Section TRANSL_LABEL.
Remark loadimm32_label:
forall r n k, tail_nolabel k (loadimm32 r n k).
Hint Resolve loadimm32_label: labels.
Remark opimm32_label:
forall op opimm r1 r2 n k,
(forall r1 r2 r3, nolabel (op r1 r2 r3)) ->
(forall r1 r2 n, nolabel (opimm r1 r2 n)) ->
tail_nolabel k (opimm32 op opimm r1 r2 n k).
Hint Resolve opimm32_label: labels.
Remark loadimm64_label:
forall r n k, tail_nolabel k (loadimm64 r n k).
Hint Resolve loadimm64_label: labels.
Remark opimm64_label:
forall op opimm r1 r2 n k,
(forall r1 r2 r3, nolabel (op r1 r2 r3)) ->
(forall r1 r2 n, nolabel (opimm r1 r2 n)) ->
tail_nolabel k (opimm64 op opimm r1 r2 n k).
Hint Resolve opimm64_label: labels.
Remark addptrofs_label:
forall r1 r2 n k, tail_nolabel k (addptrofs r1 r2 n k).
Hint Resolve addptrofs_label: labels.
Remark transl_cond_float_nolabel:
forall c r1 r2 r3 insn normal,
transl_cond_float c r1 r2 r3 = (insn, normal) -> nolabel insn.
Remark transl_cond_single_nolabel:
forall c r1 r2 r3 insn normal,
transl_cond_single c r1 r2 r3 = (insn, normal) -> nolabel insn.
Remark transl_cbranch_label:
forall cond args lbl k c,
transl_cbranch cond args lbl k = OK c -> tail_nolabel k c.
Remark transl_cond_op_label:
forall cond args r k c,
transl_cond_op cond r 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 indexed_memory_access_label:
forall (mk_instr: ireg -> offset -> instruction) base ofs k,
(forall r o, nolabel (mk_instr r o)) ->
tail_nolabel k (indexed_memory_access mk_instr base ofs k).
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 src base ofs ty k c,
storeind src base ofs ty k = OK c -> tail_nolabel k c.
Remark loadind_ptr_label:
forall base ofs dst k, tail_nolabel k (loadind_ptr base ofs dst k).
Remark storeind_ptr_label:
forall src base ofs k, tail_nolabel k (storeind_ptr src base ofs k).
Remark transl_memory_access_label:
forall (mk_instr: ireg -> offset -> instruction) addr args k c,
(forall r o, nolabel (mk_instr r o)) ->
transl_memory_access mk_instr addr args k = OK c ->
tail_nolabel k c.
Remark make_epilogue_label:
forall f k, tail_nolabel k (make_epilogue f k).
End TRANSL_LABEL.
(** A valid branch in a piece of Mach code translates to a valid ``go to'' transition in the generated Asm 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 Asm code pointed by the PC register is the translation of the current Mach code sequence. - Mach register values and Asm 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)
(DXP: ep = true -> rs#X30 = 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_opt_steps_goto:
forall s fb f rs1 i c ep tf tc m1' m2 m2' sp ms2 lbl c',
match_stack ge s ->
Mem.extends m2 m2' ->
Genv.find_funct_ptr ge fb = Some (Internal f) ->
Mach.find_label lbl f.(Mach.fn_code) = Some c' ->
transl_code_at_pc ge (rs1 PC) fb f (i :: c) ep tf tc ->
it1_is_parent ep i = false ->
(forall k c (TR: transl_instr f i ep k = OK c),
exists jmp, exists k', exists rs2,
exec_straight_opt tge tf c rs1 m1' (jmp :: k') rs2 m2'
/\ agree ms2 sp rs2
/\ exec_instr tge tf jmp rs2 m2' = goto_label tf lbl rs2 m2') ->
exists st',
plus step tge (State rs1 m1') E0 st' /\
match_states (Mach.State s fb sp c' ms2 m2) st'.