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context_facts.v
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Require Import List.
Import ListNotations.
Require Import bbv.Word.
Require Import Nat.
Require Import Coq.NArith.NArith.
Require Import FORVES2.octagon.
Import Octagon.
Require Import FORVES2.constants.
Import Constants.
Require Import FORVES2.constraints.
Import Constraints.
Require Import FORVES2.context.
Import Context.
Require Import FORVES2.symbolic_state.
Import SymbolicState.
Require Import FORVES2.symbolic_state_eval.
Import SymbolicStateEval.
Require Import FORVES2.stack_operation_instructions.
Import StackOpInstrs.
Require Import FORVES2.symbolic_execution_soundness.
Import SymbolicExecutionSoundness.
Module ContextFacts.
Lemma chk_eq_wrt_ctx_snd:
forall ctx sv1 sv2,
chk_eq_wrt_ctx ctx sv1 sv2 = true ->
forall model mem strg exts maxidx sb ops,
is_model (ctx_cs ctx) model = true ->
exists v,
eval_sstack_val sv1 model mem strg exts maxidx sb ops = Some v /\
eval_sstack_val sv2 model mem strg exts maxidx sb ops = Some v.
Proof.
intros ctx sv1 sv2 H_chkr model mem strg exts maxidx sb ops H_is_model.
destruct ctx as [cs [chkr H_chkr_snd]].
simpl in H_is_model.
destruct sv1 as [val1 | var1 | fvar1 ] eqn:E_sv1;
destruct sv2 as [val2 | var2 | fvar2 ] eqn:E_sv2;
try discriminate.
(* val1 = val2 *)
+ unfold chk_eq_wrt_ctx in H_chkr.
unfold sstack_val_to_cliteral in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_EQ (C_VAL (wordToN val1)) (C_VAL (wordToN val2))) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_Val val1 model mem strg exts maxidx sb ops) as H_eval_val1.
pose proof (eval_sstack_val_Val val2 model mem strg exts maxidx sb ops) as H_eval_val2.
rewrite N.eqb_eq in H_chkr_snd.
apply wordToN_inj in H_chkr_snd.
rewrite H_eval_val1.
rewrite H_eval_val2.
exists val1.
rewrite H_chkr_snd.
split; try reflexivity.
(* val = var *)
+ unfold chk_eq_wrt_ctx in H_chkr.
unfold sstack_val_to_cliteral in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_EQ (C_VAL (wordToN val1)) (C_VAR var2)) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_Val val1 model mem strg exts maxidx sb ops) as H_eval_val1.
pose proof (eval_sstack_val_InVar var2 model mem strg exts maxidx sb ops) as H_eval_var2.
rewrite N.eqb_eq in H_chkr_snd.
apply wordToN_inj in H_chkr_snd.
rewrite H_eval_val1.
rewrite H_eval_var2.
exists val1.
rewrite H_chkr_snd.
split; try reflexivity.
(* var = val *)
+ unfold chk_eq_wrt_ctx in H_chkr.
unfold sstack_val_to_cliteral in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_EQ (C_VAR var1) (C_VAL (wordToN val2))) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_InVar var1 model mem strg exts maxidx sb ops) as H_eval_var1.
pose proof (eval_sstack_val_Val val2 model mem strg exts maxidx sb ops) as H_eval_val2.
rewrite N.eqb_eq in H_chkr_snd.
apply wordToN_inj in H_chkr_snd.
rewrite H_eval_var1.
rewrite H_eval_val2.
exists val2.
rewrite H_chkr_snd.
split; try reflexivity.
(* var = var *)
+ unfold chk_eq_wrt_ctx in H_chkr.
unfold sstack_val_to_cliteral in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_EQ (C_VAR var1) (C_VAR var2)) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_InVar var1 model mem strg exts maxidx sb ops) as H_eval_var1.
pose proof (eval_sstack_val_InVar var2 model mem strg exts maxidx sb ops) as H_eval_var2.
rewrite N.eqb_eq in H_chkr_snd.
apply wordToN_inj in H_chkr_snd.
rewrite H_eval_var1.
rewrite H_eval_var2.
exists (model var1).
rewrite H_chkr_snd.
split; try reflexivity.
Qed.
Lemma chk_lt_wrt_ctx_snd:
forall ctx sv1 sv2,
chk_lt_wrt_ctx ctx sv1 sv2 = true ->
forall model mem strg exts maxidx sb ops,
is_model (ctx_cs ctx) model = true ->
exists v1 v2,
eval_sstack_val sv1 model mem strg exts maxidx sb ops = Some v1 /\
eval_sstack_val sv2 model mem strg exts maxidx sb ops = Some v2 /\
(wordToN v1 < wordToN v2)%N.
Proof.
intros ctx sv1 sv2 H_chkr model mem strg exts maxidx sb ops H_is_model.
destruct ctx as [cs [chkr H_chkr_snd]].
simpl in H_is_model.
destruct sv1 as [val1 | var1 | fvar1 ] eqn:E_sv1;
destruct sv2 as [val2 | var2 | fvar2 ] eqn:E_sv2;
try discriminate.
(* val1 = val2 *)
+ unfold chk_lt_wrt_ctx in H_chkr.
unfold sstack_val_to_cliteral in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_LT (C_VAL (wordToN val1)) (C_VAL (wordToN val2))) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_Val val1 model mem strg exts maxidx sb ops) as H_eval_val1.
pose proof (eval_sstack_val_Val val2 model mem strg exts maxidx sb ops) as H_eval_val2.
rewrite N.ltb_lt in H_chkr_snd.
rewrite H_eval_val1.
rewrite H_eval_val2.
exists val1.
exists val2.
repeat split; try reflexivity.
apply H_chkr_snd.
(* val = var *)
+ unfold chk_lt_wrt_ctx in H_chkr.
unfold sstack_val_to_cliteral in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_LT (C_VAL (wordToN val1)) (C_VAR var2)) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_Val val1 model mem strg exts maxidx sb ops) as H_eval_val1.
pose proof (eval_sstack_val_InVar var2 model mem strg exts maxidx sb ops) as H_eval_var2.
rewrite N.ltb_lt in H_chkr_snd.
rewrite H_eval_val1.
rewrite H_eval_var2.
exists val1.
exists (model var2).
repeat split; try reflexivity.
apply H_chkr_snd.
(* var = val *)
+ unfold chk_lt_wrt_ctx in H_chkr.
unfold sstack_val_to_cliteral in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_LT (C_VAR var1) (C_VAL (wordToN val2))) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_InVar var1 model mem strg exts maxidx sb ops) as H_eval_var1.
pose proof (eval_sstack_val_Val val2 model mem strg exts maxidx sb ops) as H_eval_val2.
rewrite N.ltb_lt in H_chkr_snd.
rewrite H_eval_var1.
rewrite H_eval_val2.
exists (model var1).
exists val2.
repeat split; try reflexivity.
apply H_chkr_snd.
(* var = var *)
+ unfold chk_lt_wrt_ctx in H_chkr.
unfold sstack_val_to_cliteral in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_LT (C_VAR var1) (C_VAR var2)) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_InVar var1 model mem strg exts maxidx sb ops) as H_eval_var1.
pose proof (eval_sstack_val_InVar var2 model mem strg exts maxidx sb ops) as H_eval_var2.
rewrite N.ltb_lt in H_chkr_snd.
rewrite H_eval_var1.
rewrite H_eval_var2.
exists (model var1).
exists (model var2).
repeat split; try reflexivity.
apply H_chkr_snd.
Qed.
Lemma chk_lt_wrt_ctx_smap_snd:
forall ctx sv1 sv2 maxidx sb,
chk_lt_wrt_ctx_smap ctx sv1 sv2 maxidx sb = true ->
forall model mem strg exts ops,
is_model (ctx_cs ctx) model = true ->
exists v1 v2,
eval_sstack_val sv1 model mem strg exts maxidx sb ops = Some v1 /\
eval_sstack_val sv2 model mem strg exts maxidx sb ops = Some v2 /\
(wordToN v1 < wordToN v2)%N.
Proof.
intros ctx sv1 sv2 maxidx sb H_chkr model mem strg exts ops H_is_model.
destruct ctx as [cs [chkr H_chkr_snd]].
simpl in H_is_model.
destruct sv1 as [val1 | var1 | fvar1 ] eqn:E_sv1;
destruct sv2 as [val2 | var2 | fvar2 ] eqn:E_sv2;
try discriminate.
(* val1 = val2 *)
+ unfold chk_lt_wrt_ctx_smap in H_chkr.
unfold sstack_val_to_cliteral_smap in H_chkr.
rewrite (follow_smap_V val1 maxidx sb) in H_chkr.
rewrite (follow_smap_V val2 maxidx sb) in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_LT (C_VAL (wordToN val1)) (C_VAL (wordToN val2))) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_Val val1 model mem strg exts maxidx sb ops) as H_eval_val1.
pose proof (eval_sstack_val_Val val2 model mem strg exts maxidx sb ops) as H_eval_val2.
rewrite N.ltb_lt in H_chkr_snd.
rewrite H_eval_val1.
rewrite H_eval_val2.
exists val1.
exists val2.
repeat split; try reflexivity.
apply H_chkr_snd.
(* val = var *)
+ unfold chk_lt_wrt_ctx_smap in H_chkr.
unfold sstack_val_to_cliteral_smap in H_chkr.
rewrite (follow_smap_V val1 maxidx sb) in H_chkr.
rewrite (follow_smap_InStackVar var2 maxidx sb) in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_LT (C_VAL (wordToN val1)) (C_VAR var2)) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_Val val1 model mem strg exts maxidx sb ops) as H_eval_val1.
pose proof (eval_sstack_val_InVar var2 model mem strg exts maxidx sb ops) as H_eval_var2.
rewrite N.ltb_lt in H_chkr_snd.
rewrite H_eval_val1.
rewrite H_eval_var2.
exists val1.
exists (model var2).
repeat split; try reflexivity.
apply H_chkr_snd.
(* val = fvar *)
+ unfold chk_lt_wrt_ctx_smap in H_chkr.
unfold sstack_val_to_cliteral_smap in H_chkr.
rewrite (follow_smap_V val1 maxidx sb) in H_chkr.
destruct (follow_in_smap (FreshVar fvar2) maxidx sb) eqn:E_follow; try destruct f eqn:E_f; try destruct smv eqn:E_smv; try destruct val eqn:E_val; simpl in H_chkr; try discriminate.
* pose proof (H_chkr_snd cs (C_LT (C_VAL (wordToN val1)) (C_VAL (wordToN val0))) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_Val val1 model mem strg exts maxidx sb ops) as H_eval_val1.
pose proof (eval_sstack_val_Val val0 model mem strg exts maxidx sb ops) as H_eval_val0.
rewrite N.ltb_lt in H_chkr_snd.
rewrite H_eval_val1.
unfold eval_sstack_val.
unfold eval_sstack_val'.
rewrite E_follow.
exists val1.
exists val0.
repeat split; try reflexivity.
apply H_chkr_snd.
* pose proof (H_chkr_snd cs (C_LT (C_VAL (wordToN val1)) (C_VAR var)) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_Val val1 model mem strg exts maxidx sb ops) as H_eval_val1.
rewrite N.ltb_lt in H_chkr_snd.
rewrite H_eval_val1.
unfold eval_sstack_val.
unfold eval_sstack_val'.
rewrite E_follow.
exists val1.
exists (model var).
repeat split; try reflexivity.
apply H_chkr_snd.
(* var = val *)
+ unfold chk_lt_wrt_ctx_smap in H_chkr.
unfold sstack_val_to_cliteral_smap in H_chkr.
rewrite (follow_smap_V val2 maxidx sb) in H_chkr.
rewrite (follow_smap_InStackVar var1 maxidx sb) in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_LT (C_VAR var1) (C_VAL (wordToN val2))) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_InVar var1 model mem strg exts maxidx sb ops) as H_eval_var1.
pose proof (eval_sstack_val_Val val2 model mem strg exts maxidx sb ops) as H_eval_val2.
rewrite N.ltb_lt in H_chkr_snd.
rewrite H_eval_var1.
rewrite H_eval_val2.
exists (model var1).
exists val2.
repeat split; try reflexivity.
apply H_chkr_snd.
(* var = var *)
+ unfold chk_lt_wrt_ctx_smap in H_chkr.
unfold sstack_val_to_cliteral_smap in H_chkr.
rewrite (follow_smap_InStackVar var1 maxidx sb) in H_chkr.
rewrite (follow_smap_InStackVar var2 maxidx sb) in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_LT (C_VAR var1) (C_VAR var2)) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_InVar var1 model mem strg exts maxidx sb ops) as H_eval_var1.
pose proof (eval_sstack_val_InVar var2 model mem strg exts maxidx sb ops) as H_eval_var2.
rewrite N.ltb_lt in H_chkr_snd.
rewrite H_eval_var1.
rewrite H_eval_var2.
exists (model var1).
exists (model var2).
repeat split; try reflexivity.
apply H_chkr_snd.
(* var = fvar *)
+ unfold chk_lt_wrt_ctx_smap in H_chkr.
unfold sstack_val_to_cliteral_smap in H_chkr.
rewrite (follow_smap_InStackVar var1 maxidx sb) in H_chkr.
destruct (follow_in_smap (FreshVar fvar2) maxidx sb) eqn:E_follow; try destruct f eqn:E_f; try destruct smv eqn:E_smv; try destruct val eqn:E_val; simpl in H_chkr; try discriminate.
* pose proof (H_chkr_snd cs (C_LT (C_VAR var1) (C_VAL (wordToN val0))) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_InVar var1 model mem strg exts maxidx sb ops) as H_eval_var1.
rewrite H_eval_var1.
unfold eval_sstack_val.
unfold eval_sstack_val'.
rewrite E_follow.
exists (model var1).
exists val0.
repeat split; try reflexivity.
rewrite N.ltb_lt in H_chkr_snd.
apply H_chkr_snd.
* pose proof (H_chkr_snd cs (C_LT (C_VAR var1) (C_VAR var)) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
rewrite N.ltb_lt in H_chkr_snd.
pose proof (eval_sstack_val_InVar var1 model mem strg exts maxidx sb ops) as H_eval_var1.
rewrite H_eval_var1.
unfold eval_sstack_val.
unfold eval_sstack_val'.
rewrite E_follow.
exists (model var1).
exists (model var).
repeat split; try reflexivity.
apply H_chkr_snd.
(* fvar = val *)
+ unfold chk_lt_wrt_ctx_smap in H_chkr.
unfold sstack_val_to_cliteral_smap in H_chkr.
rewrite (follow_smap_V val2 maxidx sb) in H_chkr.
destruct (follow_in_smap (FreshVar fvar1) maxidx sb) eqn:E_follow; try destruct f eqn:E_f; try destruct smv eqn:E_smv; try destruct val eqn:E_val; simpl in H_chkr; try discriminate.
* pose proof (H_chkr_snd cs (C_LT (C_VAL (wordToN val0)) (C_VAL (wordToN val2))) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
rewrite N.ltb_lt in H_chkr_snd.
pose proof (eval_sstack_val_Val val2 model mem strg exts maxidx sb ops) as H_eval_val2.
rewrite H_eval_val2.
unfold eval_sstack_val.
unfold eval_sstack_val'.
rewrite E_follow.
exists val0.
exists val2.
repeat split; try reflexivity.
apply H_chkr_snd.
* pose proof (H_chkr_snd cs (C_LT (C_VAR var) (C_VAL (wordToN val2))) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_Val val2 model mem strg exts maxidx sb ops) as H_eval_val2.
rewrite N.ltb_lt in H_chkr_snd.
rewrite H_eval_val2.
unfold eval_sstack_val.
unfold eval_sstack_val'.
rewrite E_follow.
exists (model var).
exists val2.
repeat split; try reflexivity.
apply H_chkr_snd.
(* fvar = var *)
+ unfold chk_lt_wrt_ctx_smap in H_chkr.
unfold sstack_val_to_cliteral_smap in H_chkr.
rewrite (follow_smap_InStackVar var2 maxidx sb) in H_chkr.
destruct (follow_in_smap (FreshVar fvar1) maxidx sb) eqn:E_follow; try destruct f eqn:E_f; try destruct smv eqn:E_smv; try destruct val eqn:E_val; simpl in H_chkr; try discriminate.
* pose proof (H_chkr_snd cs (C_LT (C_VAL (wordToN val0)) (C_VAR var2)) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
rewrite N.ltb_lt in H_chkr_snd.
pose proof (eval_sstack_val_InVar var2 model mem strg exts maxidx sb ops) as H_eval_var2.
rewrite H_eval_var2.
unfold eval_sstack_val.
unfold eval_sstack_val'.
rewrite E_follow.
exists val0.
exists (model var2).
repeat split; try reflexivity.
apply H_chkr_snd.
* pose proof (H_chkr_snd cs (C_LT (C_VAR var) (C_VAR var2)) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
rewrite N.ltb_lt in H_chkr_snd.
pose proof (eval_sstack_val_InVar var2 model mem strg exts maxidx sb ops) as H_eval_var2.
rewrite H_eval_var2.
unfold eval_sstack_val.
unfold eval_sstack_val'.
rewrite E_follow.
exists (model var).
exists (model var2).
repeat split; try reflexivity.
apply H_chkr_snd.
(* fvar = fvar *)
+ unfold chk_lt_wrt_ctx_smap in H_chkr.
unfold sstack_val_to_cliteral_smap in H_chkr.
destruct (follow_in_smap (FreshVar fvar1) maxidx sb) eqn:E_follow1; try destruct f eqn:E_f; try destruct smv eqn:E_smv; try destruct val eqn:E_val;
destruct (follow_in_smap (FreshVar fvar2) maxidx sb) eqn:E_follow2; try destruct f0 eqn:E_f0; try destruct smv0 eqn:E_smv0; try destruct val1 eqn:E_val1; try simpl in H_chkr; try discriminate.
* pose proof (H_chkr_snd cs (C_LT (C_VAL (wordToN val0)) (C_VAL (wordToN val2))) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
rewrite N.ltb_lt in H_chkr_snd.
unfold eval_sstack_val.
unfold eval_sstack_val'.
rewrite E_follow1.
rewrite E_follow2.
exists val0.
exists val2.
repeat split; try reflexivity.
apply H_chkr_snd.
* pose proof (H_chkr_snd cs (C_LT (C_VAL (wordToN val0)) (C_VAR var)) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
rewrite N.ltb_lt in H_chkr_snd.
unfold eval_sstack_val.
unfold eval_sstack_val'.
rewrite E_follow1.
rewrite E_follow2.
exists val0.
exists (model var).
repeat split; try reflexivity.
apply H_chkr_snd.
* destruct val0 eqn:E_val0; simpl in H_chkr; try discriminate.
** pose proof (H_chkr_snd cs (C_LT (C_VAR var) (C_VAL (wordToN val1))) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
rewrite N.ltb_lt in H_chkr_snd.
unfold eval_sstack_val.
unfold eval_sstack_val'.
rewrite E_follow1.
rewrite E_follow2.
exists (model var).
exists val1.
repeat split; try reflexivity.
apply H_chkr_snd.
** pose proof (H_chkr_snd cs (C_LT (C_VAR var) (C_VAR var0)) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
rewrite N.ltb_lt in H_chkr_snd.
unfold eval_sstack_val.
unfold eval_sstack_val'.
rewrite E_follow1.
rewrite E_follow2.
exists (model var).
exists (model var0).
repeat split; try reflexivity.
apply H_chkr_snd.
Qed.
Lemma chk_neq_wrt_ctx_snd:
forall ctx sv1 sv2,
chk_neq_wrt_ctx ctx sv1 sv2 = true ->
forall model mem strg exts maxidx sb ops,
is_model (ctx_cs ctx) model = true ->
exists v1 v2,
eval_sstack_val sv1 model mem strg exts maxidx sb ops = Some v1 /\
eval_sstack_val sv2 model mem strg exts maxidx sb ops = Some v2 /\
weqb v1 v2 = false.
Proof.
intros ctx sv1 sv2 H_imp_chkr model mem strg exts maxidx sb ops H_is_model.
unfold chk_neq_wrt_ctx in H_imp_chkr.
apply Bool.orb_true_iff in H_imp_chkr.
destruct H_imp_chkr as [H_imp_chkr | H_imp_chkr].
+ pose proof (chk_lt_wrt_ctx_snd ctx sv1 sv2 H_imp_chkr model mem strg exts maxidx sb ops H_is_model) as H_chk_lt_wrt_ctx_snd.
destruct H_chk_lt_wrt_ctx_snd as [v1' [v2' [ H_eval_sv1 [H_eval_sv2 H_wlt_v1'_v2']]]].
exists v1'.
exists v2'.
rewrite H_eval_sv1.
rewrite H_eval_sv2.
repeat split; try reflexivity.
apply weqb_ne.
intuition.
rewrite H in H_wlt_v1'_v2'.
unfold wlt in H_wlt_v1'_v2'.
intuition.
+ pose proof (chk_lt_wrt_ctx_snd ctx sv2 sv1 H_imp_chkr model mem strg exts maxidx sb ops H_is_model) as H_chk_lt_wrt_ctx_snd.
destruct H_chk_lt_wrt_ctx_snd as [v2' [v1' [ H_eval_sv2 [H_eval_sv1 H_wlt_v2'_v1']]]].
exists v1'.
exists v2'.
rewrite H_eval_sv1.
rewrite H_eval_sv2.
repeat split; try reflexivity.
apply weqb_ne.
intuition.
rewrite H in H_wlt_v2'_v1'.
unfold wlt in H_wlt_v2'_v1'.
intuition.
Qed.
Lemma chk_lt_lshift_wrt_ctx_snd:
forall ctx sv1 sv2 shift,
chk_lt_lshift_wrt_ctx ctx sv1 sv2 shift = true ->
forall model mem strg exts maxidx sb ops,
is_model (ctx_cs ctx) model = true ->
exists v1 v2,
eval_sstack_val sv1 model mem strg exts maxidx sb ops = Some v1 /\
eval_sstack_val sv2 model mem strg exts maxidx sb ops = Some v2 /\
((wordToN v1)+shift < (wordToN v2))%N.
Proof.
intros ctx sv1 sv2 shift H_chkr model mem strg exts maxidx sb ops H_is_model.
destruct ctx as [cs [chkr H_chkr_snd]].
simpl in H_is_model.
destruct sv1 as [val1 | var1 | fvar1 ] eqn:E_sv1;
destruct sv2 as [val2 | var2 | fvar2 ] eqn:E_sv2;
try discriminate.
(* val1 = val2 *)
+ unfold chk_lt_lshift_wrt_ctx in H_chkr.
unfold sstack_val_to_cliteral in H_chkr.
unfold sstack_val_to_cliteral_w_shift in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_LT (C_VAL (wordToN val1 + shift)) (C_VAL (wordToN val2))) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_Val val1 model mem strg exts maxidx sb ops) as H_eval_val1.
pose proof (eval_sstack_val_Val val2 model mem strg exts maxidx sb ops) as H_eval_val2.
rewrite N.ltb_lt in H_chkr_snd.
rewrite H_eval_val1.
rewrite H_eval_val2.
exists val1.
exists val2.
repeat split; try reflexivity.
apply H_chkr_snd.
(* val = var *)
+ unfold chk_lt_lshift_wrt_ctx in H_chkr.
unfold sstack_val_to_cliteral in H_chkr.
unfold sstack_val_to_cliteral_w_shift in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_LT (C_VAL (wordToN val1 + shift)) (C_VAR var2)) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_Val val1 model mem strg exts maxidx sb ops) as H_eval_val1.
pose proof (eval_sstack_val_InVar var2 model mem strg exts maxidx sb ops) as H_eval_var2.
rewrite N.ltb_lt in H_chkr_snd.
rewrite H_eval_val1.
rewrite H_eval_var2.
exists val1.
exists (model var2).
repeat split; try reflexivity.
apply H_chkr_snd.
(* var = val *)
+ unfold chk_lt_lshift_wrt_ctx in H_chkr.
unfold sstack_val_to_cliteral in H_chkr.
unfold sstack_val_to_cliteral_w_shift in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_LT (C_VAR_DELTA var1 shift) (C_VAL (wordToN val2))) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_InVar var1 model mem strg exts maxidx sb ops) as H_eval_var1.
pose proof (eval_sstack_val_Val val2 model mem strg exts maxidx sb ops) as H_eval_val2.
rewrite N.ltb_lt in H_chkr_snd.
rewrite H_eval_var1.
rewrite H_eval_val2.
exists (model var1).
exists val2.
repeat split; try reflexivity.
apply H_chkr_snd.
(* var = var *)
+ unfold chk_lt_lshift_wrt_ctx in H_chkr.
unfold sstack_val_to_cliteral in H_chkr.
unfold sstack_val_to_cliteral_w_shift in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_LT (C_VAR_DELTA var1 shift) (C_VAR var2)) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_InVar var1 model mem strg exts maxidx sb ops) as H_eval_var1.
pose proof (eval_sstack_val_InVar var2 model mem strg exts maxidx sb ops) as H_eval_var2.
rewrite N.ltb_lt in H_chkr_snd.
rewrite H_eval_var1.
rewrite H_eval_var2.
exists (model var1).
exists (model var2).
repeat split; try reflexivity.
apply H_chkr_snd.
Qed.
Lemma chk_lt_rshift_wrt_ctx_snd:
forall ctx sv1 sv2 shift,
chk_lt_rshift_wrt_ctx ctx sv1 sv2 shift = true ->
forall model mem strg exts maxidx sb ops,
is_model (ctx_cs ctx) model = true ->
exists v1 v2,
eval_sstack_val sv1 model mem strg exts maxidx sb ops = Some v1 /\
eval_sstack_val sv2 model mem strg exts maxidx sb ops = Some v2 /\
((wordToN v1) < (wordToN v2) + shift)%N.
Proof.
intros ctx sv1 sv2 shift H_chkr model mem strg exts maxidx sb ops H_is_model.
destruct ctx as [cs [chkr H_chkr_snd]].
simpl in H_is_model.
destruct sv1 as [val1 | var1 | fvar1 ] eqn:E_sv1;
destruct sv2 as [val2 | var2 | fvar2 ] eqn:E_sv2;
try discriminate.
(* val1 = val2 *)
+ unfold chk_lt_rshift_wrt_ctx in H_chkr.
unfold sstack_val_to_cliteral in H_chkr.
unfold sstack_val_to_cliteral_w_shift in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_LT (C_VAL (wordToN val1)) (C_VAL (wordToN val2 + shift))) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_Val val1 model mem strg exts maxidx sb ops) as H_eval_val1.
pose proof (eval_sstack_val_Val val2 model mem strg exts maxidx sb ops) as H_eval_val2.
rewrite N.ltb_lt in H_chkr_snd.
rewrite H_eval_val1.
rewrite H_eval_val2.
exists val1.
exists val2.
repeat split; try reflexivity.
apply H_chkr_snd.
(* val = var *)
+ unfold chk_lt_rshift_wrt_ctx in H_chkr.
unfold sstack_val_to_cliteral in H_chkr.
unfold sstack_val_to_cliteral_w_shift in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_LT (C_VAL (wordToN val1)) (C_VAR_DELTA var2 shift)) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_Val val1 model mem strg exts maxidx sb ops) as H_eval_val1.
pose proof (eval_sstack_val_InVar var2 model mem strg exts maxidx sb ops) as H_eval_var2.
rewrite N.ltb_lt in H_chkr_snd.
rewrite H_eval_val1.
rewrite H_eval_var2.
exists val1.
exists (model var2).
repeat split; try reflexivity.
apply H_chkr_snd.
(* var = val *)
+ unfold chk_lt_rshift_wrt_ctx in H_chkr.
unfold sstack_val_to_cliteral in H_chkr.
unfold sstack_val_to_cliteral_w_shift in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_LT (C_VAR var1) (C_VAL (wordToN val2 + shift))) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_InVar var1 model mem strg exts maxidx sb ops) as H_eval_var1.
pose proof (eval_sstack_val_Val val2 model mem strg exts maxidx sb ops) as H_eval_val2.
rewrite N.ltb_lt in H_chkr_snd.
rewrite H_eval_var1.
rewrite H_eval_val2.
exists (model var1).
exists val2.
repeat split; try reflexivity.
apply H_chkr_snd.
(* var = var *)
+ unfold chk_lt_rshift_wrt_ctx in H_chkr.
unfold sstack_val_to_cliteral in H_chkr.
unfold sstack_val_to_cliteral_w_shift in H_chkr.
simpl in H_chkr.
pose proof (H_chkr_snd cs (C_LT (C_VAR var1) (C_VAR_DELTA var2 shift)) H_chkr) as H_chkr_snd.
unfold is_model in H_is_model.
unfold imply in H_chkr_snd.
pose proof (H_chkr_snd model H_is_model) as H_chkr_snd.
unfold satisfies_single_constraint in H_chkr_snd.
unfold cliteral_to_nat in H_chkr_snd.
pose proof (eval_sstack_val_InVar var1 model mem strg exts maxidx sb ops) as H_eval_var1.
pose proof (eval_sstack_val_InVar var2 model mem strg exts maxidx sb ops) as H_eval_var2.
rewrite N.ltb_lt in H_chkr_snd.
rewrite H_eval_var1.
rewrite H_eval_var2.
exists (model var1).
exists (model var2).