Require Import mathcomp.ssreflect.ssreflect.
Require Import CTL_def hilbert.
Import IC.

Set Implicit Arguments.
Import Prenex Implicits.

Translation of History Rules to Hilbert Refutations


Definition AU_ (u s t : form) := AU (s :/\: u) (t :/\: u).

Lemma AUH_hil (s t u v : form) :
  prv (u :/\: t ---> Bot) ->
  prv (u ---> s ---> ~~: AX (AU_ (~~: u :/\: v) s t)) ->
  prv (u ---> AU_ v s t ---> Bot).

Definition EU_ (H s t : form) := EU (s :/\: H) (t :/\: H).

Lemma ARH_hil (s t C H : form) :
  prv (t ---> ~~: C) ->
  prv (EX (EU_ (~~: C :/\: H) s t) ---> s ---> ~~: C) ->
  prv (C ---> EU_ H s t ---> fF).