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Commit 6aac0120 authored by Robbert Krebbers's avatar Robbert Krebbers
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A type class for plainly. 

Based on an earlier MR by @jung.
parent e39f72fe
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_CoqProject
View file @ 6aac0120
......@@ -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
......
theories/base_logic/derived.v
View file @ 6aac0120
......@@ -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.
......
theories/base_logic/lib/fancy_updates.v
View file @ 6aac0120
......@@ -30,6 +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 $]".
Qed.
Instance uPred_bi_fupd `{invG Σ} : BiFUpd (uPredSI (iResUR Σ)) :=
{| bi_fupd_mixin := uPred_fupd_mixin |}.
Instance uPred_bi_bupd_fupd `{invG Σ} : BiBUpdFUpd (uPredSI (iResUR Σ)).
Proof. rewrite /BiBUpdFUpd uPred_fupd_eq. by iIntros (E P) ">? [$ $] !> !>". Qed.
Instance uPred_bi_fupd_plainly `{invG Σ} : BiFUpdPlainly (uPredSI (iResUR Σ)).
Proof.
split.
- iIntros (E1 E2 E2' P Q ? (E3&->&HE)%subseteq_disjoint_union_L) "HQP HQ".
rewrite uPred_fupd_eq /uPred_fupd_def ownE_op //. iIntros "H".
iMod ("HQ" with "H") as ">(Hws & [HE1 HE3] & HQ)"; iModIntro.
......@@ -39,9 +49,4 @@ Proof.
- rewrite uPred_fupd_eq /uPred_fupd_def. iIntros (E P ?) "HP [Hw HE]".
iAssert (▷ ◇ P)%I with "[-]" as "#$"; last by iFrame.
iNext. by iMod ("HP" with "[$]") as "(_ & _ & HP)".
Qed.
Instance uPred_bi_fupd `{invG Σ} : BiFUpd (uPredSI (iResUR Σ)) :=
{| bi_fupd_mixin := uPred_fupd_mixin |}.
Instance uPred_bi_bupd_fupd `{invG Σ} : BiBUpdFUpd (uPredSI (iResUR Σ)).
Proof. rewrite /BiBUpdFUpd uPred_fupd_eq. by iIntros (E P) ">? [$ $] !> !>". Qed.
Qed.
\ No newline at end of file
theories/base_logic/upred.v
View file @ 6aac0120
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) |}.
......@@ -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 +544,46 @@ 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.
......@@ -593,12 +602,16 @@ Proof.
{ by rewrite assoc -(dist_le _ _ _ _ Hx); last lia. }
exists (x' ⋅ 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.
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.
......@@ -630,7 +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 *)
......@@ -684,7 +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.
......
theories/bi/bi.v
View file @ 6aac0120
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
*)
theories/bi/big_op.v
View file @ 6aac0120
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.
theories/bi/derived_connectives.v
View file @ 6aac0120
......@@ -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 "□ P" := (bi_affinely (bi_persistently P))%I
(at level 20, right associativity) : bi_scope.
Notation "■ 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 "□? 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 "□? p P") : bi_scope.
Notation "■? p P" := (bi_affinely_if p (bi_plainly_if p P))%I