logic.v 18.1 KB
 Robbert Krebbers committed Nov 11, 2015 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 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 85 86 87 88 89 90 91 92 93 94 ``````Require Import cmra cofe_instances. Local Hint Extern 1 (_ ≼ _) => etransitivity; [eassumption|]. Local Hint Extern 1 (_ ≼ _) => etransitivity; [|eassumption]. Local Hint Extern 10 (_ ≤ _) => omega. Structure iProp (A : cmraT) : Type := IProp { iprop_holds :> nat → A -> Prop; iprop_ne x1 x2 n : iprop_holds n x1 → x1 ={n}= x2 → iprop_holds n x2; iprop_weaken x1 x2 n1 n2 : x1 ≼ x2 → n2 ≤ n1 → validN n2 x2 → iprop_holds n1 x1 → iprop_holds n2 x2 }. Add Printing Constructor iProp. Instance: Params (@iprop_holds) 3. Instance iprop_equiv (A : cmraT) : Equiv (iProp A) := λ P Q, ∀ x n, validN n x → P n x ↔ Q n x. Instance iprop_dist (A : cmraT) : Dist (iProp A) := λ n P Q, ∀ x n', n' < n → validN n' x → P n' x ↔ Q n' x. Program Instance iprop_compl (A : cmraT) : Compl (iProp A) := λ c, {| iprop_holds n x := c (S n) n x |}. Next Obligation. by intros A c x y n ??; simpl in *; apply iprop_ne with x. Qed. Next Obligation. intros A c x1 x2 n1 n2 ????; simpl in *. apply (chain_cauchy c (S n2) (S n1)); eauto using iprop_weaken, cmra_valid_le. Qed. Instance iprop_cofe (A : cmraT) : Cofe (iProp A). Proof. split. * intros P Q; split; [by intros HPQ n x i ??; apply HPQ|]. intros HPQ x n ?; apply HPQ with (S n); auto. * intros n; split. + by intros P x i. + by intros P Q HPQ x i ??; symmetry; apply HPQ. + by intros P Q Q' HP HQ x i ??; transitivity (Q i x); [apply HP|apply HQ]. * intros n P Q HPQ x i ??; apply HPQ; auto. * intros P Q x i ??; lia. * intros c n x i ??; apply (chain_cauchy c (S i) n); auto. Qed. Instance iprop_holds_ne {A} (P : iProp A) n : Proper (dist n ==> iff) (P n). Proof. intros x1 x2 Hx; split; eauto using iprop_ne. Qed. Instance iprop_holds_proper {A} (P : iProp A) n : Proper ((≡) ==> iff) (P n). Proof. by intros x1 x2 Hx; apply iprop_holds_ne, equiv_dist. Qed. Definition iPropC (A : cmraT) : cofeT := CofeT (iProp A). (** functor *) Program Definition iProp_map {A B : cmraT} (f: B -n> A) `{!CMRAPreserving f} (P : iProp A) : iProp B := {| iprop_holds n x := P n (f x) |}. Next Obligation. by intros A B f ? P y1 y2 n ? Hy; simpl; rewrite <-Hy. Qed. Next Obligation. by intros A B f ? P y1 y2 n i ???; simpl; apply iprop_weaken; auto; apply validN_preserving || apply included_preserving. Qed. (** logical entailement *) Instance iprop_entails {A} : SubsetEq (iProp A) := λ P Q, ∀ x n, validN n x → P n x → Q n x. (** logical connectives *) Program Definition iprop_const {A} (P : Prop) : iProp A := {| iprop_holds n x := P |}. Solve Obligations with done. Program Definition iprop_and {A} (P Q : iProp A) : iProp A := {| iprop_holds n x := P n x ∧ Q n x |}. Solve Obligations with naive_solver eauto 2 using iprop_ne, iprop_weaken. Program Definition iprop_or {A} (P Q : iProp A) : iProp A := {| iprop_holds n x := P n x ∨ Q n x |}. Solve Obligations with naive_solver eauto 2 using iprop_ne, iprop_weaken. Program Definition iprop_impl {A} (P Q : iProp A) : iProp A := {| iprop_holds n x := ∀ x' n', x ≼ x' → n' ≤ n → validN n' x' → P n' x' → Q n' x' |}. Next Obligation. intros A P Q x1' x1 n1 HPQ Hx1 x2 n2 ????. destruct (cmra_included_dist_l x1 x2 x1' n1) as (x2'&?&Hx2); auto. assert (x2' ={n2}= x2) as Hx2' by (by apply dist_le with n1). assert (validN n2 x2') by (by rewrite Hx2'); rewrite <-Hx2'. by apply HPQ, iprop_weaken with x2' n2, iprop_ne with x2. Qed. Next Obligation. naive_solver eauto 2 with lia. Qed. Program Definition iprop_forall {A B} (P : A → iProp B) : iProp B := {| iprop_holds n x := ∀ a, P a n x |}. Solve Obligations with naive_solver eauto 2 using iprop_ne, iprop_weaken. Program Definition iprop_exist {A B} (P : A → iProp B) : iProp B := {| iprop_holds n x := ∃ a, P a n x |}. Solve Obligations with naive_solver eauto 2 using iprop_ne, iprop_weaken. Program Definition iprop_eq {A} {B : cofeT} (b1 b2 : B) : iProp A := {| iprop_holds n x := b1 ={n}= b2 |}. Solve Obligations with naive_solver eauto 2 using (dist_le (A:=B)). Program Definition iprop_sep {A} (P Q : iProp A) : iProp A := {| iprop_holds n x := ∃ x1 x2, x ={n}= x1 ⋅ x2 ∧ P n x1 ∧ Q n x2 |}. Next Obligation. `````` Robbert Krebbers committed Nov 11, 2015 95 `````` by intros A P Q x y n (x1&x2&?&?&?) Hxy; exists x1, x2; rewrite <-Hxy. `````` Robbert Krebbers committed Nov 11, 2015 96 97 98 99 100 101 102 ``````Qed. Next Obligation. intros A P Q x y n1 n2 Hxy ?? (x1&x2&Hx&?&?). assert (∃ x2', y ={n2}= x1 ⋅ x2' ∧ x2 ≼ x2') as (x2'&Hy&?). { rewrite ra_included_spec in Hxy; destruct Hxy as [z Hy]. exists (x2 ⋅ z); split; eauto using ra_included_l. apply dist_le with n1; auto. by rewrite (associative op), <-Hx, Hy. } `````` Robbert Krebbers committed Nov 11, 2015 103 `````` exists x1, x2'; split_ands; auto. `````` Robbert Krebbers committed Nov 11, 2015 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 `````` * apply iprop_weaken with x1 n1; auto. by apply cmra_valid_op_l with x2'; rewrite <-Hy. * apply iprop_weaken with x2 n1; auto. by apply cmra_valid_op_r with x1; rewrite <-Hy. Qed. Program Definition iprop_wand {A} (P Q : iProp A) : iProp A := {| iprop_holds n x := ∀ x' n', n' ≤ n → validN n' (x ⋅ x') → P n' x' → Q n' (x ⋅ x') |}. Next Obligation. intros A P Q x1 x2 n1 HPQ Hx x3 n2 ???; simpl in *. rewrite <-(dist_le _ _ _ _ Hx) by done; apply HPQ; auto. by rewrite (dist_le _ _ _ n2 Hx). Qed. Next Obligation. intros A P Q x1 x2 n1 n2 ??? HPQ x3 n3 ???; simpl in *. apply iprop_weaken with (x1 ⋅ x3) n3; auto using ra_preserving_r. apply HPQ; auto. apply cmra_valid_included with (x2 ⋅ x3); auto using ra_preserving_r. Qed. Program Definition iprop_later {A} (P : iProp A) : iProp A := {| iprop_holds n x := match n return _ with 0 => True | S n' => P n' x end |}. Next Obligation. intros A P ?? [|n]; eauto using iprop_ne,(dist_le (A:=A)). Qed. Next Obligation. intros A P x1 x2 [|n1] [|n2] ????; auto with lia. apply iprop_weaken with x1 n1; eauto using cmra_valid_S. Qed. Program Definition iprop_always {A} (P : iProp A) : iProp A := {| iprop_holds n x := P n (unit x) |}. Next Obligation. by intros A P x1 x2 n ? Hx; simpl in *; rewrite <-Hx. Qed. Next Obligation. intros A P x1 x2 n1 n2 ????; eapply iprop_weaken with (unit x1) n1; auto using ra_unit_preserving, cmra_unit_valid. Qed. `````` Robbert Krebbers committed Nov 12, 2015 140 141 142 143 144 145 146 147 148 149 150 ``````Program Definition iprop_own {A : cmraT} (a : A) : iProp A := {| iprop_holds n x := ∃ a', x ={n}= a ⋅ a' |}. Next Obligation. by intros A a x1 x2 n [a' Hx] ?; exists a'; rewrite <-Hx. Qed. Next Obligation. intros A a x1 x n1 n2; rewrite ra_included_spec; intros [x2 Hx] ?? [a' Hx1]. exists (a' ⋅ x2). by rewrite (associative op), <-(dist_le _ _ _ _ Hx1), Hx. Qed. Program Definition iprop_valid {A : cmraT} (a : A) : iProp A := {| iprop_holds n x := validN n a |}. Solve Obligations with naive_solver eauto 2 using cmra_valid_le. `````` Robbert Krebbers committed Nov 11, 2015 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 ``````Definition iprop_fixpoint {A} (P : iProp A → iProp A) `{!Contractive P} : iProp A := fixpoint P (iprop_const True). Delimit Scope iprop_scope with I. Bind Scope iprop_scope with iProp. Arguments iprop_holds {_} _%I _ _. Notation "'False'" := (iprop_const False) : iprop_scope. Notation "'True'" := (iprop_const True) : iprop_scope. Infix "∧" := iprop_and : iprop_scope. Infix "∨" := iprop_or : iprop_scope. Infix "→" := iprop_impl : iprop_scope. Infix "★" := iprop_sep (at level 80, right associativity) : iprop_scope. Infix "-★" := iprop_wand (at level 90) : iprop_scope. Notation "∀ x .. y , P" := (iprop_forall (λ x, .. (iprop_forall (λ y, P)) ..)) : iprop_scope. Notation "∃ x .. y , P" := (iprop_exist (λ x, .. (iprop_exist (λ y, P)) ..)) : iprop_scope. Notation "▷ P" := (iprop_later P) (at level 20) : iprop_scope. Notation "□ P" := (iprop_always P) (at level 20) : iprop_scope. Section logic. Context {A : cmraT}. Implicit Types P Q : iProp A. Global Instance iprop_preorder : PreOrder ((⊆) : relation (iProp A)). Proof. split. by intros P x i. by intros P Q Q' HP HQ x i ??; apply HQ, HP. Qed. Lemma iprop_equiv_spec P Q : P ≡ Q ↔ P ⊆ Q ∧ Q ⊆ P. Proof. split. * intros HPQ; split; intros x i; apply HPQ. * by intros [HPQ HQP]; intros x i ?; split; [apply HPQ|apply HQP]. Qed. (** Non-expansiveness *) Global Instance iprop_const_proper : Proper (iff ==> (≡)) (@iprop_const A). Proof. intros P Q HPQ ???; apply HPQ. Qed. Global Instance iprop_and_ne n : Proper (dist n ==> dist n ==> dist n) (@iprop_and A). Proof. intros P P' HP Q Q' HQ; split; intros [??]; split; by apply HP || by apply HQ. Qed. `````` Robbert Krebbers committed Nov 12, 2015 193 194 ``````Global Instance iprop_and_proper : Proper ((≡) ==> (≡) ==> (≡)) (@iprop_and A) := ne_proper_2 _. `````` Robbert Krebbers committed Nov 11, 2015 195 196 197 198 199 200 ``````Global Instance iprop_or_ne n : Proper (dist n ==> dist n ==> dist n) (@iprop_or A). Proof. intros P P' HP Q Q' HQ; split; intros [?|?]; first [by left; apply HP | by right; apply HQ]. Qed. `````` Robbert Krebbers committed Nov 12, 2015 201 202 ``````Global Instance iprop_or_proper : Proper ((≡) ==> (≡) ==> (≡)) (@iprop_or A) := ne_proper_2 _. `````` Robbert Krebbers committed Nov 11, 2015 203 204 205 206 207 ``````Global Instance iprop_impl_ne n : Proper (dist n ==> dist n ==> dist n) (@iprop_impl A). Proof. intros P P' HP Q Q' HQ; split; intros HPQ x' n'' ????; apply HQ,HPQ,HP; auto. Qed. `````` Robbert Krebbers committed Nov 12, 2015 208 209 ``````Global Instance iprop_impl_proper : Proper ((≡) ==> (≡) ==> (≡)) (@iprop_impl A) := ne_proper_2 _. `````` Robbert Krebbers committed Nov 11, 2015 210 211 212 213 ``````Global Instance iprop_sep_ne n : Proper (dist n ==> dist n ==> dist n) (@iprop_sep A). Proof. intros P P' HP Q Q' HQ x n' ? Hx'; split; intros (x1&x2&Hx&?&?); `````` Robbert Krebbers committed Nov 11, 2015 214 `````` exists x1, x2; rewrite Hx in Hx'; split_ands; try apply HP; try apply HQ; `````` Robbert Krebbers committed Nov 11, 2015 215 216 `````` eauto using cmra_valid_op_l, cmra_valid_op_r. Qed. `````` Robbert Krebbers committed Nov 12, 2015 217 218 ``````Global Instance iprop_sep_proper : Proper ((≡) ==> (≡) ==> (≡)) (@iprop_sep A) := ne_proper_2 _. `````` Robbert Krebbers committed Nov 11, 2015 219 220 221 222 223 224 ``````Global Instance iprop_wand_ne n : Proper (dist n ==> dist n ==> dist n) (@iprop_wand A). Proof. intros P P' HP Q Q' HQ x n' ??; split; intros HPQ x' n'' ???; apply HQ, HPQ, HP; eauto using cmra_valid_op_r. Qed. `````` Robbert Krebbers committed Nov 12, 2015 225 226 ``````Global Instance iprop_wand_proper : Proper ((≡) ==> (≡) ==> (≡)) (@iprop_wand A) := ne_proper_2 _. `````` Robbert Krebbers committed Nov 11, 2015 227 228 229 230 231 232 233 ``````Global Instance iprop_eq_ne {B : cofeT} n : Proper (dist n ==> dist n ==> dist n) (@iprop_eq A B). Proof. intros x x' Hx y y' Hy z n'; split; intros; simpl in *. * by rewrite <-(dist_le _ _ _ _ Hx), <-(dist_le _ _ _ _ Hy) by auto. * by rewrite (dist_le _ _ _ _ Hx), (dist_le _ _ _ _ Hy) by auto. Qed. `````` Robbert Krebbers committed Nov 12, 2015 234 235 ``````Global Instance iprop_eq_proper {B : cofeT} : Proper ((≡) ==> (≡) ==> (≡)) (@iprop_eq A B) := ne_proper_2 _. `````` Robbert Krebbers committed Nov 11, 2015 236 237 238 ``````Global Instance iprop_forall_ne {B : cofeT} : Proper (pointwise_relation _ (dist n) ==> dist n) (@iprop_forall B A). Proof. by intros n P1 P2 HP12 x n'; split; intros HP a; apply HP12. Qed. `````` Robbert Krebbers committed Nov 12, 2015 239 240 241 ``````Global Instance iprop_forall_proper {B : cofeT} : Proper (pointwise_relation _ (≡) ==> (≡)) (@iprop_forall B A). Proof. by intros P1 P2 HP12 x n'; split; intros HP a; apply HP12. Qed. `````` Robbert Krebbers committed Nov 11, 2015 242 243 244 245 246 ``````Global Instance iprop_exists_ne {B : cofeT} : Proper (pointwise_relation _ (dist n) ==> dist n) (@iprop_exist B A). Proof. by intros n P1 P2 HP12 x n'; split; intros [a HP]; exists a; apply HP12. Qed. `````` Robbert Krebbers committed Nov 12, 2015 247 248 249 250 251 ``````Global Instance iprop_exist_proper {B : cofeT} : Proper (pointwise_relation _ (≡) ==> (≡)) (@iprop_exist B A). Proof. by intros P1 P2 HP12 x n'; split; intros [a HP]; exists a; apply HP12. Qed. `````` Robbert Krebbers committed Nov 11, 2015 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 `````` (** Introduction and elimination rules *) Lemma iprop_True_intro P : P ⊆ True%I. Proof. done. Qed. Lemma iprop_False_elim P : False%I ⊆ P. Proof. by intros x n ?. Qed. Lemma iprop_and_elim_l P Q : (P ∧ Q)%I ⊆ P. Proof. by intros x n ? [??]. Qed. Lemma iprop_and_elim_r P Q : (P ∧ Q)%I ⊆ Q. Proof. by intros x n ? [??]. Qed. Lemma iprop_and_intro R P Q : R ⊆ P → R ⊆ Q → R ⊆ (P ∧ Q)%I. Proof. intros HP HQ x n ??; split. by apply HP. by apply HQ. Qed. Lemma iprop_or_intro_l P Q : P ⊆ (P ∨ Q)%I. Proof. by left. Qed. Lemma iprop_or_intro_r P Q : Q ⊆ (P ∨ Q)%I. Proof. by right. Qed. Lemma iprop_or_elim R P Q : P ⊆ R → Q ⊆ R → (P ∨ Q)%I ⊆ R. Proof. intros HP HQ x n ? [?|?]. by apply HP. by apply HQ. Qed. Lemma iprop_impl_intro P Q R : (R ∧ P)%I ⊆ Q → R ⊆ (P → Q)%I. Proof. intros HQ x n ?? x' n' ????; apply HQ; naive_solver eauto using iprop_weaken. Qed. Lemma iprop_impl_elim P Q : ((P → Q) ∧ P)%I ⊆ Q. Proof. by intros x n ? [HQ HP]; apply HQ. Qed. Lemma iprop_forall_intro P `(Q: B → iProp A): (∀ b, P ⊆ Q b) → P ⊆ (∀ b, Q b)%I. Proof. by intros HPQ x n ?? b; apply HPQ. Qed. Lemma iprop_forall_elim `(P : B → iProp A) b : (∀ b, P b)%I ⊆ P b. Proof. intros x n ? HP; apply HP. Qed. Lemma iprop_exist_intro `(P : B → iProp A) b : P b ⊆ (∃ b, P b)%I. Proof. by intros x n ??; exists b. Qed. Lemma iprop_exist_elim `(P : B → iProp A) Q : (∀ b, P b ⊆ Q) → (∃ b, P b)%I ⊆ Q. Proof. by intros HPQ x n ? [b ?]; apply HPQ with b. Qed. (* BI connectives *) Lemma iprop_sep_elim_l P Q : (P ★ Q)%I ⊆ P. Proof. intros x n Hvalid (x1&x2&Hx&?&?); rewrite Hx in Hvalid |- *. by apply iprop_weaken with x1 n; auto using ra_included_l. Qed. Global Instance iprop_sep_left_id : LeftId (≡) True%I (@iprop_sep A). Proof. intros P x n Hvalid; split. * intros (x1&x2&Hx&_&?); rewrite Hx in Hvalid |- *. apply iprop_weaken with x2 n; auto using ra_included_r. `````` Robbert Krebbers committed Nov 11, 2015 296 `````` * by intros ?; exists (unit x), x; rewrite ra_unit_l. `````` Robbert Krebbers committed Nov 11, 2015 297 298 299 300 ``````Qed. Global Instance iprop_sep_commutative : Commutative (≡) (@iprop_sep A). Proof. by intros P Q x n ?; split; `````` Robbert Krebbers committed Nov 11, 2015 301 `````` intros (x1&x2&?&?&?); exists x2, x1; rewrite (commutative op). `````` Robbert Krebbers committed Nov 11, 2015 302 303 304 305 ``````Qed. Global Instance iprop_sep_associative : Associative (≡) (@iprop_sep A). Proof. intros P Q R x n ?; split. `````` Robbert Krebbers committed Nov 11, 2015 306 `````` * intros (x1&x2&Hx&?&y1&y2&Hy&?&?); exists (x1 ⋅ y1), y2; split_ands; auto. `````` Robbert Krebbers committed Nov 11, 2015 307 `````` + by rewrite <-(associative op), <-Hy, <-Hx. `````` Robbert Krebbers committed Nov 11, 2015 308 309 `````` + by exists x1, y1. * intros (x1&x2&Hx&(y1&y2&Hy&?&?)&?); exists y1, (y2 ⋅ x2); split_ands; auto. `````` Robbert Krebbers committed Nov 11, 2015 310 `````` + by rewrite (associative op), <-Hy, <-Hx. `````` Robbert Krebbers committed Nov 11, 2015 311 `````` + by exists y2, x2. `````` Robbert Krebbers committed Nov 11, 2015 312 313 314 315 ``````Qed. Lemma iprop_wand_intro P Q R : (R ★ P)%I ⊆ Q → R ⊆ (P -★ Q)%I. Proof. intros HPQ x n ?? x' n' ???; apply HPQ; auto. `````` Robbert Krebbers committed Nov 11, 2015 316 `````` exists x, x'; split_ands; auto. `````` Robbert Krebbers committed Nov 11, 2015 317 318 319 320 321 322 323 324 `````` eapply iprop_weaken with x n; eauto using cmra_valid_op_l. Qed. Lemma iprop_wand_elim P Q : ((P -★ Q) ★ P)%I ⊆ Q. Proof. by intros x n Hvalid (x1&x2&Hx&HPQ&?); rewrite Hx in Hvalid |- *; apply HPQ. Qed. Lemma iprop_sep_or P Q R : ((P ∨ Q) ★ R)%I ≡ ((P ★ R) ∨ (Q ★ R))%I. Proof. `````` Robbert Krebbers committed Nov 11, 2015 325 326 `````` split; [by intros (x1&x2&Hx&[?|?]&?); [left|right]; exists x1, x2|]. intros [(x1&x2&Hx&?&?)|(x1&x2&Hx&?&?)]; exists x1, x2; split_ands; `````` Robbert Krebbers committed Nov 11, 2015 327 328 329 `````` first [by left | by right | done]. Qed. Lemma iprop_sep_and P Q R : ((P ∧ Q) ★ R)%I ⊆ ((P ★ R) ∧ (Q ★ R))%I. `````` Robbert Krebbers committed Nov 11, 2015 330 ``````Proof. by intros x n ? (x1&x2&Hx&[??]&?); split; exists x1, x2. Qed. `````` Robbert Krebbers committed Nov 11, 2015 331 332 333 ``````Lemma iprop_sep_exist {B} (P : B → iProp A) Q : ((∃ b, P b) ★ Q)%I ≡ (∃ b, P b ★ Q)%I. Proof. `````` Robbert Krebbers committed Nov 11, 2015 334 335 `````` split; [by intros (x1&x2&Hx&[b ?]&?); exists b, x1, x2|]. intros [b (x1&x2&Hx&?&?)]; exists x1, x2; split_ands; by try exists b. `````` Robbert Krebbers committed Nov 11, 2015 336 337 338 ``````Qed. Lemma iprop_sep_forall `(P : B → iProp A) Q : ((∀ b, P b) ★ Q)%I ⊆ (∀ b, P b ★ Q)%I. `````` Robbert Krebbers committed Nov 11, 2015 339 ``````Proof. by intros x n ? (x1&x2&Hx&?&?); intros b; exists x1, x2. Qed. `````` Robbert Krebbers committed Nov 11, 2015 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 `````` (* Later *) Global Instance iprop_later_contractive : Contractive (@iprop_later A). Proof. intros n P Q HPQ x [|n'] ??; simpl; [done|]. apply HPQ; eauto using cmra_valid_S. Qed. Lemma iprop_later_weaken P : P ⊆ (▷ P)%I. Proof. intros x [|n] ??; simpl in *; [done|]. apply iprop_weaken with x (S n); auto using cmra_valid_S. Qed. Lemma iprop_lub P : (▷ P → P)%I ⊆ P. Proof. intros x n ? HP; induction n as [|n IH]; [by apply HP|]. apply HP, IH, iprop_weaken with x (S n); eauto using cmra_valid_S. Qed. Lemma iprop_later_impl P Q : (▷ (P → Q))%I ⊆ (▷ P → ▷ Q)%I. Proof. intros x [|n] ? HPQ x' [|n'] ???; auto with lia. apply HPQ; auto using cmra_valid_S. Qed. Lemma iprop_later_and P Q : (▷ (P ∧ Q))%I ≡ (▷ P ∧ ▷ Q)%I. Proof. by intros x [|n]; split. Qed. Lemma iprop_later_or P Q : (▷ (P ∨ Q))%I ≡ (▷ P ∨ ▷ Q)%I. Proof. intros x [|n]; simpl; tauto. Qed. Lemma iprop_later_forall `(P : B → iProp A) : (▷ ∀ b, P b)%I ≡ (∀ b, ▷ P b)%I. Proof. by intros x [|n]. Qed. Lemma iprop_later_exist `(P : B → iProp A) : (∃ b, ▷ P b)%I ⊆ (▷ ∃ b, P b)%I. Proof. by intros x [|n]. Qed. Lemma iprop_later_exist' `{Inhabited B} (P : B → iProp A) : (▷ ∃ b, P b)%I ≡ (∃ b, ▷ P b)%I. Proof. intros x [|n]; split; try done. by destruct (_ : Inhabited B) as [b]; exists b. Qed. Lemma iprop_later_sep P Q : (▷ (P ★ Q))%I ≡ (▷ P ★ ▷ Q)%I. Proof. intros x n ?; split. `````` Robbert Krebbers committed Nov 11, 2015 379 `````` * destruct n as [|n]; simpl; [by exists x, x|intros (x1&x2&Hx&?&?)]. `````` Robbert Krebbers committed Nov 11, 2015 380 381 `````` destruct (cmra_extend_op x x1 x2 n) as ([y1 y2]&Hx'&Hy1&Hy2); auto using cmra_valid_S; simpl in *. `````` Robbert Krebbers committed Nov 11, 2015 382 `````` exists y1, y2; split; [by rewrite Hx'|by rewrite Hy1, Hy2]. `````` Robbert Krebbers committed Nov 11, 2015 383 `````` * destruct n as [|n]; simpl; [done|intros (x1&x2&Hx&?&?)]. `````` Robbert Krebbers committed Nov 11, 2015 384 `````` exists x1, x2; eauto using (dist_S (A:=A)). `````` Robbert Krebbers committed Nov 11, 2015 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 ``````Qed. (* Always *) Lemma iprop_always_necessity P : (□ P)%I ⊆ P. Proof. intros x n ??; apply iprop_weaken with (unit x) n;auto using ra_included_unit. Qed. Lemma iprop_always_intro P Q : (□ P)%I ⊆ Q → (□ P)%I ⊆ (□ Q)%I. Proof. intros HPQ x n ??; apply HPQ; simpl in *; auto using cmra_unit_valid. by rewrite ra_unit_idempotent. Qed. Lemma iprop_always_impl P Q : (□ (P → Q))%I ⊆ (□P → □Q)%I. Proof. intros x n ? HPQ x' n' ???. apply HPQ; auto using ra_unit_preserving, cmra_unit_valid. Qed. Lemma iprop_always_and P Q : (□ (P ∧ Q))%I ≡ (□ P ∧ □ Q)%I. Proof. done. Qed. Lemma iprop_always_or P Q : (□ (P ∨ Q))%I ≡ (□ P ∨ □ Q)%I. Proof. done. Qed. Lemma iprop_always_forall `(P : B → iProp A) : (□ ∀ b, P b)%I ≡ (∀ b, □ P b)%I. Proof. done. Qed. Lemma iprop_always_exist `(P : B → iProp A) : (□ ∃ b, P b)%I ≡ (∃ b, □ P b)%I. Proof. done. Qed. Lemma iprop_always_and_always_box P Q : (□ P ∧ Q)%I ⊆ (□ P ★ Q)%I. Proof. `````` Robbert Krebbers committed Nov 11, 2015 412 `````` intros x n ? [??]; exists (unit x), x; simpl in *. `````` Robbert Krebbers committed Nov 11, 2015 413 414 415 `````` by rewrite ra_unit_l, ra_unit_idempotent. Qed. `````` Robbert Krebbers committed Nov 12, 2015 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 ``````(* Own *) Lemma iprop_own_op (a1 a2 : A) : iprop_own (a1 ⋅ a2) ≡ (iprop_own a1 ★ iprop_own a2)%I. Proof. intros x n ?; split. * intros [z ?]; exists a1, (a2 ⋅ z); split; [by rewrite (associative op)|]. split. by exists (unit a1); rewrite ra_unit_r. by exists z. * intros (y1&y2&Hx&[z1 Hy1]&[z2 Hy2]); exists (z1 ⋅ z2). rewrite (associative op), <-(commutative op z1), <-!(associative op), <-Hy2. by rewrite (associative op), (commutative op z1), <-Hy1. Qed. Lemma iprop_own_valid (a : A) : iprop_own a ⊆ iprop_valid a. Proof. intros x n Hv [a' Hx]; simpl; rewrite Hx in Hv; eauto using cmra_valid_op_l. Qed. `````` Robbert Krebbers committed Nov 11, 2015 432 433 434 435 436 ``````(* Fix *) Lemma iprop_fixpoint_unfold (P : iProp A → iProp A) `{!Contractive P} : iprop_fixpoint P ≡ P (iprop_fixpoint P). Proof. apply fixpoint_unfold. Qed. End logic.``````