Section TRANSL_LABEL.
Remark mk_mov_label:
forall rd rs k c, mk_mov rd rs k = OK c -> tail_nolabel k c.
Hint Resolve mk_mov_label: labels.
Remark mk_shrximm_label:
forall n k c, mk_shrximm n k = OK c -> tail_nolabel k c.
Hint Resolve mk_shrximm_label: labels.
Remark mk_shrxlimm_label:
forall n k c, mk_shrxlimm n k = OK c -> tail_nolabel k c.
Hint Resolve mk_shrxlimm_label: labels.
Remark mk_intconv_label:
forall f r1 r2 k c, mk_intconv f r1 r2 k = OK c ->
(forall r r', nolabel (f r r')) ->
tail_nolabel k c.
Hint Resolve mk_intconv_label: labels.
Remark mk_storebyte_label:
forall addr r k c, mk_storebyte addr r k = OK c -> tail_nolabel k c.
Hint Resolve mk_storebyte_label: labels.
Remark loadind_label:
forall base ofs ty dst k c,
loadind base ofs ty dst k = OK c ->
tail_nolabel k c.
Remark storeind_label:
forall base ofs ty src k c,
storeind src base ofs ty k = OK c ->
tail_nolabel k c.
Remark mk_setcc_base_label:
forall xc rd k,
tail_nolabel k (mk_setcc_base xc rd k).
Remark mk_setcc_label:
forall xc rd k,
tail_nolabel k (mk_setcc xc rd k).
Remark mk_jcc_label:
forall xc lbl' k,
tail_nolabel k (mk_jcc xc lbl' k).
Remark mk_sel_label:
forall xc rd r2 k c,
mk_sel xc rd r2 k = OK c ->
tail_nolabel k c.
Remark transl_cond_label:
forall cond args k c,
transl_cond cond args k = OK c ->
tail_nolabel k c.
Remark transl_op_label:
forall op args r k c,
transl_op op args r k = OK c ->
tail_nolabel k c.
Remark transl_load_label:
forall chunk addr args dest k c,
transl_load chunk addr args dest k = OK c ->
tail_nolabel k c.
Remark transl_store_label:
forall chunk addr args src k c,
transl_store chunk addr args src k = OK c ->
tail_nolabel k c.
End TRANSL_LABEL.
(** A valid branch in a piece of Mach code translates to a valid ``go to'' transition in the generated PPC code. *)
(** Existence of return addresses *)
(** * 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_states: Mach.state -> Asm.state -> Prop :=
| match_states_intro:
forall s fb sp c ep ms m m' rs f tf tc
(STACKS: match_stack ge s)
(FIND: Genv.find_funct_ptr ge fb = Some (Internal f))
(MEXT: Mem.extends m m')
(AT: transl_code_at_pc ge (rs PC) fb f c ep tf tc)
(AG: agree ms sp rs)
(AXP: ep = true -> rs#RAX = parent_sp s),
match_states (Mach.State s fb sp c ms m)
(Asm.State rs m')
| match_states_call:
forall s fb ms m m' rs
(STACKS: match_stack ge s)
(MEXT: Mem.extends m m')
(AG: agree ms (parent_sp s) rs)
(ATPC: rs PC = Vptr fb Ptrofs.zero)
(ATLR: rs RA = parent_ra s),
match_states (Mach.Callstate s fb ms m)
(Asm.State rs m')
| match_states_return:
forall s ms m m' rs
(STACKS: match_stack ge s)
(MEXT: Mem.extends m m')
(AG: agree ms (parent_sp s) rs)
(ATPC: rs PC = parent_ra s),
match_states (Mach.Returnstate s ms m)
(Asm.State rs m').

Lemma exec_straight_steps:
forall s fb f rs1 i c ep tf tc m1' m2 m2' sp ms2,
match_stack ge s ->
Mem.extends m2 m2' ->
Genv.find_funct_ptr ge fb = Some (Internal f) ->
transl_code_at_pc ge (rs1 PC) fb f (i :: c) ep tf tc ->
(forall k c (TR: transl_instr f i ep k = OK c),
exists rs2,
exec_straight tge tf c rs1 m1' k rs2 m2'
/\ agree ms2 sp rs2
/\ (it1_is_parent ep i = true -> rs2#RAX = parent_sp s)) ->
exists st',
plus step tge (State rs1 m1') E0 st' /\
match_states (Mach.State s fb sp c ms2 m2) st'.