source: LTS/frame_variable_pass.ma @ 3563

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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 "Simulation.ma".
include "variable.ma".
include alias "arithmetics/nat.ma".

definition update_fun : ∀A : DeqSet.∀B : Type[0].(A → B) → A → B → A → B ≝ 
λA,B,f,a,b,x.if x == a then b else f x.

let rec map_exp (e : expr) (map : variable → variable) on e : expr ≝ 
match e with
[  Var v ⇒ Var (map v)
| Const n ⇒ Const n
| Plus e1 e2 ⇒ Plus (map_exp e1 map) (map_exp e2 map) 
| Minus e1 e2 ⇒ Minus (map_exp e1 map) (map_exp e2 map)
].

let rec map_cond (c : condition) (map : variable → variable) on c : condition ≝ 
match c with
[Eq e1 e2 ⇒ Eq (map_exp e1 map) (map_exp e2 map)
| Not c1 ⇒ Not (map_cond c1 map)
].

let rec trans_imp_instr (i : imp_Instructions) (map : variable → variable) on i : frame_Instructions  ≝ 
match i with
[ EMPTY ⇒ EMPTY …
| RETURN rt ⇒ RETURN … rt
| SEQ seq opt_l i ⇒ 
    match seq with
    [ sNop ⇒ SEQ … sNop opt_l (trans_imp_instr i map)
    | sAss v e ⇒ SEQ frame_state_params flat_labels (sAss (map v) (map_exp e map)) opt_l (trans_imp_instr i map)
    ]
| COND cond ltrue i_true lfalse i_false instr ⇒ 
    COND frame_state_params flat_labels (map_cond cond map) ltrue (trans_imp_instr i_true map) lfalse (trans_imp_instr i_false map) 
           (trans_imp_instr instr map)
| LOOP cond ltrue i_true lfalse instr ⇒ 
    LOOP frame_state_params flat_labels (map_cond cond map) ltrue (trans_imp_instr i_true map) lfalse  
           (trans_imp_instr instr map)
| CALL f act_p opt_l instr ⇒ 
      CALL frame_state_params flat_labels f 〈(\fst act_p),(map_exp (\snd act_p) map)〉 opt_l (trans_imp_instr instr map)
| IO lin io lout instr ⇒ ?
].
cases io qed.


let rec All2 (A,B : Type[0]) (f : A → B → Prop) (l1 : list A) (l2 : list B) on l1 : Prop ≝ 
match l1 with
[ nil ⇒ match l2 with [nil ⇒ True | cons _ _ ⇒ False]
| cons x xs ⇒ match l2 with [nil ⇒ False | cons y ys ⇒ f x y ∧ All2 … f xs ys]
].

definition environment_rel : (variable → variable) → environment → activation_frame → Prop ≝ 
λmap,st1,st2.
 All … (λx.let 〈v,val〉 ≝ x in read_frame … st2 (to_shift + (map v)) O = return val)
  st1.

definition store_rel : (variable → variable) → store_t → frame_store_t → Prop ≝ 
λmap,st1,st2.
All2 … (λel1,el2.\snd el1 = \snd el2 ∧ environment_rel map (\fst el1) (\fst el2))
 st1 (\fst st2) ∧ \snd st2 = 〈O,O〉.

definition trans_imp_prog : (variable → variable) → Program imp_env_params imp_instr_params flat_labels → 
Program frame_env_params imp_instr_params flat_labels ≝ λmap,prog.
mk_Program ??? 
(foldr … 
   (λx,tail.(mk_env_item … 
               (mk_signature frame_env_params … (f_name … (f_sig … x)) it (f_ret …(f_sig … x))) 
               (f_lab … x) (trans_imp_instr (f_body … x) map)) :: tail) (nil ?) (env … prog))
(trans_imp_instr (main … prog) map).

definition imp_variable_pass_rel ≝  
λmap : variable → variable.
λprog : Program imp_state_params imp_state_params imp_state_params.
let t_prog ≝ trans_imp_prog map prog in
mk_relations flat_labels (operational_semantics imp_state_params imp_sem_state_params prog)
 (operational_semantics frame_state_params frame_sem_state_params t_prog) 
(λs1,s2.store_rel map (store … s1) (store … s2) ∧ code … s2 = trans_imp_instr (code … s1) map ∧  
cont … s2 = foldr … (λx,rem.〈\fst x,(trans_imp_instr (\snd x) map)〉 :: rem) (nil ?) (cont … s1))
(λ_.λ_.True).

lemma eval_cond_ok:
 ∀m: variable → variable.
  ∀e: expr.
  let e' ≝ map_exp e m in
   ∀env: environment.
    ∀env': activation_frame.
     environment_rel m env env' →
      ∀val.
       sem_expr env e = Some … val →
        frame_sem_expr env' e' = Some … val.
 #m #e elim e
 [ #v #env elim env
   [ #env' #_ #val #abs normalize in abs; destruct
   | * #var #val #tl #IH #env' * #RF #Htl #val'
     whd in ⊢ (??%? → ?); inversion (?==?) #Hv_var normalize nodelta
     [ #Hval_val' destruct >(\P Hv_var) assumption
     | #Hread_tl @IH // assumption ]]
 | normalize //
 |*: #e1 #e2 #IH1 #IH2 #env #env' #Rel #v
   change with (!x ← ?; ? = return ?) in match (sem_expr ?? = Some ??);
   #H cases (bind_inversion ????? H) -H #v1 * #He1
   #H cases (bind_inversion ????? H) -H #v2 * #He2
   #H normalize in H; destruct
   change with (m_bind ?????) in ⊢ (??%?);
   >(IH1 … He1) // >(IH2 … He2) // ]
qed.

definition simulation_imp_frame ≝ 
λprog : Program imp_state_params imp_state_params imp_state_params.
λmap : variable → variable.
λprf : All … (λx.map (f_pars imp_state_params imp_state_params (f_sig … x)) = O) (env … prog).
mk_simulation_conditions … (imp_variable_pass_rel map prog) ???????????.
cases daemon (*TODO*)
qed.

(*
[ #s1 #s2 ** #H1 #H2 #H3 whd in match (as_initial ???); (*store iniziali in relazione*) cases daemon
| cases daemon
| cases daemon
| #s1 #s2 #s1' #H inversion H
  [ #s11 #s22 * #act #instr #cont #EQcode_s11 #EQcont_s11 #EQ #EQ1 destruct #EQstore #Hio1 #Hio2
    #act_no_ret #EQ1 #EQ2 #EQ3 destruct #_ ** #Hstore #EQcode #EQcont #_ #_
    %{(mk_state … (code … s22) (cont … s22) (store … s2) (io_info … s22))}
    %{(t_ind … (t_base …))} 
    [2: @(cost_act … (None ?)) 
    | @hide_prf whd applyS(empty) [9: <EQcode assumption |2: normalize % * |3: % |4: cases daemon (*da mettere a posto io *)
    |5: % |6: % |7: % |8: <EQcont assumption |*:]
    | % [ % // % // ] %1 %2 [cases daemon] %1 cases daemon
    ]
  |*: cases daemon  
  ]
|*: cases daemon
]
qed.*)