Library Asmgenproof

Correctness proof for PPC generation: main proof.





Require Import Coqlib.

Require Import Maps.

Require Import Errors.

Require Import AST.

Require Import Integers.

Require Import Floats.

Require Import Values.

Require Import Mem.

Require Import Events.

Require Import Globalenvs.

Require Import Smallstep.

Require Import Op.

Require Import Locations.

Require Import Mach.

Require Import Machconcr.

Require Import Machtyping.

Require Import Asm.

Require Import Asmgen.

Require Import Asmgenretaddr.

Require Import Asmgenproof1.




Section PRESERVATION.




Variable prog: Mach.program.

Variable tprog: Asm.program.

Hypothesis TRANSF: transf_program prog = Errors.OK tprog.




Let ge := Genv.globalenv prog.

Let tge := Genv.globalenv tprog.




Lemma symbols_preserved:

  forall id, Genv.find_symbol tge id = Genv.find_symbol ge id.






Lemma functions_translated:

  forall b f,

  Genv.find_funct_ptr ge b = Some f ->

  exists tf, Genv.find_funct_ptr tge b = Some tf /\ transf_fundef f = Errors.OK tf.

Proof

  (Genv.find_funct_ptr_transf_partial transf_fundef TRANSF).




Lemma functions_transl:

  forall f b,

  Genv.find_funct_ptr ge b = Some (Internal f) ->

  Genv.find_funct_ptr tge b = Some (Internal (transl_function f)).






Lemma functions_transl_no_overflow:

  forall b f,

  Genv.find_funct_ptr ge b = Some (Internal f) ->

  list_length_z (transl_function f) <= Int.max_unsigned.

Properties of control flow





Lemma find_instr_in:

  forall c pos i,

  find_instr pos c = Some i -> In i c.






Lemma find_instr_tail:

  forall c1 i c2 pos,

  code_tail pos c1 (i :: c2) ->

  find_instr pos c1 = Some i.






Remark code_tail_bounds:

  forall fn ofs i c,

  code_tail ofs fn (i :: c) -> 0 <= ofs < list_length_z fn.






Lemma code_tail_next:

  forall fn ofs i c,

  code_tail ofs fn (i :: c) ->

  code_tail (ofs + 1) fn c.






Lemma code_tail_next_int:

  forall fn ofs i c,

  list_length_z fn <= Int.max_unsigned ->

  code_tail (Int.unsigned ofs) fn (i :: c) ->

  code_tail (Int.unsigned (Int.add ofs Int.one)) fn c.

transl_code_at_pc pc fn c holds if the code pointer pc points within the PPC code generated by translating Mach function fn, and c is the tail of the generated code at the position corresponding to the code pointer pc.





Inductive transl_code_at_pc: val -> block -> Mach.function -> Mach.code -> Prop :=

  transl_code_at_pc_intro:

    forall b ofs f c,

    Genv.find_funct_ptr ge b = Some (Internal f) ->

    code_tail (Int.unsigned ofs) (transl_function f) (transl_code f c) ->

    transl_code_at_pc (Vptr b ofs) b f c.

The following lemmas show that straight-line executions (predicate exec_straight) correspond to correct PPC executions (predicate exec_steps) under adequate transl_code_at_pc hypotheses.





Lemma exec_straight_steps_1:

  forall fn c rs m c' rs' m',

  exec_straight tge fn c rs m c' rs' m' ->

  list_length_z fn <= Int.max_unsigned ->

  forall b ofs,

  rs#PC = Vptr b ofs ->

  Genv.find_funct_ptr tge b = Some (Internal fn) ->

  code_tail (Int.unsigned ofs) fn c ->

  plus step tge (State rs m) E0 (State rs' m').






Lemma exec_straight_steps_2:

  forall fn c rs m c' rs' m',

  exec_straight tge fn c rs m c' rs' m' ->

  list_length_z fn <= Int.max_unsigned ->

  forall b ofs,

  rs#PC = Vptr b ofs ->

  Genv.find_funct_ptr tge b = Some (Internal fn) ->

  code_tail (Int.unsigned ofs) fn c ->

  exists ofs',

     rs'#PC = Vptr b ofs'

  /\ code_tail (Int.unsigned ofs') fn c'.






Lemma exec_straight_exec:

  forall fb f c c' rs m rs' m',

  transl_code_at_pc (rs PC) fb f c ->

  exec_straight tge (transl_function f)

                (transl_code f c) rs m c' rs' m' ->

  plus step tge (State rs m) E0 (State rs' m').






Lemma exec_straight_at:

  forall fb f c c' rs m rs' m',

  transl_code_at_pc (rs PC) fb f c ->

  exec_straight tge (transl_function f)

                (transl_code f c) rs m (transl_code f c') rs' m' ->

  transl_code_at_pc (rs' PC) fb f c'.

Correctness of the return addresses predicted by PPCgen.return_address_offset.





Remark code_tail_no_bigger:

  forall pos c1 c2, code_tail pos c1 c2 -> (length c2 <= length c1)%nat.






Remark code_tail_unique:

  forall fn c pos pos',

  code_tail pos fn c -> code_tail pos' fn c -> pos = pos'.






Lemma return_address_offset_correct:

  forall b ofs fb f c ofs',

  transl_code_at_pc (Vptr b ofs) fb f c ->

  return_address_offset f c ofs' ->

  ofs' = ofs.

The find_label function returns the code tail starting at the given label. A connection with code_tail is then established.





Fixpoint find_label (lbl: label) (c: code) {struct c} : option code :=

  match c with

  | nil => None

  | instr :: c' =>

      if is_label lbl instr then Some c' else find_label lbl c'

  end.




Lemma label_pos_code_tail:

  forall lbl c pos c',

  find_label lbl c = Some c' ->

  exists pos',

  label_pos lbl pos c = Some pos' 

  /\ code_tail (pos' - pos) c c'

  /\ pos < pos' <= pos + list_length_z c.

The following lemmas show that the translation from Mach to PPC preserves labels, in the sense that the following diagram commutes:

                          translation
        Mach code ------------------------ PPC instr sequence
            |                                          |
            | Mach.find_label lbl       find_label lbl |
            |                                          |
            v                                          v
        Mach code tail ------------------- PPC instr seq tail
                          translation

The proof demands many boring lemmas showing that PPC constructor functions do not introduce new labels.





Section TRANSL_LABEL.




Variable lbl: label.




Remark loadimm_label:

  forall r n k, find_label lbl (loadimm r n k) = find_label lbl k.



Hint Rewrite loadimm_label: labels.




Remark addimm_1_label:

  forall r1 r2 n k, find_label lbl (addimm_1 r1 r2 n k) = find_label lbl k.



Remark addimm_2_label:

  forall r1 r2 n k, find_label lbl (addimm_2 r1 r2 n k) = find_label lbl k.



Remark addimm_label:

  forall r1 r2 n k, find_label lbl (addimm r1 r2 n k) = find_label lbl k.



Hint Rewrite addimm_label: labels.




Remark andimm_label:

  forall r1 r2 n k, find_label lbl (andimm r1 r2 n k) = find_label lbl k.



Hint Rewrite andimm_label: labels.




Remark orimm_label:

  forall r1 r2 n k, find_label lbl (orimm r1 r2 n k) = find_label lbl k.



Hint Rewrite orimm_label: labels.




Remark xorimm_label:

  forall r1 r2 n k, find_label lbl (xorimm r1 r2 n k) = find_label lbl k.



Hint Rewrite xorimm_label: labels.




Remark loadind_aux_label:

  forall base ofs ty dst k, find_label lbl (loadind_aux base ofs ty dst :: k) = find_label lbl k.



Remark loadind_label:

  forall base ofs ty dst k, find_label lbl (loadind base ofs ty dst k) = find_label lbl k.



Hint Rewrite loadind_label: labels.

Remark storeind_aux_label:

  forall base ofs ty dst k, find_label lbl (storeind_aux base ofs ty dst :: k) = find_label lbl k.



Remark storeind_label:

  forall base ofs ty src k, find_label lbl (storeind base src ofs ty k) = find_label lbl k.



Hint Rewrite storeind_label: labels.

Remark floatcomp_label:

  forall cmp r1 r2 k, find_label lbl (floatcomp cmp r1 r2 k) = find_label lbl k.






Remark transl_cond_label:

  forall cond args k, find_label lbl (transl_cond cond args k) = find_label lbl k.



Hint Rewrite transl_cond_label: labels.




Remark transl_op_cmp_label:

  forall c args r k,

  find_label lbl

    (match classify_condition c args with

     | condition_ge0 _ a _ =>

        Prlwinm (ireg_of r) (ireg_of a) Int.one Int.one

        :: Pxori (ireg_of r) (ireg_of r) (Cint Int.one) :: k

     | condition_lt0 _ a _ =>

        Prlwinm (ireg_of r) (ireg_of a) Int.one Int.one :: k

     | condition_default _ _ =>

        transl_cond c args

          (Pmfcrbit (ireg_of r) (fst (crbit_for_cond c))

           :: (if snd (crbit_for_cond c)

               then k

               else Pxori (ireg_of r) (ireg_of r) (Cint Int.one) :: k))

    end) = find_label lbl k.



Hint Rewrite transl_op_cmp_label: labels.




Remark transl_op_label:

  forall op args r k, find_label lbl (transl_op op args r k) = find_label lbl k.



Hint Rewrite transl_op_label: labels.




Remark transl_load_store_label:

  forall (mk1: constant -> ireg -> instruction) (mk2: ireg -> ireg -> instruction)

         addr args k,

  (forall c r, is_label lbl (mk1 c r) = false) ->

  (forall r1 r2, is_label lbl (mk2 r1 r2) = false) ->

  find_label lbl (transl_load_store mk1 mk2 addr args k) = find_label lbl k.



Hint Rewrite transl_load_store_label: labels.




Lemma transl_instr_label:

  forall f i k,

  find_label lbl (transl_instr f i k) = 

    if Mach.is_label lbl i then Some k else find_label lbl k.






Lemma transl_code_label:

  forall f c,

  find_label lbl (transl_code f c) = 

    option_map (transl_code f) (Mach.find_label lbl c).






Lemma transl_find_label:

  forall f,

  find_label lbl (transl_function f) = 

    option_map (transl_code f) (Mach.find_label lbl f.(fn_code)).






End TRANSL_LABEL.

A valid branch in a piece of Mach code translates to a valid ``go to'' transition in the generated PPC code.





Lemma find_label_goto_label:

  forall f lbl rs m c' b ofs,

  Genv.find_funct_ptr ge b = Some (Internal f) ->

  rs PC = Vptr b ofs ->

  Mach.find_label lbl f.(fn_code) = Some c' ->

  exists rs',

    goto_label (transl_function f) lbl rs m = OK rs' m  

  /\ transl_code_at_pc (rs' PC) b f c'

  /\ forall r, r <> PC -> rs'#r = rs#r.

Proof of semantic preservation

Semantic preservation is proved using simulation diagrams of the following form.

           st1 --------------- st2
            |                   |
           t|                  *|t
            |                   |
            v                   v
           st1'--------------- st2'

The invariant is the match_states predicate below, which includes:

The PPC code pointed by the PC register is the translation of the current Mach code sequence.
Mach register values and PPC register values agree.





Inductive match_stack: list Machconcr.stackframe -> Prop :=

  | match_stack_nil:

      match_stack nil

  | match_stack_cons: forall fb sp ra c s f,

      Genv.find_funct_ptr ge fb = Some (Internal f) ->

      wt_function f ->

      incl c f.(fn_code) ->

      transl_code_at_pc ra fb f c ->

      match_stack s ->

      match_stack (Stackframe fb sp ra c :: s).




Inductive match_states: Machconcr.state -> Asm.state -> Prop :=

  | match_states_intro:

      forall s fb sp c ms m rs f

        (STACKS: match_stack s)

        (FIND: Genv.find_funct_ptr ge fb = Some (Internal f))

        (WTF: wt_function f)

        (INCL: incl c f.(fn_code))

        (AT: transl_code_at_pc (rs PC) fb f c)

        (AG: agree ms sp rs),

      match_states (Machconcr.State s fb sp c ms m)

                   (Asm.State rs m)

  | match_states_call:

      forall s fb ms m rs

        (STACKS: match_stack s)

        (AG: agree ms (parent_sp s) rs)

        (ATPC: rs PC = Vptr fb Int.zero)

        (ATLR: rs LR = parent_ra s),

      match_states (Machconcr.Callstate s fb ms m)

                   (Asm.State rs m)

  | match_states_return:

      forall s ms m rs

        (STACKS: match_stack s)

        (AG: agree ms (parent_sp s) rs)

        (ATPC: rs PC = parent_ra s),

      match_states (Machconcr.Returnstate s ms m)

                   (Asm.State rs m).




Lemma exec_straight_steps:

  forall s fb sp m1 f c1 rs1 c2 m2 ms2,

  match_stack s ->

  Genv.find_funct_ptr ge fb = Some (Internal f) ->

  wt_function f ->

  incl c2 f.(fn_code) ->

  transl_code_at_pc (rs1 PC) fb f c1 ->

  (exists rs2,

     exec_straight tge (transl_function f) (transl_code f c1) rs1 m1 (transl_code f c2) rs2 m2

     /\ agree ms2 sp rs2) ->

  exists st',

  plus step tge (State rs1 m1) E0 st' /\

  match_states (Machconcr.State s fb sp c2 ms2 m2) st'.

We need to show that, in the simulation diagram, we cannot take infinitely many Mach transitions that correspond to zero transitions on the PPC side. Actually, all Mach transitions correspond to at least one PPC transition, except the transition from Machconcr.Returnstate to Machconcr.State. So, the following integer measure will suffice to rule out the unwanted behaviour.





Definition measure (s: Machconcr.state) : nat :=

  match s with

  | Machconcr.State _ _ _ _ _ _ => 0%nat

  | Machconcr.Callstate _ _ _ _ => 0%nat

  | Machconcr.Returnstate _ _ _ => 1%nat

  end.

We show the simulation diagram by case analysis on the Mach transition on the left. Since the proof is large, we break it into one lemma per transition.





Definition exec_instr_prop (s1: Machconcr.state) (t: trace) (s2: Machconcr.state) : Prop :=

  forall s1' (MS: match_states s1 s1'),

  (exists s2', plus step tge s1' t s2' /\ match_states s2 s2')

  \/ (measure s2 < measure s1 /\ t = E0 /\ match_states s2 s1')%nat.




Lemma exec_Mlabel_prop:

  forall (s : list stackframe) (fb : block) (sp : val)

         (lbl : Mach.label) (c : list Mach.instruction) (ms : Mach.regset)

         (m : mem),

  exec_instr_prop (Machconcr.State s fb sp (Mlabel lbl :: c) ms m) E0

                  (Machconcr.State s fb sp c ms m).






Lemma exec_Mgetstack_prop:

  forall (s : list stackframe) (fb : block) (sp : val) (ofs : int)

         (ty : typ) (dst : mreg) (c : list Mach.instruction)

         (ms : Mach.regset) (m : mem) (v : val),

  load_stack m sp ty ofs = Some v ->

  exec_instr_prop (Machconcr.State s fb sp (Mgetstack ofs ty dst :: c) ms m) E0

                  (Machconcr.State s fb sp c (Regmap.set dst v ms) m).






Lemma exec_Msetstack_prop:

  forall (s : list stackframe) (fb : block) (sp : val) (src : mreg)

         (ofs : int) (ty : typ) (c : list Mach.instruction)

         (ms : mreg -> val) (m m' : mem),

  store_stack m sp ty ofs (ms src) = Some m' ->

  exec_instr_prop (Machconcr.State s fb sp (Msetstack src ofs ty :: c) ms m) E0

                  (Machconcr.State s fb sp c ms m').






Lemma exec_Mgetparam_prop:

  forall (s : list stackframe) (fb : block) (f: Mach.function) (sp parent : val)

         (ofs : int) (ty : typ) (dst : mreg) (c : list Mach.instruction)

         (ms : Mach.regset) (m : mem) (v : val),

  Genv.find_funct_ptr ge fb = Some (Internal f) ->

  load_stack m sp Tint f.(fn_link_ofs) = Some parent ->

  load_stack m parent ty ofs = Some v ->

  exec_instr_prop (Machconcr.State s fb sp (Mgetparam ofs ty dst :: c) ms m) E0

                  (Machconcr.State s fb sp c (Regmap.set dst v ms) m).






Lemma exec_Mop_prop:

  forall (s : list stackframe) (fb : block) (sp : val) (op : operation)

         (args : list mreg) (res : mreg) (c : list Mach.instruction)

         (ms : mreg -> val) (m : mem) (v : val),

  eval_operation ge sp op ms ## args = Some v ->

  exec_instr_prop (Machconcr.State s fb sp (Mop op args res :: c) ms m) E0

                  (Machconcr.State s fb sp c (Regmap.set res v ms) m).






Lemma exec_Mload_prop:

  forall (s : list stackframe) (fb : block) (sp : val)

         (chunk : memory_chunk) (addr : addressing) (args : list mreg)

         (dst : mreg) (c : list Mach.instruction) (ms : mreg -> val)

         (m : mem) (a v : val),

  eval_addressing ge sp addr ms ## args = Some a ->

  loadv chunk m a = Some v ->

  exec_instr_prop (Machconcr.State s fb sp (Mload chunk addr args dst :: c) ms m)

               E0 (Machconcr.State s fb sp c (Regmap.set dst v ms) m).






Lemma exec_Mstore_prop:

  forall (s : list stackframe) (fb : block) (sp : val)

         (chunk : memory_chunk) (addr : addressing) (args : list mreg)

         (src : mreg) (c : list Mach.instruction) (ms : mreg -> val)

         (m m' : mem) (a : val),

  eval_addressing ge sp addr ms ## args = Some a ->

  storev chunk m a (ms src) = Some m' ->

  exec_instr_prop (Machconcr.State s fb sp (Mstore chunk addr args src :: c) ms m) E0

                  (Machconcr.State s fb sp c ms m').






Lemma exec_Mcall_prop:

  forall (s : list stackframe) (fb : block) (sp : val)

         (sig : signature) (ros : mreg + ident) (c : Mach.code)

         (ms : Mach.regset) (m : mem) (f : function) (f' : block)

         (ra : int),

  find_function_ptr ge ros ms = Some f' ->

  Genv.find_funct_ptr ge fb = Some (Internal f) ->

  return_address_offset f c ra ->

  exec_instr_prop (Machconcr.State s fb sp (Mcall sig ros :: c) ms m) E0

                  (Callstate (Stackframe fb sp (Vptr fb ra) c :: s) f' ms m).






Lemma exec_Mtailcall_prop:

  forall (s : list stackframe) (fb stk : block) (soff : int)

         (sig : signature) (ros : mreg + ident) (c : list Mach.instruction)

         (ms : Mach.regset) (m : mem) (f: Mach.function) (f' : block),

  find_function_ptr ge ros ms = Some f' ->

  Genv.find_funct_ptr ge fb = Some (Internal f) ->

  load_stack m (Vptr stk soff) Tint f.(fn_link_ofs) = Some (parent_sp s) ->

  load_stack m (Vptr stk soff) Tint f.(fn_retaddr_ofs) = Some (parent_ra s) ->

  exec_instr_prop

          (Machconcr.State s fb (Vptr stk soff) (Mtailcall sig ros :: c) ms m) E0

          (Callstate s f' ms (free m stk)).






Lemma exec_Mgoto_prop:

  forall (s : list stackframe) (fb : block) (f : function) (sp : val)

         (lbl : Mach.label) (c : list Mach.instruction) (ms : Mach.regset)

         (m : mem) (c' : Mach.code),

  Genv.find_funct_ptr ge fb = Some (Internal f) ->

  Mach.find_label lbl (fn_code f) = Some c' ->

  exec_instr_prop (Machconcr.State s fb sp (Mgoto lbl :: c) ms m) E0

                  (Machconcr.State s fb sp c' ms m).






Lemma exec_Mcond_true_prop:

  forall (s : list stackframe) (fb : block) (f : function) (sp : val)

         (cond : condition) (args : list mreg) (lbl : Mach.label)

         (c : list Mach.instruction) (ms : mreg -> val) (m : mem)

         (c' : Mach.code),

  eval_condition cond ms ## args = Some true ->

  Genv.find_funct_ptr ge fb = Some (Internal f) ->

  Mach.find_label lbl (fn_code f) = Some c' ->

  exec_instr_prop (Machconcr.State s fb sp (Mcond cond args lbl :: c) ms m) E0

                  (Machconcr.State s fb sp c' ms m).






Lemma exec_Mcond_false_prop:

  forall (s : list stackframe) (fb : block) (sp : val)

         (cond : condition) (args : list mreg) (lbl : Mach.label)

         (c : list Mach.instruction) (ms : mreg -> val) (m : mem),

  eval_condition cond ms ## args = Some false ->

  exec_instr_prop (Machconcr.State s fb sp (Mcond cond args lbl :: c) ms m) E0

                  (Machconcr.State s fb sp c ms m).






Lemma exec_Mjumptable_prop:

  forall (s : list stackframe) (fb : block) (f : function) (sp : val)

         (arg : mreg) (tbl : list Mach.label) (c : list Mach.instruction)

         (rs : mreg -> val) (m : mem) (n : int) (lbl : Mach.label)

         (c' : Mach.code),

  rs arg = Vint n ->

  list_nth_z tbl (Int.signed n) = Some lbl ->

  Genv.find_funct_ptr ge fb = Some (Internal f) ->

  Mach.find_label lbl (fn_code f) = Some c' ->

  exec_instr_prop

    (Machconcr.State s fb sp (Mjumptable arg tbl :: c) rs m) E0

    (Machconcr.State s fb sp c' rs m).






Lemma exec_Mreturn_prop:

  forall (s : list stackframe) (fb stk : block) (soff : int)

         (c : list Mach.instruction) (ms : Mach.regset) (m : mem) (f: Mach.function),

  Genv.find_funct_ptr ge fb = Some (Internal f) ->

  load_stack m (Vptr stk soff) Tint f.(fn_link_ofs) = Some (parent_sp s) ->

  load_stack m (Vptr stk soff) Tint f.(fn_retaddr_ofs) = Some (parent_ra s) ->

  exec_instr_prop (Machconcr.State s fb (Vptr stk soff) (Mreturn :: c) ms m) E0

                  (Returnstate s ms (free m stk)).






Hypothesis wt_prog: wt_program prog.




Lemma exec_function_internal_prop:

  forall (s : list stackframe) (fb : block) (ms : Mach.regset)

         (m : mem) (f : function) (m1 m2 m3 : mem) (stk : block),

  Genv.find_funct_ptr ge fb = Some (Internal f) ->

  alloc m (- fn_framesize f) (fn_stacksize f) = (m1, stk) ->

  let sp := Vptr stk (Int.repr (- fn_framesize f)) in

  store_stack m1 sp Tint f.(fn_link_ofs) (parent_sp s) = Some m2 ->

  store_stack m2 sp Tint f.(fn_retaddr_ofs) (parent_ra s) = Some m3 ->

  exec_instr_prop (Machconcr.Callstate s fb ms m) E0

                  (Machconcr.State s fb sp (fn_code f) ms m3).






Lemma exec_function_external_prop:

  forall (s : list stackframe) (fb : block) (ms : Mach.regset)

         (m : mem) (t0 : trace) (ms' : RegEq.t -> val)

         (ef : external_function) (args : list val) (res : val),

  Genv.find_funct_ptr ge fb = Some (External ef) ->

  event_match ef args t0 res ->

  Machconcr.extcall_arguments ms m (parent_sp s) (ef_sig ef) args ->

  ms' = Regmap.set (Conventions.loc_result (ef_sig ef)) res ms ->

  exec_instr_prop (Machconcr.Callstate s fb ms m)

               t0 (Machconcr.Returnstate s ms' m).






Lemma exec_return_prop:

  forall (s : list stackframe) (fb : block) (sp ra : val)

         (c : Mach.code) (ms : Mach.regset) (m : mem),

  exec_instr_prop (Machconcr.Returnstate (Stackframe fb sp ra c :: s) ms m) E0

                  (Machconcr.State s fb sp c ms m).






Theorem transf_instr_correct:

  forall s1 t s2, Machconcr.step ge s1 t s2 ->

  exec_instr_prop s1 t s2.

Proof

  (Machconcr.step_ind ge exec_instr_prop

           exec_Mlabel_prop

           exec_Mgetstack_prop

           exec_Msetstack_prop

           exec_Mgetparam_prop

           exec_Mop_prop

           exec_Mload_prop

           exec_Mstore_prop

           exec_Mcall_prop

           exec_Mtailcall_prop

           exec_Mgoto_prop

           exec_Mcond_true_prop

           exec_Mcond_false_prop

           exec_Mjumptable_prop

           exec_Mreturn_prop

           exec_function_internal_prop

           exec_function_external_prop

           exec_return_prop).




Lemma transf_initial_states:

  forall st1, Machconcr.initial_state prog st1 ->

  exists st2, Asm.initial_state tprog st2 /\ match_states st1 st2.






Lemma transf_final_states:

  forall st1 st2 r, 

  match_states st1 st2 -> Machconcr.final_state st1 r -> Asm.final_state st2 r.






Theorem transf_program_correct:

  forall (beh: program_behavior), not_wrong beh ->

  Machconcr.exec_program prog beh -> Asm.exec_program tprog beh.






End PRESERVATION.