mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Coalgebra.Convolution

Goal state before:
pp: R : Type u_1
A : Type u_3
B : Type u_4
C : Type u_5
inst✝⁷ : CommSemiring R
inst✝⁶ : Semiring A
inst✝⁵ : Algebra R A
inst✝⁴ : Semiring B
inst✝³ : Algebra R B
inst✝² : AddCommMonoid C
inst✝¹ : Module R C
inst✝ : CoalgebraStruct R C
h : A →ₐ[R] B
f g : WithConv (C →ₗ[R] A)
⊢ h.toLinearMap ∘ₗ (f * g).ofConv = (toConv (h.toLinearMap ∘ₗ f.ofConv) * toConv (h.toLinearMap ∘ₗ g.ofConv)).ofConv

Tactic:
by
  simp [convMul_def, map_comp, ← comp_assoc, AlgHom.comp_mul']

Goal state after:
Module: Mathlib.Data.W.Basic

Goal state before:
pp: α : Type u_1
β : α → Type u_2
inst✝¹ : (a : α) → Fintype (β a)
inst✝ : (a : α) → Encodable (β a)
f : WType' β 0 → Empty :=
  fun x ↦
    match x with
    | ⟨val, h⟩ => ⋯.elim
x : Empty
⊢ WType' β 0

Tactic:
cases x

Goal state after:
Module: Mathlib.LinearAlgebra.Dimension.ErdosKaplansky

Goal state before:
pp: K : Type u
inst✝ : DivisionRing K
aleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K)
card_K : ℵ₀ < #K
this : Module.rank K (ℕ → K) < #K
ιK : Type u
bK : Module.Basis ιK K (ℕ → K)
L : Subfield K := Subfield.closure (Set.range fun i ↦ bK i.1 i.2)
⊢ #↥L < #K

Tactic:
refine (Subfield.cardinalMk_closure_le_max _).trans_lt
      (max_lt_iff.mpr ⟨mk_range_le.trans_lt ?_, card_K⟩)

Goal state after:
K : Type u
inst✝ : DivisionRing K
aleph0_le : ℵ₀ ≤ Module.rank K (ℕ → K)
card_K : ℵ₀ < #K
this : Module.rank K (ℕ → K) < #K
ιK : Type u
bK : Module.Basis ιK K (ℕ → K)
L : Subfield K := Subfield.closure (Set.range fun i ↦ bK i.1 i.2)
⊢ #(ιK × ℕ) < #K
Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic

Goal state before:
pp: C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
X Y Y' Z : C
inst✝ : ExactPairing Y Y'
f : X ⟶ Y ⊗ Z
⊢ (fun f ↦ (λ_ X).inv ≫ η_ Y Y' ▷ X ≫ (α_ Y Y' X).hom ≫ Y ◁ f)
      ((fun f ↦ Y' ◁ f ≫ (α_ Y' Y Z).inv ≫ ε_ Y Y' ▷ Z ≫ (λ_ Z).hom) f) =
    f

Tactic:
by
    calc
      _ = 𝟙 _ ⊗≫ (η_ Y Y' ▷ X ≫ (Y ⊗ Y') ◁ f) ⊗≫ Y ◁ ε_ Y Y' ▷ Z ⊗≫ 𝟙 _ := by
        monoidal
      _ = f ⊗≫ (η_ Y Y' ▷ Y ⊗≫ Y ◁ ε_ Y Y') ▷ Z ⊗≫ 𝟙 _ := by
        rw [← whisker_exchange]; monoidal
      _ = f := by
        rw [evaluation_coevaluation'']; monoidal

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

Goal state before:
pp: case mp
K : Type u_13
K₁ : Type u_14
K₂ : Type u_15
V : Type u_16
V₁ : Type u_17
V₂ : Type u_18
inst✝⁸ : Field K
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module K V
inst✝⁵ : Field K₁
inst✝⁴ : AddCommGroup V₁
inst✝³ : Module K₁ V₁
inst✝² : Field K₂
inst✝¹ : AddCommGroup V₂
inst✝ : Module K₂ V₂
I₁ : K₁ →+* K
I₂ : K₂ →+* K
B : V₁ →ₛₗ[I₁] V₂ →ₛₗ[I₂] V
x : V₁
y : V₂
a : K₁
ha : a ≠ 0
H : (B x) y = 0
⊢ (B (a • x)) y = 0

Tactic:
rw [map_smulₛₗ₂, H, smul_zero]

Goal state after:
Module: Mathlib.LinearAlgebra.Contraction

Goal state before:
pp: R : Type u_2
M : Type u_3
inst✝² : CommSemiring R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f : Dual R M
m : M
⊢ Dual.transpose ((dualTensorHom R M M) (f ⊗ₜ[R] m)) =
    (dualTensorHom R (Dual R M) (Dual R M)) ((Dual.eval R M) m ⊗ₜ[R] f)

Tactic:
ext f' m'
  simp only [Dual.transpose_apply, coe_comp, Function.comp_apply, dualTensorHom_apply,
    map_smulₛₗ, RingHom.id_apply, smul_eq_mul, Dual.eval_apply,
    LinearMap.smul_apply]
  exact mul_comm _ _

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

Goal state before:
pp: case mp
R : Type u_1
R₁ : Type u_2
M₁ : Type u_6
inst✝⁵ : CommRing R
inst✝⁴ : CommSemiring R₁
inst✝³ : AddCommMonoid M₁
inst✝² : Module R₁ M₁
I : R₁ →+* R
inst✝¹ : NoZeroDivisors R
inst✝ : CharZero R
B : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R
⊢ B.IsAlt → B = -B.flip

Tactic:
intro h

Goal state after:
case mp
R : Type u_1
R₁ : Type u_2
M₁ : Type u_6
inst✝⁵ : CommRing R
inst✝⁴ : CommSemiring R₁
inst✝³ : AddCommMonoid M₁
inst✝² : Module R₁ M₁
I : R₁ →+* R
inst✝¹ : NoZeroDivisors R
inst✝ : CharZero R
B : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R
h : B.IsAlt
⊢ B = -B.flip
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: case mpr
R : Type u_1
R₁ : Type u_2
M₁ : Type u_6
inst✝⁵ : CommRing R
inst✝⁴ : CommSemiring R₁
inst✝³ : AddCommMonoid M₁
inst✝² : Module R₁ M₁
I : R₁ →+* R
inst✝¹ : NoZeroDivisors R
inst✝ : CharZero R
B : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R
⊢ B = -B.flip → B.IsAlt

Tactic:
intro h

Goal state after:
case mpr
R : Type u_1
R₁ : Type u_2
M₁ : Type u_6
inst✝⁵ : CommRing R
inst✝⁴ : CommSemiring R₁
inst✝³ : AddCommMonoid M₁
inst✝² : Module R₁ M₁
I : R₁ →+* R
inst✝¹ : NoZeroDivisors R
inst✝ : CharZero R
B : M₁ →ₛₗ[I] M₁ →ₛₗ[I] R
h : B = -B.flip
⊢ B.IsAlt
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: case mp
R : Type u_1
M : Type u_5
M₁ : Type u_6
M₃ : Type u_8
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid M₁
inst✝² : Module R M₁
inst✝¹ : AddCommMonoid M₃
inst✝ : Module R M₃
I : R →+* R
B : M →ₗ[R] M →ₛₗ[I] M₃
B' : M₁ →ₗ[R] M₁ →ₛₗ[I] M₃
f : M →ₗ[R] M₁
g : M₁ →ₗ[R] M
⊢ B.IsAdjointPair B' ⇑f ⇑g → B' ∘ₗ f = B.compl₂ g

Tactic:
intro h

Goal state after:
case mp
R : Type u_1
M : Type u_5
M₁ : Type u_6
M₃ : Type u_8
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid M₁
inst✝² : Module R M₁
inst✝¹ : AddCommMonoid M₃
inst✝ : Module R M₃
I : R →+* R
B : M →ₗ[R] M →ₛₗ[I] M₃
B' : M₁ →ₗ[R] M₁ →ₛₗ[I] M₃
f : M →ₗ[R] M₁
g : M₁ →ₗ[R] M
h : B.IsAdjointPair B' ⇑f ⇑g
⊢ B' ∘ₗ f = B.compl₂ g
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: case mpr
R : Type u_1
M : Type u_5
M₁ : Type u_6
M₃ : Type u_8
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid M₁
inst✝² : Module R M₁
inst✝¹ : AddCommMonoid M₃
inst✝ : Module R M₃
I : R →+* R
B : M →ₗ[R] M →ₛₗ[I] M₃
B' : M₁ →ₗ[R] M₁ →ₛₗ[I] M₃
f : M →ₗ[R] M₁
g : M₁ →ₗ[R] M
⊢ B' ∘ₗ f = B.compl₂ g → B.IsAdjointPair B' ⇑f ⇑g

Tactic:
intro h

Goal state after:
case mpr
R : Type u_1
M : Type u_5
M₁ : Type u_6
M₃ : Type u_8
inst✝⁶ : CommSemiring R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : AddCommMonoid M₁
inst✝² : Module R M₁
inst✝¹ : AddCommMonoid M₃
inst✝ : Module R M₃
I : R →+* R
B : M →ₗ[R] M →ₛₗ[I] M₃
B' : M₁ →ₗ[R] M₁ →ₛₗ[I] M₃
f : M →ₗ[R] M₁
g : M₁ →ₗ[R] M
h : B' ∘ₗ f = B.compl₂ g
⊢ B.IsAdjointPair B' ⇑f ⇑g
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: case mp
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:
intro h

Goal state after:
case mp
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 : B.SeparatingLeft
⊢ ∀ (x : M₁), B x = 0 → x = 0
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: case mpr
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
⊢ (∀ (x : M₁), B x = 0 → x = 0) → B.SeparatingLeft

Tactic:
intro h

Goal state after:
case mpr
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
⊢ B.SeparatingLeft
Module: Mathlib.CategoryTheory.Limits.Constructions.Equalizers

Goal state before:
pp: case zero
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)).ι.app WalkingParallelPair.zero ≫
      (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 WalkingParallelPair.zero

Tactic:
simp only [Functor.mapCocone_ι_app, Cocone.ofCofork_ι, Category.id_comp,
              eqToHom_refl, Category.assoc, Functor.map_comp, Cofork.ofπ_ι_app, pushout.inl_desc,
              PreservesPushout.inl_iso_inv_assoc]

Goal state after:
case zero
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.map (K.map WalkingParallelPairHom.left) ≫ c.ι.app WalkingParallelPair.one = c.ι.app WalkingParallelPair.zero
Module: Mathlib.CategoryTheory.Limits.Constructions.Equalizers

Goal state before:
pp: case one
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)).ι.app WalkingParallelPair.one ≫
      (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 WalkingParallelPair.one

Tactic:
simp only [Functor.mapCocone_ι_app, Cocone.ofCofork_ι, Category.id_comp,
              eqToHom_refl, Category.assoc, Functor.map_comp, Cofork.ofπ_ι_app, pushout.inl_desc,
              PreservesPushout.inl_iso_inv_assoc]

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
X Y : C
f g : X ⟶ Y
inst✝² : HasEqualizer f g
inst✝¹ : HasEqualizer (G.map f) (G.map g)
inst✝ : PreservesLimit (parallelPair f g) G
⊢ (iso G f g).inv ≫ G.map (equalizer.ι f g) = equalizer.ι (G.map f) (G.map g)

Tactic:
← Iso.cancel_iso_hom_left (PreservesEqualizer.iso G f g),

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
D : Type u₂
inst✝³ : Category.{v₂, u₂} D
G : C ⥤ D
X Y : C
f g : X ⟶ Y
inst✝² : HasEqualizer f g
inst✝¹ : HasEqualizer (G.map f) (G.map g)
inst✝ : PreservesLimit (parallelPair f g) G
⊢ (iso G f g).hom ≫ (iso G f g).inv ≫ G.map (equalizer.ι f g) = (iso G f g).hom ≫ equalizer.ι (G.map f) (G.map g)
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: case mp
R : Type u_1
M : Type u_5
inst✝⁵ : CommRing R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
B : LinearMap.BilinForm R M
inst✝ : IsTorsionFree R M
hp : ∀ (x : M), x ≠ 0 → 0 < (B x) x
x y : M
hle : ∀ (z : M), 0 ≤ (B z) z
⊢ ¬LinearIndependent R ![x, y] → (B x) x * (B y) y ≤ (B x) y * (B y) x

Tactic:
intro h

Goal state after:
case mp
R : Type u_1
M : Type u_5
inst✝⁵ : CommRing R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
B : LinearMap.BilinForm R M
inst✝ : IsTorsionFree R M
hp : ∀ (x : M), x ≠ 0 → 0 < (B x) x
x y : M
hle : ∀ (z : M), 0 ≤ (B z) z
h : ¬LinearIndependent R ![x, y]
⊢ (B x) x * (B y) y ≤ (B x) y * (B y) x
Module: Mathlib.LinearAlgebra.SesquilinearForm.Basic

Goal state before:
pp: case mpr
R : Type u_1
M : Type u_5
inst✝⁵ : CommRing R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
B : LinearMap.BilinForm R M
inst✝ : IsTorsionFree R M
hp : ∀ (x : M), x ≠ 0 → 0 < (B x) x
x y : M
hle : ∀ (z : M), 0 ≤ (B z) z
⊢ (B x) x * (B y) y ≤ (B x) y * (B y) x → ¬LinearIndependent R ![x, y]

Tactic:
intro h

Goal state after:
case mpr
R : Type u_1
M : Type u_5
inst✝⁵ : CommRing R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
B : LinearMap.BilinForm R M
inst✝ : IsTorsionFree R M
hp : ∀ (x : M), x ≠ 0 → 0 < (B x) x
x y : M
hle : ∀ (z : M), 0 ≤ (B z) z
h : (B x) x * (B y) y ≤ (B x) y * (B y) x
⊢ ¬LinearIndependent R ![x, y]
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: case right.mk
K : Type u_1
V₁ : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V₁
inst✝ : Module K V₁
W : Subspace K V₁
y✝ : V₁ ⧸ W
x : V₁
⊢ (∀ (x_1 : ↥(dualAnnihilator W)), x_1 x = 0) → ∀ φ ∈ dualAnnihilator W, φ x = 0

Tactic:
SetLike.forall

Goal state after:
case right.mk
K : Type u_1
V₁ : Type u_2
inst✝² : Field K
inst✝¹ : AddCommGroup V₁
inst✝ : Module K V₁
W : Subspace K V₁
y✝ : V₁ ⧸ W
x : V₁
⊢ (∀ (x_1 : Dual K V₁) (h : x_1 ∈ dualAnnihilator W), ⟨x_1, h⟩ x = 0) → ∀ φ ∈ dualAnnihilator W, φ x = 0
Module: Mathlib.LinearAlgebra.Dual.Lemmas

Goal state before:
pp: ι : Type u_1
α : Type u_2
F : Type u_3
inst✝¹ : Finite ι
inst✝ : Field F
f : ι → α → F
⊢ span F (Set.range (flip f)) = ⊤ ↔ LinearIndependent F f

Tactic:
rw [linearIndependent_iff_ker, ← Submodule.map_eq_top_iff (e := Finsupp.llift F F F ι),
    ← Subspace.dualCoannihilator_dualAnnihilator_eq (W := map ..), dualAnnihilator_eq_top_iff]

Goal state after:
ι : Type u_1
α : Type u_2
F : Type u_3
inst✝¹ : Finite ι
inst✝ : Field F
f : ι → α → F
⊢ (map (↑(Finsupp.llift F F F ι)) (span F (Set.range (flip f)))).dualCoannihilator = ⊥ ↔
    (Finsupp.linearCombination F f).ker = ⊥
Module: Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
D : Type u₂
inst✝⁴ : Category.{v₂, u₂} D
inst✝³ : HasCoequalizers C
inst✝² : HasFiniteCoproducts C
G : C ⥤ D
inst✝¹ : PreservesColimitsOfShape WalkingParallelPair G
inst✝ : PreservesFiniteCoproducts G
⊢ ∀ (J : Type) [inst : SmallCategory J] [FinCategory J], PreservesColimitsOfShape J G

Tactic:
by
    intro J sJ fJ
    apply preservesColimit_of_preservesCoequalizers_and_coproduct

Goal state after:
Module: Mathlib.Algebra.Category.Grp.EpiMono

Goal state before:
pp: A B : GrpCat
f : A ⟶ B
x : ↑B
hx : x ∈ (Hom.hom f).range
⊢ (h x) ∞ = ∞

Tactic:
by
  change ((τ).symm.trans (g x)).trans τ _ = _
  simp only [Equiv.coe_trans, Function.comp_apply]
  rw [τ_symm_apply_infinity, g_apply_fromCoset]
  exact τ_apply_fromCoset' f x hx

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Basic

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
f : S₁ ⟶ S₂
⊢ IsIso f.τ₁ ∧ IsIso f.τ₂ ∧ IsIso f.τ₃ → IsIso f

Tactic:
rintro ⟨_, _, _⟩

Goal state after:
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
f : S₁ ⟶ S₂
left✝¹ : IsIso f.τ₁
left✝ : IsIso f.τ₂
right✝ : IsIso f.τ₃
⊢ IsIso f
Module: Mathlib.Algebra.Category.Grp.EpiMono

Goal state before:
pp: A B : GrpCat
f✝ : A ⟶ B
X✝ Y✝ : GrpCat
f : X✝ ⟶ Y✝
e : Epi f
⊢ Epi ((forget GrpCat).map f)

Tactic:
by rwa [epi_iff_surjective, ← CategoryTheory.ofHom_epi_iff_surjective] at e

Goal state after:
Module: Mathlib.Algebra.Category.Grp.EpiMono

Goal state before:
pp: A B : CommGrpCat
f✝ : A ⟶ B
X✝ Y✝ : CommGrpCat
f : X✝ ⟶ Y✝
e : Epi f
⊢ Epi ((forget CommGrpCat).map f)

Tactic:
by rwa [epi_iff_surjective, ← CategoryTheory.ofHom_epi_iff_surjective] at e

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: case mp
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
⊢ QuasiIso φ → IsIso (homologyMap φ)

Tactic:
intro h

Goal state after:
case mp
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h : QuasiIso φ
⊢ IsIso (homologyMap φ)
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: case mpr
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
⊢ IsIso (homologyMap φ) → QuasiIso φ

Tactic:
intro h

Goal state after:
case mpr
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h : IsIso (homologyMap φ)
⊢ QuasiIso φ
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: case mp
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
γ : LeftHomologyMapData φ h₁ h₂
⊢ IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv) → IsIso γ.φH

Tactic:
intro h

Goal state after:
case mp
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
γ : LeftHomologyMapData φ h₁ h₂
h : IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv)
⊢ IsIso γ.φH
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: case mpr
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
γ : LeftHomologyMapData φ h₁ h₂
⊢ IsIso γ.φH → IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv)

Tactic:
intro h

Goal state after:
case mpr
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
γ : LeftHomologyMapData φ h₁ h₂
h : IsIso γ.φH
⊢ IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv)
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: case mp
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
⊢ IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv) → IsIso γ.φH

Tactic:
intro h

Goal state after:
case mp
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
h : IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv)
⊢ IsIso γ.φH
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: case mpr
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
⊢ IsIso γ.φH → IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv)

Tactic:
intro h

Goal state after:
case mpr
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ : RightHomologyMapData φ h₁ h₂
h : IsIso γ.φH
⊢ IsIso (h₁.homologyIso.hom ≫ γ.φH ≫ h₂.homologyIso.inv)
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ QuasiIso φ ↔ IsIso (leftHomologyMap' φ h₁ h₂)

Tactic:
have γ : LeftHomologyMapData φ h₁ h₂ := default
  rw [γ.quasiIso_iff, γ.leftHomologyMap'_eq]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: case mp
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
γ : HomologyMapData φ S₁.homologyData S₂.homologyData
⊢ IsIso γ.left.φH.op → IsIso γ.left.φH

Tactic:
intro h

Goal state after:
case mp
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
γ : HomologyMapData φ S₁.homologyData S₂.homologyData
h : IsIso γ.left.φH.op
⊢ IsIso γ.left.φH
Module: Mathlib.Algebra.Homology.ShortComplex.QuasiIso

Goal state before:
pp: case mpr
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
γ : HomologyMapData φ S₁.homologyData S₂.homologyData
⊢ IsIso γ.left.φH → IsIso γ.left.φH.op

Tactic:
intro h

Goal state after:
case mpr
C : Type u_1
inst✝³ : Category.{v_1, u_1} C
inst✝² : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
inst✝¹ : S₁.HasHomology
inst✝ : S₂.HasHomology
φ : S₁ ⟶ S₂
γ : HomologyMapData φ S₁.homologyData S₂.homologyData
h : IsIso γ.left.φH
⊢ IsIso γ.left.φH.op
Module: Mathlib.Algebra.Homology.ShortComplex.Homology

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.HomologyData
h₂ : S₂.HomologyData
γ₁ γ₂ : HomologyMapData φ h₁ h₂
eq : γ₁ = γ₂
⊢ γ₁.left.φH = γ₂.left.φH

Tactic:
by rw [eq]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
γ₁ γ₂ : LeftHomologyMapData φ h₁ h₂
eq : γ₁ = γ₂
⊢ γ₁.φH = γ₂.φH

Tactic:
by rw [eq]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
γ₁ γ₂ : LeftHomologyMapData φ h₁ h₂
eq : γ₁ = γ₂
⊢ γ₁.φK = γ₂.φK

Tactic:
by rw [eq]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ₁ γ₂ : RightHomologyMapData φ h₁ h₂
eq : γ₁ = γ₂
⊢ γ₁.φH = γ₂.φH

Tactic:
by rw [eq]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.RightHomology

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₁ : S₁.RightHomologyData
h₂ : S₂.RightHomologyData
γ₁ γ₂ : RightHomologyMapData φ h₁ h₂
eq : γ₁ = γ₂
⊢ γ₁.φQ = γ₂.φQ

Tactic:
by rw [eq]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.LeftHomology

Goal state before:
pp: case refine_1
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
S₁ S₂ : ShortComplex C
φ : S₁ ⟶ S₂
h₂✝ : IsIso φ.τ₂
h₃ : Mono φ.τ₃
h₁ : S₁.LeftHomologyData
h₂ : S₂.LeftHomologyData
⊢ (h₂.i ≫ inv φ.τ₂) ≫ S₁.g = 0

Tactic:
simp only [assoc, ← cancel_mono φ.τ₃, zero_comp, ← φ.comm₂₃, IsIso.inv_hom_id_assoc, h₂.wi]

Goal state after:
Module: Mathlib.Algebra.Homology.ShortComplex.Abelian

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
inst✝² : Abelian C
S : ShortComplex C
D : Type u'
inst✝¹ : Category.{v', u'} D
inst✝ : HasZeroMorphisms D
γ : kernel S.g ⟶ cokernel S.f := kernel.ι S.g ≫ cokernel.π S.f
g' : cokernel S.f ⟶ S.X₃ := cokernel.desc S.f S.g ⋯
⊢ γ ≫ g' = 0

Tactic:
by simp [γ, g']

Goal state after: