Module: Mathlib.GroupTheory.HNNExtension

Goal state before:
pp: case pos.cons.refl.inl
G : Type u_1
inst✝ : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
d : TransversalPair G A B
g✝ : G
w✝ : NormalWord d
h1✝ : w✝.head ∈ d.set { val := -↑1, inv := -↑1⁻¹, val_inv := ⋯, inv_val := ⋯ }
h2✝ :
  ∀ u' ∈ Option.map Prod.fst w✝.toList.head?,
    w✝.head ∈ toSubgroup A B { val := -↑1, inv := -↑1⁻¹, val_inv := ⋯, inv_val := ⋯ } →
      { val := -↑1, inv := -↑1⁻¹, val_inv := ⋯, inv_val := ⋯ } = u'
hcan : Cancels 1 (cons g✝ { val := -↑1, inv := -↑1⁻¹, val_inv := ⋯, inv_val := ⋯ } w✝ h1✝ h2✝)
a✝ :
  cons g✝ { val := -↑1, inv := -↑1⁻¹, val_inv := ⋯, inv_val := ⋯ } w✝ h1✝ h2✝ = w✝ →
    ReducedWord.prod φ
        (unitsSMulWithCancel φ 1 (cons g✝ { val := -↑1, inv := -↑1⁻¹, val_inv := ⋯, inv_val := ⋯ } w✝ h1✝ h2✝)
            hcan).toReducedWord =
      t ^ ↑1 *
        ReducedWord.prod φ (cons g✝ { val := -↑1, inv := -↑1⁻¹, val_inv := ⋯, inv_val := ⋯ } w✝ h1✝ h2✝).toReducedWord
⊢ of ↑((toSubgroupEquiv φ 1) ⟨g✝, ⋯⟩) * ReducedWord.prod φ w✝.toReducedWord =
    t ^ ↑1 * (of g✝ * ((t ^ ↑1)⁻¹ * ReducedWord.prod φ w✝.toReducedWord))

pp: case pos.cons.refl.inr
G : Type u_1
inst✝ : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
d : TransversalPair G A B
g✝ : G
w✝ : NormalWord d
h1✝ : w✝.head ∈ d.set { val := -↑(-1), inv := -↑(-1)⁻¹, val_inv := ⋯, inv_val := ⋯ }
h2✝ :
  ∀ u' ∈ Option.map Prod.fst w✝.toList.head?,
    w✝.head ∈ toSubgroup A B { val := -↑(-1), inv := -↑(-1)⁻¹, val_inv := ⋯, inv_val := ⋯ } →
      { val := -↑(-1), inv := -↑(-1)⁻¹, val_inv := ⋯, inv_val := ⋯ } = u'
hcan : Cancels (-1) (cons g✝ { val := -↑(-1), inv := -↑(-1)⁻¹, val_inv := ⋯, inv_val := ⋯ } w✝ h1✝ h2✝)
a✝ :
  cons g✝ { val := -↑(-1), inv := -↑(-1)⁻¹, val_inv := ⋯, inv_val := ⋯ } w✝ h1✝ h2✝ = w✝ →
    ReducedWord.prod φ
        (unitsSMulWithCancel φ (-1) (cons g✝ { val := -↑(-1), inv := -↑(-1)⁻¹, val_inv := ⋯, inv_val := ⋯ } w✝ h1✝ h2✝)
            hcan).toReducedWord =
      t ^ ↑(-1) *
        ReducedWord.prod φ
          (cons g✝ { val := -↑(-1), inv := -↑(-1)⁻¹, val_inv := ⋯, inv_val := ⋯ } w✝ h1✝ h2✝).toReducedWord
⊢ of ↑((toSubgroupEquiv φ (-1)) ⟨g✝, ⋯⟩) * ReducedWord.prod φ w✝.toReducedWord =
    t ^ ↑(-1) * (of g✝ * ((t ^ ↑(-1))⁻¹ * ReducedWord.prod φ w✝.toReducedWord))

Tactic:
· simp [equiv_eq_conj, mul_assoc]

Goal state after:
case pos.cons.refl.inr
G : Type u_1
inst✝ : Group G
A B : Subgroup G
φ : ↥A ≃* ↥B
d : TransversalPair G A B
g✝ : G
w✝ : NormalWord d
h1✝ : w✝.head ∈ d.set { val := -↑(-1), inv := -↑(-1)⁻¹, val_inv := ⋯, inv_val := ⋯ }
h2✝ :
  ∀ u' ∈ Option.map Prod.fst w✝.toList.head?,
    w✝.head ∈ toSubgroup A B { val := -↑(-1), inv := -↑(-1)⁻¹, val_inv := ⋯, inv_val := ⋯ } →
      { val := -↑(-1), inv := -↑(-1)⁻¹, val_inv := ⋯, inv_val := ⋯ } = u'
hcan : Cancels (-1) (cons g✝ { val := -↑(-1), inv := -↑(-1)⁻¹, val_inv := ⋯, inv_val := ⋯ } w✝ h1✝ h2✝)
a✝ :
  cons g✝ { val := -↑(-1), inv := -↑(-1)⁻¹, val_inv := ⋯, inv_val := ⋯ } w✝ h1✝ h2✝ = w✝ →
    ReducedWord.prod φ
        (unitsSMulWithCancel φ (-1) (cons g✝ { val := -↑(-1), inv := -↑(-1)⁻¹, val_inv := ⋯, inv_val := ⋯ } w✝ h1✝ h2✝)
            hcan).toReducedWord =
      t ^ ↑(-1) *
        ReducedWord.prod φ
          (cons g✝ { val := -↑(-1), inv := -↑(-1)⁻¹, val_inv := ⋯, inv_val := ⋯ } w✝ h1✝ h2✝).toReducedWord
⊢ of ↑((toSubgroupEquiv φ (-1)) ⟨g✝, ⋯⟩) * ReducedWord.prod φ w✝.toReducedWord =
    t ^ ↑(-1) * (of g✝ * ((t ^ ↑(-1))⁻¹ * ReducedWord.prod φ w✝.toReducedWord))