l3lab/ntp-mathlib huggingface dataset:
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    

Proof state:
case load
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝ : σ → σ
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
h : q₁ ∈ stmts₁ (load a✝ q)
hs : SupportsStmt S (load a✝ q)
⊢ SupportsStmt S q₁

Next tactic:
simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    

Proof state:
case branch
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝² : σ → Bool
a✝¹ a✝ : Stmt₂
a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → SupportsStmt S a✝¹ → SupportsStmt S q₁
a_ih✝ : q₁ ∈ stmts₁ a✝ → SupportsStmt S a✝ → SupportsStmt S q₁
h : q₁ ∈ stmts₁ (branch a✝² a✝¹ a✝)
hs : SupportsStmt S (branch a✝² a✝¹ a✝)
⊢ SupportsStmt S q₁

Next tactic:
simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    

Proof state:
case goto
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝ : σ → Λ
h : q₁ ∈ stmts₁ (goto a✝)
hs : SupportsStmt S (goto a✝)
⊢ SupportsStmt S q₁

Next tactic:
simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    

Proof state:
case halt
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
h : q₁ ∈ stmts₁ halt
hs : SupportsStmt S halt
⊢ SupportsStmt S q₁

Next tactic:
simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  

Proof state:
case push
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
k✝ : K
a✝ : σ → Γ k✝
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ = push k✝ a✝ q ∨ q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁
case peek
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
k✝ : K
a✝ : σ → Option (Γ k✝) → σ
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ = peek k✝ a✝ q ∨ q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁
case pop
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
k✝ : K
a✝ : σ → Option (Γ k✝) → σ
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ = pop k✝ a✝ q ∨ q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁
case load
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝ : σ → σ
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁
case branch
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝² : σ → Bool
a✝¹ a✝ : Stmt₂
a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → SupportsStmt S a✝¹ → SupportsStmt S q₁
a_ih✝ : q₁ ∈ stmts₁ a✝ → SupportsStmt S a✝ → SupportsStmt S q₁
hs : SupportsStmt S a✝¹ ∧ SupportsStmt S a✝
h : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝
⊢ SupportsStmt S q₁
case goto
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝ : σ → Λ
hs : ∀ (v : σ), a✝ v ∈ S
h : q₁ = goto a✝
⊢ SupportsStmt S q₁
case halt
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
hs : True
h : q₁ = halt
⊢ SupportsStmt S q₁

Next tactic:
iterate 4 rcases h with (rfl | h) <;> [exact hs; exact IH h hs]
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  iterate 4 

Proof state:
case push
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
k✝ : K
a✝ : σ → Γ k✝
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ = push k✝ a✝ q ∨ q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁
case peek
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
k✝ : K
a✝ : σ → Option (Γ k✝) → σ
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ = peek k✝ a✝ q ∨ q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁
case pop
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
k✝ : K
a✝ : σ → Option (Γ k✝) → σ
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ = pop k✝ a✝ q ∨ q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁
case load
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝ : σ → σ
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁
case branch
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝² : σ → Bool
a✝¹ a✝ : Stmt₂
a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → SupportsStmt S a✝¹ → SupportsStmt S q₁
a_ih✝ : q₁ ∈ stmts₁ a✝ → SupportsStmt S a✝ → SupportsStmt S q₁
hs : SupportsStmt S a✝¹ ∧ SupportsStmt S a✝
h : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝
⊢ SupportsStmt S q₁
case goto
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝ : σ → Λ
hs : ∀ (v : σ), a✝ v ∈ S
h : q₁ = goto a✝
⊢ SupportsStmt S q₁
case halt
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
hs : True
h : q₁ = halt
⊢ SupportsStmt S q₁

Next tactic:
rcases h with (rfl | h) <;> [exact hs; exact IH h hs]
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  iterate 4 

Proof state:
case push
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
k✝ : K
a✝ : σ → Γ k✝
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ = push k✝ a✝ q ∨ q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁

Next tactic:
rcases h with (rfl | h)
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  iterate 4 rcases h with (rfl | h) <;> [

Proof state:
case push.inl
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
k✝ : K
a✝ : σ → Γ k✝
q : Stmt₂
hs : SupportsStmt S q
IH : push k✝ a✝ q ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S (push k✝ a✝ q)
⊢ SupportsStmt S (push k✝ a✝ q)

Next tactic:
exact hs
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  iterate 4 rcases h with (rfl | h) <;> [exact hs; 

Proof state:
case push.inr
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
k✝ : K
a✝ : σ → Γ k✝
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁

Next tactic:
exact IH h hs
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  iterate 4 

Proof state:
case peek
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
k✝ : K
a✝ : σ → Option (Γ k✝) → σ
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ = peek k✝ a✝ q ∨ q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁
case pop
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
k✝ : K
a✝ : σ → Option (Γ k✝) → σ
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ = pop k✝ a✝ q ∨ q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁
case load
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝ : σ → σ
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁
case branch
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝² : σ → Bool
a✝¹ a✝ : Stmt₂
a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → SupportsStmt S a✝¹ → SupportsStmt S q₁
a_ih✝ : q₁ ∈ stmts₁ a✝ → SupportsStmt S a✝ → SupportsStmt S q₁
hs : SupportsStmt S a✝¹ ∧ SupportsStmt S a✝
h : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝
⊢ SupportsStmt S q₁
case goto
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝ : σ → Λ
hs : ∀ (v : σ), a✝ v ∈ S
h : q₁ = goto a✝
⊢ SupportsStmt S q₁
case halt
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
hs : True
h : q₁ = halt
⊢ SupportsStmt S q₁

Next tactic:
rcases h with (rfl | h) <;> [exact hs; exact IH h hs]
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  iterate 4 

Proof state:
case peek
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
k✝ : K
a✝ : σ → Option (Γ k✝) → σ
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ = peek k✝ a✝ q ∨ q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁

Next tactic:
rcases h with (rfl | h)
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  iterate 4 rcases h with (rfl | h) <;> [

Proof state:
case peek.inl
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
k✝ : K
a✝ : σ → Option (Γ k✝) → σ
q : Stmt₂
hs : SupportsStmt S q
IH : peek k✝ a✝ q ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S (peek k✝ a✝ q)
⊢ SupportsStmt S (peek k✝ a✝ q)

Next tactic:
exact hs
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  iterate 4 rcases h with (rfl | h) <;> [exact hs; 

Proof state:
case peek.inr
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
k✝ : K
a✝ : σ → Option (Γ k✝) → σ
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁

Next tactic:
exact IH h hs
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  iterate 4 

Proof state:
case pop
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
k✝ : K
a✝ : σ → Option (Γ k✝) → σ
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ = pop k✝ a✝ q ∨ q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁
case load
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝ : σ → σ
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁
case branch
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝² : σ → Bool
a✝¹ a✝ : Stmt₂
a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → SupportsStmt S a✝¹ → SupportsStmt S q₁
a_ih✝ : q₁ ∈ stmts₁ a✝ → SupportsStmt S a✝ → SupportsStmt S q₁
hs : SupportsStmt S a✝¹ ∧ SupportsStmt S a✝
h : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝
⊢ SupportsStmt S q₁
case goto
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝ : σ → Λ
hs : ∀ (v : σ), a✝ v ∈ S
h : q₁ = goto a✝
⊢ SupportsStmt S q₁
case halt
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
hs : True
h : q₁ = halt
⊢ SupportsStmt S q₁

Next tactic:
rcases h with (rfl | h) <;> [exact hs; exact IH h hs]
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  iterate 4 

Proof state:
case pop
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
k✝ : K
a✝ : σ → Option (Γ k✝) → σ
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ = pop k✝ a✝ q ∨ q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁

Next tactic:
rcases h with (rfl | h)
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  iterate 4 rcases h with (rfl | h) <;> [

Proof state:
case pop.inl
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
k✝ : K
a✝ : σ → Option (Γ k✝) → σ
q : Stmt₂
hs : SupportsStmt S q
IH : pop k✝ a✝ q ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S (pop k✝ a✝ q)
⊢ SupportsStmt S (pop k✝ a✝ q)

Next tactic:
exact hs
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  iterate 4 rcases h with (rfl | h) <;> [exact hs; 

Proof state:
case pop.inr
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
k✝ : K
a✝ : σ → Option (Γ k✝) → σ
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁

Next tactic:
exact IH h hs
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  iterate 4 

Proof state:
case load
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝ : σ → σ
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁
case branch
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝² : σ → Bool
a✝¹ a✝ : Stmt₂
a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → SupportsStmt S a✝¹ → SupportsStmt S q₁
a_ih✝ : q₁ ∈ stmts₁ a✝ → SupportsStmt S a✝ → SupportsStmt S q₁
hs : SupportsStmt S a✝¹ ∧ SupportsStmt S a✝
h : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝
⊢ SupportsStmt S q₁
case goto
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝ : σ → Λ
hs : ∀ (v : σ), a✝ v ∈ S
h : q₁ = goto a✝
⊢ SupportsStmt S q₁
case halt
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
hs : True
h : q₁ = halt
⊢ SupportsStmt S q₁

Next tactic:
rcases h with (rfl | h) <;> [exact hs; exact IH h hs]
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  iterate 4 

Proof state:
case load
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝ : σ → σ
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁

Next tactic:
rcases h with (rfl | h)
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  iterate 4 rcases h with (rfl | h) <;> [

Proof state:
case load.inl
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
a✝ : σ → σ
q : Stmt₂
hs : SupportsStmt S q
IH : load a✝ q ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S (load a✝ q)
⊢ SupportsStmt S (load a✝ q)

Next tactic:
exact hs
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  iterate 4 rcases h with (rfl | h) <;> [exact hs; 

Proof state:
case load.inr
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝ : σ → σ
q : Stmt₂
IH : q₁ ∈ stmts₁ q → SupportsStmt S q → SupportsStmt S q₁
hs : SupportsStmt S q
h : q₁ ∈ stmts₁ q
⊢ SupportsStmt S q₁

Next tactic:
exact IH h hs
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  iterate 4 rcases h with (rfl | h) <;> [exact hs; exact IH h hs]
  

Proof state:
case branch
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝² : σ → Bool
a✝¹ a✝ : Stmt₂
a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → SupportsStmt S a✝¹ → SupportsStmt S q₁
a_ih✝ : q₁ ∈ stmts₁ a✝ → SupportsStmt S a✝ → SupportsStmt S q₁
hs : SupportsStmt S a✝¹ ∧ SupportsStmt S a✝
h : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝
⊢ SupportsStmt S q₁
case goto
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
a✝ : σ → Λ
hs : ∀ (v : σ), a✝ v ∈ S
h : q₁ = goto a✝
⊢ SupportsStmt S q₁
case halt
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁ : Stmt₂
hs : True
h : q₁ = halt
⊢ SupportsStmt S q₁

Next tactic:
case branch f q₁ q₂ IH₁ IH₂ => rcases h with (rfl | h | h); exacts [hs, IH₁ h hs.1, IH₂ h hs.2]
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  iterate 4 rcases h with (rfl | h) <;> [exact hs; exact IH h hs]
  

Proof state:
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁✝ : Stmt₂
f : σ → Bool
q₁ q₂ : Stmt₂
IH₁ : q₁✝ ∈ stmts₁ q₁ → SupportsStmt S q₁ → SupportsStmt S q₁✝
IH₂ : q₁✝ ∈ stmts₁ q₂ → SupportsStmt S q₂ → SupportsStmt S q₁✝
hs : SupportsStmt S q₁ ∧ SupportsStmt S q₂
h : q₁✝ = branch f q₁ q₂ ∨ q₁✝ ∈ stmts₁ q₁ ∨ q₁✝ ∈ stmts₁ q₂
⊢ SupportsStmt S q₁✝

Next tactic:
case branch f q₁ q₂ IH₁ IH₂ => rcases h with (rfl | h | h); exacts [hs, IH₁ h hs.1, IH₂ h hs.2]
theorem stmts₁_supportsStmt_mono {S : Finset Λ} {q₁ q₂ : Stmt₂} (h : q₁ ∈ stmts₁ q₂)
    (hs : SupportsStmt S q₂) : SupportsStmt S q₁ := by
  induction' q₂ with _ _ q IH _ _ q IH _ _ q IH _ q IH <;>
    simp only [stmts₁, SupportsStmt, Finset.mem_insert, Finset.mem_union, Finset.mem_singleton]
      at h hs
  iterate 4 rcases h with (rfl | h) <;> [exact hs; exact IH h hs]
  case branch f q₁ q₂ IH₁ IH₂ => 

Proof state:
K : Type u_1
inst✝ : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
σ : Type u_4
S : Finset Λ
q₁✝ : Stmt₂
f : σ → Bool
q₁ q₂ : Stmt₂
IH₁ : q₁✝ ∈ stmts₁ q₁ → SupportsStmt S q₁ → SupportsStmt S q₁✝
IH₂ : q₁✝ ∈ stmts₁ q₂ → SupportsStmt S q₂ → SupportsStmt S q₁✝
hs : SupportsStmt S q₁ ∧ SupportsStmt S q₂
h : q₁✝ = branch f q₁ q₂ ∨ q₁✝ ∈ stmts₁ q₁ ∨ q₁✝ ∈ stmts₁ q₂
⊢ SupportsStmt S q₁✝

Next tactic:
rcases h with (rfl | h | h)