From 528b255fe9c00eb53ffe4acb38fcc4b2dcc3356e Mon Sep 17 00:00:00 2001 From: Ralf Jung <jung@mpi-sws.org> Date: Wed, 21 Mar 2018 15:35:25 +0100 Subject: [PATCH] split class_instances into BI and SBI instanced --- _CoqProject | 3 +- theories/bi/lib/fractional.v | 2 +- ...class_instances.v => class_instances_bi.v} | 712 +---------------- theories/proofmode/class_instances_sbi.v | 715 ++++++++++++++++++ theories/proofmode/monpred.v | 2 +- theories/proofmode/tactics.v | 2 +- 6 files changed, 721 insertions(+), 715 deletions(-) rename theories/proofmode/{class_instances.v => class_instances_bi.v} (60%) create mode 100644 theories/proofmode/class_instances_sbi.v diff --git a/_CoqProject b/_CoqProject index eb0723eca..ee5700f03 100644 --- a/_CoqProject +++ b/_CoqProject @@ -101,7 +101,8 @@ theories/proofmode/sel_patterns.v theories/proofmode/tactics.v theories/proofmode/notation.v theories/proofmode/classes.v -theories/proofmode/class_instances.v +theories/proofmode/class_instances_bi.v +theories/proofmode/class_instances_sbi.v theories/proofmode/monpred.v theories/proofmode/modalities.v theories/proofmode/modality_instances.v diff --git a/theories/bi/lib/fractional.v b/theories/bi/lib/fractional.v index 192db2027..cef06cc0f 100644 --- a/theories/bi/lib/fractional.v +++ b/theories/bi/lib/fractional.v @@ -1,5 +1,5 @@ From iris.bi Require Export bi. -From iris.proofmode Require Import classes class_instances. +From iris.proofmode Require Import classes class_instances_bi. Set Default Proof Using "Type". Class Fractional {PROP : bi} (Φ : Qp → PROP) := diff --git a/theories/proofmode/class_instances.v b/theories/proofmode/class_instances_bi.v similarity index 60% rename from theories/proofmode/class_instances.v rename to theories/proofmode/class_instances_bi.v index 8fc614bf6..720d877e8 100644 --- a/theories/proofmode/class_instances.v +++ b/theories/proofmode/class_instances_bi.v @@ -1,6 +1,6 @@ From stdpp Require Import nat_cancel. From iris.bi Require Import bi tactics. -From iris.proofmode Require Export modality_instances classes. +From iris.proofmode Require Import modality_instances classes. Set Default Proof Using "Type". Import bi. @@ -1099,713 +1099,3 @@ Global Instance as_emp_valid_embed `{BiEmbed PROP PROP'} (φ : Prop) (P : PROP) AsEmpValid0 φ P → AsEmpValid φ ⎡P⎤. Proof. rewrite /AsEmpValid0 /AsEmpValid=> _ ->. rewrite embed_emp_valid //. Qed. End bi_instances. - -Section sbi_instances. -Context {PROP : sbi}. -Implicit Types P Q R : PROP. - -(* FromAssumption *) -Global Instance from_assumption_later p P Q : - FromAssumption p P Q → KnownRFromAssumption p P (▷ Q)%I. -Proof. rewrite /KnownRFromAssumption /FromAssumption=>->. apply later_intro. Qed. -Global Instance from_assumption_laterN n p P Q : - FromAssumption p P Q → KnownRFromAssumption p P (▷^n Q)%I. -Proof. rewrite /KnownRFromAssumption /FromAssumption=>->. apply laterN_intro. Qed. -Global Instance from_assumption_except_0 p P Q : - FromAssumption p P Q → KnownRFromAssumption p P (◇ Q)%I. -Proof. rewrite /KnownRFromAssumption /FromAssumption=>->. apply except_0_intro. Qed. - -Global Instance from_assumption_bupd `{BiBUpd PROP} p P Q : - FromAssumption p P Q → KnownRFromAssumption p P (|==> Q). -Proof. rewrite /KnownRFromAssumption /FromAssumption=>->. apply bupd_intro. Qed. -Global Instance from_assumption_fupd `{BiBUpdFUpd PROP} E p P Q : - FromAssumption p P (|==> Q) → KnownRFromAssumption p P (|={E}=> Q)%I. -Proof. rewrite /KnownRFromAssumption /FromAssumption=>->. apply bupd_fupd. Qed. - -Global Instance from_assumption_plainly_l_true `{BiPlainly PROP} P Q : - FromAssumption true P Q → KnownLFromAssumption true (■P) Q. -Proof. - rewrite /KnownLFromAssumption /FromAssumption /= =><-. - rewrite intuitionistically_plainly_elim //. -Qed. -Global Instance from_assumption_plainly_l_false `{BiPlainly PROP, BiAffine PROP} P Q : - FromAssumption true P Q → KnownLFromAssumption false (■P) Q. -Proof. - rewrite /KnownLFromAssumption /FromAssumption /= =><-. - rewrite plainly_elim_persistently intuitionistically_persistently //. -Qed. - -(* FromPure *) -Global Instance from_pure_internal_eq af {A : ofeT} (a b : A) : - @FromPure PROP af (a ≡ b) (a ≡ b). -Proof. by rewrite /FromPure pure_internal_eq affinely_if_elim. Qed. -Global Instance from_pure_later a P φ : FromPure a P φ → FromPure a (▷ P) φ. -Proof. rewrite /FromPure=> ->. apply later_intro. Qed. -Global Instance from_pure_laterN a n P φ : FromPure a P φ → FromPure a (▷^n P) φ. -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 `{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 `{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). -Proof. - rewrite /IntoWand /= => HR. - by rewrite !later_intuitionistically_if_2 -later_wand HR. -Qed. -Global Instance into_wand_later_args p q R P Q : - IntoWand p q R P Q → IntoWand' p q R (▷ P) (▷ Q). -Proof. - rewrite /IntoWand' /IntoWand /= => HR. - by rewrite !later_intuitionistically_if_2 - (later_intro (□?p R)%I) -later_wand HR. -Qed. -Global Instance into_wand_laterN n p q R P Q : - IntoWand p q R P Q → IntoWand p q (▷^n R) (▷^n P) (▷^n Q). -Proof. - rewrite /IntoWand /= => HR. - by rewrite !laterN_intuitionistically_if_2 -laterN_wand HR. -Qed. -Global Instance into_wand_laterN_args n p q R P Q : - IntoWand p q R P Q → IntoWand' p q R (▷^n P) (▷^n Q). -Proof. - rewrite /IntoWand' /IntoWand /= => HR. - by rewrite !laterN_intuitionistically_if_2 - (laterN_intro _ (□?p R)%I) -laterN_wand HR. -Qed. - -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 !intuitionistically_if_elim HR. - apply wand_intro_l. by rewrite bupd_sep wand_elim_r. -Qed. -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 intuitionistically_if_elim HR. - apply wand_intro_l. by rewrite bupd_frame_l wand_elim_r. -Qed. -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 intuitionistically_if_elim bupd_wand_r. -Qed. - -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 !intuitionistically_if_elim HR. - apply wand_intro_l. by rewrite fupd_sep wand_elim_r. -Qed. -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 intuitionistically_if_elim HR. - apply wand_intro_l. by rewrite fupd_frame_l wand_elim_r. -Qed. -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 intuitionistically_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. rewrite /IntoWand /= intuitionistically_plainly_elim //. Qed. -Global Instance into_wand_plainly_false `{BiPlainly PROP} q R P Q : - Absorbing R → IntoWand false q R P Q → IntoWand false q (■R) P Q. -Proof. intros ?. by rewrite /IntoWand plainly_elim. Qed. - -(* FromAnd *) -Global Instance from_and_later P Q1 Q2 : - FromAnd P Q1 Q2 → FromAnd (▷ P) (▷ Q1) (▷ Q2). -Proof. rewrite /FromAnd=> <-. by rewrite later_and. Qed. -Global Instance from_and_laterN n P Q1 Q2 : - FromAnd P Q1 Q2 → FromAnd (▷^n P) (▷^n Q1) (▷^n Q2). -Proof. rewrite /FromAnd=> <-. by rewrite laterN_and. Qed. -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). -Proof. rewrite /FromSep=> <-. by rewrite later_sep. Qed. -Global Instance from_sep_laterN n P Q1 Q2 : - FromSep P Q1 Q2 → FromSep (▷^n P) (▷^n Q1) (▷^n Q2). -Proof. rewrite /FromSep=> <-. by rewrite laterN_sep. Qed. -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 `{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 `{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). -Proof. - rewrite /IntoAnd=> HP. apply intuitionistically_if_intro'. - by rewrite later_intuitionistically_if_2 HP - intuitionistically_if_elim later_and. -Qed. -Global Instance into_and_laterN n p P Q1 Q2 : - IntoAnd p P Q1 Q2 → IntoAnd p (▷^n P) (▷^n Q1) (▷^n Q2). -Proof. - rewrite /IntoAnd=> HP. apply intuitionistically_if_intro'. - by rewrite laterN_intuitionistically_if_2 HP - intuitionistically_if_elim laterN_and. -Qed. -Global Instance into_and_except_0 p P Q1 Q2 : - IntoAnd p P Q1 Q2 → IntoAnd p (◇ P) (◇ Q1) (◇ Q2). -Proof. - rewrite /IntoAnd=> HP. apply intuitionistically_if_intro'. - by rewrite except_0_intuitionistically_if_2 HP - intuitionistically_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 -[(□ ■P)%I]intuitionistically_idemp intuitionistically_plainly =>->. - rewrite [(□ (_ ∧ _))%I]intuitionistically_elim //. - - intros ->. by rewrite plainly_and. -Qed. - -(* IntoSep *) -Global Instance into_sep_later P Q1 Q2 : - IntoSep P Q1 Q2 → IntoSep (▷ P) (▷ Q1) (▷ Q2). -Proof. rewrite /IntoSep=> ->. by rewrite later_sep. Qed. -Global Instance into_sep_laterN n P Q1 Q2 : - IntoSep P Q1 Q2 → IntoSep (▷^n P) (▷^n Q1) (▷^n Q2). -Proof. rewrite /IntoSep=> ->. by rewrite laterN_sep. Qed. -Global Instance into_sep_except_0 P Q1 Q2 : - IntoSep P Q1 Q2 → IntoSep (◇ P) (◇ Q1) (◇ Q2). -Proof. rewrite /IntoSep=> ->. by rewrite except_0_sep. Qed. - -(* FIXME: This instance is overly specific, generalize it. *) -Global Instance into_sep_affinely_later `{!Timeless (emp%I : PROP)} P Q1 Q2 : - IntoSep P Q1 Q2 → Affine Q1 → Affine Q2 → - IntoSep (<affine> ▷ P) (<affine> ▷ Q1) (<affine> ▷ Q2). -Proof. - rewrite /IntoSep /= => -> ??. - rewrite -{1}(affine_affinely Q1) -{1}(affine_affinely Q2) later_sep !later_affinely_1. - rewrite -except_0_sep /sbi_except_0 affinely_or. apply or_elim, affinely_elim. - rewrite -(idemp bi_and (<affine> ▷ False)%I) persistent_and_sep_1. - 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 (■P) (■Q1) (■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). -Proof. rewrite /FromOr=><-. by rewrite later_or. Qed. -Global Instance from_or_laterN n P Q1 Q2 : - FromOr P Q1 Q2 → FromOr (▷^n P) (▷^n Q1) (▷^n Q2). -Proof. rewrite /FromOr=><-. by rewrite laterN_or. Qed. -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 `{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 `{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). -Proof. rewrite /IntoOr=>->. by rewrite later_or. Qed. -Global Instance into_or_laterN n P Q1 Q2 : - IntoOr P Q1 Q2 → IntoOr (▷^n P) (▷^n Q1) (▷^n Q2). -Proof. rewrite /IntoOr=>->. by rewrite laterN_or. Qed. -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. -Proof. - rewrite /FromExist=> <-. apply exist_elim=>x. apply later_mono, exist_intro. -Qed. -Global Instance from_exist_laterN {A} n P (Φ : A → PROP) : - FromExist P Φ → FromExist (▷^n P) (λ a, ▷^n (Φ a))%I. -Proof. - rewrite /FromExist=> <-. apply exist_elim=>x. apply laterN_mono, exist_intro. -Qed. -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 `{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 `{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. -Proof. rewrite /IntoExist=> HP ?. by rewrite HP later_exist. Qed. -Global Instance into_exist_laterN {A} n P (Φ : A → PROP) : - IntoExist P Φ → Inhabited A → IntoExist (▷^n P) (λ a, ▷^n (Φ a))%I. -Proof. rewrite /IntoExist=> HP ?. by rewrite HP laterN_exist. Qed. -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. -Proof. rewrite /IntoForall=> HP. by rewrite HP later_forall. Qed. -Global Instance into_forall_except_0 {A} P (Φ : A → PROP) : - IntoForall P Φ → IntoForall (◇ P) (λ a, ◇ (Φ a))%I. -Proof. rewrite /IntoForall=> HP. by rewrite HP except_0_forall. Qed. -Global Instance into_forall_impl_pure φ P Q : - FromPureT false P φ → IntoForall (P → Q) (λ _ : φ, Q). -Proof. - rewrite /FromPureT /FromPure /IntoForall=> -[φ' [-> <-]]. - by rewrite pure_impl_forall. -Qed. -Global Instance into_forall_wand_pure φ P Q : - FromPureT true P φ → IntoForall (P -∗ Q) (λ _ : φ, Q). -Proof. - rewrite /FromPureT /FromPure /IntoForall=> -[φ' [-> <-]] /=. - apply forall_intro=>? /=. - 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. -Proof. rewrite /FromForall=> <-. by rewrite later_forall. Qed. -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 (◇ P). -Proof. by rewrite /IsExcept0 except_0_idemp. Qed. -Global Instance is_except_0_later P : IsExcept0 (▷ P). -Proof. by rewrite /IsExcept0 except_0_later. Qed. -Global Instance is_except_0_embed `{SbiEmbed PROP 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 `{BiFUpd PROP} E1 E2 P : - IsExcept0 (|={E1,E2}=> P). -Proof. by rewrite /IsExcept0 except_0_fupd. Qed. - -(* FromModal *) -Global Instance from_modal_later P : - FromModal (modality_laterN 1) (▷^1 P) (▷ P) P. -Proof. by rewrite /FromModal. Qed. -Global Instance from_modal_laterN n P : - FromModal (modality_laterN n) (▷^n P) (▷^n P) P. -Proof. by rewrite /FromModal. Qed. -Global Instance from_modal_except_0 P : FromModal modality_id (◇ P) (◇ P) P. -Proof. by rewrite /FromModal /= -except_0_intro. Qed. - -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 `{BiFUpd PROP} : - FromModal modality_id (|={E}=> P) (|={E}=> P) P. -Proof. by rewrite /FromModal /= -fupd_intro. Qed. - -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 embed_laterN. Qed. - -Global Instance from_modal_plainly `{BiPlainly PROP} P : - FromModal modality_plainly (■P) (■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) : - @IntoInternalEq PROP A (x ≡ y) x y. -Proof. by rewrite /IntoInternalEq. Qed. -Global Instance into_internal_eq_affinely {A : ofeT} (x y : A) P : - IntoInternalEq P x y → IntoInternalEq (<affine> P) x y. -Proof. rewrite /IntoInternalEq=> ->. by rewrite affinely_elim. Qed. -Global Instance into_internal_eq_intuitionistically {A : ofeT} (x y : A) P : - IntoInternalEq P x y → IntoInternalEq (□ P) x y. -Proof. rewrite /IntoInternalEq=> ->. by rewrite intuitionistically_elim. Qed. -Global Instance into_internal_eq_absorbingly {A : ofeT} (x y : A) P : - IntoInternalEq P x y → IntoInternalEq (<absorb> P) x y. -Proof. rewrite /IntoInternalEq=> ->. by rewrite absorbingly_internal_eq. Qed. -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 (<pers> P) x y. -Proof. rewrite /IntoInternalEq=> ->. by rewrite persistently_elim. Qed. -Global Instance into_internal_eq_embed - `{SbiEmbed PROP PROP'} {A : ofeT} (x y : A) P : - IntoInternalEq P x y → IntoInternalEq ⎡P⎤ x y. -Proof. rewrite /IntoInternalEq=> ->. by rewrite embed_internal_eq. Qed. - -(* IntoExcept0 *) -Global Instance into_except_0_except_0 P : IntoExcept0 (◇ P) P. -Proof. by rewrite /IntoExcept0. Qed. -Global Instance into_except_0_later P : Timeless P → IntoExcept0 (▷ P) P. -Proof. by rewrite /IntoExcept0. Qed. -Global Instance into_except_0_later_if p P : Timeless P → IntoExcept0 (▷?p P) P. -Proof. rewrite /IntoExcept0. destruct p; auto using except_0_intro. Qed. - -Global Instance into_except_0_affinely P Q : - IntoExcept0 P Q → IntoExcept0 (<affine> P) (<affine> Q). -Proof. rewrite /IntoExcept0=> ->. by rewrite except_0_affinely_2. Qed. -Global Instance into_except_0_intuitionistically P Q : - IntoExcept0 P Q → IntoExcept0 (□ P) (□ Q). -Proof. rewrite /IntoExcept0=> ->. by rewrite except_0_intuitionistically_2. Qed. -Global Instance into_except_0_absorbingly P Q : - IntoExcept0 P Q → IntoExcept0 (<absorb> P) (<absorb> Q). -Proof. rewrite /IntoExcept0=> ->. by rewrite except_0_absorbingly. Qed. -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 (<pers> P) (<pers> Q). -Proof. rewrite /IntoExcept0=> ->. by rewrite except_0_persistently. Qed. -Global Instance into_except_0_embed `{SbiEmbed PROP PROP'} P Q : - IntoExcept0 P Q → IntoExcept0 ⎡P⎤ ⎡Q⎤. -Proof. rewrite /IntoExcept0=> ->. by rewrite embed_except_0. Qed. - -(* ElimModal *) -Global Instance elim_modal_timeless P Q : - IntoExcept0 P P' → IsExcept0 Q → ElimModal True P P' Q Q. -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_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 `{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 `{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 `{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. - -Global Instance elim_modal_embed_fupd_goal `{BiEmbedFUpd PROP PROP'} - φ E1 E2 E3 (P P' : PROP') (Q Q' : PROP) : - ElimModal φ P P' (|={E1,E3}=> ⎡Q⎤)%I (|={E2,E3}=> ⎡Q'⎤)%I → - ElimModal φ P P' ⎡|={E1,E3}=> Q⎤ ⎡|={E2,E3}=> Q'⎤. -Proof. by rewrite /ElimModal !embed_fupd. Qed. -Global Instance elim_modal_embed_fupd_hyp `{BiEmbedFUpd PROP PROP'} - φ E1 E2 (P : PROP) (P' Q Q' : PROP') : - ElimModal φ (|={E1,E2}=> ⎡P⎤)%I P' Q Q' → - ElimModal φ ⎡|={E1,E2}=> P⎤ P' Q Q'. -Proof. by rewrite /ElimModal embed_fupd. Qed. - -(* AddModal *) -(* High priority to add a ▷ rather than a ◇ when P is timeless. *) -Global Instance add_modal_later_except_0 P Q : - Timeless P → AddModal (▷ P) P (◇ Q) | 0. -Proof. - intros. rewrite /AddModal (except_0_intro (_ -∗ _)%I) (timeless P). - by rewrite -except_0_sep wand_elim_r except_0_idemp. -Qed. -Global Instance add_modal_later P Q : - Timeless P → AddModal (▷ P) P (▷ Q) | 0. -Proof. - intros. rewrite /AddModal (except_0_intro (_ -∗ _)%I) (timeless P). - by rewrite -except_0_sep wand_elim_r except_0_later. -Qed. -Global Instance add_modal_except_0 P Q : AddModal (◇ P) P (◇ Q) | 1. -Proof. - intros. rewrite /AddModal (except_0_intro (_ -∗ _)%I). - by rewrite -except_0_sep wand_elim_r except_0_idemp. -Qed. -Global Instance add_modal_except_0_later P Q : AddModal (◇ P) P (▷ Q) | 1. -Proof. - intros. rewrite /AddModal (except_0_intro (_ -∗ _)%I). - by rewrite -except_0_sep wand_elim_r except_0_later. -Qed. - -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 `{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. - -Global Instance add_modal_embed_fupd_goal `{BiEmbedFUpd PROP PROP'} - E1 E2 (P P' : PROP') (Q : PROP) : - AddModal P P' (|={E1,E2}=> ⎡Q⎤)%I → AddModal P P' ⎡|={E1,E2}=> Q⎤. -Proof. by rewrite /AddModal !embed_fupd. Qed. - -(* Frame *) -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' → Frame p (a ≡ b) ⎡P⎤ Q'. -Proof. rewrite /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. -Global Instance make_laterN_0 P : MakeLaterN 0 P P | 0. -Proof. by rewrite /MakeLaterN. Qed. -Global Instance make_laterN_1 P : MakeLaterN 1 P (▷ P) | 2. -Proof. by rewrite /MakeLaterN. Qed. -Global Instance make_laterN_default P : MakeLaterN n P (▷^n P) | 100. -Proof. by rewrite /MakeLaterN. Qed. - -Global Instance frame_later p R R' P Q Q' : - NoBackTrack (MaybeIntoLaterN true 1 R' R) → - Frame p R P Q → MakeLaterN 1 Q Q' → Frame p R' (▷ P) Q'. -Proof. - rewrite /Frame /MakeLaterN /MaybeIntoLaterN=>-[->] <- <-. - by rewrite later_intuitionistically_if_2 later_sep. -Qed. -Global Instance frame_laterN p n R R' P Q Q' : - NoBackTrack (MaybeIntoLaterN true n R' R) → - Frame p R P Q → MakeLaterN n Q Q' → Frame p R' (▷^n P) Q'. -Proof. - rewrite /Frame /MakeLaterN /MaybeIntoLaterN=>-[->] <- <-. - by rewrite laterN_intuitionistically_if_2 laterN_sep. -Qed. - -Global Instance frame_bupd `{BiBUpd PROP} p R P Q : - Frame p R P Q → Frame p R (|==> P) (|==> Q). -Proof. rewrite /Frame=><-. by rewrite bupd_frame_l. Qed. -Global Instance frame_fupd `{BiFUpd PROP} p E1 E2 R P Q : - Frame p R P Q → Frame p R (|={E1,E2}=> P) (|={E1,E2}=> Q). -Proof. rewrite /Frame=><-. by rewrite fupd_frame_l. Qed. - -Global Instance make_except_0_True : @KnownMakeExcept0 PROP True True. -Proof. by rewrite /KnownMakeExcept0 /MakeExcept0 except_0_True. Qed. -Global Instance make_except_0_default P : MakeExcept0 P (◇ P) | 100. -Proof. by rewrite /MakeExcept0. Qed. - -Global Instance frame_except_0 p R P Q Q' : - Frame p R P Q → MakeExcept0 Q Q' → Frame p R (◇ P) Q'. -Proof. - rewrite /Frame /MakeExcept0=><- <-. - by rewrite except_0_sep -(except_0_intro (□?p R)%I). -Qed. - -(* IntoLater *) -Global Instance into_laterN_0 only_head P : IntoLaterN only_head 0 P P. -Proof. by rewrite /IntoLaterN /MaybeIntoLaterN. Qed. -Global Instance into_laterN_later only_head n n' m' P Q lQ : - NatCancel n 1 n' m' → - (** If canceling has failed (i.e. [1 = m']), we should make progress deeper - into [P], as such, we continue with the [IntoLaterN] class, which is required - to make progress. If canceling has succeeded, we do not need to make further - progress, but there may still be a left-over (i.e. [n']) to cancel more deeply - into [P], as such, we continue with [MaybeIntoLaterN]. *) - TCIf (TCEq 1 m') (IntoLaterN only_head n' P Q) (MaybeIntoLaterN only_head n' P Q) → - MakeLaterN m' Q lQ → - IntoLaterN only_head n (▷ P) lQ | 2. -Proof. - rewrite /MakeLaterN /IntoLaterN /MaybeIntoLaterN /NatCancel. - move=> Hn [_ ->|->] <-; - by rewrite -later_laterN -laterN_plus -Hn Nat.add_comm. -Qed. -Global Instance into_laterN_laterN only_head n m n' m' P Q lQ : - NatCancel n m n' m' → - TCIf (TCEq m m') (IntoLaterN only_head n' P Q) (MaybeIntoLaterN only_head n' P Q) → - MakeLaterN m' Q lQ → - IntoLaterN only_head n (▷^m P) lQ | 1. -Proof. - rewrite /MakeLaterN /IntoLaterN /MaybeIntoLaterN /NatCancel. - move=> Hn [_ ->|->] <-; by rewrite -!laterN_plus -Hn Nat.add_comm. -Qed. - -Global Instance into_laterN_and_l n P1 P2 Q1 Q2 : - IntoLaterN false n P1 Q1 → MaybeIntoLaterN false n P2 Q2 → - IntoLaterN false n (P1 ∧ P2) (Q1 ∧ Q2) | 10. -Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> -> ->. by rewrite laterN_and. Qed. -Global Instance into_laterN_and_r n P P2 Q2 : - IntoLaterN false n P2 Q2 → IntoLaterN false n (P ∧ P2) (P ∧ Q2) | 11. -Proof. - rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_and -(laterN_intro _ P). -Qed. - -Global Instance into_laterN_forall {A} n (Φ Ψ : A → PROP) : - (∀ x, IntoLaterN false n (Φ x) (Ψ x)) → - IntoLaterN false n (∀ x, Φ x) (∀ x, Ψ x). -Proof. rewrite /IntoLaterN /MaybeIntoLaterN laterN_forall=> ?. by apply forall_mono. Qed. -Global Instance into_laterN_exist {A} n (Φ Ψ : A → PROP) : - (∀ x, IntoLaterN false n (Φ x) (Ψ x)) → - IntoLaterN false n (∃ x, Φ x) (∃ x, Ψ x). -Proof. rewrite /IntoLaterN /MaybeIntoLaterN -laterN_exist_2=> ?. by apply exist_mono. Qed. - -Global Instance into_laterN_or_l n P1 P2 Q1 Q2 : - IntoLaterN false n P1 Q1 → MaybeIntoLaterN false n P2 Q2 → - IntoLaterN false n (P1 ∨ P2) (Q1 ∨ Q2) | 10. -Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> -> ->. by rewrite laterN_or. Qed. -Global Instance into_laterN_or_r n P P2 Q2 : - IntoLaterN false n P2 Q2 → - IntoLaterN false n (P ∨ P2) (P ∨ Q2) | 11. -Proof. - rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_or -(laterN_intro _ P). -Qed. - -Global Instance into_later_affinely n P Q : - IntoLaterN false n P Q → IntoLaterN false n (<affine> P) (<affine> Q). -Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_affinely_2. Qed. -Global Instance into_later_intuitionistically n P Q : - IntoLaterN false n P Q → IntoLaterN false n (□ P) (□ Q). -Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_intuitionistically_2. Qed. -Global Instance into_later_absorbingly n P Q : - IntoLaterN false n P Q → IntoLaterN false n (<absorb> P) (<absorb> Q). -Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_absorbingly. Qed. -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 (<pers> P) (<pers> Q). -Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_persistently. Qed. -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 embed_laterN. Qed. - -Global Instance into_laterN_sep_l n P1 P2 Q1 Q2 : - IntoLaterN false n P1 Q1 → MaybeIntoLaterN false n P2 Q2 → - IntoLaterN false n (P1 ∗ P2) (Q1 ∗ Q2) | 10. -Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> -> ->. by rewrite laterN_sep. Qed. -Global Instance into_laterN_sep_r n P P2 Q2 : - IntoLaterN false n P2 Q2 → - IntoLaterN false n (P ∗ P2) (P ∗ Q2) | 11. -Proof. - rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_sep -(laterN_intro _ P). -Qed. - -Global Instance into_laterN_big_sepL n {A} (Φ Ψ : nat → A → PROP) (l: list A) : - (∀ x k, IntoLaterN false n (Φ k x) (Ψ k x)) → - IntoLaterN false n ([∗ list] k ↦ x ∈ l, Φ k x) ([∗ list] k ↦ x ∈ l, Ψ k x). -Proof. - rewrite /IntoLaterN /MaybeIntoLaterN=> ?. - rewrite big_opL_commute. by apply big_sepL_mono. -Qed. -Global Instance into_laterN_big_sepM n `{Countable K} {A} - (Φ Ψ : K → A → PROP) (m : gmap K A) : - (∀ x k, IntoLaterN false n (Φ k x) (Ψ k x)) → - IntoLaterN false n ([∗ map] k ↦ x ∈ m, Φ k x) ([∗ map] k ↦ x ∈ m, Ψ k x). -Proof. - rewrite /IntoLaterN /MaybeIntoLaterN=> ?. - rewrite big_opM_commute. by apply big_sepM_mono. -Qed. -Global Instance into_laterN_big_sepS n `{Countable A} - (Φ Ψ : A → PROP) (X : gset A) : - (∀ x, IntoLaterN false n (Φ x) (Ψ x)) → - IntoLaterN false n ([∗ set] x ∈ X, Φ x) ([∗ set] x ∈ X, Ψ x). -Proof. - rewrite /IntoLaterN /MaybeIntoLaterN=> ?. - rewrite big_opS_commute. by apply big_sepS_mono. -Qed. -Global Instance into_laterN_big_sepMS n `{Countable A} - (Φ Ψ : A → PROP) (X : gmultiset A) : - (∀ x, IntoLaterN false n (Φ x) (Ψ x)) → - IntoLaterN false n ([∗ mset] x ∈ X, Φ x) ([∗ mset] x ∈ X, Ψ x). -Proof. - rewrite /IntoLaterN /MaybeIntoLaterN=> ?. - rewrite big_opMS_commute. by apply big_sepMS_mono. -Qed. -End sbi_instances. diff --git a/theories/proofmode/class_instances_sbi.v b/theories/proofmode/class_instances_sbi.v new file mode 100644 index 000000000..c2001de07 --- /dev/null +++ b/theories/proofmode/class_instances_sbi.v @@ -0,0 +1,715 @@ +From stdpp Require Import nat_cancel. +From iris.bi Require Import bi tactics. +From iris.proofmode Require Import modality_instances classes class_instances_bi. +Set Default Proof Using "Type". +Import bi. + +Section sbi_instances. +Context {PROP : sbi}. +Implicit Types P Q R : PROP. + +(* FromAssumption *) +Global Instance from_assumption_later p P Q : + FromAssumption p P Q → KnownRFromAssumption p P (▷ Q)%I. +Proof. rewrite /KnownRFromAssumption /FromAssumption=>->. apply later_intro. Qed. +Global Instance from_assumption_laterN n p P Q : + FromAssumption p P Q → KnownRFromAssumption p P (▷^n Q)%I. +Proof. rewrite /KnownRFromAssumption /FromAssumption=>->. apply laterN_intro. Qed. +Global Instance from_assumption_except_0 p P Q : + FromAssumption p P Q → KnownRFromAssumption p P (◇ Q)%I. +Proof. rewrite /KnownRFromAssumption /FromAssumption=>->. apply except_0_intro. Qed. + +Global Instance from_assumption_bupd `{BiBUpd PROP} p P Q : + FromAssumption p P Q → KnownRFromAssumption p P (|==> Q). +Proof. rewrite /KnownRFromAssumption /FromAssumption=>->. apply bupd_intro. Qed. +Global Instance from_assumption_fupd `{BiBUpdFUpd PROP} E p P Q : + FromAssumption p P (|==> Q) → KnownRFromAssumption p P (|={E}=> Q)%I. +Proof. rewrite /KnownRFromAssumption /FromAssumption=>->. apply bupd_fupd. Qed. + +Global Instance from_assumption_plainly_l_true `{BiPlainly PROP} P Q : + FromAssumption true P Q → KnownLFromAssumption true (■P) Q. +Proof. + rewrite /KnownLFromAssumption /FromAssumption /= =><-. + rewrite intuitionistically_plainly_elim //. +Qed. +Global Instance from_assumption_plainly_l_false `{BiPlainly PROP, BiAffine PROP} P Q : + FromAssumption true P Q → KnownLFromAssumption false (■P) Q. +Proof. + rewrite /KnownLFromAssumption /FromAssumption /= =><-. + rewrite plainly_elim_persistently intuitionistically_persistently //. +Qed. + +(* FromPure *) +Global Instance from_pure_internal_eq af {A : ofeT} (a b : A) : + @FromPure PROP af (a ≡ b) (a ≡ b). +Proof. by rewrite /FromPure pure_internal_eq affinely_if_elim. Qed. +Global Instance from_pure_later a P φ : FromPure a P φ → FromPure a (▷ P) φ. +Proof. rewrite /FromPure=> ->. apply later_intro. Qed. +Global Instance from_pure_laterN a n P φ : FromPure a P φ → FromPure a (▷^n P) φ. +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 `{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 `{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). +Proof. + rewrite /IntoWand /= => HR. + by rewrite !later_intuitionistically_if_2 -later_wand HR. +Qed. +Global Instance into_wand_later_args p q R P Q : + IntoWand p q R P Q → IntoWand' p q R (▷ P) (▷ Q). +Proof. + rewrite /IntoWand' /IntoWand /= => HR. + by rewrite !later_intuitionistically_if_2 + (later_intro (□?p R)%I) -later_wand HR. +Qed. +Global Instance into_wand_laterN n p q R P Q : + IntoWand p q R P Q → IntoWand p q (▷^n R) (▷^n P) (▷^n Q). +Proof. + rewrite /IntoWand /= => HR. + by rewrite !laterN_intuitionistically_if_2 -laterN_wand HR. +Qed. +Global Instance into_wand_laterN_args n p q R P Q : + IntoWand p q R P Q → IntoWand' p q R (▷^n P) (▷^n Q). +Proof. + rewrite /IntoWand' /IntoWand /= => HR. + by rewrite !laterN_intuitionistically_if_2 + (laterN_intro _ (□?p R)%I) -laterN_wand HR. +Qed. + +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 !intuitionistically_if_elim HR. + apply wand_intro_l. by rewrite bupd_sep wand_elim_r. +Qed. +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 intuitionistically_if_elim HR. + apply wand_intro_l. by rewrite bupd_frame_l wand_elim_r. +Qed. +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 intuitionistically_if_elim bupd_wand_r. +Qed. + +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 !intuitionistically_if_elim HR. + apply wand_intro_l. by rewrite fupd_sep wand_elim_r. +Qed. +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 intuitionistically_if_elim HR. + apply wand_intro_l. by rewrite fupd_frame_l wand_elim_r. +Qed. +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 intuitionistically_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. rewrite /IntoWand /= intuitionistically_plainly_elim //. Qed. +Global Instance into_wand_plainly_false `{BiPlainly PROP} q R P Q : + Absorbing R → IntoWand false q R P Q → IntoWand false q (■R) P Q. +Proof. intros ?. by rewrite /IntoWand plainly_elim. Qed. + +(* FromAnd *) +Global Instance from_and_later P Q1 Q2 : + FromAnd P Q1 Q2 → FromAnd (▷ P) (▷ Q1) (▷ Q2). +Proof. rewrite /FromAnd=> <-. by rewrite later_and. Qed. +Global Instance from_and_laterN n P Q1 Q2 : + FromAnd P Q1 Q2 → FromAnd (▷^n P) (▷^n Q1) (▷^n Q2). +Proof. rewrite /FromAnd=> <-. by rewrite laterN_and. Qed. +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). +Proof. rewrite /FromSep=> <-. by rewrite later_sep. Qed. +Global Instance from_sep_laterN n P Q1 Q2 : + FromSep P Q1 Q2 → FromSep (▷^n P) (▷^n Q1) (▷^n Q2). +Proof. rewrite /FromSep=> <-. by rewrite laterN_sep. Qed. +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 `{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 `{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). +Proof. + rewrite /IntoAnd=> HP. apply intuitionistically_if_intro'. + by rewrite later_intuitionistically_if_2 HP + intuitionistically_if_elim later_and. +Qed. +Global Instance into_and_laterN n p P Q1 Q2 : + IntoAnd p P Q1 Q2 → IntoAnd p (▷^n P) (▷^n Q1) (▷^n Q2). +Proof. + rewrite /IntoAnd=> HP. apply intuitionistically_if_intro'. + by rewrite laterN_intuitionistically_if_2 HP + intuitionistically_if_elim laterN_and. +Qed. +Global Instance into_and_except_0 p P Q1 Q2 : + IntoAnd p P Q1 Q2 → IntoAnd p (◇ P) (◇ Q1) (◇ Q2). +Proof. + rewrite /IntoAnd=> HP. apply intuitionistically_if_intro'. + by rewrite except_0_intuitionistically_if_2 HP + intuitionistically_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 -[(□ ■P)%I]intuitionistically_idemp intuitionistically_plainly =>->. + rewrite [(□ (_ ∧ _))%I]intuitionistically_elim //. + - intros ->. by rewrite plainly_and. +Qed. + +(* IntoSep *) +Global Instance into_sep_later P Q1 Q2 : + IntoSep P Q1 Q2 → IntoSep (▷ P) (▷ Q1) (▷ Q2). +Proof. rewrite /IntoSep=> ->. by rewrite later_sep. Qed. +Global Instance into_sep_laterN n P Q1 Q2 : + IntoSep P Q1 Q2 → IntoSep (▷^n P) (▷^n Q1) (▷^n Q2). +Proof. rewrite /IntoSep=> ->. by rewrite laterN_sep. Qed. +Global Instance into_sep_except_0 P Q1 Q2 : + IntoSep P Q1 Q2 → IntoSep (◇ P) (◇ Q1) (◇ Q2). +Proof. rewrite /IntoSep=> ->. by rewrite except_0_sep. Qed. + +(* FIXME: This instance is overly specific, generalize it. *) +Global Instance into_sep_affinely_later `{!Timeless (emp%I : PROP)} P Q1 Q2 : + IntoSep P Q1 Q2 → Affine Q1 → Affine Q2 → + IntoSep (<affine> ▷ P) (<affine> ▷ Q1) (<affine> ▷ Q2). +Proof. + rewrite /IntoSep /= => -> ??. + rewrite -{1}(affine_affinely Q1) -{1}(affine_affinely Q2) later_sep !later_affinely_1. + rewrite -except_0_sep /sbi_except_0 affinely_or. apply or_elim, affinely_elim. + rewrite -(idemp bi_and (<affine> ▷ False)%I) persistent_and_sep_1. + 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 (■P) (■Q1) (■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). +Proof. rewrite /FromOr=><-. by rewrite later_or. Qed. +Global Instance from_or_laterN n P Q1 Q2 : + FromOr P Q1 Q2 → FromOr (▷^n P) (▷^n Q1) (▷^n Q2). +Proof. rewrite /FromOr=><-. by rewrite laterN_or. Qed. +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 `{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 `{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). +Proof. rewrite /IntoOr=>->. by rewrite later_or. Qed. +Global Instance into_or_laterN n P Q1 Q2 : + IntoOr P Q1 Q2 → IntoOr (▷^n P) (▷^n Q1) (▷^n Q2). +Proof. rewrite /IntoOr=>->. by rewrite laterN_or. Qed. +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. +Proof. + rewrite /FromExist=> <-. apply exist_elim=>x. apply later_mono, exist_intro. +Qed. +Global Instance from_exist_laterN {A} n P (Φ : A → PROP) : + FromExist P Φ → FromExist (▷^n P) (λ a, ▷^n (Φ a))%I. +Proof. + rewrite /FromExist=> <-. apply exist_elim=>x. apply laterN_mono, exist_intro. +Qed. +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 `{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 `{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. +Proof. rewrite /IntoExist=> HP ?. by rewrite HP later_exist. Qed. +Global Instance into_exist_laterN {A} n P (Φ : A → PROP) : + IntoExist P Φ → Inhabited A → IntoExist (▷^n P) (λ a, ▷^n (Φ a))%I. +Proof. rewrite /IntoExist=> HP ?. by rewrite HP laterN_exist. Qed. +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. +Proof. rewrite /IntoForall=> HP. by rewrite HP later_forall. Qed. +Global Instance into_forall_except_0 {A} P (Φ : A → PROP) : + IntoForall P Φ → IntoForall (◇ P) (λ a, ◇ (Φ a))%I. +Proof. rewrite /IntoForall=> HP. by rewrite HP except_0_forall. Qed. +Global Instance into_forall_impl_pure φ P Q : + FromPureT false P φ → IntoForall (P → Q) (λ _ : φ, Q). +Proof. + rewrite /FromPureT /FromPure /IntoForall=> -[φ' [-> <-]]. + by rewrite pure_impl_forall. +Qed. +Global Instance into_forall_wand_pure φ P Q : + FromPureT true P φ → IntoForall (P -∗ Q) (λ _ : φ, Q). +Proof. + rewrite /FromPureT /FromPure /IntoForall=> -[φ' [-> <-]] /=. + apply forall_intro=>? /=. + 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. +Proof. rewrite /FromForall=> <-. by rewrite later_forall. Qed. +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 (◇ P). +Proof. by rewrite /IsExcept0 except_0_idemp. Qed. +Global Instance is_except_0_later P : IsExcept0 (▷ P). +Proof. by rewrite /IsExcept0 except_0_later. Qed. +Global Instance is_except_0_embed `{SbiEmbed PROP 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 `{BiFUpd PROP} E1 E2 P : + IsExcept0 (|={E1,E2}=> P). +Proof. by rewrite /IsExcept0 except_0_fupd. Qed. + +(* FromModal *) +Global Instance from_modal_later P : + FromModal (modality_laterN 1) (▷^1 P) (▷ P) P. +Proof. by rewrite /FromModal. Qed. +Global Instance from_modal_laterN n P : + FromModal (modality_laterN n) (▷^n P) (▷^n P) P. +Proof. by rewrite /FromModal. Qed. +Global Instance from_modal_except_0 P : FromModal modality_id (◇ P) (◇ P) P. +Proof. by rewrite /FromModal /= -except_0_intro. Qed. + +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 `{BiFUpd PROP} : + FromModal modality_id (|={E}=> P) (|={E}=> P) P. +Proof. by rewrite /FromModal /= -fupd_intro. Qed. + +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 embed_laterN. Qed. + +Global Instance from_modal_plainly `{BiPlainly PROP} P : + FromModal modality_plainly (■P) (■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) : + @IntoInternalEq PROP A (x ≡ y) x y. +Proof. by rewrite /IntoInternalEq. Qed. +Global Instance into_internal_eq_affinely {A : ofeT} (x y : A) P : + IntoInternalEq P x y → IntoInternalEq (<affine> P) x y. +Proof. rewrite /IntoInternalEq=> ->. by rewrite affinely_elim. Qed. +Global Instance into_internal_eq_intuitionistically {A : ofeT} (x y : A) P : + IntoInternalEq P x y → IntoInternalEq (□ P) x y. +Proof. rewrite /IntoInternalEq=> ->. by rewrite intuitionistically_elim. Qed. +Global Instance into_internal_eq_absorbingly {A : ofeT} (x y : A) P : + IntoInternalEq P x y → IntoInternalEq (<absorb> P) x y. +Proof. rewrite /IntoInternalEq=> ->. by rewrite absorbingly_internal_eq. Qed. +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 (<pers> P) x y. +Proof. rewrite /IntoInternalEq=> ->. by rewrite persistently_elim. Qed. +Global Instance into_internal_eq_embed + `{SbiEmbed PROP PROP'} {A : ofeT} (x y : A) P : + IntoInternalEq P x y → IntoInternalEq ⎡P⎤ x y. +Proof. rewrite /IntoInternalEq=> ->. by rewrite embed_internal_eq. Qed. + +(* IntoExcept0 *) +Global Instance into_except_0_except_0 P : IntoExcept0 (◇ P) P. +Proof. by rewrite /IntoExcept0. Qed. +Global Instance into_except_0_later P : Timeless P → IntoExcept0 (▷ P) P. +Proof. by rewrite /IntoExcept0. Qed. +Global Instance into_except_0_later_if p P : Timeless P → IntoExcept0 (▷?p P) P. +Proof. rewrite /IntoExcept0. destruct p; auto using except_0_intro. Qed. + +Global Instance into_except_0_affinely P Q : + IntoExcept0 P Q → IntoExcept0 (<affine> P) (<affine> Q). +Proof. rewrite /IntoExcept0=> ->. by rewrite except_0_affinely_2. Qed. +Global Instance into_except_0_intuitionistically P Q : + IntoExcept0 P Q → IntoExcept0 (□ P) (□ Q). +Proof. rewrite /IntoExcept0=> ->. by rewrite except_0_intuitionistically_2. Qed. +Global Instance into_except_0_absorbingly P Q : + IntoExcept0 P Q → IntoExcept0 (<absorb> P) (<absorb> Q). +Proof. rewrite /IntoExcept0=> ->. by rewrite except_0_absorbingly. Qed. +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 (<pers> P) (<pers> Q). +Proof. rewrite /IntoExcept0=> ->. by rewrite except_0_persistently. Qed. +Global Instance into_except_0_embed `{SbiEmbed PROP PROP'} P Q : + IntoExcept0 P Q → IntoExcept0 ⎡P⎤ ⎡Q⎤. +Proof. rewrite /IntoExcept0=> ->. by rewrite embed_except_0. Qed. + +(* ElimModal *) +Global Instance elim_modal_timeless P Q : + IntoExcept0 P P' → IsExcept0 Q → ElimModal True P P' Q Q. +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_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 `{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 `{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 `{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. + +Global Instance elim_modal_embed_fupd_goal `{BiEmbedFUpd PROP PROP'} + φ E1 E2 E3 (P P' : PROP') (Q Q' : PROP) : + ElimModal φ P P' (|={E1,E3}=> ⎡Q⎤)%I (|={E2,E3}=> ⎡Q'⎤)%I → + ElimModal φ P P' ⎡|={E1,E3}=> Q⎤ ⎡|={E2,E3}=> Q'⎤. +Proof. by rewrite /ElimModal !embed_fupd. Qed. +Global Instance elim_modal_embed_fupd_hyp `{BiEmbedFUpd PROP PROP'} + φ E1 E2 (P : PROP) (P' Q Q' : PROP') : + ElimModal φ (|={E1,E2}=> ⎡P⎤)%I P' Q Q' → + ElimModal φ ⎡|={E1,E2}=> P⎤ P' Q Q'. +Proof. by rewrite /ElimModal embed_fupd. Qed. + +(* AddModal *) +(* High priority to add a ▷ rather than a ◇ when P is timeless. *) +Global Instance add_modal_later_except_0 P Q : + Timeless P → AddModal (▷ P) P (◇ Q) | 0. +Proof. + intros. rewrite /AddModal (except_0_intro (_ -∗ _)%I) (timeless P). + by rewrite -except_0_sep wand_elim_r except_0_idemp. +Qed. +Global Instance add_modal_later P Q : + Timeless P → AddModal (▷ P) P (▷ Q) | 0. +Proof. + intros. rewrite /AddModal (except_0_intro (_ -∗ _)%I) (timeless P). + by rewrite -except_0_sep wand_elim_r except_0_later. +Qed. +Global Instance add_modal_except_0 P Q : AddModal (◇ P) P (◇ Q) | 1. +Proof. + intros. rewrite /AddModal (except_0_intro (_ -∗ _)%I). + by rewrite -except_0_sep wand_elim_r except_0_idemp. +Qed. +Global Instance add_modal_except_0_later P Q : AddModal (◇ P) P (▷ Q) | 1. +Proof. + intros. rewrite /AddModal (except_0_intro (_ -∗ _)%I). + by rewrite -except_0_sep wand_elim_r except_0_later. +Qed. + +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 `{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. + +Global Instance add_modal_embed_fupd_goal `{BiEmbedFUpd PROP PROP'} + E1 E2 (P P' : PROP') (Q : PROP) : + AddModal P P' (|={E1,E2}=> ⎡Q⎤)%I → AddModal P P' ⎡|={E1,E2}=> Q⎤. +Proof. by rewrite /AddModal !embed_fupd. Qed. + +(* Frame *) +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' → Frame p (a ≡ b) ⎡P⎤ Q'. +Proof. rewrite /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. +Global Instance make_laterN_0 P : MakeLaterN 0 P P | 0. +Proof. by rewrite /MakeLaterN. Qed. +Global Instance make_laterN_1 P : MakeLaterN 1 P (▷ P) | 2. +Proof. by rewrite /MakeLaterN. Qed. +Global Instance make_laterN_default P : MakeLaterN n P (▷^n P) | 100. +Proof. by rewrite /MakeLaterN. Qed. + +Global Instance frame_later p R R' P Q Q' : + NoBackTrack (MaybeIntoLaterN true 1 R' R) → + Frame p R P Q → MakeLaterN 1 Q Q' → Frame p R' (▷ P) Q'. +Proof. + rewrite /Frame /MakeLaterN /MaybeIntoLaterN=>-[->] <- <-. + by rewrite later_intuitionistically_if_2 later_sep. +Qed. +Global Instance frame_laterN p n R R' P Q Q' : + NoBackTrack (MaybeIntoLaterN true n R' R) → + Frame p R P Q → MakeLaterN n Q Q' → Frame p R' (▷^n P) Q'. +Proof. + rewrite /Frame /MakeLaterN /MaybeIntoLaterN=>-[->] <- <-. + by rewrite laterN_intuitionistically_if_2 laterN_sep. +Qed. + +Global Instance frame_bupd `{BiBUpd PROP} p R P Q : + Frame p R P Q → Frame p R (|==> P) (|==> Q). +Proof. rewrite /Frame=><-. by rewrite bupd_frame_l. Qed. +Global Instance frame_fupd `{BiFUpd PROP} p E1 E2 R P Q : + Frame p R P Q → Frame p R (|={E1,E2}=> P) (|={E1,E2}=> Q). +Proof. rewrite /Frame=><-. by rewrite fupd_frame_l. Qed. + +Global Instance make_except_0_True : @KnownMakeExcept0 PROP True True. +Proof. by rewrite /KnownMakeExcept0 /MakeExcept0 except_0_True. Qed. +Global Instance make_except_0_default P : MakeExcept0 P (◇ P) | 100. +Proof. by rewrite /MakeExcept0. Qed. + +Global Instance frame_except_0 p R P Q Q' : + Frame p R P Q → MakeExcept0 Q Q' → Frame p R (◇ P) Q'. +Proof. + rewrite /Frame /MakeExcept0=><- <-. + by rewrite except_0_sep -(except_0_intro (□?p R)%I). +Qed. + +(* IntoLater *) +Global Instance into_laterN_0 only_head P : IntoLaterN only_head 0 P P. +Proof. by rewrite /IntoLaterN /MaybeIntoLaterN. Qed. +Global Instance into_laterN_later only_head n n' m' P Q lQ : + NatCancel n 1 n' m' → + (** If canceling has failed (i.e. [1 = m']), we should make progress deeper + into [P], as such, we continue with the [IntoLaterN] class, which is required + to make progress. If canceling has succeeded, we do not need to make further + progress, but there may still be a left-over (i.e. [n']) to cancel more deeply + into [P], as such, we continue with [MaybeIntoLaterN]. *) + TCIf (TCEq 1 m') (IntoLaterN only_head n' P Q) (MaybeIntoLaterN only_head n' P Q) → + MakeLaterN m' Q lQ → + IntoLaterN only_head n (▷ P) lQ | 2. +Proof. + rewrite /MakeLaterN /IntoLaterN /MaybeIntoLaterN /NatCancel. + move=> Hn [_ ->|->] <-; + by rewrite -later_laterN -laterN_plus -Hn Nat.add_comm. +Qed. +Global Instance into_laterN_laterN only_head n m n' m' P Q lQ : + NatCancel n m n' m' → + TCIf (TCEq m m') (IntoLaterN only_head n' P Q) (MaybeIntoLaterN only_head n' P Q) → + MakeLaterN m' Q lQ → + IntoLaterN only_head n (▷^m P) lQ | 1. +Proof. + rewrite /MakeLaterN /IntoLaterN /MaybeIntoLaterN /NatCancel. + move=> Hn [_ ->|->] <-; by rewrite -!laterN_plus -Hn Nat.add_comm. +Qed. + +Global Instance into_laterN_and_l n P1 P2 Q1 Q2 : + IntoLaterN false n P1 Q1 → MaybeIntoLaterN false n P2 Q2 → + IntoLaterN false n (P1 ∧ P2) (Q1 ∧ Q2) | 10. +Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> -> ->. by rewrite laterN_and. Qed. +Global Instance into_laterN_and_r n P P2 Q2 : + IntoLaterN false n P2 Q2 → IntoLaterN false n (P ∧ P2) (P ∧ Q2) | 11. +Proof. + rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_and -(laterN_intro _ P). +Qed. + +Global Instance into_laterN_forall {A} n (Φ Ψ : A → PROP) : + (∀ x, IntoLaterN false n (Φ x) (Ψ x)) → + IntoLaterN false n (∀ x, Φ x) (∀ x, Ψ x). +Proof. rewrite /IntoLaterN /MaybeIntoLaterN laterN_forall=> ?. by apply forall_mono. Qed. +Global Instance into_laterN_exist {A} n (Φ Ψ : A → PROP) : + (∀ x, IntoLaterN false n (Φ x) (Ψ x)) → + IntoLaterN false n (∃ x, Φ x) (∃ x, Ψ x). +Proof. rewrite /IntoLaterN /MaybeIntoLaterN -laterN_exist_2=> ?. by apply exist_mono. Qed. + +Global Instance into_laterN_or_l n P1 P2 Q1 Q2 : + IntoLaterN false n P1 Q1 → MaybeIntoLaterN false n P2 Q2 → + IntoLaterN false n (P1 ∨ P2) (Q1 ∨ Q2) | 10. +Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> -> ->. by rewrite laterN_or. Qed. +Global Instance into_laterN_or_r n P P2 Q2 : + IntoLaterN false n P2 Q2 → + IntoLaterN false n (P ∨ P2) (P ∨ Q2) | 11. +Proof. + rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_or -(laterN_intro _ P). +Qed. + +Global Instance into_later_affinely n P Q : + IntoLaterN false n P Q → IntoLaterN false n (<affine> P) (<affine> Q). +Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_affinely_2. Qed. +Global Instance into_later_intuitionistically n P Q : + IntoLaterN false n P Q → IntoLaterN false n (□ P) (□ Q). +Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_intuitionistically_2. Qed. +Global Instance into_later_absorbingly n P Q : + IntoLaterN false n P Q → IntoLaterN false n (<absorb> P) (<absorb> Q). +Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_absorbingly. Qed. +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 (<pers> P) (<pers> Q). +Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_persistently. Qed. +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 embed_laterN. Qed. + +Global Instance into_laterN_sep_l n P1 P2 Q1 Q2 : + IntoLaterN false n P1 Q1 → MaybeIntoLaterN false n P2 Q2 → + IntoLaterN false n (P1 ∗ P2) (Q1 ∗ Q2) | 10. +Proof. rewrite /IntoLaterN /MaybeIntoLaterN=> -> ->. by rewrite laterN_sep. Qed. +Global Instance into_laterN_sep_r n P P2 Q2 : + IntoLaterN false n P2 Q2 → + IntoLaterN false n (P ∗ P2) (P ∗ Q2) | 11. +Proof. + rewrite /IntoLaterN /MaybeIntoLaterN=> ->. by rewrite laterN_sep -(laterN_intro _ P). +Qed. + +Global Instance into_laterN_big_sepL n {A} (Φ Ψ : nat → A → PROP) (l: list A) : + (∀ x k, IntoLaterN false n (Φ k x) (Ψ k x)) → + IntoLaterN false n ([∗ list] k ↦ x ∈ l, Φ k x) ([∗ list] k ↦ x ∈ l, Ψ k x). +Proof. + rewrite /IntoLaterN /MaybeIntoLaterN=> ?. + rewrite big_opL_commute. by apply big_sepL_mono. +Qed. +Global Instance into_laterN_big_sepM n `{Countable K} {A} + (Φ Ψ : K → A → PROP) (m : gmap K A) : + (∀ x k, IntoLaterN false n (Φ k x) (Ψ k x)) → + IntoLaterN false n ([∗ map] k ↦ x ∈ m, Φ k x) ([∗ map] k ↦ x ∈ m, Ψ k x). +Proof. + rewrite /IntoLaterN /MaybeIntoLaterN=> ?. + rewrite big_opM_commute. by apply big_sepM_mono. +Qed. +Global Instance into_laterN_big_sepS n `{Countable A} + (Φ Ψ : A → PROP) (X : gset A) : + (∀ x, IntoLaterN false n (Φ x) (Ψ x)) → + IntoLaterN false n ([∗ set] x ∈ X, Φ x) ([∗ set] x ∈ X, Ψ x). +Proof. + rewrite /IntoLaterN /MaybeIntoLaterN=> ?. + rewrite big_opS_commute. by apply big_sepS_mono. +Qed. +Global Instance into_laterN_big_sepMS n `{Countable A} + (Φ Ψ : A → PROP) (X : gmultiset A) : + (∀ x, IntoLaterN false n (Φ x) (Ψ x)) → + IntoLaterN false n ([∗ mset] x ∈ X, Φ x) ([∗ mset] x ∈ X, Ψ x). +Proof. + rewrite /IntoLaterN /MaybeIntoLaterN=> ?. + rewrite big_opMS_commute. by apply big_sepMS_mono. +Qed. +End sbi_instances. diff --git a/theories/proofmode/monpred.v b/theories/proofmode/monpred.v index e7ae94493..d3dbe02ff 100644 --- a/theories/proofmode/monpred.v +++ b/theories/proofmode/monpred.v @@ -1,6 +1,6 @@ From iris.bi Require Export monpred. From iris.bi Require Import plainly. -From iris.proofmode Require Import tactics class_instances. +From iris.proofmode Require Import tactics modality_instances. Class MakeMonPredAt {I : biIndex} {PROP : bi} (i : I) (P : monPred I PROP) (𓟠: PROP) := diff --git a/theories/proofmode/tactics.v b/theories/proofmode/tactics.v index f4475fcfb..908b01fdb 100644 --- a/theories/proofmode/tactics.v +++ b/theories/proofmode/tactics.v @@ -3,7 +3,7 @@ From iris.proofmode Require Import base intro_patterns spec_patterns sel_pattern From iris.bi Require Export bi. From stdpp Require Import namespaces. From iris.proofmode Require Export classes notation. -From iris.proofmode Require Import class_instances. +From iris.proofmode Require Import class_instances_bi class_instances_sbi. From stdpp Require Import hlist pretty. Set Default Proof Using "Type". Export ident. -- GitLab