Library iris.bi.embedding

From iris.algebra Require Import monoid.
From iris.bi Require Import interface derived_laws_later big_op.
From iris.bi Require Import plainly updates internal_eq.
From iris.prelude Require Import options.

Set Default Proof Using "Type*".

Class Embed (A B : Type) := embed : A B.
Global Arguments embed {_ _ _} _%I : simpl never.
Notation "⎡ P ⎤" := (embed P) : bi_scope.
Global Instance: Params (@embed) 3 := {}.
Typeclasses Opaque embed.

Global Hint Mode Embed ! - : typeclass_instances.
Global Hint Mode Embed - ! : typeclass_instances.

Record BiEmbedMixin (PROP1 PROP2 : bi) `(Embed PROP1 PROP2) := {
  bi_embed_mixin_ne : NonExpansive (embed (A:=PROP1) (B:=PROP2));
  bi_embed_mixin_mono : Proper ((⊢) ==> (⊢)) (embed (A:=PROP1) (B:=PROP2));
  bi_embed_mixin_emp_valid_inj (P : PROP1) :
    (⊢@{PROP2} P) P;
  
The following axiom expresses that the embedding is injective in the OFE sense. Instead of this axiom being expressed in terms of siProp or externally (i.e., as Inj (dist n) (dist n) embed), it is expressed using the internal equality of any other BI PROP'. This is more general, as we do not have any machinary to embed siProp into a BI with internal equality.
  bi_embed_mixin_interal_inj `{BiInternalEq PROP'} (P Q : PROP1) :
    P Q ⊢@{PROP'} (P Q);
  bi_embed_mixin_emp_2 : emp ⊢@{PROP2} emp;
  bi_embed_mixin_impl_2 (P Q : PROP1) :
    (P Q) ⊢@{PROP2} P Q;
  bi_embed_mixin_forall_2 A (Φ : A PROP1) :
    ( x, Φ x) ⊢@{PROP2} x, Φ x;
  bi_embed_mixin_exist_1 A (Φ : A PROP1) :
     x, Φ x ⊢@{PROP2} x, Φ x;
  bi_embed_mixin_sep (P Q : PROP1) :
    P Q ⊣⊢@{PROP2} P Q;
  bi_embed_mixin_wand_2 (P Q : PROP1) :
    (P -∗ Q) ⊢@{PROP2} P -∗ Q;
  bi_embed_mixin_persistently (P : PROP1) :
    <pers> P ⊣⊢@{PROP2} <pers> P
}.

Class BiEmbed (PROP1 PROP2 : bi) := {
  bi_embed_embed :> Embed PROP1 PROP2;
  bi_embed_mixin : BiEmbedMixin PROP1 PROP2 bi_embed_embed;
}.
Global Hint Mode BiEmbed ! - : typeclass_instances.
Global Hint Mode BiEmbed - ! : typeclass_instances.
Global Arguments bi_embed_embed : simpl never.

Class BiEmbedEmp (PROP1 PROP2 : bi) `{!BiEmbed PROP1 PROP2} :=
  embed_emp_1 : emp : PROP1 emp.
Global Hint Mode BiEmbedEmp ! - - : typeclass_instances.
Global Hint Mode BiEmbedEmp - ! - : typeclass_instances.

Class BiEmbedLater (PROP1 PROP2 : bi) `{!BiEmbed PROP1 PROP2} :=
  embed_later P : P ⊣⊢@{PROP2} P.
Global Hint Mode BiEmbedLater ! - - : typeclass_instances.
Global Hint Mode BiEmbedLater - ! - : typeclass_instances.

Class BiEmbedInternalEq (PROP1 PROP2 : bi)
    `{!BiEmbed PROP1 PROP2, !BiInternalEq PROP1, !BiInternalEq PROP2} :=
  embed_internal_eq_1 (A : ofe) (x y : A) : x y ⊢@{PROP2} x y.
Global Hint Mode BiEmbedInternalEq ! - - - - : typeclass_instances.
Global Hint Mode BiEmbedInternalEq - ! - - - : typeclass_instances.

Class BiEmbedBUpd (PROP1 PROP2 : bi)
    `{!BiEmbed PROP1 PROP2, !BiBUpd PROP1, !BiBUpd PROP2} :=
  embed_bupd P : |==> P ⊣⊢@{PROP2} |==> P.
Global Hint Mode BiEmbedBUpd - ! - - - : typeclass_instances.
Global Hint Mode BiEmbedBUpd ! - - - - : typeclass_instances.

Class BiEmbedFUpd (PROP1 PROP2 : bi)
    `{!BiEmbed PROP1 PROP2, !BiFUpd PROP1, !BiFUpd PROP2} :=
  embed_fupd E1 E2 P : |={E1,E2}=> P ⊣⊢@{PROP2} |={E1,E2}=> P.
Global Hint Mode BiEmbedFUpd - ! - - - : typeclass_instances.
Global Hint Mode BiEmbedFUpd ! - - - - : typeclass_instances.

Class BiEmbedPlainly (PROP1 PROP2 : bi)
    `{!BiEmbed PROP1 PROP2, !BiPlainly PROP1, !BiPlainly PROP2} :=
  embed_plainly (P : PROP1) : P ⊣⊢@{PROP2} P.
Global Hint Mode BiEmbedPlainly - ! - - - : typeclass_instances.
Global Hint Mode BiEmbedPlainly ! - - - - : typeclass_instances.

Section embed_laws.
  Context `{BiEmbed PROP1 PROP2}.
  Local Notation embed := (embed (A:=PROP1) (B:=PROP2)).
  Local Notation "⎡ P ⎤" := (embed P) : bi_scope.
  Implicit Types P : PROP1.

  Global Instance embed_ne : NonExpansive embed.
  Proof. eapply bi_embed_mixin_ne, bi_embed_mixin. Qed.
  Global Instance embed_mono : Proper ((⊢) ==> (⊢)) embed.
  Proof. eapply bi_embed_mixin_mono, bi_embed_mixin. Qed.
  Lemma embed_emp_valid_inj P : (⊢@{PROP2} P) P.
  Proof. eapply bi_embed_mixin_emp_valid_inj, bi_embed_mixin. Qed.
  Lemma embed_interal_inj `{!BiInternalEq PROP'} (P Q : PROP1) :
    P Q ⊢@{PROP'} (P Q).
  Proof. eapply bi_embed_mixin_interal_inj, bi_embed_mixin. Qed.
  Lemma embed_emp_2 : emp emp.
  Proof. eapply bi_embed_mixin_emp_2, bi_embed_mixin. Qed.
  Lemma embed_impl_2 P Q : (P Q) P Q.
  Proof. eapply bi_embed_mixin_impl_2, bi_embed_mixin. Qed.
  Lemma embed_forall_2 A (Φ : A PROP1) : ( x, Φ x) x, Φ x.
  Proof. eapply bi_embed_mixin_forall_2, bi_embed_mixin. Qed.
  Lemma embed_exist_1 A (Φ : A PROP1) : x, Φ x x, Φ x.
  Proof. eapply bi_embed_mixin_exist_1, bi_embed_mixin. Qed.
  Lemma embed_sep P Q : P Q ⊣⊢ P Q.
  Proof. eapply bi_embed_mixin_sep, bi_embed_mixin. Qed.
  Lemma embed_wand_2 P Q : (P -∗ Q) P -∗ Q.
  Proof. eapply bi_embed_mixin_wand_2, bi_embed_mixin. Qed.
  Lemma embed_persistently P : <pers> P ⊣⊢ <pers> P.
  Proof. eapply bi_embed_mixin_persistently, bi_embed_mixin. Qed.
End embed_laws.

Section embed.
  Context `{BiEmbed PROP1 PROP2}.
  Local Notation embed := (embed (A:=PROP1) (B:=PROP2)).
  Local Notation "⎡ P ⎤" := (embed P) : bi_scope.
  Implicit Types P Q R : PROP1.

  Global Instance embed_proper : Proper ((≡) ==> (≡)) embed.
  Proof. apply (ne_proper _). Qed.
  Global Instance embed_flip_mono : Proper (flip (⊢) ==> flip (⊢)) embed.
  Proof. solve_proper. Qed.
  Global Instance embed_entails_inj : Inj (⊢) (⊢) embed.
  Proof.
    moveP Q /bi.entails_wand. rewrite embed_wand_2.
    by move⇒ /embed_emp_valid_inj /bi.wand_entails.
  Qed.

  Global Instance embed_inj : Inj (≡) (≡) embed.
  Proof.
    intros P Q EQ. apply bi.equiv_entails, conj; apply (inj embed); rewrite EQ //.
  Qed.

  Lemma embed_emp_valid (P : PROP1) : ( P) ( P).
  Proof.
    rewrite /bi_emp_valid. splitHP.
    - by apply embed_emp_valid_inj.
    - by rewrite embed_emp_2 HP.
  Qed.

  Lemma embed_emp `{!BiEmbedEmp PROP1 PROP2} : emp ⊣⊢ emp.
  Proof. apply (anti_symm _); eauto using embed_emp_1, embed_emp_2. Qed.

  Lemma embed_forall A (Φ : A PROP1) : x, Φ x ⊣⊢ x, Φ x.
  Proof.
    apply bi.equiv_entails; split; [|apply embed_forall_2].
    apply bi.forall_intro=>?. by rewrite bi.forall_elim.
  Qed.
  Lemma embed_exist A (Φ : A PROP1) : x, Φ x ⊣⊢ x, Φ x.
  Proof.
    apply bi.equiv_entails; split; [apply embed_exist_1|].
    apply bi.exist_elim=>?. by rewrite -bi.exist_intro.
  Qed.
  Lemma embed_and P Q : P Q ⊣⊢ P Q.
  Proof. rewrite !bi.and_alt embed_forall. by f_equiv=>-[]. Qed.
  Lemma embed_or P Q : P Q ⊣⊢ P Q.
  Proof. rewrite !bi.or_alt embed_exist. by f_equiv=>-[]. Qed.
  Lemma embed_impl P Q : P Q ⊣⊢ (P Q).
  Proof.
    apply bi.equiv_entails; split; [|apply embed_impl_2].
    apply bi.impl_intro_l. by rewrite -embed_and bi.impl_elim_r.
  Qed.
  Lemma embed_wand P Q : P -∗ Q ⊣⊢ (P -∗ Q).
  Proof.
    apply bi.equiv_entails; split; [|apply embed_wand_2].
    apply bi.wand_intro_l. by rewrite -embed_sep bi.wand_elim_r.
  Qed.
  Lemma embed_pure φ : φ ⊣⊢ φ.
  Proof.
    rewrite (@bi.pure_alt PROP1) (@bi.pure_alt PROP2) embed_exist.
    do 2 f_equiv. apply bi.equiv_entails. split; [apply bi.True_intro|].
    rewrite -(_ : (emp emp : PROP1) True) ?embed_impl;
      last apply bi.True_intro.
    apply bi.impl_intro_l. by rewrite right_id.
  Qed.

  Lemma embed_iff P Q : P Q ⊣⊢ (P Q).
  Proof. by rewrite embed_and !embed_impl. Qed.
  Lemma embed_wand_iff P Q : P ∗-∗ Q ⊣⊢ (P ∗-∗ Q).
  Proof. by rewrite embed_and !embed_wand. Qed.
  Lemma embed_affinely_2 P : <affine> P <affine> P.
  Proof. by rewrite embed_and -embed_emp_2. Qed.
  Lemma embed_affinely `{!BiEmbedEmp PROP1 PROP2} P : <affine> P ⊣⊢ <affine> P.
  Proof. by rewrite /bi_intuitionistically embed_and embed_emp. Qed.
  Lemma embed_absorbingly P : <absorb> P ⊣⊢ <absorb> P.
  Proof. by rewrite embed_sep embed_pure. Qed.
  Lemma embed_intuitionistically_2 P : P P.
  Proof. by rewrite /bi_intuitionistically -embed_affinely_2 embed_persistently. Qed.
  Lemma embed_intuitionistically `{!BiEmbedEmp PROP1 PROP2} P : P ⊣⊢ P.
  Proof. by rewrite /bi_intuitionistically embed_affinely embed_persistently. Qed.

  Lemma embed_persistently_if P b : <pers>?b P ⊣⊢ <pers>?b P.
  Proof. destruct b; simpl; auto using embed_persistently. Qed.
  Lemma embed_affinely_if_2 P b : <affine>?b P <affine>?b P.
  Proof. destruct b; simpl; auto using embed_affinely_2. Qed.
  Lemma embed_affinely_if `{!BiEmbedEmp PROP1 PROP2} P b :
    <affine>?b P ⊣⊢ <affine>?b P.
  Proof. destruct b; simpl; auto using embed_affinely. Qed.
  Lemma embed_absorbingly_if b P : <absorb>?b P ⊣⊢ <absorb>?b P.
  Proof. destruct b; simpl; auto using embed_absorbingly. Qed.
  Lemma embed_intuitionistically_if_2 P b : □?b P □?b P.
  Proof. destruct b; simpl; auto using embed_intuitionistically_2. Qed.
  Lemma embed_intuitionistically_if `{!BiEmbedEmp PROP1 PROP2} P b :
    □?b P ⊣⊢ □?b P.
  Proof. destruct b; simpl; auto using embed_intuitionistically. Qed.

  Global Instance embed_persistent P : Persistent P Persistent P.
  Proof. intros ?. by rewrite /Persistent -embed_persistently -persistent. Qed.
  Global Instance embed_affine `{!BiEmbedEmp PROP1 PROP2} P : Affine P Affine P.
  Proof. intros ?. by rewrite /Affine (affine P) embed_emp. Qed.
  Global Instance embed_absorbing P : Absorbing P Absorbing P.
  Proof. intros ?. by rewrite /Absorbing -embed_absorbingly absorbing. Qed.

  Global Instance embed_and_homomorphism :
    MonoidHomomorphism bi_and bi_and (≡) embed.
  Proof.
    by split; [split|]; try apply _;
      [setoid_rewrite embed_and|rewrite embed_pure].
  Qed.
  Global Instance embed_or_homomorphism :
    MonoidHomomorphism bi_or bi_or (≡) embed.
  Proof.
    by split; [split|]; try apply _;
      [setoid_rewrite embed_or|rewrite embed_pure].
  Qed.

  Global Instance embed_sep_entails_homomorphism :
    MonoidHomomorphism bi_sep bi_sep (flip (⊢)) embed.
  Proof.
    split; [split|]; simpl; try apply _;
      [by setoid_rewrite embed_sep|by rewrite embed_emp_2].
  Qed.

  Lemma embed_big_sepL_2 {A} (Φ : nat A PROP1) l :
    ([∗ list] kx l, Φ k x) [∗ list] kx l, Φ k x.
  Proof. apply (big_opL_commute (R:=flip (⊢)) _). Qed.
  Lemma embed_big_sepM_2 `{Countable K} {A} (Φ : K A PROP1) (m : gmap K A) :
    ([∗ map] kx m, Φ k x) [∗ map] kx m, Φ k x.
  Proof. apply (big_opM_commute (R:=flip (⊢)) _). Qed.
  Lemma embed_big_sepS_2 `{Countable A} (Φ : A PROP1) (X : gset A) :
    ([∗ set] y X, Φ y) [∗ set] y X, Φ y.
  Proof. apply (big_opS_commute (R:=flip (⊢)) _). Qed.
  Lemma embed_big_sepMS_2 `{Countable A} (Φ : A PROP1) (X : gmultiset A) :
    ([∗ mset] y X, Φ y) [∗ mset] y X, Φ y.
  Proof. apply (big_opMS_commute (R:=flip (⊢)) _). Qed.

  Section big_ops_emp.
    Context `{!BiEmbedEmp PROP1 PROP2}.

    Global Instance embed_sep_homomorphism :
      MonoidHomomorphism bi_sep bi_sep (≡) embed.
    Proof.
      by split; [split|]; try apply _;
        [setoid_rewrite embed_sep|rewrite embed_emp].
    Qed.

    Lemma embed_big_sepL {A} (Φ : nat A PROP1) l :
      [∗ list] kx l, Φ k x ⊣⊢ [∗ list] kx l, Φ k x.
    Proof. apply (big_opL_commute _). Qed.
    Lemma embed_big_sepM `{Countable K} {A} (Φ : K A PROP1) (m : gmap K A) :
      [∗ map] kx m, Φ k x ⊣⊢ [∗ map] kx m, Φ k x.
    Proof. apply (big_opM_commute _). Qed.
    Lemma embed_big_sepS `{Countable A} (Φ : A PROP1) (X : gset A) :
      [∗ set] y X, Φ y ⊣⊢ [∗ set] y X, Φ y.
    Proof. apply (big_opS_commute _). Qed.
    Lemma embed_big_sepMS `{Countable A} (Φ : A PROP1) (X : gmultiset A) :
      [∗ mset] y X, Φ y ⊣⊢ [∗ mset] y X, Φ y.
    Proof. apply (big_opMS_commute _). Qed.
  End big_ops_emp.

  Section later.
    Context `{!BiEmbedLater PROP1 PROP2}.

    Lemma embed_laterN n P : ▷^n P ⊣⊢ ▷^n P.
    Proof. induction n=>//=. rewrite embed_later. by f_equiv. Qed.
    Lemma embed_except_0 P : P ⊣⊢ P.
    Proof. by rewrite embed_or embed_later embed_pure. Qed.

    Global Instance embed_timeless P : Timeless P Timeless P.
    Proof.
      intros ?. by rewrite /Timeless -embed_except_0 -embed_later timeless.
    Qed.
  End later.

  Section internal_eq.
    Context `{!BiInternalEq PROP1, !BiInternalEq PROP2, !BiEmbedInternalEq PROP1 PROP2}.

    Lemma embed_internal_eq (A : ofe) (x y : A) : x y ⊣⊢@{PROP2} x y.
    Proof.
      apply bi.equiv_entails; split; [apply embed_internal_eq_1|].
      etrans; [apply (internal_eq_rewrite x y (λ y, x y%I)); solve_proper|].
      rewrite -(internal_eq_refl True%I) embed_pure.
      eapply bi.impl_elim; [done|]. apply bi.True_intro.
    Qed.
  End internal_eq.

  Section plainly.
    Context `{!BiPlainly PROP1, !BiPlainly PROP2, !BiEmbedPlainly PROP1 PROP2}.

    Lemma embed_plainly_if p P : ■?p P ⊣⊢ ■?p P.
    Proof. destruct p; simpl; auto using embed_plainly. Qed.

    Lemma embed_plain (P : PROP1) : Plain P Plain (PROP:=PROP2) P.
    Proof. intros ?. by rewrite /Plain {1}(plain P) embed_plainly. Qed.
  End plainly.
End embed.

Global Hint Extern 4 (Plain _) ⇒
  notypeclasses refine (embed_plain _ _) : typeclass_instances.

Transitive embedding: this constructs an embedding of PROP1 into PROP3 by combining the embeddings of PROP1 into PROP2 and PROP2 into PROP3. Note that declaring these instances globally can make TC search ambiguous or diverging. These are only defined so that a user can conveniently use them to manually combine embeddings.
Section embed_embed.
  Context `{BiEmbed PROP1 PROP2, BiEmbed PROP2 PROP3}.

  Local Instance embed_embed : Embed PROP1 PROP3 := λ P, P %I.

  Lemma embed_embedding_mixin : BiEmbedMixin PROP1 PROP3 embed_embed.
  Proof.
    split; unfold embed, embed_embed.
    - solve_proper.
    - solve_proper.
    - intros P. by rewrite !embed_emp_valid.
    - intros PROP' ? P Q. by rewrite !embed_interal_inj.
    - by rewrite -!embed_emp_2.
    - intros P Q. by rewrite -!embed_impl.
    - intros A Φ. by rewrite -!embed_forall.
    - intros A Φ. by rewrite -!embed_exist.
    - intros P Q. by rewrite -!embed_sep.
    - intros P Q. by rewrite -!embed_wand.
    - intros P. by rewrite -!embed_persistently.
  Qed.
  Local Instance embed_bi_embed : BiEmbed PROP1 PROP3 :=
    {| bi_embed_mixin := embed_embedding_mixin |}.

  Lemma embed_embed_alt (P : PROP1) : P ⊣⊢@{PROP3} P .
  Proof. done. Qed.

  Lemma embed_embed_emp :
    BiEmbedEmp PROP1 PROP2 BiEmbedEmp PROP2 PROP3
    BiEmbedEmp PROP1 PROP3.
  Proof. rewrite /BiEmbedEmp !embed_embed_alt. by intros → →. Qed.
  Lemma embed_embed_later :
    BiEmbedLater PROP1 PROP2 BiEmbedLater PROP2 PROP3
    BiEmbedLater PROP1 PROP3.
  Proof. intros ?? P. by rewrite !embed_embed_alt !embed_later. Qed.
  Lemma embed_embed_internal_eq
      `{!BiInternalEq PROP1, !BiInternalEq PROP2, !BiInternalEq PROP3} :
    BiEmbedInternalEq PROP1 PROP2 BiEmbedInternalEq PROP2 PROP3
    BiEmbedInternalEq PROP1 PROP3.
  Proof. intros ?? A x y. by rewrite !embed_embed_alt !embed_internal_eq. Qed.
  Lemma embed_embed_bupd `{!BiBUpd PROP1, !BiBUpd PROP2, !BiBUpd PROP3} :
    BiEmbedBUpd PROP1 PROP2 BiEmbedBUpd PROP2 PROP3
    BiEmbedBUpd PROP1 PROP3.
  Proof. intros ?? P. by rewrite !embed_embed_alt !embed_bupd. Qed.
  Lemma embed_embed_fupd `{!BiFUpd PROP1, !BiFUpd PROP2, !BiFUpd PROP3} :
    BiEmbedFUpd PROP1 PROP2 BiEmbedFUpd PROP2 PROP3
    BiEmbedFUpd PROP1 PROP3.
  Proof. intros ?? E1 E2 P. by rewrite !embed_embed_alt !embed_fupd. Qed.
  Lemma embed_embed_plainly
      `{!BiPlainly PROP1, !BiPlainly PROP2, !BiPlainly PROP3} :
    BiEmbedPlainly PROP1 PROP2 BiEmbedPlainly PROP2 PROP3
    BiEmbedPlainly PROP1 PROP3.
  Proof. intros ?? P. by rewrite !embed_embed_alt !embed_plainly. Qed.
End embed_embed.