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Commit 8076fe39 authored by Robbert Krebbers's avatar Robbert Krebbers
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Total weakest preconditions for heap_lang.

parent f7dc0953
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_CoqProject
View file @ 8076fe39
......@@ -79,6 +79,7 @@ theories/heap_lang/lib/ticket_lock.v
theories/heap_lang/lib/counter.v
theories/heap_lang/proofmode.v
theories/heap_lang/adequacy.v
theories/heap_lang/total_adequacy.v
theories/proofmode/base.v
theories/proofmode/tokens.v
theories/proofmode/coq_tactics.v
......
theories/heap_lang/lifting.v
View file @ 8076fe39
From iris.base_logic Require Export gen_heap.
From iris.program_logic Require Export weakestpre lifting.
From iris.program_logic Require Import ectx_lifting.
From iris.program_logic Require Export weakestpre.
From iris.program_logic Require Import ectx_lifting total_ectx_lifting.
From iris.heap_lang Require Export lang.
From iris.heap_lang Require Import tactics.
From iris.proofmode Require Import tactics.
From stdpp Require Import fin_maps.
Set Default Proof Using "Type".
Import uPred.
(** Basic rules for language operations. *)
Class heapG Σ := HeapG {
heapG_invG : invG Σ;
......@@ -39,8 +36,6 @@ Ltac inv_head_step :=
repeat match goal with
| _ => progress simplify_map_eq/= (* simplify memory stuff *)
| H : to_val _ = Some _ |- _ => apply of_to_val in H
| H : _ = of_val ?v |- _ =>
is_var v; destruct v; first[discriminate H|injection H as H]
| H : head_step ?e _ _ _ _ |- _ =>
try (is_var e; fail 1); (* inversion yields many goals if [e] is a variable
and can thus better be avoided. *)
......@@ -54,22 +49,6 @@ Local Hint Constructors head_step.
Local Hint Resolve alloc_fresh.
Local Hint Resolve to_of_val.
Section lifting.
Context `{heapG Σ}.
Implicit Types P Q : iProp Σ.
Implicit Types Φ : val → iProp Σ.
Implicit Types efs : list expr.
Implicit Types σ : state.
(** Base axioms for core primitives of the language: Stateless reductions *)
Lemma wp_fork s E e Φ :
▷ Φ (LitV LitUnit) ∗ ▷ WP e @ s; ⊤ {{ _, True }} ⊢ WP Fork e @ s; E {{ Φ }}.
Proof.
rewrite -(wp_lift_pure_det_head_step (Fork e) (Lit LitUnit) [e]) //=; eauto.
- by rewrite -step_fupd_intro // later_sep -(wp_value _ _ _ (Lit _)) // right_id.
- intros; inv_head_step; eauto.
Qed.
Local Ltac solve_exec_safe := intros; subst; do 3 eexists; econstructor; eauto.
Local Ltac solve_exec_puredet := simpl; intros; by inv_head_step.
Local Ltac solve_pure_exec :=
......@@ -79,48 +58,69 @@ Local Ltac solve_pure_exec :=
Class AsRec (e : expr) (f x : binder) (erec : expr) :=
as_rec : e = Rec f x erec.
Global Instance AsRec_rec f x e : AsRec (Rec f x e) f x e := eq_refl.
Global Instance AsRec_rec_locked_val v f x e :
Instance AsRec_rec f x e : AsRec (Rec f x e) f x e := eq_refl.
Instance AsRec_rec_locked_val v f x e :
AsRec (of_val v) f x e → AsRec (of_val (locked v)) f x e.
Proof. by unlock. Qed.
Global Instance pure_rec f x (erec e1 e2 : expr)
Instance pure_rec f x (erec e1 e2 : expr)
`{!AsVal e2, AsRec e1 f x erec, Closed (f :b: x :b: []) erec} :
PureExec True (App e1 e2) (subst' x e2 (subst' f e1 erec)).
Proof. unfold AsRec in *; solve_pure_exec. Qed.
Global Instance pure_unop op e v v' `{!IntoVal e v} :
Instance pure_unop op e v v' `{!IntoVal e v} :
PureExec (un_op_eval op v = Some v') (UnOp op e) (of_val v').
Proof. solve_pure_exec. Qed.
Global Instance pure_binop op e1 e2 v1 v2 v' `{!IntoVal e1 v1, !IntoVal e2 v2} :
Instance pure_binop op e1 e2 v1 v2 v' `{!IntoVal e1 v1, !IntoVal e2 v2} :
PureExec (bin_op_eval op v1 v2 = Some v') (BinOp op e1 e2) (of_val v').
Proof. solve_pure_exec. Qed.
Global Instance pure_if_true e1 e2 :
PureExec True (If (Lit (LitBool true)) e1 e2) e1.
Instance pure_if_true e1 e2 : PureExec True (If (Lit (LitBool true)) e1 e2) e1.
Proof. solve_pure_exec. Qed.
Global Instance pure_if_false e1 e2 :
PureExec True (If (Lit (LitBool false)) e1 e2) e2.
Instance pure_if_false e1 e2 : PureExec True (If (Lit (LitBool false)) e1 e2) e2.
Proof. solve_pure_exec. Qed.
Global Instance pure_fst e1 e2 v1 `{!IntoVal e1 v1, !AsVal e2} :
Instance pure_fst e1 e2 v1 `{!IntoVal e1 v1, !AsVal e2} :
PureExec True (Fst (Pair e1 e2)) e1.
Proof. solve_pure_exec. Qed.
Global Instance pure_snd e1 e2 v2 `{!AsVal e1, !IntoVal e2 v2} :
Instance pure_snd e1 e2 v2 `{!AsVal e1, !IntoVal e2 v2} :
PureExec True (Snd (Pair e1 e2)) e2.
Proof. solve_pure_exec. Qed.
Global Instance pure_case_inl e0 v e1 e2 `{!IntoVal e0 v} :
Instance pure_case_inl e0 v e1 e2 `{!IntoVal e0 v} :
PureExec True (Case (InjL e0) e1 e2) (App e1 e0).
Proof. solve_pure_exec. Qed.
Global Instance pure_case_inr e0 v e1 e2 `{!IntoVal e0 v} :
Instance pure_case_inr e0 v e1 e2 `{!IntoVal e0 v} :
PureExec True (Case (InjR e0) e1 e2) (App e2 e0).
Proof. solve_pure_exec. Qed.
Section lifting.
Context `{heapG Σ}.
Implicit Types P Q : iProp Σ.
Implicit Types Φ : val → iProp Σ.
Implicit Types efs : list expr.
Implicit Types σ : state.
(** Base axioms for core primitives of the language: Stateless reductions *)
Lemma wp_fork s E e Φ :
▷ Φ (LitV LitUnit) ∗ ▷ WP e @ s; ⊤ {{ _, True }} ⊢ WP Fork e @ s; E {{ Φ }}.
Proof.
iIntros "[HΦ He]".
iApply wp_lift_pure_det_head_step; [auto|intros; inv_head_step; eauto|].
iModIntro; iNext; iIntros "!> /= {$He}". by iApply wp_value.
Qed.
Lemma twp_fork s E e Φ :
Φ (LitV LitUnit) ∗ WP e @ s; ⊤ [{ _, True }] ⊢ WP Fork e @ s; E [{ Φ }].
Proof.
iIntros "[HΦ He]".
iApply twp_lift_pure_det_head_step; [auto|intros; inv_head_step; eauto|].
iIntros "!> /= {$He}". by iApply twp_value.
Qed.
(** Heap *)
Lemma wp_alloc s E e v :
IntoVal e v →
......@@ -132,6 +132,16 @@ Proof.
iMod (@gen_heap_alloc with "Hσ") as "[Hσ Hl]"; first done.
iModIntro; iSplit=> //. iFrame. by iApply "HΦ".
Qed.
Lemma twp_alloc s E e v :
IntoVal e v →
[[{ True }]] Alloc e @ s; E [[{ l, RET LitV (LitLoc l); l ↦ v }]].
Proof.
iIntros (<-%of_to_val Φ) "_ HΦ". iApply twp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1) "Hσ !>"; iSplit; first by auto.
iIntros (v2 σ2 efs Hstep); inv_head_step.
iMod (@gen_heap_alloc with "Hσ") as "[Hσ Hl]"; first done.
iModIntro; iSplit=> //. iFrame. by iApply "HΦ".
Qed.
Lemma wp_load s E l q v :
{{{ ▷ l ↦{q} v }}} Load (Lit (LitLoc l)) @ s; E {{{ RET v; l ↦{q} v }}}.
......@@ -142,6 +152,15 @@ Proof.
iNext; iIntros (v2 σ2 efs Hstep); inv_head_step.
iModIntro; iSplit=> //. iFrame. by iApply "HΦ".
Qed.
Lemma twp_load s E l q v :
[[{ l ↦{q} v }]] Load (Lit (LitLoc l)) @ s; E [[{ RET v; l ↦{q} v }]].
Proof.
iIntros (Φ) "Hl HΦ". iApply twp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1) "Hσ !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iSplit; first by eauto.
iIntros (v2 σ2 efs Hstep); inv_head_step.
iModIntro; iSplit=> //. iFrame. by iApply "HΦ".
Qed.
Lemma wp_store s E l v' e v :
IntoVal e v →
......@@ -154,6 +173,17 @@ Proof.
iMod (@gen_heap_update with "Hσ Hl") as "[$ Hl]".
iModIntro. iSplit=>//. by iApply "HΦ".
Qed.
Lemma twp_store s E l v' e v :
IntoVal e v →
[[{ l ↦ v' }]] Store (Lit (LitLoc l)) e @ s; E [[{ RET LitV LitUnit; l ↦ v }]].
Proof.
iIntros (<-%of_to_val Φ) "Hl HΦ".
iApply twp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1) "Hσ !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iSplit; first by eauto. iIntros (v2 σ2 efs Hstep); inv_head_step.
iMod (@gen_heap_update with "Hσ Hl") as "[$ Hl]".
iModIntro. iSplit=>//. by iApply "HΦ".
Qed.
Lemma wp_cas_fail s E l q v' e1 v1 e2 :
IntoVal e1 v1 → AsVal e2 → v' ≠ v1 →
......@@ -166,6 +196,17 @@ Proof.
iSplit; first by eauto. iNext; iIntros (v2' σ2 efs Hstep); inv_head_step.
iModIntro; iSplit=> //. iFrame. by iApply "HΦ".
Qed.
Lemma twp_cas_fail s E l q v' e1 v1 e2 :
IntoVal e1 v1 → AsVal e2 → v' ≠ v1 →
[[{ l ↦{q} v' }]] CAS (Lit (LitLoc l)) e1 e2 @ s; E
[[{ RET LitV (LitBool false); l ↦{q} v' }]].
Proof.
iIntros (<-%of_to_val [v2 <-%of_to_val] ? Φ) "Hl HΦ".
iApply twp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1) "Hσ !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iSplit; first by eauto. iIntros (v2' σ2 efs Hstep); inv_head_step.
iModIntro; iSplit=> //. iFrame. by iApply "HΦ".
Qed.
Lemma wp_cas_suc s E l e1 v1 e2 v2 :
IntoVal e1 v1 → IntoVal e2 v2 →
......@@ -179,6 +220,18 @@ Proof.
iMod (@gen_heap_update with "Hσ Hl") as "[$ Hl]".
iModIntro. iSplit=>//. by iApply "HΦ".
Qed.
Lemma twp_cas_suc s E l e1 v1 e2 v2 :
IntoVal e1 v1 → IntoVal e2 v2 →
[[{ l ↦ v1 }]] CAS (Lit (LitLoc l)) e1 e2 @ s; E
[[{ RET LitV (LitBool true); l ↦ v2 }]].
Proof.
iIntros (<-%of_to_val <-%of_to_val Φ) "Hl HΦ".
iApply twp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1) "Hσ !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iSplit; first by eauto. iIntros (v2' σ2 efs Hstep); inv_head_step.
iMod (@gen_heap_update with "Hσ Hl") as "[$ Hl]".
iModIntro. iSplit=>//. by iApply "HΦ".
Qed.
Lemma wp_faa s E l i1 e2 i2 :
IntoVal e2 (LitV (LitInt i2)) →
......@@ -192,4 +245,16 @@ Proof.
iMod (@gen_heap_update with "Hσ Hl") as "[$ Hl]".
iModIntro. iSplit=>//. by iApply "HΦ".
Qed.
Lemma twp_faa s E l i1 e2 i2 :
IntoVal e2 (LitV (LitInt i2)) →
[[{ l ↦ LitV (LitInt i1) }]] FAA (Lit (LitLoc l)) e2 @ s; E
[[{ RET LitV (LitInt i1); l ↦ LitV (LitInt (i1 + i2)) }]].
Proof.
iIntros (<-%of_to_val Φ) "Hl HΦ".
iApply twp_lift_atomic_head_step_no_fork; auto.
iIntros (σ1) "Hσ !>". iDestruct (@gen_heap_valid with "Hσ Hl") as %?.
iSplit; first by eauto. iIntros (v2' σ2 efs Hstep); inv_head_step.
iMod (@gen_heap_update with "Hσ Hl") as "[$ Hl]".
iModIntro. iSplit=>//. by iApply "HΦ".
Qed.
End lifting.
theories/heap_lang/proofmode.v
View file @ 8076fe39
From iris.program_logic Require Export weakestpre.
From iris.program_logic Require Export weakestpre total_weakestpre.
From iris.proofmode Require Import coq_tactics.
From iris.proofmode Require Export tactics.
From iris.heap_lang Require Export tactics lifting.
......@@ -9,10 +9,15 @@ Lemma tac_wp_expr_eval `{heapG Σ} Δ s E Φ e e' :
e = e' →
envs_entails Δ (WP e' @ s; E {{ Φ }}) → envs_entails Δ (WP e @ s; E {{ Φ }}).
Proof. by intros ->. Qed.
Lemma tac_twp_expr_eval `{heapG Σ} Δ s E Φ e e' :
e = e' →
envs_entails Δ (WP e' @ s; E [{ Φ }]) → envs_entails Δ (WP e @ s; E [{ Φ }]).
Proof. by intros ->. Qed.
Tactic Notation "wp_expr_eval" tactic(t) :=
try iStartProof;
try (eapply tac_wp_expr_eval; [t; reflexivity|]).
try (first [eapply tac_wp_expr_eval|eapply tac_twp_expr_eval];
[t; reflexivity|]).
Ltac wp_expr_simpl := wp_expr_eval simpl.
Ltac wp_expr_simpl_subst := wp_expr_eval simpl_subst.
......@@ -25,16 +30,28 @@ Lemma tac_wp_pure `{heapG Σ} Δ Δ' s E e1 e2 φ Φ :
envs_entails Δ (WP e1 @ s; E {{ Φ }}).
Proof.
rewrite /envs_entails=> ??? HΔ'. rewrite into_laterN_env_sound /=.
rewrite HΔ' -wp_pure_step_later //.
rewrite HΔ' -lifting.wp_pure_step_later //.
Qed.
Lemma tac_twp_pure `{heapG Σ} Δ s E e1 e2 φ Φ :
PureExec φ e1 e2 →
φ →
envs_entails Δ (WP e2 @ s; E [{ Φ }]) →
envs_entails Δ (WP e1 @ s; E [{ Φ }]).
Proof.
rewrite /envs_entails=> ?? ->. rewrite -total_lifting.twp_pure_step //.
Qed.
Lemma tac_wp_value `{heapG Σ} Δ s E Φ e v :
IntoVal e v →
envs_entails Δ (Φ v) → envs_entails Δ (WP e @ s; E {{ Φ }}).
Proof. rewrite /envs_entails=> ? ->. by apply wp_value. Qed.
Lemma tac_twp_value `{heapG Σ} Δ s E Φ e v :
IntoVal e v →
envs_entails Δ (Φ v) → envs_entails Δ (WP e @ s; E [{ Φ }]).
Proof. rewrite /envs_entails=> ? ->. by apply twp_value. Qed.
Ltac wp_value_head :=
eapply tac_wp_value;
first [eapply tac_wp_value || eapply tac_twp_value];
[apply _
|iEval (lazy beta; simpl of_val)].
......@@ -52,6 +69,16 @@ Tactic Notation "wp_pure" open_constr(efoc) :=
|wp_expr_simpl_subst; try wp_value_head (* new goal *)
])
|| fail "wp_pure: cannot find" efoc "in" e "or" efoc "is not a reduct"
| |- envs_entails _ (twp ?s ?E ?e ?Q) =>
let e := eval simpl in e in
reshape_expr e ltac:(fun K e' =>
unify e' efoc;
eapply (tac_twp_pure _ _ _ (fill K e'));
[apply _ (* PureExec *)
|try fast_done (* The pure condition for PureExec *)
|wp_expr_simpl_subst; try wp_value_head (* new goal *)
])
|| fail "wp_pure: cannot find" efoc "in" e "or" efoc "is not a reduct"
| _ => fail "wp_pure: not a 'wp'"
end.
......@@ -74,12 +101,22 @@ Lemma tac_wp_bind `{heapG Σ} K Δ s E Φ e f :
envs_entails Δ (WP e @ s; E {{ v, WP f (of_val v) @ s; E {{ Φ }} }})%I →
envs_entails Δ (WP fill K e @ s; E {{ Φ }}).
Proof. rewrite /envs_entails=> -> ->. by apply: wp_bind. Qed.
Lemma tac_twp_bind `{heapG Σ} K Δ s E Φ e f :
f = (λ e, fill K e) → (* as an eta expanded hypothesis so that we can `simpl` it *)
envs_entails Δ (WP e @ s; E [{ v, WP f (of_val v) @ s; E [{ Φ }] }])%I →
envs_entails Δ (WP fill K e @ s; E [{ Φ }]).
Proof. rewrite /envs_entails=> -> ->. by apply: twp_bind. Qed.
Ltac wp_bind_core K :=
lazymatch eval hnf in K with
| [] => idtac
| _ => eapply (tac_wp_bind K); [simpl; reflexivity|lazy beta]
end.
Ltac twp_bind_core K :=
lazymatch eval hnf in K with
| [] => idtac
| _ => eapply (tac_twp_bind K); [simpl; reflexivity|lazy beta]
end.
Tactic Notation "wp_bind" open_constr(efoc) :=
iStartProof;
......@@ -87,6 +124,9 @@ Tactic Notation "wp_bind" open_constr(efoc) :=
| |- envs_entails _ (wp ?s ?E ?e ?Q) =>
reshape_expr e ltac:(fun K e' => unify e' efoc; wp_bind_core K)
|| fail "wp_bind: cannot find" efoc "in" e
| |- envs_entails _ (twp ?s ?E ?e ?Q) =>
reshape_expr e ltac:(fun K e' => unify e' efoc; twp_bind_core K)
|| fail "wp_bind: cannot find" efoc "in" e
| _ => fail "wp_bind: not a 'wp'"
end.
......@@ -111,6 +151,19 @@ Proof.
destruct (HΔ l) as (Δ''&?&HΔ'). rewrite envs_app_sound //; simpl.
by rewrite right_id HΔ'.
Qed.
Lemma tac_twp_alloc Δ s E j K e v Φ :
IntoVal e v →
(∀ l, ∃ Δ',
envs_app false (Esnoc Enil j (l ↦ v)) Δ = Some Δ' ∧
envs_entails Δ' (WP fill K (Lit (LitLoc l)) @ s; E [{ Φ }])) →
envs_entails Δ (WP fill K (Alloc e) @ s; E [{ Φ }]).
Proof.
rewrite /envs_entails=> ? HΔ.
rewrite -twp_bind. eapply wand_apply; first exact: twp_alloc.
rewrite left_id. apply forall_intro=> l.
destruct (HΔ l) as (Δ'&?&HΔ'). rewrite envs_app_sound //; simpl.
by rewrite right_id HΔ'.
Qed.
Lemma tac_wp_load Δ Δ' s E i K l q v Φ :
IntoLaterNEnvs 1 Δ Δ' →
......@@ -123,6 +176,16 @@ Proof.
rewrite into_laterN_env_sound -later_sep envs_lookup_split //; simpl.
by apply later_mono, sep_mono_r, wand_mono.
Qed.
Lemma tac_twp_load Δ s E i K l q v Φ :
envs_lookup i Δ = Some (false, l ↦{q} v)%I →
envs_entails Δ (WP fill K (of_val v) @ s; E [{ Φ }]) →
envs_entails Δ (WP fill K (Load (Lit (LitLoc l))) @ s; E [{ Φ }]).
Proof.
rewrite /envs_entails=> ??.
rewrite -twp_bind. eapply wand_apply; first exact: twp_load.
rewrite envs_lookup_split //; simpl.
by apply sep_mono_r, wand_mono.
Qed.
Lemma tac_wp_store Δ Δ' Δ'' s E i K l v e v' Φ :
IntoVal e v' →
......@@ -137,6 +200,17 @@ Proof.
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_twp_store Δ Δ' s E i K l v e v' Φ :
IntoVal e v' →
envs_lookup i Δ = Some (false, l ↦ v)%I →
envs_simple_replace i false (Esnoc Enil i (l ↦ v')) Δ = Some Δ' →
envs_entails Δ' (WP fill K (Lit LitUnit) @ s; E [{ Φ }]) →
envs_entails Δ (WP fill K (Store (Lit (LitLoc l)) e) @ s; E [{ Φ }]).
Proof.
intros. rewrite -twp_bind. eapply wand_apply; first by eapply twp_store.
rewrite envs_simple_replace_sound //; simpl.
rewrite right_id. by apply sep_mono_r, wand_mono.
Qed.
Lemma tac_wp_cas_fail Δ Δ' s E i K l q v e1 v1 e2 Φ :
IntoVal e1 v1 → AsVal e2 →
......@@ -150,6 +224,15 @@ Proof.
rewrite into_laterN_env_sound -later_sep envs_lookup_split //; simpl.
by apply later_mono, sep_mono_r, wand_mono.
Qed.
Lemma tac_twp_cas_fail Δ s E i K l q v e1 v1 e2 Φ :
IntoVal e1 v1 → AsVal e2 →
envs_lookup i Δ = Some (false, l ↦{q} v)%I → v ≠ v1 →
envs_entails Δ (WP fill K (Lit (LitBool false)) @ s; E [{ Φ }]) →
envs_entails Δ (WP fill K (CAS (Lit (LitLoc l)) e1 e2) @ s; E [{ Φ }]).
Proof.
intros. rewrite -twp_bind. eapply wand_apply; first exact: twp_cas_fail.
rewrite envs_lookup_split //; simpl. by apply sep_mono_r, wand_mono.
Qed.
Lemma tac_wp_cas_suc Δ Δ' Δ'' s E i K l v e1 v1 e2 v2 Φ :
IntoVal e1 v1 → IntoVal e2 v2 →
......@@ -164,6 +247,17 @@ Proof.
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_twp_cas_suc Δ Δ' s E i K l v e1 v1 e2 v2 Φ :
IntoVal e1 v1 → IntoVal e2 v2 →
envs_lookup i Δ = Some (false, l ↦ v)%I → v = v1 →
envs_simple_replace i false (Esnoc Enil i (l ↦ v2)) Δ = Some Δ' →
envs_entails Δ' (WP fill K (Lit (LitBool true)) @ s; E [{ Φ }]) →
envs_entails Δ (WP fill K (CAS (Lit (LitLoc l)) e1 e2) @ s; E [{ Φ }]).
Proof.
intros; subst. rewrite -twp_bind. eapply wand_apply; first exact: twp_cas_suc.
rewrite envs_simple_replace_sound //; simpl.
rewrite right_id. by apply sep_mono_r, wand_mono.
Qed.
Lemma tac_wp_faa Δ Δ' Δ'' s E i K l i1 e2 i2 Φ :
IntoVal e2 (LitV (LitInt i2)) →
......@@ -178,6 +272,18 @@ Proof.
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_twp_faa Δ Δ' s E i K l i1 e2 i2 Φ :
IntoVal e2 (LitV (LitInt i2)) →
envs_lookup i Δ = Some (false, l ↦ LitV (LitInt i1))%I →
envs_simple_replace i false (Esnoc Enil i (l ↦ LitV (LitInt (i1 + i2)))) Δ = Some Δ' →
envs_entails Δ' (WP fill K (Lit (LitInt i1)) @ s; E [{ Φ }]) →
envs_entails Δ (WP fill K (FAA (Lit (LitLoc l)) e2) @ s; E [{ Φ }]).
Proof.
rewrite /envs_entails=> ????; subst.
rewrite -twp_bind. eapply wand_apply; first exact: (twp_faa _ _ _ i1 _ i2).
rewrite envs_simple_replace_sound //; simpl.
rewrite right_id. by apply sep_mono_r, wand_mono.
Qed.
End heap.
Tactic Notation "wp_apply" open_constr(lem) :=
......@@ -189,10 +295,21 @@ Tactic Notation "wp_apply" open_constr(lem) :=
lazymatch iTypeOf H with
| Some (_,?P) => fail "wp_apply: cannot apply" P
end
| |- envs_entails _ (twp ?s ?E ?e ?Q) =>
reshape_expr e ltac:(fun K e' =>
twp_bind_core K; iApplyHyp H; wp_expr_simpl) ||
lazymatch iTypeOf H with
| Some (_,?P) => fail "wp_apply: cannot apply" P
end
| _ => fail "wp_apply: not a 'wp'"
end).
Tactic Notation "wp_alloc" ident(l) "as" constr(H) :=
let finish _ :=
first [intros l | fail 1 "wp_alloc:" l "not fresh"];
eexists; split;
[env_cbv; reflexivity || fail "wp_alloc:" H "not fresh"
|wp_expr_simpl; try wp_value_head] in
iStartProof;
lazymatch goal with
| |- envs_entails _ (wp ?s ?E ?e ?Q) =>
......@@ -201,10 +318,13 @@ Tactic Notation "wp_alloc" ident(l) "as" constr(H) :=
eapply (tac_wp_alloc _ _ _ _ H K); [apply _|..])
|fail 1 "wp_alloc: cannot find 'Alloc' in" e];
[apply _
|first [intros l | fail 1 "wp_alloc:" l "not fresh"];
eexists; split;
[env_cbv; reflexivity || fail "wp_alloc:" H "not fresh"
|wp_expr_simpl; try wp_value_head]]
|finish ()]
| |- envs_entails _ (twp ?s ?E ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' =>
eapply (tac_twp_alloc _ _ _ H K); [apply _|..])
|fail 1 "wp_alloc: cannot find 'Alloc' in" e];
finish ()
| _ => fail "wp_alloc: not a 'wp'"
end.
......@@ -212,6 +332,9 @@ Tactic Notation "wp_alloc" ident(l) :=
let H := iFresh in wp_alloc l as H.
Tactic Notation "wp_load" :=
let solve_mapsto _ :=
let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wp_load: cannot find" l "↦ ?" in
iStartProof;
lazymatch goal with
| |- envs_entails _ (wp ?s ?E ?e ?Q) =>
......@@ -219,13 +342,23 @@ Tactic Notation "wp_load" :=
[reshape_expr e ltac:(fun K e' => eapply (tac_wp_load _ _ _ _ _ K))
|fail 1 "wp_load: cannot find 'Load' in" e];
[apply _
|let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wp_load: cannot find" l "↦ ?"
|solve_mapsto ()
|wp_expr_simpl; try wp_value_head]
| |- envs_entails _ (twp ?s ?E ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' => eapply (tac_twp_load _ _ _ _ K))
|fail 1 "wp_load: cannot find 'Load' in" e];
[solve_mapsto ()
|wp_expr_simpl; try wp_value_head]
| _ => fail "wp_load: not a 'wp'"
end.
Tactic Notation "wp_store" :=
let solve_mapsto _ :=
let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wp_store: cannot find" l "↦ ?" in
let finish _ :=
wp_expr_simpl; try first [wp_pure (Seq (Lit LitUnit) _)|wp_value_head] in
iStartProof;
lazymatch goal with
| |- envs_entails _ (wp ?s ?E ?e ?Q) =>
......@@ -234,14 +367,24 @@ Tactic Notation "wp_store" :=
eapply (tac_wp_store _ _ _ _ _ _ K); [apply _|..])
|fail 1 "wp_store: cannot find 'Store' in" e];
[apply _
|let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wp_store: cannot find" l "↦ ?"
|solve_mapsto ()
|env_cbv; reflexivity
|wp_expr_simpl; try first [wp_pure (Seq (Lit LitUnit) _)|wp_value_head]]
|finish ()]
| |- envs_entails _ (twp ?s ?E ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' =>
eapply (tac_twp_store _ _ _ _ _ K); [apply _|..])
|fail 1 "wp_store: cannot find 'Store' in" e];
[solve_mapsto ()
|env_cbv; reflexivity
|finish ()]
| _ => fail "wp_store: not a 'wp'"
end.
Tactic Notation "wp_cas_fail" :=
let solve_mapsto _ :=
let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wp_cas_fail: cannot find" l "↦ ?" in
iStartProof;
lazymatch goal with
| |- envs_entails _ (wp ?s ?E ?e ?Q) =>
......@@ -250,14 +393,24 @@ Tactic Notation "wp_cas_fail" :=
eapply (tac_wp_cas_fail _ _ _ _ _ K); [apply _|apply _|..])
|fail 1 "wp_cas_fail: cannot find 'CAS' in" e];
[apply _
|let l := match goal with |- _ = Some (_, (?l ↦{_} _)%I) => l end in
iAssumptionCore || fail "wp_cas_fail: cannot find" l "↦ ?"
|solve_mapsto ()
|try congruence
|simpl; try wp_value_head]
| |- envs_entails _ (twp ?s ?E ?e ?Q) =>
first
[reshape_expr e ltac:(fun K e' =>
eapply (tac_twp_cas_fail _ _ _ _ K); [apply _|apply _|..])