Add monotone resource algebra to iris_staging

_CoqProject

@@ -172,6 +172,7 @@ iris_heap_lang/lib/array.v
 iris_staging/algebra/list.v
 iris_staging/base_logic/algebra.v
 iris_staging/heap_lang/interpreter.v
+iris_staging/algebra/monotone.v
 
 iris_deprecated/base_logic/auth.v
 iris_deprecated/base_logic/sts.v

iris_staging/algebra/monotone.v

+From iris.algebra Require Export cmra.
+From iris.algebra Require Import updates local_updates.
+From iris.prelude Require Import options.
+
+Local Arguments pcore _ _ !_ /.
+Local Arguments cmra_pcore _ !_ /.
+Local Arguments validN _ _ _ !_ /.
+Local Arguments valid _ _  !_ /.
+Local Arguments cmra_validN _ _ !_ /.
+Local Arguments cmra_valid _  !_ /.
+
+(* Given a preorder relation R on a type A we construct a resource algebra mra R
+   and an injection principal : A -> mra R such that:
+   R x y iff principal x ≼ principal y
+   where ≼ is the extension order of mra R resource algebra. This is exactly
+   what the lemma [principal_included] shows.
+
+   This resource algebra is useful for reasoning about monotonicity.
+   See the following paper for more details:
+
+   Reasoning About Monotonicity in Separation Logic
+   Amin Timany and Lars Birkedal
+   in Certified Programs and Proofs (CPP) 2021
+*)
+
+Definition mra {A : Type} (R : relation A) : Type := list A.
+Definition principal {A : Type} (R : relation A) (a : A) : mra R := [a].
+
+(* OFE *)
+Section ofe.
+  Context {A : Type} {R : relation A}.
+  Implicit Types a b : A.
+  Implicit Types x y : mra R.
+
+  Local Definition below (a : A) (x : mra R) := ∃ b, b ∈ x ∧ R a b.
+
+  Local Lemma below_app a x y : below a (x ++ y) ↔ below a x ∨ below a y.
+  Proof.
+    split.
+    - intros (b & [|]%elem_of_app & ?); [left|right]; exists b; eauto.
+    - intros [(b & ? & ?)|(b & ? & ?)]; exists b; rewrite elem_of_app; eauto.
+  Qed.
+
+  Local Lemma below_principal a b : below a (principal R b) ↔ R a b.
+  Proof.
+    split.
+    - intros (c & ->%elem_of_list_singleton & ?); done.
+    - intros Hab; exists b; split; first apply elem_of_list_singleton; done.
+  Qed.
+
+  Local Instance mra_equiv : Equiv (mra R) :=
+    λ x y, ∀ a, below a x ↔ below a y.
+
+  Local Instance mra_equiv_equiv : Equivalence mra_equiv.
+  Proof.
+    split; [by firstorder|by firstorder|].
+    intros ??? Heq1 Heq2 ?; split; intros ?;
+      [apply Heq2; apply Heq1|apply Heq1; apply Heq2]; done.
+  Qed.
+
+  Canonical Structure mraO := discreteO (mra R).
+End ofe.
+
+Global Arguments mraO [_] _.
+
+(* CMRA *)
+Section cmra.
+  Context {A : Type} {R : relation A}.
+  Implicit Types a b : A.
+  Implicit Types x y : mra R.
+
+  Local Instance mra_valid : Valid (mra R) := λ x, True.
+  Local Instance mra_validN : ValidN (mra R) := λ n x, True.
+  Local Program Instance mra_op : Op (mra R) := λ x y, x ++ y.
+  Local Instance mra_pcore : PCore (mra R) := Some.
+
+  Lemma mra_cmra_mixin : CmraMixin (mra R).
+  Proof.
+    apply discrete_cmra_mixin; first apply _.
+    apply ra_total_mixin.
+    - eauto.
+    - intros ??? Heq a; specialize (Heq a); rewrite !below_app; firstorder.
+    - intros ?; done.
+    - done.
+    - intros ????; rewrite !below_app; firstorder.
+    - intros ???; rewrite !below_app; firstorder.
+    - rewrite /core /pcore /=; intros ??; rewrite below_app; firstorder.
+    - done.
+    - intros ? ? [? ?]; eexists _; done.
+    - done.
+  Qed.
+
+  Canonical Structure mraR : cmra := Cmra (mra R) mra_cmra_mixin.
+
+  Global Instance mra_cmra_total : CmraTotal mraR.
+  Proof. rewrite /CmraTotal; eauto. Qed.
+  Global Instance mra_core_id (x : mra R) : CoreId x.
+  Proof. by constructor. Qed.
+
+  Global Instance mra_cmra_discrete : CmraDiscrete mraR.
+  Proof. split; last done. intros ? ?; done. Qed.
+
+  Local Instance mra_unit : Unit (mra R) := @nil A.
+  Lemma auth_ucmra_mixin : UcmraMixin (mra R).
+  Proof. split; done. Qed.
+
+  Canonical Structure mraUR := Ucmra (mra R) auth_ucmra_mixin.
+
+  Lemma mra_idemp (x : mra R) : x ⋅ x ≡ x.
+  Proof. intros a; rewrite below_app; naive_solver. Qed.
+
+  Lemma mra_included (x y : mra R) : x ≼ y ↔ y ≡ x ⋅ y.
+  Proof.
+    split; [|by intros ?; exists y].
+    intros [z ->]; rewrite assoc mra_idemp; done.
+  Qed.
+
+  Lemma principal_R_opN_base `{!Transitive R} n x y :
+    (∀ b, b ∈ y → ∃ c, c ∈ x ∧ R b c) → y ⋅ x ≡{n}≡ x.
+  Proof.
+    intros HR; split; rewrite /op /mra_op below_app; [|by firstorder].
+    intros [(c & (d & Hd1 & Hd2)%HR & Hc2)|]; [|done].
+    exists d; split; [|transitivity c]; done.
+  Qed.
+
+  Lemma principal_R_opN `{!Transitive R} n a b :
+    R a b → principal R a ⋅ principal R b ≡{n}≡ principal R b.
+  Proof.
+    intros; apply principal_R_opN_base; intros c; rewrite /principal.
+    setoid_rewrite elem_of_list_singleton => ->; eauto.
+  Qed.
+
+  Lemma principal_R_op `{!Transitive R} a b :
+    R a b → principal R a ⋅ principal R b ≡ principal R b.
+  Proof. by intros ? ?; apply (principal_R_opN 0). Qed.
+
+  Lemma principal_op_RN n a b x :
+    R a a → principal R a ⋅ x ≡{n}≡ principal R b → R a b.
+  Proof.
+    intros Ha HR.
+    destruct (HR a) as [[z [HR1%elem_of_list_singleton HR2]] _];
+      last by subst; eauto.
+    rewrite /op /mra_op /principal below_app below_principal; auto.
+  Qed.
+
+  Lemma principal_op_R' a b x :
+    R a a → principal R a ⋅ x ≡ principal R b → R a b.
+  Proof. intros ? ?; eapply (principal_op_RN 0); eauto. Qed.
+
+  Lemma principal_op_R `{!Reflexive R} a b x :
+    principal R a ⋅ x ≡ principal R b → R a b.
+  Proof. intros; eapply principal_op_R'; eauto. Qed.
+
+  Lemma principal_includedN `{!PreOrder R} n a b :
+    principal R a ≼{n} principal R b ↔ R a b.
+  Proof.
+    split.
+    - intros [z Hz]; eapply principal_op_RN; first done. by rewrite Hz.
+    - intros ?; exists (principal R b); rewrite principal_R_opN; eauto.
+  Qed.
+
+  Lemma principal_included `{!PreOrder R} a b :
+    principal R a ≼ principal R b ↔ R a b.
+  Proof. apply (principal_includedN 0). Qed.
+
+  Lemma mra_local_update_grow `{!Transitive R} a x b:
+    R a b →
+    (principal R a, x) ~l~> (principal R b, principal R b).
+  Proof.
+    intros Hana Hanb.
+    apply local_update_unital_discrete.
+    intros z _ Habz.
+    split; first done.
+    intros w; split.
+    - intros (y & ->%elem_of_list_singleton & Hy2).
+      exists b; split; first constructor; done.
+    - intros (y & [->|Hy1]%elem_of_cons & Hy2).
+      + exists b; split; first constructor; done.
+      + exists b; split; first constructor.
+        specialize (Habz w) as [_ [c [->%elem_of_list_singleton Hc2]]].
+        { exists y; split; first (by apply elem_of_app; right); eauto. }
+        etrans; eauto.
+  Qed.
+
+  Lemma mra_local_update_get_frag `{!PreOrder R} a b:
+    R b a →
+    (principal R a, ε) ~l~> (principal R a, principal R b).
+  Proof.
+    intros Hana.
+    apply local_update_unital_discrete.
+    intros z _; rewrite left_id; intros <-.
+    split; first done.
+    apply mra_included; by apply principal_included.
+  Qed.
+
+End cmra.
+
+Global Arguments mraR {_} _.
+Global Arguments mraUR {_} _.
+
+
+(* Might be useful if the type of elements is an OFE. *)
+Section mra_over_ofe.
+  Context {A : ofe} {R : relation A}.
+  Implicit Types a b : A.
+  Implicit Types x y : mra R.
+
+  Global Instance principal_ne
+         `{!∀ n, Proper ((dist n) ==> (dist n) ==> iff) R} :
+    NonExpansive (principal R).
+  Proof. intros n a1 a2 Ha; split; rewrite !below_principal !Ha; done. Qed.
+
+  Global Instance principal_proper
+         `{!∀ n, Proper ((dist n) ==> (dist n) ==> iff) R} :
+    Proper ((≡) ==> (≡)) (principal R) := ne_proper _.
+
+  Lemma principal_inj_related a b :
+    principal R a ≡ principal R b → R a a → R a b.
+  Proof.
+    intros Hab ?.
+    destruct (Hab a) as [[? [?%elem_of_list_singleton ?]] _];
+      last by subst; auto.
+    exists a; rewrite /principal elem_of_list_singleton; done.
+  Qed.
+
+  Lemma principal_inj_general a b :
+    principal R a ≡ principal R b →
+    R a a → R b b → (R a b → R b a → a ≡ b) → a ≡ b.
+  Proof. intros ??? Has; apply Has; apply principal_inj_related; auto. Qed.
+
+  Global Instance principal_inj_instance `{!Reflexive R} `{!AntiSymm (≡) R} :
+    Inj (≡) (≡) (principal R).
+  Proof. intros ???; apply principal_inj_general; auto. Qed.
+
+  Global Instance principal_injN `{!Reflexive R} {Has : AntiSymm (≡) R} n :
+    Inj (dist n) (dist n) (principal R).
+  Proof.
+    intros x y Hxy%discrete_iff; last apply _.
+    eapply equiv_dist; revert Hxy; apply inj; apply _.
+  Qed.
+
+End mra_over_ofe.

iris_staging/base_logic/algebra.v

 (* This is just an integration file for [iris_staging.algebra.list];
 both should be stabilized together. *)
 From iris.algebra Require Import cmra.
-From iris.staging.algebra Require Import list.
+From iris.staging.algebra Require Import list monotone.
 From iris.base_logic Require Import bi derived.
 From iris.prelude Require Import options.
 
@@ -19,4 +19,16 @@ Section list_cmra.
   Lemma list_validI l : ✓ l ⊣⊢ ∀ i, ✓ (l !! i).
   Proof. uPred.unseal; constructor=> n x ?. apply list_lookup_validN. Qed.
 End list_cmra.
+
+Section monotone.
+  Lemma monotone_equivI {A : ofe} (R : relation A)
+        `{!(∀ n : nat, Proper (dist n ==> dist n ==> iff) R)}
+        `{!Reflexive R} `{!AntiSymm (≡) R} a b :
+    principal R a ≡ principal R b ⊣⊢ (a ≡ b).
+  Proof.
+    uPred.unseal. do 2 split; intros ?.
+    - exact: principal_injN.
+    - exact: principal_ne.
+  Qed.
+End monotone.
 End upred.