(** * 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').

(* Specifications: *)
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#X30 = parent_sp s)) ->
exists st',
plus step tge (State rs1 m1') E0 st' /\
match_states (Mach.State s fb sp c ms2 m2) st'.
Lemma exec_straight_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 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'.
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'.