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sstack_val_cmp_impl_soundness.v
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Require Import Arith.
Require Import Nat.
Require Import Bool.
Require Import bbv.Word.
Require Import Coq.NArith.NArith.
Require Import List.
Import ListNotations.
Require Import FORVES2.constants.
Import Constants.
Require Import FORVES2.program.
Import Program.
Require Import FORVES2.execution_state.
Import ExecutionState.
Require Import FORVES2.stack_operation_instructions.
Import StackOpInstrs.
Require Import FORVES2.misc.
Import Misc.
Require Import FORVES2.symbolic_state.
Import SymbolicState.
Require Import FORVES2.symbolic_state_eval.
Import SymbolicStateEval.
Require Import FORVES2.symbolic_state_eval_facts.
Import SymbolicStateEvalFacts.
Require Import FORVES2.valid_symbolic_state.
Import ValidSymbolicState.
Require Import FORVES2.valid_symbolic_state.
Import ValidSymbolicState.
Require Import FORVES2.symbolic_state_cmp.
Import SymbolicStateCmp.
Require Import FORVES2.symbolic_execution_soundness.
Import SymbolicExecutionSoundness.
Require Import FORVES2.sstack_val_cmp_impl.
Import SStackValCmpImpl.
Require Import FORVES2.eval_common.
Import EvalCommon.
Require Import FORVES2.symbolic_state_dec.
Import SymbolicStateDec.
Require Import FORVES2.constraints.
Import Constraints.
Require Import FORVES2.context.
Import Context.
Require Import FORVES2.context_facts.
Import ContextFacts.
Module SStackValCmpImplSoundness.
(* compare_sstack_val_trivial *)
Lemma trivial_compare_sstack_val_d0_snd:
sstack_val_cmp_fail_for_d_eq_0 trivial_compare_sstack_val.
Proof.
unfold sstack_val_cmp_fail_for_d_eq_0.
intros.
unfold trivial_compare_sstack_val.
reflexivity.
Qed.
Lemma trivial_compare_sstack_val_snd:
safe_sstack_value_cmp_wrt_others trivial_compare_sstack_val.
Proof.
unfold safe_sstack_value_cmp_wrt_others.
intros d smemory_cmp sstorage_cmp sha3_cmp.
unfold safe_sstack_val_cmp_ext_2_d.
unfold safe_sstack_val_cmp_ext_1_d.
intros H_safe_smemory_cmp H_safe_sstorage_cmp H_safe_sha3_cmp.
intros d' H_d'_le_Sd'.
unfold safe_sstack_val_cmp.
intros ctx sv1 sv2 maxidx1 bs1 maxidx2 bs2 ops.
intros H_valid_sv1 H_valid_sv2 H_valid_bs1 H_valid_bs2 H_cmp.
intros model mem strg exts H_is_model.
unfold trivial_compare_sstack_val in H_cmp.
destruct d' as [|d''] eqn:E_d; try discriminate.
unfold eval_sstack_val.
destruct sv1 as [w1 | n1 | n1]; destruct sv2 as [w2 | n2 | n2] eqn:E_sv2; try discriminate.
+ apply weqb_sound in H_cmp.
pose proof (eval_sstack_val'_Val w1 model mem strg exts maxidx1 bs1 ops) as H_eval_w1.
pose proof (eval_sstack_val'_Val w2 model mem strg exts maxidx2 bs2 ops) as H_eval_w2.
rewrite H_eval_w1.
rewrite H_eval_w2.
rewrite H_cmp.
exists w2.
split; reflexivity.
+ pose proof (eval_sstack_val'_InVar n1 model mem strg exts maxidx1 bs1 ops) as H_eval_instkv_n1.
pose proof (eval_sstack_val'_InVar n2 model mem strg exts maxidx2 bs2 ops) as H_eval_instkv_n2.
rewrite H_eval_instkv_n1.
rewrite H_eval_instkv_n2.
apply Nat.eqb_eq in H_cmp.
rewrite H_cmp.
exists (model n2).
split; try reflexivity.
+ destruct (n1 =? n2) eqn:E_n1_eq_n2; try discriminate.
destruct (maxidx1 =? maxidx2) eqn:E_maxidx1_eq_maxidx2; try discriminate.
destruct (sbindings_eq_dec bs1 bs2) eqn:E_bs1_eq_bs2; try discriminate.
apply Nat.eqb_eq in E_n1_eq_n2.
apply Nat.eqb_eq in E_maxidx1_eq_maxidx2.
rewrite E_n1_eq_n2.
rewrite E_maxidx1_eq_maxidx2.
rewrite e.
pose proof (eval_sstack_val'_succ (S maxidx2) (FreshVar n2) model mem strg exts maxidx2 bs2 ops H_valid_sv2 H_valid_bs2 (gt_Sn_n maxidx2)) as H_eval_sv2.
destruct H_eval_sv2 as [v2 H_eval_sv2].
rewrite H_eval_sv2.
exists v2.
split; try reflexivity.
Qed.
(* compare_sstack_val *)
Lemma basic_compare_sstack_val_d0_snd:
sstack_val_cmp_fail_for_d_eq_0 basic_compare_sstack_val.
Proof.
unfold sstack_val_cmp_fail_for_d_eq_0.
intros.
simpl.
reflexivity.
Qed.
Lemma basic_compare_sstack_val_snd:
safe_sstack_value_cmp_wrt_others basic_compare_sstack_val.
Proof.
unfold safe_sstack_value_cmp_wrt_others.
induction d as [|d' IHd'].
- intros smemory_cmp sstorage_cmp sha3_cmp H_safe_smemory_cmp H_safe_sstorage_cmp H_safe_sha3_cmp.
unfold safe_sstack_val_cmp_ext_2_d.
unfold safe_sstack_val_cmp_ext_1_d.
intros d' H_d'_le_1.
unfold safe_sstack_val_cmp.
intros ctx sv1 sv2 maxidx1 sb1 maxidx2 sb2 ops H_valid_sv1 H_valid_sv2 H_valid_sb1 H_valid_sb2 H_cmp_sv1_sv2.
apply Nat.leb_le in H_d'_le_1 as H_d'_le_1_leb.
intros model mem strg exts H_is_model.
unfold eval_sstack_val.
unfold eval_sstack_val'. fold eval_sstack_val'.
destruct d' as [|d''] eqn:E_d'; try discriminate; destruct d'' as [|d'''] eqn:E_d''; try discriminate.
unfold basic_compare_sstack_val in H_cmp_sv1_sv2.
pose proof (follow_in_smap_suc sb1 sv1 maxidx1 ops H_valid_sv1 H_valid_sb1) as H_follow_suc_sv1.
destruct H_follow_suc_sv1 as [smv1 [maxidx1' [sb1' [H_follow_suc_sv1 _]]]].
pose proof (follow_in_smap_suc sb2 sv2 maxidx2 ops H_valid_sv2 H_valid_sb2) as H_follow_suc_sv2.
destruct H_follow_suc_sv2 as [smv2 [maxidx2' [sb2' [H_follow_suc_sv2 _]]]].
rewrite H_follow_suc_sv1.
rewrite H_follow_suc_sv2.
rewrite H_follow_suc_sv1 in H_cmp_sv1_sv2.
rewrite H_follow_suc_sv2 in H_cmp_sv1_sv2.
pose proof (valid_follow_in_smap sb1 sv1 maxidx1 ops smv1 maxidx1' sb1' H_valid_sv1 H_valid_sb1 H_follow_suc_sv1) as H_follow_valid_sv1.
pose proof (valid_follow_in_smap sb2 sv2 maxidx2 ops smv2 maxidx2' sb2' H_valid_sv2 H_valid_sb2 H_follow_suc_sv2) as H_follow_valid_sv2.
destruct smv1 eqn:E_smv1; destruct smv2 eqn:E_smv2; try discriminate.
+ destruct val; destruct val0; try discriminate.
* apply weqb_sound in H_cmp_sv1_sv2.
rewrite H_cmp_sv1_sv2.
exists val0.
split; reflexivity.
* pose proof (chk_eq_wrt_ctx_snd ctx (Val val) (InVar var) H_cmp_sv1_sv2 model mem strg exts maxidx1 sb1 ops H_is_model) as H_ctx_sv1'_eq_sv2'_0.
pose proof (chk_eq_wrt_ctx_snd ctx (Val val) (InVar var) H_cmp_sv1_sv2 model mem strg exts maxidx2 sb2 ops H_is_model) as H_ctx_sv1'_eq_sv2'_1.
destruct H_ctx_sv1'_eq_sv2'_0 as [v1 [H_ctx_sv1'_eq_sv2'_0_0 H_ctx_sv1'_eq_sv2'_0_1]].
destruct H_ctx_sv1'_eq_sv2'_1 as [v2 [H_ctx_sv1'_eq_sv2'_1_0 H_ctx_sv1'_eq_sv2'_1_1]].
pose proof (eval_sstack_val_Val val model mem strg exts maxidx1 sb1 ops) as H_eval_val_0.
pose proof (eval_sstack_val_Val val model mem strg exts maxidx2 sb2 ops) as H_eval_val_1.
pose proof (eval_sstack_val_InVar var model mem strg exts maxidx1 sb1 ops) as H_eval_var_0.
pose proof (eval_sstack_val_InVar var model mem strg exts maxidx2 sb2 ops) as H_eval_var_1.
rewrite H_eval_val_0 in H_ctx_sv1'_eq_sv2'_0_0.
rewrite H_eval_var_0 in H_ctx_sv1'_eq_sv2'_0_1.
injection H_ctx_sv1'_eq_sv2'_0_0 as H_ctx_sv1'_eq_sv2'_0_0.
injection H_ctx_sv1'_eq_sv2'_0_1 as H_ctx_sv1'_eq_sv2'_0_1.
exists v1.
rewrite H_ctx_sv1'_eq_sv2'_0_1.
rewrite H_ctx_sv1'_eq_sv2'_0_0.
split; reflexivity.
* pose proof (chk_eq_wrt_ctx_snd ctx (InVar var) (Val val) H_cmp_sv1_sv2 model mem strg exts maxidx1 sb1 ops H_is_model) as H_ctx_sv1'_eq_sv2'_0.
pose proof (chk_eq_wrt_ctx_snd ctx (InVar var) (Val val) H_cmp_sv1_sv2 model mem strg exts maxidx2 sb2 ops H_is_model) as H_ctx_sv1'_eq_sv2'_1.
destruct H_ctx_sv1'_eq_sv2'_0 as [v1 [H_ctx_sv1'_eq_sv2'_0_0 H_ctx_sv1'_eq_sv2'_0_1]].
destruct H_ctx_sv1'_eq_sv2'_1 as [v2 [H_ctx_sv1'_eq_sv2'_1_0 H_ctx_sv1'_eq_sv2'_1_1]].
pose proof (eval_sstack_val_Val val model mem strg exts maxidx1 sb1 ops) as H_eval_val_0.
pose proof (eval_sstack_val_Val val model mem strg exts maxidx2 sb2 ops) as H_eval_val_1.
pose proof (eval_sstack_val_InVar var model mem strg exts maxidx1 sb1 ops) as H_eval_var_0.
pose proof (eval_sstack_val_InVar var model mem strg exts maxidx2 sb2 ops) as H_eval_var_1.
rewrite H_eval_var_0 in H_ctx_sv1'_eq_sv2'_0_0.
rewrite H_eval_val_0 in H_ctx_sv1'_eq_sv2'_0_1.
injection H_ctx_sv1'_eq_sv2'_0_0 as H_ctx_sv1'_eq_sv2'_0_0.
injection H_ctx_sv1'_eq_sv2'_0_1 as H_ctx_sv1'_eq_sv2'_0_1.
exists v1.
rewrite H_ctx_sv1'_eq_sv2'_0_1.
rewrite H_ctx_sv1'_eq_sv2'_0_0.
split; reflexivity.
* apply orb_prop in H_cmp_sv1_sv2.
destruct H_cmp_sv1_sv2 as [H_cmp_sv1_sv2 | H_cmp_sv1_sv2].
** apply Nat.eqb_eq in H_cmp_sv1_sv2.
rewrite H_cmp_sv1_sv2.
exists (model var0).
split; reflexivity.
** pose proof (chk_eq_wrt_ctx_snd ctx (InVar var) (InVar var0) H_cmp_sv1_sv2 model mem strg exts maxidx1 sb1 ops H_is_model) as H_ctx_sv1'_eq_sv2'_0.
pose proof (chk_eq_wrt_ctx_snd ctx (InVar var) (InVar var0) H_cmp_sv1_sv2 model mem strg exts maxidx2 sb2 ops H_is_model) as H_ctx_sv1'_eq_sv2'_1.
destruct H_ctx_sv1'_eq_sv2'_0 as [v1 [H_ctx_sv1'_eq_sv2'_0_0 H_ctx_sv1'_eq_sv2'_0_1]].
destruct H_ctx_sv1'_eq_sv2'_1 as [v2 [H_ctx_sv1'_eq_sv2'_1_0 H_ctx_sv1'_eq_sv2'_1_1]].
pose proof (eval_sstack_val_InVar var model mem strg exts maxidx1 sb1 ops) as H_eval_var_0.
pose proof (eval_sstack_val_InVar var0 model mem strg exts maxidx1 sb1 ops) as H_eval_var_1.
pose proof (eval_sstack_val_InVar var model mem strg exts maxidx2 sb2 ops) as H_eval_var_0_1.
pose proof (eval_sstack_val_InVar var0 model mem strg exts maxidx2 sb2 ops) as H_eval_var_1_1.
rewrite H_eval_var_1 in H_ctx_sv1'_eq_sv2'_0_1.
rewrite H_eval_var_0 in H_ctx_sv1'_eq_sv2'_0_0.
injection H_ctx_sv1'_eq_sv2'_0_0 as H_ctx_sv1'_eq_sv2'_0_0.
injection H_ctx_sv1'_eq_sv2'_0_1 as H_ctx_sv1'_eq_sv2'_0_1.
exists v1.
rewrite H_ctx_sv1'_eq_sv2'_0_1.
rewrite H_ctx_sv1'_eq_sv2'_0_0.
split; reflexivity.
+ apply andb_prop in H_cmp_sv1_sv2 as [ H_cmp_sv1_sv2_0 H_cmp_sv1_sv2_1].
apply N.eqb_eq in H_cmp_sv1_sv2_0.
apply N.eqb_eq in H_cmp_sv1_sv2_1.
rewrite H_cmp_sv1_sv2_0.
rewrite H_cmp_sv1_sv2_1.
exists (get_tags_exts exts cat0 val0). split; reflexivity.
+ destruct (label =?i label0) eqn:E_eqb_label_label0; try discriminate.
apply eqb_stack_op_instr_eq in E_eqb_label_label0 as E_eq_label_label0.
rewrite <- E_eq_label_label0.
rewrite <- E_eq_label_label0 in H_follow_suc_sv2.
destruct (ops label) eqn:E_ops_label.
destruct args as [|a args'] eqn:E_args; destruct args0 as [|a0 args0'] eqn:E_args_0.
* simpl in H_cmp_sv1_sv2.
simpl.
destruct n.
** exists (f exts []). split; reflexivity.
** simpl in H_follow_valid_sv1.
simpl in H_follow_valid_sv2.
destruct H_follow_valid_sv1 as [H_follow_valid_sv1_0 _].
unfold valid_stack_op_instr in H_follow_valid_sv1_0.
rewrite E_ops_label in H_follow_valid_sv1_0.
destruct H_follow_valid_sv1_0 as [H_follow_valid_sv1_0 _].
discriminate H_follow_valid_sv1_0.
* simpl in H_cmp_sv1_sv2. destruct H_comm; discriminate.
* simpl in H_cmp_sv1_sv2. destruct H_comm; try discriminate. destruct args'; try discriminate.
destruct args'; try discriminate.
* simpl in H_cmp_sv1_sv2. destruct H_comm; try discriminate.
destruct args' as [| a' args'']; try discriminate.
destruct args'' as [| a'' args''']; try discriminate.
destruct args0' as [| a0' args0'']; try discriminate.
destruct args0'' as [| a0'' args0''']; try discriminate.
+ unfold safe_sha3_cmp_ext_d in H_safe_sha3_cmp.
pose proof (H_safe_sha3_cmp 0 (Nat.le_refl 0)) as H_safe_sha3_cmp.
simpl in H_safe_sha3_cmp.
unfold safe_sha3_cmp in H_safe_sha3_cmp.
simpl in H_follow_valid_sv1.
destruct H_follow_valid_sv1 as [ [H_follow_valid_offset [H_follow_valid_size H_follow_valid_smem]] [H_follow_valid_sb1' H_maxidx1_gt_maxidx1']].
pose proof (H_maxidx1_gt_maxidx1' (eq_refl true)) as H_maxidx1_gt_maxidx1'.
simpl in H_follow_valid_sv2.
destruct H_follow_valid_sv2 as [ [H_follow_valid_offset0 [H_follow_valid_size0 H_follow_valid_smem0]] [H_follow_valid_sb2' H_maxidx2_gt_maxidx2']].
pose proof (H_maxidx2_gt_maxidx2' (eq_refl true)) as H_maxidx2_gt_maxidx2'.
pose proof (H_safe_sha3_cmp ctx offset size smem offset0 size0 smem0 maxidx1' sb1' maxidx2' sb2' ops H_follow_valid_offset H_follow_valid_size H_follow_valid_offset0 H_follow_valid_size0 H_follow_valid_sb1' H_follow_valid_sb2' H_follow_valid_smem H_follow_valid_smem0 H_cmp_sv1_sv2 model mem strg exts H_is_model) as H_safe_sha3_cmp.
destruct H_safe_sha3_cmp as [coffset [csize [mem1 [coffset0 [csize0 [mem2 [v [H_eval_smem [H_eval_smem0 [H_eval_offset [H_eval_size [H_eval_offset0 [H_eval_size0 [H_sha3_mem1 H_sha3_mem2]]]]]]]]]]]]]].
unfold eval_smemory in H_eval_smem.
destruct (map_option (EvalCommon.instantiate_memory_update (fun sv : sstack_val => eval_sstack_val sv model mem strg exts maxidx1' sb1' ops)) smem) as [updates|] eqn:H_eval_smem_0; try discriminate.
unfold eval_sstack_val in H_eval_smem_0.
assert (H_maxidx1_ge_S_maxidx1': S maxidx1' <= maxidx1). intuition.
pose proof (instantiate_memory_update_mapo_preserved_when_depth_ext_le smem (S maxidx1') maxidx1 model mem strg exts maxidx1' sb1' ops updates H_maxidx1_ge_S_maxidx1' H_eval_smem_0) as H_eval_smem_0_ext.
rewrite H_eval_smem_0_ext.
unfold eval_smemory in H_eval_smem0.
destruct (map_option (EvalCommon.instantiate_memory_update (fun sv : sstack_val => eval_sstack_val sv model mem strg exts maxidx2' sb2' ops)) smem0) as [updates0|] eqn:H_eval_smem0_0; try discriminate.
unfold eval_sstack_val in H_eval_smem0_0.
assert (H_maxidx2_ge_S_maxidx2': S maxidx2' <= maxidx2). intuition.
pose proof (instantiate_memory_update_mapo_preserved_when_depth_ext_le smem0 (S maxidx2') maxidx2 model mem strg exts maxidx2' sb2' ops updates0 H_maxidx2_ge_S_maxidx2' H_eval_smem0_0) as H_eval_smem0_0_ext.
rewrite H_eval_smem0_0_ext.
unfold eval_sstack_val in H_eval_offset.
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx1') maxidx1 maxidx1' sb1' offset coffset model mem strg exts ops H_maxidx1_ge_S_maxidx1' H_eval_offset) as H_eval_offset_0.
rewrite H_eval_offset_0.
unfold eval_sstack_val in H_eval_size.
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx1') maxidx1 maxidx1' sb1' size csize model mem strg exts ops H_maxidx1_ge_S_maxidx1' H_eval_size) as H_eval_size_0.
rewrite H_eval_size_0.
unfold eval_sstack_val in H_eval_offset0.
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx2') maxidx2 maxidx2' sb2' offset0 coffset0 model mem strg exts ops H_maxidx2_ge_S_maxidx2' H_eval_offset0) as H_eval_offset0_0.
rewrite H_eval_offset0_0.
unfold eval_sstack_val in H_eval_size0.
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx2') maxidx2 maxidx2' sb2' size0 csize0 model mem strg exts ops H_maxidx2_ge_S_maxidx2' H_eval_size0) as H_eval_size0_0.
rewrite H_eval_size0_0.
injection H_eval_smem as H_eval_smem.
rewrite H_eval_smem.
injection H_eval_smem0 as H_eval_smem0.
rewrite H_eval_smem0.
rewrite H_sha3_mem1.
rewrite H_sha3_mem2.
exists v.
split; reflexivity.
- intros smemory_cmp sstorage_cmp sha3_cmp H_safe_smemory_cmp H_safe_sstorage_cmp H_safe_sha3_cmp.
unfold safe_sstack_val_cmp_ext_2_d.
unfold safe_sstack_val_cmp_ext_1_d.
unfold safe_sstack_val_cmp.
intros d'0 H_d'0_le_SS_d'.
intros ctx sv1 sv2 maxidx1 sb1 maxidx2 sb2 ops H_valid_sv1 H_valid_sv2 H_valid_sb1 H_valid_sb2 H_cmp_sv1_sv2.
intros model mem strg exts H_is_model.
destruct d'0; try discriminate.
simpl in H_cmp_sv1_sv2.
unfold eval_sstack_val.
unfold eval_sstack_val'. fold eval_sstack_val'.
assert(H_d'_le_Sd': d' <= S d'). intuition.
pose proof (safe_smemory_cmp_ext_d_lt smemory_cmp (basic_compare_sstack_val smemory_cmp sstorage_cmp sha3_cmp) d' (S d') H_d'_le_Sd' H_safe_smemory_cmp) as H_safe_smemory_cmp_d'.
pose proof (safe_sstorage_cmp_ext_d_lt sstorage_cmp (basic_compare_sstack_val smemory_cmp sstorage_cmp sha3_cmp) d' (S d') H_d'_le_Sd' H_safe_sstorage_cmp) as H_safe_sstorgae_cmp_d'.
pose proof (safe_sha3_cmp_ext_d_lt sha3_cmp (basic_compare_sstack_val smemory_cmp sstorage_cmp sha3_cmp) d' (S d') H_d'_le_Sd' H_safe_sha3_cmp) as H_safe_sha3_cmp_d'.
pose proof (IHd' smemory_cmp sstorage_cmp sha3_cmp H_safe_smemory_cmp_d' H_safe_sstorgae_cmp_d' H_safe_sha3_cmp_d') as H_safe_sstack_value_cmp_cmp_Sd'.
assert(H_d'0_le_Sd': d'0 <= S d'). intuition.
pose proof (safe_sstack_val_cmp_ext_2_d_le basic_compare_sstack_val smemory_cmp sstorage_cmp sha3_cmp d'0 (S d') H_d'0_le_Sd' H_safe_sstack_value_cmp_cmp_Sd') as H_safe_sstack_value_cmp_cmp_d'0.
pose proof (follow_in_smap_suc sb1 sv1 maxidx1 ops H_valid_sv1 H_valid_sb1) as H_follow_suc_sv1.
destruct H_follow_suc_sv1 as [smv1 [maxidx1' [sb1' [H_follow_suc_sv1 _]]]].
pose proof (follow_in_smap_suc sb2 sv2 maxidx2 ops H_valid_sv2 H_valid_sb2) as H_follow_suc_sv2.
destruct H_follow_suc_sv2 as [smv2 [maxidx2' [sb2' [H_follow_suc_sv2 _]]]].
rewrite H_follow_suc_sv1.
rewrite H_follow_suc_sv2.
rewrite H_follow_suc_sv1 in H_cmp_sv1_sv2.
rewrite H_follow_suc_sv2 in H_cmp_sv1_sv2.
pose proof (valid_follow_in_smap sb1 sv1 maxidx1 ops smv1 maxidx1' sb1' H_valid_sv1 H_valid_sb1 H_follow_suc_sv1) as H_follow_valid_sv1.
pose proof (valid_follow_in_smap sb2 sv2 maxidx2 ops smv2 maxidx2' sb2' H_valid_sv2 H_valid_sb2 H_follow_suc_sv2) as H_follow_valid_sv2.
destruct smv1 eqn:E_smv1; destruct smv2 eqn:E_smv2; try discriminate.
+ destruct val; destruct val0; try discriminate.
* apply weqb_sound in H_cmp_sv1_sv2.
rewrite H_cmp_sv1_sv2.
exists val0.
split; reflexivity.
* pose proof (chk_eq_wrt_ctx_snd ctx (Val val) (InVar var) H_cmp_sv1_sv2 model mem strg exts maxidx1 sb1 ops H_is_model) as H_ctx_sv1'_eq_sv2'_0.
pose proof (chk_eq_wrt_ctx_snd ctx (Val val) (InVar var) H_cmp_sv1_sv2 model mem strg exts maxidx2 sb2 ops H_is_model) as H_ctx_sv1'_eq_sv2'_1.
destruct H_ctx_sv1'_eq_sv2'_0 as [v1 [H_ctx_sv1'_eq_sv2'_0_0 H_ctx_sv1'_eq_sv2'_0_1]].
destruct H_ctx_sv1'_eq_sv2'_1 as [v2 [H_ctx_sv1'_eq_sv2'_1_0 H_ctx_sv1'_eq_sv2'_1_1]].
pose proof (eval_sstack_val_Val val model mem strg exts maxidx1 sb1 ops) as H_eval_val_0.
pose proof (eval_sstack_val_Val val model mem strg exts maxidx2 sb2 ops) as H_eval_val_1.
pose proof (eval_sstack_val_InVar var model mem strg exts maxidx1 sb1 ops) as H_eval_var_0.
pose proof (eval_sstack_val_InVar var model mem strg exts maxidx2 sb2 ops) as H_eval_var_1.
rewrite H_eval_val_0 in H_ctx_sv1'_eq_sv2'_0_0.
rewrite H_eval_var_0 in H_ctx_sv1'_eq_sv2'_0_1.
injection H_ctx_sv1'_eq_sv2'_0_0 as H_ctx_sv1'_eq_sv2'_0_0.
injection H_ctx_sv1'_eq_sv2'_0_1 as H_ctx_sv1'_eq_sv2'_0_1.
exists v1.
rewrite H_ctx_sv1'_eq_sv2'_0_1.
rewrite H_ctx_sv1'_eq_sv2'_0_0.
split; reflexivity.
* pose proof (chk_eq_wrt_ctx_snd ctx (InVar var) (Val val) H_cmp_sv1_sv2 model mem strg exts maxidx1 sb1 ops H_is_model) as H_ctx_sv1'_eq_sv2'_0.
pose proof (chk_eq_wrt_ctx_snd ctx (InVar var) (Val val) H_cmp_sv1_sv2 model mem strg exts maxidx2 sb2 ops H_is_model) as H_ctx_sv1'_eq_sv2'_1.
destruct H_ctx_sv1'_eq_sv2'_0 as [v1 [H_ctx_sv1'_eq_sv2'_0_0 H_ctx_sv1'_eq_sv2'_0_1]].
destruct H_ctx_sv1'_eq_sv2'_1 as [v2 [H_ctx_sv1'_eq_sv2'_1_0 H_ctx_sv1'_eq_sv2'_1_1]].
pose proof (eval_sstack_val_Val val model mem strg exts maxidx1 sb1 ops) as H_eval_val_0.
pose proof (eval_sstack_val_Val val model mem strg exts maxidx2 sb2 ops) as H_eval_val_1.
pose proof (eval_sstack_val_InVar var model mem strg exts maxidx1 sb1 ops) as H_eval_var_0.
pose proof (eval_sstack_val_InVar var model mem strg exts maxidx2 sb2 ops) as H_eval_var_1.
rewrite H_eval_var_0 in H_ctx_sv1'_eq_sv2'_0_0.
rewrite H_eval_val_0 in H_ctx_sv1'_eq_sv2'_0_1.
injection H_ctx_sv1'_eq_sv2'_0_0 as H_ctx_sv1'_eq_sv2'_0_0.
injection H_ctx_sv1'_eq_sv2'_0_1 as H_ctx_sv1'_eq_sv2'_0_1.
exists v1.
rewrite H_ctx_sv1'_eq_sv2'_0_1.
rewrite H_ctx_sv1'_eq_sv2'_0_0.
split; reflexivity.
* apply orb_prop in H_cmp_sv1_sv2.
destruct H_cmp_sv1_sv2 as [H_cmp_sv1_sv2 | H_cmp_sv1_sv2].
** apply Nat.eqb_eq in H_cmp_sv1_sv2.
rewrite H_cmp_sv1_sv2.
exists (model var0).
split; reflexivity.
** pose proof (chk_eq_wrt_ctx_snd ctx (InVar var) (InVar var0) H_cmp_sv1_sv2 model mem strg exts maxidx1 sb1 ops H_is_model) as H_ctx_sv1'_eq_sv2'_0.
pose proof (chk_eq_wrt_ctx_snd ctx (InVar var) (InVar var0) H_cmp_sv1_sv2 model mem strg exts maxidx2 sb2 ops H_is_model) as H_ctx_sv1'_eq_sv2'_1.
destruct H_ctx_sv1'_eq_sv2'_0 as [v1 [H_ctx_sv1'_eq_sv2'_0_0 H_ctx_sv1'_eq_sv2'_0_1]].
destruct H_ctx_sv1'_eq_sv2'_1 as [v2 [H_ctx_sv1'_eq_sv2'_1_0 H_ctx_sv1'_eq_sv2'_1_1]].
pose proof (eval_sstack_val_InVar var model mem strg exts maxidx1 sb1 ops) as H_eval_var_0.
pose proof (eval_sstack_val_InVar var0 model mem strg exts maxidx1 sb1 ops) as H_eval_var_1.
pose proof (eval_sstack_val_InVar var model mem strg exts maxidx2 sb2 ops) as H_eval_var_0_1.
pose proof (eval_sstack_val_InVar var0 model mem strg exts maxidx2 sb2 ops) as H_eval_var_1_1.
rewrite H_eval_var_1 in H_ctx_sv1'_eq_sv2'_0_1.
rewrite H_eval_var_0 in H_ctx_sv1'_eq_sv2'_0_0.
injection H_ctx_sv1'_eq_sv2'_0_0 as H_ctx_sv1'_eq_sv2'_0_0.
injection H_ctx_sv1'_eq_sv2'_0_1 as H_ctx_sv1'_eq_sv2'_0_1.
exists v1.
rewrite H_ctx_sv1'_eq_sv2'_0_1.
rewrite H_ctx_sv1'_eq_sv2'_0_0.
split; reflexivity.
+ apply andb_prop in H_cmp_sv1_sv2 as [H_cmp_sv1_sv2_0 H_cmp_sv1_sv2_1].
apply N.eqb_eq in H_cmp_sv1_sv2_0.
apply N.eqb_eq in H_cmp_sv1_sv2_1.
rewrite H_cmp_sv1_sv2_0.
rewrite H_cmp_sv1_sv2_1.
exists (get_tags_exts exts cat0 val0). split; reflexivity.
+ fold eval_sstack_val' in H_cmp_sv1_sv2. destruct (label =?i label0) eqn:E_eqb_label_label0; try discriminate.
apply eqb_stack_op_instr_eq in E_eqb_label_label0 as E_eq_label_label0.
rewrite <- E_eq_label_label0.
rewrite <- E_eq_label_label0 in H_follow_suc_sv2.
rewrite <- E_eq_label_label0 in H_follow_valid_sv2.
destruct H_follow_valid_sv1 as [H_follow_valid_sv1_0 [H_follow_valid_sv1_1 H_follow_valid_sv1_2]].
simpl in H_follow_valid_sv1_0.
unfold valid_stack_op_instr in H_follow_valid_sv1_0.
destruct (ops label) eqn:E_ops_label.
destruct H_follow_valid_sv1_0 as [H_follow_valid_sv1_0_0 H_follow_valid_sv1_0_1].
destruct H_follow_valid_sv2 as [H_follow_valid_sv2_0 [H_follow_valid_sv2_1 H_follow_valid_sv2_2]].
simpl in H_follow_valid_sv2_0.
unfold valid_stack_op_instr in H_follow_valid_sv2_0.
rewrite E_ops_label in H_follow_valid_sv2_0.
destruct H_follow_valid_sv2_0 as [H_follow_valid_sv2_0_0 H_follow_valid_sv2_0_1].
apply Nat.eqb_eq in H_follow_valid_sv1_0_0 as H_follow_valid_sv1_0_0_eq.
apply Nat.eqb_eq in H_follow_valid_sv2_0_0 as H_follow_valid_sv2_0_0_eq.
rewrite H_follow_valid_sv1_0_0_eq.
rewrite H_follow_valid_sv2_0_0_eq.
assert(H_follow_valid_sv1_0_0_eq' := H_follow_valid_sv1_0_0_eq).
rewrite <- H_follow_valid_sv2_0_0 in H_follow_valid_sv1_0_0_eq'.
unfold safe_sstack_val_cmp_ext_2_d in H_safe_sstack_value_cmp_cmp_d'0.
unfold safe_sstack_val_cmp_ext_1_d in H_safe_sstack_value_cmp_cmp_d'0.
unfold safe_sstack_val_cmp in H_safe_sstack_value_cmp_cmp_d'0.
assert(H_fldr:
forall args1 args2,
valid_sstack maxidx1' args1 ->
valid_sstack maxidx2' args2 ->
fold_right_two_lists (fun e1 e2 : sstack_val => basic_compare_sstack_val smemory_cmp sstorage_cmp sha3_cmp d'0 ctx e1 e2 maxidx1' sb1' maxidx2' sb2' ops) args1 args2 = true ->
exists args',
map_option (fun sv' : sstack_val => eval_sstack_val' (S maxidx1') sv' model mem strg exts maxidx1' sb1' ops) args1 = Some args' /\
map_option (fun sv' : sstack_val => eval_sstack_val' (S maxidx2') sv' model mem strg exts maxidx2' sb2' ops) args2 = Some args').
(* staring proof of assert *)
* induction args1 as [|a1_1 args1'].
** destruct args2 as [|a1_2 args2']; try discriminate.
intros.
simpl.
exists [].
split; reflexivity.
** destruct args2 as [|a1_2 args2']; try discriminate.
intros H_valid_sstack_args1 H_valid_sstack_args2 H_fldr.
unfold fold_right_two_lists in H_fldr.
rewrite <- fold_right_two_lists_ho in H_fldr.
destruct (basic_compare_sstack_val smemory_cmp sstorage_cmp sha3_cmp d'0 ctx a1_1 a1_2
maxidx1' sb1' maxidx2' sb2' ops) eqn:E_cmp_a1_1_a1_2; try discriminate.
simpl in H_valid_sstack_args1.
destruct H_valid_sstack_args1 as [H_valid_a1_1 H_valid_arg1'].
simpl in H_valid_sstack_args2.
destruct H_valid_sstack_args2 as [H_valid_a1_2 H_valid_arg2'].
pose proof (H_safe_sstack_value_cmp_cmp_d'0 d'0 (Nat.le_refl d'0) ctx a1_1 a1_2 maxidx1' sb1' maxidx2' sb2' ops H_valid_a1_1 H_valid_a1_2 H_follow_valid_sv1_1 H_follow_valid_sv2_1 E_cmp_a1_1_a1_2 model mem strg exts H_is_model) as H_eval_sstack_value_a1_1_a1_2.
destruct H_eval_sstack_value_a1_1_a1_2 as [v [H_eval_sstack_value_a1_1 H_eval_sstack_value_a1_2]].
unfold eval_sstack_val in H_eval_sstack_value_a1_1.
unfold eval_sstack_val in H_eval_sstack_value_a1_2.
unfold map_option.
repeat rewrite <- map_option_ho.
rewrite H_eval_sstack_value_a1_1.
rewrite H_eval_sstack_value_a1_2.
pose proof (IHargs1' args2' H_valid_arg1' H_valid_arg2' H_fldr) as H_mapo.
destruct H_mapo as [args' [H_mapo_args1' H_mapo_args2']].
rewrite H_mapo_args1'.
rewrite H_mapo_args2'.
exists (v :: args').
split; reflexivity.
(* ending proof of assert *)
* destruct (fold_right_two_lists
(fun e1 e2 : sstack_val =>
basic_compare_sstack_val smemory_cmp sstorage_cmp sha3_cmp d'0 ctx e1 e2 maxidx1' sb1' maxidx2' sb2'
ops) args args0) eqn:E_fldr.
** pose proof (H_fldr args args0 H_follow_valid_sv1_0_1 H_follow_valid_sv2_0_1 E_fldr) as H_fldr_0.
destruct H_fldr_0 as [args' [H_fldr_0_0 H_fldr_0_1]].
simpl in H_follow_valid_sv2_2.
pose proof (H_follow_valid_sv2_2 (eq_refl true)) as H_follow_valid_sv2_2.
simpl in H_follow_valid_sv1_2.
pose proof (H_follow_valid_sv1_2 (eq_refl true)) as H_follow_valid_sv1_2.
assert (H_maxidx1_ge_S_maxidx1': maxidx1 >= S maxidx1'). intuition.
pose proof (eval_sstack_val'_mapo_preserved_when_depth_ext_le args (S maxidx1') maxidx1 model mem strg exts maxidx1' sb1' ops args' H_maxidx1_ge_S_maxidx1' H_fldr_0_0) as H_fldr_0_0_0.
rewrite H_fldr_0_0_0.
assert (H_maxidx2_ge_S_maxidx2': maxidx2 >= S maxidx2'). intuition.
pose proof (eval_sstack_val'_mapo_preserved_when_depth_ext_le args0 (S maxidx2') maxidx2 model mem strg exts maxidx2' sb2' ops args' H_maxidx2_ge_S_maxidx2' H_fldr_0_1) as H_fldr_0_1_0.
rewrite H_fldr_0_1_0.
exists (f exts args').
split; reflexivity.
** destruct H_comm as [H_f_comm_proof|]; try discriminate.
destruct args as [|a1 args]; try discriminate.
destruct args as [|a2 args]; try discriminate.
destruct args; try discriminate.
destruct args0 as [|b1 args0]; try discriminate.
destruct args0 as [|b2 args0]; try discriminate.
destruct args0; try discriminate.
destruct (basic_compare_sstack_val smemory_cmp sstorage_cmp sha3_cmp d'0 ctx a1 b2 maxidx1' sb1' maxidx2' sb2' ops) eqn:E_cmp_a1_b2; try discriminate.
simpl in H_follow_valid_sv1_0_1.
destruct H_follow_valid_sv1_0_1 as [H_follow_valid_a1 [H_follow_valid_a2 _]].
simpl in H_follow_valid_sv2_0_1.
destruct H_follow_valid_sv2_0_1 as [H_follow_valid_b1 [H_follow_valid_b2 _]].
pose proof (H_safe_sstack_value_cmp_cmp_d'0 d'0 (Nat.le_refl d'0) ctx a1 b2 maxidx1' sb1' maxidx2' sb2' ops H_follow_valid_a1 H_follow_valid_b2 H_follow_valid_sv1_1 H_follow_valid_sv2_1 E_cmp_a1_b2 model mem strg exts H_is_model) as H_safe_sstack_value_cmp_a1_b2.
unfold eval_sstack_val in H_safe_sstack_value_cmp_a1_b2.
destruct H_safe_sstack_value_cmp_a1_b2 as [v1 [H_safe_sstack_value_cmp_a1 H_safe_sstack_value_cmp_b2]].
pose proof (H_safe_sstack_value_cmp_cmp_d'0 d'0 (Nat.le_refl d'0) ctx a2 b1 maxidx1' sb1' maxidx2' sb2' ops H_follow_valid_a2 H_follow_valid_b1 H_follow_valid_sv1_1 H_follow_valid_sv2_1 H_cmp_sv1_sv2 model mem strg exts H_is_model) as H_safe_sstack_value_cmp_a2_b1.
unfold eval_sstack_val in H_safe_sstack_value_cmp_a2_b1.
destruct H_safe_sstack_value_cmp_a2_b1 as [v2 [H_safe_sstack_value_cmp_a2 H_safe_sstack_value_cmp_b1]].
assert(H_Smaxidx1'_le_maxidx1: S maxidx1' <= maxidx1). intuition.
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx1') maxidx1 maxidx1' sb1' a1 v1 model mem strg exts ops H_Smaxidx1'_le_maxidx1 H_safe_sstack_value_cmp_a1) as H_safe_sstack_value_cmp_a1_ext.
assert(H_Smaxidx2'_le_maxidx2: S maxidx2' <= maxidx2). intuition.
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx2') maxidx2 maxidx2' sb2' b2 v1 model mem strg exts ops H_Smaxidx2'_le_maxidx2 H_safe_sstack_value_cmp_b2) as H_safe_sstack_value_cmp_b2_ext.
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx1') maxidx1 maxidx1' sb1' a2 v2 model mem strg exts ops H_Smaxidx1'_le_maxidx1 H_safe_sstack_value_cmp_a2) as H_safe_sstack_value_cmp_a2_ext.
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx2') maxidx2 maxidx2' sb2' b1 v2 model mem strg exts ops H_Smaxidx2'_le_maxidx2 H_safe_sstack_value_cmp_b1) as H_safe_sstack_value_cmp_b1_ext.
unfold map_option.
rewrite H_safe_sstack_value_cmp_a1_ext.
rewrite H_safe_sstack_value_cmp_b2_ext.
rewrite H_safe_sstack_value_cmp_a2_ext.
rewrite H_safe_sstack_value_cmp_b1_ext.
exists (f exts [v1; v2]).
unfold commutative_op in H_f_comm_proof.
rewrite H_f_comm_proof.
split; reflexivity.
+ destruct (basic_compare_sstack_val smemory_cmp sstorage_cmp sha3_cmp d'0 ctx offset offset0 maxidx1' sb1' maxidx2' sb2' ops) eqn:E_cmp_offset_offset0; try discriminate.
simpl in H_follow_valid_sv1.
destruct H_follow_valid_sv1 as [[H_valid_offset H_valid_smem] [H_valid_sb1' H_maxidx1_gt_maxidx1']].
pose proof (H_maxidx1_gt_maxidx1' (eq_refl true)) as H_maxidx1_gt_maxidx1'.
simpl in H_follow_valid_sv2.
destruct H_follow_valid_sv2 as [[H_valid_offset0 H_valid_smem0] [H_valid_sb2' H_maxidx2_gt_maxidx2']].
pose proof (H_maxidx2_gt_maxidx2' (eq_refl true)) as H_maxidx2_gt_maxidx2'.
pose proof (H_safe_sstack_value_cmp_cmp_d'0 d'0 (Nat.le_refl d'0) ctx offset offset0 maxidx1' sb1' maxidx2' sb2' ops H_valid_offset H_valid_offset0 H_valid_sb1' H_valid_sb2' E_cmp_offset_offset0 model mem strg exts H_is_model) as H_eval_offset_offset0.
unfold eval_sstack_val in H_eval_offset_offset0.
destruct H_eval_offset_offset0 as [v [H_eval_offset H_eval_offset0]].
assert (H_Smaxidx1'_le_maxidx1: S maxidx1' <= maxidx1). intuition.
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx1') maxidx1 maxidx1' sb1' offset v model mem strg exts ops H_Smaxidx1'_le_maxidx1 H_eval_offset) as H_eval_soffset_ext.
rewrite H_eval_soffset_ext.
assert (H_Smaxidx2'_le_maxidx2: S maxidx2' <= maxidx2). intuition.
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx2') maxidx2 maxidx2' sb2' offset0 v model mem strg exts ops H_Smaxidx2'_le_maxidx2 H_eval_offset0) as H_eval_soffset0_ext.
rewrite H_eval_soffset0_ext.
unfold safe_smemory_cmp_ext_d in H_safe_smemory_cmp.
unfold safe_smemory_cmp in H_safe_smemory_cmp.
assert(H_d'0_le_d': d'0 <= S d'). intuition.
pose proof (H_safe_smemory_cmp d'0 H_d'0_le_d' ctx smem smem0 maxidx1' sb1' maxidx2' sb2' ops H_valid_sb1' H_valid_sb2' H_valid_smem H_valid_smem0 H_cmp_sv1_sv2 model mem strg exts H_is_model) as H_safe_smemory_cmp_0.
destruct H_safe_smemory_cmp_0 as [mem' [H_eval_mem H_eval_mem0]].
unfold eval_smemory in H_eval_mem.
unfold eval_smemory in H_eval_mem0.
destruct (map_option
(instantiate_memory_update
(fun sv : sstack_val =>
eval_sstack_val sv model mem strg exts maxidx1'
sb1' ops)) smem) as [updates|] eqn:E_mapo_mem; try discriminate.
unfold eval_sstack_val in E_mapo_mem.
destruct (map_option
(instantiate_memory_update
(fun sv : sstack_val =>
eval_sstack_val sv model mem strg exts maxidx2'
sb2' ops)) smem0) as [updates0|] eqn:E_mapo_mem0; try discriminate.
unfold eval_sstack_val in E_mapo_mem0.
pose proof (instantiate_memory_update_mapo_preserved_when_depth_ext_le smem (S maxidx1') maxidx1 model mem strg exts maxidx1' sb1' ops updates H_Smaxidx1'_le_maxidx1 E_mapo_mem) as E_mapo_mem_ext.
rewrite E_mapo_mem_ext.
pose proof (instantiate_memory_update_mapo_preserved_when_depth_ext_le smem0 (S maxidx2') maxidx2 model mem strg exts maxidx2' sb2' ops updates0 H_Smaxidx2'_le_maxidx2 E_mapo_mem0) as E_mapo_mem0_ext.
rewrite E_mapo_mem0_ext.
injection H_eval_mem0 as H_eval_mem0.
injection H_eval_mem as H_eval_mem.
rewrite H_eval_mem0.
rewrite H_eval_mem.
exists (concrete_interpreter.ConcreteInterpreter.mload mem' v).
split; reflexivity.
+ destruct (basic_compare_sstack_val smemory_cmp sstorage_cmp sha3_cmp d'0 ctx key key0 maxidx1' sb1' maxidx2' sb2' ops) eqn:E_cmp_key_key0; try discriminate.
simpl in H_follow_valid_sv1.
destruct H_follow_valid_sv1 as [[H_valid_key H_valid_sstrg] [H_valid_sb1' H_maxidx1_gt_maxidx1']].
pose proof (H_maxidx1_gt_maxidx1' (eq_refl true)) as H_maxidx1_gt_maxidx1'.
simpl in H_follow_valid_sv2.
destruct H_follow_valid_sv2 as [[H_valid_key0 H_valid_sstrg0] [H_valid_sb2' H_maxidx2_gt_maxidx2']].
pose proof (H_maxidx2_gt_maxidx2' (eq_refl true)) as H_maxidx2_gt_maxidx2'.
pose proof (H_safe_sstack_value_cmp_cmp_d'0 d'0 (Nat.le_refl d'0) ctx key key0 maxidx1' sb1' maxidx2' sb2' ops H_valid_key H_valid_key0 H_valid_sb1' H_valid_sb2' E_cmp_key_key0 model mem strg exts H_is_model) as H_eval_key_key0.
unfold eval_sstack_val in H_eval_key_key0.
destruct H_eval_key_key0 as [v [H_eval_key H_eval_key0]].
assert (H_Smaxidx1'_le_maxidx1: S maxidx1' <= maxidx1). intuition.
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx1') maxidx1 maxidx1' sb1' key v model mem strg exts ops H_Smaxidx1'_le_maxidx1 H_eval_key) as H_eval_skey_ext.
rewrite H_eval_skey_ext.
assert (H_Smaxidx2'_le_maxidx2: S maxidx2' <= maxidx2). intuition.
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx2') maxidx2 maxidx2' sb2' key0 v model mem strg exts ops H_Smaxidx2'_le_maxidx2 H_eval_key0) as H_eval_skey0_ext.
rewrite H_eval_skey0_ext.
unfold safe_sstorage_cmp_ext_d in H_safe_sstorage_cmp.
unfold safe_sstorage_cmp in H_safe_sstorage_cmp.
assert(H_d'0_le_d': d'0 <= S d'). intuition.
pose proof (H_safe_sstorage_cmp d'0 H_d'0_le_d' ctx sstrg sstrg0 maxidx1' sb1' maxidx2' sb2' ops H_valid_sb1' H_valid_sb2' H_valid_sstrg H_valid_sstrg0 H_cmp_sv1_sv2 model mem strg exts H_is_model) as H_safe_sstorage_cmp_0.
destruct H_safe_sstorage_cmp_0 as [strg' [H_eval_sstrg H_eval_sstrg0]].
unfold eval_sstorage in H_eval_sstrg.
unfold eval_sstorage in H_eval_sstrg0.
destruct (map_option
(instantiate_storage_update
(fun sv : sstack_val =>
eval_sstack_val sv model mem strg exts maxidx1'
sb1' ops)) sstrg) as [updates|] eqn:E_mapo_strg; try discriminate.
unfold eval_sstack_val in E_mapo_strg.
destruct (map_option
(instantiate_storage_update
(fun sv : sstack_val =>
eval_sstack_val sv model mem strg exts maxidx2'
sb2' ops)) sstrg0) as [updates0|] eqn:E_mapo_strg0; try discriminate.
unfold eval_sstack_val in E_mapo_strg0.
pose proof (instantiate_storage_update_mapo_preserved_when_depth_ext_le sstrg (S maxidx1') maxidx1 model mem strg exts maxidx1' sb1' ops updates H_Smaxidx1'_le_maxidx1 E_mapo_strg) as E_mapo_strg_ext.
rewrite E_mapo_strg_ext.
pose proof (instantiate_storage_update_mapo_preserved_when_depth_ext_le sstrg0 (S maxidx2') maxidx2 model mem strg exts maxidx2' sb2' ops updates0 H_Smaxidx2'_le_maxidx2 E_mapo_strg0) as E_mapo_strg0_ext.
rewrite E_mapo_strg0_ext.
injection H_eval_sstrg0 as H_eval_sstrg0.
injection H_eval_sstrg as H_eval_sstrg.
rewrite H_eval_sstrg0.
rewrite H_eval_sstrg.
exists (concrete_interpreter.ConcreteInterpreter.sload strg' v).
split; reflexivity.
+ assert(H_d'0_le_d': d'0 <= S d'). intuition.
assert (H_Smaxidx1'_le_maxidx1: S maxidx1' <= maxidx1). intuition.
assert (H_Smaxidx2'_le_maxidx2: S maxidx2' <= maxidx2). intuition.
simpl in H_follow_valid_sv1.
destruct H_follow_valid_sv1 as [[H_valid_offset [H_valid_size H_valid_smem]] [H_valid_sb1' H_maxidx1_gt_maxidx1']].
pose proof (H_maxidx1_gt_maxidx1' (eq_refl true)) as H_maxidx1_gt_maxidx1'.
simpl in H_follow_valid_sv2.
destruct H_follow_valid_sv2 as [[H_valid_offset0 [H_valid_size0 H_valid_smem0]] [H_valid_sb2' H_maxidx2_gt_maxidx2']].
pose proof (H_maxidx2_gt_maxidx2' (eq_refl true)) as H_maxidx2_gt_maxidx2'.
destruct (if
basic_compare_sstack_val smemory_cmp sstorage_cmp sha3_cmp d'0 ctx offset offset0 maxidx1' sb1'
maxidx2' sb2' ops
then
if
basic_compare_sstack_val smemory_cmp sstorage_cmp sha3_cmp d'0 ctx size size0 maxidx1' sb1'
maxidx2' sb2' ops
then
smemory_cmp (basic_compare_sstack_val smemory_cmp sstorage_cmp sha3_cmp d'0) ctx smem smem0
maxidx1' sb1' maxidx2' sb2' ops
else false
else false) eqn:E_std_sha3.
* destruct (basic_compare_sstack_val smemory_cmp sstorage_cmp sha3_cmp d'0 ctx offset offset0 maxidx1' sb1' maxidx2' sb2' ops) eqn:E_cmp_offset_offset0; try discriminate.
destruct (basic_compare_sstack_val smemory_cmp sstorage_cmp sha3_cmp d'0 ctx size size0 maxidx1' sb1' maxidx2' sb2' ops) eqn:E_cmp_size_size0; try discriminate.
pose proof (H_safe_sstack_value_cmp_cmp_d'0 d'0 (Nat.le_refl d'0) ctx offset offset0 maxidx1' sb1' maxidx2' sb2' ops H_valid_offset H_valid_offset0 H_valid_sb1' H_valid_sb2' E_cmp_offset_offset0 model mem strg exts H_is_model) as H_eval_offset_offset0.
unfold eval_sstack_val in H_eval_offset_offset0.
destruct H_eval_offset_offset0 as [v [H_eval_offset H_eval_offset0]].
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx1') maxidx1 maxidx1' sb1' offset v model mem strg exts ops H_Smaxidx1'_le_maxidx1 H_eval_offset) as H_eval_soffset_ext.
rewrite H_eval_soffset_ext.
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx2') maxidx2 maxidx2' sb2' offset0 v model mem strg exts ops H_Smaxidx2'_le_maxidx2 H_eval_offset0) as H_eval_soffset0_ext.
rewrite H_eval_soffset0_ext.
pose proof (H_safe_sstack_value_cmp_cmp_d'0 d'0 (Nat.le_refl d'0) ctx size size0 maxidx1' sb1' maxidx2' sb2' ops H_valid_size H_valid_size0 H_valid_sb1' H_valid_sb2' E_cmp_size_size0 model mem strg exts H_is_model) as H_eval_size_size0.
unfold eval_sstack_val in H_eval_size_size0.
destruct H_eval_size_size0 as [v' [H_eval_size H_eval_size0]].
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx1') maxidx1 maxidx1' sb1' size v' model mem strg exts ops H_Smaxidx1'_le_maxidx1 H_eval_size) as H_eval_ssize_ext.
rewrite H_eval_ssize_ext.
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx2') maxidx2 maxidx2' sb2' size0 v' model mem strg exts ops H_Smaxidx2'_le_maxidx2 H_eval_size0) as H_eval_ssize0_ext.
rewrite H_eval_ssize0_ext.
unfold safe_smemory_cmp_ext_d in H_safe_smemory_cmp.
unfold safe_smemory_cmp in H_safe_smemory_cmp.
pose proof (H_safe_smemory_cmp d'0 H_d'0_le_d' ctx smem smem0 maxidx1' sb1' maxidx2' sb2' ops H_valid_sb1' H_valid_sb2' H_valid_smem H_valid_smem0 E_std_sha3 model mem strg exts H_is_model) as H_safe_smemory_cmp_0.
destruct H_safe_smemory_cmp_0 as [mem' [H_eval_mem H_eval_mem0]].
unfold eval_smemory in H_eval_mem.
unfold eval_smemory in H_eval_mem0.
destruct (map_option
(instantiate_memory_update
(fun sv : sstack_val =>
eval_sstack_val sv model mem strg exts maxidx1'
sb1' ops)) smem) as [updates|] eqn:E_mapo_mem; try discriminate.
unfold eval_sstack_val in E_mapo_mem.
destruct (map_option
(instantiate_memory_update
(fun sv : sstack_val =>
eval_sstack_val sv model mem strg exts maxidx2'
sb2' ops)) smem0) as [updates0|] eqn:E_mapo_mem0; try discriminate.
unfold eval_sstack_val in E_mapo_mem0.
pose proof (instantiate_memory_update_mapo_preserved_when_depth_ext_le smem (S maxidx1') maxidx1 model mem strg exts maxidx1' sb1' ops updates H_Smaxidx1'_le_maxidx1 E_mapo_mem) as E_mapo_mem_ext.
rewrite E_mapo_mem_ext.
pose proof (instantiate_memory_update_mapo_preserved_when_depth_ext_le smem0 (S maxidx2') maxidx2 model mem strg exts maxidx2' sb2' ops updates0 H_Smaxidx2'_le_maxidx2 E_mapo_mem0) as E_mapo_mem0_ext.
rewrite E_mapo_mem0_ext.
injection H_eval_mem0 as H_eval_mem0.
injection H_eval_mem as H_eval_mem.
rewrite H_eval_mem0.
rewrite H_eval_mem.
exists (get_sha3_info_op (get_keccak256_exts exts) (wordToNat v') (concrete_interpreter.ConcreteInterpreter.mload' mem' v (wordToNat v'))).
split; reflexivity.
* unfold safe_sha3_cmp_ext_d in H_safe_sha3_cmp.
unfold safe_sha3_cmp in H_safe_sha3_cmp.
pose proof (H_safe_sha3_cmp d'0 H_d'0_le_d' ctx offset size smem offset0 size0 smem0 maxidx1' sb1' maxidx2' sb2' ops H_valid_offset H_valid_size H_valid_offset0 H_valid_size0 H_valid_sb1' H_valid_sb2' H_valid_smem H_valid_smem0 H_cmp_sv1_sv2 model mem strg exts H_is_model) as H_safe_sha3_cmp_0.
destruct H_safe_sha3_cmp_0 as [coffset [csize [mem1 [coffset0 [csize0 [mem2 [v [H_eval_smem [H_eval_smem0 [H_eval_offset [H_eval_size [H_eval_offset0 [H_eval_size0 [H_sha3_mem1 H_sha3_mem2]]]]]]]]]]]]]].
unfold eval_smemory in H_eval_smem.
destruct (map_option (EvalCommon.instantiate_memory_update (fun sv : sstack_val => eval_sstack_val sv model mem strg exts maxidx1' sb1' ops)) smem) as [updates|] eqn:H_eval_smem_0; try discriminate.
unfold eval_sstack_val in H_eval_smem_0.
assert (H_maxidx1_ge_S_maxidx1': S maxidx1' <= maxidx1). intuition.
pose proof (instantiate_memory_update_mapo_preserved_when_depth_ext_le smem (S maxidx1') maxidx1 model mem strg exts maxidx1' sb1' ops updates H_maxidx1_ge_S_maxidx1' H_eval_smem_0) as H_eval_smem_0_ext.
rewrite H_eval_smem_0_ext.
unfold eval_smemory in H_eval_smem0.
destruct (map_option (EvalCommon.instantiate_memory_update (fun sv : sstack_val => eval_sstack_val sv model mem strg exts maxidx2' sb2' ops)) smem0) as [updates0|] eqn:H_eval_smem0_0; try discriminate.
unfold eval_sstack_val in H_eval_smem0_0.
assert (H_maxidx2_ge_S_maxidx2': S maxidx2' <= maxidx2). intuition.
pose proof (instantiate_memory_update_mapo_preserved_when_depth_ext_le smem0 (S maxidx2') maxidx2 model mem strg exts maxidx2' sb2' ops updates0 H_maxidx2_ge_S_maxidx2' H_eval_smem0_0) as H_eval_smem0_0_ext.
rewrite H_eval_smem0_0_ext.
unfold eval_sstack_val in H_eval_offset.
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx1') maxidx1 maxidx1' sb1' offset coffset model mem strg exts ops H_maxidx1_ge_S_maxidx1' H_eval_offset) as H_eval_offset_0.
rewrite H_eval_offset_0.
unfold eval_sstack_val in H_eval_size.
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx1') maxidx1 maxidx1' sb1' size csize model mem strg exts ops H_maxidx1_ge_S_maxidx1' H_eval_size) as H_eval_size_0.
rewrite H_eval_size_0.
unfold eval_sstack_val in H_eval_offset0.
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx2') maxidx2 maxidx2' sb2' offset0 coffset0 model mem strg exts ops H_maxidx2_ge_S_maxidx2' H_eval_offset0) as H_eval_offset0_0.
rewrite H_eval_offset0_0.
unfold eval_sstack_val in H_eval_size0.
pose proof (eval_sstack_val'_preserved_when_depth_extended_le (S maxidx2') maxidx2 maxidx2' sb2' size0 csize0 model mem strg exts ops H_maxidx2_ge_S_maxidx2' H_eval_size0) as H_eval_size0_0.
rewrite H_eval_size0_0.
injection H_eval_smem as H_eval_smem.
rewrite H_eval_smem.
injection H_eval_smem0 as H_eval_smem0.
rewrite H_eval_smem0.
rewrite H_sha3_mem1.
rewrite H_sha3_mem2.
exists v.
split; reflexivity.
Qed.
Lemma basic_compare_sstack_val_w_eq_chk_d0_snd:
sstack_val_cmp_fail_for_d_eq_0 basic_compare_sstack_val_w_eq_chk.
Proof.
unfold sstack_val_cmp_fail_for_d_eq_0.
intros.
simpl.
reflexivity.
Qed.
Lemma basic_compare_sstack_val_w_eq_chk_snd:
safe_sstack_value_cmp_wrt_others basic_compare_sstack_val_w_eq_chk.
Proof.
unfold safe_sstack_value_cmp_wrt_others.
induction d as [|d' IHd'].
- intros smemory_cmp sstorage_cmp sha3_cmp H_safe_smemory_cmp H_safe_sstorage_cmp H_safe_sha3_cmp.
unfold safe_sstack_val_cmp_ext_2_d.
unfold safe_sstack_val_cmp_ext_1_d.
intros d' H_d'_le_1.
unfold safe_sstack_val_cmp.
intros ctx sv1 sv2 maxidx1 sb1 maxidx2 sb2 ops H_valid_sv1 H_valid_sv2 H_valid_sb1 H_valid_sb2 H_cmp_sv1_sv2.
apply Nat.leb_le in H_d'_le_1 as H_d'_le_1_leb.
destruct d' as [|d''] eqn:E_d'; try discriminate; destruct d'' as [|d'''] eqn:E_d''; try discriminate.
unfold basic_compare_sstack_val_w_eq_chk in H_cmp_sv1_sv2.
intros model mem strg exts H_is_model.
unfold eval_sstack_val.
destruct(trivial_compare_sstack_val smemory_cmp sstorage_cmp sha3_cmp 1 ctx sv1 sv2 maxidx1 sb1 maxidx2 sb2 ops) eqn:E_trivial_chk.
(* when calling trivial check *)
unfold trivial_compare_sstack_val in E_trivial_chk.
destruct sv1 as [w1 | n1 | n1]; destruct sv2 as [w2 | n2 | n2] eqn:E_sv2; try discriminate.
(* Val *)
apply weqb_sound in E_trivial_chk.
pose proof (eval_sstack_val'_Val w1 model mem strg exts maxidx1 sb1 ops) as H_eval_w1.
pose proof (eval_sstack_val'_Val w2 model mem strg exts maxidx2 sb2 ops) as H_eval_w2.
rewrite H_eval_w1.
rewrite H_eval_w2.
rewrite E_trivial_chk.
exists w2.
split; reflexivity.
(* InVar *)
pose proof (eval_sstack_val'_InVar n1 model mem strg exts maxidx1 sb1 ops) as H_eval_instkv_n1.
pose proof (eval_sstack_val'_InVar n2 model mem strg exts maxidx2 sb2 ops) as H_eval_instkv_n2.
rewrite H_eval_instkv_n1.
rewrite H_eval_instkv_n2.
apply Nat.eqb_eq in E_trivial_chk.
rewrite E_trivial_chk.
exists (model n2).
split; try reflexivity.
(* FreshVar *)
destruct (n1 =? n2) eqn:E_n1_eq_n2; try discriminate.
destruct (maxidx1 =? maxidx2) eqn:E_maxidx1_eq_maxidx2; try discriminate.
destruct (sbindings_eq_dec sb1 sb2) eqn:E_bs1_eq_bs2; try discriminate.
apply Nat.eqb_eq in E_n1_eq_n2.
apply Nat.eqb_eq in E_maxidx1_eq_maxidx2.
rewrite E_n1_eq_n2.
rewrite E_maxidx1_eq_maxidx2.
rewrite e.
pose proof (eval_sstack_val'_succ (S maxidx2) (FreshVar n2) model mem strg exts maxidx2 sb2 ops H_valid_sv2 H_valid_sb2 (gt_Sn_n maxidx2)) as H_eval_sv2.
destruct H_eval_sv2 as [v2 H_eval_sv2].
rewrite H_eval_sv2.
exists v2.
split; try reflexivity.
(* end of call to trivial chk *)
unfold eval_sstack_val'. fold eval_sstack_val'.
unfold basic_compare_sstack_val_w_eq_chk in H_cmp_sv1_sv2.
pose proof (follow_in_smap_suc sb1 sv1 maxidx1 ops H_valid_sv1 H_valid_sb1) as H_follow_suc_sv1.
destruct H_follow_suc_sv1 as [smv1 [maxidx1' [sb1' [H_follow_suc_sv1 _]]]].
pose proof (follow_in_smap_suc sb2 sv2 maxidx2 ops H_valid_sv2 H_valid_sb2) as H_follow_suc_sv2.
destruct H_follow_suc_sv2 as [smv2 [maxidx2' [sb2' [H_follow_suc_sv2 _]]]].
rewrite H_follow_suc_sv1.
rewrite H_follow_suc_sv2.
rewrite H_follow_suc_sv1 in H_cmp_sv1_sv2.
rewrite H_follow_suc_sv2 in H_cmp_sv1_sv2.
pose proof (valid_follow_in_smap sb1 sv1 maxidx1 ops smv1 maxidx1' sb1' H_valid_sv1 H_valid_sb1 H_follow_suc_sv1) as H_follow_valid_sv1.
pose proof (valid_follow_in_smap sb2 sv2 maxidx2 ops smv2 maxidx2' sb2' H_valid_sv2 H_valid_sb2 H_follow_suc_sv2) as H_follow_valid_sv2.
destruct smv1 eqn:E_smv1; destruct smv2 eqn:E_smv2; try discriminate.
+ destruct val; destruct val0; try discriminate.
* apply weqb_sound in H_cmp_sv1_sv2.
rewrite H_cmp_sv1_sv2.
exists val0.
split; reflexivity.
* pose proof (chk_eq_wrt_ctx_snd ctx (Val val) (InVar var) H_cmp_sv1_sv2 model mem strg exts maxidx1 sb1 ops H_is_model) as H_ctx_sv1'_eq_sv2'_0.
pose proof (chk_eq_wrt_ctx_snd ctx (Val val) (InVar var) H_cmp_sv1_sv2 model mem strg exts maxidx2 sb2 ops H_is_model) as H_ctx_sv1'_eq_sv2'_1.
destruct H_ctx_sv1'_eq_sv2'_0 as [v1 [H_ctx_sv1'_eq_sv2'_0_0 H_ctx_sv1'_eq_sv2'_0_1]].
destruct H_ctx_sv1'_eq_sv2'_1 as [v2 [H_ctx_sv1'_eq_sv2'_1_0 H_ctx_sv1'_eq_sv2'_1_1]].
pose proof (eval_sstack_val_Val val model mem strg exts maxidx1 sb1 ops) as H_eval_val_0.
pose proof (eval_sstack_val_Val val model mem strg exts maxidx2 sb2 ops) as H_eval_val_1.
pose proof (eval_sstack_val_InVar var model mem strg exts maxidx1 sb1 ops) as H_eval_var_0.
pose proof (eval_sstack_val_InVar var model mem strg exts maxidx2 sb2 ops) as H_eval_var_1.
rewrite H_eval_val_0 in H_ctx_sv1'_eq_sv2'_0_0.
rewrite H_eval_var_0 in H_ctx_sv1'_eq_sv2'_0_1.
injection H_ctx_sv1'_eq_sv2'_0_0 as H_ctx_sv1'_eq_sv2'_0_0.
injection H_ctx_sv1'_eq_sv2'_0_1 as H_ctx_sv1'_eq_sv2'_0_1.
exists v1.
rewrite H_ctx_sv1'_eq_sv2'_0_1.
rewrite H_ctx_sv1'_eq_sv2'_0_0.
split; reflexivity.
* pose proof (chk_eq_wrt_ctx_snd ctx (InVar var) (Val val) H_cmp_sv1_sv2 model mem strg exts maxidx1 sb1 ops H_is_model) as H_ctx_sv1'_eq_sv2'_0.
pose proof (chk_eq_wrt_ctx_snd ctx (InVar var) (Val val) H_cmp_sv1_sv2 model mem strg exts maxidx2 sb2 ops H_is_model) as H_ctx_sv1'_eq_sv2'_1.
destruct H_ctx_sv1'_eq_sv2'_0 as [v1 [H_ctx_sv1'_eq_sv2'_0_0 H_ctx_sv1'_eq_sv2'_0_1]].
destruct H_ctx_sv1'_eq_sv2'_1 as [v2 [H_ctx_sv1'_eq_sv2'_1_0 H_ctx_sv1'_eq_sv2'_1_1]].
pose proof (eval_sstack_val_Val val model mem strg exts maxidx1 sb1 ops) as H_eval_val_0.
pose proof (eval_sstack_val_Val val model mem strg exts maxidx2 sb2 ops) as H_eval_val_1.
pose proof (eval_sstack_val_InVar var model mem strg exts maxidx1 sb1 ops) as H_eval_var_0.
pose proof (eval_sstack_val_InVar var model mem strg exts maxidx2 sb2 ops) as H_eval_var_1.
rewrite H_eval_var_0 in H_ctx_sv1'_eq_sv2'_0_0.
rewrite H_eval_val_0 in H_ctx_sv1'_eq_sv2'_0_1.
injection H_ctx_sv1'_eq_sv2'_0_0 as H_ctx_sv1'_eq_sv2'_0_0.
injection H_ctx_sv1'_eq_sv2'_0_1 as H_ctx_sv1'_eq_sv2'_0_1.
exists v1.
rewrite H_ctx_sv1'_eq_sv2'_0_1.
rewrite H_ctx_sv1'_eq_sv2'_0_0.
split; reflexivity.
* apply orb_prop in H_cmp_sv1_sv2.
destruct H_cmp_sv1_sv2 as [H_cmp_sv1_sv2 | H_cmp_sv1_sv2].
** apply Nat.eqb_eq in H_cmp_sv1_sv2.
rewrite H_cmp_sv1_sv2.
exists (model var0).
split; reflexivity.
** pose proof (chk_eq_wrt_ctx_snd ctx (InVar var) (InVar var0) H_cmp_sv1_sv2 model mem strg exts maxidx1 sb1 ops H_is_model) as H_ctx_sv1'_eq_sv2'_0.
pose proof (chk_eq_wrt_ctx_snd ctx (InVar var) (InVar var0) H_cmp_sv1_sv2 model mem strg exts maxidx2 sb2 ops H_is_model) as H_ctx_sv1'_eq_sv2'_1.
destruct H_ctx_sv1'_eq_sv2'_0 as [v1 [H_ctx_sv1'_eq_sv2'_0_0 H_ctx_sv1'_eq_sv2'_0_1]].
destruct H_ctx_sv1'_eq_sv2'_1 as [v2 [H_ctx_sv1'_eq_sv2'_1_0 H_ctx_sv1'_eq_sv2'_1_1]].
pose proof (eval_sstack_val_InVar var model mem strg exts maxidx1 sb1 ops) as H_eval_var_0.
pose proof (eval_sstack_val_InVar var0 model mem strg exts maxidx1 sb1 ops) as H_eval_var_1.
pose proof (eval_sstack_val_InVar var model mem strg exts maxidx2 sb2 ops) as H_eval_var_0_1.
pose proof (eval_sstack_val_InVar var0 model mem strg exts maxidx2 sb2 ops) as H_eval_var_1_1.
rewrite H_eval_var_1 in H_ctx_sv1'_eq_sv2'_0_1.
rewrite H_eval_var_0 in H_ctx_sv1'_eq_sv2'_0_0.
injection H_ctx_sv1'_eq_sv2'_0_0 as H_ctx_sv1'_eq_sv2'_0_0.
injection H_ctx_sv1'_eq_sv2'_0_1 as H_ctx_sv1'_eq_sv2'_0_1.
exists v1.