, 12689], [12694, 12703], [12728, 12783], [12800, 12831], [12842, 12880], [12896, 12927], [12938, 12976], [12992, 13311], [19894, 19967], [40908, 40959], [42125, 42191], [42238, 42239], [42509, 42511], [42540, 42559], [42592, 42593], [42607, 42622], [42648, 42655], [42736, 42774], [42784, 42785], [42889, 42890], [42893, 43002], [43043, 43055], [43062, 43071], [43124, 43137], [43188, 43215], [43226, 43249], [43256, 43258], [43260, 43263], [43302, 43311], [43335, 43359], [43389, 43395], [43443, 43470], [43482, 43519], [43561, 43583], [43596, 43599], [43610, 43615], [43639, 43641], [43643, 43647], [43698, 43700], [43703, 43704], [43710, 43711], [43715, 43738], [43742, 43967], [44003, 44015], [44026, 44031], [55204, 55215], [55239, 55242], [55292, 55295], [57344, 63743], [64046, 64047], [64110, 64111], [64218, 64255], [64263, 64274], [64280, 64284], [64434, 64466], [64830, 64847], [64912, 64913], [64968, 65007], [65020, 65135], [65277, 65295], [65306, 65312], [65339, 65344], [65371, 65381], [65471, 65473], [65480, 65481], [65488, 65489], [65496, 65497]]; for (i = 0; i < ranges.length; i++) { start = ranges[i][0]; end = ranges[i][1]; for (j = start; j <= end; j++) { result[j] = true; } } return result; })(); function splitQuery(query) { var result = []; var start = -1; for (var i = 0; i < query.length; i++) { if (splitChars[query.charCodeAt(i)]) { if (start !== -1) { result.push(query.slice(start, i)); start = -1; } } else if (start === -1) { start = i; } } if (start !== -1) { result.push(query.slice(start)); } return result; }
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-
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@@ -8,20 +8,18 @@* :license: BSD, see LICENSE for details. * */ "use strict"; /** * Simple result scoring code. */ if (typeof Scorer === "undefined") { if (!Scorer) { /** * Simple result scoring code. */ var Scorer = { // Implement the following function to further tweak the score for each result // The function takes a result array [docname, title, anchor, descr, score, filename] // The function takes a result array [filename, title, anchor, descr, score] // and returns the new score. /* score: result => { const [docname, title, anchor, descr, score, filename] = result return score score: function(result) { return result[4]; }, */
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@@ -30,11 +28,9 @@ if (typeof Scorer === "undefined") {// or matches in the last dotted part of the object name objPartialMatch: 6, // Additive scores depending on the priority of the object objPrio: { 0: 15, // used to be importantResults 1: 5, // used to be objectResults 2: -5, // used to be unimportantResults }, objPrio: {0: 15, // used to be importantResults 1: 5, // used to be objectResults 2: -5}, // used to be unimportantResults // Used when the priority is not in the mapping. objPrioDefault: 0,
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@@ -43,455 +39,456 @@ if (typeof Scorer === "undefined") {partialTitle: 7, // query found in terms term: 5, partialTerm: 2, partialTerm: 2 }; } const _removeChildren = (element) => { while (element && element.lastChild) element.removeChild(element.lastChild); }; /** * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping */ const _escapeRegExp = (string) => string.replace(/[.*+\-?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string const _displayItem = (item, highlightTerms, searchTerms) => { const docBuilder = DOCUMENTATION_OPTIONS.BUILDER; const docUrlRoot = DOCUMENTATION_OPTIONS.URL_ROOT; const docFileSuffix = DOCUMENTATION_OPTIONS.FILE_SUFFIX; const docLinkSuffix = DOCUMENTATION_OPTIONS.LINK_SUFFIX; const showSearchSummary = DOCUMENTATION_OPTIONS.SHOW_SEARCH_SUMMARY; const [docName, title, anchor, descr] = item; let listItem = document.createElement("li"); let requestUrl; let linkUrl; if (docBuilder === "dirhtml") { // dirhtml builder let dirname = docName + "/"; if (dirname.match(/\/index\/$/)) dirname = dirname.substring(0, dirname.length - 6); else if (dirname === "index/") dirname = ""; requestUrl = docUrlRoot + dirname; linkUrl = requestUrl; } else { // normal html builders requestUrl = docUrlRoot + docName + docFileSuffix; linkUrl = docName + docLinkSuffix; } const params = new URLSearchParams(); params.set("highlight", [...highlightTerms].join(" ")); let linkEl = listItem.appendChild(document.createElement("a")); linkEl.href = linkUrl + "?" + params.toString() + anchor; linkEl.innerHTML = title; if (descr) listItem.appendChild(document.createElement("span")).innerText = " (" + descr + ")"; else if (showSearchSummary) fetch(requestUrl) .then((responseData) => responseData.text()) .then((data) => { if (data) listItem.appendChild( Search.makeSearchSummary(data, searchTerms, highlightTerms) ); }); Search.output.appendChild(listItem); }; const _finishSearch = (resultCount) => { Search.stopPulse(); Search.title.innerText = _("Search Results"); if (!resultCount) Search.status.innerText = Documentation.gettext( "Your search did not match any documents. Please make sure that all words are spelled correctly and that you've selected enough categories." ); else Search.status.innerText = _( `Search finished, found ${resultCount} page(s) matching the search query.` ); }; const _displayNextItem = ( results, resultCount, 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); setTimeout( () => _displayNextItem(results, resultCount, highlightTerms, searchTerms), 5 ); if (!splitQuery) { function splitQuery(query) { return query.split(/\s+/); } // search finished, update title and status message else _finishSearch(resultCount); }; /** * Default splitQuery function. Can be overridden in ``sphinx.search`` with a * custom function per language. * * The regular expression works by splitting the string on consecutive characters * that are not Unicode letters, numbers, underscores, or emoji characters. * This is the same as ``\W+`` in Python, preserving the surrogate pair area. */ if (typeof splitQuery === "undefined") { var splitQuery = (query) => query .split(/[^\p{Letter}\p{Number}_\p{Emoji_Presentation}]+/gu) .filter(term => term) // remove remaining empty strings } /** * Search Module */ const Search = { _index: null, _queued_query: null, _pulse_status: -1, htmlToText: (htmlString) => { const htmlElement = document .createRange() .createContextualFragment(htmlString); _removeChildren(htmlElement.querySelectorAll(".headerlink")); const docContent = htmlElement.querySelector('[role="main"]'); if (docContent !== undefined) return docContent.textContent; console.warn( "Content block not found. Sphinx search tries to obtain it via '[role=main]'. Could you check your theme or template." ); return ""; var Search = { _index : null, _queued_query : null, _pulse_status : -1, htmlToText : function(htmlString) { var virtualDocument = document.implementation.createHTMLDocument('virtual'); var htmlElement = $(htmlString, virtualDocument); htmlElement.find('.headerlink').remove(); docContent = htmlElement.find('[role=main]')[0]; if(docContent === undefined) { console.warn("Content block not found. Sphinx search tries to obtain it " + "via '[role=main]'. Could you check your theme or template."); return ""; } return docContent.textContent || docContent.innerText; }, init: () => { const query = new URLSearchParams(window.location.search).get("q"); document .querySelectorAll('input[name="q"]') .forEach((el) => (el.value = query)); if (query) Search.performSearch(query); init : function() { var params = $.getQueryParameters(); if (params.q) { var query = params.q[0]; $('input[name="q"]')[0].value = query; this.performSearch(query); } }, loadIndex: (url) => (document.body.appendChild(document.createElement("script")).src = url), loadIndex : function(url) { $.ajax({type: "GET", url: url, data: null, dataType: "script", cache: true, complete: function(jqxhr, textstatus) { if (textstatus != "success") { document.getElementById("searchindexloader").src = url; } }}); }, setIndex: (index) => { Search._index = index; if (Search._queued_query !== null) { const query = Search._queued_query; Search._queued_query = null; Search.query(query); setIndex : function(index) { var q; this._index = index; if ((q = this._queued_query) !== null) { this._queued_query = null; Search.query(q); } }, hasIndex: () => Search._index !== null, deferQuery: (query) => (Search._queued_query = query), hasIndex : function() { return this._index !== null; }, stopPulse: () => (Search._pulse_status = -1), deferQuery : function(query) { this._queued_query = query; }, startPulse: () => { if (Search._pulse_status >= 0) return; stopPulse : function() { this._pulse_status = 0; }, const pulse = () => { startPulse : function() { if (this._pulse_status >= 0) return; function pulse() { var i; Search._pulse_status = (Search._pulse_status + 1) % 4; Search.dots.innerText = ".".repeat(Search._pulse_status); if (Search._pulse_status >= 0) window.setTimeout(pulse, 500); }; var dotString = ''; for (i = 0; i < Search._pulse_status; i++) dotString += '.'; Search.dots.text(dotString); if (Search._pulse_status > -1) window.setTimeout(pulse, 500); } pulse(); }, /** * perform a search for something (or wait until index is loaded) */ performSearch: (query) => { performSearch : function(query) { // create the required interface elements const searchText = document.createElement("h2"); searchText.textContent = _("Searching"); const searchSummary = document.createElement("p"); searchSummary.classList.add("search-summary"); searchSummary.innerText = ""; const searchList = document.createElement("ul"); searchList.classList.add("search"); const out = document.getElementById("search-results"); Search.title = out.appendChild(searchText); Search.dots = Search.title.appendChild(document.createElement("span")); Search.status = out.appendChild(searchSummary); Search.output = out.appendChild(searchList); const searchProgress = document.getElementById("search-progress"); // Some themes don't use the search progress node if (searchProgress) { searchProgress.innerText = _("Preparing search..."); } Search.startPulse(); this.out = $('#search-results'); this.title = $('<h2>' + _('Searching') + '</h2>').appendTo(this.out); this.dots = $('<span></span>').appendTo(this.title); this.status = $('<p class="search-summary"> </p>').appendTo(this.out); this.output = $('<ul class="search"/>').appendTo(this.out); $('#search-progress').text(_('Preparing search...')); this.startPulse(); // index already loaded, the browser was quick! if (Search.hasIndex()) Search.query(query); else Search.deferQuery(query); if (this.hasIndex()) this.query(query); else this.deferQuery(query); }, /** * execute search (requires search index to be loaded) */ query: (query) => { // stem the search terms and add them to the correct list const stemmer = new Stemmer(); const searchTerms = new Set(); const excludedTerms = new Set(); const highlightTerms = new Set(); const objectTerms = new Set(splitQuery(query.toLowerCase().trim())); splitQuery(query.trim()).forEach((queryTerm) => { const queryTermLower = queryTerm.toLowerCase(); // maybe skip this "word" // stopwords array is from language_data.js if ( stopwords.indexOf(queryTermLower) !== -1 || queryTerm.match(/^\d+$/) ) return; query : function(query) { var i; // stem the searchterms and add them to the correct list var stemmer = new Stemmer(); var searchterms = []; var excluded = []; var hlterms = []; var tmp = splitQuery(query); var objectterms = []; for (i = 0; i < tmp.length; i++) { if (tmp[i] !== "") { objectterms.push(tmp[i].toLowerCase()); } if ($u.indexOf(stopwords, tmp[i].toLowerCase()) != -1 || tmp[i] === "") { // skip this "word" continue; } // stem the word let word = stemmer.stemWord(queryTermLower); var word = stemmer.stemWord(tmp[i].toLowerCase()); // prevent stemmer from cutting word smaller than two chars if(word.length < 3 && tmp[i].length >= 3) { word = tmp[i]; } var toAppend; // select the correct list if (word[0] === "-") excludedTerms.add(word.substr(1)); if (word[0] == '-') { toAppend = excluded; word = word.substr(1); } else { searchTerms.add(word); highlightTerms.add(queryTermLower); toAppend = searchterms; hlterms.push(tmp[i].toLowerCase()); } }); // only add if not already in the list if (!$u.contains(toAppend, word)) toAppend.push(word); } var highlightstring = '?highlight=' + $.urlencode(hlterms.join(" ")); // console.debug("SEARCH: searching for:"); // console.info("required: ", [...searchTerms]); // console.info("excluded: ", [...excludedTerms]); // console.debug('SEARCH: searching for:'); // console.info('required: ', searchterms); // console.info('excluded: ', excluded); // prepare search var terms = this._index.terms; var titleterms = this._index.titleterms; // array of [docname, title, anchor, descr, score, filename] let results = []; _removeChildren(document.getElementById("search-progress")); // array of [filename, title, anchor, descr, score] var results = []; $('#search-progress').empty(); // lookup as object objectTerms.forEach((term) => results.push(...Search.performObjectSearch(term, objectTerms)) ); for (i = 0; i < objectterms.length; i++) { var others = [].concat(objectterms.slice(0, i), objectterms.slice(i+1, objectterms.length)); results = results.concat(this.performObjectSearch(objectterms[i], others)); } // lookup as search terms in fulltext results.push(...Search.performTermsSearch(searchTerms, excludedTerms)); results = results.concat(this.performTermsSearch(searchterms, excluded, terms, titleterms)); // let the scorer override scores with a custom scoring function if (Scorer.score) results.forEach((item) => (item[4] = Scorer.score(item))); if (Scorer.score) { for (i = 0; i < results.length; i++) results[i][4] = Scorer.score(results[i]); } // now sort the results by score (in opposite order of appearance, since the // display function below uses pop() to retrieve items) and then // alphabetically results.sort((a, b) => { const leftScore = a[4]; const rightScore = b[4]; if (leftScore === rightScore) { results.sort(function(a, b) { var left = a[4]; var right = b[4]; if (left > right) { return 1; } else if (left < right) { return -1; } else { // same score: sort alphabetically const leftTitle = a[1].toLowerCase(); const rightTitle = b[1].toLowerCase(); if (leftTitle === rightTitle) return 0; return leftTitle > rightTitle ? -1 : 1; // inverted is intentional left = a[1].toLowerCase(); right = b[1].toLowerCase(); return (left > right) ? -1 : ((left < right) ? 1 : 0); } return leftScore > rightScore ? 1 : -1; }); // remove duplicate search results // note the reversing of results, so that in the case of duplicates, the highest-scoring entry is kept let seen = new Set(); results = results.reverse().reduce((acc, result) => { let resultStr = result.slice(0, 4).concat([result[5]]).map(v => String(v)).join(','); if (!seen.has(resultStr)) { acc.push(result); seen.add(resultStr); } return acc; }, []); results = results.reverse(); // for debugging //Search.lastresults = results.slice(); // a copy // console.info("search results:", Search.lastresults); //console.info('search results:', Search.lastresults); // print the results _displayNextItem(results, results.length, highlightTerms, searchTerms); var resultCount = results.length; function displayNextItem() { // results left, load the summary and display it if (results.length) { var item = results.pop(); var listItem = $('<li></li>'); var requestUrl = ""; var linkUrl = ""; if (DOCUMENTATION_OPTIONS.BUILDER === 'dirhtml') { // dirhtml builder var dirname = item[0] + '/'; if (dirname.match(/\/index\/$/)) { dirname = dirname.substring(0, dirname.length-6); } else if (dirname == 'index/') { dirname = ''; } requestUrl = DOCUMENTATION_OPTIONS.URL_ROOT + dirname; linkUrl = requestUrl; } else { // normal html builders requestUrl = DOCUMENTATION_OPTIONS.URL_ROOT + item[0] + DOCUMENTATION_OPTIONS.FILE_SUFFIX; linkUrl = item[0] + DOCUMENTATION_OPTIONS.LINK_SUFFIX; } listItem.append($('<a/>').attr('href', linkUrl + highlightstring + item[2]).html(item[1])); if (item[3]) { listItem.append($('<span> (' + item[3] + ')</span>')); Search.output.append(listItem); setTimeout(function() { displayNextItem(); }, 5); } else if (DOCUMENTATION_OPTIONS.HAS_SOURCE) { $.ajax({url: requestUrl, dataType: "text", complete: function(jqxhr, textstatus) { var data = jqxhr.responseText; if (data !== '' && data !== undefined) { var summary = Search.makeSearchSummary(data, searchterms, hlterms); if (summary) { listItem.append(summary); } } Search.output.append(listItem); setTimeout(function() { displayNextItem(); }, 5); }}); } else { // no source available, just display title Search.output.append(listItem); setTimeout(function() { displayNextItem(); }, 5); } } // search finished, update title and status message else { Search.stopPulse(); Search.title.text(_('Search Results')); if (!resultCount) Search.status.text(_('Your search did not match any documents. 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else if (parts.slice(-1)[0].indexOf(object) > -1) score += Scorer.objPartialMatch; // matches in last name const objName = objNames[match[1]][2]; const title = titles[match[0]]; // If more than one term searched for, we require other words to be // found in the name/title/description const otherTerms = new Set(objectTerms); otherTerms.delete(object); if (otherTerms.size > 0) { const haystack = `${prefix} ${name} ${objName} ${title}`.toLowerCase(); if ( [...otherTerms].some((otherTerm) => haystack.indexOf(otherTerm) < 0) ) return; performObjectSearch : function(object, otherterms) { var filenames = this._index.filenames; var docnames = this._index.docnames; var objects = this._index.objects; var objnames = this._index.objnames; var titles = this._index.titles; var i; var results = []; for (var prefix in objects) { for (var iMatch = 0; iMatch != objects[prefix].length; ++iMatch) { var match = objects[prefix][iMatch]; var name = match[4]; var fullname = (prefix ? prefix + '.' : '') + name; var fullnameLower = fullname.toLowerCase() if (fullnameLower.indexOf(object) > -1) { var score = 0; var parts = fullnameLower.split('.'); // check for different match types: exact matches of full name or // "last name" (i.e. last dotted part) if (fullnameLower == object || parts[parts.length - 1] == object) { score += Scorer.objNameMatch; // matches in last name } else if (parts[parts.length - 1].indexOf(object) > -1) { score += Scorer.objPartialMatch; } var objname = objnames[match[1]][2]; var title = titles[match[0]]; // If more than one term searched for, we require other words to be // found in the name/title/description if (otherterms.length > 0) { var haystack = (prefix + ' ' + name + ' ' + objname + ' ' + title).toLowerCase(); var allfound = true; for (i = 0; i < otherterms.length; i++) { if (haystack.indexOf(otherterms[i]) == -1) { allfound = false; break; } } if (!allfound) { continue; } } var descr = objname + _(', in ') + title; var anchor = match[3]; if (anchor === '') anchor = fullname; else if (anchor == '-') anchor = objnames[match[1]][1] + '-' + fullname; // add custom score for some objects according to scorer if (Scorer.objPrio.hasOwnProperty(match[2])) { score += Scorer.objPrio[match[2]]; } else { score += Scorer.objPrioDefault; } results.push([docnames[match[0]], fullname, '#'+anchor, descr, score, filenames[match[0]]]); } } } let anchor = match[3]; if (anchor === "") anchor = fullname; else if (anchor === "-") anchor = objNames[match[1]][1] + "-" + fullname; const descr = objName + _(", in ") + title; // add custom score for some objects according to scorer if (Scorer.objPrio.hasOwnProperty(match[2])) score += Scorer.objPrio[match[2]]; else score += Scorer.objPrioDefault; results.push([ docNames[match[0]], fullname, "#" + anchor, descr, score, filenames[match[0]], ]); }; Object.keys(objects).forEach((prefix) => objects[prefix].forEach((array) => objectSearchCallback(prefix, array) ) ); return results; }, /** * See https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions */ escapeRegExp : function(string) { return string.replace(/[.*+\-?^${}()|[\]\\]/g, '\\$&'); // $& means the whole matched string }, /** * search for full-text terms in the index */ performTermsSearch: (searchTerms, excludedTerms) => { // prepare search const terms = Search._index.terms; const titleTerms = Search._index.titleterms; const docNames = Search._index.docnames; const filenames = Search._index.filenames; const titles = Search._index.titles; performTermsSearch : function(searchterms, excluded, terms, titleterms) { var docnames = this._index.docnames; var filenames = this._index.filenames; var titles = this._index.titles; const scoreMap = new Map(); const fileMap = new Map(); var i, j, file; var fileMap = {}; var scoreMap = {}; var results = []; // perform the search on the required terms searchTerms.forEach((word) => { const files = []; const arr = [ { files: terms[word], score: Scorer.term }, { files: titleTerms[word], score: Scorer.title }, for (i = 0; i < searchterms.length; i++) { var word = searchterms[i]; var files = []; var _o = [ {files: terms[word], score: Scorer.term}, {files: titleterms[word], score: Scorer.title} ]; // add support for partial matches if (word.length > 2) { const escapedWord = _escapeRegExp(word); Object.keys(terms).forEach((term) => { if (term.match(escapedWord) && !terms[word]) arr.push({ files: terms[term], score: Scorer.partialTerm }); }); Object.keys(titleTerms).forEach((term) => { if (term.match(escapedWord) && !titleTerms[word]) arr.push({ files: titleTerms[word], score: Scorer.partialTitle }); }); var word_regex = this.escapeRegExp(word); for (var w in terms) { if (w.match(word_regex) && !terms[word]) { _o.push({files: terms[w], score: Scorer.partialTerm}) } } for (var w in titleterms) { if (w.match(word_regex) && !titleterms[word]) { _o.push({files: titleterms[w], score: Scorer.partialTitle}) } } } // no match but word was a required one if (arr.every((record) => record.files === undefined)) return; if ($u.every(_o, function(o){return o.files === undefined;})) { break; } // found search word in contents arr.forEach((record) => { if (record.files === undefined) return; let recordFiles = record.files; if (recordFiles.length === undefined) recordFiles = [recordFiles]; files.push(...recordFiles); // set score for the word in each file recordFiles.forEach((file) => { if (!scoreMap.has(file)) scoreMap.set(file, {}); scoreMap.get(file)[word] = record.score; }); $u.each(_o, function(o) { var _files = o.files; if (_files === undefined) return if (_files.length === undefined) _files = [_files]; files = files.concat(_files); // set score for the word in each file to Scorer.term for (j = 0; j < _files.length; j++) { file = _files[j]; if (!(file in scoreMap)) scoreMap[file] = {}; scoreMap[file][word] = o.score; } }); // create the mapping files.forEach((file) => { if (fileMap.has(file) && fileMap.get(file).indexOf(word) === -1) fileMap.get(file).push(word); else fileMap.set(file, [word]); }); }); for (j = 0; j < files.length; j++) { file = files[j]; if (file in fileMap && fileMap[file].indexOf(word) === -1) fileMap[file].push(word); else fileMap[file] = [word]; } } // now check if the files don't contain excluded terms const results = []; for (const [file, wordList] of fileMap) { // check if all requirements are matched for (file in fileMap) { var valid = true; // as search terms with length < 3 are discarded const filteredTermCount = [...searchTerms].filter( (term) => term.length > 2 ).length; // check if all requirements are matched var filteredTermCount = // as search terms with length < 3 are discarded: ignore searchterms.filter(function(term){return term.length > 2}).length if ( wordList.length !== searchTerms.size && wordList.length !== filteredTermCount ) continue; fileMap[file].length != searchterms.length && fileMap[file].length != filteredTermCount ) continue; // ensure that none of the excluded terms is in the search result if ( [...excludedTerms].some( (term) => terms[term] === file || titleTerms[term] === file || (terms[term] || []).includes(file) || (titleTerms[term] || []).includes(file) ) ) break; for (i = 0; i < excluded.length; i++) { if (terms[excluded[i]] == file || titleterms[excluded[i]] == file || $u.contains(terms[excluded[i]] || [], file) || $u.contains(titleterms[excluded[i]] || [], file)) { valid = false; break; } } // select one (max) score for the file. const score = Math.max(...wordList.map((w) => scoreMap.get(file)[w])); // add result to the result list results.push([ docNames[file], titles[file], "", null, score, filenames[file], ]); // if we have still a valid result we can add it to the result list if (valid) { // select one (max) score for the file. // for better ranking, we should calculate ranking by using words statistics like basic tf-idf... var score = $u.max($u.map(fileMap[file], function(w){return scoreMap[file][w]})); results.push([docnames[file], titles[file], '', null, score, filenames[file]]); } } return results; },
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@@ -1,8 +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="viewport" content="width=device-width, initial-scale=1.0" /> <title>Mathematics in Lean — Mathematics in Lean 0.1 documentation</title> <link rel="stylesheet" href="_static/pygments.css" type="text/css" />
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@@ -79,8 +77,8 @@<div role="main" class="document" itemscope="itemscope" itemtype="http://schema.org/Article"> <div itemprop="articleBody"> <section id="mathematics-in-lean"> <h1>Mathematics in Lean<a class="headerlink" href="#mathematics-in-lean" title="Permalink to this heading"></a></h1> <div class="section" id="mathematics-in-lean"> <h1>Mathematics in Lean<a class="headerlink" href="#mathematics-in-lean" title="Permalink to this headline"></a></h1> <div class="toctree-wrapper compound"> <ul> <li class="toctree-l1"><a class="reference internal" href="C01_Introduction.html">1. Introduction</a><ul>
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_nhds_uniqu:7,tendsto_pow_attop_nhds_0_of_lt_1:7,tendsto_right_iff:7,tendsto_subseq:7,term:[0,1,2,3,4,5,7],termin:0,terminolog:5,ters:6,test:4,text:[0,3,5,7],textbook:[0,2],th:4,than:[0,1,2,3,4,5,6,7,8],thank:[6,7],thei:[0,1,2,3,4,5,6,7,8],them:[0,1,2,3,4,5,6,7],theme:3,themselv:[0,4,5],theorem:[0,2,4,5,7,8,9,11],theoret:[2,3,7],theori:[0,1,2,3,6,7,8,11],therefor:[1,4,5,7],thereof:0,thi:[0,1,2,3,4,5,6,7,8,9],thing:[0,1,2,3,4,5,6,7],think:[0,1,2,3,4,5,6,7,8],third:[1,2,3,5,7],thorough:0,those:[0,1,2,3,5,6,7,8],though:[2,3,4,5],three:[1,2,4,5,7],through:[0,1,3,4,5,6,7],thu:[1,3,4,5,7],thumb:4,tighter:[1,3],tightli:5,time:[1,4,5,6,7,8,9],timesav:1,tiresom:7,to_addit:6,to_localinvers:8,to_real:9,tofun:[5,6],togeth:[1,2,3,4,5,7],tomuloneclass:6,too:[0,1,3,5,6,7],tool:[0,1,5,6],top:[0,6,7,8],topic:[4,6],topolog:[1,5,6,8,11],topological_spac:7,topological_structur:7,topologicalspac:[7,8],toreal:9,total:1,tour:[7,8],trace:6,tradeoff:3,train:[1,6],tran:[2,4,5,6],trans_assoc:5,trans_refl:5,transform:[2,3],transit:[1,2,5,7],translat:[4,6,7],trap:6,treat:[2,3,5,6,7,8],tri:[1,2,4,6],triangl:[1,2,5],trick:[1,2,4,6,8],tricki:[2,3,4,6],trigger:6,tripl:3,trivial:[3,4,7],troubl:[2,3,5],truncat:4,truth:[1,4],tupl:5,turn:[2,3,4,5,6,7,8],tutori:0,twice:[0,1,7],two:[0,1,2,3,4,5,6,7,9],two_l:4,two_le_of_mod_4_eq_3:4,two_mul:1,type:[0,1,2,3,4,5,6,7,8,9],typeclass:6,typic:1,u:[2,3,5,7,9],ubf:2,ubfa:2,ubg:2,ubgb:2,ultim:0,un:3,unbundl:6,unchang:[1,2],uncom:1,uncount:7,uncurri:7,under:[5,6,7],underli:[0,1,5,6],underneath:[1,3],underscor:[1,2,3],understand:[0,2,3,4,5,6,7],understood:[4,7],unfold:[1,2,3,4,5,7],unicod:[1,3,6],unif:6,uniform:[3,7,8],uniformcontinu:7,uniformcontinuous_iff:7,uniformli:8,union:[1,3,4,5,6,8,9],union_def:3,uniqu:[1,2,3,4,5,7],unit:[5,6,7],univ:[3,7,8,9],univ_set:7,univers:[1,3,5,7,11],unknown:6,unless:[1,2,5],unlik:[3,4],unnecessari:3,unpack:[2,3],unpleas:[6,7],unrel:6,unshad:3,unspecifi:6,unsurprisingli:3,until:[1,4,5,6,7],up:[0,2,3,4,5,6,7],updat:0,upper:[1,2],us:[0,2,3,4,5,6,7,8,9,11],usabl:6,usag:1,user:[0,6],usual:[1,2,3,4,5,7,9],v:[2,3,5,7],val:5,valid:2,valu:[1,2,3,4,5,6,7,8,9],varepsilon:[2,7],variabl:[1,2,3,4,5,6,7,8,9],variant:[1,2,7],variat:[1,3,4,5,7,8],varieti:7,variou:[2,3,5,6,9],vastli:2,ve:[6,7],vector:[1,3,6,8],verbos:3,veri:[0,2,4,6,7,9],verifi:2,versa:[1,3],version:[0,1,2,3,4,5,6,7,8,9],vertic:[2,5],vi:5,via:7,vice:[1,3],view:[0,5,6,7],visibl:1,vocabulari:3,volum:9,vs:[0,1,2,3,5],vscode:0,w:[1,6],wa:[1,3],wai:[0,1,2,3,4,5,6,7,9],want:[0,1,2,3,4,5,6,7,8,9],wasn:7,we:[0,1,2,3,4,5,6,7,8,9],weak:6,weaker:[1,2],weav:5,web:[0,1,4],weight:5,weightedaverag:5,weird:[6,7],weirder:8,welcom:[0,1],well:[0,1,2,3,4,5,6,7],were:[2,4,5,7],what:[0,1,2,3,4,5,6,7],whatev:[0,2,4],whatsnew:6,when:[0,1,2,3,4,5,6,7,8],whenev:[1,2,4,5],where:[0,1,2,3,4,5,6,7,8,9],wherea:[0,1,2,3,5,7],wherebi:[5,7],whether:[1,2,3,4,6,7],which:[0,1,2,3,4,5,6,7,8,9],who:[0,7],whole:5,whose:[1,2,4,5,6,7,8],why:[0,2,4,6,7],window:[0,1,2,5],wise:5,wish:7,with_top:8,within:[1,2,7],without:[1,2,4,5,6,7,8],withtop:8,wlog:3,won:[2,4,6,7],word:[1,2,3,4,5,7],work:[0,1,2,3,4,5,6,7,8],world:0,worri:[1,3,4,5],worth:[1,2,5],would:[1,2,3,4,5,6,7,8],wouldn:6,wrap:[6,7],write:[0,1,2,4,5,6,7,8,9],written:[1,2,3,4,5,6,7,8],wrong:3,wrote:4,x:[0,1,2,3,4,5,6,7,8,9],x_0:7,x_1:[2,3],x_2:[2,3],x_i:7,x_nonneg:5,xa:[3,5],xai:3,xb:5,xeq:[2,3],xgt:2,xlt:2,xltz:2,xmem:3,xnt:3,xnu:3,xpo:3,xs:[2,3],xstu:3,xsu:3,xt:3,xtu:3,xu:3,xy:[2,5],y:[0,1,2,3,4,5,6,7,8,9],y_nonneg:5,ya:5,yb:5,yball:7,yeq:2,yet:[2,3,4,5,6,7],yield:[1,2,3,4,5,7],ylim:7,ylt:2,you:[0,1,2,3,4,5,6,7,8,9],your:[0,1,2,3,5,6,7],yourself:[2,5],ypo:3,z:[0,1,2,5,6,7,8,9],z_nonneg:5,za:5,zb:5,zermelo:2,zero:[2,3,4,5,6,7,8,9],zero_add:[1,4,5,6],zero_def:5,zero_dvd_iff:4,zero_im:5,zero_l:[4,5],zero_lt_on:[2,4],zero_lt_two:5,zero_mul:[1,4,5,6],zero_r:5,zero_smul:6,zf:2,zlty:2,zsmul:6,zulip:[0,4]},titles:["<span class=\"section-number\">1. </span>Introduction","<span class=\"section-number\">2. </span>Basics","<span class=\"section-number\">3. </span>Logic","<span class=\"section-number\">4. </span>Sets and Functions","<span class=\"section-number\">5. </span>Elementary Number Theory","<span class=\"section-number\">6. </span>Structures","<span class=\"section-number\">7. </span>Hierarchies","<span class=\"section-number\">8. </span>Topology","<span class=\"section-number\">9. </span>Differential Calculus","<span class=\"section-number\">10. </span>Integration and Measure Theory","Index","Mathematics in Lean"],titleterms:{"function":[3,7],"schr\u00f6der":3,The:[2,3],about:1,algebra:[1,5],appli:1,asymptot:8,ball:7,basic:[1,6],bernstein:3,bi:2,build:5,calcul:1,calculu:8,close:7,compact:7,comparison:8,complet:7,conjunct:2,continu:[7,8],converg:[2,7],countabl:7,defin:5,differenti:8,disjunct:2,elementari:[4,8,9],exampl:1,existenti:2,fact:1,filter:7,fundament:7,gaussian:5,get:0,hierarchi:6,ident:1,implic:2,index:10,induct:4,infinit:4,integ:5,integr:9,introduct:0,irrat:4,lean:11,lemma:1,linear:8,logic:2,mani:4,map:8,mathemat:11,measur:9,metric:7,more:1,morphism:6,negat:2,norm:8,number:4,object:6,open:7,overview:0,prime:4,prove:1,quantifi:2,recurs:4,root:4,rw:1,separ:7,sequenc:2,set:[3,7],space:[7,8],start:0,structur:[1,5],sub:6,theorem:[1,3],theori:[4,9],topolog:7,uniformli:7,univers:2,us:1}})
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