-- Mathlib/GroupTheory/HNNExtension.lean
theorem unitsSMul_neg (u : ℤˣ) (w : NormalWord d) :
    unitsSMul φ (-u) (unitsSMul φ u w) = w

Proof state:
case neg.cons.refl
G : Type u_1
inst✝ : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
d : TransversalPair G A B
u : ℤˣ
g : G
w : NormalWord d
h1 : w.head ∈ d.set { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ }
h2 :
  ∀ u' ∈ Option.map Prod.fst w.toList.head?,
    w.head ∈ toSubgroup A B { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } →
      { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } = u'
hcan✝ :
  ¬Cancels (-u)
      (if h : Cancels u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2) then
        unitsSMulWithCancel φ u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2) h
      else
        let g' := unitsSMulGroup φ d u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2).head;
        cons (↑g'.1) u
          ((↑g'.2 * (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2).head⁻¹) •
            cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2)
          ⋯ ⋯)
hcan2 : Cancels u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2)
hcan :
  ¬Cancels (-u)
      (unitsSMulWithCancel φ u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2) hcan2)
this : (⋯.equiv w.head).1 = 1
⊢ cons (↑((toSubgroupEquiv φ (-u)) (⋯.equiv (↑((toSubgroupEquiv φ u) ⟨g, ⋯⟩) * w.head)).1)) (-u)
      ((↑(⋯.equiv (↑((toSubgroupEquiv φ u) ⟨g, ⋯⟩) * w.head)).2 * (w.head⁻¹ * (↑((toSubgroupEquiv φ u) ⟨g, ⋯⟩))⁻¹)) •
        ↑((toSubgroupEquiv φ u) ⟨g, ⋯⟩) • w)
      ⋯ ⋯ =
    cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2

Tactic:
apply NormalWord.ext

Result:
case neg.cons.refl.h1
G : Type u_1
inst✝ : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
d : TransversalPair G A B
u : ℤˣ
g : G
w : NormalWord d
h1 : w.head ∈ d.set { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ }
h2 :
  ∀ u' ∈ Option.map Prod.fst w.toList.head?,
    w.head ∈ toSubgroup A B { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } →
      { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } = u'
hcan✝ :
  ¬Cancels (-u)
      (if h : Cancels u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2) then
        unitsSMulWithCancel φ u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2) h
      else
        let g' := unitsSMulGroup φ d u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2).head;
        cons (↑g'.1) u
          ((↑g'.2 * (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2).head⁻¹) •
            cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2)
          ⋯ ⋯)
hcan2 : Cancels u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2)
hcan :
  ¬Cancels (-u)
      (unitsSMulWithCancel φ u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2) hcan2)
this : (⋯.equiv w.head).1 = 1
⊢ (cons (↑((toSubgroupEquiv φ (-u)) (⋯.equiv (↑((toSubgroupEquiv φ u) ⟨g, ⋯⟩) * w.head)).1)) (-u)
        ((↑(⋯.equiv (↑((toSubgroupEquiv φ u) ⟨g, ⋯⟩) * w.head)).2 * (w.head⁻¹ * (↑((toSubgroupEquiv φ u) ⟨g, ⋯⟩))⁻¹)) •
          ↑((toSubgroupEquiv φ u) ⟨g, ⋯⟩) • w)
        ⋯ ⋯).head =
    (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2).head

case neg.cons.refl.h2
G : Type u_1
inst✝ : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
d : TransversalPair G A B
u : ℤˣ
g : G
w : NormalWord d
h1 : w.head ∈ d.set { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ }
h2 :
  ∀ u' ∈ Option.map Prod.fst w.toList.head?,
    w.head ∈ toSubgroup A B { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } →
      { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } = u'
hcan✝ :
  ¬Cancels (-u)
      (if h : Cancels u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2) then
        unitsSMulWithCancel φ u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2) h
      else
        let g' := unitsSMulGroup φ d u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2).head;
        cons (↑g'.1) u
          ((↑g'.2 * (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2).head⁻¹) •
            cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2)
          ⋯ ⋯)
hcan2 : Cancels u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2)
hcan :
  ¬Cancels (-u)
      (unitsSMulWithCancel φ u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2) hcan2)
this : (⋯.equiv w.head).1 = 1
⊢ (cons (↑((toSubgroupEquiv φ (-u)) (⋯.equiv (↑((toSubgroupEquiv φ u) ⟨g, ⋯⟩) * w.head)).1)) (-u)
        ((↑(⋯.equiv (↑((toSubgroupEquiv φ u) ⟨g, ⋯⟩) * w.head)).2 * (w.head⁻¹ * (↑((toSubgroupEquiv φ u) ⟨g, ⋯⟩))⁻¹)) •
          ↑((toSubgroupEquiv φ u) ⟨g, ⋯⟩) • w)
        ⋯ ⋯).toList =
    (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2).toList

Proof state:
case neg.cons.refl.h1
G : Type u_1
inst✝ : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
d : TransversalPair G A B
u : ℤˣ
g : G
w : NormalWord d
h1 : w.head ∈ d.set { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ }
h2 :
  ∀ u' ∈ Option.map Prod.fst w.toList.head?,
    w.head ∈ toSubgroup A B { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } →
      { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } = u'
hcan✝ :
  ¬Cancels (-u)
      (if h : Cancels u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2) then
        unitsSMulWithCancel φ u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2) h
      else
        let g' := unitsSMulGroup φ d u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2).head;
        cons (↑g'.1) u
          ((↑g'.2 * (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2).head⁻¹) •
            cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2)
          ⋯ ⋯)
hcan2 : Cancels u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2)
hcan :
  ¬Cancels (-u)
      (unitsSMulWithCancel φ u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2) hcan2)
this : (⋯.equiv w.head).1 = 1
⊢ ↑((toSubgroupEquiv φ (-u)) (⋯.equiv (↑((toSubgroupEquiv φ u) ⟨g, ⋯⟩) * w.head)).1) = g

Tactic:
conv_lhs => erw [IsComplement.equiv_mul_left]

Result:
case neg.cons.refl.h1
G : Type u_1
inst✝ : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
d : TransversalPair G A B
u : ℤˣ
g : G
w : NormalWord d
h1 : w.head ∈ d.set { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ }
h2 :
  ∀ u' ∈ Option.map Prod.fst w.toList.head?,
    w.head ∈ toSubgroup A B { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } →
      { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } = u'
hcan✝ :
  ¬Cancels (-u)
      (if h : Cancels u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2) then
        unitsSMulWithCancel φ u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2) h
      else
        let g' := unitsSMulGroup φ d u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2).head;
        cons (↑g'.1) u
          ((↑g'.2 * (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2).head⁻¹) •
            cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2)
          ⋯ ⋯)
hcan2 : Cancels u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2)
hcan :
  ¬Cancels (-u)
      (unitsSMulWithCancel φ u (cons g { val := -↑u, inv := -↑u⁻¹, val_inv := ⋯, inv_val := ⋯ } w h1 h2) hcan2)
this : (⋯.equiv w.head).1 = 1
⊢ ↑((toSubgroupEquiv φ (-u)) ((toSubgroupEquiv φ u) ⟨g, ⋯⟩ * (⋯.equiv w.head).1, (⋯.equiv w.head).2).1) = g