source: src/RTLabs/RTLabs_semantics.ma @ 2649

include "basics/lists/list.ma".

include "common/Errors.ma".
include "common/Globalenvs.ma".
include "common/IO.ma".
include "common/SmallstepExec.ma".

include "RTLabs/RTLabs_syntax.ma".

definition genv ≝ genv_t (fundef internal_function).

record frame : Type[0] ≝
{ func   : internal_function
; locals : register_env val
; next   : label
; next_ok: present ?? (f_graph func) next
; sp     : block
; retdst : option register
}.

definition adv : ∀l:label. ∀f:frame. present ?? (f_graph (func f)) l → frame ≝
λl,f,p. mk_frame (func f) (locals f) l p (sp f) (retdst f).

inductive state : Type[0] ≝
| State : 
   ∀   f : frame.
   ∀  fs : list frame.
   ∀   m : mem.
   state
| Callstate :
   ∀  fd : fundef internal_function.
   ∀args : list val.
   ∀ dst : option register.
   ∀ stk : list frame.
   ∀   m : mem.
   state
| Returnstate :
   ∀ rtv : option val.
   ∀ dst : option register.
   ∀ stk : list frame.
   ∀   m : mem.
   state
| Finalstate :
   ∀   r : int.
   state
.

definition build_state ≝ λf.λfs.λm.λn.λp. State (adv n f p) fs m.

definition next_instruction : frame → statement ≝
λf. lookup_present ?? (f_graph (func f)) (next f) (next_ok f).

definition init_locals : list (register × typ) → register_env val ≝
foldr ?? (λidt,en. let 〈id,ty〉 ≝ idt in add RegisterTag val en id Vundef) (empty_map ??).

definition reg_store ≝
λreg,v,locals.
  update RegisterTag val locals reg v.

let rec params_store (rs:list (register × typ)) (vs:list val) (locals : register_env val) : res (register_env val) ≝
match rs with
[ nil ⇒ match vs with [ nil ⇒ OK ? locals | _ ⇒ Error ? (msg WrongNumberOfParameters)]
| cons rt rst ⇒ match vs with [ nil ⇒ Error ? (msg WrongNumberOfParameters) | cons v vst ⇒
  let 〈r,ty〉 ≝ rt in
  let locals' ≝ add RegisterTag val locals r v in
  params_store rst vst locals'
] ].


definition reg_retrieve : register_env ? → register → res val ≝
λlocals,reg. opt_to_res … [MSG BadRegister; CTX ? reg] (lookup ?? locals reg).

definition eval_statement : genv → state → IO io_out io_in (trace × state) ≝
λge,st.
match st return λ_. IO ??? with
[ State f fs m ⇒
  let s ≝ next_instruction f in 
  match s return λs. labels_present ? s → IO ??? with
  [ St_skip l ⇒ λH. return 〈E0, build_state f fs m l ?〉
  | St_cost cl l ⇒ λH. return 〈Echarge cl, build_state f fs m l ?〉
  | St_const _ r cst l ⇒ λH. 
      ! v ← opt_to_res … (msg FailedConstant) (eval_constant ? (find_symbol … ge) (sp f) cst);
      ! locals ← reg_store r v (locals f);
      return 〈E0, State (mk_frame (func f) locals l ? (sp f) (retdst f)) fs m〉
  | St_op1 _ _ op dst src l ⇒ λH.
      ! v ← reg_retrieve (locals f) src;
      ! v' ← opt_to_res … (msg FailedOp) (eval_unop ?? op v);
      ! locals ← reg_store dst v' (locals f);
      return 〈E0, State (mk_frame (func f) locals l ? (sp f) (retdst f)) fs m〉
  | St_op2 _ _ _  op dst src1 src2 l ⇒ λH.
      ! v1 ← reg_retrieve (locals f) src1;
      ! v2 ← reg_retrieve (locals f) src2;
      ! v' ← opt_to_res … (msg FailedOp) (eval_binop m ??? op v1 v2);
      ! locals ← reg_store dst v' (locals f);
      return 〈E0, State (mk_frame (func f) locals l ? (sp f) (retdst f)) fs m〉
  | St_load chunk addr dst l ⇒ λH.
      ! vaddr ← reg_retrieve (locals f) addr;
      ! v ← opt_to_res … (msg FailedLoad) (loadv chunk m vaddr);
      ! locals ← reg_store dst v (locals f);
      return 〈E0, State (mk_frame (func f) locals l ? (sp f) (retdst f)) fs m〉
  | St_store chunk addr src l ⇒ λH.
      ! vaddr ← reg_retrieve (locals f) addr;
      ! v ← reg_retrieve (locals f) src;
      ! m' ← opt_to_res … (msg FailedStore) (storev chunk m vaddr v);
      return 〈E0, build_state f fs m' l ?〉
  | St_call_id id args dst l ⇒ λH.
      ! b ← opt_to_res … [MSG MissingSymbol; CTX ? id] (find_symbol … ge id);
      ! fd ← opt_to_res … [MSG BadFunction; CTX ? id] (find_funct_ptr … ge b);
      ! vs ← m_list_map … (reg_retrieve (locals f)) args;
      return 〈E0, Callstate fd vs dst (adv l f ?::fs) m〉
  | St_call_ptr frs args dst l ⇒ λH.
      ! fv ← reg_retrieve (locals f) frs;
      ! fd ← opt_to_res … (msg BadFunction) (find_funct … ge fv);
      ! vs ← m_list_map … (reg_retrieve (locals f)) args;
      return 〈E0, Callstate fd vs dst (adv l f ?::fs) m〉
(*
  | St_tailcall_id id args ⇒ λH.
      ! b ← opt_to_res … [MSG MissingSymbol; CTX ? id] (find_symbol … ge id);
      ! fd ← opt_to_res … [MSG BadFunction; CTX ? id] (find_funct_ptr … ge b);
      ! vs ← m_list_map … (reg_retrieve (locals f)) args;
      return 〈E0, Callstate fd vs (retdst f) fs (free m (sp f))〉
  | St_tailcall_ptr frs args ⇒ λH.
      ! fv ← reg_retrieve (locals f) frs;
      ! fd ← opt_to_res … (msg BadFunction) (find_funct … ge fv);
      ! vs ← m_list_map … (reg_retrieve (locals f)) args;
      return 〈E0, Callstate fd vs (retdst f) fs (free m (sp f))〉
*)
  | St_cond src ltrue lfalse ⇒ λH.
      ! v ← reg_retrieve (locals f) src;
      ! b ← eval_bool_of_val v;
      return 〈E0, build_state f fs m (if b then ltrue else lfalse) ?〉
(*
  | St_jumptable r ls ⇒ λH.
      ! v ← reg_retrieve (locals f) r;
      match v with
      [ Vint _ i ⇒
          match nth_opt ? (nat_of_bitvector ? i) ls return λx. nth_opt ??? = x → ? with
          [ None ⇒ λ_. Error ? (msg BadJumpTable)
          | Some l ⇒ λE. return 〈E0, build_state f fs m l ?〉
          ] (refl ??)
      | _ ⇒ Error ? (msg BadJumpValue)
      ]
*)
  | St_return ⇒ λH.
      ! v ← match f_result (func f) with
            [ None ⇒ return (None ?)
            | Some rt ⇒
                let 〈r,ty〉 ≝ rt in
                ! v ← reg_retrieve (locals f) r;
                return (Some ? v)
            ];
      return 〈E0, Returnstate v (retdst f) fs (free m (sp f))〉
  ] (f_closed (func f) (next f) s ?)
| Callstate fd params dst fs m ⇒
    match fd return λ_. IO ??? with
    [ Internal fn ⇒
        ! locals ← params_store (f_params fn) params (init_locals (f_locals fn));
        (* CSC: XXX alignment call (concrete_stacksize in OCaml) is missing
           here *)
        let 〈m', sp〉 ≝ alloc m 0 (f_stacksize fn) (* XData *) in
        return 〈E0, State (mk_frame fn locals (f_entry fn) ? sp dst) fs m'〉
    | External fn ⇒
        ! evargs ← err_to_io … (check_eventval_list params (sig_args (ef_sig fn)));
        ! evres ← do_io (ef_id fn) evargs (proj_sig_res (ef_sig fn));
        return 〈Eextcall (ef_id fn) evargs (mk_eventval ? evres), Returnstate (Some ? (mk_val ? evres) (*FIXME should allow None *)) dst fs m〉
    ]
| Returnstate v dst fs m ⇒
    match fs with
    [ nil ⇒ 
        match v with
        [ Some v' ⇒
          match v' with
          [ Vint sz r ⇒ match sz return λsz. bvint sz → monad Res ? with
                        [ I32 ⇒ λr. return 〈E0, Finalstate r〉
                        | _ ⇒ λ_. Error ? (msg ReturnMismatch)
                        ] r
          | _ ⇒ Error ? (msg ReturnMismatch)
          ]
        | _ ⇒ Error ? (msg ReturnMismatch)
        ]
    | cons f fs' ⇒
        ! locals ← match dst with
                   [ None ⇒ match v with [ None ⇒ OK ? (locals f)
                                         | _ ⇒ Error ? (msg ReturnMismatch) ]
                   | Some d ⇒ match v with [ None ⇒ Error ? (msg ReturnMismatch)
                                           | Some v' ⇒ reg_store d v' (locals f) ] ];
        return 〈E0, State (mk_frame (func f) locals (next f) ? (sp f) (retdst f)) fs' m〉
    ]
| Finalstate r ⇒ Error ? (msg FinalState) (* Already in final state *)
]. try @(next_ok f) try @lookup_lookup_present try @H
[ cases b [ @(proj1 ?? H) | @(proj2 ?? H) ]
| cases (f_entry fn) //
] qed.

definition RTLabs_is_final : state → option int ≝
λs. match s with
[ Finalstate r ⇒ Some ? r
| _ ⇒ None ?
].

definition RTLabs_exec : trans_system io_out io_in ≝
  mk_trans_system … ? (λ_.RTLabs_is_final) eval_statement.

definition make_global : RTLabs_program → genv ≝
λp. globalenv … (λx.[Init_space x]) p.

definition make_initial_state : RTLabs_program → res state ≝
λp.
  let ge ≝ make_global p in
  do m ← init_mem … (λx.[Init_space x]) p;
  let main ≝ prog_main ?? p in
  do b ← opt_to_res ? [MSG MissingSymbol; CTX ? main] (find_symbol … ge main);
  do f ← opt_to_res ? [MSG BadFunction; CTX ? main] (find_funct_ptr … ge b);
  OK ? (Callstate f (nil ?) (None ?) (nil ?) m).

definition RTLabs_fullexec : fullexec io_out io_in ≝
  mk_fullexec … RTLabs_exec make_global make_initial_state.


(* Some lemmas about the semantics. *)

(* Note parts of frames and states that are preserved. *)

inductive frame_rel : frame → frame → Prop ≝
| mk_frame_rel :
  ∀func,locals,next,next_ok,sp,retdst,locals',next',next_ok'. frame_rel
   (mk_frame func locals next next_ok sp retdst)
   (mk_frame func locals' next' next_ok' sp retdst)
.

inductive state_rel : genv → state → state → Prop ≝
| normal : ∀ge,f,f',fs,m,m'. frame_rel f f' → state_rel ge (State f fs m) (State f' fs m')
| to_call : ∀ge,f,fs,m,fd,args,f',dst. frame_rel f f' → ∀b. find_funct_ptr ? ge b = Some ? fd → state_rel ge (State f fs m) (Callstate fd args dst (f'::fs) m)
(*
| tail_call : ∀ge,f,fs,m,fd,args,dst,m'. ∀b. find_funct_ptr ?? ge b = Some ? fd → state_rel ge (State f fs m) (Callstate fd args dst fs m')
*)
| from_call : ∀ge,fn,locals,next,nok,sp,fs,m,args,dst,m'. state_rel ge (Callstate (Internal ? fn) args dst fs m) (State (mk_frame fn locals next nok sp dst) fs m')
| to_ret : ∀ge,f,fs,m,rtv,dst,m'. state_rel ge (State f fs m) (Returnstate rtv dst fs m')
| from_ret : ∀ge,f,fs,rtv,dst,f',m. next f = next f' → frame_rel f f' → state_rel ge (Returnstate rtv dst (f::fs) m) (State f' fs m)
| finish : ∀ge,r,dst,m. state_rel ge (Returnstate (Some ? (Vint I32 r)) dst [ ] m) (Finalstate r)
.

lemma bind_ok : ∀A,B,e,f,v. ∀P:B → Type[0].
  (∀a. e = OK A a → f a = OK ? v → P v) →
  (match e with [ OK v ⇒ f v | Error m ⇒ Error ? m ] = OK ? v → P v).
#A #B *
[ #a #f #v #P #H #E whd in E:(??%?); @(H a) //
| #m #f #v #P #H #E whd in E:(??%?); destruct
] qed.

lemma eval_preserves : ∀ge,s,tr,s'.
  eval_statement ge s = Value ??? 〈tr,s'〉 →
  state_rel ge s s'.
#ge *
[ * #func #locals #next #next_ok #sp #retdst #fs #m
  #tr #s'
  whd in ⊢ (??%? → ?);
  generalize in ⊢ (??(?%)? → ?);
  cases (next_instruction ?)
  [ #l #LP whd in ⊢ (??%? → ?); #E destruct % %
  | #c #l #LP whd in ⊢ (??%? → ?); #E destruct % %
  | #t #r #c #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev @bind_ok #locals' #Eloc #E whd in E:(??%?); destruct % %
  | #t1 #t2 #op #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev @bind_ok #v' #Ev'  @bind_ok #loc #Eloc #E whd in E:(??%?); destruct % %
  | #t1 #t2 #t' #op #r1 #r2 #r3 #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #v' #Ev'  @bind_ok #loc #Eloc #E whd in E:(??%?); destruct % %
  | #ch #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev @bind_ok #v' #Ev'  @bind_ok #loc #Eloc #E whd in E:(??%?); destruct % %
  | #ch #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev @bind_ok #v' #Ev'  @bind_ok #loc #Eloc #E whd in E:(??%?); destruct % %
  | #id #rs #or #l #LP whd in ⊢ (??%? → ?); @bind_res_value #b #Eb @bind_ok #fd #Efd  @bind_ok #vs #Evs #E whd in E:(??%?); destruct %2 [ % | @b | cases (find_funct_ptr ???) in Efd ⊢ %; normalize [2:#fd'] #E' destruct @refl ] 
  | #r #rs #or #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev @bind_ok #fd #Efd  @bind_ok #vs #Evs #E whd in E:(??%?); destruct cases (find_funct_find_funct_ptr ?? v ??) [ #b * #Ev' #Efn %2 [ % | @b | @Efn ] | ||| cases (find_funct ???) in Efd ⊢ %; [2:#x] normalize #E' destruct @refl ]
(*
  | #id #rs #LP whd in ⊢ (??%? → ?); @bind_res_value #b #Eb @bind_ok #fd #Efd  @bind_ok #vs #Evs #E whd in E:(??%?); destruct %3 [ @b | cases (find_funct_ptr ????) in Efd ⊢ %; [2:#x] normalize #E' destruct @refl ]
  | #r #rs #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev @bind_ok #fd #Efd  @bind_ok #vs #Evs #E whd in E:(??%?); destruct cases (find_funct_find_funct_ptr ?? v ??) [ #r * #b * #c * #Ev' #Efn %3 [ @b | @Efn ] | ||| cases (find_funct ????) in Efd ⊢ %; [2:#x] normalize #E' destruct @refl ]
*)
  | #r #l1 #l2 #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev @bind_ok #b #Eb #E whd in E:(??%?); destruct % %
(*  | #r #ls #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev cases v [ #E whd in E:(??%?); destruct | #sz #i whd in ⊢ (??%? → ?); generalize in ⊢ (??(?%)? → ?); cases (nth_opt ?? ls) in ⊢ (∀e:???%. ??(match % with [_ ⇒ ?|_ ⇒ ?]?)? → ?); [ #e #E whd in E:(??%?); destruct | #l' #e #E whd in E:(??%?); destruct % % ] | *: [ 1,3: #vl ] #E whd in E:(??%?); destruct ]*)
  | #LP whd in ⊢ (??%? → ?); @bind_res_value #v cases (f_result func) [ 2: * #r #t whd in ⊢ (??%? → ?); @bind_ok #v0 #Ev0 ] #E whd in E:(??%?); #E' whd in E':(??%?); destruct %4
  ]
| * #fn #args #retdst #stk #m #tr #s'
  whd in ⊢ (??%? → ?);
  [ @bind_res_value #loc #Eloc cases (alloc m O ? (*?*)) #m' #b whd in ⊢ (??%? → ?);
    #E destruct %3
  | @bindIO_value #evargs #Eargs whd in ⊢ (??%? → ?); #E destruct
  ]
| #v #r * [2: #f #fs ] #m #tr #s'
  whd in ⊢ (??%? → ?);
  [ @bind_res_value #loc #Eloc #E whd in E:(??%?); destruct
    %5 cases f #func #locals #next #next_ok #sp #retdst %
  | cases v [ normalize #E destruct | * [ 2: * #r normalize #E destruct // | *: normalize #a try #b destruct ] ]
  ]
| #r #tr #s' normalize #E destruct 
] qed.

(* Show that, in principle at least, we can calculate which function we called
   on a transition. The proof is a functional inversion similar to eval_preserves,
   above. *)

definition func_block_of_exec : ∀ge,s,t,fd,args,dst,fs,m.
  eval_statement ge s = Value ??? 〈t,Callstate fd args dst fs m〉 →
  Σb:block. find_funct_ptr … ge b = Some ? fd.
#ge *
[ * #func #locals #next #nok #sp #dst #fs #m #tr #fd #args #dst' #fs' #m'
  whd in ⊢ (??%? → ?); generalize in ⊢ (??(?%)? → ?);
  cases (next_instruction ?)
  (* Function call cases. *)
  [ 8,9: #x #rs #or #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev @bind_ok #fd' #Efd @bind_ok #vs #Evs #E normalize in E; destruct
    [ %{v} @Efd
    | cases v in Efd;
      [ 4: * #b #off #Efd %{b} whd in Efd:(??(???%)?); cases (eq_offset ??) in Efd;
           [ #E @E | #E normalize in E; destruct ]
      | *: normalize #A try #B try #C destruct
      ]
    ]
  (* and everything else cannot occur, but we need to work through the
     definition to find the resulting state to demonstrate that. *)
  | #l #LP whd in ⊢ (??%? → ?); #E destruct
  | #c #l #LP whd in ⊢ (??%? → ?); #E destruct
  | #t #r #c #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev @bind_ok #locals' #Eloc #E whd in E:(??%?); destruct
  | #t1 #t2 #op #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev @bind_ok #v' #Ev'  @bind_ok #loc #Eloc #E whd in E:(??%?); destruct
  | #t1 #t2 #t' #op #r1 #r2 #r3 #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v1 #Ev1 @bind_ok #v2 #Ev2 @bind_ok #v' #Ev'  @bind_ok #loc #Eloc #E whd in E:(??%?); destruct
  | #ch #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev @bind_ok #v' #Ev'  @bind_ok #loc #Eloc #E whd in E:(??%?); destruct
  | #ch #r1 #r2 #l #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev @bind_ok #v' #Ev'  @bind_ok #loc #Eloc #E whd in E:(??%?); destruct
  | #r #l1 #l2 #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev @bind_ok #b #Eb #E whd in E:(??%?); destruct
(*  | #r #ls #LP whd in ⊢ (??%? → ?); @bind_res_value #v #Ev cases v [ #E whd in E:(??%?); destruct | #sz #i whd in ⊢ (??%? → ?); generalize in ⊢ (??(?%)? → ?); cases (nth_opt ?? ls) in ⊢ (∀e:???%. ??(match % with [_ ⇒ ?|_ ⇒ ?]?)? → ?); [ #e #E whd in E:(??%?); destruct | #l' #e #E whd in E:(??%?); destruct ] | *: [1,3: #vl] #E whd in E:(??%?); destruct ]*)
  | #LP whd in ⊢ (??%? → ?); @bind_res_value #v cases (f_result func) [ 2: * #r #t whd in ⊢ (??%? → ?); @bind_ok #v0 #Ev0 ] #E whd in E:(??%?); #E' whd in E':(??%?); destruct
  ]
| * #fn #args #retdst #stk #m #tr #fd' #args' #dst' #fs' #m'
  whd in ⊢ (??%? → ?);
  [ @bind_res_value #loc #Eloc cases (alloc m O ?(*?*)) #m' #b whd in ⊢ (??%? → ?);
    #E destruct
  | @bindIO_value #evargs #Eargs whd in ⊢ (??%? → ?); #E destruct
  ]
| #v #r * [2: #f #fs ] #m #tr #fd' #args' #dst' #fs' #m'
  whd in ⊢ (??%? → ?);
  [ @bind_res_value #loc #Eloc #E whd in E:(??%?); destruct
  | cases v [ normalize #E destruct | * [ 2: * #r normalize #E destruct | *: normalize #a try #b destruct ] ]
  ]
| #r #tr #fd' #args' #dst' #fs' #m'
  normalize #E destruct 
] qed.

lemma final_cannot_move : ∀ge,s.
  RTLabs_is_final s ≠ None ? →
  ∃err. eval_statement ge s = Wrong ??? err.
#ge *
[ #f #fs #m * #F cases (F ?) @refl
| #a #b #c #d #e * #F cases (F ?) @refl
| #a #b #c #d * #F cases (F ?) @refl
| #r #F % [2: @refl | skip ]
] qed.

lemma step_not_final : ∀ge,s,tr,s'.
  eval_statement ge s = Value … 〈tr,s'〉 →
  RTLabs_is_final s = None ?.
#ge #s #tr #s' #EX
lapply (final_cannot_move ge s)
cases (RTLabs_is_final s)
[ // | #r #F cases (F ?) [ #m #E >E in EX; #EX destruct | % #E destruct ]
qed.

lemma initial_state_is_not_final : ∀p,s.
  make_initial_state p = OK ? s →
  RTLabs_is_final s = None ?.
#p #s whd in ⊢ (??%? → ?);
@bind_ok #m #Em
@bind_ok #b #Eb
@bind_ok #f #Ef
#E destruct
@refl
qed.