-- Mathlib/GroupTheory/CoprodI.lean
theorem mem_equivPair_tail_iff {i j : ι} {w : Word M} (m : M i) :
    (⟨i, m⟩ ∈ (equivPair j w).tail.toList) ↔ ⟨i, m⟩ ∈ w.toList.tail
      ∨ i ≠ j ∧ ∃ h : w.toList ≠ [], w.toList.head h = ⟨i, m⟩

Proof state:
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 j : ι
m : M i
tail : Word M
ih :
  ⟨i, m⟩ ∈
      (↑(consRecOn tail ⟨{ head := 1, tail := empty, fstIdx_ne := ⋯ }, ⋯⟩ fun j_1 m w h1 h2 x =>
              if ij : j = j_1 then ⟨{ head := ⋯ ▸ m, tail := w, fstIdx_ne := ⋯ }, ⋯⟩
              else ⟨{ head := 1, tail := cons m w h1 h2, fstIdx_ne := ⋯ }, ⋯⟩)).tail.toList ↔
    ⟨i, m⟩ ∈ tail.toList.tail ∨ ¬i = j ∧ ∃ (h : ¬tail.toList = []), tail.toList.head h = ⟨i, m⟩
g : M j
h1 : tail.fstIdx ≠ some j
h2 : g ≠ 1
⊢ ⟨i, m⟩ ∈ (↑⟨{ head := ⋯ ▸ g, tail := tail, fstIdx_ne := ⋯ }, ⋯⟩).tail.toList ↔
    ⟨i, m⟩ ∈ (cons g tail h1 h2).toList.tail ∨
      ¬i = j ∧ ∃ (h : ¬(cons g tail h1 h2).toList = []), (cons g tail h1 h2).toList.head h = ⟨i, m⟩

Tactic:
by_cases hij : j = i <;> simp_all

Result:
no goals

Proof state:
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 j : ι
w : Word M
m : M i
⊢ ⟨i, m⟩ ∈ ((equivPair j) w).tail.toList ↔
    ⟨i, m⟩ ∈ w.toList.tail ∨ i ≠ j ∧ ∃ (h : w.toList ≠ []), w.toList.head h = ⟨i, m⟩

Tactic:
simp only [equivPair, equivPairAux, ne_eq, Equiv.coe_fn_mk]

Result:
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 j : ι
w : Word M
m : M i
⊢ ⟨i, m⟩ ∈
      (↑(consRecOn w ⟨{ head := 1, tail := empty, fstIdx_ne := ⋯ }, ⋯⟩ fun j_1 m w h1 h2 x =>
              if ij : j = j_1 then ⟨{ head := ⋯ ▸ m, tail := w, fstIdx_ne := ⋯ }, ⋯⟩
              else ⟨{ head := 1, tail := cons m w h1 h2, fstIdx_ne := ⋯ }, ⋯⟩)).tail.toList ↔
    ⟨i, m⟩ ∈ w.toList.tail ∨ ¬i = j ∧ ∃ (h : ¬w.toList = []), w.toList.head h = ⟨i, m⟩

Proof state:
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 j : ι
m : M i
k : ι
g : M k
tail : Word M
h1 : tail.fstIdx ≠ some k
h2 : g ≠ 1
ih :
  ⟨i, m⟩ ∈
      (↑(consRecOn tail ⟨{ head := 1, tail := empty, fstIdx_ne := ⋯ }, ⋯⟩ fun j_1 m w h1 h2 x =>
              if ij : j = j_1 then ⟨{ head := ⋯ ▸ m, tail := w, fstIdx_ne := ⋯ }, ⋯⟩
              else ⟨{ head := 1, tail := cons m w h1 h2, fstIdx_ne := ⋯ }, ⋯⟩)).tail.toList ↔
    ⟨i, m⟩ ∈ tail.toList.tail ∨ ¬i = j ∧ ∃ (h : ¬tail.toList = []), tail.toList.head h = ⟨i, m⟩
h : ¬j = k
⊢ ⟨i, m⟩ ∈ (↑⟨{ head := 1, tail := cons g tail h1 h2, fstIdx_ne := ⋯ }, ⋯⟩).tail.toList ↔
    ⟨i, m⟩ ∈ (cons g tail h1 h2).toList.tail ∨
      ¬i = j ∧ ∃ (h : ¬(cons g tail h1 h2).toList = []), (cons g tail h1 h2).toList.head h = ⟨i, m⟩

Tactic:
by_cases hik : i = k

Result:
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 j : ι
m : M i
k : ι
g : M k
tail : Word M
h1 : tail.fstIdx ≠ some k
h2 : g ≠ 1
ih :
  ⟨i, m⟩ ∈
      (↑(consRecOn tail ⟨{ head := 1, tail := empty, fstIdx_ne := ⋯ }, ⋯⟩ fun j_1 m w h1 h2 x =>
              if ij : j = j_1 then ⟨{ head := ⋯ ▸ m, tail := w, fstIdx_ne := ⋯ }, ⋯⟩
              else ⟨{ head := 1, tail := cons m w h1 h2, fstIdx_ne := ⋯ }, ⋯⟩)).tail.toList ↔
    ⟨i, m⟩ ∈ tail.toList.tail ∨ ¬i = j ∧ ∃ (h : ¬tail.toList = []), tail.toList.head h = ⟨i, m⟩
h : ¬j = k
hik : i = k
⊢ ⟨i, m⟩ ∈ (↑⟨{ head := 1, tail := cons g tail h1 h2, fstIdx_ne := ⋯ }, ⋯⟩).tail.toList ↔
    ⟨i, m⟩ ∈ (cons g tail h1 h2).toList.tail ∨
      ¬i = j ∧ ∃ (h : ¬(cons g tail h1 h2).toList = []), (cons g tail h1 h2).toList.head h = ⟨i, m⟩

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 j : ι
m : M i
k : ι
g : M k
tail : Word M
h1 : tail.fstIdx ≠ some k
h2 : g ≠ 1
ih :
  ⟨i, m⟩ ∈
      (↑(consRecOn tail ⟨{ head := 1, tail := empty, fstIdx_ne := ⋯ }, ⋯⟩ fun j_1 m w h1 h2 x =>
              if ij : j = j_1 then ⟨{ head := ⋯ ▸ m, tail := w, fstIdx_ne := ⋯ }, ⋯⟩
              else ⟨{ head := 1, tail := cons m w h1 h2, fstIdx_ne := ⋯ }, ⋯⟩)).tail.toList ↔
    ⟨i, m⟩ ∈ tail.toList.tail ∨ ¬i = j ∧ ∃ (h : ¬tail.toList = []), tail.toList.head h = ⟨i, m⟩
h : ¬j = k
hik : ¬i = k
⊢ ⟨i, m⟩ ∈ (↑⟨{ head := 1, tail := cons g tail h1 h2, fstIdx_ne := ⋯ }, ⋯⟩).tail.toList ↔
    ⟨i, m⟩ ∈ (cons g tail h1 h2).toList.tail ∨
      ¬i = j ∧ ∃ (h : ¬(cons g tail h1 h2).toList = []), (cons g tail h1 h2).toList.head h = ⟨i, m⟩

Proof state:
case h_cons
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 j : ι
m : M i
k : ι
g : M k
tail : Word M
h1 : tail.fstIdx ≠ some k
h2 : g ≠ 1
ih :
  ⟨i, m⟩ ∈
      (↑(consRecOn tail ⟨{ head := 1, tail := empty, fstIdx_ne := ⋯ }, ⋯⟩ fun j_1 m w h1 h2 x =>
              if ij : j = j_1 then ⟨{ head := ⋯ ▸ m, tail := w, fstIdx_ne := ⋯ }, ⋯⟩
              else ⟨{ head := 1, tail := cons m w h1 h2, fstIdx_ne := ⋯ }, ⋯⟩)).tail.toList ↔
    ⟨i, m⟩ ∈ tail.toList.tail ∨ ¬i = j ∧ ∃ (h : ¬tail.toList = []), tail.toList.head h = ⟨i, m⟩
⊢ ⟨i, m⟩ ∈
      (↑(if ij : j = k then ⟨{ head := ⋯ ▸ g, tail := tail, fstIdx_ne := ⋯ }, ⋯⟩
            else ⟨{ head := 1, tail := cons g tail h1 h2, fstIdx_ne := ⋯ }, ⋯⟩)).tail.toList ↔
    ⟨i, m⟩ ∈ (cons g tail h1 h2).toList.tail ∨
      ¬i = j ∧ ∃ (h : ¬(cons g tail h1 h2).toList = []), (cons g tail h1 h2).toList.head h = ⟨i, m⟩

Tactic:
split_ifs with h

Result:
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 j : ι
m : M i
k : ι
g : M k
tail : Word M
h1 : tail.fstIdx ≠ some k
h2 : g ≠ 1
ih :
  ⟨i, m⟩ ∈
      (↑(consRecOn tail ⟨{ head := 1, tail := empty, fstIdx_ne := ⋯ }, ⋯⟩ fun j_1 m w h1 h2 x =>
              if ij : j = j_1 then ⟨{ head := ⋯ ▸ m, tail := w, fstIdx_ne := ⋯ }, ⋯⟩
              else ⟨{ head := 1, tail := cons m w h1 h2, fstIdx_ne := ⋯ }, ⋯⟩)).tail.toList ↔
    ⟨i, m⟩ ∈ tail.toList.tail ∨ ¬i = j ∧ ∃ (h : ¬tail.toList = []), tail.toList.head h = ⟨i, m⟩
h : j = k
⊢ ⟨i, m⟩ ∈ (↑⟨{ head := ⋯ ▸ g, tail := tail, fstIdx_ne := ⋯ }, ⋯⟩).tail.toList ↔
    ⟨i, m⟩ ∈ (cons g tail h1 h2).toList.tail ∨
      ¬i = j ∧ ∃ (h : ¬(cons g tail h1 h2).toList = []), (cons g tail h1 h2).toList.head h = ⟨i, m⟩

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 j : ι
m : M i
k : ι
g : M k
tail : Word M
h1 : tail.fstIdx ≠ some k
h2 : g ≠ 1
ih :
  ⟨i, m⟩ ∈
      (↑(consRecOn tail ⟨{ head := 1, tail := empty, fstIdx_ne := ⋯ }, ⋯⟩ fun j_1 m w h1 h2 x =>
              if ij : j = j_1 then ⟨{ head := ⋯ ▸ m, tail := w, fstIdx_ne := ⋯ }, ⋯⟩
              else ⟨{ head := 1, tail := cons m w h1 h2, fstIdx_ne := ⋯ }, ⋯⟩)).tail.toList ↔
    ⟨i, m⟩ ∈ tail.toList.tail ∨ ¬i = j ∧ ∃ (h : ¬tail.toList = []), tail.toList.head h = ⟨i, m⟩
h : ¬j = k
⊢ ⟨i, m⟩ ∈ (↑⟨{ head := 1, tail := cons g tail h1 h2, fstIdx_ne := ⋯ }, ⋯⟩).tail.toList ↔
    ⟨i, m⟩ ∈ (cons g tail h1 h2).toList.tail ∨
      ¬i = j ∧ ∃ (h : ¬(cons g tail h1 h2).toList = []), (cons g tail h1 h2).toList.head h = ⟨i, m⟩