class_instances.v

From iris.proofmode Require Export classes.
From iris.algebra Require Import gmap.
From iris.base_logic Require Import big_op.
Import uPred.

Section classes.
Context {M : ucmraT}.
Implicit Types P Q R : uPred M.

(* FromAssumption *)
Global Instance from_assumption_exact p P : FromAssumption p P P.
Proof. destruct p; by rewrite /FromAssumption /= ?always_elim. Qed.
Global Instance from_assumption_always_l p P Q :
  FromAssumption p P Q → FromAssumption p (□ P) Q.
Proof. rewrite /FromAssumption=><-. by rewrite always_elim. Qed.
Global Instance from_assumption_always_r P Q :
  FromAssumption true P Q → FromAssumption true P (□ Q).
Proof. rewrite /FromAssumption=><-. by rewrite always_always. Qed.
Global Instance from_assumption_bupd p P Q :
  FromAssumption p P Q → FromAssumption p P (|==> Q)%I.
Proof. rewrite /FromAssumption=>->. apply bupd_intro. Qed.

(* IntoPure *)
Global Instance into_pure_pure φ : @IntoPure M (■ φ) φ.
Proof. done. Qed.
Global Instance into_pure_eq {A : cofeT} (a b : A) :
  Timeless a → @IntoPure M (a ≡ b) (a ≡ b).
Proof. intros. by rewrite /IntoPure timeless_eq. Qed.
Global Instance into_pure_cmra_valid `{CMRADiscrete A} (a : A) :
  @IntoPure M (✓ a) (✓ a).
Proof. by rewrite /IntoPure discrete_valid. Qed.

(* FromPure *)
Global Instance from_pure_pure φ : @FromPure M (■ φ) φ.
Proof. done. Qed.
Global Instance from_pure_eq {A : cofeT} (a b : A) : @FromPure M (a ≡ b) (a ≡ b).
Proof. rewrite /FromPure. eapply pure_elim; [done|]=> ->. apply eq_refl'. Qed.
Global Instance from_pure_cmra_valid {A : cmraT} (a : A) :
  @FromPure M (✓ a) (✓ a).
Proof.
  rewrite /FromPure. eapply pure_elim; [done|]=> ?.
  rewrite -cmra_valid_intro //. auto with I.
Qed.
Global Instance from_pure_bupd P φ : FromPure P φ → FromPure (|==> P) φ.
Proof. rewrite /FromPure=> ->. apply bupd_intro. Qed.

(* IntoPersistentP *)
Global Instance into_persistentP_always_trans P Q :
  IntoPersistentP P Q → IntoPersistentP (□ P) Q | 0.
Proof. rewrite /IntoPersistentP=> ->. by rewrite always_always. Qed.
Global Instance into_persistentP_always P : IntoPersistentP (□ P) P | 1.
Proof. done. Qed.
Global Instance into_persistentP_persistent P :
  PersistentP P → IntoPersistentP P P | 100.
Proof. done. Qed.

(* IntoLater *)
Global Instance into_later_default P : IntoLater P P | 1000.
Proof. apply later_intro. Qed.
Global Instance into_later_later P : IntoLater (▷ P) P.
Proof. done. Qed.
Global Instance into_later_and P1 P2 Q1 Q2 :
  IntoLater P1 Q1 → IntoLater P2 Q2 → IntoLater (P1 ∧ P2) (Q1 ∧ Q2).
Proof. intros ??; red. by rewrite later_and; apply and_mono. Qed.
Global Instance into_later_or P1 P2 Q1 Q2 :
  IntoLater P1 Q1 → IntoLater P2 Q2 → IntoLater (P1 ∨ P2) (Q1 ∨ Q2).
Proof. intros ??; red. by rewrite later_or; apply or_mono. Qed.
Global Instance into_later_sep P1 P2 Q1 Q2 :
  IntoLater P1 Q1 → IntoLater P2 Q2 → IntoLater (P1 ★ P2) (Q1 ★ Q2).
Proof. intros ??; red. by rewrite later_sep; apply sep_mono. Qed.

Global Instance into_later_big_sepM `{Countable K} {A}
    (Φ Ψ : K → A → uPred M) (m : gmap K A) :
  (∀ x k, IntoLater (Φ k x) (Ψ k x)) →
  IntoLater ([★ map] k ↦ x ∈ m, Φ k x) ([★ map] k ↦ x ∈ m, Ψ k x).
Proof.
  rewrite /IntoLater=> ?. rewrite big_sepM_later; by apply big_sepM_mono.
Qed.
Global Instance into_later_big_sepS `{Countable A}
    (Φ Ψ : A → uPred M) (X : gset A) :
  (∀ x, IntoLater (Φ x) (Ψ x)) →
  IntoLater ([★ set] x ∈ X, Φ x) ([★ set] x ∈ X, Ψ x).
Proof.
  rewrite /IntoLater=> ?. rewrite big_sepS_later; by apply big_sepS_mono.
Qed.

(* FromLater *)
Global Instance from_later_later P : FromLater (▷ P) P.
Proof. done. Qed.
Global Instance from_later_and P1 P2 Q1 Q2 :
  FromLater P1 Q1 → FromLater P2 Q2 → FromLater (P1 ∧ P2) (Q1 ∧ Q2).
Proof. intros ??; red. by rewrite later_and; apply and_mono. Qed.
Global Instance from_later_or P1 P2 Q1 Q2 :
  FromLater P1 Q1 → FromLater P2 Q2 → FromLater (P1 ∨ P2) (Q1 ∨ Q2).
Proof. intros ??; red. by rewrite later_or; apply or_mono. Qed.
Global Instance from_later_sep P1 P2 Q1 Q2 :
  FromLater P1 Q1 → FromLater P2 Q2 → FromLater (P1 ★ P2) (Q1 ★ Q2).
Proof. intros ??; red. by rewrite later_sep; apply sep_mono. Qed.

(* IntoWand *)
Global Instance into_wand_wand P Q Q' :
  FromAssumption false Q Q' → IntoWand (P -★ Q) P Q'.
Proof. by rewrite /FromAssumption /IntoWand /= => ->. Qed.
Global Instance into_wand_impl P Q Q' :
  FromAssumption false Q Q' → IntoWand (P → Q) P Q'.
Proof. rewrite /FromAssumption /IntoWand /= => ->. by rewrite impl_wand. Qed.
Global Instance into_wand_iff_l P Q : IntoWand (P ↔ Q) P Q.
Proof. by apply and_elim_l', impl_wand. Qed.
Global Instance into_wand_iff_r P Q : IntoWand (P ↔ Q) Q P.
Proof. apply and_elim_r', impl_wand. Qed.
Global Instance into_wand_always R P Q : IntoWand R P Q → IntoWand (□ R) P Q.
Proof. rewrite /IntoWand=> ->. apply always_elim. Qed.
Global Instance into_wand_bupd R P Q :
  IntoWand R P Q → IntoWand R (|==> P) (|==> Q) | 100.
Proof. rewrite /IntoWand=>->. apply wand_intro_l. by rewrite bupd_wand_r. Qed.

(* FromAnd *)
Global Instance from_and_and P1 P2 : FromAnd (P1 ∧ P2) P1 P2.
Proof. done. Qed.
Global Instance from_and_sep_persistent_l P1 P2 :
  PersistentP P1 → FromAnd (P1 ★ P2) P1 P2 | 9.
Proof. intros. by rewrite /FromAnd always_and_sep_l. Qed.
Global Instance from_and_sep_persistent_r P1 P2 :
  PersistentP P2 → FromAnd (P1 ★ P2) P1 P2 | 10.
Proof. intros. by rewrite /FromAnd always_and_sep_r. Qed.
Global Instance from_and_always P Q1 Q2 :
  FromAnd P Q1 Q2 → FromAnd (□ P) (□ Q1) (□ Q2).
Proof. rewrite /FromAnd=> <-. by rewrite always_and. Qed.
Global Instance from_and_later P Q1 Q2 :
  FromAnd P Q1 Q2 → FromAnd (▷ P) (▷ Q1) (▷ Q2).
Proof. rewrite /FromAnd=> <-. by rewrite later_and. Qed.

(* FromSep *)
Global Instance from_sep_sep P1 P2 : FromSep (P1 ★ P2) P1 P2 | 100.
Proof. done. Qed.
Global Instance from_sep_ownM (a b1 b2 : M) :
  FromOp a b1 b2 →
  FromSep (uPred_ownM a) (uPred_ownM b1) (uPred_ownM b2).
Proof. intros. by rewrite /FromSep -ownM_op from_op. Qed.
Global Instance from_sep_always P Q1 Q2 :
  FromSep P Q1 Q2 → FromSep (□ P) (□ Q1) (□ Q2).
Proof. rewrite /FromSep=> <-. by rewrite always_sep. Qed.
Global Instance from_sep_later P Q1 Q2 :
  FromSep P Q1 Q2 → FromSep (▷ P) (▷ Q1) (▷ Q2).
Proof. rewrite /FromSep=> <-. by rewrite later_sep. Qed.
Global Instance from_sep_bupd P Q1 Q2 :
  FromSep P Q1 Q2 → FromSep (|==> P) (|==> Q1) (|==> Q2).
Proof. rewrite /FromSep=><-. apply bupd_sep. Qed.

Global Instance from_sep_big_sepM
    `{Countable K} {A} (Φ Ψ1 Ψ2 : K → A → uPred M) m :
  (∀ k x, FromSep (Φ k x) (Ψ1 k x) (Ψ2 k x)) →
  FromSep ([★ map] k ↦ x ∈ m, Φ k x)
    ([★ map] k ↦ x ∈ m, Ψ1 k x) ([★ map] k ↦ x ∈ m, Ψ2 k x).
Proof.
  rewrite /FromSep=> ?. rewrite -big_sepM_sepM. by apply big_sepM_mono.
Qed.
Global Instance from_sep_big_sepS `{Countable A} (Φ Ψ1 Ψ2 : A → uPred M) X :
  (∀ x, FromSep (Φ x) (Ψ1 x) (Ψ2 x)) →
  FromSep ([★ set] x ∈ X, Φ x) ([★ set] x ∈ X, Ψ1 x) ([★ set] x ∈ X, Ψ2 x).
Proof.
  rewrite /FromSep=> ?. rewrite -big_sepS_sepS. by apply big_sepS_mono.
Qed.

(* FromOp *)
Global Instance from_op_op {A : cmraT} (a b : A) : FromOp (a ⋅ b) a b.
Proof. by rewrite /FromOp. Qed.
Global Instance from_op_persistent {A : cmraT} (a : A) :
  Persistent a → FromOp a a a.
Proof. intros. by rewrite /FromOp -(persistent_dup a). Qed.
Global Instance from_op_pair {A B : cmraT} (a b1 b2 : A) (a' b1' b2' : B) :
  FromOp a b1 b2 → FromOp a' b1' b2' →
  FromOp (a,a') (b1,b1') (b2,b2').
Proof. by constructor. Qed.
Global Instance from_op_Some {A : cmraT} (a : A) b1 b2 :
  FromOp a b1 b2 → FromOp (Some a) (Some b1) (Some b2).
Proof. by constructor. Qed.

(* IntoOp *)
Global Instance into_op_op {A : cmraT} (a b : A) : IntoOp (a ⋅ b) a b.
Proof. by rewrite /IntoOp. Qed.
Global Instance into_op_persistent {A : cmraT} (a : A) :
  Persistent a → IntoOp a a a.
Proof. intros; apply (persistent_dup a). Qed.
Global Instance into_op_pair {A B : cmraT} (a b1 b2 : A) (a' b1' b2' : B) :
  IntoOp a b1 b2 → IntoOp a' b1' b2' →
  IntoOp (a,a') (b1,b1') (b2,b2').
Proof. by constructor. Qed.
Global Instance into_op_Some {A : cmraT} (a : A) b1 b2 :
  IntoOp a b1 b2 → IntoOp (Some a) (Some b1) (Some b2).
Proof. by constructor. Qed.

(* IntoAnd *)
Global Instance into_and_sep p P Q : IntoAnd p (P ★ Q) P Q.
Proof. by apply mk_into_and_sep. Qed.
Global Instance into_and_ownM p (a b1 b2 : M) :
  IntoOp a b1 b2 →
  IntoAnd p (uPred_ownM a) (uPred_ownM b1) (uPred_ownM b2).
Proof. intros. apply mk_into_and_sep. by rewrite (into_op a) ownM_op. Qed.

Global Instance into_and_and P Q : IntoAnd true (P ∧ Q) P Q.
Proof. done. Qed.
Global Instance into_and_and_persistent_l P Q :
  PersistentP P → IntoAnd false (P ∧ Q) P Q.
Proof. intros; by rewrite /IntoAnd /= always_and_sep_l. Qed.
Global Instance into_and_and_persistent_r P Q :
  PersistentP Q → IntoAnd false (P ∧ Q) P Q.
Proof. intros; by rewrite /IntoAnd /= always_and_sep_r. Qed.

Global Instance into_and_later p P Q1 Q2 :
  IntoAnd p P Q1 Q2 → IntoAnd p (▷ P) (▷ Q1) (▷ Q2).
Proof. rewrite /IntoAnd=>->. destruct p; by rewrite ?later_and ?later_sep. Qed.

Global Instance into_and_big_sepM
    `{Countable K} {A} (Φ Ψ1 Ψ2 : K → A → uPred M) p m :
  (∀ k x, IntoAnd p (Φ k x) (Ψ1 k x) (Ψ2 k x)) →
  IntoAnd p ([★ map] k ↦ x ∈ m, Φ k x)
    ([★ map] k ↦ x ∈ m, Ψ1 k x) ([★ map] k ↦ x ∈ m, Ψ2 k x).
Proof.
  rewrite /IntoAnd=> HΦ. destruct p.
  - apply and_intro; apply big_sepM_mono; auto.
    + intros k x ?. by rewrite HΦ and_elim_l.
    + intros k x ?. by rewrite HΦ and_elim_r.
  - rewrite -big_sepM_sepM. apply big_sepM_mono; auto.
Qed.
Global Instance into_and_big_sepS `{Countable A} (Φ Ψ1 Ψ2 : A → uPred M) p X :
  (∀ x, IntoAnd p (Φ x) (Ψ1 x) (Ψ2 x)) →
  IntoAnd p ([★ set] x ∈ X, Φ x) ([★ set] x ∈ X, Ψ1 x) ([★ set] x ∈ X, Ψ2 x).
Proof.
  rewrite /IntoAnd=> HΦ. destruct p.
  - apply and_intro; apply big_sepS_mono; auto.
    + intros x ?. by rewrite HΦ and_elim_l.
    + intros x ?. by rewrite HΦ and_elim_r.
  - rewrite -big_sepS_sepS. apply big_sepS_mono; auto.
Qed.

(* Frame *)
Global Instance frame_here R : Frame R R True.
Proof. by rewrite /Frame right_id. Qed.
Global Instance frame_here_pure φ Q : FromPure Q φ → Frame (■ φ) Q True.
Proof. rewrite /FromPure /Frame=> ->. by rewrite right_id. Qed.

Class MakeSep (P Q PQ : uPred M) := make_sep : P ★ Q ⊣⊢ PQ.
Global Instance make_sep_true_l P : MakeSep True P P.
Proof. by rewrite /MakeSep left_id. Qed.
Global Instance make_sep_true_r P : MakeSep P True P.
Proof. by rewrite /MakeSep right_id. Qed.
Global Instance make_sep_default P Q : MakeSep P Q (P ★ Q) | 100.
Proof. done. Qed.
Global Instance frame_sep_l R P1 P2 Q Q' :
  Frame R P1 Q → MakeSep Q P2 Q' → Frame R (P1 ★ P2) Q' | 9.
Proof. rewrite /Frame /MakeSep => <- <-. by rewrite assoc. Qed.
Global Instance frame_sep_r R P1 P2 Q Q' :
  Frame R P2 Q → MakeSep P1 Q Q' → Frame R (P1 ★ P2) Q' | 10.
Proof. rewrite /Frame /MakeSep => <- <-. by rewrite assoc (comm _ R) assoc. Qed.

Class MakeAnd (P Q PQ : uPred M) := make_and : P ∧ Q ⊣⊢ PQ.
Global Instance make_and_true_l P : MakeAnd True P P.
Proof. by rewrite /MakeAnd left_id. Qed.
Global Instance make_and_true_r P : MakeAnd P True P.
Proof. by rewrite /MakeAnd right_id. Qed.
Global Instance make_and_default P Q : MakeAnd P Q (P ∧ Q) | 100.
Proof. done. Qed.
Global Instance frame_and_l R P1 P2 Q Q' :
  Frame R P1 Q → MakeAnd Q P2 Q' → Frame R (P1 ∧ P2) Q' | 9.
Proof. rewrite /Frame /MakeAnd => <- <-; eauto 10 with I. Qed.
Global Instance frame_and_r R P1 P2 Q Q' :
  Frame R P2 Q → MakeAnd P1 Q Q' → Frame R (P1 ∧ P2) Q' | 10.
Proof. rewrite /Frame /MakeAnd => <- <-; eauto 10 with I. Qed.

Class MakeOr (P Q PQ : uPred M) := make_or : P ∨ Q ⊣⊢ PQ.
Global Instance make_or_true_l P : MakeOr True P True.
Proof. by rewrite /MakeOr left_absorb. Qed.
Global Instance make_or_true_r P : MakeOr P True True.
Proof. by rewrite /MakeOr right_absorb. Qed.
Global Instance make_or_default P Q : MakeOr P Q (P ∨ Q) | 100.
Proof. done. Qed.
Global Instance frame_or R P1 P2 Q1 Q2 Q :
  Frame R P1 Q1 → Frame R P2 Q2 → MakeOr Q1 Q2 Q → Frame R (P1 ∨ P2) Q.
Proof. rewrite /Frame /MakeOr => <- <- <-. by rewrite -sep_or_l. Qed.

Global Instance frame_wand R P1 P2 Q2 :
  Frame R P2 Q2 → Frame R (P1 -★ P2) (P1 -★ Q2).
Proof.
  rewrite /Frame=> ?. apply wand_intro_l.
  by rewrite assoc (comm _ P1) -assoc wand_elim_r.
Qed.

Class MakeLater (P lP : uPred M) := make_later : ▷ P ⊣⊢ lP.
Global Instance make_later_true : MakeLater True True.
Proof. by rewrite /MakeLater later_True. Qed.
Global Instance make_later_default P : MakeLater P (▷ P) | 100.
Proof. done. Qed.

Global Instance frame_later R R' P Q Q' :
  IntoLater R' R → Frame R P Q → MakeLater Q Q' → Frame R' (▷ P) Q'.
Proof.
  rewrite /Frame /MakeLater /IntoLater=>-> <- <-. by rewrite later_sep.
Qed.

Class MakeExcept0 (P Q : uPred M) := make_except_0 : ◇ P ⊣⊢ Q.
Global Instance make_except_0_True : MakeExcept0 True True.
Proof. by rewrite /MakeExcept0 except_0_True. Qed.
Global Instance make_except_0_default P : MakeExcept0 P (◇ P) | 100.
Proof. done. Qed.

Global Instance frame_except_0 R P Q Q' :
  Frame R P Q → MakeExcept0 Q Q' → Frame R (◇ P) Q'.
Proof.
  rewrite /Frame /MakeExcept0=><- <-.
  by rewrite except_0_sep -(except_0_intro R).
Qed.

Global Instance frame_exist {A} R (Φ Ψ : A → uPred M) :
  (∀ a, Frame R (Φ a) (Ψ a)) → Frame R (∃ x, Φ x) (∃ x, Ψ x).
Proof. rewrite /Frame=> ?. by rewrite sep_exist_l; apply exist_mono. Qed.
Global Instance frame_forall {A} R (Φ Ψ : A → uPred M) :
  (∀ a, Frame R (Φ a) (Ψ a)) → Frame R (∀ x, Φ x) (∀ x, Ψ x).
Proof. rewrite /Frame=> ?. by rewrite sep_forall_l; apply forall_mono. Qed.

Global Instance frame_bupd R P Q : Frame R P Q → Frame R (|==> P) (|==> Q).
Proof. rewrite /Frame=><-. by rewrite bupd_frame_l. Qed.

(* FromOr *)
Global Instance from_or_or P1 P2 : FromOr (P1 ∨ P2) P1 P2.
Proof. done. Qed.
Global Instance from_or_bupd P Q1 Q2 :
  FromOr P Q1 Q2 → FromOr (|==> P) (|==> Q1) (|==> Q2).
Proof. rewrite /FromOr=><-. apply or_elim; apply bupd_mono; auto with I. Qed.

(* IntoOr *)
Global Instance into_or_or P Q : IntoOr (P ∨ Q) P Q.
Proof. done. Qed.
Global Instance into_or_later P Q1 Q2 :
  IntoOr P Q1 Q2 → IntoOr (▷ P) (▷ Q1) (▷ Q2).
Proof. rewrite /IntoOr=>->. by rewrite later_or. Qed.

(* FromExist *)
Global Instance from_exist_exist {A} (Φ : A → uPred M): FromExist (∃ a, Φ a) Φ.
Proof. done. Qed.
Global Instance from_exist_bupd {A} P (Φ : A → uPred M) :
  FromExist P Φ → FromExist (|==> P) (λ a, |==> Φ a)%I.
Proof.
  rewrite /FromExist=><-. apply exist_elim=> a. by rewrite -(exist_intro a).
Qed.
Global Instance from_exist_later {A} P (Φ : A → uPred M) :
  FromExist P Φ → FromExist (▷ P) (λ a, ▷ (Φ a))%I.
Proof. rewrite /FromExist=> <-. apply exist_elim=>x. apply later_mono, exist_intro. Qed.

(* IntoExist *)
Global Instance into_exist_exist {A} (Φ : A → uPred M) : IntoExist (∃ a, Φ a) Φ.
Proof. done. Qed.
Global Instance into_exist_later {A} P (Φ : A → uPred M) :
  IntoExist P Φ → Inhabited A → IntoExist (▷ P) (λ a, ▷ (Φ a))%I.
Proof. rewrite /IntoExist=> HP ?. by rewrite HP later_exist. Qed.
Global Instance into_exist_always {A} P (Φ : A → uPred M) :
  IntoExist P Φ → IntoExist (□ P) (λ a, □ (Φ a))%I.
Proof. rewrite /IntoExist=> HP. by rewrite HP always_exist. Qed.

(* IntoModal *)
Global Instance into_modal_later P : IntoModal P (▷ P).
Proof. apply later_intro. Qed.
Global Instance into_modal_bupd P : IntoModal P (|==> P).
Proof. apply bupd_intro. Qed.
Global Instance into_modal_except_0 P : IntoModal P (◇ P).
Proof. apply except_0_intro. Qed.

(* ElimModal *)
Global Instance elim_modal_wand P P' Q Q' R :
  ElimModal P P' Q Q' → ElimModal P P' (R -★ Q) (R -★ Q').
Proof.
  rewrite /ElimModal=> H. apply wand_intro_r.
  by rewrite wand_curry -assoc (comm _ P') -wand_curry wand_elim_l.
Qed.
Global Instance forall_modal_wand {A} P P' (Φ Ψ : A → uPred M) :
  (∀ x, ElimModal P P' (Φ x) (Ψ x)) → ElimModal P P' (∀ x, Φ x) (∀ x, Ψ x).
Proof.
  rewrite /ElimModal=> H. apply forall_intro=> a. by rewrite (forall_elim a).
Qed.

Global Instance elim_modal_bupd P Q : ElimModal (|==> P) P (|==> Q) (|==> Q).
Proof. by rewrite /ElimModal bupd_frame_r wand_elim_r bupd_trans. Qed.

Global Instance elim_modal_except_0 P Q : IsExcept0 Q → ElimModal (◇ P) P Q Q.
Proof.
  intros. rewrite /ElimModal (except_0_intro (_ -★ _)).
  by rewrite -except_0_sep wand_elim_r.
Qed.
Global Instance elim_modal_timeless_bupd P Q :
  TimelessP P → IsExcept0 Q → ElimModal (▷ P) P Q Q.
Proof.
  intros. rewrite /ElimModal (except_0_intro (_ -★ _)) (timelessP P).
  by rewrite -except_0_sep wand_elim_r.
Qed.

Global Instance is_except_0_except_0 P : IsExcept0 (◇ P).
Proof. by rewrite /IsExcept0 except_0_idemp. Qed.
Global Instance is_except_0_later P : IsExcept0 (▷ P).
Proof. by rewrite /IsExcept0 except_0_later. Qed.
Global Instance is_except_0_bupd P : IsExcept0 P → IsExcept0 (|==> P).
Proof.
  rewrite /IsExcept0=> HP.
  by rewrite -{2}HP -(except_0_idemp P) -except_0_bupd -(except_0_intro P).
Qed.
End classes.