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

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
I : Type u_3
inst✝ : Finite I
e : I → R
he : OrthogonalIdempotents e
this : Pairwise fun i j ↦ IsCoprime (Ideal.span {1 - e i}) (Ideal.span {1 - e j})
x : R
⊢ (RingHom.pi fun i ↦ Ideal.Quotient.mk (Ideal.span {1 - e i})) x =
    (Ideal.quotientInfToPiQuotient fun i ↦ Ideal.span {1 - e i}) ((Ideal.Quotient.mk (⨅ i, Ideal.span {1 - e i})) x)

Tactic:
ext i; simp [Ideal.quotientInfToPiQuotient]

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

Goal state before:
pp: R : Type v
M : Type w
inst✝² : CommRing R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f : End R M
μ : R
k : ℕ
h : f.HasUnifEigenvalue μ ↑k
⊢ k ≠ 0

Tactic:
rintro rfl
  simp [HasUnifEigenvalue, Nat.cast_zero, genEigenspace_zero] at h

Goal state after:
Module: Mathlib.RingTheory.Idempotents

Goal state before:
pp: case he₁.«0».«1»
R : Type u_1
inst✝ : CommRing R
e f : R
he : IsIdempotentElem e
hf : IsIdempotentElem f
hef₁ : e + f = 1
hef₂ : e * f = 0
o : Fin 2 → R :=
  fun i ↦
    match i with
    | 0 => e
    | 1 => f
hij : True
⊢ (1 - o 0) * (1 - o 1) = 0

Tactic:
simp [o, mul_comm, hef₂, ← hef₁]

Goal state after:
Module: Mathlib.RingTheory.Idempotents

Goal state before:
pp: case he₁.«1».«0»
R : Type u_1
inst✝ : CommRing R
e f : R
he : IsIdempotentElem e
hf : IsIdempotentElem f
hef₁ : e + f = 1
hef₂ : e * f = 0
o : Fin 2 → R :=
  fun i ↦
    match i with
    | 0 => e
    | 1 => f
hij : True
⊢ (1 - o 1) * (1 - o 0) = 0

Tactic:
simp [o, mul_comm, hef₂, ← hef₁]

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Ideal

Goal state before:
pp: R : Type u_1
inst✝² : Ring R
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
I : TwoSidedIdeal R
M : Matrix n n R
Mmem : M ∈ (matrix n I).jacobson
p q : n
⊢ M p q ∈ I.jacobson

Tactic:
mem_jacobson_iff

Goal state after:
R : Type u_1
inst✝² : Ring R
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
I : TwoSidedIdeal R
M : Matrix n n R
Mmem : M ∈ (matrix n I).jacobson
p q : n
⊢ ∀ (y : R), ∃ z, z * y * M p q + z - 1 ∈ I
Module: Mathlib.LinearAlgebra.Matrix.Ideal

Goal state before:
pp: R : Type u_1
inst✝² : Ring R
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
I : TwoSidedIdeal R
M : Matrix n n R
p q : n
Mmem : M * single q p 1 ∈ (matrix n I).jacobson
⊢ ∀ (y : R), ∃ z, z * y * M p q + z - 1 ∈ I

Tactic:
mem_jacobson_iff

Goal state after:
R : Type u_1
inst✝² : Ring R
n : Type u_2
inst✝¹ : Fintype n
inst✝ : DecidableEq n
I : TwoSidedIdeal R
M : Matrix n n R
p q : n
Mmem : ∀ (y : Matrix n n R), ∃ z, z * y * (M * single q p 1) + z - 1 ∈ matrix n I
⊢ ∀ (y : R), ∃ z, z * y * M p q + z - 1 ∈ I
Module: Mathlib.Algebra.DirectSum.LinearMap

Goal state before:
pp: case neg
ι : Type u_1
R : Type u_2
M : Type u_3
inst✝⁵ : CommRing R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
N : ι → Submodule R M
inst✝² : DecidableEq ι
κ : ι → Type u_4
inst✝¹ : (i : ι) → Fintype (κ i)
inst✝ : (i : ι) → DecidableEq (κ i)
s : Finset ι
h : IsInternal fun i ↦ N ↑i
b : (i : ↥s) → Basis (κ ↑i) R ↥(N ↑i)
σ : ι → ι
hσ : ∀ (i : ι), σ i ≠ i
f : End R M
hf : ∀ (i : ι), MapsTo ⇑f ↑(N i) ↑(N (σ i))
hN : ∀ i ∉ s, N i = ⊥
i : ↥s
k : κ ↑i
hi : σ ↑i ∉ s
⊢ ((h.collectedBasis b).repr (f ↑((b i) k))) ⟨i, k⟩ = 0

Tactic:
· suffices f (b i k) = 0 by simp [this]
    simpa [hN _ hi] using hf i <| Subtype.mem (b i k)

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
w : List (α × Bool)
⊢ reduce (invRev w) = invRev (reduce w)

Tactic:
apply reduce.min
  rw [← red_invRev_iff, invRev_invRev]
  apply Red.reduce_left
  have : Red (invRev (invRev w)) (invRev (reduce (invRev w))) := reduce.red.invRev
  rwa [invRev_invRev] at this

Goal state after:
Module: Mathlib.Algebra.FreeMonoid.FreeSemigroup

Goal state before:
pp: α : Type u_1
x : FreeSemigroup α
⊢ toFreeMonoid x ≠ 1

Tactic:
induction x with simp

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Eisenstein.Basic

Goal state before:
pp: R : Type u
inst✝ : CommRing R
𝓟 : Ideal R
f : R[X]
hf : f.IsWeaklyEisensteinAt 𝓟
x : R
hroot : ∑ i ∈ range (f.natDegree + 1), f.coeff i * x ^ i = 0
hmo : f.Monic
i : ℕ
hi : f.natDegree ≤ i
k : ℕ
hk : i = f.natDegree + k
⊢ x ^ f.natDegree ∈ 𝓟

Tactic:
Finset.range_add_one,

Goal state after:
R : Type u
inst✝ : CommRing R
𝓟 : Ideal R
f : R[X]
hf : f.IsWeaklyEisensteinAt 𝓟
x : R
hroot : ∑ i ∈ insert f.natDegree (range f.natDegree), f.coeff i * x ^ i = 0
hmo : f.Monic
i : ℕ
hi : f.natDegree ≤ i
k : ℕ
hk : i = f.natDegree + k
⊢ x ^ f.natDegree ∈ 𝓟
Module: Mathlib.RingTheory.IntegralClosure.IntegrallyClosed

Goal state before:
pp: case refine_2.refine_2
R : Type u_4
S : Type u_5
T : Type u_6
U : Type u_7
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : CommRing T
inst✝⁹ : CommRing U
inst✝⁸ : Algebra R T
inst✝⁷ : Algebra S T
inst✝⁶ : Algebra R U
inst✝⁵ : Algebra S U
inst✝⁴ : Algebra T U
inst✝³ : IsScalarTower S T U
inst✝² : IsScalarTower R T U
inst✝¹ : IsIntegralClosure S R T
inst✝ : IsIntegrallyClosedIn T U
x : U
⊢ (∃ y, (algebraMap S U) y = x) → IsIntegral R x

Tactic:
rintro ⟨x, rfl⟩

Goal state after:
case refine_2.refine_2
R : Type u_4
S : Type u_5
T : Type u_6
U : Type u_7
inst✝¹² : CommRing R
inst✝¹¹ : CommRing S
inst✝¹⁰ : CommRing T
inst✝⁹ : CommRing U
inst✝⁸ : Algebra R T
inst✝⁷ : Algebra S T
inst✝⁶ : Algebra R U
inst✝⁵ : Algebra S U
inst✝⁴ : Algebra T U
inst✝³ : IsScalarTower S T U
inst✝² : IsScalarTower R T U
inst✝¹ : IsIntegralClosure S R T
inst✝ : IsIntegrallyClosedIn T U
x : S
⊢ IsIntegral R ((algebraMap S U) x)
Module: Mathlib.Algebra.Group.Action.Equidecomp

Goal state before:
pp: X : Type u_1
G : Type u_2
inst✝ : SMul G X
⊢ Injective toPartialEquiv

Tactic:
intro ⟨_, _, _⟩

Goal state after:
X : Type u_1
G : Type u_2
inst✝ : SMul G X
toPartialEquiv✝ : PartialEquiv X X
w✝ : Finset G
h✝ : IsDecompOn (↑toPartialEquiv✝) toPartialEquiv✝.source w✝
⊢ ∀ ⦃a₂ : Equidecomp X G⦄,
    { toPartialEquiv := toPartialEquiv✝, isDecompOn' := ⋯ }.toPartialEquiv = a₂.toPartialEquiv →
      { toPartialEquiv := toPartialEquiv✝, isDecompOn' := ⋯ } = a₂
Module: Mathlib.Algebra.Group.Action.Pointwise.Finset

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : DecidableEq γ
inst✝² : SMul α γ
inst✝¹ : SMul β γ
inst✝ : SMulCommClass α β γ
a : α
s : Finset β
t : Finset γ
⊢ ↑(a • s • t) = ↑(s • a • t)

Tactic:
by simp only [coe_smul_finset, coe_smul, smul_comm]

Goal state after:
Module: Mathlib.Algebra.Group.NatPowAssoc

Goal state before:
pp: R : Type u_2
inst✝² : NonAssocSemiring R
inst✝¹ : Pow R ℕ
inst✝ : NatPowAssoc R
n m : ℕ
⊢ ↑(n ^ m) = ↑n ^ m

Tactic:
induction m with

Goal state after:
Module: Mathlib.CategoryTheory.FintypeCat

Goal state before:
pp: X Y : Skeleton
x✝ : IsIsomorphic X Y
h : X ≅ Y
⊢ Function.RightInverse (fun x ↦ (h.inv { down := x }).down) fun x ↦ (h.hom { down := x }).down

Tactic:
intro a
            change ULift.down _ = _
            rw [ULift.up_down]
            change ((h.inv ≫ h.hom) _).down = _
            simp
            rfl

Goal state after:
Module: Mathlib.CategoryTheory.Idempotents.Basic

Goal state before:
pp: case right
C : Type u_1
inst✝ : Category.{v_1, u_1} C
a✝ : IsIdempotentComplete C
X : C
p : X ⟶ X
hp : p ≫ p = p
Y : C
i : Y ⟶ X
e : X ⟶ Y
h₁ : i ≫ e = 𝟙 Y
h₂ : e ≫ i = p
s : Fork (𝟙 X) p
m : s.pt ⟶ (Fork.ofι i ⋯).pt
hm : m ≫ (Fork.ofι i ⋯).ι = s.ι
⊢ m = s.ι ≫ e

Tactic:
Fork.ι_ofι

Goal state after:
case right
C : Type u_1
inst✝ : Category.{v_1, u_1} C
a✝ : IsIdempotentComplete C
X : C
p : X ⟶ X
hp : p ≫ p = p
Y : C
i : Y ⟶ X
e : X ⟶ Y
h₁ : i ≫ e = 𝟙 Y
h₂ : e ≫ i = p
s : Fork (𝟙 X) p
m : s.pt ⟶ (Fork.ofι i ⋯).pt
hm : m ≫ i = s.ι
⊢ m = s.ι ≫ e
Module: Mathlib.Order.Category.NonemptyFinLinOrd

Goal state before:
pp: X Y : NonemptyFinLinOrd
f : X ⟶ Y
I : NonemptyFinLinOrd := of ↑(⇑(ConcreteCategory.hom f) '' ⊤)
e : X ⟶ I := ofHom { toFun := fun x ↦ ⟨(ConcreteCategory.hom f) x, ⋯⟩, monotone' := ⋯ }
⊢ Nonempty (StrongEpiMonoFactorisation f)

Tactic:
let m : I ⟶ Y := ofHom ⟨fun y => y.1, by tauto⟩

Goal state after:
X Y : NonemptyFinLinOrd
f : X ⟶ Y
I : NonemptyFinLinOrd := of ↑(⇑(ConcreteCategory.hom f) '' ⊤)
e : X ⟶ I := ofHom { toFun := fun x ↦ ⟨(ConcreteCategory.hom f) x, ⋯⟩, monotone' := ⋯ }
m : I ⟶ Y := ofHom { toFun := fun y ↦ ↑y, monotone' := ⋯ }
⊢ Nonempty (StrongEpiMonoFactorisation f)
Module: Mathlib.CategoryTheory.Idempotents.Karoubi

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
P Q : Karoubi C
f g : P ⟶ Q
⊢ f = g ↔ f.f = g.f

Tactic:
constructor
  · intro h
    rw [h]
  · apply Hom.ext

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

Goal state before:
pp: case inl
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
H : i ≤ j.castSucc
k : Fin (⦋n + 1⦌.len + 1)
hik : i ≤ k
⊢ j.succ.predAbove (i.castSucc.succAbove k) = i.succAbove (j.predAbove k)

Tactic:
Fin.succAbove_of_le_castSucc _ _ (Fin.castSucc_le_castSucc_iff.mpr hik),

Goal state after:
case inl
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
H : i ≤ j.castSucc
k : Fin (⦋n + 1⦌.len + 1)
hik : i ≤ k
⊢ j.succ.predAbove k.succ = i.succAbove (j.predAbove k)
Module: Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter

Goal state before:
pp: i n m : ℕ
j : Fin (m + 2)
hi : n + i = m
hj : ↑j ≤ i
⊢ δ₀Iter i hi ≫ δ j = δ₀Iter (i + 1) ⋯

Tactic:
by
  refine ConcreteCategory.hom_ext _ _ (fun k ↦ ?_)
  ext
  rw [dsimp% ConcreteCategory.comp_apply (δ₀Iter i hi) (δ j), coe_δ,
    δ₀Iter_apply .., δ₀Iter_apply ..,
    Fin.succAbove_of_le_castSucc _ _ (by grind)]
  simp [add_assoc]

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

Goal state before:
pp: case succ
j : ℕ
hj :
  ∀ {n : ℕ} (i : Fin (n + 2)) {m : ℕ} (i' : Fin (m + 2)) (h : m + j = n),
    j < ↑i → ↑i = ↑i' + j → δ i ≫ σ₀Iter j ⋯ = σ₀Iter j ⋯ ≫ δ i'
n : ℕ
i : Fin (n + 2)
m : ℕ
i' : Fin (m + 2)
h : m + (j + 1) = n
hi' : j + 1 < ↑i
hi'' : ↑i = ↑i' + (j + 1)
⊢ δ i ≫ σ₀Iter j ⋯ ≫ σ 0 = σ₀Iter (j + 1) ⋯ ≫ δ i'

Tactic:
σ₀Iter_succ_assoc _,

Goal state after:
case succ
j : ℕ
hj :
  ∀ {n : ℕ} (i : Fin (n + 2)) {m : ℕ} (i' : Fin (m + 2)) (h : m + j = n),
    j < ↑i → ↑i = ↑i' + j → δ i ≫ σ₀Iter j ⋯ = σ₀Iter j ⋯ ≫ δ i'
n : ℕ
i : Fin (n + 2)
m : ℕ
i' : Fin (m + 2)
h : m + (j + 1) = n
hi' : j + 1 < ↑i
hi'' : ↑i = ↑i' + (j + 1)
⊢ δ i ≫ σ₀Iter j ⋯ ≫ σ 0 = σ₀Iter j ⋯ ≫ σ 0 ≫ δ i'
Module: Mathlib.AlgebraicTopology.SimplicialSet.Dimension

Goal state before:
pp: X : SSet
d : ℕ
inst✝ : X.HasDimensionLT d
n : ℕ
hn : d ≤ n
⊢ X.nonDegenerate n = ∅

Tactic:
by
  simp [nonDegenerate, X.degenerate_eq_univ_of_hasDimensionLT d n hn]

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

Goal state before:
pp: X Y : SSet
f : X ⟶ Y
⊢ (∀ (x : SimplexCategoryᵒᵖ), (range f).obj x = ⊤.obj x) ↔ ∀ (k : SimplexCategoryᵒᵖ), Epi (f.app k)

Tactic:
simp only [epi_iff_surjective, Subfunctor.range_obj, Subfunctor.top_obj,
    Set.top_eq_univ, Set.range_eq_univ]

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

Goal state before:
pp: case neg
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)
hf : SplitEpi f₁
α : Fin (m + 1) → Fin (n + 1) :=
  fun y ↦ if y = (SimplexCategory.Hom.toOrderHom f₁) x then x else (SimplexCategory.Hom.toOrderHom hf.section_) y
y : Fin (m + 1)
hy : ¬y = (SimplexCategory.Hom.toOrderHom f₁) x
⊢ (SimplexCategory.Hom.toOrderHom f₁) ((SimplexCategory.Hom.toOrderHom hf.section_) y) = y

Tactic:
· apply SimplexCategory.congr_toOrderHom_apply hf.id

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

Goal state before:
pp: X : SSet
x y : X.S
f : ⦋x.dim⦌ ⟶ ⦋y.dim⦌
inst✝ : Epi f
hf : (ConcreteCategory.hom (X.map f.op)) y.simplex = x.simplex
this : IsSplitEpi f
⊢ (ConcreteCategory.hom (X.map (section_ f).op)) x.simplex = y.simplex

Tactic:
simp [← hf, ← Functor.map_comp_apply, ← op_comp]

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

Goal state before:
pp: case left
n m : ℕ
f₁ : unop (op ⦋m⦌) ⟶ ⦋n⦌
h₁✝ : objEquiv.symm f₁ ∈ Δ[n].nonDegenerate m
f₂ : unop (op ⦋m⦌) ⟶ ⦋n⦌
h₂✝ : objEquiv.symm f₂ ∈ Δ[n].nonDegenerate m
h₁ : Function.Injective ⇑(Hom.toOrderHom f₁)
h₂ : Function.Injective ⇑(Hom.toOrderHom f₂)
h₃ : Finset.image (⇑(Hom.toOrderHom f₁)) univ = Finset.image (⇑(Hom.toOrderHom f₂)) univ
⊢ f₁ = f₂

Tactic:
apply SimplexCategory.Hom.ext

Goal state after:
case left
n m : ℕ
f₁ : unop (op ⦋m⦌) ⟶ ⦋n⦌
h₁✝ : objEquiv.symm f₁ ∈ Δ[n].nonDegenerate m
f₂ : unop (op ⦋m⦌) ⟶ ⦋n⦌
h₂✝ : objEquiv.symm f₂ ∈ Δ[n].nonDegenerate m
h₁ : Function.Injective ⇑(Hom.toOrderHom f₁)
h₂ : Function.Injective ⇑(Hom.toOrderHom f₂)
h₃ : Finset.image (⇑(Hom.toOrderHom f₁)) univ = Finset.image (⇑(Hom.toOrderHom f₂)) univ
⊢ Hom.toOrderHom f₁ = Hom.toOrderHom f₂
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: Δ : SimplexCategory
n : ℕ
φ : (stdSimplex.obj Δ).left _⦋n⦌
⊢ objEquiv
      ((ConcreteCategory.hom ((↾fun f ↦ objEquiv.symm (shift (objEquiv f))) ≫ (stdSimplex.obj Δ).left.δ 0)).toFun φ) =
    objEquiv ((ConcreteCategory.hom (𝟙 ((stdSimplex.obj Δ).left _⦋n⦌))).toFun φ)

Tactic:
apply SimplexCategory.Hom.ext

Goal state after:
Δ : SimplexCategory
n : ℕ
φ : (stdSimplex.obj Δ).left _⦋n⦌
⊢ SimplexCategory.Hom.toOrderHom
      (objEquiv
        ((ConcreteCategory.hom ((↾fun f ↦ objEquiv.symm (shift (objEquiv f))) ≫ (stdSimplex.obj Δ).left.δ 0)).toFun
          φ)) =
    SimplexCategory.Hom.toOrderHom (objEquiv ((ConcreteCategory.hom (𝟙 ((stdSimplex.obj Δ).left _⦋n⦌))).toFun φ))
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: Δ : SimplexCategory
n : ℕ
i : Fin (n + 2)
φ : (stdSimplex.obj Δ).left _⦋n + 1⦌
⊢ objEquiv
      ((ConcreteCategory.hom ((↾fun f ↦ objEquiv.symm (shift (objEquiv f))) ≫ (stdSimplex.obj Δ).left.δ i.succ)).toFun
        φ) =
    objEquiv
      ((ConcreteCategory.hom ((stdSimplex.obj Δ).left.δ i ≫ ↾fun f ↦ objEquiv.symm (shift (objEquiv f)))).toFun φ)

Tactic:
apply SimplexCategory.Hom.ext

Goal state after:
Δ : SimplexCategory
n : ℕ
i : Fin (n + 2)
φ : (stdSimplex.obj Δ).left _⦋n + 1⦌
⊢ SimplexCategory.Hom.toOrderHom
      (objEquiv
        ((ConcreteCategory.hom ((↾fun f ↦ objEquiv.symm (shift (objEquiv f))) ≫ (stdSimplex.obj Δ).left.δ i.succ)).toFun
          φ)) =
    SimplexCategory.Hom.toOrderHom
      (objEquiv
        ((ConcreteCategory.hom ((stdSimplex.obj Δ).left.δ i ≫ ↾fun f ↦ objEquiv.symm (shift (objEquiv f)))).toFun φ))
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: Δ : SimplexCategory
n : ℕ
i : Fin (n + 1)
φ : (stdSimplex.obj Δ).left _⦋n⦌
⊢ objEquiv
      ((ConcreteCategory.hom ((↾fun f ↦ objEquiv.symm (shift (objEquiv f))) ≫ (stdSimplex.obj Δ).left.σ i.succ)).toFun
        φ) =
    objEquiv
      ((ConcreteCategory.hom ((stdSimplex.obj Δ).left.σ i ≫ ↾fun f ↦ objEquiv.symm (shift (objEquiv f)))).toFun φ)

Tactic:
apply SimplexCategory.Hom.ext

Goal state after:
Δ : SimplexCategory
n : ℕ
i : Fin (n + 1)
φ : (stdSimplex.obj Δ).left _⦋n⦌
⊢ SimplexCategory.Hom.toOrderHom
      (objEquiv
        ((ConcreteCategory.hom ((↾fun f ↦ objEquiv.symm (shift (objEquiv f))) ≫ (stdSimplex.obj Δ).left.σ i.succ)).toFun
          φ)) =
    SimplexCategory.Hom.toOrderHom
      (objEquiv
        ((ConcreteCategory.hom ((stdSimplex.obj Δ).left.σ i ≫ ↾fun f ↦ objEquiv.symm (shift (objEquiv f)))).toFun φ))
Module: Mathlib.Algebra.Homology.TotalComplex

Goal state before:
pp: case pos
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)
⊢ K.d₂ c₁₂ i₁ i₂ i₁₂' ≫ K.D₂ c₁₂ i₁₂' i₁₂'' = 0

Tactic:
totalAux.d₂_eq K c₁₂ i₁ h₃ i₁₂'

Goal state after:
case pos
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)
⊢ (c₁.ε₂ c₂ c₁₂ (i₁, i₂) •
        (K.X i₁).d i₂ (c₂.next i₂) ≫ K.toGradedObject.ιMapObj (c₁.π c₂ c₁₂) (i₁, c₂.next i₂) i₁₂' ?pos✝) ≫
      K.D₂ c₁₂ i₁₂' i₁₂'' =
    0

case pos
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)
⊢ c₁.π c₂ c₁₂ (i₁, c₂.next i₂) = i₁₂'
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
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 : c₁.Rel i₁ (c₁.next i₁)
h₄ : c₂.Rel i₂ (c₂.next i₂)
h₅ : c₁.π c₂ c₁₂ (i₁, c₂.next i₂) = i₁₂'
⊢ c₁₂.next (c₁.π c₂ c₁₂ (i₁, c₂.next i₂)) = c₁₂.next i₁₂'

Tactic:
h₅

Goal state after:
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 : c₁.Rel i₁ (c₁.next i₁)
h₄ : c₂.Rel i₂ (c₂.next i₂)
h₅ : c₁.π c₂ c₁₂ (i₁, c₂.next i₂) = i₁₂'
⊢ c₁₂.next i₁₂' = c₁₂.next i₁₂'
Module: Mathlib.Algebra.Homology.TotalComplexSymmetry

Goal state before:
pp: C : Type u_1
I₁ : Type u_2
I₂ : Type u_3
J : Type u_4
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Preadditive C
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c : ComplexShape J
inst✝⁶ : TotalComplexShape c₁ c₂ c
inst✝⁵ : TotalComplexShape c₂ c₁ c
inst✝⁴ : TotalComplexShapeSymmetry c₁ c₂ c
inst✝³ : K.HasTotal c
inst✝² : DecidableEq J
inst✝¹ : TotalComplexShapeSymmetry c₂ c₁ c
inst✝ : TotalComplexShapeSymmetrySymmetry c₁ c₂ c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
⊢ c₂.σ c₁ c i₂ i₁ • K.flip.ιTotal c i₂ i₁ j ⋯ = K.flip.flip.ιTotal c i₁ i₂ j h ≫ (K.totalFlipIso c).inv.f j

Tactic:
dsimp only [flip_flip]

Goal state after:
C : Type u_1
I₁ : Type u_2
I₂ : Type u_3
J : Type u_4
inst✝⁸ : Category.{v_1, u_1} C
inst✝⁷ : Preadditive C
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
c : ComplexShape J
inst✝⁶ : TotalComplexShape c₁ c₂ c
inst✝⁵ : TotalComplexShape c₂ c₁ c
inst✝⁴ : TotalComplexShapeSymmetry c₁ c₂ c
inst✝³ : K.HasTotal c
inst✝² : DecidableEq J
inst✝¹ : TotalComplexShapeSymmetry c₂ c₁ c
inst✝ : TotalComplexShapeSymmetrySymmetry c₁ c₂ c
j : J
i₁ : I₁
i₂ : I₂
h : c₁.π c₂ c (i₁, i₂) = j
⊢ c₂.σ c₁ c i₂ i₁ • K.flip.ιTotal c i₂ i₁ j ⋯ = K.ιTotal c i₁ i₂ j h ≫ (K.totalFlipIso c).inv.f j
Module: Mathlib.CategoryTheory.ObjectProperty.Orthogonal

Goal state before:
pp: C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : HasZeroMorphisms C
P : ObjectProperty C
X✝ Y✝ : C
e : X✝ ≅ Y✝
h : P.rightOrthogonal X✝
X : C
f : X ⟶ Y✝
hX : P X
⊢ f ≫ e.inv = 0

Tactic:
exact h _ hX

Goal state after:
Module: Mathlib.Algebra.Homology.HomotopyCategory.HomComplexCohomology

Goal state before:
pp: case refine_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
y : CohomologyClass K L n
x : Cocycle K L n
h : toHom (mk x) = toHom y
⊢ mk x = y

Tactic:
obtain ⟨y, rfl⟩ := y.mk_surjective

Goal state after:
case refine_1
C : Type u
inst✝¹ : Category.{v, u} C
inst✝ : Preadditive C
K L : CochainComplex C ℤ
n : ℤ
x y : Cocycle K L n
h : toHom (mk x) = toHom (mk y)
⊢ mk x = mk y
Module: Mathlib.CategoryTheory.Triangulated.Orthogonal

Goal state before:
pp: case refine_1
C : Type u
inst✝⁹ : Category.{v, u} C
D : Type u'
inst✝⁸ : Category.{v', u'} D
P : ObjectProperty C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : P.IsTriangulated
inst✝¹ : IsTriangulated C
Y : C
hY : P.trW.isLocal Y
L : C ⥤ D
inst✝ : L.IsLocalization P.trW
X : C
f₁ f₂ : X ⟶ Y
hf : L.map f₁ = L.map f₂
⊢ f₁ = f₂

Tactic:
rw [MorphismProperty.map_eq_iff_precomp L P.trW] at hf

Goal state after:
case refine_1
C : Type u
inst✝⁹ : Category.{v, u} C
D : Type u'
inst✝⁸ : Category.{v', u'} D
P : ObjectProperty C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : P.IsTriangulated
inst✝¹ : IsTriangulated C
Y : C
hY : P.trW.isLocal Y
L : C ⥤ D
inst✝ : L.IsLocalization P.trW
X : C
f₁ f₂ : X ⟶ Y
hf : ∃ X_1 s, ∃ (_ : P.trW s), s ≫ f₁ = s ≫ f₂
⊢ f₁ = f₂
Module: Mathlib.CategoryTheory.Triangulated.Orthogonal

Goal state before:
pp: case refine_1
C : Type u
inst✝⁹ : Category.{v, u} C
D : Type u'
inst✝⁸ : Category.{v', u'} D
P : ObjectProperty C
inst✝⁷ : HasZeroObject C
inst✝⁶ : HasShift C ℤ
inst✝⁵ : Preadditive C
inst✝⁴ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝³ : Pretriangulated C
inst✝² : P.IsTriangulated
inst✝¹ : IsTriangulated C
X : C
hX : P.trW.isColocal X
L : C ⥤ D
inst✝ : L.IsLocalization P.trW
Y : C
f₁ f₂ : X ⟶ Y
Z : C
s : Y ⟶ Z
hs : P.trW s
eq : f₁ ≫ s = f₂ ≫ s
⊢ f₁ = f₂

Tactic:
exact (hX _ hs).1 eq

Goal state after:
Module: Mathlib.Algebra.Homology.Embedding.ExtendHomotopy

Goal state before:
pp: case pos
ι : Type u_1
ι' : Type u_2
c : ComplexShape ι
c' : ComplexShape ι'
C : Type u_3
inst✝³ : Category.{v_1, u_3} C
inst✝² : HasZeroObject C
inst✝¹ : Preadditive C
K L : HomologicalComplex C c
f g : K ⟶ L
h : Homotopy f g
e : c.Embedding c'
inst✝ : e.IsRelIff
i : ι
hi : c.Rel (c.prev i) i
hi' : c'.Rel (e.f (c.prev i)) (e.f i)
⊢ (K.extendXIso e ⋯).hom ≫ (prevD i) h.hom ≫ (L.extendXIso e ⋯).inv = (prevD (e.f i)) (extend.hom e h.hom)

Tactic:
simp [prevD_eq _ hi, prevD_eq _ hi', extend.hom_eq _ _ rfl rfl,
            extend_d_eq _ _ rfl rfl]

Goal state after:
Module: Mathlib.CategoryTheory.Triangulated.TStructure.Basic

Goal state before:
pp: case mpr
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
inst✝⁴ : Preadditive C
inst✝³ : HasZeroObject C
inst✝² : HasShift C ℤ
inst✝¹ : ∀ (n : ℤ), (shiftFunctor C n).Additive
inst✝ : Pretriangulated C
t : TStructure C
X : C
n a n' : ℤ
hn' : a + n' = n
⊢ t.IsGE X n → t.IsGE ((shiftFunctor C a).obj X) n'

Tactic:
intro
    exact t.isGE_shift X n a n' hn'

Goal state after:
Module: Mathlib.CategoryTheory.MorphismProperty.LiftingProperty

Goal state before:
pp: C : Type u
inst✝ : Category.{v, u} C
T : MorphismProperty C
J : Type u_1
A : J → C
⊢ ∀ (X₂ : J → C) [inst : Limits.HasCoproduct A] [inst_1 : Limits.HasCoproduct X₂] (f : (j : J) → A j ⟶ X₂ j),
    (∀ (j : J), T.llp (f j)) → T.llp (Limits.Sigma.map f)

Tactic:
B

Goal state after:
C : Type u
inst✝ : Category.{v, u} C
T : MorphismProperty C
J : Type u_1
A B : J → C
⊢ ∀ [inst : Limits.HasCoproduct A] [inst_1 : Limits.HasCoproduct B] (f : (j : J) → A j ⟶ B j),
    (∀ (j : J), T.llp (f j)) → T.llp (Limits.Sigma.map f)