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(*       ___                                                              *)
(*      ||M||                                                             *)
(*      ||A||       A project by Andrea Asperti                           *)
(*      ||T||                                                             *)
(*      ||I||       Developers:                                           *)
(*      ||T||         The HELM team.                                      *)
(*      ||A||         http://helm.cs.unibo.it                             *)
(*      \   /                                                             *)
(*       \ /        This file is distributed under the terms of the       *)
(*        v         GNU General Public License Version 2                  *)
(*                                                                        *)
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include "basics/types.ma".
include "basics/deqsets.ma".
include "Traces.ma".

record concrete_transition_sys (p : label_params) : Type[2] ≝ 
{ trans_sys :> abstract_status p
}.

record abstract_transition_sys : Type[1] ≝ 
{ abs_state_t :> Type[0]
; abs_instr : monoid
; act_abs : monoid_action abs_instr abs_state_t
}.

notation "〚 term 19 C 〛 term 90 A" with precedence 10 for @{ 'sem $C $A }.
interpretation "act_abs" 'sem f x = (act ?? (act_abs ?) f x).

record galois_rel (p : label_params) 
                   (C : concrete_transition_sys p) 
                   (A : abstract_transition_sys) : Type[0] ≝ 
{ rel_galois :2> C → A → Prop
}.

record galois_connection (p : label_params) : Type[2] ≝ 
{ concr : concrete_transition_sys p
; abs_d : abstract_transition_sys
; galois_rel:> galois_rel p concr abs_d
}.

record abstract_interpretation (p : label_params) : Type[2] ≝
{ galois_conn:> galois_connection p
; instr_t : Type[0]
; fetch : concr … galois_conn → option instr_t
; instr_map : instr_t → abs_instr … (abs_d … galois_conn)
; instr_map_ok :
   ∀s,s': concr … galois_conn. ∀a: abs_d … galois_conn.∀l,i.
    as_execute … l s s' → 
     fetch … s = Some ? i →
      ∀I. instr_map … i = I →
       galois_conn s a → galois_conn s' (〚I〛 a)
}.