Theorem: SimplexCategory.δ_comp_δ
File: Mathlib/AlgebraicTopology/SimplexCategory.lean

Proof steps:
state_before: n : ℕ
i j : Fin (n + 2)
H : i ≤ j
⊢ δ i ≫ δ j.succ = δ j ≫ δ i.castSucc
state_after: case a.h.h.h
n : ℕ
i j : Fin (n + 2)
H : i ≤ j
k : Fin ([n].len + 1)
⊢ ↑((Hom.toOrderHom (δ i ≫ δ j.succ)) k) = ↑((Hom.toOrderHom (δ j ≫ δ i.castSucc)) k)
tactic: ext k
state_before: case a.h.h.h
n : ℕ
i j : Fin (n + 2)
H : i ≤ j
k : Fin ([n].len + 1)
⊢ ↑((Hom.toOrderHom (δ i ≫ δ j.succ)) k) = ↑((Hom.toOrderHom (δ j ≫ δ i.castSucc)) k)
state_after: case a.h.h.h
n : ℕ
i j : Fin (n + 2)
H : i ≤ j
k : Fin ([n].len + 1)
⊢ ↑(if (if k.castSucc < i then k.castSucc else k.succ).castSucc < j.succ then
        (if k.castSucc < i then k.castSucc else k.succ).castSucc
      else (if k.castSucc < i then k.castSucc else k.succ).succ) =
    ↑(if (if k.castSucc < j then k.castSucc else k.succ).castSucc < i.castSucc then
        (if k.castSucc < j then k.castSucc else k.succ).castSucc
      else (if k.castSucc < j then k.castSucc else k.succ).succ)
tactic: dsimp [δ, Fin.succAbove]
state_before: case a.h.h.h
n : ℕ
i j : Fin (n + 2)
H : i ≤ j
k : Fin ([n].len + 1)
⊢ ↑(if (if k.castSucc < i then k.castSucc else k.succ).castSucc < j.succ then
        (if k.castSucc < i then k.castSucc else k.succ).castSucc
      else (if k.castSucc < i then k.castSucc else k.succ).succ) =
    ↑(if (if k.castSucc < j then k.castSucc else k.succ).castSucc < i.castSucc then
        (if k.castSucc < j then k.castSucc else k.succ).castSucc
      else (if k.castSucc < j then k.castSucc else k.succ).succ)
state_after: case a.h.h.h.mk
n : ℕ
j : Fin (n + 2)
k : Fin ([n].len + 1)
i : ℕ
isLt✝ : i < n + 2
H : ⟨i, isLt✝⟩ ≤ j
⊢ ↑(if (if k.castSucc < ⟨i, isLt✝⟩ then k.castSucc else k.succ).castSucc < j.succ then
        (if k.castSucc < ⟨i, isLt✝⟩ then k.castSucc else k.succ).castSucc
      else (if k.castSucc < ⟨i, isLt✝⟩ then k.castSucc else k.succ).succ) =
    ↑(if (if k.castSucc < j then k.castSucc else k.succ).castSucc < ⟨i, isLt✝⟩.castSucc then
        (if k.castSucc < j then k.castSucc else k.succ).castSucc
      else (if k.castSucc < j then k.castSucc else k.succ).succ)
tactic: rcases i with ⟨i, _⟩
state_before: case a.h.h.h.mk
n : ℕ
j : Fin (n + 2)
k : Fin ([n].len + 1)
i : ℕ
isLt✝ : i < n + 2
H : ⟨i, isLt✝⟩ ≤ j
⊢ ↑(if (if k.castSucc < ⟨i, isLt✝⟩ then k.castSucc else k.succ).castSucc < j.succ then
        (if k.castSucc < ⟨i, isLt✝⟩ then k.castSucc else k.succ).castSucc
      else (if k.castSucc < ⟨i, isLt✝⟩ then k.castSucc else k.succ).succ) =
    ↑(if (if k.castSucc < j then k.castSucc else k.succ).castSucc < ⟨i, isLt✝⟩.castSucc then
        (if k.castSucc < j then k.castSucc else k.succ).castSucc
      else (if k.castSucc < j then k.castSucc else k.succ).succ)
state_after: case a.h.h.h.mk.mk
n : ℕ
k : Fin ([n].len + 1)
i : ℕ
isLt✝¹ : i < n + 2
j : ℕ
isLt✝ : j < n + 2
H : ⟨i, isLt✝¹⟩ ≤ ⟨j, isLt✝⟩
⊢ ↑(if (if k.castSucc < ⟨i, isLt✝¹⟩ then k.castSucc else k.succ).castSucc < ⟨j, isLt✝⟩.succ then
        (if k.castSucc < ⟨i, isLt✝¹⟩ then k.castSucc else k.succ).castSucc
      else (if k.castSucc < ⟨i, isLt✝¹⟩ then k.castSucc else k.succ).succ) =
    ↑(if (if k.castSucc < ⟨j, isLt✝⟩ then k.castSucc else k.succ).castSucc < ⟨i, isLt✝¹⟩.castSucc then
        (if k.castSucc < ⟨j, isLt✝⟩ then k.castSucc else k.succ).castSucc
      else (if k.castSucc < ⟨j, isLt✝⟩ then k.castSucc else k.succ).succ)
tactic: rcases j with ⟨j, _⟩
state_before: case a.h.h.h.mk.mk
n : ℕ
k : Fin ([n].len + 1)
i : ℕ
isLt✝¹ : i < n + 2
j : ℕ
isLt✝ : j < n + 2
H : ⟨i, isLt✝¹⟩ ≤ ⟨j, isLt✝⟩
⊢ ↑(if (if k.castSucc < ⟨i, isLt✝¹⟩ then k.castSucc else k.succ).castSucc < ⟨j, isLt✝⟩.succ then
        (if k.castSucc < ⟨i, isLt✝¹⟩ then k.castSucc else k.succ).castSucc
      else (if k.castSucc < ⟨i, isLt✝¹⟩ then k.castSucc else k.succ).succ) =
    ↑(if (if k.castSucc < ⟨j, isLt✝⟩ then k.castSucc else k.succ).castSucc < ⟨i, isLt✝¹⟩.castSucc then
        (if k.castSucc < ⟨j, isLt✝⟩ then k.castSucc else k.succ).castSucc
      else (if k.castSucc < ⟨j, isLt✝⟩ then k.castSucc else k.succ).succ)
state_after: case a.h.h.h.mk.mk.mk
n i : ℕ
isLt✝² : i < n + 2
j : ℕ
isLt✝¹ : j < n + 2
H : ⟨i, isLt✝²⟩ ≤ ⟨j, isLt✝¹⟩
k : ℕ
isLt✝ : k < [n].len + 1
⊢ ↑(if
          (if ⟨k, isLt✝⟩.castSucc < ⟨i, isLt✝²⟩ then ⟨k, isLt✝⟩.castSucc else ⟨k, isLt✝⟩.succ).castSucc <
            ⟨j, isLt✝¹⟩.succ then
        (if ⟨k, isLt✝⟩.castSucc < ⟨i, isLt✝²⟩ then ⟨k, isLt✝⟩.castSucc else ⟨k, isLt✝⟩.succ).castSucc
      else (if ⟨k, isLt✝⟩.castSucc < ⟨i, isLt✝²⟩ then ⟨k, isLt✝⟩.castSucc else ⟨k, isLt✝⟩.succ).succ) =
    ↑(if
          (if ⟨k, isLt✝⟩.castSucc < ⟨j, isLt✝¹⟩ then ⟨k, isLt✝⟩.castSucc else ⟨k, isLt✝⟩.succ).castSucc <
            ⟨i, isLt✝²⟩.castSucc then
        (if ⟨k, isLt✝⟩.castSucc < ⟨j, isLt✝¹⟩ then ⟨k, isLt✝⟩.castSucc else ⟨k, isLt✝⟩.succ).castSucc
      else (if ⟨k, isLt✝⟩.castSucc < ⟨j, isLt✝¹⟩ then ⟨k, isLt✝⟩.castSucc else ⟨k, isLt✝⟩.succ).succ)
tactic: rcases k with ⟨k, _⟩