derived.v 46.6 KB
 Robbert Krebbers committed Oct 25, 2016 1 ``````From iris.base_logic Require Export primitive. `````` Ralf Jung committed Jan 05, 2017 2 ``````Set Default Proof Using "Type". `````` Robbert Krebbers committed Dec 13, 2016 3 ``````Import upred.uPred primitive.uPred. `````` Robbert Krebbers committed Oct 25, 2016 4 5 6 7 8 `````` Definition uPred_iff {M} (P Q : uPred M) : uPred M := ((P → Q) ∧ (Q → P))%I. Instance: Params (@uPred_iff) 1. Infix "↔" := uPred_iff : uPred_scope. `````` Robbert Krebbers committed Nov 27, 2016 9 10 11 12 13 14 15 16 17 18 ``````Definition uPred_laterN {M} (n : nat) (P : uPred M) : uPred M := Nat.iter n uPred_later P. Instance: Params (@uPred_laterN) 2. Notation "▷^ n P" := (uPred_laterN n P) (at level 20, n at level 9, P at level 20, format "▷^ n P") : uPred_scope. Notation "▷? p P" := (uPred_laterN (Nat.b2n p) P) (at level 20, p at level 9, P at level 20, format "▷? p P") : uPred_scope. `````` Robbert Krebbers committed Oct 25, 2017 19 ``````Definition uPred_persistently_if {M} (p : bool) (P : uPred M) : uPred M := `````` Robbert Krebbers committed Oct 25, 2016 20 `````` (if p then □ P else P)%I. `````` Robbert Krebbers committed Oct 25, 2017 21 22 23 ``````Instance: Params (@uPred_persistently_if) 2. Arguments uPred_persistently_if _ !_ _/. Notation "□? p P" := (uPred_persistently_if p P) `````` Robbert Krebbers committed Nov 27, 2016 24 `````` (at level 20, p at level 9, P at level 20, format "□? p P"). `````` Robbert Krebbers committed Oct 25, 2016 25 `````` `````` Robbert Krebbers committed Oct 25, 2016 26 27 ``````Definition uPred_except_0 {M} (P : uPred M) : uPred M := ▷ False ∨ P. Notation "◇ P" := (uPred_except_0 P) `````` Robbert Krebbers committed Oct 25, 2016 28 `````` (at level 20, right associativity) : uPred_scope. `````` Robbert Krebbers committed Oct 25, 2016 29 30 ``````Instance: Params (@uPred_except_0) 1. Typeclasses Opaque uPred_except_0. `````` Robbert Krebbers committed Oct 25, 2016 31 `````` `````` Robbert Krebbers committed Oct 25, 2017 32 ``````Class Timeless {M} (P : uPred M) := timelessP : ▷ P ⊢ ◇ P. `````` Robbert Krebbers committed Oct 25, 2016 33 ``````Arguments timelessP {_} _ {_}. `````` Robbert Krebbers committed Oct 25, 2017 34 35 ``````Hint Mode Timeless + ! : typeclass_instances. Instance: Params (@Timeless) 1. `````` Robbert Krebbers committed Oct 25, 2016 36 `````` `````` Robbert Krebbers committed Oct 25, 2017 37 38 39 40 ``````Class Persistent {M} (P : uPred M) := persistent : P ⊢ □ P. Arguments persistent {_} _ {_}. Hint Mode Persistent + ! : typeclass_instances. Instance: Params (@Persistent) 1. `````` Robbert Krebbers committed Oct 25, 2016 41 `````` `````` Robbert Krebbers committed Dec 13, 2016 42 ``````Module uPred. `````` Robbert Krebbers committed Oct 25, 2016 43 44 45 46 47 48 49 50 51 52 ``````Section derived. Context {M : ucmraT}. Implicit Types φ : Prop. Implicit Types P Q : uPred M. Implicit Types A : Type. Notation "P ⊢ Q" := (@uPred_entails M P%I Q%I). (* Force implicit argument M *) Notation "P ⊣⊢ Q" := (equiv (A:=uPred M) P%I Q%I). (* Force implicit argument M *) (* Derived logical stuff *) Lemma False_elim P : False ⊢ P. `````` Robbert Krebbers committed Nov 22, 2016 53 ``````Proof. by apply (pure_elim' False). Qed. `````` Robbert Krebbers committed Oct 25, 2016 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 ``````Lemma True_intro P : P ⊢ True. Proof. by apply pure_intro. Qed. Lemma and_elim_l' P Q R : (P ⊢ R) → P ∧ Q ⊢ R. Proof. by rewrite and_elim_l. Qed. Lemma and_elim_r' P Q R : (Q ⊢ R) → P ∧ Q ⊢ R. Proof. by rewrite and_elim_r. Qed. Lemma or_intro_l' P Q R : (P ⊢ Q) → P ⊢ Q ∨ R. Proof. intros ->; apply or_intro_l. Qed. Lemma or_intro_r' P Q R : (P ⊢ R) → P ⊢ Q ∨ R. Proof. intros ->; apply or_intro_r. Qed. Lemma exist_intro' {A} P (Ψ : A → uPred M) a : (P ⊢ Ψ a) → P ⊢ ∃ a, Ψ a. Proof. intros ->; apply exist_intro. Qed. Lemma forall_elim' {A} P (Ψ : A → uPred M) : (P ⊢ ∀ a, Ψ a) → ∀ a, P ⊢ Ψ a. Proof. move=> HP a. by rewrite HP forall_elim. Qed. Hint Resolve pure_intro. Hint Resolve or_elim or_intro_l' or_intro_r'. Hint Resolve and_intro and_elim_l' and_elim_r'. Hint Immediate True_intro False_elim. Lemma impl_intro_l P Q R : (Q ∧ P ⊢ R) → P ⊢ Q → R. Proof. intros HR; apply impl_intro_r; rewrite -HR; auto. Qed. Lemma impl_elim_l P Q : (P → Q) ∧ P ⊢ Q. Proof. apply impl_elim with P; auto. Qed. Lemma impl_elim_r P Q : P ∧ (P → Q) ⊢ Q. Proof. apply impl_elim with P; auto. Qed. Lemma impl_elim_l' P Q R : (P ⊢ Q → R) → P ∧ Q ⊢ R. Proof. intros; apply impl_elim with Q; auto. Qed. Lemma impl_elim_r' P Q R : (Q ⊢ P → R) → P ∧ Q ⊢ R. Proof. intros; apply impl_elim with P; auto. Qed. `````` 85 ``````Lemma impl_entails P Q : (P → Q)%I → P ⊢ Q. `````` Robbert Krebbers committed Oct 25, 2016 86 ``````Proof. intros HPQ; apply impl_elim with P; rewrite -?HPQ; auto. Qed. `````` 87 88 ``````Lemma entails_impl P Q : (P ⊢ Q) → (P → Q)%I. Proof. intro. apply impl_intro_l. auto. Qed. `````` Robbert Krebbers committed Oct 25, 2016 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 `````` Lemma and_mono P P' Q Q' : (P ⊢ Q) → (P' ⊢ Q') → P ∧ P' ⊢ Q ∧ Q'. Proof. auto. Qed. Lemma and_mono_l P P' Q : (P ⊢ Q) → P ∧ P' ⊢ Q ∧ P'. Proof. by intros; apply and_mono. Qed. Lemma and_mono_r P P' Q' : (P' ⊢ Q') → P ∧ P' ⊢ P ∧ Q'. Proof. by apply and_mono. Qed. Lemma or_mono P P' Q Q' : (P ⊢ Q) → (P' ⊢ Q') → P ∨ P' ⊢ Q ∨ Q'. Proof. auto. Qed. Lemma or_mono_l P P' Q : (P ⊢ Q) → P ∨ P' ⊢ Q ∨ P'. Proof. by intros; apply or_mono. Qed. Lemma or_mono_r P P' Q' : (P' ⊢ Q') → P ∨ P' ⊢ P ∨ Q'. Proof. by apply or_mono. Qed. Lemma impl_mono P P' Q Q' : (Q ⊢ P) → (P' ⊢ Q') → (P → P') ⊢ Q → Q'. Proof. intros HP HQ'; apply impl_intro_l; rewrite -HQ'. apply impl_elim with P; eauto. Qed. Lemma forall_mono {A} (Φ Ψ : A → uPred M) : (∀ a, Φ a ⊢ Ψ a) → (∀ a, Φ a) ⊢ ∀ a, Ψ a. Proof. intros HP. apply forall_intro=> a; rewrite -(HP a); apply forall_elim. Qed. Lemma exist_mono {A} (Φ Ψ : A → uPred M) : (∀ a, Φ a ⊢ Ψ a) → (∃ a, Φ a) ⊢ ∃ a, Ψ a. Proof. intros HΦ. apply exist_elim=> a; rewrite (HΦ a); apply exist_intro. Qed. Global Instance and_mono' : Proper ((⊢) ==> (⊢) ==> (⊢)) (@uPred_and M). Proof. by intros P P' HP Q Q' HQ; apply and_mono. Qed. Global Instance and_flip_mono' : Proper (flip (⊢) ==> flip (⊢) ==> flip (⊢)) (@uPred_and M). Proof. by intros P P' HP Q Q' HQ; apply and_mono. Qed. Global Instance or_mono' : Proper ((⊢) ==> (⊢) ==> (⊢)) (@uPred_or M). Proof. by intros P P' HP Q Q' HQ; apply or_mono. Qed. Global Instance or_flip_mono' : Proper (flip (⊢) ==> flip (⊢) ==> flip (⊢)) (@uPred_or M). Proof. by intros P P' HP Q Q' HQ; apply or_mono. Qed. Global Instance impl_mono' : Proper (flip (⊢) ==> (⊢) ==> (⊢)) (@uPred_impl M). Proof. by intros P P' HP Q Q' HQ; apply impl_mono. Qed. `````` Robbert Krebbers committed Oct 28, 2016 131 132 133 ``````Global Instance impl_flip_mono' : Proper ((⊢) ==> flip (⊢) ==> flip (⊢)) (@uPred_impl M). Proof. by intros P P' HP Q Q' HQ; apply impl_mono. Qed. `````` Robbert Krebbers committed Oct 25, 2016 134 135 136 ``````Global Instance forall_mono' A : Proper (pointwise_relation _ (⊢) ==> (⊢)) (@uPred_forall M A). Proof. intros P1 P2; apply forall_mono. Qed. `````` Robbert Krebbers committed Oct 28, 2016 137 138 139 ``````Global Instance forall_flip_mono' A : Proper (pointwise_relation _ (flip (⊢)) ==> flip (⊢)) (@uPred_forall M A). Proof. intros P1 P2; apply forall_mono. Qed. `````` Robbert Krebbers committed Oct 25, 2016 140 ``````Global Instance exist_mono' A : `````` Jacques-Henri Jourdan committed Aug 07, 2017 141 `````` Proper (pointwise_relation _ (⊢) ==> (⊢)) (@uPred_exist M A). `````` Robbert Krebbers committed Oct 28, 2016 142 143 144 ``````Proof. intros P1 P2; apply exist_mono. Qed. Global Instance exist_flip_mono' A : Proper (pointwise_relation _ (flip (⊢)) ==> flip (⊢)) (@uPred_exist M A). `````` Robbert Krebbers committed Oct 25, 2016 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 ``````Proof. intros P1 P2; apply exist_mono. Qed. Global Instance and_idem : IdemP (⊣⊢) (@uPred_and M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance or_idem : IdemP (⊣⊢) (@uPred_or M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance and_comm : Comm (⊣⊢) (@uPred_and M). Proof. intros P Q; apply (anti_symm (⊢)); auto. Qed. Global Instance True_and : LeftId (⊣⊢) True%I (@uPred_and M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance and_True : RightId (⊣⊢) True%I (@uPred_and M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance False_and : LeftAbsorb (⊣⊢) False%I (@uPred_and M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance and_False : RightAbsorb (⊣⊢) False%I (@uPred_and M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance True_or : LeftAbsorb (⊣⊢) True%I (@uPred_or M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance or_True : RightAbsorb (⊣⊢) True%I (@uPred_or M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance False_or : LeftId (⊣⊢) False%I (@uPred_or M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance or_False : RightId (⊣⊢) False%I (@uPred_or M). Proof. intros P; apply (anti_symm (⊢)); auto. Qed. Global Instance and_assoc : Assoc (⊣⊢) (@uPred_and M). Proof. intros P Q R; apply (anti_symm (⊢)); auto. Qed. Global Instance or_comm : Comm (⊣⊢) (@uPred_or M). Proof. intros P Q; apply (anti_symm (⊢)); auto. Qed. Global Instance or_assoc : Assoc (⊣⊢) (@uPred_or M). Proof. intros P Q R; apply (anti_symm (⊢)); auto. Qed. Global Instance True_impl : LeftId (⊣⊢) True%I (@uPred_impl M). Proof. intros P; apply (anti_symm (⊢)). - by rewrite -(left_id True%I uPred_and (_ → _)%I) impl_elim_r. - by apply impl_intro_l; rewrite left_id. Qed. `````` Robbert Krebbers committed Nov 21, 2016 181 182 183 184 185 ``````Lemma False_impl P : (False → P) ⊣⊢ True. Proof. apply (anti_symm (⊢)); [by auto|]. apply impl_intro_l. rewrite left_absorb. auto. Qed. `````` Robbert Krebbers committed Oct 25, 2016 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 `````` Lemma exists_impl_forall {A} P (Ψ : A → uPred M) : ((∃ x : A, Ψ x) → P) ⊣⊢ ∀ x : A, Ψ x → P. Proof. apply equiv_spec; split. - apply forall_intro=>x. by rewrite -exist_intro. - apply impl_intro_r, impl_elim_r', exist_elim=>x. apply impl_intro_r. by rewrite (forall_elim x) impl_elim_r. Qed. Lemma or_and_l P Q R : P ∨ Q ∧ R ⊣⊢ (P ∨ Q) ∧ (P ∨ R). Proof. apply (anti_symm (⊢)); first auto. do 2 (apply impl_elim_l', or_elim; apply impl_intro_l); auto. Qed. Lemma or_and_r P Q R : P ∧ Q ∨ R ⊣⊢ (P ∨ R) ∧ (Q ∨ R). Proof. by rewrite -!(comm _ R) or_and_l. Qed. Lemma and_or_l P Q R : P ∧ (Q ∨ R) ⊣⊢ P ∧ Q ∨ P ∧ R. Proof. apply (anti_symm (⊢)); last auto. apply impl_elim_r', or_elim; apply impl_intro_l; auto. Qed. Lemma and_or_r P Q R : (P ∨ Q) ∧ R ⊣⊢ P ∧ R ∨ Q ∧ R. Proof. by rewrite -!(comm _ R) and_or_l. Qed. Lemma and_exist_l {A} P (Ψ : A → uPred M) : P ∧ (∃ a, Ψ a) ⊣⊢ ∃ a, P ∧ Ψ a. Proof. apply (anti_symm (⊢)). - apply impl_elim_r'. apply exist_elim=>a. apply impl_intro_l. by rewrite -(exist_intro a). - apply exist_elim=>a. apply and_intro; first by rewrite and_elim_l. by rewrite -(exist_intro a) and_elim_r. Qed. Lemma and_exist_r {A} P (Φ: A → uPred M) : (∃ a, Φ a) ∧ P ⊣⊢ ∃ a, Φ a ∧ P. Proof. rewrite -(comm _ P) and_exist_l. apply exist_proper=>a. by rewrite comm. Qed. `````` Robbert Krebbers committed Nov 17, 2016 222 223 224 225 226 227 228 ``````Lemma or_exist {A} (Φ Ψ : A → uPred M) : (∃ a, Φ a ∨ Ψ a) ⊣⊢ (∃ a, Φ a) ∨ (∃ a, Ψ a). Proof. apply (anti_symm (⊢)). - apply exist_elim=> a. by rewrite -!(exist_intro a). - apply or_elim; apply exist_elim=> a; rewrite -(exist_intro a); auto. Qed. `````` Robbert Krebbers committed Oct 25, 2016 229 `````` `````` Ralf Jung committed Nov 22, 2016 230 ``````Lemma pure_elim φ Q R : (Q ⊢ ⌜φ⌝) → (φ → Q ⊢ R) → Q ⊢ R. `````` Robbert Krebbers committed Nov 22, 2016 231 232 233 234 ``````Proof. intros HQ HQR. rewrite -(idemp uPred_and Q) {1}HQ. apply impl_elim_l', pure_elim'=> ?. by apply entails_impl, HQR. Qed. `````` Ralf Jung committed Nov 22, 2016 235 ``````Lemma pure_mono φ1 φ2 : (φ1 → φ2) → ⌜φ1⌝ ⊢ ⌜φ2⌝. `````` Robbert Krebbers committed Oct 25, 2016 236 237 238 ``````Proof. intros; apply pure_elim with φ1; eauto. Qed. Global Instance pure_mono' : Proper (impl ==> (⊢)) (@uPred_pure M). Proof. intros φ1 φ2; apply pure_mono. Qed. `````` Jacques-Henri Jourdan committed Aug 07, 2017 239 240 ``````Global Instance pure_flip_mono : Proper (flip impl ==> flip (⊢)) (@uPred_pure M). Proof. intros φ1 φ2; apply pure_mono. Qed. `````` Ralf Jung committed Nov 22, 2016 241 ``````Lemma pure_iff φ1 φ2 : (φ1 ↔ φ2) → ⌜φ1⌝ ⊣⊢ ⌜φ2⌝. `````` Robbert Krebbers committed Oct 25, 2016 242 ``````Proof. intros [??]; apply (anti_symm _); auto using pure_mono. Qed. `````` Ralf Jung committed Nov 22, 2016 243 ``````Lemma pure_intro_l φ Q R : φ → (⌜φ⌝ ∧ Q ⊢ R) → Q ⊢ R. `````` Robbert Krebbers committed Oct 25, 2016 244 ``````Proof. intros ? <-; auto using pure_intro. Qed. `````` Ralf Jung committed Nov 22, 2016 245 ``````Lemma pure_intro_r φ Q R : φ → (Q ∧ ⌜φ⌝ ⊢ R) → Q ⊢ R. `````` Robbert Krebbers committed Oct 25, 2016 246 ``````Proof. intros ? <-; auto. Qed. `````` Ralf Jung committed Nov 22, 2016 247 ``````Lemma pure_intro_impl φ Q R : φ → (Q ⊢ ⌜φ⌝ → R) → Q ⊢ R. `````` Robbert Krebbers committed Oct 25, 2016 248 ``````Proof. intros ? ->. eauto using pure_intro_l, impl_elim_r. Qed. `````` Ralf Jung committed Nov 22, 2016 249 ``````Lemma pure_elim_l φ Q R : (φ → Q ⊢ R) → ⌜φ⌝ ∧ Q ⊢ R. `````` Robbert Krebbers committed Oct 25, 2016 250 ``````Proof. intros; apply pure_elim with φ; eauto. Qed. `````` Ralf Jung committed Nov 22, 2016 251 ``````Lemma pure_elim_r φ Q R : (φ → Q ⊢ R) → Q ∧ ⌜φ⌝ ⊢ R. `````` Robbert Krebbers committed Oct 25, 2016 252 ``````Proof. intros; apply pure_elim with φ; eauto. Qed. `````` Robbert Krebbers committed Nov 21, 2016 253 `````` `````` Ralf Jung committed Nov 22, 2016 254 ``````Lemma pure_True (φ : Prop) : φ → ⌜φ⌝ ⊣⊢ True. `````` Robbert Krebbers committed Oct 25, 2016 255 ``````Proof. intros; apply (anti_symm _); auto. Qed. `````` Ralf Jung committed Nov 22, 2016 256 ``````Lemma pure_False (φ : Prop) : ¬φ → ⌜φ⌝ ⊣⊢ False. `````` Robbert Krebbers committed Nov 21, 2016 257 ``````Proof. intros; apply (anti_symm _); eauto using pure_elim. Qed. `````` Robbert Krebbers committed Oct 25, 2016 258 `````` `````` Ralf Jung committed Nov 22, 2016 259 ``````Lemma pure_and φ1 φ2 : ⌜φ1 ∧ φ2⌝ ⊣⊢ ⌜φ1⌝ ∧ ⌜φ2⌝. `````` Robbert Krebbers committed Oct 25, 2016 260 261 262 263 264 ``````Proof. apply (anti_symm _). - eapply pure_elim=> // -[??]; auto. - eapply (pure_elim φ1); [auto|]=> ?. eapply (pure_elim φ2); auto. Qed. `````` Ralf Jung committed Nov 22, 2016 265 ``````Lemma pure_or φ1 φ2 : ⌜φ1 ∨ φ2⌝ ⊣⊢ ⌜φ1⌝ ∨ ⌜φ2⌝. `````` Robbert Krebbers committed Oct 25, 2016 266 267 268 269 270 ``````Proof. apply (anti_symm _). - eapply pure_elim=> // -[?|?]; auto. - apply or_elim; eapply pure_elim; eauto. Qed. `````` Ralf Jung committed Nov 22, 2016 271 ``````Lemma pure_impl φ1 φ2 : ⌜φ1 → φ2⌝ ⊣⊢ (⌜φ1⌝ → ⌜φ2⌝). `````` Robbert Krebbers committed Oct 25, 2016 272 273 274 275 ``````Proof. apply (anti_symm _). - apply impl_intro_l. rewrite -pure_and. apply pure_mono. naive_solver. - rewrite -pure_forall_2. apply forall_intro=> ?. `````` Robbert Krebbers committed Nov 21, 2016 276 `````` by rewrite -(left_id True uPred_and (_→_))%I (pure_True φ1) // impl_elim_r. `````` Robbert Krebbers committed Oct 25, 2016 277 ``````Qed. `````` Ralf Jung committed Nov 22, 2016 278 ``````Lemma pure_forall {A} (φ : A → Prop) : ⌜∀ x, φ x⌝ ⊣⊢ ∀ x, ⌜φ x⌝. `````` Robbert Krebbers committed Oct 25, 2016 279 280 281 282 ``````Proof. apply (anti_symm _); auto using pure_forall_2. apply forall_intro=> x. eauto using pure_mono. Qed. `````` Ralf Jung committed Nov 22, 2016 283 ``````Lemma pure_exist {A} (φ : A → Prop) : ⌜∃ x, φ x⌝ ⊣⊢ ∃ x, ⌜φ x⌝. `````` Robbert Krebbers committed Oct 25, 2016 284 285 286 287 288 289 ``````Proof. apply (anti_symm _). - eapply pure_elim=> // -[x ?]. rewrite -(exist_intro x); auto. - apply exist_elim=> x. eauto using pure_mono. Qed. `````` Ralf Jung committed Nov 22, 2016 290 ``````Lemma internal_eq_refl' {A : ofeT} (a : A) P : P ⊢ a ≡ a. `````` Robbert Krebbers committed Oct 25, 2016 291 292 ``````Proof. rewrite (True_intro P). apply internal_eq_refl. Qed. Hint Resolve internal_eq_refl'. `````` Ralf Jung committed Nov 22, 2016 293 ``````Lemma equiv_internal_eq {A : ofeT} P (a b : A) : a ≡ b → P ⊢ a ≡ b. `````` Robbert Krebbers committed Oct 25, 2016 294 ``````Proof. by intros ->. Qed. `````` Ralf Jung committed Nov 22, 2016 295 ``````Lemma internal_eq_sym {A : ofeT} (a b : A) : a ≡ b ⊢ b ≡ a. `````` Robbert Krebbers committed Oct 25, 2016 296 ``````Proof. apply (internal_eq_rewrite a b (λ b, b ≡ a)%I); auto. solve_proper. Qed. `````` Ralf Jung committed Dec 05, 2016 297 298 299 ``````Lemma internal_eq_rewrite_contractive {A : ofeT} a b (Ψ : A → uPred M) P {HΨ : Contractive Ψ} : (P ⊢ ▷ (a ≡ b)) → (P ⊢ Ψ a) → P ⊢ Ψ b. Proof. `````` Robbert Krebbers committed Dec 05, 2016 300 301 `````` move: HΨ=> /contractiveI HΨ Heq ?. apply (internal_eq_rewrite (Ψ a) (Ψ b) id _)=>//=. by rewrite -HΨ. `````` Ralf Jung committed Dec 05, 2016 302 ``````Qed. `````` Robbert Krebbers committed Oct 25, 2016 303 `````` `````` Ralf Jung committed Nov 22, 2016 304 ``````Lemma pure_impl_forall φ P : (⌜φ⌝ → P) ⊣⊢ (∀ _ : φ, P). `````` Robbert Krebbers committed Nov 20, 2016 305 306 ``````Proof. apply (anti_symm _). `````` Robbert Krebbers committed Nov 21, 2016 307 `````` - apply forall_intro=> ?. by rewrite pure_True // left_id. `````` Robbert Krebbers committed Nov 20, 2016 308 309 `````` - apply impl_intro_l, pure_elim_l=> Hφ. by rewrite (forall_elim Hφ). Qed. `````` Ralf Jung committed Nov 22, 2016 310 ``````Lemma pure_alt φ : ⌜φ⌝ ⊣⊢ ∃ _ : φ, True. `````` Robbert Krebbers committed Oct 25, 2016 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 ``````Proof. apply (anti_symm _). - eapply pure_elim; eauto=> H. rewrite -(exist_intro H); auto. - by apply exist_elim, pure_intro. Qed. Lemma and_alt P Q : P ∧ Q ⊣⊢ ∀ b : bool, if b then P else Q. Proof. apply (anti_symm _); first apply forall_intro=> -[]; auto. apply and_intro. by rewrite (forall_elim true). by rewrite (forall_elim false). Qed. Lemma or_alt P Q : P ∨ Q ⊣⊢ ∃ b : bool, if b then P else Q. Proof. apply (anti_symm _); last apply exist_elim=> -[]; auto. apply or_elim. by rewrite -(exist_intro true). by rewrite -(exist_intro false). Qed. `````` Ralf Jung committed Jan 27, 2017 327 ``````Global Instance iff_ne : NonExpansive2 (@uPred_iff M). `````` Robbert Krebbers committed Oct 25, 2016 328 329 330 331 332 333 ``````Proof. unfold uPred_iff; solve_proper. Qed. Global Instance iff_proper : Proper ((⊣⊢) ==> (⊣⊢) ==> (⊣⊢)) (@uPred_iff M) := ne_proper_2 _. Lemma iff_refl Q P : Q ⊢ P ↔ P. Proof. rewrite /uPred_iff; apply and_intro; apply impl_intro_l; auto. Qed. `````` 334 ``````Lemma iff_equiv P Q : (P ↔ Q)%I → (P ⊣⊢ Q). `````` Robbert Krebbers committed Oct 25, 2016 335 336 ``````Proof. intros HPQ; apply (anti_symm (⊢)); `````` 337 `````` apply impl_entails; rewrite /uPred_valid HPQ /uPred_iff; auto. `````` Robbert Krebbers committed Oct 25, 2016 338 ``````Qed. `````` 339 ``````Lemma equiv_iff P Q : (P ⊣⊢ Q) → (P ↔ Q)%I. `````` Robbert Krebbers committed Oct 25, 2016 340 ``````Proof. intros ->; apply iff_refl. Qed. `````` Robbert Krebbers committed Oct 25, 2016 341 ``````Lemma internal_eq_iff P Q : P ≡ Q ⊢ P ↔ Q. `````` Robbert Krebbers committed Oct 25, 2016 342 ``````Proof. `````` Robbert Krebbers committed Oct 25, 2016 343 344 `````` apply (internal_eq_rewrite P Q (λ Q, P ↔ Q))%I; first solve_proper; auto using iff_refl. `````` Robbert Krebbers committed Oct 25, 2016 345 346 347 348 ``````Qed. (* Derived BI Stuff *) Hint Resolve sep_mono. `````` Robbert Krebbers committed Nov 03, 2016 349 ``````Lemma sep_mono_l P P' Q : (P ⊢ Q) → P ∗ P' ⊢ Q ∗ P'. `````` Robbert Krebbers committed Oct 25, 2016 350 ``````Proof. by intros; apply sep_mono. Qed. `````` Robbert Krebbers committed Nov 03, 2016 351 ``````Lemma sep_mono_r P P' Q' : (P' ⊢ Q') → P ∗ P' ⊢ P ∗ Q'. `````` Robbert Krebbers committed Oct 25, 2016 352 353 354 355 356 357 ``````Proof. by apply sep_mono. Qed. Global Instance sep_mono' : Proper ((⊢) ==> (⊢) ==> (⊢)) (@uPred_sep M). Proof. by intros P P' HP Q Q' HQ; apply sep_mono. Qed. Global Instance sep_flip_mono' : Proper (flip (⊢) ==> flip (⊢) ==> flip (⊢)) (@uPred_sep M). Proof. by intros P P' HP Q Q' HQ; apply sep_mono. Qed. `````` Robbert Krebbers committed Nov 03, 2016 358 ``````Lemma wand_mono P P' Q Q' : (Q ⊢ P) → (P' ⊢ Q') → (P -∗ P') ⊢ Q -∗ Q'. `````` Robbert Krebbers committed Oct 25, 2016 359 360 361 362 363 ``````Proof. intros HP HQ; apply wand_intro_r. rewrite HP -HQ. by apply wand_elim_l'. Qed. Global Instance wand_mono' : Proper (flip (⊢) ==> (⊢) ==> (⊢)) (@uPred_wand M). Proof. by intros P P' HP Q Q' HQ; apply wand_mono. Qed. `````` Robbert Krebbers committed Oct 28, 2016 364 365 366 ``````Global Instance wand_flip_mono' : Proper ((⊢) ==> flip (⊢) ==> flip (⊢)) (@uPred_wand M). Proof. by intros P P' HP Q Q' HQ; apply wand_mono. Qed. `````` Robbert Krebbers committed Oct 25, 2016 367 368 369 370 371 372 373 374 375 376 377 378 `````` Global Instance sep_comm : Comm (⊣⊢) (@uPred_sep M). Proof. intros P Q; apply (anti_symm _); auto using sep_comm'. Qed. Global Instance sep_assoc : Assoc (⊣⊢) (@uPred_sep M). Proof. intros P Q R; apply (anti_symm _); auto using sep_assoc'. by rewrite !(comm _ P) !(comm _ _ R) sep_assoc'. Qed. Global Instance True_sep : LeftId (⊣⊢) True%I (@uPred_sep M). Proof. intros P; apply (anti_symm _); auto using True_sep_1, True_sep_2. Qed. Global Instance sep_True : RightId (⊣⊢) True%I (@uPred_sep M). Proof. by intros P; rewrite comm left_id. Qed. `````` Robbert Krebbers committed Nov 03, 2016 379 ``````Lemma sep_elim_l P Q : P ∗ Q ⊢ P. `````` Robbert Krebbers committed Oct 25, 2016 380 ``````Proof. by rewrite (True_intro Q) right_id. Qed. `````` Robbert Krebbers committed Nov 03, 2016 381 382 383 ``````Lemma sep_elim_r P Q : P ∗ Q ⊢ Q. Proof. by rewrite (comm (∗))%I; apply sep_elim_l. Qed. Lemma sep_elim_l' P Q R : (P ⊢ R) → P ∗ Q ⊢ R. `````` Robbert Krebbers committed Oct 25, 2016 384 ``````Proof. intros ->; apply sep_elim_l. Qed. `````` Robbert Krebbers committed Nov 03, 2016 385 ``````Lemma sep_elim_r' P Q R : (Q ⊢ R) → P ∗ Q ⊢ R. `````` Robbert Krebbers committed Oct 25, 2016 386 387 ``````Proof. intros ->; apply sep_elim_r. Qed. Hint Resolve sep_elim_l' sep_elim_r'. `````` 388 ``````Lemma sep_intro_True_l P Q R : P%I → (R ⊢ Q) → R ⊢ P ∗ Q. `````` Robbert Krebbers committed Oct 25, 2016 389 ``````Proof. by intros; rewrite -(left_id True%I uPred_sep R); apply sep_mono. Qed. `````` 390 ``````Lemma sep_intro_True_r P Q R : (R ⊢ P) → Q%I → R ⊢ P ∗ Q. `````` Robbert Krebbers committed Oct 25, 2016 391 ``````Proof. by intros; rewrite -(right_id True%I uPred_sep R); apply sep_mono. Qed. `````` 392 ``````Lemma sep_elim_True_l P Q R : P → (P ∗ R ⊢ Q) → R ⊢ Q. `````` Robbert Krebbers committed Oct 25, 2016 393 ``````Proof. by intros HP; rewrite -HP left_id. Qed. `````` 394 ``````Lemma sep_elim_True_r P Q R : P → (R ∗ P ⊢ Q) → R ⊢ Q. `````` Robbert Krebbers committed Oct 25, 2016 395 ``````Proof. by intros HP; rewrite -HP right_id. Qed. `````` Robbert Krebbers committed Nov 03, 2016 396 ``````Lemma wand_intro_l P Q R : (Q ∗ P ⊢ R) → P ⊢ Q -∗ R. `````` Robbert Krebbers committed Oct 25, 2016 397 ``````Proof. rewrite comm; apply wand_intro_r. Qed. `````` Robbert Krebbers committed Nov 03, 2016 398 ``````Lemma wand_elim_l P Q : (P -∗ Q) ∗ P ⊢ Q. `````` Robbert Krebbers committed Oct 25, 2016 399 ``````Proof. by apply wand_elim_l'. Qed. `````` Robbert Krebbers committed Nov 03, 2016 400 ``````Lemma wand_elim_r P Q : P ∗ (P -∗ Q) ⊢ Q. `````` Robbert Krebbers committed Oct 25, 2016 401 ``````Proof. rewrite (comm _ P); apply wand_elim_l. Qed. `````` Robbert Krebbers committed Nov 03, 2016 402 ``````Lemma wand_elim_r' P Q R : (Q ⊢ P -∗ R) → P ∗ Q ⊢ R. `````` Robbert Krebbers committed Oct 25, 2016 403 ``````Proof. intros ->; apply wand_elim_r. Qed. `````` Robbert Krebbers committed Nov 03, 2016 404 ``````Lemma wand_apply P Q R S : (P ⊢ Q -∗ R) → (S ⊢ P ∗ Q) → S ⊢ R. `````` Ralf Jung committed Nov 01, 2016 405 ``````Proof. intros HR%wand_elim_l' HQ. by rewrite HQ. Qed. `````` Robbert Krebbers committed Nov 03, 2016 406 ``````Lemma wand_frame_l P Q R : (Q -∗ R) ⊢ P ∗ Q -∗ P ∗ R. `````` Robbert Krebbers committed Oct 25, 2016 407 ``````Proof. apply wand_intro_l. rewrite -assoc. apply sep_mono_r, wand_elim_r. Qed. `````` Robbert Krebbers committed Nov 03, 2016 408 ``````Lemma wand_frame_r P Q R : (Q -∗ R) ⊢ Q ∗ P -∗ R ∗ P. `````` Robbert Krebbers committed Oct 25, 2016 409 ``````Proof. `````` Robbert Krebbers committed Nov 03, 2016 410 `````` apply wand_intro_l. rewrite ![(_ ∗ P)%I]comm -assoc. `````` Robbert Krebbers committed Oct 25, 2016 411 412 `````` apply sep_mono_r, wand_elim_r. Qed. `````` Robbert Krebbers committed Nov 03, 2016 413 ``````Lemma wand_diag P : (P -∗ P) ⊣⊢ True. `````` Robbert Krebbers committed Oct 25, 2016 414 ``````Proof. apply (anti_symm _); auto. apply wand_intro_l; by rewrite right_id. Qed. `````` Robbert Krebbers committed Nov 03, 2016 415 ``````Lemma wand_True P : (True -∗ P) ⊣⊢ P. `````` Robbert Krebbers committed Oct 25, 2016 416 417 ``````Proof. apply (anti_symm _); last by auto using wand_intro_l. `````` 418 `````` eapply sep_elim_True_l; last by apply wand_elim_r. done. `````` Robbert Krebbers committed Oct 25, 2016 419 ``````Qed. `````` 420 ``````Lemma wand_entails P Q : (P -∗ Q)%I → P ⊢ Q. `````` Robbert Krebbers committed Oct 25, 2016 421 422 423 ``````Proof. intros HPQ. eapply sep_elim_True_r; first exact: HPQ. by rewrite wand_elim_r. Qed. `````` 424 425 ``````Lemma entails_wand P Q : (P ⊢ Q) → (P -∗ Q)%I. Proof. intro. apply wand_intro_l. auto. Qed. `````` Robbert Krebbers committed Nov 03, 2016 426 ``````Lemma wand_curry P Q R : (P -∗ Q -∗ R) ⊣⊢ (P ∗ Q -∗ R). `````` Robbert Krebbers committed Oct 25, 2016 427 428 429 430 431 432 ``````Proof. apply (anti_symm _). - apply wand_intro_l. by rewrite (comm _ P) -assoc !wand_elim_r. - do 2 apply wand_intro_l. by rewrite assoc (comm _ Q) wand_elim_r. Qed. `````` Robbert Krebbers committed Nov 03, 2016 433 ``````Lemma sep_and P Q : (P ∗ Q) ⊢ (P ∧ Q). `````` Robbert Krebbers committed Oct 25, 2016 434 ``````Proof. auto. Qed. `````` Robbert Krebbers committed Nov 03, 2016 435 ``````Lemma impl_wand P Q : (P → Q) ⊢ P -∗ Q. `````` Robbert Krebbers committed Oct 25, 2016 436 ``````Proof. apply wand_intro_r, impl_elim with P; auto. Qed. `````` Ralf Jung committed Nov 22, 2016 437 ``````Lemma pure_elim_sep_l φ Q R : (φ → Q ⊢ R) → ⌜φ⌝ ∗ Q ⊢ R. `````` Robbert Krebbers committed Oct 25, 2016 438 ``````Proof. intros; apply pure_elim with φ; eauto. Qed. `````` Ralf Jung committed Nov 22, 2016 439 ``````Lemma pure_elim_sep_r φ Q R : (φ → Q ⊢ R) → Q ∗ ⌜φ⌝ ⊢ R. `````` Robbert Krebbers committed Oct 25, 2016 440 441 442 443 444 445 446 ``````Proof. intros; apply pure_elim with φ; eauto. Qed. Global Instance sep_False : LeftAbsorb (⊣⊢) False%I (@uPred_sep M). Proof. intros P; apply (anti_symm _); auto. Qed. Global Instance False_sep : RightAbsorb (⊣⊢) False%I (@uPred_sep M). Proof. intros P; apply (anti_symm _); auto. Qed. `````` Jacques-Henri Jourdan committed Dec 23, 2016 447 ``````Lemma entails_equiv_and P Q : (P ⊣⊢ Q ∧ P) ↔ (P ⊢ Q). `````` Robbert Krebbers committed Dec 27, 2016 448 ``````Proof. split. by intros ->; auto. intros; apply (anti_symm _); auto. Qed. `````` Robbert Krebbers committed Nov 03, 2016 449 ``````Lemma sep_and_l P Q R : P ∗ (Q ∧ R) ⊢ (P ∗ Q) ∧ (P ∗ R). `````` Robbert Krebbers committed Oct 25, 2016 450 ``````Proof. auto. Qed. `````` Robbert Krebbers committed Nov 03, 2016 451 ``````Lemma sep_and_r P Q R : (P ∧ Q) ∗ R ⊢ (P ∗ R) ∧ (Q ∗ R). `````` Robbert Krebbers committed Oct 25, 2016 452 ``````Proof. auto. Qed. `````` Robbert Krebbers committed Nov 03, 2016 453 ``````Lemma sep_or_l P Q R : P ∗ (Q ∨ R) ⊣⊢ (P ∗ Q) ∨ (P ∗ R). `````` Robbert Krebbers committed Oct 25, 2016 454 455 456 457 ``````Proof. apply (anti_symm (⊢)); last by eauto 8. apply wand_elim_r', or_elim; apply wand_intro_l; auto. Qed. `````` Robbert Krebbers committed Nov 03, 2016 458 ``````Lemma sep_or_r P Q R : (P ∨ Q) ∗ R ⊣⊢ (P ∗ R) ∨ (Q ∗ R). `````` Robbert Krebbers committed Oct 25, 2016 459 ``````Proof. by rewrite -!(comm _ R) sep_or_l. Qed. `````` Robbert Krebbers committed Nov 03, 2016 460 ``````Lemma sep_exist_l {A} P (Ψ : A → uPred M) : P ∗ (∃ a, Ψ a) ⊣⊢ ∃ a, P ∗ Ψ a. `````` Robbert Krebbers committed Oct 25, 2016 461 462 463 464 465 466 ``````Proof. intros; apply (anti_symm (⊢)). - apply wand_elim_r', exist_elim=>a. apply wand_intro_l. by rewrite -(exist_intro a). - apply exist_elim=> a; apply sep_mono; auto using exist_intro. Qed. `````` Robbert Krebbers committed Nov 03, 2016 467 ``````Lemma sep_exist_r {A} (Φ: A → uPred M) Q: (∃ a, Φ a) ∗ Q ⊣⊢ ∃ a, Φ a ∗ Q. `````` Robbert Krebbers committed Oct 25, 2016 468 ``````Proof. setoid_rewrite (comm _ _ Q); apply sep_exist_l. Qed. `````` Robbert Krebbers committed Nov 03, 2016 469 ``````Lemma sep_forall_l {A} P (Ψ : A → uPred M) : P ∗ (∀ a, Ψ a) ⊢ ∀ a, P ∗ Ψ a. `````` Robbert Krebbers committed Oct 25, 2016 470 ``````Proof. by apply forall_intro=> a; rewrite forall_elim. Qed. `````` Robbert Krebbers committed Nov 03, 2016 471 ``````Lemma sep_forall_r {A} (Φ : A → uPred M) Q : (∀ a, Φ a) ∗ Q ⊢ ∀ a, Φ a ∗ Q. `````` Robbert Krebbers committed Oct 25, 2016 472 473 474 ``````Proof. by apply forall_intro=> a; rewrite forall_elim. Qed. (* Always derived *) `````` Robbert Krebbers committed Oct 25, 2017 475 476 477 478 479 480 ``````Hint Resolve persistently_mono persistently_elim. Global Instance persistently_mono' : Proper ((⊢) ==> (⊢)) (@uPred_persistently M). Proof. intros P Q; apply persistently_mono. Qed. Global Instance persistently_flip_mono' : Proper (flip (⊢) ==> flip (⊢)) (@uPred_persistently M). Proof. intros P Q; apply persistently_mono. Qed. `````` Robbert Krebbers committed Oct 25, 2016 481 `````` `````` Robbert Krebbers committed Oct 25, 2017 482 483 484 485 ``````Lemma persistently_intro' P Q : (□ P ⊢ Q) → □ P ⊢ □ Q. Proof. intros <-. apply persistently_idemp_2. Qed. Lemma persistently_idemp P : □ □ P ⊣⊢ □ P. Proof. apply (anti_symm _); auto using persistently_idemp_2. Qed. `````` Robbert Krebbers committed Oct 25, 2016 486 `````` `````` Robbert Krebbers committed Oct 25, 2017 487 ``````Lemma persistently_pure φ : □ ⌜φ⌝ ⊣⊢ ⌜φ⌝. `````` Robbert Krebbers committed Jun 13, 2017 488 489 490 491 ``````Proof. apply (anti_symm _); auto. apply pure_elim'=> Hφ. trans (∀ x : False, □ True : uPred M)%I; [by apply forall_intro|]. `````` Robbert Krebbers committed Oct 25, 2017 492 `````` rewrite persistently_forall_2. auto using persistently_mono, pure_intro. `````` Robbert Krebbers committed Jun 13, 2017 493 ``````Qed. `````` Robbert Krebbers committed Oct 25, 2017 494 ``````Lemma persistently_forall {A} (Ψ : A → uPred M) : (□ ∀ a, Ψ a) ⊣⊢ (∀ a, □ Ψ a). `````` Robbert Krebbers committed Oct 25, 2016 495 ``````Proof. `````` Robbert Krebbers committed Oct 25, 2017 496 `````` apply (anti_symm _); auto using persistently_forall_2. `````` Robbert Krebbers committed Oct 25, 2016 497 498 `````` apply forall_intro=> x. by rewrite (forall_elim x). Qed. `````` Robbert Krebbers committed Oct 25, 2017 499 ``````Lemma persistently_exist {A} (Ψ : A → uPred M) : (□ ∃ a, Ψ a) ⊣⊢ (∃ a, □ Ψ a). `````` Robbert Krebbers committed Oct 25, 2016 500 ``````Proof. `````` Robbert Krebbers committed Oct 25, 2017 501 `````` apply (anti_symm _); auto using persistently_exist_1. `````` Robbert Krebbers committed Oct 25, 2016 502 503 `````` apply exist_elim=> x. by rewrite (exist_intro x). Qed. `````` Robbert Krebbers committed Oct 25, 2017 504 505 506 507 508 ``````Lemma persistently_and P Q : □ (P ∧ Q) ⊣⊢ □ P ∧ □ Q. Proof. rewrite !and_alt persistently_forall. by apply forall_proper=> -[]. Qed. Lemma persistently_or P Q : □ (P ∨ Q) ⊣⊢ □ P ∨ □ Q. Proof. rewrite !or_alt persistently_exist. by apply exist_proper=> -[]. Qed. Lemma persistently_impl P Q : □ (P → Q) ⊢ □ P → □ Q. `````` Robbert Krebbers committed Oct 25, 2016 509 ``````Proof. `````` Robbert Krebbers committed Oct 25, 2017 510 511 `````` apply impl_intro_l; rewrite -persistently_and. apply persistently_mono, impl_elim with P; auto. `````` Robbert Krebbers committed Oct 25, 2016 512 ``````Qed. `````` Robbert Krebbers committed Oct 25, 2017 513 ``````Lemma persistently_internal_eq {A:ofeT} (a b : A) : □ (a ≡ b) ⊣⊢ a ≡ b. `````` Robbert Krebbers committed Oct 25, 2016 514 ``````Proof. `````` Robbert Krebbers committed Oct 25, 2017 515 `````` apply (anti_symm (⊢)); auto using persistently_elim. `````` Robbert Krebbers committed Oct 25, 2016 516 `````` apply (internal_eq_rewrite a b (λ b, □ (a ≡ b))%I); auto. `````` Robbert Krebbers committed Oct 25, 2016 517 `````` { intros n; solve_proper. } `````` Robbert Krebbers committed Oct 25, 2017 518 `````` rewrite -(internal_eq_refl a) persistently_pure; auto. `````` Robbert Krebbers committed Oct 25, 2016 519 520 ``````Qed. `````` Robbert Krebbers committed Oct 25, 2017 521 522 523 524 525 526 ``````Lemma persistently_and_sep_l' P Q : □ P ∧ Q ⊣⊢ □ P ∗ Q. Proof. apply (anti_symm (⊢)); auto using persistently_and_sep_l_1. Qed. Lemma persistently_and_sep_r' P Q : P ∧ □ Q ⊣⊢ P ∗ □ Q. Proof. by rewrite !(comm _ P) persistently_and_sep_l'. Qed. Lemma persistently_sep_dup' P : □ P ⊣⊢ □ P ∗ □ P. Proof. by rewrite -persistently_and_sep_l' idemp. Qed. `````` Robbert Krebbers committed Jun 13, 2017 527 `````` `````` Robbert Krebbers committed Oct 25, 2017 528 ``````Lemma persistently_and_sep P Q : □ (P ∧ Q) ⊣⊢ □ (P ∗ Q). `````` Robbert Krebbers committed Jun 13, 2017 529 530 ``````Proof. apply (anti_symm (⊢)); auto. `````` Robbert Krebbers committed Oct 25, 2017 531 `````` rewrite -{1}persistently_idemp persistently_and persistently_and_sep_l'; auto. `````` Robbert Krebbers committed Jun 13, 2017 532 ``````Qed. `````` Robbert Krebbers committed Oct 25, 2017 533 534 ``````Lemma persistently_sep P Q : □ (P ∗ Q) ⊣⊢ □ P ∗ □ Q. Proof. by rewrite -persistently_and_sep -persistently_and_sep_l' persistently_and. Qed. `````` Robbert Krebbers committed Oct 25, 2016 535 `````` `````` Robbert Krebbers committed Oct 25, 2017 536 537 538 ``````Lemma persistently_wand P Q : □ (P -∗ Q) ⊢ □ P -∗ □ Q. Proof. by apply wand_intro_r; rewrite -persistently_sep wand_elim_l. Qed. Lemma persistently_wand_impl P Q : □ (P -∗ Q) ⊣⊢ □ (P → Q). `````` Robbert Krebbers committed Oct 25, 2016 539 540 ``````Proof. apply (anti_symm (⊢)); [|by rewrite -impl_wand]. `````` Robbert Krebbers committed Oct 25, 2017 541 542 `````` apply persistently_intro', impl_intro_r. by rewrite persistently_and_sep_l' persistently_elim wand_elim_l. `````` Robbert Krebbers committed Oct 25, 2016 543 ``````Qed. `````` Robbert Krebbers committed Oct 25, 2017 544 ``````Lemma wand_impl_persistently P Q : ((□ P) -∗ Q) ⊣⊢ ((□ P) → Q). `````` Ralf Jung committed Aug 23, 2017 545 546 ``````Proof. apply (anti_symm (⊢)); [|by rewrite -impl_wand]. `````` Robbert Krebbers committed Oct 25, 2017 547 `````` apply impl_intro_l. by rewrite persistently_and_sep_l' wand_elim_r. `````` Ralf Jung committed Aug 23, 2017 548 ``````Qed. `````` Robbert Krebbers committed Oct 25, 2017 549 550 551 552 ``````Lemma persistently_entails_l' P Q : (P ⊢ □ Q) → P ⊢ □ Q ∗ P. Proof. intros; rewrite -persistently_and_sep_l'; auto. Qed. Lemma persistently_entails_r' P Q : (P ⊢ □ Q) → P ⊢ P ∗ □ Q. Proof. intros; rewrite -persistently_and_sep_r'; auto. Qed. `````` Robbert Krebbers committed Oct 25, 2016 553 `````` `````` Robbert Krebbers committed Oct 25, 2017 554 555 ``````Lemma persistently_laterN n P : □ ▷^n P ⊣⊢ ▷^n □ P. Proof. induction n as [|n IH]; simpl; auto. by rewrite persistently_later IH. Qed. `````` Robbert Krebbers committed Nov 27, 2016 556 `````` `````` Robbert Krebbers committed May 12, 2017 557 558 559 560 ``````Lemma wand_alt P Q : (P -∗ Q) ⊣⊢ ∃ R, R ∗ □ (P ∗ R → Q). Proof. apply (anti_symm (⊢)). - rewrite -(right_id True%I uPred_sep (P -∗ Q)%I) -(exist_intro (P -∗ Q)%I). `````` Robbert Krebbers committed Oct 25, 2017 561 `````` apply sep_mono_r. rewrite -persistently_pure. apply persistently_mono, impl_intro_l. `````` Robbert Krebbers committed May 12, 2017 562 `````` by rewrite wand_elim_r right_id. `````` Robbert Krebbers committed Oct 25, 2017 563 564 `````` - apply exist_elim=> R. apply wand_intro_l. rewrite assoc -persistently_and_sep_r'. by rewrite persistently_elim impl_elim_r. `````` Robbert Krebbers committed May 12, 2017 565 566 567 568 569 ``````Qed. Lemma impl_alt P Q : (P → Q) ⊣⊢ ∃ R, R ∧ □ (P ∧ R -∗ Q). Proof. apply (anti_symm (⊢)). - rewrite -(right_id True%I uPred_and (P → Q)%I) -(exist_intro (P → Q)%I). `````` Robbert Krebbers committed Oct 25, 2017 570 `````` apply and_mono_r. rewrite -persistently_pure. apply persistently_mono, wand_intro_l. `````` Robbert Krebbers committed May 12, 2017 571 `````` by rewrite impl_elim_r right_id. `````` Robbert Krebbers committed Oct 25, 2017 572 573 `````` - apply exist_elim=> R. apply impl_intro_l. rewrite assoc persistently_and_sep_r'. by rewrite persistently_elim wand_elim_r. `````` Robbert Krebbers committed May 12, 2017 574 ``````Qed. `````` Robbert Krebbers committed Nov 27, 2016 575 `````` `````` Robbert Krebbers committed Oct 25, 2016 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 ``````(* Later derived *) Lemma later_proper P Q : (P ⊣⊢ Q) → ▷ P ⊣⊢ ▷ Q. Proof. by intros ->. Qed. Hint Resolve later_mono later_proper. Global Instance later_mono' : Proper ((⊢) ==> (⊢)) (@uPred_later M). Proof. intros P Q; apply later_mono. Qed. Global Instance later_flip_mono' : Proper (flip (⊢) ==> flip (⊢)) (@uPred_later M). Proof. intros P Q; apply later_mono. Qed. Lemma later_intro P : P ⊢ ▷ P. Proof. rewrite -(and_elim_l (▷ P) P) -(löb (▷ P ∧ P)). apply impl_intro_l. by rewrite {1}(and_elim_r (▷ P)). Qed. Lemma later_True : ▷ True ⊣⊢ True. Proof. apply (anti_symm (⊢)); auto using later_intro. Qed. Lemma later_forall {A} (Φ : A → uPred M) : (▷ ∀ a, Φ a) ⊣⊢ (∀ a, ▷ Φ a). Proof. apply (anti_symm _); auto using later_forall_2. apply forall_intro=> x. by rewrite (forall_elim x). Qed. `````` Robbert Krebbers committed Sep 27, 2017 599 600 ``````Lemma later_exist_2 {A} (Φ : A → uPred M) : (∃ a, ▷ Φ a) ⊢ ▷ (∃ a, Φ a). Proof. apply exist_elim; eauto using exist_intro. Qed. `````` Robbert Krebbers committed Oct 25, 2016 601 602 603 ``````Lemma later_exist `{Inhabited A} (Φ : A → uPred M) : ▷ (∃ a, Φ a) ⊣⊢ (∃ a, ▷ Φ a). Proof. `````` Robbert Krebbers committed Sep 27, 2017 604 `````` apply: anti_symm; [|apply later_exist_2]. `````` Robbert Krebbers committed Oct 25, 2016 605 606 607 608 609 610 611 612 613 `````` rewrite later_exist_false. apply or_elim; last done. rewrite -(exist_intro inhabitant); auto. Qed. Lemma later_and P Q : ▷ (P ∧ Q) ⊣⊢ ▷ P ∧ ▷ Q. Proof. rewrite !and_alt later_forall. by apply forall_proper=> -[]. Qed. Lemma later_or P Q : ▷ (P ∨ Q) ⊣⊢ ▷ P ∨ ▷ Q. Proof. rewrite !or_alt later_exist. by apply exist_proper=> -[]. Qed. Lemma later_impl P Q : ▷ (P → Q) ⊢ ▷ P → ▷ Q. Proof. apply impl_intro_l; rewrite -later_and; eauto using impl_elim. Qed. `````` Robbert Krebbers committed Nov 03, 2016 614 ``````Lemma later_wand P Q : ▷ (P -∗ Q) ⊢ ▷ P -∗ ▷ Q. `````` Robbert Krebbers committed Oct 25, 2016 615 616 617 618 619 ``````Proof. apply wand_intro_r; rewrite -later_sep; eauto using wand_elim_l. Qed. Lemma later_iff P Q : ▷ (P ↔ Q) ⊢ ▷ P ↔ ▷ Q. Proof. by rewrite /uPred_iff later_and !later_impl. Qed. `````` Robbert Krebbers committed Nov 27, 2016 620 ``````(* Iterated later modality *) `````` Ralf Jung committed Jan 27, 2017 621 ``````Global Instance laterN_ne m : NonExpansive (@uPred_laterN M m). `````` Robbert Krebbers committed Nov 27, 2016 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 ``````Proof. induction m; simpl. by intros ???. solve_proper. Qed. Global Instance laterN_proper m : Proper ((⊣⊢) ==> (⊣⊢)) (@uPred_laterN M m) := ne_proper _. Lemma laterN_0 P : ▷^0 P ⊣⊢ P. Proof. done. Qed. Lemma later_laterN n P : ▷^(S n) P ⊣⊢ ▷ ▷^n P. Proof. done. Qed. Lemma laterN_later n P : ▷^(S n) P ⊣⊢ ▷^n ▷ P. Proof. induction n; simpl; auto. Qed. Lemma laterN_plus n1 n2 P : ▷^(n1 + n2) P ⊣⊢ ▷^n1 ▷^n2 P. Proof. induction n1; simpl; auto. Qed. Lemma laterN_le n1 n2 P : n1 ≤ n2 → ▷^n1 P ⊢ ▷^n2 P. Proof. induction 1; simpl; by rewrite -?later_intro. Qed. Lemma laterN_mono n P Q : (P ⊢ Q) → ▷^n P ⊢ ▷^n Q. Proof. induction n; simpl; auto. Qed. Global Instance laterN_mono' n : Proper ((⊢) ==> (⊢)) (@uPred_laterN M n). Proof. intros P Q; apply laterN_mono. Qed. Global Instance laterN_flip_mono' n : Proper (flip (⊢) ==> flip (⊢)) (@uPred_laterN M n). Proof. intros P Q; apply laterN_mono. Qed. Lemma laterN_intro n P : P ⊢ ▷^n P. Proof. induction n as [|n IH]; simpl; by rewrite -?later_intro. Qed. Lemma laterN_True n : ▷^n True ⊣⊢ True. Proof. apply (anti_symm (⊢)); auto using laterN_intro. Qed. Lemma laterN_forall {A} n (Φ : A → uPred M) : (▷^n ∀ a, Φ a) ⊣⊢ (∀ a, ▷^n Φ a). Proof. induction n as [|n IH]; simpl; rewrite -?later_forall; auto. Qed. `````` Robbert Krebbers committed Sep 27, 2017 652 653 ``````Lemma laterN_exist_2 {A} n (Φ : A → uPred M) : (∃ a, ▷^n Φ a) ⊢ ▷^n (∃ a, Φ a). Proof. apply exist_elim; eauto using exist_intro, laterN_mono. Qed. `````` Robbert Krebbers committed Nov 27, 2016 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 ``````Lemma laterN_exist `{Inhabited A} n (Φ : A → uPred M) : (▷^n ∃ a, Φ a) ⊣⊢ ∃ a, ▷^n Φ a. Proof. induction n as [|n IH]; simpl; rewrite -?later_exist; auto. Qed. Lemma laterN_and n P Q : ▷^n (P ∧ Q) ⊣⊢ ▷^n P ∧ ▷^n Q. Proof. induction n as [|n IH]; simpl; rewrite -?later_and; auto. Qed. Lemma laterN_or n P Q : ▷^n (P ∨ Q) ⊣⊢ ▷^n P ∨ ▷^n Q. Proof. induction n as [|n IH]; simpl; rewrite -?later_or; auto. Qed. Lemma laterN_impl n P Q : ▷^n (P → Q) ⊢ ▷^n P → ▷^n Q. Proof. apply impl_intro_l; rewrite -laterN_and; eauto using impl_elim, laterN_mono. Qed. Lemma laterN_sep n P Q : ▷^n (P ∗ Q) ⊣⊢ ▷^n P ∗ ▷^n Q. Proof. induction n as [|n IH]; simpl; rewrite -?later_sep; auto. Qed. Lemma laterN_wand n P Q : ▷^n (P -∗ Q) ⊢ ▷^n P -∗ ▷^n Q. Proof. apply wand_intro_r; rewrite -laterN_sep; eauto using wand_elim_l,laterN_mono. Qed. Lemma laterN_iff n P Q : ▷^n (P ↔ Q) ⊢ ▷^n P ↔ ▷^n Q. Proof. by rewrite /uPred_iff laterN_and !laterN_impl. Qed. `````` Robbert Krebbers committed Oct 25, 2017 674 675 ``````(* Conditional persistently *) Global Instance persistently_if_ne p : NonExpansive (@uPred_persistently_if M p). `````` Robbert Krebbers committed Oct 25, 2016 676 ``````Proof. solve_proper. Qed. `````` Robbert Krebbers committed Oct 25, 2017 677 ``````Global Instance persistently_if_proper p : Proper ((⊣⊢) ==> (⊣⊢)) (@uPred_persistently_if M p). `````` Robbert Krebbers committed Oct 25, 2016 678 ``````Proof. solve_proper. Qed. `````` Robbert Krebbers committed Oct 25, 2017 679 ``````Global Instance persistently_if_mono p : Proper ((⊢) ==> (⊢)) (@uPred_persistently_if M p). `````` Robbert Krebbers committed Oct 25, 2016 680 681 ``````Proof. solve_proper. Qed. `````` Robbert Krebbers committed Oct 25, 2017 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 ``````Lemma persistently_if_elim p P : □?p P ⊢ P. Proof. destruct p; simpl; auto using persistently_elim. Qed. Lemma persistently_elim_if p P : □ P ⊢ □?p P. Proof. destruct p; simpl; auto using persistently_elim. Qed. Lemma persistently_if_pure p φ : □?p ⌜φ⌝ ⊣⊢ ⌜φ⌝. Proof. destruct p; simpl; auto using persistently_pure. Qed. Lemma persistently_if_and p P Q : □?p (P ∧ Q) ⊣⊢ □?p P ∧ □?p Q. Proof. destruct p; simpl; auto using persistently_and. Qed. Lemma persistently_if_or p P Q : □?p (P ∨ Q) ⊣⊢ □?p P ∨ □?p Q. Proof. destruct p; simpl; auto using persistently_or. Qed. Lemma persistently_if_exist {A} p (Ψ : A → uPred M) : (□?p ∃ a, Ψ a) ⊣⊢ ∃ a, □?p Ψ a. Proof. destruct p; simpl; auto using persistently_exist. Qed. Lemma persistently_if_sep p P Q : □?p (P ∗ Q) ⊣⊢ □?p P ∗ □?p Q. Proof. destruct p; simpl; auto using persistently_sep. Qed. Lemma persistently_if_later p P : □?p ▷ P ⊣⊢ ▷ □?p P. Proof. destruct p; simpl; auto using persistently_later. Qed. Lemma persistently_if_laterN p n P : □?p ▷^n P ⊣⊢ ▷^n □?p P. Proof. destruct p; simpl; auto using persistently_laterN. Qed. `````` Robbert Krebbers committed Oct 25, 2016 701 702 `````` (* True now *) `````` Ralf Jung committed Jan 27, 2017 703 ``````Global Instance except_0_ne : NonExpansive (@uPred_except_0 M). `````` Robbert Krebbers committed Oct 25, 2016 704 ``````Proof. solve_proper. Qed. `````` Robbert Krebbers committed Oct 25, 2016 705 ``````Global Instance except_0_proper : Proper ((⊣⊢) ==> (⊣⊢)) (@uPred_except_0 M). `````` Robbert Krebbers committed Oct 25, 2016 706 ``````Proof. solve_proper. Qed. `````` Robbert Krebbers committed Oct 25, 2016 707 ``````Global Instance except_0_mono' : Proper ((⊢) ==> (⊢)) (@uPred_except_0 M). `````` Robbert Krebbers committed Oct 25, 2016 708 ``````Proof. solve_proper. Qed. `````` Robbert Krebbers committed Oct 25, 2016 709 710 ``````Global Instance except_0_flip_mono' : Proper (flip (⊢) ==> flip (⊢)) (@uPred_except_0 M). `````` Robbert Krebbers committed Oct 25, 2016 711 712 ``````Proof. solve_proper. Qed. `````` Robbert Krebbers committed Oct 25, 2016 713 714 715 ``````Lemma except_0_intro P : P ⊢ ◇ P. Proof. rewrite /uPred_except_0; auto. Qed. Lemma except_0_mono P Q : (P ⊢ Q) → ◇ P ⊢ ◇ Q. `````` Robbert Krebbers committed Oct 25, 2016 716 ``````Proof. by intros ->. Qed. `````` Robbert Krebbers committed Oct 25, 2016 717 718 719 720 721 722 723 724 725 ``````Lemma except_0_idemp P : ◇ ◇ P ⊢ ◇ P. Proof. rewrite /uPred_except_0; auto. Qed. Lemma except_0_True : ◇ True ⊣⊢ True. Proof. rewrite /uPred_except_0. apply (anti_symm _); auto. Qed. Lemma except_0_or P Q : ◇ (P ∨ Q) ⊣⊢ ◇ P ∨ ◇ Q. Proof. rewrite /uPred_except_0. apply (anti_symm _); auto. Qed. Lemma except_0_and P Q : ◇ (P ∧ Q) ⊣⊢ ◇ P ∧ ◇ Q. Proof. by rewrite /uPred_except_0 or_and_l. Qed. `````` Robbert Krebbers committed Nov 03, 2016 726 ``````Lemma except_0_sep P Q : ◇ (P ∗ Q) ⊣⊢ ◇ P ∗ ◇ Q. `````` Robbert Krebbers committed Oct 25, 2016 727 728 ``````Proof. rewrite /uPred_except_0. apply (anti_symm _). `````` Robbert Krebbers committed Oct 25, 2016 729 `````` - apply or_elim; last by auto. `````` Robbert Krebbers committed Oct 25, 2017 730 `````` by rewrite -!or_intro_l -persistently_pure -persistently_later -persistently_sep_dup'. `````` Robbert Krebbers committed Oct 25, 2016 731 732 `````` - rewrite sep_or_r sep_elim_l sep_or_l; auto. Qed. `````` Robbert Krebbers committed Oct 25, 2016 733 ``````Lemma except_0_forall {A} (Φ : A → uPred M) : ◇ (∀ a, Φ a) ⊢ ∀ a, ◇ Φ a. `````` Robbert Krebbers committed Oct 25, 2016 734 ``````Proof. apply forall_intro=> a. by rewrite (forall_elim a). Qed. `````` Robbert Krebbers committed May 12, 2017 735 ``````Lemma except_0_exist_2 {A} (Φ : A → uPred M) : (∃ a, ◇ Φ a) ⊢ ◇ ∃ a, Φ a. `````` Robbert Krebbers committed Oct 25, 2016 736 ``````Proof. apply exist_elim=> a. by rewrite (exist_intro a). Qed. `````` Robbert Krebbers committed May 12, 2017 737 738 739 740 741 742 743 ``````Lemma except_0_exist `{Inhabited A} (Φ : A → uPred M) : ◇ (∃ a, Φ a) ⊣⊢ (∃ a, ◇ Φ a). Proof. apply (anti_symm _); [|by apply except_0_exist_2]. apply or_elim. - rewrite -(exist_intro inhabitant). by apply or_intro_l. - apply exist_mono=> a. apply except_0_intro. Qed. `````` Robbert Krebbers committed Oct 25, 2016 744 745 ``````Lemma except_0_later P : ◇ ▷ P ⊢ ▷ P. Proof. by rewrite /uPred_except_0 -later_or False_or. Qed. `````` Robbert Krebbers committed Oct 25, 2017 746 747 748 749 ``````Lemma except_0_persistently P : ◇ □ P ⊣⊢ □ ◇ P. Proof. by rewrite /uPred_except_0 persistently_or persistently_later persistently_pure. Qed. Lemma except_0_persistently_if p P : ◇ □?p P ⊣⊢ □?p ◇ P. Proof. destruct p; simpl; auto using except_0_persistently. Qed. `````` Robbert Krebbers committed Nov 03, 2016 750 ``````Lemma except_0_frame_l P Q : P ∗ ◇ Q ⊢ ◇ (P ∗ Q). `````` Robbert Krebbers committed Oct 25, 2016 751 ``````Proof. by rewrite {1}(except_0_intro P) except_0_sep. Qed. `````` Robbert Krebbers committed Nov 03, 2016 752 ``````Lemma except_0_frame_r P Q : ◇ P ∗ Q ⊢ ◇ (P ∗ Q). `````` Robbert Krebbers committed Oct 25, 2016 753 ``````Proof. by rewrite {1}(except_0_intro Q) except_0_sep. Qed. `````` Robbert Krebbers committed Oct 25, 2016 754 755 `````` (* Own and valid derived *) `````` Robbert Krebbers committed Oct 25, 2017