have myFresh := keepFreshH β x_not_in
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)
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))
σ1 : Formula :=
  dis
    (List.map
      (fun x_1 =>
        match x_1 with
        | (F, δ) => if δ ≠ ε then Con ((~⌈⌈δ⌉⌉~·x) :: F) else ⊥)
      (H β))
W : Type
M : KripkeModel W
w : W
this :
  repl_in_F x ρ (dis (List.map (fun Fδ => if Fδ.2 = [] then Formula.bottom else Con ((~⌈⌈Fδ.2⌉⌉~·x) :: Fδ.1)) (H β))) =
    dis
      (List.map
        (fun Fδ => if Fδ.2 = [] then repl_in_F x ρ Formula.bottom else repl_in_F x ρ (Con ((~⌈⌈Fδ.2⌉⌉~·x) :: Fδ.1)))
        (H β))
⊢ dis (List.map (fun Fδ => if Fδ.2 = [] then Formula.bottom else repl_in_F x ρ (Con ((~⌈⌈Fδ.2⌉⌉~·x) :: Fδ.1))) (H β)) =
    dis (List.map (fun Fδ => if Fδ.2 = [] then Formula.bottom else Con ((~⌈⌈Fδ.2⌉⌉~ρ) :: Fδ.1)) (H β))
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)
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))
σ1 : Formula :=
  dis
    (List.map
      (fun x_1 =>
        match x_1 with
        | (F, δ) => if δ ≠ ε then Con ((~⌈⌈δ⌉⌉~·x) :: F) else ⊥)
      (H β))
W : Type
M : KripkeModel W
w : W
this :
  repl_in_F x ρ (dis (List.map (fun Fδ => if Fδ.2 = [] then Formula.bottom else Con ((~⌈⌈Fδ.2⌉⌉~·x) :: Fδ.1)) (H β))) =
    dis
      (List.map
        (fun Fδ => if Fδ.2 = [] then repl_in_F x ρ Formula.bottom else repl_in_F x ρ (Con ((~⌈⌈Fδ.2⌉⌉~·x) :: Fδ.1)))
        (H β))
myFresh : ∀ (F : List Formula) (δ : List Program), (F, δ) ∈ H β → x ∉ voc F ∧ x ∉ voc δ
⊢ dis (List.map (fun Fδ => if Fδ.2 = [] then Formula.bottom else repl_in_F x ρ (Con ((~⌈⌈Fδ.2⌉⌉~·x) :: Fδ.1))) (H β)) =
    dis (List.map (fun Fδ => if Fδ.2 = [] then Formula.bottom else Con ((~⌈⌈Fδ.2⌉⌉~ρ) :: Fδ.1)) (H β))