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(** A simpler simulation relation that hides most of the physical state,
except for the call ID stack.
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Useful whenever the resources we own are strong enough to carry us from step to step.

When your goal is simple, to make it stateful just do [intros σs σt].
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To go the other direction, [apply sim_simplify NEW_POST]. Then you will likely
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want to clean some stuff from your context.
*)

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From stbor.sim Require Export body instance.
From stbor.sim Require Import refl_step.
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Definition fun_post_simple
  initial_call_id_stack (r: resUR) (n: nat) vs (css: call_id_stack) vt cst :=
  ( c, cst = c::initial_call_id_stack  end_call_sat r c) 
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  rrel vrel r vs vt.
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Definition sim_simple fs ft r n es css et cst
  (Φ : resUR  nat  result  call_id_stack  result  call_id_stack  Prop) : Prop :=
   σs σt, σs.(scs) = css  σt.(scs) = cst 
    r { n , fs , ft } ( es , σs )  ( et , σt ) :
    (λ r n vs' σs' vt' σt', Φ r n vs' σs'.(scs) vt' σt'.(scs)).

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Notation "r ⊨ˢ{ n , fs , ft } ( es , css ) ≥ ( et , cst ) : Φ" :=
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  (sim_simple fs ft r n%nat es%E css et%E cst Φ)
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  (at level 70, es, et at next level,
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   format "'[hv' r  '/' ⊨ˢ{ n , fs , ft }  '/  ' '[ ' ( es ,  css ) ']'  '/' ≥  '/  ' '[ ' ( et ,  cst ) ']'  '/' :  Φ ']'").

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Section simple.
Implicit Types Φ: resUR  nat  result  call_id_stack  result  call_id_stack  Prop.

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(* FIXME: does this [apply]? *)
Lemma sim_simplify
  (Φnew: resUR  nat  result  call_id_stack  result  call_id_stack  Prop)
  (Φ: resUR  nat  result  state  result  state  Prop)
  r n fs ft es σs et σt :
  ( r n vs σs vt σt, Φnew r n vs σs.(scs) vt σt.(scs)  Φ r n vs σs vt σt) 
  r ⊨ˢ{ n , fs , ft } (es, σs.(scs))  (et, σt.(scs)) : Φnew 
  r { n , fs , ft } (es, σs)  (et, σt) : Φ.
Proof.
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  intros HΦ HH. eapply sim_local_body_post_mono; last exact: HH.
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  apply HΦ.
Qed.
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Lemma sim_simplify'
  (Φnew: resUR  nat  result  call_id_stack  result  call_id_stack  Prop)
  (Φ: resUR  nat  result  state  result  state  Prop)
  r n fs ft es σs et σt css cst :
  ( r n vs σs vt σt, Φnew r n vs σs.(scs) vt σt.(scs)  Φ r n vs σs vt σt) 
  σs.(scs) = css 
  σt.(scs) = cst 
  r ⊨ˢ{ n , fs , ft } (es, css)  (et, cst) : Φnew 
  r { n , fs , ft } (es, σs)  (et, σt) : Φ.
Proof.
  intros HΦ <-<-. eapply sim_simplify. done.
Qed.
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Lemma sim_simple_frame_core Φ r n fs ft es css et cst :
  r ⊨ˢ{ n , fs , ft }
    (es, css)  (et, cst)
  : (λ r' n' es' css' et' cst', Φ (core r  r') n' es' css' et' cst') 
  r ⊨ˢ{ n , fs , ft } (es, css)  (et, cst) : Φ.
Proof.
  intros Hold σs σt <-<-. eapply sim_body_frame_core. auto.
Qed.

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Lemma sim_simple_post_mono Φ1 Φ2 r n fs ft es css et cst :
  Φ1 <6= Φ2 
  r ⊨ˢ{ n , fs , ft } (es, css)  (et, cst) : Φ1 
  r ⊨ˢ{ n , fs , ft } (es, css)  (et, cst) : Φ2.
Proof.
  intros HΦ Hold σs σt <-<-.
  eapply sim_local_body_post_mono; last exact: Hold.
  auto.
Qed.

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(** Simple proof for a function taking one argument. *)
(* TODO: make the two call stacks the same. *)
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Lemma sim_fun_simple1 n (esf etf: function) :
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  length (esf.(fun_args)) = 1%nat 
  length (etf.(fun_args)) = 1%nat 
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  ( fs ft rf es css et cst vs vt,
    sim_local_funs_lookup fs ft 
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    vrel rf vs vt 
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    subst_l (esf.(fun_args)) [Val vs] (esf.(fun_body)) = Some es 
    subst_l (etf.(fun_args)) [Val vt] (etf.(fun_body)) = Some et 
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    rf ⊨ˢ{n,fs,ft} (InitCall es, css)  (InitCall et, cst) : fun_post_simple cst) 
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  ⊨ᶠ esf  etf.
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Proof.
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  intros Hls Hlt HH fs ft Hlook rf es et vls vlt σs σt FREL SUBSTs SUBSTt. exists n.
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  move:(subst_l_is_Some_length _ _ _ _ SUBSTs) (subst_l_is_Some_length _ _ _ _ SUBSTt).
  rewrite Hls Hlt.
  destruct vls as [|vs []]; [done| |done].
  destruct vlt as [|vt []]; [done| |done].
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  inversion FREL as [|???? RREL]. intros _ _. simplify_eq.
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  eapply sim_simplify; last by eapply HH. done.
Qed.
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Lemma sim_simple_bind fs ft
  (Ks: list (ectxi_language.ectx_item (bor_ectxi_lang fs)))
  (Kt: list (ectxi_language.ectx_item (bor_ectxi_lang ft)))
  es et r n css cst Φ :
  r ⊨ˢ{n,fs,ft} (es, css)  (et, cst)
    : (λ r' n' es' css' et' cst',
        r' ⊨ˢ{n',fs,ft} (fill Ks es', css')  (fill Kt et', cst') : Φ) 
  r ⊨ˢ{n,fs,ft} (fill Ks es, css)  (fill Kt et, cst) : Φ.
Proof.
  intros HH σs σt <-<-. apply sim_body_bind.
  eapply sim_local_body_post_mono; last exact: HH.
  clear. simpl. intros r n vs σs vt σt HH. exact: HH.
Qed.

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Lemma sim_simple_val fs ft r n (vs vt: value) css cst Φ :
  Φ r n vs css vt cst 
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  r ⊨ˢ{n,fs,ft} (vs, css)  (vt, cst) : Φ.
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Proof.
  intros HH σs σt <-<-. eapply (sim_body_result _ _ _ _ vs vt). done.
Qed.

Lemma sim_simple_place fs ft r n ls lt ts tt tys tyt css cst Φ :
  Φ r n (PlaceR ls ts tys) css (PlaceR lt tt tyt) cst 
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  r ⊨ˢ{n,fs,ft} (Place ls ts tys, css)  (Place lt tt tyt, cst) : Φ.
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Proof.
  intros HH σs σt <-<-. eapply (sim_body_result _ _ _ _ (PlaceR _ _ _) (PlaceR _ _ _)). done.
Qed.

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(** * Administrative *)
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Lemma sim_simple_init_call fs ft r n es css et cst Φ :
  ( c: call_id,
    let r' := res_callState c (csOwned ) in
    r  r' ⊨ˢ{n,fs,ft} (es, c::cst)  (et, c::cst) : Φ) 
  r ⊨ˢ{n,fs,ft} (InitCall es, css)  (InitCall et, cst) : Φ.
Proof.
  intros HH σs σt <-<-. apply sim_body_init_call=>/= ?.
  apply HH; done.
Qed.

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(* [Val <$> _] in the goal makes this not work with [apply], but
we'd need tactic support for anything better. *)
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Lemma sim_simple_call n' rls rlt rv fs ft r r' n fid css cst Φ :
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  sim_local_funs_lookup fs ft 
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  Forall2 (rrel vrel rv) rls rlt 
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  r  r'  rv 
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  ( rret vs vt vls vlt,
    rls = ValR <$> vls  rlt = ValR <$> vlt 
    vrel rret vs vt 
    r'  rret ⊨ˢ{n',fs,ft} (Val vs, css)  (Val vt, cst) : Φ) 
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  r ⊨ˢ{n,fs,ft}
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    (Call #[ScFnPtr fid] (of_result <$> rls), css)  (Call #[ScFnPtr fid] (of_result <$> rlt), cst) : Φ.
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Proof.
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  intros Hfns Hrel Hres HH σs σt <-<-. rewrite Hres.
  apply: sim_body_step_over_call.
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  - done.
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  - done.
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  - intros. exists n'. eapply sim_body_res_proper; last apply: HH; try done.
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Qed.
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(** * Memory *)
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Lemma sim_simple_alloc_local fs ft r n T css cst Φ :
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  ( (l: loc) (tg: nat),
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    let r' := res_mapsto l (tsize T)  (init_stack (Tagged tg)) in
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    Φ (r  r') n (PlaceR l (Tagged tg) T) css (PlaceR l (Tagged tg) T) cst) 
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  r ⊨ˢ{n,fs,ft} (Alloc T, css)  (Alloc T, cst) : Φ.
Proof.
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  intros HH σs σt <-<-. apply sim_body_alloc_local=>/=. exact: HH.
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Qed.

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Lemma sim_simple_write_local fs ft r r' n l tg ty v v' css cst Φ :
  tsize ty = 1%nat 
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  r  r'  res_mapsto l 1 v' (init_stack (Tagged tg)) 
  ( s, v = [s]  Φ (r'  res_mapsto l 1 s (init_stack (Tagged tg))) n (ValR [%S]) css (ValR [%S]) cst) 
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  r ⊨ˢ{n,fs,ft}
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    (Place l (Tagged tg) ty <- #v, css)  (Place l (Tagged tg) ty <- #v, cst)
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  : Φ.
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Proof. intros Hty Hres HH σs σt <-<-. eapply sim_body_write_local_1; eauto. Qed.
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Lemma sim_simple_retag_local fs ft r r' r'' rs n l s' s tg m ty css cst Φ :
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  r  r'  res_mapsto l 1 s (init_stack (Tagged tg)) 
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  arel rs s' s 
  r'  r''  rs 
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  ( l_inner tg_inner hplt,
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    let s_new := ScPtr l_inner (Tagged tg_inner) in
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    let tk := match m with Mutable => tkUnique | Immutable => tkPub end in
    match m with 
    | Mutable => is_Some (hplt !! l_inner)
    | Immutable => if is_freeze ty then is_Some (hplt !! l_inner) else True
    end 
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    Φ (r''  rs  res_mapsto l 1 s_new (init_stack (Tagged tg))  res_tag tg_inner tk hplt) n (ValR [%S]) css (ValR [%S]) cst) 
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  r ⊨ˢ{n,fs,ft}
    (Retag (Place l (Tagged tg) (Reference (RefPtr m) ty)) Default, css)
  
    (Retag (Place l (Tagged tg) (Reference (RefPtr m) ty)) Default, cst)
  : Φ.
Proof.
Admitted.

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(** * Pure *)
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Lemma sim_simple_let fs ft r n x (vs1 vt1: expr) es2 et2 css cst Φ :
  terminal vs1  terminal vt1 
  r ⊨ˢ{n,fs,ft} (subst' x vs1 es2, css)  (subst' x vt1 et2, cst) : Φ 
  r ⊨ˢ{n,fs,ft} (let: x := vs1 in es2, css)  ((let: x := vt1 in et2), cst) : Φ.
Proof. intros ?? HH σs σt <-<-. apply sim_body_let; eauto. Qed.

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Lemma sim_simple_let_val fs ft r n x (vs1 vt1: value) es2 et2 css cst Φ :
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  r ⊨ˢ{n,fs,ft} (subst' x vs1 es2, css)  (subst' x vt1 et2, cst) : Φ 
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  r ⊨ˢ{n,fs,ft} (let: x := vs1 in es2, css)  ((let: x := vt1 in et2), cst) : Φ.
Proof. intros HH σs σt <-<-. apply sim_body_let; eauto. Qed.

Lemma sim_simple_let_place fs ft r n x ls lt ts tt tys tyt es2 et2 css cst Φ :
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  r ⊨ˢ{n,fs,ft} (subst' x (Place ls ts tys) es2, css)  (subst' x (Place lt tt tyt) et2, cst) : Φ 
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  r ⊨ˢ{n,fs,ft} (let: x := Place ls ts tys in es2, css)  ((let: x := Place lt tt tyt in et2), cst) : Φ.
Proof. intros HH σs σt <-<-. apply sim_body_let; eauto. Qed.

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Lemma sim_simple_ref fs ft r n (pl: result) css cst Φ :
  ( l t T, pl = PlaceR l t T 
    Φ r n (ValR [ScPtr l t]) css (ValR [ScPtr l t]) cst) 
  r ⊨ˢ{n,fs,ft} ((& pl)%E, css)  ((& pl)%E, cst) : Φ.
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Proof. intros HH σs σt <-<-. apply sim_body_ref; eauto. Qed.

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Lemma sim_simple_deref fs ft r n (rf: result) T css cst Φ :
  ( l t, rf = ValR [ScPtr l t] 
    Φ r n (PlaceR l t T) css (PlaceR l t T) cst) 
  r ⊨ˢ{n,fs,ft} (Deref rf T, css)  (Deref rf T, cst) : Φ.
Proof. intros HH σs σt <-<-. apply sim_body_deref; eauto. Qed.
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Lemma sim_simple_var fs ft r n css cst var Φ :
  r ⊨ˢ{n,fs,ft} (Var var, css)  (Var var, cst) : Φ.
Proof. intros σs σt <-<-. apply sim_body_var; eauto. Qed.

Lemma sim_simple_proj fs ft r n (v i: result) css cst Φ :
  ( vi vv iv, v = ValR vv  i = ValR [ScInt iv] 
    vv !! (Z.to_nat iv) = Some vi  0  iv 
    Φ r n (ValR [vi]) css (ValR [vi]) cst) 
  r ⊨ˢ{n,fs,ft} (Proj v i, css)  (Proj v i, cst) : Φ.
Proof. intros HH σs σt <-<-. apply sim_body_proj; eauto. Qed.

Lemma sim_simple_conc fs ft r n (r1 r2: result) css cst Φ :
  ( v1 v2, r1 = ValR v1  r2 = ValR v2 
    Φ r n (ValR (v1 ++ v2)) css (ValR (v1 ++ v2)) cst) 
  r ⊨ˢ{n,fs,ft} (Conc r1 r2, css)  (Conc r1 r2, cst) : Φ.
Proof. intros HH σs σt <-<-. apply sim_body_conc; eauto. Qed.

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End simple.