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Result

Found 342 declarations mentioning Path. Of these, only the first 200 are shown.

• Path 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] (x y : X) : Type u_1
• Path.refl 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] (x : X) : Path x x
• Path.instTopologicalSpace 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} : TopologicalSpace (Path x y)
• Path.simps.apply 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) : ↑unitInterval → X
• Path.instFunLike 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} : FunLike (Path x y) (↑unitInterval) X
• Path.symm 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) : Path y x
• Path.extend 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) : C(ℝ, X)
• Path.symm_bijective 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} : Function.Bijective Path.symm
• Path.trans 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y z : X} (γ : Path x y) (γ' : Path y z) : Path x z
• Path.instHasUncurryPath 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {α : Type u_4} {x y : α → X} : Function.HasUncurry ((a : α) → Path (x a) (y a)) (α × ↑unitInterval) X
• Path.refl_symm 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a : X} : (Path.refl a).symm = Path.refl a
• Path.cast 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) {x' y' : X} (hx : x' = x) (hy : y' = y) : Path x' y'
• Path.inv 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a b : X} [Inv X] [ContinuousInv X] (γ : Path a b) : Path a⁻¹ b⁻¹
• Path.neg 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a b : X} [Neg X] [ContinuousNeg X] (γ : Path a b) : Path (-a) (-b)
• Path.map 📋 Mathlib.Topology.Path
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y : X} (γ : Path x y) {f : X → Y} (h : Continuous f) : Path (f x) (f y)
• Path.refl_trans_refl 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a : X} : (Path.refl a).trans (Path.refl a) = Path.refl a
• Path.symm_symm 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) : γ.symm.symm = γ
• Path.cast_rfl_rfl 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) : γ.cast ⋯ ⋯ = γ
• Path.continuous_symm 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} : Continuous Path.symm
• Path.toContinuousMap 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (self : Path x y) : C(↑unitInterval, X)
• Path.pi 📋 Mathlib.Topology.Path
  {ι : Type u_3} {χ : ι → Type u_4} [(i : ι) → TopologicalSpace (χ i)] {as bs : (i : ι) → χ i} (γ : (i : ι) → Path (as i) (bs i)) : Path as bs
• Path.refl_apply 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] (x : X) (x✝ : ↑unitInterval) : (Path.refl x) x✝ = x
• Path.map_id 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) : γ.map ⋯ = γ
• Path.continuousMapClass 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} : ContinuousMapClass (Path x y) (↑unitInterval) X
• Path.prod 📋 Mathlib.Topology.Path
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {a₁ a₂ : X} {b₁ b₂ : Y} (γ₁ : Path a₁ a₂) (γ₂ : Path b₁ b₂) : Path (a₁, b₁) (a₂, b₂)
• Path.bijective_cast 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y x' y' : X} (hx : x' = x) (hy : y' = y) : Function.Bijective fun x_1 => x_1.cast hx hy
• Path.refl_range 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a : X} : Set.range ⇑(Path.refl a) = {a}
• Path.source_mem_range 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) : x ∈ Set.range ⇑γ
• Path.target_mem_range 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) : y ∈ Set.range ⇑γ
• Path.instContinuousEvalElemRealUnitInterval 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} : ContinuousEval (Path x y) (↑unitInterval) X
• Path.ofLine 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} {f : ℝ → X} (hf : ContinuousOn f unitInterval) (h₀ : f 0 = x) (h₁ : f 1 = y) : Path x y
• Path.add 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] [Add X] [ContinuousAdd X] {a₁ b₁ a₂ b₂ : X} (γ₁ : Path a₁ b₁) (γ₂ : Path a₂ b₂) : Path (a₁ + a₂) (b₁ + b₂)
• Path.mul 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] [Mul X] [ContinuousMul X] {a₁ b₁ a₂ b₂ : X} (γ₁ : Path a₁ b₁) (γ₂ : Path a₂ b₂) : Path (a₁ * a₂) (b₁ * b₂)
• Path.source 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) : γ 0 = x
• Path.target 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) : γ 1 = y
• Path.continuous_extend 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) : Continuous ⇑γ.extend
• Path.extend_one 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) : γ.extend 1 = y
• Path.extend_zero 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) : γ.extend 0 = x
• Path.id 📋 Mathlib.Topology.Path
  : Path 0 1
• Path.continuous 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) : Continuous ⇑γ
• Path.extend_cast 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y x' y' : X} (γ : Path x y) (hx : x' = x) (hy : y' = y) : (γ.cast hx hy).extend = γ.extend
• Path.map' 📋 Mathlib.Topology.Path
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y : X} (γ : Path x y) {f : X → Y} (h : ContinuousOn f (Set.range ⇑γ)) : Path (f x) (f y)
• Path.extend_of_le_zero 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a b : X} (γ : Path a b) {t : ℝ} (ht : t ≤ 0) : γ.extend t = a
• Path.extend_of_one_le 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a b : X} (γ : Path a b) {t : ℝ} (ht : 1 ≤ t) : γ.extend t = b
• Path.trans_symm 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y z : X} (γ : Path x y) (γ' : Path y z) : (γ.trans γ').symm = γ'.symm.trans γ.symm
• Path.source' 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (self : Path x y) : self.toFun 0 = x
• Path.target' 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (self : Path x y) : self.toFun 1 = y
• Path.cast_symm 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a₁ a₂ b₁ b₂ : X} (γ : Path a₂ b₂) (ha : a₁ = a₂) (hb : b₁ = b₂) : γ.symm.cast hb ha = (γ.cast ha hb).symm
• Path.exists_congr 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x₁ x₂ y₁ y₂ : X} {p : Path x₁ y₁ → Prop} (hx : x₁ = x₂) (hy : y₁ = y₂) : (∃ γ, p γ) ↔ ∃ γ, p (γ.cast hx hy)
• Path.map_symm 📋 Mathlib.Topology.Path
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y : X} (γ : Path x y) {f : X → Y} (h : Continuous f) : (γ.map h).symm = γ.symm.map h
• Path.ext 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} {γ₁ γ₂ : Path x y} : ⇑γ₁ = ⇑γ₂ → γ₁ = γ₂
• Path.symm_range 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a b : X} (γ : Path a b) : Set.range ⇑γ.symm = Set.range ⇑γ
• Path.ext_iff 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} {γ₁ γ₂ : Path x y} : γ₁ = γ₂ ↔ ⇑γ₁ = ⇑γ₂
• Path.extend_range 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a b : X} (γ : Path a b) : Set.range ⇑γ.extend = Set.range ⇑γ
• Path.ofLine_extend 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) : Path.ofLine ⋯ ⋯ ⋯ = γ
• Path.symm_apply 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) (a✝ : ↑unitInterval) : γ.symm a✝ = (⇑γ ∘ unitInterval.symm) a✝
• Path.cast_coe 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) {x' y' : X} (hx : x' = x) (hy : y' = y) : ⇑(γ.cast hx hy) = ⇑γ
• Continuous.path_trans 📋 Mathlib.Topology.Path
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y z : X} {f : Y → Path x y} {g : Y → Path y z} : Continuous f → Continuous g → Continuous fun t => (f t).trans (g t)
• Path.truncate 📋 Mathlib.Topology.Path
  {X : Type u_4} [TopologicalSpace X] {a b : X} (γ : Path a b) (t₀ t₁ : ℝ) : Path (γ.extend (min t₀ t₁)) (γ.extend t₁)
• Path.image_extend_of_subset 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) {s : Set ℝ} (h : unitInterval ⊆ s) : ⇑γ.extend '' s = Set.range ⇑γ
• Path.truncateOfLE 📋 Mathlib.Topology.Path
  {X : Type u_4} [TopologicalSpace X] {a b : X} (γ : Path a b) {t₀ t₁ : ℝ} (h : t₀ ≤ t₁) : Path (γ.extend t₀) (γ.extend t₁)
• Path.cast_trans 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a₁ a₂ b₁ b₂ c₁ c₂ : X} (γ : Path a₂ b₂) (γ' : Path b₂ c₂) (ha : a₁ = a₂) (hb : b₁ = b₂) (hc : c₁ = c₂) : (γ.trans γ').cast ha hc = (γ.cast ha hb).trans (γ'.cast hb hc)
• Path.ofLine_mem 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} {f : ℝ → X} (hf : ContinuousOn f unitInterval) (h₀ : f 0 = x) (h₁ : f 1 = y) (t : ↑unitInterval) : (Path.ofLine hf h₀ h₁) t ∈ f '' unitInterval
• Path.continuous_uncurry_iff 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} {Y : Type u_4} [TopologicalSpace Y] {g : Y → Path x y} : Continuous ↿g ↔ Continuous g
• Path.map_trans 📋 Mathlib.Topology.Path
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y z : X} (γ : Path x y) (γ' : Path y z) {f : X → Y} (h : Continuous f) : (γ.trans γ').map h = (γ.map h).trans (γ'.map h)
• Path.inv_apply 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a b : X} [Inv X] [ContinuousInv X] (γ : Path a b) (a✝ : ↑unitInterval) : γ.inv a✝ = (γ a✝)⁻¹
• Path.neg_apply 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a b : X} [Neg X] [ContinuousNeg X] (γ : Path a b) (a✝ : ↑unitInterval) : γ.neg a✝ = -γ a✝
• Path.map_coe 📋 Mathlib.Topology.Path
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y : X} (γ : Path x y) {f : X → Y} (h : Continuous f) : ⇑(γ.map h) = f ∘ ⇑γ
• Path.map_map 📋 Mathlib.Topology.Path
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y : X} (γ : Path x y) {Z : Type u_4} [TopologicalSpace Z] {f : X → Y} (hf : Continuous f) {g : Y → Z} (hg : Continuous g) : (γ.map hf).map hg = γ.map ⋯
• Path.extend_symm 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) : ⇑γ.symm.extend = fun x_1 => γ.extend (1 - x_1)
• Path.extend_symm_apply 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) (t : ℝ) : γ.symm.extend t = γ.extend (1 - t)
• Path.continuous_trans 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y z : X} : Continuous fun ρ => ρ.1.trans ρ.2
• Path.reparam_id 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) : γ.reparam id ⋯ ⋯ ⋯ = γ
• Path.coe_toContinuousMap 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) : ⇑γ.toContinuousMap = ⇑γ
• Path.trans_pi_eq_pi_trans 📋 Mathlib.Topology.Path
  {ι : Type u_3} {χ : ι → Type u_4} [(i : ι) → TopologicalSpace (χ i)] {as bs cs : (i : ι) → χ i} (γ₀ : (i : ι) → Path (as i) (bs i)) (γ₁ : (i : ι) → Path (bs i) (cs i)) : (Path.pi γ₀).trans (Path.pi γ₁) = Path.pi fun i => (γ₀ i).trans (γ₁ i)
• Path.extend_apply 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a b : X} (γ : Path a b) {t : ℝ} (ht : t ∈ Set.Icc 0 1) : γ.extend t = γ ⟨t, ht⟩
• Path.pi_coe 📋 Mathlib.Topology.Path
  {ι : Type u_3} {χ : ι → Type u_4} [(i : ι) → TopologicalSpace (χ i)] {as bs : (i : ι) → χ i} (γ : (i : ι) → Path (as i) (bs i)) : ⇑(Path.pi γ) = fun t i => (γ i) t
• Path.extend_extends' 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a b : X} (γ : Path a b) (t : ↑(Set.Icc 0 1)) : γ.extend ↑t = γ t
• Path.trans_range 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a b c : X} (γ₁ : Path a b) (γ₂ : Path b c) : Set.range ⇑(γ₁.trans γ₂) = Set.range ⇑γ₁ ∪ Set.range ⇑γ₂
• Path.mk 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (toContinuousMap : C(↑unitInterval, X)) (source' : toContinuousMap.toFun 0 = x) (target' : toContinuousMap.toFun 1 = y) : Path x y
• Continuous.pathExtend 📋 Mathlib.Topology.Path
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {x y : X} {γ : Y → Path x y} {f : Y → ℝ} (hγ : Continuous ↿γ) (hf : Continuous f) : Continuous fun t => (γ t).extend (f t)
• Path.id_apply 📋 Mathlib.Topology.Path
  (a : ↑unitInterval) : Path.id a = a
• Path.coe_mk_mk 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (f : ↑unitInterval → X) (h₁ : Continuous f) (h₂ : f 0 = x) (h₃ : f 1 = y) : ⇑{ toFun := f, continuous_toFun := h₁, source' := h₂, target' := h₃ } = f
• Path.trans_prod_eq_prod_trans 📋 Mathlib.Topology.Path
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {a₁ a₂ a₃ : X} {b₁ b₂ b₃ : Y} (γ₁ : Path a₁ a₂) (δ₁ : Path a₂ a₃) (γ₂ : Path b₁ b₂) (δ₂ : Path b₂ b₃) : (γ₁.prod γ₂).trans (δ₁.prod δ₂) = (γ₁.trans δ₁).prod (γ₂.trans δ₂)
• Path.reparam 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) (f : ↑unitInterval → ↑unitInterval) (hfcont : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) : Path x y
• Path.symm_continuous_family 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {ι : Type u_4} [TopologicalSpace ι] {a b : ι → X} (γ : (t : ι) → Path (a t) (b t)) (h : Continuous ↿γ) : Continuous ↿fun t => (γ t).symm
• Path.prod_coe 📋 Mathlib.Topology.Path
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {a₁ a₂ : X} {b₁ b₂ : Y} (γ₁ : Path a₁ a₂) (γ₂ : Path b₁ b₂) : ⇑(γ₁.prod γ₂) = fun t => (γ₁ t, γ₂ t)
• Path.continuous_uncurry_extend_of_continuous_family 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {ι : Type u_4} [TopologicalSpace ι] {a b : ι → X} (γ : (t : ι) → Path (a t) (b t)) (h : Continuous ↿γ) : Continuous ↿fun t => ⇑(γ t).extend
• Path.add_apply 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] [Add X] [ContinuousAdd X] {a₁ b₁ a₂ b₂ : X} (γ₁ : Path a₁ b₁) (γ₂ : Path a₂ b₂) (a✝ : ↑unitInterval) : (γ₁.add γ₂) a✝ = γ₁ a✝ + γ₂ a✝
• Path.mul_apply 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] [Mul X] [ContinuousMul X] {a₁ b₁ a₂ b₂ : X} (γ₁ : Path a₁ b₁) (γ₂ : Path a₂ b₂) (a✝ : ↑unitInterval) : (γ₁.mul γ₂) a✝ = γ₁ a✝ * γ₂ a✝
• Path.refl_reparam 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x : X} {f : ↑unitInterval → ↑unitInterval} (hfcont : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) : (Path.refl x).reparam f hfcont hf₀ hf₁ = Path.refl x
• Path.extend_trans_of_le_half 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y z : X} (γ₁ : Path x y) (γ₂ : Path y z) {t : ℝ} (ht : t ≤ 1 / 2) : (γ₁.trans γ₂).extend t = γ₁.extend (2 * t)
• Path.extend_trans_of_half_le 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y z : X} (γ₁ : Path x y) (γ₂ : Path y z) {t : ℝ} (ht : 1 / 2 ≤ t) : (γ₁.trans γ₂).extend t = γ₂.extend (2 * t - 1)
• ContinuousAt.pathExtend 📋 Mathlib.Topology.Path
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {g : Y → ℝ} {l r : Y → X} (γ : (y : Y) → Path (l y) (r y)) {y : Y} (hγ : ContinuousAt ↿γ (y, Set.projIcc 0 1 ⋯ (g y))) (hg : ContinuousAt g y) : ContinuousAt (fun i => (γ i).extend (g i)) y
• Path.range_reparam 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) {f : ↑unitInterval → ↑unitInterval} (hfcont : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) : Set.range ⇑(γ.reparam f hfcont hf₀ hf₁) = Set.range ⇑γ
• Path.coe_reparam 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (γ : Path x y) {f : ↑unitInterval → ↑unitInterval} (hfcont : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) : ⇑(γ.reparam f hfcont hf₀ hf₁) = ⇑γ ∘ f
• Path.coe_mk' 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} (f : C(↑unitInterval, X)) (h₁ : f.toFun 0 = x) (h₂ : f.toFun 1 = y) : ⇑{ toContinuousMap := f, source' := h₁, target' := h₂ } = ⇑f
• Path.truncate_const_continuous_family 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a b : X} (γ : Path a b) (t : ℝ) : Continuous ↿(γ.truncate t)
• Path.truncate_range 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a b : X} (γ : Path a b) {t₀ t₁ : ℝ} : Set.range ⇑(γ.truncate t₀ t₁) ⊆ Set.range ⇑γ
• Path.trans_continuous_family 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {ι : Type u_4} [TopologicalSpace ι] {a b c : ι → X} (γ₁ : (t : ι) → Path (a t) (b t)) (h₁ : Continuous ↿γ₁) (γ₂ : (t : ι) → Path (b t) (c t)) (h₂ : Continuous ↿γ₂) : Continuous ↿fun t => (γ₁ t).trans (γ₂ t)
• Path.truncate_zero_one 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a b : X} (γ : Path a b) : γ.truncate 0 1 = γ.cast ⋯ ⋯
• Path.truncate_one_one 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a b : X} (γ : Path a b) : γ.truncate 1 1 = (Path.refl b).cast ⋯ ⋯
• Path.truncate_zero_zero 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a b : X} (γ : Path a b) : γ.truncate 0 0 = (Path.refl a).cast ⋯ ⋯
• Path.truncate_self 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a b : X} (γ : Path a b) (t : ℝ) : γ.truncate t t = (Path.refl (γ.extend t)).cast ⋯ ⋯
• Path.range_coe 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y : X} : Set.range toContinuousMap = {f | f 0 = x ∧ f 1 = y}
• Filter.Tendsto.pathExtend 📋 Mathlib.Topology.Path
  {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {l r : Y → X} {y : Y} {l₁ : Filter ℝ} {l₂ : Filter X} {γ : (y : Y) → Path (l y) (r y)} (hγ : Filter.Tendsto (↿γ) (nhds y ×ˢ Filter.map (Set.projIcc 0 1 ⋯) l₁) l₂) : Filter.Tendsto (↿fun x => ⇑(γ x).extend) (nhds y ×ˢ l₁) l₂
• Path.truncate_continuous_family 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {a b : X} (γ : Path a b) : Continuous fun x => (γ.truncate x.1 x.2.1) x.2.2
• Path.trans_apply 📋 Mathlib.Topology.Path
  {X : Type u_1} [TopologicalSpace X] {x y z : X} (γ : Path x y) (γ' : Path y z) (t : ↑unitInterval) : (γ.trans γ') t = if h : ↑t ≤ 1 / 2 then γ ⟨2 * ↑t, ⋯⟩ else γ' ⟨2 * ↑t - 1, ⋯⟩
• PathConnectedSpace.somePath 📋 Mathlib.Topology.Connected.PathConnected
  {X : Type u_1} [TopologicalSpace X] [PathConnectedSpace X] (x y : X) : Path x y
• Joined.somePath 📋 Mathlib.Topology.Connected.PathConnected
  {X : Type u_1} [TopologicalSpace X] {x y : X} (h : Joined x y) : Path x y
• JoinedIn.somePath 📋 Mathlib.Topology.Connected.PathConnected
  {X : Type u_1} [TopologicalSpace X] {x y : X} {F : Set X} (h : JoinedIn F x y) : Path x y
• ZerothHomotopy.sound 📋 Mathlib.Topology.Connected.PathConnected
  {X : Type u_1} [TopologicalSpace X] {x y : X} (p : Path x y) : ZerothHomotopy.mk x = ZerothHomotopy.mk y
• ZerothHomotopy.lift 📋 Mathlib.Topology.Connected.PathConnected
  {X : Type u_1} [TopologicalSpace X] {T : Type u_4} (f : X → T) (hf : ∀ ⦃x y : X⦄ (x_1 : Path x y), f x = f y) : ZerothHomotopy X → T
• ZerothHomotopy.lift_mk 📋 Mathlib.Topology.Connected.PathConnected
  {X : Type u_1} [TopologicalSpace X] {T : Type u_4} (f : X → T) (hf : ∀ ⦃x y : X⦄ (x_1 : Path x y), f x = f y) (x : X) : ZerothHomotopy.lift f hf (ZerothHomotopy.mk x) = f x
• JoinedIn.somePath_mem 📋 Mathlib.Topology.Connected.PathConnected
  {X : Type u_1} [TopologicalSpace X] {x y : X} {F : Set X} (h : JoinedIn F x y) (t : ↑unitInterval) : h.somePath t ∈ F
• PathConnectedSpace.exists_path_through_family 📋 Mathlib.Topology.Connected.PathConnected
  {X : Type u_1} [TopologicalSpace X] [PathConnectedSpace X] {n : ℕ} (p : Fin (n + 1) → X) : ∃ γ, ∀ (i : Fin (n + 1)), p i ∈ Set.range ⇑γ
• PathConnectedSpace.exists_path_through_family' 📋 Mathlib.Topology.Connected.PathConnected
  {X : Type u_1} [TopologicalSpace X] [PathConnectedSpace X] {n : ℕ} (p : Fin (n + 1) → X) : ∃ γ t, ∀ (i : Fin (n + 1)), γ (t i) = p i
• IsPathConnected.exists_path_through_family 📋 Mathlib.Topology.Connected.PathConnected
  {X : Type u_1} [TopologicalSpace X] {n : ℕ} {s : Set X} (h : IsPathConnected s) (p : Fin (n + 1) → X) (hp : ∀ (i : Fin (n + 1)), p i ∈ s) : ∃ γ, Set.range ⇑γ ⊆ s ∧ ∀ (i : Fin (n + 1)), p i ∈ Set.range ⇑γ
• IsPathConnected.exists_path_through_family' 📋 Mathlib.Topology.Connected.PathConnected
  {X : Type u_1} [TopologicalSpace X] {n : ℕ} {s : Set X} (h : IsPathConnected s) (p : Fin (n + 1) → X) (hp : ∀ (i : Fin (n + 1)), p i ∈ s) : ∃ γ t, (∀ (t : ↑unitInterval), γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), γ (t i) = p i
• Path.segment 📋 Mathlib.Analysis.Convex.PathConnected
  {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] (a b : E) : Path a b
• Path.segment_same 📋 Mathlib.Analysis.Convex.PathConnected
  {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] (a : E) : Path.segment a a = Path.refl a
• Path.segment_symm 📋 Mathlib.Analysis.Convex.PathConnected
  {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] (a b : E) : (Path.segment a b).symm = Path.segment b a
• Path.segment_injective_of_ne 📋 Mathlib.Analysis.Convex.PathConnected
  {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] {a b : E} (hne : a ≠ b) : Function.Injective ⇑(Path.segment a b)
• Path.cast_segment 📋 Mathlib.Analysis.Convex.PathConnected
  {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] {a b c d : E} (hac : c = a) (hbd : d = b) : (Path.segment a b).cast hac hbd = Path.segment c d
• Path.range_segment 📋 Mathlib.Analysis.Convex.PathConnected
  {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] (a b : E) : Set.range ⇑(Path.segment a b) = segment ℝ a b
• segment_image_Ico 📋 Mathlib.Analysis.Convex.PathConnected
  {x y : ℝ} (h : x < y) : ⇑(Path.segment x y) '' Set.Ico 0 1 = Set.Ico x y
• segment_image_Ioc 📋 Mathlib.Analysis.Convex.PathConnected
  {x y : ℝ} (h : x < y) : ⇑(Path.segment x y) '' Set.Ioc 0 1 = Set.Ioc x y
• Path.segment_add_segment 📋 Mathlib.Analysis.Convex.PathConnected
  {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] (a b c d : E) : (Path.segment a b).add (Path.segment c d) = Path.segment (a + c) (b + d)
• Path.segment_apply 📋 Mathlib.Analysis.Convex.PathConnected
  {E : Type u_1} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul ℝ E] (a b : E) (t : ↑unitInterval) : (Path.segment a b) t = (AffineMap.lineMap a b) ↑t
• Path.Homotopic.setoid 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] (x₀ x₁ : X) : Setoid (Path x₀ x₁)
• Path.Homotopic.Quotient.mk 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) : Path.Homotopic.Quotient x₀ x₁
• Path.Homotopic 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} (p₀ p₁ : Path x₀ x₁) : Prop
• Path.Homotopy 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} (p₀ p₁ : Path x₀ x₁) : Type u
• Path.Homotopic.equivalence 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} : Equivalence Path.Homotopic
• Path.Homotopic.instIsEquiv 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} : IsEquiv (Path x₀ x₁) Path.Homotopic
• Path.Homotopic.refl 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) : p.Homotopic p
• Path.Homotopy.refl 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) : p.Homotopy p
• Path.Homotopic.Quotient.mk_surjective 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} : Function.Surjective Path.Homotopic.Quotient.mk
• Path.Homotopic.symm 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} ⦃p₀ p₁ : Path x₀ x₁⦄ (h : p₀.Homotopic p₁) : p₁.Homotopic p₀
• Path.Homotopy.symm 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) : p₁.Homotopy p₀
• Path.toHomotopyConst 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} (p : Path x₀ x₁) : (ContinuousMap.const Y x₀).Homotopy (ContinuousMap.const Y x₁)
• Path.Homotopy.eval 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) (t : ↑unitInterval) : Path x₀ x₁
• Path.Homotopic.Quotient.ind 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x y : X} {motive : Path.Homotopic.Quotient x y → Prop} (mk : ∀ (a : Path x y), motive (Path.Homotopic.Quotient.mk a)) (q : Path.Homotopic.Quotient x y) : motive q
• Path.Homotopy.symm_bijective 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} : Function.Bijective Path.Homotopy.symm
• Path.Homotopic.Quotient.mk_symm 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} (P : Path x₀ x₁) : Path.Homotopic.Quotient.mk P.symm = (Path.Homotopic.Quotient.mk P).symm
• Path.Homotopic.symm₂ 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p q : Path x₀ x₁} (h : p.Homotopic q) : p.symm.Homotopic q.symm
• Path.Homotopy.symm₂ 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p q : Path x₀ x₁} (F : p.Homotopy q) : p.symm.Homotopy q.symm
• Path.Homotopic.Quotient.exact 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p q : Path x₀ x₁} (h : Path.Homotopic.Quotient.mk p = Path.Homotopic.Quotient.mk q) : p.Homotopic q
• Path.Homotopic.Quotient.mk''_eq_mk 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) : Quotient.mk'' p = Path.Homotopic.Quotient.mk p
• Path.Homotopic.Quotient.mk'_eq_mk 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) : Quotient.mk' p = Path.Homotopic.Quotient.mk p
• Path.Homotopic.trans 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} ⦃p₀ p₁ p₂ : Path x₀ x₁⦄ (h₀ : p₀.Homotopic p₁) (h₁ : p₁.Homotopic p₂) : p₀.Homotopic p₂
• Path.Homotopy.trans 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ p₂ : Path x₀ x₁} (F : p₀.Homotopy p₁) (G : p₁.Homotopy p₂) : p₀.Homotopy p₂
• Path.Homotopic.Quotient.eq 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p q : Path x₀ x₁} : Path.Homotopic.Quotient.mk p = Path.Homotopic.Quotient.mk q ↔ p.Homotopic q
• Path.Homotopy.symm_symm 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) : F.symm.symm = F
• Path.Homotopy.eval_one 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) : F.eval 1 = p₁
• Path.Homotopy.eval_zero 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) : F.eval 0 = p₀
• Path.Homotopic.Quotient.mk_trans 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ x₂ : X} (P₀ : Path x₀ x₁) (P₁ : Path x₁ x₂) : Path.Homotopic.Quotient.mk (P₀.trans P₁) = (Path.Homotopic.Quotient.mk P₀).trans (Path.Homotopic.Quotient.mk P₁)
• Path.Homotopy.cast 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ q₀ q₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) (h₀ : p₀ = q₀) (h₁ : p₁ = q₁) : q₀.Homotopy q₁
• Path.Homotopic.Quotient.mk_cast 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x y : X} (P : Path x y) {x' y' : X} (hx : x' = x) (hy : y' = y) : Path.Homotopic.Quotient.mk (P.cast hx hy) = (Path.Homotopic.Quotient.mk P).cast hx hy
• Path.Homotopic.pathCast 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ x₂ x₃ : X} {p q : Path x₀ x₁} (hpq : p.Homotopic q) (hsource : x₂ = x₀) (htarget : x₃ = x₁) : (p.cast hsource htarget).Homotopic (q.cast hsource htarget)
• Path.Homotopy.pathCast 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x x' y y' : X} {p q : Path x y} (F : p.Homotopy q) (hx : x' = x) (hy : y' = y) : (p.cast hx hy).Homotopy (q.cast hx hy)
• Path.Homotopic.Quotient.ind₂ 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {Y : Type u_1} [TopologicalSpace Y] {x₀ y₀ : X} {x₁ y₁ : Y} {motive : Path.Homotopic.Quotient x₀ y₀ → Path.Homotopic.Quotient x₁ y₁ → Prop} (mk : ∀ (a : Path x₀ y₀) (b : Path x₁ y₁), motive (Path.Homotopic.Quotient.mk a) (Path.Homotopic.Quotient.mk b)) (q₀ : Path.Homotopic.Quotient x₀ y₀) (q₁ : Path.Homotopic.Quotient x₁ y₁) : motive q₀ q₁
• ContinuousMap.Homotopy.evalAt 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f g : C(X, Y)} (H : f.Homotopy g) (x : X) : Path (f x) (g x)
• Path.Homotopic.hcomp 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ x₂ : X} {p₀ p₁ : Path x₀ x₁} {q₀ q₁ : Path x₁ x₂} (hp : p₀.Homotopic p₁) (hq : q₀.Homotopic q₁) : (p₀.trans q₀).Homotopic (p₁.trans q₁)
• Path.Homotopy.hcomp 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ x₂ : X} {p₀ q₀ : Path x₀ x₁} {p₁ q₁ : Path x₁ x₂} (F : p₀.Homotopy q₀) (G : p₁.Homotopy q₁) : (p₀.trans p₁).Homotopy (q₀.trans q₁)
• Path.Homotopy.symm_trans 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ p₂ : Path x₀ x₁} (F : p₀.Homotopy p₁) (G : p₁.Homotopy p₂) : (F.trans G).symm = G.symm.trans F.symm
• Path.Homotopic.hpath_hext 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ x₂ x₃ : X} {p₁ : Path x₀ x₁} {p₂ : Path x₂ x₃} (hp : ∀ (t : ↑unitInterval), p₁ t = p₂ t) : ⟦p₁⟧ ≍ ⟦p₂⟧
• Path.toHomotopyConst_apply 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} (p : Path x₀ x₁) (a✝ : ↑unitInterval × Y) : p.toHomotopyConst a✝ = p a✝.1
• Path.Homotopic.map 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} {p q : Path x₀ x₁} (h : p.Homotopic q) (f : C(X, Y)) : (p.map ⋯).Homotopic (q.map ⋯)
• Path.Homotopy.map 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} {p q : Path x₀ x₁} (F : p.Homotopy q) (f : C(X, Y)) : (p.map ⋯).Homotopy (q.map ⋯)
• Path.Homotopic.Quotient.mk_map 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} (P₀ : Path x₀ x₁) (f : C(X, Y)) : Path.Homotopic.Quotient.mk (P₀.map ⋯) = (Path.Homotopic.Quotient.mk P₀).map f
• Path.Homotopy.reparam 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) (f : ↑unitInterval → ↑unitInterval) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) : p.Homotopy (p.reparam f hf hf₀ hf₁)
• ContinuousMap.Homotopy.evalAt_apply 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f g : C(X, Y)} (H : f.Homotopy g) (x : X) (t : ↑unitInterval) : (H.evalAt x) t = H (t, x)
• Path.Homotopy.coeFn_injective 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} : Function.Injective DFunLike.coe
• Path.Homotopy.source 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) (t : ↑unitInterval) : F (t, 0) = x₀
• Path.Homotopy.target 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) (t : ↑unitInterval) : F (t, 1) = x₁
• Path.Homotopy.hcomp_half 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ x₂ : X} {p₀ q₀ : Path x₀ x₁} {p₁ q₁ : Path x₁ x₂} (F : p₀.Homotopy q₀) (G : p₁.Homotopy q₁) (t : ↑unitInterval) : (F.hcomp G) (t, ⟨1 / 2, ⋯⟩) = x₁
• Path.Homotopy.eval_apply 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) (t a : ↑unitInterval) : (F.eval t) a = (F.curry t) a
• Path.Homotopy.refl_apply 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) (x : ↑unitInterval × ↑unitInterval) : (Path.Homotopy.refl p) x = p x.2
• Path.Homotopy.symm₂_apply 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p q : Path x₀ x₁} (F : p.Homotopy q) (x : ↑unitInterval × ↑unitInterval) : F.symm₂ x = F (x.1, unitInterval.symm x.2)
• Path.Homotopy.symm_apply 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) (x : ↑unitInterval × ↑unitInterval) : F.symm x = F (unitInterval.symm x.1, x.2)
• Path.Homotopy.cast_apply 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ q₀ q₁ : Path x₀ x₁} (F : p₀.Homotopy p₁) (h₀ : p₀ = q₀) (h₁ : p₁ = q₁) (a : ↑unitInterval × ↑unitInterval) : (F.cast h₀ h₁) a = F a
• Path.Homotopy.map_apply 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {x₀ x₁ : X} {p q : Path x₀ x₁} (F : p.Homotopy q) (f : C(X, Y)) (a✝ : ↑unitInterval × ↑unitInterval) : (F.map f) a✝ = (⇑f ∘ ⇑F) a✝
• Path.Homotopy.hcomp_apply 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ x₂ : X} {p₀ q₀ : Path x₀ x₁} {p₁ q₁ : Path x₁ x₂} (F : p₀.Homotopy q₀) (G : p₁.Homotopy q₁) (x : ↑unitInterval × ↑unitInterval) : (F.hcomp G) x = if h : ↑x.2 ≤ 1 / 2 then (F.eval x.1) ⟨2 * ↑x.2, ⋯⟩ else (G.eval x.1) ⟨2 * ↑x.2 - 1, ⋯⟩
• Path.Homotopy.trans_apply 📋 Mathlib.Topology.Homotopy.Path
  {X : Type u} [TopologicalSpace X] {x₀ x₁ : X} {p₀ p₁ p₂ : Path x₀ x₁} (F : p₀.Homotopy p₁) (G : p₁.Homotopy p₂) (x : ↑unitInterval × ↑unitInterval) : (F.trans G) x = if h : ↑x.1 ≤ 1 / 2 then F (⟨2 * ↑x.1, ⋯⟩, x.2) else G (⟨2 * ↑x.1 - 1, ⋯⟩, x.2)
• Path.Homotopic.refl_trans 📋 Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic
  {X : Type u_1} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) : ((Path.refl x₀).trans p).Homotopic p
• Path.Homotopic.trans_refl 📋 Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic
  {X : Type u_1} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) : (p.trans (Path.refl x₁)).Homotopic p
• Path.Homotopy.reflTrans 📋 Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic
  {X : Type u_1} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) : ((Path.refl x₀).trans p).Homotopy p
• Path.Homotopy.transRefl 📋 Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic
  {X : Type u_1} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) : (p.trans (Path.refl x₁)).Homotopy p
• Path.Homotopic.symm_trans 📋 Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic
  {X : Type u_1} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) : (p.symm.trans p).Homotopic (Path.refl x₁)
• Path.Homotopic.trans_symm 📋 Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic
  {X : Type u_1} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) : (p.trans p.symm).Homotopic (Path.refl x₀)
• Path.Homotopy.reflSymmTrans 📋 Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic
  {X : Type u_1} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) : (Path.refl x₁).Homotopy (p.symm.trans p)
• Path.Homotopy.reflTransSymm 📋 Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic
  {X : Type u_1} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) : (Path.refl x₀).Homotopy (p.trans p.symm)
• Path.Homotopic.trans_assoc 📋 Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic
  {X : Type u_1} [TopologicalSpace X] {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) : ((p.trans q).trans r).Homotopic (p.trans (q.trans r))
• Path.Homotopy.transAssoc 📋 Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic
  {X : Type u_1} [TopologicalSpace X] {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) : ((p.trans q).trans r).Homotopy (p.trans (q.trans r))
• FundamentalGroupoid.id_eq_path_refl 📋 Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic
  {X : Type u_1} [TopologicalSpace X] (x : FundamentalGroupoid X) : CategoryTheory.CategoryStruct.id x = ⟦Path.refl x.as⟧
• FundamentalGroupoid.fromPath_eq_iff_homotopic 📋 Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic
  {X : Type u_1} [TopologicalSpace X] {x₀ x₁ : X} (f g : Path x₀ x₁) : FundamentalGroupoid.fromPath (Path.Homotopic.Quotient.mk f) = FundamentalGroupoid.fromPath (Path.Homotopic.Quotient.mk g) ↔ f.Homotopic g
• Path.Homotopy.trans_refl_reparam 📋 Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic
  {X : Type u_1} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) : p.trans (Path.refl x₁) = p.reparam (fun t => ⟨Path.Homotopy.transReflReparamAux t, ⋯⟩) Path.Homotopy.trans_refl_reparam._proof_1 ⋯ ⋯
• Path.Homotopy.trans_assoc_reparam 📋 Mathlib.AlgebraicTopology.FundamentalGroupoid.Basic
  {X : Type u_1} [TopologicalSpace X] {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) : (p.trans q).trans r = (p.trans (q.trans r)).reparam (fun t => ⟨Path.Homotopy.transAssocReparamAux t, ⋯⟩) Path.Homotopy.trans_assoc_reparam._proof_1 ⋯ ⋯
• FundamentalGroup.fundamentalGroupMulEquivOfPath 📋 Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup
  {X : Type u_1} [TopologicalSpace X] {x₀ x₁ : X} (p : Path x₀ x₁) : FundamentalGroup X x₀ ≃* FundamentalGroup X x₁

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision 69fae59