theorem tr_supports {S : Finset Λ} (ss : TM1.Supports M S) :
    TM0.Supports (tr M) ↑(trStmts M S) := by
  constructor
  · apply Finset.mem_product.2
    constructor
    · simp only [default, TM1.stmts, Finset.mem_insertNone, Option.mem_def, Option.some_inj,
        forall_eq', Finset.mem_biUnion]
      exact ⟨_, ss.1, TM1.stmts₁_self⟩
    · apply Finset.mem_univ
  · intro q a q' s h₁ h₂
    rcases q with ⟨_ | q, v⟩; · cases h₁
    cases' q' with q' v'
    simp only [trStmts, Finset.mem_coe] at h₂ ⊢
    rw [Finset.mem_product] at h₂ ⊢
    simp only [Finset.mem_univ, and_true_iff] at h₂ ⊢
    cases q'; · exact Multiset.mem_cons_self _ _
    simp only [tr, Option.mem_def] at h₁
    have := TM1.stmts_supportsStmt ss h₂
    revert this; induction q generalizing v <;> intro hs
    case move d q =>
      cases h₁; refine' TM1.stmts_trans _ h₂
      unfold TM1.stmts₁
      exact Finset.mem_insert_of_mem TM1.stmts₁_self
    case write b q =>
      cases h₁; refine' TM1.stmts_trans _ h₂
      unfold TM1.stmts₁
      exact Finset.mem_insert_of_mem TM1.stmts₁_self
    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
    

Proof state:
case right.mk.some.mk.some.branch
Γ : Type u_1
inst✝³ : Inhabited Γ
Λ : Type u_2
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_1
inst✝³ : Inhabited Γ
Λ : Type u_2
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_1
inst✝³ : Inhabited Γ
Λ : Type u_2
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

Next tactic:
case branch p q₁ q₂ IH₁ IH₂ =>
      cases h : p a v <;> rw [trAux, h] at h₁
      · refine' IH₂ _ (TM1.stmts_trans _ h₂) h₁ hs.2
        unfold TM1.stmts₁
        exact Finset.mem_insert_of_mem (Finset.mem_union_right _ TM1.stmts₁_self)
      · refine' IH₁ _ (TM1.stmts_trans _ h₂) h₁ hs.1
        unfold TM1.stmts₁
        exact Finset.mem_insert_of_mem (Finset.mem_union_left _ TM1.stmts₁_self)