l3lab/ntp-mathlib huggingface dataset:
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    

Proof state:
case branch
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝² : Γ → σ → Bool
a✝¹ a✝ : Stmt₁
a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → q₀ ∈ stmts₁ a✝¹
a_ih✝ : q₁ ∈ stmts₁ a✝ → q₀ ∈ stmts₁ a✝
h₁₂ : q₁ ∈ insert (branch a✝² a✝¹ a✝) (stmts₁ a✝¹ ∪ stmts₁ a✝)
⊢ q₀ ∈ insert (branch a✝² a✝¹ a✝) (stmts₁ a✝¹ ∪ stmts₁ a✝)

Next tactic:
simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    

Proof state:
case goto
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Γ → σ → Λ
h₁₂ : q₁ ∈ {goto a✝}
⊢ q₀ ∈ {goto a✝}

Next tactic:
simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    

Proof state:
case halt
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
h₁₂ : q₁ ∈ {halt}
⊢ q₀ ∈ {halt}

Next tactic:
simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  

Proof state:
case move
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Dir
q : Stmt₁
IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q
h₁₂ : q₁ = move a✝ q ∨ q₁ ∈ stmts₁ q
⊢ q₀ ∈ insert (move a✝ q) (stmts₁ q)
case write
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Γ → σ → Γ
q : Stmt₁
IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q
h₁₂ : q₁ = write a✝ q ∨ q₁ ∈ stmts₁ q
⊢ q₀ ∈ insert (write a✝ q) (stmts₁ q)
case load
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Γ → σ → σ
q : Stmt₁
IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q
h₁₂ : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q
⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q)
case branch
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝² : Γ → σ → Bool
a✝¹ a✝ : Stmt₁
a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → q₀ ∈ stmts₁ a✝¹
a_ih✝ : q₁ ∈ stmts₁ a✝ → q₀ ∈ stmts₁ a✝
h₁₂ : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝
⊢ q₀ ∈ insert (branch a✝² a✝¹ a✝) (stmts₁ a✝¹ ∪ stmts₁ a✝)
case goto
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Γ → σ → Λ
h₁₂ : q₁ = goto a✝
⊢ q₀ ∈ {goto a✝}
case halt
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
h₁₂ : q₁ = halt
⊢ q₀ ∈ {halt}

Next tactic:
iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (IH h₁₂)
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    

Proof state:
case move
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Dir
q : Stmt₁
IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q
h₁₂ : q₁ = move a✝ q ∨ q₁ ∈ stmts₁ q
⊢ q₀ ∈ insert (move a✝ q) (stmts₁ q)
case write
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Γ → σ → Γ
q : Stmt₁
IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q
h₁₂ : q₁ = write a✝ q ∨ q₁ ∈ stmts₁ q
⊢ q₀ ∈ insert (write a✝ q) (stmts₁ q)
case load
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Γ → σ → σ
q : Stmt₁
IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q
h₁₂ : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q
⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q)
case branch
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝² : Γ → σ → Bool
a✝¹ a✝ : Stmt₁
a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → q₀ ∈ stmts₁ a✝¹
a_ih✝ : q₁ ∈ stmts₁ a✝ → q₀ ∈ stmts₁ a✝
h₁₂ : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝
⊢ q₀ ∈ insert (branch a✝² a✝¹ a✝) (stmts₁ a✝¹ ∪ stmts₁ a✝)
case goto
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Γ → σ → Λ
h₁₂ : q₁ = goto a✝
⊢ q₀ ∈ {goto a✝}
case halt
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
h₁₂ : q₁ = halt
⊢ q₀ ∈ {halt}

Next tactic:
rcases h₁₂ with (rfl | h₁₂)
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · 

Proof state:
case move.inl
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₀ : Stmt₁
a✝ : Dir
q : Stmt₁
h₀₁ : q₀ ∈ stmts₁ (move a✝ q)
IH : move a✝ q ∈ stmts₁ q → q₀ ∈ stmts₁ q
⊢ q₀ ∈ insert (move a✝ q) (stmts₁ q)

Next tactic:
unfold stmts₁ at h₀₁
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · unfold stmts₁ at h₀₁
      

Proof state:
case move.inl
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₀ : Stmt₁
a✝ : Dir
q : Stmt₁
IH : move a✝ q ∈ stmts₁ q → q₀ ∈ stmts₁ q
h₀₁ : q₀ ∈ insert (move a✝ q) (stmts₁ q)
⊢ q₀ ∈ insert (move a✝ q) (stmts₁ q)

Next tactic:
exact h₀₁
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · 

Proof state:
case move.inr
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Dir
q : Stmt₁
IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q
h₁₂ : q₁ ∈ stmts₁ q
⊢ q₀ ∈ insert (move a✝ q) (stmts₁ q)

Next tactic:
exact Finset.mem_insert_of_mem (IH h₁₂)
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    

Proof state:
case write
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Γ → σ → Γ
q : Stmt₁
IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q
h₁₂ : q₁ = write a✝ q ∨ q₁ ∈ stmts₁ q
⊢ q₀ ∈ insert (write a✝ q) (stmts₁ q)
case load
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Γ → σ → σ
q : Stmt₁
IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q
h₁₂ : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q
⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q)
case branch
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝² : Γ → σ → Bool
a✝¹ a✝ : Stmt₁
a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → q₀ ∈ stmts₁ a✝¹
a_ih✝ : q₁ ∈ stmts₁ a✝ → q₀ ∈ stmts₁ a✝
h₁₂ : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝
⊢ q₀ ∈ insert (branch a✝² a✝¹ a✝) (stmts₁ a✝¹ ∪ stmts₁ a✝)
case goto
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Γ → σ → Λ
h₁₂ : q₁ = goto a✝
⊢ q₀ ∈ {goto a✝}
case halt
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
h₁₂ : q₁ = halt
⊢ q₀ ∈ {halt}

Next tactic:
rcases h₁₂ with (rfl | h₁₂)
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · 

Proof state:
case write.inl
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₀ : Stmt₁
a✝ : Γ → σ → Γ
q : Stmt₁
h₀₁ : q₀ ∈ stmts₁ (write a✝ q)
IH : write a✝ q ∈ stmts₁ q → q₀ ∈ stmts₁ q
⊢ q₀ ∈ insert (write a✝ q) (stmts₁ q)

Next tactic:
unfold stmts₁ at h₀₁
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · unfold stmts₁ at h₀₁
      

Proof state:
case write.inl
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₀ : Stmt₁
a✝ : Γ → σ → Γ
q : Stmt₁
IH : write a✝ q ∈ stmts₁ q → q₀ ∈ stmts₁ q
h₀₁ : q₀ ∈ insert (write a✝ q) (stmts₁ q)
⊢ q₀ ∈ insert (write a✝ q) (stmts₁ q)

Next tactic:
exact h₀₁
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · 

Proof state:
case write.inr
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Γ → σ → Γ
q : Stmt₁
IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q
h₁₂ : q₁ ∈ stmts₁ q
⊢ q₀ ∈ insert (write a✝ q) (stmts₁ q)

Next tactic:
exact Finset.mem_insert_of_mem (IH h₁₂)
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    

Proof state:
case load
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Γ → σ → σ
q : Stmt₁
IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q
h₁₂ : q₁ = load a✝ q ∨ q₁ ∈ stmts₁ q
⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q)
case branch
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝² : Γ → σ → Bool
a✝¹ a✝ : Stmt₁
a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → q₀ ∈ stmts₁ a✝¹
a_ih✝ : q₁ ∈ stmts₁ a✝ → q₀ ∈ stmts₁ a✝
h₁₂ : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝
⊢ q₀ ∈ insert (branch a✝² a✝¹ a✝) (stmts₁ a✝¹ ∪ stmts₁ a✝)
case goto
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Γ → σ → Λ
h₁₂ : q₁ = goto a✝
⊢ q₀ ∈ {goto a✝}
case halt
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
h₁₂ : q₁ = halt
⊢ q₀ ∈ {halt}

Next tactic:
rcases h₁₂ with (rfl | h₁₂)
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · 

Proof state:
case load.inl
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₀ : Stmt₁
a✝ : Γ → σ → σ
q : Stmt₁
h₀₁ : q₀ ∈ stmts₁ (load a✝ q)
IH : load a✝ q ∈ stmts₁ q → q₀ ∈ stmts₁ q
⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q)

Next tactic:
unfold stmts₁ at h₀₁
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · unfold stmts₁ at h₀₁
      

Proof state:
case load.inl
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₀ : Stmt₁
a✝ : Γ → σ → σ
q : Stmt₁
IH : load a✝ q ∈ stmts₁ q → q₀ ∈ stmts₁ q
h₀₁ : q₀ ∈ insert (load a✝ q) (stmts₁ q)
⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q)

Next tactic:
exact h₀₁
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · 

Proof state:
case load.inr
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Γ → σ → σ
q : Stmt₁
IH : q₁ ∈ stmts₁ q → q₀ ∈ stmts₁ q
h₁₂ : q₁ ∈ stmts₁ q
⊢ q₀ ∈ insert (load a✝ q) (stmts₁ q)

Next tactic:
exact Finset.mem_insert_of_mem (IH h₁₂)
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (IH h₁₂)
  

Proof state:
case branch
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝² : Γ → σ → Bool
a✝¹ a✝ : Stmt₁
a_ih✝¹ : q₁ ∈ stmts₁ a✝¹ → q₀ ∈ stmts₁ a✝¹
a_ih✝ : q₁ ∈ stmts₁ a✝ → q₀ ∈ stmts₁ a✝
h₁₂ : q₁ = branch a✝² a✝¹ a✝ ∨ q₁ ∈ stmts₁ a✝¹ ∨ q₁ ∈ stmts₁ a✝
⊢ q₀ ∈ insert (branch a✝² a✝¹ a✝) (stmts₁ a✝¹ ∪ stmts₁ a✝)
case goto
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Γ → σ → Λ
h₁₂ : q₁ = goto a✝
⊢ q₀ ∈ {goto a✝}
case halt
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
h₁₂ : q₁ = halt
⊢ q₀ ∈ {halt}

Next tactic:
case branch p q₁ q₂ IH₁ IH₂ =>
    rcases h₁₂ with (rfl | h₁₂ | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (Finset.mem_union_left _ <| IH₁ h₁₂)
    · exact Finset.mem_insert_of_mem (Finset.mem_union_right _ <| IH₂ h₁₂)
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (IH h₁₂)
  

Proof state:
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁✝ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁✝
p : Γ → σ → Bool
q₁ q₂ : Stmt₁
IH₁ : q₁✝ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁
IH₂ : q₁✝ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂
h₁₂ : q₁✝ = branch p q₁ q₂ ∨ q₁✝ ∈ stmts₁ q₁ ∨ q₁✝ ∈ stmts₁ q₂
⊢ q₀ ∈ insert (branch p q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂)

Next tactic:
case branch p q₁ q₂ IH₁ IH₂ =>
    rcases h₁₂ with (rfl | h₁₂ | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (Finset.mem_union_left _ <| IH₁ h₁₂)
    · exact Finset.mem_insert_of_mem (Finset.mem_union_right _ <| IH₂ h₁₂)
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (IH h₁₂)
  case branch p q₁ q₂ IH₁ IH₂ =>
    

Proof state:
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁✝ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁✝
p : Γ → σ → Bool
q₁ q₂ : Stmt₁
IH₁ : q₁✝ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁
IH₂ : q₁✝ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂
h₁₂ : q₁✝ = branch p q₁ q₂ ∨ q₁✝ ∈ stmts₁ q₁ ∨ q₁✝ ∈ stmts₁ q₂
⊢ q₀ ∈ insert (branch p q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂)

Next tactic:
rcases h₁₂ with (rfl | h₁₂ | h₁₂)
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (IH h₁₂)
  case branch p q₁ q₂ IH₁ IH₂ =>
    rcases h₁₂ with (rfl | h₁₂ | h₁₂)
    · 

Proof state:
case inl
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₀ : Stmt₁
p : Γ → σ → Bool
q₁ q₂ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ (branch p q₁ q₂)
IH₁ : branch p q₁ q₂ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁
IH₂ : branch p q₁ q₂ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂
⊢ q₀ ∈ insert (branch p q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂)

Next tactic:
unfold stmts₁ at h₀₁
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (IH h₁₂)
  case branch p q₁ q₂ IH₁ IH₂ =>
    rcases h₁₂ with (rfl | h₁₂ | h₁₂)
    · unfold stmts₁ at h₀₁
      

Proof state:
case inl
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₀ : Stmt₁
p : Γ → σ → Bool
q₁ q₂ : Stmt₁
IH₁ : branch p q₁ q₂ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁
IH₂ : branch p q₁ q₂ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂
h₀₁ : q₀ ∈ insert (branch p q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂)
⊢ q₀ ∈ insert (branch p q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂)

Next tactic:
exact h₀₁
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (IH h₁₂)
  case branch p q₁ q₂ IH₁ IH₂ =>
    rcases h₁₂ with (rfl | h₁₂ | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · 

Proof state:
case inr.inl
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁✝ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁✝
p : Γ → σ → Bool
q₁ q₂ : Stmt₁
IH₁ : q₁✝ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁
IH₂ : q₁✝ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂
h₁₂ : q₁✝ ∈ stmts₁ q₁
⊢ q₀ ∈ insert (branch p q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂)

Next tactic:
exact Finset.mem_insert_of_mem (Finset.mem_union_left _ <| IH₁ h₁₂)
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (IH h₁₂)
  case branch p q₁ q₂ IH₁ IH₂ =>
    rcases h₁₂ with (rfl | h₁₂ | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (Finset.mem_union_left _ <| IH₁ h₁₂)
    · 

Proof state:
case inr.inr
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁✝ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁✝
p : Γ → σ → Bool
q₁ q₂ : Stmt₁
IH₁ : q₁✝ ∈ stmts₁ q₁ → q₀ ∈ stmts₁ q₁
IH₂ : q₁✝ ∈ stmts₁ q₂ → q₀ ∈ stmts₁ q₂
h₁₂ : q₁✝ ∈ stmts₁ q₂
⊢ q₀ ∈ insert (branch p q₁ q₂) (stmts₁ q₁ ∪ stmts₁ q₂)

Next tactic:
exact Finset.mem_insert_of_mem (Finset.mem_union_right _ <| IH₂ h₁₂)
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (IH h₁₂)
  case branch p q₁ q₂ IH₁ IH₂ =>
    rcases h₁₂ with (rfl | h₁₂ | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (Finset.mem_union_left _ <| IH₁ h₁₂)
    · exact Finset.mem_insert_of_mem (Finset.mem_union_right _ <| IH₂ h₁₂)
  

Proof state:
case goto
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
a✝ : Γ → σ → Λ
h₁₂ : q₁ = goto a✝
⊢ q₀ ∈ {goto a✝}
case halt
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
h₁₂ : q₁ = halt
⊢ q₀ ∈ {halt}

Next tactic:
case goto l => subst h₁₂; exact h₀₁
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (IH h₁₂)
  case branch p q₁ q₂ IH₁ IH₂ =>
    rcases h₁₂ with (rfl | h₁₂ | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (Finset.mem_union_left _ <| IH₁ h₁₂)
    · exact Finset.mem_insert_of_mem (Finset.mem_union_right _ <| IH₂ h₁₂)
  

Proof state:
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
l : Γ → σ → Λ
h₁₂ : q₁ = goto l
⊢ q₀ ∈ {goto l}

Next tactic:
case goto l => subst h₁₂; exact h₀₁
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (IH h₁₂)
  case branch p q₁ q₂ IH₁ IH₂ =>
    rcases h₁₂ with (rfl | h₁₂ | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (Finset.mem_union_left _ <| IH₁ h₁₂)
    · exact Finset.mem_insert_of_mem (Finset.mem_union_right _ <| IH₂ h₁₂)
  case goto l => 

Proof state:
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
l : Γ → σ → Λ
h₁₂ : q₁ = goto l
⊢ q₀ ∈ {goto l}

Next tactic:
subst h₁₂
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (IH h₁₂)
  case branch p q₁ q₂ IH₁ IH₂ =>
    rcases h₁₂ with (rfl | h₁₂ | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (Finset.mem_union_left _ <| IH₁ h₁₂)
    · exact Finset.mem_insert_of_mem (Finset.mem_union_right _ <| IH₂ h₁₂)
  case goto l => subst h₁₂; 

Proof state:
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₀ : Stmt₁
l : Γ → σ → Λ
h₀₁ : q₀ ∈ stmts₁ (goto l)
⊢ q₀ ∈ {goto l}

Next tactic:
exact h₀₁
theorem stmts₁_trans {q₁ q₂ : Stmt₁} : q₁ ∈ stmts₁ q₂ → stmts₁ q₁ ⊆ stmts₁ q₂ := by
  intro h₁₂ q₀ h₀₁
  induction' q₂ with _ q IH _ q IH _ q IH <;> simp only [stmts₁] at h₁₂ ⊢ <;>
    simp only [Finset.mem_insert, Finset.mem_union, Finset.mem_singleton] at h₁₂
  iterate 3
    rcases h₁₂ with (rfl | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (IH h₁₂)
  case branch p q₁ q₂ IH₁ IH₂ =>
    rcases h₁₂ with (rfl | h₁₂ | h₁₂)
    · unfold stmts₁ at h₀₁
      exact h₀₁
    · exact Finset.mem_insert_of_mem (Finset.mem_union_left _ <| IH₁ h₁₂)
    · exact Finset.mem_insert_of_mem (Finset.mem_union_right _ <| IH₂ h₁₂)
  case goto l => subst h₁₂; exact h₀₁
  

Proof state:
case halt
Γ : Type u_1
inst✝ : Inhabited Γ
Λ : Type u_2
σ : Type u_3
q₁ q₀ : Stmt₁
h₀₁ : q₀ ∈ stmts₁ q₁
h₁₂ : q₁ = halt
⊢ q₀ ∈ {halt}

Next tactic:
case halt => subst h₁₂; exact h₀₁