Theorem: Turing.TM1to0.tr_supports
File: Mathlib/Computability/TuringMachine.lean

tactic: constructor
state_before: Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
⊢ TM0.Supports (tr M) ↑(trStmts M S)

tactic: apply Finset.mem_product.2
state_before: case left
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
⊢ default ∈ ↑(trStmts M S)

tactic: constructor
state_before: case left
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
⊢ default.fst ∈ TM1.stmts M S ∧ default.snd ∈ Finset.univ

tactic: simp only [default, TM1.stmts, Finset.mem_insertNone, Option.mem_def, Option.some_inj,
  forall_eq', Finset.mem_biUnion]
state_before: case left.left
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
⊢ default.fst ∈ TM1.stmts M S

tactic: exact ⟨_, ss.1, TM1.stmts₁_self⟩
state_before: case left.left
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
⊢ ∃ a, a ∈ S ∧ M default ∈ TM1.stmts₁ (M a)

tactic: apply Finset.mem_univ
state_before: case left.right
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
⊢ default.snd ∈ Finset.univ

tactic: intro q a q' s h₁ h₂
state_before: case right
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
⊢ ∀ {q : Λ'₁₀} {a : Γ} {q' : Λ'₁₀} {s : Stmt₀}, (q', s) ∈ tr M q a → q ∈ ↑(trStmts M S) → q' ∈ ↑(trStmts M S)

tactic: rcases q with ⟨_ | q, v⟩
state_before: case right
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
q : Λ'₁₀
a : Γ
q' : Λ'₁₀
s : Stmt₀
h₁ : (q', s) ∈ tr M q a
h₂ : q ∈ ↑(trStmts M S)
⊢ q' ∈ ↑(trStmts M S)

tactic: cases' q' with q' v'
state_before: case right.mk.some
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
q' : Λ'₁₀
s : Stmt₀
v : σ
q : Stmt₁
h₁ : (q', s) ∈ tr M (some q, v) a
h₂ : (some q, v) ∈ ↑(trStmts M S)
⊢ q' ∈ ↑(trStmts M S)

tactic: simp only [trStmts, Finset.mem_coe] at h₂⊢
state_before: case right.mk.some.mk
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v : σ
q : Stmt₁
h₂ : (some q, v) ∈ ↑(trStmts M S)
q' : Option Stmt₁
v' : σ
h₁ : ((q', v'), s) ∈ tr M (some q, v) a
⊢ (q', v') ∈ ↑(trStmts M S)

tactic: rw [Finset.mem_product] at h₂⊢
state_before: case right.mk.some.mk
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v : σ
q : Stmt₁
q' : Option Stmt₁
v' : σ
h₁ : ((q', v'), s) ∈ tr M (some q, v) a
h₂ : (some q, v) ∈ TM1.stmts M S ×ˢ Finset.univ
⊢ (q', v') ∈ TM1.stmts M S ×ˢ Finset.univ

tactic: simp only [Finset.mem_univ, and_true_iff] at h₂⊢
state_before: case right.mk.some.mk
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v : σ
q : Stmt₁
q' : Option Stmt₁
v' : σ
h₁ : ((q', v'), s) ∈ tr M (some q, v) a
h₂ : (some q, v).fst ∈ TM1.stmts M S ∧ (some q, v).snd ∈ Finset.univ
⊢ (q', v').fst ∈ TM1.stmts M S ∧ (q', v').snd ∈ Finset.univ

tactic: cases q'
state_before: case right.mk.some.mk
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v : σ
q : Stmt₁
q' : Option Stmt₁
v' : σ
h₁ : ((q', v'), s) ∈ tr M (some q, v) a
h₂ : some q ∈ TM1.stmts M S
⊢ q' ∈ TM1.stmts M S

tactic: simp only [tr, Option.mem_def] at h₁
state_before: case right.mk.some.mk.some
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v : σ
q : Stmt₁
v' : σ
h₂ : some q ∈ TM1.stmts M S
val✝ : Stmt₁
h₁ : ((some val✝, v'), s) ∈ tr M (some q, v) a
⊢ some val✝ ∈ TM1.stmts M S

tactic: have := TM1.stmts_supportsStmt ss h₂
state_before: case right.mk.some.mk.some
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v : σ
q : Stmt₁
v' : σ
h₂ : some q ∈ TM1.stmts M S
val✝ : Stmt₁
h₁ : some (trAux M a q v) = some ((some val✝, v'), s)
⊢ some val✝ ∈ TM1.stmts M S

tactic: revert this
state_before: case right.mk.some.mk.some
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v : σ
q : Stmt₁
v' : σ
h₂ : some q ∈ TM1.stmts M S
val✝ : Stmt₁
h₁ : some (trAux M a q v) = some ((some val✝, v'), s)
this : TM1.SupportsStmt S q
⊢ some val✝ ∈ TM1.stmts M S

tactic: induction q generalizing v <;> intro hs
state_before: case right.mk.some.mk.some
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v : σ
q : Stmt₁
v' : σ
h₂ : some q ∈ TM1.stmts M S
val✝ : Stmt₁
h₁ : some (trAux M a q v) = some ((some val✝, v'), s)
⊢ TM1.SupportsStmt S q → some val✝ ∈ TM1.stmts M S

tactic: case move d q =>
  cases h₁; refine' TM1.stmts_trans _ h₂
  unfold TM1.stmts₁
  exact Finset.mem_insert_of_mem TM1.stmts₁_self
state_before: case right.mk.some.mk.some.move
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
a✝¹ : Dir
a✝ : Stmt₁
a_ih✝ :
  ∀ (v : σ),
    some a✝ ∈ TM1.stmts M S →
      some (trAux M a a✝ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S a✝ → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.move a✝¹ a✝) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.move a✝¹ a✝) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.move a✝¹ a✝)
⊢ some val✝ ∈ TM1.stmts M S

case right.mk.some.mk.some.write
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
a✝¹ : Γ → σ → Γ
a✝ : Stmt₁
a_ih✝ :
  ∀ (v : σ),
    some a✝ ∈ TM1.stmts M S →
      some (trAux M a a✝ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S a✝ → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.write a✝¹ a✝) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.write a✝¹ a✝) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.write a✝¹ a✝)
⊢ some val✝ ∈ TM1.stmts M S

case right.mk.some.mk.some.load
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
a✝¹ : Γ → σ → σ
a✝ : Stmt₁
a_ih✝ :
  ∀ (v : σ),
    some a✝ ∈ TM1.stmts M S →
      some (trAux M a a✝ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S a✝ → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.load a✝¹ a✝) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.load a✝¹ a✝) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.load a✝¹ a✝)
⊢ some val✝ ∈ TM1.stmts M S

case right.mk.some.mk.some.branch
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
a✝² : Γ → σ → Bool
a✝¹ a✝ : Stmt₁
a_ih✝¹ :
  ∀ (v : σ),
    some a✝¹ ∈ TM1.stmts M S →
      some (trAux M a a✝¹ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S a✝¹ → some val✝ ∈ TM1.stmts M S
a_ih✝ :
  ∀ (v : σ),
    some a✝ ∈ TM1.stmts M S →
      some (trAux M a a✝ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S a✝ → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.branch a✝² a✝¹ a✝) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.branch a✝² a✝¹ a✝) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.branch a✝² a✝¹ a✝)
⊢ some val✝ ∈ TM1.stmts M S

case right.mk.some.mk.some.goto
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
a✝ : Γ → σ → Λ
v : σ
h₂ : some (TM1.Stmt.goto a✝) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.goto a✝) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.goto a✝)
⊢ some val✝ ∈ TM1.stmts M S

case right.mk.some.mk.some.halt
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
v : σ
h₂ : some TM1.Stmt.halt ∈ TM1.stmts M S
h₁ : some (trAux M a TM1.Stmt.halt v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S TM1.Stmt.halt
⊢ some val✝ ∈ TM1.stmts M S

tactic: case write b q =>
  cases h₁; refine' TM1.stmts_trans _ h₂
  unfold TM1.stmts₁
  exact Finset.mem_insert_of_mem TM1.stmts₁_self
state_before: case right.mk.some.mk.some.write
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
a✝¹ : Γ → σ → Γ
a✝ : Stmt₁
a_ih✝ :
  ∀ (v : σ),
    some a✝ ∈ TM1.stmts M S →
      some (trAux M a a✝ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S a✝ → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.write a✝¹ a✝) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.write a✝¹ a✝) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.write a✝¹ a✝)
⊢ some val✝ ∈ TM1.stmts M S

case right.mk.some.mk.some.load
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
a✝¹ : Γ → σ → σ
a✝ : Stmt₁
a_ih✝ :
  ∀ (v : σ),
    some a✝ ∈ TM1.stmts M S →
      some (trAux M a a✝ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S a✝ → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.load a✝¹ a✝) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.load a✝¹ a✝) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.load a✝¹ a✝)
⊢ some val✝ ∈ TM1.stmts M S

case right.mk.some.mk.some.branch
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
a✝² : Γ → σ → Bool
a✝¹ a✝ : Stmt₁
a_ih✝¹ :
  ∀ (v : σ),
    some a✝¹ ∈ TM1.stmts M S →
      some (trAux M a a✝¹ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S a✝¹ → some val✝ ∈ TM1.stmts M S
a_ih✝ :
  ∀ (v : σ),
    some a✝ ∈ TM1.stmts M S →
      some (trAux M a a✝ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S a✝ → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.branch a✝² a✝¹ a✝) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.branch a✝² a✝¹ a✝) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.branch a✝² a✝¹ a✝)
⊢ some val✝ ∈ TM1.stmts M S

case right.mk.some.mk.some.goto
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
a✝ : Γ → σ → Λ
v : σ
h₂ : some (TM1.Stmt.goto a✝) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.goto a✝) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.goto a✝)
⊢ some val✝ ∈ TM1.stmts M S

case right.mk.some.mk.some.halt
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
v : σ
h₂ : some TM1.Stmt.halt ∈ TM1.stmts M S
h₁ : some (trAux M a TM1.Stmt.halt v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S TM1.Stmt.halt
⊢ some val✝ ∈ TM1.stmts M S

tactic: case load b q IH =>
  refine' IH _ (TM1.stmts_trans _ h₂) h₁ hs
  unfold TM1.stmts₁
  exact Finset.mem_insert_of_mem TM1.stmts₁_self
state_before: case right.mk.some.mk.some.load
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
a✝¹ : Γ → σ → σ
a✝ : Stmt₁
a_ih✝ :
  ∀ (v : σ),
    some a✝ ∈ TM1.stmts M S →
      some (trAux M a a✝ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S a✝ → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.load a✝¹ a✝) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.load a✝¹ a✝) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.load a✝¹ a✝)
⊢ some val✝ ∈ TM1.stmts M S

case right.mk.some.mk.some.branch
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
a✝² : Γ → σ → Bool
a✝¹ a✝ : Stmt₁
a_ih✝¹ :
  ∀ (v : σ),
    some a✝¹ ∈ TM1.stmts M S →
      some (trAux M a a✝¹ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S a✝¹ → some val✝ ∈ TM1.stmts M S
a_ih✝ :
  ∀ (v : σ),
    some a✝ ∈ TM1.stmts M S →
      some (trAux M a a✝ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S a✝ → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.branch a✝² a✝¹ a✝) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.branch a✝² a✝¹ a✝) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.branch a✝² a✝¹ a✝)
⊢ some val✝ ∈ TM1.stmts M S

case right.mk.some.mk.some.goto
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
a✝ : Γ → σ → Λ
v : σ
h₂ : some (TM1.Stmt.goto a✝) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.goto a✝) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.goto a✝)
⊢ some val✝ ∈ TM1.stmts M S

case right.mk.some.mk.some.halt
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
v : σ
h₂ : some TM1.Stmt.halt ∈ TM1.stmts M S
h₁ : some (trAux M a TM1.Stmt.halt v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S TM1.Stmt.halt
⊢ some val✝ ∈ TM1.stmts M S

tactic: case goto l =>
  cases h₁
  exact Finset.some_mem_insertNone.2 (Finset.mem_biUnion.2 ⟨_, hs _ _, TM1.stmts₁_self⟩)
state_before: case right.mk.some.mk.some.goto
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
a✝ : Γ → σ → Λ
v : σ
h₂ : some (TM1.Stmt.goto a✝) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.goto a✝) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.goto a✝)
⊢ some val✝ ∈ TM1.stmts M S

case right.mk.some.mk.some.halt
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
v : σ
h₂ : some TM1.Stmt.halt ∈ TM1.stmts M S
h₁ : some (trAux M a TM1.Stmt.halt v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S TM1.Stmt.halt
⊢ some val✝ ∈ TM1.stmts M S

tactic: case halt => cases h₁
state_before: case right.mk.some.mk.some.halt
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
v : σ
h₂ : some TM1.Stmt.halt ∈ TM1.stmts M S
h₁ : some (trAux M a TM1.Stmt.halt v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S TM1.Stmt.halt
⊢ some val✝ ∈ TM1.stmts M S

tactic: cases h₁
state_before: case right.mk.none
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
q' : Λ'₁₀
s : Stmt₀
v : σ
h₁ : (q', s) ∈ tr M (none, v) a
h₂ : (none, v) ∈ ↑(trStmts M S)
⊢ q' ∈ ↑(trStmts M S)

tactic: exact Multiset.mem_cons_self _ _
state_before: case right.mk.some.mk.none
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v : σ
q : Stmt₁
v' : σ
h₂ : some q ∈ TM1.stmts M S
h₁ : ((none, v'), s) ∈ tr M (some q, v) a
⊢ none ∈ TM1.stmts M S

tactic: cases h₁
state_before: Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
a✝ : Dir
d : Stmt₁
q :
  ∀ (v : σ),
    some d ∈ TM1.stmts M S →
      some (trAux M a d v) = some ((some val✝, v'), s) → TM1.SupportsStmt S d → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.move a✝ d) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.move a✝ d) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.move a✝ d)
⊢ some val✝ ∈ TM1.stmts M S

tactic: refine' TM1.stmts_trans _ h₂
state_before: case refl
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
v' : σ
val✝ : Stmt₁
a✝ : Dir
h₂ : some (TM1.Stmt.move a✝ val✝) ∈ TM1.stmts M S
hs : TM1.SupportsStmt S (TM1.Stmt.move a✝ val✝)
q :
  ∀ (v : σ),
    some val✝ ∈ TM1.stmts M S →
      some (trAux M a val✝ v) = some ((some val✝, v'), move a✝) → TM1.SupportsStmt S val✝ → some val✝ ∈ TM1.stmts M S
⊢ some val✝ ∈ TM1.stmts M S

tactic: unfold TM1.stmts₁
state_before: case refl
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
v' : σ
val✝ : Stmt₁
a✝ : Dir
h₂ : some (TM1.Stmt.move a✝ val✝) ∈ TM1.stmts M S
hs : TM1.SupportsStmt S (TM1.Stmt.move a✝ val✝)
q :
  ∀ (v : σ),
    some val✝ ∈ TM1.stmts M S →
      some (trAux M a val✝ v) = some ((some val✝, v'), move a✝) → TM1.SupportsStmt S val✝ → some val✝ ∈ TM1.stmts M S
⊢ val✝ ∈ TM1.stmts₁ (TM1.Stmt.move a✝ val✝)

tactic: exact Finset.mem_insert_of_mem TM1.stmts₁_self
state_before: case refl
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
v' : σ
val✝ : Stmt₁
a✝ : Dir
h₂ : some (TM1.Stmt.move a✝ val✝) ∈ TM1.stmts M S
hs : TM1.SupportsStmt S (TM1.Stmt.move a✝ val✝)
q :
  ∀ (v : σ),
    some val✝ ∈ TM1.stmts M S →
      some (trAux M a val✝ v) = some ((some val✝, v'), move a✝) → TM1.SupportsStmt S val✝ → some val✝ ∈ TM1.stmts M S
⊢ val✝ ∈ insert (TM1.Stmt.move a✝ val✝) (TM1.stmts₁ val✝)

tactic: cases h₁
state_before: Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
a✝ : Γ → σ → Γ
b : Stmt₁
q :
  ∀ (v : σ),
    some b ∈ TM1.stmts M S →
      some (trAux M a b v) = some ((some val✝, v'), s) → TM1.SupportsStmt S b → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.write a✝ b) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.write a✝ b) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.write a✝ b)
⊢ some val✝ ∈ TM1.stmts M S

tactic: refine' TM1.stmts_trans _ h₂
state_before: case refl
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
v' : σ
val✝ : Stmt₁
a✝ : Γ → σ → Γ
h₂ : some (TM1.Stmt.write a✝ val✝) ∈ TM1.stmts M S
hs : TM1.SupportsStmt S (TM1.Stmt.write a✝ val✝)
q :
  ∀ (v : σ),
    some val✝ ∈ TM1.stmts M S →
      some (trAux M a val✝ v) = some ((some val✝, v'), write (a✝ a v')) →
        TM1.SupportsStmt S val✝ → some val✝ ∈ TM1.stmts M S
⊢ some val✝ ∈ TM1.stmts M S

tactic: unfold TM1.stmts₁
state_before: case refl
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
v' : σ
val✝ : Stmt₁
a✝ : Γ → σ → Γ
h₂ : some (TM1.Stmt.write a✝ val✝) ∈ TM1.stmts M S
hs : TM1.SupportsStmt S (TM1.Stmt.write a✝ val✝)
q :
  ∀ (v : σ),
    some val✝ ∈ TM1.stmts M S →
      some (trAux M a val✝ v) = some ((some val✝, v'), write (a✝ a v')) →
        TM1.SupportsStmt S val✝ → some val✝ ∈ TM1.stmts M S
⊢ val✝ ∈ TM1.stmts₁ (TM1.Stmt.write a✝ val✝)

tactic: exact Finset.mem_insert_of_mem TM1.stmts₁_self
state_before: case refl
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
v' : σ
val✝ : Stmt₁
a✝ : Γ → σ → Γ
h₂ : some (TM1.Stmt.write a✝ val✝) ∈ TM1.stmts M S
hs : TM1.SupportsStmt S (TM1.Stmt.write a✝ val✝)
q :
  ∀ (v : σ),
    some val✝ ∈ TM1.stmts M S →
      some (trAux M a val✝ v) = some ((some val✝, v'), write (a✝ a v')) →
        TM1.SupportsStmt S val✝ → some val✝ ∈ TM1.stmts M S
⊢ val✝ ∈ insert (TM1.Stmt.write a✝ val✝) (TM1.stmts₁ val✝)

tactic: refine' IH _ (TM1.stmts_trans _ h₂) h₁ hs
state_before: Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
b : Γ → σ → σ
q : Stmt₁
IH :
  ∀ (v : σ),
    some q ∈ TM1.stmts M S →
      some (trAux M a q v) = some ((some val✝, v'), s) → TM1.SupportsStmt S q → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.load b q) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.load b q) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.load b q)
⊢ some val✝ ∈ TM1.stmts M S

tactic: unfold TM1.stmts₁
state_before: Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
b : Γ → σ → σ
q : Stmt₁
IH :
  ∀ (v : σ),
    some q ∈ TM1.stmts M S →
      some (trAux M a q v) = some ((some val✝, v'), s) → TM1.SupportsStmt S q → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.load b q) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.load b q) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.load b q)
⊢ q ∈ TM1.stmts₁ (TM1.Stmt.load b q)

tactic: exact Finset.mem_insert_of_mem TM1.stmts₁_self
state_before: Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
b : Γ → σ → σ
q : Stmt₁
IH :
  ∀ (v : σ),
    some q ∈ TM1.stmts M S →
      some (trAux M a q v) = some ((some val✝, v'), s) → TM1.SupportsStmt S q → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.load b q) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.load b q) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.load b q)
⊢ q ∈ insert (TM1.Stmt.load b q) (TM1.stmts₁ q)

tactic: cases h : p a v <;> rw [trAux, h] at h₁
state_before: Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
p : Γ → σ → Bool
q₁ q₂ : Stmt₁
IH₁ :
  ∀ (v : σ),
    some q₁ ∈ TM1.stmts M S →
      some (trAux M a q₁ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S q₁ → some val✝ ∈ TM1.stmts M S
IH₂ :
  ∀ (v : σ),
    some q₂ ∈ TM1.stmts M S →
      some (trAux M a q₂ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S q₂ → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.branch p q₁ q₂) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.branch p q₁ q₂) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.branch p q₁ q₂)
⊢ some val✝ ∈ TM1.stmts M S

tactic: refine' IH₂ _ (TM1.stmts_trans _ h₂) h₁ hs.2
state_before: case false
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
p : Γ → σ → Bool
q₁ q₂ : Stmt₁
IH₁ :
  ∀ (v : σ),
    some q₁ ∈ TM1.stmts M S →
      some (trAux M a q₁ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S q₁ → some val✝ ∈ TM1.stmts M S
IH₂ :
  ∀ (v : σ),
    some q₂ ∈ TM1.stmts M S →
      some (trAux M a q₂ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S q₂ → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.branch p q₁ q₂) ∈ TM1.stmts M S
h₁ : some (bif false then trAux M a q₁ v else trAux M a q₂ v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.branch p q₁ q₂)
h : p a v = false
⊢ some val✝ ∈ TM1.stmts M S

tactic: unfold TM1.stmts₁
state_before: case false
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
p : Γ → σ → Bool
q₁ q₂ : Stmt₁
IH₁ :
  ∀ (v : σ),
    some q₁ ∈ TM1.stmts M S →
      some (trAux M a q₁ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S q₁ → some val✝ ∈ TM1.stmts M S
IH₂ :
  ∀ (v : σ),
    some q₂ ∈ TM1.stmts M S →
      some (trAux M a q₂ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S q₂ → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.branch p q₁ q₂) ∈ TM1.stmts M S
h₁ : some (bif false then trAux M a q₁ v else trAux M a q₂ v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.branch p q₁ q₂)
h : p a v = false
⊢ q₂ ∈ TM1.stmts₁ (TM1.Stmt.branch p q₁ q₂)

tactic: exact Finset.mem_insert_of_mem (Finset.mem_union_right _ TM1.stmts₁_self)
state_before: case false
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
p : Γ → σ → Bool
q₁ q₂ : Stmt₁
IH₁ :
  ∀ (v : σ),
    some q₁ ∈ TM1.stmts M S →
      some (trAux M a q₁ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S q₁ → some val✝ ∈ TM1.stmts M S
IH₂ :
  ∀ (v : σ),
    some q₂ ∈ TM1.stmts M S →
      some (trAux M a q₂ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S q₂ → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.branch p q₁ q₂) ∈ TM1.stmts M S
h₁ : some (bif false then trAux M a q₁ v else trAux M a q₂ v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.branch p q₁ q₂)
h : p a v = false
⊢ q₂ ∈ insert (TM1.Stmt.branch p q₁ q₂) (TM1.stmts₁ q₁ ∪ TM1.stmts₁ q₂)

tactic: refine' IH₁ _ (TM1.stmts_trans _ h₂) h₁ hs.1
state_before: case true
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
p : Γ → σ → Bool
q₁ q₂ : Stmt₁
IH₁ :
  ∀ (v : σ),
    some q₁ ∈ TM1.stmts M S →
      some (trAux M a q₁ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S q₁ → some val✝ ∈ TM1.stmts M S
IH₂ :
  ∀ (v : σ),
    some q₂ ∈ TM1.stmts M S →
      some (trAux M a q₂ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S q₂ → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.branch p q₁ q₂) ∈ TM1.stmts M S
h₁ : some (bif true then trAux M a q₁ v else trAux M a q₂ v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.branch p q₁ q₂)
h : p a v = true
⊢ some val✝ ∈ TM1.stmts M S

tactic: unfold TM1.stmts₁
state_before: case true
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
p : Γ → σ → Bool
q₁ q₂ : Stmt₁
IH₁ :
  ∀ (v : σ),
    some q₁ ∈ TM1.stmts M S →
      some (trAux M a q₁ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S q₁ → some val✝ ∈ TM1.stmts M S
IH₂ :
  ∀ (v : σ),
    some q₂ ∈ TM1.stmts M S →
      some (trAux M a q₂ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S q₂ → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.branch p q₁ q₂) ∈ TM1.stmts M S
h₁ : some (bif true then trAux M a q₁ v else trAux M a q₂ v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.branch p q₁ q₂)
h : p a v = true
⊢ q₁ ∈ TM1.stmts₁ (TM1.Stmt.branch p q₁ q₂)

tactic: exact Finset.mem_insert_of_mem (Finset.mem_union_left _ TM1.stmts₁_self)
state_before: case true
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
p : Γ → σ → Bool
q₁ q₂ : Stmt₁
IH₁ :
  ∀ (v : σ),
    some q₁ ∈ TM1.stmts M S →
      some (trAux M a q₁ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S q₁ → some val✝ ∈ TM1.stmts M S
IH₂ :
  ∀ (v : σ),
    some q₂ ∈ TM1.stmts M S →
      some (trAux M a q₂ v) = some ((some val✝, v'), s) → TM1.SupportsStmt S q₂ → some val✝ ∈ TM1.stmts M S
v : σ
h₂ : some (TM1.Stmt.branch p q₁ q₂) ∈ TM1.stmts M S
h₁ : some (bif true then trAux M a q₁ v else trAux M a q₂ v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.branch p q₁ q₂)
h : p a v = true
⊢ q₁ ∈ insert (TM1.Stmt.branch p q₁ q₂) (TM1.stmts₁ q₁ ∪ TM1.stmts₁ q₂)

tactic: cases h₁
state_before: Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
l : Γ → σ → Λ
v : σ
h₂ : some (TM1.Stmt.goto l) ∈ TM1.stmts M S
h₁ : some (trAux M a (TM1.Stmt.goto l) v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S (TM1.Stmt.goto l)
⊢ some val✝ ∈ TM1.stmts M S

tactic: exact Finset.some_mem_insertNone.2 (Finset.mem_biUnion.2 ⟨_, hs _ _, TM1.stmts₁_self⟩)
state_before: case refl
Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
v' : σ
l : Γ → σ → Λ
h₂ : some (TM1.Stmt.goto l) ∈ TM1.stmts M S
hs : TM1.SupportsStmt S (TM1.Stmt.goto l)
⊢ some (M (l a v')) ∈ TM1.stmts M S

tactic: cases h₁
state_before: Γ : Type u_2
inst✝³ : Inhabited Γ
Λ : Type u_1
inst✝² : Inhabited Λ
σ : Type u_3
inst✝¹ : Inhabited σ
M : Λ → Stmt₁
inst✝ : Fintype σ
S : Finset Λ
ss : TM1.Supports M S
a : Γ
s : Stmt₀
v' : σ
val✝ : Stmt₁
v : σ
h₂ : some TM1.Stmt.halt ∈ TM1.stmts M S
h₁ : some (trAux M a TM1.Stmt.halt v) = some ((some val✝, v'), s)
hs : TM1.SupportsStmt S TM1.Stmt.halt
⊢ some val✝ ∈ TM1.stmts M S