l3lab/ntp-mathlib huggingface dataset:
theorem codeSupp'_supports {S c k} (H : codeSupp c k ⊆ S) : Supports (codeSupp' c k) S := by
  induction c generalizing k
  iterate 3
    exact trStmts₁_supports (trNormal_supports H) (Finset.Subset.trans (codeSupp_self _ _) H)
  case cons f fs IHf IHfs =>
    have H' := H; simp only [codeSupp_cons, Finset.union_subset_iff] at H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHf H'.2, _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun h => _
    · simp only [codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2.1, h.2.2.2, H.2⟩
    refine' supports_union.2 ⟨IHfs _, _⟩
    · rw [codeSupp, contSupp_cons₁] at H'
      exact Finset.union_subset_right (Finset.union_subset_right H'.2)
    exact
      trStmts₁_supports (head_supports <| Finset.union_subset_right H)
        (Finset.union_subset_right h)
  case comp f g IHf IHg =>
    have H' := H; rw [codeSupp_comp] at H'; have H' := Finset.union_subset_right H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    

Proof state:
S : Finset Λ'
f g : Code
IHf : ∀ {k : Cont'}, codeSupp f k ⊆ S → Supports (codeSupp' f k) S
IHg : ∀ {k : Cont'}, codeSupp g k ⊆ S → Supports (codeSupp' g k) S
k : Cont'
H : codeSupp (Code.comp f g) k ⊆ S
H'✝ : trStmts₁ (trNormal (Code.comp f g) k) ∪ codeSupp g (Cont'.comp f k) ⊆ S
H' : codeSupp g (Cont'.comp f k) ⊆ S
h : codeSupp' g (Cont'.comp f k) ∪ (trStmts₁ (trNormal f k) ∪ codeSupp' f k) ⊆ S
⊢ Supports (codeSupp' g (Cont'.comp f k) ∪ (trStmts₁ (trNormal f k) ∪ codeSupp' f k)) S

Next tactic:
refine' supports_union.2 ⟨IHg H', _⟩
theorem codeSupp'_supports {S c k} (H : codeSupp c k ⊆ S) : Supports (codeSupp' c k) S := by
  induction c generalizing k
  iterate 3
    exact trStmts₁_supports (trNormal_supports H) (Finset.Subset.trans (codeSupp_self _ _) H)
  case cons f fs IHf IHfs =>
    have H' := H; simp only [codeSupp_cons, Finset.union_subset_iff] at H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHf H'.2, _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun h => _
    · simp only [codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2.1, h.2.2.2, H.2⟩
    refine' supports_union.2 ⟨IHfs _, _⟩
    · rw [codeSupp, contSupp_cons₁] at H'
      exact Finset.union_subset_right (Finset.union_subset_right H'.2)
    exact
      trStmts₁_supports (head_supports <| Finset.union_subset_right H)
        (Finset.union_subset_right h)
  case comp f g IHf IHg =>
    have H' := H; rw [codeSupp_comp] at H'; have H' := Finset.union_subset_right H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHg H', _⟩
    

Proof state:
S : Finset Λ'
f g : Code
IHf : ∀ {k : Cont'}, codeSupp f k ⊆ S → Supports (codeSupp' f k) S
IHg : ∀ {k : Cont'}, codeSupp g k ⊆ S → Supports (codeSupp' g k) S
k : Cont'
H : codeSupp (Code.comp f g) k ⊆ S
H'✝ : trStmts₁ (trNormal (Code.comp f g) k) ∪ codeSupp g (Cont'.comp f k) ⊆ S
H' : codeSupp g (Cont'.comp f k) ⊆ S
h : codeSupp' g (Cont'.comp f k) ∪ (trStmts₁ (trNormal f k) ∪ codeSupp' f k) ⊆ S
⊢ Supports (trStmts₁ (trNormal f k) ∪ codeSupp' f k) S

Next tactic:
refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun _ => _
theorem codeSupp'_supports {S c k} (H : codeSupp c k ⊆ S) : Supports (codeSupp' c k) S := by
  induction c generalizing k
  iterate 3
    exact trStmts₁_supports (trNormal_supports H) (Finset.Subset.trans (codeSupp_self _ _) H)
  case cons f fs IHf IHfs =>
    have H' := H; simp only [codeSupp_cons, Finset.union_subset_iff] at H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHf H'.2, _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun h => _
    · simp only [codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2.1, h.2.2.2, H.2⟩
    refine' supports_union.2 ⟨IHfs _, _⟩
    · rw [codeSupp, contSupp_cons₁] at H'
      exact Finset.union_subset_right (Finset.union_subset_right H'.2)
    exact
      trStmts₁_supports (head_supports <| Finset.union_subset_right H)
        (Finset.union_subset_right h)
  case comp f g IHf IHg =>
    have H' := H; rw [codeSupp_comp] at H'; have H' := Finset.union_subset_right H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHg H', _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun _ => _
    · 

Proof state:
case refine'_1
S : Finset Λ'
f g : Code
IHf : ∀ {k : Cont'}, codeSupp f k ⊆ S → Supports (codeSupp' f k) S
IHg : ∀ {k : Cont'}, codeSupp g k ⊆ S → Supports (codeSupp' g k) S
k : Cont'
H : codeSupp (Code.comp f g) k ⊆ S
H'✝ : trStmts₁ (trNormal (Code.comp f g) k) ∪ codeSupp g (Cont'.comp f k) ⊆ S
H' : codeSupp g (Cont'.comp f k) ⊆ S
h : codeSupp' g (Cont'.comp f k) ∪ (trStmts₁ (trNormal f k) ∪ codeSupp' f k) ⊆ S
⊢ codeSupp f k ⊆ S

Next tactic:
simp only [codeSupp', codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
theorem codeSupp'_supports {S c k} (H : codeSupp c k ⊆ S) : Supports (codeSupp' c k) S := by
  induction c generalizing k
  iterate 3
    exact trStmts₁_supports (trNormal_supports H) (Finset.Subset.trans (codeSupp_self _ _) H)
  case cons f fs IHf IHfs =>
    have H' := H; simp only [codeSupp_cons, Finset.union_subset_iff] at H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHf H'.2, _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun h => _
    · simp only [codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2.1, h.2.2.2, H.2⟩
    refine' supports_union.2 ⟨IHfs _, _⟩
    · rw [codeSupp, contSupp_cons₁] at H'
      exact Finset.union_subset_right (Finset.union_subset_right H'.2)
    exact
      trStmts₁_supports (head_supports <| Finset.union_subset_right H)
        (Finset.union_subset_right h)
  case comp f g IHf IHg =>
    have H' := H; rw [codeSupp_comp] at H'; have H' := Finset.union_subset_right H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHg H', _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun _ => _
    · simp only [codeSupp', codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      

Proof state:
case refine'_1
S : Finset Λ'
f g : Code
IHf : ∀ {k : Cont'}, codeSupp f k ⊆ S → Supports (codeSupp' f k) S
IHg : ∀ {k : Cont'}, codeSupp g k ⊆ S → Supports (codeSupp' g k) S
k : Cont'
H'✝ : trStmts₁ (trNormal (Code.comp f g) k) ∪ codeSupp g (Cont'.comp f k) ⊆ S
H' : codeSupp g (Cont'.comp f k) ⊆ S
h : codeSupp' g (Cont'.comp f k) ⊆ S ∧ trStmts₁ (trNormal f k) ⊆ S ∧ codeSupp' f k ⊆ S
H :
  (trStmts₁ (trNormal (Code.comp f g) k) ⊆ S ∧
      codeSupp' g (Cont'.comp f k) ⊆ S ∧ trStmts₁ (trNormal f k) ⊆ S ∧ codeSupp' f k ⊆ S) ∧
    contSupp k ⊆ S
⊢ codeSupp' f k ⊆ S ∧ contSupp k ⊆ S

Next tactic:
exact ⟨h.2.2, H.2⟩
theorem codeSupp'_supports {S c k} (H : codeSupp c k ⊆ S) : Supports (codeSupp' c k) S := by
  induction c generalizing k
  iterate 3
    exact trStmts₁_supports (trNormal_supports H) (Finset.Subset.trans (codeSupp_self _ _) H)
  case cons f fs IHf IHfs =>
    have H' := H; simp only [codeSupp_cons, Finset.union_subset_iff] at H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHf H'.2, _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun h => _
    · simp only [codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2.1, h.2.2.2, H.2⟩
    refine' supports_union.2 ⟨IHfs _, _⟩
    · rw [codeSupp, contSupp_cons₁] at H'
      exact Finset.union_subset_right (Finset.union_subset_right H'.2)
    exact
      trStmts₁_supports (head_supports <| Finset.union_subset_right H)
        (Finset.union_subset_right h)
  case comp f g IHf IHg =>
    have H' := H; rw [codeSupp_comp] at H'; have H' := Finset.union_subset_right H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHg H', _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun _ => _
    · simp only [codeSupp', codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2, H.2⟩
    

Proof state:
case refine'_2
S : Finset Λ'
f g : Code
IHf : ∀ {k : Cont'}, codeSupp f k ⊆ S → Supports (codeSupp' f k) S
IHg : ∀ {k : Cont'}, codeSupp g k ⊆ S → Supports (codeSupp' g k) S
k : Cont'
H : codeSupp (Code.comp f g) k ⊆ S
H'✝ : trStmts₁ (trNormal (Code.comp f g) k) ∪ codeSupp g (Cont'.comp f k) ⊆ S
H' : codeSupp g (Cont'.comp f k) ⊆ S
h : codeSupp' g (Cont'.comp f k) ∪ (trStmts₁ (trNormal f k) ∪ codeSupp' f k) ⊆ S
x✝ : codeSupp' f k ⊆ S
⊢ Supports (codeSupp' f k) S

Next tactic:
exact IHf (Finset.union_subset_right H')
theorem codeSupp'_supports {S c k} (H : codeSupp c k ⊆ S) : Supports (codeSupp' c k) S := by
  induction c generalizing k
  iterate 3
    exact trStmts₁_supports (trNormal_supports H) (Finset.Subset.trans (codeSupp_self _ _) H)
  case cons f fs IHf IHfs =>
    have H' := H; simp only [codeSupp_cons, Finset.union_subset_iff] at H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHf H'.2, _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun h => _
    · simp only [codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2.1, h.2.2.2, H.2⟩
    refine' supports_union.2 ⟨IHfs _, _⟩
    · rw [codeSupp, contSupp_cons₁] at H'
      exact Finset.union_subset_right (Finset.union_subset_right H'.2)
    exact
      trStmts₁_supports (head_supports <| Finset.union_subset_right H)
        (Finset.union_subset_right h)
  case comp f g IHf IHg =>
    have H' := H; rw [codeSupp_comp] at H'; have H' := Finset.union_subset_right H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHg H', _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun _ => _
    · simp only [codeSupp', codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2, H.2⟩
    exact IHf (Finset.union_subset_right H')
  

Proof state:
case case
S : Finset Λ'
a✝¹ a✝ : Code
a_ih✝¹ : ∀ {k : Cont'}, codeSupp a✝¹ k ⊆ S → Supports (codeSupp' a✝¹ k) S
a_ih✝ : ∀ {k : Cont'}, codeSupp a✝ k ⊆ S → Supports (codeSupp' a✝ k) S
k : Cont'
H : codeSupp (Code.case a✝¹ a✝) k ⊆ S
⊢ Supports (codeSupp' (Code.case a✝¹ a✝) k) S
case fix
S : Finset Λ'
a✝ : Code
a_ih✝ : ∀ {k : Cont'}, codeSupp a✝ k ⊆ S → Supports (codeSupp' a✝ k) S
k : Cont'
H : codeSupp (Code.fix a✝) k ⊆ S
⊢ Supports (codeSupp' (Code.fix a✝) k) S

Next tactic:
case case f g IHf IHg =>
    have H' := H; simp only [codeSupp_case, Finset.union_subset_iff] at H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun _ => _
    exact supports_union.2 ⟨IHf H'.2.1, IHg H'.2.2⟩
theorem codeSupp'_supports {S c k} (H : codeSupp c k ⊆ S) : Supports (codeSupp' c k) S := by
  induction c generalizing k
  iterate 3
    exact trStmts₁_supports (trNormal_supports H) (Finset.Subset.trans (codeSupp_self _ _) H)
  case cons f fs IHf IHfs =>
    have H' := H; simp only [codeSupp_cons, Finset.union_subset_iff] at H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHf H'.2, _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun h => _
    · simp only [codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2.1, h.2.2.2, H.2⟩
    refine' supports_union.2 ⟨IHfs _, _⟩
    · rw [codeSupp, contSupp_cons₁] at H'
      exact Finset.union_subset_right (Finset.union_subset_right H'.2)
    exact
      trStmts₁_supports (head_supports <| Finset.union_subset_right H)
        (Finset.union_subset_right h)
  case comp f g IHf IHg =>
    have H' := H; rw [codeSupp_comp] at H'; have H' := Finset.union_subset_right H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHg H', _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun _ => _
    · simp only [codeSupp', codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2, H.2⟩
    exact IHf (Finset.union_subset_right H')
  

Proof state:
S : Finset Λ'
f g : Code
IHf : ∀ {k : Cont'}, codeSupp f k ⊆ S → Supports (codeSupp' f k) S
IHg : ∀ {k : Cont'}, codeSupp g k ⊆ S → Supports (codeSupp' g k) S
k : Cont'
H : codeSupp (Code.case f g) k ⊆ S
⊢ Supports (codeSupp' (Code.case f g) k) S

Next tactic:
case case f g IHf IHg =>
    have H' := H; simp only [codeSupp_case, Finset.union_subset_iff] at H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun _ => _
    exact supports_union.2 ⟨IHf H'.2.1, IHg H'.2.2⟩
theorem codeSupp'_supports {S c k} (H : codeSupp c k ⊆ S) : Supports (codeSupp' c k) S := by
  induction c generalizing k
  iterate 3
    exact trStmts₁_supports (trNormal_supports H) (Finset.Subset.trans (codeSupp_self _ _) H)
  case cons f fs IHf IHfs =>
    have H' := H; simp only [codeSupp_cons, Finset.union_subset_iff] at H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHf H'.2, _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun h => _
    · simp only [codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2.1, h.2.2.2, H.2⟩
    refine' supports_union.2 ⟨IHfs _, _⟩
    · rw [codeSupp, contSupp_cons₁] at H'
      exact Finset.union_subset_right (Finset.union_subset_right H'.2)
    exact
      trStmts₁_supports (head_supports <| Finset.union_subset_right H)
        (Finset.union_subset_right h)
  case comp f g IHf IHg =>
    have H' := H; rw [codeSupp_comp] at H'; have H' := Finset.union_subset_right H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHg H', _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun _ => _
    · simp only [codeSupp', codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2, H.2⟩
    exact IHf (Finset.union_subset_right H')
  case case f g IHf IHg =>
    

Proof state:
S : Finset Λ'
f g : Code
IHf : ∀ {k : Cont'}, codeSupp f k ⊆ S → Supports (codeSupp' f k) S
IHg : ∀ {k : Cont'}, codeSupp g k ⊆ S → Supports (codeSupp' g k) S
k : Cont'
H : codeSupp (Code.case f g) k ⊆ S
⊢ Supports (codeSupp' (Code.case f g) k) S

Next tactic:
have H' := H
theorem codeSupp'_supports {S c k} (H : codeSupp c k ⊆ S) : Supports (codeSupp' c k) S := by
  induction c generalizing k
  iterate 3
    exact trStmts₁_supports (trNormal_supports H) (Finset.Subset.trans (codeSupp_self _ _) H)
  case cons f fs IHf IHfs =>
    have H' := H; simp only [codeSupp_cons, Finset.union_subset_iff] at H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHf H'.2, _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun h => _
    · simp only [codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2.1, h.2.2.2, H.2⟩
    refine' supports_union.2 ⟨IHfs _, _⟩
    · rw [codeSupp, contSupp_cons₁] at H'
      exact Finset.union_subset_right (Finset.union_subset_right H'.2)
    exact
      trStmts₁_supports (head_supports <| Finset.union_subset_right H)
        (Finset.union_subset_right h)
  case comp f g IHf IHg =>
    have H' := H; rw [codeSupp_comp] at H'; have H' := Finset.union_subset_right H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHg H', _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun _ => _
    · simp only [codeSupp', codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2, H.2⟩
    exact IHf (Finset.union_subset_right H')
  case case f g IHf IHg =>
    have H' := H; 

Proof state:
S : Finset Λ'
f g : Code
IHf : ∀ {k : Cont'}, codeSupp f k ⊆ S → Supports (codeSupp' f k) S
IHg : ∀ {k : Cont'}, codeSupp g k ⊆ S → Supports (codeSupp' g k) S
k : Cont'
H H' : codeSupp (Code.case f g) k ⊆ S
⊢ Supports (codeSupp' (Code.case f g) k) S

Next tactic:
simp only [codeSupp_case, Finset.union_subset_iff] at H'
theorem codeSupp'_supports {S c k} (H : codeSupp c k ⊆ S) : Supports (codeSupp' c k) S := by
  induction c generalizing k
  iterate 3
    exact trStmts₁_supports (trNormal_supports H) (Finset.Subset.trans (codeSupp_self _ _) H)
  case cons f fs IHf IHfs =>
    have H' := H; simp only [codeSupp_cons, Finset.union_subset_iff] at H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHf H'.2, _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun h => _
    · simp only [codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2.1, h.2.2.2, H.2⟩
    refine' supports_union.2 ⟨IHfs _, _⟩
    · rw [codeSupp, contSupp_cons₁] at H'
      exact Finset.union_subset_right (Finset.union_subset_right H'.2)
    exact
      trStmts₁_supports (head_supports <| Finset.union_subset_right H)
        (Finset.union_subset_right h)
  case comp f g IHf IHg =>
    have H' := H; rw [codeSupp_comp] at H'; have H' := Finset.union_subset_right H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHg H', _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun _ => _
    · simp only [codeSupp', codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2, H.2⟩
    exact IHf (Finset.union_subset_right H')
  case case f g IHf IHg =>
    have H' := H; simp only [codeSupp_case, Finset.union_subset_iff] at H'
    

Proof state:
S : Finset Λ'
f g : Code
IHf : ∀ {k : Cont'}, codeSupp f k ⊆ S → Supports (codeSupp' f k) S
IHg : ∀ {k : Cont'}, codeSupp g k ⊆ S → Supports (codeSupp' g k) S
k : Cont'
H : codeSupp (Code.case f g) k ⊆ S
H' : trStmts₁ (trNormal (Code.case f g) k) ⊆ S ∧ codeSupp f k ⊆ S ∧ codeSupp g k ⊆ S
⊢ Supports (codeSupp' (Code.case f g) k) S

Next tactic:
refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun _ => _
theorem codeSupp'_supports {S c k} (H : codeSupp c k ⊆ S) : Supports (codeSupp' c k) S := by
  induction c generalizing k
  iterate 3
    exact trStmts₁_supports (trNormal_supports H) (Finset.Subset.trans (codeSupp_self _ _) H)
  case cons f fs IHf IHfs =>
    have H' := H; simp only [codeSupp_cons, Finset.union_subset_iff] at H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHf H'.2, _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun h => _
    · simp only [codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2.1, h.2.2.2, H.2⟩
    refine' supports_union.2 ⟨IHfs _, _⟩
    · rw [codeSupp, contSupp_cons₁] at H'
      exact Finset.union_subset_right (Finset.union_subset_right H'.2)
    exact
      trStmts₁_supports (head_supports <| Finset.union_subset_right H)
        (Finset.union_subset_right h)
  case comp f g IHf IHg =>
    have H' := H; rw [codeSupp_comp] at H'; have H' := Finset.union_subset_right H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHg H', _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun _ => _
    · simp only [codeSupp', codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2, H.2⟩
    exact IHf (Finset.union_subset_right H')
  case case f g IHf IHg =>
    have H' := H; simp only [codeSupp_case, Finset.union_subset_iff] at H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun _ => _
    

Proof state:
S : Finset Λ'
f g : Code
IHf : ∀ {k : Cont'}, codeSupp f k ⊆ S → Supports (codeSupp' f k) S
IHg : ∀ {k : Cont'}, codeSupp g k ⊆ S → Supports (codeSupp' g k) S
k : Cont'
H : codeSupp (Code.case f g) k ⊆ S
H' : trStmts₁ (trNormal (Code.case f g) k) ⊆ S ∧ codeSupp f k ⊆ S ∧ codeSupp g k ⊆ S
x✝ : codeSupp' f k ∪ codeSupp' g k ⊆ S
⊢ Supports (codeSupp' f k ∪ codeSupp' g k) S

Next tactic:
exact supports_union.2 ⟨IHf H'.2.1, IHg H'.2.2⟩
theorem codeSupp'_supports {S c k} (H : codeSupp c k ⊆ S) : Supports (codeSupp' c k) S := by
  induction c generalizing k
  iterate 3
    exact trStmts₁_supports (trNormal_supports H) (Finset.Subset.trans (codeSupp_self _ _) H)
  case cons f fs IHf IHfs =>
    have H' := H; simp only [codeSupp_cons, Finset.union_subset_iff] at H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHf H'.2, _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun h => _
    · simp only [codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2.1, h.2.2.2, H.2⟩
    refine' supports_union.2 ⟨IHfs _, _⟩
    · rw [codeSupp, contSupp_cons₁] at H'
      exact Finset.union_subset_right (Finset.union_subset_right H'.2)
    exact
      trStmts₁_supports (head_supports <| Finset.union_subset_right H)
        (Finset.union_subset_right h)
  case comp f g IHf IHg =>
    have H' := H; rw [codeSupp_comp] at H'; have H' := Finset.union_subset_right H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHg H', _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun _ => _
    · simp only [codeSupp', codeSupp, Finset.union_subset_iff, contSupp] at h H ⊢
      exact ⟨h.2.2, H.2⟩
    exact IHf (Finset.union_subset_right H')
  case case f g IHf IHg =>
    have H' := H; simp only [codeSupp_case, Finset.union_subset_iff] at H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun _ => _
    exact supports_union.2 ⟨IHf H'.2.1, IHg H'.2.2⟩
  

Proof state:
case fix
S : Finset Λ'
a✝ : Code
a_ih✝ : ∀ {k : Cont'}, codeSupp a✝ k ⊆ S → Supports (codeSupp' a✝ k) S
k : Cont'
H : codeSupp (Code.fix a✝) k ⊆ S
⊢ Supports (codeSupp' (Code.fix a✝) k) S

Next tactic:
case fix f IHf =>
    have H' := H; simp only [codeSupp_fix, Finset.union_subset_iff] at H'
    refine' trStmts₁_supports' (trNormal_supports H) (Finset.union_subset_left H) fun h => _
    refine' supports_union.2 ⟨IHf H'.2, _⟩
    refine' trStmts₁_supports' (trNormal_supports _) (Finset.union_subset_right h) fun _ => _
    · simp only [codeSupp', codeSupp, Finset.union_subset_iff, contSupp, trStmts₁,
        Finset.insert_subset_iff] at h H ⊢
      exact ⟨h.1, ⟨H.1.1, h⟩, H.2⟩
    exact supports_singleton.2 (ret_supports <| Finset.union_subset_right H)