-- Module: Mathlib.Combinatorics.Tiling.Tile
Goal:
G : Type u_1
X : Type u_2
ιₚ : Type u_3
inst : Group G
inst_1 : MulAction G X
ps : Protoset G X ιₚ
pt₁ pt₂ : PlacedTile ps
h :
  pt₁.index = pt₂.index ∧
    Quotient.mk
          (QuotientGroup.leftRel
            (Subgroup.map (Subgroup.subtype (MulAction.stabilizer G (↑((↑ps : ιₚ → Prototile G X) pt₁.index) : Set X)))
              ((↑ps : ιₚ → Prototile G X) pt₁.index).symmetries)) ⁻¹'
        {pt₁.groupElts} =
      Quotient.mk
          (QuotientGroup.leftRel
            (Subgroup.map (Subgroup.subtype (MulAction.stabilizer G (↑((↑ps : ιₚ → Prototile G X) pt₂.index) : Set X)))
              ((↑ps : ιₚ → Prototile G X) pt₂.index).symmetries)) ⁻¹'
        {pt₂.groupElts}
hi : pt₁.index = pt₂.index
hq :
  Quotient.mk
        (QuotientGroup.leftRel
          (Subgroup.map (Subgroup.subtype (MulAction.stabilizer G (↑((↑ps : ιₚ → Prototile G X) pt₁.index) : Set X)))
            ((↑ps : ιₚ → Prototile G X) pt₁.index).symmetries)) ⁻¹'
      {pt₁.groupElts} =
    Quotient.mk
        (QuotientGroup.leftRel
          (Subgroup.map (Subgroup.subtype (MulAction.stabilizer G (↑((↑ps : ιₚ → Prototile G X) pt₂.index) : Set X)))
            ((↑ps : ιₚ → Prototile G X) pt₂.index).symmetries)) ⁻¹'
      {pt₂.groupElts}
refine_2 :
  ∀ (i₁ : ιₚ)
    (g₁ :
      G ⧸
        Subgroup.map (Subgroup.subtype (MulAction.stabilizer G (↑((↑ps : ιₚ → Prototile G X) i₁) : Set X)))
          ((↑ps : ιₚ → Prototile G X) i₁).symmetries),
    { index := i₁, groupElts := g₁ : PlacedTile ps }.index = pt₂.index ∧
        Quotient.mk
              (QuotientGroup.leftRel
                (Subgroup.map
                  (Subgroup.subtype
                    (MulAction.stabilizer G
                      (↑((↑ps : ιₚ → Prototile G X) { index := i₁, groupElts := g₁ : PlacedTile ps }.index) : Set X)))
                  ((↑ps : ιₚ → Prototile G X) { index := i₁, groupElts := g₁ : PlacedTile ps }.index).symmetries)) ⁻¹'
            {{ index := i₁, groupElts := g₁ : PlacedTile ps }.groupElts} =
          Quotient.mk
              (QuotientGroup.leftRel
                (Subgroup.map
                  (Subgroup.subtype (MulAction.stabilizer G (↑((↑ps : ιₚ → Prototile G X) pt₂.index) : Set X)))
                  ((↑ps : ιₚ → Prototile G X) pt₂.index).symmetries)) ⁻¹'
            {pt₂.groupElts} →
      ∀ (hi : { index := i₁, groupElts := g₁ : PlacedTile ps }.index = pt₂.index)
        (hq :
          Quotient.mk
                (QuotientGroup.leftRel
                  (Subgroup.map
                    (Subgroup.subtype
                      (MulAction.stabilizer G
                        (↑((↑ps : ιₚ → Prototile G X) { index := i₁, groupElts := g₁ : PlacedTile ps }.index) : Set X)))
                    ((↑ps : ιₚ → Prototile G X) { index := i₁, groupElts := g₁ : PlacedTile ps }.index).symmetries)) ⁻¹'
              {{ index := i₁, groupElts := g₁ : PlacedTile ps }.groupElts} =
            Quotient.mk
                (QuotientGroup.leftRel
                  (Subgroup.map
                    (Subgroup.subtype (MulAction.stabilizer G (↑((↑ps : ιₚ → Prototile G X) pt₂.index) : Set X)))
                    ((↑ps : ιₚ → Prototile G X) pt₂.index).symmetries)) ⁻¹'
              {pt₂.groupElts}),
        { index := i₁, groupElts := g₁ : PlacedTile ps } = pt₂
⊢ pt₁ = pt₂

Tactic:
rcases pt₁ with ⟨i₁, g₁⟩