ofe.v 50 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. *) `````` Robbert Krebbers committed Nov 11, 2015 14 15 ``````(** Unbundeled version *) 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 46 `````` (** Bundeled 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 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 Oct 25, 2017 164 165 `````` 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 166 `````` `````` 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 `````` Proof. transitivity (c n); first by apply conv_compl. symmetry. `````` Ralf Jung committed Jun 20, 2018 184 `````` apply chain_cauchy. lia. `````` Ralf Jung committed Feb 29, 2016 185 `````` Qed. `````` Robbert Krebbers committed Apr 13, 2017 186 `````` `````` Robbert Krebbers committed Oct 25, 2017 187 `````` Lemma discrete_iff n (x : A) `{!Discrete x} y : x ≡ y ↔ x ≡{n}≡ y. `````` Robbert Krebbers committed Feb 24, 2016 188 `````` Proof. `````` Robbert Krebbers committed Oct 25, 2017 189 `````` split; intros; auto. apply (discrete _), dist_le with n; auto with lia. `````` Robbert Krebbers committed Feb 24, 2016 190 `````` Qed. `````` Robbert Krebbers committed Oct 25, 2017 191 `````` Lemma discrete_iff_0 n (x : A) `{!Discrete x} y : x ≡{0}≡ y ↔ x ≡{n}≡ y. `````` Robbert Krebbers committed Nov 28, 2017 192 `````` Proof. by rewrite -!discrete_iff. Qed. `````` Robbert Krebbers committed Feb 09, 2017 193 ``````End ofe. `````` Robbert Krebbers committed Nov 11, 2015 194 `````` `````` Robbert Krebbers committed Dec 02, 2016 195 ``````(** Contractive functions *) `````` Robbert Krebbers committed Aug 17, 2017 196 ``````Definition dist_later `{Dist A} (n : nat) (x y : A) : Prop := `````` Robbert Krebbers committed Dec 05, 2016 197 `````` match n with 0 => True | S n => x ≡{n}≡ y end. `````` Robbert Krebbers committed Aug 17, 2017 198 ``````Arguments dist_later _ _ !_ _ _ /. `````` Robbert Krebbers committed Dec 05, 2016 199 `````` `````` Robbert Krebbers committed Aug 17, 2017 200 ``````Global Instance dist_later_equivalence (A : ofeT) n : Equivalence (@dist_later A _ n). `````` Robbert Krebbers committed Dec 05, 2016 201 202 ``````Proof. destruct n as [|n]. by split. apply dist_equivalence. Qed. `````` Ralf Jung committed Feb 22, 2017 203 204 205 ``````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 206 207 208 209 210 211 212 213 214 215 216 ``````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 217 ``````Notation Contractive f := (∀ n, Proper (dist_later n ==> dist n) f). `````` Robbert Krebbers committed Dec 02, 2016 218 `````` `````` Ralf Jung committed Nov 22, 2016 219 ``````Instance const_contractive {A B : ofeT} (x : A) : Contractive (@const A B x). `````` Robbert Krebbers committed Mar 06, 2016 220 221 ``````Proof. by intros n y1 y2. Qed. `````` Robbert Krebbers committed Dec 02, 2016 222 ``````Section contractive. `````` Ralf Jung committed Jan 25, 2017 223 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Dec 02, 2016 224 225 226 227 `````` 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 228 `````` Proof. by apply (_ : Contractive f). Qed. `````` Robbert Krebbers committed Dec 02, 2016 229 `````` Lemma contractive_S n x y : x ≡{n}≡ y → f x ≡{S n}≡ f y. `````` Robbert Krebbers committed Dec 05, 2016 230 `````` Proof. intros. by apply (_ : Contractive f). Qed. `````` Robbert Krebbers committed Dec 02, 2016 231 `````` `````` Ralf Jung committed Jan 27, 2017 232 233 `````` 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 234 235 236 237 `````` Global Instance contractive_proper : Proper ((≡) ==> (≡)) f | 100. Proof. apply (ne_proper _). Qed. End contractive. `````` Robbert Krebbers committed Dec 05, 2016 238 239 ``````Ltac f_contractive := match goal with `````` Robbert Krebbers committed Aug 17, 2017 240 241 242 `````` | |- ?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 243 244 `````` end; try match goal with `````` Robbert Krebbers committed Aug 17, 2017 245 `````` | |- @dist_later ?A _ ?n ?x ?y => `````` Ralf Jung committed Mar 01, 2017 246 `````` destruct n as [|n]; [exact I|change (@dist A _ n x y)] `````` Robbert Krebbers committed Dec 05, 2016 247 `````` end; `````` Robbert Krebbers committed Aug 17, 2017 248 `````` try simple apply reflexivity. `````` Robbert Krebbers committed Dec 05, 2016 249 `````` `````` Robbert Krebbers committed Aug 17, 2017 250 251 ``````Ltac solve_contractive := solve_proper_core ltac:(fun _ => first [f_contractive | f_equiv]). `````` Robbert Krebbers committed Nov 22, 2015 252 `````` `````` Robbert Krebbers committed Mar 09, 2017 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 ``````(** 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. `````` Robbert Krebbers committed Oct 25, 2017 270 `````` Lemma limit_preserving_discrete (P : A → Prop) : `````` Robbert Krebbers committed Mar 09, 2017 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 `````` 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 293 ``````(** Fixpoint *) `````` Ralf Jung committed Nov 22, 2016 294 ``````Program Definition fixpoint_chain {A : ofeT} `{Inhabited A} (f : A → A) `````` Robbert Krebbers committed Feb 10, 2016 295 `````` `{!Contractive f} : chain A := {| chain_car i := Nat.iter (S i) f inhabitant |}. `````` Robbert Krebbers committed Nov 11, 2015 296 ``````Next Obligation. `````` Robbert Krebbers committed Mar 06, 2016 297 `````` intros A ? f ? n. `````` Ralf Jung committed Jun 20, 2018 298 `````` induction n as [|n IH]=> -[|i] //= ?; try lia. `````` Robbert Krebbers committed Feb 17, 2016 299 `````` - apply (contractive_0 f). `````` Ralf Jung committed Jun 20, 2018 300 `````` - apply (contractive_S f), IH; auto with lia. `````` Robbert Krebbers committed Nov 11, 2015 301 ``````Qed. `````` Robbert Krebbers committed Mar 18, 2016 302 `````` `````` Ralf Jung committed Nov 22, 2016 303 ``````Program Definition fixpoint_def `{Cofe A, Inhabited A} (f : A → A) `````` Robbert Krebbers committed Nov 17, 2015 304 `````` `{!Contractive f} : A := compl (fixpoint_chain f). `````` Ralf Jung committed Jan 11, 2017 305 ``````Definition fixpoint_aux : seal (@fixpoint_def). by eexists. Qed. `````` Ralf Jung committed Mar 05, 2018 306 307 ``````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 308 309 `````` Section fixpoint. `````` Ralf Jung committed Nov 22, 2016 310 `````` Context `{Cofe A, Inhabited A} (f : A → A) `{!Contractive f}. `````` Robbert Krebbers committed Aug 21, 2016 311 `````` `````` Robbert Krebbers committed Nov 17, 2015 312 `````` Lemma fixpoint_unfold : fixpoint f ≡ f (fixpoint f). `````` Robbert Krebbers committed Nov 11, 2015 313 `````` Proof. `````` Robbert Krebbers committed Mar 18, 2016 314 315 `````` apply equiv_dist=>n. rewrite fixpoint_eq /fixpoint_def (conv_compl n (fixpoint_chain f)) //. `````` Robbert Krebbers committed Feb 12, 2016 316 `````` induction n as [|n IH]; simpl; eauto using contractive_0, contractive_S. `````` Robbert Krebbers committed Nov 11, 2015 317 `````` Qed. `````` Robbert Krebbers committed Aug 21, 2016 318 319 320 `````` Lemma fixpoint_unique (x : A) : x ≡ f x → x ≡ fixpoint f. Proof. `````` Robbert Krebbers committed Aug 22, 2016 321 322 323 `````` 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 324 325 `````` Qed. `````` Robbert Krebbers committed Nov 17, 2015 326 `````` Lemma fixpoint_ne (g : A → A) `{!Contractive g} n : `````` Ralf Jung committed Feb 10, 2016 327 `````` (∀ z, f z ≡{n}≡ g z) → fixpoint f ≡{n}≡ fixpoint g. `````` Robbert Krebbers committed Nov 11, 2015 328 `````` Proof. `````` Robbert Krebbers committed Mar 18, 2016 329 `````` intros Hfg. rewrite fixpoint_eq /fixpoint_def `````` Robbert Krebbers committed Feb 18, 2016 330 `````` (conv_compl n (fixpoint_chain f)) (conv_compl n (fixpoint_chain g)) /=. `````` Robbert Krebbers committed Feb 10, 2016 331 332 `````` 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 333 `````` Qed. `````` Robbert Krebbers committed Nov 17, 2015 334 335 `````` Lemma fixpoint_proper (g : A → A) `{!Contractive g} : (∀ x, f x ≡ g x) → fixpoint f ≡ fixpoint g. `````` Robbert Krebbers committed Nov 11, 2015 336 `````` Proof. setoid_rewrite equiv_dist; naive_solver eauto using fixpoint_ne. Qed. `````` Jacques-Henri Jourdan committed Dec 23, 2016 337 338 `````` Lemma fixpoint_ind (P : A → Prop) : `````` Jacques-Henri Jourdan committed Dec 23, 2016 339 `````` Proper ((≡) ==> impl) P → `````` Jacques-Henri Jourdan committed Dec 23, 2016 340 `````` (∃ x, P x) → (∀ x, P x → P (f x)) → `````` Robbert Krebbers committed Mar 09, 2017 341 `````` LimitPreserving P → `````` Jacques-Henri Jourdan committed Dec 23, 2016 342 343 344 345 `````` 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 346 `````` { intros n. rewrite /chcar. induction n as [|n IH]=> -[|i] //=; `````` Ralf Jung committed Jun 20, 2018 347 `````` eauto using contractive_0, contractive_S with lia. } `````` Jacques-Henri Jourdan committed Dec 23, 2016 348 `````` set (fp2 := compl {| chain_cauchy := Hcauch |}). `````` Robbert Krebbers committed Mar 09, 2017 349 350 351 352 `````` 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 353 `````` apply Hlim=> n /=. by apply Nat_iter_ind. `````` Jacques-Henri Jourdan committed Dec 23, 2016 354 `````` Qed. `````` Robbert Krebbers committed Nov 11, 2015 355 356 ``````End fixpoint. `````` Robbert Krebbers committed Mar 09, 2017 357 `````` `````` Ralf Jung committed Jan 25, 2017 358 359 360 ``````(** 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 361 `````` `````` Ralf Jung committed Jan 25, 2017 362 ``````Section fixpointK. `````` Ralf Jung committed Jan 25, 2017 363 `````` Local Set Default Proof Using "Type*". `````` Robbert Krebbers committed Jan 25, 2017 364 `````` Context `{Cofe A, Inhabited A} (f : A → A) (k : nat). `````` Ralf Jung committed Feb 23, 2017 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 `````` 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 388 389 `````` Let f_proper : Proper ((≡) ==> (≡)) f := ne_proper f. `````` Ralf Jung committed Feb 23, 2017 390 `````` Local Existing Instance f_proper. `````` Ralf Jung committed Jan 25, 2017 391 `````` `````` Ralf Jung committed Jan 25, 2017 392 `````` Lemma fixpointK_unfold : fixpointK k f ≡ f (fixpointK k f). `````` Ralf Jung committed Jan 25, 2017 393 `````` Proof. `````` Robbert Krebbers committed Jan 25, 2017 394 395 `````` symmetry. rewrite /fixpointK. apply fixpoint_unique. by rewrite -Nat_iter_S_r Nat_iter_S -fixpoint_unfold. `````` Ralf Jung committed Jan 25, 2017 396 397 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 398 `````` Lemma fixpointK_unique (x : A) : x ≡ f x → x ≡ fixpointK k f. `````` Ralf Jung committed Jan 25, 2017 399 `````` Proof. `````` Robbert Krebbers committed Jan 25, 2017 400 401 `````` intros Hf. apply fixpoint_unique. clear f_contractive. induction k as [|k' IH]=> //=. by rewrite -IH. `````` Ralf Jung committed Jan 25, 2017 402 403 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 404 `````` Section fixpointK_ne. `````` Robbert Krebbers committed Jan 25, 2017 405 `````` Context (g : A → A) `{g_contractive : !Contractive (Nat.iter k g)}. `````` Ralf Jung committed Jan 27, 2017 406 `````` Context {g_ne : NonExpansive g}. `````` Ralf Jung committed Jan 25, 2017 407 `````` `````` Ralf Jung committed Jan 25, 2017 408 `````` Lemma fixpointK_ne n : (∀ z, f z ≡{n}≡ g z) → fixpointK k f ≡{n}≡ fixpointK k g. `````` Ralf Jung committed Jan 25, 2017 409 `````` Proof. `````` Robbert Krebbers committed Jan 25, 2017 410 411 412 `````` 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 413 414 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 415 416 417 `````` 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 418 419 420 421 `````` Lemma fixpointK_ind (P : A → Prop) : Proper ((≡) ==> impl) P → (∃ x, P x) → (∀ x, P x → P (f x)) → `````` Robbert Krebbers committed Mar 09, 2017 422 `````` LimitPreserving P → `````` Ralf Jung committed Feb 21, 2017 423 424 `````` P (fixpointK k f). Proof. `````` Robbert Krebbers committed Mar 09, 2017 425 `````` intros. rewrite /fixpointK. apply fixpoint_ind; eauto. `````` Robbert Krebbers committed Mar 11, 2017 426 `````` intros; apply Nat_iter_ind; auto. `````` Ralf Jung committed Feb 21, 2017 427 `````` Qed. `````` Ralf Jung committed Jan 25, 2017 428 ``````End fixpointK. `````` Ralf Jung committed Jan 25, 2017 429 `````` `````` Robbert Krebbers committed Dec 05, 2016 430 ``````(** Mutual fixpoints *) `````` Ralf Jung committed Jan 25, 2017 431 ``````Section fixpointAB. `````` 432 433 `````` Local Unset Default Proof Using. `````` Robbert Krebbers committed Dec 05, 2016 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 `````` 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 475 ``````End fixpointAB. `````` Robbert Krebbers committed Dec 05, 2016 476 `````` `````` Ralf Jung committed Jan 25, 2017 477 ``````Section fixpointAB_ne. `````` Robbert Krebbers committed Dec 05, 2016 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 `````` 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 509 ``````End fixpointAB_ne. `````` Robbert Krebbers committed Dec 05, 2016 510 `````` `````` Robbert Krebbers committed Jul 25, 2016 511 ``````(** Non-expansive function space *) `````` Ralf Jung committed Nov 22, 2016 512 513 ``````Record ofe_mor (A B : ofeT) : Type := CofeMor { ofe_mor_car :> A → B; `````` Ralf Jung committed Jan 27, 2017 514 `````` ofe_mor_ne : NonExpansive ofe_mor_car `````` Robbert Krebbers committed Nov 11, 2015 515 516 ``````}. Arguments CofeMor {_ _} _ {_}. `````` Ralf Jung committed Nov 22, 2016 517 518 ``````Add Printing Constructor ofe_mor. Existing Instance ofe_mor_ne. `````` Robbert Krebbers committed Nov 11, 2015 519 `````` `````` Robbert Krebbers committed Jun 17, 2016 520 521 522 523 ``````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 524 525 526 527 528 529 530 ``````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 531 532 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 533 `````` - intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|]. `````` Robbert Krebbers committed Feb 18, 2016 534 `````` intros Hfg k; apply equiv_dist=> n; apply Hfg. `````` Robbert Krebbers committed Feb 17, 2016 535 `````` - intros n; split. `````` Robbert Krebbers committed Jan 14, 2016 536 537 `````` + by intros f x. + by intros f g ? x. `````` Ralf Jung committed Feb 20, 2016 538 `````` + by intros f g h ?? x; trans (g x). `````` Robbert Krebbers committed Feb 17, 2016 539 `````` - by intros n f g ? x; apply dist_S. `````` Robbert Krebbers committed Jan 14, 2016 540 `````` Qed. `````` Ralf Jung committed Nov 22, 2016 541 542 543 544 545 546 547 548 549 550 551 `````` 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. `````` Jacques-Henri Jourdan committed Jul 28, 2017 552 `````` Global Program Instance ofe_mor_cofe `{Cofe B} : Cofe ofe_morC := `````` Ralf Jung committed Nov 22, 2016 553 554 555 556 557 `````` {| 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 558 `````` `````` Ralf Jung committed Jan 27, 2017 559 560 561 `````` 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 562 563 564 `````` 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 565 `````` Proof. done. Qed. `````` Ralf Jung committed Nov 22, 2016 566 ``````End ofe_mor. `````` Robbert Krebbers committed Jan 14, 2016 567 `````` `````` Ralf Jung committed Nov 22, 2016 568 ``````Arguments ofe_morC : clear implicits. `````` Robbert Krebbers committed Jul 25, 2016 569 ``````Notation "A -n> B" := `````` Ralf Jung committed Nov 22, 2016 570 571 `````` (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 572 `````` Inhabited (A -n> B) := populate (λne _, inhabitant). `````` Robbert Krebbers committed Nov 11, 2015 573 `````` `````` Ralf Jung committed Mar 17, 2016 574 ``````(** Identity and composition and constant function *) `````` Robbert Krebbers committed Nov 11, 2015 575 ``````Definition cid {A} : A -n> A := CofeMor id. `````` Maxime Dénès committed Jan 24, 2019 576 ``````Instance: Params (@cid) 1 := {}. `````` Ralf Jung committed Nov 22, 2016 577 ``````Definition cconst {A B : ofeT} (x : B) : A -n> B := CofeMor (const x). `````` Maxime Dénès committed Jan 24, 2019 578 ``````Instance: Params (@cconst) 2 := {}. `````` Robbert Krebbers committed Mar 02, 2016 579 `````` `````` Robbert Krebbers committed Nov 11, 2015 580 581 ``````Definition ccompose {A B C} (f : B -n> C) (g : A -n> B) : A -n> C := CofeMor (f ∘ g). `````` Maxime Dénès committed Jan 24, 2019 582 ``````Instance: Params (@ccompose) 3 := {}. `````` Robbert Krebbers committed Nov 11, 2015 583 ``````Infix "◎" := ccompose (at level 40, left associativity). `````` Ralf Jung committed Nov 16, 2017 584 585 586 ``````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 587 `````` `````` Ralf Jung committed Mar 02, 2016 588 ``````(* Function space maps *) `````` Ralf Jung committed Nov 22, 2016 589 ``````Definition ofe_mor_map {A A' B B'} (f : A' -n> A) (g : B -n> B') `````` Ralf Jung committed Mar 02, 2016 590 `````` (h : A -n> B) : A' -n> B' := g ◎ h ◎ f. `````` Ralf Jung committed Nov 22, 2016 591 592 ``````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 593 ``````Proof. intros ??? ??? ???. by repeat apply ccompose_ne. Qed. `````` Ralf Jung committed Mar 02, 2016 594 `````` `````` Ralf Jung committed Nov 22, 2016 595 596 ``````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 597 598 ``````Instance ofe_morC_map_ne {A A' B B'} : NonExpansive2 (@ofe_morC_map A A' B B'). `````` Ralf Jung committed Mar 02, 2016 599 ``````Proof. `````` Ralf Jung committed Jan 27, 2017 600 `````` intros n f f' Hf g g' Hg ?. rewrite /= /ofe_mor_map. `````` Robbert Krebbers committed Mar 02, 2016 601 `````` by repeat apply ccompose_ne. `````` Ralf Jung committed Mar 02, 2016 602 603 ``````Qed. `````` Robbert Krebbers committed Nov 11, 2015 604 ``````(** unit *) `````` Robbert Krebbers committed Jan 14, 2016 605 606 ``````Section unit. Instance unit_dist : Dist unit := λ _ _ _, True. `````` Ralf Jung committed Nov 22, 2016 607 `````` Definition unit_ofe_mixin : OfeMixin unit. `````` Robbert Krebbers committed Jan 14, 2016 608 `````` Proof. by repeat split; try exists 0. Qed. `````` Ralf Jung committed Nov 22, 2016 609 `````` Canonical Structure unitC : ofeT := OfeT unit unit_ofe_mixin. `````` Robbert Krebbers committed Nov 28, 2016 610 `````` `````` Ralf Jung committed Nov 22, 2016 611 612 `````` Global Program Instance unit_cofe : Cofe unitC := { compl x := () }. Next Obligation. by repeat split; try exists 0. Qed. `````` Robbert Krebbers committed Nov 28, 2016 613 `````` `````` Robbert Krebbers committed Oct 25, 2017 614 `````` Global Instance unit_ofe_discrete : OfeDiscrete unitC. `````` Robbert Krebbers committed Jan 31, 2016 615 `````` Proof. done. Qed. `````` Robbert Krebbers committed Jan 14, 2016 616 ``````End unit. `````` Robbert Krebbers committed Nov 11, 2015 617 618 `````` (** Product *) `````` Robbert Krebbers committed Jan 14, 2016 619 ``````Section product. `````` Ralf Jung committed Nov 22, 2016 620 `````` Context {A B : ofeT}. `````` Robbert Krebbers committed Jan 14, 2016 621 622 623 `````` Instance prod_dist : Dist (A * B) := λ n, prod_relation (dist n) (dist n). Global Instance pair_ne : `````` Ralf Jung committed Jan 27, 2017 624 625 626 `````` 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 627 `````` Definition prod_ofe_mixin : OfeMixin (A * B). `````` Robbert Krebbers committed Jan 14, 2016 628 629 `````` Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 630 `````` - intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation. `````` Robbert Krebbers committed Jan 14, 2016 631 `````` rewrite !equiv_dist; naive_solver. `````` Robbert Krebbers committed Feb 17, 2016 632 633 `````` - apply _. - by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S. `````` Robbert Krebbers committed Jan 14, 2016 634 `````` Qed. `````` Ralf Jung committed Nov 22, 2016 635 636 637 638 639 640 641 642 643 `````` 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. `````` Robbert Krebbers committed Oct 25, 2017 644 645 646 `````` 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 647 648 `````` Global Instance prod_ofe_discrete : OfeDiscrete A → OfeDiscrete B → OfeDiscrete prodC. `````` Robbert Krebbers committed Feb 24, 2016 649 `````` Proof. intros ?? [??]; apply _. Qed. `````` Robbert Krebbers committed Jan 14, 2016 650 651 652 653 654 ``````End product. Arguments prodC : clear implicits. Typeclasses Opaque prod_dist. `````` Ralf Jung committed Nov 22, 2016 655 ``````Instance prod_map_ne {A A' B B' : ofeT} n : `````` Robbert Krebbers committed Nov 11, 2015 656 657 658 659 660 `````` 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 661 662 663 ``````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 664 `````` `````` Robbert Krebbers committed Mar 02, 2016 665 666 ``````(** Functors *) Structure cFunctor := CFunctor { `````` Ralf Jung committed Nov 22, 2016 667 `````` cFunctor_car : ofeT → ofeT → ofeT; `````` Robbert Krebbers committed Mar 02, 2016 668 669 `````` 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 670 671 `````` cFunctor_ne {A1 A2 B1 B2} : NonExpansive (@cFunctor_map A1 A2 B1 B2); `````` Ralf Jung committed Nov 22, 2016 672 `````` cFunctor_id {A B : ofeT} (x : cFunctor_car A B) : `````` Robbert Krebbers committed Mar 02, 2016 673 674 675 676 677 `````` 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 678 ``````Existing Instance cFunctor_ne. `````` Maxime Dénès committed Jan 24, 2019 679 ``````Instance: Params (@cFunctor_map) 5 := {}. `````` Robbert Krebbers committed Mar 02, 2016 680 `````` `````` Ralf Jung committed Mar 07, 2016 681 682 683 ``````Delimit Scope cFunctor_scope with CF. Bind Scope cFunctor_scope with cFunctor. `````` Ralf Jung committed Mar 07, 2016 684 685 ``````Class cFunctorContractive (F : cFunctor) := cFunctor_contractive A1 A2 B1 B2 :> Contractive (@cFunctor_map F A1 A2 B1 B2). `````` Ralf Jung committed Feb 19, 2019 686 ``````Hint Mode cFunctorContractive ! : typeclass_instances. `````` Ralf Jung committed Mar 07, 2016 687 `````` `````` Ralf Jung committed Nov 22, 2016 688 ``````Definition cFunctor_diag (F: cFunctor) (A: ofeT) : ofeT := cFunctor_car F A A. `````` Robbert Krebbers committed Mar 02, 2016 689 690 ``````Coercion cFunctor_diag : cFunctor >-> Funclass. `````` Ralf Jung committed Nov 22, 2016 691 ``````Program Definition constCF (B : ofeT) : cFunctor := `````` Robbert Krebbers committed Mar 02, 2016 692 693 `````` {| cFunctor_car A1 A2 := B; cFunctor_map A1 A2 B1 B2 f := cid |}. Solve Obligations with done. `````` Ralf Jung committed Jan 06, 2017 694 ``````Coercion constCF : ofeT >-> cFunctor. `````` Robbert Krebbers committed Mar 02, 2016 695 `````` `````` Ralf Jung committed Mar 07, 2016 696 ``````Instance constCF_contractive B : cFunctorContractive (constCF B). `````` Robbert Krebbers committed Mar 07, 2016 697 ``````Proof. rewrite /cFunctorContractive; apply _. Qed. `````` Ralf Jung committed Mar 07, 2016 698 699 700 701 `````` 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 702 ``````Notation "∙" := idCF : cFunctor_scope. `````` Ralf Jung committed Mar 07, 2016 703 `````` `````` Robbert Krebbers committed Mar 02, 2016 704 705 706 707 708 ``````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 709 710 711 ``````Next Obligation. intros ?? A1 A2 B1 B2 n ???; by apply prodC_map_ne; apply cFunctor_ne. Qed. `````` Robbert Krebbers committed Mar 02, 2016 712 713 714 715 716 ``````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 717 ``````Notation "F1 * F2" := (prodCF F1%CF F2%CF) : cFunctor_scope. `````` Robbert Krebbers committed Mar 02, 2016 718 `````` `````` Ralf Jung committed Mar 07, 2016 719 720 721 722 723 724 725 726 ``````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 Nov 22, 2016 727 ``````Program Definition ofe_morCF (F1 F2 : cFunctor) : cFunctor := {| `````` Robbert Krebbers committed Jul 25, 2016 728 `````` cFunctor_car A B := cFunctor_car F1 B A -n> cFunctor_car F2 A B; `````` Ralf Jung committed Mar 02, 2016 729 `````` cFunctor_map A1 A2 B1 B2 fg := `````` Ralf Jung committed Nov 22, 2016 730 `````` ofe_morC_map (cFunctor_map F1 (fg.2, fg.1)) (cFunctor_map F2 fg) `````` Ralf Jung committed Mar 02, 2016 731 ``````|}. `````` Robbert Krebbers committed Mar 07, 2016 732 733 ``````Next Obligation. intros F1 F2 A1 A2 B1 B2 n [f g] [f' g'] Hfg; simpl in *. `````` Ralf Jung committed Nov 22, 2016 734 `````` apply ofe_morC_map_ne; apply cFunctor_ne; split; by apply Hfg. `````` Robbert Krebbers committed Mar 07, 2016 735 ``````Qed. `````` Ralf Jung committed Mar 02, 2016 736 ``````Next Obligation. `````` Robbert Krebbers committed Mar 02, 2016 737 738 `````` intros F1 F2 A B [f ?] ?; simpl. rewrite /= !cFunctor_id. apply (ne_proper f). apply cFunctor_id. `````` Ralf Jung committed Mar 02, 2016 739 740 ``````Qed. Next Obligation. `````` Robbert Krebbers committed Mar 02, 2016 741 742 `````` intros F1 F2 A1 A2 A3 B1 B2 B3 f g f' g' [h ?] ?; simpl in *. rewrite -!cFunctor_compose. do 2 apply (ne_proper _). apply cFunctor_compose. `````` Ralf Jung committed Mar 02, 2016 743 ``````Qed. `````` Ralf Jung committed Jan 06, 2017 744 ``````Notation "F1 -n> F2" := (ofe_morCF F1%CF F2%CF) : cFunctor_scope. `````` Ralf Jung committed Mar 02, 2016 745 `````` `````` Ralf Jung committed Nov 22, 2016 746 ``````Instance ofe_morCF_contractive F1 F2 : `````` Ralf Jung committed Mar 07, 2016 747 `````` cFunctorContractive F1 → cFunctorContractive F2 → `````` Ralf Jung committed Nov 22, 2016 748 `` cFunctorContractive ``