let σ0 : Formula := dis $
  (H β).map (fun (F,δ) => if δ = ε then Con F else ⊥)
case mp
ψ : Formula
β : Program
ρ : Formula := dis (List.map (fun Fδ => Con (Yset Fδ ψ)) (H (∗β)))
right_to_left_claim : ∀ (W : Type) (M : KripkeModel W) (w : W), evaluate M w ρ → evaluate M w (~⌈∗β⌉ψ)
W : Type
M : KripkeModel W
w : W
x : ℕ := freshVarProg β
x_not_in : x ∉ voc β
⊢ evaluate M w (~⌈∗β⌉ψ) → evaluate M w ρ
case mp
ψ : Formula
β : Program
ρ : Formula := dis (List.map (fun Fδ => Con (Yset Fδ ψ)) (H (∗β)))
right_to_left_claim : ∀ (W : Type) (M : KripkeModel W) (w : W), evaluate M w ρ → evaluate M w (~⌈∗β⌉ψ)
W : Type
M : KripkeModel W
w : W
x : ℕ := freshVarProg β
x_not_in : x ∉ voc β
σ0 : Formula :=
  dis
    (List.map
      (fun x =>
        match x with
        | (F, δ) => if δ = ε then Con F else ⊥)
      (H β))
⊢ evaluate M w (~⌈∗β⌉ψ) → evaluate M w ρ