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

Goal state before:
pp: R : Type v
M : Type w
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
f : End R M
μ₁ μ₂ : R
hμ : μ₁ ≠ μ₂
k✝ l✝ : ℕ∞
k : ℕ
property✝¹ : ↑k ≤ k✝
l : ℕ
property✝ : ↑l ≤ l✝
a✝ : Nontrivial M
p : Submodule R M := (f.genEigenspace μ₁) ↑k ⊓ (f.genEigenspace μ₂) ↑l
hp : Nontrivial ↥p
f₁ : End R ↥p := LinearMap.restrict (f - (algebraMap R (End R M)) μ₁) ⋯
f₂ : End R ↥p := LinearMap.restrict (f - (algebraMap R (End R M)) μ₂) ⋯
this : IsNilpotent ((algebraMap R (End R ↥p)) (μ₁ - μ₂))
hf₁₂ : f₂ - f₁ = (algebraMap R (End R ↥p)) (μ₁ - μ₂)
⊢ False

Tactic:
IsNilpotent.map_iff (FaithfulSMul.algebraMap_injective R (End R p)),

Goal state after:
R : Type v
M : Type w
inst✝⁴ : CommRing R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R M
f : End R M
μ₁ μ₂ : R
hμ : μ₁ ≠ μ₂
k✝ l✝ : ℕ∞
k : ℕ
property✝¹ : ↑k ≤ k✝
l : ℕ
property✝ : ↑l ≤ l✝
a✝ : Nontrivial M
p : Submodule R M := (f.genEigenspace μ₁) ↑k ⊓ (f.genEigenspace μ₂) ↑l
hp : Nontrivial ↥p
f₁ : End R ↥p := LinearMap.restrict (f - (algebraMap R (End R M)) μ₁) ⋯
f₂ : End R ↥p := LinearMap.restrict (f - (algebraMap R (End R M)) μ₂) ⋯
this : IsNilpotent (μ₁ - μ₂)
hf₁₂ : f₂ - f₁ = (algebraMap R (End R ↥p)) (μ₁ - μ₂)
⊢ False
Module: Mathlib.LinearAlgebra.Trace

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
M : Type v
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
f g : M →ₗ[R] M
⊢ (trace R M) (f * g) = (trace R M) (g * f)

Tactic:
by
  classical
  by_cases H : ∃ s : Finset M, Nonempty (Basis s R M)
  · let ⟨s, ⟨b⟩⟩ := H
    simp_rw [trace_eq_matrix_trace R b, LinearMap.toMatrix_mul]
    apply Matrix.trace_mul_comm
  · rw [trace, dif_neg H, LinearMap.zero_apply, LinearMap.zero_apply]

Goal state after:
Module: Mathlib.LinearAlgebra.Trace

Goal state before:
pp: R : Type u_2
M : Type u_3
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f g : End R M
⊢ (trace R M) (g * f) - (trace R M) (g * f) = 0

Tactic:
exact sub_self _

Goal state after:
Module: Mathlib.GroupTheory.FreeGroup.Reduce

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
x : FreeGroup α
⊢ x⁻¹.norm = x.norm

Tactic:
by
  simp only [norm, toWord_inv, invRev_length]

Goal state after:
Module: Mathlib.Algebra.FreeAlgebra.Cardinality

Goal state before:
pp: R : Type u
inst✝² : CommSemiring R
A : Type v
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
H : lift.{u, v} #↑(Set.range ⇑((FreeAlgebra.lift R) Subtype.val)) ≤ #(FreeAlgebra R { x // x ∈ s })
⊢ lift.{u, v} #↥(adjoin R s) ≤ max (max (lift.{v, u} #R) (lift.{u, v} #↑s)) ℵ₀

Tactic:
Algebra.adjoin_eq_range_freeAlgebra_lift

Goal state after:
R : Type u
inst✝² : CommSemiring R
A : Type v
inst✝¹ : Semiring A
inst✝ : Algebra R A
s : Set A
H : lift.{u, v} #↑(Set.range ⇑((FreeAlgebra.lift R) Subtype.val)) ≤ #(FreeAlgebra R { x // x ∈ s })
⊢ lift.{u, v} #↥((FreeAlgebra.lift R) Subtype.val).range ≤ max (max (lift.{v, u} #R) (lift.{u, v} #↑s)) ℵ₀
Module: Mathlib.Algebra.Group.Subgroup.Order

Goal state before:
pp: case refine_2
G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
φ : G →* H
hφ : Function.Surjective ⇑φ
M : Subgroup H
hM✝ : IsCoatom M
hM : ∀ (b : Subgroup H), M < b → b = ⊤
⊢ ∀ (b : Subgroup G), comap φ M < b → b = ⊤

Tactic:
intro K hK

Goal state after:
case refine_2
G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
φ : G →* H
hφ : Function.Surjective ⇑φ
M : Subgroup H
hM✝ : IsCoatom M
hM : ∀ (b : Subgroup H), M < b → b = ⊤
K : Subgroup G
hK : comap φ M < K
⊢ K = ⊤
Module: Mathlib.Algebra.Group.Subgroup.Order

Goal state before:
pp: case refine_2
G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
φ : G →* H
hφ : Function.Surjective ⇑φ
M : Subgroup H
hM✝ : IsCoatom M
K : Subgroup G
hM : comap φ M < comap φ (map φ K) → comap φ (map φ K) = comap φ ⊤
hK : comap φ M < K
⊢ K = ⊤

Tactic:
comap_map_eq_self ((M.ker_le_comap φ).trans hK.le),

Goal state after:
case refine_2
G : Type u_1
inst✝¹ : Group G
H : Type u_2
inst✝ : Group H
φ : G →* H
hφ : Function.Surjective ⇑φ
M : Subgroup H
hM✝ : IsCoatom M
K : Subgroup G
hM : comap φ M < K → K = comap φ ⊤
hK : comap φ M < K
⊢ K = ⊤
Module: Mathlib.Algebra.Group.Irreducible.Indecomposable

Goal state before:
pp: ι : Type u_1
G : Type u_3
S : Type u_4
inst✝⁴ : CommGroup G
inst✝³ : LinearOrder S
inst✝² : InvolutiveInv ι
inst✝¹ : CommGroup S
inst✝ : IsOrderedMonoid S
v : ι → G
hv_inv : ∀ (i : ι), v i⁻¹ = (v i)⁻¹
f : G →* S
i : ι
hi✝ : i ∈ {i | 1 < (invMonoidHom.comp f) (v i)}
hi' :
  ∀ j ∈ {i | 1 < (invMonoidHom.comp f) (v i)},
    ∀ k ∈ {i | 1 < (invMonoidHom.comp f) (v i)}, v i = v j * v k → v j = 1 ∨ v k = 1
j : ι
hj : j ∈ {i | 1 < f (v i)}
k : ι
hk : k ∈ {i | 1 < f (v i)}
hi : v i = v j⁻¹ * v k⁻¹
⊢ j⁻¹ ∈ {i | 1 < (invMonoidHom.comp f) (v i)}

Tactic:
simpa [hv_inv]

Goal state after:
Module: Mathlib.Algebra.Group.Irreducible.Indecomposable

Goal state before:
pp: ι : Type u_1
G : Type u_3
S : Type u_4
inst✝⁴ : CommGroup G
inst✝³ : LinearOrder S
inst✝² : InvolutiveInv ι
inst✝¹ : CommGroup S
inst✝ : IsOrderedMonoid S
v : ι → G
hv_inv : ∀ (i : ι), v i⁻¹ = (v i)⁻¹
f : G →* S
i : ι
hi✝ : i ∈ {i | 1 < (invMonoidHom.comp f) (v i)}
hi' :
  ∀ j ∈ {i | 1 < (invMonoidHom.comp f) (v i)},
    ∀ k ∈ {i | 1 < (invMonoidHom.comp f) (v i)}, v i = v j * v k → v j = 1 ∨ v k = 1
j : ι
hj : j ∈ {i | 1 < f (v i)}
k : ι
hk : k ∈ {i | 1 < f (v i)}
hi : v i = v j⁻¹ * v k⁻¹
⊢ k⁻¹ ∈ {i | 1 < (invMonoidHom.comp f) (v i)}

Tactic:
simpa [hv_inv]

Goal state after:
Module: Mathlib.Algebra.Group.Irreducible.Indecomposable

Goal state before:
pp: ι : Type u_1
G : Type u_3
S : Type u_4
inst✝⁴ : CommGroup G
inst✝³ : LinearOrder S
inst✝² : InvolutiveInv ι
inst✝¹ : CommGroup S
inst✝ : IsOrderedMonoid S
v : ι → G
f : G →* S
s : Set ι
hf : ∀ i ∈ s, 1 < f (v i)
i : ι
hv_one : v i ≠ 1
hv_inv : v i⁻¹ = (v i)⁻¹
this :
  ∀ {ι : Type u_1} {G : Type u_3} {S : Type u_4} [inst : CommGroup G] [inst_1 : LinearOrder S]
    [inst_2 : InvolutiveInv ι] [inst_3 : CommGroup S] [IsOrderedMonoid S] (v : ι → G) (f : G →* S) (s : Set ι),
    (∀ i ∈ s, 1 < f (v i)) →
      ∀ (i : ι),
        v i ≠ 1 →
          v i⁻¹ = (v i)⁻¹ → v i ∈ closure (v '' s) ∨ (v i)⁻¹ ∈ closure (v '' s) → v i ∈ closure (v '' s) → f (v i) ≠ 1
hi : v i ∉ closure (v '' s)
hi' : (v i)⁻¹ ∈ closure (v '' s)
⊢ v i⁻¹ ≠ 1

Tactic:
simpa [hv_inv]

Goal state after:
Module: Mathlib.Algebra.Group.Irreducible.Indecomposable

Goal state before:
pp: ι : Type u_1
G : Type u_3
S : Type u_4
inst✝⁴ : CommGroup G
inst✝³ : LinearOrder S
inst✝² : InvolutiveInv ι
inst✝¹ : CommGroup S
inst✝ : IsOrderedMonoid S
v : ι → G
f : G →* S
s : Set ι
hf : ∀ i ∈ s, 1 < f (v i)
i : ι
hv_one : v i ≠ 1
hv_inv : v i⁻¹ = (v i)⁻¹
this :
  ∀ {ι : Type u_1} {G : Type u_3} {S : Type u_4} [inst : CommGroup G] [inst_1 : LinearOrder S]
    [inst_2 : InvolutiveInv ι] [inst_3 : CommGroup S] [IsOrderedMonoid S] (v : ι → G) (f : G →* S) (s : Set ι),
    (∀ i ∈ s, 1 < f (v i)) →
      ∀ (i : ι),
        v i ≠ 1 →
          v i⁻¹ = (v i)⁻¹ → v i ∈ closure (v '' s) ∨ (v i)⁻¹ ∈ closure (v '' s) → v i ∈ closure (v '' s) → f (v i) ≠ 1
hi : v i ∉ closure (v '' s)
hi' : (v i)⁻¹ ∈ closure (v '' s)
⊢ v i⁻¹ ∈ closure (v '' s)

Tactic:
simpa [hv_inv]

Goal state after:
Module: Mathlib.Algebra.GroupWithZero.Range

Goal state before:
pp: case h
A : Type u_1
B : Type u_2
inst✝¹ : MonoidWithZero A
inst✝ : MonoidWithZero B
f : A →*₀ B
a✝ b✝ : Bˣ
y : A
hy : f y = ↑a✝
y' : A
hy' : f y' = ↑b✝
⊢ f (y * y') = ↑(a✝ * b✝)

Tactic:
rw [map_mul, hy, hy', Units.val_mul]

Goal state after:
Module: Mathlib.CategoryTheory.FintypeCat

Goal state before:
pp: n : ℕ
⊢ Fintype.card (incl.obj (mk n)).obj = n

Tactic:
convert! Finset.card_fin n

Goal state after:
n : ℕ
⊢ Fintype.card (incl.obj (mk n)).obj = Finset.univ.card
Module: Mathlib.Order.Category.NonemptyFinLinOrd

Goal state before:
pp: case mp
A B : NonemptyFinLinOrd
f : A ⟶ B
a✝ : Epi f
m : ↑B.toLinOrd
hm : ∀ (a : ↑A.toLinOrd), (ConcreteCategory.hom f) a ≠ m
Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2))
p₁ : B ⟶ Y := ofHom { toFun := fun b ↦ if b < m then { down := 0 } else { down := 1 }, monotone' := ⋯ }
⊢ False

Tactic:
let p₂ : B ⟶ Y := ofHom
      ⟨fun b => if b ≤ m then ULift.up 0 else ULift.up 1, fun x₁ x₂ h => by
        simp only
        split_ifs with h₁ h₂ h₂
        any_goals apply Fin.zero_le
        · exfalso
          exact h₁ (h.trans h₂)
        · rfl⟩

Goal state after:
case mp
A B : NonemptyFinLinOrd
f : A ⟶ B
a✝ : Epi f
m : ↑B.toLinOrd
hm : ∀ (a : ↑A.toLinOrd), (ConcreteCategory.hom f) a ≠ m
Y : NonemptyFinLinOrd := of (ULift.{u, 0} (Fin 2))
p₁ : B ⟶ Y := ofHom { toFun := fun b ↦ if b < m then { down := 0 } else { down := 1 }, monotone' := ⋯ }
p₂ : B ⟶ Y := ofHom { toFun := fun b ↦ if b ≤ m then { down := 0 } else { down := 1 }, monotone' := ⋯ }
⊢ False
Module: Mathlib.AlgebraicTopology.MooreComplex

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Abelian C
X : SimplicialObject C
n : ℕ
⊢ underlying.obj (objX X (n + 1 + 1)) ⟶ underlying.obj (objX X (n + 1))

Tactic:
refine factorThru _ (arrow _ ≫ X.δ (0 : Fin (n + 3))) ?_
    -- We now need to show that it factors!
    -- A morphism factors through an intersection of subobjects if it factors through each.
    refine (finset_inf_factors _).mpr fun i _ => ?_
    -- A morphism `f` factors through the kernel of `g` exactly if `f ≫ g = 0`.
    apply kernelSubobject_factors
    dsimp [objX]
    -- Use a simplicial identity
    rw [Category.assoc, ← Fin.castSucc_zero, ← X.δ_comp_δ (Fin.zero_le i.succ)]
    -- We can rewrite the arrow out of the intersection of all the kernels as a composition
    -- of a morphism we don't care about with the arrow out of the kernel of `X.δ i.succ.succ`.
    rw [← factorThru_arrow _ _ (finset_inf_arrow_factors Finset.univ _ i.succ (by simp)),
      Category.assoc, kernelSubobject_arrow_comp_assoc, zero_comp, comp_zero]

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplexCategory.Basic

Goal state before:
pp: n m : SimplexCategory
f : n ⟶ m
⊢ Epi f ↔ Function.Surjective ⇑(Hom.toOrderHom f)

Tactic:
rw [← Functor.epi_map_iff_epi skeletalEquivalence.functor]

Goal state after:
n m : SimplexCategory
f : n ⟶ m
⊢ Epi (skeletalEquivalence.functor.map f) ↔ Function.Surjective ⇑(Hom.toOrderHom f)
Module: Mathlib.AlgebraicTopology.CechNerve

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ∀ (n : ℕ) (f : Arrow C), HasWidePullback f.right (fun x ↦ f.left) fun x ↦ f.hom
X : Augmented C
Y : Arrow C
⊢ ∀ {Y' : Arrow C} (f : Augmented.toArrow.obj X ⟶ Y) (g : Y ⟶ Y'),
    { toFun := equivalenceLeftToRight X Y', invFun := equivalenceRightToLeft X Y', left_inv := ⋯, right_inv := ⋯ }
        (f ≫ g) =
      { toFun := equivalenceLeftToRight X Y, invFun := equivalenceRightToLeft X Y, left_inv := ⋯, right_inv := ⋯ } f ≫
        augmentedCechNerve.map g

Tactic:
Y'

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : ∀ (n : ℕ) (f : Arrow C), HasWidePullback f.right (fun x ↦ f.left) fun x ↦ f.hom
X : Augmented C
Y Y' : Arrow C
⊢ ∀ (f : Augmented.toArrow.obj X ⟶ Y) (g : Y ⟶ Y'),
    { toFun := equivalenceLeftToRight X Y', invFun := equivalenceRightToLeft X Y', left_inv := ⋯, right_inv := ⋯ }
        (f ≫ g) =
      { toFun := equivalenceLeftToRight X Y, invFun := equivalenceRightToLeft X Y, left_inv := ⋯, right_inv := ⋯ } f ≫
        augmentedCechNerve.map g
Module: Mathlib.AlgebraicTopology.CechNerve

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
f : Arrow C
inst✝ : ∀ (n : ℕ), HasWidePushout f.left (fun x ↦ f.right) fun x ↦ f.hom
x y : SimplexCategory
g : x ⟶ y
j : Fin (x.len + 1)
⊢ f.hom ≫ WidePushout.ι (fun x ↦ f.hom) ((SimplexCategory.Hom.toOrderHom g) j) = WidePushout.head fun x ↦ f.hom

Tactic:
← WidePushout.arrow_ι

Goal state after:
C : Type u
inst✝¹ : Category.{v, u} C
f : Arrow C
inst✝ : ∀ (n : ℕ), HasWidePushout f.left (fun x ↦ f.right) fun x ↦ f.hom
x y : SimplexCategory
g : x ⟶ y
j : Fin (x.len + 1)
⊢ f.hom ≫ WidePushout.ι (fun x ↦ f.hom) ((SimplexCategory.Hom.toOrderHom g) j) =
    f.hom ≫ WidePushout.ι (fun x ↦ f.hom) ?j

case j
C : Type u
inst✝¹ : Category.{v, u} C
f : Arrow C
inst✝ : ∀ (n : ℕ), HasWidePushout f.left (fun x ↦ f.right) fun x ↦ f.hom
x y : SimplexCategory
g : x ⟶ y
j : Fin (x.len + 1)
⊢ Fin (y.len + 1)
Module: Mathlib.AlgebraicTopology.CechNerve

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
f✝ : Arrow C
inst✝² : ∀ (n : ℕ), HasWidePushout f✝.left (fun x ↦ f✝.right) fun x ↦ f✝.hom
f g : Arrow C
inst✝¹ : ∀ (n : ℕ), HasWidePushout f.left (fun x ↦ f.right) fun x ↦ f.hom
inst✝ : ∀ (n : ℕ), HasWidePushout g.left (fun x ↦ g.right) fun x ↦ g.hom
F : f ⟶ g
n : SimplexCategory
i : Fin (n.len + 1)
⊢ Hom.left F ≫ g.hom ≫ WidePushout.ι (fun x ↦ g.hom) i = Hom.left F ≫ WidePushout.head fun x ↦ g.hom

Tactic:
← WidePushout.arrow_ι

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
f✝ : Arrow C
inst✝² : ∀ (n : ℕ), HasWidePushout f✝.left (fun x ↦ f✝.right) fun x ↦ f✝.hom
f g : Arrow C
inst✝¹ : ∀ (n : ℕ), HasWidePushout f.left (fun x ↦ f.right) fun x ↦ f.hom
inst✝ : ∀ (n : ℕ), HasWidePushout g.left (fun x ↦ g.right) fun x ↦ g.hom
F : f ⟶ g
n : SimplexCategory
i : Fin (n.len + 1)
⊢ Hom.left F ≫ g.hom ≫ WidePushout.ι (fun x ↦ g.hom) i = Hom.left F ≫ g.hom ≫ WidePushout.ι (fun x ↦ g.hom) ?j

case j
C : Type u
inst✝³ : Category.{v, u} C
f✝ : Arrow C
inst✝² : ∀ (n : ℕ), HasWidePushout f✝.left (fun x ↦ f✝.right) fun x ↦ f✝.hom
f g : Arrow C
inst✝¹ : ∀ (n : ℕ), HasWidePushout f.left (fun x ↦ f.right) fun x ↦ f.hom
inst✝ : ∀ (n : ℕ), HasWidePushout g.left (fun x ↦ g.right) fun x ↦ g.hom
F : f ⟶ g
n : SimplexCategory
i : Fin (n.len + 1)
⊢ Fin (n.len + 1)
Module: Mathlib.AlgebraicTopology.SimplicialSet.Degenerate

Goal state before:
pp: case mp
X : SSet
n m : ℕ
x✝ : m < n
x : X _⦋n⦌
⊢ (∃ f, ∃ (_ : Epi f), opObjEquiv (opObjEquiv.symm x) ∈ Set.range ⇑(ConcreteCategory.hom (X.map f.op))) →
    ∃ f, ∃ (_ : Epi f), opObjEquiv.symm x ∈ Set.range ⇑(ConcreteCategory.hom (X.op.map f.op))

Tactic:
rintro ⟨f, _, y, rfl⟩

Goal state after:
case mp
X : SSet
n m : ℕ
x✝ : m < n
f : ⦋n⦌ ⟶ ⦋m⦌
w✝ : Epi f
y : X _⦋m⦌
⊢ ∃ f_1,
    ∃ (_ : Epi f_1),
      opObjEquiv.symm ((ConcreteCategory.hom (X.map f.op)) y) ∈ Set.range ⇑(ConcreteCategory.hom (X.op.map f_1.op))
Module: Mathlib.AlgebraicTopology.SimplicialSet.Degenerate

Goal state before:
pp: case mpr
X : SSet
n : ℕ
x : X.op _⦋n⦌
m : ℕ
x✝ : m < n
⊢ (∃ f, ∃ (_ : Epi f), x ∈ Set.range ⇑(ConcreteCategory.hom (X.op.map f.op))) →
    ∃ f, ∃ (_ : Epi f), opObjEquiv x ∈ Set.range ⇑(ConcreteCategory.hom (X.map f.op))

Tactic:
rintro ⟨f, _, y, rfl⟩

Goal state after:
case mpr
X : SSet
n m : ℕ
x✝ : m < n
f : ⦋n⦌ ⟶ ⦋m⦌
w✝ : Epi f
y : X.op _⦋m⦌
⊢ ∃ f_1,
    ∃ (_ : Epi f_1),
      opObjEquiv ((ConcreteCategory.hom (X.op.map f.op)) y) ∈ Set.range ⇑(ConcreteCategory.hom (X.map f_1.op))
Module: Mathlib.AlgebraicTopology.SimplicialSet.Degenerate

Goal state before:
pp: X : SSet
n : ℕ
x : X.op _⦋n⦌
⊢ opObjEquiv x ∈ X.degenerate n ↔ x ∈ X.op.degenerate n

Tactic:
simp only [mem_degenerate_iff]
  refine exists_congr (fun m ↦ exists_congr (fun _ ↦ ?_))
  constructor
  · obtain ⟨x, rfl⟩ := opObjEquiv.symm.surjective x
    rintro ⟨f, _, y, rfl⟩
    exact ⟨SimplexCategory.rev.map f, inferInstance, opObjEquiv.symm y, by simp [op_map]⟩
  · rintro ⟨f, _, y, rfl⟩
    exact ⟨SimplexCategory.rev.map f, inferInstance, opObjEquiv y, by simp [op_map]⟩

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.NonDegenerateSimplices

Goal state before:
pp: X : SSet
x y : X.N
h : x.subcomplex = y.subcomplex
⊢ x = y

Tactic:
simp [le_antisymm_iff, le_iff, h]

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.Degenerate

Goal state before:
pp: X : SSet
n : ℕ
x✝ : X _⦋n⦌
m : ℕ
f₁ : ⦋n⦌ ⟶ ⦋m⦌
inst✝ : Epi f₁
y₁ : ↑(X.nonDegenerate m)
hy₁ : x✝ = (ConcreteCategory.hom (X.map f₁.op)) ↑y₁
f₂ : ⦋n⦌ ⟶ ⦋m⦌
y₂ : ↑(X.nonDegenerate m)
hy₂ : x✝ = (ConcreteCategory.hom (X.map f₂.op)) ↑y₂
x : Fin (⦋n⦌.len + 1)
this : ∃ hf₁, (SimplexCategory.Hom.toOrderHom hf₁.section_) ((SimplexCategory.Hom.toOrderHom f₁) x) = x
⊢ (SimplexCategory.Hom.toOrderHom f₁) x = (SimplexCategory.Hom.toOrderHom f₂) x

Tactic:
by
    obtain ⟨hf₁, hf₁'⟩ := this
    dsimp at hf₁'
    simpa [g, hf₁'] using (SimplexCategory.congr_toOrderHom_apply (g_eq_id hy₁ hy₂ hf₁)
      (f₁.toOrderHom x)).symm

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.CompStructTruncated

Goal state before:
pp: X Y : Truncated 2
x : X.obj (Opposite.op { obj := ⦋0⦌, property := _proof_1 })
f : X ⟶ Y
⊢ ((id x).map f).edge = (id ((ConcreteCategory.hom (f.app (Opposite.op { obj := ⦋0⦌, property := _proof_1 }))) x)).edge

Tactic:
simp [NatTrans.naturality_apply]

Goal state after:
Module: Mathlib.Algebra.Homology.AlternatingConst

Goal state before:
pp: C : Type u_1
inst✝² : Category.{v_1, u_1} C
inst✝¹ : Preadditive C
inst✝ : HasZeroObject C
X : C
⊢ ∀ (i j : ℕ),
    (ComplexShape.down ℕ).Rel i j →
      (Iso.refl (((Functor.const SimplexCategoryᵒᵖ ⋙ alternatingFaceMapComplex C).obj X).X i)).hom ≫
          (ChainComplex.alternatingConst.obj X).d i j =
        ((Functor.const SimplexCategoryᵒᵖ ⋙ alternatingFaceMapComplex C).obj X).d i j ≫
          (Iso.refl (((Functor.const SimplexCategoryᵒᵖ ⋙ alternatingFaceMapComplex C).obj X).X j)).hom

Tactic:
by
    rintro _ i rfl
    simp [SimplicialObject.δ, ← Finset.sum_smul, Fin.sum_neg_one_pow, Nat.even_add_one,
      -Nat.not_even_iff_odd]

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n i : ℕ
x✝¹ x✝ : Δ[n] _⦋i⦌
h : (fun x j ↦ (Hom.toOrderHom (objEquiv x)) j) x✝¹ = (fun x j ↦ (Hom.toOrderHom (objEquiv x)) j) x✝
⊢ objEquiv x✝¹ = objEquiv x✝

Tactic:
by ext : 3; apply congr_fun h

Goal state after:
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: case neg
n : ℕ
Δ : SimplexCategory
f : ⦋n⦌ ⟶ Δ
i₁ i₂ : Fin (⦋n + 1⦌.len + 1)
hi : i₁ ≤ i₂
h₁ : ¬i₁ = 0
⊢ shiftFun (⇑(SimplexCategory.Hom.toOrderHom f)) i₁ ≤ shiftFun (⇑(SimplexCategory.Hom.toOrderHom f)) i₂

Tactic:
· have h₂ : i₂ ≠ 0 := by
            rintro rfl
            exact h₁ (le_antisymm hi (Fin.zero_le _))
          obtain ⟨j₁, hj₁⟩ := Fin.eq_succ_of_ne_zero h₁
          obtain ⟨j₂, hj₂⟩ := Fin.eq_succ_of_ne_zero h₂
          subst hj₁ hj₂
          simpa only [shiftFun_succ] using f.toOrderHom.monotone (Fin.succ_le_succ_iff.mp hi)

Goal state after:
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
f : Arrow C
inst✝ : ∀ (n : ℕ), HasWidePullback f.right (fun x ↦ f.left) fun x ↦ f.hom
S : SplitEpi f.hom
n : ℕ
i : Fin ((unop (op ⦋n + 1⦌)).len + 1)
⊢ Fin.cases ((WidePullback.base fun x ↦ f.hom) ≫ S.section_) (WidePullback.π fun x ↦ f.hom) i ≫ f.hom =
    WidePullback.base fun x ↦ f.hom

Tactic:
cases i using Fin.cases <;> simp

Goal state after:
Module: Mathlib.Algebra.Homology.TotalComplex

Goal state before:
pp: C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
i₁ : I₁
i₂ : I₂
i₁₂ : I₁₂
h : ¬c₂.Rel i₂ (c₂.next i₂)
⊢ K.d₂ c₁₂ i₁ i₂ i₁₂ = 0

Tactic:
dsimp [d₂]
  rw [HomologicalComplex.shape _ _ _ h, zero_comp, smul_zero]

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject.Trifunctor

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₁₂ : Type u_5
C₂₃ : Type u_6
inst✝⁶ : Category.{v_1, u_1} C₁
inst✝⁵ : Category.{v_2, u_2} C₂
inst✝⁴ : Category.{v_3, u_3} C₃
inst✝³ : Category.{v_4, u_4} C₄
inst✝² : Category.{v_5, u_5} C₁₂
inst✝¹ : Category.{v_6, u_6} C₂₃
F : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
J : Type u_10
p : I₁ × I₂ × I₃ → J
X₁ : GradedObject I₁ C₁
inst✝ :
  ∀ (X₂ : GradedObject I₂ C₂) (X₃ : GradedObject I₃ C₃), ((((mapTrifunctor F I₁ I₂ I₃).obj X₁).obj X₂).obj X₃).HasMap p
X₂ : GradedObject I₂ C₂
X₃ : GradedObject I₃ C₃
j : J
i₁ : I₁
i₂ : I₂
i₃ : I₃
h : p (i₁, i₂, i₃) = j
⊢ ιMapTrifunctorMapObj F p X₁ X₂ X₃ i₁ i₂ i₃ j h ≫ mapTrifunctorMapMap F p (𝟙 X₁) (𝟙 X₂) (𝟙 X₃) j =
    ιMapTrifunctorMapObj F p X₁ X₂ X₃ i₁ i₂ i₃ j h ≫ 𝟙 (mapTrifunctorMapObj F p X₁ X₂ X₃) j

Tactic:
simp only [ι_mapTrifunctorMapMap, categoryOfGradedObjects_id, Functor.map_id,
          NatTrans.id_app, id_comp, comp_id]

Goal state after:
Module: Mathlib.CategoryTheory.GradedObject.Trifunctor

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₃ : Type u_3
C₄ : Type u_4
C₁₂ : Type u_5
C₂₃ : Type u_6
inst✝⁷ : Category.{v_1, u_1} C₁
inst✝⁶ : Category.{v_2, u_2} C₂
inst✝⁵ : Category.{v_3, u_3} C₃
inst✝⁴ : Category.{v_4, u_4} C₄
inst✝³ : Category.{v_5, u_5} C₁₂
inst✝² : Category.{v_6, u_6} C₂₃
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
J : Type u_10
r : I₁ × I₂ × I₃ → J
ρ₁₂ : BifunctorComp₁₂IndexData r
X₁ : GradedObject I₁ C₁
X₂ : GradedObject I₂ C₂
X₃ : GradedObject I₃ C₃
inst✝¹ : (((mapBifunctor F₁₂ I₁ I₂).obj X₁).obj X₂).HasMap ρ₁₂.p
inst✝ : (((mapBifunctor G ρ₁₂.I₁₂ I₃).obj (mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂)).obj X₃).HasMap ρ₁₂.q
i₁ : I₁
i₂ : I₂
i₃ : I₃
j : J
h : r (i₁, i₂, i₃) = j
i₁₂ : ρ₁₂.I₁₂
h₁₂ : ρ₁₂.p (i₁, i₂) = i₁₂
⊢ ρ₁₂.q (i₁₂, i₃) = j

Tactic:
rw [← h₁₂, ← h, ← ρ₁₂.hpq]

Goal state after:
Module: Mathlib.Algebra.Homology.TotalComplex

Goal state before:
pp: case neg
C : Type u_1
inst✝⁴ : Category.{v_1, u_1} C
inst✝³ : Preadditive C
I₁ : Type u_2
I₂ : Type u_3
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c₁₂ : ComplexShape I₁₂
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : DecidableEq I₁₂
inst✝ : K.HasTotal c₁₂
i₁₂ i₁₂' i₁₂'' : I₁₂
h₁ : c₁₂.Rel i₁₂ i₁₂'
h₂ : c₁₂.Rel i₁₂' i₁₂''
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c₁₂ (i₁, i₂) = i₁₂
h₃ : c₂.Rel i₂ (c₂.next i₂)
h₄ : ¬c₂.Rel (c₂.next i₂) (c₂.next (c₂.next i₂))
⊢ c₁.ε₂ c₂ c₁₂ (i₁, i₂) • (K.X i₁).d i₂ (c₂.next i₂) ≫ K.d₂ c₁₂ i₁ (c₂.next i₂) i₁₂'' = 0

Tactic:
rw [K.d₂_eq_zero c₁₂ _ _ _ h₄, comp_zero, smul_zero]

Goal state after:
Module: Mathlib.Algebra.Homology.BifunctorAssociator

Goal state before:
pp: C₁ : Type u_1
C₂ : Type u_2
C₁₂ : Type u_3
C₃ : Type u_5
C₄ : Type u_6
inst✝²⁰ : Category.{v_1, u_1} C₁
inst✝¹⁹ : Category.{v_2, u_2} C₂
inst✝¹⁸ : Category.{v_3, u_5} C₃
inst✝¹⁷ : Category.{v_4, u_6} C₄
inst✝¹⁶ : Category.{v_5, u_3} C₁₂
inst✝¹⁵ : HasZeroMorphisms C₁
inst✝¹⁴ : HasZeroMorphisms C₂
inst✝¹³ : HasZeroMorphisms C₃
inst✝¹² : Preadditive C₁₂
inst✝¹¹ : Preadditive C₄
F₁₂ : C₁ ⥤ C₂ ⥤ C₁₂
G : C₁₂ ⥤ C₃ ⥤ C₄
inst✝¹⁰ : F₁₂.PreservesZeroMorphisms
inst✝⁹ : ∀ (X₁ : C₁), (F₁₂.obj X₁).PreservesZeroMorphisms
inst✝⁸ : G.Additive
inst✝⁷ : ∀ (X₁₂ : C₁₂), (G.obj X₁₂).PreservesZeroMorphisms
ι₁ : Type u_7
ι₂ : Type u_8
ι₃ : Type u_9
ι₁₂ : Type u_10
ι₄ : Type u_12
inst✝⁶ : DecidableEq ι₄
c₁ : ComplexShape ι₁
c₂ : ComplexShape ι₂
c₃ : ComplexShape ι₃
K₁ : HomologicalComplex C₁ c₁
K₂ : HomologicalComplex C₂ c₂
K₃ : HomologicalComplex C₃ c₃
c₁₂ : ComplexShape ι₁₂
c₄ : ComplexShape ι₄
inst✝⁵ : TotalComplexShape c₁ c₂ c₁₂
inst✝⁴ : TotalComplexShape c₁₂ c₃ c₄
inst✝³ : K₁.HasMapBifunctor K₂ F₁₂ c₁₂
inst✝² : DecidableEq ι₁₂
inst✝¹ : (K₁.mapBifunctor K₂ F₁₂ c₁₂).HasMapBifunctor K₃ G c₄
inst✝ : HasGoodTrifunctor₁₂Obj F₁₂ G K₁ K₂ K₃ c₁₂ c₄
j : ι₄
A : C₄
f :
  (i₁ : ι₁) →
    (i₂ : ι₂) →
      (i₃ : ι₃) → c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j → ((G.obj ((F₁₂.obj (K₁.X i₁)).obj (K₂.X i₂))).obj (K₃.X i₃) ⟶ A)
i₁ : ι₁
i₂ : ι₂
i₃ : ι₃
h : c₁.r c₂ c₃ c₁₂ c₄ (i₁, i₂, i₃) = j
⊢ ι F₁₂ G K₁ K₂ K₃ c₁₂ c₄ i₁ i₂ i₃ j h ≫ mapBifunctor₁₂Desc f = f i₁ i₂ i₃ h

Tactic:
apply GradedObject.ι_mapBifunctor₁₂BifunctorDesc

Goal state after:
Module: Mathlib.Algebra.Homology.CochainComplexOpposite

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
f g : K ⟶ L
h : Homotopy f g
p q : ℤ
hpq : ¬(ComplexShape.up ℤ).Rel q p
⊢ (h.hom (-q) (-p)).op = 0

Tactic:
rw [h.zero, op_zero]

Goal state after:
case a
C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
f g : K ⟶ L
h : Homotopy f g
p q : ℤ
hpq : ¬(ComplexShape.up ℤ).Rel q p
⊢ ¬(ComplexShape.up ℤ).Rel (-p) (-q)