ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AbsConvexOpenSets.coe_nhds
theorem coe_nhds (s : AbsConvexOpenSets 𝕜 E) : (s : Set E) ∈ 𝓝 (0 : E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace E] [AddCommMonoid E] [Zero E] [SeminormedRing 𝕜] [SMul 𝕜 E] [SMul ℝ E] (s : AbsConvexOpenSets 𝕜 E)
Goal: ↑s ∈ 𝓝 0
-- AbsConvexOpenSets.coe_balanced
theorem coe_balanced (s : AbsConvexOpenSets 𝕜 E) : Balanced 𝕜 (s : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace E] [AddCommMonoid E] [Zero E] [SeminormedRing 𝕜] [SMul 𝕜 E] [SMul ℝ E] (s : AbsConvexOpenSets 𝕜 E)
Goal: Balanced 𝕜 ↑s
-- AbsConvexOpenSets.coe_convex
theorem coe_convex (s : AbsConvexOpenSets 𝕜 E) : Convex ℝ (s : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [TopologicalSpace E] [AddCommMonoid E] [Zero E] [SeminormedRing 𝕜] [SMul 𝕜 E] [SMul ℝ E] (s : AbsConvexOpenSets 𝕜 E)
Goal: Convex ℝ ↑s
-- gaugeSeminormFamily_ball
theorem gaugeSeminormFamily_ball (s : AbsConvexOpenSets 𝕜 E) :
    (gaugeSeminormFamily 𝕜 E s).ball 0 1 = (s : Set E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [Module ℝ E] [IsScalarTower ℝ 𝕜 E] [ContinuousSMul ℝ E] (s : AbsConvexOpenSets 𝕜 E)
Goal: (gaugeSeminormFamily 𝕜 E s).ball 0 1 = ↑s
-- with_gaugeSeminormFamily
theorem with_gaugeSeminormFamily : WithSeminorms (gaugeSeminormFamily 𝕜 E) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [AddCommGroup E] [TopologicalSpace E] [Module 𝕜 E] [Module ℝ E] [IsScalarTower ℝ 𝕜 E] [ContinuousSMul ℝ E] [TopologicalAddGroup E] [ContinuousSMul 𝕜 E] [SMulCommClass ℝ 𝕜 E] [LocallyConvexSpace ℝ E]
Goal: WithSeminorms (gaugeSeminormFamily 𝕜 E)
-- LinearMap.coe_toSeminorm
theorem coe_toSeminorm {f : E →ₗ[𝕜] 𝕜} : ⇑f.toSeminorm = fun x => ‖f x‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : E →ₗ[𝕜] 𝕜}
Goal: ⇑f.toSeminorm = fun x => ‖f x‖
-- LinearMap.toSeminorm_apply
@[simp]
theorem toSeminorm_apply {f : E →ₗ[𝕜] 𝕜} {x : E} : f.toSeminorm x = ‖f x‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : E →ₗ[𝕜] 𝕜} {x : E}
Goal: f.toSeminorm x = ‖f x‖
-- LinearMap.toSeminorm_ball_zero
theorem toSeminorm_ball_zero {f : E →ₗ[𝕜] 𝕜} {r : ℝ} :
    Seminorm.ball f.toSeminorm 0 r = { x : E | ‖f x‖ < r } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] {f : E →ₗ[𝕜] 𝕜} {r : ℝ}
Goal: f.toSeminorm.ball 0 r = {x | ‖f x‖ < r}
-- LinearMap.toSeminorm_comp
theorem toSeminorm_comp (f : F →ₗ[𝕜] 𝕜) (g : E →ₗ[𝕜] F) :
    f.toSeminorm.comp g = (f.comp g).toSeminorm 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [AddCommGroup F] [Module 𝕜 F] (f : F →ₗ[𝕜] 𝕜) (g : E →ₗ[𝕜] F)
Goal: f.toSeminorm.comp g = (f ∘ₗ g).toSeminorm
-- LinearMap.toSeminormFamily_apply
@[simp]
theorem toSeminormFamily_apply {B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜} {x y} : (B.toSeminormFamily y) x = ‖B x y‖ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [AddCommGroup F] [Module 𝕜 F] {B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜} {x : E} {y : F}
Goal: (B.toSeminormFamily y) x = ‖(B x) y‖
-- LinearMap.hasBasis_weakBilin
theorem LinearMap.hasBasis_weakBilin (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) :
    (𝓝 (0 : WeakBilin B)).HasBasis B.toSeminormFamily.basisSets _root_.id 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [AddCommGroup F] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜)
Goal: (𝓝 0).HasBasis B.toSeminormFamily.basisSets id
-- LinearMap.weakBilin_withSeminorms
theorem LinearMap.weakBilin_withSeminorms (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) :
    WithSeminorms (LinearMap.toSeminormFamily B : F → Seminorm 𝕜 (WeakBilin B)) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NormedField 𝕜] [AddCommGroup E] [Module 𝕜 E] [AddCommGroup F] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜)
Goal: WithSeminorms B.toSeminormFamily
-- LinearMap.polar_mem_iff
theorem polar_mem_iff (s : Set E) (y : F) : y ∈ B.polar s ↔ ∀ x ∈ s, ‖B x y‖ ≤ 1 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NormedCommRing 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) (s : Set E) (y : F)
Goal: y ∈ B.polar s ↔ ∀ x ∈ s, ‖(B x) y‖ ≤ 1
-- LinearMap.polar_mem
theorem polar_mem (s : Set E) (y : F) (hy : y ∈ B.polar s) : ∀ x ∈ s, ‖B x y‖ ≤ 1 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NormedCommRing 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) (s : Set E) (y : F) (hy : y ∈ B.polar s) (x : E)
Goal: x ∈ s → ‖(B x) y‖ ≤ 1
-- LinearMap.zero_mem_polar
@[simp]
theorem zero_mem_polar (s : Set E) : (0 : F) ∈ B.polar s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NormedCommRing 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) (s : Set E)
Goal: 0 ∈ B.polar s
-- LinearMap.polar_eq_iInter
theorem polar_eq_iInter {s : Set E} : B.polar s = ⋂ x ∈ s, { y : F | ‖B x y‖ ≤ 1 } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NormedCommRing 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) {s : Set E}
Goal: B.polar s = ⋂ x ∈ s, {y | ‖(B x) y‖ ≤ 1}
-- LinearMap.polar_gc
theorem polar_gc :
    GaloisConnection (OrderDual.toDual ∘ B.polar) (B.flip.polar ∘ OrderDual.ofDual) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NormedCommRing 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜)
Goal: GaloisConnection (⇑OrderDual.toDual ∘ B.polar) (B.flip.polar ∘ ⇑OrderDual.ofDual)
-- LinearMap.polar_iUnion
@[simp]
theorem polar_iUnion {ι} {s : ι → Set E} : B.polar (⋃ i, s i) = ⋂ i, B.polar (s i) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NormedCommRing 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) {ι : Sort u_4} {s : ι → Set E}
Goal: B.polar (⋃ i, s i) = ⋂ i, B.polar (s i)
-- LinearMap.polar_union
@[simp]
theorem polar_union {s t : Set E} : B.polar (s ∪ t) = B.polar s ∩ B.polar t 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NormedCommRing 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) {s t : Set E}
Goal: B.polar (s ∪ t) = B.polar s ∩ B.polar t
-- LinearMap.polar_antitone
theorem polar_antitone : Antitone (B.polar : Set E → Set F) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NormedCommRing 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜)
Goal: Antitone B.polar
-- LinearMap.polar_empty
@[simp]
theorem polar_empty : B.polar ∅ = Set.univ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NormedCommRing 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜)
Goal: B.polar ∅ = Set.univ
-- LinearMap.polar_zero
@[simp]
theorem polar_zero : B.polar ({0} : Set E) = Set.univ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NormedCommRing 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜)
Goal: B.polar {0} = Set.univ
-- LinearMap.subset_bipolar
theorem subset_bipolar (s : Set E) : s ⊆ B.flip.polar (B.polar s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NormedCommRing 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) (s : Set E)
Goal: s ⊆ B.flip.polar (B.polar s)
-- LinearMap.tripolar_eq_polar
@[simp]
theorem tripolar_eq_polar (s : Set E) : B.polar (B.flip.polar (B.polar s)) = B.polar s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NormedCommRing 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) (s : Set E)
Goal: B.polar (B.flip.polar (B.polar s)) = B.polar s
-- LinearMap.polar_weak_closed
theorem polar_weak_closed (s : Set E) : IsClosed[WeakBilin.instTopologicalSpace B.flip]
    (B.polar s) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NormedCommRing 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) (s : Set E)
Goal: IsClosed (B.polar s)
-- LinearMap.polar_univ
theorem polar_univ (h : SeparatingRight B) : B.polar Set.univ = {(0 : F)} 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) (h : B.SeparatingRight)
Goal: B.polar Set.univ = {0}
-- LinearMap.polar_subMulAction
theorem polar_subMulAction {S : Type*} [SetLike S E] [SMulMemClass S 𝕜 E] (m : S) :
    B.polar m = { y | ∀ x ∈ m, B x y = 0 } 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [AddCommMonoid E] [AddCommMonoid F] [Module 𝕜 E] [Module 𝕜 F] (B : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜) {S : Type u_4} [SetLike S E] [SMulMemClass S 𝕜 E] (m : S)
Goal: B.polar ↑m = {y | ∀ x ∈ m, (B x) y = 0}
-- AntitoneOn.integral_le_sum
theorem AntitoneOn.integral_le_sum (hf : AntitoneOn f (Icc x₀ (x₀ + a))) :
    (∫ x in x₀..x₀ + a, f x) ≤ ∑ i ∈ Finset.range a, f (x₀ + i) 
Arguments: {x₀ : ℝ} {a : ℕ} {f : ℝ → ℝ} (hf : AntitoneOn f (Set.Icc x₀ (x₀ + ↑a)))
Goal: ∫ (x : ℝ) in x₀..x₀ + ↑a, f x ≤ ∑ i ∈ Finset.range a, f (x₀ + ↑i)
-- AntitoneOn.integral_le_sum_Ico
theorem AntitoneOn.integral_le_sum_Ico (hab : a ≤ b) (hf : AntitoneOn f (Set.Icc a b)) :
    (∫ x in a..b, f x) ≤ ∑ x ∈ Finset.Ico a b, f x 
Arguments: {a b : ℕ} {f : ℝ → ℝ} (hab : a ≤ b) (hf : AntitoneOn f (Set.Icc ↑a ↑b))
Goal: ∫ (x : ℝ) in ↑a..↑b, f x ≤ ∑ x ∈ Finset.Ico a b, f ↑x
-- AntitoneOn.sum_le_integral
theorem AntitoneOn.sum_le_integral (hf : AntitoneOn f (Icc x₀ (x₀ + a))) :
    (∑ i ∈ Finset.range a, f (x₀ + (i + 1 : ℕ))) ≤ ∫ x in x₀..x₀ + a, f x 
Arguments: {x₀ : ℝ} {a : ℕ} {f : ℝ → ℝ} (hf : AntitoneOn f (Set.Icc x₀ (x₀ + ↑a)))
Goal: ∑ i ∈ Finset.range a, f (x₀ + ↑(i + 1)) ≤ ∫ (x : ℝ) in x₀..x₀ + ↑a, f x
-- AntitoneOn.sum_le_integral_Ico
theorem AntitoneOn.sum_le_integral_Ico (hab : a ≤ b) (hf : AntitoneOn f (Set.Icc a b)) :
    (∑ i ∈ Finset.Ico a b, f (i + 1 : ℕ)) ≤ ∫ x in a..b, f x 
Arguments: {a b : ℕ} {f : ℝ → ℝ} (hab : a ≤ b) (hf : AntitoneOn f (Set.Icc ↑a ↑b))
Goal: ∑ i ∈ Finset.Ico a b, f ↑(i + 1) ≤ ∫ (x : ℝ) in ↑a..↑b, f x
-- MonotoneOn.sum_le_integral
theorem MonotoneOn.sum_le_integral (hf : MonotoneOn f (Icc x₀ (x₀ + a))) :
    (∑ i ∈ Finset.range a, f (x₀ + i)) ≤ ∫ x in x₀..x₀ + a, f x 
Arguments: {x₀ : ℝ} {a : ℕ} {f : ℝ → ℝ} (hf : MonotoneOn f (Set.Icc x₀ (x₀ + ↑a)))
Goal: ∑ i ∈ Finset.range a, f (x₀ + ↑i) ≤ ∫ (x : ℝ) in x₀..x₀ + ↑a, f x
-- MonotoneOn.sum_le_integral_Ico
theorem MonotoneOn.sum_le_integral_Ico (hab : a ≤ b) (hf : MonotoneOn f (Set.Icc a b)) :
    ∑ x ∈ Finset.Ico a b, f x ≤ ∫ x in a..b, f x 
Arguments: {a b : ℕ} {f : ℝ → ℝ} (hab : a ≤ b) (hf : MonotoneOn f (Set.Icc ↑a ↑b))
Goal: ∑ x ∈ Finset.Ico a b, f ↑x ≤ ∫ (x : ℝ) in ↑a..↑b, f x
-- MonotoneOn.integral_le_sum
theorem MonotoneOn.integral_le_sum (hf : MonotoneOn f (Icc x₀ (x₀ + a))) :
    (∫ x in x₀..x₀ + a, f x) ≤ ∑ i ∈ Finset.range a, f (x₀ + (i + 1 : ℕ)) 
Arguments: {x₀ : ℝ} {a : ℕ} {f : ℝ → ℝ} (hf : MonotoneOn f (Set.Icc x₀ (x₀ + ↑a)))
Goal: ∫ (x : ℝ) in x₀..x₀ + ↑a, f x ≤ ∑ i ∈ Finset.range a, f (x₀ + ↑(i + 1))
-- MonotoneOn.integral_le_sum_Ico
theorem MonotoneOn.integral_le_sum_Ico (hab : a ≤ b) (hf : MonotoneOn f (Set.Icc a b)) :
    (∫ x in a..b, f x) ≤ ∑ i ∈ Finset.Ico a b, f (i + 1 : ℕ) 
Arguments: {a b : ℕ} {f : ℝ → ℝ} (hab : a ≤ b) (hf : MonotoneOn f (Set.Icc ↑a ↑b))
Goal: ∫ (x : ℝ) in ↑a..↑b, f x ≤ ∑ i ∈ Finset.Ico a b, f ↑(i + 1)
-- AffineIsometry.norm_map
structure AffineIsometry extends P →ᵃ[𝕜] P₂ where
  norm_map : ∀ x : V, ‖linear x‖ = ‖x‖
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (self : P →ᵃⁱ[𝕜] P₂) (x : V)
Goal: ‖self.linear x‖ = ‖x‖
-- AffineIsometry.linear_eq_linearIsometry
@[simp]
theorem linear_eq_linearIsometry : f.linear = f.linearIsometry.toLinearMap 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂)
Goal: f.linear = f.linearIsometry.toLinearMap
-- AffineIsometry.coe_toAffineMap
@[simp]
theorem coe_toAffineMap : ⇑f.toAffineMap = f 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂)
Goal: ⇑f.toAffineMap = ⇑f
-- AffineIsometry.toAffineMap_injective
theorem toAffineMap_injective : Injective (toAffineMap : (P →ᵃⁱ[𝕜] P₂) → P →ᵃ[𝕜] P₂) 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂]
Goal: Function.Injective AffineIsometry.toAffineMap
-- AffineIsometry.coeFn_injective
theorem coeFn_injective : @Injective (P →ᵃⁱ[𝕜] P₂) (P → P₂) (↑) 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂]
Goal: Function.Injective DFunLike.coe
-- AffineIsometry.ext
@[ext]
theorem ext {f g : P →ᵃⁱ[𝕜] P₂} (h : ∀ x, f x = g x) : f = g 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] {f g : P →ᵃⁱ[𝕜] P₂} (h : ∀ (x : P), f x = g x)
Goal: f = g
-- LinearIsometry.coe_toAffineIsometry
@[simp]
theorem coe_toAffineIsometry : ⇑(f.toAffineIsometry : V →ᵃⁱ[𝕜] V₂) = f 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] (f : V →ₗᵢ[𝕜] V₂)
Goal: ⇑f.toAffineIsometry = ⇑f
-- LinearIsometry.toAffineIsometry_linearIsometry
@[simp]
theorem toAffineIsometry_linearIsometry : f.toAffineIsometry.linearIsometry = f 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] (f : V →ₗᵢ[𝕜] V₂)
Goal: f.toAffineIsometry.linearIsometry = f
-- LinearIsometry.toAffineIsometry_toAffineMap
@[simp]
theorem toAffineIsometry_toAffineMap : f.toAffineIsometry.toAffineMap = f.toLinearMap.toAffineMap 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] (f : V →ₗᵢ[𝕜] V₂)
Goal: f.toAffineIsometry.toAffineMap = f.toAffineMap
-- AffineIsometry.map_vadd
@[simp]
theorem map_vadd (p : P) (v : V) : f (v +ᵥ p) = f.linearIsometry v +ᵥ f p 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (p : P) (v : V)
Goal: f (v +ᵥ p) = f.linearIsometry v +ᵥ f p
-- AffineIsometry.map_vsub
@[simp]
theorem map_vsub (p1 p2 : P) : f.linearIsometry (p1 -ᵥ p2) = f p1 -ᵥ f p2 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (p1 p2 : P)
Goal: f.linearIsometry (p1 -ᵥ p2) = f p1 -ᵥ f p2
-- AffineIsometry.dist_map
@[simp]
theorem dist_map (x y : P) : dist (f x) (f y) = dist x y 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (x y : P)
Goal: dist (f x) (f y) = dist x y
-- AffineIsometry.nndist_map
@[simp]
theorem nndist_map (x y : P) : nndist (f x) (f y) = nndist x y 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (x y : P)
Goal: nndist (f x) (f y) = nndist x y
-- AffineIsometry.edist_map
@[simp]
theorem edist_map (x y : P) : edist (f x) (f y) = edist x y 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (x y : P)
Goal: edist (f x) (f y) = edist x y
-- AffineIsometry.isometry
protected theorem isometry : Isometry f 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂)
Goal: Isometry ⇑f
-- AffineIsometry.injective
protected theorem injective : Injective f₁ 
Arguments: {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁'] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f₁ : P₁' →ᵃⁱ[𝕜] P₂)
Goal: Function.Injective ⇑f₁
-- AffineIsometry.map_eq_iff
@[simp]
theorem map_eq_iff {x y : P₁'} : f₁ x = f₁ y ↔ x = y 
Arguments: {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁'] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f₁ : P₁' →ᵃⁱ[𝕜] P₂) {x y : P₁'}
Goal: f₁ x = f₁ y ↔ x = y
-- AffineIsometry.map_ne
theorem map_ne {x y : P₁'} (h : x ≠ y) : f₁ x ≠ f₁ y 
Arguments: {𝕜 : Type u_1} {V₁' : Type u_4} {V₂ : Type u_5} {P₁' : Type u_9} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V₁'] [NormedSpace 𝕜 V₁'] [MetricSpace P₁'] [NormedAddTorsor V₁' P₁'] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f₁ : P₁' →ᵃⁱ[𝕜] P₂) {x y : P₁'} (h : x ≠ y)
Goal: f₁ x ≠ f₁ y
-- AffineIsometry.lipschitz
protected theorem lipschitz : LipschitzWith 1 f 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂)
Goal: LipschitzWith 1 ⇑f
-- AffineIsometry.antilipschitz
protected theorem antilipschitz : AntilipschitzWith 1 f 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂)
Goal: AntilipschitzWith 1 ⇑f
-- AffineIsometry.continuous
@[continuity]
protected theorem continuous : Continuous f 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂)
Goal: Continuous ⇑f
-- AffineIsometry.ediam_image
theorem ediam_image (s : Set P) : EMetric.diam (f '' s) = EMetric.diam s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (s : Set P)
Goal: EMetric.diam (⇑f '' s) = EMetric.diam s
-- AffineIsometry.ediam_range
theorem ediam_range : EMetric.diam (range f) = EMetric.diam (univ : Set P) 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂)
Goal: EMetric.diam (Set.range ⇑f) = EMetric.diam Set.univ
-- AffineIsometry.diam_image
theorem diam_image (s : Set P) : Metric.diam (f '' s) = Metric.diam s 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) (s : Set P)
Goal: Metric.diam (⇑f '' s) = Metric.diam s
-- AffineIsometry.diam_range
theorem diam_range : Metric.diam (range f) = Metric.diam (univ : Set P) 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂)
Goal: Metric.diam (Set.range ⇑f) = Metric.diam Set.univ
-- AffineIsometry.comp_continuous_iff
@[simp]
theorem comp_continuous_iff {α : Type*} [TopologicalSpace α] {g : α → P} :
    Continuous (f ∘ g) ↔ Continuous g 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {P : Type u_10} {P₂ : Type u_11} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] (f : P →ᵃⁱ[𝕜] P₂) {α : Type u_14} [TopologicalSpace α] {g : α → P}
Goal: Continuous (⇑f ∘ g) ↔ Continuous g
-- AffineIsometry.coe_id
@[simp]
theorem coe_id : ⇑(id : P →ᵃⁱ[𝕜] P) = _root_.id 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P]
Goal: ⇑AffineIsometry.id = id
-- AffineIsometry.id_apply
@[simp]
theorem id_apply (x : P) : (AffineIsometry.id : P →ᵃⁱ[𝕜] P) x = x 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] (x : P)
Goal: AffineIsometry.id x = x
-- AffineIsometry.id_toAffineMap
@[simp]
theorem id_toAffineMap : (id.toAffineMap : P →ᵃ[𝕜] P) = AffineMap.id 𝕜 P 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {P : Type u_10} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P]
Goal: AffineIsometry.id.toAffineMap = AffineMap.id 𝕜 P
-- AffineIsometry.coe_comp
@[simp]
theorem coe_comp (g : P₂ →ᵃⁱ[𝕜] P₃) (f : P →ᵃⁱ[𝕜] P₂) : ⇑(g.comp f) = g ∘ f 
Arguments: {𝕜 : Type u_1} {V : Type u_2} {V₂ : Type u_5} {V₃ : Type u_6} {P : Type u_10} {P₂ : Type u_11} {P₃ : Type u_12} [NormedField 𝕜] [SeminormedAddCommGroup V] [NormedSpace 𝕜 V] [PseudoMetricSpace P] [NormedAddTorsor V P] [SeminormedAddCommGroup V₂] [NormedSpace 𝕜 V₂] [PseudoMetricSpace P₂] [NormedAddTorsor V₂ P₂] [SeminormedAddCommGroup V₃] [NormedSpace 𝕜 V₃] [PseudoMetricSpace P₃] [NormedAddTorsor V₃ P₃] (g : P₂ →ᵃⁱ[𝕜] P₃) (f : P →ᵃⁱ[𝕜] P₂)
Goal: ⇑(g.comp f) = ⇑g ∘ ⇑f