Require Import mathcomp.ssreflect.ssreflect.
Require Import CTL_def hilbert.
Import IC.
Set Implicit Arguments.
Import Prenex Implicits.
Require Import CTL_def hilbert.
Import IC.
Set Implicit Arguments.
Import Prenex Implicits.
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).