induction d <;> simp[*]
L : Language
Δ✝ Δ₁ Δ₂ Γ✝ : Sequent L
p✝ q✝ r : SyntacticFormula L
C C' : Set (SyntacticFormula L)
Δ Γ : Sequent L
p q : SyntacticFormula L
d :  ⊢¹ Δ
ss : C ⊆ C'
⊢ CutRestricted C d → CutRestricted C' d
case or
L : Language
Δ✝¹ Δ₁ Δ₂ Γ✝ : Sequent L
p✝¹ q✝¹ r : SyntacticFormula L
C C' : Set (SyntacticFormula L)
Δ Γ : Sequent L
p q : SyntacticFormula L
ss : C ⊆ C'
Δ✝ : List (SyntacticFormula L)
p✝ q✝ : SyntacticFormula L
a✝ : Derivation (p✝ :: q✝ :: Δ✝)
a_ih✝ : CutRestricted C a✝ → CutRestricted C' a✝
⊢ CutRestricted C a✝ → CutRestricted C' a✝

case and
L : Language
Δ✝¹ Δ₁ Δ₂ Γ✝ : Sequent L
p✝¹ q✝¹ r : SyntacticFormula L
C C' : Set (SyntacticFormula L)
Δ Γ : Sequent L
p q : SyntacticFormula L
ss : C ⊆ C'
Δ✝ : List (SyntacticFormula L)
p✝ q✝ : SyntacticFormula L
a✝¹ : Derivation (p✝ :: Δ✝)
a✝ : Derivation (q✝ :: Δ✝)
a_ih✝¹ : CutRestricted C a✝¹ → CutRestricted C' a✝¹
a_ih✝ : CutRestricted C a✝ → CutRestricted C' a✝
⊢ CutRestricted C a✝¹ → CutRestricted C a✝ → CutRestricted C' a✝¹ ∧ CutRestricted C' a✝

case all
L : Language
Δ✝¹ Δ₁ Δ₂ Γ✝ : Sequent L
p✝¹ q✝ r : SyntacticFormula L
C C' : Set (SyntacticFormula L)
Δ Γ : Sequent L
p q : SyntacticFormula L
ss : C ⊆ C'
Δ✝ : List (SyntacticSemiformula L 0)
p✝ : Semiformula L ℕ (0 + 1)
a✝ : Derivation ((Rew.hom Rew.free) p✝ :: Δ✝⁺)
a_ih✝ : CutRestricted C a✝ → CutRestricted C' a✝
⊢ CutRestricted C a✝ → CutRestricted C' a✝

case ex
L : Language
Δ✝¹ Δ₁ Δ₂ Γ✝ : Sequent L
p✝¹ q✝ r : SyntacticFormula L
C C' : Set (SyntacticFormula L)
Δ Γ : Sequent L
p q : SyntacticFormula L
ss : C ⊆ C'
Δ✝ : List (SyntacticFormula L)
p✝ : Semiformula L ℕ (Nat.succ 0)
t✝ : Semiterm L ℕ 0
a✝ : Derivation ([→ t✝].hom p✝ :: Δ✝)
a_ih✝ : CutRestricted C a✝ → CutRestricted C' a✝
⊢ CutRestricted C a✝ → CutRestricted C' a✝

case wk
L : Language
Δ✝¹ Δ₁ Δ₂ Γ✝¹ : Sequent L
p✝ q✝ r : SyntacticFormula L
C C' : Set (SyntacticFormula L)
Δ Γ : Sequent L
p q : SyntacticFormula L
ss : C ⊆ C'
Δ✝ Γ✝ : Sequent L
a✝¹ : Derivation Δ✝
a✝ : Δ✝ ⊆ Γ✝
a_ih✝ : CutRestricted C a✝¹ → CutRestricted C' a✝¹
⊢ CutRestricted C a✝¹ → CutRestricted C' a✝¹

case cut
L : Language
Δ✝¹ Δ₁ Δ₂ Γ✝ : Sequent L
p✝¹ q✝ r : SyntacticFormula L
C C' : Set (SyntacticFormula L)
Δ Γ : Sequent L
p q : SyntacticFormula L
ss : C ⊆ C'
Δ✝ : List (SyntacticFormula L)
p✝ : SyntacticFormula L
a✝¹ : Derivation (p✝ :: Δ✝)
a✝ : Derivation (~p✝ :: Δ✝)
a_ih✝¹ : CutRestricted C a✝¹ → CutRestricted C' a✝¹
a_ih✝ : CutRestricted C a✝ → CutRestricted C' a✝
⊢ p✝ ∈ C → CutRestricted C a✝¹ → CutRestricted C a✝ → p✝ ∈ C' ∧ CutRestricted C' a✝¹ ∧ CutRestricted C' a✝