ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- OreLocalization.smul_one_oreDiv_one_smul
@[to_additive]
theorem smul_one_oreDiv_one_smul (r : R) (x : X[S⁻¹]) :
    ((r • 1 : M) /ₒ (1 : S)) • x = r • x 
Arguments: {R : Type u_1} {M : Type u_3} {X : Type u_4} [Monoid M] {S : Submonoid M} [OreLocalization.OreSet S] [MulAction M X] [SMul R X] [SMul R M] [IsScalarTower R M M] [IsScalarTower R M X] (r : R) (x : OreLocalization S X)
Goal: (r • 1 /ₒ 1) • x = r • x
-- AddOreLocalization.vadd_zero_oreSub_zero_vadd
@[to_additive]
theorem smul_one_oreDiv_one_smul (r : R) (x : X[S⁻¹]) :
    ((r • 1 : M) /ₒ (1 : S)) • x = r • x 
Arguments: {R : Type u_1} {M : Type u_3} {X : Type u_4} [AddMonoid M] {S : AddSubmonoid M} [AddOreLocalization.AddOreSet S] [AddAction M X] [VAdd R X] [VAdd R M] [VAddAssocClass R M M] [VAddAssocClass R M X] (r : R) (x : AddOreLocalization S X)
Goal: r +ᵥ 0 -ₒ 0 +ᵥ x = r +ᵥ x
-- OreLocalization.smul_oreDiv_one
@[to_additive]
theorem smul_oreDiv_one (r : R) (x : X) : r • (x /ₒ (1 : S)) = (r • x) /ₒ (1 : S) 
Arguments: {R : Type u_1} {M : Type u_3} {X : Type u_4} [Monoid M] {S : Submonoid M} [OreLocalization.OreSet S] [MulAction M X] [SMul R X] [SMul R M] [IsScalarTower R M M] [IsScalarTower R M X] (r : R) (x : X)
Goal: r • (x /ₒ 1) = r • x /ₒ 1
-- AddOreLocalization.vadd_oreSub_zero
@[to_additive]
theorem smul_oreDiv_one (r : R) (x : X) : r • (x /ₒ (1 : S)) = (r • x) /ₒ (1 : S) 
Arguments: {R : Type u_1} {M : Type u_3} {X : Type u_4} [AddMonoid M] {S : AddSubmonoid M} [AddOreLocalization.AddOreSet S] [AddAction M X] [VAdd R X] [VAdd R M] [VAddAssocClass R M M] [VAddAssocClass R M X] (r : R) (x : X)
Goal: r +ᵥ (x -ₒ 0) = r +ᵥ x -ₒ 0
-- OreLocalization.oreDiv_mul_oreDiv_comm
@[to_additive]
theorem oreDiv_mul_oreDiv_comm {r₁ r₂ : R} {s₁ s₂ : S} :
    r₁ /ₒ s₁ * (r₂ /ₒ s₂) = r₁ * r₂ /ₒ (s₁ * s₂) 
Arguments: {R : Type u_1} [CommMonoid R] {S : Submonoid R} [OreLocalization.OreSet S] {r₁ r₂ : R} {s₁ s₂ : ↥S}
Goal: r₁ /ₒ s₁ * (r₂ /ₒ s₂) = r₁ * r₂ /ₒ (s₁ * s₂)
-- AddOreLocalization.oreSub_add_oreSub_comm
@[to_additive]
theorem oreDiv_mul_oreDiv_comm {r₁ r₂ : R} {s₁ s₂ : S} :
    r₁ /ₒ s₁ * (r₂ /ₒ s₂) = r₁ * r₂ /ₒ (s₁ * s₂) 
Arguments: {R : Type u_1} [AddCommMonoid R] {S : AddSubmonoid R} [AddOreLocalization.AddOreSet S] {r₁ r₂ : R} {s₁ s₂ : ↥S}
Goal: r₁ -ₒ s₁ + (r₂ -ₒ s₂) = r₁ + r₂ -ₒ (s₁ + s₂)
-- OreLocalization.zero_def
protected theorem zero_def : (0 : X[S⁻¹]) = 0 /ₒ 1 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {X : Type u_2} [Zero X] [MulAction R X]
Goal: 0 = 0 /ₒ 1
-- OreLocalization.zero_oreDiv'
@[simp]
theorem zero_oreDiv' (s : S) : (0 : R) /ₒ s = 0 
Arguments: {R : Type u_1} [MonoidWithZero R] {S : Submonoid R} [OreLocalization.OreSet S] (s : ↥S)
Goal: 0 /ₒ s = 0
-- _private.Mathlib.RingTheory.OreLocalization.Basic.0.OreLocalization.add''_char
private theorem add''_char (r₁ : X) (s₁ : S) (r₂ : X) (s₂ : S) (rb : R) (sb : R)
    (hb : sb * s₁ = rb * s₂) (h : sb * s₁ ∈ S) :
    add'' r₁ s₁ r₂ s₂ = (sb • r₁ + rb • r₂) /ₒ ⟨sb * s₁, h⟩ 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {X : Type u_2} [AddMonoid X] [DistribMulAction R X] (r₁ : X) (s₁ : ↥S) (r₂ : X) (s₂ : ↥S) (rb sb : R) (hb : sb * ↑s₁ = rb * ↑s₂) (h : sb * ↑s₁ ∈ S)
Goal: OreLocalization.add'' r₁ s₁ r₂ s₂ = (sb • r₁ + rb • r₂) /ₒ ⟨sb * ↑s₁, h⟩
-- OreLocalization.oreDiv_add_oreDiv
theorem oreDiv_add_oreDiv {r r' : X} {s s' : S} :
    r /ₒ s + r' /ₒ s' =
      (oreDenom (s : R) s' • r + oreNum (s : R) s' • r') /ₒ (oreDenom (s : R) s' * s) 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {X : Type u_2} [AddMonoid X] [DistribMulAction R X] {r r' : X} {s s' : ↥S}
Goal: r /ₒ s + r' /ₒ s' =
  (OreLocalization.oreDenom (↑s) s' • r + OreLocalization.oreNum (↑s) s' • r') /ₒ (OreLocalization.oreDenom (↑s) s' * s)
-- OreLocalization.oreDiv_add_char'
theorem oreDiv_add_char' {r r' : X} (s s' : S) (rb : R) (sb : R)
    (h : sb * s = rb * s') (h' : sb * s ∈ S) :
    r /ₒ s + r' /ₒ s' = (sb • r + rb • r') /ₒ ⟨sb * s, h'⟩ 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {X : Type u_2} [AddMonoid X] [DistribMulAction R X] {r r' : X} (s s' : ↥S) (rb sb : R) (h : sb * ↑s = rb * ↑s') (h' : sb * ↑s ∈ S)
Goal: r /ₒ s + r' /ₒ s' = (sb • r + rb • r') /ₒ ⟨sb * ↑s, h'⟩
-- OreLocalization.oreDiv_add_char
theorem oreDiv_add_char {r r' : X} (s s' : S) (rb : R) (sb : S) (h : sb * s = rb * s') :
    r /ₒ s + r' /ₒ s' = (sb • r + rb • r') /ₒ (sb * s) 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {X : Type u_2} [AddMonoid X] [DistribMulAction R X] {r r' : X} (s s' : ↥S) (rb : R) (sb : ↥S) (h : ↑sb * ↑s = rb * ↑s')
Goal: r /ₒ s + r' /ₒ s' = (sb • r + rb • r') /ₒ (sb * s)
-- OreLocalization.add_oreDiv
@[simp]
theorem add_oreDiv {r r' : X} {s : S} : r /ₒ s + r' /ₒ s = (r + r') /ₒ s 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {X : Type u_2} [AddMonoid X] [DistribMulAction R X] {r r' : X} {s : ↥S}
Goal: r /ₒ s + r' /ₒ s = (r + r') /ₒ s
-- OreLocalization.add_assoc
protected theorem add_assoc (x y z : X[S⁻¹]) : x + y + z = x + (y + z) 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {X : Type u_2} [AddMonoid X] [DistribMulAction R X] (x y z : OreLocalization S X)
Goal: x + y + z = x + (y + z)
-- OreLocalization.zero_oreDiv
@[simp]
theorem zero_oreDiv (s : S) : (0 : X) /ₒ s = 0 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {X : Type u_2} [AddMonoid X] [DistribMulAction R X] (s : ↥S)
Goal: 0 /ₒ s = 0
-- OreLocalization.zero_add
protected theorem zero_add (x : X[S⁻¹]) : 0 + x = x 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {X : Type u_2} [AddMonoid X] [DistribMulAction R X] (x : OreLocalization S X)
Goal: 0 + x = x
-- OreLocalization.add_zero
protected theorem add_zero (x : X[S⁻¹]) : x + 0 = x 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {X : Type u_2} [AddMonoid X] [DistribMulAction R X] (x : OreLocalization S X)
Goal: x + 0 = x
-- OreLocalization.smul_zero
protected theorem smul_zero (x : R[S⁻¹]) : x • (0 : X[S⁻¹]) = 0 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {X : Type u_2} [AddMonoid X] [DistribMulAction R X] (x : OreLocalization S R)
Goal: x • 0 = 0
-- OreLocalization.smul_add
protected theorem smul_add (z : R[S⁻¹]) (x y : X[S⁻¹]) :
    z • (x + y) = z • x + z • y 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {X : Type u_2} [AddMonoid X] [DistribMulAction R X] (z : OreLocalization S R) (x y : OreLocalization S X)
Goal: z • (x + y) = z • x + z • y
-- OreLocalization.add_comm
protected theorem add_comm (x y : X[S⁻¹]) : x + y = y + x 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {X : Type u_2} [AddCommMonoid X] [DistribMulAction R X] (x y : OreLocalization S X)
Goal: x + y = y + x
-- OreLocalization.neg_def
@[simp]
protected theorem neg_def (r : X) (s : S) : -(r /ₒ s) = -r /ₒ s 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {X : Type u_2} [AddGroup X] [DistribMulAction R X] (r : X) (s : ↥S)
Goal: -(r /ₒ s) = -r /ₒ s
-- OreLocalization.add_left_neg
protected theorem add_left_neg (x : X[S⁻¹]) : -x + x = 0 
Arguments: {R : Type u_1} [Monoid R] {S : Submonoid R} [OreLocalization.OreSet S] {X : Type u_2} [AddGroup X] [DistribMulAction R X] (x : OreLocalization S X)
Goal: -x + x = 0
-- Ideal.finrank_quotient_eq_sum
theorem finrank_quotient_eq_sum {ι} [Fintype ι] (b : Basis ι R S) [Nontrivial F]
    [∀ i, Module.Free F (R ⧸ span ({I.smithCoeffs b hI i} : Set R))]
    [∀ i, Module.Finite F (R ⧸ span ({I.smithCoeffs b hI i} : Set R))] :
    FiniteDimensional.finrank F (S ⧸ I) =
      ∑ i, FiniteDimensional.finrank F (R ⧸ span ({I.smithCoeffs b hI i} : Set R)) 
Arguments: {R : Type u_2} {S : Type u_3} [CommRing R] [CommRing S] [Algebra R S] [IsDomain R] [IsPrincipalIdealRing R] [IsDomain S] (F : Type u_4) [CommRing F] [Algebra F R] [Algebra F S] [IsScalarTower F R S] {I : Ideal S} (hI : I ≠ ⊥) {ι : Type u_5} [Fintype ι] (b : Basis ι R S) [Nontrivial F] [∀ (i : ι), Module.Free F (R ⧸ Ideal.span {Ideal.smithCoeffs b I hI i})] [∀ (i : ι), Module.Finite F (R ⧸ Ideal.span {Ideal.smithCoeffs b I hI i})]
Goal: FiniteDimensional.finrank F (S ⧸ I) = ∑ i : ι, FiniteDimensional.finrank F (R ⧸ Ideal.span {Ideal.smithCoeffs b I hI i})
-- Orientation.map_apply
@[simp]
theorem Orientation.map_apply (e : M ≃ₗ[R] N) (v : M [⋀^ι]→ₗ[R] R) (hv : v ≠ 0) :
    Orientation.map ι e (rayOfNeZero _ v hv) =
      rayOfNeZero _ (v.compLinearMap e.symm) (mt (v.compLinearEquiv_eq_zero_iff e.symm).mp hv) 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] (ι : Type u_4) (e : M ≃ₗ[R] N) (v : M [⋀^ι]→ₗ[R] R) (hv : v ≠ 0)
Goal: (Orientation.map ι e) (rayOfNeZero R v hv) = rayOfNeZero R (v.compLinearMap ↑e.symm) ⋯
-- Orientation.map_refl
@[simp]
theorem Orientation.map_refl : (Orientation.map ι <| LinearEquiv.refl R M) = Equiv.refl _ 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (ι : Type u_4)
Goal: Orientation.map ι (LinearEquiv.refl R M) = Equiv.refl (Orientation R M ι)
-- Orientation.map_symm
@[simp]
theorem Orientation.map_symm (e : M ≃ₗ[R] N) :
    (Orientation.map ι e).symm = Orientation.map ι e.symm 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] (ι : Type u_4) (e : M ≃ₗ[R] N)
Goal: (Orientation.map ι e).symm = Orientation.map ι e.symm
-- Orientation.reindex_apply
@[simp]
theorem Orientation.reindex_apply (e : ι ≃ ι') (v : M [⋀^ι]→ₗ[R] R) (hv : v ≠ 0) :
    Orientation.reindex R M e (rayOfNeZero _ v hv) =
      rayOfNeZero _ (v.domDomCongr e) (mt (v.domDomCongr_eq_zero_iff e).mp hv) 
Arguments: (R : Type u_1) [StrictOrderedCommSemiring R] (M : Type u_2) [AddCommMonoid M] [Module R M] {ι : Type u_4} {ι' : Type u_5} (e : ι ≃ ι') (v : M [⋀^ι]→ₗ[R] R) (hv : v ≠ 0)
Goal: (Orientation.reindex R M e) (rayOfNeZero R v hv) = rayOfNeZero R (AlternatingMap.domDomCongr e v) ⋯
-- Orientation.reindex_refl
@[simp]
theorem Orientation.reindex_refl : (Orientation.reindex R M <| Equiv.refl ι) = Equiv.refl _ 
Arguments: (R : Type u_1) [StrictOrderedCommSemiring R] (M : Type u_2) [AddCommMonoid M] [Module R M] {ι : Type u_4}
Goal: Orientation.reindex R M (Equiv.refl ι) = Equiv.refl (Orientation R M ι)
-- Orientation.reindex_symm
@[simp]
theorem Orientation.reindex_symm (e : ι ≃ ι') :
    (Orientation.reindex R M e).symm = Orientation.reindex R M e.symm 
Arguments: (R : Type u_1) [StrictOrderedCommSemiring R] (M : Type u_2) [AddCommMonoid M] [Module R M] {ι : Type u_4} {ι' : Type u_5} (e : ι ≃ ι')
Goal: (Orientation.reindex R M e).symm = Orientation.reindex R M e.symm
-- Orientation.map_positiveOrientation_of_isEmpty
@[simp]
theorem Orientation.map_positiveOrientation_of_isEmpty [IsEmpty ι] (f : M ≃ₗ[R] N) :
    Orientation.map ι f positiveOrientation = positiveOrientation 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] {N : Type u_3} [AddCommMonoid N] [Module R N] (ι : Type u_4) [IsEmpty ι] (f : M ≃ₗ[R] N)
Goal: (Orientation.map ι f) positiveOrientation = positiveOrientation
-- Orientation.map_of_isEmpty
@[simp]
theorem Orientation.map_of_isEmpty [IsEmpty ι] (x : Orientation R M ι) (f : M ≃ₗ[R] M) :
    Orientation.map ι f x = x 
Arguments: {R : Type u_1} [StrictOrderedCommSemiring R] {M : Type u_2} [AddCommMonoid M] [Module R M] (ι : Type u_4) [IsEmpty ι] (x : Orientation R M ι) (f : M ≃ₗ[R] M)
Goal: (Orientation.map ι f) x = x
-- Orientation.map_neg
@[simp]
protected theorem Orientation.map_neg {ι : Type*} (f : M ≃ₗ[R] N) (x : Orientation R M ι) :
    Orientation.map ι f (-x) = -Orientation.map ι f x 
Arguments: {R : Type u_1} [StrictOrderedCommRing R] {M : Type u_2} {N : Type u_3} [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] {ι : Type u_4} (f : M ≃ₗ[R] N) (x : Orientation R M ι)
Goal: (Orientation.map ι f) (-x) = -(Orientation.map ι f) x
-- Orientation.reindex_neg
@[simp]
protected theorem Orientation.reindex_neg {ι ι' : Type*} (e : ι ≃ ι') (x : Orientation R M ι) :
    Orientation.reindex R M e (-x) = -Orientation.reindex R M e x 
Arguments: {R : Type u_1} [StrictOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_4} {ι' : Type u_5} (e : ι ≃ ι') (x : Orientation R M ι)
Goal: (Orientation.reindex R M e) (-x) = -(Orientation.reindex R M e) x
-- Basis.map_orientation_eq_det_inv_smul
theorem map_orientation_eq_det_inv_smul [Finite ι] (e : Basis ι R M) (x : Orientation R M ι)
    (f : M ≃ₗ[R] M) : Orientation.map ι f x = (LinearEquiv.det f)⁻¹ • x 
Arguments: {R : Type u_1} [StrictOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_4} [Finite ι] (e : Basis ι R M) (x : Orientation R M ι) (f : M ≃ₗ[R] M)
Goal: (Orientation.map ι f) x = (LinearEquiv.det f)⁻¹ • x
-- Basis.orientation_map
theorem orientation_map (e : Basis ι R M) (f : M ≃ₗ[R] N) :
    (e.map f).orientation = Orientation.map ι f e.orientation 
Arguments: {R : Type u_1} [StrictOrderedCommRing R] {M : Type u_2} {N : Type u_3} [AddCommGroup M] [AddCommGroup N] [Module R M] [Module R N] {ι : Type u_4} [Fintype ι] [DecidableEq ι] (e : Basis ι R M) (f : M ≃ₗ[R] N)
Goal: (e.map f).orientation = (Orientation.map ι f) e.orientation
-- Basis.orientation_reindex
theorem orientation_reindex (e : Basis ι R M) (eι : ι ≃ ι') :
    (e.reindex eι).orientation = Orientation.reindex R M eι e.orientation 
Arguments: {R : Type u_1} [StrictOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_4} {ι' : Type u_5} [Fintype ι] [DecidableEq ι] [Fintype ι'] [DecidableEq ι'] (e : Basis ι R M) (eι : ι ≃ ι')
Goal: (e.reindex eι).orientation = (Orientation.reindex R M eι) e.orientation
-- Basis.orientation_unitsSMul
theorem orientation_unitsSMul (e : Basis ι R M) (w : ι → Units R) :
    (e.unitsSMul w).orientation = (∏ i, w i)⁻¹ • e.orientation 
Arguments: {R : Type u_1} [StrictOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_4} [Fintype ι] [DecidableEq ι] (e : Basis ι R M) (w : ι → Rˣ)
Goal: (e.unitsSMul w).orientation = (∏ i : ι, w i)⁻¹ • e.orientation
-- Basis.orientation_isEmpty
@[simp]
theorem orientation_isEmpty [IsEmpty ι] (b : Basis ι R M) :
    b.orientation = positiveOrientation 
Arguments: {R : Type u_1} [StrictOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_4} [Fintype ι] [DecidableEq ι] [IsEmpty ι] (b : Basis ι R M)
Goal: b.orientation = positiveOrientation
-- Orientation.eq_or_eq_neg_of_isEmpty
theorem eq_or_eq_neg_of_isEmpty [IsEmpty ι] (o : Orientation R M ι) :
    o = positiveOrientation ∨ o = -positiveOrientation 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_3} [IsEmpty ι] (o : Orientation R M ι)
Goal: o = positiveOrientation ∨ o = -positiveOrientation
-- Basis.orientation_eq_iff_det_pos
theorem orientation_eq_iff_det_pos (e₁ e₂ : Basis ι R M) :
    e₁.orientation = e₂.orientation ↔ 0 < e₁.det e₂ 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_3} [Fintype ι] [DecidableEq ι] (e₁ e₂ : Basis ι R M)
Goal: e₁.orientation = e₂.orientation ↔ 0 < e₁.det ⇑e₂
-- Basis.orientation_eq_or_eq_neg
theorem orientation_eq_or_eq_neg (e : Basis ι R M) (x : Orientation R M ι) :
    x = e.orientation ∨ x = -e.orientation 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_3} [Fintype ι] [DecidableEq ι] (e : Basis ι R M) (x : Orientation R M ι)
Goal: x = e.orientation ∨ x = -e.orientation
-- Basis.orientation_ne_iff_eq_neg
theorem orientation_ne_iff_eq_neg (e : Basis ι R M) (x : Orientation R M ι) :
    x ≠ e.orientation ↔ x = -e.orientation 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_3} [Fintype ι] [DecidableEq ι] (e : Basis ι R M) (x : Orientation R M ι)
Goal: x ≠ e.orientation ↔ x = -e.orientation
-- Basis.orientation_comp_linearEquiv_eq_iff_det_pos
theorem orientation_comp_linearEquiv_eq_iff_det_pos (e : Basis ι R M) (f : M ≃ₗ[R] M) :
    (e.map f).orientation = e.orientation ↔ 0 < LinearMap.det (f : M →ₗ[R] M) 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_3} [Fintype ι] [DecidableEq ι] (e : Basis ι R M) (f : M ≃ₗ[R] M)
Goal: (e.map f).orientation = e.orientation ↔ 0 < LinearMap.det ↑f
-- Basis.orientation_comp_linearEquiv_eq_neg_iff_det_neg
theorem orientation_comp_linearEquiv_eq_neg_iff_det_neg (e : Basis ι R M) (f : M ≃ₗ[R] M) :
    (e.map f).orientation = -e.orientation ↔ LinearMap.det (f : M →ₗ[R] M) < 0 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_3} [Fintype ι] [DecidableEq ι] (e : Basis ι R M) (f : M ≃ₗ[R] M)
Goal: (e.map f).orientation = -e.orientation ↔ LinearMap.det ↑f < 0
-- Basis.orientation_neg_single
@[simp]
theorem orientation_neg_single (e : Basis ι R M) (i : ι) :
    (e.unitsSMul (Function.update 1 i (-1))).orientation = -e.orientation 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_3} [Fintype ι] [DecidableEq ι] (e : Basis ι R M) (i : ι)
Goal: (e.unitsSMul (Function.update 1 i (-1))).orientation = -e.orientation
-- Basis.orientation_adjustToOrientation
@[simp]
theorem orientation_adjustToOrientation [Nonempty ι] (e : Basis ι R M)
    (x : Orientation R M ι) : (e.adjustToOrientation x).orientation = x 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_3} [Fintype ι] [DecidableEq ι] [Nonempty ι] (e : Basis ι R M) (x : Orientation R M ι)
Goal: (e.adjustToOrientation x).orientation = x
-- Basis.adjustToOrientation_apply_eq_or_eq_neg
theorem adjustToOrientation_apply_eq_or_eq_neg [Nonempty ι] (e : Basis ι R M)
    (x : Orientation R M ι) (i : ι) :
    e.adjustToOrientation x i = e i ∨ e.adjustToOrientation x i = -e i 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_3} [Fintype ι] [DecidableEq ι] [Nonempty ι] (e : Basis ι R M) (x : Orientation R M ι) (i : ι)
Goal: (e.adjustToOrientation x) i = e i ∨ (e.adjustToOrientation x) i = -e i
-- Basis.det_adjustToOrientation
theorem det_adjustToOrientation [Nonempty ι] (e : Basis ι R M)
    (x : Orientation R M ι) :
    (e.adjustToOrientation x).det = e.det ∨ (e.adjustToOrientation x).det = -e.det 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_3} [Fintype ι] [DecidableEq ι] [Nonempty ι] (e : Basis ι R M) (x : Orientation R M ι)
Goal: (e.adjustToOrientation x).det = e.det ∨ (e.adjustToOrientation x).det = -e.det
-- Basis.abs_det_adjustToOrientation
@[simp]
theorem abs_det_adjustToOrientation [Nonempty ι] (e : Basis ι R M)
    (x : Orientation R M ι) (v : ι → M) : |(e.adjustToOrientation x).det v| = |e.det v| 
Arguments: {R : Type u_1} [LinearOrderedCommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_3} [Fintype ι] [DecidableEq ι] [Nonempty ι] (e : Basis ι R M) (x : Orientation R M ι) (v : ι → M)
Goal: |(e.adjustToOrientation x).det v| = |e.det v|
-- Orientation.eq_or_eq_neg
theorem eq_or_eq_neg (x₁ x₂ : Orientation R M ι) (h : Fintype.card ι = finrank R M) :
    x₁ = x₂ ∨ x₁ = -x₂ 
Arguments: {R : Type u_1} [LinearOrderedField R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_3} [Fintype ι] [_i : FiniteDimensional R M] (x₁ x₂ : Orientation R M ι) (h : Fintype.card ι = FiniteDimensional.finrank R M)
Goal: x₁ = x₂ ∨ x₁ = -x₂
-- Orientation.ne_iff_eq_neg
theorem ne_iff_eq_neg (x₁ x₂ : Orientation R M ι) (h : Fintype.card ι = finrank R M) :
    x₁ ≠ x₂ ↔ x₁ = -x₂ 
Arguments: {R : Type u_1} [LinearOrderedField R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_3} [Fintype ι] [_i : FiniteDimensional R M] (x₁ x₂ : Orientation R M ι) (h : Fintype.card ι = FiniteDimensional.finrank R M)
Goal: x₁ ≠ x₂ ↔ x₁ = -x₂
-- Orientation.map_eq_det_inv_smul
theorem map_eq_det_inv_smul (x : Orientation R M ι) (f : M ≃ₗ[R] M)
    (h : Fintype.card ι = finrank R M) : Orientation.map ι f x = (LinearEquiv.det f)⁻¹ • x 
Arguments: {R : Type u_1} [LinearOrderedField R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_3} [Fintype ι] [_i : FiniteDimensional R M] (x : Orientation R M ι) (f : M ≃ₗ[R] M) (h : Fintype.card ι = FiniteDimensional.finrank R M)
Goal: (Orientation.map ι f) x = (LinearEquiv.det f)⁻¹ • x
-- Orientation.map_eq_iff_det_pos
theorem map_eq_iff_det_pos (x : Orientation R M ι) (f : M ≃ₗ[R] M)
    (h : Fintype.card ι = finrank R M) :
    Orientation.map ι f x = x ↔ 0 < LinearMap.det (f : M →ₗ[R] M) 
Arguments: {R : Type u_1} [LinearOrderedField R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_3} [Fintype ι] [_i : FiniteDimensional R M] (x : Orientation R M ι) (f : M ≃ₗ[R] M) (h : Fintype.card ι = FiniteDimensional.finrank R M)
Goal: (Orientation.map ι f) x = x ↔ 0 < LinearMap.det ↑f
-- Orientation.map_eq_neg_iff_det_neg
theorem map_eq_neg_iff_det_neg (x : Orientation R M ι) (f : M ≃ₗ[R] M)
    (h : Fintype.card ι = finrank R M) :
    Orientation.map ι f x = -x ↔ LinearMap.det (f : M →ₗ[R] M) < 0 
Arguments: {R : Type u_1} [LinearOrderedField R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_3} [Fintype ι] (x : Orientation R M ι) (f : M ≃ₗ[R] M) (h : Fintype.card ι = FiniteDimensional.finrank R M)
Goal: (Orientation.map ι f) x = -x ↔ LinearMap.det ↑f < 0
-- Orientation.someBasis_orientation
@[simp]
theorem someBasis_orientation [Nonempty ι] [DecidableEq ι] (x : Orientation R M ι)
    (h : Fintype.card ι = finrank R M) : (x.someBasis h).orientation = x 
Arguments: {R : Type u_1} [LinearOrderedField R] {M : Type u_2} [AddCommGroup M] [Module R M] {ι : Type u_3} [Fintype ι] [_i : FiniteDimensional R M] [Nonempty ι] [DecidableEq ι] (x : Orientation R M ι) (h : Fintype.card ι = FiniteDimensional.finrank R M)
Goal: (x.someBasis h).orientation = x
-- QuadraticModuleCat.cliffordAlgebra_obj_carrier
@[simps]
def QuadraticModuleCat.cliffordAlgebra : QuadraticModuleCat.{u} R ⥤ AlgebraCat.{u} R 
Arguments: {R : Type u} [CommRing R] (M : QuadraticModuleCat R)
Goal: ↑(QuadraticModuleCat.cliffordAlgebra.obj M) = CliffordAlgebra M.form
-- QuadraticModuleCat.cliffordAlgebra_map
@[simps]
def QuadraticModuleCat.cliffordAlgebra : QuadraticModuleCat.{u} R ⥤ AlgebraCat.{u} R 
Arguments: {R : Type u} [CommRing R] {_M _N : QuadraticModuleCat R} (f : _M ⟶ _N)
Goal: QuadraticModuleCat.cliffordAlgebra.map f = CliffordAlgebra.map f.toIsometry
-- CliffordAlgebra.star_def
theorem star_def (x : CliffordAlgebra Q) : star x = reverse (involute x) 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (x : CliffordAlgebra Q)
Goal: star x = CliffordAlgebra.reverse (CliffordAlgebra.involute x)
-- CliffordAlgebra.star_def'
theorem star_def' (x : CliffordAlgebra Q) : star x = involute (reverse x) 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (x : CliffordAlgebra Q)
Goal: star x = CliffordAlgebra.involute (CliffordAlgebra.reverse x)
-- CliffordAlgebra.star_ι
@[simp]
theorem star_ι (m : M) : star (ι Q m) = -ι Q m 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (m : M)
Goal: star ((CliffordAlgebra.ι Q) m) = -(CliffordAlgebra.ι Q) m
-- CliffordAlgebra.star_smul
@[simp]
theorem star_smul (r : R) (x : CliffordAlgebra Q) : star (r • x) = r • star x 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (r : R) (x : CliffordAlgebra Q)
Goal: star (r • x) = r • star x
-- CliffordAlgebra.star_algebraMap
@[simp]
theorem star_algebraMap (r : R) :
    star (algebraMap R (CliffordAlgebra Q) r) = algebraMap R (CliffordAlgebra Q) r 
Arguments: {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (r : R)
Goal: star ((algebraMap R (CliffordAlgebra Q)) r) = (algebraMap R (CliffordAlgebra Q)) r