Module: Mathlib.GroupTheory.CoprodI

Goal state before:
pp: case pos
ι : Type u_1
M : ι → Type u_2
inst✝² : (i : ι) → Monoid (M i)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
i head : ι
m✝ : M head
w✝ : Word M
h1✝ : w✝.fstIdx ≠ some head
h2✝ : m✝ ≠ 1
a✝ :
  ({
          toFun := fun w ↦
            ↑(consRecOn w ⟨{ head := 1, tail := empty, fstIdx_ne := ⋯ }, ⋯⟩ fun j m w h1 h2 x ↦
                if ij : i = j then ⟨{ head := ⋯ ▸ m, tail := w, fstIdx_ne := ⋯ }, ⋯⟩
                else ⟨{ head := 1, tail := cons m w h1 h2, fstIdx_ne := ⋯ }, ⋯⟩),
          invFun := rcons, left_inv := ⋯, right_inv := ⋯ }
        w✝).head =
    if h : ∃ (h : w✝.toList ≠ []), (w✝.toList.head h).fst = i then ⋯ ▸ (w✝.toList.head ⋯).snd else 1
hi : i = head
⊢ ({
          toFun := fun w ↦
            ↑(consRecOn w ⟨{ head := 1, tail := empty, fstIdx_ne := ⋯ }, ⋯⟩ fun j m w h1 h2 x ↦
                if ij : i = j then ⟨{ head := ⋯ ▸ m, tail := w, fstIdx_ne := ⋯ }, ⋯⟩
                else ⟨{ head := 1, tail := cons m w h1 h2, fstIdx_ne := ⋯ }, ⋯⟩),
          invFun := rcons, left_inv := ⋯, right_inv := ⋯ }
        (cons m✝ w✝ h1✝ h2✝)).head =
    if h : ∃ (h : (cons m✝ w✝ h1✝ h2✝).toList ≠ []), ((cons m✝ w✝ h1✝ h2✝).toList.head h).fst = i then
      ⋯ ▸ ((cons m✝ w✝ h1✝ h2✝).toList.head ⋯).snd
    else 1

pp: case neg
ι : Type u_1
M : ι → Type u_2
inst✝² : (i : ι) → Monoid (M i)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
i head : ι
m✝ : M head
w✝ : Word M
h1✝ : w✝.fstIdx ≠ some head
h2✝ : m✝ ≠ 1
a✝ :
  ({
          toFun := fun w ↦
            ↑(consRecOn w ⟨{ head := 1, tail := empty, fstIdx_ne := ⋯ }, ⋯⟩ fun j m w h1 h2 x ↦
                if ij : i = j then ⟨{ head := ⋯ ▸ m, tail := w, fstIdx_ne := ⋯ }, ⋯⟩
                else ⟨{ head := 1, tail := cons m w h1 h2, fstIdx_ne := ⋯ }, ⋯⟩),
          invFun := rcons, left_inv := ⋯, right_inv := ⋯ }
        w✝).head =
    if h : ∃ (h : w✝.toList ≠ []), (w✝.toList.head h).fst = i then ⋯ ▸ (w✝.toList.head ⋯).snd else 1
hi : ¬i = head
⊢ ({
          toFun := fun w ↦
            ↑(consRecOn w ⟨{ head := 1, tail := empty, fstIdx_ne := ⋯ }, ⋯⟩ fun j m w h1 h2 x ↦
                if ij : i = j then ⟨{ head := ⋯ ▸ m, tail := w, fstIdx_ne := ⋯ }, ⋯⟩
                else ⟨{ head := 1, tail := cons m w h1 h2, fstIdx_ne := ⋯ }, ⋯⟩),
          invFun := rcons, left_inv := ⋯, right_inv := ⋯ }
        (cons m✝ w✝ h1✝ h2✝)).head =
    if h : ∃ (h : (cons m✝ w✝ h1✝ h2✝).toList ≠ []), ((cons m✝ w✝ h1✝ h2✝).toList.head h).fst = i then
      ⋯ ▸ ((cons m✝ w✝ h1✝ h2✝).toList.head ⋯).snd
    else 1

Tactic:
· subst hi; simp

Goal state after:
case neg
ι : Type u_1
M : ι → Type u_2
inst✝² : (i : ι) → Monoid (M i)
inst✝¹ : DecidableEq ι
inst✝ : (i : ι) → DecidableEq (M i)
i head : ι
m✝ : M head
w✝ : Word M
h1✝ : w✝.fstIdx ≠ some head
h2✝ : m✝ ≠ 1
a✝ :
  ({
          toFun := fun w ↦
            ↑(consRecOn w ⟨{ head := 1, tail := empty, fstIdx_ne := ⋯ }, ⋯⟩ fun j m w h1 h2 x ↦
                if ij : i = j then ⟨{ head := ⋯ ▸ m, tail := w, fstIdx_ne := ⋯ }, ⋯⟩
                else ⟨{ head := 1, tail := cons m w h1 h2, fstIdx_ne := ⋯ }, ⋯⟩),
          invFun := rcons, left_inv := ⋯, right_inv := ⋯ }
        w✝).head =
    if h : ∃ (h : w✝.toList ≠ []), (w✝.toList.head h).fst = i then ⋯ ▸ (w✝.toList.head ⋯).snd else 1
hi : ¬i = head
⊢ ({
          toFun := fun w ↦
            ↑(consRecOn w ⟨{ head := 1, tail := empty, fstIdx_ne := ⋯ }, ⋯⟩ fun j m w h1 h2 x ↦
                if ij : i = j then ⟨{ head := ⋯ ▸ m, tail := w, fstIdx_ne := ⋯ }, ⋯⟩
                else ⟨{ head := 1, tail := cons m w h1 h2, fstIdx_ne := ⋯ }, ⋯⟩),
          invFun := rcons, left_inv := ⋯, right_inv := ⋯ }
        (cons m✝ w✝ h1✝ h2✝)).head =
    if h : ∃ (h : (cons m✝ w✝ h1✝ h2✝).toList ≠ []), ((cons m✝ w✝ h1✝ h2✝).toList.head h).fst = i then
      ⋯ ▸ ((cons m✝ w✝ h1✝ h2✝).toList.head ⋯).snd
    else 1