upred.v 31.5 KB
 Robbert Krebbers committed Oct 30, 2017 1 ``````From iris.algebra Require Export cmra updates. `````` Jacques-Henri Jourdan committed Dec 14, 2017 2 ``````From iris.bi Require Export derived_connectives updates. `````` Robbert Krebbers committed Dec 02, 2017 3 ``````From stdpp Require Import finite. `````` Ralf Jung committed Jan 05, 2017 4 ``````Set Default Proof Using "Type". `````` Robbert Krebbers committed Oct 30, 2017 5 6 7 ``````Local Hint Extern 1 (_ ≼ _) => etrans; [eassumption|]. Local Hint Extern 1 (_ ≼ _) => etrans; [|eassumption]. Local Hint Extern 10 (_ ≤ _) => omega. `````` Robbert Krebbers committed Oct 25, 2016 8 `````` `````` Ralf Jung committed Jan 05, 2017 9 10 11 12 13 ``````(** The basic definition of the uPred type, its metric and functor laws. You probably do not want to import this file. Instead, import base_logic.base_logic; that will also give you all the primitive and many derived laws for the logic. *) `````` Ralf Jung committed Dec 08, 2017 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 ``````(* A good way of understanding this definition of the uPred OFE is to consider the OFE uPred0 of monotonous SProp predicates. That is, uPred0 is the OFE of non-expansive functions from M to SProp that are monotonous with respect to CMRA inclusion. This notion of monotonicity has to be stated in the SProp logic. Together with the usual closedness property of SProp, this gives exactly uPred_mono. Then, we quotient uPred0 *in the sProp logic* with respect to equivalence on valid elements of M. That is, we quotient with respect to the following *sProp* equivalence relation: P1 ≡ P2 := ∀ x, ✓ x → (P1(x) ↔ P2(x)) (1) When seen from the ambiant logic, obtaining this quotient requires definig both a custom Equiv and Dist. It is worth noting that this equivalence relation admits canonical representatives. More precisely, one can show that every equivalence class contains exactly one element P0 such that: `````` Ralf Jung committed Dec 08, 2017 32 33 34 35 36 `````` ∀ x, (✓ x → P0(x)) → P0(x) (2) (Again, this assertion has to be understood in sProp). Intuitively, this says that P0 trivially holds whenever the resource is invalid. Starting from any element P, one can find this canonical representative by choosing: `````` Ralf Jung committed Dec 08, 2017 37 38 39 40 41 42 43 44 45 46 47 48 `````` P0(x) := ✓ x → P(x) (3) Hence, as an alternative definition of uPred, we could use the set of canonical representatives (i.e., the subtype of monotonous sProp predicates that verify (2)). This alternative definition would save us from using a quotient. However, the definitions of the various connectives would get more complicated, because we have to make sure they all verify (2), which sometimes requires some adjustments. We would moreover need to prove one more property for every logical connective. *) `````` Robbert Krebbers committed Oct 25, 2016 49 50 ``````Record uPred (M : ucmraT) : Type := IProp { uPred_holds :> nat → M → Prop; `````` Jacques-Henri Jourdan committed Apr 04, 2017 51 `````` `````` Jacques-Henri Jourdan committed Dec 07, 2017 52 53 `````` uPred_mono n1 n2 x1 x2 : uPred_holds n1 x1 → x1 ≼{n1} x2 → n2 ≤ n1 → uPred_holds n2 x2 `````` Robbert Krebbers committed Oct 25, 2016 54 55 56 57 58 ``````}. Arguments uPred_holds {_} _ _ _ : simpl never. Add Printing Constructor uPred. Instance: Params (@uPred_holds) 3. `````` Robbert Krebbers committed Oct 30, 2017 59 ``````Bind Scope bi_scope with uPred. `````` Robbert Krebbers committed Oct 25, 2016 60 61 62 63 64 65 66 67 68 69 70 ``````Arguments uPred_holds {_} _%I _ _. Section cofe. Context {M : ucmraT}. Inductive uPred_equiv' (P Q : uPred M) : Prop := { uPred_in_equiv : ∀ n x, ✓{n} x → P n x ↔ Q n x }. Instance uPred_equiv : Equiv (uPred M) := uPred_equiv'. Inductive uPred_dist' (n : nat) (P Q : uPred M) : Prop := { uPred_in_dist : ∀ n' x, n' ≤ n → ✓{n'} x → P n' x ↔ Q n' x }. Instance uPred_dist : Dist (uPred M) := uPred_dist'. `````` Ralf Jung committed Nov 22, 2016 71 `````` Definition uPred_ofe_mixin : OfeMixin (uPred M). `````` Robbert Krebbers committed Oct 25, 2016 72 73 74 75 76 77 78 79 80 81 82 83 `````` Proof. split. - intros P Q; split. + by intros HPQ n; split=> i x ??; apply HPQ. + intros HPQ; split=> n x ?; apply HPQ with n; auto. - intros n; split. + by intros P; split=> x i. + by intros P Q HPQ; split=> x i ??; symmetry; apply HPQ. + intros P Q Q' HP HQ; split=> i x ??. by trans (Q i x);[apply HP|apply HQ]. - intros n P Q HPQ; split=> i x ??; apply HPQ; auto. Qed. `````` Ralf Jung committed Nov 22, 2016 84 85 86 `````` Canonical Structure uPredC : ofeT := OfeT (uPred M) uPred_ofe_mixin. Program Definition uPred_compl : Compl uPredC := λ c, `````` Jacques-Henri Jourdan committed Dec 06, 2017 87 `````` {| uPred_holds n x := ∀ n', n' ≤ n → ✓{n'}x → c n' n' x |}. `````` Ralf Jung committed Nov 22, 2016 88 `````` Next Obligation. `````` Jacques-Henri Jourdan committed Dec 07, 2017 89 90 91 `````` move=> /= c n1 n2 x1 x2 HP Hx12 Hn12 n3 Hn23 Hv. eapply uPred_mono. eapply HP, cmra_validN_includedN, cmra_includedN_le=>//; lia. eapply cmra_includedN_le=>//; lia. done. `````` Ralf Jung committed Nov 22, 2016 92 93 94 `````` Qed. Global Program Instance uPred_cofe : Cofe uPredC := {| compl := uPred_compl |}. Next Obligation. `````` Jacques-Henri Jourdan committed Dec 06, 2017 95 96 `````` intros n c; split=>i x Hin Hv. etrans; [|by symmetry; apply (chain_cauchy c i n)]. split=>H; [by apply H|]. `````` Jacques-Henri Jourdan committed Dec 07, 2017 97 `````` repeat intro. apply (chain_cauchy c n' i)=>//. by eapply uPred_mono. `````` Ralf Jung committed Nov 22, 2016 98 `````` Qed. `````` Robbert Krebbers committed Oct 25, 2016 99 100 101 102 103 104 105 106 107 108 109 110 111 ``````End cofe. Arguments uPredC : clear implicits. Instance uPred_ne {M} (P : uPred M) n : Proper (dist n ==> iff) (P n). Proof. intros x1 x2 Hx; split=> ?; eapply uPred_mono; eauto; by rewrite Hx. Qed. Instance uPred_proper {M} (P : uPred M) n : Proper ((≡) ==> iff) (P n). Proof. by intros x1 x2 Hx; apply uPred_ne, equiv_dist. Qed. Lemma uPred_holds_ne {M} (P Q : uPred M) n1 n2 x : P ≡{n2}≡ Q → n2 ≤ n1 → ✓{n2} x → Q n1 x → P n2 x. Proof. `````` Jacques-Henri Jourdan committed Dec 07, 2017 112 `````` intros [Hne] ???. eapply Hne; try done. eauto using uPred_mono, cmra_validN_le. `````` Robbert Krebbers committed Oct 25, 2016 113 114 ``````Qed. `````` Ralf Jung committed Dec 08, 2017 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 ``````(* Equivalence to the definition of uPred in the appendix. *) Lemma uPred_alt {M : ucmraT} (P: nat → M → Prop) : (∀ n1 n2 x1 x2, P n1 x1 → x1 ≼{n1} x2 → n2 ≤ n1 → P n2 x2) ↔ ( (∀ x n1 n2, n2 ≤ n1 → P n1 x → P n2 x) (* Pointwise down-closed *) ∧ (∀ n x1 x2, x1 ≡{n}≡ x2 → ∀ m, m ≤ n → P m x1 ↔ P m x2) (* Non-expansive *) ∧ (∀ n x1 x2, x1 ≼{n} x2 → ∀ m, m ≤ n → P m x1 → P m x2) (* Monotonicity *) ). Proof. (* Provide this lemma to eauto. *) assert (∀ n1 n2 (x1 x2 : M), n2 ≤ n1 → x1 ≡{n1}≡ x2 → x1 ≼{n2} x2). { intros ????? H. eapply cmra_includedN_le; last done. by rewrite H. } (* Now go ahead. *) split. - intros Hupred. repeat split; eauto using cmra_includedN_le. - intros (Hdown & _ & Hmono) **. eapply Hmono; [done..|]. eapply Hdown; done. `````` Robbert Krebbers committed Oct 25, 2016 130 131 132 133 ``````Qed. (** functor *) Program Definition uPred_map {M1 M2 : ucmraT} (f : M2 -n> M1) `````` Robbert Krebbers committed Oct 25, 2017 134 `````` `{!CmraMorphism f} (P : uPred M1) : `````` Robbert Krebbers committed Oct 25, 2016 135 `````` uPred M2 := {| uPred_holds n x := P n (f x) |}. `````` 136 ``````Next Obligation. naive_solver eauto using uPred_mono, cmra_morphism_monotoneN. Qed. `````` Robbert Krebbers committed Oct 25, 2016 137 138 `````` Instance uPred_map_ne {M1 M2 : ucmraT} (f : M2 -n> M1) `````` Robbert Krebbers committed Oct 25, 2017 139 `````` `{!CmraMorphism f} n : Proper (dist n ==> dist n) (uPred_map f). `````` Robbert Krebbers committed Oct 25, 2016 140 141 ``````Proof. intros x1 x2 Hx; split=> n' y ??. `````` 142 `````` split; apply Hx; auto using cmra_morphism_validN. `````` Robbert Krebbers committed Oct 25, 2016 143 144 145 146 ``````Qed. Lemma uPred_map_id {M : ucmraT} (P : uPred M): uPred_map cid P ≡ P. Proof. by split=> n x ?. Qed. Lemma uPred_map_compose {M1 M2 M3 : ucmraT} (f : M1 -n> M2) (g : M2 -n> M3) `````` Robbert Krebbers committed Oct 25, 2017 147 `````` `{!CmraMorphism f, !CmraMorphism g} (P : uPred M3): `````` Robbert Krebbers committed Oct 25, 2016 148 149 150 `````` uPred_map (g ◎ f) P ≡ uPred_map f (uPred_map g P). Proof. by split=> n x Hx. Qed. Lemma uPred_map_ext {M1 M2 : ucmraT} (f g : M1 -n> M2) `````` Robbert Krebbers committed Oct 25, 2017 151 `````` `{!CmraMorphism f} `{!CmraMorphism g}: `````` Robbert Krebbers committed Oct 25, 2016 152 153 `````` (∀ x, f x ≡ g x) → ∀ x, uPred_map f x ≡ uPred_map g x. Proof. intros Hf P; split=> n x Hx /=; by rewrite /uPred_holds /= Hf. Qed. `````` Robbert Krebbers committed Oct 25, 2017 154 ``````Definition uPredC_map {M1 M2 : ucmraT} (f : M2 -n> M1) `{!CmraMorphism f} : `````` Robbert Krebbers committed Oct 25, 2016 155 156 `````` uPredC M1 -n> uPredC M2 := CofeMor (uPred_map f : uPredC M1 → uPredC M2). Lemma uPredC_map_ne {M1 M2 : ucmraT} (f g : M2 -n> M1) `````` Robbert Krebbers committed Oct 25, 2017 157 `````` `{!CmraMorphism f, !CmraMorphism g} n : `````` Robbert Krebbers committed Oct 25, 2016 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 `````` f ≡{n}≡ g → uPredC_map f ≡{n}≡ uPredC_map g. Proof. by intros Hfg P; split=> n' y ??; rewrite /uPred_holds /= (dist_le _ _ _ _(Hfg y)); last lia. Qed. Program Definition uPredCF (F : urFunctor) : cFunctor := {| cFunctor_car A B := uPredC (urFunctor_car F B A); cFunctor_map A1 A2 B1 B2 fg := uPredC_map (urFunctor_map F (fg.2, fg.1)) |}. Next Obligation. intros F A1 A2 B1 B2 n P Q HPQ. apply uPredC_map_ne, urFunctor_ne; split; by apply HPQ. Qed. Next Obligation. intros F A B P; simpl. rewrite -{2}(uPred_map_id P). apply uPred_map_ext=>y. by rewrite urFunctor_id. Qed. Next Obligation. intros F A1 A2 A3 B1 B2 B3 f g f' g' P; simpl. rewrite -uPred_map_compose. apply uPred_map_ext=>y; apply urFunctor_compose. Qed. Instance uPredCF_contractive F : urFunctorContractive F → cFunctorContractive (uPredCF F). Proof. `````` Robbert Krebbers committed Dec 05, 2016 184 185 `````` intros ? A1 A2 B1 B2 n P Q HPQ. apply uPredC_map_ne, urFunctor_contractive. destruct n; split; by apply HPQ. `````` Robbert Krebbers committed Oct 25, 2016 186 187 188 189 190 ``````Qed. (** logical entailement *) Inductive uPred_entails {M} (P Q : uPred M) : Prop := { uPred_in_entails : ∀ n x, ✓{n} x → P n x → Q n x }. `````` Jacques-Henri Jourdan committed Dec 07, 2017 191 ``````Hint Resolve uPred_mono : uPred_def. `````` Robbert Krebbers committed Oct 25, 2016 192 `````` `````` Robbert Krebbers committed Oct 30, 2017 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 ``````(** logical connectives *) Program Definition uPred_pure_def {M} (φ : Prop) : uPred M := {| uPred_holds n x := φ |}. Solve Obligations with done. Definition uPred_pure_aux : seal (@uPred_pure_def). by eexists. Qed. Definition uPred_pure {M} := unseal uPred_pure_aux M. Definition uPred_pure_eq : @uPred_pure = @uPred_pure_def := seal_eq uPred_pure_aux. Definition uPred_emp {M} : uPred M := uPred_pure True. Program Definition uPred_and_def {M} (P Q : uPred M) : uPred M := {| uPred_holds n x := P n x ∧ Q n x |}. Solve Obligations with naive_solver eauto 2 with uPred_def. Definition uPred_and_aux : seal (@uPred_and_def). by eexists. Qed. Definition uPred_and {M} := unseal uPred_and_aux M. Definition uPred_and_eq: @uPred_and = @uPred_and_def := seal_eq uPred_and_aux. Program Definition uPred_or_def {M} (P Q : uPred M) : uPred M := {| uPred_holds n x := P n x ∨ Q n x |}. Solve Obligations with naive_solver eauto 2 with uPred_def. Definition uPred_or_aux : seal (@uPred_or_def). by eexists. Qed. Definition uPred_or {M} := unseal uPred_or_aux M. Definition uPred_or_eq: @uPred_or = @uPred_or_def := seal_eq uPred_or_aux. Program Definition uPred_impl_def {M} (P Q : uPred M) : uPred M := {| uPred_holds n x := ∀ n' x', x ≼ x' → n' ≤ n → ✓{n'} x' → P n' x' → Q n' x' |}. Next Obligation. `````` Jacques-Henri Jourdan committed Dec 21, 2017 222 `````` intros M P Q n1 n1' x1 x1' HPQ [x2 Hx1'] Hn1 n2 x3 [x4 Hx3] ?; simpl in *. `````` Robbert Krebbers committed Oct 30, 2017 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 `````` rewrite Hx3 (dist_le _ _ _ _ Hx1'); auto. intros ??. eapply HPQ; auto. exists (x2 ⋅ x4); by rewrite assoc. Qed. Definition uPred_impl_aux : seal (@uPred_impl_def). by eexists. Qed. Definition uPred_impl {M} := unseal uPred_impl_aux M. Definition uPred_impl_eq : @uPred_impl = @uPred_impl_def := seal_eq uPred_impl_aux. Program Definition uPred_forall_def {M A} (Ψ : A → uPred M) : uPred M := {| uPred_holds n x := ∀ a, Ψ a n x |}. Solve Obligations with naive_solver eauto 2 with uPred_def. Definition uPred_forall_aux : seal (@uPred_forall_def). by eexists. Qed. Definition uPred_forall {M A} := unseal uPred_forall_aux M A. Definition uPred_forall_eq : @uPred_forall = @uPred_forall_def := seal_eq uPred_forall_aux. Program Definition uPred_exist_def {M A} (Ψ : A → uPred M) : uPred M := {| uPred_holds n x := ∃ a, Ψ a n x |}. Solve Obligations with naive_solver eauto 2 with uPred_def. Definition uPred_exist_aux : seal (@uPred_exist_def). by eexists. Qed. Definition uPred_exist {M A} := unseal uPred_exist_aux M A. Definition uPred_exist_eq: @uPred_exist = @uPred_exist_def := seal_eq uPred_exist_aux. Program Definition uPred_internal_eq_def {M} {A : ofeT} (a1 a2 : A) : uPred M := {| uPred_holds n x := a1 ≡{n}≡ a2 |}. Solve Obligations with naive_solver eauto 2 using (dist_le (A:=A)). Definition uPred_internal_eq_aux : seal (@uPred_internal_eq_def). by eexists. Qed. Definition uPred_internal_eq {M A} := unseal uPred_internal_eq_aux M A. Definition uPred_internal_eq_eq: @uPred_internal_eq = @uPred_internal_eq_def := seal_eq uPred_internal_eq_aux. Program Definition uPred_sep_def {M} (P Q : uPred M) : uPred M := {| uPred_holds n x := ∃ x1 x2, x ≡{n}≡ x1 ⋅ x2 ∧ P n x1 ∧ Q n x2 |}. Next Obligation. `````` Jacques-Henri Jourdan committed Dec 21, 2017 257 `````` intros M P Q n1 n2 x y (x1&x2&Hx&?&?) [z Hy] Hn. `````` Robbert Krebbers committed Oct 30, 2017 258 `````` exists x1, (x2 ⋅ z); split_and?; eauto using uPred_mono, cmra_includedN_l. `````` Jacques-Henri Jourdan committed Dec 21, 2017 259 `````` eapply dist_le, Hn. by rewrite Hy Hx assoc. `````` Robbert Krebbers committed Oct 30, 2017 260 261 262 263 264 265 266 267 268 ``````Qed. Definition uPred_sep_aux : seal (@uPred_sep_def). by eexists. Qed. Definition uPred_sep {M} := unseal uPred_sep_aux M. Definition uPred_sep_eq: @uPred_sep = @uPred_sep_def := seal_eq uPred_sep_aux. Program Definition uPred_wand_def {M} (P Q : uPred M) : uPred M := {| uPred_holds n x := ∀ n' x', n' ≤ n → ✓{n'} (x ⋅ x') → P n' x' → Q n' (x ⋅ x') |}. Next Obligation. `````` Jacques-Henri Jourdan committed Dec 21, 2017 269 270 `````` intros M P Q n1 n1' x1 x1' HPQ ? Hn n3 x3 ???; simpl in *. eapply uPred_mono with n3 (x1 ⋅ x3); `````` Robbert Krebbers committed Oct 30, 2017 271 272 273 274 275 276 277 `````` eauto using cmra_validN_includedN, cmra_monoN_r, cmra_includedN_le. Qed. Definition uPred_wand_aux : seal (@uPred_wand_def). by eexists. Qed. Definition uPred_wand {M} := unseal uPred_wand_aux M. Definition uPred_wand_eq : @uPred_wand = @uPred_wand_def := seal_eq uPred_wand_aux. `````` Jacques-Henri Jourdan committed Nov 03, 2017 278 279 280 ``````(* Equivalently, this could be `∀ y, P n y`. That's closer to the intuition of "embedding the step-indexed logic in Iris", but the two are equivalent because Iris is afine. The following is easier to work with. *) `````` Jacques-Henri Jourdan committed Nov 03, 2017 281 282 ``````Program Definition uPred_plainly_def {M} (P : uPred M) : uPred M := {| uPred_holds n x := P n ε |}. `````` Jacques-Henri Jourdan committed Dec 21, 2017 283 ``````Solve Obligations with naive_solver eauto using uPred_mono, ucmra_unit_validN. `````` Jacques-Henri Jourdan committed Nov 03, 2017 284 285 286 287 288 ``````Definition uPred_plainly_aux : seal (@uPred_plainly_def). by eexists. Qed. Definition uPred_plainly {M} := unseal uPred_plainly_aux M. Definition uPred_plainly_eq : @uPred_plainly = @uPred_plainly_def := seal_eq uPred_plainly_aux. `````` Robbert Krebbers committed Oct 30, 2017 289 290 291 292 293 294 295 296 297 298 299 300 301 ``````Program Definition uPred_persistently_def {M} (P : uPred M) : uPred M := {| uPred_holds n x := P n (core x) |}. Next Obligation. intros M; naive_solver eauto using uPred_mono, @cmra_core_monoN. Qed. Definition uPred_persistently_aux : seal (@uPred_persistently_def). by eexists. Qed. Definition uPred_persistently {M} := unseal uPred_persistently_aux M. Definition uPred_persistently_eq : @uPred_persistently = @uPred_persistently_def := seal_eq uPred_persistently_aux. Program Definition uPred_later_def {M} (P : uPred M) : uPred M := {| uPred_holds n x := match n return _ with 0 => True | S n' => P n' x end |}. Next Obligation. `````` Jacques-Henri Jourdan committed Dec 21, 2017 302 `````` intros M P [|n1] [|n2] x1 x2; eauto using uPred_mono, cmra_includedN_S with lia. `````` Robbert Krebbers committed Oct 30, 2017 303 304 305 306 307 308 309 310 311 ``````Qed. Definition uPred_later_aux : seal (@uPred_later_def). by eexists. Qed. Definition uPred_later {M} := unseal uPred_later_aux M. Definition uPred_later_eq : @uPred_later = @uPred_later_def := seal_eq uPred_later_aux. Program Definition uPred_ownM_def {M : ucmraT} (a : M) : uPred M := {| uPred_holds n x := a ≼{n} x |}. Next Obligation. `````` Jacques-Henri Jourdan committed Dec 21, 2017 312 313 `````` intros M a n1 n2 x1 x [a' Hx1] [x2 Hx] Hn. eapply cmra_includedN_le=>//. exists (a' ⋅ x2). by rewrite Hx(assoc op) Hx1. `````` Robbert Krebbers committed Oct 30, 2017 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 ``````Qed. Definition uPred_ownM_aux : seal (@uPred_ownM_def). by eexists. Qed. Definition uPred_ownM {M} := unseal uPred_ownM_aux M. Definition uPred_ownM_eq : @uPred_ownM = @uPred_ownM_def := seal_eq uPred_ownM_aux. Program Definition uPred_cmra_valid_def {M} {A : cmraT} (a : A) : uPred M := {| uPred_holds n x := ✓{n} a |}. Solve Obligations with naive_solver eauto 2 using cmra_validN_le. Definition uPred_cmra_valid_aux : seal (@uPred_cmra_valid_def). by eexists. Qed. Definition uPred_cmra_valid {M A} := unseal uPred_cmra_valid_aux M A. Definition uPred_cmra_valid_eq : @uPred_cmra_valid = @uPred_cmra_valid_def := seal_eq uPred_cmra_valid_aux. Program Definition uPred_bupd_def {M} (Q : uPred M) : uPred M := {| uPred_holds n x := ∀ k yf, k ≤ n → ✓{k} (x ⋅ yf) → ∃ x', ✓{k} (x' ⋅ yf) ∧ Q k x' |}. Next Obligation. `````` Jacques-Henri Jourdan committed Dec 21, 2017 332 `````` intros M Q n1 n2 x1 x2 HQ [x3 Hx] Hn k yf Hk. `````` Robbert Krebbers committed Oct 30, 2017 333 334 335 `````` rewrite (dist_le _ _ _ _ Hx); last lia. intros Hxy. destruct (HQ k (x3 ⋅ yf)) as (x'&?&?); [auto|by rewrite assoc|]. exists (x' ⋅ x3); split; first by rewrite -assoc. `````` Jacques-Henri Jourdan committed Dec 21, 2017 336 `````` eauto using uPred_mono, cmra_includedN_l. `````` Robbert Krebbers committed Oct 30, 2017 337 ``````Qed. `````` Jacques-Henri Jourdan committed Dec 11, 2017 338 339 340 341 ``````Definition uPred_bupd_aux {M} : seal (@uPred_bupd_def M). by eexists. Qed. Instance uPred_bupd {M} : BUpd (uPred M) := unseal uPred_bupd_aux. Definition uPred_bupd_eq {M} : @bupd _ uPred_bupd = @uPred_bupd_def M := seal_eq uPred_bupd_aux. `````` Robbert Krebbers committed Oct 30, 2017 342 343 344 345 346 `````` Module uPred_unseal. Definition unseal_eqs := (uPred_pure_eq, uPred_and_eq, uPred_or_eq, uPred_impl_eq, uPred_forall_eq, uPred_exist_eq, uPred_internal_eq_eq, uPred_sep_eq, uPred_wand_eq, `````` Jacques-Henri Jourdan committed Nov 03, 2017 347 `````` uPred_plainly_eq, uPred_persistently_eq, uPred_later_eq, uPred_ownM_eq, `````` Jacques-Henri Jourdan committed Dec 11, 2017 348 `````` uPred_cmra_valid_eq, @uPred_bupd_eq). `````` Robbert Krebbers committed Nov 14, 2017 349 350 351 ``````Ltac unseal := (* Coq unfold is used to circumvent bug #5699 in rewrite /foo *) unfold bi_emp; simpl; unfold uPred_emp, bi_pure, bi_and, bi_or, bi_impl, bi_forall, bi_exist, `````` Jacques-Henri Jourdan committed Dec 04, 2017 352 `````` bi_internal_eq, bi_sep, bi_wand, bi_plainly, bi_persistently, sbi_later; simpl; `````` Robbert Krebbers committed Nov 14, 2017 353 354 355 `````` unfold sbi_emp, sbi_pure, sbi_and, sbi_or, sbi_impl, sbi_forall, sbi_exist, sbi_internal_eq, sbi_sep, sbi_wand, sbi_plainly, sbi_persistently; simpl; rewrite !unseal_eqs /=. `````` Robbert Krebbers committed Oct 30, 2017 356 357 358 359 360 ``````End uPred_unseal. Import uPred_unseal. Local Arguments uPred_holds {_} !_ _ _ /. `````` Ralf Jung committed Dec 20, 2017 361 ``````Lemma uPred_bi_mixin (M : ucmraT) : BiMixin (ofe_mixin_of (uPred M)) `````` Robbert Krebbers committed Oct 30, 2017 362 363 `````` uPred_entails uPred_emp uPred_pure uPred_and uPred_or uPred_impl (@uPred_forall M) (@uPred_exist M) (@uPred_internal_eq M) `````` Jacques-Henri Jourdan committed Nov 03, 2017 364 `````` uPred_sep uPred_wand uPred_plainly uPred_persistently. `````` Robbert Krebbers committed Oct 30, 2017 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 ``````Proof. split. - (* PreOrder uPred_entails *) split. + by intros P; split=> x i. + by intros P Q Q' HP HQ; split=> x i ??; apply HQ, HP. - (* (P ⊣⊢ Q) ↔ (P ⊢ Q) ∧ (Q ⊢ P) *) intros P Q. split. + intros HPQ; split; split=> x i; apply HPQ. + intros [HPQ HQP]; split=> x n; by split; [apply HPQ|apply HQP]. - (* Proper (iff ==> dist n) (@uPred_pure M) *) intros φ1 φ2 Hφ. by unseal; split=> -[|n] ?; try apply Hφ. - (* NonExpansive2 uPred_and *) intros n P P' HP Q Q' HQ; unseal; split=> x n' ??. split; (intros [??]; split; [by apply HP|by apply HQ]). - (* NonExpansive2 uPred_or *) intros n P P' HP Q Q' HQ; split=> x n' ??. unseal; split; (intros [?|?]; [left; by apply HP|right; by apply HQ]). - (* NonExpansive2 uPred_impl *) intros n P P' HP Q Q' HQ; split=> x n' ??. unseal; split; intros HPQ x' n'' ????; apply HQ, HPQ, HP; auto. - (* Proper (pointwise_relation A (dist n) ==> dist n) uPred_forall *) by intros A n Ψ1 Ψ2 HΨ; unseal; split=> n' x; split; intros HP a; apply HΨ. - (* Proper (pointwise_relation A (dist n) ==> dist n) uPred_exist *) intros A n Ψ1 Ψ2 HΨ. unseal; split=> n' x ??; split; intros [a ?]; exists a; by apply HΨ. - (* NonExpansive2 uPred_sep *) intros n P P' HP Q Q' HQ; split=> n' x ??. unseal; split; intros (x1&x2&?&?&?); ofe_subst x; exists x1, x2; split_and!; try (apply HP || apply HQ); eauto using cmra_validN_op_l, cmra_validN_op_r. - (* NonExpansive2 uPred_wand *) intros n P P' HP Q Q' HQ; split=> n' x ??. unseal; split; intros HPQ x' n'' ???; apply HQ, HPQ, HP; eauto using cmra_validN_op_r. `````` Jacques-Henri Jourdan committed Nov 03, 2017 400 401 402 `````` - (* NonExpansive uPred_plainly *) intros n P1 P2 HP. unseal; split=> n' x; split; apply HP; eauto using @ucmra_unit_validN. `````` Robbert Krebbers committed Oct 30, 2017 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 `````` - (* NonExpansive uPred_persistently *) intros n P1 P2 HP. unseal; split=> n' x; split; apply HP; eauto using @cmra_core_validN. - (* NonExpansive2 (@uPred_internal_eq M A) *) intros A n x x' Hx y y' Hy; split=> n' z; unseal; split; intros; simpl in *. + by rewrite -(dist_le _ _ _ _ Hx) -?(dist_le _ _ _ _ Hy); auto. + by rewrite (dist_le _ _ _ _ Hx) ?(dist_le _ _ _ _ Hy); auto. - (* φ → P ⊢ ⌜φ⌝ *) intros P φ ?. unseal; by split. - (* (φ → True ⊢ P) → ⌜φ⌝ ⊢ P *) intros φ P. unseal=> HP; split=> n x ??. by apply HP. - (* (∀ x : A, ⌜φ x⌝) ⊢ ⌜∀ x : A, φ x⌝ *) by unseal. - (* P ∧ Q ⊢ P *) intros P Q. unseal; by split=> n x ? [??]. - (* P ∧ Q ⊢ Q *) intros P Q. unseal; by split=> n x ? [??]. - (* (P ⊢ Q) → (P ⊢ R) → P ⊢ Q ∧ R *) intros P Q R HQ HR; unseal; split=> n x ??; by split; [apply HQ|apply HR]. - (* P ⊢ P ∨ Q *) intros P Q. unseal; split=> n x ??; left; auto. - (* Q ⊢ P ∨ Q *) intros P Q. unseal; split=> n x ??; right; auto. - (* (P ⊢ R) → (Q ⊢ R) → P ∨ Q ⊢ R *) intros P Q R HP HQ. unseal; split=> n x ? [?|?]. by apply HP. by apply HQ. - (* (P ∧ Q ⊢ R) → P ⊢ Q → R. *) intros P Q R. unseal => HQ; split=> n x ?? n' x' ????. apply HQ; `````` Jacques-Henri Jourdan committed Dec 21, 2017 430 `````` naive_solver eauto using uPred_mono, cmra_included_includedN. `````` Robbert Krebbers committed Oct 30, 2017 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 `````` - (* (P ⊢ Q → R) → P ∧ Q ⊢ R *) intros P Q R. unseal=> HP; split=> n x ? [??]. apply HP with n x; auto. - (* (∀ a, P ⊢ Ψ a) → P ⊢ ∀ a, Ψ a *) intros A P Ψ. unseal; intros HPΨ; split=> n x ?? a; by apply HPΨ. - (* (∀ a, Ψ a) ⊢ Ψ a *) intros A Ψ a. unseal; split=> n x ? HP; apply HP. - (* Ψ a ⊢ ∃ a, Ψ a *) intros A Ψ a. unseal; split=> n x ??; by exists a. - (* (∀ a, Ψ a ⊢ Q) → (∃ a, Ψ a) ⊢ Q *) intros A Ψ Q. unseal; intros HΨ; split=> n x ? [a ?]; by apply HΨ with a. - (* P ⊢ a ≡ a *) intros A P a. unseal; by split=> n x ?? /=. - (* a ≡ b ⊢ Ψ a → Ψ b *) intros A a b Ψ Hnonexp. unseal; split=> n x ? Hab n' x' ??? HΨ. eapply Hnonexp with n a; auto. - (* (∀ x, f x ≡ g x) ⊢ f ≡ g *) by unseal. - (* `x ≡ `y ⊢ x ≡ y *) by unseal. - (* Discrete a → a ≡ b ⊣⊢ ⌜a ≡ b⌝ *) intros A a b ?. unseal; split=> n x ?; by apply (discrete_iff n). - (* (P ⊢ Q) → (P' ⊢ Q') → P ∗ P' ⊢ Q ∗ Q' *) intros P P' Q Q' HQ HQ'; unseal. split; intros n' x ? (x1&x2&?&?&?); exists x1,x2; ofe_subst x; eauto 7 using cmra_validN_op_l, cmra_validN_op_r, uPred_in_entails. - (* P ⊢ emp ∗ P *) intros P. rewrite /uPred_emp. unseal; split=> n x ?? /=. exists (core x), x. by rewrite cmra_core_l. - (* emp ∗ P ⊢ P *) intros P. unseal; split; intros n x ? (x1&x2&?&_&?); ofe_subst; eauto using uPred_mono, cmra_includedN_r. - (* P ∗ Q ⊢ Q ∗ P *) intros P Q. unseal; split; intros n x ? (x1&x2&?&?&?). exists x2, x1; by rewrite (comm op). - (* (P ∗ Q) ∗ R ⊢ P ∗ (Q ∗ R) *) intros P Q R. unseal; split; intros n x ? (x1&x2&Hx&(y1&y2&Hy&?&?)&?). exists y1, (y2 ⋅ x2); split_and?; auto. + by rewrite (assoc op) -Hy -Hx. + by exists y2, x2. - (* (P ∗ Q ⊢ R) → P ⊢ Q -∗ R *) intros P Q R. unseal=> HPQR; split=> n x ?? n' x' ???; apply HPQR; auto. exists x, x'; split_and?; auto. `````` Jacques-Henri Jourdan committed Dec 21, 2017 473 `````` eapply uPred_mono; eauto using cmra_validN_op_l. `````` Robbert Krebbers committed Oct 30, 2017 474 475 476 `````` - (* (P ⊢ Q -∗ R) → P ∗ Q ⊢ R *) intros P Q R. unseal=> HPQR. split; intros n x ? (?&?&?&?&?). ofe_subst. eapply HPQR; eauto using cmra_validN_op_l. `````` Jacques-Henri Jourdan committed Nov 03, 2017 477 478 479 480 481 482 483 484 485 486 487 488 489 `````` - (* (P ⊢ Q) → bi_plainly P ⊢ bi_plainly Q *) intros P QR HP. unseal; split=> n x ? /=. by apply HP, ucmra_unit_validN. - (* bi_plainly P ⊢ bi_persistently P *) unseal; split; simpl; eauto using uPred_mono, @ucmra_unit_leastN. - (* bi_plainly P ⊢ bi_plainly (bi_plainly P) *) unseal; split=> n x ?? //. - (* (∀ a, bi_plainly (Ψ a)) ⊢ bi_plainly (∀ a, Ψ a) *) by unseal. - (* bi_plainly ((P → Q) ∧ (Q → P)) ⊢ P ≡ Q *) unseal; split=> n x ? /= HPQ; split=> n' x' ? HP; split; eapply HPQ; eauto using @ucmra_unit_least. - (* (bi_plainly P → bi_persistently Q) ⊢ bi_persistently (bi_plainly P → Q) *) unseal; split=> /= n x ? HPQ n' x' ????. `````` Jacques-Henri Jourdan committed Dec 21, 2017 490 `````` eapply uPred_mono with n' (core x)=>//; [|by apply cmra_included_includedN]. `````` Jacques-Henri Jourdan committed Nov 03, 2017 491 492 493 `````` apply (HPQ n' x); eauto using cmra_validN_le. - (* (bi_plainly P → bi_plainly Q) ⊢ bi_plainly (bi_plainly P → Q) *) unseal; split=> /= n x ? HPQ n' x' ????. `````` Jacques-Henri Jourdan committed Dec 21, 2017 494 `````` eapply uPred_mono with n' ε=>//; [|by apply cmra_included_includedN]. `````` Jacques-Henri Jourdan committed Nov 03, 2017 495 496 497 498 499 500 501 `````` apply (HPQ n' x); eauto using cmra_validN_le. - (* P ⊢ bi_plainly emp (ADMISSIBLE) *) by unseal. - (* bi_plainly P ∗ Q ⊢ bi_plainly P *) intros P Q. move: (uPred_persistently P)=> P'. unseal; split; intros n x ? (x1&x2&?&?&_); ofe_subst; eauto using uPred_mono, cmra_includedN_l. `````` Jacques-Henri Jourdan committed Nov 02, 2017 502 `````` - (* (P ⊢ Q) → bi_persistently P ⊢ bi_persistently Q *) `````` Robbert Krebbers committed Oct 30, 2017 503 `````` intros P QR HP. unseal; split=> n x ? /=. by apply HP, cmra_core_validN. `````` Jacques-Henri Jourdan committed Nov 02, 2017 504 `````` - (* bi_persistently P ⊢ bi_persistently (bi_persistently P) *) `````` Robbert Krebbers committed Oct 30, 2017 505 `````` intros P. unseal; split=> n x ?? /=. by rewrite cmra_core_idemp. `````` Jacques-Henri Jourdan committed Nov 03, 2017 506 507 `````` - (* bi_plainly (bi_persistently P) ⊢ bi_plainly P (ADMISSIBLE) *) intros P. unseal; split=> n x ?? /=. by rewrite -(core_id_core ε). `````` Jacques-Henri Jourdan committed Nov 02, 2017 508 `````` - (* (∀ a, bi_persistently (Ψ a)) ⊢ bi_persistently (∀ a, Ψ a) *) `````` Robbert Krebbers committed Oct 30, 2017 509 `````` by unseal. `````` Jacques-Henri Jourdan committed Nov 02, 2017 510 `````` - (* bi_persistently (∃ a, Ψ a) ⊢ ∃ a, bi_persistently (Ψ a) *) `````` Robbert Krebbers committed Oct 30, 2017 511 `````` by unseal. `````` Jacques-Henri Jourdan committed Nov 02, 2017 512 `````` - (* bi_persistently P ∗ Q ⊢ bi_persistently P (ADMISSIBLE) *) `````` Robbert Krebbers committed Oct 30, 2017 513 514 515 `````` intros P Q. move: (uPred_persistently P)=> P'. unseal; split; intros n x ? (x1&x2&?&?&_); ofe_subst; eauto using uPred_mono, cmra_includedN_l. `````` Jacques-Henri Jourdan committed Nov 02, 2017 516 `````` - (* bi_persistently P ∧ Q ⊢ (emp ∧ P) ∗ Q *) `````` Robbert Krebbers committed Oct 30, 2017 517 518 `````` intros P Q. unseal; split=> n x ? [??]; simpl in *. exists (core x), x; rewrite ?cmra_core_l; auto. `````` Robbert Krebbers committed Oct 30, 2017 519 520 ``````Qed. `````` Ralf Jung committed Dec 20, 2017 521 ``````Lemma uPred_sbi_mixin (M : ucmraT) : SbiMixin `````` Robbert Krebbers committed Oct 30, 2017 522 `````` uPred_entails uPred_pure uPred_or uPred_impl `````` Robbert Krebbers committed Oct 30, 2017 523 `````` (@uPred_forall M) (@uPred_exist M) (@uPred_internal_eq M) `````` Jacques-Henri Jourdan committed Nov 03, 2017 524 `````` uPred_sep uPred_plainly uPred_persistently uPred_later. `````` Robbert Krebbers committed Oct 30, 2017 525 526 ``````Proof. split. `````` Jacques-Henri Jourdan committed Dec 04, 2017 527 `````` - (* Contractive sbi_later *) `````` Robbert Krebbers committed Oct 30, 2017 528 529 530 531 532 533 534 535 536 537 538 `````` unseal; intros [|n] P Q HPQ; split=> -[|n'] x ?? //=; try omega. apply HPQ; eauto using cmra_validN_S. - (* Next x ≡ Next y ⊢ ▷ (x ≡ y) *) by unseal. - (* ▷ (x ≡ y) ⊢ Next x ≡ Next y *) by unseal. - (* (P ⊢ Q) → ▷ P ⊢ ▷ Q *) intros P Q. unseal=> HP; split=>-[|n] x ??; [done|apply HP; eauto using cmra_validN_S]. - (* (▷ P → P) ⊢ P *) intros P. unseal; split=> n x ? HP; induction n as [|n IH]; [by apply HP|]. `````` Jacques-Henri Jourdan committed Dec 21, 2017 539 `````` apply HP, IH, uPred_mono with (S n) x; eauto using cmra_validN_S. `````` Robbert Krebbers committed Oct 30, 2017 540 541 542 543 544 545 546 547 548 549 550 551 552 `````` - (* (∀ a, ▷ Φ a) ⊢ ▷ ∀ a, Φ a *) intros A Φ. unseal; by split=> -[|n] x. - (* (▷ ∃ a, Φ a) ⊢ ▷ False ∨ (∃ a, ▷ Φ a) *) intros A Φ. unseal; split=> -[|[|n]] x /=; eauto. - (* ▷ (P ∗ Q) ⊢ ▷ P ∗ ▷ Q *) intros P Q. unseal; split=> -[|n] x ? /=. { by exists x, (core x); rewrite cmra_core_r. } intros (x1&x2&Hx&?&?); destruct (cmra_extend n x x1 x2) as (y1&y2&Hx'&Hy1&Hy2); eauto using cmra_validN_S; simpl in *. exists y1, y2; split; [by rewrite Hx'|by rewrite Hy1 Hy2]. - (* ▷ P ∗ ▷ Q ⊢ ▷ (P ∗ Q) *) intros P Q. unseal; split=> -[|n] x ? /=; [done|intros (x1&x2&Hx&?&?)]. exists x1, x2; eauto using dist_S. `````` Jacques-Henri Jourdan committed Nov 03, 2017 553 554 555 556 `````` - (* ▷ bi_plainly P ⊢ bi_plainly (▷ P) *) by unseal. - (* bi_plainly (▷ P) ⊢ ▷ bi_plainly P *) by unseal. `````` Jacques-Henri Jourdan committed Nov 02, 2017 557 `````` - (* ▷ bi_persistently P ⊢ bi_persistently (▷ P) *) `````` Robbert Krebbers committed Oct 30, 2017 558 `````` by unseal. `````` Jacques-Henri Jourdan committed Nov 02, 2017 559 `````` - (* bi_persistently (▷ P) ⊢ ▷ bi_persistently P *) `````` Robbert Krebbers committed Oct 30, 2017 560 561 562 563 `````` by unseal. - (* ▷ P ⊢ ▷ False ∨ (▷ False → P) *) intros P. unseal; split=> -[|n] x ? /= HP; [by left|right]. intros [|n'] x' ????; [|done]. `````` Jacques-Henri Jourdan committed Dec 21, 2017 564 `````` eauto using uPred_mono, cmra_included_includedN. `````` Robbert Krebbers committed Oct 30, 2017 565 566 567 568 569 570 571 572 573 574 575 576 ``````Qed. Canonical Structure uPredI (M : ucmraT) : bi := {| bi_ofe_mixin := ofe_mixin_of (uPred M); bi_bi_mixin := uPred_bi_mixin M |}. Canonical Structure uPredSI (M : ucmraT) : sbi := {| sbi_ofe_mixin := ofe_mixin_of (uPred M); sbi_bi_mixin := uPred_bi_mixin M; sbi_sbi_mixin := uPred_sbi_mixin M |}. Coercion uPred_valid {M} : uPred M → Prop := bi_valid. (* Latest notation *) Notation "✓ x" := (uPred_cmra_valid x) (at level 20) : bi_scope. `````` Robbert Krebbers committed Oct 25, 2016 577 `````` `````` Robbert Krebbers committed Dec 13, 2016 578 ``````Module uPred. `````` Robbert Krebbers committed Oct 30, 2017 579 580 ``````Include uPred_unseal. Section uPred. `````` Robbert Krebbers committed Oct 25, 2016 581 ``````Context {M : ucmraT}. `````` Robbert Krebbers committed Oct 30, 2017 582 ``````Implicit Types φ : Prop. `````` Robbert Krebbers committed Oct 25, 2016 583 ``````Implicit Types P Q : uPred M. `````` Robbert Krebbers committed Oct 30, 2017 584 585 586 ``````Implicit Types A : Type. Arguments uPred_holds {_} !_ _ _ /. Hint Immediate uPred_in_entails. `````` Robbert Krebbers committed Oct 25, 2016 587 `````` `````` Robbert Krebbers committed Oct 30, 2017 588 ``````Global Instance ownM_ne : NonExpansive (@uPred_ownM M). `````` Robbert Krebbers committed Oct 25, 2016 589 ``````Proof. `````` Robbert Krebbers committed Oct 30, 2017 590 591 `````` intros n a b Ha. unseal; split=> n' x ? /=. by rewrite (dist_le _ _ _ _ Ha); last lia. `````` Robbert Krebbers committed Oct 25, 2016 592 ``````Qed. `````` Robbert Krebbers committed Oct 30, 2017 593 ``````Global Instance ownM_proper: Proper ((≡) ==> (⊣⊢)) (@uPred_ownM M) := ne_proper _. `````` Robbert Krebbers committed Oct 25, 2016 594 `````` `````` Robbert Krebbers committed Oct 30, 2017 595 596 ``````Global Instance cmra_valid_ne {A : cmraT} : NonExpansive (@uPred_cmra_valid M A). `````` Robbert Krebbers committed Oct 25, 2016 597 ``````Proof. `````` Robbert Krebbers committed Oct 30, 2017 598 599 `````` intros n a b Ha; unseal; split=> n' x ? /=. by rewrite (dist_le _ _ _ _ Ha); last lia. `````` Robbert Krebbers committed Oct 25, 2016 600 ``````Qed. `````` Robbert Krebbers committed Oct 30, 2017 601 602 603 ``````Global Instance cmra_valid_proper {A : cmraT} : Proper ((≡) ==> (⊣⊢)) (@uPred_cmra_valid M A) := ne_proper _. `````` Jacques-Henri Jourdan committed Dec 04, 2017 604 ``````(** BI instances *) `````` Jacques-Henri Jourdan committed Dec 04, 2017 605 `````` `````` Jacques-Henri Jourdan committed Dec 04, 2017 606 607 608 609 610 ``````Global Instance uPred_affine : BiAffine (uPredI M) | 0. Proof. intros P. rewrite /Affine. by apply bi.pure_intro. Qed. Global Instance uPred_plainly_exist_1 : BiPlainlyExist (uPredI M). Proof. unfold BiPlainlyExist. by unseal. Qed. `````` Jacques-Henri Jourdan committed Dec 04, 2017 611 `````` `````` Robbert Krebbers committed Oct 30, 2017 612 ``````(** Limits *) `````` Robbert Krebbers committed Mar 09, 2017 613 614 ``````Lemma entails_lim (cP cQ : chain (uPredC M)) : (∀ n, cP n ⊢ cQ n) → compl cP ⊢ compl cQ. `````` Ralf Jung committed Dec 21, 2016 615 ``````Proof. `````` Robbert Krebbers committed Mar 09, 2017 616 `````` intros Hlim; split=> n m ? HP. `````` Ralf Jung committed Dec 21, 2016 617 618 619 `````` eapply uPred_holds_ne, Hlim, HP; eauto using conv_compl. Qed. `````` Robbert Krebbers committed Oct 30, 2017 620 621 622 623 624 625 626 627 628 629 630 ``````(* Own *) Lemma ownM_op (a1 a2 : M) : uPred_ownM (a1 ⋅ a2) ⊣⊢ uPred_ownM a1 ∗ uPred_ownM a2. Proof. rewrite /bi_sep /=; unseal. split=> n x ?; split. - intros [z ?]; exists a1, (a2 ⋅ z); split; [by rewrite (assoc op)|]. split. by exists (core a1); rewrite cmra_core_r. by exists z. - intros (y1&y2&Hx&[z1 Hy1]&[z2 Hy2]); exists (z1 ⋅ z2). by rewrite (assoc op _ z1) -(comm op z1) (assoc op z1) -(assoc op _ a2) (comm op z1) -Hy1 -Hy2. Qed. `````` Jacques-Henri Jourdan committed Nov 02, 2017 631 632 ``````Lemma persistently_ownM_core (a : M) : uPred_ownM a ⊢ bi_persistently (uPred_ownM (core a)). `````` Robbert Krebbers committed Oct 30, 2017 633 634 635 636 ``````Proof. rewrite /bi_persistently /=. unseal. split=> n x Hx /=. by apply cmra_core_monoN. Qed. `````` Robbert Krebbers committed Dec 02, 2017 637 ``````Lemma ownM_unit : bi_valid (uPred_ownM (ε:M)). `````` Robbert Krebbers committed Oct 30, 2017 638 639 640 ``````Proof. unseal; split=> n x ??; by exists x; rewrite left_id. Qed. Lemma later_ownM (a : M) : ▷ uPred_ownM a ⊢ ∃ b, uPred_ownM b ∧ ▷ (a ≡ b). Proof. `````` Jacques-Henri Jourdan committed Dec 04, 2017 641 `````` rewrite /bi_and /sbi_later /bi_exist /bi_internal_eq /=; unseal. `````` Robbert Krebbers committed Oct 30, 2017 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 `````` split=> -[|n] x /= ? Hax; first by eauto using ucmra_unit_leastN. destruct Hax as [y ?]. destruct (cmra_extend n x a y) as (a'&y'&Hx&?&?); auto using cmra_validN_S. exists a'. rewrite Hx. eauto using cmra_includedN_l. Qed. (* Valid *) Lemma discrete_valid {A : cmraT} `{!CmraDiscrete A} (a : A) : ✓ a ⊣⊢ (⌜✓ a⌝ : uPred M). Proof. unseal. split=> n x _. by rewrite /= -cmra_discrete_valid_iff. Qed. Lemma ownM_valid (a : M) : uPred_ownM a ⊢ ✓ a. Proof. unseal; split=> n x Hv [a' ?]; ofe_subst; eauto using cmra_validN_op_l. Qed. Lemma cmra_valid_intro {A : cmraT} (a : A) : ✓ a → bi_valid (PROP:=uPredI M) (✓ a). Proof. unseal=> ?; split=> n x ? _ /=; by apply cmra_valid_validN. Qed. Lemma cmra_valid_elim {A : cmraT} (a : A) : ¬ ✓{0} a → ✓ a ⊢ (False : uPred M). Proof. intros Ha. unseal. split=> n x ??; apply Ha, cmra_validN_le with n; auto. Qed. `````` Jacques-Henri Jourdan committed Nov 03, 2017 663 ``````Lemma plainly_cmra_valid_1 {A : cmraT} (a : A) : ✓ a ⊢ bi_plainly (✓ a : uPred M). `````` Robbert Krebbers committed Oct 30, 2017 664 665 666 667 668 669 670 671 672 673 ``````Proof. by unseal. Qed. Lemma cmra_valid_weaken {A : cmraT} (a b : A) : ✓ (a ⋅ b) ⊢ (✓ a : uPred M). Proof. unseal; split=> n x _; apply cmra_validN_op_l. Qed. Lemma prod_validI {A B : cmraT} (x : A * B) : ✓ x ⊣⊢ (✓ x.1 ∧ ✓ x.2 : uPred M). Proof. by unseal. Qed. Lemma option_validI {A : cmraT} (mx : option A) : ✓ mx ⊣⊢ match mx with Some x => ✓ x | None => True : uPred M end. Proof. unseal. by destruct mx. Qed. `````` Robbert Krebbers committed Dec 02, 2017 674 675 676 677 ``````Lemma ofe_fun_validI `{Finite A} {B : A → ucmraT} (g : ofe_fun B) : (✓ g : uPred M) ⊣⊢ ∀ i, ✓ g i. Proof. by uPred.unseal. Qed. `````` Robbert Krebbers committed Oct 30, 2017 678 ``````(* Basic update modality *) `````` Jacques-Henri Jourdan committed Jan 18, 2018 679 ``````Global Instance bupd_facts : BUpdFacts (uPredI M). `````` Robbert Krebbers committed Oct 30, 2017 680 ``````Proof. `````` Jacques-Henri Jourdan committed Jan 18, 2018 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 `````` split. - intros n P Q HPQ. unseal; split=> n' x; split; intros HP k yf ??; destruct (HP k yf) as (x'&?&?); auto; exists x'; split; auto; apply HPQ; eauto using cmra_validN_op_l. - unseal. split=> n x ? HP k yf ?; exists x; split; first done. apply uPred_mono with n x; eauto using cmra_validN_op_l. - unseal. intros HPQ; split=> n x ? HP k yf ??. destruct (HP k yf) as (x'&?&?); eauto. exists x'; split; eauto using uPred_in_entails, cmra_validN_op_l. - unseal; split; naive_solver. - unseal. split; intros n x ? (x1&x2&Hx&HP&?) k yf ??. destruct (HP k (x2 ⋅ yf)) as (x'&?&?); eauto. { by rewrite assoc -(dist_le _ _ _ _ Hx); last lia. } exists (x' ⋅ x2); split; first by rewrite -assoc. exists x', x2. eauto using uPred_mono, cmra_validN_op_l, cmra_validN_op_r. - unseal; split => n x Hnx /= Hng. destruct (Hng n ε) as [? [_ Hng']]; try rewrite right_id; auto. eapply uPred_mono; eauto using ucmra_unit_leastN. `````` Robbert Krebbers committed Oct 30, 2017 700 ``````Qed. `````` Jacques-Henri Jourdan committed Jan 18, 2018 701 `````` `````` Robbert Krebbers committed Oct 30, 2017 702 703 704 705 706 707 708 709 710 711 ``````Lemma bupd_ownM_updateP x (Φ : M → Prop) : x ~~>: Φ → uPred_ownM x ==∗ ∃ y, ⌜Φ y⌝ ∧ uPred_ownM y. Proof. intros Hup. unseal. split=> n x2 ? [x3 Hx] k yf ??. destruct (Hup k (Some (x3 ⋅ yf))) as (y&?&?); simpl in *. { rewrite /= assoc -(dist_le _ _ _ _ Hx); auto. } exists (y ⋅ x3); split; first by rewrite -assoc. exists y; eauto using cmra_includedN_l. Qed. End uPred. `````` Robbert Krebbers committed Dec 13, 2016 712 ``End uPred.``