ofe.v 47.8 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 11, 2015 3 `````` `````` Ralf Jung committed Nov 22, 2016 4 ``````(** This files defines (a shallow embedding of) the category of OFEs: `````` Ralf Jung committed Feb 16, 2016 5 6 7 8 9 10 11 12 `````` 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. This makes writing such functions much easier. It turns out that it many cases, we do not even need non-expansiveness. *) `````` Robbert Krebbers committed Nov 11, 2015 13 14 ``````(** Unbundeled version *) Class Dist A := dist : nat → relation A. `````` Robbert Krebbers committed Nov 12, 2015 15 ``````Instance: Params (@dist) 3. `````` Ralf Jung committed Feb 10, 2016 16 17 ``````Notation "x ≡{ n }≡ y" := (dist n x y) (at level 70, n at next level, format "x ≡{ n }≡ y"). `````` Robbert Krebbers committed Feb 13, 2016 18 ``````Hint Extern 0 (_ ≡{_}≡ _) => reflexivity. `````` Ralf Jung committed Feb 10, 2016 19 ``````Hint Extern 0 (_ ≡{_}≡ _) => symmetry; assumption. `````` Ralf Jung committed Jan 27, 2017 20 21 ``````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 22 `````` `````` Robbert Krebbers committed Feb 09, 2017 23 ``````Tactic Notation "ofe_subst" ident(x) := `````` Robbert Krebbers committed Jan 13, 2016 24 `````` repeat match goal with `````` Robbert Krebbers committed Feb 17, 2016 25 `````` | _ => progress simplify_eq/= `````` Robbert Krebbers committed Jan 13, 2016 26 27 28 `````` | 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 29 ``````Tactic Notation "ofe_subst" := `````` Robbert Krebbers committed Nov 17, 2015 30 `````` repeat match goal with `````` Robbert Krebbers committed Feb 17, 2016 31 `````` | _ => progress simplify_eq/= `````` Robbert Krebbers committed Dec 21, 2015 32 33 `````` | 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 34 `````` end. `````` Robbert Krebbers committed Nov 11, 2015 35 `````` `````` Ralf Jung committed Nov 22, 2016 36 ``````Record OfeMixin A `{Equiv A, Dist A} := { `````` Ralf Jung committed Feb 10, 2016 37 `````` mixin_equiv_dist x y : x ≡ y ↔ ∀ n, x ≡{n}≡ y; `````` Robbert Krebbers committed Jan 14, 2016 38 `````` mixin_dist_equivalence n : Equivalence (dist n); `````` Ralf Jung committed Nov 22, 2016 39 `````` mixin_dist_S n x y : x ≡{S n}≡ y → x ≡{n}≡ y `````` Robbert Krebbers committed Nov 11, 2015 40 41 42 ``````}. (** Bundeled version *) `````` Ralf Jung committed Nov 22, 2016 43 44 45 46 47 ``````Structure ofeT := OfeT' { ofe_car :> Type; ofe_equiv : Equiv ofe_car; ofe_dist : Dist ofe_car; ofe_mixin : OfeMixin ofe_car; `````` Robbert Krebbers committed Jun 15, 2016 48 `````` _ : Type `````` Robbert Krebbers committed Nov 11, 2015 49 ``````}. `````` Ralf Jung committed Nov 22, 2016 50 51 52 53 54 55 56 57 58 ``````Arguments OfeT' _ {_ _} _ _. Notation OfeT A m := (OfeT' A m A). 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 59 `````` `````` Robbert Krebbers committed Feb 09, 2017 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 ``````(** 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 different places (see for example the constructors [CMRAT] and [UCMRAT] in the 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 85 ``````(** Lifting properties from the mixin *) `````` Ralf Jung committed Nov 22, 2016 86 87 ``````Section ofe_mixin. Context {A : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 88 `````` Implicit Types x y : A. `````` Ralf Jung committed Feb 10, 2016 89 `````` Lemma equiv_dist x y : x ≡ y ↔ ∀ n, x ≡{n}≡ y. `````` Ralf Jung committed Nov 22, 2016 90 `````` Proof. apply (mixin_equiv_dist _ (ofe_mixin A)). Qed. `````` Robbert Krebbers committed Jan 14, 2016 91 `````` Global Instance dist_equivalence n : Equivalence (@dist A _ n). `````` Ralf Jung committed Nov 22, 2016 92 `````` Proof. apply (mixin_dist_equivalence _ (ofe_mixin A)). Qed. `````` Ralf Jung committed Feb 10, 2016 93 `````` Lemma dist_S n x y : x ≡{S n}≡ y → x ≡{n}≡ y. `````` Ralf Jung committed Nov 22, 2016 94 95 `````` Proof. apply (mixin_dist_S _ (ofe_mixin A)). Qed. End ofe_mixin. `````` Robbert Krebbers committed Jan 14, 2016 96 `````` `````` Robbert Krebbers committed May 28, 2016 97 98 ``````Hint Extern 1 (_ ≡{_}≡ _) => apply equiv_dist; assumption. `````` Ralf Jung committed Dec 21, 2016 99 ``````(** Discrete OFEs and Timeless elements *) `````` Ralf Jung committed Mar 15, 2016 100 ``````(* TODO: On paper, We called these "discrete elements". I think that makes `````` Ralf Jung committed Mar 07, 2016 101 `````` more sense. *) `````` Robbert Krebbers committed Jan 22, 2017 102 103 104 ``````Class Timeless {A : ofeT} (x : A) := timeless y : x ≡{0}≡ y → x ≡ y. Arguments timeless {_} _ {_} _ _. Hint Mode Timeless + ! : typeclass_instances. `````` Robbert Krebbers committed Feb 11, 2017 105 ``````Instance: Params (@Timeless) 1. `````` Robbert Krebbers committed Jan 22, 2017 106 `````` `````` Ralf Jung committed Nov 22, 2016 107 108 109 110 111 112 113 114 115 116 ``````Class Discrete (A : ofeT) := discrete_timeless (x : A) :> Timeless x. (** 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 Feb 24, 2016 128 `````` `````` Robbert Krebbers committed Feb 09, 2017 129 ``````Lemma compl_chain_map `{Cofe A, Cofe B} (f : A → B) c `(NonExpansive f) : `````` Jacques-Henri Jourdan committed Jan 05, 2017 130 131 132 `````` compl (chain_map f c) ≡ f (compl c). Proof. apply equiv_dist=>n. by rewrite !conv_compl. Qed. `````` Ralf Jung committed Mar 01, 2017 133 134 135 136 137 138 139 140 ``````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 141 ``````(** General properties *) `````` Robbert Krebbers committed Feb 09, 2017 142 ``````Section ofe. `````` Ralf Jung committed Nov 22, 2016 143 `````` Context {A : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 144 `````` Implicit Types x y : A. `````` Robbert Krebbers committed Feb 09, 2017 145 `````` Global Instance ofe_equivalence : Equivalence ((≡) : relation A). `````` Robbert Krebbers committed Nov 11, 2015 146 147 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 148 149 `````` - by intros x; rewrite equiv_dist. - by intros x y; rewrite !equiv_dist. `````` Ralf Jung committed Feb 20, 2016 150 `````` - by intros x y z; rewrite !equiv_dist; intros; trans y. `````` Robbert Krebbers committed Nov 11, 2015 151 `````` Qed. `````` Robbert Krebbers committed Jan 14, 2016 152 `````` Global Instance dist_ne n : Proper (dist n ==> dist n ==> iff) (@dist A _ n). `````` Robbert Krebbers committed Nov 11, 2015 153 154 `````` Proof. intros x1 x2 ? y1 y2 ?; split; intros. `````` Ralf Jung committed Feb 20, 2016 155 156 `````` - by trans x1; [|trans y1]. - by trans x2; [|trans y2]. `````` Robbert Krebbers committed Nov 11, 2015 157 `````` Qed. `````` Robbert Krebbers committed Jan 14, 2016 158 `````` Global Instance dist_proper n : Proper ((≡) ==> (≡) ==> iff) (@dist A _ n). `````` Robbert Krebbers committed Nov 11, 2015 159 `````` Proof. `````` Robbert Krebbers committed Jan 13, 2016 160 `````` by move => x1 x2 /equiv_dist Hx y1 y2 /equiv_dist Hy; rewrite (Hx n) (Hy n). `````` Robbert Krebbers committed Nov 11, 2015 161 162 163 `````` Qed. Global Instance dist_proper_2 n x : Proper ((≡) ==> iff) (dist n x). Proof. by apply dist_proper. Qed. `````` Robbert Krebbers committed Feb 11, 2017 164 165 166 `````` Global Instance Timeless_proper : Proper ((≡) ==> iff) (@Timeless A). Proof. intros x y Hxy. rewrite /Timeless. by setoid_rewrite Hxy. Qed. `````` Robbert Krebbers committed Feb 18, 2016 167 `````` Lemma dist_le n n' x y : x ≡{n}≡ y → n' ≤ n → x ≡{n'}≡ y. `````` Robbert Krebbers committed Nov 11, 2015 168 `````` Proof. induction 2; eauto using dist_S. Qed. `````` Ralf Jung committed Feb 29, 2016 169 170 `````` Lemma dist_le' n n' x y : n' ≤ n → x ≡{n}≡ y → x ≡{n'}≡ y. Proof. intros; eauto using dist_le. Qed. `````` Robbert Krebbers committed Feb 11, 2017 171 172 `````` Instance ne_proper {B : ofeT} (f : A → B) `{!NonExpansive f} : Proper ((≡) ==> (≡)) f | 100. `````` Robbert Krebbers committed Nov 11, 2015 173 `````` Proof. by intros x1 x2; rewrite !equiv_dist; intros Hx n; rewrite (Hx n). Qed. `````` Robbert Krebbers committed Feb 11, 2017 174 `````` Instance ne_proper_2 {B C : ofeT} (f : A → B → C) `{!NonExpansive2 f} : `````` Robbert Krebbers committed Nov 11, 2015 175 176 177 `````` Proper ((≡) ==> (≡) ==> (≡)) f | 100. Proof. unfold Proper, respectful; setoid_rewrite equiv_dist. `````` Robbert Krebbers committed Jan 13, 2016 178 `````` by intros x1 x2 Hx y1 y2 Hy n; rewrite (Hx n) (Hy n). `````` Robbert Krebbers committed Nov 11, 2015 179 `````` Qed. `````` Robbert Krebbers committed Feb 24, 2016 180 `````` `````` Ralf Jung committed Nov 22, 2016 181 `````` Lemma conv_compl' `{Cofe A} n (c : chain A) : compl c ≡{n}≡ c (S n). `````` Ralf Jung committed Feb 29, 2016 182 183 184 185 `````` Proof. transitivity (c n); first by apply conv_compl. symmetry. apply chain_cauchy. omega. Qed. `````` Robbert Krebbers committed Feb 24, 2016 186 187 `````` Lemma timeless_iff n (x : A) `{!Timeless x} y : x ≡ y ↔ x ≡{n}≡ y. Proof. `````` Robbert Krebbers committed May 28, 2016 188 `````` split; intros; auto. apply (timeless _), dist_le with n; auto with lia. `````` Robbert Krebbers committed Feb 24, 2016 189 `````` Qed. `````` Robbert Krebbers committed Feb 09, 2017 190 ``````End ofe. `````` Robbert Krebbers committed Nov 11, 2015 191 `````` `````` Robbert Krebbers committed Dec 02, 2016 192 ``````(** Contractive functions *) `````` Robbert Krebbers committed Dec 05, 2016 193 194 195 196 197 198 199 ``````Definition dist_later {A : ofeT} (n : nat) (x y : A) : Prop := match n with 0 => True | S n => x ≡{n}≡ y end. Arguments dist_later _ !_ _ _ /. Global Instance dist_later_equivalence A n : Equivalence (@dist_later A n). Proof. destruct n as [|n]. by split. apply dist_equivalence. Qed. `````` Ralf Jung committed Feb 22, 2017 200 201 202 ``````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 203 204 205 206 207 208 209 210 211 212 213 ``````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 214 ``````Notation Contractive f := (∀ n, Proper (dist_later n ==> dist n) f). `````` Robbert Krebbers committed Dec 02, 2016 215 `````` `````` Ralf Jung committed Nov 22, 2016 216 ``````Instance const_contractive {A B : ofeT} (x : A) : Contractive (@const A B x). `````` Robbert Krebbers committed Mar 06, 2016 217 218 ``````Proof. by intros n y1 y2. Qed. `````` Robbert Krebbers committed Dec 02, 2016 219 ``````Section contractive. `````` Ralf Jung committed Jan 25, 2017 220 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Dec 02, 2016 221 222 223 224 `````` 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 225 `````` Proof. by apply (_ : Contractive f). Qed. `````` Robbert Krebbers committed Dec 02, 2016 226 `````` Lemma contractive_S n x y : x ≡{n}≡ y → f x ≡{S n}≡ f y. `````` Robbert Krebbers committed Dec 05, 2016 227 `````` Proof. intros. by apply (_ : Contractive f). Qed. `````` Robbert Krebbers committed Dec 02, 2016 228 `````` `````` Ralf Jung committed Jan 27, 2017 229 230 `````` 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 231 232 233 234 `````` Global Instance contractive_proper : Proper ((≡) ==> (≡)) f | 100. Proof. apply (ne_proper _). Qed. End contractive. `````` Robbert Krebbers committed Dec 05, 2016 235 236 237 238 239 240 241 ``````Ltac f_contractive := match goal with | |- ?f _ ≡{_}≡ ?f _ => apply (_ : Proper (dist_later _ ==> _) f) | |- ?f _ _ ≡{_}≡ ?f _ _ => apply (_ : Proper (dist_later _ ==> _ ==> _) f) | |- ?f _ _ ≡{_}≡ ?f _ _ => apply (_ : Proper (_ ==> dist_later _ ==> _) f) end; try match goal with `````` Jacques-Henri Jourdan committed Dec 26, 2016 242 `````` | |- @dist_later ?A ?n ?x ?y => `````` Ralf Jung committed Mar 01, 2017 243 `````` destruct n as [|n]; [exact I|change (@dist A _ n x y)] `````` Robbert Krebbers committed Dec 05, 2016 244 245 246 `````` end; try reflexivity. `````` Ralf Jung committed Feb 22, 2017 247 ``````Ltac solve_contractive := solve_proper_core ltac:(fun _ => first [f_contractive | f_equiv]). `````` Robbert Krebbers committed Nov 22, 2015 248 `````` `````` Robbert Krebbers committed Mar 09, 2017 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 ``````(** 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. Global Instance limit_preserving_const (P : Prop) : LimitPreserving (λ _, P). Proof. intros c HP. apply (HP 0). Qed. Lemma limit_preserving_timeless (P : A → Prop) : 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. End limit_preserving. `````` Robbert Krebbers committed Nov 11, 2015 289 ``````(** Fixpoint *) `````` Ralf Jung committed Nov 22, 2016 290 ``````Program Definition fixpoint_chain {A : ofeT} `{Inhabited A} (f : A → A) `````` Robbert Krebbers committed Feb 10, 2016 291 `````` `{!Contractive f} : chain A := {| chain_car i := Nat.iter (S i) f inhabitant |}. `````` Robbert Krebbers committed Nov 11, 2015 292 ``````Next Obligation. `````` Robbert Krebbers committed Mar 06, 2016 293 `````` intros A ? f ? n. `````` Robbert Krebbers committed Dec 05, 2016 294 `````` induction n as [|n IH]=> -[|i] //= ?; try omega. `````` Robbert Krebbers committed Feb 17, 2016 295 296 `````` - apply (contractive_0 f). - apply (contractive_S f), IH; auto with omega. `````` Robbert Krebbers committed Nov 11, 2015 297 ``````Qed. `````` Robbert Krebbers committed Mar 18, 2016 298 `````` `````` Ralf Jung committed Nov 22, 2016 299 ``````Program Definition fixpoint_def `{Cofe A, Inhabited A} (f : A → A) `````` Robbert Krebbers committed Nov 17, 2015 300 `````` `{!Contractive f} : A := compl (fixpoint_chain f). `````` Ralf Jung committed Jan 11, 2017 301 302 303 ``````Definition fixpoint_aux : seal (@fixpoint_def). by eexists. Qed. Definition fixpoint {A AC AiH} f {Hf} := unseal fixpoint_aux A AC AiH f Hf. Definition fixpoint_eq : @fixpoint = @fixpoint_def := seal_eq fixpoint_aux. `````` Robbert Krebbers committed Nov 11, 2015 304 305 `````` Section fixpoint. `````` Ralf Jung committed Nov 22, 2016 306 `````` Context `{Cofe A, Inhabited A} (f : A → A) `{!Contractive f}. `````` Robbert Krebbers committed Aug 21, 2016 307 `````` `````` Robbert Krebbers committed Nov 17, 2015 308 `````` Lemma fixpoint_unfold : fixpoint f ≡ f (fixpoint f). `````` Robbert Krebbers committed Nov 11, 2015 309 `````` Proof. `````` Robbert Krebbers committed Mar 18, 2016 310 311 `````` apply equiv_dist=>n. rewrite fixpoint_eq /fixpoint_def (conv_compl n (fixpoint_chain f)) //. `````` Robbert Krebbers committed Feb 12, 2016 312 `````` induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S. `````` Robbert Krebbers committed Nov 11, 2015 313 `````` Qed. `````` Robbert Krebbers committed Aug 21, 2016 314 315 316 `````` Lemma fixpoint_unique (x : A) : x ≡ f x → x ≡ fixpoint f. Proof. `````` Robbert Krebbers committed Aug 22, 2016 317 318 319 `````` 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 320 321 `````` Qed. `````` Robbert Krebbers committed Nov 17, 2015 322 `````` Lemma fixpoint_ne (g : A → A) `{!Contractive g} n : `````` Ralf Jung committed Feb 10, 2016 323 `````` (∀ z, f z ≡{n}≡ g z) → fixpoint f ≡{n}≡ fixpoint g. `````` Robbert Krebbers committed Nov 11, 2015 324 `````` Proof. `````` Robbert Krebbers committed Mar 18, 2016 325 `````` intros Hfg. rewrite fixpoint_eq /fixpoint_def `````` Robbert Krebbers committed Feb 18, 2016 326 `````` (conv_compl n (fixpoint_chain f)) (conv_compl n (fixpoint_chain g)) /=. `````` Robbert Krebbers committed Feb 10, 2016 327 328 `````` 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 329 `````` Qed. `````` Robbert Krebbers committed Nov 17, 2015 330 331 `````` Lemma fixpoint_proper (g : A → A) `{!Contractive g} : (∀ x, f x ≡ g x) → fixpoint f ≡ fixpoint g. `````` Robbert Krebbers committed Nov 11, 2015 332 `````` Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_ne. Qed. `````` Jacques-Henri Jourdan committed Dec 23, 2016 333 334 `````` Lemma fixpoint_ind (P : A → Prop) : `````` Jacques-Henri Jourdan committed Dec 23, 2016 335 `````` Proper ((≡) ==> impl) P → `````` Jacques-Henri Jourdan committed Dec 23, 2016 336 `````` (∃ x, P x) → (∀ x, P x → P (f x)) → `````` Robbert Krebbers committed Mar 09, 2017 337 `````` LimitPreserving P → `````` Jacques-Henri Jourdan committed Dec 23, 2016 338 339 340 341 `````` 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 342 343 `````` { intros n. rewrite /chcar. induction n as [|n IH]=> -[|i] //=; eauto using contractive_0, contractive_S with omega. } `````` Jacques-Henri Jourdan committed Dec 23, 2016 344 `````` set (fp2 := compl {| chain_cauchy := Hcauch |}). `````` Robbert Krebbers committed Mar 09, 2017 345 346 347 348 `````` 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 349 `````` apply Hlim=> n /=. by apply Nat_iter_ind. `````` Jacques-Henri Jourdan committed Dec 23, 2016 350 `````` Qed. `````` Robbert Krebbers committed Nov 11, 2015 351 352 ``````End fixpoint. `````` Robbert Krebbers committed Mar 09, 2017 353 `````` `````` Ralf Jung committed Jan 25, 2017 354 355 356 ``````(** 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 357 `````` `````` Ralf Jung committed Jan 25, 2017 358 ``````Section fixpointK. `````` Ralf Jung committed Jan 25, 2017 359 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Jan 25, 2017 360 `````` Context `{Cofe A, Inhabited A} (f : A → A) (k : nat). `````` Ralf Jung committed Feb 23, 2017 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 `````` 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 384 385 `````` Let f_proper : Proper ((≡) ==> (≡)) f := ne_proper f. `````` Ralf Jung committed Feb 23, 2017 386 `````` Local Existing Instance f_proper. `````` Ralf Jung committed Jan 25, 2017 387 `````` `````` Ralf Jung committed Jan 25, 2017 388 `````` Lemma fixpointK_unfold : fixpointK k f ≡ f (fixpointK k f). `````` Ralf Jung committed Jan 25, 2017 389 `````` Proof. `````` Robbert Krebbers committed Jan 25, 2017 390 391 `````` symmetry. rewrite /fixpointK. apply fixpoint_unique. by rewrite -Nat_iter_S_r Nat_iter_S -fixpoint_unfold. `````` Ralf Jung committed Jan 25, 2017 392 393 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 394 `````` Lemma fixpointK_unique (x : A) : x ≡ f x → x ≡ fixpointK k f. `````` Ralf Jung committed Jan 25, 2017 395 `````` Proof. `````` Robbert Krebbers committed Jan 25, 2017 396 397 `````` intros Hf. apply fixpoint_unique. clear f_contractive. induction k as [|k' IH]=> //=. by rewrite -IH. `````` Ralf Jung committed Jan 25, 2017 398 399 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 400 `````` Section fixpointK_ne. `````` Robbert Krebbers committed Jan 25, 2017 401 `````` Context (g : A → A) `{g_contractive : !Contractive (Nat.iter k g)}. `````` Ralf Jung committed Jan 27, 2017 402 `````` Context {g_ne : NonExpansive g}. `````` Ralf Jung committed Jan 25, 2017 403 `````` `````` Ralf Jung committed Jan 25, 2017 404 `````` Lemma fixpointK_ne n : (∀ z, f z ≡{n}≡ g z) → fixpointK k f ≡{n}≡ fixpointK k g. `````` Ralf Jung committed Jan 25, 2017 405 `````` Proof. `````` Robbert Krebbers committed Jan 25, 2017 406 407 408 `````` 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 409 410 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 411 412 413 `````` 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 414 415 416 417 `````` Lemma fixpointK_ind (P : A → Prop) : Proper ((≡) ==> impl) P → (∃ x, P x) → (∀ x, P x → P (f x)) → `````` Robbert Krebbers committed Mar 09, 2017 418 `````` LimitPreserving P → `````` Ralf Jung committed Feb 21, 2017 419 420 `````` P (fixpointK k f). Proof. `````` Robbert Krebbers committed Mar 09, 2017 421 `````` intros. rewrite /fixpointK. apply fixpoint_ind; eauto. `````` Robbert Krebbers committed Mar 11, 2017 422 `````` intros; apply Nat_iter_ind; auto. `````` Ralf Jung committed Feb 21, 2017 423 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 424 ``````End fixpointK. `````` Ralf Jung committed Jan 25, 2017 425 `````` `````` Robbert Krebbers committed Dec 05, 2016 426 ``````(** Mutual fixpoints *) `````` Ralf Jung committed Jan 25, 2017 427 ``````Section fixpointAB. `````` 428 429 `````` Local Unset Default Proof Using. `````` Robbert Krebbers committed Dec 05, 2016 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 `````` 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 471 ``````End fixpointAB. `````` Robbert Krebbers committed Dec 05, 2016 472 `````` `````` Ralf Jung committed Jan 25, 2017 473 ``````Section fixpointAB_ne. `````` Robbert Krebbers committed Dec 05, 2016 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 `````` 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 505 ``````End fixpointAB_ne. `````` Robbert Krebbers committed Dec 05, 2016 506 `````` `````` Robbert Krebbers committed Jul 25, 2016 507 ``````(** Function space *) `````` Ralf Jung committed Nov 22, 2016 508 ``````(* We make [ofe_fun] a definition so that we can register it as a canonical `````` Robbert Krebbers committed Aug 05, 2016 509 ``````structure. *) `````` Ralf Jung committed Nov 22, 2016 510 ``````Definition ofe_fun (A : Type) (B : ofeT) := A → B. `````` Robbert Krebbers committed Jul 25, 2016 511 `````` `````` Ralf Jung committed Nov 22, 2016 512 513 514 515 516 ``````Section ofe_fun. Context {A : Type} {B : ofeT}. Instance ofe_fun_equiv : Equiv (ofe_fun A B) := λ f g, ∀ x, f x ≡ g x. Instance ofe_fun_dist : Dist (ofe_fun A B) := λ n f g, ∀ x, f x ≡{n}≡ g x. Definition ofe_fun_ofe_mixin : OfeMixin (ofe_fun A B). `````` Robbert Krebbers committed Jul 25, 2016 517 518 519 520 521 522 523 524 525 526 `````` Proof. split. - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|]. intros Hfg k; apply equiv_dist=> n; apply Hfg. - intros n; split. + by intros f x. + by intros f g ? x. + by intros f g h ?? x; trans (g x). - by intros n f g ? x; apply dist_S. Qed. `````` Ralf Jung committed Nov 22, 2016 527 `````` Canonical Structure ofe_funC := OfeT (ofe_fun A B) ofe_fun_ofe_mixin. `````` Robbert Krebbers committed Jul 25, 2016 528 `````` `````` Ralf Jung committed Nov 22, 2016 529 530 531 532 533 534 535 536 537 `````` Program Definition ofe_fun_chain `(c : chain ofe_funC) (x : A) : chain B := {| chain_car n := c n x |}. Next Obligation. intros c x n i ?. by apply (chain_cauchy c). Qed. Global Program Instance ofe_fun_cofe `{Cofe B} : Cofe ofe_funC := { compl c x := compl (ofe_fun_chain c x) }. Next Obligation. intros ? n c x. apply (conv_compl n (ofe_fun_chain c x)). Qed. End ofe_fun. Arguments ofe_funC : clear implicits. `````` Robbert Krebbers committed Jul 25, 2016 538 ``````Notation "A -c> B" := `````` Ralf Jung committed Nov 22, 2016 539 540 `````` (ofe_funC A B) (at level 99, B at level 200, right associativity). Instance ofe_fun_inhabited {A} {B : ofeT} `{Inhabited B} : `````` Robbert Krebbers committed Jul 25, 2016 541 542 `````` Inhabited (A -c> B) := populate (λ _, inhabitant). `````` Robbert Krebbers committed Jul 25, 2016 543 ``````(** Non-expansive function space *) `````` Ralf Jung committed Nov 22, 2016 544 545 ``````Record ofe_mor (A B : ofeT) : Type := CofeMor { ofe_mor_car :> A → B; `````` Ralf Jung committed Jan 27, 2017 546 `````` ofe_mor_ne : NonExpansive ofe_mor_car `````` Robbert Krebbers committed Nov 11, 2015 547 548 ``````}. Arguments CofeMor {_ _} _ {_}. `````` Ralf Jung committed Nov 22, 2016 549 550 ``````Add Printing Constructor ofe_mor. Existing Instance ofe_mor_ne. `````` Robbert Krebbers committed Nov 11, 2015 551 `````` `````` Robbert Krebbers committed Jun 17, 2016 552 553 554 555 ``````Notation "'λne' x .. y , t" := (@CofeMor _ _ (λ x, .. (@CofeMor _ _ (λ y, t) _) ..) _) (at level 200, x binder, y binder, right associativity). `````` Ralf Jung committed Nov 22, 2016 556 557 558 559 560 561 562 ``````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 563 564 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 565 `````` - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|]. `````` Robbert Krebbers committed Feb 18, 2016 566 `````` intros Hfg k; apply equiv_dist=> n; apply Hfg. `````` Robbert Krebbers committed Feb 17, 2016 567 `````` - intros n; split. `````` Robbert Krebbers committed Jan 14, 2016 568 569 `````` + by intros f x. + by intros f g ? x. `````` Ralf Jung committed Feb 20, 2016 570 `````` + by intros f g h ?? x; trans (g x). `````` Robbert Krebbers committed Feb 17, 2016 571 `````` - by intros n f g ? x; apply dist_S. `````` Robbert Krebbers committed Jan 14, 2016 572 `````` Qed. `````` Ralf Jung committed Nov 22, 2016 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 `````` Canonical Structure ofe_morC := OfeT (ofe_mor A B) ofe_mor_ofe_mixin. Program Definition ofe_mor_chain (c : chain ofe_morC) (x : A) : chain B := {| chain_car n := c n x |}. Next Obligation. intros c x n i ?. by apply (chain_cauchy c). Qed. Program Definition ofe_mor_compl `{Cofe B} : Compl ofe_morC := λ c, {| 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. Global Program Instance ofe_more_cofe `{Cofe B} : Cofe ofe_morC := {| 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 590 `````` `````` Ralf Jung committed Jan 27, 2017 591 592 593 `````` 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 594 595 596 `````` 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 597 `````` Proof. done. Qed. `````` Ralf Jung committed Nov 22, 2016 598 ``````End ofe_mor. `````` Robbert Krebbers committed Jan 14, 2016 599 `````` `````` Ralf Jung committed Nov 22, 2016 600 ``````Arguments ofe_morC : clear implicits. `````` Robbert Krebbers committed Jul 25, 2016 601 ``````Notation "A -n> B" := `````` Ralf Jung committed Nov 22, 2016 602 603 `````` (ofe_morC A B) (at level 99, B at level 200, right associativity). Instance ofe_mor_inhabited {A B : ofeT} `{Inhabited B} : `````` Robbert Krebbers committed Jul 25, 2016 604 `````` Inhabited (A -n> B) := populate (λne _, inhabitant). `````` Robbert Krebbers committed Nov 11, 2015 605 `````` `````` Ralf Jung committed Mar 17, 2016 606 ``````(** Identity and composition and constant function *) `````` Robbert Krebbers committed Nov 11, 2015 607 608 ``````Definition cid {A} : A -n> A := CofeMor id. Instance: Params (@cid) 1. `````` Ralf Jung committed Nov 22, 2016 609 ``````Definition cconst {A B : ofeT} (x : B) : A -n> B := CofeMor (const x). `````` Ralf Jung committed Mar 17, 2016 610 ``````Instance: Params (@cconst) 2. `````` Robbert Krebbers committed Mar 02, 2016 611 `````` `````` Robbert Krebbers committed Nov 11, 2015 612 613 614 615 616 ``````Definition ccompose {A B C} (f : B -n> C) (g : A -n> B) : A -n> C := CofeMor (f ∘ g). Instance: Params (@ccompose) 3. Infix "◎" := ccompose (at level 40, left associativity). Lemma ccompose_ne {A B C} (f1 f2 : B -n> C) (g1 g2 : A -n> B) n : `````` Ralf Jung committed Feb 10, 2016 617 `````` f1 ≡{n}≡ f2 → g1 ≡{n}≡ g2 → f1 ◎ g1 ≡{n}≡ f2 ◎ g2. `````` Robbert Krebbers committed Jan 13, 2016 618 ``````Proof. by intros Hf Hg x; rewrite /= (Hg x) (Hf (g2 x)). Qed. `````` Robbert Krebbers committed Nov 11, 2015 619 `````` `````` Ralf Jung committed Mar 02, 2016 620 ``````(* Function space maps *) `````` Ralf Jung committed Nov 22, 2016 621 ``````Definition ofe_mor_map {A A' B B'} (f : A' -n> A) (g : B -n> B') `````` Ralf Jung committed Mar 02, 2016 622 `````` (h : A -n> B) : A' -n> B' := g ◎ h ◎ f. `````` Ralf Jung committed Nov 22, 2016 623 624 ``````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 625 ``````Proof. intros ??? ??? ???. by repeat apply ccompose_ne. Qed. `````` Ralf Jung committed Mar 02, 2016 626 `````` `````` Ralf Jung committed Nov 22, 2016 627 628 ``````Definition ofe_morC_map {A A' B B'} (f : A' -n> A) (g : B -n> B') : (A -n> B) -n> (A' -n> B') := CofeMor (ofe_mor_map f g). `````` Ralf Jung committed Jan 27, 2017 629 630 ``````Instance ofe_morC_map_ne {A A' B B'} : NonExpansive2 (@ofe_morC_map A A' B B'). `````` Ralf Jung committed Mar 02, 2016 631 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 632 `````` intros n f f' Hf g g' Hg ?. rewrite /= /ofe_mor_map. `````` Robbert Krebbers committed Mar 02, 2016 633 `````` by repeat apply ccompose_ne. `````` Ralf Jung committed Mar 02, 2016 634 635 ``````Qed. `````` Robbert Krebbers committed Nov 11, 2015 636 ``````(** unit *) `````` Robbert Krebbers committed Jan 14, 2016 637 638 ``````Section unit. Instance unit_dist : Dist unit := λ _ _ _, True. `````` Ralf Jung committed Nov 22, 2016 639 `````` Definition unit_ofe_mixin : OfeMixin unit. `````` Robbert Krebbers committed Jan 14, 2016 640 `````` Proof. by repeat split; try exists 0. Qed. `````` Ralf Jung committed Nov 22, 2016 641 `````` Canonical Structure unitC : ofeT := OfeT unit unit_ofe_mixin. `````` Robbert Krebbers committed Nov 28, 2016 642 `````` `````` Ralf Jung committed Nov 22, 2016 643 644 `````` Global Program Instance unit_cofe : Cofe unitC := { compl x := () }. Next Obligation. by repeat split; try exists 0. Qed. `````` Robbert Krebbers committed Nov 28, 2016 645 646 `````` Global Instance unit_discrete_cofe : Discrete unitC. `````` Robbert Krebbers committed Jan 31, 2016 647 `````` Proof. done. Qed. `````` Robbert Krebbers committed Jan 14, 2016 648 ``````End unit. `````` Robbert Krebbers committed Nov 11, 2015 649 650 `````` (** Product *) `````` Robbert Krebbers committed Jan 14, 2016 651 ``````Section product. `````` Ralf Jung committed Nov 22, 2016 652 `````` Context {A B : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 653 654 655 `````` Instance prod_dist : Dist (A * B) := λ n, prod_relation (dist n) (dist n). Global Instance pair_ne : `````` Ralf Jung committed Jan 27, 2017 656 657 658 `````` 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 659 `````` Definition prod_ofe_mixin : OfeMixin (A * B). `````` Robbert Krebbers committed Jan 14, 2016 660 661 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 662 `````` - intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation. `````` Robbert Krebbers committed Jan 14, 2016 663 `````` rewrite !equiv_dist; naive_solver. `````` Robbert Krebbers committed Feb 17, 2016 664 665 `````` - apply _. - by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S. `````` Robbert Krebbers committed Jan 14, 2016 666 `````` Qed. `````` Ralf Jung committed Nov 22, 2016 667 668 669 670 671 672 673 674 675 `````` Canonical Structure prodC : ofeT := OfeT (A * B) prod_ofe_mixin. Global Program Instance prod_cofe `{Cofe A, Cofe B} : Cofe prodC := { 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. `````` Jacques-Henri Jourdan committed Jun 15, 2016 676 677 678 `````` Global Instance prod_timeless (x : A * B) : Timeless (x.1) → Timeless (x.2) → Timeless x. Proof. by intros ???[??]; split; apply (timeless _). Qed. `````` Robbert Krebbers committed Feb 24, 2016 679 680 `````` Global Instance prod_discrete_cofe : Discrete A → Discrete B → Discrete prodC. Proof. intros ?? [??]; apply _. Qed. `````` Robbert Krebbers committed Jan 14, 2016 681 682 683 684 685 ``````End product. Arguments prodC : clear implicits. Typeclasses Opaque prod_dist. `````` Ralf Jung committed Nov 22, 2016 686 ``````Instance prod_map_ne {A A' B B' : ofeT} n : `````` Robbert Krebbers committed Nov 11, 2015 687 688 689 690 691 `````` 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. Definition prodC_map {A A' B B'} (f : A -n> A') (g : B -n> B') : prodC A B -n> prodC A' B' := CofeMor (prod_map f g). `````` Ralf Jung committed Jan 27, 2017 692 693 694 ``````Instance prodC_map_ne {A A' B B'} : NonExpansive2 (@prodC_map A A' B B'). Proof. intros n f f' Hf g g' Hg [??]; split; [apply Hf|apply Hg]. Qed. `````` Robbert Krebbers committed Nov 11, 2015 695 `````` `````` Robbert Krebbers committed Mar 02, 2016 696 697 ``````(** Functors *) Structure cFunctor := CFunctor { `````` Ralf Jung committed Nov 22, 2016 698 `````` cFunctor_car : ofeT → ofeT → ofeT; `````` Robbert Krebbers committed Mar 02, 2016 699 700 `````` cFunctor_map {A1 A2 B1 B2} : ((A2 -n> A1) * (B1 -n> B2)) → cFunctor_car A1 B1 -n> cFunctor_car A2 B2; `````` Ralf Jung committed Jan 27, 2017 701 702 `````` cFunctor_ne {A1 A2 B1 B2} : NonExpansive (@cFunctor_map A1 A2 B1 B2); `````` Ralf Jung committed Nov 22, 2016 703 `````` cFunctor_id {A B : ofeT} (x : cFunctor_car A B) : `````` Robbert Krebbers committed Mar 02, 2016 704 705 706 707 708 `````` cFunctor_map (cid,cid) x ≡ x; cFunctor_compose {A1 A2 A3 B1 B2 B3} (f : A2 -n> A1) (g : A3 -n> A2) (f' : B1 -n> B2) (g' : B2 -n> B3) x : cFunctor_map (f◎g, g'◎f') x ≡ cFunctor_map (g,g') (cFunctor_map (f,f') x) }. `````` Robbert Krebbers committed Mar 07, 2016 709 ``````Existing Instance cFunctor_ne. `````` Robbert Krebbers committed Mar 02, 2016 710 711 ``````Instance: Params (@cFunctor_map) 5. `````` Ralf Jung committed Mar 07, 2016 712 713 714 ``````Delimit Scope cFunctor_scope with CF. Bind Scope cFunctor_scope with cFunctor. `````` Ralf Jung committed Mar 07, 2016 715 716 717 ``````Class cFunctorContractive (F : cFunctor) := cFunctor_contractive A1 A2 B1 B2 :> Contractive (@cFunctor_map F A1 A2 B1 B2). `````` Ralf Jung committed Nov 22, 2016 718 ``````Definition cFunctor_diag (F: cFunctor) (A: ofeT) : ofeT := cFunctor_car F A A. `````` Robbert Krebbers committed Mar 02, 2016 719 720 ``````Coercion cFunctor_diag : cFunctor >-> Funclass. `````` Ralf Jung committed Nov 22, 2016 721 ``````Program Definition constCF (B : ofeT) : cFunctor := `````` Robbert Krebbers committed Mar 02, 2016 722 723 `````` {| cFunctor_car A1 A2 := B; cFunctor_map A1 A2 B1 B2 f := cid |}. Solve Obligations with done. `````` Ralf Jung committed Jan 06, 2017 724 ``````Coercion constCF : ofeT >-> cFunctor. `````` Robbert Krebbers committed Mar 02, 2016 725 `````` `````` Ralf Jung committed Mar 07, 2016 726 ``````Instance constCF_contractive B : cFunctorContractive (constCF B). `````` Robbert Krebbers committed Mar 07, 2016 727 ``````Proof. rewrite /cFunctorContractive; apply _. Qed. `````` Ralf Jung committed Mar 07, 2016 728 729 730 731 `````` Program Definition idCF : cFunctor := {| cFunctor_car A1 A2 := A2; cFunctor_map A1 A2 B1 B2 f := f.2 |}. Solve Obligations with done. `````` Ralf Jung committed Jan 06, 2017 732 ``````Notation "∙" := idCF : cFunctor_scope. `````` Ralf Jung committed Mar 07, 2016 733 `````` `````` Robbert Krebbers committed Mar 02, 2016 734 735 736 737 738 ``````Program Definition prodCF (F1 F2 : cFunctor) : cFunctor := {| cFunctor_car A B := prodC (cFunctor_car F1 A B) (cFunctor_car F2 A B); cFunctor_map A1 A2 B1 B2 fg := prodC_map (cFunctor_map F1 fg) (cFunctor_map F2 fg) |}. `````` Robbert Krebbers committed Mar 07, 2016 739 740 741 ``````Next Obligation. intros ?? A1 A2 B1 B2 n ???; by apply prodC_map_ne; apply cFunctor_ne. Qed. `````` Robbert Krebbers committed Mar 02, 2016 742 743 744 745 746 ``````Next Obligation. by intros F1 F2 A B [??]; rewrite /= !cFunctor_id. Qed. Next Obligation. intros F1 F2 A1 A2 A3 B1 B2 B3 f g f' g' [??]; simpl. by rewrite !cFunctor_compose. Qed. `````` Ralf Jung committed Jan 06, 2017 747 ``````Notation "F1 * F2" := (prodCF F1%CF F2%CF) : cFunctor_scope. `````` Robbert Krebbers committed Mar 02, 2016 748 `````` `````` Ralf Jung committed Mar 07, 2016 749 750 751 752 753 754 755 756 ``````Instance prodCF_contractive F1 F2 : cFunctorContractive F1 → cFunctorContractive F2 → cFunctorContractive (prodCF F1 F2). Proof. intros ?? A1 A2 B1 B2 n ???; by apply prodC_map_ne; apply cFunctor_contractive. Qed. `````` Ralf Jung committed Jan 27, 2017 757 758 759 ``````Instance compose_ne {A} {B B' : ofeT} (f : B -n> B') : NonExpansive (compose f : (A -c> B) → A -c> B'). Proof. intros n g g' Hf x; simpl. by rewrite (Hf x). Qed. `````` Jacques-Henri Jourdan committed Oct 05, 2016 760 `````` `````` Ralf Jung committed Nov 22, 2016 761 ``````Definition ofe_funC_map {A B B'} (f : B -n> B') : (A -c> B) -n> (A -c> B') := `````` Jacques-Henri Jourdan committed Oct 05, 2016 762 `````` @CofeMor (_ -c> _) (_ -c> _) (compose f) _. `````` Ralf Jung committed Jan 27, 2017 763 764 765 ``````Instance ofe_funC_map_ne {A B B'} : NonExpansive (@ofe_funC_map A B B'). Proof. intros n f f' Hf g x. apply Hf. Qed. `````` Jacques-Henri Jourdan committed Oct 05, 2016 766 `````` `````` Ralf Jung committed Nov 22, 2016 767 768 769 ``````Program Definition ofe_funCF (T : Type) (F : cFunctor) : cFunctor := {| cFunctor_car A B := ofe_funC T (cFunctor_car F A B); cFunctor_map A1 A2 B1 B2 fg := ofe_funC_map (cFunctor_map F fg) `````` Jacques-Henri Jourdan committed Oct 05, 2016 770 771 ``````|}. Next Obligation. `````` Ralf Jung committed Nov 22, 2016 772 `````` intros ?? A1 A2 B1 B2 n ???; by apply ofe_funC_map_ne; apply cFunctor_ne. `````` Jacques-Henri Jourdan committed Oct 05, 2016 773 774 775 776 777 778 ``````Qed. Next Obligation. intros F1 F2 A B ??. by rewrite /= /compose /= !cFunctor_id. Qed. Next Obligation. intros T F A1 A2 A3 B1 B2 B3 f g f' g' ??; simpl. by rewrite !cFunctor_compose. Qed. `````` Ralf Jung committed Jan 06, 2017 779 ``````Notation "T -c> F" := (ofe_funCF T%type F%CF) : cFunctor_scope. `````` Jacques-Henri Jourdan committed Oct 05, 2016 780 `````` `````` Ralf Jung committed Nov 22, 2016 781 782 ``````Instance ofe_funCF_contractive (T : Type) (F : cFunctor) : cFunctorContractive F → cFunctorContractive (ofe_funCF T F). `````` Jacques-Henri Jourdan committed Oct 05, 2016 783 784 ``````Proof. intros ?? A1 A2 B1 B2 n ???; `````` Ralf Jung committed Nov 22, 2016 785 `````` by apply ofe_funC_map_ne; apply cFunctor_contractive. `````` Jacques-Henri Jourdan committed Oct 05, 2016 786 787 ``````Qed. `````` Ralf Jung committed Nov 22, 2016 788 ``````Program Definition ofe_morCF (F1 F2 : cFunctor) : cFunctor := {| `````` Robbert Krebbers committed Jul 25, 2016 789 `````` cFunctor_car A B := cFunctor_car F1 B A -n> cFunctor_car F2 A B; `````` Ralf Jung committed Mar 02, 2016 790 `````` cFunctor_map A1 A2 B1 B2 fg := `````` Ralf Jung committed Nov 22, 2016 791 `````` ofe_morC_map (cFunctor_map F1 (fg.2, fg.1)) (cFunctor_map F2 fg) `````` Ralf Jung committed Mar 02, 2016 792 ``````|}. `````` Robbert Krebbers committed Mar 07, 2016 793 794 ``````Next Obligation. intros F1 F2 A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *. `````` Ralf Jung committed Nov 22, 2016 795 `````` apply ofe_morC_map_ne; apply cFunctor_ne; split; by apply Hfg. `````` Robbert Krebbers committed Mar 07, 2016 796 ``````Qed. `````` Ralf Jung committed Mar 02, 2016 797 ``````Next Obligation. `````` Robbert Krebbers committed Mar 02, 2016 798 799 `` intros F1 F2 A B [f ``