Module: Mathlib.Computability.TuringMachine.StackTuringMachine

Goal state before:
pp: case refine_3
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
p✝ : σ → Bool
q₁✝ q₂✝ : TM2.Stmt Γ Λ σ
IH₁ :
  TM2.SupportsStmt S q₁✝ →
    (∀ x ∈ trStmts₁ q₁✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q₁✝) ∧ ∀ l' ∈ trStmts₁ q₁✝, TM1.SupportsStmt (trSupp M S) (tr M l')
IH₂ :
  TM2.SupportsStmt S q₂✝ →
    (∀ x ∈ trStmts₁ q₂✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q₂✝) ∧ ∀ l' ∈ trStmts₁ q₂✝, TM1.SupportsStmt (trSupp M S) (tr M l')
ss' : TM2.SupportsStmt S (TM2.Stmt.branch p✝ q₁✝ q₂✝)
sub : ∀ x ∈ trStmts₁ q₁✝ ∪ trStmts₁ q₂✝, x ∈ trSupp M S
IH₁₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₁✝)
IH₁₂ : ∀ l' ∈ trStmts₁ q₁✝, TM1.SupportsStmt (trSupp M S) (tr M l')
IH₂₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₂✝)
IH₂₂ : ∀ l' ∈ trStmts₁ q₂✝, TM1.SupportsStmt (trSupp M S) (tr M l')
l : Λ' K Γ Λ σ
h : l ∈ trStmts₁ q₁✝ ∪ trStmts₁ q₂✝
⊢ TM1.SupportsStmt (trSupp M S) (tr M l)

Tactic:
rcases Finset.mem_union.1 h with (h | h)

Goal state after:
case refine_3.inl
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
p✝ : σ → Bool
q₁✝ q₂✝ : TM2.Stmt Γ Λ σ
IH₁ :
  TM2.SupportsStmt S q₁✝ →
    (∀ x ∈ trStmts₁ q₁✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q₁✝) ∧ ∀ l' ∈ trStmts₁ q₁✝, TM1.SupportsStmt (trSupp M S) (tr M l')
IH₂ :
  TM2.SupportsStmt S q₂✝ →
    (∀ x ∈ trStmts₁ q₂✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q₂✝) ∧ ∀ l' ∈ trStmts₁ q₂✝, TM1.SupportsStmt (trSupp M S) (tr M l')
ss' : TM2.SupportsStmt S (TM2.Stmt.branch p✝ q₁✝ q₂✝)
sub : ∀ x ∈ trStmts₁ q₁✝ ∪ trStmts₁ q₂✝, x ∈ trSupp M S
IH₁₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₁✝)
IH₁₂ : ∀ l' ∈ trStmts₁ q₁✝, TM1.SupportsStmt (trSupp M S) (tr M l')
IH₂₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₂✝)
IH₂₂ : ∀ l' ∈ trStmts₁ q₂✝, TM1.SupportsStmt (trSupp M S) (tr M l')
l : Λ' K Γ Λ σ
h✝ : l ∈ trStmts₁ q₁✝ ∪ trStmts₁ q₂✝
h : l ∈ trStmts₁ q₁✝
⊢ TM1.SupportsStmt (trSupp M S) (tr M l)

case refine_3.inr
K : Type u_1
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
inst✝¹ : DecidableEq K
M : Λ → TM2.Stmt Γ Λ σ
inst✝ : Inhabited Λ
S : Finset Λ
ss : TM2.Supports M S
p✝ : σ → Bool
q₁✝ q₂✝ : TM2.Stmt Γ Λ σ
IH₁ :
  TM2.SupportsStmt S q₁✝ →
    (∀ x ∈ trStmts₁ q₁✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q₁✝) ∧ ∀ l' ∈ trStmts₁ q₁✝, TM1.SupportsStmt (trSupp M S) (tr M l')
IH₂ :
  TM2.SupportsStmt S q₂✝ →
    (∀ x ∈ trStmts₁ q₂✝, x ∈ trSupp M S) →
      TM1.SupportsStmt (trSupp M S) (trNormal q₂✝) ∧ ∀ l' ∈ trStmts₁ q₂✝, TM1.SupportsStmt (trSupp M S) (tr M l')
ss' : TM2.SupportsStmt S (TM2.Stmt.branch p✝ q₁✝ q₂✝)
sub : ∀ x ∈ trStmts₁ q₁✝ ∪ trStmts₁ q₂✝, x ∈ trSupp M S
IH₁₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₁✝)
IH₁₂ : ∀ l' ∈ trStmts₁ q₁✝, TM1.SupportsStmt (trSupp M S) (tr M l')
IH₂₁ : TM1.SupportsStmt (trSupp M S) (trNormal q₂✝)
IH₂₂ : ∀ l' ∈ trStmts₁ q₂✝, TM1.SupportsStmt (trSupp M S) (tr M l')
l : Λ' K Γ Λ σ
h✝ : l ∈ trStmts₁ q₁✝ ∪ trStmts₁ q₂✝
h : l ∈ trStmts₁ q₂✝
⊢ TM1.SupportsStmt (trSupp M S) (tr M l)