-- Mathlib/Computability/TMToPartrec.lean
theorem trStmts₁_supports {S q} (H₁ : (q : Λ').Supports S) (HS₁ : trStmts₁ q ⊆ S) :
    Supports (trStmts₁ q) S

Proof state:
S : Finset Λ'
q : Λ'
H₁ : Λ'.Supports S q
HS₁ : trStmts₁ q ⊆ S
W : ∀ {q : Λ'}, q ∈ trStmts₁ q
⊢ Supports (trStmts₁ q) S

Tactic:
induction q with
| move _ _ _ q q_ih => _ | clear _ _ q q_ih => _ | copy q q_ih => _ | push _ _ q q_ih => _
| read q q_ih => _ | succ q q_ih => _ | pred q₁ q₂ q₁_ih q₂_ih => _ | ret => _ <;>
  simp [trStmts₁, -Finset.singleton_subset_iff] at HS₁ ⊢

Result:
case move
S : Finset Λ'
W : ∀ {q : Λ'}, q ∈ trStmts₁ q
p✝ : Γ' → Bool
k₁✝ k₂✝ : K'
q : Λ'
q_ih : Λ'.Supports S q → trStmts₁ q ⊆ S → Supports (trStmts₁ q) S
H₁ : Λ'.Supports S (Λ'.move p✝ k₁✝ k₂✝ q)
HS₁ : insert (Λ'.move p✝ k₁✝ k₂✝ q) (trStmts₁ q) ⊆ S
⊢ Supports (insert (Λ'.move p✝ k₁✝ k₂✝ q) (trStmts₁ q)) S

case clear
S : Finset Λ'
W : ∀ {q : Λ'}, q ∈ trStmts₁ q
p✝ : Γ' → Bool
k✝ : K'
q : Λ'
q_ih : Λ'.Supports S q → trStmts₁ q ⊆ S → Supports (trStmts₁ q) S
H₁ : Λ'.Supports S (Λ'.clear p✝ k✝ q)
HS₁ : insert (Λ'.clear p✝ k✝ q) (trStmts₁ q) ⊆ S
⊢ Supports (insert (Λ'.clear p✝ k✝ q) (trStmts₁ q)) S

case copy
S : Finset Λ'
W : ∀ {q : Λ'}, q ∈ trStmts₁ q
q : Λ'
q_ih : Λ'.Supports S q → trStmts₁ q ⊆ S → Supports (trStmts₁ q) S
H₁ : Λ'.Supports S q.copy
HS₁ : insert q.copy (trStmts₁ q) ⊆ S
⊢ Supports (insert q.copy (trStmts₁ q)) S

case push
S : Finset Λ'
W : ∀ {q : Λ'}, q ∈ trStmts₁ q
k✝ : K'
s✝ : Option Γ' → Option Γ'
q : Λ'
q_ih : Λ'.Supports S q → trStmts₁ q ⊆ S → Supports (trStmts₁ q) S
H₁ : Λ'.Supports S (Λ'.push k✝ s✝ q)
HS₁ : insert (Λ'.push k✝ s✝ q) (trStmts₁ q) ⊆ S
⊢ Supports (insert (Λ'.push k✝ s✝ q) (trStmts₁ q)) S

case read
S : Finset Λ'
W : ∀ {q : Λ'}, q ∈ trStmts₁ q
q : Option Γ' → Λ'
q_ih : ∀ (a : Option Γ'), Λ'.Supports S (q a) → trStmts₁ (q a) ⊆ S → Supports (trStmts₁ (q a)) S
H₁ : Λ'.Supports S (Λ'.read q)
HS₁ : insert (Λ'.read q) (Finset.univ.biUnion fun s => trStmts₁ (q s)) ⊆ S
⊢ Supports (insert (Λ'.read q) (Finset.univ.biUnion fun s => trStmts₁ (q s))) S

case succ
S : Finset Λ'
W : ∀ {q : Λ'}, q ∈ trStmts₁ q
q : Λ'
q_ih : Λ'.Supports S q → trStmts₁ q ⊆ S → Supports (trStmts₁ q) S
H₁ : Λ'.Supports S q.succ
HS₁ : insert q.succ (insert (unrev q) (trStmts₁ q)) ⊆ S
⊢ Supports (insert q.succ (insert (unrev q) (trStmts₁ q))) S

case pred
S : Finset Λ'
W : ∀ {q : Λ'}, q ∈ trStmts₁ q
q₁ q₂ : Λ'
q₁_ih : Λ'.Supports S q₁ → trStmts₁ q₁ ⊆ S → Supports (trStmts₁ q₁) S
q₂_ih : Λ'.Supports S q₂ → trStmts₁ q₂ ⊆ S → Supports (trStmts₁ q₂) S
H₁ : Λ'.Supports S (q₁.pred q₂)
HS₁ : insert (q₁.pred q₂) (insert (unrev q₂) (trStmts₁ q₁ ∪ trStmts₁ q₂)) ⊆ S
⊢ Supports (insert (q₁.pred q₂) (insert (unrev q₂) (trStmts₁ q₁ ∪ trStmts₁ q₂))) S

case ret
S : Finset Λ'
W : ∀ {q : Λ'}, q ∈ trStmts₁ q
k✝ : Cont'
H₁ : Λ'.Supports S (Λ'.ret k✝)
HS₁ : {Λ'.ret k✝} ⊆ S
⊢ Supports {Λ'.ret k✝} S

Proof state:
case move.intro
S : Finset Λ'
W : ∀ {q : Λ'}, q ∈ trStmts₁ q
p✝ : Γ' → Bool
k₁✝ k₂✝ : K'
q : Λ'
q_ih : Λ'.Supports S q → trStmts₁ q ⊆ S → Supports (trStmts₁ q) S
H₁ : Λ'.Supports S (Λ'.move p✝ k₁✝ k₂✝ q)
HS₁ : insert (Λ'.move p✝ k₁✝ k₂✝ q) (trStmts₁ q) ⊆ S
h₁ : Λ'.move p✝ k₁✝ k₂✝ q ∈ S
h₂ : trStmts₁ q ⊆ S
h₃ : q ∈ S
⊢ Supports (insert (Λ'.move p✝ k₁✝ k₂✝ q) (trStmts₁ q)) S

Tactic:
exact supports_insert.2 ⟨⟨fun _ => h₃, fun _ => h₁⟩, q_ih H₁ h₂⟩

Result:
no goals

Proof state:
case pred.intro
S : Finset Λ'
W : ∀ {q : Λ'}, q ∈ trStmts₁ q
q₁ q₂ : Λ'
q₁_ih : Λ'.Supports S q₁ → trStmts₁ q₁ ⊆ S → Supports (trStmts₁ q₁) S
q₂_ih : Λ'.Supports S q₂ → trStmts₁ q₂ ⊆ S → Supports (trStmts₁ q₂) S
H₁ : Λ'.Supports S (q₁.pred q₂)
HS₁ : insert (q₁.pred q₂) (insert (unrev q₂) (trStmts₁ q₁ ∪ trStmts₁ q₂)) ⊆ S
h₁ : q₁.pred q₂ ∈ S
h₂ : insert (unrev q₂) (trStmts₁ q₁ ∪ trStmts₁ q₂) ⊆ S
⊢ Supports (insert (q₁.pred q₂) (insert (unrev q₂) (trStmts₁ q₁ ∪ trStmts₁ q₂))) S

Tactic:
simp [Finset.subset_iff] at h₂

Result:
case pred.intro
S : Finset Λ'
W : ∀ {q : Λ'}, q ∈ trStmts₁ q
q₁ q₂ : Λ'
q₁_ih : Λ'.Supports S q₁ → trStmts₁ q₁ ⊆ S → Supports (trStmts₁ q₁) S
q₂_ih : Λ'.Supports S q₂ → trStmts₁ q₂ ⊆ S → Supports (trStmts₁ q₂) S
H₁ : Λ'.Supports S (q₁.pred q₂)
HS₁ : insert (q₁.pred q₂) (insert (unrev q₂) (trStmts₁ q₁ ∪ trStmts₁ q₂)) ⊆ S
h₁ : q₁.pred q₂ ∈ S
h₂ : unrev q₂ ∈ S ∧ ∀ (a : Λ'), a ∈ trStmts₁ q₁ ∨ a ∈ trStmts₁ q₂ → a ∈ S
⊢ Supports (insert (q₁.pred q₂) (insert (unrev q₂) (trStmts₁ q₁ ∪ trStmts₁ q₂))) S

Proof state:
case read.intro
S : Finset Λ'
W : ∀ {q : Λ'}, q ∈ trStmts₁ q
q : Option Γ' → Λ'
q_ih : ∀ (a : Option Γ'), Λ'.Supports S (q a) → trStmts₁ (q a) ⊆ S → Supports (trStmts₁ (q a)) S
H₁ : Λ'.Supports S (Λ'.read q)
HS₁ : insert (Λ'.read q) (Finset.univ.biUnion fun s => trStmts₁ (q s)) ⊆ S
h₁ : Λ'.read q ∈ S
h₂ : ∀ ⦃x : Λ'⦄ (x_1 : Option Γ'), x ∈ trStmts₁ (q x_1) → x ∈ S
⊢ Supports (Finset.univ.biUnion fun s => trStmts₁ (q s)) S

Tactic:
exact supports_biUnion.2 fun _ => q_ih _ (H₁ _) fun _ h => h₂ _ h

Result:
no goals

Proof state:
case read.intro
S : Finset Λ'
W : ∀ {q : Λ'}, q ∈ trStmts₁ q
q : Option Γ' → Λ'
q_ih : ∀ (a : Option Γ'), Λ'.Supports S (q a) → trStmts₁ (q a) ⊆ S → Supports (trStmts₁ (q a)) S
H₁ : Λ'.Supports S (Λ'.read q)
HS₁ : insert (Λ'.read q) (Finset.univ.biUnion fun s => trStmts₁ (q s)) ⊆ S
h₁ : Λ'.read q ∈ S
h₂ : ∀ ⦃x : Λ'⦄ (x_1 : Option Γ'), x ∈ trStmts₁ (q x_1) → x ∈ S
⊢ Supports (insert (Λ'.read q) (Finset.univ.biUnion fun s => trStmts₁ (q s))) S

Tactic:
refine supports_insert.2 ⟨fun _ => h₂ _ W, ?_⟩

Result:
case read.intro
S : Finset Λ'
W : ∀ {q : Λ'}, q ∈ trStmts₁ q
q : Option Γ' → Λ'
q_ih : ∀ (a : Option Γ'), Λ'.Supports S (q a) → trStmts₁ (q a) ⊆ S → Supports (trStmts₁ (q a)) S
H₁ : Λ'.Supports S (Λ'.read q)
HS₁ : insert (Λ'.read q) (Finset.univ.biUnion fun s => trStmts₁ (q s)) ⊆ S
h₁ : Λ'.read q ∈ S
h₂ : ∀ ⦃x : Λ'⦄ (x_1 : Option Γ'), x ∈ trStmts₁ (q x_1) → x ∈ S
⊢ Supports (Finset.univ.biUnion fun s => trStmts₁ (q s)) S

Proof state:
case pred.intro.refine_2
S : Finset Λ'
W : ∀ {q : Λ'}, q ∈ trStmts₁ q
q₁ q₂ : Λ'
q₁_ih : Λ'.Supports S q₁ → trStmts₁ q₁ ⊆ S → Supports (trStmts₁ q₁) S
q₂_ih : Λ'.Supports S q₂ → trStmts₁ q₂ ⊆ S → Supports (trStmts₁ q₂) S
H₁ : Λ'.Supports S (q₁.pred q₂)
HS₁ : insert (q₁.pred q₂) (insert (unrev q₂) (trStmts₁ q₁ ∪ trStmts₁ q₂)) ⊆ S
h₁ : q₁.pred q₂ ∈ S
h₂ : unrev q₂ ∈ S ∧ ∀ (a : Λ'), a ∈ trStmts₁ q₁ ∨ a ∈ trStmts₁ q₂ → a ∈ S
⊢ Supports (trStmts₁ q₂) S

Tactic:
exact q₂_ih H₁.2 fun _ h => h₂.2 _ (Or.inr h)

Result:
no goals

Proof state:
case pred.intro.refine_1
S : Finset Λ'
W : ∀ {q : Λ'}, q ∈ trStmts₁ q
q₁ q₂ : Λ'
q₁_ih : Λ'.Supports S q₁ → trStmts₁ q₁ ⊆ S → Supports (trStmts₁ q₁) S
q₂_ih : Λ'.Supports S q₂ → trStmts₁ q₂ ⊆ S → Supports (trStmts₁ q₂) S
H₁ : Λ'.Supports S (q₁.pred q₂)
HS₁ : insert (q₁.pred q₂) (insert (unrev q₂) (trStmts₁ q₁ ∪ trStmts₁ q₂)) ⊆ S
h₁ : q₁.pred q₂ ∈ S
h₂ : unrev q₂ ∈ S ∧ ∀ (a : Λ'), a ∈ trStmts₁ q₁ ∨ a ∈ trStmts₁ q₂ → a ∈ S
⊢ Supports (trStmts₁ q₁) S

Tactic:
exact q₁_ih H₁.1 fun _ h => h₂.2 _ (Or.inl h)

Result:
no goals