ofe.v 66.1 KB
 Robbert Krebbers committed Mar 10, 2016 1 ``````From iris.algebra Require Export base. `````` Ralf Jung committed Jan 05, 2017 2 ``````Set Default Proof Using "Type". `````` Robbert Krebbers committed Nov 14, 2017 3 ``````Set Primitive Projections. `````` Robbert Krebbers committed Nov 11, 2015 4 `````` `````` Ralf Jung committed Nov 22, 2016 5 ``````(** This files defines (a shallow embedding of) the category of OFEs: `````` Ralf Jung committed Feb 16, 2016 6 7 8 9 `````` Complete ordered families of equivalences. This is a cartesian closed category, and mathematically speaking, the entire development lives in this category. However, we will generally prefer to work with raw Coq functions plus some registered Proper instances for non-expansiveness. `````` Jacques-Henri Jourdan committed Dec 23, 2017 10 `````` This makes writing such functions much easier. It turns out that it many `````` Ralf Jung committed Feb 16, 2016 11 12 13 `````` cases, we do not even need non-expansiveness. *) `````` Paolo G. Giarrusso committed Jun 24, 2019 14 ``````(** Unbundled version *) `````` Robbert Krebbers committed Nov 11, 2015 15 ``````Class Dist A := dist : nat → relation A. `````` Maxime Dénès committed Jan 24, 2019 16 ``````Instance: Params (@dist) 3 := {}. `````` Ralf Jung committed Feb 10, 2016 17 18 ``````Notation "x ≡{ n }≡ y" := (dist n x y) (at level 70, n at next level, format "x ≡{ n }≡ y"). `````` Robbert Krebbers committed Dec 12, 2018 19 20 21 ``````Notation "x ≡{ n }@{ A }≡ y" := (dist (A:=A) n x y) (at level 70, n at next level, only parsing). `````` Tej Chajed committed Nov 29, 2018 22 23 ``````Hint Extern 0 (_ ≡{_}≡ _) => reflexivity : core. Hint Extern 0 (_ ≡{_}≡ _) => symmetry; assumption : core. `````` Ralf Jung committed Jan 27, 2017 24 25 ``````Notation NonExpansive f := (∀ n, Proper (dist n ==> dist n) f). Notation NonExpansive2 f := (∀ n, Proper (dist n ==> dist n ==> dist n) f). `````` Robbert Krebbers committed Jan 13, 2016 26 `````` `````` Robbert Krebbers committed Feb 09, 2017 27 ``````Tactic Notation "ofe_subst" ident(x) := `````` Robbert Krebbers committed Jan 13, 2016 28 `````` repeat match goal with `````` Robbert Krebbers committed Feb 17, 2016 29 `````` | _ => progress simplify_eq/= `````` Robbert Krebbers committed Jan 13, 2016 30 31 32 `````` | H:@dist ?A ?d ?n x _ |- _ => setoid_subst_aux (@dist A d n) x | H:@dist ?A ?d ?n _ x |- _ => symmetry in H;setoid_subst_aux (@dist A d n) x end. `````` Robbert Krebbers committed Feb 09, 2017 33 ``````Tactic Notation "ofe_subst" := `````` Robbert Krebbers committed Nov 17, 2015 34 `````` repeat match goal with `````` Robbert Krebbers committed Feb 17, 2016 35 `````` | _ => progress simplify_eq/= `````` Robbert Krebbers committed Dec 21, 2015 36 37 `````` | H:@dist ?A ?d ?n ?x _ |- _ => setoid_subst_aux (@dist A d n) x | H:@dist ?A ?d ?n _ ?x |- _ => symmetry in H;setoid_subst_aux (@dist A d n) x `````` Robbert Krebbers committed Nov 17, 2015 38 `````` end. `````` Robbert Krebbers committed Nov 11, 2015 39 `````` `````` Robbert Krebbers committed Nov 14, 2017 40 41 42 43 44 ``````Record OfeMixin A `{Equiv A, Dist A} := { mixin_equiv_dist x y : x ≡ y ↔ ∀ n, x ≡{n}≡ y; mixin_dist_equivalence n : Equivalence (dist n); mixin_dist_S n x y : x ≡{S n}≡ y → x ≡{n}≡ y }. `````` Robbert Krebbers committed Nov 11, 2015 45 `````` `````` Paolo G. Giarrusso committed Jun 24, 2019 46 ``````(** Bundled version *) `````` Robbert Krebbers committed Nov 14, 2017 47 ``````Structure ofeT := OfeT { `````` Ralf Jung committed Nov 22, 2016 48 49 50 `````` ofe_car :> Type; ofe_equiv : Equiv ofe_car; ofe_dist : Dist ofe_car; `````` Robbert Krebbers committed Nov 14, 2017 51 `````` ofe_mixin : OfeMixin ofe_car `````` Robbert Krebbers committed Nov 11, 2015 52 ``````}. `````` Robbert Krebbers committed Nov 14, 2017 53 ``````Arguments OfeT _ {_ _} _. `````` Ralf Jung committed Nov 22, 2016 54 55 56 57 58 59 60 ``````Add Printing Constructor ofeT. Hint Extern 0 (Equiv _) => eapply (@ofe_equiv _) : typeclass_instances. Hint Extern 0 (Dist _) => eapply (@ofe_dist _) : typeclass_instances. Arguments ofe_car : simpl never. Arguments ofe_equiv : simpl never. Arguments ofe_dist : simpl never. Arguments ofe_mixin : simpl never. `````` Robbert Krebbers committed Jan 14, 2016 61 `````` `````` Robbert Krebbers committed Feb 09, 2017 62 63 64 ``````(** When declaring instances of subclasses of OFE (like CMRAs and unital CMRAs) we need Coq to *infer* the canonical OFE instance of a given type and take the mixin out of it. This makes sure we do not use two different OFE instances in `````` Robbert Krebbers committed Oct 25, 2017 65 ``````different places (see for example the constructors [CmraT] and [UcmraT] in the `````` Robbert Krebbers committed Feb 09, 2017 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 ``````file [cmra.v].) In order to infer the OFE instance, we use the definition [ofe_mixin_of'] which is inspired by the [clone] trick in ssreflect. It works as follows, when type checking [@ofe_mixin_of' A ?Ac id] Coq faces a unification problem: ofe_car ?Ac ~ A which will resolve [?Ac] to the canonical OFE instance corresponding to [A]. The definition [@ofe_mixin_of' A ?Ac id] will then provide the corresponding mixin. Note that type checking of [ofe_mixin_of' A id] will fail when [A] does not have a canonical OFE instance. The notation [ofe_mixin_of A] that we define on top of [ofe_mixin_of' A id] hides the [id] and normalizes the mixin to head normal form. The latter is to ensure that we do not end up with redundant canonical projections to the mixin, i.e. them all being of the shape [ofe_mixin_of' A id]. *) Definition ofe_mixin_of' A {Ac : ofeT} (f : Ac → A) : OfeMixin Ac := ofe_mixin Ac. Notation ofe_mixin_of A := ltac:(let H := eval hnf in (ofe_mixin_of' A id) in exact H) (only parsing). `````` Robbert Krebbers committed Jan 14, 2016 87 ``````(** Lifting properties from the mixin *) `````` Ralf Jung committed Nov 22, 2016 88 89 ``````Section ofe_mixin. Context {A : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 90 `````` Implicit Types x y : A. `````` Ralf Jung committed Feb 10, 2016 91 `````` Lemma equiv_dist x y : x ≡ y ↔ ∀ n, x ≡{n}≡ y. `````` Ralf Jung committed Nov 22, 2016 92 `````` Proof. apply (mixin_equiv_dist _ (ofe_mixin A)). Qed. `````` Robbert Krebbers committed Jan 14, 2016 93 `````` Global Instance dist_equivalence n : Equivalence (@dist A _ n). `````` Ralf Jung committed Nov 22, 2016 94 `````` Proof. apply (mixin_dist_equivalence _ (ofe_mixin A)). Qed. `````` Ralf Jung committed Feb 10, 2016 95 `````` Lemma dist_S n x y : x ≡{S n}≡ y → x ≡{n}≡ y. `````` Ralf Jung committed Nov 22, 2016 96 97 `````` Proof. apply (mixin_dist_S _ (ofe_mixin A)). Qed. End ofe_mixin. `````` Robbert Krebbers committed Jan 14, 2016 98 `````` `````` Tej Chajed committed Nov 29, 2018 99 ``````Hint Extern 1 (_ ≡{_}≡ _) => apply equiv_dist; assumption : core. `````` Robbert Krebbers committed May 28, 2016 100 `````` `````` Robbert Krebbers committed Oct 25, 2017 101 102 103 104 ``````(** Discrete OFEs and discrete OFE elements *) Class Discrete {A : ofeT} (x : A) := discrete y : x ≡{0}≡ y → x ≡ y. Arguments discrete {_} _ {_} _ _. Hint Mode Discrete + ! : typeclass_instances. `````` Maxime Dénès committed Jan 24, 2019 105 ``````Instance: Params (@Discrete) 1 := {}. `````` Robbert Krebbers committed Oct 25, 2017 106 `````` `````` Robbert Krebbers committed Oct 25, 2017 107 ``````Class OfeDiscrete (A : ofeT) := ofe_discrete_discrete (x : A) :> Discrete x. `````` Ralf Jung committed Nov 22, 2016 108 109 110 111 112 113 114 115 116 `````` (** OFEs with a completion *) Record chain (A : ofeT) := { chain_car :> nat → A; chain_cauchy n i : n ≤ i → chain_car i ≡{n}≡ chain_car n }. Arguments chain_car {_} _ _. Arguments chain_cauchy {_} _ _ _ _. `````` Robbert Krebbers committed Dec 05, 2016 117 ``````Program Definition chain_map {A B : ofeT} (f : A → B) `````` Ralf Jung committed Jan 27, 2017 118 `````` `{!NonExpansive f} (c : chain A) : chain B := `````` Robbert Krebbers committed Dec 05, 2016 119 120 121 `````` {| chain_car n := f (c n) |}. Next Obligation. by intros A B f Hf c n i ?; apply Hf, chain_cauchy. Qed. `````` Ralf Jung committed Nov 22, 2016 122 123 124 125 126 127 ``````Notation Compl A := (chain A%type → A). Class Cofe (A : ofeT) := { compl : Compl A; conv_compl n c : compl c ≡{n}≡ c n; }. Arguments compl : simpl never. `````` Robbert Krebbers committed Jun 21, 2019 128 ``````Hint Mode Cofe ! : typeclass_instances. `````` Robbert Krebbers committed Feb 24, 2016 129 `````` `````` Robbert Krebbers committed Feb 09, 2017 130 ``````Lemma compl_chain_map `{Cofe A, Cofe B} (f : A → B) c `(NonExpansive f) : `````` Jacques-Henri Jourdan committed Jan 05, 2017 131 132 133 `````` compl (chain_map f c) ≡ f (compl c). Proof. apply equiv_dist=>n. by rewrite !conv_compl. Qed. `````` Ralf Jung committed Mar 01, 2017 134 135 136 137 138 139 140 141 ``````Program Definition chain_const {A : ofeT} (a : A) : chain A := {| chain_car n := a |}. Next Obligation. by intros A a n i _. Qed. Lemma compl_chain_const {A : ofeT} `{!Cofe A} (a : A) : compl (chain_const a) ≡ a. Proof. apply equiv_dist=>n. by rewrite conv_compl. Qed. `````` Robbert Krebbers committed Nov 11, 2015 142 ``````(** General properties *) `````` Robbert Krebbers committed Feb 09, 2017 143 ``````Section ofe. `````` Ralf Jung committed Nov 22, 2016 144 `````` Context {A : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 145 `````` Implicit Types x y : A. `````` Robbert Krebbers committed Feb 09, 2017 146 `````` Global Instance ofe_equivalence : Equivalence ((≡) : relation A). `````` Robbert Krebbers committed Nov 11, 2015 147 148 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 149 150 `````` - by intros x; rewrite equiv_dist. - by intros x y; rewrite !equiv_dist. `````` Ralf Jung committed Feb 20, 2016 151 `````` - by intros x y z; rewrite !equiv_dist; intros; trans y. `````` Robbert Krebbers committed Nov 11, 2015 152 `````` Qed. `````` Robbert Krebbers committed Jan 14, 2016 153 `````` Global Instance dist_ne n : Proper (dist n ==> dist n ==> iff) (@dist A _ n). `````` Robbert Krebbers committed Nov 11, 2015 154 155 `````` Proof. intros x1 x2 ? y1 y2 ?; split; intros. `````` Ralf Jung committed Feb 20, 2016 156 157 `````` - by trans x1; [|trans y1]. - by trans x2; [|trans y2]. `````` Robbert Krebbers committed Nov 11, 2015 158 `````` Qed. `````` Robbert Krebbers committed Jan 14, 2016 159 `````` Global Instance dist_proper n : Proper ((≡) ==> (≡) ==> iff) (@dist A _ n). `````` Robbert Krebbers committed Nov 11, 2015 160 `````` Proof. `````` Robbert Krebbers committed Jan 13, 2016 161 `````` by move => x1 x2 /equiv_dist Hx y1 y2 /equiv_dist Hy; rewrite (Hx n) (Hy n). `````` Robbert Krebbers committed Nov 11, 2015 162 163 164 `````` Qed. Global Instance dist_proper_2 n x : Proper ((≡) ==> iff) (dist n x). Proof. by apply dist_proper. Qed. `````` Robbert Krebbers committed Oct 25, 2017 165 166 `````` Global Instance Discrete_proper : Proper ((≡) ==> iff) (@Discrete A). Proof. intros x y Hxy. rewrite /Discrete. by setoid_rewrite Hxy. Qed. `````` Robbert Krebbers committed Feb 11, 2017 167 `````` `````` Robbert Krebbers committed Feb 18, 2016 168 `````` Lemma dist_le n n' x y : x ≡{n}≡ y → n' ≤ n → x ≡{n'}≡ y. `````` Robbert Krebbers committed Nov 11, 2015 169 `````` Proof. induction 2; eauto using dist_S. Qed. `````` Ralf Jung committed Feb 29, 2016 170 171 `````` Lemma dist_le' n n' x y : n' ≤ n → x ≡{n}≡ y → x ≡{n'}≡ y. Proof. intros; eauto using dist_le. Qed. `````` Paolo G. Giarrusso committed May 27, 2020 172 173 174 175 176 177 `````` (** [ne_proper] and [ne_proper_2] are not instances to improve efficiency of type class search during setoid rewriting. Instances of [NonExpansive{,2}] are hence accompanied by instances of [Proper] built using these lemmas. *) Lemma ne_proper {B : ofeT} (f : A → B) `{!NonExpansive f} : Proper ((≡) ==> (≡)) f. `````` Robbert Krebbers committed Nov 11, 2015 178 `````` Proof. by intros x1 x2; rewrite !equiv_dist; intros Hx n; rewrite (Hx n). Qed. `````` Paolo G. Giarrusso committed May 27, 2020 179 180 `````` Lemma ne_proper_2 {B C : ofeT} (f : A → B → C) `{!NonExpansive2 f} : Proper ((≡) ==> (≡) ==> (≡)) f. `````` Robbert Krebbers committed Nov 11, 2015 181 182 `````` Proof. unfold Proper, respectful; setoid_rewrite equiv_dist. `````` Robbert Krebbers committed Jan 13, 2016 183 `````` by intros x1 x2 Hx y1 y2 Hy n; rewrite (Hx n) (Hy n). `````` Robbert Krebbers committed Nov 11, 2015 184 `````` Qed. `````` Robbert Krebbers committed Feb 24, 2016 185 `````` `````` Ralf Jung committed Nov 22, 2016 186 `````` Lemma conv_compl' `{Cofe A} n (c : chain A) : compl c ≡{n}≡ c (S n). `````` Ralf Jung committed Feb 29, 2016 187 188 `````` Proof. transitivity (c n); first by apply conv_compl. symmetry. `````` Ralf Jung committed Jun 20, 2018 189 `````` apply chain_cauchy. lia. `````` Ralf Jung committed Feb 29, 2016 190 `````` Qed. `````` Robbert Krebbers committed Apr 13, 2017 191 `````` `````` Robbert Krebbers committed Oct 25, 2017 192 `````` Lemma discrete_iff n (x : A) `{!Discrete x} y : x ≡ y ↔ x ≡{n}≡ y. `````` Robbert Krebbers committed Feb 24, 2016 193 `````` Proof. `````` Robbert Krebbers committed Oct 25, 2017 194 `````` split; intros; auto. apply (discrete _), dist_le with n; auto with lia. `````` Robbert Krebbers committed Feb 24, 2016 195 `````` Qed. `````` Robbert Krebbers committed Oct 25, 2017 196 `````` Lemma discrete_iff_0 n (x : A) `{!Discrete x} y : x ≡{0}≡ y ↔ x ≡{n}≡ y. `````` Robbert Krebbers committed Nov 28, 2017 197 `````` Proof. by rewrite -!discrete_iff. Qed. `````` Robbert Krebbers committed Feb 09, 2017 198 ``````End ofe. `````` Robbert Krebbers committed Nov 11, 2015 199 `````` `````` Robbert Krebbers committed Dec 02, 2016 200 ``````(** Contractive functions *) `````` Robbert Krebbers committed Aug 17, 2017 201 ``````Definition dist_later `{Dist A} (n : nat) (x y : A) : Prop := `````` Robbert Krebbers committed Dec 05, 2016 202 `````` match n with 0 => True | S n => x ≡{n}≡ y end. `````` Robbert Krebbers committed Aug 17, 2017 203 ``````Arguments dist_later _ _ !_ _ _ /. `````` Robbert Krebbers committed Dec 05, 2016 204 `````` `````` Robbert Krebbers committed Aug 17, 2017 205 ``````Global Instance dist_later_equivalence (A : ofeT) n : Equivalence (@dist_later A _ n). `````` Robbert Krebbers committed Dec 05, 2016 206 207 ``````Proof. destruct n as [|n]. by split. apply dist_equivalence. Qed. `````` Ralf Jung committed Feb 22, 2017 208 209 210 ``````Lemma dist_dist_later {A : ofeT} n (x y : A) : dist n x y → dist_later n x y. Proof. intros Heq. destruct n; first done. exact: dist_S. Qed. `````` Ralf Jung committed Mar 01, 2017 211 212 213 214 215 216 217 218 219 220 221 ``````Lemma dist_later_dist {A : ofeT} n (x y : A) : dist_later (S n) x y → dist n x y. Proof. done. Qed. (* We don't actually need this lemma (as our tactics deal with this through other means), but technically speaking, this is the reason why pre-composing a non-expansive function to a contractive function preserves contractivity. *) Lemma ne_dist_later {A B : ofeT} (f : A → B) : NonExpansive f → ∀ n, Proper (dist_later n ==> dist_later n) f. Proof. intros Hf [|n]; last exact: Hf. hnf. by intros. Qed. `````` Robbert Krebbers committed Dec 05, 2016 222 ``````Notation Contractive f := (∀ n, Proper (dist_later n ==> dist n) f). `````` Robbert Krebbers committed Dec 02, 2016 223 `````` `````` Ralf Jung committed Nov 22, 2016 224 ``````Instance const_contractive {A B : ofeT} (x : A) : Contractive (@const A B x). `````` Robbert Krebbers committed Mar 06, 2016 225 226 ``````Proof. by intros n y1 y2. Qed. `````` Robbert Krebbers committed Dec 02, 2016 227 ``````Section contractive. `````` Ralf Jung committed Jan 25, 2017 228 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Dec 02, 2016 229 230 231 232 `````` Context {A B : ofeT} (f : A → B) `{!Contractive f}. Implicit Types x y : A. Lemma contractive_0 x y : f x ≡{0}≡ f y. `````` Robbert Krebbers committed Dec 05, 2016 233 `````` Proof. by apply (_ : Contractive f). Qed. `````` Robbert Krebbers committed Dec 02, 2016 234 `````` Lemma contractive_S n x y : x ≡{n}≡ y → f x ≡{S n}≡ f y. `````` Robbert Krebbers committed Dec 05, 2016 235 `````` Proof. intros. by apply (_ : Contractive f). Qed. `````` Robbert Krebbers committed Dec 02, 2016 236 `````` `````` Ralf Jung committed Jan 27, 2017 237 238 `````` Global Instance contractive_ne : NonExpansive f | 100. Proof. by intros n x y ?; apply dist_S, contractive_S. Qed. `````` Robbert Krebbers committed Dec 02, 2016 239 240 241 242 `````` Global Instance contractive_proper : Proper ((≡) ==> (≡)) f | 100. Proof. apply (ne_proper _). Qed. End contractive. `````` Robbert Krebbers committed Dec 05, 2016 243 244 ``````Ltac f_contractive := match goal with `````` Robbert Krebbers committed Aug 17, 2017 245 246 247 `````` | |- ?f _ ≡{_}≡ ?f _ => simple apply (_ : Proper (dist_later _ ==> _) f) | |- ?f _ _ ≡{_}≡ ?f _ _ => simple apply (_ : Proper (dist_later _ ==> _ ==> _) f) | |- ?f _ _ ≡{_}≡ ?f _ _ => simple apply (_ : Proper (_ ==> dist_later _ ==> _) f) `````` Robbert Krebbers committed Dec 05, 2016 248 249 `````` end; try match goal with `````` Robbert Krebbers committed Aug 17, 2017 250 `````` | |- @dist_later ?A _ ?n ?x ?y => `````` Ralf Jung committed Mar 01, 2017 251 `````` destruct n as [|n]; [exact I|change (@dist A _ n x y)] `````` Robbert Krebbers committed Dec 05, 2016 252 `````` end; `````` Robbert Krebbers committed Aug 17, 2017 253 `````` try simple apply reflexivity. `````` Robbert Krebbers committed Dec 05, 2016 254 `````` `````` Robbert Krebbers committed Aug 17, 2017 255 256 ``````Ltac solve_contractive := solve_proper_core ltac:(fun _ => first [f_contractive | f_equiv]). `````` Robbert Krebbers committed Nov 22, 2015 257 `````` `````` Robbert Krebbers committed Mar 09, 2017 258 259 260 261 262 263 264 265 266 267 268 269 270 271 ``````(** Limit preserving predicates *) Class LimitPreserving `{!Cofe A} (P : A → Prop) : Prop := limit_preserving (c : chain A) : (∀ n, P (c n)) → P (compl c). Hint Mode LimitPreserving + + ! : typeclass_instances. Section limit_preserving. Context `{Cofe A}. (* These are not instances as they will never fire automatically... but they can still be helpful in proving things to be limit preserving. *) Lemma limit_preserving_ext (P Q : A → Prop) : (∀ x, P x ↔ Q x) → LimitPreserving P → LimitPreserving Q. Proof. intros HP Hlimit c ?. apply HP, Hlimit=> n; by apply HP. Qed. `````` Robbert Krebbers committed Jun 21, 2019 272 `````` Global Instance limit_preserving_const (P : Prop) : LimitPreserving (λ _ : A, P). `````` Robbert Krebbers committed Mar 09, 2017 273 274 `````` Proof. intros c HP. apply (HP 0). Qed. `````` Robbert Krebbers committed Oct 25, 2017 275 `````` Lemma limit_preserving_discrete (P : A → Prop) : `````` Robbert Krebbers committed Mar 09, 2017 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 `````` Proper (dist 0 ==> impl) P → LimitPreserving P. Proof. intros PH c Hc. by rewrite (conv_compl 0). Qed. Lemma limit_preserving_and (P1 P2 : A → Prop) : LimitPreserving P1 → LimitPreserving P2 → LimitPreserving (λ x, P1 x ∧ P2 x). Proof. intros Hlim1 Hlim2 c Hc. split. apply Hlim1, Hc. apply Hlim2, Hc. Qed. Lemma limit_preserving_impl (P1 P2 : A → Prop) : Proper (dist 0 ==> impl) P1 → LimitPreserving P2 → LimitPreserving (λ x, P1 x → P2 x). Proof. intros Hlim1 Hlim2 c Hc HP1. apply Hlim2=> n; apply Hc. eapply Hlim1, HP1. apply dist_le with n; last lia. apply (conv_compl n). Qed. Lemma limit_preserving_forall {B} (P : B → A → Prop) : (∀ y, LimitPreserving (P y)) → LimitPreserving (λ x, ∀ y, P y x). Proof. intros Hlim c Hc y. by apply Hlim. Qed. `````` Robbert Krebbers committed Mar 31, 2020 296 297 298 299 300 301 302 `````` Lemma limit_preserving_equiv `{!Cofe B} (f g : A → B) : NonExpansive f → NonExpansive g → LimitPreserving (λ x, f x ≡ g x). Proof. intros Hf Hg c Hc. apply equiv_dist=> n. by rewrite -!compl_chain_map !conv_compl /= Hc. Qed. `````` Robbert Krebbers committed Mar 09, 2017 303 304 ``````End limit_preserving. `````` Robbert Krebbers committed Nov 11, 2015 305 ``````(** Fixpoint *) `````` Ralf Jung committed Nov 22, 2016 306 ``````Program Definition fixpoint_chain {A : ofeT} `{Inhabited A} (f : A → A) `````` Robbert Krebbers committed Feb 10, 2016 307 `````` `{!Contractive f} : chain A := {| chain_car i := Nat.iter (S i) f inhabitant |}. `````` Robbert Krebbers committed Nov 11, 2015 308 ``````Next Obligation. `````` Robbert Krebbers committed Mar 06, 2016 309 `````` intros A ? f ? n. `````` Ralf Jung committed Jun 20, 2018 310 `````` induction n as [|n IH]=> -[|i] //= ?; try lia. `````` Robbert Krebbers committed Feb 17, 2016 311 `````` - apply (contractive_0 f). `````` Ralf Jung committed Jun 20, 2018 312 `````` - apply (contractive_S f), IH; auto with lia. `````` Robbert Krebbers committed Nov 11, 2015 313 ``````Qed. `````` Robbert Krebbers committed Mar 18, 2016 314 `````` `````` Ralf Jung committed Nov 22, 2016 315 ``````Program Definition fixpoint_def `{Cofe A, Inhabited A} (f : A → A) `````` Robbert Krebbers committed Nov 17, 2015 316 `````` `{!Contractive f} : A := compl (fixpoint_chain f). `````` Paolo G. Giarrusso committed Mar 31, 2020 317 ``````Definition fixpoint_aux : seal (@fixpoint_def). Proof. by eexists. Qed. `````` Ralf Jung committed Mar 05, 2018 318 319 ``````Definition fixpoint {A AC AiH} f {Hf} := fixpoint_aux.(unseal) A AC AiH f Hf. Definition fixpoint_eq : @fixpoint = @fixpoint_def := fixpoint_aux.(seal_eq). `````` Robbert Krebbers committed Nov 11, 2015 320 321 `````` Section fixpoint. `````` Ralf Jung committed Nov 22, 2016 322 `````` Context `{Cofe A, Inhabited A} (f : A → A) `{!Contractive f}. `````` Robbert Krebbers committed Aug 21, 2016 323 `````` `````` Robbert Krebbers committed Nov 17, 2015 324 `````` Lemma fixpoint_unfold : fixpoint f ≡ f (fixpoint f). `````` Robbert Krebbers committed Nov 11, 2015 325 `````` Proof. `````` Robbert Krebbers committed Mar 18, 2016 326 327 `````` apply equiv_dist=>n. rewrite fixpoint_eq /fixpoint_def (conv_compl n (fixpoint_chain f)) //. `````` Robbert Krebbers committed Feb 12, 2016 328 `````` induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S. `````` Robbert Krebbers committed Nov 11, 2015 329 `````` Qed. `````` Robbert Krebbers committed Aug 21, 2016 330 331 332 `````` Lemma fixpoint_unique (x : A) : x ≡ f x → x ≡ fixpoint f. Proof. `````` Robbert Krebbers committed Aug 22, 2016 333 334 335 `````` rewrite !equiv_dist=> Hx n. induction n as [|n IH]; simpl in *. - rewrite Hx fixpoint_unfold; eauto using contractive_0. - rewrite Hx fixpoint_unfold. apply (contractive_S _), IH. `````` Robbert Krebbers committed Aug 21, 2016 336 337 `````` Qed. `````` Robbert Krebbers committed Nov 17, 2015 338 `````` Lemma fixpoint_ne (g : A → A) `{!Contractive g} n : `````` Ralf Jung committed Feb 10, 2016 339 `````` (∀ z, f z ≡{n}≡ g z) → fixpoint f ≡{n}≡ fixpoint g. `````` Robbert Krebbers committed Nov 11, 2015 340 `````` Proof. `````` Robbert Krebbers committed Mar 18, 2016 341 `````` intros Hfg. rewrite fixpoint_eq /fixpoint_def `````` Robbert Krebbers committed Feb 18, 2016 342 `````` (conv_compl n (fixpoint_chain f)) (conv_compl n (fixpoint_chain g)) /=. `````` Robbert Krebbers committed Feb 10, 2016 343 344 `````` induction n as [|n IH]; simpl in *; [by rewrite !Hfg|]. rewrite Hfg; apply contractive_S, IH; auto using dist_S. `````` Robbert Krebbers committed Nov 11, 2015 345 `````` Qed. `````` Robbert Krebbers committed Nov 17, 2015 346 347 `````` Lemma fixpoint_proper (g : A → A) `{!Contractive g} : (∀ x, f x ≡ g x) → fixpoint f ≡ fixpoint g. `````` Robbert Krebbers committed Nov 11, 2015 348 `````` Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_ne. Qed. `````` Jacques-Henri Jourdan committed Dec 23, 2016 349 350 `````` Lemma fixpoint_ind (P : A → Prop) : `````` Jacques-Henri Jourdan committed Dec 23, 2016 351 `````` Proper ((≡) ==> impl) P → `````` Jacques-Henri Jourdan committed Dec 23, 2016 352 `````` (∃ x, P x) → (∀ x, P x → P (f x)) → `````` Robbert Krebbers committed Mar 09, 2017 353 `````` LimitPreserving P → `````` Jacques-Henri Jourdan committed Dec 23, 2016 354 355 356 357 `````` P (fixpoint f). Proof. intros ? [x Hx] Hincr Hlim. set (chcar i := Nat.iter (S i) f x). assert (Hcauch : ∀ n i : nat, n ≤ i → chcar i ≡{n}≡ chcar n). `````` Robbert Krebbers committed Mar 09, 2017 358 `````` { intros n. rewrite /chcar. induction n as [|n IH]=> -[|i] //=; `````` Ralf Jung committed Jun 20, 2018 359 `````` eauto using contractive_0, contractive_S with lia. } `````` Jacques-Henri Jourdan committed Dec 23, 2016 360 `````` set (fp2 := compl {| chain_cauchy := Hcauch |}). `````` Robbert Krebbers committed Mar 09, 2017 361 362 363 364 `````` assert (f fp2 ≡ fp2). { apply equiv_dist=>n. rewrite /fp2 (conv_compl n) /= /chcar. induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S. } rewrite -(fixpoint_unique fp2) //. `````` Robbert Krebbers committed Mar 11, 2017 365 `````` apply Hlim=> n /=. by apply Nat_iter_ind. `````` Jacques-Henri Jourdan committed Dec 23, 2016 366 `````` Qed. `````` Robbert Krebbers committed Nov 11, 2015 367 368 ``````End fixpoint. `````` Robbert Krebbers committed Mar 09, 2017 369 `````` `````` Ralf Jung committed Jan 25, 2017 370 371 372 ``````(** Fixpoint of f when f^k is contractive. **) Definition fixpointK `{Cofe A, Inhabited A} k (f : A → A) `{!Contractive (Nat.iter k f)} := fixpoint (Nat.iter k f). `````` Ralf Jung committed Jan 25, 2017 373 `````` `````` Ralf Jung committed Jan 25, 2017 374 ``````Section fixpointK. `````` Ralf Jung committed Jan 25, 2017 375 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Jan 25, 2017 376 `````` Context `{Cofe A, Inhabited A} (f : A → A) (k : nat). `````` Ralf Jung committed Feb 23, 2017 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 `````` Context {f_contractive : Contractive (Nat.iter k f)} {f_ne : NonExpansive f}. (* Note than f_ne is crucial here: there are functions f such that f^2 is contractive, but f is not non-expansive. Consider for example f: SPred → SPred (where SPred is "downclosed sets of natural numbers"). Define f (using informative excluded middle) as follows: f(N) = N (where N is the set of all natural numbers) f({0, ..., n}) = {0, ... n-1} if n is even (so n-1 is at least -1, in which case we return the empty set) f({0, ..., n}) = {0, ..., n+2} if n is odd In other words, if we consider elements of SPred as ordinals, then we decreaste odd finite ordinals by 1 and increase even finite ordinals by 2. f is not non-expansive: Consider f({0}) = ∅ and f({0,1}) = f({0,1,2,3}). The arguments are clearly 0-equal, but the results are not. Now consider g := f^2. We have g(N) = N g({0, ..., n}) = {0, ... n+1} if n is even g({0, ..., n}) = {0, ..., n+4} if n is odd g is contractive. All outputs contain 0, so they are all 0-equal. Now consider two n-equal inputs. We have to show that the outputs are n+1-equal. Either they both do not contain n in which case they have to be fully equal and hence so are the results. Or else they both contain n, so the results will both contain n+1, so the results are n+1-equal. *) `````` Robbert Krebbers committed Jan 25, 2017 400 401 `````` Let f_proper : Proper ((≡) ==> (≡)) f := ne_proper f. `````` Ralf Jung committed Feb 23, 2017 402 `````` Local Existing Instance f_proper. `````` Ralf Jung committed Jan 25, 2017 403 `````` `````` Ralf Jung committed Jan 25, 2017 404 `````` Lemma fixpointK_unfold : fixpointK k f ≡ f (fixpointK k f). `````` Ralf Jung committed Jan 25, 2017 405 `````` Proof. `````` Robbert Krebbers committed Jan 25, 2017 406 407 `````` symmetry. rewrite /fixpointK. apply fixpoint_unique. by rewrite -Nat_iter_S_r Nat_iter_S -fixpoint_unfold. `````` Ralf Jung committed Jan 25, 2017 408 409 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 410 `````` Lemma fixpointK_unique (x : A) : x ≡ f x → x ≡ fixpointK k f. `````` Ralf Jung committed Jan 25, 2017 411 `````` Proof. `````` Robbert Krebbers committed Jan 25, 2017 412 413 `````` intros Hf. apply fixpoint_unique. clear f_contractive. induction k as [|k' IH]=> //=. by rewrite -IH. `````` Ralf Jung committed Jan 25, 2017 414 415 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 416 `````` Section fixpointK_ne. `````` Robbert Krebbers committed Jan 25, 2017 417 `````` Context (g : A → A) `{g_contractive : !Contractive (Nat.iter k g)}. `````` Ralf Jung committed Jan 27, 2017 418 `````` Context {g_ne : NonExpansive g}. `````` Ralf Jung committed Jan 25, 2017 419 `````` `````` Ralf Jung committed Jan 25, 2017 420 `````` Lemma fixpointK_ne n : (∀ z, f z ≡{n}≡ g z) → fixpointK k f ≡{n}≡ fixpointK k g. `````` Ralf Jung committed Jan 25, 2017 421 `````` Proof. `````` Robbert Krebbers committed Jan 25, 2017 422 423 424 `````` rewrite /fixpointK=> Hfg /=. apply fixpoint_ne=> z. clear f_contractive g_contractive. induction k as [|k' IH]=> //=. by rewrite IH Hfg. `````` Ralf Jung committed Jan 25, 2017 425 426 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 427 428 429 `````` Lemma fixpointK_proper : (∀ z, f z ≡ g z) → fixpointK k f ≡ fixpointK k g. Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpointK_ne. Qed. End fixpointK_ne. `````` Ralf Jung committed Feb 21, 2017 430 431 432 433 `````` Lemma fixpointK_ind (P : A → Prop) : Proper ((≡) ==> impl) P → (∃ x, P x) → (∀ x, P x → P (f x)) → `````` Robbert Krebbers committed Mar 09, 2017 434 `````` LimitPreserving P → `````` Ralf Jung committed Feb 21, 2017 435 436 `````` P (fixpointK k f). Proof. `````` Robbert Krebbers committed Mar 09, 2017 437 `````` intros. rewrite /fixpointK. apply fixpoint_ind; eauto. `````` Robbert Krebbers committed Mar 11, 2017 438 `````` intros; apply Nat_iter_ind; auto. `````` Ralf Jung committed Feb 21, 2017 439 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 440 ``````End fixpointK. `````` Ralf Jung committed Jan 25, 2017 441 `````` `````` Robbert Krebbers committed Dec 05, 2016 442 ``````(** Mutual fixpoints *) `````` Ralf Jung committed Jan 25, 2017 443 ``````Section fixpointAB. `````` 444 445 `````` Local Unset Default Proof Using. `````` Robbert Krebbers committed Dec 05, 2016 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 `````` Context `{Cofe A, Cofe B, !Inhabited A, !Inhabited B}. Context (fA : A → B → A). Context (fB : A → B → B). Context `{∀ n, Proper (dist_later n ==> dist n ==> dist n) fA}. Context `{∀ n, Proper (dist_later n ==> dist_later n ==> dist n) fB}. Local Definition fixpoint_AB (x : A) : B := fixpoint (fB x). Local Instance fixpoint_AB_contractive : Contractive fixpoint_AB. Proof. intros n x x' Hx; rewrite /fixpoint_AB. apply fixpoint_ne=> y. by f_contractive. Qed. Local Definition fixpoint_AA (x : A) : A := fA x (fixpoint_AB x). Local Instance fixpoint_AA_contractive : Contractive fixpoint_AA. Proof. solve_contractive. Qed. Definition fixpoint_A : A := fixpoint fixpoint_AA. Definition fixpoint_B : B := fixpoint_AB fixpoint_A. Lemma fixpoint_A_unfold : fA fixpoint_A fixpoint_B ≡ fixpoint_A. Proof. by rewrite {2}/fixpoint_A (fixpoint_unfold _). Qed. Lemma fixpoint_B_unfold : fB fixpoint_A fixpoint_B ≡ fixpoint_B. Proof. by rewrite {2}/fixpoint_B /fixpoint_AB (fixpoint_unfold _). Qed. Instance: Proper ((≡) ==> (≡) ==> (≡)) fA. Proof. apply ne_proper_2=> n x x' ? y y' ?. f_contractive; auto using dist_S. Qed. Instance: Proper ((≡) ==> (≡) ==> (≡)) fB. Proof. apply ne_proper_2=> n x x' ? y y' ?. f_contractive; auto using dist_S. Qed. Lemma fixpoint_A_unique p q : fA p q ≡ p → fB p q ≡ q → p ≡ fixpoint_A. Proof. intros HfA HfB. rewrite -HfA. apply fixpoint_unique. rewrite /fixpoint_AA. f_equiv=> //. apply fixpoint_unique. by rewrite HfA HfB. Qed. Lemma fixpoint_B_unique p q : fA p q ≡ p → fB p q ≡ q → q ≡ fixpoint_B. Proof. intros. apply fixpoint_unique. by rewrite -fixpoint_A_unique. Qed. `````` Ralf Jung committed Jan 25, 2017 487 ``````End fixpointAB. `````` Robbert Krebbers committed Dec 05, 2016 488 `````` `````` Ralf Jung committed Jan 25, 2017 489 ``````Section fixpointAB_ne. `````` Robbert Krebbers committed Dec 05, 2016 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 `````` Context `{Cofe A, Cofe B, !Inhabited A, !Inhabited B}. Context (fA fA' : A → B → A). Context (fB fB' : A → B → B). Context `{∀ n, Proper (dist_later n ==> dist n ==> dist n) fA}. Context `{∀ n, Proper (dist_later n ==> dist n ==> dist n) fA'}. Context `{∀ n, Proper (dist_later n ==> dist_later n ==> dist n) fB}. Context `{∀ n, Proper (dist_later n ==> dist_later n ==> dist n) fB'}. Lemma fixpoint_A_ne n : (∀ x y, fA x y ≡{n}≡ fA' x y) → (∀ x y, fB x y ≡{n}≡ fB' x y) → fixpoint_A fA fB ≡{n}≡ fixpoint_A fA' fB'. Proof. intros HfA HfB. apply fixpoint_ne=> z. rewrite /fixpoint_AA /fixpoint_AB HfA. f_equiv. by apply fixpoint_ne. Qed. Lemma fixpoint_B_ne n : (∀ x y, fA x y ≡{n}≡ fA' x y) → (∀ x y, fB x y ≡{n}≡ fB' x y) → fixpoint_B fA fB ≡{n}≡ fixpoint_B fA' fB'. Proof. intros HfA HfB. apply fixpoint_ne=> z. rewrite HfB. f_contractive. apply fixpoint_A_ne; auto using dist_S. Qed. Lemma fixpoint_A_proper : (∀ x y, fA x y ≡ fA' x y) → (∀ x y, fB x y ≡ fB' x y) → fixpoint_A fA fB ≡ fixpoint_A fA' fB'. Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_A_ne. Qed. Lemma fixpoint_B_proper : (∀ x y, fA x y ≡ fA' x y) → (∀ x y, fB x y ≡ fB' x y) → fixpoint_B fA fB ≡ fixpoint_B fA' fB'. Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_B_ne. Qed. `````` Ralf Jung committed Jan 25, 2017 521 ``````End fixpointAB_ne. `````` Robbert Krebbers committed Dec 05, 2016 522 `````` `````` Robbert Krebbers committed Jul 25, 2016 523 ``````(** Non-expansive function space *) `````` Robbert Krebbers committed Jun 16, 2019 524 ``````Record ofe_mor (A B : ofeT) : Type := OfeMor { `````` Ralf Jung committed Nov 22, 2016 525 `````` ofe_mor_car :> A → B; `````` Ralf Jung committed Jan 27, 2017 526 `````` ofe_mor_ne : NonExpansive ofe_mor_car `````` Robbert Krebbers committed Nov 11, 2015 527 ``````}. `````` Robbert Krebbers committed Jun 16, 2019 528 ``````Arguments OfeMor {_ _} _ {_}. `````` Ralf Jung committed Nov 22, 2016 529 530 ``````Add Printing Constructor ofe_mor. Existing Instance ofe_mor_ne. `````` Robbert Krebbers committed Nov 11, 2015 531 `````` `````` Robbert Krebbers committed Jun 17, 2016 532 ``````Notation "'λne' x .. y , t" := `````` Robbert Krebbers committed Jun 16, 2019 533 `````` (@OfeMor _ _ (λ x, .. (@OfeMor _ _ (λ y, t) _) ..) _) `````` Robbert Krebbers committed Jun 17, 2016 534 535 `````` (at level 200, x binder, y binder, right associativity). `````` Ralf Jung committed Nov 22, 2016 536 537 538 539 540 541 542 ``````Section ofe_mor. Context {A B : ofeT}. Global Instance ofe_mor_proper (f : ofe_mor A B) : Proper ((≡) ==> (≡)) f. Proof. apply ne_proper, ofe_mor_ne. Qed. Instance ofe_mor_equiv : Equiv (ofe_mor A B) := λ f g, ∀ x, f x ≡ g x. Instance ofe_mor_dist : Dist (ofe_mor A B) := λ n f g, ∀ x, f x ≡{n}≡ g x. Definition ofe_mor_ofe_mixin : OfeMixin (ofe_mor A B). `````` Robbert Krebbers committed Jan 14, 2016 543 544 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 545 `````` - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|]. `````` Robbert Krebbers committed Feb 18, 2016 546 `````` intros Hfg k; apply equiv_dist=> n; apply Hfg. `````` Robbert Krebbers committed Feb 17, 2016 547 `````` - intros n; split. `````` Robbert Krebbers committed Jan 14, 2016 548 549 `````` + by intros f x. + by intros f g ? x. `````` Ralf Jung committed Feb 20, 2016 550 `````` + by intros f g h ?? x; trans (g x). `````` Robbert Krebbers committed Feb 17, 2016 551 `````` - by intros n f g ? x; apply dist_S. `````` Robbert Krebbers committed Jan 14, 2016 552 `````` Qed. `````` Robbert Krebbers committed Jun 16, 2019 553 `````` Canonical Structure ofe_morO := OfeT (ofe_mor A B) ofe_mor_ofe_mixin. `````` Ralf Jung committed Nov 22, 2016 554 `````` `````` Robbert Krebbers committed Jun 16, 2019 555 `````` Program Definition ofe_mor_chain (c : chain ofe_morO) `````` Ralf Jung committed Nov 22, 2016 556 557 `````` (x : A) : chain B := {| chain_car n := c n x |}. Next Obligation. intros c x n i ?. by apply (chain_cauchy c). Qed. `````` Robbert Krebbers committed Jun 16, 2019 558 `````` Program Definition ofe_mor_compl `{Cofe B} : Compl ofe_morO := λ c, `````` Ralf Jung committed Nov 22, 2016 559 560 561 562 563 `````` {| ofe_mor_car x := compl (ofe_mor_chain c x) |}. Next Obligation. intros ? c n x y Hx. by rewrite (conv_compl n (ofe_mor_chain c x)) (conv_compl n (ofe_mor_chain c y)) /= Hx. Qed. `````` Robbert Krebbers committed Jun 16, 2019 564 `````` Global Program Instance ofe_mor_cofe `{Cofe B} : Cofe ofe_morO := `````` Ralf Jung committed Nov 22, 2016 565 566 567 568 569 `````` {| compl := ofe_mor_compl |}. Next Obligation. intros ? n c x; simpl. by rewrite (conv_compl n (ofe_mor_chain c x)) /=. Qed. `````` Robbert Krebbers committed Jan 14, 2016 570 `````` `````` Ralf Jung committed Jan 27, 2017 571 572 573 `````` Global Instance ofe_mor_car_ne : NonExpansive2 (@ofe_mor_car A B). Proof. intros n f g Hfg x y Hx; rewrite Hx; apply Hfg. Qed. `````` Ralf Jung committed Nov 22, 2016 574 575 576 `````` Global Instance ofe_mor_car_proper : Proper ((≡) ==> (≡) ==> (≡)) (@ofe_mor_car A B) := ne_proper_2 _. Lemma ofe_mor_ext (f g : ofe_mor A B) : f ≡ g ↔ ∀ x, f x ≡ g x. `````` Robbert Krebbers committed Jan 14, 2016 577 `````` Proof. done. Qed. `````` Ralf Jung committed Nov 22, 2016 578 ``````End ofe_mor. `````` Robbert Krebbers committed Jan 14, 2016 579 `````` `````` Robbert Krebbers committed Jun 16, 2019 580 ``````Arguments ofe_morO : clear implicits. `````` Robbert Krebbers committed Jul 25, 2016 581 ``````Notation "A -n> B" := `````` Robbert Krebbers committed Jun 16, 2019 582 `````` (ofe_morO A B) (at level 99, B at level 200, right associativity). `````` Ralf Jung committed Nov 22, 2016 583 ``````Instance ofe_mor_inhabited {A B : ofeT} `{Inhabited B} : `````` Robbert Krebbers committed Jul 25, 2016 584 `````` Inhabited (A -n> B) := populate (λne _, inhabitant). `````` Robbert Krebbers committed Nov 11, 2015 585 `````` `````` Ralf Jung committed Mar 17, 2016 586 ``````(** Identity and composition and constant function *) `````` Robbert Krebbers committed Jun 16, 2019 587 ``````Definition cid {A} : A -n> A := OfeMor id. `````` Maxime Dénès committed Jan 24, 2019 588 ``````Instance: Params (@cid) 1 := {}. `````` Robbert Krebbers committed Jun 16, 2019 589 ``````Definition cconst {A B : ofeT} (x : B) : A -n> B := OfeMor (const x). `````` Maxime Dénès committed Jan 24, 2019 590 ``````Instance: Params (@cconst) 2 := {}. `````` Robbert Krebbers committed Mar 02, 2016 591 `````` `````` Robbert Krebbers committed Nov 11, 2015 592 ``````Definition ccompose {A B C} `````` Robbert Krebbers committed Jun 16, 2019 593 `````` (f : B -n> C) (g : A -n> B) : A -n> C := OfeMor (f ∘ g). `````` Maxime Dénès committed Jan 24, 2019 594 ``````Instance: Params (@ccompose) 3 := {}. `````` Robbert Krebbers committed Nov 11, 2015 595 ``````Infix "◎" := ccompose (at level 40, left associativity). `````` Ralf Jung committed Nov 16, 2017 596 597 598 ``````Global Instance ccompose_ne {A B C} : NonExpansive2 (@ccompose A B C). Proof. intros n ?? Hf g1 g2 Hg x. rewrite /= (Hg x) (Hf (g2 x)) //. Qed. `````` Robbert Krebbers committed Nov 11, 2015 599 `````` `````` Ralf Jung committed Mar 02, 2016 600 ``````(* Function space maps *) `````` Ralf Jung committed Nov 22, 2016 601 ``````Definition ofe_mor_map {A A' B B'} (f : A' -n> A) (g : B -n> B') `````` Ralf Jung committed Mar 02, 2016 602 `````` (h : A -n> B) : A' -n> B' := g ◎ h ◎ f. `````` Ralf Jung committed Nov 22, 2016 603 604 ``````Instance ofe_mor_map_ne {A A' B B'} n : Proper (dist n ==> dist n ==> dist n ==> dist n) (@ofe_mor_map A A' B B'). `````` Robbert Krebbers committed Mar 02, 2016 605 ``````Proof. intros ??? ??? ???. by repeat apply ccompose_ne. Qed. `````` Ralf Jung committed Mar 02, 2016 606 `````` `````` Robbert Krebbers committed Jun 16, 2019 607 608 609 610 ``````Definition ofe_morO_map {A A' B B'} (f : A' -n> A) (g : B -n> B') : (A -n> B) -n> (A' -n> B') := OfeMor (ofe_mor_map f g). Instance ofe_morO_map_ne {A A' B B'} : NonExpansive2 (@ofe_morO_map A A' B B'). `````` Ralf Jung committed Mar 02, 2016 611 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 612 `````` intros n f f' Hf g g' Hg ?. rewrite /= /ofe_mor_map. `````` Robbert Krebbers committed Mar 02, 2016 613 `````` by repeat apply ccompose_ne. `````` Ralf Jung committed Mar 02, 2016 614 615 ``````Qed. `````` Robbert Krebbers committed Apr 01, 2020 616 ``````(** * Unit type *) `````` Robbert Krebbers committed Jan 14, 2016 617 618 ``````Section unit. Instance unit_dist : Dist unit := λ _ _ _, True. `````` Ralf Jung committed Nov 22, 2016 619 `````` Definition unit_ofe_mixin : OfeMixin unit. `````` Robbert Krebbers committed Jan 14, 2016 620 `````` Proof. by repeat split; try exists 0. Qed. `````` Robbert Krebbers committed Jun 16, 2019 621 `````` Canonical Structure unitO : ofeT := OfeT unit unit_ofe_mixin. `````` Robbert Krebbers committed Nov 28, 2016 622 `````` `````` Robbert Krebbers committed Jun 16, 2019 623 `````` Global Program Instance unit_cofe : Cofe unitO := { compl x := () }. `````` Ralf Jung committed Nov 22, 2016 624 `````` Next Obligation. by repeat split; try exists 0. Qed. `````` Robbert Krebbers committed Nov 28, 2016 625 `````` `````` Robbert Krebbers committed Jun 16, 2019 626 `````` Global Instance unit_ofe_discrete : OfeDiscrete unitO. `````` Robbert Krebbers committed Jan 31, 2016 627 `````` Proof. done. Qed. `````` Robbert Krebbers committed Jan 14, 2016 628 ``````End unit. `````` Robbert Krebbers committed Nov 11, 2015 629 `````` `````` Robbert Krebbers committed Apr 01, 2020 630 ``````(** * Empty type *) `````` Ralf Jung committed Aug 26, 2019 631 632 633 634 635 636 637 638 639 640 641 642 643 ``````Section empty. Instance Empty_set_dist : Dist Empty_set := λ _ _ _, True. Definition Empty_set_ofe_mixin : OfeMixin Empty_set. Proof. by repeat split; try exists 0. Qed. Canonical Structure Empty_setO : ofeT := OfeT Empty_set Empty_set_ofe_mixin. Global Program Instance Empty_set_cofe : Cofe Empty_setO := { compl x := x 0 }. Next Obligation. by repeat split; try exists 0. Qed. Global Instance Empty_set_ofe_discrete : OfeDiscrete Empty_setO. Proof. done. Qed. End empty. `````` Robbert Krebbers committed Apr 01, 2020 644 ``````(** * Product type *) `````` Robbert Krebbers committed Jan 14, 2016 645 ``````Section product. `````` Ralf Jung committed Nov 22, 2016 646 `````` Context {A B : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 647 648 649 `````` Instance prod_dist : Dist (A * B) := λ n, prod_relation (dist n) (dist n). Global Instance pair_ne : `````` Ralf Jung committed Jan 27, 2017 650 651 652 `````` NonExpansive2 (@pair A B) := _. Global Instance fst_ne : NonExpansive (@fst A B) := _. Global Instance snd_ne : NonExpansive (@snd A B) := _. `````` Ralf Jung committed Nov 22, 2016 653 `````` Definition prod_ofe_mixin : OfeMixin (A * B). `````` Robbert Krebbers committed Jan 14, 2016 654 655 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 656 `````` - intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation. `````` Robbert Krebbers committed Jan 14, 2016 657 `````` rewrite !equiv_dist; naive_solver. `````` Robbert Krebbers committed Feb 17, 2016 658 659 `````` - apply _. - by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S. `````` Robbert Krebbers committed Jan 14, 2016 660 `````` Qed. `````` Robbert Krebbers committed Jun 16, 2019 661 `````` Canonical Structure prodO : ofeT := OfeT (A * B) prod_ofe_mixin. `````` Ralf Jung committed Nov 22, 2016 662 `````` `````` Robbert Krebbers committed Jun 16, 2019 663 `````` Global Program Instance prod_cofe `{Cofe A, Cofe B} : Cofe prodO := `````` Ralf Jung committed Nov 22, 2016 664 665 666 667 668 669 `````` { compl c := (compl (chain_map fst c), compl (chain_map snd c)) }. Next Obligation. intros ?? n c; split. apply (conv_compl n (chain_map fst c)). apply (conv_compl n (chain_map snd c)). Qed. `````` Robbert Krebbers committed Oct 25, 2017 670 671 672 `````` Global Instance prod_discrete (x : A * B) : Discrete (x.1) → Discrete (x.2) → Discrete x. Proof. by intros ???[??]; split; apply (discrete _). Qed. `````` Robbert Krebbers committed Oct 25, 2017 673 `````` Global Instance prod_ofe_discrete : `````` Robbert Krebbers committed Jun 16, 2019 674 `````` OfeDiscrete A → OfeDiscrete B → OfeDiscrete prodO. `````` Robbert Krebbers committed Feb 24, 2016 675 `````` Proof. intros ?? [??]; apply _. Qed. `````` Robbert Krebbers committed Jan 14, 2016 676 677 ``````End product. `````` Robbert Krebbers committed Jun 16, 2019 678 ``````Arguments prodO : clear implicits. `````` Robbert Krebbers committed Jan 14, 2016 679 680 ``````Typeclasses Opaque prod_dist. `````` Ralf Jung committed Nov 22, 2016 681 ``````Instance prod_map_ne {A A' B B' : ofeT} n : `````` Robbert Krebbers committed Nov 11, 2015 682 683 684 `````` Proper ((dist n ==> dist n) ==> (dist n ==> dist n) ==> dist n ==> dist n) (@prod_map A A' B B'). Proof. by intros f f' Hf g g' Hg ?? [??]; split; [apply Hf|apply Hg]. Qed. `````` Robbert Krebbers committed Jun 16, 2019 685 686 687 688 ``````Definition prodO_map {A A' B B'} (f : A -n> A') (g : B -n> B') : prodO A B -n> prodO A' B' := OfeMor (prod_map f g). Instance prodO_map_ne {A A' B B'} : NonExpansive2 (@prodO_map A A' B B'). `````` Ralf Jung committed Jan 27, 2017 689 ``````Proof. intros n f f' Hf g g' Hg [??]; split; [apply Hf|apply Hg]. Qed. `````` Robbert Krebbers committed Nov 11, 2015 690 `````` `````` Robbert Krebbers committed Apr 01, 2020 691 ``````(** * COFE → OFE Functors *) `````` Robbert Krebbers committed Jun 16, 2019 692 693 694 695 ``````Record oFunctor := OFunctor { oFunctor_car : ∀ A `{!Cofe A} B `{!Cofe B}, ofeT; oFunctor_map `{!Cofe A1, !Cofe A2, !Cofe B1, !Cofe B2} : ((A2 -n> A1) * (B1 -n> B2)) → oFunctor_car A1 B1 -n> oFunctor_car A2 B2; `````` Robbert Krebbers committed Apr 02, 2020 696 `````` oFunctor_map_ne `{!Cofe A1, !Cofe A2, !Cofe B1, !Cofe B2} : `````` Robbert Krebbers committed Jun 16, 2019 697 `````` NonExpansive (@oFunctor_map A1 _ A2 _ B1 _ B2 _); `````` Robbert Krebbers committed Apr 02, 2020 698 `````` oFunctor_map_id `{!Cofe A, !Cofe B} (x : oFunctor_car A B) : `````` Robbert Krebbers committed Jun 16, 2019 699 `````` oFunctor_map (cid,cid) x ≡ x; `````` Robbert Krebbers committed Apr 02, 2020 700 `````` oFunctor_map_compose `{!Cofe A1, !Cofe A2, !Cofe A3, !Cofe B1, !Cofe B2, !Cofe B3} `````` Robbert Krebbers committed Mar 02, 2016 701 `````` (f : A2 -n> A1) (g : A3 -n> A2) (f' : B1 -n> B2) (g' : B2 -n> B3) x : `````` Robbert Krebbers committed Jun 16, 2019 702 `````` oFunctor_map (f◎g, g'◎f') x ≡ oFunctor_map (g,g') (oFunctor_map (f,f') x) `````` Robbert Krebbers committed Mar 02, 2016 703 ``````}. `````` Robbert Krebbers committed Apr 02, 2020 704 ``````Existing Instance oFunctor_map_ne. `````` Robbert Krebbers committed Jun 16, 2019 705 ``````Instance: Params (@oFunctor_map) 9 := {}. `````` Robbert Krebbers committed Mar 02, 2016 706 `````` `````` Ralf Jung committed Aug 12, 2020 707 ``````Declare Scope oFunctor_scope. `````` Robbert Krebbers committed Jun 16, 2019 708 709 ``````Delimit Scope oFunctor_scope with OF. Bind Scope oFunctor_scope with oFunctor. `````` Ralf Jung committed Mar 07, 2016 710 `````` `````` Robbert Krebbers committed Jun 16, 2019 711 ``````Class oFunctorContractive (F : oFunctor) := `````` Robbert Krebbers committed Apr 02, 2020 712 `````` oFunctor_map_contractive `{!Cofe A1, !Cofe A2, !Cofe B1, !Cofe B2} :> `````` Robbert Krebbers committed Jun 16, 2019 713 714 `````` Contractive (@oFunctor_map F A1 _ A2 _ B1 _ B2 _). Hint Mode oFunctorContractive ! : typeclass_instances. `````` Ralf Jung committed Mar 07, 2016 715 `````` `````` Robbert Krebbers committed Apr 01, 2020 716 717 ``````(** Not a coercion due to the [Cofe] type class argument, and to avoid ambiguous coercion paths, see https://gitlab.mpi-sws.org/iris/iris/issues/240. *) `````` Robbert Krebbers committed Apr 01, 2020 718 ``````Definition oFunctor_apply (F: oFunctor) (A: ofeT) `{!Cofe A} : ofeT := `````` Robbert Krebbers committed Jun 16, 2019 719 `````` oFunctor_car F A A. `````` Robbert Krebbers committed Mar 02, 2016 720 `````` `````` Robbert Krebbers committed Apr 02, 2020 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 ``````Program Definition oFunctor_oFunctor_compose (F1 F2 : oFunctor) `{!∀ `{Cofe A, Cofe B}, Cofe (oFunctor_car F2 A B)} : oFunctor := {| oFunctor_car A _ B _ := oFunctor_car F1 (oFunctor_car F2 B A) (oFunctor_car F2 A B); oFunctor_map A1 _ A2 _ B1 _ B2 _ 'fg := oFunctor_map F1 (oFunctor_map F2 (fg.2,fg.1),oFunctor_map F2 fg) |}. Next Obligation. intros F1 F2 ? A1 ? A2 ? B1 ? B2 ? n [f1 g1] [f2 g2] [??]; simpl in *. apply oFunctor_map_ne; split; apply oFunctor_map_ne; by split. Qed. Next Obligation. intros F1 F2 ? A ? B ? x; simpl in *. rewrite -{2}(oFunctor_map_id F1 x). apply equiv_dist=> n. apply oFunctor_map_ne. split=> y /=; by rewrite !oFunctor_map_id. Qed. Next Obligation. intros F1 F2 ? A1 ? A2 ? A3 ? B1 ? B2 ? B3 ? f g f' g' x; simpl in *. rewrite -oFunctor_map_compose. apply equiv_dist=> n. apply oFunctor_map_ne. split=> y /=; by rewrite !oFunctor_map_compose. Qed. Instance oFunctor_oFunctor_compose_contractive_1 (F1 F2 : oFunctor) `{!∀ `{Cofe A, Cofe B}, Cofe (oFunctor_car F2 A B)} : oFunctorContractive F1 → oFunctorContractive (oFunctor_oFunctor_compose F1 F2). Proof. intros ? A1 ? A2 ? B1 ? B2 ? n [f1 g1] [f2 g2] Hfg; simpl in *. f_contractive; destruct Hfg; split; simpl in *; apply oFunctor_map_ne; by split. Qed. Instance oFunctor_oFunctor_compose_contractive_2 (F1 F2 : oFunctor) `{!∀ `{Cofe A, Cofe B}, Cofe (oFunctor_car F2 A B)} : oFunctorContractive F2 → oFunc``````