upred.v 31.4 KB
 Robbert Krebbers committed Oct 30, 2017 1 ``````From iris.algebra Require Export cmra updates. `````` Robbert Krebbers committed Mar 03, 2018 2 ``````From iris.bi Require Export derived_connectives updates plainly. `````` 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 ``````(** 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. `````` Ralf Jung committed Mar 05, 2018 198 ``````Definition uPred_pure {M} := uPred_pure_aux.(unseal) M. `````` Robbert Krebbers committed Oct 30, 2017 199 ``````Definition uPred_pure_eq : `````` Ralf Jung committed Mar 05, 2018 200 `````` @uPred_pure = @uPred_pure_def := uPred_pure_aux.(seal_eq). `````` Robbert Krebbers committed Oct 30, 2017 201 202 203 204 205 206 207 `````` 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. `````` Ralf Jung committed Mar 05, 2018 208 209 ``````Definition uPred_and {M} := uPred_and_aux.(unseal) M. Definition uPred_and_eq: @uPred_and = @uPred_and_def := uPred_and_aux.(seal_eq). `````` Robbert Krebbers committed Oct 30, 2017 210 211 212 213 214 `````` 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. `````` Ralf Jung committed Mar 05, 2018 215 216 ``````Definition uPred_or {M} := uPred_or_aux.(unseal) M. Definition uPred_or_eq: @uPred_or = @uPred_or_def := uPred_or_aux.(seal_eq). `````` Robbert Krebbers committed Oct 30, 2017 217 218 219 220 221 `````` 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 `````` 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. `````` Ralf Jung committed Mar 05, 2018 227 ``````Definition uPred_impl {M} := uPred_impl_aux.(unseal) M. `````` Robbert Krebbers committed Oct 30, 2017 228 ``````Definition uPred_impl_eq : `````` Ralf Jung committed Mar 05, 2018 229 `````` @uPred_impl = @uPred_impl_def := uPred_impl_aux.(seal_eq). `````` Robbert Krebbers committed Oct 30, 2017 230 231 232 233 234 `````` 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. `````` Ralf Jung committed Mar 05, 2018 235 ``````Definition uPred_forall {M A} := uPred_forall_aux.(unseal) M A. `````` Robbert Krebbers committed Oct 30, 2017 236 ``````Definition uPred_forall_eq : `````` Ralf Jung committed Mar 05, 2018 237 `````` @uPred_forall = @uPred_forall_def := uPred_forall_aux.(seal_eq). `````` Robbert Krebbers committed Oct 30, 2017 238 239 240 241 242 `````` 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. `````` Ralf Jung committed Mar 05, 2018 243 244 ``````Definition uPred_exist {M A} := uPred_exist_aux.(unseal) M A. Definition uPred_exist_eq: @uPred_exist = @uPred_exist_def := uPred_exist_aux.(seal_eq). `````` Robbert Krebbers committed Oct 30, 2017 245 246 247 248 249 `````` 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. `````` Ralf Jung committed Mar 05, 2018 250 ``````Definition uPred_internal_eq {M A} := uPred_internal_eq_aux.(unseal) M A. `````` Robbert Krebbers committed Oct 30, 2017 251 ``````Definition uPred_internal_eq_eq: `````` Ralf Jung committed Mar 05, 2018 252 `````` @uPred_internal_eq = @uPred_internal_eq_def := uPred_internal_eq_aux.(seal_eq). `````` Robbert Krebbers committed Oct 30, 2017 253 254 255 256 `````` 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 ``````Qed. Definition uPred_sep_aux : seal (@uPred_sep_def). by eexists. Qed. `````` Ralf Jung committed Mar 05, 2018 262 263 ``````Definition uPred_sep {M} := uPred_sep_aux.(unseal) M. Definition uPred_sep_eq: @uPred_sep = @uPred_sep_def := uPred_sep_aux.(seal_eq). `````` Robbert Krebbers committed Oct 30, 2017 264 265 266 267 268 `````` 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 `````` eauto using cmra_validN_includedN, cmra_monoN_r, cmra_includedN_le. Qed. Definition uPred_wand_aux : seal (@uPred_wand_def). by eexists. Qed. `````` Ralf Jung committed Mar 05, 2018 274 ``````Definition uPred_wand {M} := uPred_wand_aux.(unseal) M. `````` Robbert Krebbers committed Oct 30, 2017 275 ``````Definition uPred_wand_eq : `````` Ralf Jung committed Mar 05, 2018 276 `````` @uPred_wand = @uPred_wand_def := uPred_wand_aux.(seal_eq). `````` Robbert Krebbers committed Oct 30, 2017 277 `````` `````` 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. *) `````` Robbert Krebbers committed Mar 03, 2018 281 ``````Program Definition uPred_plainly_def {M} : Plainly (uPred M) := λ P, `````` Jacques-Henri Jourdan committed Nov 03, 2017 282 `````` {| 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. `````` Ralf Jung committed Mar 05, 2018 284 285 286 287 ``````Definition uPred_plainly_aux : seal (@uPred_plainly_def). by eexists. Qed. Definition uPred_plainly {M} := uPred_plainly_aux.(unseal) M. Definition uPred_plainly_eq : @uPred_plainly = @uPred_plainly_def := uPred_plainly_aux.(seal_eq). `````` Jacques-Henri Jourdan committed Nov 03, 2017 288 `````` `````` Robbert Krebbers committed Oct 30, 2017 289 290 291 292 293 294 ``````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. `````` Ralf Jung committed Mar 05, 2018 295 ``````Definition uPred_persistently {M} := uPred_persistently_aux.(unseal) M. `````` Robbert Krebbers committed Oct 30, 2017 296 ``````Definition uPred_persistently_eq : `````` Ralf Jung committed Mar 05, 2018 297 `````` @uPred_persistently = @uPred_persistently_def := uPred_persistently_aux.(seal_eq). `````` Robbert Krebbers committed Oct 30, 2017 298 299 300 301 `````` 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 ``````Qed. Definition uPred_later_aux : seal (@uPred_later_def). by eexists. Qed. `````` Ralf Jung committed Mar 05, 2018 305 ``````Definition uPred_later {M} := uPred_later_aux.(unseal) M. `````` Robbert Krebbers committed Oct 30, 2017 306 ``````Definition uPred_later_eq : `````` Ralf Jung committed Mar 05, 2018 307 `````` @uPred_later = @uPred_later_def := uPred_later_aux.(seal_eq). `````` Robbert Krebbers committed Oct 30, 2017 308 309 310 311 `````` 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 ``````Qed. Definition uPred_ownM_aux : seal (@uPred_ownM_def). by eexists. Qed. `````` Ralf Jung committed Mar 05, 2018 316 ``````Definition uPred_ownM {M} := uPred_ownM_aux.(unseal) M. `````` Robbert Krebbers committed Oct 30, 2017 317 ``````Definition uPred_ownM_eq : `````` Ralf Jung committed Mar 05, 2018 318 `````` @uPred_ownM = @uPred_ownM_def := uPred_ownM_aux.(seal_eq). `````` Robbert Krebbers committed Oct 30, 2017 319 320 321 322 323 `````` 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. `````` Ralf Jung committed Mar 05, 2018 324 ``````Definition uPred_cmra_valid {M A} := uPred_cmra_valid_aux.(unseal) M A. `````` Robbert Krebbers committed Oct 30, 2017 325 ``````Definition uPred_cmra_valid_eq : `````` Ralf Jung committed Mar 05, 2018 326 `````` @uPred_cmra_valid = @uPred_cmra_valid_def := uPred_cmra_valid_aux.(seal_eq). `````` Robbert Krebbers committed Oct 30, 2017 327 328 329 330 331 `````` 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 ``````Definition uPred_bupd_aux {M} : seal (@uPred_bupd_def M). by eexists. Qed. `````` Ralf Jung committed Mar 05, 2018 339 ``````Definition uPred_bupd {M} : BUpd (uPred M) := uPred_bupd_aux.(unseal). `````` Jacques-Henri Jourdan committed Dec 11, 2017 340 ``````Definition uPred_bupd_eq {M} : `````` Ralf Jung committed Mar 05, 2018 341 `````` @bupd _ uPred_bupd = @uPred_bupd_def M := uPred_bupd_aux.(seal_eq). `````` 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 ``````Ltac unseal := (* Coq unfold is used to circumvent bug #5699 in rewrite /foo *) `````` Robbert Krebbers committed Mar 03, 2018 350 `````` unfold bi_emp; simpl; unfold sbi_emp; simpl; `````` Robbert Krebbers committed Nov 14, 2017 351 `````` unfold uPred_emp, bi_pure, bi_and, bi_or, bi_impl, bi_forall, bi_exist, `````` Robbert Krebbers committed Mar 03, 2018 352 `````` bi_sep, bi_wand, bi_persistently, sbi_internal_eq, sbi_later; simpl; `````` Robbert Krebbers committed Nov 14, 2017 353 `````` unfold sbi_emp, sbi_pure, sbi_and, sbi_or, sbi_impl, sbi_forall, sbi_exist, `````` Robbert Krebbers committed Mar 03, 2018 354 `````` sbi_internal_eq, sbi_sep, sbi_wand, sbi_persistently; simpl; `````` Ralf Jung committed Mar 05, 2018 355 `````` unfold plainly, bi_plainly_plainly; simpl; `````` Robbert Krebbers committed Nov 14, 2017 356 `````` rewrite !unseal_eqs /=. `````` Robbert Krebbers committed Oct 30, 2017 357 358 359 360 361 ``````End uPred_unseal. Import uPred_unseal. Local Arguments uPred_holds {_} !_ _ _ /. `````` Jacques-Henri Jourdan committed Feb 02, 2018 362 363 364 ``````Lemma uPred_bi_mixin (M : ucmraT) : BiMixin uPred_entails uPred_emp uPred_pure uPred_and uPred_or uPred_impl `````` Robbert Krebbers committed Mar 03, 2018 365 `````` (@uPred_forall M) (@uPred_exist M) uPred_sep uPred_wand `````` Jacques-Henri Jourdan committed Feb 02, 2018 366 `````` uPred_persistently. `````` Robbert Krebbers committed Oct 30, 2017 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 ``````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. - (* NonExpansive uPred_persistently *) intros n P1 P2 HP. unseal; split=> n' x; split; apply HP; eauto using @cmra_core_validN. - (* 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 425 `````` naive_solver eauto using uPred_mono, cmra_included_includedN. `````` Robbert Krebbers committed Oct 30, 2017 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 `````` - (* (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 ⊢ 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 457 `````` eapply uPred_mono; eauto using cmra_validN_op_l. `````` Robbert Krebbers committed Oct 30, 2017 458 459 460 `````` - (* (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. `````` Robbert Krebbers committed Mar 04, 2018 461 `````` - (* (P ⊢ Q) → P ⊢ Q *) `````` Robbert Krebbers committed Oct 30, 2017 462 `````` intros P QR HP. unseal; split=> n x ? /=. by apply HP, cmra_core_validN. `````` Robbert Krebbers committed Mar 04, 2018 463 `````` - (* P ⊢ P *) `````` Robbert Krebbers committed Oct 30, 2017 464 `````` intros P. unseal; split=> n x ?? /=. by rewrite cmra_core_idemp. `````` Robbert Krebbers committed Mar 04, 2018 465 `````` - (* P ⊢ emp (ADMISSIBLE) *) `````` Robbert Krebbers committed Mar 03, 2018 466 `````` by unseal. `````` Robbert Krebbers committed Mar 04, 2018 467 `````` - (* (∀ a, (Ψ a)) ⊢ (∀ a, Ψ a) *) `````` Robbert Krebbers committed Oct 30, 2017 468 `````` by unseal. `````` Robbert Krebbers committed Mar 04, 2018 469 `````` - (* (∃ a, Ψ a) ⊢ ∃ a, (Ψ a) *) `````` Robbert Krebbers committed Oct 30, 2017 470 `````` by unseal. `````` Robbert Krebbers committed Mar 04, 2018 471 `````` - (* P ∗ Q ⊢ P (ADMISSIBLE) *) `````` Robbert Krebbers committed Oct 30, 2017 472 473 474 `````` intros P Q. move: (uPred_persistently P)=> P'. unseal; split; intros n x ? (x1&x2&?&?&_); ofe_subst; eauto using uPred_mono, cmra_includedN_l. `````` Robbert Krebbers committed Mar 04, 2018 475 `````` - (* P ∧ Q ⊢ P ∗ Q *) `````` Robbert Krebbers committed Oct 30, 2017 476 477 `````` intros P Q. unseal; split=> n x ? [??]; simpl in *. exists (core x), x; rewrite ?cmra_core_l; auto. `````` Robbert Krebbers committed Oct 30, 2017 478 479 ``````Qed. `````` Robbert Krebbers committed Mar 03, 2018 480 481 482 483 ``````Lemma uPred_sbi_mixin (M : ucmraT) : SbiMixin uPred_entails uPred_pure uPred_or uPred_impl (@uPred_forall M) (@uPred_exist M) uPred_sep uPred_persistently (@uPred_internal_eq M) uPred_later. `````` Robbert Krebbers committed Oct 30, 2017 484 485 ``````Proof. split. `````` Jacques-Henri Jourdan committed Dec 04, 2017 486 `````` - (* Contractive sbi_later *) `````` Robbert Krebbers committed Oct 30, 2017 487 488 `````` unseal; intros [|n] P Q HPQ; split=> -[|n'] x ?? //=; try omega. apply HPQ; eauto using cmra_validN_S. `````` Jacques-Henri Jourdan committed Feb 02, 2018 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 `````` - (* 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. - (* 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). `````` Robbert Krebbers committed Oct 30, 2017 504 505 506 507 508 509 510 511 512 `````` - (* 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 513 `````` apply HP, IH, uPred_mono with (S n) x; eauto using cmra_validN_S. `````` Robbert Krebbers committed Oct 30, 2017 514 515 516 517 518 519 520 521 522 523 524 525 526 `````` - (* (∀ 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. `````` Robbert Krebbers committed Mar 04, 2018 527 `````` - (* ▷ P ⊢ ▷ P *) `````` Robbert Krebbers committed Oct 30, 2017 528 `````` by unseal. `````` Robbert Krebbers committed Mar 04, 2018 529 `````` - (* ▷ P ⊢ ▷ P *) `````` Robbert Krebbers committed Oct 30, 2017 530 531 532 533 `````` 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 534 `````` eauto using uPred_mono, cmra_included_includedN. `````` Robbert Krebbers committed Oct 30, 2017 535 536 537 538 539 540 541 542 543 544 545 546 ``````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 547 `````` `````` Robbert Krebbers committed Mar 03, 2018 548 549 550 551 552 553 554 555 ``````Lemma uPred_plainly_mixin M : BiPlainlyMixin (uPredSI M) uPred_plainly. Proof. split. - (* NonExpansive uPred_plainly *) intros n P1 P2 HP. unseal; split=> n' x; split; apply HP; eauto using @ucmra_unit_validN. - (* (P ⊢ Q) → ■ P ⊢ ■ Q *) intros P QR HP. unseal; split=> n x ? /=. by apply HP, ucmra_unit_validN. `````` Robbert Krebbers committed Mar 04, 2018 556 `````` - (* ■ P ⊢ P *) `````` Robbert Krebbers committed Mar 03, 2018 557 558 559 560 561 `````` unseal; split; simpl; eauto using uPred_mono, @ucmra_unit_leastN. - (* ■ P ⊢ ■ ■ P *) unseal; split=> n x ?? //. - (* (∀ a, ■ (Ψ a)) ⊢ ■ (∀ a, Ψ a) *) by unseal. `````` Robbert Krebbers committed Mar 04, 2018 562 `````` - (* (■ P → Q) ⊢ (■ P → Q) *) `````` Robbert Krebbers committed Mar 03, 2018 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 `````` unseal; split=> /= n x ? HPQ n' x' ????. eapply uPred_mono with n' (core x)=>//; [|by apply cmra_included_includedN]. apply (HPQ n' x); eauto using cmra_validN_le. - (* (■ P → ■ Q) ⊢ ■ (■ P → Q) *) unseal; split=> /= n x ? HPQ n' x' ????. eapply uPred_mono with n' ε=>//; [|by apply cmra_included_includedN]. apply (HPQ n' x); eauto using cmra_validN_le. - (* P ⊢ ■ emp (ADMISSIBLE) *) by unseal. - (* ■ P ∗ Q ⊢ ■ P *) intros P Q. move: (uPred_persistently P)=> P'. unseal; split; intros n x ? (x1&x2&?&?&_); ofe_subst; eauto using uPred_mono, cmra_includedN_l. - (* ■ ((P -∗ Q) ∧ (Q -∗ P)) ⊢ P ≡ Q *) unseal; split=> n x ? /= HPQ. split=> n' x' ??. move: HPQ=> [] /(_ n' x'); rewrite !left_id=> ?. move=> /(_ n' x'); rewrite !left_id=> ?. naive_solver. - (* ▷ ■ P ⊢ ■ ▷ P *) by unseal. - (* ■ ▷ P ⊢ ▷ ■ P *) by unseal. Qed. Instance uPred_plainlyC M : BiPlainly (uPredSI M) := {| bi_plainly_mixin := uPred_plainly_mixin M |}. `````` Robbert Krebbers committed Mar 03, 2018 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 ``````Lemma uPred_bupd_mixin M : BiBUpdMixin (uPredI M) uPred_bupd. Proof. 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. Qed. Instance uPred_bi_bupd M : BiBUpd (uPredI M) := {| bi_bupd_mixin := uPred_bupd_mixin M |}. `````` Robbert Krebbers committed Mar 03, 2018 609 610 611 612 613 614 615 ``````Instance uPred_bi_bupd_plainly M : BiBUpdPlainly (uPredSI M). Proof. rewrite /BiBUpdPlainly. unseal; split => n x Hnx /= Hng. destruct (Hng n ε) as [? [_ Hng']]; try rewrite right_id; auto. eapply uPred_mono; eauto using ucmra_unit_leastN. Qed. `````` Robbert Krebbers committed Dec 13, 2016 616 ``````Module uPred. `````` Robbert Krebbers committed Oct 30, 2017 617 618 ``````Include uPred_unseal. Section uPred. `````` Robbert Krebbers committed Oct 25, 2016 619 ``````Context {M : ucmraT}. `````` Robbert Krebbers committed Oct 30, 2017 620 ``````Implicit Types φ : Prop. `````` Robbert Krebbers committed Oct 25, 2016 621 ``````Implicit Types P Q : uPred M. `````` Robbert Krebbers committed Oct 30, 2017 622 623 624 ``````Implicit Types A : Type. Arguments uPred_holds {_} !_ _ _ /. Hint Immediate uPred_in_entails. `````` Robbert Krebbers committed Oct 25, 2016 625 `````` `````` Robbert Krebbers committed Oct 30, 2017 626 ``````Global Instance ownM_ne : NonExpansive (@uPred_ownM M). `````` Robbert Krebbers committed Oct 25, 2016 627 ``````Proof. `````` Robbert Krebbers committed Oct 30, 2017 628 629 `````` intros n a b Ha. unseal; split=> n' x ? /=. by rewrite (dist_le _ _ _ _ Ha); last lia. `````` Robbert Krebbers committed Oct 25, 2016 630 ``````Qed. `````` Robbert Krebbers committed Oct 30, 2017 631 ``````Global Instance ownM_proper: Proper ((≡) ==> (⊣⊢)) (@uPred_ownM M) := ne_proper _. `````` Robbert Krebbers committed Oct 25, 2016 632 `````` `````` Robbert Krebbers committed Oct 30, 2017 633 634 ``````Global Instance cmra_valid_ne {A : cmraT} : NonExpansive (@uPred_cmra_valid M A). `````` Robbert Krebbers committed Oct 25, 2016 635 ``````Proof. `````` Robbert Krebbers committed Oct 30, 2017 636 637 `````` intros n a b Ha; unseal; split=> n' x ? /=. by rewrite (dist_le _ _ _ _ Ha); last lia. `````` Robbert Krebbers committed Oct 25, 2016 638 ``````Qed. `````` Robbert Krebbers committed Oct 30, 2017 639 640 641 ``````Global Instance cmra_valid_proper {A : cmraT} : Proper ((≡) ==> (⊣⊢)) (@uPred_cmra_valid M A) := ne_proper _. `````` Jacques-Henri Jourdan committed Dec 04, 2017 642 ``````(** BI instances *) `````` Jacques-Henri Jourdan committed Dec 04, 2017 643 `````` `````` Jacques-Henri Jourdan committed Dec 04, 2017 644 645 646 ``````Global Instance uPred_affine : BiAffine (uPredI M) | 0. Proof. intros P. rewrite /Affine. by apply bi.pure_intro. Qed. `````` Robbert Krebbers committed Mar 03, 2018 647 ``````Global Instance uPred_plainly_exist_1 : BiPlainlyExist (uPredSI M). `````` Jacques-Henri Jourdan committed Dec 04, 2017 648 ``````Proof. unfold BiPlainlyExist. by unseal. Qed. `````` Jacques-Henri Jourdan committed Dec 04, 2017 649 `````` `````` Robbert Krebbers committed Oct 30, 2017 650 ``````(** Limits *) `````` Robbert Krebbers committed Mar 09, 2017 651 652 ``````Lemma entails_lim (cP cQ : chain (uPredC M)) : (∀ n, cP n ⊢ cQ n) → compl cP ⊢ compl cQ. `````` Ralf Jung committed Dec 21, 2016 653 ``````Proof. `````` Robbert Krebbers committed Mar 09, 2017 654 `````` intros Hlim; split=> n m ? HP. `````` Ralf Jung committed Dec 21, 2016 655 656 657 `````` eapply uPred_holds_ne, Hlim, HP; eauto using conv_compl. Qed. `````` Robbert Krebbers committed Oct 30, 2017 658 659 660 661 662 663 664 665 666 667 668 ``````(* 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. `````` Robbert Krebbers committed Mar 04, 2018 669 ``````Lemma persistently_ownM_core (a : M) : uPred_ownM a ⊢ uPred_ownM (core a). `````` Robbert Krebbers committed Oct 30, 2017 670 671 672 673 ``````Proof. rewrite /bi_persistently /=. unseal. split=> n x Hx /=. by apply cmra_core_monoN. Qed. `````` Robbert Krebbers committed Dec 02, 2017 674 ``````Lemma ownM_unit : bi_valid (uPred_ownM (ε:M)). `````` Robbert Krebbers committed Oct 30, 2017 675 676 677 ``````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 Feb 02, 2018 678 `````` rewrite /bi_and /sbi_later /bi_exist /sbi_internal_eq /=; unseal. `````` Robbert Krebbers committed Oct 30, 2017 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 `````` 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. `````` Robbert Krebbers committed Mar 03, 2018 700 ``````Lemma plainly_cmra_valid_1 {A : cmraT} (a : A) : ✓ a ⊢ ■ (✓ a : uPred M). `````` Robbert Krebbers committed Oct 30, 2017 701 702 703 704 705 706 707 708 709 710 ``````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. `````` Jacques-Henri Jourdan committed Feb 20, 2018 711 ``````Lemma ofe_fun_validI `{B : A → ucmraT} (g : ofe_fun B) : `````` Robbert Krebbers committed Dec 02, 2017 712 713 714 `````` (✓ g : uPred M) ⊣⊢ ∀ i, ✓ g i. Proof. by uPred.unseal. Qed. `````` Robbert Krebbers committed Oct 30, 2017 715 716 717 718 719 720 721 722 723 724 ``````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 725 ``End uPred.``