bedrock.lang.cpp.logic.cptr
(*
* Copyright (c) 2021 BedRock Systems, Inc.
* This software is distributed under the terms of the BedRock Open-Source License.
* See the LICENSE-BedRock file in the repository root for details.
*)
Require Import bedrock.lang.proofmode.proofmode.
Require Import bedrock.lang.cpp.semantics.
Require Import bedrock.lang.cpp.logic.pred.
Require Import bedrock.lang.cpp.logic.wp.
Section defs.
Context `{Σ : cpp_logic}.
* Copyright (c) 2021 BedRock Systems, Inc.
* This software is distributed under the terms of the BedRock Open-Source License.
* See the LICENSE-BedRock file in the repository root for details.
*)
Require Import bedrock.lang.proofmode.proofmode.
Require Import bedrock.lang.cpp.semantics.
Require Import bedrock.lang.cpp.logic.pred.
Require Import bedrock.lang.cpp.logic.wp.
Section defs.
Context `{Σ : cpp_logic}.
Function specifications written in weakest pre-condition style.
In other words, fs_spec asserts that arguments satisfy the function's
precondition and that the continuation coincides with the function's
postcondition.
Record function_spec : Type :=
{ fs_cc : calling_conv
; fs_arity : function_arity
; fs_return : type
; fs_arguments : list type
; fs_spec : list ptr -d> (ptr -> mpred) -d> mpredO
}.
#[global] Instance function_spec_inhabited : Inhabited function_spec :=
populate (Build_function_spec inhabitant inhabitant inhabitant inhabitant inhabitant).
Definition type_of_spec (fs : function_spec) : type :=
normalize_type (Tfunction {| ft_cc:=fs.(fs_cc) ; ft_arity:=fs.(fs_arity) ; ft_return := fs.(fs_return) ; ft_params := fs.(fs_arguments) |}).
Lemma cc_type_of_spec fs1 fs2 :
type_of_spec fs1 = type_of_spec fs2 →
fs1.(fs_cc) = fs2.(fs_cc).
Proof.
destruct fs1, fs2. intros [=]. by simplify_eq/=.
Qed.
Lemma length_type_of_spec fs1 fs2 :
type_of_spec fs1 = type_of_spec fs2 →
length (fs_arguments fs1) = length (fs_arguments fs2).
Proof.
destruct fs1, fs2; rewrite /type_of_spec/=; intros [= _ _ _ Hmap].
erewrite <-length_map, Hmap.
by rewrite length_map.
Qed.
Section ofe.
Implicit Types P Q : function_spec.
Instance function_spec_equiv : Equiv function_spec := fun P Q =>
type_of_spec P = type_of_spec Q /\
P.(fs_spec) ≡ Q.(fs_spec).
Instance function_spec_dist : Dist function_spec := fun (n : nat) P Q =>
type_of_spec P = type_of_spec Q /\
P.(fs_spec) ≡{n}≡ Q.(fs_spec).
Lemma function_spec_ofe_mixin : OfeMixin function_spec.
Proof.
split.
- intros P Q. split.
+ intros [] n. split. done. by apply equiv_dist.
+ intros HPQ. split; first by destruct (HPQ 0).
apply equiv_dist=>n. by destruct (HPQ n).
- intros n. split.
+ done.
+ by intros P Q [].
+ intros P Q R [Ht1 HPQ] [Ht2 HQR]. split.
* by rewrite Ht1 Ht2.
* etrans. by apply HPQ. done.
- intros n m P Q [] ?. split; first done. by eapply dist_lt.
Qed.
Canonical Structure function_specO := Ofe function_spec function_spec_ofe_mixin.
#[global] Instance type_of_spec_ne (n : nat) :
Proper (dist n ==> eq) type_of_spec.
Proof. by intros P Q [? _]. Qed.
#[global] Instance fs_spec_ne : NonExpansive fs_spec.
Proof. by intros n P Q [_ ?]. Qed.
#[global] Instance fs_cc_ne (n : nat) : Proper (dist n ==> eq) fs_cc.
Proof. move=>P Q /type_of_spec_ne. exact: cc_type_of_spec. Qed.
#[global] Instance length_fs_arguments_ne (n : nat) :
Proper (dist n ==> eq) (fun P => length (fs_arguments P)).
Proof. move=>P Q /type_of_spec_ne. exact: length_type_of_spec. Qed.
End ofe.
#[global,program] Instance function_spec_cofe : Cofe function_specO := {|
compl c := {|
fs_cc := (c 0).(fs_cc);
fs_arity := (c 0).(fs_arity);
fs_return := (c 0).(fs_return);
fs_arguments := (c 0).(fs_arguments);
fs_spec := compl (chain_map fs_spec c);
|}
|}.
Next Obligation.
intros n c. split; simpl.
- destruct (chain_cauchy c 0 n) as [-> _]. lia. done.
- intros. apply conv_compl.
Qed.
Lemma fs_equivI_type P Q :
P ≡ Q ⊢@{mpredI} [| type_of_spec P = type_of_spec Q |].
Proof.
rewrite only_provable_equiv.
constructor => ?.
rewrite monPred_at_internal_eq monPred_at_and monPred_at_emp monPred_at_pure.
repeat uPred.unseal. constructor => n x ?. by intros [? _].
Qed.
Lemma fs_equivI_spec P Q vs K :
P ≡ Q ⊢@{mpredI} P.(fs_spec) vs K ≡ Q.(fs_spec) vs K.
Proof.
constructor => ?. rewrite !monPred_at_internal_eq.
repeat uPred.unseal. constructor=>n x ? [_ HPQ]. apply HPQ.
Qed.
Lemma fs_equivI_intro P Q :
type_of_spec P = type_of_spec Q ->
Forall vs K, P.(fs_spec) vs K ≡ Q.(fs_spec) vs K ⊢@{mpredI} P ≡ Q.
Proof.
intros.
constructor => ?.
repeat setoid_rewrite monPred_at_forall.
repeat setoid_rewrite monPred_at_internal_eq.
repeat uPred.unseal. constructor=>x ?; by split.
Qed.
Lemma fs_equivI P Q :
P ≡ Q ⊣⊢@{mpredI}
[| type_of_spec P = type_of_spec Q |] **
Forall vs K, P.(fs_spec) vs K ≡ Q.(fs_spec) vs K.
Proof.
split'.
- iIntros "?". iSplit; first by rewrite fs_equivI_type.
iIntros (vs K). by rewrite fs_equivI_spec.
- iIntros "[% ?]". by rewrite fs_equivI_intro.
Qed.
{ fs_cc : calling_conv
; fs_arity : function_arity
; fs_return : type
; fs_arguments : list type
; fs_spec : list ptr -d> (ptr -> mpred) -d> mpredO
}.
#[global] Instance function_spec_inhabited : Inhabited function_spec :=
populate (Build_function_spec inhabitant inhabitant inhabitant inhabitant inhabitant).
Definition type_of_spec (fs : function_spec) : type :=
normalize_type (Tfunction {| ft_cc:=fs.(fs_cc) ; ft_arity:=fs.(fs_arity) ; ft_return := fs.(fs_return) ; ft_params := fs.(fs_arguments) |}).
Lemma cc_type_of_spec fs1 fs2 :
type_of_spec fs1 = type_of_spec fs2 →
fs1.(fs_cc) = fs2.(fs_cc).
Proof.
destruct fs1, fs2. intros [=]. by simplify_eq/=.
Qed.
Lemma length_type_of_spec fs1 fs2 :
type_of_spec fs1 = type_of_spec fs2 →
length (fs_arguments fs1) = length (fs_arguments fs2).
Proof.
destruct fs1, fs2; rewrite /type_of_spec/=; intros [= _ _ _ Hmap].
erewrite <-length_map, Hmap.
by rewrite length_map.
Qed.
Section ofe.
Implicit Types P Q : function_spec.
Instance function_spec_equiv : Equiv function_spec := fun P Q =>
type_of_spec P = type_of_spec Q /\
P.(fs_spec) ≡ Q.(fs_spec).
Instance function_spec_dist : Dist function_spec := fun (n : nat) P Q =>
type_of_spec P = type_of_spec Q /\
P.(fs_spec) ≡{n}≡ Q.(fs_spec).
Lemma function_spec_ofe_mixin : OfeMixin function_spec.
Proof.
split.
- intros P Q. split.
+ intros [] n. split. done. by apply equiv_dist.
+ intros HPQ. split; first by destruct (HPQ 0).
apply equiv_dist=>n. by destruct (HPQ n).
- intros n. split.
+ done.
+ by intros P Q [].
+ intros P Q R [Ht1 HPQ] [Ht2 HQR]. split.
* by rewrite Ht1 Ht2.
* etrans. by apply HPQ. done.
- intros n m P Q [] ?. split; first done. by eapply dist_lt.
Qed.
Canonical Structure function_specO := Ofe function_spec function_spec_ofe_mixin.
#[global] Instance type_of_spec_ne (n : nat) :
Proper (dist n ==> eq) type_of_spec.
Proof. by intros P Q [? _]. Qed.
#[global] Instance fs_spec_ne : NonExpansive fs_spec.
Proof. by intros n P Q [_ ?]. Qed.
#[global] Instance fs_cc_ne (n : nat) : Proper (dist n ==> eq) fs_cc.
Proof. move=>P Q /type_of_spec_ne. exact: cc_type_of_spec. Qed.
#[global] Instance length_fs_arguments_ne (n : nat) :
Proper (dist n ==> eq) (fun P => length (fs_arguments P)).
Proof. move=>P Q /type_of_spec_ne. exact: length_type_of_spec. Qed.
End ofe.
#[global,program] Instance function_spec_cofe : Cofe function_specO := {|
compl c := {|
fs_cc := (c 0).(fs_cc);
fs_arity := (c 0).(fs_arity);
fs_return := (c 0).(fs_return);
fs_arguments := (c 0).(fs_arguments);
fs_spec := compl (chain_map fs_spec c);
|}
|}.
Next Obligation.
intros n c. split; simpl.
- destruct (chain_cauchy c 0 n) as [-> _]. lia. done.
- intros. apply conv_compl.
Qed.
Lemma fs_equivI_type P Q :
P ≡ Q ⊢@{mpredI} [| type_of_spec P = type_of_spec Q |].
Proof.
rewrite only_provable_equiv.
constructor => ?.
rewrite monPred_at_internal_eq monPred_at_and monPred_at_emp monPred_at_pure.
repeat uPred.unseal. constructor => n x ?. by intros [? _].
Qed.
Lemma fs_equivI_spec P Q vs K :
P ≡ Q ⊢@{mpredI} P.(fs_spec) vs K ≡ Q.(fs_spec) vs K.
Proof.
constructor => ?. rewrite !monPred_at_internal_eq.
repeat uPred.unseal. constructor=>n x ? [_ HPQ]. apply HPQ.
Qed.
Lemma fs_equivI_intro P Q :
type_of_spec P = type_of_spec Q ->
Forall vs K, P.(fs_spec) vs K ≡ Q.(fs_spec) vs K ⊢@{mpredI} P ≡ Q.
Proof.
intros.
constructor => ?.
repeat setoid_rewrite monPred_at_forall.
repeat setoid_rewrite monPred_at_internal_eq.
repeat uPred.unseal. constructor=>x ?; by split.
Qed.
Lemma fs_equivI P Q :
P ≡ Q ⊣⊢@{mpredI}
[| type_of_spec P = type_of_spec Q |] **
Forall vs K, P.(fs_spec) vs K ≡ Q.(fs_spec) vs K.
Proof.
split'.
- iIntros "?". iSplit; first by rewrite fs_equivI_type.
iIntros (vs K). by rewrite fs_equivI_spec.
- iIntros "[% ?]". by rewrite fs_equivI_intro.
Qed.
mpred implication on function_spec.
Here, Q is a lower-level spec, and P is a derived/higher-level spec.
fs_impl and fs_entails are mostly used to derive
properties like properness and non-expansiveness for cptrR.
Note that function_spec and co (fs_impl and fs_entails)
do not package the usually expected properties for
weakest-precondition.
cptrR gets those properties from axioms for wp_fptr.
Definition fs_impl (Q P : function_spec) : mpred :=
[| type_of_spec P = type_of_spec Q |] ∗
□ ∀ vs K, P.(fs_spec) vs K -∗ Q.(fs_spec) vs K.
Lemma fs_impl_reflexive P : |-- fs_impl P P.
Proof. iSplit; auto. Qed.
Lemma fs_impl_transitive P Q R : fs_impl P Q |-- fs_impl Q R -* fs_impl P R.
Proof.
rewrite /fs_impl; iIntros "(-> & #H1) (-> & #H2)".
iSplit; first done.
iIntros "!>" (vs K) "Y".
iApply ("H1" with "(H2 Y)").
Qed.
Definition fs_entails (P Q : function_spec) : Prop := |-- fs_impl P Q.
#[global] Instance fs_entails_preorder : PreOrder fs_entails.
Proof.
split; rewrite /fs_entails.
- intro x. exact: fs_impl_reflexive.
- intros ? ? ? H1 H2. by iApply (fs_impl_transitive).
Qed.
#[global] Instance : RewriteRelation fs_entails := {}.
(* mpred bi-implication on function_spec *)
Definition fs_equiv (P Q : function_spec) : mpred :=
[| type_of_spec P = type_of_spec Q |] ∗
□ ∀ vs K, P.(fs_spec) vs K ∗-∗ Q.(fs_spec) vs K.
Lemma fs_equivI_equiv P Q : P ≡ Q |-- fs_equiv P Q.
Proof.
rewrite fs_equivI /fs_equiv. f_equiv.
rewrite -(bi.intuitionistic_intuitionistically (Forall _, _)).
repeat f_equiv. rewrite prop_ext plainly_elim_persistently.
by iIntros "#$".
Qed.
Lemma fs_equiv_equivI P Q : ■ fs_equiv P Q |-- P ≡ Q.
Proof.
iIntros "[% #EQ]". iApply fs_equivI_intro; first done.
iIntros (vs K). iApply prop_ext. iModIntro. iApply "EQ".
Qed.
Lemma fs_equiv_split P Q : fs_equiv P Q -|- fs_impl P Q ** fs_impl Q P.
Proof.
rewrite /fs_equiv /fs_impl; iSplit.
- iIntros "(-> & #W)"; repeat iSplit => //;
iIntros "!>" (??) "A"; iApply ("W" with "A").
- iIntros "((-> & #W1) & (_ & #W2))".
iSplit => //; iIntros "!>" (??); iSplit;
[by iApply "W2" | by iApply "W1"].
Qed.
Lemma function_spec_equiv_iff P Q : P ≡ Q <-> |-- fs_equiv P Q.
Proof.
enough (HEQ : P ≡ Q <-> ⊢@{mpredI} P ≡ Q).
{ rewrite {}HEQ. split.
- by rewrite fs_equivI_equiv.
- rewrite /bi_emp_valid {2}plainly_emp_2=>->. by rewrite fs_equiv_equivI. }
split.
- intros->. apply internal_eq_refl.
- rewrite monPred_internal_eq_unfold => /embed_emp_valid_inj.
apply uPred.internal_eq_soundness.
Qed.
Lemma function_spec_equiv_split P Q : P ≡ Q ↔ fs_entails P Q /\ fs_entails Q P.
Proof.
rewrite /fs_entails function_spec_equiv_iff fs_equiv_split; split.
{ by intros H; split; iDestruct H as "[??]". }
{ intros [H1 H2]. iDestruct H1 as "$". iDestruct H2 as "$". }
Qed.
(* this is the core definition that the program logic will be based on.
it is really an assertion about assembly.
*)
Definition cptrR_def {resolve : genv} (fs : function_spec) : Rep :=
as_Rep (fun p =>
strict_valid_ptr p **
□ (Forall vs Q,
fs.(fs_spec) vs Q -*
wp_fptr resolve.(genv_tu).(types) (type_of_spec fs) p vs Q)).
Definition cptrR_aux : seal (@cptrR_def). Proof. by eexists. Qed.
Definition cptrR := cptrR_aux.(unseal).
Definition cptrR_eq : @cptrR = _ := cptrR_aux.(seal_eq).
#[global] Hint Opaque cptrR : typeclass_instances.
[| type_of_spec P = type_of_spec Q |] ∗
□ ∀ vs K, P.(fs_spec) vs K -∗ Q.(fs_spec) vs K.
Lemma fs_impl_reflexive P : |-- fs_impl P P.
Proof. iSplit; auto. Qed.
Lemma fs_impl_transitive P Q R : fs_impl P Q |-- fs_impl Q R -* fs_impl P R.
Proof.
rewrite /fs_impl; iIntros "(-> & #H1) (-> & #H2)".
iSplit; first done.
iIntros "!>" (vs K) "Y".
iApply ("H1" with "(H2 Y)").
Qed.
Definition fs_entails (P Q : function_spec) : Prop := |-- fs_impl P Q.
#[global] Instance fs_entails_preorder : PreOrder fs_entails.
Proof.
split; rewrite /fs_entails.
- intro x. exact: fs_impl_reflexive.
- intros ? ? ? H1 H2. by iApply (fs_impl_transitive).
Qed.
#[global] Instance : RewriteRelation fs_entails := {}.
(* mpred bi-implication on function_spec *)
Definition fs_equiv (P Q : function_spec) : mpred :=
[| type_of_spec P = type_of_spec Q |] ∗
□ ∀ vs K, P.(fs_spec) vs K ∗-∗ Q.(fs_spec) vs K.
Lemma fs_equivI_equiv P Q : P ≡ Q |-- fs_equiv P Q.
Proof.
rewrite fs_equivI /fs_equiv. f_equiv.
rewrite -(bi.intuitionistic_intuitionistically (Forall _, _)).
repeat f_equiv. rewrite prop_ext plainly_elim_persistently.
by iIntros "#$".
Qed.
Lemma fs_equiv_equivI P Q : ■ fs_equiv P Q |-- P ≡ Q.
Proof.
iIntros "[% #EQ]". iApply fs_equivI_intro; first done.
iIntros (vs K). iApply prop_ext. iModIntro. iApply "EQ".
Qed.
Lemma fs_equiv_split P Q : fs_equiv P Q -|- fs_impl P Q ** fs_impl Q P.
Proof.
rewrite /fs_equiv /fs_impl; iSplit.
- iIntros "(-> & #W)"; repeat iSplit => //;
iIntros "!>" (??) "A"; iApply ("W" with "A").
- iIntros "((-> & #W1) & (_ & #W2))".
iSplit => //; iIntros "!>" (??); iSplit;
[by iApply "W2" | by iApply "W1"].
Qed.
Lemma function_spec_equiv_iff P Q : P ≡ Q <-> |-- fs_equiv P Q.
Proof.
enough (HEQ : P ≡ Q <-> ⊢@{mpredI} P ≡ Q).
{ rewrite {}HEQ. split.
- by rewrite fs_equivI_equiv.
- rewrite /bi_emp_valid {2}plainly_emp_2=>->. by rewrite fs_equiv_equivI. }
split.
- intros->. apply internal_eq_refl.
- rewrite monPred_internal_eq_unfold => /embed_emp_valid_inj.
apply uPred.internal_eq_soundness.
Qed.
Lemma function_spec_equiv_split P Q : P ≡ Q ↔ fs_entails P Q /\ fs_entails Q P.
Proof.
rewrite /fs_entails function_spec_equiv_iff fs_equiv_split; split.
{ by intros H; split; iDestruct H as "[??]". }
{ intros [H1 H2]. iDestruct H1 as "$". iDestruct H2 as "$". }
Qed.
(* this is the core definition that the program logic will be based on.
it is really an assertion about assembly.
*)
Definition cptrR_def {resolve : genv} (fs : function_spec) : Rep :=
as_Rep (fun p =>
strict_valid_ptr p **
□ (Forall vs Q,
fs.(fs_spec) vs Q -*
wp_fptr resolve.(genv_tu).(types) (type_of_spec fs) p vs Q)).
Definition cptrR_aux : seal (@cptrR_def). Proof. by eexists. Qed.
Definition cptrR := cptrR_aux.(unseal).
Definition cptrR_eq : @cptrR = _ := cptrR_aux.(seal_eq).
#[global] Hint Opaque cptrR : typeclass_instances.
A version of fs_impl and fs_entails with fupd.
These are used in stating strong wp properties for cptrR,
like cptrR_fs_impl_fupd and cptrR_fs_entails_fupd (see below).
Definition fs_impl_fupd (Q P : function_spec) : mpred :=
[| type_of_spec P = type_of_spec Q |] ∗
□ ∀ vs K, P.(fs_spec) vs K -∗ |={top}=> Q.(fs_spec) vs (λ v, |={top}=> K v).
Definition fs_entails_fupd (P Q : function_spec) : Prop := |-- fs_impl_fupd P Q.
End defs.
#[global] Instance: Params (@cptrR) 4 := {}.
#[global] Arguments cptrR {_ _ Σ resolve} _ : rename.
Section with_cpp.
Context `{Σ : cpp_logic} {resolve : genv}.
#[global] Instance cptrR_persistent {s} : Persistent (cptrR s).
Proof. rewrite cptrR_eq. apply _. Qed.
#[global] Instance cptrR_affine {s} : Affine (cptrR s).
Proof. rewrite cptrR_eq. apply _. Qed.
Lemma cptrR_strict_valid_observe (p : ptr) f : Observe (strict_valid_ptr p) (_at p (cptrR f)).
Proof.
apply observe_intro_persistent; refine _.
rewrite cptrR_eq/cptrR_def.
rewrite _at_as_Rep.
iIntros "[$ _]".
Qed.
(* NOTE this should become an instance. *)
Lemma cptrR_valid_observe (p : ptr) f : Observe (valid_ptr p) (_at p (cptrR f)).
Proof. apply observe_strict_valid_valid; apply cptrR_strict_valid_observe. Qed.
#[local] Opaque type_of_spec.
Lemma cptrR_fs_impl f g :
pureR (fs_impl f g) |-- cptrR f -* cptrR g.
Proof.
rewrite cptrR_eq/cptrR_def /pureR /as_Rep.
constructor => p; rewrite Rep_wand_force; iIntros "#(%ty & fs_impl)" => /=.
iIntros "(val & #rest)"; iFrame.
rewrite ty. iModIntro. iIntros (vs Q) "fs_g".
iApply "rest".
by iApply "fs_impl".
Qed.
Lemma cptrR_fs_impl_fupd f g :
pureR (fs_impl_fupd f g) |-- cptrR f -* cptrR g.
Proof.
rewrite cptrR_eq/cptrR_def /pureR /as_Rep.
constructor => p; rewrite Rep_wand_force; iIntros "#(%ty & fs_impl)" => /=.
iIntros "(val & #rest)"; iFrame.
rewrite ty. iModIntro. iIntros (vs Q) "fs_g".
iApply wp_fptr_fupd. iApply fupd_spec.
iApply "rest".
by iApply "fs_impl".
Qed.
#[local] Transparent type_of_spec.
(* TODO: Proper wrt genv_leq. *)
#[global] Instance cptrR_ne : NonExpansive cptrR.
Proof.
intros n P Q HPQ. rewrite cptrR_eq/cptrR_def.
apply as_Rep_ne=>p. (do 2!f_equiv). do 5 f_equiv. by apply fs_spec_ne.
f_equiv. apply HPQ.
Qed.
#[global] Instance cptrR_proper : Proper (equiv ==> equiv) cptrR.
Proof. exact: ne_proper. Qed.
#[global] Instance cptrR_mono : Proper (fs_entails ==> (⊢)) cptrR.
Proof.
intros ??; rewrite /fs_entails/flip => impl. iApply cptrR_fs_impl.
by rewrite -impl pureR_emp.
Qed.
Lemma _at_cptrR_mono (p : ptr) (spec spec' : function_spec) :
fs_entails spec spec' ->
p |-> cptrR spec |-- p |-> cptrR spec'.
Proof. intros; rewrite cptrR_mono; eauto. Qed.
#[global] Instance cptrR_flip_mono : Proper (flip fs_entails ==> flip (⊢)) cptrR.
Proof. by intros ?? <-. Qed.
Lemma cptrR_mono_fupd : Proper (fs_entails_fupd ==> (⊢)) cptrR.
Proof.
intros ??; rewrite /fs_entails_fupd/flip => impl. iApply cptrR_fs_impl_fupd.
by rewrite -impl pureR_emp.
Qed.
Lemma cptrR_flip_mono_fupd : Proper (flip fs_entails_fupd ==> flip (⊢)) cptrR.
Proof. repeat intro. by rewrite -cptrR_mono_fupd. Qed.
End with_cpp.
#[global] Instance Persistent_spec `{cpp_logic} {resolve:genv} p s :
Persistent (_at p (cptrR s)) := _.
#[global] Instance Affine_spec `{cpp_logic} {resolve:genv} p s :
Affine (_at p (cptrR s)) := _.
[| type_of_spec P = type_of_spec Q |] ∗
□ ∀ vs K, P.(fs_spec) vs K -∗ |={top}=> Q.(fs_spec) vs (λ v, |={top}=> K v).
Definition fs_entails_fupd (P Q : function_spec) : Prop := |-- fs_impl_fupd P Q.
End defs.
#[global] Instance: Params (@cptrR) 4 := {}.
#[global] Arguments cptrR {_ _ Σ resolve} _ : rename.
Section with_cpp.
Context `{Σ : cpp_logic} {resolve : genv}.
#[global] Instance cptrR_persistent {s} : Persistent (cptrR s).
Proof. rewrite cptrR_eq. apply _. Qed.
#[global] Instance cptrR_affine {s} : Affine (cptrR s).
Proof. rewrite cptrR_eq. apply _. Qed.
Lemma cptrR_strict_valid_observe (p : ptr) f : Observe (strict_valid_ptr p) (_at p (cptrR f)).
Proof.
apply observe_intro_persistent; refine _.
rewrite cptrR_eq/cptrR_def.
rewrite _at_as_Rep.
iIntros "[$ _]".
Qed.
(* NOTE this should become an instance. *)
Lemma cptrR_valid_observe (p : ptr) f : Observe (valid_ptr p) (_at p (cptrR f)).
Proof. apply observe_strict_valid_valid; apply cptrR_strict_valid_observe. Qed.
#[local] Opaque type_of_spec.
Lemma cptrR_fs_impl f g :
pureR (fs_impl f g) |-- cptrR f -* cptrR g.
Proof.
rewrite cptrR_eq/cptrR_def /pureR /as_Rep.
constructor => p; rewrite Rep_wand_force; iIntros "#(%ty & fs_impl)" => /=.
iIntros "(val & #rest)"; iFrame.
rewrite ty. iModIntro. iIntros (vs Q) "fs_g".
iApply "rest".
by iApply "fs_impl".
Qed.
Lemma cptrR_fs_impl_fupd f g :
pureR (fs_impl_fupd f g) |-- cptrR f -* cptrR g.
Proof.
rewrite cptrR_eq/cptrR_def /pureR /as_Rep.
constructor => p; rewrite Rep_wand_force; iIntros "#(%ty & fs_impl)" => /=.
iIntros "(val & #rest)"; iFrame.
rewrite ty. iModIntro. iIntros (vs Q) "fs_g".
iApply wp_fptr_fupd. iApply fupd_spec.
iApply "rest".
by iApply "fs_impl".
Qed.
#[local] Transparent type_of_spec.
(* TODO: Proper wrt genv_leq. *)
#[global] Instance cptrR_ne : NonExpansive cptrR.
Proof.
intros n P Q HPQ. rewrite cptrR_eq/cptrR_def.
apply as_Rep_ne=>p. (do 2!f_equiv). do 5 f_equiv. by apply fs_spec_ne.
f_equiv. apply HPQ.
Qed.
#[global] Instance cptrR_proper : Proper (equiv ==> equiv) cptrR.
Proof. exact: ne_proper. Qed.
#[global] Instance cptrR_mono : Proper (fs_entails ==> (⊢)) cptrR.
Proof.
intros ??; rewrite /fs_entails/flip => impl. iApply cptrR_fs_impl.
by rewrite -impl pureR_emp.
Qed.
Lemma _at_cptrR_mono (p : ptr) (spec spec' : function_spec) :
fs_entails spec spec' ->
p |-> cptrR spec |-- p |-> cptrR spec'.
Proof. intros; rewrite cptrR_mono; eauto. Qed.
#[global] Instance cptrR_flip_mono : Proper (flip fs_entails ==> flip (⊢)) cptrR.
Proof. by intros ?? <-. Qed.
Lemma cptrR_mono_fupd : Proper (fs_entails_fupd ==> (⊢)) cptrR.
Proof.
intros ??; rewrite /fs_entails_fupd/flip => impl. iApply cptrR_fs_impl_fupd.
by rewrite -impl pureR_emp.
Qed.
Lemma cptrR_flip_mono_fupd : Proper (flip fs_entails_fupd ==> flip (⊢)) cptrR.
Proof. repeat intro. by rewrite -cptrR_mono_fupd. Qed.
End with_cpp.
#[global] Instance Persistent_spec `{cpp_logic} {resolve:genv} p s :
Persistent (_at p (cptrR s)) := _.
#[global] Instance Affine_spec `{cpp_logic} {resolve:genv} p s :
Affine (_at p (cptrR s)) := _.