ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- AffineBasis.affineCombination_coord_eq_self
@[simp]
theorem affineCombination_coord_eq_self [Fintype ι] (q : P) :
    (Finset.univ.affineCombination k b fun i => b.coord i q) = q 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P) [Fintype ι] (q : P)
Goal: ((Finset.affineCombination k Finset.univ ⇑b) fun i => (b.coord i) q) = q
-- AffineBasis.linear_combination_coord_eq_self
@[simp]
theorem linear_combination_coord_eq_self [Fintype ι] (b : AffineBasis ι k V) (v : V) :
    ∑ i, b.coord i v • b i = v 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} [AddCommGroup V] [Ring k] [Module k V] [Fintype ι] (b : AffineBasis ι k V) (v : V)
Goal: ∑ i : ι, (b.coord i) v • b i = v
-- AffineBasis.ext_elem
theorem ext_elem [Finite ι] {q₁ q₂ : P} (h : ∀ i, b.coord i q₁ = b.coord i q₂) : q₁ = q₂ 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P) [Finite ι] {q₁ q₂ : P} (h : ∀ (i : ι), (b.coord i) q₁ = (b.coord i) q₂)
Goal: q₁ = q₂
-- AffineBasis.coe_coord_of_subsingleton_eq_one
@[simp]
theorem coe_coord_of_subsingleton_eq_one [Subsingleton ι] (i : ι) : (b.coord i : P → k) = 1 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P) [Subsingleton ι] (i : ι)
Goal: ⇑(b.coord i) = 1
-- AffineBasis.surjective_coord
theorem surjective_coord [Nontrivial ι] (i : ι) : Function.Surjective <| b.coord i 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P) [Nontrivial ι] (i : ι)
Goal: Function.Surjective ⇑(b.coord i)
-- AffineBasis.coords_apply
@[simp]
theorem coords_apply (q : P) (i : ι) : b.coords q i = b.coord i q 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (b : AffineBasis ι k P) (q : P) (i : ι)
Goal: b.coords q i = (b.coord i) q
-- AffineBasis.coe_vadd
@[simp, norm_cast] lemma coe_vadd (v : V) (b : AffineBasis ι k P) : ⇑(v +ᵥ b) = v +ᵥ ⇑b 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (v : V) (b : AffineBasis ι k P)
Goal: ⇑(v +ᵥ b) = v +ᵥ ⇑b
-- AffineBasis.basisOf_vadd
@[simp] lemma basisOf_vadd (v : V) (b : AffineBasis ι k P) : (v +ᵥ b).basisOf = b.basisOf 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] (v : V) (b : AffineBasis ι k P)
Goal: (v +ᵥ b).basisOf = b.basisOf
-- AffineBasis.coord_vadd
@[simp] lemma coord_vadd (v : V) (b : AffineBasis ι k P) :
    (v +ᵥ b).coord i = (b.coord i).comp (AffineEquiv.constVAdd k P v).symm 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [Ring k] [Module k V] {i : ι} (v : V) (b : AffineBasis ι k P)
Goal: (v +ᵥ b).coord i = (b.coord i).comp ↑(AffineEquiv.constVAdd k P v).symm
-- AffineBasis.coord_apply_centroid
@[simp]
theorem coord_apply_centroid [CharZero k] (b : AffineBasis ι k P) {s : Finset ι} {i : ι}
    (hi : i ∈ s) : b.coord i (s.centroid k b) = (s.card : k)⁻¹ 
Arguments: {ι : Type u_1} {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [DivisionRing k] [Module k V] [CharZero k] (b : AffineBasis ι k P) {s : Finset ι} {i : ι} (hi : i ∈ s)
Goal: (b.coord i) (Finset.centroid k s ⇑b) = (↑s.card)⁻¹
-- AffineBasis.exists_affine_subbasis
theorem exists_affine_subbasis {t : Set P} (ht : affineSpan k t = ⊤) :
    ∃ s ⊆ t, ∃ b : AffineBasis s k P, ⇑b = ((↑) : s → P) 
Arguments: {k : Type u_3} {V : Type u_4} {P : Type u_5} [AddCommGroup V] [AffineSpace V P] [DivisionRing k] [Module k V] {t : Set P} (ht : affineSpan k t = ⊤)
Goal: ∃ s ⊆ t, ∃ b, ⇑b = Subtype.val
-- AffineBasis.exists_affineBasis
theorem exists_affineBasis : ∃ (s : Set P) (b : AffineBasis (↥s) k P), ⇑b = ((↑) : s → P) 
Arguments: (k : Type u_3) (V : Type u_4) (P : Type u_5) [AddCommGroup V] [AffineSpace V P] [DivisionRing k] [Module k V]
Goal: ∃ s b, ⇑b = Subtype.val
-- ContinuousAffineEquiv.continuous_toFun
structure ContinuousAffineEquiv (k P₁ P₂ : Type*) {V₁ V₂ : Type*} [Ring k]
    [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁]
    [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₂]
    extends P₁ ≃ᵃ[k] P₂ where
  continuous_toFun : Continuous toFun := by continuity
  continuous_invFun : Continuous invFun := by continuity
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_4} {V₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₂] (self : P₁ ≃ᵃL[k] P₂)
Goal: Continuous (↑self).toFun
-- ContinuousAffineEquiv.continuous_invFun
structure ContinuousAffineEquiv (k P₁ P₂ : Type*) {V₁ V₂ : Type*} [Ring k]
    [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁]
    [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₂]
    extends P₁ ≃ᵃ[k] P₂ where
  continuous_toFun : Continuous toFun := by continuity
  continuous_invFun : Continuous invFun := by continuity
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_4} {V₂ : Type u_5} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₂] (self : P₁ ≃ᵃL[k] P₂)
Goal: Continuous (↑self).invFun
-- ContinuousAffineEquiv.toAffineEquiv_injective
theorem toAffineEquiv_injective : Injective (toAffineEquiv : (P₁ ≃ᵃL[k] P₂) → P₁ ≃ᵃ[k] P₂) 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂]
Goal: Function.Injective ContinuousAffineEquiv.toAffineEquiv
-- ContinuousAffineEquiv.coe_injective
theorem coe_injective : Function.Injective ((↑) : (P₁ ≃ᵃL[k] P₂) → P₁ ≃ᵃ[k] P₂) 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂]
Goal: Function.Injective ContinuousAffineEquiv.toAffineEquiv
-- ContinuousAffineEquiv.coe_coe
@[simp, norm_cast]
theorem coe_coe (e : P₁ ≃ᵃL[k] P₂) : ⇑(e : P₁ ≃ᵃ[k] P₂) = e 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂)
Goal: ⇑↑e = ⇑e
-- ContinuousAffineEquiv.coe_toEquiv
@[simp]
theorem coe_toEquiv (e : P₁ ≃ᵃL[k] P₂) : ⇑e.toEquiv = e 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂)
Goal: ⇑(↑e).toEquiv = ⇑e
-- ContinuousAffineEquiv.ext
@[ext]
theorem ext {e e' : P₁ ≃ᵃL[k] P₂} (h : ∀ x, e x = e' x) : e = e' 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] {e e' : P₁ ≃ᵃL[k] P₂} (h : ∀ (x : P₁), e x = e' x)
Goal: e = e'
-- ContinuousAffineEquiv.ext_iff
theorem ext_iff {e e' : P₁ ≃ᵃL[k] P₂} : e = e' ↔ ∀ x, e x = e' x 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] {e e' : P₁ ≃ᵃL[k] P₂}
Goal: e = e' ↔ ∀ (x : P₁), e x = e' x
-- ContinuousAffineEquiv.continuous
@[continuity]
protected theorem continuous (e : P₁ ≃ᵃL[k] P₂) : Continuous e 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂)
Goal: Continuous ⇑e
-- ContinuousAffineEquiv.coe_refl
@[simp]
theorem coe_refl : ⇑(refl k P₁) = id 
Arguments: {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁]
Goal: ⇑(ContinuousAffineEquiv.refl k P₁) = id
-- ContinuousAffineEquiv.refl_apply
@[simp]
theorem refl_apply (x : P₁) : refl k P₁ x = x 
Arguments: {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] (x : P₁)
Goal: (ContinuousAffineEquiv.refl k P₁) x = x
-- ContinuousAffineEquiv.toAffineEquiv_refl
@[simp]
theorem toAffineEquiv_refl : (refl k P₁).toAffineEquiv = AffineEquiv.refl k P₁ 
Arguments: {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁]
Goal: ↑(ContinuousAffineEquiv.refl k P₁) = AffineEquiv.refl k P₁
-- ContinuousAffineEquiv.toEquiv_refl
@[simp]
theorem toEquiv_refl : (refl k P₁).toEquiv = Equiv.refl P₁ 
Arguments: {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁]
Goal: (↑(ContinuousAffineEquiv.refl k P₁)).toEquiv = Equiv.refl P₁
-- ContinuousAffineEquiv.symm_toAffineEquiv
@[simp]
theorem symm_toAffineEquiv (e : P₁ ≃ᵃL[k] P₂) : e.toAffineEquiv.symm = e.symm.toAffineEquiv 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂)
Goal: (↑e).symm = ↑e.symm
-- ContinuousAffineEquiv.symm_toEquiv
@[simp]
theorem symm_toEquiv (e : P₁ ≃ᵃL[k] P₂) : e.toEquiv.symm = e.symm.toEquiv 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂)
Goal: (↑e).symm = (↑e.symm).toEquiv
-- ContinuousAffineEquiv.apply_symm_apply
@[simp]
theorem apply_symm_apply (e : P₁ ≃ᵃL[k] P₂) (p : P₂) : e (e.symm p) = p 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂) (p : P₂)
Goal: e (e.symm p) = p
-- ContinuousAffineEquiv.symm_apply_apply
@[simp]
theorem symm_apply_apply (e : P₁ ≃ᵃL[k] P₂) (p : P₁) : e.symm (e p) = p 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂) (p : P₁)
Goal: e.symm (e p) = p
-- ContinuousAffineEquiv.apply_eq_iff_eq_symm_apply
theorem apply_eq_iff_eq_symm_apply (e : P₁ ≃ᵃL[k] P₂) {p₁ p₂} : e p₁ = p₂ ↔ p₁ = e.symm p₂ 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂) {p₁ : P₁} {p₂ : P₂}
Goal: e p₁ = p₂ ↔ p₁ = e.symm p₂
-- ContinuousAffineEquiv.apply_eq_iff_eq
theorem apply_eq_iff_eq (e : P₁ ≃ᵃL[k] P₂) {p₁ p₂ : P₁} : e p₁ = e p₂ ↔ p₁ = p₂ 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂) {p₁ p₂ : P₁}
Goal: e p₁ = e p₂ ↔ p₁ = p₂
-- ContinuousAffineEquiv.symm_symm
@[simp]
theorem symm_symm (e : P₁ ≃ᵃL[k] P₂) : e.symm.symm = e 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂)
Goal: e.symm.symm = e
-- ContinuousAffineEquiv.symm_symm_apply
theorem symm_symm_apply (e : P₁ ≃ᵃL[k] P₂) (x : P₁) : e.symm.symm x = e x 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂) (x : P₁)
Goal: e.symm.symm x = e x
-- ContinuousAffineEquiv.symm_apply_eq
theorem symm_apply_eq (e : P₁ ≃ᵃL[k] P₂)  {x y} : e.symm x = y ↔ x = e y 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂) {x : P₂} {y : P₁}
Goal: e.symm x = y ↔ x = e y
-- ContinuousAffineEquiv.eq_symm_apply
theorem eq_symm_apply (e : P₁ ≃ᵃL[k] P₂) {x y} : y = e.symm x ↔ e y = x 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂) {x : P₂} {y : P₁}
Goal: y = e.symm x ↔ e y = x
-- ContinuousAffineEquiv.image_symm
@[simp]
theorem image_symm (f : P₁ ≃ᵃL[k] P₂) (s : Set P₂) : f.symm '' s = f ⁻¹' s 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (f : P₁ ≃ᵃL[k] P₂) (s : Set P₂)
Goal: ⇑f.symm '' s = ⇑f ⁻¹' s
-- ContinuousAffineEquiv.preimage_symm
@[simp]
theorem preimage_symm (f : P₁ ≃ᵃL[k] P₂) (s : Set P₁) : f.symm ⁻¹' s = f '' s 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (f : P₁ ≃ᵃL[k] P₂) (s : Set P₁)
Goal: ⇑f.symm ⁻¹' s = ⇑f '' s
-- ContinuousAffineEquiv.bijective
protected theorem bijective (e : P₁ ≃ᵃL[k] P₂) : Bijective e 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂)
Goal: Function.Bijective ⇑e
-- ContinuousAffineEquiv.surjective
protected theorem surjective (e : P₁ ≃ᵃL[k] P₂) : Surjective e 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂)
Goal: Function.Surjective ⇑e
-- ContinuousAffineEquiv.injective
protected theorem injective (e : P₁ ≃ᵃL[k] P₂) : Injective e 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂)
Goal: Function.Injective ⇑e
-- ContinuousAffineEquiv.image_eq_preimage
protected theorem image_eq_preimage (e : P₁ ≃ᵃL[k] P₂) (s : Set P₁) : e '' s = e.symm ⁻¹' s 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂) (s : Set P₁)
Goal: ⇑e '' s = ⇑e.symm ⁻¹' s
-- ContinuousAffineEquiv.image_symm_eq_preimage
protected theorem image_symm_eq_preimage (e : P₁ ≃ᵃL[k] P₂) (s : Set P₂) :
    e.symm '' s = e ⁻¹' s 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂) (s : Set P₂)
Goal: ⇑e.symm '' s = ⇑e ⁻¹' s
-- ContinuousAffineEquiv.image_preimage
@[simp]
theorem image_preimage (e : P₁ ≃ᵃL[k] P₂) (s : Set P₂) : e '' (e ⁻¹' s) = s 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂) (s : Set P₂)
Goal: ⇑e '' (⇑e ⁻¹' s) = s
-- ContinuousAffineEquiv.preimage_image
@[simp]
theorem preimage_image (e : P₁ ≃ᵃL[k] P₂) (s : Set P₁) : e ⁻¹' (e '' s) = s 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂) (s : Set P₁)
Goal: ⇑e ⁻¹' (⇑e '' s) = s
-- ContinuousAffineEquiv.symm_image_image
theorem symm_image_image (e : P₁ ≃ᵃL[k] P₂) (s : Set P₁) : e.symm '' (e '' s) = s 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂) (s : Set P₁)
Goal: ⇑e.symm '' (⇑e '' s) = s
-- ContinuousAffineEquiv.image_symm_image
theorem image_symm_image (e : P₁ ≃ᵃL[k] P₂) (s : Set P₂) : e '' (e.symm '' s) = s 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂) (s : Set P₂)
Goal: ⇑e '' (⇑e.symm '' s) = s
-- ContinuousAffineEquiv.refl_symm
@[simp]
theorem refl_symm : (refl k P₁).symm = refl k P₁ 
Arguments: {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁]
Goal: (ContinuousAffineEquiv.refl k P₁).symm = ContinuousAffineEquiv.refl k P₁
-- ContinuousAffineEquiv.symm_refl
@[simp]
theorem symm_refl : (refl k P₁).symm = refl k P₁ 
Arguments: {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁]
Goal: (ContinuousAffineEquiv.refl k P₁).symm = ContinuousAffineEquiv.refl k P₁
-- ContinuousAffineEquiv.coe_trans
@[simp]
theorem coe_trans (e : P₁ ≃ᵃL[k] P₂) (e' : P₂ ≃ᵃL[k] P₃) : ⇑(e.trans e') = e' ∘ e 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {P₃ : Type u_4} {V₁ : Type u_6} {V₂ : Type u_7} {V₃ : Type u_8} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [AddCommGroup V₃] [Module k V₃] [AddTorsor V₃ P₃] [TopologicalSpace P₁] [TopologicalSpace P₂] [TopologicalSpace P₃] (e : P₁ ≃ᵃL[k] P₂) (e' : P₂ ≃ᵃL[k] P₃)
Goal: ⇑(e.trans e') = ⇑e' ∘ ⇑e
-- ContinuousAffineEquiv.trans_apply
@[simp]
theorem trans_apply (e : P₁ ≃ᵃL[k] P₂) (e' : P₂ ≃ᵃL[k] P₃) (p : P₁) : e.trans e' p = e' (e p) 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {P₃ : Type u_4} {V₁ : Type u_6} {V₂ : Type u_7} {V₃ : Type u_8} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [AddCommGroup V₃] [Module k V₃] [AddTorsor V₃ P₃] [TopologicalSpace P₁] [TopologicalSpace P₂] [TopologicalSpace P₃] (e : P₁ ≃ᵃL[k] P₂) (e' : P₂ ≃ᵃL[k] P₃) (p : P₁)
Goal: (e.trans e') p = e' (e p)
-- ContinuousAffineEquiv.trans_assoc
theorem trans_assoc (e₁ : P₁ ≃ᵃL[k] P₂) (e₂ : P₂ ≃ᵃL[k] P₃) (e₃ : P₃ ≃ᵃL[k] P₄) :
    (e₁.trans e₂).trans e₃ = e₁.trans (e₂.trans e₃) 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {P₃ : Type u_4} {P₄ : Type u_5} {V₁ : Type u_6} {V₂ : Type u_7} {V₃ : Type u_8} {V₄ : Type u_9} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [AddCommGroup V₃] [Module k V₃] [AddTorsor V₃ P₃] [AddCommGroup V₄] [Module k V₄] [AddTorsor V₄ P₄] [TopologicalSpace P₁] [TopologicalSpace P₂] [TopologicalSpace P₃] [TopologicalSpace P₄] (e₁ : P₁ ≃ᵃL[k] P₂) (e₂ : P₂ ≃ᵃL[k] P₃) (e₃ : P₃ ≃ᵃL[k] P₄)
Goal: (e₁.trans e₂).trans e₃ = e₁.trans (e₂.trans e₃)
-- ContinuousAffineEquiv.trans_refl
@[simp]
theorem trans_refl (e : P₁ ≃ᵃL[k] P₂) : e.trans (refl k P₂) = e 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂)
Goal: e.trans (ContinuousAffineEquiv.refl k P₂) = e
-- ContinuousAffineEquiv.refl_trans
@[simp]
theorem refl_trans (e : P₁ ≃ᵃL[k] P₂) : (refl k P₁).trans e = e 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂)
Goal: (ContinuousAffineEquiv.refl k P₁).trans e = e
-- ContinuousAffineEquiv.self_trans_symm
@[simp]
theorem self_trans_symm (e : P₁ ≃ᵃL[k] P₂) : e.trans e.symm = refl k P₁ 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂)
Goal: e.trans e.symm = ContinuousAffineEquiv.refl k P₁
-- ContinuousAffineEquiv.symm_trans_self
@[simp]
theorem symm_trans_self (e : P₁ ≃ᵃL[k] P₂) : e.symm.trans e = refl k P₂ 
Arguments: {k : Type u_1} {P₁ : Type u_2} {P₂ : Type u_3} {V₁ : Type u_6} {V₂ : Type u_7} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂] [TopologicalSpace P₁] [TopologicalSpace P₂] (e : P₁ ≃ᵃL[k] P₂)
Goal: e.symm.trans e = ContinuousAffineEquiv.refl k P₂
-- ContinuousLinearEquiv.coe_toContinuousAffineEquiv
@[simp]
theorem _root_.ContinuousLinearEquiv.coe_toContinuousAffineEquiv (e : E ≃L[k] F) :
    ⇑e.toContinuousAffineEquiv = e 
Arguments: {k : Type u_1} [Ring k] {E : Type u_10} {F : Type u_11} [AddCommGroup E] [Module k E] [TopologicalSpace E] [AddCommGroup F] [Module k F] [TopologicalSpace F] (e : E ≃L[k] F)
Goal: ⇑e.toContinuousAffineEquiv = ⇑e
-- ContinuousAffineEquiv.constVAdd_coe
lemma constVAdd_coe [ContinuousConstVAdd V₁ P₁] (v : V₁) :
    (constVAdd k P₁ v).toAffineEquiv = .constVAdd k P₁ v 
Arguments: {k : Type u_1} {P₁ : Type u_2} {V₁ : Type u_6} [Ring k] [AddCommGroup V₁] [Module k V₁] [AddTorsor V₁ P₁] [TopologicalSpace P₁] [ContinuousConstVAdd V₁ P₁] (v : V₁)
Goal: ↑(ContinuousAffineEquiv.constVAdd k P₁ v) = AffineEquiv.constVAdd k P₁ v
-- LinearMap.BilinForm.IsRefl.eq_zero
theorem eq_zero : ∀ {x y : M}, B x y = 0 → B y x = 0 
Arguments: {R : Type u_1} {M : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] {B : LinearMap.BilinForm R M} (H : B.IsRefl) {x y : M}
Goal: (B x) y = 0 → (B y) x = 0
-- LinearMap.BilinForm.IsRefl.neg
protected theorem neg {B : BilinForm R₁ M₁} (hB : B.IsRefl) : (-B).IsRefl 
Arguments: {R₁ : Type u_3} {M₁ : Type u_4} [CommRing R₁] [AddCommGroup M₁] [Module R₁ M₁] {B : LinearMap.BilinForm R₁ M₁} (hB : B.IsRefl)
Goal: (-B).IsRefl