WBLogic.heap_lang.proofmode
From iris.proofmode Require Import coq_tactics reduction spec_patterns.
From iris.proofmode Require Export tactics.
From iris.program_logic Require Import atomic.
From iris.heap_lang Require Export tactics proofmode.
From WBLogic.heap_lang Require Export derived_laws.
From iris.heap_lang Require Import notation.
From iris.prelude Require Import options.
Import uPred.
Lemma tac_wbwp_expr_eval `{!wbheapGS Σ} Δ E out Φ e e' :
(∀ (e'':=e'), e = e'') →
envs_entails Δ (WBWP e' @ out; E {{ Φ }}) → envs_entails Δ (WBWP e @ out; E {{ Φ }}).
Proof. by intros ->. Qed.
Tactic Notation "wbwp_expr_eval" tactic3(t) :=
iStartProof;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
notypeclasses refine (tac_wbwp_expr_eval _ _ _ _ e _ _ _);
[let x := fresh in intros x; t; unfold x; notypeclasses refine eq_refl|]
| _ => fail "wbwp_expr_eval: not a 'wbwp'"
end.
Ltac wbwp_expr_simpl := wbwp_expr_eval simpl.
Lemma tac_wbwp_pure `{!wbheapGS Σ} Δ Δ' E out K e1 e2 φ n Φ :
PureExec φ n e1 e2 →
φ →
MaybeIntoLaterNEnvs n Δ Δ' →
envs_entails Δ' (WBWP (fill K e2) @ out; E {{ Φ }}) →
envs_entails Δ (WBWP (fill K e1) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ??? HΔ'. rewrite into_laterN_env_sound /=.
(* We want pure_exec_fill to be available to TC search locally. *)
pose proof @pure_exec_fill.
rewrite HΔ' -lifting.wbwp_pure_step_later //.
iIntros "Hwp !> _" => //.
Qed.
Lemma tac_wbwp_pure_credit `{!wbheapGS Σ} Δ Δ' E out j K e1 e2 ϕ Φ :
PureExec ϕ 1 e1 e2 →
ϕ →
MaybeIntoLaterNEnvs 1 Δ Δ' →
match envs_app false (Esnoc Enil j (£ 1)) Δ' with
| Some Δ'' =>
envs_entails Δ'' (WBWP fill K e2 @ out; E {{ Φ }})
| None => False
end →
envs_entails Δ (WBWP (fill K e1) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ??? HΔ.
pose proof @pure_exec_fill.
rewrite -lifting.wbwp_pure_step_later; last done.
rewrite into_laterN_env_sound /=. apply later_mono.
destruct (envs_app _ _ _) as [Δ''|] eqn:HΔ'; [ | contradiction ].
rewrite envs_app_sound //; simpl.
rewrite right_id. apply wand_intro_r. by rewrite wand_elim_l.
Qed.
Lemma tac_wbwp_value_nofupd `{!wbheapGS Σ} Δ E out Φ v :
envs_entails Δ (Φ v) → envs_entails Δ (WBWP (Val v) @ out; E {{ Φ }}).
Proof. rewrite envs_entails_unseal=> ->. by apply wbwp_value. Qed.
Lemma tac_wbwp_value `{!wbheapGS Σ} Δ E out (Φ : val → iPropI Σ) v :
envs_entails Δ (|={E}=> Φ v) → envs_entails Δ (WBWP (Val v) @ out; E {{ Φ }}).
Proof. rewrite envs_entails_unseal=> ->. iApply wbwp_value_fupd. Qed.
From iris.proofmode Require Export tactics.
From iris.program_logic Require Import atomic.
From iris.heap_lang Require Export tactics proofmode.
From WBLogic.heap_lang Require Export derived_laws.
From iris.heap_lang Require Import notation.
From iris.prelude Require Import options.
Import uPred.
Lemma tac_wbwp_expr_eval `{!wbheapGS Σ} Δ E out Φ e e' :
(∀ (e'':=e'), e = e'') →
envs_entails Δ (WBWP e' @ out; E {{ Φ }}) → envs_entails Δ (WBWP e @ out; E {{ Φ }}).
Proof. by intros ->. Qed.
Tactic Notation "wbwp_expr_eval" tactic3(t) :=
iStartProof;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
notypeclasses refine (tac_wbwp_expr_eval _ _ _ _ e _ _ _);
[let x := fresh in intros x; t; unfold x; notypeclasses refine eq_refl|]
| _ => fail "wbwp_expr_eval: not a 'wbwp'"
end.
Ltac wbwp_expr_simpl := wbwp_expr_eval simpl.
Lemma tac_wbwp_pure `{!wbheapGS Σ} Δ Δ' E out K e1 e2 φ n Φ :
PureExec φ n e1 e2 →
φ →
MaybeIntoLaterNEnvs n Δ Δ' →
envs_entails Δ' (WBWP (fill K e2) @ out; E {{ Φ }}) →
envs_entails Δ (WBWP (fill K e1) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ??? HΔ'. rewrite into_laterN_env_sound /=.
(* We want pure_exec_fill to be available to TC search locally. *)
pose proof @pure_exec_fill.
rewrite HΔ' -lifting.wbwp_pure_step_later //.
iIntros "Hwp !> _" => //.
Qed.
Lemma tac_wbwp_pure_credit `{!wbheapGS Σ} Δ Δ' E out j K e1 e2 ϕ Φ :
PureExec ϕ 1 e1 e2 →
ϕ →
MaybeIntoLaterNEnvs 1 Δ Δ' →
match envs_app false (Esnoc Enil j (£ 1)) Δ' with
| Some Δ'' =>
envs_entails Δ'' (WBWP fill K e2 @ out; E {{ Φ }})
| None => False
end →
envs_entails Δ (WBWP (fill K e1) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ??? HΔ.
pose proof @pure_exec_fill.
rewrite -lifting.wbwp_pure_step_later; last done.
rewrite into_laterN_env_sound /=. apply later_mono.
destruct (envs_app _ _ _) as [Δ''|] eqn:HΔ'; [ | contradiction ].
rewrite envs_app_sound //; simpl.
rewrite right_id. apply wand_intro_r. by rewrite wand_elim_l.
Qed.
Lemma tac_wbwp_value_nofupd `{!wbheapGS Σ} Δ E out Φ v :
envs_entails Δ (Φ v) → envs_entails Δ (WBWP (Val v) @ out; E {{ Φ }}).
Proof. rewrite envs_entails_unseal=> ->. by apply wbwp_value. Qed.
Lemma tac_wbwp_value `{!wbheapGS Σ} Δ E out (Φ : val → iPropI Σ) v :
envs_entails Δ (|={E}=> Φ v) → envs_entails Δ (WBWP (Val v) @ out; E {{ Φ }}).
Proof. rewrite envs_entails_unseal=> ->. iApply wbwp_value_fupd. Qed.
Simplify the goal if it is WP of a value.
If the postcondition already allows a fupd, do not add a second one.
But otherwise, *do* add a fupd. This ensures that all the lemmas applied
here are bidirectional, so we never will make a goal unprovable.
Ltac wbwp_value_head :=
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out (Val _) (λ _, fupd ?E _ _)) =>
eapply tac_wp_value_nofupd
| |- envs_entails _ (wbwp ?E ?out (Val _) (λ _, wp _ ?E _ _)) =>
eapply tac_wbwp_value_nofupd
| |- envs_entails _ (wbwp ?E ?out (Val _) _) =>
eapply tac_wp_value
end.
Ltac wbwp_finish :=
wbwp_expr_simpl; (* simplify occurences of subst/fill *)
try wbwp_value_head; (* in case we have reached a value, get rid of the WP *)
pm_prettify. (* prettify ▷s caused by MaybeIntoLaterNEnvs and
λs caused by wp_value *)
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out (Val _) (λ _, fupd ?E _ _)) =>
eapply tac_wp_value_nofupd
| |- envs_entails _ (wbwp ?E ?out (Val _) (λ _, wp _ ?E _ _)) =>
eapply tac_wbwp_value_nofupd
| |- envs_entails _ (wbwp ?E ?out (Val _) _) =>
eapply tac_wp_value
end.
Ltac wbwp_finish :=
wbwp_expr_simpl; (* simplify occurences of subst/fill *)
try wbwp_value_head; (* in case we have reached a value, get rid of the WP *)
pm_prettify. (* prettify ▷s caused by MaybeIntoLaterNEnvs and
λs caused by wp_value *)
The argument efoc can be used to specify the construct that should be
reduced. For example, you can write wp_pure (EIf _ _ _), which will search
for an EIf _ _ _ in the expression, and reduce it.
The use of open_constr in this tactic is essential. It will convert all holes
(i.e. _s) into evars, that later get unified when an occurences is found
(see unify e' efoc in the code below).
Tactic Notation "wbwp_pure" open_constr(efoc) :=
iStartProof;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
let e := eval simpl in e in
reshape_expr e ltac:(fun K e' =>
unify e' efoc;
eapply (tac_wbwp_pure _ _ _ _ K e');
[tc_solve (* PureExec *)
|try solve_vals_compare_safe (* The pure condition for PureExec --
handles trivial goals, including vals_compare_safe *)
|tc_solve (* IntoLaters *)
|wbwp_finish (* new goal *)
])
|| fail "wbwp_pure: cannot find" efoc "in" e "or" efoc "is not a redex"
| _ => fail "wbwp_pure: not a 'wbwp'"
end.
Tactic Notation "wbwp_pure" :=
wbwp_pure _.
Tactic Notation "wbwp_pure" open_constr(efoc) "credit:" constr(H) :=
iStartProof;
let Htmp := iFresh in
let finish _ :=
pm_reduce;
(iDestructHyp Htmp as H || fail 2 "wbwp_pure:" H "is not fresh");
wp_finish
in
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
let e := eval simpl in e in
reshape_expr e ltac:(fun K e' =>
unify e' efoc;
eapply (tac_wbwp_pure_credit _ _ _ _ Htmp K e');
[tc_solve (* PureExec *)
|try solve_vals_compare_safe (* The pure condition for PureExec --
handles trivial goals, including vals_compare_safe *)
|tc_solve (* IntoLaters *)
|finish () (* new goal *)
])
|| fail "wbwp_pure: cannot find" efoc "in" e "or" efoc "is not a redex"
| _ => fail "wbwp_pure: not a 'wp'"
end.
Tactic Notation "wbwp_pure" "credit:" constr(H) :=
wbwp_pure _ credit: H.
(* TODO: do this in one go, without repeat. *)
Ltac wbwp_pures :=
iStartProof;
first [ (* The `;` makes sure that no side-condition magically spawns. *)
progress repeat (wbwp_pure _; [])
| wbwp_finish (* In case wp_pure never ran, make sure we do the usual cleanup. *)
].
iStartProof;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
let e := eval simpl in e in
reshape_expr e ltac:(fun K e' =>
unify e' efoc;
eapply (tac_wbwp_pure _ _ _ _ K e');
[tc_solve (* PureExec *)
|try solve_vals_compare_safe (* The pure condition for PureExec --
handles trivial goals, including vals_compare_safe *)
|tc_solve (* IntoLaters *)
|wbwp_finish (* new goal *)
])
|| fail "wbwp_pure: cannot find" efoc "in" e "or" efoc "is not a redex"
| _ => fail "wbwp_pure: not a 'wbwp'"
end.
Tactic Notation "wbwp_pure" :=
wbwp_pure _.
Tactic Notation "wbwp_pure" open_constr(efoc) "credit:" constr(H) :=
iStartProof;
let Htmp := iFresh in
let finish _ :=
pm_reduce;
(iDestructHyp Htmp as H || fail 2 "wbwp_pure:" H "is not fresh");
wp_finish
in
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
let e := eval simpl in e in
reshape_expr e ltac:(fun K e' =>
unify e' efoc;
eapply (tac_wbwp_pure_credit _ _ _ _ Htmp K e');
[tc_solve (* PureExec *)
|try solve_vals_compare_safe (* The pure condition for PureExec --
handles trivial goals, including vals_compare_safe *)
|tc_solve (* IntoLaters *)
|finish () (* new goal *)
])
|| fail "wbwp_pure: cannot find" efoc "in" e "or" efoc "is not a redex"
| _ => fail "wbwp_pure: not a 'wp'"
end.
Tactic Notation "wbwp_pure" "credit:" constr(H) :=
wbwp_pure _ credit: H.
(* TODO: do this in one go, without repeat. *)
Ltac wbwp_pures :=
iStartProof;
first [ (* The `;` makes sure that no side-condition magically spawns. *)
progress repeat (wbwp_pure _; [])
| wbwp_finish (* In case wp_pure never ran, make sure we do the usual cleanup. *)
].
Unlike wp_pures, the tactics wp_rec and wp_lam should also reduce
lambdas/recs that are hidden behind a definition, i.e. they should use
AsRecV_recv as a proper instance instead of a Hint Extern.
We achieve this by putting AsRecV_recv in the current environment so that it
can be used as an instance by the typeclass resolution system. We then perform
the reduction, and finally we clear this new hypothesis.
Tactic Notation "wbwp_rec" :=
let H := fresh in
assert (H := AsRecV_recv);
wbwp_pure (App _ _);
clear H.
Tactic Notation "wbwp_if" := wbwp_pure (If _ _ _).
Tactic Notation "wbwp_if_true" := wbwp_pure (If (LitV (LitBool true)) _ _).
Tactic Notation "wbwp_if_false" := wbwp_pure (If (LitV (LitBool false)) _ _).
Tactic Notation "wbwp_unop" := wbwp_pure (UnOp _ _).
Tactic Notation "wbwp_binop" := wbwp_pure (BinOp _ _ _).
Tactic Notation "wbwp_op" := wbwp_unop || wbwp_binop.
Tactic Notation "wbwp_lam" := wbwp_rec.
Tactic Notation "wbwp_let" := wbwp_pure (Rec BAnon (BNamed _) _); wbwp_lam.
Tactic Notation "wbwp_seq" := wbwp_pure (Rec BAnon BAnon _); wbwp_lam.
Tactic Notation "wbwp_proj" := wbwp_pure (Fst _) || wbwp_pure (Snd _).
Tactic Notation "wbwp_case" := wbwp_pure (Case _ _ _).
Tactic Notation "wbwp_match" := wbwp_case; wp_pure (Rec _ _ _); wbwp_lam.
Tactic Notation "wbwp_inj" := wbwp_pure (InjL _) || wbwp_pure (InjR _).
Tactic Notation "wbwp_pair" := wbwp_pure (Pair _ _).
Tactic Notation "wbwp_closure" := wbwp_pure (Rec _ _ _).
Lemma tac_wbwp_bind `{!wbheapGS Σ} K Δ E out Φ e f :
f = (λ e, fill K e) → (* as an eta expanded hypothesis so that we can `simpl` it *)
envs_entails Δ (WBWP e @ out; E {{ v, WBWP f (Val v) @ out; E {{ Φ }} }})%I →
envs_entails Δ (WBWP fill K e @ out; E {{ Φ }}).
Proof. rewrite envs_entails_unseal=> -> ->. by apply: wbwp_bind. Qed.
Ltac wbwp_bind_core K :=
lazymatch eval hnf in K with
| [] => idtac
| _ => eapply (tac_wbwp_bind K); [simpl; reflexivity|reduction.pm_prettify]
end.
Tactic Notation "wbwp_bind" open_constr(efoc) :=
iStartProof;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
first [ reshape_expr e ltac:(fun K e' => unify e' efoc; wbwp_bind_core K)
| fail 1 "wp_bind: cannot find" efoc "in" e ]
| _ => fail "wp_bind: not a 'wp'"
end.
let H := fresh in
assert (H := AsRecV_recv);
wbwp_pure (App _ _);
clear H.
Tactic Notation "wbwp_if" := wbwp_pure (If _ _ _).
Tactic Notation "wbwp_if_true" := wbwp_pure (If (LitV (LitBool true)) _ _).
Tactic Notation "wbwp_if_false" := wbwp_pure (If (LitV (LitBool false)) _ _).
Tactic Notation "wbwp_unop" := wbwp_pure (UnOp _ _).
Tactic Notation "wbwp_binop" := wbwp_pure (BinOp _ _ _).
Tactic Notation "wbwp_op" := wbwp_unop || wbwp_binop.
Tactic Notation "wbwp_lam" := wbwp_rec.
Tactic Notation "wbwp_let" := wbwp_pure (Rec BAnon (BNamed _) _); wbwp_lam.
Tactic Notation "wbwp_seq" := wbwp_pure (Rec BAnon BAnon _); wbwp_lam.
Tactic Notation "wbwp_proj" := wbwp_pure (Fst _) || wbwp_pure (Snd _).
Tactic Notation "wbwp_case" := wbwp_pure (Case _ _ _).
Tactic Notation "wbwp_match" := wbwp_case; wp_pure (Rec _ _ _); wbwp_lam.
Tactic Notation "wbwp_inj" := wbwp_pure (InjL _) || wbwp_pure (InjR _).
Tactic Notation "wbwp_pair" := wbwp_pure (Pair _ _).
Tactic Notation "wbwp_closure" := wbwp_pure (Rec _ _ _).
Lemma tac_wbwp_bind `{!wbheapGS Σ} K Δ E out Φ e f :
f = (λ e, fill K e) → (* as an eta expanded hypothesis so that we can `simpl` it *)
envs_entails Δ (WBWP e @ out; E {{ v, WBWP f (Val v) @ out; E {{ Φ }} }})%I →
envs_entails Δ (WBWP fill K e @ out; E {{ Φ }}).
Proof. rewrite envs_entails_unseal=> -> ->. by apply: wbwp_bind. Qed.
Ltac wbwp_bind_core K :=
lazymatch eval hnf in K with
| [] => idtac
| _ => eapply (tac_wbwp_bind K); [simpl; reflexivity|reduction.pm_prettify]
end.
Tactic Notation "wbwp_bind" open_constr(efoc) :=
iStartProof;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
first [ reshape_expr e ltac:(fun K e' => unify e' efoc; wbwp_bind_core K)
| fail 1 "wp_bind: cannot find" efoc "in" e ]
| _ => fail "wp_bind: not a 'wp'"
end.
Heap tactics
Section heap.
Context `{!wbheapGS Σ}.
Implicit Types P Q : iProp Σ.
Implicit Types Φ : val → iProp Σ.
Implicit Types Δ : envs (uPredI (iResUR Σ)).
Implicit Types v : val.
Implicit Types z : Z.
Lemma tac_wbwp_allocN Δ Δ' E out j K v n Φ :
(0 < n)%Z →
MaybeIntoLaterNEnvs 1 Δ Δ' →
(∀ l,
match envs_app false (Esnoc Enil j (array l (DfracOwn 1) (replicate (Z.to_nat n) v))) Δ' with
| Some Δ'' =>
envs_entails Δ'' (WBWP fill K (Val $ LitV $ LitLoc l) @ out; E {{ Φ }})
| None => False
end) →
envs_entails Δ (WBWP fill K (AllocN (Val $ LitV $ LitInt n) (Val v)) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ? ? HΔ.
rewrite -wbwp_bind. eapply wand_apply; first by iApply wbwp_allocN.
rewrite left_id into_laterN_env_sound; apply later_mono, forall_intro=> l.
specialize (HΔ l).
destruct (envs_app _ _ _) as [Δ''|] eqn:HΔ'; [ | contradiction ].
rewrite envs_app_sound //; simpl.
apply wand_intro_l. by rewrite (sep_elim_l (l ↦∗ _)%I) right_id wand_elim_r.
Qed.
Lemma tac_wbwp_alloc Δ Δ' E out j K v Φ :
MaybeIntoLaterNEnvs 1 Δ Δ' →
(∀ l,
match envs_app false (Esnoc Enil j (l ↦ v)) Δ' with
| Some Δ'' =>
envs_entails Δ'' (WBWP fill K (Val $ LitV l) @ out; E {{ Φ }})
| None => False
end) →
envs_entails Δ (WBWP fill K (Alloc (Val v)) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ? HΔ.
rewrite -wbwp_bind. eapply wand_apply; first by iApply wbwp_alloc.
rewrite left_id into_laterN_env_sound; apply later_mono, forall_intro=> l.
specialize (HΔ l).
destruct (envs_app _ _ _) as [Δ''|] eqn:HΔ'; [ | contradiction ].
rewrite envs_app_sound //; simpl.
apply wand_intro_l. by rewrite (sep_elim_l (l ↦ v)%I) right_id wand_elim_r.
Qed.
Lemma tac_wbwp_free Δ Δ' E out i K l v Φ :
MaybeIntoLaterNEnvs 1 Δ Δ' →
envs_lookup i Δ' = Some (false, l ↦ v)%I →
(let Δ'' := envs_delete false i false Δ' in
envs_entails Δ'' (WBWP fill K (Val $ LitV LitUnit) @ out; E {{ Φ }})) →
envs_entails Δ (WBWP fill K (Free (LitV l)) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ? Hlk Hfin.
rewrite -wbwp_bind. eapply wand_apply; first by iApply wbwp_free.
rewrite into_laterN_env_sound -later_sep envs_lookup_split //; simpl.
rewrite -Hfin wand_elim_r (envs_lookup_sound' _ _ _ _ _ Hlk).
apply later_mono, sep_mono_r, wand_intro_r. rewrite right_id //.
Qed.
Lemma tac_wbwp_load Δ Δ' E out i K b l q v Φ :
MaybeIntoLaterNEnvs 1 Δ Δ' →
envs_lookup i Δ' = Some (b, l ↦{q} v)%I →
envs_entails Δ' (WBWP fill K (Val v) @ out; E {{ Φ }}) →
envs_entails Δ (WBWP fill K (Load (LitV l)) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ?? Hi.
rewrite -wbwp_bind. eapply wand_apply; first by iApply wbwp_load.
rewrite into_laterN_env_sound -later_sep envs_lookup_split //; simpl.
apply later_mono.
destruct b; simpl.
* iIntros "[#$ He]". iIntros "_". iApply Hi. iApply "He". iFrame "#".
* by apply sep_mono_r, wand_mono.
Qed.
Lemma tac_wbwp_store Δ Δ' E out i K l v v' Φ :
MaybeIntoLaterNEnvs 1 Δ Δ' →
envs_lookup i Δ' = Some (false, l ↦ v)%I →
match envs_simple_replace i false (Esnoc Enil i (l ↦ v')) Δ' with
| Some Δ'' => envs_entails Δ'' (WBWP fill K (Val $ LitV LitUnit) @ out; E {{ Φ }})
| None => False
end →
envs_entails Δ (WBWP fill K (Store (LitV l) (Val v')) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ???.
destruct (envs_simple_replace _ _ _) as [Δ''|] eqn:HΔ''; [ | contradiction ].
rewrite -wbwp_bind. eapply wand_apply; first by iApply wbwp_store.
rewrite into_laterN_env_sound -later_sep envs_simple_replace_sound //; simpl.
rewrite right_id. by apply later_mono, sep_mono_r, wand_mono.
Qed.
Lemma tac_wbwp_xchg Δ Δ' E out i K l v v' Φ :
MaybeIntoLaterNEnvs 1 Δ Δ' →
envs_lookup i Δ' = Some (false, l ↦ v)%I →
match envs_simple_replace i false (Esnoc Enil i (l ↦ v')) Δ' with
| Some Δ'' => envs_entails Δ'' (WBWP fill K (Val $ v) @ out; E {{ Φ }})
| None => False
end →
envs_entails Δ (WBWP fill K (Xchg (LitV l) (Val v')) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ???.
destruct (envs_simple_replace _ _ _) as [Δ''|] eqn:HΔ''; [ | contradiction ].
rewrite -wbwp_bind. eapply wand_apply; first by iApply wbwp_xchg.
rewrite into_laterN_env_sound -later_sep envs_simple_replace_sound //; simpl.
rewrite right_id.
by apply later_mono, sep_mono_r, wand_mono.
Qed.
Lemma tac_wbwp_cmpxchg Δ Δ' E out i K l v v1 v2 Φ :
MaybeIntoLaterNEnvs 1 Δ Δ' →
envs_lookup i Δ' = Some (false, l ↦ v)%I →
vals_compare_safe v v1 →
match envs_simple_replace i false (Esnoc Enil i (l ↦ v2)) Δ' with
| Some Δ'' =>
v = v1 →
envs_entails Δ'' (WBWP fill K (Val $ PairV v (LitV $ LitBool true)) @ out; E {{ Φ }})
| None => False
end →
(v ≠ v1 →
envs_entails Δ' (WBWP fill K (Val $ PairV v (LitV $ LitBool false)) @ out; E {{ Φ }})) →
envs_entails Δ (WBWP fill K (CmpXchg (LitV l) (Val v1) (Val v2)) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ??? Hsuc Hfail.
destruct (envs_simple_replace _ _ _ _) as [Δ''|] eqn:HΔ''; [ | contradiction ].
destruct (decide (v = v1)) as [Heq|Hne].
- rewrite -wbwp_bind. eapply wand_apply.
{ iApply wbwp_cmpxchg_suc; eauto. }
rewrite into_laterN_env_sound -later_sep /= {1}envs_simple_replace_sound //; simpl.
apply later_mono, sep_mono_r. rewrite right_id. apply wand_mono; auto.
- rewrite -wbwp_bind. eapply wand_apply.
{ iApply wbwp_cmpxchg_fail; eauto. }
rewrite into_laterN_env_sound -later_sep /= {1}envs_lookup_split //; simpl.
apply later_mono, sep_mono_r. apply wand_mono; auto.
Qed.
Lemma tac_wbwp_cmpxchg_fail Δ Δ' E out i K l q v v1 v2 Φ :
MaybeIntoLaterNEnvs 1 Δ Δ' →
envs_lookup i Δ' = Some (false, l ↦{q} v)%I →
v ≠ v1 → vals_compare_safe v v1 →
envs_entails Δ' (WBWP fill K (Val $ PairV v (LitV $ LitBool false)) @ out; E {{ Φ }}) →
envs_entails Δ (WBWP fill K (CmpXchg (LitV l) v1 v2) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ?????.
rewrite -wbwp_bind. eapply wand_apply; first by iApply wbwp_cmpxchg_fail.
rewrite into_laterN_env_sound -later_sep envs_lookup_split //; simpl.
by apply later_mono, sep_mono_r, wand_mono.
Qed.
Lemma tac_wbwp_cmpxchg_suc Δ Δ' E out i K l v v1 v2 Φ :
MaybeIntoLaterNEnvs 1 Δ Δ' →
envs_lookup i Δ' = Some (false, l ↦ v)%I →
v = v1 → vals_compare_safe v v1 →
match envs_simple_replace i false (Esnoc Enil i (l ↦ v2)) Δ' with
| Some Δ'' =>
envs_entails Δ'' (WBWP fill K (Val $ PairV v (LitV $ LitBool true)) @ out; E {{ Φ }})
| None => False
end →
envs_entails Δ (WBWP fill K (CmpXchg (LitV l) v1 v2) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ?????; subst.
destruct (envs_simple_replace _ _ _) as [Δ''|] eqn:HΔ''; [ | contradiction ].
rewrite -wbwp_bind. eapply wand_apply.
{ iApply wbwp_cmpxchg_suc; eauto. }
rewrite into_laterN_env_sound -later_sep envs_simple_replace_sound //; simpl.
rewrite right_id. by apply later_mono, sep_mono_r, wand_mono.
Qed.
Lemma tac_wbwp_faa Δ Δ' E out i K l z1 z2 Φ :
MaybeIntoLaterNEnvs 1 Δ Δ' →
envs_lookup i Δ' = Some (false, l ↦ LitV z1)%I →
match envs_simple_replace i false (Esnoc Enil i (l ↦ LitV (LitInt (z1 + z2)))) Δ' with
| Some Δ'' => envs_entails Δ'' (WBWP fill K (Val $ LitV z1) @ out; E {{ Φ }})
| None => False
end →
envs_entails Δ (WBWP fill K (FAA (LitV l) (LitV z2)) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ???.
destruct (envs_simple_replace _ _ _) as [Δ''|] eqn:HΔ''; [ | contradiction ].
rewrite -wbwp_bind. eapply wand_apply; first by iApply (wbwp_faa _ _ _ z1 z2).
rewrite into_laterN_env_sound -later_sep envs_simple_replace_sound //; simpl.
rewrite right_id. by apply later_mono, sep_mono_r, wand_mono.
Qed.
End heap.
Context `{!wbheapGS Σ}.
Implicit Types P Q : iProp Σ.
Implicit Types Φ : val → iProp Σ.
Implicit Types Δ : envs (uPredI (iResUR Σ)).
Implicit Types v : val.
Implicit Types z : Z.
Lemma tac_wbwp_allocN Δ Δ' E out j K v n Φ :
(0 < n)%Z →
MaybeIntoLaterNEnvs 1 Δ Δ' →
(∀ l,
match envs_app false (Esnoc Enil j (array l (DfracOwn 1) (replicate (Z.to_nat n) v))) Δ' with
| Some Δ'' =>
envs_entails Δ'' (WBWP fill K (Val $ LitV $ LitLoc l) @ out; E {{ Φ }})
| None => False
end) →
envs_entails Δ (WBWP fill K (AllocN (Val $ LitV $ LitInt n) (Val v)) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ? ? HΔ.
rewrite -wbwp_bind. eapply wand_apply; first by iApply wbwp_allocN.
rewrite left_id into_laterN_env_sound; apply later_mono, forall_intro=> l.
specialize (HΔ l).
destruct (envs_app _ _ _) as [Δ''|] eqn:HΔ'; [ | contradiction ].
rewrite envs_app_sound //; simpl.
apply wand_intro_l. by rewrite (sep_elim_l (l ↦∗ _)%I) right_id wand_elim_r.
Qed.
Lemma tac_wbwp_alloc Δ Δ' E out j K v Φ :
MaybeIntoLaterNEnvs 1 Δ Δ' →
(∀ l,
match envs_app false (Esnoc Enil j (l ↦ v)) Δ' with
| Some Δ'' =>
envs_entails Δ'' (WBWP fill K (Val $ LitV l) @ out; E {{ Φ }})
| None => False
end) →
envs_entails Δ (WBWP fill K (Alloc (Val v)) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ? HΔ.
rewrite -wbwp_bind. eapply wand_apply; first by iApply wbwp_alloc.
rewrite left_id into_laterN_env_sound; apply later_mono, forall_intro=> l.
specialize (HΔ l).
destruct (envs_app _ _ _) as [Δ''|] eqn:HΔ'; [ | contradiction ].
rewrite envs_app_sound //; simpl.
apply wand_intro_l. by rewrite (sep_elim_l (l ↦ v)%I) right_id wand_elim_r.
Qed.
Lemma tac_wbwp_free Δ Δ' E out i K l v Φ :
MaybeIntoLaterNEnvs 1 Δ Δ' →
envs_lookup i Δ' = Some (false, l ↦ v)%I →
(let Δ'' := envs_delete false i false Δ' in
envs_entails Δ'' (WBWP fill K (Val $ LitV LitUnit) @ out; E {{ Φ }})) →
envs_entails Δ (WBWP fill K (Free (LitV l)) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ? Hlk Hfin.
rewrite -wbwp_bind. eapply wand_apply; first by iApply wbwp_free.
rewrite into_laterN_env_sound -later_sep envs_lookup_split //; simpl.
rewrite -Hfin wand_elim_r (envs_lookup_sound' _ _ _ _ _ Hlk).
apply later_mono, sep_mono_r, wand_intro_r. rewrite right_id //.
Qed.
Lemma tac_wbwp_load Δ Δ' E out i K b l q v Φ :
MaybeIntoLaterNEnvs 1 Δ Δ' →
envs_lookup i Δ' = Some (b, l ↦{q} v)%I →
envs_entails Δ' (WBWP fill K (Val v) @ out; E {{ Φ }}) →
envs_entails Δ (WBWP fill K (Load (LitV l)) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ?? Hi.
rewrite -wbwp_bind. eapply wand_apply; first by iApply wbwp_load.
rewrite into_laterN_env_sound -later_sep envs_lookup_split //; simpl.
apply later_mono.
destruct b; simpl.
* iIntros "[#$ He]". iIntros "_". iApply Hi. iApply "He". iFrame "#".
* by apply sep_mono_r, wand_mono.
Qed.
Lemma tac_wbwp_store Δ Δ' E out i K l v v' Φ :
MaybeIntoLaterNEnvs 1 Δ Δ' →
envs_lookup i Δ' = Some (false, l ↦ v)%I →
match envs_simple_replace i false (Esnoc Enil i (l ↦ v')) Δ' with
| Some Δ'' => envs_entails Δ'' (WBWP fill K (Val $ LitV LitUnit) @ out; E {{ Φ }})
| None => False
end →
envs_entails Δ (WBWP fill K (Store (LitV l) (Val v')) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ???.
destruct (envs_simple_replace _ _ _) as [Δ''|] eqn:HΔ''; [ | contradiction ].
rewrite -wbwp_bind. eapply wand_apply; first by iApply wbwp_store.
rewrite into_laterN_env_sound -later_sep envs_simple_replace_sound //; simpl.
rewrite right_id. by apply later_mono, sep_mono_r, wand_mono.
Qed.
Lemma tac_wbwp_xchg Δ Δ' E out i K l v v' Φ :
MaybeIntoLaterNEnvs 1 Δ Δ' →
envs_lookup i Δ' = Some (false, l ↦ v)%I →
match envs_simple_replace i false (Esnoc Enil i (l ↦ v')) Δ' with
| Some Δ'' => envs_entails Δ'' (WBWP fill K (Val $ v) @ out; E {{ Φ }})
| None => False
end →
envs_entails Δ (WBWP fill K (Xchg (LitV l) (Val v')) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ???.
destruct (envs_simple_replace _ _ _) as [Δ''|] eqn:HΔ''; [ | contradiction ].
rewrite -wbwp_bind. eapply wand_apply; first by iApply wbwp_xchg.
rewrite into_laterN_env_sound -later_sep envs_simple_replace_sound //; simpl.
rewrite right_id.
by apply later_mono, sep_mono_r, wand_mono.
Qed.
Lemma tac_wbwp_cmpxchg Δ Δ' E out i K l v v1 v2 Φ :
MaybeIntoLaterNEnvs 1 Δ Δ' →
envs_lookup i Δ' = Some (false, l ↦ v)%I →
vals_compare_safe v v1 →
match envs_simple_replace i false (Esnoc Enil i (l ↦ v2)) Δ' with
| Some Δ'' =>
v = v1 →
envs_entails Δ'' (WBWP fill K (Val $ PairV v (LitV $ LitBool true)) @ out; E {{ Φ }})
| None => False
end →
(v ≠ v1 →
envs_entails Δ' (WBWP fill K (Val $ PairV v (LitV $ LitBool false)) @ out; E {{ Φ }})) →
envs_entails Δ (WBWP fill K (CmpXchg (LitV l) (Val v1) (Val v2)) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ??? Hsuc Hfail.
destruct (envs_simple_replace _ _ _ _) as [Δ''|] eqn:HΔ''; [ | contradiction ].
destruct (decide (v = v1)) as [Heq|Hne].
- rewrite -wbwp_bind. eapply wand_apply.
{ iApply wbwp_cmpxchg_suc; eauto. }
rewrite into_laterN_env_sound -later_sep /= {1}envs_simple_replace_sound //; simpl.
apply later_mono, sep_mono_r. rewrite right_id. apply wand_mono; auto.
- rewrite -wbwp_bind. eapply wand_apply.
{ iApply wbwp_cmpxchg_fail; eauto. }
rewrite into_laterN_env_sound -later_sep /= {1}envs_lookup_split //; simpl.
apply later_mono, sep_mono_r. apply wand_mono; auto.
Qed.
Lemma tac_wbwp_cmpxchg_fail Δ Δ' E out i K l q v v1 v2 Φ :
MaybeIntoLaterNEnvs 1 Δ Δ' →
envs_lookup i Δ' = Some (false, l ↦{q} v)%I →
v ≠ v1 → vals_compare_safe v v1 →
envs_entails Δ' (WBWP fill K (Val $ PairV v (LitV $ LitBool false)) @ out; E {{ Φ }}) →
envs_entails Δ (WBWP fill K (CmpXchg (LitV l) v1 v2) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ?????.
rewrite -wbwp_bind. eapply wand_apply; first by iApply wbwp_cmpxchg_fail.
rewrite into_laterN_env_sound -later_sep envs_lookup_split //; simpl.
by apply later_mono, sep_mono_r, wand_mono.
Qed.
Lemma tac_wbwp_cmpxchg_suc Δ Δ' E out i K l v v1 v2 Φ :
MaybeIntoLaterNEnvs 1 Δ Δ' →
envs_lookup i Δ' = Some (false, l ↦ v)%I →
v = v1 → vals_compare_safe v v1 →
match envs_simple_replace i false (Esnoc Enil i (l ↦ v2)) Δ' with
| Some Δ'' =>
envs_entails Δ'' (WBWP fill K (Val $ PairV v (LitV $ LitBool true)) @ out; E {{ Φ }})
| None => False
end →
envs_entails Δ (WBWP fill K (CmpXchg (LitV l) v1 v2) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ?????; subst.
destruct (envs_simple_replace _ _ _) as [Δ''|] eqn:HΔ''; [ | contradiction ].
rewrite -wbwp_bind. eapply wand_apply.
{ iApply wbwp_cmpxchg_suc; eauto. }
rewrite into_laterN_env_sound -later_sep envs_simple_replace_sound //; simpl.
rewrite right_id. by apply later_mono, sep_mono_r, wand_mono.
Qed.
Lemma tac_wbwp_faa Δ Δ' E out i K l z1 z2 Φ :
MaybeIntoLaterNEnvs 1 Δ Δ' →
envs_lookup i Δ' = Some (false, l ↦ LitV z1)%I →
match envs_simple_replace i false (Esnoc Enil i (l ↦ LitV (LitInt (z1 + z2)))) Δ' with
| Some Δ'' => envs_entails Δ'' (WBWP fill K (Val $ LitV z1) @ out; E {{ Φ }})
| None => False
end →
envs_entails Δ (WBWP fill K (FAA (LitV l) (LitV z2)) @ out; E {{ Φ }}).
Proof.
rewrite envs_entails_unseal=> ???.
destruct (envs_simple_replace _ _ _) as [Δ''|] eqn:HΔ''; [ | contradiction ].
rewrite -wbwp_bind. eapply wand_apply; first by iApply (wbwp_faa _ _ _ z1 z2).
rewrite into_laterN_env_sound -later_sep envs_simple_replace_sound //; simpl.
rewrite right_id. by apply later_mono, sep_mono_r, wand_mono.
Qed.
End heap.
The tactic wp_apply_core lem tac_suc tac_fail evaluates lem to a
hypothesis H that can be applied, and then runs wp_bind_core K; tac_suc H
for every possible evaluation context K.
TC resolution of lem premises happens *after* tac_suc H got executed.
- The tactic tac_suc should do iApplyHyp H to actually apply the hypothesis, but can perform other operations in addition (see wp_apply and awp_apply below).
- The tactic tac_fail cont is called when tac_suc H fails for all evaluation contexts K, and can perform further operations before invoking cont to try again.
Ltac wbwp_apply_core lem tac_suc tac_fail := first
[iPoseProofCore lem as false (fun H =>
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
reshape_expr e ltac:(fun K e' =>
wbwp_bind_core K; tac_suc H)
| _ => fail 1 "wbwp_apply: not a 'wp'"
end)
|tac_fail ltac:(fun _ => wp_apply_core lem tac_suc tac_fail)
|let P := type of lem in
fail "wbwp_apply: cannot apply" lem ":" P ].
Tactic Notation "wbwp_apply" open_constr(lem) :=
wbwp_apply_core lem ltac:(fun H => iApplyHyp H; try iNext; try wbwp_expr_simpl)
ltac:(fun cont => fail).
Tactic Notation "wbwp_smart_apply" open_constr(lem) :=
wbwp_apply_core lem ltac:(fun H => iApplyHyp H; try iNext; try wbwp_expr_simpl)
ltac:(fun cont => wbwp_pure _; []; cont ()).
(* (** Tactic tailored for atomic triples: the first, simple one just runs *)
(* iAuIntro on the goal, as atomic triples always have an atomic update as their *)
(* premise. The second one additionaly does some framing: it gets rid of Hs from *)
(* the context, reducing clutter. You get them all back in the continuation of the *)
(* atomic operation. *) *)
(* Tactic Notation "awp_apply" open_constr(lem) := *)
(* wp_apply_core lem ltac:(fun H => iApplyHyp H) ltac:(fun cont => fail); *)
(* last iAuIntro. *)
(* Tactic Notation "awp_apply" open_constr(lem) "without" constr(Hs) := *)
(* (* Convert "list of hypothesis" into specialization pattern. *) *)
(* let Hs := words Hs in *)
(* let Hs := eval vm_compute in (INamed <*)
(* wp_apply_core lem *)
(* ltac:(fun H => *)
(* iApply (wp_frame_wand with *)
(* SGoal $ SpecGoal GSpatial false [] Hs false); iAccu|iApplyHyp H) *)
(* ltac:(fun cont => fail); *)
(* last iAuIntro. *)
Tactic Notation "wbwp_alloc" ident(l) "as" constr(H) :=
let Htmp := iFresh in
let finish _ :=
first [intros l | fail 1 "wbwp_alloc:" l "not fresh"];
pm_reduce;
lazymatch goal with
| |- False => fail 1 "wbwp_alloc:" H "not fresh"
| _ => iDestructHyp Htmp as H; wbwp_finish
end in
wbwp_pures;
[iPoseProofCore lem as false (fun H =>
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
reshape_expr e ltac:(fun K e' =>
wbwp_bind_core K; tac_suc H)
| _ => fail 1 "wbwp_apply: not a 'wp'"
end)
|tac_fail ltac:(fun _ => wp_apply_core lem tac_suc tac_fail)
|let P := type of lem in
fail "wbwp_apply: cannot apply" lem ":" P ].
Tactic Notation "wbwp_apply" open_constr(lem) :=
wbwp_apply_core lem ltac:(fun H => iApplyHyp H; try iNext; try wbwp_expr_simpl)
ltac:(fun cont => fail).
Tactic Notation "wbwp_smart_apply" open_constr(lem) :=
wbwp_apply_core lem ltac:(fun H => iApplyHyp H; try iNext; try wbwp_expr_simpl)
ltac:(fun cont => wbwp_pure _; []; cont ()).
(* (** Tactic tailored for atomic triples: the first, simple one just runs *)
(* iAuIntro on the goal, as atomic triples always have an atomic update as their *)
(* premise. The second one additionaly does some framing: it gets rid of Hs from *)
(* the context, reducing clutter. You get them all back in the continuation of the *)
(* atomic operation. *) *)
(* Tactic Notation "awp_apply" open_constr(lem) := *)
(* wp_apply_core lem ltac:(fun H => iApplyHyp H) ltac:(fun cont => fail); *)
(* last iAuIntro. *)
(* Tactic Notation "awp_apply" open_constr(lem) "without" constr(Hs) := *)
(* (* Convert "list of hypothesis" into specialization pattern. *) *)
(* let Hs := words Hs in *)
(* let Hs := eval vm_compute in (INamed <*)
(* wp_apply_core lem *)
(* ltac:(fun H => *)
(* iApply (wp_frame_wand with *)
(* SGoal $ SpecGoal GSpatial false [] Hs false); iAccu|iApplyHyp H) *)
(* ltac:(fun cont => fail); *)
(* last iAuIntro. *)
Tactic Notation "wbwp_alloc" ident(l) "as" constr(H) :=
let Htmp := iFresh in
let finish _ :=
first [intros l | fail 1 "wbwp_alloc:" l "not fresh"];
pm_reduce;
lazymatch goal with
| |- False => fail 1 "wbwp_alloc:" H "not fresh"
| _ => iDestructHyp Htmp as H; wbwp_finish
end in
wbwp_pures;
The code first tries to use allocation lemma for a single reference,
ie, tac_wp_alloc (respectively, tac_twp_alloc).
If that fails, it tries to use the lemma tac_wp_allocN
(respectively, tac_twp_allocN) for allocating an array.
Notice that we could have used the array allocation lemma also for single
references. However, that would produce the resource l ↦∗ v instead of
l ↦ v for single references. These are logically equivalent assertions
but are not equal.
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
let process_single _ :=
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_alloc _ _ _ _ Htmp K))
|fail 1 "wbwp_alloc: cannot find 'Alloc' in" e];
[tc_solve
|finish ()]
in
let process_array _ :=
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_allocN _ _ _ _ Htmp K))
|fail 1 "wbwp_alloc: cannot find 'Alloc' in" e];
[idtac|tc_solve
|finish ()]
in (process_single ()) || (process_array ())
| _ => fail "wbwp_alloc: not a 'wbwp'"
end.
Tactic Notation "wbwp_alloc" ident(l) :=
wbwp_alloc l as "?".
Tactic Notation "wbwp_free" :=
let solve_mapsto _ :=
let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wbwp_free: cannot find" l "↦ ?" in
wbwp_pures;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_free _ _ _ _ _ K))
|fail 1 "wbwp_free: cannot find 'Free' in" e];
[tc_solve
|solve_mapsto ()
|pm_reduce; wbwp_finish]
| _ => fail "wbwp_free: not a 'wbwp'"
end.
Tactic Notation "wbwp_load" :=
let solve_mapsto _ :=
let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wbwp_load: cannot find" l "↦ ?" in
wbwp_pures;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_load _ _ _ _ _ K))
|fail 1 "wbwp_load: cannot find 'Load' in" e];
[tc_solve
|solve_mapsto ()
|wbwp_finish]
| _ => fail "wbwp_load: not a 'wbwp'"
end.
Tactic Notation "wbwp_store" :=
let solve_mapsto _ :=
let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wbwp_store: cannot find" l "↦ ?" in
wbwp_pures;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_store _ _ _ _ _ K))
|fail 1 "wbwp_store: cannot find 'Store' in" e];
[tc_solve
|solve_mapsto ()
|pm_reduce; first [wbwp_seq|wbwp_finish]]
| _ => fail "wbwp_store: not a 'wbwp'"
end.
Tactic Notation "wbwp_xchg" :=
let solve_mapsto _ :=
let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wbwp_xchg: cannot find" l "↦ ?" in
wbwp_pures;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_xchg _ _ _ _ _ K))
|fail 1 "wbwp_xchg: cannot find 'Xchg' in" e];
[tc_solve
|solve_mapsto ()
|pm_reduce; first [wbwp_seq|wbwp_finish]]
| _ => fail "wbwp_xchg: not a 'wbwp'"
end.
Tactic Notation "wbwp_cmpxchg" "as" simple_intropattern(H1) "|" simple_intropattern(H2) :=
let solve_mapsto _ :=
let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wbwp_cmpxchg: cannot find" l "↦ ?" in
wbwp_pures;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_cmpxchg _ _ _ _ _ K))
|fail 1 "wbwp_cmpxchg: cannot find 'CmpXchg' in" e];
[tc_solve
|solve_mapsto ()
|try solve_vals_compare_safe
|pm_reduce; intros H1; wbwp_finish
|intros H2; wbwp_finish]
| _ => fail "wbwp_cmpxchg: not a 'wbwp'"
end.
Tactic Notation "wbwp_cmpxchg_fail" :=
let solve_mapsto _ :=
let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wbwp_cmpxchg_fail: cannot find" l "↦ ?" in
wbwp_pures;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_cmpxchg_fail _ _ _ _ _ K))
|fail 1 "wbwp_cmpxchg_fail: cannot find 'CmpXchg' in" e];
[tc_solve
|solve_mapsto ()
|try (simpl; congruence) (* value inequality *)
|try solve_vals_compare_safe
|wbwp_finish]
| _ => fail "wbwp_cmpxchg_fail: not a 'wbwp'"
end.
Tactic Notation "wbwp_cmpxchg_suc" :=
let solve_mapsto _ :=
let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wbwp_cmpxchg_suc: cannot find" l "↦ ?" in
wbwp_pures;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_cmpxchg_suc _ _ _ _ _ K))
|fail 1 "wbwp_cmpxchg_suc: cannot find 'CmpXchg' in" e];
[tc_solve
|solve_mapsto ()
|try (simpl; congruence) (* value equality *)
|try solve_vals_compare_safe
|pm_reduce; wbwp_finish]
| _ => fail "wbwp_cmpxchg_suc: not a 'wbwp'"
end.
Tactic Notation "wbwp_faa" :=
let solve_mapsto _ :=
let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wbwp_faa: cannot find" l "↦ ?" in
wp_pures;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_faa _ _ _ _ _ K))
|fail 1 "wbwp_faa: cannot find 'FAA' in" e];
[tc_solve
|solve_mapsto ()
|pm_reduce; wbwp_finish]
| _ => fail "wbwp_faa: not a 'wbwp'"
end.
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
let process_single _ :=
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_alloc _ _ _ _ Htmp K))
|fail 1 "wbwp_alloc: cannot find 'Alloc' in" e];
[tc_solve
|finish ()]
in
let process_array _ :=
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_allocN _ _ _ _ Htmp K))
|fail 1 "wbwp_alloc: cannot find 'Alloc' in" e];
[idtac|tc_solve
|finish ()]
in (process_single ()) || (process_array ())
| _ => fail "wbwp_alloc: not a 'wbwp'"
end.
Tactic Notation "wbwp_alloc" ident(l) :=
wbwp_alloc l as "?".
Tactic Notation "wbwp_free" :=
let solve_mapsto _ :=
let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wbwp_free: cannot find" l "↦ ?" in
wbwp_pures;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_free _ _ _ _ _ K))
|fail 1 "wbwp_free: cannot find 'Free' in" e];
[tc_solve
|solve_mapsto ()
|pm_reduce; wbwp_finish]
| _ => fail "wbwp_free: not a 'wbwp'"
end.
Tactic Notation "wbwp_load" :=
let solve_mapsto _ :=
let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wbwp_load: cannot find" l "↦ ?" in
wbwp_pures;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_load _ _ _ _ _ K))
|fail 1 "wbwp_load: cannot find 'Load' in" e];
[tc_solve
|solve_mapsto ()
|wbwp_finish]
| _ => fail "wbwp_load: not a 'wbwp'"
end.
Tactic Notation "wbwp_store" :=
let solve_mapsto _ :=
let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wbwp_store: cannot find" l "↦ ?" in
wbwp_pures;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_store _ _ _ _ _ K))
|fail 1 "wbwp_store: cannot find 'Store' in" e];
[tc_solve
|solve_mapsto ()
|pm_reduce; first [wbwp_seq|wbwp_finish]]
| _ => fail "wbwp_store: not a 'wbwp'"
end.
Tactic Notation "wbwp_xchg" :=
let solve_mapsto _ :=
let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wbwp_xchg: cannot find" l "↦ ?" in
wbwp_pures;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_xchg _ _ _ _ _ K))
|fail 1 "wbwp_xchg: cannot find 'Xchg' in" e];
[tc_solve
|solve_mapsto ()
|pm_reduce; first [wbwp_seq|wbwp_finish]]
| _ => fail "wbwp_xchg: not a 'wbwp'"
end.
Tactic Notation "wbwp_cmpxchg" "as" simple_intropattern(H1) "|" simple_intropattern(H2) :=
let solve_mapsto _ :=
let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wbwp_cmpxchg: cannot find" l "↦ ?" in
wbwp_pures;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_cmpxchg _ _ _ _ _ K))
|fail 1 "wbwp_cmpxchg: cannot find 'CmpXchg' in" e];
[tc_solve
|solve_mapsto ()
|try solve_vals_compare_safe
|pm_reduce; intros H1; wbwp_finish
|intros H2; wbwp_finish]
| _ => fail "wbwp_cmpxchg: not a 'wbwp'"
end.
Tactic Notation "wbwp_cmpxchg_fail" :=
let solve_mapsto _ :=
let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wbwp_cmpxchg_fail: cannot find" l "↦ ?" in
wbwp_pures;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_cmpxchg_fail _ _ _ _ _ K))
|fail 1 "wbwp_cmpxchg_fail: cannot find 'CmpXchg' in" e];
[tc_solve
|solve_mapsto ()
|try (simpl; congruence) (* value inequality *)
|try solve_vals_compare_safe
|wbwp_finish]
| _ => fail "wbwp_cmpxchg_fail: not a 'wbwp'"
end.
Tactic Notation "wbwp_cmpxchg_suc" :=
let solve_mapsto _ :=
let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wbwp_cmpxchg_suc: cannot find" l "↦ ?" in
wbwp_pures;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_cmpxchg_suc _ _ _ _ _ K))
|fail 1 "wbwp_cmpxchg_suc: cannot find 'CmpXchg' in" e];
[tc_solve
|solve_mapsto ()
|try (simpl; congruence) (* value equality *)
|try solve_vals_compare_safe
|pm_reduce; wbwp_finish]
| _ => fail "wbwp_cmpxchg_suc: not a 'wbwp'"
end.
Tactic Notation "wbwp_faa" :=
let solve_mapsto _ :=
let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wbwp_faa: cannot find" l "↦ ?" in
wp_pures;
lazymatch goal with
| |- envs_entails _ (wbwp ?E ?out ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' => eapply (tac_wbwp_faa _ _ _ _ _ K))
|fail 1 "wbwp_faa: cannot find 'FAA' in" e];
[tc_solve
|solve_mapsto ()
|pm_reduce; wbwp_finish]
| _ => fail "wbwp_faa: not a 'wbwp'"
end.