mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n m : ℕ
x✝ : { x // x ∈ Δ[n].nonDegenerate m }
x : Δ[n] _⦋m⦌
hx : Function.Injective ⇑(Hom.toOrderHom (objEquiv x))
⊢ #(Finset.image (⇑(Hom.toOrderHom (objEquiv x))) univ) = m + 1

Tactic:
]

Goal state after:
n m : ℕ
x✝ : { x // x ∈ Δ[n].nonDegenerate m }
x : Δ[n] _⦋m⦌
hx : Function.Injective ⇑(Hom.toOrderHom (objEquiv x))
⊢ #(Finset.image (⇑(Hom.toOrderHom (objEquiv x))) univ) = m + 1
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i : Fin (n + 1)
k : Fin (n + 1 + 1)
hk : i.succ.rev = k
⊢ X.left.δ₀Iter ↑i ⋯ ≫ X.left.δ 0 ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑i ⋯ =
    X.left.δ₀Iter ↑i ⋯ ≫ X.left.δ 0 ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑i ⋯

Tactic:
]

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

Goal state before:
pp: n m : ℕ
x✝ : { x // x ∈ Δ[n].nonDegenerate m }
x : Δ[n] _⦋m⦌
hx : Function.Injective ⇑(Hom.toOrderHom (objEquiv x))
⊢ m + 1 = m + 1

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : ↑j.succ ≤ ↑i
k : Fin (n + 1)
hk : j.rev = k
⊢ ∃ l, ↑i = ↑j + 1 + l

Tactic:
]

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : ↑j.succ ≤ ↑i
k : Fin (n + 1)
hk : j.rev = k
⊢ ∃ l, ↑i = ↑j + 1 + l
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
⊢ Set.range ⇑(Hom.toOrderHom f₁) = Set.range ⇑(Hom.toOrderHom f₂)

Tactic:
]

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
⊢ Set.range ⇑(Hom.toOrderHom f₁) = Set.range ⇑(Hom.toOrderHom f₂)
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: case right.refine_1
n m : ℕ
S : Finset (Fin (n + 1))
hS : #S = m + 1
e : Fin (m + 1) ≃o ↥S := monoEquivOfFin ↥S ⋯
⊢ Function.Injective
    ⇑(Hom.toOrderHom (objEquiv (objMk ((OrderHom.Subtype.val fun x ↦ x ∈ S).comp e.toOrderEmbedding.toOrderHom))))

Tactic:
]

Goal state after:
case right.refine_1
n m : ℕ
S : Finset (Fin (n + 1))
hS : #S = m + 1
e : Fin (m + 1) ≃o ↥S := monoEquivOfFin ↥S ⋯
⊢ Function.Injective
    ⇑(Hom.toOrderHom (objEquiv (objMk ((OrderHom.Subtype.val fun x ↦ x ∈ S).comp e.toOrderEmbedding.toOrderHom))))
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: case left
n d : ℕ
x : ↑(Δ[n].nonDegenerate d)
this : Function.Injective ⇑(Hom.toOrderHom (objEquiv ↑x))
⊢ Function.Injective fun i ↦ ⟨↑x i, ⋯⟩

Tactic:
]

Goal state after:
case left
n d : ℕ
x : ↑(Δ[n].nonDegenerate d)
this : Function.Injective ⇑(Hom.toOrderHom (objEquiv ↑x))
⊢ Function.Injective fun i ↦ ⟨↑x i, ⋯⟩
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: case right
n d : ℕ
x : ↑(Δ[n].nonDegenerate d)
j : Fin (n + 1)
hj✝ : j ∈ ↑(nonDegenerateEquiv' x)
hj : ∃ i, ↑x i = j
⊢ ∃ a, (fun i ↦ ⟨↑x i, ⋯⟩) a = ⟨j, hj✝⟩

Tactic:
]

Goal state after:
case right
n d : ℕ
x : ↑(Δ[n].nonDegenerate d)
j : Fin (n + 1)
hj✝ : j ∈ ↑(nonDegenerateEquiv' x)
hj : ∃ i, ↑x i = j
⊢ ∃ a, (fun i ↦ ⟨↑x i, ⋯⟩) a = ⟨j, hj✝⟩
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : j.castSucc < i
k : Fin (n + 1)
hk : j.rev = k
l : ℕ
hl : ↑i = ↑j + 1 + l
this :
  ed.s (↑k + 1) ≫ X.left.δ ⟨l + 1 + 1, ⋯⟩ =
    X.left.δ ⟨l + 1, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.homotopy._proof_24 n i j k hk l hl⟩ ≫
      ed.s ↑k
⊢ X.left.δ₀Iter ↑j.castSucc ⋯ ≫
      X.left.δ ⟨l + 1, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.homotopy._proof_24 n i j k hk l hl⟩ ≫
        ed.s ↑k ≫ X.left.σ₀Iter ↑j.castSucc ⋯ =
    X.left.δ i ≫ X.left.δ₀Iter ↑j ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j ⋯

Tactic:
← X.left.δ_δ₀Iter'_assoc _ _ i (by grind)

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : j.castSucc < i
k : Fin (n + 1)
hk : j.rev = k
l : ℕ
hl : ↑i = ↑j + 1 + l
this :
  ed.s (↑k + 1) ≫ X.left.δ ⟨l + 1 + 1, ⋯⟩ =
    X.left.δ ⟨l + 1, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.homotopy._proof_24 n i j k hk l hl⟩ ≫
      ed.s ↑k
⊢ X.left.δ i ≫ X.left.δ₀Iter ↑j.castSucc ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j.castSucc ⋯ =
    X.left.δ i ≫ X.left.δ₀Iter ↑j ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j ⋯
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n d : ℕ
x : ↑(Δ[n].nonDegenerate d)
this : Function.Injective ⇑(Hom.toOrderHom (objEquiv ↑x))
⊢ ∀ {a b : Fin (d + 1)},
    (Equiv.ofBijective (fun i ↦ ⟨↑x i, ⋯⟩) ⋯) a ≤ (Equiv.ofBijective (fun i ↦ ⟨↑x i, ⋯⟩) ⋯) b ↔ a ≤ b

Tactic:
]

Goal state after:
n d : ℕ
x : ↑(Δ[n].nonDegenerate d)
this : Function.Injective ⇑(Hom.toOrderHom (objEquiv ↑x))
⊢ ∀ {a b : Fin (d + 1)},
    (Equiv.ofBijective (fun i ↦ ⟨↑x i, ⋯⟩) ⋯) a ≤ (Equiv.ofBijective (fun i ↦ ⟨↑x i, ⋯⟩) ⋯) b ↔ a ≤ b
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : j.castSucc < i
k : Fin (n + 1)
hk : j.rev = k
l : ℕ
hl : ↑i = ↑j + 1 + l
this :
  ed.s (↑k + 1) ≫ X.left.δ ⟨l + 1 + 1, ⋯⟩ =
    X.left.δ ⟨l + 1, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.homotopy._proof_24 n i j k hk l hl⟩ ≫
      ed.s ↑k
⊢ X.left.δ i ≫ X.left.δ₀Iter ↑j.castSucc ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j.castSucc ⋯ =
    X.left.δ i ≫ X.left.δ₀Iter ↑j ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j ⋯

Tactic:
]

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i : Fin (n + 2)
j : Fin (n + 1)
hij : j.castSucc < i
k : Fin (n + 1)
hk : j.rev = k
l : ℕ
hl : ↑i = ↑j + 1 + l
this :
  ed.s (↑k + 1) ≫ X.left.δ ⟨l + 1 + 1, ⋯⟩ =
    X.left.δ ⟨l + 1, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.homotopy._proof_24 n i j k hk l hl⟩ ≫
      ed.s ↑k
⊢ X.left.δ i ≫ X.left.δ₀Iter ↑j.castSucc ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j.castSucc ⋯ =
    X.left.δ i ≫ X.left.δ₀Iter ↑j ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j ⋯
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: case mp
n d : ℕ
x : ↑(Δ[n].nonDegenerate d)
this : Function.Injective ⇑(Hom.toOrderHom (objEquiv ↑x))
a b : Fin (d + 1)
⊢ ¬↑x b < ↑x a → ¬b < a

Tactic:
]

Goal state after:
case mp
n d : ℕ
x : ↑(Δ[n].nonDegenerate d)
this : Function.Injective ⇑(Hom.toOrderHom (objEquiv ↑x))
a b : Fin (d + 1)
⊢ ¬↑x b < ↑x a → ¬b < a
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i j : Fin (n + 1)
hij : i ≤ j
k : Fin (n + 1)
hk : j.rev = k
⊢ X.left.δ₀Iter ↑j ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j ⋯ ≫ X.left.σ i.castSucc =
    X.left.σ i ≫ X.left.δ₀Iter ↑j.succ ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j.succ ⋯

Tactic:
]

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i j : Fin (n + 1)
hij : i ≤ j
k : Fin (n + 1)
hk : j.rev = k
⊢ X.left.δ₀Iter ↑j ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j ⋯ ≫ X.left.σ i.castSucc =
    X.left.σ i ≫ X.left.δ₀Iter ↑j.succ ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter ↑j.succ ⋯
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n d : ℕ
A : Δ[n].Subcomplex
x : ↑(Δ[n].nonDegenerate d)
⊢ ↑x ∈ A.obj (op ⦋d⦌) ↔ ↑x ∈ A.obj (op ⦋d⦌)

Tactic:
]

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

Goal state before:
pp: n : SimplexCategory
d : SimplexCategoryᵒᵖ
⊢ Finite (unop d ⟶ n)

Tactic:
]

Goal state after:
n : SimplexCategory
d : SimplexCategoryᵒᵖ
⊢ Finite (unop d ⟶ n)
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: X : SSet
n : ℕ
f : Δ[n] ⟶ X
⊢ (Subcomplex.range f).toSSet.Finite

Tactic:
]

Goal state after:
X : SSet
n : ℕ
f : Δ[n] ⟶ X
⊢ (Subcomplex.range f).toSSet.Finite
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n : ℕ
S : Finset (Fin (n + 1))
d : ℕ
hd : #S ≤ d
hm : #S = 0
⊢ #S = 0

Tactic:
]

Goal state after:
n : ℕ
S : Finset (Fin (n + 1))
d : ℕ
hd : #S ≤ d
hm : #S = 0
⊢ #S = 0
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: case zero
n d : ℕ
hd : #∅ ≤ d
hm : #∅ = 0
⊢ ⊥.toSSet.HasDimensionLT d

Tactic:
]

Goal state after:
case zero
n d : ℕ
hd : #∅ ≤ d
hm : #∅ = 0
⊢ ⊥.toSSet.HasDimensionLT d
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: case succ
n : ℕ
S : Finset (Fin (n + 1))
d : ℕ
hd : #S ≤ d
m : ℕ
hm : #S = m + 1
⊢ Δ[m].HasDimensionLT d

Tactic:
]

Goal state after:
case succ
n : ℕ
S : Finset (Fin (n + 1))
d : ℕ
hd : #S ≤ d
m : ℕ
hm : #S = m + 1
⊢ Δ[m].HasDimensionLT d
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i j : Fin (n + 1)
hij : i ≤ j
k : Fin (n + 1)
hk : j.rev = k
⊢ X.left.δ₀Iter ↑j ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter (↑j + 1) ⋯ = X.left.δ₀Iter ↑j ⋯ ≫ ed.s ↑k ≫ X.left.σ₀Iter (↑j + 1) ⋯

Tactic:
]

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

Goal state before:
pp: n : ℕ
⊢ StrictMono ⇑(objEquiv.symm (𝟙 (unop (op ⦋n⦌))))

Tactic:
]

Goal state after:
n : ℕ
⊢ StrictMono ⇑(objEquiv.symm (𝟙 (unop (op ⦋n⦌))))
Module: Mathlib.AlgebraicTopology.ExtraDegeneracy

Goal state before:
pp: C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i j : Fin (n + 1)
hij : j ≤ i
k : Fin (n + 1)
hk : j.rev = k
l : ℕ
hl : ↑j + l = ↑i
⊢ X.left.δ₀Iter ↑j ⋯ ≫
      ed.s ↑k ≫
        X.left.σ ⟨l + 1, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.homotopy._proof_43 n i j k hk l hl⟩ ≫
          X.left.σ₀Iter ↑j ⋯ =
    X.left.δ₀Iter (↑j.castSucc)
        (CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.homotopy._proof_32 n j k hk) ≫
      ed.s ↑k ≫
        X.left.σ
            ⟨l, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.homotopy._proof_41 n i j k hk l hl⟩.succ ≫
          X.left.σ₀Iter ↑j.castSucc ⋯

Tactic:
]

Goal state after:
C : Type u_1
inst✝ : Category.{v_1, u_1} C
X : Augmented C
ed : X.ExtraDegeneracy
n : ℕ
i j : Fin (n + 1)
hij : j ≤ i
k : Fin (n + 1)
hk : j.rev = k
l : ℕ
hl : ↑j + l = ↑i
⊢ X.left.δ₀Iter ↑j ⋯ ≫
      ed.s ↑k ≫
        X.left.σ ⟨l + 1, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.homotopy._proof_43 n i j k hk l hl⟩ ≫
          X.left.σ₀Iter ↑j ⋯ =
    X.left.δ₀Iter (↑j.castSucc)
        (CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.homotopy._proof_32 n j k hk) ≫
      ed.s ↑k ≫
        X.left.σ
            ⟨l, CategoryTheory.SimplicialObject.Augmented.ExtraDegeneracy.homotopy._proof_41 n i j k hk l hl⟩.succ ≫
          X.left.σ₀Iter ↑j.castSucc ⋯
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n : SimplexCategory
d d' : SimplexCategoryᵒᵖ
f : d ⟶ d'
g : (stdSimplex.obj n).op.obj d
i : Fin (d'.1.len + 1)
⊢ ((opObjEquiv (opObjEquiv.symm ((ConcreteCategory.hom ((stdSimplex.obj n).map (rev.map f.unop).op)) (opObjEquiv g))))
        i.rev).rev =
    ((ConcreteCategory.hom ((stdSimplex.obj n).map f)) (stdSimplex.opObjEquiv g)) i

Tactic:
]

Goal state after:
n : SimplexCategory
d d' : SimplexCategoryᵒᵖ
f : d ⟶ d'
g : (stdSimplex.obj n).op.obj d
i : Fin (d'.1.len + 1)
⊢ ((opObjEquiv (opObjEquiv.symm ((ConcreteCategory.hom ((stdSimplex.obj n).map (rev.map f.unop).op)) (opObjEquiv g))))
        i.rev).rev =
    ((ConcreteCategory.hom ((stdSimplex.obj n).map f)) (stdSimplex.opObjEquiv g)) i
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n : SimplexCategory
d d' : SimplexCategoryᵒᵖ
f : d ⟶ d'
g : (stdSimplex.obj n).op.obj d
i : Fin (d'.1.len + 1)
⊢ (stdSimplex.opObjEquiv ((ConcreteCategory.hom ((stdSimplex.obj n).op.map f)) g)) i =
    ((ConcreteCategory.hom ((stdSimplex.obj n).map f)) (stdSimplex.opObjEquiv g)) i

Tactic:
rw [stdSimplex.opObjEquiv_apply, op_map]

Goal state after:
n : SimplexCategory
d d' : SimplexCategoryᵒᵖ
f : d ⟶ d'
g : (stdSimplex.obj n).op.obj d
i : Fin (d'.1.len + 1)
⊢ ((opObjEquiv (opObjEquiv.symm ((ConcreteCategory.hom ((stdSimplex.obj n).map (rev.map f.unop).op)) (opObjEquiv g))))
        i.rev).rev =
    ((ConcreteCategory.hom ((stdSimplex.obj n).map f)) (stdSimplex.opObjEquiv g)) i
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n : SimplexCategory
d d' : SimplexCategoryᵒᵖ
f : d ⟶ d'
g : (stdSimplex.obj n).op.obj d
i : Fin (d'.1.len + 1)
⊢ (((ConcreteCategory.hom ((stdSimplex.obj n).map (rev.map f.unop).op)) (opObjEquiv g)) i.rev).rev =
    ((ConcreteCategory.hom ((stdSimplex.obj n).map f)) (stdSimplex.opObjEquiv g)) i

Tactic:
]

Goal state after:
n : SimplexCategory
d d' : SimplexCategoryᵒᵖ
f : d ⟶ d'
g : (stdSimplex.obj n).op.obj d
i : Fin (d'.1.len + 1)
⊢ (((ConcreteCategory.hom ((stdSimplex.obj n).map (rev.map f.unop).op)) (opObjEquiv g)) i.rev).rev =
    ((ConcreteCategory.hom ((stdSimplex.obj n).map f)) (stdSimplex.opObjEquiv g)) i
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: n : SimplexCategory
d d' : SimplexCategoryᵒᵖ
f : d ⟶ d'
g : (stdSimplex.obj n).op.obj d
i : Fin (d'.1.len + 1)
⊢ ((opObjEquiv g) ((ConcreteCategory.hom f.unop) i.rev.rev).rev).rev =
    ((ConcreteCategory.hom ((stdSimplex.obj n).map f)) (stdSimplex.opObjEquiv g)) i

Tactic:
]

Goal state after:
n : SimplexCategory
d d' : SimplexCategoryᵒᵖ
f : d ⟶ d'
g : (stdSimplex.obj n).op.obj d
i : Fin (d'.1.len + 1)
⊢ ((opObjEquiv g) ((ConcreteCategory.hom f.unop) i.rev.rev).rev).rev =
    ((ConcreteCategory.hom ((stdSimplex.obj n).map f)) (stdSimplex.opObjEquiv g)) i
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: X : SSet
n : ℕ
x : X _⦋n⦌
⊢ range (toOfSimplex x) = ⊤

Tactic:
]

Goal state after:
X : SSet
n : ℕ
x : X _⦋n⦌
⊢ range (toOfSimplex x) = ⊤
Module: Mathlib.Algebra.Homology.Augment

Goal state before:
pp: V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
i : ℕ
⊢ 0 = 0

pp: case a
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
i : ℕ
⊢ ¬(ComplexShape.up ℕ).Rel 0 (i + 2)

Tactic:
]

Goal state after:
case a
V : Type u
inst✝¹ : Category.{v, u} V
inst✝ : HasZeroMorphisms V
C : CochainComplex V ℕ
i : ℕ
⊢ ¬(ComplexShape.up ℕ).Rel 0 (i + 2)
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: X : SSet
n : ℕ
x : X _⦋n⦌
m : SimplexCategoryᵒᵖ
u : op ⦋n⦌ ⟶ m
⊢ ↑({ toFun := fun x_1 ↦ ⟨(ConcreteCategory.hom (X.map u)) ↑x_1, ⋯⟩ } ⟨yonedaEquiv.toFun (yonedaEquiv.invFun x), ⋯⟩) =
    ↑⟨(ConcreteCategory.hom (X.map u)) (yonedaEquiv.toFun (yonedaEquiv.invFun x)), ⋯⟩

Tactic:
]

Goal state after:
X : SSet
n : ℕ
x : X _⦋n⦌
m : SimplexCategoryᵒᵖ
u : op ⦋n⦌ ⟶ m
⊢ ↑({ toFun := fun x_1 ↦ ⟨(ConcreteCategory.hom (X.map u)) ↑x_1, ⋯⟩ } ⟨yonedaEquiv.toFun (yonedaEquiv.invFun x), ⋯⟩) =
    ↑⟨(ConcreteCategory.hom (X.map u)) (yonedaEquiv.toFun (yonedaEquiv.invFun x)), ⋯⟩
Module: Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex

Goal state before:
pp: case mp
X : SSet
n : ℕ
x : X _⦋n⦌
a✝ : IsIso (toOfSimplex x)
⊢ Mono (toOfSimplex x ≫ (ofSimplex x).ι)

Tactic:
]

Goal state after:
case mp
X : SSet
n : ℕ
x : X _⦋n⦌
a✝ : IsIso (toOfSimplex x)
⊢ Mono (toOfSimplex x ≫ (ofSimplex x).ι)
Module: Mathlib.Algebra.Homology.ComplexShapeSigns

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₁₂ : ComplexShape I₁₂
inst✝ : TotalComplexShape c₁ c₂ c₁₂
i₁ i₁' : I₁
i₂ i₂' : I₂
h₁ : c₁.Rel i₁ i₁'
h₂ : c₂.Rel i₂ i₂'
⊢ c₁.ε₁ c₂ c₁₂ (i₁, i₂) * (c₁.ε₂ c₂ c₁₂ (i₁, i₂) * (c₁.ε₁ c₂ c₁₂ (i₁, i₂') * c₁.ε₁ c₂ c₁₂ (i₁, i₂'))) =
    -c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')

Tactic:
]

Goal state after:
I₁ : Type u_1
I₂ : Type u_2
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₁₂ : ComplexShape I₁₂
inst✝ : TotalComplexShape c₁ c₂ c₁₂
i₁ i₁' : I₁
i₂ i₂' : I₂
h₁ : c₁.Rel i₁ i₁'
h₂ : c₂.Rel i₂ i₂'
⊢ c₁.ε₁ c₂ c₁₂ (i₁, i₂) * (c₁.ε₂ c₂ c₁₂ (i₁, i₂) * (c₁.ε₁ c₂ c₁₂ (i₁, i₂') * c₁.ε₁ c₂ c₁₂ (i₁, i₂'))) =
    -c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')
Module: Mathlib.Algebra.Homology.ComplexShapeSigns

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₁₂ : Type u_4
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₁₂ : ComplexShape I₁₂
inst✝ : TotalComplexShape c₁ c₂ c₁₂
i₁ i₁' : I₁
i₂ i₂' : I₂
h₁ : c₁.Rel i₁ i₁'
h₂ : c₂.Rel i₂ i₂'
⊢ c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂') = c₁.ε₂ c₂ c₁₂ (i₁', i₂) * c₁.ε₁ c₂ c₁₂ (i₁, i₂')

Tactic:
]

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

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₃ : Type u_3
I₁₂ : Type u_4
I₂₃ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₃ : ComplexShape I₃
c₁₂ : ComplexShape I₁₂
c₂₃ : ComplexShape I₂₃
c✝ : ComplexShape J
inst✝² : TotalComplexShape c₁ c₂ c₁₂
I : Type u_7
inst✝¹ : AddMonoid I
c : ComplexShape I
inst✝ : c.TensorSigns
i₁✝ i₁'✝ i₂✝ i₂'✝ : I
h : c.Rel i₁✝ i₁'✝
x✝ : c.Rel i₂✝ i₂'✝
⊢ c.ε i₁✝ = c.ε i₁✝

Tactic:
]

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

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₃ : Type u_3
I₁₂ : Type u_4
I₂₃ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₃ : ComplexShape I₃
c₁₂ : ComplexShape I₁₂
c₂₃ : ComplexShape I₂₃
c✝ : ComplexShape J
inst✝² : TotalComplexShape c₁ c₂ c₁₂
I : Type u_7
inst✝¹ : AddMonoid I
c : ComplexShape I
inst✝ : c.TensorSigns
q : ℕ
⊢ (-1) ^ q = (-1) ^ q

Tactic:
]

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

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₃ : Type u_3
I₁₂ : Type u_4
I₂₃ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₃ : ComplexShape I₃
c₁₂ : ComplexShape I₁₂
c₂₃ : ComplexShape I₂₃
c✝ : ComplexShape J
inst✝² : TotalComplexShape c₁ c₂ c₁₂
I : Type u_7
inst✝¹ : AddMonoid I
c : ComplexShape I
inst✝ : c.TensorSigns
p : ℤ
⊢ -p.negOnePow = -p.negOnePow

Tactic:
]

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

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₃ : Type u_3
I₁₂ : Type u_4
I₂₃ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₃ : ComplexShape I₃
c₁₂ : ComplexShape I₁₂
c₂₃ : ComplexShape I₂₃
c✝ : ComplexShape J
inst✝² : TotalComplexShape c₁ c₂ c₁₂
I : Type u_7
inst✝¹ : AddMonoid I
c : ComplexShape I
inst✝ : c.TensorSigns
x✝² x✝¹ x✝ : I
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₃ : Type u_3
I₁₂ : Type u_4
I₂₃ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₃ : ComplexShape I₃
c₁₂ : ComplexShape I₁₂
c₂₃ : ComplexShape I₂₃
c✝ : ComplexShape J
inst✝² : TotalComplexShape c₁ c₂ c₁₂
I : Type u_7
inst✝¹ : AddMonoid I
c : ComplexShape I
inst✝ : c.TensorSigns
x✝² x✝¹ x✝ : I
⊢ c.ε x✝² = c.ε x✝²

Tactic:
]

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✝⁵ : Category.{v_1, u_1} C₁
inst✝⁴ : Category.{v_2, u_2} C₂
inst✝³ : Category.{v_3, u_3} C₃
inst✝² : Category.{v_4, u_4} C₄
inst✝¹ : Category.{v_5, u_5} C₁₂
inst✝ : Category.{v_6, u_6} C₂₃
F F' : C₁ ⥤ C₂ ⥤ C₃ ⥤ C₄
I₁ : Type u_7
I₂ : Type u_8
I₃ : Type u_9
X₁ Y₁ : GradedObject I₁ C₁
φ : X₁ ⟶ Y₁
X₂ Y₂ : GradedObject I₂ C₂
ψ : X₂ ⟶ Y₂
X₃ : GradedObject I₃ C₃
x : I₁ × I₂ × I₃
⊢ (F.map (φ x.1)).app (X₂ x.2.1) ≫ (F.obj (Y₁ x.1)).map (ψ x.2.1) =
    (F.map (φ x.1)).app (X₂ x.2.1) ≫ (F.obj (Y₁ x.1)).map (ψ x.2.1)

Tactic:
]

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

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₃ : Type u_3
I₁₂ : Type u_4
I₂₃ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₃ : ComplexShape I₃
c₁₂ : ComplexShape I₁₂
c₂₃ : ComplexShape I₂₃
c✝ : ComplexShape J
inst✝² : TotalComplexShape c₁ c₂ c₁₂
I : Type u_7
inst✝¹ : AddMonoid I
c : ComplexShape I
inst✝ : c.TensorSigns
x✝² x✝¹ x✝ : I
⊢ c.ε x✝² * c.ε x✝¹ = c.ε x✝² * c.ε x✝¹

Tactic:
]

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

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₃ : Type u_3
I₁₂ : Type u_4
I₂₃ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₃ : ComplexShape I₃
c₁₂ : ComplexShape I₁₂
c₂₃ : ComplexShape I₂₃
c : ComplexShape J
inst✝ : TotalComplexShape c₁ c₂ c₁₂
i₁✝ i₁'✝ : I₂
i₂✝ i₂'✝ : I₁
h₂ : c₂.Rel i₁✝ i₁'✝
h₁ : c₁.Rel i₂✝ i₂'✝
⊢ ((match (i₁✝, i₂✝) with
      | (i₂, i₁) => c₁.ε₁ c₂ c₁₂ (i₁, i₂)) *
      match (i₁✝, i₂'✝) with
      | (i₂, i₁) => c₁.ε₂ c₂ c₁₂ (i₁, i₂)) =
    c₁.ε₁ c₂ c₁₂ (i₂✝, i₁✝) * c₁.ε₂ c₂ c₁₂ (i₂'✝, i₁✝)

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
I₁ : Type u_2
I₂ : Type u_3
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K L : HomologicalComplex₂ C c₁ c₂
f : K.toGradedObject ⟶ L.toGradedObject
comm₁ : ∀ (i₁ i₁' : I₁) (i₂ : I₂), c₁.Rel i₁ i₁' → f (i₁, i₂) ≫ (L.d i₁ i₁').f i₂ = (K.d i₁ i₁').f i₂ ≫ f (i₁', i₂)
comm₂ : ∀ (i₁ : I₁) (i₂ i₂' : I₂), c₂.Rel i₂ i₂' → f (i₁, i₂) ≫ (L.X i₁).d i₂ i₂' = (K.X i₁).d i₂ i₂' ≫ f (i₁, i₂')
i₁ i₁' : I₁
h₁ : c₁.Rel i₁ i₁'
⊢ { f := fun i₂ ↦ f (i₁, i₂), comm' := ⋯ } ≫ L.d i₁ i₁' = K.d i₁ i₁' ≫ { f := fun i₂ ↦ f (i₁', i₂), comm' := ⋯ }

Tactic:
by
    ext i₂
    exact comm₁ i₁ i₁' i₂ h₁

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
I₁ : Type u_2
I₂ : Type u_3
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
i₁ i₁' : I₁
h : ¬c₁.Rel i₁ i₁'
i₂ : I₂
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: C : Type u_1
inst✝¹ : Category.{v_1, u_1} C
inst✝ : HasZeroMorphisms C
I₁ : Type u_2
I₂ : Type u_3
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
K : HomologicalComplex₂ C c₁ c₂
i₁ i₁' i₁'' : I₁
i₂ : I₂
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₃ : Type u_3
I₁₂ : Type u_4
I₂₃ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₃ : ComplexShape I₃
c₁₂ : ComplexShape I₁₂
c₂₃ : ComplexShape I₂₃
c : ComplexShape J
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : TotalComplexShape c₂ c₁ c₁₂
inst✝ : TotalComplexShapeSymmetry c₁ c₂ c₁₂
p q : ℤ
⊢ (p * q).negOnePow = (p * q).negOnePow

Tactic:
]

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

Goal state before:
pp: I₁ : Type u_1
I₂ : Type u_2
I₃ : Type u_3
I₁₂ : Type u_4
I₂₃ : Type u_5
J : Type u_6
c₁ : ComplexShape I₁
c₂ : ComplexShape I₂
c₃ : ComplexShape I₃
c₁₂ : ComplexShape I₁₂
c₂₃ : ComplexShape I₂₃
c : ComplexShape J
inst✝² : TotalComplexShape c₁ c₂ c₁₂
inst✝¹ : TotalComplexShape c₂ c₁ c₁₂
inst✝ : TotalComplexShapeSymmetry c₁ c₂ c₁₂
p q : ℤ
⊢ (p * q + p).negOnePow = (p * q + p).negOnePow

Tactic:
]

Goal state after: