mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.Contraction

Goal state before:
pp: R : Type u_2
M : Type u_3
P : Type u_5
Q : Type u_6
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid P
inst✝⁵ : AddCommMonoid Q
inst✝⁴ : Module R M
inst✝³ : Module R P
inst✝² : Module R Q
inst✝¹ : Projective R M
inst✝ : Module.Finite R M
⊢ ↑(lTensorHomEquivHomLTensor R M P Q) = lTensorHomToHomLTensor (RingHom.id R) M P Q

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.Equalizers

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
D : Type u'
inst✝² : Category.{v', u'} D
G : C ⥤ D
inst✝¹ : HasBinaryProducts C
inst✝ : HasPullbacks C
F : WalkingParallelPair ⥤ C
⊢ ∀ (s : Cone F) (m : s.pt ⟶ (equalizerCone F).pt),
    (∀ (j : WalkingParallelPair), m ≫ (equalizerCone F).π.app j = s.π.app j) →
      m = pullback.lift (s.π.app WalkingParallelPair.zero) (s.π.app WalkingParallelPair.zero) ⋯

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Contraction

Goal state before:
pp: R : Type u_2
M : Type u_3
P : Type u_5
Q : Type u_6
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid P
inst✝⁵ : AddCommMonoid Q
inst✝⁴ : Module R M
inst✝³ : Module R P
inst✝² : Module R Q
inst✝¹ : Projective R M
inst✝ : Module.Finite R M
⊢ ↑(rTensorHomEquivHomRTensor R M P Q) = rTensorHomToHomRTensor (RingHom.id R) M P Q

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : CommSemiring R₁
inst✝⁴ : AddCommMonoid M₁
inst✝³ : Module R₁ M₁
inst✝² : CommSemiring R₂
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R₂ M₂
I₁ : R₁ →+* R
I₂ : R₂ →+* R
B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M
h : ∀ (x : M₁), B x = 0 → x = 0
x : M₁
hB : ∀ (y : M₂), (B x) y = 0
⊢ B x = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Contraction

Goal state before:
pp: R : Type u_2
M : Type u_3
P : Type u_5
Q : Type u_6
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid P
inst✝⁵ : AddCommMonoid Q
inst✝⁴ : Module R M
inst✝³ : Module R P
inst✝² : Module R Q
inst✝¹ : Projective R M
inst✝ : Module.Finite R M
x : P ⊗[R] (M →ₗ[R] Q)
⊢ (lTensorHomEquivHomLTensor R M P Q) x = (lTensorHomToHomLTensor (RingHom.id R) M P Q) x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : CommSemiring R₁
inst✝⁴ : AddCommMonoid M₁
inst✝³ : Module R₁ M₁
inst✝² : CommSemiring R₂
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R₂ M₂
I₁ : R₁ →+* R
I₂ : R₂ →+* R
B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M
⊢ B.SeparatingLeft ↔ ∀ (x : M₁), B x = 0 → x = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : CommSemiring R₁
inst✝⁴ : AddCommMonoid M₁
inst✝³ : Module R₁ M₁
inst✝² : CommSemiring R₂
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R₂ M₂
I₁ : R₁ →+* R
I₂ : R₂ →+* R
B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M
⊢ B.SeparatingRight ↔ ∀ (y : M₂), B.flip y = 0 → y = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Contraction

Goal state before:
pp: R : Type u_2
M : Type u_3
P : Type u_5
Q : Type u_6
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : AddCommMonoid P
inst✝⁵ : AddCommMonoid Q
inst✝⁴ : Module R M
inst✝³ : Module R P
inst✝² : Module R Q
inst✝¹ : Projective R M
inst✝ : Module.Finite R M
x : (M →ₗ[R] P) ⊗[R] Q
⊢ (rTensorHomEquivHomRTensor R M P Q) x = (rTensorHomToHomRTensor (RingHom.id R) M P Q) x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
R₁ : Type u_2
R₂ : Type u_3
M : Type u_5
M₁ : Type u_6
M₂ : Type u_7
inst✝⁸ : CommSemiring R
inst✝⁷ : AddCommMonoid M
inst✝⁶ : Module R M
inst✝⁵ : CommSemiring R₁
inst✝⁴ : AddCommMonoid M₁
inst✝³ : Module R₁ M₁
inst✝² : CommSemiring R₂
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R₂ M₂
I₁ : R₁ →+* R
I₂ : R₂ →+* R
B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M
⊢ B.SeparatingRight ↔ B.flip.ker = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.Equalizers

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u'
inst✝⁴ : Category.{v', u'} D
G : C ⥤ D
inst✝³ : HasBinaryProducts C
inst✝² : HasPullbacks C
inst✝¹ : PreservesLimitsOfShape (Discrete WalkingPair) G
inst✝ : PreservesLimitsOfShape WalkingCospan G
K : WalkingParallelPair ⥤ C
c : Cone (K ⋙ G)
⊢ c.pt ⟶ (G.mapCone (equalizerCone K)).pt

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
M : Type u_5
M₁ : Type u_6
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup M₁
inst✝ : Module R M₁
B : M →ₗ[R] M →ₗ[R] M₁
hB : B.IsRefl
⊢ B.Nondegenerate ↔ B.SeparatingLeft

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Contraction

Goal state before:
pp: R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
Q : Type u_6
inst✝¹² : CommSemiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : AddCommMonoid N
inst✝⁹ : AddCommMonoid P
inst✝⁸ : AddCommMonoid Q
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Projective R M
inst✝² : Module.Finite R M
inst✝¹ : Projective R N
inst✝ : Module.Finite R N
⊢ ↑(homTensorHomEquiv R M N P Q) = homTensorHomMap (RingHom.id R) M N P Q

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
M : Type u_5
M₁ : Type u_6
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup M₁
inst✝ : Module R M₁
B : M →ₗ[R] M →ₗ[R] M₁
hB : B.IsRefl
⊢ B.Nondegenerate ↔ B.SeparatingRight

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.Equalizers

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u'
inst✝⁴ : Category.{v', u'} D
G : C ⥤ D
inst✝³ : HasBinaryProducts C
inst✝² : HasPullbacks C
inst✝¹ : PreservesLimitsOfShape (Discrete WalkingPair) G
inst✝ : PreservesLimitsOfShape WalkingCospan G
K : WalkingParallelPair ⥤ C
c : Cone (K ⋙ G)
j : WalkingParallelPair
⊢ (pullback.lift (c.π.app WalkingParallelPair.zero) (c.π.app WalkingParallelPair.zero) ⋯ ≫
        (PreservesPullback.iso G (prod.lift (𝟙 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left))
            (prod.lift (𝟙 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))).inv) ≫
      (G.mapCone (equalizerCone K)).π.app j =
    c.π.app j

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
M : Type u_5
M₁ : Type u_6
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup M₁
inst✝ : Module R M₁
B : M →ₗ[R] M →ₗ[R] M₁
W : Submodule R M
hW : (B.domRestrict₁₂ W W).Nondegenerate
x : M
hx : x ∈ W
hx' : x ∈ B.ker
x' : ↥W := ⟨x, hx⟩
this : x' = 0
⊢ x = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Contraction

Goal state before:
pp: R : Type u_2
M : Type u_3
N : Type u_4
P : Type u_5
Q : Type u_6
inst✝¹² : CommSemiring R
inst✝¹¹ : AddCommMonoid M
inst✝¹⁰ : AddCommMonoid N
inst✝⁹ : AddCommMonoid P
inst✝⁸ : AddCommMonoid Q
inst✝⁷ : Module R M
inst✝⁶ : Module R N
inst✝⁵ : Module R P
inst✝⁴ : Module R Q
inst✝³ : Projective R M
inst✝² : Module.Finite R M
inst✝¹ : Projective R N
inst✝ : Module.Finite R N
x : (M →ₗ[R] P) ⊗[R] (N →ₗ[R] Q)
⊢ (homTensorHomEquiv R M N P Q) x = (homTensorHomMap (RingHom.id R) M N P Q) x

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Contraction

Goal state before:
pp: R : Type u_7
M : Type u_8
N : Type u_9
ι : Type u_10
κ : Type u_11
inst✝⁸ : DecidableEq ι
inst✝⁷ : DecidableEq κ
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype κ
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
b : Basis ι R M
c : Basis κ R N
f : Dual R (M ⊗[R] N)
⊢ (dualDistribInvOfBasis b c) f = ∑ i, ∑ j, f (b i ⊗ₜ[R] c j) • b.dualBasis i ⊗ₜ[R] c.dualBasis j

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
M : Type u_5
M₁ : Type u_6
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : AddCommGroup M₁
inst✝ : Module R M₁
B : M →ₗ[R] M →ₗ[R] M₁
W : Submodule R M
hW : (B.domRestrict₁₂ W W).Nondegenerate
⊢ Disjoint W B.ker

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
M : Type u_5
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : M →ₗ[R] M →ₗ[R] R
hB : B.IsSymm
W : Submodule R M
hW : IsCompl W B.ker
hB' : (B.domRestrict₁₂ W W).IsRefl
x : M
hx : x ∈ W
hx' : ∀ (y : ↥W), ((B.domRestrict₁₂ W W) ⟨x, hx⟩) y = 0
⊢ ∀ y ∈ W, (B x) y = 0

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.Equalizers

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u'
inst✝⁴ : Category.{v', u'} D
G : C ⥤ D
inst✝³ : HasBinaryProducts C
inst✝² : HasPullbacks C
inst✝¹ : PreservesLimitsOfShape (Discrete WalkingPair) G
inst✝ : PreservesLimitsOfShape WalkingCospan G
K : WalkingParallelPair ⥤ C
s : Cone (K ⋙ G)
m : s.pt ⟶ (G.mapCone (equalizerCone K)).pt
h : ∀ (j : WalkingParallelPair), m ≫ (G.mapCone (equalizerCone K)).π.app j = s.π.app j
⊢ m =
    pullback.lift (s.π.app WalkingParallelPair.zero) (s.π.app WalkingParallelPair.zero) ⋯ ≫
      (PreservesPullback.iso G (prod.lift (𝟙 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.left))
          (prod.lift (𝟙 (K.obj WalkingParallelPair.zero)) (K.map WalkingParallelPairHom.right))).inv

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
M : Type u_5
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : M →ₗ[R] M →ₗ[R] R
hB : B.IsSymm
W : Submodule R M
hW : IsCompl W B.ker
hB' : (B.domRestrict₁₂ W W).IsRefl
x : M
hx : x ∈ W
hx' : ∀ y ∈ W, (B x) y = 0
y : M
⊢ ∃ u ∈ W, ∃ v ∈ B.ker, u + v = y

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.Equalizers

Goal state before:
pp: C : Type u
inst✝² : Category.{v, u} C
inst✝¹ : HasBinaryCoproducts C
inst✝ : HasPushouts C
F : WalkingParallelPair ⥤ C
⊢ pushoutInl F =
    pushout.inr (coprod.desc (𝟙 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.left))
      (coprod.desc (𝟙 (F.obj WalkingParallelPair.one)) (F.map WalkingParallelPairHom.right))

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
M : Type u_5
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : M →ₗ[R] M →ₗ[R] R
hB : B.IsSymm
W : Submodule R M
hW : IsCompl W B.ker
hB' : (B.domRestrict₁₂ W W).IsRefl
x : M
hx : x ∈ W
hx' : ∀ y ∈ W, (B x) y = 0
u : M
hu : u ∈ W
v : M
hv : v ∈ B.ker
this : (B x) u = 0
⊢ (B x) (u + v) = 0 (u + v)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
M : Type u_5
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : M →ₗ[R] M →ₗ[R] R
hB : B.IsSymm
W : Submodule R M
hW : IsCompl W B.ker
hB' : (B.domRestrict₁₂ W W).IsRefl
x : M
hx : x ∈ W
hx' : ∀ y ∈ W, (B x) y = 0
⊢ x ∈ W ⊓ B.ker

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: R : Type u_1
M : Type u_5
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
B : M →ₗ[R] M →ₗ[R] R
hB : B.IsSymm
W : Submodule R M
hW : IsCompl W B.ker
⊢ (B.domRestrict₁₂ W W).Nondegenerate

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.Equalizers

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
D : Type u'
inst✝² : Category.{v', u'} D
G : C ⥤ D
inst✝¹ : HasBinaryCoproducts C
inst✝ : HasPushouts C
F : WalkingParallelPair ⥤ C
⊢ F.map WalkingParallelPairHom.left ≫ pushoutInl F = F.map WalkingParallelPairHom.right ≫ pushoutInl F

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.Equalizers

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
D : Type u'
inst✝² : Category.{v', u'} D
G : C ⥤ D
inst✝¹ : HasBinaryCoproducts C
inst✝ : HasPushouts C
F : WalkingParallelPair ⥤ C
⊢ ∀ (s : Cocone F) (j : WalkingParallelPair),
    (coequalizerCocone F).ι.app j ≫ pushout.desc (s.ι.app WalkingParallelPair.one) (s.ι.app WalkingParallelPair.one) ⋯ =
      s.ι.app j

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.Equalizers

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
D : Type u'
inst✝² : Category.{v', u'} D
G : C ⥤ D
inst✝¹ : HasBinaryCoproducts C
inst✝ : HasPushouts C
F : WalkingParallelPair ⥤ C
c : Cocone F
m : (coequalizerCocone F).pt ⟶ c.pt
J : ∀ (j : WalkingParallelPair), (coequalizerCocone F).ι.app j ≫ m = c.ι.app j
⊢ pushoutInl F ≫ m = c.ι.app WalkingParallelPair.one

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: n : Type u_19
R : Type u_20
M : Type u_21
M₁ : Type u_22
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid M₁
inst✝² : Module R M
inst✝¹ : Module R M₁
I I' : R →+* R
B : M →ₛₗ[I] M →ₛₗ[I'] M₁
inst✝ : Nontrivial R
v : Basis n R M
h : B.IsOrthoᵢ ⇑v
hB : B.SeparatingLeft
i : n
ho : (B (v i)) (v i) = 0
vi : n →₀ R
j : n
this : (B (v i)) (v j) = 0
⊢ I' (vi j) • (B (v i)) (v j) = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: n : Type u_19
R : Type u_20
M : Type u_21
M₁ : Type u_22
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid M₁
inst✝² : Module R M
inst✝¹ : Module R M₁
I I' : R →+* R
B : M →ₛₗ[I] M →ₛₗ[I'] M₁
inst✝ : Nontrivial R
v : Basis n R M
h : B.IsOrthoᵢ ⇑v
hB : B.SeparatingLeft
i : n
⊢ (B (v i)) (v i) ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Contraction

Goal state before:
pp: R : Type u_7
M : Type u_8
N : Type u_9
ι : Type u_10
κ : Type u_11
inst✝⁸ : DecidableEq ι
inst✝⁷ : DecidableEq κ
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype κ
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
b : Basis ι R M
c : Basis κ R N
⊢ dualDistrib R M N ∘ₗ dualDistribInvOfBasis b c = LinearMap.id

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.Equalizers

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
D : Type u'
inst✝² : Category.{v', u'} D
G : C ⥤ D
inst✝¹ : HasBinaryCoproducts C
inst✝ : HasPushouts C
F : WalkingParallelPair ⥤ C
⊢ ∀ (s : Cocone F) (m : (coequalizerCocone F).pt ⟶ s.pt),
    (∀ (j : WalkingParallelPair), (coequalizerCocone F).ι.app j ≫ m = s.ι.app j) →
      m = pushout.desc (s.ι.app WalkingParallelPair.one) (s.ι.app WalkingParallelPair.one) ⋯

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: n : Type u_19
R : Type u_20
M : Type u_21
M₁ : Type u_22
inst✝⁵ : CommSemiring R
inst✝⁴ : AddCommMonoid M
inst✝³ : AddCommMonoid M₁
inst✝² : Module R M
inst✝¹ : Module R M₁
I I' : R →+* R
B : M →ₛₗ[I] M →ₛₗ[I'] M₁
inst✝ : Nontrivial R
v : Basis n R M
h : B.IsOrthoᵢ ⇑v
hB : B.SeparatingRight
i : n
⊢ (B (v i)) (v i) ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: K : Type uK
V : Type uV
inst✝² : Field K
inst✝¹ : AddCommGroup V
inst✝ : Module K V
⊢ Nonempty (V ≃ₗ[K] Dual K V) ↔ FiniteDimensional K V

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: R : Type uR
M : Type uM
ι : Type uι
inst✝³ : CommSemiring R
inst✝² : AddCommMonoid M
inst✝¹ : Module R M
inst✝ : Finite ι
b : Basis ι R M
⊢ Module.rank R (Dual R M) = lift.{uR, uM} (Module.rank R M)

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: V : Type uV
inst✝³ : AddCommMonoid V
R : Type u_1
inst✝² : Semiring R
inst✝¹ : Module R V
inst✝ : Projective R V
v : V
hv : v = 0
⊢ ∀ (φ : Dual R V), φ v = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: V : Type uV
inst✝³ : AddCommMonoid V
R : Type u_1
inst✝² : Semiring R
inst✝¹ : Module R V
inst✝ : Projective R V
v : V
⊢ (∀ (φ : Dual R V), φ v = 0) ↔ v = 0

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: K : Type uK
V : Type uV
inst✝³ : CommSemiring K
inst✝² : AddCommMonoid V
inst✝¹ : Module K V
inst✝ : Projective K V
⊢ Nontrivial (Dual K V) ↔ Nontrivial V

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.Equalizers

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u'
inst✝⁴ : Category.{v', u'} D
G : C ⥤ D
inst✝³ : HasBinaryCoproducts C
inst✝² : HasPushouts C
inst✝¹ : PreservesColimitsOfShape (Discrete WalkingPair) G
inst✝ : PreservesColimitsOfShape WalkingSpan G
K : WalkingParallelPair ⥤ C
c : Cocone (K ⋙ G)
⊢ (G.mapCocone (coequalizerCocone K)).pt ⟶ c.pt

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: K : Type uK
V : Type uV
inst✝³ : CommSemiring K
inst✝² : AddCommMonoid V
inst✝¹ : Module K V
inst✝ : Free K V
⊢ Module.Finite K (Dual K V) ↔ Module.Finite K V

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Constructions.Equalizers

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u'
inst✝⁴ : Category.{v', u'} D
G : C ⥤ D
inst✝³ : HasBinaryCoproducts C
inst✝² : HasPushouts C
inst✝¹ : PreservesColimitsOfShape (Discrete WalkingPair) G
inst✝ : PreservesColimitsOfShape WalkingSpan G
K : WalkingParallelPair ⥤ C
c : Cocone (K ⋙ G)
j : WalkingParallelPair
⊢ (G.mapCocone (coequalizerCocone K)).ι.app j ≫
      (PreservesPushout.iso G (coprod.desc (𝟙 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.left))
            (coprod.desc (𝟙 (K.obj WalkingParallelPair.one)) (K.map WalkingParallelPairHom.right))).inv ≫
        pushout.desc (c.ι.app WalkingParallelPair.one) (c.ι.app WalkingParallelPair.one) ⋯ =
    c.ι.app j

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Contraction

Goal state before:
pp: R : Type u_7
M : Type u_8
N : Type u_9
ι : Type u_10
κ : Type u_11
inst✝⁸ : DecidableEq ι
inst✝⁷ : DecidableEq κ
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype κ
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
b : Basis ι R M
c : Basis κ R N
⊢ dualDistribInvOfBasis b c ∘ₗ dualDistrib R M N = LinearMap.id

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Contraction

Goal state before:
pp: case refine_2
ι✝ : Type u_1
R✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
P : Type u_5
Q : Type u_6
R : Type u_7
M : Type u_8
N : Type u_9
ι : Type u_10
κ : Type u_11
inst✝⁸ : DecidableEq ι
inst✝⁷ : DecidableEq κ
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype κ
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
b : Basis ι R M
c : Basis κ R N
⊢ dualDistribInvOfBasis b c ∘ₗ dualDistrib R M N = LinearMap.id

Tactic:
· exact dualDistrib_dualDistribInvOfBasis_right_inverse _ _

Goal state after:
Module: Mathlib.LinearAlgebra.Contraction

Goal state before:
pp: ι✝ : Type u_1
R✝ : Type u_2
M✝ : Type u_3
N✝ : Type u_4
P : Type u_5
Q : Type u_6
R : Type u_7
M : Type u_8
N : Type u_9
ι : Type u_10
κ : Type u_11
inst✝⁸ : DecidableEq ι
inst✝⁷ : DecidableEq κ
inst✝⁶ : Fintype ι
inst✝⁵ : Fintype κ
inst✝⁴ : CommSemiring R
inst✝³ : AddCommMonoid M
inst✝² : AddCommMonoid N
inst✝¹ : Module R M
inst✝ : Module R N
b : Basis ι R M
c : Basis κ R N
⊢ Dual R M ⊗[R] Dual R N ≃ₗ[R] Dual R (M ⊗[R] N)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
G : C ⥤ D
X Y : C
f g : X ⟶ Y
Z : C
π : Y ⟶ Z
q : IsSplitCoequalizer f g π
F : C ⥤ D
⊢ F.map f ≫ F.map π = F.map g ≫ F.map π

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
G : C ⥤ D
X Y : C
f g : X ⟶ Y
Z : C
π : Y ⟶ Z
q : IsSplitCoequalizer f g π
F : C ⥤ D
⊢ F.map q.rightSection ≫ F.map π = 𝟙 (F.obj Z)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
G : C ⥤ D
X Y : C
f g : X ⟶ Y
Z : C
π : Y ⟶ Z
q : IsSplitCoequalizer f g π
F : C ⥤ D
⊢ F.map q.leftSection ≫ F.map g = 𝟙 (F.obj Y)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
G : C ⥤ D
X Y : C
f g : X ⟶ Y
Z : C
π : Y ⟶ Z
q : IsSplitCoequalizer f g π
F : C ⥤ D
⊢ F.map q.leftSection ≫ F.map f = F.map π ≫ F.map q.rightSection

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
G : C ⥤ D
X Y : C
f g : X ⟶ Y
Z : C
h : Y ⟶ Z
t : IsSplitCoequalizer f g h
s : Cofork f g
⊢ t.asCofork.π ≫ t.rightSection ≫ s.π = s.π

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.SplitCoequalizer

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
G : C ⥤ D
X Y : C
f g : X ⟶ Y
Z : C
h : Y ⟶ Z
t : IsSplitCoequalizer f g h
s : Cofork f g
m✝ : t.asCofork.pt ⟶ s.pt
hm : t.asCofork.π ≫ m✝ = s.π
⊢ m✝ = t.rightSection ≫ s.π

Tactic:
by

Goal state after:
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: K : Type uK
V : Type uV
inst✝¹⁰ : CommSemiring K
inst✝⁹ : AddCommMonoid V
inst✝⁸ : Module K V
inst✝⁷ : Projective K V
R : Type u_1
M : Type u_2
N : Type u_3
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : Module R M
inst✝² : Module R N
inst✝¹ : IsReflexive R M
inst✝ : IsReflexive R N
e : Dual R (Dual R (M × N)) ≃ₗ[R] Dual R (Dual R M) × Dual R (Dual R N) :=
  (dualProdDualEquivDual R M N).dualMap ≪≫ₗ (dualProdDualEquivDual R (Dual R M) (Dual R N)).symm
⊢ Dual.eval R (M × N) = ↑e.symm ∘ₗ (Dual.eval R M).prodMap (Dual.eval R N)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.SplitEqualizer

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
G : C ⥤ D
X Y : C
f g : X ⟶ Y
W : C
ι : W ⟶ X
q : IsSplitEqualizer f g ι
F : C ⥤ D
⊢ F.map ι ≫ F.map f = F.map ι ≫ F.map g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.SplitEqualizer

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
G : C ⥤ D
X Y : C
f g : X ⟶ Y
W : C
ι : W ⟶ X
q : IsSplitEqualizer f g ι
F : C ⥤ D
⊢ F.map ι ≫ F.map q.leftRetraction = 𝟙 (F.obj W)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.SplitEqualizer

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
G : C ⥤ D
X Y : C
f g : X ⟶ Y
W : C
ι : W ⟶ X
q : IsSplitEqualizer f g ι
F : C ⥤ D
⊢ F.map g ≫ F.map q.rightRetraction = 𝟙 (F.obj X)

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.SplitEqualizer

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
G : C ⥤ D
X Y : C
f g : X ⟶ Y
W : C
ι : W ⟶ X
q : IsSplitEqualizer f g ι
F : C ⥤ D
⊢ F.map f ≫ F.map q.rightRetraction = F.map q.leftRetraction ≫ F.map ι

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Shapes.SplitEqualizer

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
D : Type u₂
inst✝ : Category.{v₂, u₂} D
G : C ⥤ D
X Y : C
f g : X ⟶ Y
W : C
h : W ⟶ X
t : IsSplitEqualizer f g h
s : Fork f g
⊢ (s.ι ≫ t.leftRetraction) ≫ t.asFork.ι = s.ι

Tactic:
by

Goal state after: