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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
⊢ ⨆ n, (skeletonOfMono i) n ≤ ⊤

Tactic:
by

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

Goal state before:
pp: n : ℕ
i : Fin (n + 1)
j k : ↑{i}ᶜ
⊢ stdSimplex.face {↑j, ↑k}ᶜ = stdSimplex.face {↑j}ᶜ ⊓ stdSimplex.face {↑k}ᶜ

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
⊢ ⨆ n, (skeletonOfMono i) n = ⊤

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d : ℕ
x : ↑(Y.nonDegenerate d)
n : ℕ
⊢ ↑x ∈ ((skeletonOfMono i) n).obj (op ⦋d⦌) ↔ ↑x ∈ Set.range ⇑(ConcreteCategory.hom (i.app (op ⦋d⦌))) ∨ d < n

Tactic:
by

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

Goal state before:
pp: n : ℕ
X : SSet
i : Fin (n + 2)
f g : Λ[n + 1, i].toSSet ⟶ X
h : ∀ (j : Fin (n + 2)) (hj : j ≠ i), ι i j hj ≫ f = ι i j hj ≫ g
⊢ f = g

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d n : ℕ
h : d < n
⊢ ((skeletonOfMono i) n).obj (op ⦋d⦌) = ⊤

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
n : ℕ
x : ↑(Y.nonDegenerate n)
hx : ↑x ∈ (Subcomplex.range i).obj (op ⦋n⦌)
⊢ Subcomplex.ofSimplex ↑x ≤ Subcomplex.range i

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
n : ℕ
x : ↑(Y.nonDegenerate n)
hx : ↑x ∉ (Subcomplex.range i).obj (op ⦋n⦌)
⊢ Subcomplex.ofSimplex ↑x ≤ Subcomplex.ofSimplex ↑x

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Types.Pushouts

Goal state before:
pp: X₁ X₂ X₃ X₄ X₅ : Type u
t : X₁ ⟶ X₂
r : X₂ ⟶ X₄
l : X₁ ⟶ X₃
b : X₃ ⟶ X₄
k : X₄ ⟶ X₅
r' : X₂ ⟶ X₅
b' : X₃ ⟶ X₅
h₁ : IsPullback t l r' b'
facr : r ≫ k = r'
facb : b ≫ k = b'
inst✝¹ : Mono r'
inst✝ : Mono k
h₂ : Set.range ⇑(ConcreteCategory.hom r) ⊔ Set.range ⇑(ConcreteCategory.hom b) = Set.univ
H :
  ∀ (x₃ y₃ : X₃),
    x₃ ∉ Set.range ⇑(ConcreteCategory.hom l) →
      y₃ ∉ Set.range ⇑(ConcreteCategory.hom l) → (ConcreteCategory.hom b') x₃ = (ConcreteCategory.hom b') y₃ → x₃ = y₃
φ : pushout t l ⟶ X₄
hφ₁ : pushout.inl t l ≫ φ = r
hφ₂ : pushout.inr t l ≫ φ = b
this : Mono (φ ≫ k)
⊢ IsIso φ

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
n : ℕ
x : ↑(Y.nonDegenerate n)
hx : ↑x ∉ (Subcomplex.range i).obj (op ⦋n⦌)
⊢ Subcomplex.ofSimplex ↑x ≤ ⨆ (_ : ↑x ∉ (Subcomplex.range i).obj (op ⦋n⦌)), Subcomplex.ofSimplex ↑x

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Types.Pushouts

Goal state before:
pp: X₁ X₂ X₃ X₄ X₅ : Type u
t : X₁ ⟶ X₂
r : X₂ ⟶ X₄
l : X₁ ⟶ X₃
b : X₃ ⟶ X₄
k : X₄ ⟶ X₅
r' : X₂ ⟶ X₅
b' : X₃ ⟶ X₅
h₁ : IsPullback t l r' b'
facr : r ≫ k = r'
facb : b ≫ k = b'
inst✝¹ : Mono r'
inst✝ : Mono k
h₂ : Set.range ⇑(ConcreteCategory.hom r) ⊔ Set.range ⇑(ConcreteCategory.hom b) = Set.univ
H :
  ∀ (x₃ y₃ : X₃),
    x₃ ∉ Set.range ⇑(ConcreteCategory.hom l) →
      y₃ ∉ Set.range ⇑(ConcreteCategory.hom l) → (ConcreteCategory.hom b') x₃ = (ConcreteCategory.hom b') y₃ → x₃ = y₃
φ : pushout t l ⟶ X₄
hφ₁ : pushout.inl t l ≫ φ = r
hφ₂ : pushout.inr t l ≫ φ = b
this✝ : Mono (φ ≫ k)
this : IsIso φ
⊢ t ≫ (Iso.refl X₂).hom = (Iso.refl X₁).hom ≫ t

Tactic:
by

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

Goal state before:
pp: X : SSet
n : ℕ
i : Fin (n + 1 + 2)
g : Λ[n + 1 + 1, i].toSSet ⟶ X
j : Fin (n + 2)
hj : j.castSucc ≠ i
k : Fin (n + 2)
hk : k.succ ≠ i
hjk : j.castSucc < k.succ
⊢ j ≤ k

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Types.Pushouts

Goal state before:
pp: X₁ X₂ X₃ X₄ X₅ : Type u
t : X₁ ⟶ X₂
r : X₂ ⟶ X₄
l : X₁ ⟶ X₃
b : X₃ ⟶ X₄
k : X₄ ⟶ X₅
r' : X₂ ⟶ X₅
b' : X₃ ⟶ X₅
h₁ : IsPullback t l r' b'
facr : r ≫ k = r'
facb : b ≫ k = b'
inst✝¹ : Mono r'
inst✝ : Mono k
h₂ : Set.range ⇑(ConcreteCategory.hom r) ⊔ Set.range ⇑(ConcreteCategory.hom b) = Set.univ
H :
  ∀ (x₃ y₃ : X₃),
    x₃ ∉ Set.range ⇑(ConcreteCategory.hom l) →
      y₃ ∉ Set.range ⇑(ConcreteCategory.hom l) → (ConcreteCategory.hom b') x₃ = (ConcreteCategory.hom b') y₃ → x₃ = y₃
φ : pushout t l ⟶ X₄
hφ₁ : pushout.inl t l ≫ φ = r
hφ₂ : pushout.inr t l ≫ φ = b
this✝ : Mono (φ ≫ k)
this : IsIso φ
⊢ l ≫ (Iso.refl X₃).hom = (Iso.refl X₁).hom ≫ l

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Types.Pushouts

Goal state before:
pp: X₁ X₂ X₃ X₄ X₅ : Type u
t : X₁ ⟶ X₂
r : X₂ ⟶ X₄
l : X₁ ⟶ X₃
b : X₃ ⟶ X₄
k : X₄ ⟶ X₅
r' : X₂ ⟶ X₅
b' : X₃ ⟶ X₅
h₁ : IsPullback t l r' b'
facr : r ≫ k = r'
facb : b ≫ k = b'
inst✝¹ : Mono r'
inst✝ : Mono k
h₂ : Set.range ⇑(ConcreteCategory.hom r) ⊔ Set.range ⇑(ConcreteCategory.hom b) = Set.univ
H :
  ∀ (x₃ y₃ : X₃),
    x₃ ∉ Set.range ⇑(ConcreteCategory.hom l) →
      y₃ ∉ Set.range ⇑(ConcreteCategory.hom l) → (ConcreteCategory.hom b') x₃ = (ConcreteCategory.hom b') y₃ → x₃ = y₃
φ : pushout t l ⟶ X₄
hφ₁ : pushout.inl t l ≫ φ = r
hφ₂ : pushout.inr t l ≫ φ = b
this✝ : Mono (φ ≫ k)
this : IsIso φ
⊢ pushout.inl t l ≫ (asIso φ).hom = (Iso.refl X₂).hom ≫ r

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Types.Pushouts

Goal state before:
pp: X₁ X₂ X₃ X₄ X₅ : Type u
t : X₁ ⟶ X₂
r : X₂ ⟶ X₄
l : X₁ ⟶ X₃
b : X₃ ⟶ X₄
k : X₄ ⟶ X₅
r' : X₂ ⟶ X₅
b' : X₃ ⟶ X₅
h₁ : IsPullback t l r' b'
facr : r ≫ k = r'
facb : b ≫ k = b'
inst✝¹ : Mono r'
inst✝ : Mono k
h₂ : Set.range ⇑(ConcreteCategory.hom r) ⊔ Set.range ⇑(ConcreteCategory.hom b) = Set.univ
H :
  ∀ (x₃ y₃ : X₃),
    x₃ ∉ Set.range ⇑(ConcreteCategory.hom l) →
      y₃ ∉ Set.range ⇑(ConcreteCategory.hom l) → (ConcreteCategory.hom b') x₃ = (ConcreteCategory.hom b') y₃ → x₃ = y₃
φ : pushout t l ⟶ X₄
hφ₁ : pushout.inl t l ≫ φ = r
hφ₂ : pushout.inr t l ≫ φ = b
this✝ : Mono (φ ≫ k)
this : IsIso φ
⊢ pushout.inr t l ≫ (asIso φ).hom = (Iso.refl X₃).hom ≫ b

Tactic:
by

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

Goal state before:
pp: n : ℕ
X : SSet
i : Fin (n + 2)
g : Λ[n + 1, i].toSSet ⟶ X
⊢ horn.IsCompatible fun j hj ↦ ι i j hj ≫ g

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Types.Pushouts

Goal state before:
pp: X₁ X₂ X₃ X₄ X₅ : Type u
t : X₁ ⟶ X₂
r : X₂ ⟶ X₄
l : X₁ ⟶ X₃
b : X₃ ⟶ X₄
k : X₄ ⟶ X₅
r' : X₂ ⟶ X₅
b' : X₃ ⟶ X₅
h₁ : IsPullback t l r' b'
facr : r ≫ k = r'
facb : b ≫ k = b'
inst✝¹ : Mono r'
inst✝ : Mono k
h₂ : Set.range ⇑(ConcreteCategory.hom r) ⊔ Set.range ⇑(ConcreteCategory.hom b) = Set.univ
H :
  ∀ (x₃ y₃ : X₃),
    x₃ ∉ Set.range ⇑(ConcreteCategory.hom l) →
      y₃ ∉ Set.range ⇑(ConcreteCategory.hom l) → (ConcreteCategory.hom b') x₃ = (ConcreteCategory.hom b') y₃ → x₃ = y₃
⊢ IsPushout t l r b

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
n : ℕ
⊢ n ≤ n + 1

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Types.Pushouts

Goal state before:
pp: X₁ X₂ X₃ X₄ : Type u
t : X₁ ⟶ X₂
r : X₂ ⟶ X₄
l : X₁ ⟶ X₃
b : X₃ ⟶ X₄
h₁ : IsPullback t l r b
inst✝ : Mono r
h₂ : Set.range ⇑(ConcreteCategory.hom r) ⊔ Set.range ⇑(ConcreteCategory.hom b) = Set.univ
H :
  ∀ (x₃ y₃ : X₃),
    x₃ ∉ Set.range ⇑(ConcreteCategory.hom l) →
      y₃ ∉ Set.range ⇑(ConcreteCategory.hom l) → (ConcreteCategory.hom b) x₃ = (ConcreteCategory.hom b) y₃ → x₃ = y₃
⊢ r ≫ 𝟙 X₄ = r

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
n : ℕ
x✝¹ : ↑(Y.nonDegenerate n)
x✝ : ↑x✝¹ ∉ (Subcomplex.range i).obj (op ⦋n⦌)
⊢ n < n + 1

Tactic:
by

Goal state after:
Module: Mathlib.CategoryTheory.Limits.Types.Pushouts

Goal state before:
pp: X₁ X₂ X₃ X₄ : Type u
t : X₁ ⟶ X₂
r : X₂ ⟶ X₄
l : X₁ ⟶ X₃
b : X₃ ⟶ X₄
h₁ : IsPullback t l r b
inst✝ : Mono r
h₂ : Set.range ⇑(ConcreteCategory.hom r) ⊔ Set.range ⇑(ConcreteCategory.hom b) = Set.univ
H :
  ∀ (x₃ y₃ : X₃),
    x₃ ∉ Set.range ⇑(ConcreteCategory.hom l) →
      y₃ ∉ Set.range ⇑(ConcreteCategory.hom l) → (ConcreteCategory.hom b) x₃ = (ConcreteCategory.hom b) y₃ → x₃ = y₃
⊢ b ≫ 𝟙 X₄ = b

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
n : ℕ
⊢ (skeletonOfMono i) (n + 1) =
    (skeletonOfMono i) n ⊔ ⨆ x, ⨆ (_ : ↑x ∉ (Subcomplex.range i).obj (op ⦋n⦌)), Subcomplex.ofSimplex ↑x

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d : ℕ
c : Cell i d
d' : ℕ
⊢ c.simplex ∈ ((skeletonOfMono i) d').obj (op ⦋d⦌) ↔
    c.simplex ∈ Set.range ⇑(ConcreteCategory.hom (i.app (op ⦋d⦌))) ∨ d < d'

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d : ℕ
c : Cell i d
⊢ Subcomplex.range c.map ≤ (skeletonOfMono i) (d + 1)

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d : ℕ
c : Cell i d
n : ℕ
y : Δ[d] _⦋n⦌
hy₁ : y ∈ Δ[d].nonDegenerate n
hy₂ : ↑⟨y, hy₁⟩ ∈ ∂Δ[d].obj (op ⦋n⦌)
⊢ ⟨y, hy₂⟩ ∈ ∂Δ[d].toSSet.nonDegenerate n

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d : ℕ
c : Cell i d
⊢ ((skeletonOfMono i) d).preimage c.map = ∂Δ[d]

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d : ℕ
c : Cell i d
⊢ Subcomplex.range (∂Δ[d].ι ≫ c.map) ≤ (skeletonOfMono i) d

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d : ℕ
⊢ d ≤ d + 1

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d : ℕ
⊢ t i d ≫ r i d = l i d ≫ b i d

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d : ℕ
c : Cell i d
⊢ c.ιSigmaBoundary ≫ t i d ≫ ((skeletonOfMono i) d).ι =
    ∂Δ[d].ι ≫ c.ιSigmaStdSimplex ≫ b i d ≫ ((skeletonOfMono i) (d + 1)).ι

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d : ℕ
c : Cell i d
⊢ c.ιSigmaBoundary ≫ l i d = ∂Δ[d].ι ≫ c.ιSigmaStdSimplex

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d : ℕ
c : Cell i d
⊢ c.ιSigmaStdSimplex ≫ b i d ≫ ((skeletonOfMono i) (d + 1)).ι = c.map

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d : ℕ
c : Cell i d
n : SimplexCategory
f : n ⟶ ⦋d⦌
⊢ ↑((ConcreteCategory.hom ((b i d).app (op n)))
        ((ConcreteCategory.hom (c.ιSigmaStdSimplex.app (op n))) (stdSimplex.objEquiv.symm f))) =
    (ConcreteCategory.hom (Y.map f.op)) c.simplex

Tactic:
by

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

Goal state before:
pp: n : ℕ
X : SSet
i : Fin (n + 2)
f : (j : Fin (n + 2)) → j ≠ i → (Δ[n] ⟶ X)
hf : horn.IsCompatible f
⊢ ∀ (a : (MultispanShape.ofLinearOrder ↑{i}ᶜ).L),
    (((CompleteLattice.MulticoequalizerDiagram.multispanIndex ⋯).toLinearOrder.map Subcomplex.toSSetFunctor).fst a ≫
        match (MultispanShape.ofLinearOrder ↑{i}ᶜ).fst a with
        | ⟨j, hj⟩ => (faceSingletonComplIso j).inv ≫ f j hj) =
      ((CompleteLattice.MulticoequalizerDiagram.multispanIndex ⋯).toLinearOrder.map Subcomplex.toSSetFunctor).snd a ≫
        match (MultispanShape.ofLinearOrder ↑{i}ᶜ).snd a with
        | ⟨j, hj⟩ => (faceSingletonComplIso j).inv ≫ f j hj

Tactic:
by

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

Goal state before:
pp: n : ℕ
X : SSet
i : Fin (n + 2)
f : (j : Fin (n + 2)) → j ≠ i → (Δ[n] ⟶ X)
hf : horn.IsCompatible f
j : Fin (n + 2)
hj : j ≠ i
⊢ ι i j hj ≫ (isColimit i).desc hf.multicofork = f j hj

Tactic:
by

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

Goal state before:
pp: ⊢ 3 ≠ 1

Tactic:
by

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

Goal state before:
pp: X : SSet
f₀ f₂ f₃ : Δ[2] ⟶ X
h₁₂ : stdSimplex.δ 2 ≫ f₀ = stdSimplex.δ 0 ≫ f₃
h₁₃ : stdSimplex.δ 1 ≫ f₀ = stdSimplex.δ 0 ≫ f₂
h₂₃ : stdSimplex.δ 2 ≫ f₂ = stdSimplex.δ 2 ≫ f₃
x✝ : (MultispanShape.ofLinearOrder ↑{1}ᶜ).R
i : Fin 4
hi : 1 ∈ {1}ᶜ
⊢ False

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d : ℕ
x✝ : SimplexCategoryᵒᵖ
n : ℕ
⊢ IsLimit (evaluation SimplexCategoryᵒᵖ (Type u) _⦋n⦌.mapCone (PullbackCone.mk (t i d) (l i d) ⋯))

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d : ℕ
n : SimplexCategory
y : Y _⦋d⦌
hy₁ : y ∈ Y.nonDegenerate d
hy₂ : ∀ (x : X _⦋d⦌), ¬(ConcreteCategory.hom (i.app (op ⦋d⦌))) x = y
f : op ⦋d⦌ ⟶ op n
hx : (ConcreteCategory.hom (Y.map f)) y ∈ ((skeletonOfMono i) (d + 1)).obj (op n)
a✝ : ⟨(ConcreteCategory.hom (Y.map f)) y, hx⟩ ∈ ⊤.obj (op n)
⊢ y ∉ Set.range ⇑(ConcreteCategory.hom (i.app (op ⦋d⦌)))

Tactic:
by

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

Goal state before:
pp: X : SSet
f₀ f₂ f₃ : Δ[2] ⟶ X
h₁₂ : stdSimplex.δ 2 ≫ f₀ = stdSimplex.δ 0 ≫ f₃
h₁₃ : stdSimplex.δ 1 ≫ f₀ = stdSimplex.δ 0 ≫ f₂
h₂₃ : stdSimplex.δ 2 ≫ f₂ = stdSimplex.δ 2 ≫ f₃
⊢ 1 < 3

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d : ℕ
⊢ Subcomplex.range (r i d) ⊔ Subcomplex.range (b i d) = ⊤

Tactic:
by

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

Goal state before:
pp: X : SSet
f₀ f₂ f₃ : Δ[2] ⟶ X
h₁₂ : stdSimplex.δ 2 ≫ f₀ = stdSimplex.δ 0 ≫ f₃
h₁₃ : stdSimplex.δ 1 ≫ f₀ = stdSimplex.δ 0 ≫ f₂
h₂₃ : stdSimplex.δ 2 ≫ f₂ = stdSimplex.δ 2 ≫ f₃
⊢ 1 < 2

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d n : ℕ
c : Cell i d
s : Δ[d] _⦋n⦌
hy :
  (ConcreteCategory.hom (c.ιSigmaStdSimplex.app (op ⦋n⦌))) s ∉ Set.range ⇑(ConcreteCategory.hom ((l i d).app (op ⦋n⦌)))
hs : s ∉ ∂Δ[d].obj (op ⦋n⦌)
⊢ (ConcreteCategory.hom (c.ιSigmaStdSimplex.app (op ⦋n⦌))) (stdSimplex.objEquiv.symm (stdSimplex.objEquiv s)) =
    (ConcreteCategory.hom (c.ιSigmaStdSimplex.app (op ⦋n⦌))) s

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d n : ℕ
y : sigmaStdSimplex i d _⦋n⦌
hy : y ∉ Set.range ⇑(ConcreteCategory.hom ((l i d).app (op ⦋n⦌)))
⊢ ∃ c f, ∃ (_ : Epi f), (ConcreteCategory.hom (c.ιSigmaStdSimplex.app (op ⦋n⦌))) (stdSimplex.objEquiv.symm f) = y

Tactic:
by

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

Goal state before:
pp: X : SSet
f₀ f₂ f₃ : Δ[2] ⟶ X
h₁₂ : stdSimplex.δ 2 ≫ f₀ = stdSimplex.δ 0 ≫ f₃
h₁₃ : stdSimplex.δ 1 ≫ f₀ = stdSimplex.δ 0 ≫ f₂
h₂₃ : stdSimplex.δ 2 ≫ f₂ = stdSimplex.δ 2 ≫ f₃
⊢ 0 < 1

Tactic:
by

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

Goal state before:
pp: X : SSet
f₀ f₂ f₃ : Δ[2] ⟶ X
h₁₂ : stdSimplex.δ 2 ≫ f₀ = stdSimplex.δ 0 ≫ f₃
h₁₃ : stdSimplex.δ 1 ≫ f₀ = stdSimplex.δ 0 ≫ f₂
h₂₃ : stdSimplex.δ 2 ≫ f₂ = stdSimplex.δ 2 ≫ f₃
x : (MultispanShape.ofLinearOrder ↑{1}ᶜ).L
⊢ (((CompleteLattice.MulticoequalizerDiagram.multispanIndex ⋯).toLinearOrder.map Subcomplex.toSSetFunctor).fst x ≫
      match (MultispanShape.ofLinearOrder ↑{1}ᶜ).fst x with
      | ⟨i, hi⟩ =>
        match i, hi with
        | 0, hi => (stdSimplex.faceSingletonComplIso 0).inv ≫ f₀
        | 1, hi => ⋯.elim
        | 2, hi => (stdSimplex.faceSingletonComplIso 2).inv ≫ f₂
        | 3, hi => (stdSimplex.faceSingletonComplIso 3).inv ≫ f₃) =
    ((CompleteLattice.MulticoequalizerDiagram.multispanIndex ⋯).toLinearOrder.map Subcomplex.toSSetFunctor).snd x ≫
      match (MultispanShape.ofLinearOrder ↑{1}ᶜ).snd x with
      | ⟨i, hi⟩ =>
        match i, hi with
        | 0, hi => (stdSimplex.faceSingletonComplIso 0).inv ≫ f₀
        | 1, hi => ⋯.elim
        | 2, hi => (stdSimplex.faceSingletonComplIso 2).inv ≫ f₂
        | 3, hi => (stdSimplex.faceSingletonComplIso 3).inv ≫ f₃

Tactic:
by

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

Goal state before:
pp: X : SSet
f₀ f₂ f₃ : Δ[2] ⟶ X
h₁₂ : stdSimplex.δ 2 ≫ f₀ = stdSimplex.δ 0 ≫ f₃
h₁₃ : stdSimplex.δ 1 ≫ f₀ = stdSimplex.δ 0 ≫ f₂
h₂₃ : stdSimplex.δ 2 ≫ f₂ = stdSimplex.δ 2 ≫ f₃
⊢ 0 ∈ {1}ᶜ

Tactic:
by

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

Goal state before:
pp: X : SSet
f₀ f₂ f₃ : Δ[2] ⟶ X
h₁₂ : stdSimplex.δ 2 ≫ f₀ = stdSimplex.δ 0 ≫ f₃
h₁₃ : stdSimplex.δ 1 ≫ f₀ = stdSimplex.δ 0 ≫ f₂
h₂₃ : stdSimplex.δ 2 ≫ f₂ = stdSimplex.δ 2 ≫ f₃
⊢ 2 ∈ {1}ᶜ

Tactic:
by

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

Goal state before:
pp: X : SSet
f₀ f₂ f₃ : Δ[2] ⟶ X
h₁₂ : stdSimplex.δ 2 ≫ f₀ = stdSimplex.δ 0 ≫ f₃
h₁₃ : stdSimplex.δ 1 ≫ f₀ = stdSimplex.δ 0 ≫ f₂
h₂₃ : stdSimplex.δ 2 ≫ f₂ = stdSimplex.δ 2 ≫ f₃
⊢ 3 ∈ {1}ᶜ

Tactic:
by

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

Goal state before:
pp: X : SSet
f₀ f₂ f₃ : Δ[2] ⟶ X
h₁₂ : stdSimplex.δ 2 ≫ f₀ = stdSimplex.δ 0 ≫ f₃
h₁₃ : stdSimplex.δ 1 ≫ f₀ = stdSimplex.δ 0 ≫ f₂
h₂₃ : stdSimplex.δ 2 ≫ f₂ = stdSimplex.δ 2 ≫ f₃
⊢ ι₀ ≫ desc f₀ f₂ f₃ h₁₂ h₁₃ h₂₃ = f₀

Tactic:
by

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

Goal state before:
pp: n : ℕ
X : Truncated (n + 1)
inst✝ : X.IsStrictSegal
d : ℕ
hd : ⦋d + 1⦌.len ≤ n + 1
x y : X.obj (op { obj := ⦋d + 1⦌, property := hd })
h :
  ∀ (i : Fin (d + 1)),
    (ConcreteCategory.hom (X.map (tr (mkOfSucc i) ⋯ hd).op)) x =
      (ConcreteCategory.hom (X.map (tr (mkOfSucc i) ⋯ hd).op)) y
⊢ X.spine (d + 1) hd x = X.spine (d + 1) hd y

Tactic:
by

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

Goal state before:
pp: case hᵥ
n : ℕ
X : Truncated (n + 1)
m : ℕ
f g : X.Path (m + 1)
h : ∀ (i : Fin (m + 1)), f.arrow i = g.arrow i
j : Fin (m + 1 + 1)
⊢ f.vertex j = g.vertex j

Tactic:
rcases Fin.eq_castSucc_or_eq_last j with ⟨k, hk⟩ | hl

Goal state after:
case hᵥ.inl
n : ℕ
X : Truncated (n + 1)
m : ℕ
f g : X.Path (m + 1)
h : ∀ (i : Fin (m + 1)), f.arrow i = g.arrow i
j : Fin (m + 1 + 1)
k : Fin (m + 1)
hk : j = k.castSucc
⊢ f.vertex j = g.vertex j

case hᵥ.inr
n : ℕ
X : Truncated (n + 1)
m : ℕ
f g : X.Path (m + 1)
h : ∀ (i : Fin (m + 1)), f.arrow i = g.arrow i
j : Fin (m + 1 + 1)
hl : j = Fin.last (m + 1)
⊢ f.vertex j = g.vertex j
Module: Mathlib.AlgebraicTopology.SimplicialSet.HornColimits

Goal state before:
pp: X : SSet
f₀ f₂ f₃ : Δ[2] ⟶ X
h₁₂ : stdSimplex.δ 2 ≫ f₀ = stdSimplex.δ 0 ≫ f₃
h₁₃ : stdSimplex.δ 1 ≫ f₀ = stdSimplex.δ 0 ≫ f₂
h₂₃ : stdSimplex.δ 2 ≫ f₂ = stdSimplex.δ 2 ≫ f₃
⊢ ι₂ ≫ desc f₀ f₂ f₃ h₁₂ h₁₃ h₂₃ = f₂

Tactic:
by

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

Goal state before:
pp: n : ℕ
X : Truncated (n + 1)
m : ℕ
f g : X.Path (m + 1)
h : ∀ (i : Fin (m + 1)), f.arrow i = g.arrow i
⊢ f = g

Tactic:
by

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

Goal state before:
pp: X : SSet
f₀ f₂ f₃ : Δ[2] ⟶ X
h₁₂ : stdSimplex.δ 2 ≫ f₀ = stdSimplex.δ 0 ≫ f₃
h₁₃ : stdSimplex.δ 1 ≫ f₀ = stdSimplex.δ 0 ≫ f₂
h₂₃ : stdSimplex.δ 2 ≫ f₂ = stdSimplex.δ 2 ≫ f₃
⊢ ι₃ ≫ desc f₀ f₂ f₃ h₁₂ h₁₃ h₂₃ = f₃

Tactic:
by

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

Goal state before:
pp: n : ℕ
X Y : Truncated (n + 1)
inst✝ : Y.IsStrictSegal
f g : X ⟶ Y
h :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := ⋯ })),
    (ConcreteCategory.hom (f.app (op { obj := ⦋1⦌, property := ⋯ }))) x =
      (ConcreteCategory.hom (g.app (op { obj := ⦋1⦌, property := ⋯ }))) x
m : ℕ
hm : ⦋m + 1⦌.len ≤ n + 1
x : X.obj (op { obj := ⦋m + 1⦌, property := hm })
i : Fin (m + 1)
⊢ (ConcreteCategory.hom (Y.map (tr (mkOfSucc i) ⋯ hm).op))
      ((ConcreteCategory.hom (f.app (op { obj := ⦋m + 1⦌, property := hm }))).toFun x) =
    (ConcreteCategory.hom (Y.map (tr (mkOfSucc i) ⋯ hm).op))
      ((ConcreteCategory.hom (g.app (op { obj := ⦋m + 1⦌, property := hm }))).toFun x)

Tactic:
by

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

Goal state before:
pp: X : SSet
f₀ f₂ f₃ : Δ[2] ⟶ X
h₁₂ : stdSimplex.δ 2 ≫ f₀ = stdSimplex.δ 0 ≫ f₃
h₁₃ : stdSimplex.δ 1 ≫ f₀ = stdSimplex.δ 0 ≫ f₂
h₂₃ : stdSimplex.δ 2 ≫ f₂ = stdSimplex.δ 2 ≫ f₃
⊢ ι₀ ≫ desc f₀ f₂ f₃ h₁₂ h₁₃ h₂₃ = f₀ ∧ ι₂ ≫ desc f₀ f₂ f₃ h₁₂ h₁₃ h₂₃ = f₂ ∧ ι₃ ≫ desc f₀ f₂ f₃ h₁₂ h₁₃ h₂₃ = f₃

Tactic:
by

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

Goal state before:
pp: X Y : SSet
i : X ⟶ Y
d : ℕ
x✝ : SimplexCategoryᵒᵖ
n : ℕ
c₁ : Cell i d
f₁ : ⦋n⦌ ⟶ ⦋d⦌
w✝¹ : Epi f₁
hx₁ :
  (ConcreteCategory.hom (c₁.ιSigmaStdSimplex.app (op ⦋n⦌))) (stdSimplex.objEquiv.symm f₁) ∉
    Set.range ⇑(ConcreteCategory.hom ((l i d).app (op ⦋n⦌)))
c₂ : Cell i d
f₂ : ⦋n⦌ ⟶ ⦋d⦌
w✝ : Epi f₂
hx₂ :
  (ConcreteCategory.hom (c₂.ιSigmaStdSimplex.app (op ⦋n⦌))) (stdSimplex.objEquiv.symm f₂) ∉
    Set.range ⇑(ConcreteCategory.hom ((l i d).app (op ⦋n⦌)))
h :
  ↑((ConcreteCategory.hom ((b i d).app (op ⦋n⦌)))
        ((ConcreteCategory.hom (c₁.ιSigmaStdSimplex.app (op ⦋n⦌))) (stdSimplex.objEquiv.symm f₁))) =
    ↑((ConcreteCategory.hom ((b i d).app (op ⦋n⦌)))
        ((ConcreteCategory.hom (c₂.ιSigmaStdSimplex.app (op ⦋n⦌))) (stdSimplex.objEquiv.symm f₂)))
⊢ (ConcreteCategory.hom (Y.map f₁.op)) c₁.simplex = (ConcreteCategory.hom (Y.map f₂.op)) c₂.simplex

Tactic:
by

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

Goal state before:
pp: n✝ : ℕ
X✝ : Truncated (n✝ + 1)
m n : ℕ
X : Truncated (n + 1)
p q : X.Path 1
h : p.vertex 1 = q.vertex 0
i : Fin 2
⊢ (ConcreteCategory.hom (((trunc (n + 1) 1 ⋯).obj X).map (tr (SimplexCategory.δ 1) Path₁._proof_1 Path₁._proof_5).op))
      (![p.arrow 0, q.arrow 0] i) =
    ![p.vertex 0, p.vertex 1, q.vertex 1] i.castSucc

Tactic:
by

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

Goal state before:
pp: ⊢ 3 ≠ 2

Tactic:
by

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

Goal state before:
pp: X : SSet
f₀ f₁ f₃ : Δ[2] ⟶ X
h₀₂ : stdSimplex.δ 2 ≫ f₁ = stdSimplex.δ 1 ≫ f₃
h₁₂ : stdSimplex.δ 2 ≫ f₀ = stdSimplex.δ 0 ≫ f₃
h₂₃ : stdSimplex.δ 0 ≫ f₀ = stdSimplex.δ 0 ≫ f₁
x✝ : (MultispanShape.ofLinearOrder ↑{2}ᶜ).R
i : Fin 4
hi : 2 ∈ {2}ᶜ
⊢ False

Tactic:
by

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

Goal state before:
pp: n : ℕ
X Y : Truncated (n + 1)
inst✝ : Y.IsStrictSegal
f g : X ⟶ Y
h :
  ∀ (x : X.obj (op { obj := ⦋1⦌, property := ⋯ })),
    (ConcreteCategory.hom (f.app (op { obj := ⦋1⦌, property := ⋯ }))) x =
      (ConcreteCategory.hom (g.app (op { obj := ⦋1⦌, property := ⋯ }))) x
⊢ f = g

Tactic:
by

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

Goal state before:
pp: n : ℕ
X : Truncated (n + 1)
sx : X.StrictSegal
m : ℕ
h : m ≤ n + 1
i : Fin (m + 1)
f : X.Path m
⊢ (ConcreteCategory.hom (X.map (tr (⦋0⦌.const ⦋m⦌ i) ⋯ h).op)) (sx.spineToSimplex m h f) = f.vertex i

Tactic:
by

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

Goal state before:
pp: n : ℕ
X : Truncated (n + 1)
sx : X.StrictSegal
m : ℕ
h : m ≤ n + 1
i : Fin m
f : X.Path m
⊢ (ConcreteCategory.hom (X.map (tr (mkOfSucc i) ⋯ h).op)) (sx.spineToSimplex m h f) = f.arrow i

Tactic:
by

Goal state after: