diff --git a/_CoqProject b/_CoqProject
index 623fce6cc6d71ffdc1b58419dff44ec34dd083dc..4ddfcad2828c5bac0ccd3d904367d4bc1cd0b650 100644
--- a/_CoqProject
+++ b/_CoqProject
@@ -28,6 +28,7 @@ theories/algebra/proofmode_classes.v
theories/bi/interface.v
theories/bi/derived_connectives.v
theories/bi/derived_laws.v
+theories/bi/plainly.v
theories/bi/big_op.v
theories/bi/updates.v
theories/bi/bi.v
diff --git a/theories/base_logic/derived.v b/theories/base_logic/derived.v
index 0972828f21cd098bb28f9c521b88b4f5955de0b5..c126c39334338cf2e5834754ecaeb6bbb9851d17 100644
--- a/theories/base_logic/derived.v
+++ b/theories/base_logic/derived.v
@@ -30,11 +30,8 @@ Global Instance ownM_mono : Proper (flip (≼) ==> (⊢)) (@uPred_ownM M).
Proof. intros a b [b' ->]. by rewrite ownM_op sep_elim_l. Qed.
Lemma ownM_unit' : uPred_ownM ε ⊣⊢ True.
Proof. apply (anti_symm _); first by apply pure_intro. apply ownM_unit. Qed.
-Lemma affinely_plainly_cmra_valid {A : cmraT} (a : A) : ■✓ a ⊣⊢ ✓ a.
-Proof.
- rewrite affine_affinely.
- apply (anti_symm _), plainly_cmra_valid_1. apply plainly_elim, _.
-Qed.
+Lemma plainly_cmra_valid {A : cmraT} (a : A) : ■✓ a ⊣⊢ ✓ a.
+Proof. apply (anti_symm _), plainly_cmra_valid_1. apply plainly_elim, _. Qed.
Lemma affinely_persistently_cmra_valid {A : cmraT} (a : A) : □ ✓ a ⊣⊢ ✓ a.
Proof.
rewrite affine_affinely. intros; apply (anti_symm _); first by rewrite persistently_elim.
diff --git a/theories/base_logic/lib/fancy_updates.v b/theories/base_logic/lib/fancy_updates.v
index 10f0a3b448d1cee136cd7316daaa3d7444bbb9bf..5861c57d92af8c66e13c17ecb848f083dbc49375 100644
--- a/theories/base_logic/lib/fancy_updates.v
+++ b/theories/base_logic/lib/fancy_updates.v
@@ -1,8 +1,7 @@
From iris.base_logic.lib Require Export own.
From stdpp Require Export coPset.
From iris.base_logic.lib Require Import wsat.
-From iris.algebra Require Import gmap.
-From iris.proofmode Require Import tactics classes.
+From iris.proofmode Require Import tactics.
Set Default Proof Using "Type".
Export invG.
Import uPred.
@@ -10,18 +9,17 @@ Import uPred.
Definition uPred_fupd_def `{invG Σ} (E1 E2 : coPset) (P : iProp Σ) : iProp Σ :=
(wsat ∗ ownE E1 ==∗ ◇ (wsat ∗ ownE E2 ∗ P))%I.
Definition uPred_fupd_aux `{invG Σ} : seal uPred_fupd_def. by eexists. Qed.
-Instance uPred_fupd `{invG Σ} : FUpd (iProp Σ):= unseal uPred_fupd_aux.
-Definition uPred_fupd_eq `{invG Σ} : fupd = uPred_fupd_def := seal_eq uPred_fupd_aux.
+Definition uPred_fupd `{invG Σ} : FUpd (iProp Σ):= unseal uPred_fupd_aux.
+Definition uPred_fupd_eq `{invG Σ} : @fupd _ uPred_fupd = uPred_fupd_def :=
+ seal_eq uPred_fupd_aux.
-Instance fupd_facts `{invG Σ} : FUpdFacts (uPredSI (iResUR Σ)).
+Lemma uPred_fupd_mixin `{invG Σ} : BiFUpdMixin (uPredSI (iResUR Σ)) uPred_fupd.
Proof.
split.
- - apply _.
- rewrite uPred_fupd_eq. solve_proper.
- intros E1 E2 P (E1''&->&?)%subseteq_disjoint_union_L.
rewrite uPred_fupd_eq /uPred_fupd_def ownE_op //.
- by iIntros "$ ($ & $ & HE) !> !> [$ $] !> !>" .
- - rewrite uPred_fupd_eq. by iIntros (E P) ">? [$ $] !> !>".
+ by iIntros "$ ($ & $ & HE) !> !> [$ $] !> !>" .
- rewrite uPred_fupd_eq. iIntros (E1 E2 P) ">H [Hw HE]". iApply "H"; by iFrame.
- rewrite uPred_fupd_eq. iIntros (E1 E2 P Q HPQ) "HP HwE". rewrite -HPQ. by iApply "HP".
- rewrite uPred_fupd_eq. iIntros (E1 E2 E3 P) "HP HwE".
@@ -32,6 +30,16 @@ Proof.
iDestruct (ownE_op' with "[HE2 HEf]") as "[? $]"; first by iFrame.
iIntros "!> !>". by iApply "HP".
- rewrite uPred_fupd_eq /uPred_fupd_def. by iIntros (????) "[HwP $]".
+Qed.
+Instance uPred_bi_fupd `{invG Σ} : BiFUpd (uPredSI (iResUR Σ)) :=
+ {| bi_fupd_mixin := uPred_fupd_mixin |}.
+
+Instance uPred_bi_bupd_fupd `{invG Σ} : BiBUpdFUpd (uPredSI (iResUR Σ)).
+Proof. rewrite /BiBUpdFUpd uPred_fupd_eq. by iIntros (E P) ">? [$ $] !> !>". Qed.
+
+Instance uPred_bi_fupd_plainly `{invG Σ} : BiFUpdPlainly (uPredSI (iResUR Σ)).
+Proof.
+ split.
- iIntros (E1 E2 E2' P Q ? (E3&->&HE)%subseteq_disjoint_union_L) "HQP HQ".
rewrite uPred_fupd_eq /uPred_fupd_def ownE_op //. iIntros "H".
iMod ("HQ" with "H") as ">(Hws & [HE1 HE3] & HQ)"; iModIntro.
@@ -41,4 +49,4 @@ Proof.
- rewrite uPred_fupd_eq /uPred_fupd_def. iIntros (E P ?) "HP [Hw HE]".
iAssert (â–· â—‡ P)%I with "[-]" as "#$"; last by iFrame.
iNext. by iMod ("HP" with "[$]") as "(_ & _ & HP)".
-Qed.
+Qed.
\ No newline at end of file
diff --git a/theories/base_logic/upred.v b/theories/base_logic/upred.v
index b7a6e372142ab0c2457de7b47a8fe8c31eeac952..4284299a89ec8ed6fa7d35fbd43edf706e338f1e 100644
--- a/theories/base_logic/upred.v
+++ b/theories/base_logic/upred.v
@@ -1,5 +1,5 @@
From iris.algebra Require Export cmra updates.
-From iris.bi Require Export derived_connectives updates.
+From iris.bi Require Export derived_connectives updates plainly.
From stdpp Require Import finite.
Set Default Proof Using "Type".
Local Hint Extern 1 (_ ≼ _) => etrans; [eassumption|].
@@ -278,13 +278,13 @@ Definition uPred_wand_eq :
(* Equivalently, this could be `∀ y, P n y`. That's closer to the intuition
of "embedding the step-indexed logic in Iris", but the two are equivalent
because Iris is afine. The following is easier to work with. *)
-Program Definition uPred_plainly_def {M} (P : uPred M) : uPred M :=
+Program Definition uPred_plainly_def {M} : Plainly (uPred M) := λ P,
{| uPred_holds n x := P n ε |}.
Solve Obligations with naive_solver eauto using uPred_mono, ucmra_unit_validN.
-Definition uPred_plainly_aux : seal (@uPred_plainly_def). by eexists. Qed.
-Definition uPred_plainly {M} := unseal uPred_plainly_aux M.
-Definition uPred_plainly_eq :
- @uPred_plainly = @uPred_plainly_def := seal_eq uPred_plainly_aux.
+Definition uPred_plainly_aux {M} : seal (@uPred_plainly_def M). by eexists. Qed.
+Definition uPred_plainly {M} := unseal (@uPred_plainly_aux M).
+Definition uPred_plainly_eq M :
+ @plainly _ uPred_plainly = @uPred_plainly_def M := seal_eq uPred_plainly_aux.
Program Definition uPred_persistently_def {M} (P : uPred M) : uPred M :=
{| uPred_holds n x := P n (core x) |}.
@@ -336,7 +336,7 @@ Next Obligation.
eauto using uPred_mono, cmra_includedN_l.
Qed.
Definition uPred_bupd_aux {M} : seal (@uPred_bupd_def M). by eexists. Qed.
-Instance uPred_bupd {M} : BUpd (uPred M) := unseal uPred_bupd_aux.
+Definition uPred_bupd {M} : BUpd (uPred M) := unseal uPred_bupd_aux.
Definition uPred_bupd_eq {M} :
@bupd _ uPred_bupd = @uPred_bupd_def M := seal_eq uPred_bupd_aux.
@@ -347,11 +347,11 @@ Definition unseal_eqs :=
uPred_plainly_eq, uPred_persistently_eq, uPred_later_eq, uPred_ownM_eq,
uPred_cmra_valid_eq, @uPred_bupd_eq).
Ltac unseal := (* Coq unfold is used to circumvent bug #5699 in rewrite /foo *)
- unfold bi_emp; simpl;
+ unfold bi_emp; simpl; unfold sbi_emp; simpl;
unfold uPred_emp, bi_pure, bi_and, bi_or, bi_impl, bi_forall, bi_exist,
- bi_sep, bi_wand, bi_plainly, bi_persistently, sbi_internal_eq, sbi_later; simpl;
+ bi_sep, bi_wand, bi_persistently, sbi_internal_eq, sbi_later; simpl;
unfold sbi_emp, sbi_pure, sbi_and, sbi_or, sbi_impl, sbi_forall, sbi_exist,
- sbi_internal_eq, sbi_sep, sbi_wand, sbi_plainly, sbi_persistently; simpl;
+ sbi_internal_eq, sbi_sep, sbi_wand, sbi_persistently; simpl;
rewrite !unseal_eqs /=.
End uPred_unseal.
Import uPred_unseal.
@@ -361,7 +361,7 @@ Local Arguments uPred_holds {_} !_ _ _ /.
Lemma uPred_bi_mixin (M : ucmraT) :
BiMixin
uPred_entails uPred_emp uPred_pure uPred_and uPred_or uPred_impl
- (@uPred_forall M) (@uPred_exist M) uPred_sep uPred_wand uPred_plainly
+ (@uPred_forall M) (@uPred_exist M) uPred_sep uPred_wand
uPred_persistently.
Proof.
split.
@@ -398,9 +398,6 @@ Proof.
intros n P P' HP Q Q' HQ; split=> n' x ??.
unseal; split; intros HPQ x' n'' ???;
apply HQ, HPQ, HP; eauto using cmra_validN_op_r.
- - (* NonExpansive uPred_plainly *)
- intros n P1 P2 HP.
- unseal; split=> n' x; split; apply HP; eauto using @ucmra_unit_validN.
- (* NonExpansive uPred_persistently *)
intros n P1 P2 HP.
unseal; split=> n' x; split; apply HP; eauto using @cmra_core_validN.
@@ -460,32 +457,12 @@ Proof.
- (* (P ⊢ Q -∗ R) → P ∗ Q ⊢ R *)
intros P Q R. unseal=> HPQR. split; intros n x ? (?&?&?&?&?). ofe_subst.
eapply HPQR; eauto using cmra_validN_op_l.
- - (* (P ⊢ Q) → bi_plainly P ⊢ bi_plainly Q *)
- intros P QR HP. unseal; split=> n x ? /=. by apply HP, ucmra_unit_validN.
- - (* bi_plainly P ⊢ bi_persistently P *)
- unseal; split; simpl; eauto using uPred_mono, @ucmra_unit_leastN.
- - (* bi_plainly P ⊢ bi_plainly (bi_plainly P) *)
- unseal; split=> n x ?? //.
- - (* (∀ a, bi_plainly (Ψ a)) ⊢ bi_plainly (∀ a, Ψ a) *)
- by unseal.
- - (* (bi_plainly P → bi_persistently Q) ⊢ bi_persistently (bi_plainly P → Q) *)
- unseal; split=> /= n x ? HPQ n' x' ????.
- eapply uPred_mono with n' (core x)=>//; [|by apply cmra_included_includedN].
- apply (HPQ n' x); eauto using cmra_validN_le.
- - (* (bi_plainly P → bi_plainly Q) ⊢ bi_plainly (bi_plainly P → Q) *)
- unseal; split=> /= n x ? HPQ n' x' ????.
- eapply uPred_mono with n' ε=>//; [|by apply cmra_included_includedN].
- apply (HPQ n' x); eauto using cmra_validN_le.
- - (* P ⊢ bi_plainly emp (ADMISSIBLE) *)
- by unseal.
- - (* bi_plainly P ∗ Q ⊢ bi_plainly P *)
- intros P Q. move: (uPred_persistently P)=> P'.
- unseal; split; intros n x ? (x1&x2&?&?&_); ofe_subst;
- eauto using uPred_mono, cmra_includedN_l.
- (* (P ⊢ Q) → bi_persistently P ⊢ bi_persistently Q *)
intros P QR HP. unseal; split=> n x ? /=. by apply HP, cmra_core_validN.
- (* bi_persistently P ⊢ bi_persistently (bi_persistently P) *)
intros P. unseal; split=> n x ?? /=. by rewrite cmra_core_idemp.
+ - (* P ⊢ bi_persistently emp (ADMISSIBLE) *)
+ by unseal.
- (* (∀ a, bi_persistently (Ψ a)) ⊢ bi_persistently (∀ a, Ψ a) *)
by unseal.
- (* bi_persistently (∃ a, Ψ a) ⊢ ∃ a, bi_persistently (Ψ a) *)
@@ -499,10 +476,10 @@ Proof.
exists (core x), x; rewrite ?cmra_core_l; auto.
Qed.
-Lemma uPred_sbi_mixin (M : ucmraT) : SbiMixin uPred_ofe_mixin
- uPred_entails uPred_pure uPred_and uPred_or uPred_impl
- (@uPred_forall M) (@uPred_exist M) uPred_sep uPred_wand
- uPred_plainly uPred_persistently (@uPred_internal_eq M) uPred_later.
+Lemma uPred_sbi_mixin (M : ucmraT) : SbiMixin
+ uPred_entails uPred_pure uPred_or uPred_impl
+ (@uPred_forall M) (@uPred_exist M) uPred_sep
+ uPred_persistently (@uPred_internal_eq M) uPred_later.
Proof.
split.
- (* Contractive sbi_later *)
@@ -523,10 +500,6 @@ Proof.
by unseal.
- (* Discrete a → a ≡ b ⊣⊢ ⌜a ≡ b⌠*)
intros A a b ?. unseal; split=> n x ?; by apply (discrete_iff n).
- - (* bi_plainly ((P -∗ Q) ∧ (Q -∗ P)) ⊢ P ≡ Q *)
- unseal; split=> n x ? /= HPQ. split=> n' x' ??.
- move: HPQ=> [] /(_ n' x'); rewrite !left_id=> ?.
- move=> /(_ n' x'); rewrite !left_id=> ?. naive_solver.
- (* Next x ≡ Next y ⊢ ▷ (x ≡ y) *)
by unseal.
- (* ▷ (x ≡ y) ⊢ Next x ≡ Next y *)
@@ -550,10 +523,6 @@ Proof.
- (* ▷ P ∗ ▷ Q ⊢ ▷ (P ∗ Q) *)
intros P Q. unseal; split=> -[|n] x ? /=; [done|intros (x1&x2&Hx&?&?)].
exists x1, x2; eauto using dist_S.
- - (* ▷ bi_plainly P ⊢ bi_plainly (▷ P) *)
- by unseal.
- - (* bi_plainly (▷ P) ⊢ ▷ bi_plainly P *)
- by unseal.
- (* ▷ bi_persistently P ⊢ bi_persistently (▷ P) *)
by unseal.
- (* bi_persistently (▷ P) ⊢ ▷ bi_persistently P *)
@@ -575,6 +544,74 @@ Coercion uPred_valid {M} : uPred M → Prop := bi_valid.
(* Latest notation *)
Notation "✓ x" := (uPred_cmra_valid x) (at level 20) : bi_scope.
+Lemma uPred_plainly_mixin M : BiPlainlyMixin (uPredSI M) uPred_plainly.
+Proof.
+ split.
+ - (* NonExpansive uPred_plainly *)
+ intros n P1 P2 HP.
+ unseal; split=> n' x; split; apply HP; eauto using @ucmra_unit_validN.
+ - (* (P ⊢ Q) → ■P ⊢ ■Q *)
+ intros P QR HP. unseal; split=> n x ? /=. by apply HP, ucmra_unit_validN.
+ - (* ■P ⊢ bi_persistently P *)
+ unseal; split; simpl; eauto using uPred_mono, @ucmra_unit_leastN.
+ - (* ■P ⊢ ■■P *)
+ unseal; split=> n x ?? //.
+ - (* (∀ a, ■(Ψ a)) ⊢ ■(∀ a, Ψ a) *)
+ by unseal.
+ - (* (■P → bi_persistently Q) ⊢ bi_persistently (■P → Q) *)
+ unseal; split=> /= n x ? HPQ n' x' ????.
+ eapply uPred_mono with n' (core x)=>//; [|by apply cmra_included_includedN].
+ apply (HPQ n' x); eauto using cmra_validN_le.
+ - (* (■P → ■Q) ⊢ ■(■P → Q) *)
+ unseal; split=> /= n x ? HPQ n' x' ????.
+ eapply uPred_mono with n' ε=>//; [|by apply cmra_included_includedN].
+ apply (HPQ n' x); eauto using cmra_validN_le.
+ - (* P ⊢ ■emp (ADMISSIBLE) *)
+ by unseal.
+ - (* ■P ∗ Q ⊢ ■P *)
+ intros P Q. move: (uPred_persistently P)=> P'.
+ unseal; split; intros n x ? (x1&x2&?&?&_); ofe_subst;
+ eauto using uPred_mono, cmra_includedN_l.
+ - (* ■((P -∗ Q) ∧ (Q -∗ P)) ⊢ P ≡ Q *)
+ unseal; split=> n x ? /= HPQ. split=> n' x' ??.
+ move: HPQ=> [] /(_ n' x'); rewrite !left_id=> ?.
+ move=> /(_ n' x'); rewrite !left_id=> ?. naive_solver.
+ - (* ▷ ■P ⊢ ■▷ P *)
+ by unseal.
+ - (* ■▷ P ⊢ ▷ ■P *)
+ by unseal.
+Qed.
+Instance uPred_plainlyC M : BiPlainly (uPredSI M) :=
+ {| bi_plainly_mixin := uPred_plainly_mixin M |}.
+
+Lemma uPred_bupd_mixin M : BiBUpdMixin (uPredI M) uPred_bupd.
+Proof.
+ split.
+ - intros n P Q HPQ.
+ unseal; split=> n' x; split; intros HP k yf ??;
+ destruct (HP k yf) as (x'&?&?); auto;
+ exists x'; split; auto; apply HPQ; eauto using cmra_validN_op_l.
+ - unseal. split=> n x ? HP k yf ?; exists x; split; first done.
+ apply uPred_mono with n x; eauto using cmra_validN_op_l.
+ - unseal. intros HPQ; split=> n x ? HP k yf ??.
+ destruct (HP k yf) as (x'&?&?); eauto.
+ exists x'; split; eauto using uPred_in_entails, cmra_validN_op_l.
+ - unseal; split; naive_solver.
+ - unseal. split; intros n x ? (x1&x2&Hx&HP&?) k yf ??.
+ destruct (HP k (x2 â‹… yf)) as (x'&?&?); eauto.
+ { by rewrite assoc -(dist_le _ _ _ _ Hx); last lia. }
+ exists (x' â‹… x2); split; first by rewrite -assoc.
+ exists x', x2. eauto using uPred_mono, cmra_validN_op_l, cmra_validN_op_r.
+Qed.
+Instance uPred_bi_bupd M : BiBUpd (uPredI M) := {| bi_bupd_mixin := uPred_bupd_mixin M |}.
+
+Instance uPred_bi_bupd_plainly M : BiBUpdPlainly (uPredSI M).
+Proof.
+ rewrite /BiBUpdPlainly. unseal; split => n x Hnx /= Hng.
+ destruct (Hng n ε) as [? [_ Hng']]; try rewrite right_id; auto.
+ eapply uPred_mono; eauto using ucmra_unit_leastN.
+Qed.
+
Module uPred.
Include uPred_unseal.
Section uPred.
@@ -606,7 +643,7 @@ Global Instance cmra_valid_proper {A : cmraT} :
Global Instance uPred_affine : BiAffine (uPredI M) | 0.
Proof. intros P. rewrite /Affine. by apply bi.pure_intro. Qed.
-Global Instance uPred_plainly_exist_1 : BiPlainlyExist (uPredI M).
+Global Instance uPred_plainly_exist_1 : BiPlainlyExist (uPredSI M).
Proof. unfold BiPlainlyExist. by unseal. Qed.
(** Limits *)
@@ -660,7 +697,7 @@ Lemma cmra_valid_elim {A : cmraT} (a : A) : ¬ ✓{0} a → ✓ a ⊢ (False : u
Proof.
intros Ha. unseal. split=> n x ??; apply Ha, cmra_validN_le with n; auto.
Qed.
-Lemma plainly_cmra_valid_1 {A : cmraT} (a : A) : ✓ a ⊢ bi_plainly (✓ a : uPred M).
+Lemma plainly_cmra_valid_1 {A : cmraT} (a : A) : ✓ a ⊢ ■(✓ a : uPred M).
Proof. by unseal. Qed.
Lemma cmra_valid_weaken {A : cmraT} (a b : A) : ✓ (a ⋅ b) ⊢ (✓ a : uPred M).
Proof. unseal; split=> n x _; apply cmra_validN_op_l. Qed.
@@ -675,30 +712,6 @@ Lemma ofe_fun_validI `{B : A → ucmraT} (g : ofe_fun B) :
(✓ g : uPred M) ⊣⊢ ∀ i, ✓ g i.
Proof. by uPred.unseal. Qed.
-(* Basic update modality *)
-Global Instance bupd_facts : BUpdFacts (uPredSI M).
-Proof.
- split.
- - intros n P Q HPQ.
- unseal; split=> n' x; split; intros HP k yf ??;
- destruct (HP k yf) as (x'&?&?); auto;
- exists x'; split; auto; apply HPQ; eauto using cmra_validN_op_l.
- - unseal. split=> n x ? HP k yf ?; exists x; split; first done.
- apply uPred_mono with n x; eauto using cmra_validN_op_l.
- - unseal. intros HPQ; split=> n x ? HP k yf ??.
- destruct (HP k yf) as (x'&?&?); eauto.
- exists x'; split; eauto using uPred_in_entails, cmra_validN_op_l.
- - unseal; split; naive_solver.
- - unseal. split; intros n x ? (x1&x2&Hx&HP&?) k yf ??.
- destruct (HP k (x2 â‹… yf)) as (x'&?&?); eauto.
- { by rewrite assoc -(dist_le _ _ _ _ Hx); last lia. }
- exists (x' â‹… x2); split; first by rewrite -assoc.
- exists x', x2. eauto using uPred_mono, cmra_validN_op_l, cmra_validN_op_r.
- - unseal; split => n x Hnx /= Hng.
- destruct (Hng n ε) as [? [_ Hng']]; try rewrite right_id; auto.
- eapply uPred_mono; eauto using ucmra_unit_leastN.
-Qed.
-
Lemma bupd_ownM_updateP x (Φ : M → Prop) :
x ~~>: Φ → uPred_ownM x ==∗ ∃ y, ⌜Φ y⌠∧ uPred_ownM y.
Proof.
diff --git a/theories/bi/bi.v b/theories/bi/bi.v
index 4cef163c554776a75c9fd03b053716d240e90d0d..d00050dc4309fbd0a9f0b4f3fbfd6b50001e244c 100644
--- a/theories/bi/bi.v
+++ b/theories/bi/bi.v
@@ -1,4 +1,4 @@
-From iris.bi Require Export derived_laws big_op updates embedding.
+From iris.bi Require Export derived_laws big_op updates plainly embedding.
Set Default Proof Using "Type".
Module Import bi.
@@ -16,4 +16,4 @@ Hint Resolve sep_elim_l sep_elim_r sep_mono : BI.
Hint Immediate True_intro False_elim : BI.
(*
Hint Immediate iff_refl internal_eq_refl' : BI.
-*)
\ No newline at end of file
+*)
diff --git a/theories/bi/big_op.v b/theories/bi/big_op.v
index 3472d004e2a2af15305b93e0d33de27ba92dbedb..d7648e9287febcd3da580f7b3a9af2a9703d6aa8 100644
--- a/theories/bi/big_op.v
+++ b/theories/bi/big_op.v
@@ -1,5 +1,6 @@
From iris.algebra Require Export big_op.
From iris.bi Require Export derived_laws.
+From iris.bi Require Import plainly.
From stdpp Require Import countable fin_collections functions.
Set Default Proof Using "Type".
@@ -125,10 +126,6 @@ Section sep_list.
⊢ ([∗ list] k↦x ∈ l, Φ k x) ∧ ([∗ list] k↦x ∈ l, Ψ k x).
Proof. auto using and_intro, big_sepL_mono, and_elim_l, and_elim_r. Qed.
- Lemma big_sepL_plainly `{BiAffine PROP} Φ l :
- bi_plainly ([∗ list] k↦x ∈ l, Φ k x) ⊣⊢ [∗ list] k↦x ∈ l, bi_plainly (Φ k x).
- Proof. apply (big_opL_commute _). Qed.
-
Lemma big_sepL_persistently `{BiAffine PROP} Φ l :
bi_persistently ([∗ list] k↦x ∈ l, Φ k x) ⊣⊢
[∗ list] k↦x ∈ l, bi_persistently (Φ k x).
@@ -163,16 +160,6 @@ Section sep_list.
apply forall_intro=> k. by rewrite (forall_elim (S k)).
Qed.
- Global Instance big_sepL_nil_plain Φ :
- Plain ([∗ list] k↦x ∈ [], Φ k x).
- Proof. simpl; apply _. Qed.
- Global Instance big_sepL_plain Φ l :
- (∀ k x, Plain (Φ k x)) → Plain ([∗ list] k↦x ∈ l, Φ k x).
- Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed.
- Global Instance big_sepL_plain_id Ps :
- TCForall Plain Ps → Plain ([∗] Ps).
- Proof. induction 1; simpl; apply _. Qed.
-
Global Instance big_sepL_nil_persistent Φ :
Persistent ([∗ list] k↦x ∈ [], Φ k x).
Proof. simpl; apply _. Qed.
@@ -278,10 +265,6 @@ Section and_list.
⊢ ([∧ list] k↦x ∈ l, Φ k x) ∧ ([∧ list] k↦x ∈ l, Ψ k x).
Proof. auto using and_intro, big_andL_mono, and_elim_l, and_elim_r. Qed.
- Lemma big_andL_plainly Φ l :
- bi_plainly ([∧ list] k↦x ∈ l, Φ k x) ⊣⊢ [∧ list] k↦x ∈ l, bi_plainly (Φ k x).
- Proof. apply (big_opL_commute _). Qed.
-
Lemma big_andL_persistently Φ l :
bi_persistently ([∧ list] k↦x ∈ l, Φ k x) ⊣⊢
[∧ list] k↦x ∈ l, bi_persistently (Φ k x).
@@ -299,19 +282,13 @@ Section and_list.
- rewrite -IH. apply forall_intro=> k; by rewrite (forall_elim (S k)).
Qed.
- Global Instance big_andL_nil_plain Φ :
- Plain ([∧ list] k↦x ∈ [], Φ k x).
- Proof. simpl; apply _. Qed.
- Global Instance big_andL_plain Φ l :
- (∀ k x, Plain (Φ k x)) → Plain ([∧ list] k↦x ∈ l, Φ k x).
- Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed.
-
Global Instance big_andL_nil_persistent Φ :
Persistent ([∧ list] k↦x ∈ [], Φ k x).
Proof. simpl; apply _. Qed.
Global Instance big_andL_persistent Φ l :
(∀ k x, Persistent (Φ k x)) → Persistent ([∧ list] k↦x ∈ l, Φ k x).
Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed.
+
End and_list.
(** ** Big ops over finite maps *)
@@ -420,10 +397,6 @@ Section gmap.
⊢ ([∗ map] k↦x ∈ m, Φ k x) ∧ ([∗ map] k↦x ∈ m, Ψ k x).
Proof. auto using and_intro, big_sepM_mono, and_elim_l, and_elim_r. Qed.
- Lemma big_sepM_plainly `{BiAffine PROP} Φ m :
- bi_plainly ([∗ map] k↦x ∈ m, Φ k x) ⊣⊢ [∗ map] k↦x ∈ m, bi_plainly (Φ k x).
- Proof. apply (big_opM_commute _). Qed.
-
Lemma big_sepM_persistently `{BiAffine PROP} Φ m :
(bi_persistently ([∗ map] k↦x ∈ m, Φ k x)) ⊣⊢
([∗ map] k↦x ∈ m, bi_persistently (Φ k x)).
@@ -464,12 +437,6 @@ Section gmap.
by rewrite pure_True // True_impl.
Qed.
- Global Instance big_sepM_empty_plain Φ : Plain ([∗ map] k↦x ∈ ∅, Φ k x).
- Proof. rewrite /big_opM map_to_list_empty. apply _. Qed.
- Global Instance big_sepM_plain Φ m :
- (∀ k x, Plain (Φ k x)) → Plain ([∗ map] k↦x ∈ m, Φ k x).
- Proof. intros. apply (big_sepL_plain _ _)=> _ [??]; apply _. Qed.
-
Global Instance big_sepM_empty_persistent Φ :
Persistent ([∗ map] k↦x ∈ ∅, Φ k x).
Proof. rewrite /big_opM map_to_list_empty. apply _. Qed.
@@ -596,10 +563,6 @@ Section gset.
([∗ set] y ∈ X, Φ y ∧ Ψ y) ⊢ ([∗ set] y ∈ X, Φ y) ∧ ([∗ set] y ∈ X, Ψ y).
Proof. auto using and_intro, big_sepS_mono, and_elim_l, and_elim_r. Qed.
- Lemma big_sepS_plainly `{BiAffine PROP} Φ X :
- bi_plainly ([∗ set] y ∈ X, Φ y) ⊣⊢ [∗ set] y ∈ X, bi_plainly (Φ y).
- Proof. apply (big_opS_commute _). Qed.
-
Lemma big_sepS_persistently `{BiAffine PROP} Φ X :
bi_persistently ([∗ set] y ∈ X, Φ y) ⊣⊢ [∗ set] y ∈ X, bi_persistently (Φ y).
Proof. apply (big_opS_commute _). Qed.
@@ -633,12 +596,6 @@ Section gset.
by rewrite pure_True ?True_impl; last set_solver.
Qed.
- Global Instance big_sepS_empty_plain Φ : Plain ([∗ set] x ∈ ∅, Φ x).
- Proof. rewrite /big_opS elements_empty. apply _. Qed.
- Global Instance big_sepS_plain Φ X :
- (∀ x, Plain (Φ x)) → Plain ([∗ set] x ∈ X, Φ x).
- Proof. rewrite /big_opS. apply _. Qed.
-
Global Instance big_sepS_empty_persistent Φ :
Persistent ([∗ set] x ∈ ∅, Φ x).
Proof. rewrite /big_opS elements_empty. apply _. Qed.
@@ -714,21 +671,11 @@ Section gmultiset.
([∗ mset] y ∈ X, Φ y ∧ Ψ y) ⊢ ([∗ mset] y ∈ X, Φ y) ∧ ([∗ mset] y ∈ X, Ψ y).
Proof. auto using and_intro, big_sepMS_mono, and_elim_l, and_elim_r. Qed.
- Lemma big_sepMS_plainly `{BiAffine PROP} Φ X :
- bi_plainly ([∗ mset] y ∈ X, Φ y) ⊣⊢ [∗ mset] y ∈ X, bi_plainly (Φ y).
- Proof. apply (big_opMS_commute _). Qed.
-
Lemma big_sepMS_persistently `{BiAffine PROP} Φ X :
bi_persistently ([∗ mset] y ∈ X, Φ y) ⊣⊢
[∗ mset] y ∈ X, bi_persistently (Φ y).
Proof. apply (big_opMS_commute _). Qed.
- Global Instance big_sepMS_empty_plain Φ : Plain ([∗ mset] x ∈ ∅, Φ x).
- Proof. rewrite /big_opMS gmultiset_elements_empty. apply _. Qed.
- Global Instance big_sepMS_plain Φ X :
- (∀ x, Plain (Φ x)) → Plain ([∗ mset] x ∈ X, Φ x).
- Proof. rewrite /big_opMS. apply _. Qed.
-
Global Instance big_sepMS_empty_persistent Φ :
Persistent ([∗ mset] x ∈ ∅, Φ x).
Proof. rewrite /big_opMS gmultiset_elements_empty. apply _. Qed.
@@ -773,6 +720,34 @@ Section list.
Global Instance big_sepL_timeless_id `{!Timeless (emp%I : PROP)} Ps :
TCForall Timeless Ps → Timeless ([∗] Ps).
Proof. induction 1; simpl; apply _. Qed.
+
+ Section plainly.
+ Context `{!BiPlainly PROP}.
+
+ Lemma big_sepL_plainly `{!BiAffine PROP} Φ l :
+ ■([∗ list] k↦x ∈ l, Φ k x) ⊣⊢ [∗ list] k↦x ∈ l, ■(Φ k x).
+ Proof. apply (big_opL_commute _). Qed.
+
+ Global Instance big_sepL_nil_plain `{!BiAffine PROP} Φ :
+ Plain ([∗ list] k↦x ∈ [], Φ k x).
+ Proof. simpl; apply _. Qed.
+
+ Global Instance big_sepL_plain `{!BiAffine PROP} Φ l :
+ (∀ k x, Plain (Φ k x)) → Plain ([∗ list] k↦x ∈ l, Φ k x).
+ Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed.
+
+ Lemma big_andL_plainly Φ l :
+ ■([∧ list] k↦x ∈ l, Φ k x) ⊣⊢ [∧ list] k↦x ∈ l, ■(Φ k x).
+ Proof. apply (big_opL_commute _). Qed.
+
+ Global Instance big_andL_nil_plain Φ :
+ Plain ([∧ list] k↦x ∈ [], Φ k x).
+ Proof. simpl; apply _. Qed.
+
+ Global Instance big_andL_plain Φ l :
+ (∀ k x, Plain (Φ k x)) → Plain ([∧ list] k↦x ∈ l, Φ k x).
+ Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed.
+ End plainly.
End list.
(** ** Big ops over finite maps *)
@@ -795,6 +770,21 @@ Section gmap.
Global Instance big_sepM_timeless `{!Timeless (emp%I : PROP)} Φ m :
(∀ k x, Timeless (Φ k x)) → Timeless ([∗ map] k↦x ∈ m, Φ k x).
Proof. intros. apply big_sepL_timeless=> _ [??]; apply _. Qed.
+
+ Section plainly.
+ Context `{!BiPlainly PROP}.
+
+ Lemma big_sepM_plainly `{BiAffine PROP} Φ m :
+ ■([∗ map] k↦x ∈ m, Φ k x) ⊣⊢ [∗ map] k↦x ∈ m, ■(Φ k x).
+ Proof. apply (big_opM_commute _). Qed.
+
+ Global Instance big_sepM_empty_plain `{BiAffine PROP} Φ :
+ Plain ([∗ map] k↦x ∈ ∅, Φ k x).
+ Proof. rewrite /big_opM map_to_list_empty. apply _. Qed.
+ Global Instance big_sepM_plain `{BiAffine PROP} Φ m :
+ (∀ k x, Plain (Φ k x)) → Plain ([∗ map] k↦x ∈ m, Φ k x).
+ Proof. intros. apply (big_sepL_plain _ _)=> _ [??]; apply _. Qed.
+ End plainly.
End gmap.
(** ** Big ops over finite sets *)
@@ -817,6 +807,20 @@ Section gset.
Global Instance big_sepS_timeless `{!Timeless (emp%I : PROP)} Φ X :
(∀ x, Timeless (Φ x)) → Timeless ([∗ set] x ∈ X, Φ x).
Proof. rewrite /big_opS. apply _. Qed.
+
+ Section plainly.
+ Context `{!BiPlainly PROP}.
+
+ Lemma big_sepS_plainly `{BiAffine PROP} Φ X :
+ ■([∗ set] y ∈ X, Φ y) ⊣⊢ [∗ set] y ∈ X, ■(Φ y).
+ Proof. apply (big_opS_commute _). Qed.
+
+ Global Instance big_sepS_empty_plain `{BiAffine PROP} Φ : Plain ([∗ set] x ∈ ∅, Φ x).
+ Proof. rewrite /big_opS elements_empty. apply _. Qed.
+ Global Instance big_sepS_plain `{BiAffine PROP} Φ X :
+ (∀ x, Plain (Φ x)) → Plain ([∗ set] x ∈ X, Φ x).
+ Proof. rewrite /big_opS. apply _. Qed.
+ End plainly.
End gset.
(** ** Big ops over finite multisets *)
@@ -839,6 +843,20 @@ Section gmultiset.
Global Instance big_sepMS_timeless `{!Timeless (emp%I : PROP)} Φ X :
(∀ x, Timeless (Φ x)) → Timeless ([∗ mset] x ∈ X, Φ x).
Proof. rewrite /big_opMS. apply _. Qed.
+
+ Section plainly.
+ Context `{!BiPlainly PROP}.
+
+ Lemma big_sepMS_plainly `{BiAffine PROP} Φ X :
+ ■([∗ mset] y ∈ X, Φ y) ⊣⊢ [∗ mset] y ∈ X, ■(Φ y).
+ Proof. apply (big_opMS_commute _). Qed.
+
+ Global Instance big_sepMS_empty_plain `{BiAffine PROP} Φ : Plain ([∗ mset] x ∈ ∅, Φ x).
+ Proof. rewrite /big_opMS gmultiset_elements_empty. apply _. Qed.
+ Global Instance big_sepMS_plain `{BiAffine PROP} Φ X :
+ (∀ x, Plain (Φ x)) → Plain ([∗ mset] x ∈ X, Φ x).
+ Proof. rewrite /big_opMS. apply _. Qed.
+ End plainly.
End gmultiset.
End sbi_big_op.
End bi.
diff --git a/theories/bi/derived_connectives.v b/theories/bi/derived_connectives.v
index 1ad05c3aaab07f29ac29662081209db05559bb61..fbd3213ec5da0785ce67333a1f60fa5836f0bbc4 100644
--- a/theories/bi/derived_connectives.v
+++ b/theories/bi/derived_connectives.v
@@ -13,12 +13,6 @@ Arguments bi_wand_iff {_} _%I _%I : simpl never.
Instance: Params (@bi_wand_iff) 1.
Infix "∗-∗" := bi_wand_iff (at level 95, no associativity) : bi_scope.
-Class Plain {PROP : bi} (P : PROP) := plain : P ⊢ bi_plainly P.
-Arguments Plain {_} _%I : simpl never.
-Arguments plain {_} _%I {_}.
-Hint Mode Plain + ! : typeclass_instances.
-Instance: Params (@Plain) 1.
-
Class Persistent {PROP : bi} (P : PROP) := persistent : P ⊢ bi_persistently P.
Arguments Persistent {_} _%I : simpl never.
Arguments persistent {_} _%I {_}.
@@ -31,8 +25,6 @@ Instance: Params (@bi_affinely) 1.
Typeclasses Opaque bi_affinely.
Notation "â–¡ P" := (bi_affinely (bi_persistently P))%I
(at level 20, right associativity) : bi_scope.
-Notation "â– P" := (bi_affinely (bi_plainly P))%I
- (at level 20, right associativity) : bi_scope.
Class Affine {PROP : bi} (Q : PROP) := affine : Q ⊢ emp.
Arguments Affine {_} _%I : simpl never.
@@ -45,11 +37,6 @@ Existing Instance absorbing_bi | 0.
Class BiPositive (PROP : bi) :=
bi_positive (P Q : PROP) : bi_affinely (P ∗ Q) ⊢ bi_affinely P ∗ Q.
-Class BiPlainlyExist (PROP : bi) :=
- plainly_exist_1 A (Ψ : A → PROP) :
- bi_plainly (∃ a, Ψ a) ⊢ ∃ a, bi_plainly (Ψ a).
-Arguments plainly_exist_1 _ {_ _} _.
-
Definition bi_absorbingly {PROP : bi} (P : PROP) : PROP := (True ∗ P)%I.
Arguments bi_absorbingly {_} _%I : simpl never.
Instance: Params (@bi_absorbingly) 1.
@@ -59,12 +46,6 @@ Class Absorbing {PROP : bi} (P : PROP) := absorbing : bi_absorbingly P ⊢ P.
Arguments Absorbing {_} _%I : simpl never.
Arguments absorbing {_} _%I.
-Definition bi_plainly_if {PROP : bi} (p : bool) (P : PROP) : PROP :=
- (if p then bi_plainly P else P)%I.
-Arguments bi_plainly_if {_} !_ _%I /.
-Instance: Params (@bi_plainly_if) 2.
-Typeclasses Opaque bi_plainly_if.
-
Definition bi_persistently_if {PROP : bi} (p : bool) (P : PROP) : PROP :=
(if p then bi_persistently P else P)%I.
Arguments bi_persistently_if {_} !_ _%I /.
@@ -79,9 +60,6 @@ Typeclasses Opaque bi_affinely_if.
Notation "â–¡? p P" := (bi_affinely_if p (bi_persistently_if p P))%I
(at level 20, p at level 9, P at level 20,
right associativity, format "â–¡? p P") : bi_scope.
-Notation "â– ? p P" := (bi_affinely_if p (bi_plainly_if p P))%I
- (at level 20, p at level 9, P at level 20,
- right associativity, format "â– ? p P") : bi_scope.
Fixpoint bi_hexist {PROP : bi} {As} : himpl As PROP → PROP :=
match As return himpl As PROP → PROP with
diff --git a/theories/bi/derived_laws.v b/theories/bi/derived_laws.v
index 8e99ebd7dbf99dc4f9aed534e10445fd9e3c1762..474bf1bb7c390a79eb113b99730bcf59f5746860 100644
--- a/theories/bi/derived_laws.v
+++ b/theories/bi/derived_laws.v
@@ -68,8 +68,6 @@ Proof.
intros Φ1 Φ2 HΦ. apply equiv_dist=> n.
apply exist_ne=> x. apply equiv_dist, HΦ.
Qed.
-Global Instance plainly_proper :
- Proper ((⊣⊢) ==> (⊣⊢)) (@bi_plainly PROP) := ne_proper _.
Global Instance persistently_proper :
Proper ((⊣⊢) ==> (⊣⊢)) (@bi_persistently PROP) := ne_proper _.
@@ -755,9 +753,6 @@ Proof.
apply persistently_mono, impl_elim with P; auto.
Qed.
-Lemma persistently_emp_intro P : P ⊢ bi_persistently emp.
-Proof. by rewrite -plainly_elim_persistently -plainly_emp_intro. Qed.
-
Lemma persistently_True_emp : bi_persistently True ⊣⊢ bi_persistently emp.
Proof. apply (anti_symm _); auto using persistently_emp_intro. Qed.
@@ -941,182 +936,6 @@ Section persistently_affinely_bi.
Qed.
End persistently_affinely_bi.
-(* Properties of the plainness modality *)
-Hint Resolve plainly_mono.
-Global Instance plainly_mono' : Proper ((⊢) ==> (⊢)) (@bi_plainly PROP).
-Proof. intros P Q; apply plainly_mono. Qed.
-Global Instance plainly_flip_mono' :
- Proper (flip (⊢) ==> flip (⊢)) (@bi_plainly PROP).
-Proof. intros P Q; apply plainly_mono. Qed.
-
-Lemma persistently_plainly P : bi_persistently (bi_plainly P) ⊣⊢ bi_plainly P.
-Proof.
- apply (anti_symm _).
- - by rewrite persistently_elim_absorbingly /bi_absorbingly comm plainly_absorbing.
- - by rewrite {1}plainly_idemp_2 plainly_elim_persistently.
-Qed.
-Lemma plainly_persistently P : bi_plainly (bi_persistently P) ⊣⊢ bi_plainly P.
-Proof.
- apply (anti_symm _).
- - rewrite -{1}(left_id True%I bi_and (bi_plainly _)) (plainly_emp_intro True%I).
- rewrite -{2}(persistently_and_emp_elim P).
- rewrite !and_alt -plainly_forall_2. by apply forall_mono=> -[].
- - by rewrite {1}plainly_idemp_2 (plainly_elim_persistently P).
-Qed.
-
-Lemma absorbingly_plainly P : bi_absorbingly (bi_plainly P) ⊣⊢ bi_plainly P.
-Proof. by rewrite -(persistently_plainly P) absorbingly_persistently. Qed.
-
-Lemma plainly_and_sep_elim P Q : bi_plainly P ∧ Q -∗ (emp ∧ P) ∗ Q.
-Proof. by rewrite plainly_elim_persistently persistently_and_sep_elim_emp. Qed.
-Lemma plainly_and_sep_assoc P Q R :
- bi_plainly P ∧ (Q ∗ R) ⊣⊢ (bi_plainly P ∧ Q) ∗ R.
-Proof. by rewrite -(persistently_plainly P) persistently_and_sep_assoc. Qed.
-Lemma plainly_and_emp_elim P : emp ∧ bi_plainly P ⊢ P.
-Proof. by rewrite plainly_elim_persistently persistently_and_emp_elim. Qed.
-Lemma plainly_elim_absorbingly P : bi_plainly P ⊢ bi_absorbingly P.
-Proof. by rewrite plainly_elim_persistently persistently_elim_absorbingly. Qed.
-Lemma plainly_elim P `{!Absorbing P} : bi_plainly P ⊢ P.
-Proof. by rewrite plainly_elim_persistently persistently_elim. Qed.
-
-Lemma plainly_idemp_1 P : bi_plainly (bi_plainly P) ⊢ bi_plainly P.
-Proof. by rewrite plainly_elim_absorbingly absorbingly_plainly. Qed.
-Lemma plainly_idemp P : bi_plainly (bi_plainly P) ⊣⊢ bi_plainly P.
-Proof. apply (anti_symm _); auto using plainly_idemp_1, plainly_idemp_2. Qed.
-
-Lemma plainly_intro' P Q : (bi_plainly P ⊢ Q) → bi_plainly P ⊢ bi_plainly Q.
-Proof. intros <-. apply plainly_idemp_2. Qed.
-
-Lemma plainly_pure φ : bi_plainly ⌜φ⌠⊣⊢ ⌜φâŒ.
-Proof.
- apply (anti_symm _); auto.
- - by rewrite plainly_elim_persistently persistently_pure.
- - apply pure_elim'=> Hφ.
- trans (∀ x : False, bi_plainly True : PROP)%I; [by apply forall_intro|].
- rewrite plainly_forall_2. by rewrite -(pure_intro _ φ).
-Qed.
-Lemma plainly_forall {A} (Ψ : A → PROP) :
- bi_plainly (∀ a, Ψ a) ⊣⊢ ∀ a, bi_plainly (Ψ a).
-Proof.
- apply (anti_symm _); auto using plainly_forall_2.
- apply forall_intro=> x. by rewrite (forall_elim x).
-Qed.
-Lemma plainly_exist_2 {A} (Ψ : A → PROP) :
- (∃ a, bi_plainly (Ψ a)) ⊢ bi_plainly (∃ a, Ψ a).
-Proof. apply exist_elim=> x. by rewrite (exist_intro x). Qed.
-Lemma plainly_exist `{BiPlainlyExist PROP} {A} (Ψ : A → PROP) :
- bi_plainly (∃ a, Ψ a) ⊣⊢ ∃ a, bi_plainly (Ψ a).
-Proof. apply (anti_symm _); auto using plainly_exist_1, plainly_exist_2. Qed.
-Lemma plainly_and P Q : bi_plainly (P ∧ Q) ⊣⊢ bi_plainly P ∧ bi_plainly Q.
-Proof. rewrite !and_alt plainly_forall. by apply forall_proper=> -[]. Qed.
-Lemma plainly_or_2 P Q : bi_plainly P ∨ bi_plainly Q ⊢ bi_plainly (P ∨ Q).
-Proof. rewrite !or_alt -plainly_exist_2. by apply exist_mono=> -[]. Qed.
-Lemma plainly_or `{BiPlainlyExist PROP} P Q :
- bi_plainly (P ∨ Q) ⊣⊢ bi_plainly P ∨ bi_plainly Q.
-Proof. rewrite !or_alt plainly_exist. by apply exist_proper=> -[]. Qed.
-Lemma plainly_impl P Q : bi_plainly (P → Q) ⊢ bi_plainly P → bi_plainly Q.
-Proof.
- apply impl_intro_l; rewrite -plainly_and.
- apply plainly_mono, impl_elim with P; auto.
-Qed.
-
-Lemma plainly_sep_dup P : bi_plainly P ⊣⊢ bi_plainly P ∗ bi_plainly P.
-Proof.
- apply (anti_symm _).
- - rewrite -{1}(idemp bi_and (bi_plainly _)).
- by rewrite -{2}(left_id emp%I _ (bi_plainly _)) plainly_and_sep_assoc and_elim_l.
- - by rewrite plainly_absorbing.
-Qed.
-
-Lemma plainly_and_sep_l_1 P Q : bi_plainly P ∧ Q ⊢ bi_plainly P ∗ Q.
-Proof. by rewrite -{1}(left_id emp%I _ Q%I) plainly_and_sep_assoc and_elim_l. Qed.
-Lemma plainly_and_sep_r_1 P Q : P ∧ bi_plainly Q ⊢ P ∗ bi_plainly Q.
-Proof. by rewrite !(comm _ P) plainly_and_sep_l_1. Qed.
-
-Lemma plainly_True_emp : bi_plainly True ⊣⊢ bi_plainly emp.
-Proof. apply (anti_symm _); auto using plainly_emp_intro. Qed.
-Lemma plainly_and_sep P Q : bi_plainly (P ∧ Q) ⊢ bi_plainly (P ∗ Q).
-Proof.
- rewrite plainly_and.
- rewrite -{1}plainly_idemp -plainly_and -{1}(left_id emp%I _ Q%I).
- by rewrite plainly_and_sep_assoc (comm bi_and) plainly_and_emp_elim.
-Qed.
-
-Lemma plainly_affinely P : bi_plainly (bi_affinely P) ⊣⊢ bi_plainly P.
-Proof. by rewrite /bi_affinely plainly_and -plainly_True_emp plainly_pure left_id. Qed.
-
-Lemma and_sep_plainly P Q :
- bi_plainly P ∧ bi_plainly Q ⊣⊢ bi_plainly P ∗ bi_plainly Q.
-Proof.
- apply (anti_symm _); auto using plainly_and_sep_l_1.
- apply and_intro.
- - by rewrite plainly_absorbing.
- - by rewrite comm plainly_absorbing.
-Qed.
-Lemma plainly_sep_2 P Q :
- bi_plainly P ∗ bi_plainly Q ⊢ bi_plainly (P ∗ Q).
-Proof. by rewrite -plainly_and_sep plainly_and -and_sep_plainly. Qed.
-Lemma plainly_sep `{BiPositive PROP} P Q :
- bi_plainly (P ∗ Q) ⊣⊢ bi_plainly P ∗ bi_plainly Q.
-Proof.
- apply (anti_symm _); auto using plainly_sep_2.
- rewrite -plainly_affinely affinely_sep -and_sep_plainly. apply and_intro.
- - by rewrite (affinely_elim_emp Q) right_id affinely_elim.
- - by rewrite (affinely_elim_emp P) left_id affinely_elim.
-Qed.
-
-Lemma plainly_wand P Q : bi_plainly (P -∗ Q) ⊢ bi_plainly P -∗ bi_plainly Q.
-Proof. apply wand_intro_r. by rewrite plainly_sep_2 wand_elim_l. Qed.
-
-Lemma plainly_entails_l P Q : (P ⊢ bi_plainly Q) → P ⊢ bi_plainly Q ∗ P.
-Proof. intros; rewrite -plainly_and_sep_l_1; auto. Qed.
-Lemma plainly_entails_r P Q : (P ⊢ bi_plainly Q) → P ⊢ P ∗ bi_plainly Q.
-Proof. intros; rewrite -plainly_and_sep_r_1; auto. Qed.
-
-Lemma plainly_impl_wand_2 P Q : bi_plainly (P -∗ Q) ⊢ bi_plainly (P → Q).
-Proof.
- apply plainly_intro', impl_intro_r.
- rewrite -{2}(left_id emp%I _ P%I) plainly_and_sep_assoc.
- by rewrite (comm bi_and) plainly_and_emp_elim wand_elim_l.
-Qed.
-
-Lemma impl_wand_plainly_2 P Q : (bi_plainly P -∗ Q) ⊢ (bi_plainly P → Q).
-Proof. apply impl_intro_l. by rewrite plainly_and_sep_l_1 wand_elim_r. Qed.
-
-Lemma valid_plainly P : bi_valid (bi_plainly P) ↔ bi_valid P.
-Proof.
- rewrite /bi_valid. split; intros HP.
- - by rewrite -(idemp bi_and emp%I) {2}HP plainly_and_emp_elim.
- - by rewrite (plainly_emp_intro emp%I) HP.
-Qed.
-
-Section plainly_affinely_bi.
- Context `{BiAffine PROP}.
-
- Lemma plainly_emp : bi_plainly emp ⊣⊢ emp.
- Proof. by rewrite -!True_emp plainly_pure. Qed.
-
- Lemma plainly_and_sep_l P Q : bi_plainly P ∧ Q ⊣⊢ bi_plainly P ∗ Q.
- Proof.
- apply (anti_symm (⊢));
- eauto using plainly_and_sep_l_1, sep_and with typeclass_instances.
- Qed.
- Lemma plainly_and_sep_r P Q : P ∧ bi_plainly Q ⊣⊢ P ∗ bi_plainly Q.
- Proof. by rewrite !(comm _ P) plainly_and_sep_l. Qed.
-
- Lemma plainly_impl_wand P Q : bi_plainly (P → Q) ⊣⊢ bi_plainly (P -∗ Q).
- Proof.
- apply (anti_symm (⊢)); auto using plainly_impl_wand_2.
- apply plainly_intro', wand_intro_l.
- by rewrite -plainly_and_sep_r plainly_elim impl_elim_r.
- Qed.
-
- Lemma impl_wand_plainly P Q : (bi_plainly P → Q) ⊣⊢ (bi_plainly P -∗ Q).
- Proof.
- apply (anti_symm (⊢)). by rewrite -impl_wand_1. by rewrite impl_wand_plainly_2.
- Qed.
-End plainly_affinely_bi.
-
(* The combined affinely persistently modality *)
Lemma affinely_persistently_elim P : □ P ⊢ P.
Proof. apply persistently_and_emp_elim. Qed.
@@ -1170,52 +989,6 @@ Proof.
- apply impl_intro_l. by rewrite persistently_and_affinely_sep_l wand_elim_r.
Qed.
-(* The combined affinely plainly modality *)
-Lemma affinely_plainly_elim P : ■P ⊢ P.
-Proof. apply plainly_and_emp_elim. Qed.
-Lemma affinely_plainly_intro' P Q : (■P ⊢ Q) → ■P ⊢ ■Q.
-Proof. intros <-. by rewrite plainly_affinely plainly_idemp. Qed.
-
-Lemma affinely_plainly_emp : ■emp ⊣⊢ emp.
-Proof.
- by rewrite -plainly_True_emp plainly_pure affinely_True_emp affinely_emp.
-Qed.
-Lemma affinely_plainly_and P Q : ■(P ∧ Q) ⊣⊢ ■P ∧ ■Q.
-Proof. by rewrite plainly_and affinely_and. Qed.
-Lemma affinely_plainly_or_2 P Q : ■P ∨ ■Q ⊢ ■(P ∨ Q).
-Proof. by rewrite -plainly_or_2 affinely_or. Qed.
-Lemma affinely_plainly_or `{BiPlainlyExist PROP} P Q : ■(P ∨ Q) ⊣⊢ ■P ∨ ■Q.
-Proof. by rewrite plainly_or affinely_or. Qed.
-Lemma affinely_plainly_exist_2 {A} (Φ : A → PROP) : (∃ x, ■Φ x) ⊢ ■(∃ x, Φ x).
-Proof. by rewrite -plainly_exist_2 affinely_exist. Qed.
-Lemma affinely_plainly_exist `{BiPlainlyExist PROP} {A} (Φ : A → PROP) :
- ■(∃ x, Φ x) ⊣⊢ ∃ x, ■Φ x.
-Proof. by rewrite plainly_exist affinely_exist. Qed.
-Lemma affinely_plainly_sep_2 P Q : ■P ∗ ■Q ⊢ ■(P ∗ Q).
-Proof. by rewrite affinely_sep_2 plainly_sep_2. Qed.
-Lemma affinely_plainly_sep `{BiPositive PROP} P Q : ■(P ∗ Q) ⊣⊢ ■P ∗ ■Q.
-Proof. by rewrite -affinely_sep -plainly_sep. Qed.
-
-Lemma affinely_plainly_idemp P : ■■P ⊣⊢ ■P.
-Proof. by rewrite plainly_affinely plainly_idemp. Qed.
-
-Lemma plainly_and_affinely_sep_l P Q : bi_plainly P ∧ Q ⊣⊢ ■P ∗ Q.
-Proof. by rewrite -(persistently_plainly P) persistently_and_affinely_sep_l. Qed.
-Lemma plainly_and_affinely_sep_r P Q : P ∧ bi_plainly Q ⊣⊢ P ∗ ■Q.
-Proof. by rewrite !(comm _ P) plainly_and_affinely_sep_l. Qed.
-Lemma and_sep_affinely_plainly P Q : ■P ∧ ■Q ⊣⊢ ■P ∗ ■Q.
-Proof.
- by rewrite -plainly_and_affinely_sep_l -affinely_and affinely_and_r.
-Qed.
-
-Lemma affinely_plainly_sep_dup P : ■P ⊣⊢ ■P ∗ ■P.
-Proof.
- by rewrite -plainly_and_affinely_sep_l affinely_and_r affinely_and idemp.
-Qed.
-
-Lemma impl_wand_affinely_plainly P Q : (bi_plainly P → Q) ⊣⊢ (■P -∗ Q).
-Proof. by rewrite -(persistently_plainly P) impl_wand_affinely_persistently. Qed.
-
(* Conditional affinely modality *)
Global Instance affinely_if_ne p : NonExpansive (@bi_affinely_if PROP p).
Proof. solve_proper. Qed.
@@ -1334,81 +1107,6 @@ Proof. destruct p; simpl; auto using affinely_persistently_sep. Qed.
Lemma affinely_persistently_if_idemp p P : □?p □?p P ⊣⊢ □?p P.
Proof. destruct p; simpl; auto using affinely_persistently_idemp. Qed.
-(* Conditional plainly *)
-Global Instance plainly_if_ne p : NonExpansive (@bi_plainly_if PROP p).
-Proof. solve_proper. Qed.
-Global Instance plainly_if_proper p :
- Proper ((⊣⊢) ==> (⊣⊢)) (@bi_plainly_if PROP p).
-Proof. solve_proper. Qed.
-Global Instance plainly_if_mono' p :
- Proper ((⊢) ==> (⊢)) (@bi_plainly_if PROP p).
-Proof. solve_proper. Qed.
-Global Instance plainly_if_flip_mono' p :
- Proper (flip (⊢) ==> flip (⊢)) (@bi_plainly_if PROP p).
-Proof. solve_proper. Qed.
-
-Lemma plainly_if_mono p P Q : (P ⊢ Q) → bi_plainly_if p P ⊢ bi_plainly_if p Q.
-Proof. by intros ->. Qed.
-
-Lemma plainly_if_pure p φ : bi_plainly_if p ⌜φ⌠⊣⊢ ⌜φâŒ.
-Proof. destruct p; simpl; auto using plainly_pure. Qed.
-Lemma plainly_if_and p P Q : bi_plainly_if p (P ∧ Q) ⊣⊢ bi_plainly_if p P ∧ bi_plainly_if p Q.
-Proof. destruct p; simpl; auto using plainly_and. Qed.
-Lemma plainly_if_or_2 p P Q :
- bi_plainly_if p P ∨ bi_plainly_if p Q ⊢ bi_plainly_if p (P ∨ Q).
-Proof. destruct p; simpl; auto using plainly_or_2. Qed.
-Lemma plainly_if_or `{BiPlainlyExist PROP} p P Q :
- bi_plainly_if p (P ∨ Q) ⊣⊢ bi_plainly_if p P ∨ bi_plainly_if p Q.
-Proof. destruct p; simpl; auto using plainly_or. Qed.
-Lemma plainly_if_exist_2 {A} p (Ψ : A → PROP) :
- (∃ a, bi_plainly_if p (Ψ a)) ⊢ bi_plainly_if p (∃ a, Ψ a).
-Proof. destruct p; simpl; auto using plainly_exist_2. Qed.
-Lemma plainly_if_exist `{BiPlainlyExist PROP} {A} p (Ψ : A → PROP) :
- bi_plainly_if p (∃ a, Ψ a) ⊣⊢ ∃ a, bi_plainly_if p (Ψ a).
-Proof. destruct p; simpl; auto using plainly_exist. Qed.
-Lemma plainly_if_sep `{BiPositive PROP} p P Q :
- bi_plainly_if p (P ∗ Q) ⊣⊢ bi_plainly_if p P ∗ bi_plainly_if p Q.
-Proof. destruct p; simpl; auto using plainly_sep. Qed.
-
-Lemma plainly_if_idemp p P :
- bi_plainly_if p (bi_plainly_if p P) ⊣⊢ bi_plainly_if p P.
-Proof. destruct p; simpl; auto using plainly_idemp. Qed.
-
-(* Conditional affinely plainly *)
-Lemma affinely_plainly_if_mono p P Q : (P ⊢ Q) → ■?p P ⊢ ■?p Q.
-Proof. by intros ->. Qed.
-
-Lemma affinely_plainly_if_elim p P : ■?p P ⊢ P.
-Proof. destruct p; simpl; auto using affinely_plainly_elim. Qed.
-Lemma affinely_plainly_affinely_plainly_if p P : ■P ⊢ ■?p P.
-Proof. destruct p; simpl; auto using affinely_plainly_elim. Qed.
-Lemma affinely_plainly_if_intro' p P Q : (■?p P ⊢ Q) → ■?p P ⊢ ■?p Q.
-Proof. destruct p; simpl; auto using affinely_plainly_intro'. Qed.
-
-Lemma affinely_plainly_if_emp p : ■?p emp ⊣⊢ emp.
-Proof. destruct p; simpl; auto using affinely_plainly_emp. Qed.
-Lemma affinely_plainly_if_and p P Q : ■?p (P ∧ Q) ⊣⊢ ■?p P ∧ ■?p Q.
-Proof. destruct p; simpl; auto using affinely_plainly_and. Qed.
-Lemma affinely_plainly_if_or_2 p P Q : ■?p P ∨ ■?p Q ⊢ ■?p (P ∨ Q).
-Proof. destruct p; simpl; auto using affinely_plainly_or_2. Qed.
-Lemma affinely_plainly_if_or `{BiPlainlyExist PROP} p P Q :
- ■?p (P ∨ Q) ⊣⊢ ■?p P ∨ ■?p Q.
-Proof. destruct p; simpl; auto using affinely_plainly_or. Qed.
-Lemma affinely_plainly_if_exist_2 {A} p (Ψ : A → PROP) :
- (∃ a, ■?p Ψ a) ⊢ ■?p ∃ a, Ψ a .
-Proof. destruct p; simpl; auto using affinely_plainly_exist_2. Qed.
-Lemma affinely_plainly_if_exist `{BiPlainlyExist PROP} {A} p (Ψ : A → PROP) :
- (■?p ∃ a, Ψ a) ⊣⊢ ∃ a, ■?p Ψ a.
-Proof. destruct p; simpl; auto using affinely_plainly_exist. Qed.
-Lemma affinely_plainly_if_sep_2 p P Q : ■?p P ∗ ■?p Q ⊢ ■?p (P ∗ Q).
-Proof. destruct p; simpl; auto using affinely_plainly_sep_2. Qed.
-Lemma affinely_plainly_if_sep `{BiPositive PROP} p P Q :
- ■?p (P ∗ Q) ⊣⊢ ■?p P ∗ ■?p Q.
-Proof. destruct p; simpl; auto using affinely_plainly_sep. Qed.
-
-Lemma affinely_plainly_if_idemp p P : ■?p ■?p P ⊣⊢ ■?p P.
-Proof. destruct p; simpl; auto using affinely_plainly_idemp. Qed.
-
(* Properties of persistent propositions *)
Global Instance Persistent_proper : Proper ((≡) ==> iff) (@Persistent PROP).
Proof. solve_proper. Qed.
@@ -1485,17 +1183,6 @@ Section persistent_bi_absorbing.
Proof. apply (anti_symm _); auto using impl_wand_1, impl_wand_2. Qed.
End persistent_bi_absorbing.
-(* Properties of plain propositions *)
-Global Instance Plain_proper : Proper ((≡) ==> iff) (@Plain PROP).
-Proof. solve_proper. Qed.
-
-Lemma plain_plainly_2 P `{!Plain P} : P ⊢ bi_plainly P.
-Proof. done. Qed.
-Lemma plain_plainly P `{!Plain P, !Absorbing P} : bi_plainly P ⊣⊢ P.
-Proof. apply (anti_symm _), plain_plainly_2, _. apply plainly_elim, _. Qed.
-Lemma plainly_intro P Q `{!Plain P} : (P ⊢ Q) → P ⊢ bi_plainly Q.
-Proof. by intros <-. Qed.
-
(* Affine instances *)
Global Instance emp_affine_l : Affine (emp%I : PROP).
Proof. by rewrite /Affine. Qed.
@@ -1555,16 +1242,13 @@ Proof. intros. by rewrite /Absorbing absorbingly_wand !absorbing -absorbingly_in
Global Instance absorbingly_absorbing P : Absorbing (bi_absorbingly P).
Proof. rewrite /bi_absorbingly. apply _. Qed.
-Global Instance plainly_absorbing P : Absorbing (bi_plainly P).
-Proof. by rewrite /Absorbing /bi_absorbingly comm plainly_absorbing. Qed.
Global Instance persistently_absorbing P : Absorbing (bi_persistently P).
Proof. by rewrite /Absorbing absorbingly_persistently. Qed.
+Global Instance persistently_if_absorbing P p :
+ Absorbing P → Absorbing (bi_persistently_if p P).
+Proof. intros; destruct p; simpl; apply _. Qed.
(* Persistence instances *)
-(* Not an instance, see the bottom of this file *)
-Lemma plain_persistent P : Plain P → Persistent P.
-Proof. intros. by rewrite /Persistent -plainly_elim_persistently. Qed.
-
Global Instance pure_persistent φ : Persistent (⌜φâŒ%I : PROP).
Proof. by rewrite /Persistent persistently_pure. Qed.
Global Instance emp_persistent : Persistent (emp%I : PROP).
@@ -1588,19 +1272,10 @@ Proof.
apply exist_mono=> x. by rewrite -!persistent.
Qed.
-Global Instance impl_persistent P Q :
- Absorbing P → Plain P → Persistent Q → Persistent (P → Q).
-Proof.
- intros. by rewrite /Persistent {2}(plain P) -persistently_impl_plainly
- -(persistent Q) (plainly_elim_absorbingly P) absorbing.
-Qed.
-
Global Instance sep_persistent P Q :
Persistent P → Persistent Q → Persistent (P ∗ Q).
Proof. intros. by rewrite /Persistent -persistently_sep_2 -!persistent. Qed.
-Global Instance plainly_persistent P : Persistent (bi_plainly P).
-Proof. by rewrite /Persistent persistently_plainly. Qed.
Global Instance persistently_persistent P : Persistent (bi_persistently P).
Proof. by rewrite /Persistent persistently_idemp. Qed.
Global Instance affinely_persistent P : Persistent P → Persistent (bi_affinely P).
@@ -1611,66 +1286,6 @@ Global Instance from_option_persistent {A} P (Ψ : A → PROP) (mx : option A) :
(∀ x, Persistent (Ψ x)) → Persistent P → Persistent (from_option Ψ P mx).
Proof. destruct mx; apply _. Qed.
-Global Instance wand_persistent P Q :
- Plain P → Persistent Q → Absorbing Q → Persistent (P -∗ Q).
-Proof.
- intros. rewrite /Persistent {2}(plain P). trans (bi_persistently (bi_plainly P → Q)).
- - rewrite -persistently_impl_plainly impl_wand_affinely_plainly -(persistent Q).
- by rewrite affinely_plainly_elim.
- - apply persistently_mono, wand_intro_l. by rewrite sep_and impl_elim_r.
-Qed.
-
-(* Plainness instances *)
-Global Instance pure_plain φ : Plain (⌜φâŒ%I : PROP).
-Proof. by rewrite /Plain plainly_pure. Qed.
-Global Instance emp_plain : Plain (emp%I : PROP).
-Proof. apply plainly_emp_intro. Qed.
-Global Instance and_plain P Q : Plain P → Plain Q → Plain (P ∧ Q).
-Proof. intros. by rewrite /Plain plainly_and -!plain. Qed.
-Global Instance or_plain P Q : Plain P → Plain Q → Plain (P ∨ Q).
-Proof. intros. by rewrite /Plain -plainly_or_2 -!plain. Qed.
-Global Instance forall_plain {A} (Ψ : A → PROP) :
- (∀ x, Plain (Ψ x)) → Plain (∀ x, Ψ x).
-Proof.
- intros. rewrite /Plain plainly_forall. apply forall_mono=> x. by rewrite -plain.
-Qed.
-Global Instance exist_plain {A} (Ψ : A → PROP) :
- (∀ x, Plain (Ψ x)) → Plain (∃ x, Ψ x).
-Proof.
- intros. rewrite /Plain -plainly_exist_2. apply exist_mono=> x. by rewrite -plain.
-Qed.
-
-Global Instance impl_plain P Q :
- Absorbing P → Plain P → Plain Q → Plain (P → Q).
-Proof.
- intros. by rewrite /Plain {2}(plain P) -plainly_impl_plainly -(plain Q)
- (plainly_elim_absorbingly P) absorbing.
-Qed.
-Global Instance wand_plain P Q :
- Plain P → Plain Q → Absorbing Q → Plain (P -∗ Q).
-Proof.
- intros. rewrite /Plain {2}(plain P). trans (bi_plainly (bi_plainly P → Q)).
- - rewrite -plainly_impl_plainly impl_wand_affinely_plainly -(plain Q).
- by rewrite affinely_plainly_elim.
- - apply plainly_mono, wand_intro_l. by rewrite sep_and impl_elim_r.
-Qed.
-Global Instance sep_plain P Q : Plain P → Plain Q → Plain (P ∗ Q).
-Proof. intros. by rewrite /Plain -plainly_sep_2 -!plain. Qed.
-
-Global Instance plainly_plain P : Plain (bi_plainly P).
-Proof. by rewrite /Plain plainly_idemp. Qed.
-Global Instance persistently_plain P : Plain P → Plain (bi_persistently P).
-Proof.
- rewrite /Plain=> HP. by rewrite {1}HP plainly_persistently persistently_plainly.
-Qed.
-Global Instance affinely_plain P : Plain P → Plain (bi_affinely P).
-Proof. rewrite /bi_affinely. apply _. Qed.
-Global Instance absorbingly_plain P : Plain P → Plain (bi_absorbingly P).
-Proof. rewrite /bi_absorbingly. apply _. Qed.
-Global Instance from_option_palin {A} P (Ψ : A → PROP) (mx : option A) :
- (∀ x, Plain (Ψ x)) → Plain P → Plain (from_option Ψ P mx).
-Proof. destruct mx; apply _. Qed.
-
(* For big ops *)
Global Instance bi_and_monoid : Monoid (@bi_and PROP) :=
{| monoid_unit := True%I |}.
@@ -1679,34 +1294,6 @@ Global Instance bi_or_monoid : Monoid (@bi_or PROP) :=
Global Instance bi_sep_monoid : Monoid (@bi_sep PROP) :=
{| monoid_unit := emp%I |}.
-Global Instance bi_plainly_and_homomorphism :
- MonoidHomomorphism bi_and bi_and (≡) (@bi_plainly PROP).
-Proof.
- split; [split|]; try apply _. apply plainly_and. apply plainly_pure.
-Qed.
-
-Global Instance bi_plainly_or_homomorphism `{BiPlainlyExist PROP} :
- MonoidHomomorphism bi_or bi_or (≡) (@bi_plainly PROP).
-Proof.
- split; [split|]; try apply _. apply plainly_or. apply plainly_pure.
-Qed.
-
-Global Instance bi_plainly_sep_weak_homomorphism `{BiPositive PROP} :
- WeakMonoidHomomorphism bi_sep bi_sep (≡) (@bi_plainly PROP).
-Proof. split; try apply _. apply plainly_sep. Qed.
-
-Global Instance bi_plainly_sep_homomorphism `{BiAffine PROP} :
- MonoidHomomorphism bi_sep bi_sep (≡) (@bi_plainly PROP).
-Proof. split. apply _. apply plainly_emp. Qed.
-
-Global Instance bi_plainly_sep_entails_weak_homomorphism :
- WeakMonoidHomomorphism bi_sep bi_sep (flip (⊢)) (@bi_plainly PROP).
-Proof. split; try apply _. intros P Q; by rewrite plainly_sep_2. Qed.
-
-Global Instance bi_plainly_sep_entails_homomorphism :
- MonoidHomomorphism bi_sep bi_sep (flip (⊢)) (@bi_plainly PROP).
-Proof. split. apply _. simpl. apply plainly_emp_intro. Qed.
-
Global Instance bi_persistently_and_homomorphism :
MonoidHomomorphism bi_and bi_and (≡) (@bi_persistently PROP).
Proof.
@@ -1779,9 +1366,6 @@ Proof.
{ intros x. symmetry; apply equiv_spec. }
apply limit_preserving_and; by apply limit_preserving_entails.
Qed.
-Global Instance limit_preserving_Plain {A:ofeT} `{Cofe A} (Φ : A → PROP) :
- NonExpansive Φ → LimitPreserving (λ x, Plain (Φ x)).
-Proof. intros. apply limit_preserving_entails; solve_proper. Qed.
Global Instance limit_preserving_Persistent {A:ofeT} `{Cofe A} (Φ : A → PROP) :
NonExpansive Φ → LimitPreserving (λ x, Persistent (Φ x)).
Proof. intros. apply limit_preserving_entails; solve_proper. Qed.
@@ -1908,48 +1492,10 @@ Proof.
apply (internal_eq_rewrite' a b (λ b, bi_persistently (a ≡ b))%I); auto.
rewrite -(internal_eq_refl emp%I a). apply persistently_emp_intro.
Qed.
-Lemma plainly_internal_eq {A:ofeT} (a b : A) : bi_plainly (a ≡ b) ⊣⊢ a ≡ b.
-Proof.
- apply (anti_symm (⊢)).
- { by rewrite plainly_elim_absorbingly absorbingly_internal_eq. }
- apply (internal_eq_rewrite' a b (λ b, bi_plainly (a ≡ b))%I); [solve_proper|done|].
- rewrite -(internal_eq_refl True%I a) plainly_pure; auto.
-Qed.
-
-Lemma plainly_alt P : bi_plainly P ⊣⊢ bi_affinely P ≡ emp.
-Proof.
- rewrite -plainly_affinely. apply (anti_symm (⊢)).
- - rewrite -prop_ext. apply plainly_mono, and_intro; apply wand_intro_l.
- + by rewrite affinely_elim_emp left_id.
- + by rewrite left_id.
- - rewrite internal_eq_sym (internal_eq_rewrite _ _ bi_plainly).
- by rewrite -plainly_True_emp plainly_pure True_impl.
-Qed.
-
-Lemma plainly_alt_absorbing P `{!Absorbing P} : bi_plainly P ⊣⊢ P ≡ True.
-Proof.
- apply (anti_symm (⊢)).
- - rewrite -prop_ext. apply plainly_mono, and_intro; apply wand_intro_l; auto.
- - rewrite internal_eq_sym (internal_eq_rewrite _ _ bi_plainly).
- by rewrite plainly_pure True_impl.
-Qed.
-
-Lemma plainly_True_alt P : bi_plainly (True -∗ P) ⊣⊢ P ≡ True.
-Proof.
- apply (anti_symm (⊢)).
- - rewrite -prop_ext. apply plainly_mono, and_intro; apply wand_intro_l; auto.
- by rewrite wand_elim_r.
- - rewrite internal_eq_sym (internal_eq_rewrite _ _
- (λ Q, bi_plainly (True -∗ Q)) ltac:(solve_proper)).
- by rewrite -entails_wand // -(plainly_emp_intro True%I) True_impl.
-Qed.
Global Instance internal_eq_absorbing {A : ofeT} (x y : A) :
Absorbing (x ≡ y : PROP)%I.
Proof. by rewrite /Absorbing absorbingly_internal_eq. Qed.
-Global Instance internal_eq_plain {A : ofeT} (a b : A) :
- Plain (a ≡ b : PROP)%I.
-Proof. by intros; rewrite /Plain plainly_internal_eq. Qed.
Global Instance internal_eq_persistent {A : ofeT} (a b : A) :
Persistent (a ≡ b : PROP)%I.
Proof. by intros; rewrite /Persistent persistently_internal_eq. Qed.
@@ -2014,25 +1560,17 @@ Lemma later_wand P Q : ▷ (P -∗ Q) ⊢ ▷ P -∗ ▷ Q.
Proof. apply wand_intro_l. by rewrite -later_sep wand_elim_r. Qed.
Lemma later_iff P Q : ▷ (P ↔ Q) ⊢ ▷ P ↔ ▷ Q.
Proof. by rewrite /bi_iff later_and !later_impl. Qed.
-Lemma later_plainly P : ▷ bi_plainly P ⊣⊢ bi_plainly (▷ P).
-Proof. apply (anti_symm _); auto using later_plainly_1, later_plainly_2. Qed.
Lemma later_persistently P : ▷ bi_persistently P ⊣⊢ bi_persistently (▷ P).
Proof. apply (anti_symm _); auto using later_persistently_1, later_persistently_2. Qed.
Lemma later_affinely_2 P : bi_affinely (▷ P) ⊢ ▷ bi_affinely P.
Proof. rewrite /bi_affinely later_and. auto using later_intro. Qed.
-Lemma later_affinely_plainly_2 P : ■▷ P ⊢ ▷ ■P.
-Proof. by rewrite -later_plainly later_affinely_2. Qed.
Lemma later_affinely_persistently_2 P : □ ▷ P ⊢ ▷ □ P.
Proof. by rewrite -later_persistently later_affinely_2. Qed.
-Lemma later_affinely_plainly_if_2 p P : ■?p ▷ P ⊢ ▷ ■?p P.
-Proof. destruct p; simpl; auto using later_affinely_plainly_2. Qed.
Lemma later_affinely_persistently_if_2 p P : □?p ▷ P ⊢ ▷ □?p P.
Proof. destruct p; simpl; auto using later_affinely_persistently_2. Qed.
Lemma later_absorbingly P : ▷ bi_absorbingly P ⊣⊢ bi_absorbingly (▷ P).
Proof. by rewrite /bi_absorbingly later_sep later_True. Qed.
-Global Instance later_plain P : Plain P → Plain (▷ P).
-Proof. intros. by rewrite /Plain -later_plainly {1}(plain P). Qed.
Global Instance later_persistent P : Persistent P → Persistent (▷ P).
Proof. intros. by rewrite /Persistent -later_persistently {1}(persistent P). Qed.
Global Instance later_absorbing P : Absorbing P → Absorbing (▷ P).
@@ -2089,25 +1627,17 @@ Lemma laterN_wand n P Q : ▷^n (P -∗ Q) ⊢ ▷^n P -∗ ▷^n Q.
Proof. apply wand_intro_l. by rewrite -laterN_sep wand_elim_r. Qed.
Lemma laterN_iff n P Q : ▷^n (P ↔ Q) ⊢ ▷^n P ↔ ▷^n Q.
Proof. by rewrite /bi_iff laterN_and !laterN_impl. Qed.
-Lemma laterN_plainly n P : ▷^n bi_plainly P ⊣⊢ bi_plainly (▷^n P).
-Proof. induction n as [|n IH]; simpl; auto. by rewrite IH later_plainly. Qed.
Lemma laterN_persistently n P : ▷^n bi_persistently P ⊣⊢ bi_persistently (▷^n P).
Proof. induction n as [|n IH]; simpl; auto. by rewrite IH later_persistently. Qed.
Lemma laterN_affinely_2 n P : bi_affinely (▷^n P) ⊢ ▷^n bi_affinely P.
Proof. rewrite /bi_affinely laterN_and. auto using laterN_intro. Qed.
-Lemma laterN_affinely_plainly_2 n P : ■▷^n P ⊢ ▷^n ■P.
-Proof. by rewrite -laterN_plainly laterN_affinely_2. Qed.
Lemma laterN_affinely_persistently_2 n P : □ ▷^n P ⊢ ▷^n □ P.
Proof. by rewrite -laterN_persistently laterN_affinely_2. Qed.
-Lemma laterN_affinely_plainly_if_2 n p P : ■?p ▷^n P ⊢ ▷^n ■?p P.
-Proof. destruct p; simpl; auto using laterN_affinely_plainly_2. Qed.
Lemma laterN_affinely_persistently_if_2 n p P : □?p ▷^n P ⊢ ▷^n □?p P.
Proof. destruct p; simpl; auto using laterN_affinely_persistently_2. Qed.
Lemma laterN_absorbingly n P : ▷^n (bi_absorbingly P) ⊣⊢ bi_absorbingly (▷^n P).
Proof. by rewrite /bi_absorbingly laterN_sep laterN_True. Qed.
-Global Instance laterN_plain n P : Plain P → Plain (▷^n P).
-Proof. induction n; apply _. Qed.
Global Instance laterN_persistent n P : Persistent P → Persistent (▷^n P).
Proof. induction n; apply _. Qed.
Global Instance laterN_absorbing n P : Absorbing P → Absorbing (▷^n P).
@@ -2170,24 +1700,14 @@ Proof.
Qed.
Lemma except_0_later P : ◇ ▷ P ⊢ ▷ P.
Proof. by rewrite /sbi_except_0 -later_or False_or. Qed.
-Lemma except_0_plainly_1 P : ◇ bi_plainly P ⊢ bi_plainly (◇ P).
-Proof. by rewrite /sbi_except_0 -plainly_or_2 -later_plainly plainly_pure. Qed.
-Lemma except_0_plainly `{BiPlainlyExist PROP} P :
- ◇ bi_plainly P ⊣⊢ bi_plainly (◇ P).
-Proof. by rewrite /sbi_except_0 plainly_or -later_plainly plainly_pure. Qed.
Lemma except_0_persistently P : ◇ bi_persistently P ⊣⊢ bi_persistently (◇ P).
Proof.
by rewrite /sbi_except_0 persistently_or -later_persistently persistently_pure.
Qed.
Lemma except_0_affinely_2 P : bi_affinely (◇ P) ⊢ ◇ bi_affinely P.
Proof. rewrite /bi_affinely except_0_and. auto using except_0_intro. Qed.
-Lemma except_0_affinely_plainly_2 `{BiPlainlyExist PROP} P : ■◇ P ⊢ ◇ ■P.
-Proof. by rewrite -except_0_plainly except_0_affinely_2. Qed.
Lemma except_0_affinely_persistently_2 P : □ ◇ P ⊢ ◇ □ P.
Proof. by rewrite -except_0_persistently except_0_affinely_2. Qed.
-Lemma except_0_affinely_plainly_if_2 `{BiPlainlyExist PROP} p P :
- ■?p ◇ P ⊢ ◇ ■?p P.
-Proof. destruct p; simpl; auto using except_0_affinely_plainly_2. Qed.
Lemma except_0_affinely_persistently_if_2 p P : □?p ◇ P ⊢ ◇ □?p P.
Proof. destruct p; simpl; auto using except_0_affinely_persistently_2. Qed.
Lemma except_0_absorbingly P : ◇ (bi_absorbingly P) ⊣⊢ bi_absorbingly (◇ P).
@@ -2205,8 +1725,6 @@ Proof.
by apply and_mono, except_0_intro.
Qed.
-Global Instance except_0_plain P : Plain P → Plain (◇ P).
-Proof. rewrite /sbi_except_0; apply _. Qed.
Global Instance except_0_persistent P : Persistent P → Persistent (◇ P).
Proof. rewrite /sbi_except_0; apply _. Qed.
Global Instance except_0_absorbing P : Absorbing P → Absorbing (◇ P).
@@ -2263,12 +1781,6 @@ Proof.
- rewrite /sbi_except_0; auto.
- apply exist_elim=> x. rewrite -(exist_intro x); auto.
Qed.
-Global Instance plainly_timeless P `{BiPlainlyExist PROP} :
- Timeless P → Timeless (bi_plainly P).
-Proof.
- intros. rewrite /Timeless /sbi_except_0 later_plainly_1.
- by rewrite (timeless P) /sbi_except_0 plainly_or {1}plainly_elim.
-Qed.
Global Instance persistently_timeless P : Timeless P → Timeless (bi_persistently P).
Proof.
intros. rewrite /Timeless /sbi_except_0 later_persistently_1.
@@ -2350,12 +1862,3 @@ Global Instance sbi_except_0_sep_entails_homomorphism :
Proof. split; try apply _. apply except_0_intro. Qed.
End sbi_derived.
End bi.
-
-(* When declared as an actual instance, [plain_persistent] will cause
-failing proof searches to take exponential time, as Coq will try to
-apply it the instance at any node in the proof search tree.
-
-To avoid that, we declare it using a [Hint Immediate], so that it will
-only be used at the leaves of the proof search tree, i.e. when the
-premise of the hint can be derived from just the current context. *)
-Hint Immediate bi.plain_persistent : typeclass_instances.
diff --git a/theories/bi/embedding.v b/theories/bi/embedding.v
index 635d39dc309c2e987f6cc96275efc176bd1a2245..00ab11806f2e78e02ff0437a1ca394ae0b13d191 100644
--- a/theories/bi/embedding.v
+++ b/theories/bi/embedding.v
@@ -1,174 +1,222 @@
From iris.algebra Require Import monoid.
-From iris.bi Require Import interface derived_laws big_op.
+From iris.bi Require Import interface derived_laws big_op plainly.
From stdpp Require Import hlist.
-(* Embeddings are often used to *define* the connectives of the
- domain BI. Hence we cannot ask it to verify the properties of
- embeddings. *)
-Class BiEmbed (A B : Type) := bi_embed : A → B.
-Arguments bi_embed {_ _ _} _%I : simpl never.
-Notation "⎡ P ⎤" := (bi_embed P) : bi_scope.
-Instance: Params (@bi_embed) 3.
-Typeclasses Opaque bi_embed.
+Class Embed (A B : Type) := embed : A → B.
+Arguments embed {_ _ _} _%I : simpl never.
+Notation "⎡ P ⎤" := (embed P) : bi_scope.
+Instance: Params (@embed) 3.
+Typeclasses Opaque embed.
+Hint Mode Embed ! - : typeclass_instances.
+Hint Mode Embed - ! : typeclass_instances.
+
+(* Mixins allow us to create instances easily without having to use Program *)
+Record BiEmbedMixin (PROP1 PROP2 : bi) `(Embed PROP1 PROP2) := {
+ bi_embed_mixin_ne : NonExpansive embed;
+ bi_embed_mixin_mono : Proper ((⊢) ==> (⊢)) embed;
+ bi_embed_mixin_entails_inj :> Inj (⊢) (⊢) embed;
+ bi_embed_mixin_emp : ⎡emp⎤ ⊣⊢ emp;
+ bi_embed_mixin_impl_2 P Q : (⎡P⎤ → ⎡Q⎤) ⊢ ⎡P → Q⎤;
+ bi_embed_mixin_forall_2 A (Φ : A → PROP1) : (∀ x, ⎡Φ x⎤) ⊢ ⎡∀ x, Φ x⎤;
+ bi_embed_mixin_exist_1 A (Φ : A → PROP1) : ⎡∃ x, Φ x⎤ ⊢ ∃ x, ⎡Φ x⎤;
+ bi_embed_mixin_sep P Q : ⎡P ∗ Q⎤ ⊣⊢ ⎡P⎤ ∗ ⎡Q⎤;
+ bi_embed_mixin_wand_2 P Q : (⎡P⎤ -∗ ⎡Q⎤) ⊢ ⎡P -∗ Q⎤;
+ bi_embed_mixin_persistently P : ⎡bi_persistently P⎤ ⊣⊢ bi_persistently ⎡P⎤
+}.
+
+Class BiEmbed (PROP1 PROP2 : bi) := {
+ bi_embed_embed :> Embed PROP1 PROP2;
+ bi_embed_mixin : BiEmbedMixin PROP1 PROP2 bi_embed_embed;
+}.
Hint Mode BiEmbed ! - : typeclass_instances.
Hint Mode BiEmbed - ! : typeclass_instances.
+Arguments bi_embed_embed : simpl never.
-Class BiEmbedding (PROP1 PROP2 : bi) `{BiEmbed PROP1 PROP2} := {
- bi_embed_ne :> NonExpansive bi_embed;
- bi_embed_mono :> Proper ((⊢) ==> (⊢)) bi_embed;
- bi_embed_entails_inj :> Inj (⊢) (⊢) bi_embed;
- bi_embed_emp : ⎡emp⎤ ⊣⊢ emp;
- bi_embed_impl_2 P Q : (⎡P⎤ → ⎡Q⎤) ⊢ ⎡P → Q⎤;
- bi_embed_forall_2 A (Φ : A → PROP1) : (∀ x, ⎡Φ x⎤) ⊢ ⎡∀ x, Φ x⎤;
- bi_embed_exist_1 A (Φ : A → PROP1) : ⎡∃ x, Φ x⎤ ⊢ ∃ x, ⎡Φ x⎤;
- bi_embed_sep P Q : ⎡P ∗ Q⎤ ⊣⊢ ⎡P⎤ ∗ ⎡Q⎤;
- bi_embed_wand_2 P Q : (⎡P⎤ -∗ ⎡Q⎤) ⊢ ⎡P -∗ Q⎤;
- bi_embed_plainly P : ⎡bi_plainly P⎤ ⊣⊢ bi_plainly ⎡P⎤;
- bi_embed_persistently P : ⎡bi_persistently P⎤ ⊣⊢ bi_persistently ⎡P⎤
+Class SbiEmbed (PROP1 PROP2 : sbi) `{BiEmbed PROP1 PROP2} := {
+ embed_internal_eq_1 (A : ofeT) (x y : A) : ⎡x ≡ y⎤ ⊢ x ≡ y;
+ embed_later P : ⎡▷ P⎤ ⊣⊢ ▷ ⎡P⎤;
+ embed_interal_inj (PROP' : sbi) (P Q : PROP1) : ⎡P⎤ ≡ ⎡Q⎤ ⊢ (P ≡ Q : PROP');
}.
-Class SbiEmbedding (PROP1 PROP2 : sbi) `{BiEmbed PROP1 PROP2} := {
- sbi_embed_bi_embed :> BiEmbedding PROP1 PROP2;
- sbi_embed_internal_eq_1 (A : ofeT) (x y : A) : ⎡x ≡ y⎤ ⊢ x ≡ y;
- sbi_embed_later P : ⎡▷ P⎤ ⊣⊢ ▷ ⎡P⎤
-}.
+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.
+ Global Instance embed_entails_inj : Inj (⊢) (⊢) embed.
+ Proof. eapply bi_embed_mixin_entails_inj, bi_embed_mixin. Qed.
+ Lemma embed_emp : ⎡emp⎤ ⊣⊢ emp.
+ Proof. eapply bi_embed_mixin_emp, 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 : ⎡bi_persistently P⎤ ⊣⊢ bi_persistently ⎡P⎤.
+ Proof. eapply bi_embed_mixin_persistently, bi_embed_mixin. Qed.
+End embed_laws.
-Section bi_embedding.
- Context `{BiEmbedding PROP1 PROP2}.
- Local Notation bi_embed := (bi_embed (A:=PROP1) (B:=PROP2)).
- Local Notation "⎡ P ⎤" := (bi_embed P) : bi_scope.
+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 bi_embed_proper : Proper ((≡) ==> (≡)) bi_embed.
+ Global Instance embed_proper : Proper ((≡) ==> (≡)) embed.
Proof. apply (ne_proper _). Qed.
- Global Instance bi_embed_flip_mono : Proper (flip (⊢) ==> flip (⊢)) bi_embed.
+ Global Instance embed_flip_mono : Proper (flip (⊢) ==> flip (⊢)) embed.
Proof. solve_proper. Qed.
- Global Instance bi_embed_inj : Inj (≡) (≡) bi_embed.
+ Global Instance embed_inj : Inj (≡) (≡) embed.
Proof.
- intros P Q EQ. apply bi.equiv_spec, conj; apply (inj bi_embed);
+ intros P Q EQ. apply bi.equiv_spec, conj; apply (inj embed);
rewrite EQ //.
Qed.
- Lemma bi_embed_valid (P : PROP1) : ⎡P⎤%I ↔ P.
+ Lemma embed_valid (P : PROP1) : ⎡P⎤%I ↔ P.
Proof.
- by rewrite /bi_valid -bi_embed_emp; split=>?; [apply (inj bi_embed)|f_equiv].
+ by rewrite /bi_valid -embed_emp; split=>?; [apply (inj embed)|f_equiv].
Qed.
- Lemma bi_embed_forall A (Φ : A → PROP1) : ⎡∀ x, Φ x⎤ ⊣⊢ ∀ x, ⎡Φ x⎤.
+ Lemma embed_forall A (Φ : A → PROP1) : ⎡∀ x, Φ x⎤ ⊣⊢ ∀ x, ⎡Φ x⎤.
Proof.
- apply bi.equiv_spec; split; [|apply bi_embed_forall_2].
+ apply bi.equiv_spec; split; [|apply embed_forall_2].
apply bi.forall_intro=>?. by rewrite bi.forall_elim.
Qed.
- Lemma bi_embed_exist A (Φ : A → PROP1) : ⎡∃ x, Φ x⎤ ⊣⊢ ∃ x, ⎡Φ x⎤.
+ Lemma embed_exist A (Φ : A → PROP1) : ⎡∃ x, Φ x⎤ ⊣⊢ ∃ x, ⎡Φ x⎤.
Proof.
- apply bi.equiv_spec; split; [apply bi_embed_exist_1|].
+ apply bi.equiv_spec; split; [apply embed_exist_1|].
apply bi.exist_elim=>?. by rewrite -bi.exist_intro.
Qed.
- Lemma bi_embed_and P Q : ⎡P ∧ Q⎤ ⊣⊢ ⎡P⎤ ∧ ⎡Q⎤.
- Proof. rewrite !bi.and_alt bi_embed_forall. by f_equiv=>-[]. Qed.
- Lemma bi_embed_or P Q : ⎡P ∨ Q⎤ ⊣⊢ ⎡P⎤ ∨ ⎡Q⎤.
- Proof. rewrite !bi.or_alt bi_embed_exist. by f_equiv=>-[]. Qed.
- Lemma bi_embed_impl P Q : ⎡P → Q⎤ ⊣⊢ (⎡P⎤ → ⎡Q⎤).
+ 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_spec; split; [|apply bi_embed_impl_2].
- apply bi.impl_intro_l. by rewrite -bi_embed_and bi.impl_elim_r.
+ apply bi.equiv_spec; split; [|apply embed_impl_2].
+ apply bi.impl_intro_l. by rewrite -embed_and bi.impl_elim_r.
Qed.
- Lemma bi_embed_wand P Q : ⎡P -∗ Q⎤ ⊣⊢ (⎡P⎤ -∗ ⎡Q⎤).
+ Lemma embed_wand P Q : ⎡P -∗ Q⎤ ⊣⊢ (⎡P⎤ -∗ ⎡Q⎤).
Proof.
- apply bi.equiv_spec; split; [|apply bi_embed_wand_2].
- apply bi.wand_intro_l. by rewrite -bi_embed_sep bi.wand_elim_r.
+ apply bi.equiv_spec; split; [|apply embed_wand_2].
+ apply bi.wand_intro_l. by rewrite -embed_sep bi.wand_elim_r.
Qed.
- Lemma bi_embed_pure φ : ⎡⌜φâŒâŽ¤ ⊣⊢ ⌜φâŒ.
+ Lemma embed_pure φ : ⎡⌜φâŒâŽ¤ ⊣⊢ ⌜φâŒ.
Proof.
- rewrite (@bi.pure_alt PROP1) (@bi.pure_alt PROP2) bi_embed_exist.
+ rewrite (@bi.pure_alt PROP1) (@bi.pure_alt PROP2) embed_exist.
do 2 f_equiv. apply bi.equiv_spec. split; [apply bi.True_intro|].
- rewrite -(_ : (emp → emp : PROP1) ⊢ True) ?bi_embed_impl;
+ rewrite -(_ : (emp → emp : PROP1) ⊢ True) ?embed_impl;
last apply bi.True_intro.
apply bi.impl_intro_l. by rewrite right_id.
Qed.
- Lemma bi_embed_iff P Q : ⎡P ↔ Q⎤ ⊣⊢ (⎡P⎤ ↔ ⎡Q⎤).
- Proof. by rewrite bi_embed_and !bi_embed_impl. Qed.
- Lemma bi_embed_wand_iff P Q : ⎡P ∗-∗ Q⎤ ⊣⊢ (⎡P⎤ ∗-∗ ⎡Q⎤).
- Proof. by rewrite bi_embed_and !bi_embed_wand. Qed.
- Lemma bi_embed_affinely P : ⎡bi_affinely P⎤ ⊣⊢ bi_affinely ⎡P⎤.
- Proof. by rewrite bi_embed_and bi_embed_emp. Qed.
- Lemma bi_embed_absorbingly P : ⎡bi_absorbingly P⎤ ⊣⊢ bi_absorbingly ⎡P⎤.
- Proof. by rewrite bi_embed_sep bi_embed_pure. Qed.
- Lemma bi_embed_plainly_if P b : ⎡bi_plainly_if b P⎤ ⊣⊢ bi_plainly_if b ⎡P⎤.
- Proof. destruct b; simpl; auto using bi_embed_plainly. Qed.
- Lemma bi_embed_persistently_if P b :
+
+ 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 P : ⎡bi_affinely P⎤ ⊣⊢ bi_affinely ⎡P⎤.
+ Proof. by rewrite embed_and embed_emp. Qed.
+ Lemma embed_absorbingly P : ⎡bi_absorbingly P⎤ ⊣⊢ bi_absorbingly ⎡P⎤.
+ Proof. by rewrite embed_sep embed_pure. Qed.
+ Lemma embed_persistently_if P b :
⎡bi_persistently_if b P⎤ ⊣⊢ bi_persistently_if b ⎡P⎤.
- Proof. destruct b; simpl; auto using bi_embed_persistently. Qed.
- Lemma bi_embed_affinely_if P b : ⎡bi_affinely_if b P⎤ ⊣⊢ bi_affinely_if b ⎡P⎤.
- Proof. destruct b; simpl; auto using bi_embed_affinely. Qed.
- Lemma bi_embed_hforall {As} (Φ : himpl As PROP1):
- ⎡bi_hforall Φ⎤ ⊣⊢ bi_hforall (hcompose bi_embed Φ).
- Proof. induction As=>//. rewrite /= bi_embed_forall. by do 2 f_equiv. Qed.
- Lemma bi_embed_hexist {As} (Φ : himpl As PROP1):
- ⎡bi_hexist Φ⎤ ⊣⊢ bi_hexist (hcompose bi_embed Φ).
- Proof. induction As=>//. rewrite /= bi_embed_exist. by do 2 f_equiv. Qed.
-
- Global Instance bi_embed_plain P : Plain P → Plain ⎡P⎤.
- Proof. intros ?. by rewrite /Plain -bi_embed_plainly -plain. Qed.
- Global Instance bi_embed_persistent P : Persistent P → Persistent ⎡P⎤.
- Proof. intros ?. by rewrite /Persistent -bi_embed_persistently -persistent. Qed.
- Global Instance bi_embed_affine P : Affine P → Affine ⎡P⎤.
- Proof. intros ?. by rewrite /Affine (affine P) bi_embed_emp. Qed.
- Global Instance bi_embed_absorbing P : Absorbing P → Absorbing ⎡P⎤.
- Proof. intros ?. by rewrite /Absorbing -bi_embed_absorbingly absorbing. Qed.
-
- Global Instance bi_embed_and_homomorphism :
- MonoidHomomorphism bi_and bi_and (≡) bi_embed.
+ Proof. destruct b; simpl; auto using embed_persistently. Qed.
+ Lemma embed_affinely_if P b : ⎡bi_affinely_if b P⎤ ⊣⊢ bi_affinely_if b ⎡P⎤.
+ Proof. destruct b; simpl; auto using embed_affinely. Qed.
+ Lemma embed_hforall {As} (Φ : himpl As PROP1):
+ ⎡bi_hforall Φ⎤ ⊣⊢ bi_hforall (hcompose embed Φ).
+ Proof. induction As=>//. rewrite /= embed_forall. by do 2 f_equiv. Qed.
+ Lemma embed_hexist {As} (Φ : himpl As PROP1):
+ ⎡bi_hexist Φ⎤ ⊣⊢ bi_hexist (hcompose embed Φ).
+ Proof. induction As=>//. rewrite /= embed_exist. by do 2 f_equiv. Qed.
+
+ Global Instance embed_persistent P : Persistent P → Persistent ⎡P⎤.
+ Proof. intros ?. by rewrite /Persistent -embed_persistently -persistent. Qed.
+ Global Instance embed_affine 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 bi_embed_and|rewrite bi_embed_pure].
+ [setoid_rewrite embed_and|rewrite embed_pure].
Qed.
- Global Instance bi_embed_or_homomorphism :
- MonoidHomomorphism bi_or bi_or (≡) bi_embed.
+ Global Instance embed_or_homomorphism :
+ MonoidHomomorphism bi_or bi_or (≡) embed.
Proof.
by split; [split|]; try apply _;
- [setoid_rewrite bi_embed_or|rewrite bi_embed_pure].
+ [setoid_rewrite embed_or|rewrite embed_pure].
Qed.
- Global Instance bi_embed_sep_homomorphism :
- MonoidHomomorphism bi_sep bi_sep (≡) bi_embed.
+ Global Instance embed_sep_homomorphism :
+ MonoidHomomorphism bi_sep bi_sep (≡) embed.
Proof.
by split; [split|]; try apply _;
- [setoid_rewrite bi_embed_sep|rewrite bi_embed_emp].
+ [setoid_rewrite embed_sep|rewrite embed_emp].
Qed.
- Lemma bi_embed_big_sepL {A} (Φ : nat → A → PROP1) l :
+ Lemma embed_big_sepL {A} (Φ : nat → A → PROP1) l :
⎡[∗ list] k↦x ∈ l, Φ k x⎤ ⊣⊢ [∗ list] k↦x ∈ l, ⎡Φ k x⎤.
Proof. apply (big_opL_commute _). Qed.
- Lemma bi_embed_big_sepM `{Countable K} {A} (Φ : K → A → PROP1) (m : gmap K A) :
+ Lemma embed_big_sepM `{Countable K} {A} (Φ : K → A → PROP1) (m : gmap K A) :
⎡[∗ map] k↦x ∈ m, Φ k x⎤ ⊣⊢ [∗ map] k↦x ∈ m, ⎡Φ k x⎤.
Proof. apply (big_opM_commute _). Qed.
- Lemma bi_embed_big_sepS `{Countable A} (Φ : A → PROP1) (X : gset A) :
+ 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 bi_embed_big_sepMS `{Countable A} (Φ : A → PROP1) (X : gmultiset A) :
+ 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 bi_embedding.
+End embed.
-Section sbi_embedding.
- Context `{SbiEmbedding PROP1 PROP2}.
+Section sbi_embed.
+ Context `{SbiEmbed PROP1 PROP2}.
Implicit Types P Q R : PROP1.
- Lemma sbi_embed_internal_eq (A : ofeT) (x y : A) : ⎡x ≡ y⎤ ⊣⊢ x ≡ y.
+ Lemma embed_internal_eq (A : ofeT) (x y : A) : ⎡x ≡ y⎤ ⊣⊢ x ≡ y.
Proof.
- apply bi.equiv_spec; split; [apply sbi_embed_internal_eq_1|].
+ apply bi.equiv_spec; split; [apply embed_internal_eq_1|].
etrans; [apply (bi.internal_eq_rewrite x y (λ y, ⎡x ≡ y⎤%I)); solve_proper|].
- rewrite -(bi.internal_eq_refl True%I) bi_embed_pure.
+ rewrite -(bi.internal_eq_refl True%I) embed_pure.
eapply bi.impl_elim; [done|]. apply bi.True_intro.
Qed.
- Lemma sbi_embed_laterN n P : ⎡▷^n P⎤ ⊣⊢ ▷^n ⎡P⎤.
- Proof. induction n=>//=. rewrite sbi_embed_later. by f_equiv. Qed.
- Lemma sbi_embed_except_0 P : ⎡◇ P⎤ ⊣⊢ ◇ ⎡P⎤.
- Proof. by rewrite bi_embed_or sbi_embed_later bi_embed_pure. Qed.
+ 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 sbi_embed_timeless P : Timeless P → Timeless ⎡P⎤.
+ Lemma embed_plainly `{!BiPlainly PROP1, !BiPlainly PROP2} P : ⎡■P⎤ ⊣⊢ ■⎡P⎤.
Proof.
- intros ?. by rewrite /Timeless -sbi_embed_except_0 -sbi_embed_later timeless.
+ rewrite !plainly_alt embed_internal_eq. apply (anti_symm _).
+ - rewrite -embed_affinely -embed_emp. apply bi.f_equiv, _.
+ - by rewrite -embed_affinely -embed_emp embed_interal_inj.
Qed.
-End sbi_embedding.
+ Lemma embed_plainly_if `{!BiPlainly PROP1, !BiPlainly PROP2} p P : ⎡■?p P⎤ ⊣⊢ ■?p ⎡P⎤.
+ Proof. destruct p; simpl; auto using embed_plainly. Qed.
+
+ Lemma embed_plain `{!BiPlainly PROP1, !BiPlainly PROP2} P : Plain P → Plain ⎡P⎤.
+ Proof. intros ?. by rewrite /Plain -embed_plainly -plain. Qed.
+
+ Global Instance embed_timeless P : Timeless P → Timeless ⎡P⎤.
+ Proof.
+ intros ?. by rewrite /Timeless -embed_except_0 -embed_later timeless.
+ Qed.
+End sbi_embed.
+
+(* Not defined using an ordinary [Instance] because the default
+[class_apply @bi_embed_plainly] shelves the [BiPlainly] premise, making proof
+search for the other premises fail. See the proof of [monPred_absolutely_plain]
+for an example where it would fail with a regular [Instance].*)
+Hint Extern 4 (Plain ⎡_⎤) => eapply @embed_plain : typeclass_instances.
diff --git a/theories/bi/interface.v b/theories/bi/interface.v
index 56ef28240860537cdbcf442ddaf8f89b986742b5..5fd4fa0e9276b76487da0cfcda71b188aac3e68b 100644
--- a/theories/bi/interface.v
+++ b/theories/bi/interface.v
@@ -9,7 +9,7 @@ Reserved Notation "P -∗ Q" (at level 99, Q at level 200, right associativity).
Reserved Notation "â–· P" (at level 20, right associativity).
Section bi_mixin.
- Context {PROP : Type} `{Dist PROP, Equiv PROP} (prop_ofe_mixin : OfeMixin PROP).
+ Context {PROP : Type} `{Dist PROP, Equiv PROP}.
Context (bi_entails : PROP → PROP → Prop).
Context (bi_emp : PROP).
Context (bi_pure : Prop → PROP).
@@ -20,7 +20,6 @@ Section bi_mixin.
Context (bi_exist : ∀ A, (A → PROP) → PROP).
Context (bi_sep : PROP → PROP → PROP).
Context (bi_wand : PROP → PROP → PROP).
- Context (bi_plainly : PROP → PROP).
Context (bi_persistently : PROP → PROP).
Context (sbi_internal_eq : ∀ A : ofeT, A → A → PROP).
Context (sbi_later : PROP → PROP).
@@ -68,7 +67,6 @@ Section bi_mixin.
Proper (pointwise_relation _ (dist n) ==> dist n) (bi_exist A);
bi_mixin_sep_ne : NonExpansive2 bi_sep;
bi_mixin_wand_ne : NonExpansive2 bi_wand;
- bi_mixin_plainly_ne : NonExpansive bi_plainly;
bi_mixin_persistently_ne : NonExpansive bi_persistently;
(* Higher-order logic *)
@@ -102,26 +100,6 @@ Section bi_mixin.
bi_mixin_wand_intro_r P Q R : (P ∗ Q ⊢ R) → P ⊢ Q -∗ R;
bi_mixin_wand_elim_l' P Q R : (P ⊢ Q -∗ R) → P ∗ Q ⊢ R;
- (* Plainly *)
- (* Note: plainly is about to be removed from this interface *)
- bi_mixin_plainly_mono P Q : (P ⊢ Q) → bi_plainly P ⊢ bi_plainly Q;
- bi_mixin_plainly_elim_persistently P : bi_plainly P ⊢ bi_persistently P;
- bi_mixin_plainly_idemp_2 P : bi_plainly P ⊢ bi_plainly (bi_plainly P);
-
- bi_mixin_plainly_forall_2 {A} (Ψ : A → PROP) :
- (∀ a, bi_plainly (Ψ a)) ⊢ bi_plainly (∀ a, Ψ a);
-
- (* The following two laws are very similar, and indeed they hold not just
- for persistently and plainly, but for any modality defined as `M P n x :=
- ∀ y, R x y → P n y`. *)
- bi_mixin_persistently_impl_plainly P Q :
- (bi_plainly P → bi_persistently Q) ⊢ bi_persistently (bi_plainly P → Q);
- bi_mixin_plainly_impl_plainly P Q :
- (bi_plainly P → bi_plainly Q) ⊢ bi_plainly (bi_plainly P → Q);
-
- bi_mixin_plainly_emp_intro P : P ⊢ bi_plainly emp;
- bi_mixin_plainly_absorbing P Q : bi_plainly P ∗ Q ⊢ bi_plainly P;
-
(* Persistently *)
(* In the ordered RA model: Holds without further assumptions. *)
bi_mixin_persistently_mono P Q :
@@ -130,8 +108,8 @@ Section bi_mixin.
bi_mixin_persistently_idemp_2 P :
bi_persistently P ⊢ bi_persistently (bi_persistently P);
- (* In the ordered RA model [P ⊢ persisently emp] (which can currently still
- be derived from the plainly axioms, which will be removed): `ε ≼ core x` *)
+ (* In the ordered RA model: `ε ≼ core x` *)
+ bi_mixin_persistently_emp_intro P : P ⊢ bi_persistently emp;
bi_mixin_persistently_forall_2 {A} (Ψ : A → PROP) :
(∀ a, bi_persistently (Ψ a)) ⊢ bi_persistently (∀ a, Ψ a);
@@ -158,8 +136,6 @@ Section bi_mixin.
sbi_mixin_fun_ext {A} {B : A → ofeT} (f g : ofe_fun B) : (∀ x, f x ≡ g x) ⊢ f ≡ g;
sbi_mixin_sig_eq {A : ofeT} (P : A → Prop) (x y : sig P) : `x ≡ `y ⊢ x ≡ y;
sbi_mixin_discrete_eq_1 {A : ofeT} (a b : A) : Discrete a → a ≡ b ⊢ ⌜a ≡ bâŒ;
- sbi_mixin_prop_ext P Q : bi_plainly ((P -∗ Q) ∧ (Q -∗ P)) ⊢
- sbi_internal_eq (OfeT PROP prop_ofe_mixin) P Q;
(* Later *)
sbi_mixin_later_eq_1 {A : ofeT} (x y : A) : Next x ≡ Next y ⊢ ▷ (x ≡ y);
@@ -173,8 +149,6 @@ Section bi_mixin.
(▷ ∃ a, Φ a) ⊢ ▷ False ∨ (∃ a, ▷ Φ a);
sbi_mixin_later_sep_1 P Q : ▷ (P ∗ Q) ⊢ ▷ P ∗ ▷ Q;
sbi_mixin_later_sep_2 P Q : ▷ P ∗ ▷ Q ⊢ ▷ (P ∗ Q);
- sbi_mixin_later_plainly_1 P : ▷ bi_plainly P ⊢ bi_plainly (▷ P);
- sbi_mixin_later_plainly_2 P : bi_plainly (▷ P) ⊢ ▷ bi_plainly P;
sbi_mixin_later_persistently_1 P :
▷ bi_persistently P ⊢ bi_persistently (▷ P);
sbi_mixin_later_persistently_2 P :
@@ -198,11 +172,10 @@ Structure bi := Bi {
bi_exist : ∀ A, (A → bi_car) → bi_car;
bi_sep : bi_car → bi_car → bi_car;
bi_wand : bi_car → bi_car → bi_car;
- bi_plainly : bi_car → bi_car;
bi_persistently : bi_car → bi_car;
bi_ofe_mixin : OfeMixin bi_car;
bi_bi_mixin : BiMixin bi_entails bi_emp bi_pure bi_and bi_or bi_impl bi_forall
- bi_exist bi_sep bi_wand bi_plainly bi_persistently;
+ bi_exist bi_sep bi_wand bi_persistently;
}.
Coercion bi_ofeC (PROP : bi) : ofeT := OfeT PROP (bi_ofe_mixin PROP).
@@ -218,7 +191,6 @@ Instance: Params (@bi_forall) 2.
Instance: Params (@bi_exist) 2.
Instance: Params (@bi_sep) 1.
Instance: Params (@bi_wand) 1.
-Instance: Params (@bi_plainly) 1.
Instance: Params (@bi_persistently) 1.
Delimit Scope bi_scope with I.
@@ -235,7 +207,6 @@ Arguments bi_forall {PROP _} _%I : simpl never, rename.
Arguments bi_exist {PROP _} _%I : simpl never, rename.
Arguments bi_sep {PROP} _%I _%I : simpl never, rename.
Arguments bi_wand {PROP} _%I _%I : simpl never, rename.
-Arguments bi_plainly {PROP} _%I : simpl never, rename.
Arguments bi_persistently {PROP} _%I : simpl never, rename.
Structure sbi := Sbi {
@@ -252,17 +223,15 @@ Structure sbi := Sbi {
sbi_exist : ∀ A, (A → sbi_car) → sbi_car;
sbi_sep : sbi_car → sbi_car → sbi_car;
sbi_wand : sbi_car → sbi_car → sbi_car;
- sbi_plainly : sbi_car → sbi_car;
sbi_persistently : sbi_car → sbi_car;
sbi_internal_eq : ∀ A : ofeT, A → A → sbi_car;
sbi_later : sbi_car → sbi_car;
sbi_ofe_mixin : OfeMixin sbi_car;
sbi_bi_mixin : BiMixin sbi_entails sbi_emp sbi_pure sbi_and sbi_or sbi_impl
- sbi_forall sbi_exist sbi_sep sbi_wand sbi_plainly
- sbi_persistently;
- sbi_sbi_mixin : SbiMixin sbi_ofe_mixin sbi_entails sbi_pure sbi_and sbi_or
- sbi_impl sbi_forall sbi_exist sbi_sep sbi_wand
- sbi_plainly sbi_persistently sbi_internal_eq sbi_later;
+ sbi_forall sbi_exist sbi_sep sbi_wand sbi_persistently;
+ sbi_sbi_mixin : SbiMixin sbi_entails sbi_pure sbi_or sbi_impl
+ sbi_forall sbi_exist sbi_sep
+ sbi_persistently sbi_internal_eq sbi_later;
}.
Instance: Params (@sbi_later) 1.
@@ -290,7 +259,6 @@ Arguments sbi_forall {PROP _} _%I : simpl never, rename.
Arguments sbi_exist {PROP _} _%I : simpl never, rename.
Arguments sbi_sep {PROP} _%I _%I : simpl never, rename.
Arguments sbi_wand {PROP} _%I _%I : simpl never, rename.
-Arguments sbi_plainly {PROP} _%I : simpl never, rename.
Arguments sbi_persistently {PROP} _%I : simpl never, rename.
Arguments sbi_internal_eq {PROP _} _ _ : simpl never, rename.
Arguments sbi_later {PROP} _%I : simpl never, rename.
@@ -366,8 +334,6 @@ Global Instance sep_ne : NonExpansive2 (@bi_sep PROP).
Proof. eapply bi_mixin_sep_ne, bi_bi_mixin. Qed.
Global Instance wand_ne : NonExpansive2 (@bi_wand PROP).
Proof. eapply bi_mixin_wand_ne, bi_bi_mixin. Qed.
-Global Instance plainly_ne : NonExpansive (@bi_plainly PROP).
-Proof. eapply bi_mixin_plainly_ne, bi_bi_mixin. Qed.
Global Instance persistently_ne : NonExpansive (@bi_persistently PROP).
Proof. eapply bi_mixin_persistently_ne, bi_bi_mixin. Qed.
@@ -424,27 +390,6 @@ Proof. eapply bi_mixin_wand_intro_r, bi_bi_mixin. Qed.
Lemma wand_elim_l' P Q R : (P ⊢ Q -∗ R) → P ∗ Q ⊢ R.
Proof. eapply bi_mixin_wand_elim_l', bi_bi_mixin. Qed.
-(* Plainly *)
-Lemma plainly_mono P Q : (P ⊢ Q) → bi_plainly P ⊢ bi_plainly Q.
-Proof. eapply bi_mixin_plainly_mono, bi_bi_mixin. Qed.
-Lemma plainly_elim_persistently P : bi_plainly P ⊢ bi_persistently P.
-Proof. eapply bi_mixin_plainly_elim_persistently, bi_bi_mixin. Qed.
-Lemma plainly_idemp_2 P : bi_plainly P ⊢ bi_plainly (bi_plainly P).
-Proof. eapply bi_mixin_plainly_idemp_2, bi_bi_mixin. Qed.
-Lemma plainly_forall_2 {A} (Ψ : A → PROP) :
- (∀ a, bi_plainly (Ψ a)) ⊢ bi_plainly (∀ a, Ψ a).
-Proof. eapply bi_mixin_plainly_forall_2, bi_bi_mixin. Qed.
-Lemma persistently_impl_plainly P Q :
- (bi_plainly P → bi_persistently Q) ⊢ bi_persistently (bi_plainly P → Q).
-Proof. eapply bi_mixin_persistently_impl_plainly, bi_bi_mixin. Qed.
-Lemma plainly_impl_plainly P Q :
- (bi_plainly P → bi_plainly Q) ⊢ bi_plainly (bi_plainly P → Q).
-Proof. eapply bi_mixin_plainly_impl_plainly, bi_bi_mixin. Qed.
-Lemma plainly_absorbing P Q : bi_plainly P ∗ Q ⊢ bi_plainly P.
-Proof. eapply (bi_mixin_plainly_absorbing bi_entails), bi_bi_mixin. Qed.
-Lemma plainly_emp_intro P : P ⊢ bi_plainly emp.
-Proof. eapply bi_mixin_plainly_emp_intro, bi_bi_mixin. Qed.
-
(* Persistently *)
Lemma persistently_mono P Q : (P ⊢ Q) → bi_persistently P ⊢ bi_persistently Q.
Proof. eapply bi_mixin_persistently_mono, bi_bi_mixin. Qed.
@@ -452,6 +397,9 @@ Lemma persistently_idemp_2 P :
bi_persistently P ⊢ bi_persistently (bi_persistently P).
Proof. eapply bi_mixin_persistently_idemp_2, bi_bi_mixin. Qed.
+Lemma persistently_emp_intro P : P ⊢ bi_persistently emp.
+Proof. eapply bi_mixin_persistently_emp_intro, bi_bi_mixin. Qed.
+
Lemma persistently_forall_2 {A} (Ψ : A → PROP) :
(∀ a, bi_persistently (Ψ a)) ⊢ bi_persistently (∀ a, Ψ a).
Proof. eapply bi_mixin_persistently_forall_2, bi_bi_mixin. Qed.
@@ -488,9 +436,6 @@ Lemma discrete_eq_1 {A : ofeT} (a b : A) :
Discrete a → a ≡ b ⊢ (⌜a ≡ b⌠: PROP).
Proof. eapply sbi_mixin_discrete_eq_1, sbi_sbi_mixin. Qed.
-Lemma prop_ext P Q : bi_plainly ((P -∗ Q) ∧ (Q -∗ P)) ⊢ P ≡ Q.
-Proof. eapply (sbi_mixin_prop_ext _ bi_entails), sbi_sbi_mixin. Qed.
-
(* Later *)
Global Instance later_contractive : Contractive (@sbi_later PROP).
Proof. eapply sbi_mixin_later_contractive, sbi_sbi_mixin. Qed.
@@ -514,14 +459,10 @@ Lemma later_sep_1 P Q : ▷ (P ∗ Q) ⊢ ▷ P ∗ ▷ Q.
Proof. eapply sbi_mixin_later_sep_1, sbi_sbi_mixin. Qed.
Lemma later_sep_2 P Q : ▷ P ∗ ▷ Q ⊢ ▷ (P ∗ Q).
Proof. eapply sbi_mixin_later_sep_2, sbi_sbi_mixin. Qed.
-Lemma later_plainly_1 P : ▷ bi_plainly P ⊢ bi_plainly (▷ P).
-Proof. eapply (sbi_mixin_later_plainly_1 _ bi_entails), sbi_sbi_mixin. Qed.
-Lemma later_plainly_2 P : bi_plainly (▷ P) ⊢ ▷ bi_plainly P.
-Proof. eapply (sbi_mixin_later_plainly_2 _ bi_entails), sbi_sbi_mixin. Qed.
Lemma later_persistently_1 P : ▷ bi_persistently P ⊢ bi_persistently (▷ P).
-Proof. eapply (sbi_mixin_later_persistently_1 _ bi_entails), sbi_sbi_mixin. Qed.
+Proof. eapply (sbi_mixin_later_persistently_1 bi_entails), sbi_sbi_mixin. Qed.
Lemma later_persistently_2 P : bi_persistently (▷ P) ⊢ ▷ bi_persistently P.
-Proof. eapply (sbi_mixin_later_persistently_2 _ bi_entails), sbi_sbi_mixin. Qed.
+Proof. eapply (sbi_mixin_later_persistently_2 bi_entails), sbi_sbi_mixin. Qed.
Lemma later_false_em P : ▷ P ⊢ ▷ False ∨ (▷ False → P).
Proof. eapply sbi_mixin_later_false_em, sbi_sbi_mixin. Qed.
diff --git a/theories/bi/lib/atomic.v b/theories/bi/lib/atomic.v
index 74eeec2015722fd87c0a1d60c970699ee52ef1f5..59792e0f078dd27789103ede34947e5b74b4471e 100644
--- a/theories/bi/lib/atomic.v
+++ b/theories/bi/lib/atomic.v
@@ -3,7 +3,7 @@ From stdpp Require Import coPset.
From iris.proofmode Require Import tactics.
Set Default Proof Using "Type".
-Definition atomic_shift {PROP: sbi} `{!FUpd PROP} {A B : Type}
+Definition atomic_shift `{BiFUpd PROP} {A B : Type}
(α : A → PROP) (* atomic pre-condition *)
(β : A → B → PROP) (* atomic post-condition *)
(Eo Em : coPset) (* outside/module masks *)
@@ -14,7 +14,7 @@ Definition atomic_shift {PROP: sbi} `{!FUpd PROP} {A B : Type}
((α x ={E∖Em, E}=∗ P) ∧ (∀ y, β x y ={E∖Em, E}=∗ Q x y)))
)%I.
-Definition atomic_update {PROP: sbi} `{!FUpd PROP} {A B : Type}
+Definition atomic_update `{BiFUpd PROP} {A B : Type}
(α : A → PROP) (* atomic pre-condition *)
(β : A → B → PROP) (* atomic post-condition *)
(Eo Em : coPset) (* outside/module masks *)
@@ -25,7 +25,7 @@ Definition atomic_update {PROP: sbi} `{!FUpd PROP} {A B : Type}
)%I.
Section lemmas.
- Context {PROP: sbi} `{FUpdFacts PROP} {A B : Type}.
+ Context `{BiFUpd PROP} {A B : Type}.
Implicit Types (α : A → PROP) (β: A → B → PROP) (P : PROP) (Q : A → B → PROP).
Lemma aupd_acc α β Eo Em Q E :
diff --git a/theories/bi/monpred.v b/theories/bi/monpred.v
index 36dcb9f0e560aa7b1c40ad81a830887d44a49796..606cc00260d7004cc6f26adc5939a6635da23d38 100644
--- a/theories/bi/monpred.v
+++ b/theories/bi/monpred.v
@@ -2,7 +2,6 @@ From stdpp Require Import coPset.
From iris.bi Require Import bi.
(** Definitions. *)
-
Structure biIndex :=
BiIndex
{ bi_index_type :> Type;
@@ -128,14 +127,28 @@ Next Obligation. solve_proper. Qed.
Definition monPred_embed_def (P : PROP) : monPred := MonPred (λ _, P) _.
Definition monPred_embed_aux : seal (@monPred_embed_def). by eexists. Qed.
-Global Instance monPred_embed : BiEmbed PROP monPred := unseal monPred_embed_aux.
-Definition monPred_embed_eq : bi_embed (A:=PROP) = _ := seal_eq _.
+Definition monPred_embed : Embed PROP monPred := unseal monPred_embed_aux.
+Definition monPred_embed_eq : @embed _ _ monPred_embed = _ := seal_eq _.
+
+Definition monPred_emp_def : monPred := MonPred (λ _, emp)%I _.
+Definition monPred_emp_aux : seal (@monPred_emp_def). by eexists. Qed.
+Definition monPred_emp := unseal monPred_emp_aux.
+Definition monPred_emp_eq : @monPred_emp = _ := seal_eq _.
+
+Definition monPred_pure_def (φ : Prop) : monPred := MonPred (λ _, ⌜φâŒ)%I _.
+Definition monPred_pure_aux : seal (@monPred_pure_def). by eexists. Qed.
+Definition monPred_pure := unseal monPred_pure_aux.
+Definition monPred_pure_eq : @monPred_pure = _ := seal_eq _.
+
+Definition monPred_absolutely_def P : monPred := MonPred (λ _, ∀ i, P i)%I _.
+Definition monPred_absolutely_aux : seal (@monPred_absolutely_def). by eexists. Qed.
+Definition monPred_absolutely := unseal monPred_absolutely_aux.
+Definition monPred_absolutely_eq : @monPred_absolutely = _ := seal_eq _.
-Definition monPred_pure (φ : Prop) : monPred := tc_opaque ⎡⌜φâŒâŽ¤%I.
-Definition monPred_emp : monPred := tc_opaque ⎡emp⎤%I.
-Definition monPred_plainly P : monPred := tc_opaque ⎡∀ i, bi_plainly (P i)⎤%I.
-Definition monPred_absolutely (P : monPred) : monPred := tc_opaque ⎡∀ i, P i⎤%I.
-Definition monPred_relatively (P : monPred) : monPred := tc_opaque ⎡∃ i, P i⎤%I.
+Definition monPred_relatively_def P : monPred := MonPred (λ _, ∃ i, P i)%I _.
+Definition monPred_relatively_aux : seal (@monPred_relatively_def). by eexists. Qed.
+Definition monPred_relatively := unseal monPred_relatively_aux.
+Definition monPred_relatively_eq : @monPred_relatively = _ := seal_eq _.
Program Definition monPred_and_def P Q : monPred :=
MonPred (λ i, P i ∧ Q i)%I _.
@@ -205,8 +218,8 @@ Definition monPred_bupd_def `{BUpd PROP} (P : monPred) : monPred :=
terms in logical terms. *)
monPred_upclosed (λ i, |==> P i)%I.
Definition monPred_bupd_aux `{BUpd PROP} : seal monPred_bupd_def. by eexists. Qed.
-Global Instance monPred_bupd `{BUpd PROP} : BUpd _ := unseal monPred_bupd_aux.
-Definition monPred_bupd_eq `{BUpd PROP} : @bupd monPred _ = _ := seal_eq _.
+Definition monPred_bupd `{BUpd PROP} : BUpd _ := unseal monPred_bupd_aux.
+Definition monPred_bupd_eq `{BUpd PROP} : @bupd _ monPred_bupd = _ := seal_eq _.
End Bi.
Arguments monPred_absolutely {_ _} _%I.
@@ -214,15 +227,21 @@ Arguments monPred_relatively {_ _} _%I.
Notation "'∀ᵢ' P" := (monPred_absolutely P) (at level 20, right associativity) : bi_scope.
Notation "'∃ᵢ' P" := (monPred_relatively P) (at level 20, right associativity) : bi_scope.
-Typeclasses Opaque monPred_pure monPred_emp monPred_plainly.
-
Section Sbi.
Context {I : biIndex} {PROP : sbi}.
Implicit Types i : I.
Notation monPred := (monPred I PROP).
Implicit Types P Q : monPred.
-Definition monPred_internal_eq (A : ofeT) (a b : A) : monPred := tc_opaque ⎡a ≡ b⎤%I.
+Definition monPred_plainly_def `{BiPlainly PROP} P : monPred := MonPred (λ _, ∀ i, ■(P i))%I _.
+Definition monPred_plainly_aux `{BiPlainly PROP} : seal monPred_plainly_def. by eexists. Qed.
+Definition monPred_plainly `{BiPlainly PROP} : Plainly _ := unseal monPred_plainly_aux.
+Definition monPred_plainly_eq `{BiPlainly PROP} : @plainly _ monPred_plainly = _ := seal_eq _.
+
+Definition monPred_internal_eq_def (A : ofeT) (a b : A) : monPred := MonPred (λ _, a ≡ b)%I _.
+Definition monPred_internal_eq_aux : seal (@monPred_internal_eq_def). by eexists. Qed.
+Definition monPred_internal_eq := unseal monPred_internal_eq_aux.
+Definition monPred_internal_eq_eq : @monPred_internal_eq = _ := seal_eq _.
Program Definition monPred_later_def P : monPred := MonPred (λ i, ▷ (P i))%I _.
Next Obligation. solve_proper. Qed.
@@ -237,8 +256,8 @@ Definition monPred_fupd_def `{FUpd PROP} (E1 E2 : coPset) (P : monPred) : monPre
terms in logical terms. *)
monPred_upclosed (λ i, |={E1,E2}=> P i)%I.
Definition monPred_fupd_aux `{FUpd PROP} : seal monPred_fupd_def. by eexists. Qed.
-Global Instance monPred_fupd `{FUpd PROP} : FUpd _ := unseal monPred_fupd_aux.
-Definition monPred_fupd_eq `{FUpd PROP} : @fupd monPred _ = _ := seal_eq _.
+Definition monPred_fupd `{FUpd PROP} : FUpd _ := unseal monPred_fupd_aux.
+Definition monPred_fupd_eq `{FUpd PROP} : @fupd _ monPred_fupd = _ := seal_eq _.
End Sbi.
Module MonPred.
@@ -246,18 +265,18 @@ Definition unseal_eqs :=
(@monPred_and_eq, @monPred_or_eq, @monPred_impl_eq,
@monPred_forall_eq, @monPred_exist_eq,
@monPred_sep_eq, @monPred_wand_eq,
- @monPred_persistently_eq, @monPred_later_eq, @monPred_in_eq,
- @monPred_embed_eq, @monPred_bupd_eq, @monPred_fupd_eq).
+ @monPred_persistently_eq, @monPred_later_eq, @monPred_internal_eq_eq, @monPred_in_eq,
+ @monPred_embed_eq, @monPred_emp_eq, @monPred_pure_eq, @monPred_plainly_eq,
+ @monPred_absolutely_eq, @monPred_relatively_eq, @monPred_bupd_eq, @monPred_fupd_eq).
Ltac unseal :=
unfold bi_affinely, bi_absorbingly, sbi_except_0, bi_pure, bi_emp,
- monPred_absolutely, monPred_relatively, monPred_upclosed, bi_and, bi_or,
+ monPred_upclosed, bi_and, bi_or,
bi_impl, bi_forall, bi_exist, sbi_internal_eq, bi_sep, bi_wand,
- bi_persistently, bi_affinely, bi_plainly, sbi_later;
+ bi_persistently, bi_affinely, sbi_later;
simpl;
unfold sbi_emp, sbi_pure, sbi_and, sbi_or, sbi_impl, sbi_forall, sbi_exist,
- sbi_internal_eq, sbi_sep, sbi_wand, sbi_persistently, sbi_plainly;
+ sbi_internal_eq, sbi_sep, sbi_wand, sbi_persistently;
simpl;
- unfold monPred_pure, monPred_emp, monPred_internal_eq, monPred_plainly; simpl;
rewrite !unseal_eqs /=.
End MonPred.
Import MonPred.
@@ -268,7 +287,7 @@ Context (I : biIndex) (PROP : bi).
Lemma monPred_bi_mixin : BiMixin (PROP:=monPred I PROP)
monPred_entails monPred_emp monPred_pure monPred_and monPred_or
monPred_impl monPred_forall monPred_exist monPred_sep monPred_wand
- monPred_plainly monPred_persistently.
+ monPred_persistently.
Proof.
split; try unseal; try by (split=> ? /=; repeat f_equiv).
- split.
@@ -306,26 +325,9 @@ Proof.
apply bi.wand_intro_r. by rewrite -HR /= !Hij.
- intros P Q R [HP]. split=> i. apply bi.wand_elim_l'.
rewrite HP /= bi.forall_elim bi.pure_impl_forall bi.forall_elim //.
- - intros P Q [?]. split=> i /=. by do 3 f_equiv.
- - intros P. split=> i /=. by rewrite bi.forall_elim bi.plainly_elim_persistently.
- - intros P. split=> i /=. repeat setoid_rewrite <-bi.plainly_forall.
- rewrite -bi.plainly_idemp_2. f_equiv. by apply bi.forall_intro=>_.
- - intros A Ψ. split=> i /=. apply bi.forall_intro=> j.
- rewrite bi.plainly_forall. apply bi.forall_intro=> a.
- by rewrite !bi.forall_elim.
- - intros P Q. split=> i /=. repeat setoid_rewrite bi.pure_impl_forall.
- repeat setoid_rewrite <-bi.plainly_forall.
- repeat setoid_rewrite bi.persistently_forall. do 4 f_equiv.
- apply bi.persistently_impl_plainly.
- - intros P Q. split=> i /=.
- repeat setoid_rewrite bi.pure_impl_forall. rewrite 2!bi.forall_elim //.
- repeat setoid_rewrite <-bi.plainly_forall.
- setoid_rewrite bi.plainly_impl_plainly. f_equiv.
- do 3 apply bi.forall_intro => ?. f_equiv. rewrite bi.forall_elim //.
- - intros P. split=> i. apply bi.forall_intro=>_. by apply bi.plainly_emp_intro.
- - intros P Q. split=> i. apply bi.sep_elim_l, _.
- intros P Q [?]. split=> i /=. by f_equiv.
- intros P. split=> i. by apply bi.persistently_idemp_2.
+ - intros P. split=> i. by apply bi.persistently_emp_intro.
- intros A Ψ. split=> i. by apply bi.persistently_forall_2.
- intros A Ψ. split=> i. by apply bi.persistently_exist_1.
- intros P Q. split=> i. apply bi.sep_elim_l, _.
@@ -333,20 +335,16 @@ Proof.
Qed.
Canonical Structure monPredI : bi :=
- Bi (monPred I PROP) monPred_dist monPred_equiv monPred_entails monPred_emp
- monPred_pure monPred_and monPred_or monPred_impl monPred_forall
- monPred_exist monPred_sep monPred_wand monPred_plainly monPred_persistently
- monPred_ofe_mixin monPred_bi_mixin.
+ {| bi_ofe_mixin := monPred_ofe_mixin; bi_bi_mixin := monPred_bi_mixin |}.
End canonical_bi.
Section canonical_sbi.
Context (I : biIndex) (PROP : sbi).
Lemma monPred_sbi_mixin :
- SbiMixin (PROP:=monPred I PROP) monPred_ofe_mixin monPred_entails monPred_pure
- monPred_and monPred_or monPred_impl monPred_forall monPred_exist
- monPred_sep monPred_wand monPred_plainly monPred_persistently
- monPred_internal_eq monPred_later.
+ SbiMixin (PROP:=monPred I PROP) monPred_entails monPred_pure
+ monPred_or monPred_impl monPred_forall monPred_exist
+ monPred_sep monPred_persistently monPred_internal_eq monPred_later.
Proof.
split; unseal.
- intros n P Q HPQ. split=> i /=.
@@ -359,11 +357,6 @@ Proof.
- intros A1 A2 f g. split=> i. by apply bi.fun_ext.
- intros A P x y. split=> i. by apply bi.sig_eq.
- intros A a b ?. split=> i. by apply bi.discrete_eq_1.
- - intros P Q. split=> i /=.
- rewrite <-(sig_monPred_sig P), <-(sig_monPred_sig Q), <-(bi.f_equiv _).
- rewrite -bi.sig_equivI /= -bi.fun_ext. f_equiv=> j.
- rewrite -bi.prop_ext !(bi.forall_elim j) !bi.pure_impl_forall
- !bi.forall_elim //.
- intros A x y. split=> i. by apply bi.later_eq_1.
- intros A x y. split=> i. by apply bi.later_eq_2.
- intros P Q [?]. split=> i. by apply bi.later_mono.
@@ -373,10 +366,6 @@ Proof.
- intros A Ψ. split=> i. by apply bi.later_exist_false.
- intros P Q. split=> i. by apply bi.later_sep_1.
- intros P Q. split=> i. by apply bi.later_sep_2.
- - intros P. split=> i /=.
- rewrite bi.later_forall. f_equiv=> j. by rewrite -bi.later_plainly_1.
- - intros P. split=> i /=.
- rewrite bi.later_forall. f_equiv=> j. by rewrite -bi.later_plainly_2.
- intros P. split=> i. by apply bi.later_persistently_1.
- intros P. split=> i. by apply bi.later_persistently_2.
- intros P. split=> i /=. rewrite -bi.forall_intro. apply bi.later_false_em.
@@ -384,12 +373,50 @@ Proof.
Qed.
Canonical Structure monPredSI : sbi :=
- Sbi (monPred I PROP) monPred_dist monPred_equiv monPred_entails monPred_emp
- monPred_pure monPred_and monPred_or monPred_impl monPred_forall
- monPred_exist monPred_sep monPred_wand monPred_plainly
- monPred_persistently monPred_internal_eq monPred_later monPred_ofe_mixin
- (bi_bi_mixin _) monPred_sbi_mixin.
+ {| sbi_ofe_mixin := monPred_ofe_mixin; sbi_bi_mixin := monPred_bi_mixin I PROP;
+ sbi_sbi_mixin := monPred_sbi_mixin |}.
+
+Lemma monPred_plainly_mixin `{BiPlainly PROP} : BiPlainlyMixin monPredSI monPred_plainly.
+Proof.
+ split; unseal.
+ - by (split=> ? /=; repeat f_equiv).
+ - intros P Q [?]. split=> i /=. by do 3 f_equiv.
+ - intros P. split=> i /=. by rewrite bi.forall_elim plainly_elim_persistently.
+ - intros P. split=> i /=. repeat setoid_rewrite <-plainly_forall.
+ rewrite -plainly_idemp_2. f_equiv. by apply bi.forall_intro=>_.
+ - intros A Ψ. split=> i /=. apply bi.forall_intro=> j.
+ rewrite plainly_forall. apply bi.forall_intro=> a.
+ by rewrite !bi.forall_elim.
+ - intros P Q. split=> i /=. repeat setoid_rewrite bi.pure_impl_forall.
+ repeat setoid_rewrite <-plainly_forall.
+ repeat setoid_rewrite bi.persistently_forall. do 4 f_equiv.
+ apply persistently_impl_plainly.
+ - intros P Q. split=> i /=.
+ repeat setoid_rewrite bi.pure_impl_forall. rewrite 2!bi.forall_elim //.
+ repeat setoid_rewrite <-plainly_forall.
+ setoid_rewrite plainly_impl_plainly. f_equiv.
+ do 3 apply bi.forall_intro => ?. f_equiv. rewrite bi.forall_elim //.
+ - intros P. split=> i /=. apply bi.forall_intro=>_. by apply plainly_emp_intro.
+ - intros P Q. split=> i. apply bi.sep_elim_l, _.
+ - intros P Q. split=> i /=.
+ rewrite <-(sig_monPred_sig P), <-(sig_monPred_sig Q), <-(bi.f_equiv _).
+ rewrite -bi.sig_equivI /= -bi.fun_ext. f_equiv=> j.
+ rewrite -prop_ext !(bi.forall_elim j) !bi.pure_impl_forall
+ !bi.forall_elim //.
+ - intros P. split=> i /=.
+ rewrite bi.later_forall. f_equiv=> j. by rewrite -later_plainly_1.
+ - intros P. split=> i /=.
+ rewrite bi.later_forall. f_equiv=> j. by rewrite -later_plainly_2.
+Qed.
+Global Instance monPred_plainlyC `{BiPlainly PROP} : BiPlainly monPredSI :=
+ {| bi_plainly_mixin := monPred_plainly_mixin |}.
+Global Instance monPred_plainly_exist `{BiPlainly PROP} `{@BiIndexBottom I bot} :
+ BiPlainlyExist PROP → BiPlainlyExist monPredSI.
+Proof.
+ split=>?/=. unseal. rewrite (bi.forall_elim bot) plainly_exist_1. do 2 f_equiv.
+ apply bi.forall_intro=>?. by do 2 f_equiv.
+Qed.
End canonical_sbi.
Class Absolute {I : biIndex} {PROP : bi} (P : monPred I PROP) :=
@@ -411,7 +438,6 @@ Implicit Types i : I.
Implicit Types P Q : monPred.
(** Instances *)
-
Global Instance monPred_at_mono :
Proper ((⊢) ==> (⊑) ==> (⊢)) monPred_at.
Proof. by move=> ?? [?] ?? ->. Qed.
@@ -433,17 +459,8 @@ Proof. split => ?. unseal. apply bi_positive. Qed.
Global Instance monPred_affine : BiAffine PROP → BiAffine monPredI.
Proof. split => ?. unseal. by apply affine. Qed.
-Global Instance monPred_plainly_exist `{BiIndexBottom bot} :
- BiPlainlyExist PROP → BiPlainlyExist monPredI.
-Proof.
- split=>?/=. unseal. rewrite (bi.forall_elim bot) plainly_exist_1. do 2 f_equiv.
- apply bi.forall_intro=>?. by do 2 f_equiv.
-Qed.
-
Global Instance monPred_at_persistent P i : Persistent P → Persistent (P i).
Proof. move => [] /(_ i). by unseal. Qed.
-Global Instance monPred_at_plain P i : Plain P → Plain (P i).
-Proof. move => [] /(_ i). unfold Plain. unseal. rewrite bi.forall_elim //. Qed.
Global Instance monPred_at_absorbing P i : Absorbing P → Absorbing (P i).
Proof. move => [] /(_ i). unfold Absorbing. by unseal. Qed.
Global Instance monPred_at_affine P i : Affine P → Affine (P i).
@@ -458,8 +475,7 @@ Global Instance monPred_in_absorbing i :
Absorbing (@monPred_in I PROP i).
Proof. unfold Absorbing. unseal. split=> ? /=. apply absorbing, _. Qed.
-
-Global Instance monPred_bi_embedding : BiEmbedding PROP monPredI.
+Definition monPred_embedding_mixin : BiEmbedMixin PROP monPredI monPred_embed.
Proof.
split; try apply _; unseal; try done.
- move =>?? /= [/(_ inhabitant) ?] //.
@@ -467,64 +483,64 @@ Proof.
by rewrite bi.forall_elim bi.pure_impl_forall bi.forall_elim.
- split=>? /=.
by rewrite bi.forall_elim bi.pure_impl_forall bi.forall_elim.
- - intros ?. split => ? /=. apply bi.equiv_spec; split.
- by apply bi.forall_intro. by rewrite bi.forall_elim.
Qed.
+Global Instance monPred_bi_embed : BiEmbed PROP monPredI :=
+ {| bi_embed_mixin := monPred_embedding_mixin |}.
+
+Lemma monPred_emp_unfold : emp%I = ⎡emp : PROP⎤%I.
+Proof. by unseal. Qed.
+Lemma monPred_pure_unfold : bi_pure = λ φ, ⎡ ⌜ φ ⌠: PROP⎤%I.
+Proof. by unseal. Qed.
+Lemma monPred_absolutely_unfold : monPred_absolutely = λ P, ⎡∀ i, P i⎤%I.
+Proof. by unseal. Qed.
+Lemma monPred_relatively_unfold : monPred_relatively = λ P, ⎡∃ i, P i⎤%I.
+Proof. by unseal. Qed.
Global Instance monPred_absolutely_ne : NonExpansive (@monPred_absolutely I PROP).
-Proof. rewrite /monPred_absolutely /=. solve_proper. Qed.
+Proof. rewrite monPred_absolutely_unfold. solve_proper. Qed.
Global Instance monPred_absolutely_proper : Proper ((≡) ==> (≡)) (@monPred_absolutely I PROP).
Proof. apply (ne_proper _). Qed.
Lemma monPred_absolutely_mono P Q : (P ⊢ Q) → (∀ᵢ P ⊢ ∀ᵢ Q).
-Proof. rewrite /monPred_absolutely /=. solve_proper. Qed.
+Proof. rewrite monPred_absolutely_unfold. solve_proper. Qed.
Global Instance monPred_absolutely_mono' : Proper ((⊢) ==> (⊢)) (@monPred_absolutely I PROP).
-Proof. rewrite /monPred_absolutely /=. solve_proper. Qed.
+Proof. intros ???. by apply monPred_absolutely_mono. Qed.
Global Instance monPred_absolutely_flip_mono' :
Proper (flip (⊢) ==> flip (⊢)) (@monPred_absolutely I PROP).
-Proof. solve_proper. Qed.
-
-Local Notation "'∀ᵢ' P" := (@monPred_absolutely I PROP P) : bi_scope.
-Local Notation "'∃ᵢ' P" := (@monPred_relatively I PROP P) : bi_scope.
+Proof. intros ???. by apply monPred_absolutely_mono. Qed.
-Global Instance monPred_absolutely_plain P : Plain P → Plain (∀ᵢ P).
-Proof. rewrite /monPred_absolutely /=. apply _. Qed.
Global Instance monPred_absolutely_persistent P : Persistent P → Persistent (∀ᵢ P).
-Proof. rewrite /monPred_absolutely /=. apply _. Qed.
+Proof. rewrite monPred_absolutely_unfold. apply _. Qed.
Global Instance monPred_absolutely_absorbing P : Absorbing P → Absorbing (∀ᵢ P).
-Proof. rewrite /monPred_absolutely /=. apply _. Qed.
+Proof. rewrite monPred_absolutely_unfold. apply _. Qed.
Global Instance monPred_absolutely_affine P : Affine P → Affine (∀ᵢ P).
-Proof. rewrite /monPred_absolutely /=. apply _. Qed.
+Proof. rewrite monPred_absolutely_unfold. apply _. Qed.
Global Instance monPred_relatively_ne : NonExpansive (@monPred_relatively I PROP).
-Proof. rewrite /monPred_relatively /=. solve_proper. Qed.
+Proof. rewrite monPred_relatively_unfold. solve_proper. Qed.
Global Instance monPred_relatively_proper : Proper ((≡) ==> (≡)) (@monPred_relatively I PROP).
Proof. apply (ne_proper _). Qed.
Lemma monPred_relatively_mono P Q : (P ⊢ Q) → (∃ᵢ P ⊢ ∃ᵢ Q).
-Proof. rewrite /monPred_relatively /=. solve_proper. Qed.
+Proof. rewrite monPred_relatively_unfold. solve_proper. Qed.
Global Instance monPred_relatively_mono' : Proper ((⊢) ==> (⊢)) (@monPred_relatively I PROP).
-Proof. rewrite /monPred_relatively /=. solve_proper. Qed.
+Proof. intros ???. by apply monPred_relatively_mono. Qed.
Global Instance monPred_relatively_flip_mono' :
Proper (flip (⊢) ==> flip (⊢)) (@monPred_relatively I PROP).
-Proof. solve_proper. Qed.
+Proof. intros ???. by apply monPred_relatively_mono. Qed.
-Global Instance monPred_relatively_plain P : Plain P → Plain (∃ᵢ P).
-Proof. rewrite /monPred_relatively /=. apply _. Qed.
Global Instance monPred_relatively_persistent P : Persistent P → Persistent (∃ᵢ P).
-Proof. rewrite /monPred_relatively /=. apply _. Qed.
+Proof. rewrite monPred_relatively_unfold. apply _. Qed.
Global Instance monPred_relatively_absorbing P : Absorbing P → Absorbing (∃ᵢ P).
-Proof. rewrite /monPred_relatively /=. apply _. Qed.
+Proof. rewrite monPred_relatively_unfold. apply _. Qed.
Global Instance monPred_relatively_affine P : Affine P → Affine (∃ᵢ P).
-Proof. rewrite /monPred_relatively /=. apply _. Qed.
+Proof. rewrite monPred_relatively_unfold. apply _. Qed.
(** monPred_at unfolding laws *)
Lemma monPred_at_embed i (P : PROP) : monPred_at ⎡P⎤ i ⊣⊢ P.
Proof. by unseal. Qed.
Lemma monPred_at_pure i (φ : Prop) : monPred_at ⌜φ⌠i ⊣⊢ ⌜φâŒ.
-Proof. by apply monPred_at_embed. Qed.
+Proof. by unseal. Qed.
Lemma monPred_at_emp i : monPred_at emp i ⊣⊢ emp.
-Proof. by apply monPred_at_embed. Qed.
-Lemma monPred_at_plainly i P : bi_plainly P i ⊣⊢ ∀ j, bi_plainly (P j).
-Proof. by apply monPred_at_embed. Qed.
+Proof. by unseal. Qed.
Lemma monPred_at_and i P Q : (P ∧ Q) i ⊣⊢ P i ∧ Q i.
Proof. by unseal. Qed.
Lemma monPred_at_or i P Q : (P ∨ Q) i ⊣⊢ P i ∨ Q i.
@@ -545,7 +561,7 @@ Lemma monPred_at_in i j : monPred_at (monPred_in j) i ⊣⊢ ⌜j ⊑ iâŒ.
Proof. by unseal. Qed.
Lemma monPred_at_absolutely i P : (∀ᵢ P) i ⊣⊢ ∀ j, P j.
Proof. by unseal. Qed.
-Lemma monPred_at_ex i P : (∃ᵢ P) i ⊣⊢ ∃ j, P j.
+Lemma monPred_at_relatively i P : (∃ᵢ P) i ⊣⊢ ∃ j, P j.
Proof. by unseal. Qed.
Lemma monPred_at_persistently_if i p P :
bi_persistently_if p P i ⊣⊢ bi_persistently_if p (P i).
@@ -565,7 +581,7 @@ Proof. unseal. rewrite bi.forall_elim bi.pure_impl_forall bi.forall_elim //. Qed
(* Laws for monPred_absolutely and of Absolute. *)
Lemma monPred_absolutely_elim P : ∀ᵢ P ⊢ P.
-Proof. unseal. split=>?. apply bi.forall_elim. Qed.
+Proof. rewrite monPred_absolutely_unfold. unseal. split=>?. apply bi.forall_elim. Qed.
Lemma monPred_absolutely_idemp P : ∀ᵢ (∀ᵢ P) ⊣⊢ ∀ᵢ P.
Proof.
apply bi.equiv_spec; split; [by apply monPred_absolutely_elim|].
@@ -601,6 +617,10 @@ Proof.
apply bi.equiv_spec; split; unseal; split=>i /=.
by rewrite (bi.forall_elim inhabitant). by apply bi.forall_intro.
Qed.
+Lemma monPred_absolutely_emp : ∀ᵢ (emp : monPred) ⊣⊢ emp.
+Proof. rewrite monPred_emp_unfold. apply monPred_absolutely_embed. Qed.
+Lemma monPred_absolutely_pure φ : ∀ᵢ (⌜ φ ⌠: monPred) ⊣⊢ ⌜ φ âŒ.
+Proof. rewrite monPred_pure_unfold. apply monPred_absolutely_embed. Qed.
Lemma monPred_relatively_intro P : P ⊢ ∃ᵢ P.
Proof. unseal. split=>?. apply bi.exist_intro. Qed.
@@ -633,27 +653,25 @@ Qed.
Lemma absolute_absolutely P `{!Absolute P} : P ⊢ ∀ᵢ P.
Proof.
- rewrite /monPred_absolutely /= bi_embed_forall. apply bi.forall_intro=>?.
+ rewrite monPred_absolutely_unfold /= embed_forall. apply bi.forall_intro=>?.
split=>?. unseal. apply absolute_at, _.
Qed.
Lemma absolute_relatively P `{!Absolute P} : ∃ᵢ P ⊢ P.
Proof.
- rewrite /monPred_relatively /= bi_embed_exist. apply bi.exist_elim=>?.
+ rewrite monPred_relatively_unfold /= embed_exist. apply bi.exist_elim=>?.
split=>?. unseal. apply absolute_at, _.
Qed.
-Global Instance emb_absolute (P : PROP) : @Absolute I PROP ⎡P⎤.
+Global Instance embed_absolute (P : PROP) : @Absolute I PROP ⎡P⎤.
Proof. intros ??. by unseal. Qed.
Global Instance pure_absolute φ : @Absolute I PROP ⌜φâŒ.
-Proof. apply emb_absolute. Qed.
+Proof. intros ??. by unseal. Qed.
Global Instance emp_absolute : @Absolute I PROP emp.
-Proof. apply emb_absolute. Qed.
-Global Instance plainly_absolute P : Absolute (bi_plainly P).
-Proof. apply emb_absolute. Qed.
+Proof. intros ??. by unseal. Qed.
Global Instance absolutely_absolute P : Absolute (∀ᵢ P).
-Proof. apply emb_absolute. Qed.
+Proof. intros ??. by unseal. Qed.
Global Instance relatively_absolute P : Absolute (∃ᵢ P).
-Proof. apply emb_absolute. Qed.
+Proof. intros ??. by unseal. Qed.
Global Instance and_absolute P Q `{!Absolute P, !Absolute Q} : Absolute (P ∧ Q).
Proof. intros i j. unseal. by rewrite !(absolute_at _ i j). Qed.
@@ -691,9 +709,6 @@ Proof. rewrite /bi_affinely. apply _. Qed.
Global Instance absorbingly_absolute P `{!Absolute P} :
Absolute (bi_absorbingly P).
Proof. rewrite /bi_absorbingly. apply _. Qed.
-Global Instance plainly_if_absolute P p `{!Absolute P} :
- Absolute (bi_plainly_if p P).
-Proof. rewrite /bi_plainly_if. destruct p; apply _. Qed.
Global Instance persistently_if_absolute P p `{!Absolute P} :
Absolute (bi_persistently_if p P).
Proof. rewrite /bi_persistently_if. destruct p; apply _. Qed.
@@ -741,19 +756,19 @@ Global Instance monPred_absolutely_monoid_and_homomorphism :
MonoidHomomorphism bi_and bi_and (≡) (@monPred_absolutely I PROP).
Proof.
split; [split|]; try apply _. apply monPred_absolutely_and.
- apply monPred_absolutely_embed.
+ apply monPred_absolutely_pure.
Qed.
Global Instance monPred_absolutely_monoid_sep_entails_homomorphism :
MonoidHomomorphism bi_sep bi_sep (flip (⊢)) (@monPred_absolutely I PROP).
Proof.
split; [split|]; try apply _. apply monPred_absolutely_sep_2.
- by rewrite monPred_absolutely_embed.
+ by rewrite monPred_absolutely_emp.
Qed.
Global Instance monPred_absolutely_monoid_sep_homomorphism `{BiIndexBottom bot} :
MonoidHomomorphism bi_sep bi_sep (≡) (@monPred_absolutely I PROP).
Proof.
split; [split|]; try apply _. apply monPred_absolutely_sep.
- by rewrite monPred_absolutely_embed.
+ by rewrite monPred_absolutely_emp.
Qed.
Lemma monPred_absolutely_big_sepL_entails {A} (Φ : nat → A → monPred) l :
@@ -802,17 +817,8 @@ Global Instance big_sepMS_absolute `{Countable A} (Φ : A → monPred)
Absolute ([∗ mset] y ∈ X, Φ y)%I.
Proof. intros ??. rewrite !monPred_at_big_sepMS. do 2 f_equiv. by apply absolute_at. Qed.
-End bi_facts.
-
-Section sbi_facts.
-Context {I : biIndex} {PROP : sbi}.
-Local Notation monPred := (monPred I PROP).
-Local Notation monPredSI := (monPredSI I PROP).
-Implicit Types i : I.
-Implicit Types P Q : monPred.
-
(** bupd *)
-Global Instance monPred_bupd_facts `{BUpdFacts PROP} : BUpdFacts monPredSI.
+Lemma monPred_bupd_mixin `{BiBUpd PROP} : BiBUpdMixin monPredI monPred_bupd.
Proof.
split; unseal; unfold monPred_bupd_def, monPred_upclosed.
(* Note: Notation is somewhat broken here. *)
@@ -825,27 +831,35 @@ Proof.
- intros P Q. split=> /= i. apply bi.forall_intro=>?.
rewrite bi.pure_impl_forall. apply bi.forall_intro=><-.
rewrite -bupd_frame_r bi.forall_elim bi.pure_impl_forall bi.forall_elim //.
- - intros P. split=> /= i. rewrite bi.forall_elim bi.pure_impl_forall
- bi.forall_elim // -bi.plainly_forall bupd_plainly bi.forall_elim //.
Qed.
+Global Instance monPred_bi_bupd `{BiBUpd PROP} : BiBUpd monPredI :=
+ {| bi_bupd_mixin := monPred_bupd_mixin |}.
-Lemma monPred_at_bupd `{BUpdFacts PROP} i P : (|==> P) i ⊣⊢ |==> P i.
+Lemma monPred_at_bupd `{BiBUpd PROP} i P : (|==> P) i ⊣⊢ |==> P i.
Proof.
unseal. setoid_rewrite bi.pure_impl_forall. apply bi.equiv_spec; split.
- rewrite !bi.forall_elim //.
- do 2 apply bi.forall_intro=>?. by do 2 f_equiv.
Qed.
-Global Instance bupd_absolute `{BUpdFacts PROP} P `{!Absolute P} :
+Global Instance bupd_absolute `{BiBUpd PROP} P `{!Absolute P} :
Absolute (|==> P)%I.
Proof. intros ??. by rewrite !monPred_at_bupd absolute_at. Qed.
-Lemma monPred_bupd_embed `{BUpdFacts PROP} (P : PROP) :
- ⎡|==> P⎤ ⊣⊢ bupd (PROP:=monPredSI) ⎡P⎤.
+Lemma monPred_bupd_embed `{BiBUpd PROP} (P : PROP) :
+ ⎡|==> P⎤ ⊣⊢ bupd (PROP:=monPredI) ⎡P⎤.
Proof.
unseal. split=>i /=. setoid_rewrite bi.pure_impl_forall. apply bi.equiv_spec; split.
- by do 2 apply bi.forall_intro=>?.
- rewrite !bi.forall_elim //.
Qed.
+End bi_facts.
+
+Section sbi_facts.
+Context {I : biIndex} {PROP : sbi}.
+Local Notation monPred := (monPred I PROP).
+Local Notation monPredSI := (monPredSI I PROP).
+Implicit Types i : I.
+Implicit Types P Q : monPred.
Global Instance monPred_at_timeless P i : Timeless P → Timeless (P i).
Proof. move => [] /(_ i). unfold Timeless. by unseal. Qed.
@@ -862,19 +876,26 @@ Proof.
by apply timeless, bi.exist_timeless.
Qed.
-Global Instance monPred_sbi_embedding : SbiEmbedding PROP monPredSI.
-Proof. split; try apply _; by unseal. Qed.
+Global Instance monPred_sbi_embed : SbiEmbed PROP monPredSI.
+Proof.
+ split; unseal=> //. intros ? P Q.
+ apply (@bi.f_equiv _ _ _ (λ P, monPred_at P inhabitant)); solve_proper.
+Qed.
-Global Instance monPred_fupd_facts `{FUpdFacts PROP} : FUpdFacts monPredSI.
+Lemma monPred_plainly_unfold `{BiPlainly PROP} : plainly = λ P, ⎡ ∀ i, ■(P i) ⎤%I.
+Proof. by unseal. Qed.
+Lemma monPred_internal_eq_unfold : @sbi_internal_eq monPredSI = λ A x y, ⎡ x ≡ y ⎤%I.
+Proof. by unseal. Qed.
+
+Lemma monPred_fupd_mixin `{BiFUpd PROP} : BiFUpdMixin monPredSI monPred_fupd.
Proof.
- split; first apply _; unseal; unfold monPred_fupd_def, monPred_upclosed.
+ split; unseal; unfold monPred_fupd_def, monPred_upclosed.
(* Note: Notation is somewhat broken here. *)
- intros E1 E2 n P Q HPQ. split=>/= i. by repeat f_equiv.
- intros E1 E2 P HE12. split=>/= i.
do 2 setoid_rewrite bi.pure_impl_forall. do 2 apply bi.forall_intro=>?.
rewrite (fupd_intro_mask E1 E2 (P i)) //. f_equiv.
do 2 apply bi.forall_intro=>?. do 2 f_equiv. by etrans.
- - intros E P. split=>/= i. by setoid_rewrite bupd_fupd.
- intros E1 E2 P. split=>/= i. etrans; [apply bi.equiv_entails, bi.except_0_forall|].
do 2 f_equiv. rewrite bi.pure_impl_forall bi.except_0_forall. do 2 f_equiv.
by apply except_0_fupd.
@@ -886,6 +907,28 @@ Proof.
- intros E1 E2 P Q. split=>/= i. setoid_rewrite bi.pure_impl_forall.
do 2 setoid_rewrite bi.sep_forall_r. setoid_rewrite fupd_frame_r.
by repeat f_equiv.
+Qed.
+Global Instance monPred_bi_fupd `{BiFUpd PROP} : BiFUpd monPredSI :=
+ {| bi_fupd_mixin := monPred_fupd_mixin |}.
+Global Instance monPred_bi_bupd_fupd `{BiBUpdFUpd PROP} : BiBUpdFUpd monPredSI.
+Proof.
+ rewrite /BiBUpdFUpd; unseal; unfold monPred_fupd_def, monPred_upclosed.
+ intros E P. split=>/= i. by setoid_rewrite bupd_fupd.
+Qed.
+
+Global Instance monPred_bi_bupd_plainly `{BiBUpdPlainly PROP} : BiBUpdPlainly monPredSI.
+Proof.
+ rewrite /BiBUpdPlainly=> P; unseal.
+ split=> /= i. rewrite bi.forall_elim bi.pure_impl_forall.
+ by rewrite bi.forall_elim // -plainly_forall bupd_plainly bi.forall_elim.
+Qed.
+
+Global Instance monPred_at_plain `{BiPlainly PROP} P i : Plain P → Plain (P i).
+Proof. move => [] /(_ i). unfold Plain. unseal. rewrite bi.forall_elim //. Qed.
+
+Global Instance monPred_bi_fupd_plainly `{BiFUpdPlainly PROP} : BiFUpdPlainly monPredSI.
+Proof.
+ split; unseal.
- intros E1 E2 E2' P Q ? HE12. split=>/= i. repeat setoid_rewrite bi.pure_impl_forall.
do 4 f_equiv. rewrite 4?bi.forall_elim // fupd_plain' //.
apply bi.wand_intro_r. rewrite bi.wand_elim_l. do 2 apply bi.forall_intro=>?.
@@ -895,13 +938,14 @@ Proof.
Qed.
(** Unfolding lemmas *)
-
+Lemma monPred_at_plainly `{BiPlainly PROP} i P : (■P) i ⊣⊢ ∀ j, ■(P j).
+Proof. by unseal. Qed.
Lemma monPred_at_internal_eq {A : ofeT} i (a b : A) :
@monPred_at I PROP (a ≡ b) i ⊣⊢ a ≡ b.
-Proof. by apply monPred_at_embed. Qed.
+Proof. rewrite monPred_internal_eq_unfold. by apply monPred_at_embed. Qed.
Lemma monPred_at_later i P : (▷ P) i ⊣⊢ ▷ P i.
Proof. by unseal. Qed.
-Lemma monPred_at_fupd `{FUpdFacts PROP} i E1 E2 P :
+Lemma monPred_at_fupd `{BiFUpd PROP} i E1 E2 P :
(|={E1,E2}=> P) i ⊣⊢ |={E1,E2}=> P i.
Proof.
unseal. setoid_rewrite bi.pure_impl_forall. apply bi.equiv_spec; split.
@@ -911,7 +955,7 @@ Qed.
Lemma monPred_at_except_0 i P : (◇ P) i ⊣⊢ ◇ P i.
Proof. by unseal. Qed.
-Lemma monPred_fupd_embed `{FUpdFacts PROP} E1 E2 (P : PROP) :
+Lemma monPred_fupd_embed `{BiFUpd PROP} E1 E2 (P : PROP) :
⎡|={E1,E2}=> P⎤ ⊣⊢ fupd E1 E2 (PROP:=monPred) ⎡P⎤.
Proof.
unseal. split=>i /=. setoid_rewrite bi.pure_impl_forall. apply bi.equiv_spec; split.
@@ -928,7 +972,17 @@ Proof.
-bi.f_equiv -bi.sig_equivI !bi.ofe_fun_equivI.
Qed.
+Global Instance monPred_absolutely_plain `{BiPlainly PROP} P : Plain P → Plain (∀ᵢ P).
+Proof. rewrite monPred_absolutely_unfold. apply _. Qed.
+Global Instance monPred_relatively_plain `{BiPlainly PROP} P : Plain P → Plain (∃ᵢ P).
+Proof. rewrite monPred_relatively_unfold. apply _. Qed.
+
(** Absolute *)
+Global Instance plainly_absolute `{BiPlainly PROP} P : Absolute (â– P).
+Proof. intros ??. by unseal. Qed.
+Global Instance plainly_if_absolute `{BiPlainly PROP} P p `{!Absolute P} :
+ Absolute (â– ?p P).
+Proof. rewrite /plainly_if. destruct p; apply _. Qed.
Global Instance internal_eq_absolute {A : ofeT} (x y : A) :
@Absolute I PROP (x ≡ y)%I.
@@ -941,8 +995,7 @@ Proof. induction n; apply _. Qed.
Global Instance except0_absolute P `{!Absolute P} : Absolute (â—‡ P)%I.
Proof. rewrite /sbi_except_0. apply _. Qed.
-Global Instance fupd_absolute E1 E2 P `{!Absolute P} `{FUpdFacts PROP} :
+Global Instance fupd_absolute E1 E2 P `{!Absolute P} `{BiFUpd PROP} :
Absolute (|={E1,E2}=> P)%I.
Proof. intros ??. by rewrite !monPred_at_fupd absolute_at. Qed.
-
End sbi_facts.
diff --git a/theories/bi/plainly.v b/theories/bi/plainly.v
new file mode 100644
index 0000000000000000000000000000000000000000..06fff510df79c80d1340fc7d52d23d2ebb68df29
--- /dev/null
+++ b/theories/bi/plainly.v
@@ -0,0 +1,528 @@
+From iris.bi Require Export derived_laws.
+From iris.algebra Require Import monoid.
+Import interface.bi derived_laws.bi.
+
+Class Plainly (A : Type) := plainly : A → A.
+Hint Mode Plainly ! : typeclass_instances.
+Instance: Params (@plainly) 2.
+Notation "â– P" := (plainly P)%I
+ (at level 20, right associativity) : bi_scope.
+
+(* Mixins allow us to create instances easily without having to use Program *)
+Record BiPlainlyMixin (PROP : sbi) `(Plainly PROP) := {
+ bi_plainly_mixin_plainly_ne : NonExpansive plainly;
+
+ bi_plainly_mixin_plainly_mono P Q : (P ⊢ Q) → ■P ⊢ ■Q;
+ bi_plainly_mixin_plainly_elim_persistently P : ■P ⊢ bi_persistently P;
+ bi_plainly_mixin_plainly_idemp_2 P : ■P ⊢ ■■P;
+
+ bi_plainly_mixin_plainly_forall_2 {A} (Ψ : A → PROP) :
+ (∀ a, ■(Ψ a)) ⊢ ■(∀ a, Ψ a);
+
+ (* The following two laws are very similar, and indeed they hold not just
+ for persistently and plainly, but for any modality defined as `M P n x :=
+ ∀ y, R x y → P n y`. *)
+ bi_plainly_mixin_persistently_impl_plainly P Q :
+ (■P → bi_persistently Q) ⊢ bi_persistently (■P → Q);
+ bi_plainly_mixin_plainly_impl_plainly P Q : (■P → ■Q) ⊢ ■(■P → Q);
+
+ bi_plainly_mixin_plainly_emp_intro P : P ⊢ ■emp;
+ bi_plainly_mixin_plainly_absorb P Q : ■P ∗ Q ⊢ ■P;
+
+ bi_plainly_mixin_prop_ext P Q : ■((P -∗ Q) ∧ (Q -∗ P)) ⊢ P ≡ Q;
+
+ bi_plainly_mixin_later_plainly_1 P : ▷ ■P ⊢ ■▷ P;
+ bi_plainly_mixin_later_plainly_2 P : ■▷ P ⊢ ▷ ■P;
+}.
+
+Class BiPlainly (PROP : sbi) := {
+ bi_plainly_plainly :> Plainly PROP;
+ bi_plainly_mixin : BiPlainlyMixin PROP bi_plainly_plainly;
+}.
+Hint Mode BiPlainly ! : typeclass_instances.
+Arguments bi_plainly_plainly : simpl never.
+
+Class BiPlainlyExist `{!BiPlainly PROP} :=
+ plainly_exist_1 A (Ψ : A → PROP) :
+ ■(∃ a, Ψ a) ⊢ ∃ a, ■(Ψ a).
+Arguments BiPlainlyExist : clear implicits.
+Arguments BiPlainlyExist _ {_}.
+Arguments plainly_exist_1 _ {_ _} _.
+Hint Mode BiPlainlyExist + ! : typeclass_instances.
+
+Section plainly_laws.
+ Context `{BiPlainly PROP}.
+ Implicit Types P Q : PROP.
+
+ Global Instance plainly_ne : NonExpansive (@plainly PROP _).
+ Proof. eapply bi_plainly_mixin_plainly_ne, bi_plainly_mixin. Qed.
+
+ Lemma plainly_mono P Q : (P ⊢ Q) → ■P ⊢ ■Q.
+ Proof. eapply bi_plainly_mixin_plainly_mono, bi_plainly_mixin. Qed.
+ Lemma plainly_elim_persistently P : ■P ⊢ bi_persistently P.
+ Proof. eapply bi_plainly_mixin_plainly_elim_persistently, bi_plainly_mixin. Qed.
+ Lemma plainly_idemp_2 P : ■P ⊢ ■■P.
+ Proof. eapply bi_plainly_mixin_plainly_idemp_2, bi_plainly_mixin. Qed.
+ Lemma plainly_forall_2 {A} (Ψ : A → PROP) : (∀ a, ■(Ψ a)) ⊢ ■(∀ a, Ψ a).
+ Proof. eapply bi_plainly_mixin_plainly_forall_2, bi_plainly_mixin. Qed.
+ Lemma persistently_impl_plainly P Q :
+ (■P → bi_persistently Q) ⊢ bi_persistently (■P → Q).
+ Proof. eapply bi_plainly_mixin_persistently_impl_plainly, bi_plainly_mixin. Qed.
+ Lemma plainly_impl_plainly P Q : (■P → ■Q) ⊢ ■(■P → Q).
+ Proof. eapply bi_plainly_mixin_plainly_impl_plainly, bi_plainly_mixin. Qed.
+ Lemma plainly_absorb P Q : ■P ∗ Q ⊢ ■P.
+ Proof. eapply bi_plainly_mixin_plainly_absorb, bi_plainly_mixin. Qed.
+ Lemma plainly_emp_intro P : P ⊢ ■emp.
+ Proof. eapply bi_plainly_mixin_plainly_emp_intro, bi_plainly_mixin. Qed.
+
+ Lemma prop_ext P Q : ■((P -∗ Q) ∧ (Q -∗ P)) ⊢ P ≡ Q.
+ Proof. eapply bi_plainly_mixin_prop_ext, bi_plainly_mixin. Qed.
+
+ Lemma later_plainly_1 P : ▷ ■P ⊢ ■(▷ P).
+ Proof. eapply bi_plainly_mixin_later_plainly_1, bi_plainly_mixin. Qed.
+ Lemma later_plainly_2 P : ■▷ P ⊢ ▷ ■P.
+ Proof. eapply bi_plainly_mixin_later_plainly_2, bi_plainly_mixin. Qed.
+End plainly_laws.
+
+(* Derived properties and connectives *)
+Class Plain `{BiPlainly PROP} (P : PROP) := plain : P ⊢ ■P.
+Arguments Plain {_ _} _%I : simpl never.
+Arguments plain {_ _} _%I {_}.
+Hint Mode Plain + ! ! : typeclass_instances.
+Instance: Params (@Plain) 1.
+
+Definition plainly_if `{!BiPlainly PROP} (p : bool) (P : PROP) : PROP :=
+ (if p then â– P else P)%I.
+Arguments plainly_if {_ _} !_ _%I /.
+Instance: Params (@plainly_if) 2.
+Typeclasses Opaque plainly_if.
+
+Notation "â– ? p P" := (plainly_if p P)%I
+ (at level 20, p at level 9, P at level 20,
+ right associativity, format "â– ? p P") : bi_scope.
+
+(* Derived laws *)
+Section plainly_derived.
+Context `{BiPlainly PROP}.
+Implicit Types P : PROP.
+
+Hint Resolve pure_intro forall_intro.
+Hint Resolve or_elim or_intro_l' or_intro_r'.
+Hint Resolve and_intro and_elim_l' and_elim_r'.
+
+Global Instance plainly_proper :
+ Proper ((⊣⊢) ==> (⊣⊢)) (@plainly PROP _) := ne_proper _.
+
+Global Instance plainly_mono' : Proper ((⊢) ==> (⊢)) (@plainly PROP _).
+Proof. intros P Q; apply plainly_mono. Qed.
+Global Instance plainly_flip_mono' :
+ Proper (flip (⊢) ==> flip (⊢)) (@plainly PROP _).
+Proof. intros P Q; apply plainly_mono. Qed.
+
+Lemma affinely_plainly_elim P : bi_affinely (■P) ⊢ P.
+Proof. by rewrite plainly_elim_persistently affinely_persistently_elim. Qed.
+
+Lemma persistently_plainly P : bi_persistently (■P) ⊣⊢ ■P.
+Proof.
+ apply (anti_symm _).
+ - by rewrite persistently_elim_absorbingly /bi_absorbingly comm plainly_absorb.
+ - by rewrite {1}plainly_idemp_2 plainly_elim_persistently.
+Qed.
+Lemma persistently_if_plainly P p : bi_persistently_if p (■P) ⊣⊢ ■P.
+Proof. destruct p; last done. exact: persistently_plainly. Qed.
+
+Lemma plainly_persistently P : ■(bi_persistently P) ⊣⊢ ■P.
+Proof.
+ apply (anti_symm _).
+ - rewrite -{1}(left_id True%I bi_and (â– _)%I) (plainly_emp_intro True%I).
+ rewrite -{2}(persistently_and_emp_elim P).
+ rewrite !and_alt -plainly_forall_2. by apply forall_mono=> -[].
+ - by rewrite {1}plainly_idemp_2 (plainly_elim_persistently P).
+Qed.
+
+Lemma absorbingly_plainly P : bi_absorbingly (■P) ⊣⊢ ■P.
+Proof. by rewrite -(persistently_plainly P) absorbingly_persistently. Qed.
+
+Lemma plainly_and_sep_elim P Q : ■P ∧ Q -∗ (emp ∧ P) ∗ Q.
+Proof. by rewrite plainly_elim_persistently persistently_and_sep_elim_emp. Qed.
+Lemma plainly_and_sep_assoc P Q R : ■P ∧ (Q ∗ R) ⊣⊢ (■P ∧ Q) ∗ R.
+Proof. by rewrite -(persistently_plainly P) persistently_and_sep_assoc. Qed.
+Lemma plainly_and_emp_elim P : emp ∧ ■P ⊢ P.
+Proof. by rewrite plainly_elim_persistently persistently_and_emp_elim. Qed.
+Lemma plainly_elim_absorbingly P : ■P ⊢ bi_absorbingly P.
+Proof. by rewrite plainly_elim_persistently persistently_elim_absorbingly. Qed.
+Lemma plainly_elim P `{!Absorbing P} : ■P ⊢ P.
+Proof. by rewrite plainly_elim_persistently persistently_elim. Qed.
+
+Lemma plainly_idemp_1 P : ■■P ⊢ ■P.
+Proof. by rewrite plainly_elim_absorbingly absorbingly_plainly. Qed.
+Lemma plainly_idemp P : ■■P ⊣⊢ ■P.
+Proof. apply (anti_symm _); auto using plainly_idemp_1, plainly_idemp_2. Qed.
+
+Lemma plainly_intro' P Q : (■P ⊢ Q) → ■P ⊢ ■Q.
+Proof. intros <-. apply plainly_idemp_2. Qed.
+
+Lemma plainly_pure φ : ■⌜φ⌠⊣⊢ bi_pure (PROP:=PROP) φ.
+Proof.
+ apply (anti_symm _); auto.
+ - by rewrite plainly_elim_persistently persistently_pure.
+ - apply pure_elim'=> Hφ.
+ trans (∀ x : False, ■True : PROP)%I; [by apply forall_intro|].
+ rewrite plainly_forall_2. by rewrite -(pure_intro _ φ).
+Qed.
+Lemma plainly_forall {A} (Ψ : A → PROP) : ■(∀ a, Ψ a) ⊣⊢ ∀ a, ■(Ψ a).
+Proof.
+ apply (anti_symm _); auto using plainly_forall_2.
+ apply forall_intro=> x. by rewrite (forall_elim x).
+Qed.
+Lemma plainly_exist_2 {A} (Ψ : A → PROP) : (∃ a, ■(Ψ a)) ⊢ ■(∃ a, Ψ a).
+Proof. apply exist_elim=> x. by rewrite (exist_intro x). Qed.
+Lemma plainly_exist `{!BiPlainlyExist PROP} {A} (Ψ : A → PROP) :
+ ■(∃ a, Ψ a) ⊣⊢ ∃ a, ■(Ψ a).
+Proof. apply (anti_symm _); auto using plainly_exist_1, plainly_exist_2. Qed.
+Lemma plainly_and P Q : ■(P ∧ Q) ⊣⊢ ■P ∧ ■Q.
+Proof. rewrite !and_alt plainly_forall. by apply forall_proper=> -[]. Qed.
+Lemma plainly_or_2 P Q : ■P ∨ ■Q ⊢ ■(P ∨ Q).
+Proof. rewrite !or_alt -plainly_exist_2. by apply exist_mono=> -[]. Qed.
+Lemma plainly_or `{!BiPlainlyExist PROP} P Q : ■(P ∨ Q) ⊣⊢ ■P ∨ ■Q.
+Proof. rewrite !or_alt plainly_exist. by apply exist_proper=> -[]. Qed.
+Lemma plainly_impl P Q : ■(P → Q) ⊢ ■P → ■Q.
+Proof.
+ apply impl_intro_l; rewrite -plainly_and.
+ apply plainly_mono, impl_elim with P; auto.
+Qed.
+
+Lemma plainly_sep_dup P : ■P ⊣⊢ ■P ∗ ■P.
+Proof.
+ apply (anti_symm _).
+ - rewrite -{1}(idemp bi_and (â– _)%I).
+ by rewrite -{2}(left_id emp%I _ (â– _)%I) plainly_and_sep_assoc and_elim_l.
+ - by rewrite plainly_absorb.
+Qed.
+
+Lemma plainly_and_sep_l_1 P Q : ■P ∧ Q ⊢ ■P ∗ Q.
+Proof. by rewrite -{1}(left_id emp%I _ Q%I) plainly_and_sep_assoc and_elim_l. Qed.
+Lemma plainly_and_sep_r_1 P Q : P ∧ ■Q ⊢ P ∗ ■Q.
+Proof. by rewrite !(comm _ P) plainly_and_sep_l_1. Qed.
+
+Lemma plainly_True_emp : ■True ⊣⊢ ■(@bi_emp PROP).
+Proof. apply (anti_symm _); eauto using plainly_mono, plainly_emp_intro. Qed.
+Lemma plainly_and_sep P Q : ■(P ∧ Q) ⊢ ■(P ∗ Q).
+Proof.
+ rewrite plainly_and.
+ rewrite -{1}plainly_idemp -plainly_and -{1}(left_id emp%I _ Q%I).
+ by rewrite plainly_and_sep_assoc (comm bi_and) plainly_and_emp_elim.
+Qed.
+
+Lemma plainly_affinely P : ■(bi_affinely P) ⊣⊢ ■P.
+Proof. by rewrite /bi_affinely plainly_and -plainly_True_emp plainly_pure left_id. Qed.
+
+Lemma and_sep_plainly P Q : ■P ∧ ■Q ⊣⊢ ■P ∗ ■Q.
+Proof.
+ apply (anti_symm _); auto using plainly_and_sep_l_1.
+ apply and_intro.
+ - by rewrite plainly_absorb.
+ - by rewrite comm plainly_absorb.
+Qed.
+Lemma plainly_sep_2 P Q : ■P ∗ ■Q ⊢ ■(P ∗ Q).
+Proof. by rewrite -plainly_and_sep plainly_and -and_sep_plainly. Qed.
+Lemma plainly_sep `{BiPositive PROP} P Q : ■(P ∗ Q) ⊣⊢ ■P ∗ ■Q.
+Proof.
+ apply (anti_symm _); auto using plainly_sep_2.
+ rewrite -(plainly_affinely (_ ∗ _)%I) affinely_sep -and_sep_plainly. apply and_intro.
+ - by rewrite (affinely_elim_emp Q) right_id affinely_elim.
+ - by rewrite (affinely_elim_emp P) left_id affinely_elim.
+Qed.
+
+Lemma plainly_wand P Q : ■(P -∗ Q) ⊢ ■P -∗ ■Q.
+Proof. apply wand_intro_r. by rewrite plainly_sep_2 wand_elim_l. Qed.
+
+Lemma plainly_entails_l P Q : (P ⊢ ■Q) → P ⊢ ■Q ∗ P.
+Proof. intros; rewrite -plainly_and_sep_l_1; auto. Qed.
+Lemma plainly_entails_r P Q : (P ⊢ ■Q) → P ⊢ P ∗ ■Q.
+Proof. intros; rewrite -plainly_and_sep_r_1; auto. Qed.
+
+Lemma plainly_impl_wand_2 P Q : ■(P -∗ Q) ⊢ ■(P → Q).
+Proof.
+ apply plainly_intro', impl_intro_r.
+ rewrite -{2}(left_id emp%I _ P%I) plainly_and_sep_assoc.
+ by rewrite (comm bi_and) plainly_and_emp_elim wand_elim_l.
+Qed.
+
+Lemma impl_wand_plainly_2 P Q : (■P -∗ Q) ⊢ (■P → Q).
+Proof. apply impl_intro_l. by rewrite plainly_and_sep_l_1 wand_elim_r. Qed.
+
+Lemma impl_wand_affinely_plainly P Q : (■P → Q) ⊣⊢ (bi_affinely (■P) -∗ Q).
+Proof. by rewrite -(persistently_plainly P) impl_wand_affinely_persistently. Qed.
+
+Section plainly_affine_bi.
+ Context `{BiAffine PROP}.
+
+ Lemma plainly_emp : ■emp ⊣⊢ bi_emp (PROP:=PROP).
+ Proof. by rewrite -!True_emp plainly_pure. Qed.
+
+ Lemma plainly_and_sep_l P Q : ■P ∧ Q ⊣⊢ ■P ∗ Q.
+ Proof.
+ apply (anti_symm (⊢));
+ eauto using plainly_and_sep_l_1, sep_and with typeclass_instances.
+ Qed.
+ Lemma plainly_and_sep_r P Q : P ∧ ■Q ⊣⊢ P ∗ ■Q.
+ Proof. by rewrite !(comm _ P) plainly_and_sep_l. Qed.
+
+ Lemma plainly_impl_wand P Q : ■(P → Q) ⊣⊢ ■(P -∗ Q).
+ Proof.
+ apply (anti_symm (⊢)); auto using plainly_impl_wand_2.
+ apply plainly_intro', wand_intro_l.
+ by rewrite -plainly_and_sep_r plainly_elim impl_elim_r.
+ Qed.
+
+ Lemma impl_wand_plainly P Q : (■P → Q) ⊣⊢ (■P -∗ Q).
+ Proof.
+ apply (anti_symm (⊢)). by rewrite -impl_wand_1. by rewrite impl_wand_plainly_2.
+ Qed.
+End plainly_affine_bi.
+
+(* Conditional plainly *)
+Global Instance plainly_if_ne p : NonExpansive (@plainly_if PROP _ p).
+Proof. solve_proper. Qed.
+Global Instance plainly_if_proper p : Proper ((⊣⊢) ==> (⊣⊢)) (@plainly_if PROP _ p).
+Proof. solve_proper. Qed.
+Global Instance plainly_if_mono' p : Proper ((⊢) ==> (⊢)) (@plainly_if PROP _ p).
+Proof. solve_proper. Qed.
+Global Instance plainly_if_flip_mono' p :
+ Proper (flip (⊢) ==> flip (⊢)) (@plainly_if PROP _ p).
+Proof. solve_proper. Qed.
+
+Lemma plainly_if_mono p P Q : (P ⊢ Q) → ■?p P ⊢ ■?p Q.
+Proof. by intros ->. Qed.
+
+Lemma plainly_if_pure p φ : ■?p ⌜φ⌠⊣⊢ bi_pure (PROP:=PROP) φ.
+Proof. destruct p; simpl; auto using plainly_pure. Qed.
+Lemma plainly_if_and p P Q : ■?p (P ∧ Q) ⊣⊢ ■?p P ∧ ■?p Q.
+Proof. destruct p; simpl; auto using plainly_and. Qed.
+Lemma plainly_if_or_2 p P Q : ■?p P ∨ ■?p Q ⊢ ■?p (P ∨ Q).
+Proof. destruct p; simpl; auto using plainly_or_2. Qed.
+Lemma plainly_if_or `{!BiPlainlyExist PROP} p P Q : ■?p (P ∨ Q) ⊣⊢ ■?p P ∨ ■?p Q.
+Proof. destruct p; simpl; auto using plainly_or. Qed.
+Lemma plainly_if_exist_2 {A} p (Ψ : A → PROP) : (∃ a, ■?p (Ψ a)) ⊢ ■?p (∃ a, Ψ a).
+Proof. destruct p; simpl; auto using plainly_exist_2. Qed.
+Lemma plainly_if_exist `{!BiPlainlyExist PROP} {A} p (Ψ : A → PROP) :
+ ■?p (∃ a, Ψ a) ⊣⊢ ∃ a, ■?p (Ψ a).
+Proof. destruct p; simpl; auto using plainly_exist. Qed.
+Lemma plainly_if_sep_2 `{!BiPositive PROP} p P Q : ■?p P ∗ ■?p Q ⊢ ■?p (P ∗ Q).
+Proof. destruct p; simpl; auto using plainly_sep_2. Qed.
+
+Lemma plainly_if_idemp p P : ■?p ■?p P ⊣⊢ ■?p P.
+Proof. destruct p; simpl; auto using plainly_idemp. Qed.
+
+(* Properties of plain propositions *)
+Global Instance Plain_proper : Proper ((≡) ==> iff) (@Plain PROP _).
+Proof. solve_proper. Qed.
+
+Lemma plain_plainly_2 P `{!Plain P} : P ⊢ ■P.
+Proof. done. Qed.
+Lemma plain_plainly P `{!Plain P, !Absorbing P} : ■P ⊣⊢ P.
+Proof. apply (anti_symm _), plain_plainly_2, _. by apply plainly_elim. Qed.
+Lemma plainly_intro P Q `{!Plain P} : (P ⊢ Q) → P ⊢ ■Q.
+Proof. by intros <-. Qed.
+
+(* Typeclass instances *)
+Global Instance plainly_absorbing P : Absorbing (â– P).
+Proof. by rewrite /Absorbing /bi_absorbingly comm plainly_absorb. Qed.
+Global Instance plainly_if_absorbing P p :
+ Absorbing P → Absorbing (plainly_if p P).
+Proof. intros; destruct p; simpl; apply _. Qed.
+
+(* Not an instance, see the bottom of this file *)
+Lemma plain_persistent P : Plain P → Persistent P.
+Proof. intros. by rewrite /Persistent -plainly_elim_persistently. Qed.
+
+(* Not an instance, see the bottom of this file *)
+Lemma impl_persistent P Q :
+ Absorbing P → Plain P → Persistent Q → Persistent (P → Q).
+Proof.
+ intros. by rewrite /Persistent {2}(plain P) -persistently_impl_plainly
+ -(persistent Q) (plainly_elim_absorbingly P) absorbing.
+Qed.
+
+Global Instance plainly_persistent P : Persistent (â– P).
+Proof. by rewrite /Persistent persistently_plainly. Qed.
+
+Global Instance wand_persistent P Q :
+ Plain P → Persistent Q → Absorbing Q → Persistent (P -∗ Q).
+Proof.
+ intros. rewrite /Persistent {2}(plain P). trans (bi_persistently (■P → Q)%I).
+ - rewrite -persistently_impl_plainly impl_wand_affinely_plainly -(persistent Q).
+ by rewrite affinely_plainly_elim.
+ - apply persistently_mono, wand_intro_l. by rewrite sep_and impl_elim_r.
+Qed.
+
+(* Instances for big operators *)
+Global Instance plainly_and_homomorphism :
+ MonoidHomomorphism bi_and bi_and (≡) (@plainly PROP _).
+Proof.
+ split; [split|]; try apply _. apply plainly_and. apply plainly_pure.
+Qed.
+
+Global Instance plainly_or_homomorphism `{!BiPlainlyExist PROP} :
+ MonoidHomomorphism bi_or bi_or (≡) (@plainly PROP _).
+Proof.
+ split; [split|]; try apply _. apply plainly_or. apply plainly_pure.
+Qed.
+
+Global Instance plainly_sep_weak_homomorphism `{!BiPositive PROP, !BiAffine PROP} :
+ WeakMonoidHomomorphism bi_sep bi_sep (≡) (@plainly PROP _).
+Proof. split; try apply _. apply plainly_sep. Qed.
+
+Global Instance plainly_sep_homomorphism `{BiAffine PROP} :
+ MonoidHomomorphism bi_sep bi_sep (≡) (@plainly PROP _).
+Proof. split. apply _. apply plainly_emp. Qed.
+
+Global Instance plainly_sep_entails_weak_homomorphism :
+ WeakMonoidHomomorphism bi_sep bi_sep (flip (⊢)) (@plainly PROP _).
+Proof. split; try apply _. intros P Q; by rewrite plainly_sep_2. Qed.
+
+Global Instance plainly_sep_entails_homomorphism `{!BiAffine PROP} :
+ MonoidHomomorphism bi_sep bi_sep (flip (⊢)) (@plainly PROP _).
+Proof. split. apply _. simpl. rewrite plainly_emp. done. Qed.
+
+Global Instance limit_preserving_Plain {A:ofeT} `{Cofe A} (Φ : A → PROP) :
+ NonExpansive Φ → LimitPreserving (λ x, Plain (Φ x)).
+Proof. intros. apply limit_preserving_entails; solve_proper. Qed.
+
+(* Plainness instances *)
+Global Instance pure_plain φ : Plain (⌜φâŒ%I : PROP).
+Proof. by rewrite /Plain plainly_pure. Qed.
+Global Instance emp_plain : Plain (PROP:=PROP) emp.
+Proof. apply plainly_emp_intro. Qed.
+Global Instance and_plain P Q : Plain P → Plain Q → Plain (P ∧ Q).
+Proof. intros. by rewrite /Plain plainly_and -!plain. Qed.
+Global Instance or_plain P Q : Plain P → Plain Q → Plain (P ∨ Q).
+Proof. intros. by rewrite /Plain -plainly_or_2 -!plain. Qed.
+Global Instance forall_plain {A} (Ψ : A → PROP) :
+ (∀ x, Plain (Ψ x)) → Plain (∀ x, Ψ x).
+Proof.
+ intros. rewrite /Plain plainly_forall. apply forall_mono=> x. by rewrite -plain.
+Qed.
+Global Instance exist_plain {A} (Ψ : A → PROP) :
+ (∀ x, Plain (Ψ x)) → Plain (∃ x, Ψ x).
+Proof.
+ intros. rewrite /Plain -plainly_exist_2. apply exist_mono=> x. by rewrite -plain.
+Qed.
+
+Global Instance impl_plain P Q : Absorbing P → Plain P → Plain Q → Plain (P → Q).
+Proof.
+ intros. by rewrite /Plain {2}(plain P) -plainly_impl_plainly -(plain Q)
+ (plainly_elim_absorbingly P) absorbing.
+Qed.
+Global Instance wand_plain P Q :
+ Plain P → Plain Q → Absorbing Q → Plain (P -∗ Q).
+Proof.
+ intros. rewrite /Plain {2}(plain P). trans (■(■P → Q))%I.
+ - rewrite -plainly_impl_plainly impl_wand_affinely_plainly -(plain Q).
+ by rewrite affinely_plainly_elim.
+ - apply plainly_mono, wand_intro_l. by rewrite sep_and impl_elim_r.
+Qed.
+Global Instance sep_plain P Q : Plain P → Plain Q → Plain (P ∗ Q).
+Proof. intros. by rewrite /Plain -plainly_sep_2 -!plain. Qed.
+
+Global Instance plainly_plain P : Plain (â– P).
+Proof. by rewrite /Plain plainly_idemp. Qed.
+Global Instance persistently_plain P : Plain P → Plain (bi_persistently P).
+Proof.
+ rewrite /Plain=> HP. rewrite {1}HP plainly_persistently persistently_plainly //.
+Qed.
+Global Instance affinely_plain P : Plain P → Plain (bi_affinely P).
+Proof. rewrite /bi_affinely. apply _. Qed.
+Global Instance absorbingly_plain P : Plain P → Plain (bi_absorbingly P).
+Proof. rewrite /bi_absorbingly. apply _. Qed.
+Global Instance from_option_plain {A} P (Ψ : A → PROP) (mx : option A) :
+ (∀ x, Plain (Ψ x)) → Plain P → Plain (from_option Ψ P mx).
+Proof. destruct mx; apply _. Qed.
+
+(* Interaction with equality *)
+Lemma plainly_internal_eq {A:ofeT} (a b : A) : plainly (A:=PROP) (a ≡ b) ⊣⊢ a ≡ b.
+Proof.
+ apply (anti_symm (⊢)).
+ { by rewrite plainly_elim. }
+ apply (internal_eq_rewrite' a b (λ b, ■(a ≡ b))%I); [solve_proper|done|].
+ rewrite -(internal_eq_refl True%I a) plainly_pure; auto.
+Qed.
+
+Lemma plainly_alt P : ■P ⊣⊢ bi_affinely P ≡ emp.
+Proof.
+ rewrite -plainly_affinely. apply (anti_symm (⊢)).
+ - rewrite -prop_ext. apply plainly_mono, and_intro; apply wand_intro_l.
+ + by rewrite affinely_elim_emp left_id.
+ + by rewrite left_id.
+ - rewrite internal_eq_sym (internal_eq_rewrite _ _ plainly).
+ by rewrite -plainly_True_emp plainly_pure True_impl.
+Qed.
+
+Lemma plainly_alt_absorbing P `{!Absorbing P} : ■P ⊣⊢ P ≡ True.
+Proof.
+ apply (anti_symm (⊢)).
+ - rewrite -prop_ext. apply plainly_mono, and_intro; apply wand_intro_l; auto.
+ - rewrite internal_eq_sym (internal_eq_rewrite _ _ plainly).
+ by rewrite plainly_pure True_impl.
+Qed.
+
+Lemma plainly_True_alt P : ■(True -∗ P) ⊣⊢ P ≡ True.
+Proof.
+ apply (anti_symm (⊢)).
+ - rewrite -prop_ext. apply plainly_mono, and_intro; apply wand_intro_l; auto.
+ by rewrite wand_elim_r.
+ - rewrite internal_eq_sym (internal_eq_rewrite _ _
+ (λ Q, ■(True -∗ Q))%I ltac:(shelve)); last solve_proper.
+ by rewrite -entails_wand // -(plainly_emp_intro True%I) True_impl.
+Qed.
+
+(* Interaction with â–· *)
+Lemma later_plainly P : ▷ ■P ⊣⊢ ■▷ P.
+Proof. apply (anti_symm _); auto using later_plainly_1, later_plainly_2. Qed.
+Lemma laterN_plainly n P : ▷^n ■P ⊣⊢ ■▷^n P.
+Proof. induction n as [|n IH]; simpl; auto. by rewrite IH later_plainly. Qed.
+
+Lemma later_plainly_if p P : ▷ ■?p P ⊣⊢ ■?p ▷ P.
+Proof. destruct p; simpl; auto using later_plainly. Qed.
+Lemma laterN_plainly_if n p P : ▷^n ■?p P ⊣⊢ ■?p (▷^n P).
+Proof. destruct p; simpl; auto using laterN_plainly. Qed.
+
+Lemma except_0_plainly_1 P : ◇ ■P ⊢ ■◇ P.
+Proof. by rewrite /sbi_except_0 -plainly_or_2 -later_plainly plainly_pure. Qed.
+Lemma except_0_plainly `{!BiPlainlyExist PROP} P : ◇ ■P ⊣⊢ ■◇ P.
+Proof. by rewrite /sbi_except_0 plainly_or -later_plainly plainly_pure. Qed.
+
+Global Instance internal_eq_plain {A : ofeT} (a b : A) :
+ Plain (a ≡ b : PROP)%I.
+Proof. by intros; rewrite /Plain plainly_internal_eq. Qed.
+
+Global Instance later_plain P : Plain P → Plain (▷ P).
+Proof. intros. by rewrite /Plain -later_plainly {1}(plain P). Qed.
+Global Instance laterN_plain n P : Plain P → Plain (▷^n P).
+Proof. induction n; apply _. Qed.
+Global Instance except_0_plain P : Plain P → Plain (◇ P).
+Proof. rewrite /sbi_except_0; apply _. Qed.
+
+Global Instance plainly_timeless P `{!BiPlainlyExist PROP} :
+ Timeless P → Timeless (■P).
+Proof.
+ intros. rewrite /Timeless /sbi_except_0 later_plainly_1.
+ by rewrite (timeless P) /sbi_except_0 plainly_or {1}plainly_elim.
+Qed.
+End plainly_derived.
+
+(* When declared as an actual instance, [plain_persistent] will cause
+failing proof searches to take exponential time, as Coq will try to
+apply it the instance at any node in the proof search tree.
+
+To avoid that, we declare it using a [Hint Immediate], so that it will
+only be used at the leaves of the proof search tree, i.e. when the
+premise of the hint can be derived from just the current context. *)
+Hint Immediate plain_persistent : typeclass_instances.
+
+(* Not defined using an ordinary [Instance] because the default
+[class_apply @impl_persistent] shelves the [BiPlainly] premise, making proof
+search for the other premises fail. See the proof of [coreP_persistent] for an
+example where it would fail with a regular [Instance].*)
+Hint Extern 4 (Persistent (_ → _)) => eapply @impl_persistent : typeclass_instances.
diff --git a/theories/bi/updates.v b/theories/bi/updates.v
index e84991291f1f38ab5290fe928f851f4f99507a9c..cb97dd5e844b843144fb2854eb39c1a27861b0ba 100644
--- a/theories/bi/updates.v
+++ b/theories/bi/updates.v
@@ -1,6 +1,8 @@
From stdpp Require Import coPset.
-From iris.bi Require Import interface derived_laws big_op.
+From iris.bi Require Import interface derived_laws big_op plainly.
+(* We first define operational type classes for the notations, and then later
+bundle these operational type classes with the laws. *)
Class BUpd (PROP : Type) : Type := bupd : PROP → PROP.
Instance : Params (@bupd) 2.
Hint Mode BUpd ! : typeclass_instances.
@@ -35,7 +37,6 @@ Notation "P ={ E }=∗ Q" := (P -∗ |={E}=> Q)
(at level 99, E at level 50, Q at level 200, only parsing) : stdpp_scope.
(** Fancy updates that take a step. *)
-
Notation "|={ E1 , E2 }â–·=> Q" := (|={E1,E2}=> (â–· |={E2,E1}=> Q))%I
(at level 99, E1, E2 at level 50, Q at level 200,
format "|={ E1 , E2 }â–·=> Q") : bi_scope.
@@ -49,26 +50,101 @@ Notation "P ={ E }▷=∗ Q" := (P ={E,E}▷=∗ Q)%I
(at level 99, E at level 50, Q at level 200,
format "P ={ E }▷=∗ Q") : bi_scope.
-(** BUpd facts *)
-
-Class BUpdFacts (PROP : sbi) `{BUpd PROP} : Prop :=
- { bupd_ne :> NonExpansive bupd;
- bupd_intro (P : PROP) : P ==∗ P;
- bupd_mono (P Q : PROP) : (P ⊢ Q) → (|==> P) ==∗ Q;
- bupd_trans (P : PROP) : (|==> |==> P) ==∗ P;
- bupd_frame_r (P R : PROP) : (|==> P) ∗ R ==∗ P ∗ R;
- bupd_plainly (P : PROP) : (|==> bi_plainly P) -∗ P }.
-Hint Mode BUpdFacts ! - : typeclass_instances.
+(** Bundled versions *)
+(* Mixins allow us to create instances easily without having to use Program *)
+Record BiBUpdMixin (PROP : bi) `(BUpd PROP) := {
+ bi_bupd_mixin_bupd_ne : NonExpansive bupd;
+ bi_bupd_mixin_bupd_intro (P : PROP) : P ==∗ P;
+ bi_bupd_mixin_bupd_mono (P Q : PROP) : (P ⊢ Q) → (|==> P) ==∗ Q;
+ bi_bupd_mixin_bupd_trans (P : PROP) : (|==> |==> P) ==∗ P;
+ bi_bupd_mixin_bupd_frame_r (P R : PROP) : (|==> P) ∗ R ==∗ P ∗ R;
+}.
+
+Record BiFUpdMixin (PROP : sbi) `(FUpd PROP) := {
+ bi_fupd_mixin_fupd_ne E1 E2 : NonExpansive (fupd E1 E2);
+ bi_fupd_mixin_fupd_intro_mask E1 E2 (P : PROP) : E2 ⊆ E1 → P ⊢ |={E1,E2}=> |={E2,E1}=> P;
+ bi_fupd_mixin_except_0_fupd E1 E2 (P : PROP) : ◇ (|={E1,E2}=> P) ={E1,E2}=∗ P;
+ bi_fupd_mixin_fupd_mono E1 E2 (P Q : PROP) : (P ⊢ Q) → (|={E1,E2}=> P) ⊢ |={E1,E2}=> Q;
+ bi_fupd_mixin_fupd_trans E1 E2 E3 (P : PROP) : (|={E1,E2}=> |={E2,E3}=> P) ⊢ |={E1,E3}=> P;
+ bi_fupd_mixin_fupd_mask_frame_r' E1 E2 Ef (P : PROP) :
+ E1 ## Ef → (|={E1,E2}=> ⌜E2 ## Ef⌠→ P) ={E1 ∪ Ef,E2 ∪ Ef}=∗ P;
+ bi_fupd_mixin_fupd_frame_r E1 E2 (P Q : PROP) : (|={E1,E2}=> P) ∗ Q ={E1,E2}=∗ P ∗ Q;
+}.
+
+Class BiBUpd (PROP : bi) := {
+ bi_bupd_bupd :> BUpd PROP;
+ bi_bupd_mixin : BiBUpdMixin PROP bi_bupd_bupd;
+}.
+Hint Mode BiBUpd ! : typeclass_instances.
+Arguments bi_bupd_bupd : simpl never.
+
+Class BiFUpd (PROP : sbi) := {
+ bi_fupd_fupd :> FUpd PROP;
+ bi_fupd_mixin : BiFUpdMixin PROP bi_fupd_fupd;
+}.
+Hint Mode BiBUpd ! : typeclass_instances.
+Arguments bi_fupd_fupd : simpl never.
+
+Class BiBUpdFUpd (PROP : sbi) `{BiBUpd PROP, BiFUpd PROP} :=
+ bupd_fupd E (P : PROP) : (|==> P) ={E}=∗ P.
+
+Class BiBUpdPlainly (PROP : sbi) `{!BiBUpd PROP, !BiPlainly PROP} :=
+ bupd_plainly (P : PROP) : (|==> ■P) -∗ P.
+
+Class BiFUpdPlainly (PROP : sbi) `{!BiFUpd PROP, !BiPlainly PROP} := {
+ fupd_plain' E1 E2 E2' (P Q : PROP) `{!Plain P} :
+ E1 ⊆ E2 →
+ (Q ={E1, E2'}=∗ P) -∗ (|={E1, E2}=> Q) ={E1}=∗ (|={E1, E2}=> Q) ∗ P;
+ later_fupd_plain E (P : PROP) `{!Plain P} :
+ (▷ |={E}=> P) ={E}=∗ ▷ ◇ P;
+}.
+
+Section bupd_laws.
+ Context `{BiBUpd PROP}.
+ Implicit Types P : PROP.
+
+ Global Instance bupd_ne : NonExpansive (@bupd PROP _).
+ Proof. eapply bi_bupd_mixin_bupd_ne, bi_bupd_mixin. Qed.
+ Lemma bupd_intro P : P ==∗ P.
+ Proof. eapply bi_bupd_mixin_bupd_intro, bi_bupd_mixin. Qed.
+ Lemma bupd_mono (P Q : PROP) : (P ⊢ Q) → (|==> P) ==∗ Q.
+ Proof. eapply bi_bupd_mixin_bupd_mono, bi_bupd_mixin. Qed.
+ Lemma bupd_trans (P : PROP) : (|==> |==> P) ==∗ P.
+ Proof. eapply bi_bupd_mixin_bupd_trans, bi_bupd_mixin. Qed.
+ Lemma bupd_frame_r (P R : PROP) : (|==> P) ∗ R ==∗ P ∗ R.
+ Proof. eapply bi_bupd_mixin_bupd_frame_r, bi_bupd_mixin. Qed.
+End bupd_laws.
+
+Section fupd_laws.
+ Context `{BiFUpd PROP}.
+ Implicit Types P : PROP.
+
+ Global Instance fupd_ne E1 E2 : NonExpansive (@fupd PROP _ E1 E2).
+ Proof. eapply bi_fupd_mixin_fupd_ne, bi_fupd_mixin. Qed.
+ Lemma fupd_intro_mask E1 E2 (P : PROP) : E2 ⊆ E1 → P ⊢ |={E1,E2}=> |={E2,E1}=> P.
+ Proof. eapply bi_fupd_mixin_fupd_intro_mask, bi_fupd_mixin. Qed.
+ Lemma except_0_fupd E1 E2 (P : PROP) : ◇ (|={E1,E2}=> P) ={E1,E2}=∗ P.
+ Proof. eapply bi_fupd_mixin_except_0_fupd, bi_fupd_mixin. Qed.
+ Lemma fupd_mono E1 E2 (P Q : PROP) : (P ⊢ Q) → (|={E1,E2}=> P) ⊢ |={E1,E2}=> Q.
+ Proof. eapply bi_fupd_mixin_fupd_mono, bi_fupd_mixin. Qed.
+ Lemma fupd_trans E1 E2 E3 (P : PROP) : (|={E1,E2}=> |={E2,E3}=> P) ⊢ |={E1,E3}=> P.
+ Proof. eapply bi_fupd_mixin_fupd_trans, bi_fupd_mixin. Qed.
+ Lemma fupd_mask_frame_r' E1 E2 Ef (P : PROP) :
+ E1 ## Ef → (|={E1,E2}=> ⌜E2 ## Ef⌠→ P) ={E1 ∪ Ef,E2 ∪ Ef}=∗ P.
+ Proof. eapply bi_fupd_mixin_fupd_mask_frame_r', bi_fupd_mixin. Qed.
+ Lemma fupd_frame_r E1 E2 (P Q : PROP) : (|={E1,E2}=> P) ∗ Q ={E1,E2}=∗ P ∗ Q.
+ Proof. eapply bi_fupd_mixin_fupd_frame_r, bi_fupd_mixin. Qed.
+End fupd_laws.
Section bupd_derived.
- Context `{BUpdFacts PROP}.
- Implicit Types P Q R: PROP.
+ Context `{BiBUpd PROP}.
+ Implicit Types P Q R : PROP.
(* FIXME: Removing the `PROP:=` diverges. *)
- Global Instance bupd_proper : Proper ((≡) ==> (≡)) (bupd (PROP:=PROP)) := ne_proper _.
+ Global Instance bupd_proper :
+ Proper ((≡) ==> (≡)) (bupd (PROP:=PROP)) := ne_proper _.
(** BUpd derived rules *)
-
Global Instance bupd_mono' : Proper ((⊢) ==> (⊢)) (bupd (PROP:=PROP)).
Proof. intros P Q; apply bupd_mono. Qed.
Global Instance bupd_flip_mono' : Proper (flip (⊢) ==> flip (⊢)) (bupd (PROP:=PROP)).
@@ -82,50 +158,30 @@ Section bupd_derived.
Proof. by rewrite bupd_frame_r bi.wand_elim_r. Qed.
Lemma bupd_sep P Q : (|==> P) ∗ (|==> Q) ==∗ P ∗ Q.
Proof. by rewrite bupd_frame_r bupd_frame_l bupd_trans. Qed.
- Lemma bupd_affinely_plainly `{BiAffine PROP} P : (|==> ■P) ⊢ P.
- Proof. by rewrite bi.affine_affinely bupd_plainly. Qed.
- Lemma bupd_plain P `{!Plain P} : (|==> P) ⊢ P.
- Proof. by rewrite {1}(plain P) bupd_plainly. Qed.
End bupd_derived.
-Lemma except_0_bupd {PROP : sbi} `{BUpdFacts PROP} (P : PROP) :
- ◇ (|==> P) ⊢ (|==> ◇ P).
-Proof.
- rewrite /sbi_except_0. apply bi.or_elim; eauto using bupd_mono, bi.or_intro_r.
- by rewrite -bupd_intro -bi.or_intro_l.
-Qed.
-
-(** FUpd facts *)
-
-(* Currently, this requires an SBI, because of [except_0_fupd] and
- [later_fupd_plain]. If need be, we can generalize this to BIs by
- extracting these propertes as lemmas to be proved by hand. *)
-Class FUpdFacts (PROP : sbi) `{BUpd PROP, FUpd PROP} : Prop :=
- { fupd_bupd_facts :> BUpdFacts PROP;
- fupd_ne E1 E2 :> NonExpansive (fupd E1 E2);
- fupd_intro_mask E1 E2 (P : PROP) : E2 ⊆ E1 → P ⊢ |={E1,E2}=> |={E2,E1}=> P;
- bupd_fupd E (P : PROP) : (|==> P) ={E}=∗ P;
- except_0_fupd E1 E2 (P : PROP) : ◇ (|={E1,E2}=> P) ={E1,E2}=∗ P;
- fupd_mono E1 E2 (P Q : PROP) : (P ⊢ Q) → (|={E1,E2}=> P) ⊢ |={E1,E2}=> Q;
- fupd_trans E1 E2 E3 (P : PROP) : (|={E1,E2}=> |={E2,E3}=> P) ⊢ |={E1,E3}=> P;
- fupd_mask_frame_r' E1 E2 Ef (P : PROP) :
- E1 ## Ef → (|={E1,E2}=> ⌜E2 ## Ef⌠→ P) ={E1 ∪ Ef,E2 ∪ Ef}=∗ P;
- fupd_frame_r E1 E2 (P Q : PROP) : (|={E1,E2}=> P) ∗ Q ={E1,E2}=∗ P ∗ Q;
- fupd_plain' E1 E2 E2' (P Q : PROP) `{!Plain P} :
- E1 ⊆ E2 →
- (Q ={E1, E2'}=∗ P) -∗ (|={E1, E2}=> Q) ={E1}=∗ (|={E1, E2}=> Q) ∗ P;
- later_fupd_plain E (P : PROP) `{!Plain P} : (▷ |={E}=> P) ={E}=∗ ▷ ◇ P }.
-
-Hint Mode FUpdFacts ! - - : typeclass_instances.
+Section bupd_derived_sbi.
+ Context {PROP : sbi} `{BiBUpd PROP}.
+ Implicit Types P Q R : PROP.
+
+ Lemma except_0_bupd P : ◇ (|==> P) ⊢ (|==> ◇ P).
+ Proof.
+ rewrite /sbi_except_0. apply bi.or_elim; eauto using bupd_mono, bi.or_intro_r.
+ by rewrite -bupd_intro -bi.or_intro_l.
+ Qed.
+
+ Lemma bupd_plain P `{BiBUpdPlainly PROP, !Plain P} : (|==> P) ⊢ P.
+ Proof. by rewrite {1}(plain P) bupd_plainly. Qed.
+End bupd_derived_sbi.
Section fupd_derived.
- Context `{FUpdFacts PROP}.
- Implicit Types P Q R: PROP.
+ Context `{BiFUpd PROP}.
+ Implicit Types P Q R : PROP.
- Global Instance fupd_proper E1 E2 : Proper ((≡) ==> (≡)) (fupd (PROP:=PROP) E1 E2) := ne_proper _.
+ Global Instance fupd_proper E1 E2 :
+ Proper ((≡) ==> (≡)) (fupd (PROP:=PROP) E1 E2) := ne_proper _.
(** FUpd derived rules *)
-
Global Instance fupd_mono' E1 E2 : Proper ((⊢) ==> (⊢)) (fupd (PROP:=PROP) E1 E2).
Proof. intros P Q; apply fupd_mono. Qed.
Global Instance fupd_flip_mono' E1 E2 :
@@ -133,7 +189,7 @@ Section fupd_derived.
Proof. intros P Q; apply fupd_mono. Qed.
Lemma fupd_intro E P : P ={E}=∗ P.
- Proof. rewrite -bupd_fupd. apply bupd_intro. Qed.
+ Proof. by rewrite {1}(fupd_intro_mask E E P) // fupd_trans. Qed.
Lemma fupd_intro_mask' E1 E2 : E2 ⊆ E1 → (|={E1,E2}=> |={E2,E1}=> bi_emp (PROP:=PROP))%I.
Proof. exact: fupd_intro_mask. Qed.
Lemma fupd_except_0 E1 E2 P : (|={E1,E2}=> ◇ P) ={E1,E2}=∗ P.
@@ -185,7 +241,7 @@ Section fupd_derived.
intros P1 P2 HP Q1 Q2 HQ. by rewrite HP HQ -fupd_sep.
Qed.
- Lemma fupd_plain E1 E2 P Q `{!Plain P} :
+ Lemma fupd_plain `{BiPlainly PROP, !BiFUpdPlainly PROP} E1 E2 P Q `{!Plain P} :
E1 ⊆ E2 → (Q -∗ P) -∗ (|={E1, E2}=> Q) ={E1}=∗ (|={E1, E2}=> Q) ∗ P.
Proof.
intros HE. rewrite -(fupd_plain' _ _ E1) //. apply bi.wand_intro_l.
@@ -193,7 +249,6 @@ Section fupd_derived.
Qed.
(** Fancy updates that take a step derived rules. *)
-
Lemma step_fupd_wand E1 E2 P Q : (|={E1,E2}▷=> P) -∗ (P -∗ Q) -∗ |={E1,E2}▷=> Q.
Proof.
apply bi.wand_intro_l.
diff --git a/theories/proofmode/class_instances.v b/theories/proofmode/class_instances.v
index 65ec13c6fc6ce20756abea1527b53a572c27ca8b..f1fe442394ac9951801c8d78800f44177b3e655c 100644
--- a/theories/proofmode/class_instances.v
+++ b/theories/proofmode/class_instances.v
@@ -39,24 +39,6 @@ Global Instance from_assumption_absorbingly_r p P Q :
FromAssumption p P Q → FromAssumption p P (bi_absorbingly Q).
Proof. rewrite /FromAssumption /= =><-. apply absorbingly_intro. Qed.
-Global Instance from_assumption_affinely_plainly_l p P Q :
- FromAssumption true P Q → FromAssumption p (■P) Q.
-Proof.
- rewrite /FromAssumption /= =><-.
- by rewrite affinely_persistently_if_elim plainly_elim_persistently.
-Qed.
-Global Instance from_assumption_plainly_l_true P Q :
- FromAssumption true P Q → FromAssumption true (bi_plainly P) Q.
-Proof.
- rewrite /FromAssumption /= =><-.
- by rewrite persistently_elim plainly_elim_persistently.
-Qed.
-Global Instance from_assumption_plainly_l_false `{BiAffine PROP} P Q :
- FromAssumption true P Q → FromAssumption false (bi_plainly P) Q.
-Proof.
- rewrite /FromAssumption /= =><-.
- by rewrite affine_affinely plainly_elim_persistently.
-Qed.
Global Instance from_assumption_affinely_persistently_l p P Q :
FromAssumption true P Q → FromAssumption p (□ P) Q.
Proof. rewrite /FromAssumption /= =><-. by rewrite affinely_persistently_if_elim. Qed.
@@ -112,14 +94,12 @@ Global Instance into_pure_affinely P φ :
Proof. rewrite /IntoPure=> ->. apply affinely_elim. Qed.
Global Instance into_pure_absorbingly P φ : IntoPure P φ → IntoPure (bi_absorbingly P) φ.
Proof. rewrite /IntoPure=> ->. by rewrite absorbingly_pure. Qed.
-Global Instance into_pure_plainly P φ : IntoPure P φ → IntoPure (bi_plainly P) φ.
-Proof. rewrite /IntoPure=> ->. apply: plainly_elim. Qed.
Global Instance into_pure_persistently P φ :
IntoPure P φ → IntoPure (bi_persistently P) φ.
Proof. rewrite /IntoPure=> ->. apply: persistently_elim. Qed.
-Global Instance into_pure_embed `{BiEmbedding PROP PROP'} P φ :
+Global Instance into_pure_embed `{BiEmbed PROP PROP'} P φ :
IntoPure P φ → IntoPure ⎡P⎤ φ.
-Proof. rewrite /IntoPure=> ->. by rewrite bi_embed_pure. Qed.
+Proof. rewrite /IntoPure=> ->. by rewrite embed_pure. Qed.
(* FromPure *)
Global Instance from_pure_pure a φ : @FromPure PROP a ⌜φ⌠φ.
@@ -173,9 +153,6 @@ Proof.
by rewrite bi.affinely_if_elim pure_wand_forall pure_impl pure_impl_forall.
Qed.
-Global Instance from_pure_plainly P φ :
- FromPure false P φ → FromPure false (bi_plainly P) φ.
-Proof. rewrite /FromPure=> <-. by rewrite plainly_pure. Qed.
Global Instance from_pure_persistently P a φ :
FromPure true P φ → FromPure a (bi_persistently P) φ.
Proof.
@@ -192,9 +169,9 @@ Proof. rewrite /FromPure /= affine_affinely //. Qed.
Global Instance from_pure_absorbingly P φ :
FromPure true P φ → FromPure false (bi_absorbingly P) φ.
Proof. rewrite /FromPure=> <- /=. apply persistent_absorbingly_affinely, _. Qed.
-Global Instance from_pure_embed `{BiEmbedding PROP PROP'} a P φ :
+Global Instance from_pure_embed `{BiEmbed PROP PROP'} a P φ :
FromPure a P φ → FromPure a ⎡P⎤ φ.
-Proof. rewrite /FromPure=> <-. by rewrite bi_embed_affinely_if bi_embed_pure. Qed.
+Proof. rewrite /FromPure=> <-. by rewrite embed_affinely_if embed_pure. Qed.
(* IntoPersistent *)
Global Instance into_persistent_persistently p P Q :
@@ -206,10 +183,10 @@ Qed.
Global Instance into_persistent_affinely p P Q :
IntoPersistent p P Q → IntoPersistent p (bi_affinely P) Q | 0.
Proof. rewrite /IntoPersistent /= => <-. by rewrite affinely_elim. Qed.
-Global Instance into_persistent_embed `{BiEmbedding PROP PROP'} p P Q :
+Global Instance into_persistent_embed `{BiEmbed PROP PROP'} p P Q :
IntoPersistent p P Q → IntoPersistent p ⎡P⎤ ⎡Q⎤ | 0.
Proof.
- rewrite /IntoPersistent -bi_embed_persistently -bi_embed_persistently_if=> -> //.
+ rewrite /IntoPersistent -embed_persistently -embed_persistently_if=> -> //.
Qed.
Global Instance into_persistent_here P : IntoPersistent true P P | 1.
Proof. by rewrite /IntoPersistent. Qed.
@@ -232,54 +209,34 @@ Global Instance from_modal_affinely_persistently_affine_bi P :
BiAffine PROP → FromModal modality_persistently (□ P) (□ P) P | 0.
Proof. intros. by rewrite /FromModal /= affine_affinely. Qed.
-Global Instance from_modal_plainly P :
- FromModal modality_plainly (bi_plainly P) (bi_plainly P) P | 2.
-Proof. by rewrite /FromModal. Qed.
-Global Instance from_modal_affinely_plainly P :
- FromModal modality_affinely_plainly (â– P) (â– P) P | 1.
-Proof. by rewrite /FromModal. Qed.
-Global Instance from_modal_affinely_plainly_affine_bi P :
- BiAffine PROP → FromModal modality_plainly (■P) (■P) P | 0.
-Proof. intros. by rewrite /FromModal /= affine_affinely. Qed.
-
Global Instance from_modal_absorbingly P :
FromModal modality_id (bi_absorbingly P) (bi_absorbingly P) P.
Proof. by rewrite /FromModal /= -absorbingly_intro. Qed.
(* When having a modality nested in an embedding, e.g. [ ⎡|==> P⎤ ], we prefer
the embedding over the modality. *)
-Global Instance from_modal_embed `{BiEmbedding PROP PROP'} (P : PROP) :
- FromModal (@modality_embed PROP PROP' _ _) ⎡P⎤ ⎡P⎤ P.
+Global Instance from_modal_embed `{BiEmbed PROP PROP'} (P : PROP) :
+ FromModal (@modality_embed PROP PROP' _) ⎡P⎤ ⎡P⎤ P.
Proof. by rewrite /FromModal. Qed.
-Global Instance from_modal_id_embed `{BiEmbedding PROP PROP'} `(sel : A) P Q :
+Global Instance from_modal_id_embed `{BiEmbed PROP PROP'} `(sel : A) P Q :
FromModal modality_id sel P Q →
FromModal modality_id sel ⎡P⎤ ⎡Q⎤ | 100.
Proof. by rewrite /FromModal /= =><-. Qed.
-Global Instance from_modal_affinely_embed `{BiEmbedding PROP PROP'} `(sel : A) P Q :
+Global Instance from_modal_affinely_embed `{BiEmbed PROP PROP'} `(sel : A) P Q :
FromModal modality_affinely sel P Q →
FromModal modality_affinely sel ⎡P⎤ ⎡Q⎤ | 100.
-Proof. rewrite /FromModal /= =><-. by rewrite bi_embed_affinely. Qed.
-Global Instance from_modal_persistently_embed `{BiEmbedding PROP PROP'} `(sel : A) P Q :
+Proof. rewrite /FromModal /= =><-. by rewrite embed_affinely. Qed.
+Global Instance from_modal_persistently_embed `{BiEmbed PROP PROP'} `(sel : A) P Q :
FromModal modality_persistently sel P Q →
FromModal modality_persistently sel ⎡P⎤ ⎡Q⎤ | 100.
-Proof. rewrite /FromModal /= =><-. by rewrite bi_embed_persistently. Qed.
-Global Instance from_modal_affinely_persistently_embed `{BiEmbedding PROP PROP'} `(sel : A) P Q :
+Proof. rewrite /FromModal /= =><-. by rewrite embed_persistently. Qed.
+Global Instance from_modal_affinely_persistently_embed `{BiEmbed PROP PROP'} `(sel : A) P Q :
FromModal modality_affinely_persistently sel P Q →
FromModal modality_affinely_persistently sel ⎡P⎤ ⎡Q⎤ | 100.
Proof.
- rewrite /FromModal /= =><-. by rewrite bi_embed_affinely bi_embed_persistently.
-Qed.
-Global Instance from_modal_plainly_embed `{BiEmbedding PROP PROP'} `(sel : A) P Q :
- FromModal modality_plainly sel P Q →
- FromModal modality_plainly sel ⎡P⎤ ⎡Q⎤ | 100.
-Proof. rewrite /FromModal /= =><-. by rewrite bi_embed_plainly. Qed.
-Global Instance from_modal_affinely_plainly_embed `{BiEmbedding PROP PROP'} `(sel : A) P Q :
- FromModal modality_affinely_plainly sel P Q →
- FromModal modality_affinely_plainly sel ⎡P⎤ ⎡Q⎤ | 100.
-Proof.
- rewrite /FromModal /= =><-. by rewrite bi_embed_affinely bi_embed_plainly.
+ rewrite /FromModal /= =><-. by rewrite embed_affinely embed_persistently.
Qed.
(* IntoWand *)
@@ -344,16 +301,6 @@ Global Instance into_wand_forall {A} p q (Φ : A → PROP) P Q x :
IntoWand p q (Φ x) P Q → IntoWand p q (∀ x, Φ x) P Q.
Proof. rewrite /IntoWand=> <-. by rewrite (forall_elim x). Qed.
-Global Instance into_wand_affinely_plainly p q R P Q :
- IntoWand p q R P Q → IntoWand p q (■R) P Q.
-Proof. by rewrite /IntoWand affinely_plainly_elim. Qed.
-Global Instance into_wand_plainly_true q R P Q :
- IntoWand true q R P Q → IntoWand true q (bi_plainly R) P Q.
-Proof. by rewrite /IntoWand /= persistently_plainly plainly_elim_persistently. Qed.
-Global Instance into_wand_plainly_false `{!BiAffine PROP} q R P Q :
- IntoWand false q R P Q → IntoWand false q (bi_plainly R) P Q.
-Proof. by rewrite /IntoWand plainly_elim. Qed.
-
Global Instance into_wand_affinely_persistently p q R P Q :
IntoWand p q R P Q → IntoWand p q (□ R) P Q.
Proof. by rewrite /IntoWand affinely_persistently_elim. Qed.
@@ -363,26 +310,26 @@ Proof. by rewrite /IntoWand /= persistently_idemp. Qed.
Global Instance into_wand_persistently_false `{!BiAffine PROP} q R P Q :
IntoWand false q R P Q → IntoWand false q (bi_persistently R) P Q.
Proof. by rewrite /IntoWand persistently_elim. Qed.
-Global Instance into_wand_embed `{BiEmbedding PROP PROP'} p q R P Q :
+Global Instance into_wand_embed `{BiEmbed PROP PROP'} p q R P Q :
IntoWand p q R P Q → IntoWand p q ⎡R⎤ ⎡P⎤ ⎡Q⎤.
Proof.
- rewrite /IntoWand -!bi_embed_persistently_if -!bi_embed_affinely_if
- -bi_embed_wand => -> //.
+ rewrite /IntoWand -!embed_persistently_if -!embed_affinely_if
+ -embed_wand => -> //.
Qed.
(* FromWand *)
Global Instance from_wand_wand P1 P2 : FromWand (P1 -∗ P2) P1 P2.
Proof. by rewrite /FromWand. Qed.
-Global Instance from_wand_embed `{BiEmbedding PROP PROP'} P Q1 Q2 :
+Global Instance from_wand_embed `{BiEmbed PROP PROP'} P Q1 Q2 :
FromWand P Q1 Q2 → FromWand ⎡P⎤ ⎡Q1⎤ ⎡Q2⎤.
-Proof. by rewrite /FromWand -bi_embed_wand => <-. Qed.
+Proof. by rewrite /FromWand -embed_wand => <-. Qed.
(* FromImpl *)
Global Instance from_impl_impl P1 P2 : FromImpl (P1 → P2) P1 P2.
Proof. by rewrite /FromImpl. Qed.
-Global Instance from_impl_embed `{BiEmbedding PROP PROP'} P Q1 Q2 :
+Global Instance from_impl_embed `{BiEmbed PROP PROP'} P Q1 Q2 :
FromImpl P Q1 Q2 → FromImpl ⎡P⎤ ⎡Q1⎤ ⎡Q2⎤.
-Proof. by rewrite /FromImpl -bi_embed_impl => <-. Qed.
+Proof. by rewrite /FromImpl -embed_impl => <-. Qed.
(* FromAnd *)
Global Instance from_and_and P1 P2 : FromAnd (P1 ∧ P2) P1 P2 | 100.
@@ -406,15 +353,6 @@ Qed.
Global Instance from_and_pure φ ψ : @FromAnd PROP ⌜φ ∧ ψ⌠⌜φ⌠⌜ψâŒ.
Proof. by rewrite /FromAnd pure_and. Qed.
-Global Instance from_and_plainly P Q1 Q2 :
- FromAnd P Q1 Q2 →
- FromAnd (bi_plainly P) (bi_plainly Q1) (bi_plainly Q2).
-Proof. rewrite /FromAnd=> <-. by rewrite plainly_and. Qed.
-Global Instance from_and_plainly_sep P Q1 Q2 :
- FromSep P Q1 Q2 →
- FromAnd (bi_plainly P) (bi_plainly Q1) (bi_plainly Q2) | 11.
-Proof. rewrite /FromAnd=> <-. by rewrite -plainly_and plainly_and_sep. Qed.
-
Global Instance from_and_persistently P Q1 Q2 :
FromAnd P Q1 Q2 →
FromAnd (bi_persistently P) (bi_persistently Q1) (bi_persistently Q2).
@@ -424,9 +362,9 @@ Global Instance from_and_persistently_sep P Q1 Q2 :
FromAnd (bi_persistently P) (bi_persistently Q1) (bi_persistently Q2) | 11.
Proof. rewrite /FromAnd=> <-. by rewrite -persistently_and persistently_and_sep. Qed.
-Global Instance from_and_embed `{BiEmbedding PROP PROP'} P Q1 Q2 :
+Global Instance from_and_embed `{BiEmbed PROP PROP'} P Q1 Q2 :
FromAnd P Q1 Q2 → FromAnd ⎡P⎤ ⎡Q1⎤ ⎡Q2⎤.
-Proof. by rewrite /FromAnd -bi_embed_and => <-. Qed.
+Proof. by rewrite /FromAnd -embed_and => <-. Qed.
Global Instance from_and_big_sepL_cons_persistent {A} (Φ : nat → A → PROP) x l :
Persistent (Φ 0 x) →
@@ -455,18 +393,14 @@ Proof. rewrite /FromSep=> <-. by rewrite affinely_sep_2. Qed.
Global Instance from_sep_absorbingly P Q1 Q2 :
FromSep P Q1 Q2 → FromSep (bi_absorbingly P) (bi_absorbingly Q1) (bi_absorbingly Q2).
Proof. rewrite /FromSep=> <-. by rewrite absorbingly_sep. Qed.
-Global Instance from_sep_plainly P Q1 Q2 :
- FromSep P Q1 Q2 →
- FromSep (bi_plainly P) (bi_plainly Q1) (bi_plainly Q2).
-Proof. rewrite /FromSep=> <-. by rewrite plainly_sep_2. Qed.
Global Instance from_sep_persistently P Q1 Q2 :
FromSep P Q1 Q2 →
FromSep (bi_persistently P) (bi_persistently Q1) (bi_persistently Q2).
Proof. rewrite /FromSep=> <-. by rewrite persistently_sep_2. Qed.
-Global Instance from_sep_embed `{BiEmbedding PROP PROP'} P Q1 Q2 :
+Global Instance from_sep_embed `{BiEmbed PROP PROP'} P Q1 Q2 :
FromSep P Q1 Q2 → FromSep ⎡P⎤ ⎡Q1⎤ ⎡Q2⎤.
-Proof. by rewrite /FromSep -bi_embed_sep => <-. Qed.
+Proof. by rewrite /FromSep -embed_sep => <-. Qed.
Global Instance from_sep_big_sepL_cons {A} (Φ : nat → A → PROP) x l :
FromSep ([∗ list] k ↦ y ∈ x :: l, Φ k y) (Φ 0 x) ([∗ list] k ↦ y ∈ l, Φ (S k) y).
@@ -511,16 +445,6 @@ Proof.
- by rewrite -affinely_and !persistently_affinely.
- intros ->. by rewrite affinely_and.
Qed.
-Global Instance into_and_plainly p P Q1 Q2 :
- IntoAnd p P Q1 Q2 →
- IntoAnd p (bi_plainly P) (bi_plainly Q1) (bi_plainly Q2).
-Proof.
- rewrite /IntoAnd /=. destruct p; simpl.
- - rewrite -plainly_and persistently_plainly -plainly_persistently
- -plainly_affinely => ->.
- by rewrite plainly_affinely plainly_persistently persistently_plainly.
- - intros ->. by rewrite plainly_and.
-Qed.
Global Instance into_and_persistently p P Q1 Q2 :
IntoAnd p P Q1 Q2 →
IntoAnd p (bi_persistently P) (bi_persistently Q1) (bi_persistently Q2).
@@ -529,11 +453,11 @@ Proof.
- by rewrite -persistently_and !persistently_idemp.
- intros ->. by rewrite persistently_and.
Qed.
-Global Instance into_and_embed `{BiEmbedding PROP PROP'} p P Q1 Q2 :
+Global Instance into_and_embed `{BiEmbed PROP PROP'} p P Q1 Q2 :
IntoAnd p P Q1 Q2 → IntoAnd p ⎡P⎤ ⎡Q1⎤ ⎡Q2⎤.
Proof.
- rewrite /IntoAnd -bi_embed_and -!bi_embed_persistently_if
- -!bi_embed_affinely_if=> -> //.
+ rewrite /IntoAnd -embed_and -!embed_persistently_if
+ -!embed_affinely_if=> -> //.
Qed.
(* IntoSep *)
@@ -567,9 +491,9 @@ Qed.
Global Instance into_sep_pure φ ψ : @IntoSep PROP ⌜φ ∧ ψ⌠⌜φ⌠⌜ψâŒ.
Proof. by rewrite /IntoSep pure_and persistent_and_sep_1. Qed.
-Global Instance into_sep_embed `{BiEmbedding PROP PROP'} P Q1 Q2 :
+Global Instance into_sep_embed `{BiEmbed PROP PROP'} P Q1 Q2 :
IntoSep P Q1 Q2 → IntoSep ⎡P⎤ ⎡Q1⎤ ⎡Q2⎤.
-Proof. rewrite /IntoSep -bi_embed_sep=> -> //. Qed.
+Proof. rewrite /IntoSep -embed_sep=> -> //. Qed.
Global Instance into_sep_affinely `{BiPositive PROP} P Q1 Q2 :
IntoSep P Q1 Q2 → IntoSep (bi_affinely P) (bi_affinely Q1) (bi_affinely Q2) | 0.
@@ -581,17 +505,6 @@ Global Instance into_sep_affinely_trim P Q1 Q2 :
IntoSep P Q1 Q2 → IntoSep (bi_affinely P) Q1 Q2 | 20.
Proof. rewrite /IntoSep /= => ->. by rewrite affinely_elim. Qed.
-Global Instance into_sep_plainly `{BiPositive PROP} P Q1 Q2 :
- IntoSep P Q1 Q2 → IntoSep (bi_plainly P) (bi_plainly Q1) (bi_plainly Q2).
-Proof. rewrite /IntoSep /= => ->. by rewrite plainly_sep. Qed.
-Global Instance into_sep_plainly_affine P Q1 Q2 :
- IntoSep P Q1 Q2 →
- TCOr (Affine Q1) (Absorbing Q2) → TCOr (Absorbing Q1) (Affine Q2) →
- IntoSep (bi_plainly P) (bi_plainly Q1) (bi_plainly Q2).
-Proof.
- rewrite /IntoSep /= => -> ??. by rewrite sep_and plainly_and plainly_and_sep_l_1.
-Qed.
-
Global Instance into_sep_persistently `{BiPositive PROP} P Q1 Q2 :
IntoSep P Q1 Q2 →
IntoSep (bi_persistently P) (bi_persistently Q1) (bi_persistently Q2).
@@ -638,16 +551,13 @@ Proof. rewrite /FromOr=> <-. by rewrite affinely_or. Qed.
Global Instance from_or_absorbingly P Q1 Q2 :
FromOr P Q1 Q2 → FromOr (bi_absorbingly P) (bi_absorbingly Q1) (bi_absorbingly Q2).
Proof. rewrite /FromOr=> <-. by rewrite absorbingly_or. Qed.
-Global Instance from_or_plainly P Q1 Q2 :
- FromOr P Q1 Q2 → FromOr (bi_plainly P) (bi_plainly Q1) (bi_plainly Q2).
-Proof. rewrite /FromOr=> <-. by rewrite -plainly_or_2. Qed.
Global Instance from_or_persistently P Q1 Q2 :
FromOr P Q1 Q2 →
FromOr (bi_persistently P) (bi_persistently Q1) (bi_persistently Q2).
Proof. rewrite /FromOr=> <-. by rewrite persistently_or. Qed.
-Global Instance from_or_embed `{BiEmbedding PROP PROP'} P Q1 Q2 :
+Global Instance from_or_embed `{BiEmbed PROP PROP'} P Q1 Q2 :
FromOr P Q1 Q2 → FromOr ⎡P⎤ ⎡Q1⎤ ⎡Q2⎤.
-Proof. by rewrite /FromOr -bi_embed_or => <-. Qed.
+Proof. by rewrite /FromOr -embed_or => <-. Qed.
(* IntoOr *)
Global Instance into_or_or P Q : IntoOr (P ∨ Q) P Q.
@@ -660,16 +570,13 @@ Proof. rewrite /IntoOr=>->. by rewrite affinely_or. Qed.
Global Instance into_or_absorbingly P Q1 Q2 :
IntoOr P Q1 Q2 → IntoOr (bi_absorbingly P) (bi_absorbingly Q1) (bi_absorbingly Q2).
Proof. rewrite /IntoOr=>->. by rewrite absorbingly_or. Qed.
-Global Instance into_or_plainly `{BiPlainlyExist PROP} P Q1 Q2 :
- IntoOr P Q1 Q2 → IntoOr (bi_plainly P) (bi_plainly Q1) (bi_plainly Q2).
-Proof. rewrite /IntoOr=>->. by rewrite plainly_or. Qed.
Global Instance into_or_persistently P Q1 Q2 :
IntoOr P Q1 Q2 →
IntoOr (bi_persistently P) (bi_persistently Q1) (bi_persistently Q2).
Proof. rewrite /IntoOr=>->. by rewrite persistently_or. Qed.
-Global Instance into_or_embed `{BiEmbedding PROP PROP'} P Q1 Q2 :
+Global Instance into_or_embed `{BiEmbed PROP PROP'} P Q1 Q2 :
IntoOr P Q1 Q2 → IntoOr ⎡P⎤ ⎡Q1⎤ ⎡Q2⎤.
-Proof. by rewrite /IntoOr -bi_embed_or => <-. Qed.
+Proof. by rewrite /IntoOr -embed_or => <-. Qed.
(* FromExist *)
Global Instance from_exist_exist {A} (Φ : A → PROP): FromExist (∃ a, Φ a) Φ.
@@ -683,15 +590,12 @@ Proof. rewrite /FromExist=> <-. by rewrite affinely_exist. Qed.
Global Instance from_exist_absorbingly {A} P (Φ : A → PROP) :
FromExist P Φ → FromExist (bi_absorbingly P) (λ a, bi_absorbingly (Φ a))%I.
Proof. rewrite /FromExist=> <-. by rewrite absorbingly_exist. Qed.
-Global Instance from_exist_plainly {A} P (Φ : A → PROP) :
- FromExist P Φ → FromExist (bi_plainly P) (λ a, bi_plainly (Φ a))%I.
-Proof. rewrite /FromExist=> <-. by rewrite -plainly_exist_2. Qed.
Global Instance from_exist_persistently {A} P (Φ : A → PROP) :
FromExist P Φ → FromExist (bi_persistently P) (λ a, bi_persistently (Φ a))%I.
Proof. rewrite /FromExist=> <-. by rewrite persistently_exist. Qed.
-Global Instance from_exist_embed `{BiEmbedding PROP PROP'} {A} P (Φ : A → PROP) :
+Global Instance from_exist_embed `{BiEmbed PROP PROP'} {A} P (Φ : A → PROP) :
FromExist P Φ → FromExist ⎡P⎤ (λ a, ⎡Φ a⎤%I).
-Proof. by rewrite /FromExist -bi_embed_exist => <-. Qed.
+Proof. by rewrite /FromExist -embed_exist => <-. Qed.
(* IntoExist *)
Global Instance into_exist_exist {A} (Φ : A → PROP) : IntoExist (∃ a, Φ a) Φ.
@@ -718,15 +622,12 @@ Qed.
Global Instance into_exist_absorbingly {A} P (Φ : A → PROP) :
IntoExist P Φ → IntoExist (bi_absorbingly P) (λ a, bi_absorbingly (Φ a))%I.
Proof. rewrite /IntoExist=> HP. by rewrite HP absorbingly_exist. Qed.
-Global Instance into_exist_plainly `{BiPlainlyExist PROP} {A} P (Φ : A → PROP) :
- IntoExist P Φ → IntoExist (bi_plainly P) (λ a, bi_plainly (Φ a))%I.
-Proof. rewrite /IntoExist=> HP. by rewrite HP plainly_exist. Qed.
Global Instance into_exist_persistently {A} P (Φ : A → PROP) :
IntoExist P Φ → IntoExist (bi_persistently P) (λ a, bi_persistently (Φ a))%I.
Proof. rewrite /IntoExist=> HP. by rewrite HP persistently_exist. Qed.
-Global Instance into_exist_embed `{BiEmbedding PROP PROP'} {A} P (Φ : A → PROP) :
+Global Instance into_exist_embed `{BiEmbed PROP PROP'} {A} P (Φ : A → PROP) :
IntoExist P Φ → IntoExist ⎡P⎤ (λ a, ⎡Φ a⎤%I).
-Proof. by rewrite /IntoExist -bi_embed_exist => <-. Qed.
+Proof. by rewrite /IntoExist -embed_exist => <-. Qed.
(* IntoForall *)
Global Instance into_forall_forall {A} (Φ : A → PROP) : IntoForall (∀ a, Φ a) Φ.
@@ -734,15 +635,12 @@ Proof. by rewrite /IntoForall. Qed.
Global Instance into_forall_affinely {A} P (Φ : A → PROP) :
IntoForall P Φ → IntoForall (bi_affinely P) (λ a, bi_affinely (Φ a))%I.
Proof. rewrite /IntoForall=> HP. by rewrite HP affinely_forall. Qed.
-Global Instance into_forall_plainly {A} P (Φ : A → PROP) :
- IntoForall P Φ → IntoForall (bi_plainly P) (λ a, bi_plainly (Φ a))%I.
-Proof. rewrite /IntoForall=> HP. by rewrite HP plainly_forall. Qed.
Global Instance into_forall_persistently {A} P (Φ : A → PROP) :
IntoForall P Φ → IntoForall (bi_persistently P) (λ a, bi_persistently (Φ a))%I.
Proof. rewrite /IntoForall=> HP. by rewrite HP persistently_forall. Qed.
-Global Instance into_forall_embed `{BiEmbedding PROP PROP'} {A} P (Φ : A → PROP) :
+Global Instance into_forall_embed `{BiEmbed PROP PROP'} {A} P (Φ : A → PROP) :
IntoForall P Φ → IntoForall ⎡P⎤ (λ a, ⎡Φ a⎤%I).
-Proof. by rewrite /IntoForall -bi_embed_forall => <-. Qed.
+Proof. by rewrite /IntoForall -embed_forall => <-. Qed.
(* FromForall *)
Global Instance from_forall_forall {A} (Φ : A → PROP) :
@@ -774,15 +672,12 @@ Global Instance from_forall_affinely `{BiAffine PROP} {A} P (Φ : A → PROP) :
Proof.
rewrite /FromForall=> <-. rewrite affine_affinely. by setoid_rewrite affinely_elim.
Qed.
-Global Instance from_forall_plainly {A} P (Φ : A → PROP) :
- FromForall P Φ → FromForall (bi_plainly P)%I (λ a, bi_plainly (Φ a))%I.
-Proof. rewrite /FromForall=> <-. by rewrite plainly_forall. Qed.
Global Instance from_forall_persistently {A} P (Φ : A → PROP) :
FromForall P Φ → FromForall (bi_persistently P)%I (λ a, bi_persistently (Φ a))%I.
Proof. rewrite /FromForall=> <-. by rewrite persistently_forall. Qed.
-Global Instance from_forall_embed `{BiEmbedding PROP PROP'} {A} P (Φ : A → PROP) :
+Global Instance from_forall_embed `{BiEmbed PROP PROP'} {A} P (Φ : A → PROP) :
FromForall P Φ → FromForall ⎡P⎤%I (λ a, ⎡Φ a⎤%I).
-Proof. by rewrite /FromForall -bi_embed_forall => <-. Qed.
+Proof. by rewrite /FromForall -embed_forall => <-. Qed.
(* IntoInv *)
Global Instance into_inv_embed {PROP' : bi} `{BiEmbed PROP PROP'} P N :
@@ -843,26 +738,26 @@ Proof.
by rewrite affinely_persistently_if_elim sep_elim_l.
Qed.
-Global Instance make_embed_pure `{BiEmbedding PROP PROP'} φ :
+Global Instance make_embed_pure `{BiEmbed PROP PROP'} φ :
KnownMakeEmbed ⌜φ⌠⌜φâŒ.
-Proof. apply bi_embed_pure. Qed.
-Global Instance make_embed_emp `{BiEmbedding PROP PROP'} :
+Proof. apply embed_pure. Qed.
+Global Instance make_embed_emp `{BiEmbed PROP PROP'} :
KnownMakeEmbed emp emp.
-Proof. apply bi_embed_emp. Qed.
-Global Instance make_embed_default `{BiEmbedding PROP PROP'} P :
+Proof. apply embed_emp. Qed.
+Global Instance make_embed_default `{BiEmbed PROP PROP'} P :
MakeEmbed P ⎡P⎤ | 100.
Proof. by rewrite /MakeEmbed. Qed.
-Global Instance frame_embed `{BiEmbedding PROP PROP'} p P Q (Q' : PROP') R :
+Global Instance frame_embed `{BiEmbed PROP PROP'} p P Q (Q' : PROP') R :
Frame p R P Q → MakeEmbed Q Q' → KnownFrame p ⎡R⎤ ⎡P⎤ Q'.
Proof.
rewrite /KnownFrame /Frame /MakeEmbed => <- <-.
- rewrite bi_embed_sep bi_embed_affinely_if bi_embed_persistently_if => //.
+ rewrite embed_sep embed_affinely_if embed_persistently_if => //.
Qed.
-Global Instance frame_pure_embed `{BiEmbedding PROP PROP'} p P Q (Q' : PROP') φ :
+Global Instance frame_pure_embed `{BiEmbed PROP PROP'} p P Q (Q' : PROP') φ :
Frame p ⌜φ⌠P Q → MakeEmbed Q Q' → KnownFrame p ⌜φ⌠⎡P⎤ Q'.
Proof.
- rewrite /KnownFrame /Frame /MakeEmbed -bi_embed_pure. apply (frame_embed p P Q).
+ rewrite /KnownFrame /Frame /MakeEmbed -embed_pure. apply (frame_embed p P Q).
Qed.
Global Instance make_sep_emp_l P : KnownLMakeSep emp P P.
@@ -1083,9 +978,9 @@ Proof.
- intros x. apply H1. revert H2. by rewrite (bi.forall_elim x).
Qed.
-Global Instance as_valid_embed `{BiEmbedding PROP PROP'} (φ : Prop) (P : PROP) :
+Global Instance as_valid_embed `{BiEmbed PROP PROP'} (φ : Prop) (P : PROP) :
AsValid0 φ P → AsValid φ ⎡P⎤.
-Proof. rewrite /AsValid0 /AsValid=> ->. rewrite bi_embed_valid //. Qed.
+Proof. rewrite /AsValid0 /AsValid=> ->. rewrite embed_valid //. Qed.
End bi_instances.
Section sbi_instances.
@@ -1103,13 +998,26 @@ Global Instance from_assumption_except_0 p P Q :
FromAssumption p P Q → FromAssumption p P (◇ Q)%I.
Proof. rewrite /FromAssumption=>->. apply except_0_intro. Qed.
-Global Instance from_assumption_bupd `{BUpdFacts PROP} p P Q :
+Global Instance from_assumption_bupd `{BiBUpd PROP} p P Q :
FromAssumption p P Q → FromAssumption p P (|==> Q).
Proof. rewrite /FromAssumption=>->. apply bupd_intro. Qed.
-Global Instance from_assumption_fupd `{FUpdFacts PROP} E p P Q :
+Global Instance from_assumption_fupd `{BiBUpdFUpd PROP} E p P Q :
FromAssumption p P (|==> Q) → FromAssumption p P (|={E}=> Q)%I.
Proof. rewrite /FromAssumption=>->. apply bupd_fupd. Qed.
+Global Instance from_assumption_plainly_l_true `{BiPlainly PROP} P Q :
+ FromAssumption true P Q → FromAssumption true (■P) Q.
+Proof.
+ rewrite /FromAssumption /= =><-.
+ by rewrite persistently_elim plainly_elim_persistently.
+Qed.
+Global Instance from_assumption_plainly_l_false `{BiPlainly PROP, BiAffine PROP} P Q :
+ FromAssumption true P Q → FromAssumption false (■P) Q.
+Proof.
+ rewrite /FromAssumption /= =><-.
+ by rewrite affine_affinely plainly_elim_persistently.
+Qed.
+
(* FromPure *)
Global Instance from_pure_internal_eq af {A : ofeT} (a b : A) :
@FromPure PROP af (a ≡ b) (a ≡ b).
@@ -1121,18 +1029,26 @@ Proof. rewrite /FromPure=> ->. apply laterN_intro. Qed.
Global Instance from_pure_except_0 a P φ : FromPure a P φ → FromPure a (◇ P) φ.
Proof. rewrite /FromPure=> ->. apply except_0_intro. Qed.
-Global Instance from_pure_bupd `{BUpdFacts PROP} a P φ :
+Global Instance from_pure_bupd `{BiBUpd PROP} a P φ :
FromPure a P φ → FromPure a (|==> P) φ.
Proof. rewrite /FromPure=> <-. apply bupd_intro. Qed.
-Global Instance from_pure_fupd `{FUpdFacts PROP} a E P φ :
+Global Instance from_pure_fupd `{BiFUpd PROP} a E P φ :
FromPure a P φ → FromPure a (|={E}=> P) φ.
Proof. rewrite /FromPure. intros <-. apply fupd_intro. Qed.
+Global Instance from_pure_plainly `{BiPlainly PROP} P φ :
+ FromPure false P φ → FromPure false (■P) φ.
+Proof. rewrite /FromPure=> <-. by rewrite plainly_pure. Qed.
+
(* IntoPure *)
Global Instance into_pure_eq {A : ofeT} (a b : A) :
Discrete a → @IntoPure PROP (a ≡ b) (a ≡ b).
Proof. intros. by rewrite /IntoPure discrete_eq. Qed.
+Global Instance into_pure_plainly `{BiPlainly PROP} P φ :
+ IntoPure P φ → IntoPure (■P) φ.
+Proof. rewrite /IntoPure=> ->. apply: plainly_elim. Qed.
+
(* IntoWand *)
Global Instance into_wand_later p q R P Q :
IntoWand p q R P Q → IntoWand p q (▷ R) (▷ P) (▷ Q).
@@ -1161,45 +1077,52 @@ Proof.
(laterN_intro _ (â–¡?p R)%I) -laterN_wand HR.
Qed.
-Global Instance into_wand_bupd `{BUpdFacts PROP} p q R P Q :
+Global Instance into_wand_bupd `{BiBUpd PROP} p q R P Q :
IntoWand false false R P Q → IntoWand p q (|==> R) (|==> P) (|==> Q).
Proof.
rewrite /IntoWand /= => HR. rewrite !affinely_persistently_if_elim HR.
apply wand_intro_l. by rewrite bupd_sep wand_elim_r.
Qed.
-Global Instance into_wand_bupd_persistent `{BUpdFacts PROP} p q R P Q :
+Global Instance into_wand_bupd_persistent `{BiBUpd PROP} p q R P Q :
IntoWand false q R P Q → IntoWand p q (|==> R) P (|==> Q).
Proof.
rewrite /IntoWand /= => HR. rewrite affinely_persistently_if_elim HR.
apply wand_intro_l. by rewrite bupd_frame_l wand_elim_r.
Qed.
-Global Instance into_wand_bupd_args `{BUpdFacts PROP} p q R P Q :
+Global Instance into_wand_bupd_args `{BiBUpd PROP} p q R P Q :
IntoWand p false R P Q → IntoWand' p q R (|==> P) (|==> Q).
Proof.
rewrite /IntoWand' /IntoWand /= => ->.
apply wand_intro_l. by rewrite affinely_persistently_if_elim bupd_wand_r.
Qed.
-Global Instance into_wand_fupd `{FUpdFacts PROP} E p q R P Q :
+Global Instance into_wand_fupd `{BiFUpd PROP} E p q R P Q :
IntoWand false false R P Q →
IntoWand p q (|={E}=> R) (|={E}=> P) (|={E}=> Q).
Proof.
rewrite /IntoWand /= => HR. rewrite !affinely_persistently_if_elim HR.
apply wand_intro_l. by rewrite fupd_sep wand_elim_r.
Qed.
-Global Instance into_wand_fupd_persistent `{FUpdFacts PROP} E1 E2 p q R P Q :
+Global Instance into_wand_fupd_persistent `{BiFUpd PROP} E1 E2 p q R P Q :
IntoWand false q R P Q → IntoWand p q (|={E1,E2}=> R) P (|={E1,E2}=> Q).
Proof.
rewrite /IntoWand /= => HR. rewrite affinely_persistently_if_elim HR.
apply wand_intro_l. by rewrite fupd_frame_l wand_elim_r.
Qed.
-Global Instance into_wand_fupd_args `{FUpdFacts PROP} E1 E2 p q R P Q :
+Global Instance into_wand_fupd_args `{BiFUpd PROP} E1 E2 p q R P Q :
IntoWand p false R P Q → IntoWand' p q R (|={E1,E2}=> P) (|={E1,E2}=> Q).
Proof.
rewrite /IntoWand' /IntoWand /= => ->.
apply wand_intro_l. by rewrite affinely_persistently_if_elim fupd_wand_r.
Qed.
+Global Instance into_wand_plainly_true `{BiPlainly PROP} q R P Q :
+ IntoWand true q R P Q → IntoWand true q (■R) P Q.
+Proof. by rewrite /IntoWand /= persistently_plainly plainly_elim_persistently. Qed.
+Global Instance into_wand_plainly_false `{BiPlainly PROP, !BiAffine PROP} q R P Q :
+ IntoWand false q R P Q → IntoWand false q (■R) P Q.
+Proof. by rewrite /IntoWand plainly_elim. Qed.
+
(* FromAnd *)
Global Instance from_and_later P Q1 Q2 :
FromAnd P Q1 Q2 → FromAnd (▷ P) (▷ Q1) (▷ Q2).
@@ -1211,6 +1134,10 @@ Global Instance from_and_except_0 P Q1 Q2 :
FromAnd P Q1 Q2 → FromAnd (◇ P) (◇ Q1) (◇ Q2).
Proof. rewrite /FromAnd=><-. by rewrite except_0_and. Qed.
+Global Instance from_and_plainly `{BiPlainly PROP} P Q1 Q2 :
+ FromAnd P Q1 Q2 → FromAnd (■P) (■Q1) (■Q2).
+Proof. rewrite /FromAnd=> <-. by rewrite plainly_and. Qed.
+
(* FromSep *)
Global Instance from_sep_later P Q1 Q2 :
FromSep P Q1 Q2 → FromSep (▷ P) (▷ Q1) (▷ Q2).
@@ -1222,13 +1149,17 @@ Global Instance from_sep_except_0 P Q1 Q2 :
FromSep P Q1 Q2 → FromSep (◇ P) (◇ Q1) (◇ Q2).
Proof. rewrite /FromSep=><-. by rewrite except_0_sep. Qed.
-Global Instance from_sep_bupd `{BUpdFacts PROP} P Q1 Q2 :
+Global Instance from_sep_bupd `{BiBUpd PROP} P Q1 Q2 :
FromSep P Q1 Q2 → FromSep (|==> P) (|==> Q1) (|==> Q2).
Proof. rewrite /FromSep=><-. apply bupd_sep. Qed.
-Global Instance from_sep_fupd `{FUpdFacts PROP} E P Q1 Q2 :
+Global Instance from_sep_fupd `{BiFUpd PROP} E P Q1 Q2 :
FromSep P Q1 Q2 → FromSep (|={E}=> P) (|={E}=> Q1) (|={E}=> Q2).
Proof. rewrite /FromSep =><-. apply fupd_sep. Qed.
+Global Instance from_sep_plainly `{BiPlainly PROP} P Q1 Q2 :
+ FromSep P Q1 Q2 → FromSep (■P) (■Q1) (■Q2).
+Proof. rewrite /FromSep=> <-. by rewrite plainly_sep_2. Qed.
+
(* IntoAnd *)
Global Instance into_and_later p P Q1 Q2 :
IntoAnd p P Q1 Q2 → IntoAnd p (▷ P) (▷ Q1) (▷ Q2).
@@ -1252,6 +1183,16 @@ Proof.
affinely_persistently_if_elim except_0_and.
Qed.
+Global Instance into_and_plainly `{BiPlainly PROP} p P Q1 Q2 :
+ IntoAnd p P Q1 Q2 → IntoAnd p (■P) (■Q1) (■Q2).
+Proof.
+ rewrite /IntoAnd /=. destruct p; simpl.
+ - rewrite -plainly_and persistently_plainly -plainly_persistently
+ -plainly_affinely => ->.
+ by rewrite plainly_affinely plainly_persistently persistently_plainly.
+ - intros ->. by rewrite plainly_and.
+Qed.
+
(* IntoSep *)
Global Instance into_sep_later P Q1 Q2 :
IntoSep P Q1 Q2 → IntoSep (▷ P) (▷ Q1) (▷ Q2).
@@ -1275,6 +1216,18 @@ Proof.
by rewrite -(False_elim Q1) -(False_elim Q2).
Qed.
+Global Instance into_sep_plainly `{BiPlainly PROP, BiPositive PROP} P Q1 Q2 :
+ IntoSep P Q1 Q2 → IntoSep (■P) (■Q1) (■Q2).
+Proof. rewrite /IntoSep /= => ->. by rewrite plainly_sep. Qed.
+
+Global Instance into_sep_plainly_affine `{BiPlainly PROP} P Q1 Q2 :
+ IntoSep P Q1 Q2 →
+ TCOr (Affine Q1) (Absorbing Q2) → TCOr (Absorbing Q1) (Affine Q2) →
+ IntoSep (â– P) (â– Q1) (â– Q2).
+Proof.
+ rewrite /IntoSep /= => -> ??. by rewrite sep_and plainly_and plainly_and_sep_l_1.
+Qed.
+
(* FromOr *)
Global Instance from_or_later P Q1 Q2 :
FromOr P Q1 Q2 → FromOr (▷ P) (▷ Q1) (▷ Q2).
@@ -1286,19 +1239,23 @@ Global Instance from_or_except_0 P Q1 Q2 :
FromOr P Q1 Q2 → FromOr (◇ P) (◇ Q1) (◇ Q2).
Proof. rewrite /FromOr=><-. by rewrite except_0_or. Qed.
-Global Instance from_or_bupd `{BUpdFacts PROP} P Q1 Q2 :
+Global Instance from_or_bupd `{BiBUpd PROP} P Q1 Q2 :
FromOr P Q1 Q2 → FromOr (|==> P) (|==> Q1) (|==> Q2).
Proof.
rewrite /FromOr=><-.
apply or_elim; apply bupd_mono; auto using or_intro_l, or_intro_r.
Qed.
-Global Instance from_or_fupd `{FUpdFacts PROP} E1 E2 P Q1 Q2 :
+Global Instance from_or_fupd `{BiFUpd PROP} E1 E2 P Q1 Q2 :
FromOr P Q1 Q2 → FromOr (|={E1,E2}=> P) (|={E1,E2}=> Q1) (|={E1,E2}=> Q2).
Proof.
rewrite /FromOr=><-. apply or_elim; apply fupd_mono;
[apply bi.or_intro_l|apply bi.or_intro_r].
Qed.
+Global Instance from_or_plainly `{BiPlainly PROP} P Q1 Q2 :
+ FromOr P Q1 Q2 → FromOr (■P) (■Q1) (■Q2).
+Proof. rewrite /FromOr=> <-. by rewrite -plainly_or_2. Qed.
+
(* IntoOr *)
Global Instance into_or_later P Q1 Q2 :
IntoOr P Q1 Q2 → IntoOr (▷ P) (▷ Q1) (▷ Q2).
@@ -1310,6 +1267,10 @@ Global Instance into_or_except_0 P Q1 Q2 :
IntoOr P Q1 Q2 → IntoOr (◇ P) (◇ Q1) (◇ Q2).
Proof. rewrite /IntoOr=>->. by rewrite except_0_or. Qed.
+Global Instance into_or_plainly `{BiPlainly PROP, BiPlainlyExist PROP} P Q1 Q2 :
+ IntoOr P Q1 Q2 → IntoOr (■P) (■Q1) (■Q2).
+Proof. rewrite /IntoOr=>->. by rewrite plainly_or. Qed.
+
(* FromExist *)
Global Instance from_exist_later {A} P (Φ : A → PROP) :
FromExist P Φ → FromExist (▷ P) (λ a, ▷ (Φ a))%I.
@@ -1325,17 +1286,21 @@ Global Instance from_exist_except_0 {A} P (Φ : A → PROP) :
FromExist P Φ → FromExist (◇ P) (λ a, ◇ (Φ a))%I.
Proof. rewrite /FromExist=> <-. by rewrite except_0_exist_2. Qed.
-Global Instance from_exist_bupd `{BUpdFacts PROP} {A} P (Φ : A → PROP) :
+Global Instance from_exist_bupd `{BiBUpd PROP} {A} P (Φ : A → PROP) :
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_fupd `{FUpdFacts PROP} {A} E1 E2 P (Φ : A → PROP) :
+Global Instance from_exist_fupd `{BiFUpd PROP} {A} E1 E2 P (Φ : A → PROP) :
FromExist P Φ → FromExist (|={E1,E2}=> P) (λ a, |={E1,E2}=> Φ a)%I.
Proof.
rewrite /FromExist=><-. apply exist_elim=> a. by rewrite -(exist_intro a).
Qed.
+Global Instance from_exist_plainly `{BiPlainly PROP} {A} P (Φ : A → PROP) :
+ FromExist P Φ → FromExist (■P) (λ a, ■(Φ a))%I.
+Proof. rewrite /FromExist=> <-. by rewrite -plainly_exist_2. Qed.
+
(* IntoExist *)
Global Instance into_exist_later {A} P (Φ : A → PROP) :
IntoExist P Φ → Inhabited A → IntoExist (▷ P) (λ a, ▷ (Φ a))%I.
@@ -1347,6 +1312,10 @@ Global Instance into_exist_except_0 {A} P (Φ : A → PROP) :
IntoExist P Φ → Inhabited A → IntoExist (◇ P) (λ a, ◇ (Φ a))%I.
Proof. rewrite /IntoExist=> HP ?. by rewrite HP except_0_exist. Qed.
+Global Instance into_exist_plainly `{BiPlainlyExist PROP} {A} P (Φ : A → PROP) :
+ IntoExist P Φ → IntoExist (■P) (λ a, ■(Φ a))%I.
+Proof. rewrite /IntoExist=> HP. by rewrite HP plainly_exist. Qed.
+
(* IntoForall *)
Global Instance into_forall_later {A} P (Φ : A → PROP) :
IntoForall P Φ → IntoForall (▷ P) (λ a, ▷ (Φ a))%I.
@@ -1368,6 +1337,10 @@ Proof.
by rewrite -(pure_intro True%I) // /bi_affinely right_id emp_wand.
Qed.
+Global Instance into_forall_plainly `{BiPlainly PROP} {A} P (Φ : A → PROP) :
+ IntoForall P Φ → IntoForall (■P) (λ a, ■(Φ a))%I.
+Proof. rewrite /IntoForall=> HP. by rewrite HP plainly_forall. Qed.
+
(* FromForall *)
Global Instance from_forall_later {A} P (Φ : A → PROP) :
FromForall P Φ → FromForall (▷ P)%I (λ a, ▷ (Φ a))%I.
@@ -1376,20 +1349,24 @@ Global Instance from_forall_except_0 {A} P (Φ : A → PROP) :
FromForall P Φ → FromForall (◇ P)%I (λ a, ◇ (Φ a))%I.
Proof. rewrite /FromForall=> <-. by rewrite except_0_forall. Qed.
+Global Instance from_forall_plainly `{BiPlainly PROP} {A} P (Φ : A → PROP) :
+ FromForall P Φ → FromForall (■P)%I (λ a, ■(Φ a))%I.
+Proof. rewrite /FromForall=> <-. by rewrite plainly_forall. Qed.
+
(* IsExcept0 *)
Global Instance is_except_0_except_0 P : IsExcept0 (â—‡ P).
Proof. by rewrite /IsExcept0 except_0_idemp. Qed.
Global Instance is_except_0_later P : IsExcept0 (â–· P).
Proof. by rewrite /IsExcept0 except_0_later. Qed.
-Global Instance is_except_0_embed `{SbiEmbedding PROP PROP'} P :
+Global Instance is_except_0_embed `{SbiEmbed PROP PROP'} P :
IsExcept0 P → IsExcept0 ⎡P⎤.
-Proof. by rewrite /IsExcept0 -sbi_embed_except_0=>->. Qed.
-Global Instance is_except_0_bupd `{BUpdFacts PROP} P : IsExcept0 P → IsExcept0 (|==> P).
+Proof. by rewrite /IsExcept0 -embed_except_0=>->. Qed.
+Global Instance is_except_0_bupd `{BiBUpd PROP} 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.
-Global Instance is_except_0_fupd `{FUpdFacts PROP} E1 E2 P :
+Global Instance is_except_0_fupd `{BiFUpd PROP} E1 E2 P :
IsExcept0 (|={E1,E2}=> P).
Proof. by rewrite /IsExcept0 except_0_fupd. Qed.
@@ -1403,17 +1380,27 @@ Proof. by rewrite /FromModal. Qed.
Global Instance from_modal_except_0 P : FromModal modality_id (â—‡ P) (â—‡ P) P.
Proof. by rewrite /FromModal /= -except_0_intro. Qed.
-Global Instance from_modal_bupd `{BUpdFacts PROP} P :
+Global Instance from_modal_bupd `{BiBUpd PROP} P :
FromModal modality_id (|==> P) (|==> P) P.
Proof. by rewrite /FromModal /= -bupd_intro. Qed.
-Global Instance from_modal_fupd E P `{FUpdFacts PROP} :
+Global Instance from_modal_fupd E P `{BiFUpd PROP} :
FromModal modality_id (|={E}=> P) (|={E}=> P) P.
Proof. by rewrite /FromModal /= -fupd_intro. Qed.
-Global Instance from_modal_later_embed `{SbiEmbedding PROP PROP'} `(sel : A) n P Q :
+Global Instance from_modal_later_embed `{SbiEmbed PROP PROP'} `(sel : A) n P Q :
FromModal (modality_laterN n) sel P Q →
FromModal (modality_laterN n) sel ⎡P⎤ ⎡Q⎤.
-Proof. rewrite /FromModal /= =><-. by rewrite sbi_embed_laterN. Qed.
+Proof. rewrite /FromModal /= =><-. by rewrite embed_laterN. Qed.
+
+Global Instance from_modal_plainly `{BiPlainly PROP} P :
+ FromModal modality_plainly (â– P) (â– P) P | 2.
+Proof. by rewrite /FromModal. Qed.
+
+Global Instance from_modal_plainly_embed
+ `{BiPlainly PROP, BiPlainly PROP', SbiEmbed PROP PROP'} `(sel : A) P Q :
+ FromModal modality_plainly sel P Q →
+ FromModal modality_plainly sel ⎡P⎤ ⎡Q⎤ | 100.
+Proof. rewrite /FromModal /= =><-. by rewrite embed_plainly. Qed.
(* IntoInternalEq *)
Global Instance into_internal_eq_internal_eq {A : ofeT} (x y : A) :
@@ -1425,16 +1412,16 @@ Proof. rewrite /IntoInternalEq=> ->. by rewrite affinely_elim. Qed.
Global Instance into_internal_eq_absorbingly {A : ofeT} (x y : A) P :
IntoInternalEq P x y → IntoInternalEq (bi_absorbingly P) x y.
Proof. rewrite /IntoInternalEq=> ->. by rewrite absorbingly_internal_eq. Qed.
-Global Instance into_internal_eq_plainly {A : ofeT} (x y : A) P :
- IntoInternalEq P x y → IntoInternalEq (bi_plainly P) x y.
+Global Instance into_internal_eq_plainly `{BiPlainly PROP} {A : ofeT} (x y : A) P :
+ IntoInternalEq P x y → IntoInternalEq (■P) x y.
Proof. rewrite /IntoInternalEq=> ->. by rewrite plainly_elim. Qed.
Global Instance into_internal_eq_persistently {A : ofeT} (x y : A) P :
IntoInternalEq P x y → IntoInternalEq (bi_persistently P) x y.
Proof. rewrite /IntoInternalEq=> ->. by rewrite persistently_elim. Qed.
Global Instance into_internal_eq_embed
- `{SbiEmbedding PROP PROP'} {A : ofeT} (x y : A) P :
+ `{SbiEmbed PROP PROP'} {A : ofeT} (x y : A) P :
IntoInternalEq P x y → IntoInternalEq ⎡P⎤ x y.
-Proof. rewrite /IntoInternalEq=> ->. by rewrite sbi_embed_internal_eq. Qed.
+Proof. rewrite /IntoInternalEq=> ->. by rewrite embed_internal_eq. Qed.
(* IntoExcept0 *)
Global Instance into_except_0_except_0 P : IntoExcept0 (â—‡ P) P.
@@ -1450,15 +1437,15 @@ Proof. rewrite /IntoExcept0=> ->. by rewrite except_0_affinely_2. Qed.
Global Instance into_except_0_absorbingly P Q :
IntoExcept0 P Q → IntoExcept0 (bi_absorbingly P) (bi_absorbingly Q).
Proof. rewrite /IntoExcept0=> ->. by rewrite except_0_absorbingly. Qed.
-Global Instance into_except_0_plainly `{BiPlainlyExist PROP} P Q :
- IntoExcept0 P Q → IntoExcept0 (bi_plainly P) (bi_plainly Q).
+Global Instance into_except_0_plainly `{BiPlainly PROP, BiPlainlyExist PROP} P Q :
+ IntoExcept0 P Q → IntoExcept0 (■P) (■Q).
Proof. rewrite /IntoExcept0=> ->. by rewrite except_0_plainly. Qed.
Global Instance into_except_0_persistently P Q :
IntoExcept0 P Q → IntoExcept0 (bi_persistently P) (bi_persistently Q).
Proof. rewrite /IntoExcept0=> ->. by rewrite except_0_persistently. Qed.
-Global Instance into_except_0_embed `{SbiEmbedding PROP PROP'} P Q :
+Global Instance into_except_0_embed `{SbiEmbed PROP PROP'} P Q :
IntoExcept0 P Q → IntoExcept0 ⎡P⎤ ⎡Q⎤.
-Proof. rewrite /IntoExcept0=> ->. by rewrite sbi_embed_except_0. Qed.
+Proof. rewrite /IntoExcept0=> ->. by rewrite embed_except_0. Qed.
(* ElimModal *)
Global Instance elim_modal_timeless P Q :
@@ -1467,21 +1454,21 @@ Proof.
intros. rewrite /ElimModal (except_0_intro (_ -∗ _)%I).
by rewrite (into_except_0 P) -except_0_sep wand_elim_r.
Qed.
-Global Instance elim_modal_bupd `{BUpdFacts PROP} P Q :
+Global Instance elim_modal_bupd `{BiBUpd PROP} P Q :
ElimModal True (|==> P) P (|==> Q) (|==> Q).
Proof. by rewrite /ElimModal bupd_frame_r wand_elim_r bupd_trans. Qed.
-Global Instance elim_modal_bupd_plain_goal `{BUpdFacts PROP} P Q :
+Global Instance elim_modal_bupd_plain_goal `{BiBUpdPlainly PROP} P Q :
Plain Q → ElimModal True (|==> P) P Q Q.
Proof. intros. by rewrite /ElimModal bupd_frame_r wand_elim_r bupd_plain. Qed.
-Global Instance elim_modal_bupd_plain `{BUpdFacts PROP} P Q :
+Global Instance elim_modal_bupd_plain `{BiBUpdPlainly PROP} P Q :
Plain P → ElimModal True (|==> P) P Q Q.
Proof. intros. by rewrite /ElimModal bupd_plain wand_elim_r. Qed.
-Global Instance elim_modal_bupd_fupd `{FUpdFacts PROP} E1 E2 P Q :
+Global Instance elim_modal_bupd_fupd `{BiBUpdFUpd PROP} E1 E2 P Q :
ElimModal True (|==> P) P (|={E1,E2}=> Q) (|={E1,E2}=> Q) | 10.
Proof.
by rewrite /ElimModal (bupd_fupd E1) fupd_frame_r wand_elim_r fupd_trans.
Qed.
-Global Instance elim_modal_fupd_fupd `{FUpdFacts PROP} E1 E2 E3 P Q :
+Global Instance elim_modal_fupd_fupd `{BiFUpd PROP} E1 E2 E3 P Q :
ElimModal True (|={E1,E2}=> P) P (|={E1,E3}=> Q) (|={E2,E3}=> Q).
Proof. by rewrite /ElimModal fupd_frame_r wand_elim_r fupd_trans. Qed.
@@ -1510,17 +1497,17 @@ Proof.
by rewrite -except_0_sep wand_elim_r except_0_later.
Qed.
-Global Instance add_modal_bupd `{BUpdFacts PROP} P Q : AddModal (|==> P) P (|==> Q).
+Global Instance add_modal_bupd `{BiBUpd PROP} P Q : AddModal (|==> P) P (|==> Q).
Proof. by rewrite /AddModal bupd_frame_r wand_elim_r bupd_trans. Qed.
-Global Instance add_modal_fupd `{FUpdFacts PROP} E1 E2 P Q :
+Global Instance add_modal_fupd `{BiFUpd PROP} E1 E2 P Q :
AddModal (|={E1}=> P) P (|={E1,E2}=> Q).
Proof. by rewrite /AddModal fupd_frame_r wand_elim_r fupd_trans. Qed.
(* Frame *)
-Global Instance frame_eq_embed `{SbiEmbedding PROP PROP'} p P Q (Q' : PROP')
+Global Instance frame_eq_embed `{SbiEmbed PROP PROP'} p P Q (Q' : PROP')
{A : ofeT} (a b : A) :
Frame p (a ≡ b) P Q → MakeEmbed Q Q' → KnownFrame p (a ≡ b) ⎡P⎤ Q'.
-Proof. rewrite /KnownFrame /Frame /MakeEmbed -sbi_embed_internal_eq. apply (frame_embed p P Q). Qed.
+Proof. rewrite /KnownFrame /Frame /MakeEmbed -embed_internal_eq. apply (frame_embed p P Q). Qed.
Global Instance make_laterN_true n : @KnownMakeLaterN PROP n True True | 0.
Proof. by rewrite /KnownMakeLaterN /MakeLaterN laterN_True. Qed.
@@ -1546,10 +1533,10 @@ Proof.
by rewrite laterN_affinely_persistently_if_2 laterN_sep.
Qed.
-Global Instance frame_bupd `{BUpdFacts PROP} p R P Q :
+Global Instance frame_bupd `{BiBUpd PROP} p R P Q :
Frame p R P Q → KnownFrame p R (|==> P) (|==> Q).
Proof. rewrite /KnownFrame /Frame=><-. by rewrite bupd_frame_l. Qed.
-Global Instance frame_fupd `{FUpdFacts PROP} p E1 E2 R P Q :
+Global Instance frame_fupd `{BiFUpd PROP} p E1 E2 R P Q :
Frame p R P Q → KnownFrame p R (|={E1,E2}=> P) (|={E1,E2}=> Q).
Proof. rewrite /KnownFrame /Frame=><-. by rewrite fupd_frame_l. Qed.
@@ -1629,15 +1616,15 @@ Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_affinely_2.
Global Instance into_later_absorbingly n P Q :
IntoLaterN false n P Q → IntoLaterN false n (bi_absorbingly P) (bi_absorbingly Q).
Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_absorbingly. Qed.
-Global Instance into_later_plainly n P Q :
- IntoLaterN false n P Q → IntoLaterN false n (bi_plainly P) (bi_plainly Q).
+Global Instance into_later_plainly `{BiPlainly PROP} n P Q :
+ IntoLaterN false n P Q → IntoLaterN false n (■P) (■Q).
Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_plainly. Qed.
Global Instance into_later_persistently n P Q :
IntoLaterN false n P Q → IntoLaterN false n (bi_persistently P) (bi_persistently Q).
Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_persistently. Qed.
-Global Instance into_later_embed`{SbiEmbedding PROP PROP'} n P Q :
+Global Instance into_later_embed`{SbiEmbed PROP PROP'} n P Q :
IntoLaterN false n P Q → IntoLaterN false n ⎡P⎤ ⎡Q⎤.
-Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite sbi_embed_laterN. Qed.
+Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite embed_laterN. Qed.
Global Instance into_laterN_sep_l n P1 P2 Q1 Q2 :
IntoLaterN false n P1 Q1 → MaybeIntoLaterN false n P2 Q2 →
diff --git a/theories/proofmode/classes.v b/theories/proofmode/classes.v
index 5786a01a7b83abacdeef05ec58eeaab2234ac439..119c45711e554f1c0e2e7b6004be20f0a00c0b2c 100644
--- a/theories/proofmode/classes.v
+++ b/theories/proofmode/classes.v
@@ -215,7 +215,8 @@ Hint Mode IntoForall + - ! - : typeclass_instances.
Class FromForall {PROP : bi} {A} (P : PROP) (Φ : A → PROP) :=
from_forall : (∀ x, Φ x) ⊢ P.
-Arguments from_forall {_ _} _ _ {_}.
+Arguments FromForall {_ _} _%I _%I : simpl never.
+Arguments from_forall {_ _} _%I _%I {_}.
Hint Mode FromForall + - ! - : typeclass_instances.
Class IsExcept0 {PROP : sbi} (Q : PROP) := is_except_0 : ◇ Q ⊢ Q.
diff --git a/theories/proofmode/modality_instances.v b/theories/proofmode/modality_instances.v
index 8fcfc827a9fa7e302292bc728f778235490eb5b9..af0e205e4b230eeb9e69a8946510fbad729997a3 100644
--- a/theories/proofmode/modality_instances.v
+++ b/theories/proofmode/modality_instances.v
@@ -34,43 +34,32 @@ Section bi_modalities.
Definition modality_affinely_persistently :=
Modality _ modality_affinely_persistently_mixin.
- Lemma modality_plainly_mixin :
- modality_mixin (@bi_plainly PROP) (MIEnvForall Plain) MIEnvClear.
- Proof.
- split; simpl; split_and?; eauto using equiv_entails_sym, plainly_intro,
- plainly_mono, plainly_and, plainly_sep_2 with typeclass_instances.
- Qed.
- Definition modality_plainly :=
- Modality _ modality_plainly_mixin.
-
- Lemma modality_affinely_plainly_mixin :
- modality_mixin (λ P : PROP, ■P)%I (MIEnvForall Plain) MIEnvIsEmpty.
- Proof.
- split; simpl; split_and?; eauto using equiv_entails_sym,
- affinely_plainly_emp, affinely_intro,
- plainly_intro, affinely_mono, plainly_mono, affinely_plainly_idemp,
- affinely_plainly_and, affinely_plainly_sep_2 with typeclass_instances.
- Qed.
- Definition modality_affinely_plainly :=
- Modality _ modality_affinely_plainly_mixin.
-
- Lemma modality_embed_mixin `{BiEmbedding PROP PROP'} :
- modality_mixin (@bi_embed PROP PROP' _)
+ Lemma modality_embed_mixin `{BiEmbed PROP PROP'} :
+ modality_mixin (@embed PROP PROP' _)
(MIEnvTransform IntoEmbed) (MIEnvTransform IntoEmbed).
Proof.
split; simpl; split_and?;
- eauto using equiv_entails_sym, bi_embed_emp, bi_embed_sep, bi_embed_and.
+ eauto using equiv_entails_sym, embed_emp, embed_sep, embed_and.
- intros P Q. rewrite /IntoEmbed=> ->.
- by rewrite bi_embed_affinely bi_embed_persistently.
+ by rewrite embed_affinely embed_persistently.
- by intros P Q ->.
Qed.
- Definition modality_embed `{BiEmbedding PROP PROP'} :=
+ Definition modality_embed `{BiEmbed PROP PROP'} :=
Modality _ modality_embed_mixin.
End bi_modalities.
Section sbi_modalities.
Context {PROP : sbi}.
+ Lemma modality_plainly_mixin `{BiPlainly PROP} :
+ modality_mixin (@plainly PROP _) (MIEnvForall Plain) MIEnvClear.
+ Proof.
+ split; simpl; split_and?; eauto using equiv_entails_sym, plainly_intro,
+ plainly_mono, plainly_and, plainly_sep_2 with typeclass_instances.
+ Qed.
+ Definition modality_plainly `{BiPlainly PROP} :=
+ Modality _ modality_plainly_mixin.
+
Lemma modality_laterN_mixin n :
modality_mixin (@sbi_laterN PROP n)
(MIEnvTransform (MaybeIntoLaterN false n)) (MIEnvTransform (MaybeIntoLaterN false n)).
diff --git a/theories/proofmode/monpred.v b/theories/proofmode/monpred.v
index d60a2635dab284e36feb439b4d75d49e25c1229b..179150e9073598be1418444aaa7af2dade29ead7 100644
--- a/theories/proofmode/monpred.v
+++ b/theories/proofmode/monpred.v
@@ -1,4 +1,5 @@
From iris.bi Require Export monpred.
+From iris.bi Require Import plainly.
From iris.proofmode Require Import tactics class_instances.
Class MakeMonPredAt {I : biIndex} {PROP : bi} (i : I)
@@ -33,6 +34,7 @@ End modalities.
Section bi.
Context {I : biIndex} {PROP : bi}.
+Local Notation monPredI := (monPredI I PROP).
Local Notation monPred := (monPred I PROP).
Local Notation MakeMonPredAt := (@MakeMonPredAt I PROP).
Implicit Types P Q R : monPred.
@@ -116,6 +118,9 @@ Global Instance make_monPred_at_in i j : MakeMonPredAt j (monPred_in i) ⌜i ⊑
Proof. by rewrite /MakeMonPredAt monPred_at_in. Qed.
Global Instance make_monPred_at_default i P : MakeMonPredAt i P (P i) | 100.
Proof. by rewrite /MakeMonPredAt. Qed.
+Global Instance make_monPred_at_bupd `{BiBUpd PROP} i P ð“Ÿ :
+ MakeMonPredAt i P 𓟠→ MakeMonPredAt i (|==> P)%I (|==> ð“Ÿ)%I.
+Proof. by rewrite /MakeMonPredAt monPred_at_bupd=> <-. Qed.
Global Instance from_assumption_make_monPred_at_l p i j P ð“Ÿ :
MakeMonPredAt i P 𓟠→ IsBiIndexRel j i → FromAssumption p (P j) ð“Ÿ.
@@ -287,21 +292,6 @@ Proof.
by rewrite monPred_at_exist.
Qed.
-Global Instance foram_forall_monPred_at_plainly i P Φ :
- (∀ i, MakeMonPredAt i P (Φ i)) →
- FromForall (bi_plainly P i) (λ j, bi_plainly (Φ j)).
-Proof.
- rewrite /FromForall /MakeMonPredAt=>H. rewrite monPred_at_plainly.
- by setoid_rewrite H.
-Qed.
-Global Instance into_forall_monPred_at_plainly i P Φ :
- (∀ i, MakeMonPredAt i P (Φ i)) →
- IntoForall (bi_plainly P i) (λ j, bi_plainly (Φ j)).
-Proof.
- rewrite /IntoForall /MakeMonPredAt=>H. rewrite monPred_at_plainly.
- by setoid_rewrite H.
-Qed.
-
Global Instance from_forall_monPred_at_absolutely P (Φ : I → PROP) i :
(∀ i, MakeMonPredAt i P (Φ i)) → FromForall ((∀ᵢ P) i)%I Φ.
Proof.
@@ -316,12 +306,12 @@ Qed.
Global Instance from_exist_monPred_at_ex P (Φ : I → PROP) i :
(∀ i, MakeMonPredAt i P (Φ i)) → FromExist ((∃ᵢ P) i) Φ.
Proof.
- rewrite /FromExist /MakeMonPredAt monPred_at_ex=>H. by setoid_rewrite <- H.
+ rewrite /FromExist /MakeMonPredAt monPred_at_relatively=>H. by setoid_rewrite <- H.
Qed.
Global Instance into_exist_monPred_at_ex P (Φ : I → PROP) i :
(∀ i, MakeMonPredAt i P (Φ i)) → IntoExist ((∃ᵢ P) i) Φ.
Proof.
- rewrite /IntoExist /MakeMonPredAt monPred_at_ex=>H. by setoid_rewrite <- H.
+ rewrite /IntoExist /MakeMonPredAt monPred_at_relatively=>H. by setoid_rewrite <- H.
Qed.
Global Instance from_forall_monPred_at {A} P (Φ : A → monPred) (Ψ : A → PROP) i :
@@ -358,7 +348,36 @@ Qed.
Global Instance into_embed_absolute P :
Absolute P → IntoEmbed P (∀ i, P i).
-Proof. rewrite /IntoEmbed=> ?. by rewrite {1}(absolute_absolutely P). Qed.
+Proof.
+ rewrite /IntoEmbed=> ?.
+ by rewrite {1}(absolute_absolutely P) monPred_absolutely_unfold.
+Qed.
+
+Global Instance elim_modal_embed_bupd_goal `{BiBUpd PROP} φ P P' ð“ ð“ ' :
+ ElimModal φ P P' (|==> ⎡ð“ ⎤)%I (|==> ⎡ð“ '⎤)%I →
+ ElimModal φ P P' ⎡|==> ð“ ⎤ ⎡|==> ð“ '⎤.
+Proof. by rewrite /ElimModal !monPred_bupd_embed. Qed.
+Global Instance elim_modal_embed_bupd_hyp `{BiBUpd PROP} φ 𓟠P' Q Q' :
+ ElimModal φ (|==> ⎡ð“ŸâŽ¤)%I P' Q Q' →
+ ElimModal φ ⎡|==> ð“ŸâŽ¤ P' Q Q'.
+Proof. by rewrite /ElimModal monPred_bupd_embed. Qed.
+
+Global Instance add_modal_embed_bupd_goal `{BiBUpd PROP} P P' ð“ :
+ AddModal P P' (|==> ⎡ð“ ⎤)%I → AddModal P P' ⎡|==> ð“ ⎤.
+Proof. by rewrite /AddModal !monPred_bupd_embed. Qed.
+
+Global Instance elim_modal_at_bupd_goal `{BiBUpd PROP} φ ð“Ÿ ð“Ÿ' Q Q' i :
+ ElimModal φ ð“Ÿ ð“Ÿ' (|==> Q i) (|==> Q' i) →
+ ElimModal φ ð“Ÿ ð“Ÿ' ((|==> Q) i) ((|==> Q') i).
+Proof. by rewrite /ElimModal !monPred_at_bupd. Qed.
+Global Instance elim_modal_at_bupd_hyp `{BiBUpd PROP} φ P ð“Ÿ' ð“ ð“ ' i:
+ ElimModal φ (|==> P i) ð“Ÿ' ð“ ð“ ' →
+ ElimModal φ ((|==> P) i) ð“Ÿ' ð“ ð“ '.
+Proof. by rewrite /ElimModal monPred_at_bupd. Qed.
+
+Global Instance add_modal_at_bupd_goal `{BiBUpd PROP} φ ð“Ÿ ð“Ÿ' Q i :
+ AddModal ð“Ÿ ð“Ÿ' (|==> Q i)%I → AddModal ð“Ÿ ð“Ÿ' ((|==> Q) i).
+Proof. by rewrite /AddModal !monPred_at_bupd. Qed.
End bi.
(* When P and/or Q are evars when doing typeclass search on [IntoWand
@@ -391,6 +410,21 @@ Implicit Types ð“Ÿ ð“ ð“¡ : PROP.
Implicit Types φ : Prop.
Implicit Types i j : I.
+Global Instance from_forall_monPred_at_plainly `{BiPlainly PROP} i P Φ :
+ (∀ i, MakeMonPredAt i P (Φ i)) →
+ FromForall ((■P) i) (λ j, ■(Φ j))%I.
+Proof.
+ rewrite /FromForall /MakeMonPredAt=>HPΦ. rewrite monPred_at_plainly.
+ by setoid_rewrite HPΦ.
+Qed.
+Global Instance into_forall_monPred_at_plainly `{BiPlainly PROP} i P Φ :
+ (∀ i, MakeMonPredAt i P (Φ i)) →
+ IntoForall ((■P) i) (λ j, ■(Φ j))%I.
+Proof.
+ rewrite /IntoForall /MakeMonPredAt=>HPΦ. rewrite monPred_at_plainly.
+ by setoid_rewrite HPΦ.
+Qed.
+
Global Instance is_except_0_monPred_at i P :
IsExcept0 P → IsExcept0 (P i).
Proof. rewrite /IsExcept0=>- [/(_ i)]. by rewrite monPred_at_except_0. Qed.
@@ -407,10 +441,7 @@ Proof. by rewrite /MakeMonPredAt monPred_at_later=><-. Qed.
Global Instance make_monPred_at_laterN i n P ð“ :
MakeMonPredAt i P ð“ → MakeMonPredAt i (â–·^n P)%I (â–·^n ð“ )%I.
Proof. rewrite /MakeMonPredAt=> <-. elim n=>//= ? <-. by rewrite monPred_at_later. Qed.
-Global Instance make_monPred_at_bupd `{BUpdFacts PROP} i P ð“Ÿ :
- MakeMonPredAt i P 𓟠→ MakeMonPredAt i (|==> P)%I (|==> ð“Ÿ)%I.
-Proof. by rewrite /MakeMonPredAt monPred_at_bupd=> <-. Qed.
-Global Instance make_monPred_at_fupd `{FUpdFacts PROP} i E1 E2 P ð“Ÿ :
+Global Instance make_monPred_at_fupd `{BiFUpd PROP} i E1 E2 P ð“Ÿ :
MakeMonPredAt i P 𓟠→ MakeMonPredAt i (|={E1,E2}=> P)%I (|={E1,E2}=> ð“Ÿ)%I.
Proof. by rewrite /MakeMonPredAt monPred_at_fupd=> <-. Qed.
@@ -440,55 +471,29 @@ Proof.
by rewrite monPred_at_later.
Qed.
-Global Instance elim_modal_embed_bupd_goal `{BUpdFacts PROP} φ P P' ð“ ð“ ' :
- ElimModal φ P P' (|==> ⎡ð“ ⎤)%I (|==> ⎡ð“ '⎤)%I →
- ElimModal φ P P' ⎡|==> ð“ ⎤ ⎡|==> ð“ '⎤.
-Proof. by rewrite /ElimModal !monPred_bupd_embed. Qed.
-Global Instance elim_modal_embed_bupd_hyp `{BUpdFacts PROP} φ 𓟠P' Q Q' :
- ElimModal φ (|==> ⎡ð“ŸâŽ¤)%I P' Q Q' →
- ElimModal φ ⎡|==> ð“ŸâŽ¤ P' Q Q'.
-Proof. by rewrite /ElimModal monPred_bupd_embed. Qed.
-
-Global Instance add_modal_embed_bupd_goal `{BUpdFacts PROP} P P' ð“ :
- AddModal P P' (|==> ⎡ð“ ⎤)%I → AddModal P P' ⎡|==> ð“ ⎤.
-Proof. by rewrite /AddModal !monPred_bupd_embed. Qed.
-
-Global Instance elim_modal_at_bupd_goal `{BUpdFacts PROP} φ ð“Ÿ ð“Ÿ' Q Q' i :
- ElimModal φ ð“Ÿ ð“Ÿ' (|==> Q i) (|==> Q' i) →
- ElimModal φ ð“Ÿ ð“Ÿ' ((|==> Q) i) ((|==> Q') i).
-Proof. by rewrite /ElimModal !monPred_at_bupd. Qed.
-Global Instance elim_modal_at_bupd_hyp `{BUpdFacts PROP} φ P ð“Ÿ' ð“ ð“ ' i:
- ElimModal φ (|==> P i) ð“Ÿ' ð“ ð“ ' →
- ElimModal φ ((|==> P) i) ð“Ÿ' ð“ ð“ '.
-Proof. by rewrite /ElimModal monPred_at_bupd. Qed.
-
-Global Instance add_modal_at_bupd_goal `{BUpdFacts PROP} φ ð“Ÿ ð“Ÿ' Q i :
- AddModal ð“Ÿ ð“Ÿ' (|==> Q i)%I → AddModal ð“Ÿ ð“Ÿ' ((|==> Q) i).
-Proof. by rewrite /AddModal !monPred_at_bupd. Qed.
-
-Global Instance elim_modal_embed_fupd_goal `{FUpdFacts PROP} φ E1 E2 E3 P P' ð“ ð“ ' :
+Global Instance elim_modal_embed_fupd_goal `{BiFUpd PROP} φ E1 E2 E3 P P' ð“ ð“ ' :
ElimModal φ P P' (|={E1,E3}=> ⎡ð“ ⎤)%I (|={E2,E3}=> ⎡ð“ '⎤)%I →
ElimModal φ P P' ⎡|={E1,E3}=> ð“ ⎤ ⎡|={E2,E3}=> ð“ '⎤.
Proof. by rewrite /ElimModal !monPred_fupd_embed. Qed.
-Global Instance elim_modal_embed_fupd_hyp `{FUpdFacts PROP} φ E1 E2 𓟠P' Q Q' :
+Global Instance elim_modal_embed_fupd_hyp `{BiFUpd PROP} φ E1 E2 𓟠P' Q Q' :
ElimModal φ (|={E1,E2}=> ⎡ð“ŸâŽ¤)%I P' Q Q' →
ElimModal φ ⎡|={E1,E2}=> ð“ŸâŽ¤ P' Q Q'.
Proof. by rewrite /ElimModal monPred_fupd_embed. Qed.
-Global Instance add_modal_embed_fupd_goal `{FUpdFacts PROP} E1 E2 P P' ð“ :
+Global Instance add_modal_embed_fupd_goal `{BiFUpd PROP} E1 E2 P P' ð“ :
AddModal P P' (|={E1,E2}=> ⎡ð“ ⎤)%I → AddModal P P' ⎡|={E1,E2}=> ð“ ⎤.
Proof. by rewrite /AddModal !monPred_fupd_embed. Qed.
-Global Instance elim_modal_at_fupd_goal `{FUpdFacts PROP} φ E1 E2 E3 ð“Ÿ ð“Ÿ' Q Q' i :
+Global Instance elim_modal_at_fupd_goal `{BiFUpd PROP} φ E1 E2 E3 ð“Ÿ ð“Ÿ' Q Q' i :
ElimModal φ ð“Ÿ ð“Ÿ' (|={E1,E3}=> Q i) (|={E2,E3}=> Q' i) →
ElimModal φ ð“Ÿ ð“Ÿ' ((|={E1,E3}=> Q) i) ((|={E2,E3}=> Q') i).
Proof. by rewrite /ElimModal !monPred_at_fupd. Qed.
-Global Instance elim_modal_at_fupd_hyp `{FUpdFacts PROP} φ E1 E2 P ð“Ÿ' ð“ ð“ ' i :
+Global Instance elim_modal_at_fupd_hyp `{BiFUpd PROP} φ E1 E2 P ð“Ÿ' ð“ ð“ ' i :
ElimModal φ (|={E1,E2}=> P i) ð“Ÿ' ð“ ð“ ' →
ElimModal φ ((|={E1,E2}=> P) i) ð“Ÿ' ð“ ð“ '.
Proof. by rewrite /ElimModal monPred_at_fupd. Qed.
-Global Instance add_modal_at_fupd_goal `{FUpdFacts PROP} E1 E2 ð“Ÿ ð“Ÿ' Q i :
+Global Instance add_modal_at_fupd_goal `{BiFUpd PROP} E1 E2 ð“Ÿ ð“Ÿ' Q i :
AddModal ð“Ÿ ð“Ÿ' (|={E1,E2}=> Q i) → AddModal ð“Ÿ ð“Ÿ' ((|={E1,E2}=> Q) i).
Proof. by rewrite /AddModal !monPred_at_fupd. Qed.
diff --git a/theories/proofmode/tactics.v b/theories/proofmode/tactics.v
index 29d90f3df3d5035436c118a77aee63dad266115e..b4bdc052d8e83a9dbf6c8afe9e1fa56bbe489e21 100644
--- a/theories/proofmode/tactics.v
+++ b/theories/proofmode/tactics.v
@@ -1963,7 +1963,7 @@ Hint Extern 0 (envs_entails _ (_ -∗ _)) => iIntros.
Hint Extern 1 (envs_entails _ (_ ∧ _)) => iSplit.
Hint Extern 1 (envs_entails _ (_ ∗ _)) => iSplit.
Hint Extern 1 (envs_entails _ (â–· _)) => iNext.
-Hint Extern 1 (envs_entails _ (bi_plainly _)) => iAlways.
+Hint Extern 1 (envs_entails _ (â– _)) => iAlways.
Hint Extern 1 (envs_entails _ (bi_persistently _)) => iAlways.
Hint Extern 1 (envs_entails _ (bi_affinely _)) => iAlways.
Hint Extern 1 (envs_entails _ (∃ _, _)) => iExists _.
diff --git a/theories/tests/proofmode_monpred.v b/theories/tests/proofmode_monpred.v
index ae928b8abccf2b7f9cc1eee8a80b5733fd3d2001..5ae50a06ebd978e0ce85aaeb0ebdf2650f0439ac 100644
--- a/theories/tests/proofmode_monpred.v
+++ b/theories/tests/proofmode_monpred.v
@@ -96,21 +96,15 @@ Section tests.
ignore hint modes. So we assume the instances in a way that
cannot be used by type class resolution, and then declare the
instance. as such. *)
- Context (BU0 : BUpd PROP * unit).
- Instance BU : BUpd PROP := fst BU0.
- Context (FU0 : FUpd PROP * unit).
- Instance FU : FUpd PROP := fst FU0.
- Context (FUF0 : FUpdFacts PROP * unit).
- Instance FUF : FUpdFacts PROP := fst FUF0.
+ Context (FU0 : BiFUpd PROP * unit).
+ Instance FU : BiFUpd PROP := fst FU0.
Lemma test_apply_fupd_intro_mask E1 E2 P :
E2 ⊆ E1 → P -∗ |={E1,E2}=> |={E2,E1}=> P.
- Proof. iIntros. by iApply fupd_intro_mask. Qed.
+ Proof. iIntros. by iApply @fupd_intro_mask. Qed.
Lemma test_apply_fupd_intro_mask_2 E1 E2 P :
E2 ⊆ E1 → P -∗ |={E1,E2}=> |={E2,E1}=> P.
- Proof.
- iIntros. iFrame. by iApply fupd_intro_mask'.
- Qed.
+ Proof. iIntros. iFrame. by iApply @fupd_intro_mask'. Qed.
Lemma test_iFrame_embed_persistent (P : PROP) (Q: monPred) :
Q ∗ □ ⎡P⎤ ⊢ Q ∗ ⎡P ∗ P⎤.
@@ -121,5 +115,4 @@ Section tests.
Lemma test_iNext_Bi P :
@bi_entails monPredI (â–· P) (â–· P).
Proof. iIntros "H". by iNext. Qed.
-
End tests.