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SoundPostcondCmdAdd.v
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SoundPostcondCmdAdd.v
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Require Import syntax.
Require Import alist.
Require Import FMapWeakList.
Require Import Classical.
Require Import Coqlib.
Require Import infrastructure.
Require Import Metatheory.
Import LLVMsyntax.
Import LLVMinfra.
Require Import opsem.
Require Import sflib.
Require Import paco.
Import Opsem.
Require Import TODO.
Require Import Exprs.
Require Import Hints.
Require Import Postcond.
Require Import Validator.
Require Import GenericValues.
Require AssnMem.
Require AssnState.
Require Import Inject.
Require Import SoundBase.
Require Import SoundReduceMaydiff.
Require Import memory_props.
Require Import TODOProof.
Require Import MemAux.
Set Implicit Arguments.
Definition cmd_is_normal (c:cmd): bool :=
match c with
| insn_malloc _ _ _ _
| insn_free _ _ _
| insn_alloca _ _ _ _
| insn_load _ _ _ _
| insn_store _ _ _ _ _
| insn_call _ _ _ _ _ _ _ => false
| _ => true
end.
Lemma normal_event
conf st0 st1 evt cmd cmds
(STEP: sInsn conf st0 st1 evt)
(CMDS: st0.(EC).(CurCmds) = cmd::cmds)
(NORMAL: cmd_is_normal cmd):
evt = events.E0.
Proof.
inv STEP; ss. inv CMDS. ss.
Qed.
Lemma AtomSetImpl_inter_empty
a l1 l2
(EMPTY: AtomSetImpl.Empty (AtomSetImpl.inter l1 l2))
(IN: a `in` l1)
:
<<NOTIN: a `notin` l2>>.
Proof.
ii. exploit EMPTY; eauto.
Qed.
Lemma AtomSetImpl_from_list_inter_is_empty
l1 l2
(INTER_EMPTY: AtomSetImpl.is_empty
(AtomSetImpl.inter (AtomSetImpl_from_list l1)
(AtomSetImpl_from_list l2)) = true)
:
List.Forall (fun x => List.Forall (fun y => x <> y) l2) l1
.
Proof.
revert l2 INTER_EMPTY. induction l1; ss. i.
apply AtomSetFacts.is_empty_iff in INTER_EMPTY.
exploit IHl1.
{ rewrite <- AtomSetFacts.is_empty_iff.
ii. eapply INTER_EMPTY.
eapply AtomSetFacts.inter_s_m_Proper; eauto.
- ii.
apply AtomSetFacts.mem_iff, AtomSetImpl_from_list_spec. right.
apply AtomSetImpl_from_list_spec, AtomSetFacts.mem_iff. ss.
- reflexivity.
}
i. econs; ss. clear -INTER_EMPTY.
hexploit AtomSetImpl_inter_empty; eauto.
{ apply AtomSetFacts.mem_iff, AtomSetImpl_from_list_spec. left. ss. }
intro A. des.
apply AtomSetFacts.not_mem_iff in A.
apply Forall_forall. ii. subst.
apply AtomSetImpl_from_list_spec in H. clarify.
Qed.
Ltac simpl_list :=
repeat match goal with
| [ H: Forall _ (_ :: _) |- _ ] => inv H
| [ H: Forall _ [] |- _ ] => clear H
end.
Ltac des_lookupAL_updateAddAL :=
repeat match goal with
| [ H: context[lookupAL ?t (updateAddAL ?t _ ?idA _) ?idB] |- _ ] =>
destruct (eq_atom_dec idB idA);
[ss; clarify; rewrite lookupAL_updateAddAL_eq in H |
ss; clarify; rewrite <- lookupAL_updateAddAL_neq in H]; ss; clarify
| [ |- context[lookupAL ?t (updateAddAL ?t _ ?idA _) ?idB] ] =>
destruct (eq_atom_dec idB idA);
[ss; clarify; rewrite lookupAL_updateAddAL_eq |
ss; clarify; rewrite <- lookupAL_updateAddAL_neq]; ss; clarify
end.
Ltac simpl_ep_set :=
repeat
match goal with
| [H: ExprPairSet.In _ (ExprPairSet.add _ _) |- _] =>
eapply ExprPairSetFacts.add_iff in H; des; clarify
end.
Ltac u := autounfold in *.
Hint Unfold AssnState.Unary.sem_idT.
Hint Unfold Cmd.get_ids.
Hint Unfold getOperandValue.
Ltac clear_true :=
repeat match goal with
| [ H: is_true true |- _ ] => clear H
| [ H: True |- _ ] => clear H
| [ H: ?x = ?x |- _ ] => clear H
end.
Lemma remove_def_from_maydiff_Subset_old
id0 inv0
:
Assertion.Subset
inv0
(remove_def_from_maydiff
id0 id0
(Assertion.update_tgt (Assertion.update_unique (add id0))
(Assertion.update_src (Assertion.update_unique (add id0)) inv0)))
.
Proof.
destruct inv0; ss.
destruct src; ss.
destruct tgt; ss.
unfold Assertion.update_src, Assertion.update_tgt,
remove_def_from_maydiff, Assertion.update_maydiff. ss.
destruct (id_dec id0 id0); clarify.
unfold Assertion.update_unique; ss.
econs; ss; try (econs; try splits; ss).
-
ii.
eapply AtomSetFacts.add_s_m. eauto.
apply AtomSetFacts.Subset_refl.
info_eauto.
-
ii.
eapply AtomSetFacts.add_s_m. eauto.
apply AtomSetFacts.Subset_refl.
info_eauto.
-
ii.
rewrite IdTSetFacts.mem_iff in *.
rewrite IdTSetFacts.remove_b in H.
des_bool; des; ss.
Qed.
Lemma remove_def_from_maydiff_Subset
id0 inv0
:
Assertion.Subset
inv0
(remove_def_from_maydiff
id0 id0 (Assertion.update_src (Assertion.update_unique (add id0)) inv0))
.
Proof.
destruct inv0; ss.
destruct src; ss.
destruct tgt; ss.
unfold Assertion.update_src, Assertion.update_tgt,
remove_def_from_maydiff, Assertion.update_maydiff. ss.
destruct (id_dec id0 id0); clarify.
unfold Assertion.update_unique; ss.
econs; ss; try (econs; try splits; ss).
-
ii.
eapply AtomSetFacts.add_s_m. eauto.
apply AtomSetFacts.Subset_refl.
info_eauto.
-
ii.
rewrite IdTSetFacts.mem_iff in *.
rewrite IdTSetFacts.remove_b in H.
des_bool; des; ss.
Qed.
Lemma add_unique_alloca
cmds_src id0 align0 inv_unary TD F B lc gn ECS0 tmn SRC_MEM als SRC_MEM_STEP tsz mb conf_src gmax
(ALLOCA: alloca TD SRC_MEM tsz gn align0 = Some (SRC_MEM_STEP, mb))
(UNIQUE: AtomSetImpl.For_all
(AssnState.Unary.sem_unique conf_src
{|
EC := {|
CurFunction := F;
CurBB := B;
CurCmds := cmds_src;
Terminator := tmn;
Locals := updateAddAL GenericValue lc id0 (blk2GV TD mb);
Allocas := mb :: als |};
ECS := ECS0;
Mem := SRC_MEM_STEP |} gmax) (Assertion.unique inv_unary))
(WF : MemProps.wf_Mem gmax (CurTargetData conf_src) SRC_MEM)
(GLOBALS : genericvalues_inject.wf_globals gmax (Globals conf_src))
(WF_LOCAL : MemProps.wf_lc SRC_MEM lc)
:
<<UNIQUE_ADD: AtomSetImpl.For_all
(AssnState.Unary.sem_unique conf_src
{|
EC := {|
CurFunction := F;
CurBB := B;
CurCmds := cmds_src;
Terminator := tmn;
Locals := updateAddAL GenericValue lc id0 (blk2GV TD mb);
Allocas := mb :: als |};
ECS := ECS0;
Mem := SRC_MEM_STEP |} gmax) (add id0 (Assertion.unique inv_unary))>>
.
Proof.
intros x xIn.
apply AtomSetFacts.add_iff in xIn.
des; [clear UNIQUE; subst id0|apply UNIQUE; auto].
econs; eauto; ss.
+ (* VAL *)
des_lookupAL_updateAddAL.
+ (* LOCALS *)
ii.
des_lookupAL_updateAddAL.
eapply locals_alloca_diffblock; ss; eauto.
+ (* MEM *)
ii. eapply mload_alloca_diffblock; eauto.
+ (* GLOBALS *)
ii. ss. clarify.
des; ss. clarify.
eapply globals_alloca_diffblock; eauto.
Qed.
Lemma postcond_cmd_add_inject_sound
m_src conf_src st0_src st1_src cmd_src cmds_src def_src uses_src
m_tgt conf_tgt st0_tgt st1_tgt cmd_tgt cmds_tgt def_tgt uses_tgt
invst1 assnmem1 inv1
invst0 assnmem0 inv0
evt
(CONF: AssnState.valid_conf m_src m_tgt conf_src conf_tgt)
(POSTCOND_CHECK: Postcond.postcond_cmd_check
cmd_src cmd_tgt def_src def_tgt uses_src uses_tgt inv1)
(STATE: AssnState.Rel.sem conf_src conf_tgt st0_src st0_tgt invst0 assnmem0 inv0)
(STATE_STEP: AssnState.Rel.sem conf_src conf_tgt st1_src st1_tgt invst1 assnmem1 inv1)
(MEM: AssnMem.Rel.sem conf_src conf_tgt st0_src.(Mem) st0_tgt.(Mem) assnmem0)
(MEM_STEP: AssnMem.Rel.sem conf_src conf_tgt st1_src.(Mem) st1_tgt.(Mem) assnmem1)
(MEM_GMAX: assnmem0.(AssnMem.Rel.gmax) = assnmem1.(AssnMem.Rel.gmax))
(STEP_SRC: sInsn conf_src st0_src st1_src evt)
(STEP_TGT: sInsn conf_tgt st0_tgt st1_tgt evt)
(CMDS_SRC: st0_src.(EC).(CurCmds) = cmd_src :: cmds_src)
(CMDS_TGT: st0_tgt.(EC).(CurCmds) = cmd_tgt :: cmds_tgt)
(NONCALL_SRC: Instruction.isCallInst cmd_src = false)
(NONCALL_TGT: Instruction.isCallInst cmd_tgt = false)
(DEF_SRC: def_src = AtomSetImpl_from_list (option_to_list (Cmd.get_def cmd_src)))
(DEF_TGT: def_tgt = AtomSetImpl_from_list (option_to_list (Cmd.get_def cmd_tgt)))
(USES_SRC: uses_src = AtomSetImpl_from_list (Cmd.get_ids cmd_src))
(USES_TGT: uses_tgt = AtomSetImpl_from_list (Cmd.get_ids cmd_tgt))
(ALLOC_INJECT: alloc_inject conf_src conf_tgt st0_src st0_tgt
st1_src st1_tgt cmd_src cmd_tgt assnmem1)
(ALLOC_PRIVATE: alloc_private conf_src conf_tgt cmd_src cmd_tgt st0_src st0_tgt
st1_src st1_tgt assnmem1)
:
<<STATE: AssnState.Rel.sem
conf_src conf_tgt st1_src st1_tgt invst1 assnmem1
(Postcond.postcond_cmd_add_inject cmd_src cmd_tgt inv1)>> /\
<<POSTCOND: Postcond.postcond_cmd_check
cmd_src cmd_tgt def_src def_tgt uses_src uses_tgt
(Postcond.postcond_cmd_add_inject cmd_src cmd_tgt inv1)>>
.
Proof.
remember (st0_src.(Mem).(Memory.Mem.nextblock)) as src_nxt.
remember (st0_tgt.(Mem).(Memory.Mem.nextblock)) as tgt_nxt.
destruct (classic (Postcond.postcond_cmd_add_inject cmd_src cmd_tgt inv1 = inv1)).
{ rewrite H in *. esplits; eauto. }
destruct cmd_src, cmd_tgt; ss.
- (* nop, alloca *)
clear ALLOC_INJECT.
unfold postcond_cmd_check in *. des_ifs; des_bool; clarify.
(* ss. clear_true. *)
(* splits; ss. *)
(* apply_all_once AtomSetImpl_from_list_inter_is_empty. *)
(* simpl_list. *)
(* inv STATE_STEP. *)
(* ((inv STEP_SRC; ss); []). *)
(* (* inv SRC. *) *)
(* inv CMDS_SRC. *)
(* econs; eauto; []. *)
(* ((inv STEP_TGT; ss); []). *)
(* (* inv TGT. *) *)
(* inv CMDS_TGT. *)
(* ss. *)
(* clear MAYDIFF. *)
(* inv TGT. *)
(* clear H H2. *)
(* clear MEM_STEP. *)
(* clear CONF. *)
(* (* inv TGT. clear LESSDEF0 NOALIAS0 UNIQUE0 PRIVATE0 WF_LOCAL0. *) *)
(* unfold alloc_private, alloc_private_unary in *. ss. *)
(* destruct ALLOC_PRIVATE as [_ ALLOC_PRIVATE]. *)
(* exploit ALLOC_PRIVATE; eauto. clear ALLOC_PRIVATE. intro ALLOC_PRIVATE. *)
(* econs; eauto; (* [|]; *) clear LESSDEF NOALIAS WF_LOCAL. *)
(* (* + clear ALLOC_PRIVATE. *) *)
(* (* clear PRIVATE. *) *)
(* (* unfold Assertion.update_src. ss. *) *)
(* (* intros ____id____ IN. *) *)
(* (* eapply AtomSetFacts.add_iff in IN. *) *)
(* (* des; [|eauto]; []. *) *)
(* (* subst. *) *)
(* (* eapply add_unique_alloca; eauto; try apply MEM; try apply STATE; *) *)
(* (* rewrite <- MEM_GMAX; try apply MEM; try apply STATE. *) *)
(* + clear UNIQUE. *)
(* clear MEM SRC STATE. *)
(* unfold Assertion.update_private. ss. *)
(* ii. rewrite IdTSetFacts.add_iff in *. des. *)
(* { (* x is alloca *) *)
(* destruct x as [[] x]; ss. *)
(* unfold IdT.lift, AssnState.Unary.sem_idT in *. ss. clarify. *)
(* exploit ALLOC_PRIVATE0; eauto. *)
(* } *)
(* { exploit PRIVATE; eauto. } *)
- (* allica, nop *)
clear ALLOC_INJECT.
unfold postcond_cmd_check in *. des_ifs; des_bool; clarify.
ss. clear_true.
splits; ss.
apply_all_once AtomSetImpl_from_list_inter_is_empty.
simpl_list.
inv STATE_STEP.
((inv STEP_SRC; ss); []).
(* inv SRC. *)
inv CMDS_SRC.
econs; eauto; [].
((inv STEP_TGT; ss); []).
(* inv TGT. *)
inv CMDS_TGT.
ss.
clear MAYDIFF.
inv SRC.
clear H H2.
clear MEM_STEP.
clear CONF.
(* inv TGT. clear LESSDEF0 NOALIAS0 UNIQUE0 PRIVATE0 WF_LOCAL0. *)
unfold alloc_private, alloc_private_unary in *. ss.
destruct ALLOC_PRIVATE as [ALLOC_PRIVATE _].
exploit ALLOC_PRIVATE; eauto. clear ALLOC_PRIVATE. intro ALLOC_PRIVATE.
econs; eauto; [|]; clear LESSDEF NOALIAS WF_LOCAL.
+ clear ALLOC_PRIVATE.
clear PRIVATE.
unfold Assertion.update_src. ss.
intros ____id____ IN.
eapply AtomSetFacts.add_iff in IN.
des; [|eauto]; [].
subst.
eapply add_unique_alloca; eauto; try apply MEM; try apply STATE;
rewrite <- MEM_GMAX; try apply MEM; try apply STATE.
+ clear UNIQUE.
clear MEM TGT STATE.
unfold Assertion.update_private. ss.
ii. rewrite IdTSetFacts.add_iff in *. des.
{ (* x is alloca *)
destruct x as [[] x]; ss.
unfold IdT.lift, AssnState.Unary.sem_idT in *. ss. clarify.
exploit ALLOC_PRIVATE0; eauto. solve_leibniz. clarify. }
{ exploit PRIVATE; eauto. }
- (* alloca, alloca *)
(*
* assnmem1 = assnmem0 + (newl_src -> newl_tgt)
* invstate.rel.sem inv0: possible (monotone w.r.t. assnmem)
* invstate.rel.sem unique (added): newl <> old locations
* newl: st0.mem.nextblock,
wf: st0's old locations < st0.mem.nextblock (genericvalues_inject.wf_sb_mi)
* invstate.rel.sem remove_def_from_maydiff (add): possible
* assnmem.rel.sem: adding (newl_src -> newl_tgt)
* assnmem.rel.le: possible
* postcond_cmd_check: monotonicity
*)
clear ALLOC_PRIVATE.
(* unfold postcond_cmd_check in POSTCOND_CHECK. des_ifs; des_bool; clarify. *)
unfold postcond_cmd_check in *. des_ifs; des_bool; clarify.
{
erewrite postcond_cmd_inject_event_Subset in Heq1; clarify.
ss.
repeat (des_bool; des); des_sumbool; clarify.
eapply remove_def_from_maydiff_Subset.
}
ss. repeat (des_bool; des; des_sumbool). subst.
clear_true.
splits; ss.
unfold remove_def_from_maydiff in *.
destruct (id_dec id0 id0); [clear_true|ss].
apply_all_once AtomSetImpl_from_list_inter_is_empty.
simpl_list.
inv STATE_STEP.
((inv STEP_SRC; ss); []).
(* inv SRC. *)
inv CMDS_SRC.
((inv STEP_TGT; ss); []).
(* inv TGT. *)
inv CMDS_TGT.
ss.
rename Mem0 into SRC_MEM.
rename Mem' into SRC_MEM_STEP.
rename Mem1 into TGT_MEM.
rename Mem'0 into TGT_MEM_STEP.
remember {| CurSystem := S; CurTargetData := TD;
CurProducts := Ps; Globals := gl; FunTable := fs |} as conf_src.
remember {| CurSystem := S0; CurTargetData := TD0;
CurProducts := Ps0; Globals := gl0; FunTable := fs0 |} as conf_tgt.
remember {|
EC := {|
CurFunction := F;
CurBB := B;
CurCmds := cmds_src;
Terminator := tmn;
Locals := updateAddAL GenericValue lc id0 (blk2GV TD mb);
Allocas := mb :: als |};
ECS := ECS0;
Mem := SRC_MEM_STEP |} as EC_src.
remember {|
EC := {|
CurFunction := F0;
CurBB := B0;
CurCmds := cmds_tgt;
Terminator := tmn0;
Locals := updateAddAL GenericValue lc0 id0 (blk2GV TD0 mb0);
Allocas := mb0 :: als0 |};
ECS := ECS1;
Mem := TGT_MEM_STEP |} as EC_tgt.
{
econs; eauto.
- (* SRC *)
inv SRC. inv MEM. inv STATE.
econs; eauto; []. ss.
eapply add_unique_alloca; eauto; try apply SRC; try apply SRC0;
rewrite <- MEM_GMAX; try apply SRC; try apply SRC0.
(* - (* TGT *) *)
(* inv TGT. inv MEM. inv STATE. *)
(* econs; eauto; []. ss. *)
(* eapply add_unique_alloca; eauto; try apply TGT; try apply TGT0; *)
(* rewrite <- MEM_GMAX; try apply TGT; try apply TGT0. *)
- (* MAYDIFF *)
inv SRC. inv TGT.
ii.
simpl in NOTIN.
rewrite IdTSetFacts.remove_b in NOTIN. repeat (des_bool; des).
{ eapply MAYDIFF; eauto. }
unfold IdTSetFacts.eqb in NOTIN.
destruct (IdTSetFacts.eq_dec (IdT.lift Tag.physical id0) id1); [|ss].
clear_true. clarify.
solve_leibniz. clarify.
u. ss.
des_lookupAL_updateAddAL.
clarify. clear_true.
esplits; eauto.
move H3 at bottom.
move H7 at bottom.
idtac.
unfold blk2GV.
unfold ptr2GV.
unfold val2GV.
Fail eapply genericvalues_inject.gv_inject_cons.
move CONF at bottom.
inv CONF. inv INJECT. simpl in TARGETDATA. clarify.
econs; eauto. { i; clarify. }
rename mb into _____________mb______________.
rename mb0 into _____________mb0______________.
move WF_LOCAL0 at bottom.
move H7 at bottom.
(* exploit MemProps.nextblock_malloc; try apply H3; []; ii; des. *)
(* exploit MemProps.nextblock_malloc; try apply H7; []; ii; des. *)
exploit MemProps.alloca_result; try apply H3; []; intro ALLOCA_RES1; des.
exploit MemProps.alloca_result; try apply H7; []; intro ALLOCA_RES2; des.
rewrite ALLOCA_RES1. rewrite ALLOCA_RES2. (* subst, clarify ruin ordering of premisses *)
move ALLOC_INJECT at bottom.
unfold alloc_inject in ALLOC_INJECT.
exploit ALLOC_INJECT; eauto; []; ii; des; clear ALLOC_INJECT.
destruct assnmem1; ss.
unfold alloc_inject_unary in *. des. ss.
des_lookupAL_updateAddAL.
econs; eauto.
- ss. clarify.
}
Qed.
Lemma opsem_assigns_getCmdID
conf st0 st1 evt
cmd cmds x
(STEP: sInsn conf st0 st1 evt)
(CMDS: st0.(EC).(CurCmds) = cmd :: cmds)
(NONCALL: Instruction.isCallInst cmd = false)
(CMD_ID: getCmdID cmd = Some x)
: exists gv, lookupAL _ st1.(EC).(Locals) x = Some gv.
Proof.
inv STEP; ss; clarify; ss; clarify;
try by esplits; apply lookupAL_updateAddAL_eq.
Qed.
Lemma wf_GVs__lessthan_undef
TD gl t gvu gv
(CONST_GV: const2GV TD gl (const_undef t) = Some gvu)
(WF_GVS: opsem_wf.OpsemPP.wf_GVs TD gv t)
: GVs.lessdef gvu gv.
Proof.
exploit const2GV_undef; eauto. i. des.
inv WF_GVS. unfold gv_chunks_match_typ in *. des_ifs.
apply all_undef_lessdef_aux; eauto.
exploit vm_matches_typ__eq__snd; eauto. i.
rewrite util.snd_split__map_snd in *. eauto.
apply vm_matches_typ__gv_has_chunk in H1. unfold gv_has_chunk in *.
eapply Forall_impl; eauto.
i. ss. des_ifs. apply genericvalues_inject.has_chunk__has_chunkb. s. ss.
Qed.
Lemma lessdef_definedness
conf st0 st1 invst evt
cmd cmds exp_pair
(STEP: sInsn conf st0 st1 evt)
(WF_CONF: opsem_wf.OpsemPP.wf_Config conf)
(WF_STATE_PREV: opsem_wf.OpsemPP.wf_State conf st0)
(CMDS: st0.(EC).(CurCmds) = cmd :: cmds)
(NONCALL: Instruction.isCallInst cmd = false)
(DEFINED: postcond_cmd_get_definedness cmd = Some exp_pair)
: AssnState.Unary.sem_lessdef conf st1 invst exp_pair.
Proof.
ii.
unfold postcond_cmd_get_definedness in *. des_ifs. ss.
unfold AssnState.Unary.sem_idT. ss.
unfold Cmd.get_def in *.
assert (CMD_TYP: lookupTypViaIDFromFdef st1.(EC).(CurFunction) i0 = Some t).
{ replace st1.(EC).(CurFunction) with st0.(EC).(CurFunction); cycle 1.
{ inv STEP; ss; by clarify. }
unfold opsem_wf.OpsemPP.wf_State in *. des_ifs. ss. des.
unfold opsem_wf.OpsemPP.wf_ExecutionContext in *.
destruct EC0. destruct WF_STATE_PREV as (_ & IN_FDEF & IN_PRODS & WF_LC & _ & CMDS_BB0).
destruct CurBB0 as [l [phis_BB0 cmds_BB0 tmn_BB0]].
eapply infrastructure_props.uniqF__lookupTypViaIDFromFdef; eauto.
- ss.
eapply infrastructure_props.uniqSystem__uniqFdef.
+ inv WF_CONF1. eauto.
+ unfold productInSystemModuleB. unfold is_true.
apply andb_true_iff. split; eauto.
- des. clarify. ss.
apply in_app. right. subst. econs. eauto.
}
assert (WF_STATE_NEXT: opsem_wf.OpsemPP.wf_State conf st1).
{ eapply opsem_wf.OpsemPP.preservation; eauto. }
assert (WF_LC : forall (gvs0 : GenericValue),
lookupAL GenericValue st1.(EC).(Locals) i0 = Some gvs0 ->
opsem_wf.OpsemPP.wf_GVs conf.(CurTargetData) gvs0 t).
{ unfold opsem_wf.OpsemPP.wf_State in WF_STATE_NEXT. des_ifs. ss.
destruct WF_STATE_NEXT as [WF_EC _].
unfold opsem_wf.OpsemPP.wf_ExecutionContext in *. destruct EC0. simpl.
destruct WF_EC as (_ & _ & _ & WF_LC & _).
unfold opsem_wf.OpsemPP.wf_lc in *. eauto.
}
exploit opsem_assigns_getCmdID; eauto. i. des.
esplits; eauto.
exploit WF_LC; eauto. intros WF_GVS.
eapply wf_GVs__lessthan_undef; eauto.
Qed.
Lemma lessdef_add_definedness
conf st0 st1 evt
cmd cmds
(WF_CONF: opsem_wf.OpsemPP.wf_Config conf)
(WF_STATE_PREV: opsem_wf.OpsemPP.wf_State conf st0)
(NONCALL: Instruction.isCallInst cmd = false)
(STEP: sInsn conf st0 st1 evt)
(CMDS: st0.(EC).(CurCmds) = cmd :: cmds)
: forall invst exp_pair lessdef
(DEFINED: postcond_cmd_get_definedness cmd = Some exp_pair)
(FORALL: ExprPairSet.For_all
(AssnState.Unary.sem_lessdef conf st1 invst)
lessdef),
ExprPairSet.For_all
(AssnState.Unary.sem_lessdef conf st1 invst)
(ExprPairSet.add exp_pair lessdef).
Proof.
ii. simpl_ep_set; ss; cycle 1.
- apply FORALL; ss.
- solve_leibniz. clarify.
exploit lessdef_definedness; eauto.
Qed.
Lemma lessdef_add
conf st invst lessdef lhs rhs
(FORALL: ExprPairSet.For_all
(AssnState.Unary.sem_lessdef conf st invst)
lessdef)
(EQ: AssnState.Unary.sem_expr conf st invst lhs = AssnState.Unary.sem_expr conf st invst rhs):
ExprPairSet.For_all
(AssnState.Unary.sem_lessdef conf st invst)
(ExprPairSet.add (lhs, rhs) (ExprPairSet.add (rhs, lhs) lessdef)).
Proof.
ii. simpl_ep_set; ss; cycle 2.
- apply FORALL; ss.
- solve_leibniz. clarify. ss.
rewrite <- EQ. esplits; eauto. apply GVs.lessdef_refl. (* TODO: reflexivity *)
- solve_leibniz. clarify. ss.
rewrite EQ. esplits; eauto. apply GVs.lessdef_refl. (* TODO: reflexivity *)
Qed.
(* TODO Move to AssnState? Unity with AssnState.Unary.sem_valueT_physical? *)
Lemma sem_list_valueT_physical
conf state invst0 sz_values1 lc gl
__INSN_ID__ mp'
(STATE: Locals (EC state) = updateAddAL GenericValue lc __INSN_ID__ mp')
(CONFIG: (Globals conf) = gl)
(POSTCOND_CHECK: Forall (fun y => __INSN_ID__ <> y)
(filter_map Value.get_ids (List.map snd sz_values1)))
:
<<EQUIV: values2GVs (CurTargetData conf) sz_values1 lc gl =
AssnState.Unary.sem_list_valueT
conf state invst0
(List.map (fun elt => (fst elt,
ValueT.lift Tag.physical (snd elt))) sz_values1)>>
.
Proof.
red.
generalize dependent lc.
generalize dependent gl.
generalize dependent POSTCOND_CHECK.
induction sz_values1; ii; ss.
destruct a; ss.
rewrite IHsz_values1; auto; cycle 1.
{ unfold compose in *. ss. destruct t; ss; simpl_list; ss. }
clear IHsz_values1.
remember
(AssnState.Unary.sem_list_valueT
conf state invst0
(List.map (fun elt => (fst elt, ValueT.lift Tag.physical (snd elt))) sz_values1)) as
X.
clear HeqX.
destruct X eqn:T; ss; des_ifs;
try (erewrite AssnState.Unary.sem_valueT_physical in Heq0; eauto; []; ii; des);
try rewrite STATE in *; u; ss;
des_ifs; simpl_list; des_lookupAL_updateAddAL.
Qed.
Lemma postcond_cmd_add_lessdef_unary_sound_alloca
conf st0 st1 cmd cmds def uses
invst0 assnmem0 inv0 gmax public
evt
(WF_CONF: opsem_wf.OpsemPP.wf_Config conf)
(WF_STATE_PREV: opsem_wf.OpsemPP.wf_State conf st0)
(POSTCOND_CHECK: AtomSetImpl.is_empty (AtomSetImpl.inter def uses))
(STATE: AssnState.Unary.sem conf st1 invst0 assnmem0 gmax public inv0)
(MEM: AssnMem.Unary.sem conf gmax public st1.(Mem) assnmem0)
(STEP: sInsn conf st0 st1 evt)
(CMDS: st0.(EC).(CurCmds) = cmd :: cmds)
(NONCALL: Instruction.isCallInst cmd = false)
(DEF: def = AtomSetImpl_from_list (option_to_list (Cmd.get_def cmd)))
(USES: uses = AtomSetImpl_from_list (Cmd.get_ids cmd))
id1 typ1 value1 align1
(GEP: cmd = insn_alloca id1 typ1 value1 align1)
:
<<STATE: AssnState.Unary.sem
conf st1 invst0 assnmem0 gmax public
(Assertion.update_lessdef (postcond_cmd_add_lessdef cmd) inv0)>>
.
Proof.
generalize (lessdef_add_definedness WF_CONF WF_STATE_PREV NONCALL STEP CMDS).
intro DEFINEDNESS.
(inv NONCALL; []); (inv STATE; []); ss; ((inv STEP; ss); []).
econs; eauto; [].
unfold postcond_cmd_add_lessdef. ss.
apply AtomSetImpl_from_list_inter_is_empty in POSTCOND_CHECK.
(* clear LESSDEF NOALIAS UNIQUE PRIVATE. *)
remember
{| CurSystem := S; CurTargetData := TD; CurProducts := Ps; Globals := gl; FunTable := fs |}
as conf.
assert(CONF1: conf.(CurTargetData) = TD).
{ rewrite Heqconf. auto. }
assert(CONF2: conf.(Globals) = gl).
{ rewrite Heqconf. auto. }
remember {|
EC := {|
CurFunction := F;
CurBB := B;
CurCmds := cs;
Terminator := tmn;
Locals := updateAddAL GenericValue lc id0 (blk2GV TD mb);
Allocas := mb :: als |};
ECS := ECS0;
Mem := Mem' |} as state.
assert(STATE1: state.(EC).(Locals) = updateAddAL GenericValue lc id0 (blk2GV TD mb)).
{ rewrite Heqstate. auto. }
assert(STATE2: state.(Mem) = Mem').
{ rewrite Heqstate. auto. }
clear Heqconf Heqstate.
inv CMDS.
(* clear MEM. *)
simpl_list.
rename id1 into __INSN_ID__.
ss. u. ss.
destruct (Decs.align_dec align1 Align.One) eqn:T; ss.
-
apply lessdef_add; [apply DEFINEDNESS; ss|];[].
{
ss. u. ss.
rewrite STATE1. des_lookupAL_updateAddAL.
exploit memory_props.MemProps.nextblock_alloca; eauto; []; ii; des.
exploit memory_props.MemProps.alloca_result; eauto; []; ii; des.
subst. ss.
unfold const2GV. unfold _const2GV.
unfold gundef.
unfold mload.
destruct (flatten_typ (CurTargetData conf) typ1) eqn:T2; ss.
erewrite MemProps.alloca_mload_aux_undef; eauto.
unfold const2GV. unfold _const2GV. unfold gundef. rewrite T2. ss.
}
-
apply lessdef_add.
+
apply lessdef_add; [apply DEFINEDNESS; ss|]; [].
{
ss. u. ss.
rewrite STATE1. des_lookupAL_updateAddAL.
exploit memory_props.MemProps.nextblock_alloca; eauto; []; ii; des.
exploit memory_props.MemProps.alloca_result; eauto; []; ii; des.
subst. ss.
unfold const2GV. unfold _const2GV.
unfold gundef.
unfold mload.
destruct (flatten_typ (CurTargetData conf) typ1) eqn:T2; ss.
erewrite MemProps.alloca_mload_aux_undef; eauto.
unfold const2GV. unfold _const2GV. unfold gundef. rewrite T2. ss.
}
+
{
ss. u. ss.
rewrite STATE1. des_lookupAL_updateAddAL.
exploit memory_props.MemProps.nextblock_alloca; eauto; []; ii; des.
exploit memory_props.MemProps.alloca_result; eauto; []; ii; des.
subst. ss.
unfold const2GV. unfold _const2GV.
unfold gundef.
unfold mload.
destruct (flatten_typ (CurTargetData conf) typ1) eqn:T2; ss.
erewrite MemProps.alloca_mload_aux_undef; eauto.
unfold const2GV. unfold _const2GV. unfold gundef. rewrite T2. ss.
}
Unshelve.
eauto.
eauto.
eauto.
Qed.
Lemma postcond_cmd_add_lessdef_unary_sound_gep
conf st0 st1 cmd cmds def uses
invst0 assnmem0 inv0 gmax public
evt
(WF_CONF: opsem_wf.OpsemPP.wf_Config conf)
(WF_STATE_PREV: opsem_wf.OpsemPP.wf_State conf st0)
(POSTCOND_CHECK: AtomSetImpl.is_empty (AtomSetImpl.inter def uses))
(STATE: AssnState.Unary.sem conf st1 invst0 assnmem0 gmax public inv0)
(MEM: AssnMem.Unary.sem conf gmax public st1.(Mem) assnmem0)
(STEP: sInsn conf st0 st1 evt)
(CMDS: st0.(EC).(CurCmds) = cmd :: cmds)
(NONCALL: Instruction.isCallInst cmd = false)
(DEF: def = AtomSetImpl_from_list (option_to_list (Cmd.get_def cmd)))
(USES: uses = AtomSetImpl_from_list (Cmd.get_ids cmd))
id1 inbounds1 typ1 value1 sz_values1 typ2
(GEP: cmd = insn_gep id1 inbounds1 typ1 value1 sz_values1 typ2)
:
<<STATE: AssnState.Unary.sem
conf st1 invst0 assnmem0 gmax public
(Assertion.update_lessdef (postcond_cmd_add_lessdef cmd) inv0)>>
.
Proof.
generalize (lessdef_add_definedness WF_CONF WF_STATE_PREV NONCALL STEP CMDS).
intro DEFINEDNESS.
(inv NONCALL; []); (inv STATE; []); ss; ((inv STEP; ss); []).
econs; eauto; [].
unfold postcond_cmd_add_lessdef. ss.
apply AtomSetImpl_from_list_inter_is_empty in POSTCOND_CHECK.
apply lessdef_add; [apply DEFINEDNESS; ss|]; [].
clear LESSDEF NOALIAS UNIQUE PRIVATE.
remember
{| CurSystem := S; CurTargetData := TD; CurProducts := Ps; Globals := gl; FunTable := fs |}
as conf.
assert(CONF1: conf.(CurTargetData) = TD).
{ rewrite Heqconf. auto. }
assert(CONF2: conf.(Globals) = gl).
{ rewrite Heqconf. auto. }
remember
{|
EC := {|
CurFunction := F;
CurBB := B;
CurCmds := cs;
Terminator := tmn;
Locals := updateAddAL GenericValue lc id0 mp';
Allocas := als |};
ECS := ECS0;
Mem := Mem0 |} as state.
assert(STATE: state.(EC).(Locals) = updateAddAL GenericValue lc id0 mp').
{ rewrite Heqstate. auto. }
clear Heqconf Heqstate.
inv CMDS.
clear MEM.
simpl_list.
rename id1 into __INSN_ID__.
ss. u. ss.
rewrite STATE. des_lookupAL_updateAddAL.
clear e.
rewrite <- H2. clear H2.
exploit sem_list_valueT_physical; eauto.
{ destruct value1; ss; simpl_list; eauto. }
ii; des.
rewrite <- x0.
rewrite AssnState.Unary.sem_valueT_physical in *.
destruct value1; ss.
- rewrite STATE; simpl_list; des_lookupAL_updateAddAL.
rewrite H.
rewrite H1.
ss.
- rewrite H.
rewrite H1.
ss.
Qed.
Lemma postcond_cmd_add_lessdef_unary_sound_select
conf st0 st1 cmd cmds def uses
invst0 assnmem0 inv0 gmax public
evt
(WF_CONF: opsem_wf.OpsemPP.wf_Config conf)
(WF_STATE_PREV: opsem_wf.OpsemPP.wf_State conf st0)
(POSTCOND_CHECK: AtomSetImpl.is_empty (AtomSetImpl.inter def uses))
(STATE: AssnState.Unary.sem conf st1 invst0 assnmem0 gmax public inv0)
(MEM: AssnMem.Unary.sem conf gmax public st1.(Mem) assnmem0)
(STEP: sInsn conf st0 st1 evt)
(CMDS: st0.(EC).(CurCmds) = cmd :: cmds)
(NONCALL: Instruction.isCallInst cmd = false)
(DEF: def = AtomSetImpl_from_list (option_to_list (Cmd.get_def cmd)))
(USES: uses = AtomSetImpl_from_list (Cmd.get_ids cmd))
id1 value_cond typ1 value1 value2
(GEP: cmd = insn_select id1 value_cond typ1 value1 value2)
:
<<STATE: AssnState.Unary.sem
conf st1 invst0 assnmem0 gmax public
(Assertion.update_lessdef (postcond_cmd_add_lessdef cmd) inv0)>>
.
Proof.
generalize (lessdef_add_definedness WF_CONF WF_STATE_PREV NONCALL STEP CMDS).
clear WF_STATE_PREV.
intro DEFINEDNESS.
(inv NONCALL; []); (inv STATE; []); ss; ((inv STEP; ss); []).
econs; eauto; [].
unfold postcond_cmd_add_lessdef. ss.
apply AtomSetImpl_from_list_inter_is_empty in POSTCOND_CHECK.
apply lessdef_add; [apply DEFINEDNESS; ss|]; [].
clear LESSDEF NOALIAS UNIQUE PRIVATE.
remember
{| CurSystem := S; CurTargetData := TD; CurProducts := Ps; Globals := gl; FunTable := fs |}
as conf.
assert(CONF1: conf.(CurTargetData) = TD).
{ rewrite Heqconf. auto. }
assert(CONF2: conf.(Globals) = gl).
{ rewrite Heqconf. auto. }
remember {|
EC := {|
CurFunction := F;
CurBB := B;
CurCmds := cs;
Terminator := tmn;
Locals := updateAddAL GenericValue lc id0 gvresult;
Allocas := als |};
ECS := ECS0;
Mem := Mem0 |} as state.
assert(STATE: state.(EC).(Locals) = updateAddAL GenericValue lc id0 gvresult).
{ rewrite Heqstate. auto. }
clear Heqconf Heqstate.
inv CMDS.
clear MEM.
simpl_list.
rename id1 into __INSN_ID__.
ss. u. ss.
rewrite STATE. des_lookupAL_updateAddAL.
clear e.
rewrite ? AssnState.Unary.sem_valueT_physical in *.
unfold SELECT in *.
destruct value_cond, value1, value2; simpl in *;
try rewrite STATE; simpl_list; des_lookupAL_updateAddAL;
try rewrite H; try rewrite H1; try rewrite H2; try rewrite INT; ss.
Qed.
Lemma postcond_cmd_add_lessdef_unary_sound
conf st0 st1 cmd cmds def uses
invst0 assnmem0 inv0 gmax public
evt
(WF_CONF: opsem_wf.OpsemPP.wf_Config conf)
(WF_STATE_PREV: opsem_wf.OpsemPP.wf_State conf st0)
(POSTCOND_CHECK: AtomSetImpl.is_empty (AtomSetImpl.inter def uses))
(STATE: AssnState.Unary.sem conf st1 invst0 assnmem0 gmax public inv0)
(MEM: AssnMem.Unary.sem conf gmax public st1.(Mem) assnmem0)
(STEP: sInsn conf st0 st1 evt)
(CMDS: st0.(EC).(CurCmds) = cmd :: cmds)
(NONCALL: Instruction.isCallInst cmd = false)
(DEF: def = AtomSetImpl_from_list (option_to_list (Cmd.get_def cmd)))
(USES: uses = AtomSetImpl_from_list (Cmd.get_ids cmd))
:
<<STATE: AssnState.Unary.sem
conf st1 invst0 assnmem0 gmax public
(Assertion.update_lessdef (postcond_cmd_add_lessdef cmd) inv0)>>
.
Proof.
generalize (lessdef_add_definedness WF_CONF WF_STATE_PREV NONCALL STEP CMDS).
intro DEFINEDNESS.
(* clear WF_CONF WF_STATE_PREV. *)
destruct cmd;
try (eapply postcond_cmd_add_lessdef_unary_sound_alloca; eauto; fail);
try (eapply postcond_cmd_add_lessdef_unary_sound_gep; eauto; fail);
try (eapply postcond_cmd_add_lessdef_unary_sound_select; eauto; fail);
ss; (inv NONCALL; []); (inv STATE; []); ss; ((inv STEP; ss); []);
try (econs; eauto; []; apply lessdef_add; [apply DEFINEDNESS;ss |]; ss;
rewrite ? AssnState.Unary.sem_valueT_physical in *; ss;
apply AtomSetImpl_from_list_inter_is_empty in POSTCOND_CHECK; [];
repeat match goal with
| [ v: value |- _ ] => destruct v
end; u; ss; simpl_list; des_lookupAL_updateAddAL; des_ifs; fail).
- (* alloca *)
clear WF_CONF WF_STATE_PREV.
clarify.
econs; eauto; [].
unfold postcond_cmd_add_lessdef. ss.
des_ifs;
repeat apply lessdef_add; ss;
(rewrite ? AssnState.Unary.sem_valueT_physical in *; ss; [];
apply AtomSetImpl_from_list_inter_is_empty in POSTCOND_CHECK; [];
repeat match goal with
| [ v: value |- _ ] => destruct v
end; u; ss; simpl_list; des_lookupAL_updateAddAL; des_ifs;
apply DEFINEDNESS; ss).
- (* load *)
clear WF_CONF WF_STATE_PREV.
econs; eauto; [].
unfold postcond_cmd_add_lessdef. ss.
des_ifs;
repeat apply lessdef_add; ss;
(rewrite ? AssnState.Unary.sem_valueT_physical in *; ss; [];
apply AtomSetImpl_from_list_inter_is_empty in POSTCOND_CHECK; [];
repeat match goal with
| [ v: value |- _ ] => destruct v
end; u; ss; simpl_list; des_lookupAL_updateAddAL; des_ifs;
apply DEFINEDNESS; ss).
- (* store *)
clear WF_CONF WF_STATE_PREV.
econs; eauto; [].
unfold postcond_cmd_add_lessdef. ss.
apply AtomSetImpl_from_list_inter_is_empty in POSTCOND_CHECK.
simpl_list.
des_ifs.
+
apply lessdef_add; [apply LESSDEF|].
clear LESSDEF NOALIAS UNIQUE PRIVATE.
ss.
destruct value1, value2; ss; u; ss; rewrite H; rewrite H0; des_lookupAL_updateAddAL;
erewrite MemProps.mstore_mload_same; eauto.
+
apply lessdef_add.
apply lessdef_add; [apply LESSDEF|].
clear LESSDEF NOALIAS UNIQUE PRIVATE.