cases Fδ_in_H
x : ℕ
F : List Formula
δ : List Program
α : Program
x_notin : x ∉ vocabOfProgram α
Fδ_in_H : F = [] ∧ δ = [] ∨ ∃ l, (∃ a b, (a, b) ∈ H α ∧ (if b = [] then [] else [(a, b ++ [∗α])]) = l) ∧ (F, δ) ∈ l
y : Formula
y_in : y ∈ F
⊢ x ∉ vocabOfFormula y
case inl
x : ℕ
F : List Formula
δ : List Program
α : Program
x_notin : x ∉ vocabOfProgram α
y : Formula
y_in : y ∈ F
h✝ : F = [] ∧ δ = []
⊢ x ∉ vocabOfFormula y

case inr
x : ℕ
F : List Formula
δ : List Program
α : Program
x_notin : x ∉ vocabOfProgram α
y : Formula
y_in : y ∈ F
h✝ : ∃ l, (∃ a b, (a, b) ∈ H α ∧ (if b = [] then [] else [(a, b ++ [∗α])]) = l) ∧ (F, δ) ∈ l
⊢ x ∉ vocabOfFormula y