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

Goal state before:
pp: case inl.inr.refine_3
m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
hx' : StrictMono ⇑(objEquiv (x.cast hd).simplex)
i : Fin d
hi✝ : i.succ ≤ min x hd
hi : i.succ = min x hd
h :
  ((objEquiv (x.cast hd).simplex) i.castSucc).1 = (k.castSucc, (x.cast hd).simplex.2 (min x hd)).1 ∧
    ((objEquiv (x.cast hd).simplex) i.castSucc).2 = (k.castSucc, (x.cast hd).simplex.2 (min x hd)).2
⊢ False

Tactic:
]

Goal state after:
case inl.inr.refine_3
m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
hx' : StrictMono ⇑(objEquiv (x.cast hd).simplex)
i : Fin d
hi✝ : i.succ ≤ min x hd
hi : i.succ = min x hd
h :
  ((objEquiv (x.cast hd).simplex) i.castSucc).1 = (k.castSucc, (x.cast hd).simplex.2 (min x hd)).1 ∧
    ((objEquiv (x.cast hd).simplex) i.castSucc).2 = (k.castSucc, (x.cast hd).simplex.2 (min x hd)).2
⊢ False
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Inner.PushoutProduct

Goal state before:
pp: X₁ Y₁ E✝ B✝ : SSet
i : X₁ ⟶ Y₁
p✝ : E✝ ⟶ B✝
inst✝¹ : Mono i
inst✝ : InnerFibration p✝
sq₁₃ : internalHom.PullbackObjObj i p✝
X✝ Y✝ : SSet
g✝ : X✝ ⟶ Y✝
n✝ : ℕ
k : Fin (n✝ + 3)
h0 : 0 < k
hn : k < Fin.last (n✝ + 2)
sq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[n✝ + 2, k].ι :=
  Functor.PushoutObjObj.ofHasPushout (curriedTensor SSet) i Λ[n✝ + 2, k].ι
E B : SSet
p : E ⟶ B
hp : innerFibrations p
sq₁₃' : internalHom.PullbackObjObj Λ[n✝ + 2, k].ι p := Functor.PullbackObjObj.ofHasPullback internalHom Λ[n✝ + 2, k].ι p
⊢ modelCategoryQuillen.I.rlp sq₁₃'.π

Tactic:
]

Goal state after:
X₁ Y₁ E✝ B✝ : SSet
i : X₁ ⟶ Y₁
p✝ : E✝ ⟶ B✝
inst✝¹ : Mono i
inst✝ : InnerFibration p✝
sq₁₃ : internalHom.PullbackObjObj i p✝
X✝ Y✝ : SSet
g✝ : X✝ ⟶ Y✝
n✝ : ℕ
k : Fin (n✝ + 3)
h0 : 0 < k
hn : k < Fin.last (n✝ + 2)
sq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[n✝ + 2, k].ι :=
  Functor.PushoutObjObj.ofHasPushout (curriedTensor SSet) i Λ[n✝ + 2, k].ι
E B : SSet
p : E ⟶ B
hp : innerFibrations p
sq₁₃' : internalHom.PullbackObjObj Λ[n✝ + 2, k].ι p := Functor.PullbackObjObj.ofHasPullback internalHom Λ[n✝ + 2, k].ι p
⊢ modelCategoryQuillen.I.rlp sq₁₃'.π
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Inner.PushoutProduct

Goal state before:
pp: X₁ Y₁ E✝ B✝ : SSet
i : X₁ ⟶ Y₁
p✝ : E✝ ⟶ B✝
inst✝¹ : Mono i
inst✝ : InnerFibration p✝
sq₁₃ : internalHom.PullbackObjObj i p✝
X✝ Y✝ : SSet
g✝ : X✝ ⟶ Y✝
n✝ : ℕ
k : Fin (n✝ + 3)
h0 : 0 < k
hn : k < Fin.last (n✝ + 2)
sq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[n✝ + 2, k].ι :=
  Functor.PushoutObjObj.ofHasPushout (curriedTensor SSet) i Λ[n✝ + 2, k].ι
E B : SSet
p : E ⟶ B
hp : innerFibrations p
sq₁₃' : internalHom.PullbackObjObj Λ[n✝ + 2, k].ι p := Functor.PullbackObjObj.ofHasPullback internalHom Λ[n✝ + 2, k].ι p
X Y : SSet
n : ℕ
⊢ HasLiftingProperty (Λ[n✝ + 2, k].unionProd ∂Δ[n]).ι p

Tactic:
]

Goal state after:
X₁ Y₁ E✝ B✝ : SSet
i : X₁ ⟶ Y₁
p✝ : E✝ ⟶ B✝
inst✝¹ : Mono i
inst✝ : InnerFibration p✝
sq₁₃ : internalHom.PullbackObjObj i p✝
X✝ Y✝ : SSet
g✝ : X✝ ⟶ Y✝
n✝ : ℕ
k : Fin (n✝ + 3)
h0 : 0 < k
hn : k < Fin.last (n✝ + 2)
sq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[n✝ + 2, k].ι :=
  Functor.PushoutObjObj.ofHasPushout (curriedTensor SSet) i Λ[n✝ + 2, k].ι
E B : SSet
p : E ⟶ B
hp : innerFibrations p
sq₁₃' : internalHom.PullbackObjObj Λ[n✝ + 2, k].ι p := Functor.PullbackObjObj.ofHasPullback internalHom Λ[n✝ + 2, k].ι p
X Y : SSet
n : ℕ
⊢ HasLiftingProperty (Λ[n✝ + 2, k].unionProd ∂Δ[n]).ι p
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Inner.PushoutProduct

Goal state before:
pp: X₁ X₂ Y₁ Y₂ : SSet
i : X₁ ⟶ Y₁
j : X₂ ⟶ Y₂
sq₁₂ : (curriedTensor SSet).PushoutObjObj i j
inst✝ : Mono i
hj : innerAnodyneExtensions j
E B : SSet
p : E ⟶ B
hp : innerFibrations p
sq₁₃ : internalHom.PullbackObjObj i p := Functor.PullbackObjObj.ofHasPullback internalHom i p
⊢ HasLiftingProperty j sq₁₃.π

Tactic:
]

Goal state after:
X₁ X₂ Y₁ Y₂ : SSet
i : X₁ ⟶ Y₁
j : X₂ ⟶ Y₂
sq₁₂ : (curriedTensor SSet).PushoutObjObj i j
inst✝ : Mono i
hj : innerAnodyneExtensions j
E B : SSet
p : E ⟶ B
hp : innerFibrations p
sq₁₃ : internalHom.PullbackObjObj i p := Functor.PullbackObjObj.ofHasPullback internalHom i p
⊢ HasLiftingProperty j sq₁₃.π
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.Inner.PushoutProduct

Goal state before:
pp: X₁ X₂ Y₁ Y₂ : SSet
i : X₁ ⟶ Y₁
j : X₂ ⟶ Y₂
sq₁₂ : (curriedTensor SSet).PushoutObjObj i j
inst✝ : Mono i
hj : innerAnodyneExtensions j
E B : SSet
p : E ⟶ B
hp : InnerFibration p
sq₁₃ : internalHom.PullbackObjObj i p := Functor.PullbackObjObj.ofHasPullback internalHom i p
⊢ InnerFibration sq₁₃.π

Tactic:
]

Goal state after:
X₁ X₂ Y₁ Y₂ : SSet
i : X₁ ⟶ Y₁
j : X₂ ⟶ Y₂
sq₁₂ : (curriedTensor SSet).PushoutObjObj i j
inst✝ : Mono i
hj : innerAnodyneExtensions j
E B : SSet
p : E ⟶ B
hp : InnerFibration p
sq₁₃ : internalHom.PullbackObjObj i p := Functor.PullbackObjObj.ofHasPullback internalHom i p
⊢ InnerFibration sq₁₃.π
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: case inl.inr.refine_3
m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
hx' : StrictMono ⇑(objEquiv (x.cast hd).simplex)
i : Fin d
hi✝ : i.succ ≤ min x hd
hi : i.succ = min x hd
h₁ : (x.cast hd).simplex.1 i.castSucc = k.castSucc
h₂ : (x.cast hd).simplex.2 i.castSucc = (x.cast hd).simplex.2 (min x hd)
⊢ (x.cast hd).simplex.1 i.castSucc = k.castSucc ∧
    (x.cast hd).simplex.1 i.succ = k.succ ∧ (x.cast hd).simplex.2 i.succ = (x.cast hd).simplex.2 i.castSucc

Tactic:
]

Goal state after:
case inl.inr.refine_3
m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
hx' : StrictMono ⇑(objEquiv (x.cast hd).simplex)
i : Fin d
hi✝ : i.succ ≤ min x hd
hi : i.succ = min x hd
h₁ : (x.cast hd).simplex.1 i.castSucc = k.castSucc
h₂ : (x.cast hd).simplex.2 i.castSucc = (x.cast hd).simplex.2 (min x hd)
⊢ (x.cast hd).simplex.1 i.castSucc = k.castSucc ∧
    (x.cast hd).simplex.1 i.succ = k.succ ∧ (x.cast hd).simplex.2 i.succ = (x.cast hd).simplex.2 i.castSucc
Module: Mathlib.CategoryTheory.Monoidal.PushoutProduct

Goal state before:
pp: C : Type u
inst✝⁶ : Category.{v, u} C
inst✝⁵ : HasPushouts C
inst✝⁴ : MonoidalCategory C
X₁ X₂ X₃ : Arrow C
W : C
inst✝³ : PreservesColimit (span (X₁.hom ▷ X₂.left) (X₁.left ◁ X₂.hom)) (tensorRight X₃.left)
inst✝² : PreservesColimit (span (X₁.hom ▷ X₂.left) (X₁.left ◁ X₂.hom)) (tensorRight X₃.right)
inst✝¹ : PreservesColimit (span (X₂.hom ▷ X₃.left) (X₂.left ◁ X₃.hom)) (tensorLeft X₁.left)
inst✝ : PreservesColimit (span (X₂.hom ▷ X₃.left) (X₂.left ◁ X₃.hom)) (tensorLeft X₁.right)
⊢ ((curriedTensor C).map (Arrow.mk (X₁.right ◁ X₂.hom)).hom).app X₃.left ≫
      pushout.inl (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
        (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom) =
    ((curriedTensor C).obj (Arrow.mk (X₁.right ◁ X₂.hom)).left).map X₃.hom ≫
      pushout.inl (((curriedTensor C).map X₁.hom).app X₂.left) (((curriedTensor C).obj X₁.left).map X₂.hom) ▷ X₃.right ≫
        pushout.inr (((curriedTensor C).map (X₁ □ X₂).hom).app X₃.left)
          (((curriedTensor C).obj (X₁ □ X₂).left).map X₃.hom)

Tactic:
simp [Limits.pushout.associator_inv_naturality_right_condition]

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: case inl.inr.refine_3
m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
hx' : StrictMono ⇑(objEquiv (x.cast hd).simplex)
i : Fin d
hi✝ : i.succ ≤ min x hd
hi : i.succ = min x hd
h₁ : (x.cast hd).simplex.1 i.castSucc = k.castSucc
h₂ : (x.cast hd).simplex.2 i.castSucc = (x.cast hd).simplex.2 (min x hd)
this : φ x hd (min x hd).succ = (objEquiv (x.cast hd).simplex) (min x hd)
⊢ (x.cast hd).simplex.1 i.succ = k.succ

Tactic:
]

Goal state after:
case inl.inr.refine_3
m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
hx' : StrictMono ⇑(objEquiv (x.cast hd).simplex)
i : Fin d
hi✝ : i.succ ≤ min x hd
hi : i.succ = min x hd
h₁ : (x.cast hd).simplex.1 i.castSucc = k.castSucc
h₂ : (x.cast hd).simplex.2 i.castSucc = (x.cast hd).simplex.2 (min x hd)
this : φ x hd (min x hd).succ = (objEquiv (x.cast hd).simplex) (min x hd)
⊢ (x.cast hd).simplex.1 i.succ = k.succ
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: case inl.inr.refine_3
m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
hx' : StrictMono ⇑(objEquiv (x.cast hd).simplex)
i : Fin d
hi✝ : i.succ ≤ min x hd
hi : i.succ = min x hd
h₁ : (x.cast hd).simplex.1 i.castSucc = k.castSucc
h₂ : (x.cast hd).simplex.2 i.castSucc = (x.cast hd).simplex.2 (min x hd)
this : φ x hd (min x hd).succ = (objEquiv (x.cast hd).simplex) (min x hd)
⊢ (x.cast hd).simplex.1 (min x hd) = ((objEquiv (x.cast hd).simplex) (min x hd)).1

Tactic:
]

Goal state after:
case inl.inr.refine_3
m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
hx' : StrictMono ⇑(objEquiv (x.cast hd).simplex)
i : Fin d
hi✝ : i.succ ≤ min x hd
hi : i.succ = min x hd
h₁ : (x.cast hd).simplex.1 i.castSucc = k.castSucc
h₂ : (x.cast hd).simplex.2 i.castSucc = (x.cast hd).simplex.2 (min x hd)
this : φ x hd (min x hd).succ = (objEquiv (x.cast hd).simplex) (min x hd)
⊢ (x.cast hd).simplex.1 (min x hd) = ((objEquiv (x.cast hd).simplex) (min x hd)).1
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: case inr.inr
m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
hx' : StrictMono ⇑(objEquiv (x.cast hd).simplex)
i : Fin (d + 1)
hi : min x hd < i
⊢ (objEquiv (x.cast hd).simplex) (i.castSucc.pred ⋯) < (objEquiv (x.cast hd).simplex) (i.succ.pred ⋯)

Tactic:
]

Goal state after:
case inr.inr
m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
hx' : StrictMono ⇑(objEquiv (x.cast hd).simplex)
i : Fin (d + 1)
hi : min x hd < i
⊢ (objEquiv (x.cast hd).simplex) (i.castSucc.pred ⋯) < (objEquiv (x.cast hd).simplex) (i.succ.pred ⋯)
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.PushoutProduct

Goal state before:
pp: X₁ Y₁ E B : SSet
i : X₁ ⟶ Y₁
p : E ⟶ B
inst✝¹ : Mono i
inst✝ : Fibration p
sq₁₃ : internalHom.PullbackObjObj i p
⊢ J.rlp sq₁₃.π

Tactic:
]

Goal state after:
X₁ Y₁ E B : SSet
i : X₁ ⟶ Y₁
p : E ⟶ B
inst✝¹ : Mono i
inst✝ : Fibration p
sq₁₃ : internalHom.PullbackObjObj i p
⊢ J.rlp sq₁₃.π
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
⊢ StrictMono ⇑{ toFun := φ x hd, monotone' := ⋯ }

Tactic:
]

Goal state after:
m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
⊢ StrictMono ⇑{ toFun := φ x hd, monotone' := ⋯ }
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.PushoutProduct

Goal state before:
pp: X₁ Y₁ E B : SSet
i : X₁ ⟶ Y₁
p : E ⟶ B
inst✝¹ : Mono i
inst✝ : Fibration p
sq₁₃ : internalHom.PullbackObjObj i p
m : ℕ
X Y : SSet
k : Fin (m + 2)
sq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[m + 1, k].ι :=
  Functor.PushoutObjObj.ofHasPushout (curriedTensor SSet) i Λ[m + 1, k].ι
⊢ HasLiftingProperty sq₁₂.ι p

Tactic:
]

Goal state after:
X₁ Y₁ E B : SSet
i : X₁ ⟶ Y₁
p : E ⟶ B
inst✝¹ : Mono i
inst✝ : Fibration p
sq₁₃ : internalHom.PullbackObjObj i p
m : ℕ
X Y : SSet
k : Fin (m + 2)
sq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[m + 1, k].ι :=
  Functor.PushoutObjObj.ofHasPushout (curriedTensor SSet) i Λ[m + 1, k].ι
⊢ HasLiftingProperty sq₁₂.ι p
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.PushoutProduct

Goal state before:
pp: X₁ Y₁ E B : SSet
i : X₁ ⟶ Y₁
p : E ⟶ B
inst✝¹ : Mono i
inst✝ : Fibration p
sq₁₃ : internalHom.PullbackObjObj i p
m : ℕ
X Y : SSet
k : Fin (m + 2)
sq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[m + 1, k].ι :=
  Functor.PushoutObjObj.ofHasPushout (curriedTensor SSet) i Λ[m + 1, k].ι
this : anodyneExtensions sq₁₂.ι
⊢ Fibration p

Tactic:
]

Goal state after:
X₁ Y₁ E B : SSet
i : X₁ ⟶ Y₁
p : E ⟶ B
inst✝¹ : Mono i
inst✝ : Fibration p
sq₁₃ : internalHom.PullbackObjObj i p
m : ℕ
X Y : SSet
k : Fin (m + 2)
sq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[m + 1, k].ι :=
  Functor.PushoutObjObj.ofHasPushout (curriedTensor SSet) i Λ[m + 1, k].ι
this : anodyneExtensions sq₁₂.ι
⊢ Fibration p
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
i : Fin (d + 1)
⊢ (objEquiv
        ((ConcreteCategory.hom (SimplicialObject.δ (Δ[m + 1] ⊗ Δ[n]) (min x hd).castSucc))
          (objEquiv.symm { toFun := φ x hd, monotone' := ⋯ })))
      i =
    (objEquiv (x.cast hd).simplex) i

Tactic:
objEquiv_δ_apply,

Goal state after:
m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
i : Fin (d + 1)
⊢ (objEquiv (objEquiv.symm { toFun := φ x hd, monotone' := ⋯ })) ((min x hd).castSucc.succAbove i) =
    (objEquiv (x.cast hd).simplex) i
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
i : Fin (d + 1)
⊢ (objEquiv (x.cast hd).simplex) i = (objEquiv (x.cast hd).simplex) i

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.PushoutProduct

Goal state before:
pp: X₁ Y₁ E✝ B✝ : SSet
i : X₁ ⟶ Y₁
p✝ : E✝ ⟶ B✝
inst✝¹ : Mono i
inst✝ : Fibration p✝
sq₁₃ : internalHom.PullbackObjObj i p✝
m : ℕ
X Y : SSet
k : Fin (m + 2)
sq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[m + 1, k].ι :=
  Functor.PushoutObjObj.ofHasPushout (curriedTensor SSet) i Λ[m + 1, k].ι
E B : SSet
p : E ⟶ B
hp : fibrations SSet p
⊢ HasLiftingProperty sq₁₂.flipTensor.ι p

Tactic:
]

Goal state after:
X₁ Y₁ E✝ B✝ : SSet
i : X₁ ⟶ Y₁
p✝ : E✝ ⟶ B✝
inst✝¹ : Mono i
inst✝ : Fibration p✝
sq₁₃ : internalHom.PullbackObjObj i p✝
m : ℕ
X Y : SSet
k : Fin (m + 2)
sq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[m + 1, k].ι :=
  Functor.PushoutObjObj.ofHasPushout (curriedTensor SSet) i Λ[m + 1, k].ι
E B : SSet
p : E ⟶ B
hp : fibrations SSet p
⊢ HasLiftingProperty sq₁₂.flipTensor.ι p
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
h : hx.simplex hd ∈ (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).obj (Opposite.op ⦋d + 1⦌)
⊢ (ConcreteCategory.hom (SimplicialObject.δ (Δ[m + 1] ⊗ Δ[n]) (min x hd).castSucc)) (hx.simplex hd) ∈
    (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).obj (Opposite.op ⦋(x.cast hd).dim⦌)

Tactic:
]

Goal state after:
m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hx : IsType₂ x
d : ℕ
hd : x.dim = d
h : hx.simplex hd ∈ (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).obj (Opposite.op ⦋d + 1⦌)
⊢ (ConcreteCategory.hom (SimplicialObject.δ (Δ[m + 1] ⊗ Δ[n]) (min x hd).castSucc)) (hx.simplex hd) ∈
    (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).obj (Opposite.op ⦋(x.cast hd).dim⦌)
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.PushoutProduct

Goal state before:
pp: X₁ Y₁ E✝ B✝ : SSet
i : X₁ ⟶ Y₁
p✝ : E✝ ⟶ B✝
inst✝¹ : Mono i
inst✝ : Fibration p✝
sq₁₃ : internalHom.PullbackObjObj i p✝
m : ℕ
X Y : SSet
k : Fin (m + 2)
sq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[m + 1, k].ι :=
  Functor.PushoutObjObj.ofHasPushout (curriedTensor SSet) i Λ[m + 1, k].ι
E B : SSet
p : E ⟶ B
hp : fibrations SSet p
sq₁₃' : internalHom.PullbackObjObj Λ[m + 1, k].ι p := Functor.PullbackObjObj.ofHasPullback internalHom Λ[m + 1, k].ι p
⊢ HasLiftingProperty i sq₁₃'.π

Tactic:
]

Goal state after:
X₁ Y₁ E✝ B✝ : SSet
i : X₁ ⟶ Y₁
p✝ : E✝ ⟶ B✝
inst✝¹ : Mono i
inst✝ : Fibration p✝
sq₁₃ : internalHom.PullbackObjObj i p✝
m : ℕ
X Y : SSet
k : Fin (m + 2)
sq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[m + 1, k].ι :=
  Functor.PushoutObjObj.ofHasPushout (curriedTensor SSet) i Λ[m + 1, k].ι
E B : SSet
p : E ⟶ B
hp : fibrations SSet p
sq₁₃' : internalHom.PullbackObjObj Λ[m + 1, k].ι p := Functor.PullbackObjObj.ofHasPullback internalHom Λ[m + 1, k].ι p
⊢ HasLiftingProperty i sq₁₃'.π
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.PushoutProduct

Goal state before:
pp: X₁ Y₁ E✝ B✝ : SSet
i : X₁ ⟶ Y₁
p✝ : E✝ ⟶ B✝
inst✝¹ : Mono i
inst✝ : Fibration p✝
sq₁₃ : internalHom.PullbackObjObj i p✝
m : ℕ
X Y : SSet
k : Fin (m + 2)
sq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[m + 1, k].ι :=
  Functor.PushoutObjObj.ofHasPushout (curriedTensor SSet) i Λ[m + 1, k].ι
E B : SSet
p : E ⟶ B
hp : fibrations SSet p
sq₁₃' : internalHom.PullbackObjObj Λ[m + 1, k].ι p := Functor.PullbackObjObj.ofHasPullback internalHom Λ[m + 1, k].ι p
⊢ I.rlp sq₁₃'.π

Tactic:
]

Goal state after:
X₁ Y₁ E✝ B✝ : SSet
i : X₁ ⟶ Y₁
p✝ : E✝ ⟶ B✝
inst✝¹ : Mono i
inst✝ : Fibration p✝
sq₁₃ : internalHom.PullbackObjObj i p✝
m : ℕ
X Y : SSet
k : Fin (m + 2)
sq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[m + 1, k].ι :=
  Functor.PushoutObjObj.ofHasPushout (curriedTensor SSet) i Λ[m + 1, k].ι
E B : SSet
p : E ⟶ B
hp : fibrations SSet p
sq₁₃' : internalHom.PullbackObjObj Λ[m + 1, k].ι p := Functor.PullbackObjObj.ofHasPullback internalHom Λ[m + 1, k].ι p
⊢ I.rlp sq₁₃'.π
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.PushoutProduct

Goal state before:
pp: X₁ Y₁ E✝ B✝ : SSet
i : X₁ ⟶ Y₁
p✝ : E✝ ⟶ B✝
inst✝¹ : Mono i
inst✝ : Fibration p✝
sq₁₃ : internalHom.PullbackObjObj i p✝
m : ℕ
X✝ Y✝ : SSet
k : Fin (m + 2)
sq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[m + 1, k].ι :=
  Functor.PushoutObjObj.ofHasPushout (curriedTensor SSet) i Λ[m + 1, k].ι
E B : SSet
p : E ⟶ B
hp : fibrations SSet p
sq₁₃' : internalHom.PullbackObjObj Λ[m + 1, k].ι p := Functor.PullbackObjObj.ofHasPullback internalHom Λ[m + 1, k].ι p
X Y : SSet
n : ℕ
⊢ HasLiftingProperty (Λ[m + 1, k].unionProd ∂Δ[n]).ι p

Tactic:
]

Goal state after:
X₁ Y₁ E✝ B✝ : SSet
i : X₁ ⟶ Y₁
p✝ : E✝ ⟶ B✝
inst✝¹ : Mono i
inst✝ : Fibration p✝
sq₁₃ : internalHom.PullbackObjObj i p✝
m : ℕ
X✝ Y✝ : SSet
k : Fin (m + 2)
sq₁₂ : (curriedTensor SSet).PushoutObjObj i Λ[m + 1, k].ι :=
  Functor.PushoutObjObj.ofHasPushout (curriedTensor SSet) i Λ[m + 1, k].ι
E B : SSet
p : E ⟶ B
hp : fibrations SSet p
sq₁₃' : internalHom.PullbackObjObj Λ[m + 1, k].ι p := Functor.PullbackObjObj.ofHasPullback internalHom Λ[m + 1, k].ι p
X Y : SSet
n : ℕ
⊢ HasLiftingProperty (Λ[m + 1, k].unionProd ∂Δ[n]).ι p
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.PushoutProduct

Goal state before:
pp: X₁ X₂ Y₁ Y₂ : SSet
i : X₁ ⟶ Y₁
j : X₂ ⟶ Y₂
sq₁₂ : (curriedTensor SSet).PushoutObjObj i j
inst✝ : Mono i
hj : anodyneExtensions j
E B : SSet
p : E ⟶ B
hp : fibrations SSet p
sq₁₃ : internalHom.PullbackObjObj i p := Functor.PullbackObjObj.ofHasPullback internalHom i p
⊢ HasLiftingProperty j sq₁₃.π

Tactic:
]

Goal state after:
X₁ X₂ Y₁ Y₂ : SSet
i : X₁ ⟶ Y₁
j : X₂ ⟶ Y₂
sq₁₂ : (curriedTensor SSet).PushoutObjObj i j
inst✝ : Mono i
hj : anodyneExtensions j
E B : SSet
p : E ⟶ B
hp : fibrations SSet p
sq₁₃ : internalHom.PullbackObjObj i p := Functor.PullbackObjObj.ofHasPullback internalHom i p
⊢ HasLiftingProperty j sq₁₃.π
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.PushoutProduct

Goal state before:
pp: X₁ X₂ Y₁ Y₂ : SSet
i : X₁ ⟶ Y₁
j : X₂ ⟶ Y₂
sq₁₂ : (curriedTensor SSet).PushoutObjObj i j
inst✝ : Mono i
hj : anodyneExtensions j
E B : SSet
p : E ⟶ B
hp : Fibration p
sq₁₃ : internalHom.PullbackObjObj i p := Functor.PullbackObjObj.ofHasPullback internalHom i p
⊢ Fibration sq₁₃.π

Tactic:
]

Goal state after:
X₁ X₂ Y₁ Y₂ : SSet
i : X₁ ⟶ Y₁
j : X₂ ⟶ Y₂
sq₁₂ : (curriedTensor SSet).PushoutObjObj i j
inst✝ : Mono i
hj : anodyneExtensions j
E B : SSet
p : E ⟶ B
hp : Fibration p
sq₁₃ : internalHom.PullbackObjObj i p := Functor.PullbackObjObj.ofHasPullback internalHom i p
⊢ Fibration sq₁₃.π
Module: Mathlib.CategoryTheory.CodiscreteCategory

Goal state before:
pp: A : Type u_1
x✝ : (Codiscrete A)ᵒᵖ
⊢ (𝟭 (Codiscrete A)ᵒᵖ).obj x✝ ≅ ((functor fun x ↦ (unop x).as) ⋙ (functor fun x ↦ (unop x).as).rightOp).obj x✝

Tactic:
by exact Iso.refl _

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: case refine_2
m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
d : ℕ
hd : x.dim = d + 1
l : Fin (d + 1)
hl : IsIndex x hd l.succ
y : Fin (d + 1)
hy : (x.cast hd).simplex.1 y.castSucc = k.succ
this : y < l
⊢ False

Tactic:
]

Goal state after:
case refine_2
m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
d : ℕ
hd : x.dim = d + 1
l : Fin (d + 1)
hl : IsIndex x hd l.succ
y : Fin (d + 1)
hy : (x.cast hd).simplex.1 y.castSucc = k.succ
this : y < l
⊢ False
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: m : ℕ
k : Fin (m + 1)
n : ℕ
x : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
d : ℕ
hd : x.dim = d + 1
l : Fin (d + 1)
hl : IsIndex x hd l.succ
t : Fin d
ht : IsIndex hl.δ ⋯ t.succ
⊢ l = l

Tactic:
]

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: m : ℕ
k : Fin (m + 1)
n : ℕ
x u : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hu : IsType₂ u
hd : x.dim = u.dim + 1
l : Fin (u.dim + 1)
hl : IsIndex x hd l.succ
i : Fin (u.dim + 2)
hu' : u.simplex = (ConcreteCategory.hom (SimplicialObject.δ (Δ[m + 1] ⊗ Δ[n]) i)) (x.cast hd).simplex
⊢ i = l.castSucc ∨ i = l.succ

Tactic:
]

Goal state after:
m : ℕ
k : Fin (m + 1)
n : ℕ
x u : (Λ[m + 1, k.castSucc].unionProd ∂Δ[n]).N
hu : IsType₂ u
hd : x.dim = u.dim + 1
l : Fin (u.dim + 1)
hl : IsIndex x hd l.succ
i : Fin (u.dim + 2)
hu' : u.simplex = (ConcreteCategory.hom (SimplicialObject.δ (Δ[m + 1] ⊗ Δ[n]) i)) (x.cast hd).simplex
⊢ i = l.castSucc ∨ i = l.succ
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: m : ℕ
k : Fin (m + 1)
n : ℕ
s : Type₁ k n
ht : IsType₂ s.δ
⊢ (ht.type₁ ⋯).x.toS = s.x.toS

Tactic:
]

Goal state after:
m : ℕ
k : Fin (m + 1)
n : ℕ
s : Type₁ k n
ht : IsType₂ s.δ
⊢ (ht.type₁ ⋯).x.toS = s.x.toS
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: m : ℕ
k : Fin (m + 1)
n : ℕ
s : Type₁ k n
ht : IsType₂ s.δ
⊢ ∃ (h : (ht.type₁ ⋯).x.dim = (s.x.cast ⋯).dim), ((ht.type₁ ⋯).x.cast h).simplex = (s.x.cast ⋯).simplex

Tactic:
]

Goal state after:
m : ℕ
k : Fin (m + 1)
n : ℕ
s : Type₁ k n
ht : IsType₂ s.δ
⊢ ∃ (h : (ht.type₁ ⋯).x.dim = (s.x.cast ⋯).dim), ((ht.type₁ ⋯).x.cast h).simplex = (s.x.cast ⋯).simplex
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: case pos
m : ℕ
k : Fin (m + 1)
n : ℕ
s : Type₁ k n
ht : IsType₂ s.δ
⊢ (k.castSucc, (s.δ.cast ⋯).simplex.2 (min s.δ ⋯)) = (objEquiv (s.x.cast ⋯).simplex) s.index.castSucc

Tactic:
]

Goal state after:
case pos
m : ℕ
k : Fin (m + 1)
n : ℕ
s : Type₁ k n
ht : IsType₂ s.δ
⊢ (k.castSucc, (s.δ.cast ⋯).simplex.2 (min s.δ ⋯)) = (objEquiv (s.x.cast ⋯).simplex) s.index.castSucc
Module: Mathlib.AlgebraicTopology.SimplicialSet.AnodyneExtensions.UnionProd

Goal state before:
pp: case pos.snd
m : ℕ
k : Fin (m + 1)
n : ℕ
s : Type₁ k n
ht : IsType₂ s.δ
⊢ (s.x.cast ⋯).simplex.2 s.index.succ = ((objEquiv (s.x.cast ⋯).simplex) s.index.castSucc).2

Tactic:
]

Goal state after:
case pos.snd
m : ℕ
k : Fin (m + 1)
n : ℕ
s : Type₁ k n
ht : IsType₂ s.δ
⊢ (s.x.cast ⋯).simplex.2 s.index.succ = ((objEquiv (s.x.cast ⋯).simplex) s.index.castSucc).2