sts.v 19.8 KB
 1 ``````From prelude Require Export sets. `````` Robbert Krebbers committed Feb 13, 2016 2 3 ``````From algebra Require Export cmra. From algebra Require Import dra. `````` Robbert Krebbers committed Nov 11, 2015 4 5 ``````Local Arguments valid _ _ !_ /. Local Arguments op _ _ !_ !_ /. `````` Ralf Jung committed Mar 08, 2016 6 ``````Local Arguments core _ _ !_ /. `````` Robbert Krebbers committed Nov 11, 2015 7 `````` `````` Robbert Krebbers committed Feb 16, 2016 8 ``````(** * Definition of STSs *) `````` Robbert Krebbers committed Feb 01, 2016 9 ``````Module sts. `````` Ralf Jung committed Feb 23, 2016 10 ``````Structure stsT := STS { `````` Ralf Jung committed Feb 15, 2016 11 12 `````` state : Type; token : Type; `````` Robbert Krebbers committed Feb 16, 2016 13 14 `````` prim_step : relation state; tok : state → set token; `````` Ralf Jung committed Feb 15, 2016 15 ``````}. `````` Ralf Jung committed Feb 16, 2016 16 ``````Arguments STS {_ _} _ _. `````` Robbert Krebbers committed Feb 16, 2016 17 18 19 20 ``````Arguments prim_step {_} _ _. Arguments tok {_} _. Notation states sts := (set (state sts)). Notation tokens sts := (set (token sts)). `````` Ralf Jung committed Feb 15, 2016 21 `````` `````` Robbert Krebbers committed Feb 16, 2016 22 23 24 ``````(** * Theory and definitions *) Section sts. Context {sts : stsT}. `````` Ralf Jung committed Feb 15, 2016 25 `````` `````` Robbert Krebbers committed Feb 16, 2016 26 27 ``````(** ** Step relations *) Inductive step : relation (state sts * tokens sts) := `````` Robbert Krebbers committed Nov 11, 2015 28 `````` | Step s1 s2 T1 T2 : `````` Ralf Jung committed Feb 17, 2016 29 `````` (* TODO: This asks for ⊥ on sets: T1 ⊥ T2 := T1 ∩ T2 ⊆ ∅. *) `````` Robbert Krebbers committed Feb 16, 2016 30 `````` prim_step s1 s2 → tok s1 ∩ T1 ≡ ∅ → tok s2 ∩ T2 ≡ ∅ → `````` Ralf Jung committed Feb 15, 2016 31 `````` tok s1 ∪ T1 ≡ tok s2 ∪ T2 → step (s1,T1) (s2,T2). `````` Robbert Krebbers committed Feb 22, 2016 32 ``````Notation steps := (rtc step). `````` Robbert Krebbers committed Feb 16, 2016 33 ``````Inductive frame_step (T : tokens sts) (s1 s2 : state sts) : Prop := `````` Robbert Krebbers committed Nov 11, 2015 34 `````` | Frame_step T1 T2 : `````` Robbert Krebbers committed Nov 16, 2015 35 `````` T1 ∩ (tok s1 ∪ T) ≡ ∅ → step (s1,T1) (s2,T2) → frame_step T s1 s2. `````` Robbert Krebbers committed Feb 16, 2016 36 37 38 `````` (** ** Closure under frame steps *) Record closed (S : states sts) (T : tokens sts) : Prop := Closed { `````` 39 `````` closed_disjoint s : s ∈ S → tok s ∩ T ≡ ∅; `````` Robbert Krebbers committed Nov 11, 2015 40 41 `````` closed_step s1 s2 : s1 ∈ S → frame_step T s1 s2 → s2 ∈ S }. `````` Robbert Krebbers committed Feb 16, 2016 42 ``````Definition up (s : state sts) (T : tokens sts) : states sts := `````` Robbert Krebbers committed Feb 24, 2016 43 `````` {[ s' | rtc (frame_step T) s s' ]}. `````` Robbert Krebbers committed Feb 16, 2016 44 ``````Definition up_set (S : states sts) (T : tokens sts) : states sts := `````` Robbert Krebbers committed Feb 16, 2016 45 `````` S ≫= λ s, up s T. `````` Robbert Krebbers committed Nov 11, 2015 46 `````` `````` Robbert Krebbers committed Feb 16, 2016 47 48 ``````(** Tactic setup *) Hint Resolve Step. `````` 49 50 51 52 ``````Hint Extern 50 (equiv (A:=set _) _ _) => set_solver : sts. Hint Extern 50 (¬equiv (A:=set _) _ _) => set_solver : sts. Hint Extern 50 (_ ∈ _) => set_solver : sts. Hint Extern 50 (_ ⊆ _) => set_solver : sts. `````` Robbert Krebbers committed Feb 16, 2016 53 54 `````` (** ** Setoids *) `````` Ralf Jung committed Feb 17, 2016 55 56 57 ``````Instance framestep_mono : Proper (flip (⊆) ==> (=) ==> (=) ==> impl) frame_step. Proof. intros ?? HT ?? <- ?? <-; destruct 1; econstructor; `````` Robbert Krebbers committed Feb 17, 2016 58 `````` eauto with sts; set_solver. `````` Ralf Jung committed Feb 17, 2016 59 ``````Qed. `````` Robbert Krebbers committed Feb 16, 2016 60 ``````Global Instance framestep_proper : Proper ((≡) ==> (=) ==> (=) ==> iff) frame_step. `````` Ralf Jung committed Feb 17, 2016 61 ``````Proof. by intros ?? [??] ??????; split; apply framestep_mono. Qed. `````` Robbert Krebbers committed Nov 16, 2015 62 ``````Instance closed_proper' : Proper ((≡) ==> (≡) ==> impl) closed. `````` Robbert Krebbers committed Nov 11, 2015 63 ``````Proof. `````` Robbert Krebbers committed Nov 16, 2015 64 `````` intros ?? HT ?? HS; destruct 1; `````` Robbert Krebbers committed Jan 13, 2016 65 `````` constructor; intros until 0; rewrite -?HS -?HT; eauto. `````` Robbert Krebbers committed Nov 11, 2015 66 ``````Qed. `````` Robbert Krebbers committed Feb 16, 2016 67 ``````Global Instance closed_proper : Proper ((≡) ==> (≡) ==> iff) closed. `````` Robbert Krebbers committed Nov 16, 2015 68 ``````Proof. by split; apply closed_proper'. Qed. `````` Robbert Krebbers committed Feb 16, 2016 69 ``````Global Instance up_preserving : Proper ((=) ==> flip (⊆) ==> (⊆)) up. `````` Robbert Krebbers committed Nov 11, 2015 70 ``````Proof. `````` 71 `````` intros s ? <- T T' HT ; apply elem_of_subseteq. `````` Robbert Krebbers committed Nov 11, 2015 72 `````` induction 1 as [|s1 s2 s3 [T1 T2]]; [constructor|]. `````` Robbert Krebbers committed Feb 24, 2016 73 `````` eapply elem_of_mkSet, rtc_l; [eapply Frame_step with T1 T2|]; eauto with sts. `````` Robbert Krebbers committed Nov 11, 2015 74 ``````Qed. `````` Robbert Krebbers committed Feb 16, 2016 75 ``````Global Instance up_proper : Proper ((=) ==> (≡) ==> (≡)) up. `````` 76 ``````Proof. by intros ??? ?? [??]; split; apply up_preserving. Qed. `````` Robbert Krebbers committed Feb 16, 2016 77 ``````Global Instance up_set_preserving : Proper ((⊆) ==> flip (⊆) ==> (⊆)) up_set. `````` Ralf Jung committed Feb 15, 2016 78 79 ``````Proof. intros S1 S2 HS T1 T2 HT. rewrite /up_set. `````` Ralf Jung committed Feb 25, 2016 80 `````` f_equiv; last done. move =>s1 s2 Hs. simpl in HT. by apply up_preserving. `````` Ralf Jung committed Feb 15, 2016 81 ``````Qed. `````` Robbert Krebbers committed Feb 16, 2016 82 ``````Global Instance up_set_proper : Proper ((≡) ==> (≡) ==> (≡)) up_set. `````` Robbert Krebbers committed Feb 16, 2016 83 ``````Proof. by intros S1 S2 [??] T1 T2 [??]; split; apply up_set_preserving. Qed. `````` Robbert Krebbers committed Feb 16, 2016 84 85 86 87 88 89 `````` (** ** Properties of closure under frame steps *) Lemma closed_steps S T s1 s2 : closed S T → s1 ∈ S → rtc (frame_step T) s1 s2 → s2 ∈ S. Proof. induction 3; eauto using closed_step. Qed. Lemma closed_op T1 T2 S1 S2 : `````` 90 `````` closed S1 T1 → closed S2 T2 → closed (S1 ∩ S2) (T1 ∪ T2). `````` Robbert Krebbers committed Feb 16, 2016 91 ``````Proof. `````` 92 `````` intros [? Hstep1] [? Hstep2]; split; [set_solver|]. `````` Robbert Krebbers committed Feb 16, 2016 93 `````` intros s3 s4; rewrite !elem_of_intersection; intros [??] [T3 T4 ?]; split. `````` Robbert Krebbers committed Feb 17, 2016 94 95 `````` - apply Hstep1 with s3, Frame_step with T3 T4; auto with sts. - apply Hstep2 with s3, Frame_step with T3 T4; auto with sts. `````` Robbert Krebbers committed Feb 16, 2016 96 97 98 99 100 ``````Qed. Lemma step_closed s1 s2 T1 T2 S Tf : step (s1,T1) (s2,T2) → closed S Tf → s1 ∈ S → T1 ∩ Tf ≡ ∅ → s2 ∈ S ∧ T2 ∩ Tf ≡ ∅ ∧ tok s2 ∩ T2 ≡ ∅. Proof. `````` 101 `````` inversion_clear 1 as [???? HR Hs1 Hs2]; intros [? Hstep]??; split_and?; auto. `````` Robbert Krebbers committed Feb 17, 2016 102 `````` - eapply Hstep with s1, Frame_step with T1 T2; auto with sts. `````` Robbert Krebbers committed Feb 17, 2016 103 `````` - set_solver -Hstep Hs1 Hs2. `````` Robbert Krebbers committed Feb 16, 2016 104 ``````Qed. `````` Ralf Jung committed Feb 20, 2016 105 106 107 108 ``````Lemma steps_closed s1 s2 T1 T2 S Tf : steps (s1,T1) (s2,T2) → closed S Tf → s1 ∈ S → T1 ∩ Tf ≡ ∅ → tok s1 ∩ T1 ≡ ∅ → s2 ∈ S ∧ T2 ∩ Tf ≡ ∅ ∧ tok s2 ∩ T2 ≡ ∅. Proof. `````` Robbert Krebbers committed Feb 22, 2016 109 110 111 112 113 `````` remember (s1,T1) as sT1 eqn:HsT1; remember (s2,T2) as sT2 eqn:HsT2. intros Hsteps; revert s1 T1 HsT1 s2 T2 HsT2. induction Hsteps as [?|? [s2 T2] ? Hstep Hsteps IH]; intros s1 T1 HsT1 s2' T2' ?????; simplify_eq; first done. destruct (step_closed s1 s2 T1 T2 S Tf) as (?&?&?); eauto. `````` Ralf Jung committed Feb 20, 2016 114 ``````Qed. `````` Robbert Krebbers committed Feb 16, 2016 115 116 `````` (** ** Properties of the closure operators *) `````` 117 ``````Lemma elem_of_up s T : s ∈ up s T. `````` Robbert Krebbers committed Nov 11, 2015 118 ``````Proof. constructor. Qed. `````` 119 ``````Lemma subseteq_up_set S T : S ⊆ up_set S T. `````` Robbert Krebbers committed Nov 11, 2015 120 ``````Proof. intros s ?; apply elem_of_bind; eauto using elem_of_up. Qed. `````` Ralf Jung committed Feb 15, 2016 121 122 ``````Lemma up_up_set s T : up s T ≡ up_set {[ s ]} T. Proof. by rewrite /up_set collection_bind_singleton. Qed. `````` Robbert Krebbers committed Dec 08, 2015 123 ``````Lemma closed_up_set S T : `````` 124 `````` (∀ s, s ∈ S → tok s ∩ T ≡ ∅) → closed (up_set S T) T. `````` Robbert Krebbers committed Nov 11, 2015 125 ``````Proof. `````` 126 `````` intros HS; unfold up_set; split. `````` Robbert Krebbers committed Feb 17, 2016 127 `````` - intros s; rewrite !elem_of_bind; intros (s'&Hstep&Hs'). `````` 128 `````` specialize (HS s' Hs'); clear Hs' S. `````` Ralf Jung committed Feb 16, 2016 129 `````` induction Hstep as [s|s1 s2 s3 [T1 T2 ? Hstep] ? IH]; first done. `````` Robbert Krebbers committed Nov 11, 2015 130 `````` inversion_clear Hstep; apply IH; clear IH; auto with sts. `````` Robbert Krebbers committed Feb 24, 2016 131 `````` - intros s1 s2; rewrite /up; set_unfold; intros (s&?&?) ?; exists s. `````` Robbert Krebbers committed Nov 11, 2015 132 133 `````` split; [eapply rtc_r|]; eauto. Qed. `````` 134 ``````Lemma closed_up s T : tok s ∩ T ≡ ∅ → closed (up s T) T. `````` Robbert Krebbers committed Nov 11, 2015 135 ``````Proof. `````` 136 `````` intros; rewrite -(collection_bind_singleton (λ s, up s T) s). `````` Robbert Krebbers committed Feb 17, 2016 137 `````` apply closed_up_set; set_solver. `````` Robbert Krebbers committed Nov 11, 2015 138 ``````Qed. `````` 139 140 ``````Lemma closed_up_set_empty S : closed (up_set S ∅) ∅. Proof. eauto using closed_up_set with sts. Qed. `````` 141 ``````Lemma closed_up_empty s : closed (up s ∅) ∅. `````` Robbert Krebbers committed Nov 11, 2015 142 ``````Proof. eauto using closed_up with sts. Qed. `````` 143 ``````Lemma up_set_empty S T : up_set S T ≡ ∅ → S ≡ ∅. `````` Robbert Krebbers committed Feb 22, 2016 144 145 ``````Proof. move:(subseteq_up_set S T). set_solver. Qed. Lemma up_set_non_empty S T : S ≢ ∅ → up_set S T ≢ ∅. `````` 146 ``````Proof. by move=>? /up_set_empty. Qed. `````` Robbert Krebbers committed Feb 22, 2016 147 148 ``````Lemma up_non_empty s T : up s T ≢ ∅. Proof. eapply non_empty_inhabited, elem_of_up. Qed. `````` 149 ``````Lemma up_closed S T : closed S T → up_set S T ≡ S. `````` Robbert Krebbers committed Nov 11, 2015 150 ``````Proof. `````` Robbert Krebbers committed Dec 08, 2015 151 `````` intros; split; auto using subseteq_up_set; intros s. `````` Robbert Krebbers committed Nov 11, 2015 152 153 154 `````` unfold up_set; rewrite elem_of_bind; intros (s'&Hstep&?). induction Hstep; eauto using closed_step. Qed. `````` Robbert Krebbers committed Feb 22, 2016 155 156 157 158 159 160 161 162 ``````Lemma up_subseteq s T S : closed S T → s ∈ S → sts.up s T ⊆ S. Proof. move=> ?? s' ?. eauto using closed_steps. Qed. Lemma up_set_subseteq S1 T S2 : closed S2 T → S1 ⊆ S2 → sts.up_set S1 T ⊆ S2. Proof. move=> ?? s [s' [? ?]]. eauto using closed_steps. Qed. End sts. Notation steps := (rtc step). End sts. `````` Robbert Krebbers committed Feb 16, 2016 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 `````` Notation stsT := sts.stsT. Notation STS := sts.STS. (** * STSs form a disjoint RA *) (* This module should never be imported, uses the module [sts] below. *) Module sts_dra. Import sts. (* The type of bounds we can give to the state of an STS. This is the type that we equip with an RA structure. *) Inductive car (sts : stsT) := | auth : state sts → set (token sts) → car sts | frag : set (state sts) → set (token sts ) → car sts. Arguments auth {_} _ _. Arguments frag {_} _ _. Section sts_dra. Context {sts : stsT}. Infix "≼" := dra_included. Implicit Types S : states sts. Implicit Types T : tokens sts. Inductive sts_equiv : Equiv (car sts) := | auth_equiv s T1 T2 : T1 ≡ T2 → auth s T1 ≡ auth s T2 | frag_equiv S1 S2 T1 T2 : T1 ≡ T2 → S1 ≡ S2 → frag S1 T1 ≡ frag S2 T2. `````` Ralf Jung committed Feb 21, 2016 189 190 ``````Global Existing Instance sts_equiv. Global Instance sts_valid : Valid (car sts) := λ x, `````` 191 192 `````` match x with | auth s T => tok s ∩ T ≡ ∅ `````` Robbert Krebbers committed Feb 22, 2016 193 194 `````` | frag S' T => closed S' T ∧ S' ≢ ∅ end. `````` Ralf Jung committed Mar 08, 2016 195 ``````Global Instance sts_core : Core (car sts) := λ x, `````` Robbert Krebbers committed Feb 16, 2016 196 197 198 199 200 201 202 203 204 205 206 `````` match x with | frag S' _ => frag (up_set S' ∅ ) ∅ | auth s _ => frag (up s ∅) ∅ end. Inductive sts_disjoint : Disjoint (car sts) := | frag_frag_disjoint S1 S2 T1 T2 : S1 ∩ S2 ≢ ∅ → T1 ∩ T2 ≡ ∅ → frag S1 T1 ⊥ frag S2 T2 | auth_frag_disjoint s S T1 T2 : s ∈ S → T1 ∩ T2 ≡ ∅ → auth s T1 ⊥ frag S T2 | frag_auth_disjoint s S T1 T2 : s ∈ S → T1 ∩ T2 ≡ ∅ → frag S T1 ⊥ auth s T2. `````` Ralf Jung committed Feb 21, 2016 207 208 ``````Global Existing Instance sts_disjoint. Global Instance sts_op : Op (car sts) := λ x1 x2, `````` Robbert Krebbers committed Feb 16, 2016 209 210 211 212 213 214 `````` match x1, x2 with | frag S1 T1, frag S2 T2 => frag (S1 ∩ S2) (T1 ∪ T2) | auth s T1, frag _ T2 => auth s (T1 ∪ T2) | frag _ T1, auth s T2 => auth s (T1 ∪ T2) | auth s T1, auth _ T2 => auth s (T1 ∪ T2)(* never happens *) end. `````` Ralf Jung committed Feb 29, 2016 215 ``````Global Instance sts_div : Div (car sts) := λ x1 x2, `````` Robbert Krebbers committed Feb 16, 2016 216 217 218 219 220 221 222 `````` match x1, x2 with | frag S1 T1, frag S2 T2 => frag (up_set S1 (T1 ∖ T2)) (T1 ∖ T2) | auth s T1, frag _ T2 => auth s (T1 ∖ T2) | frag _ T2, auth s T1 => auth s (T1 ∖ T2) (* never happens *) | auth s T1, auth _ T2 => frag (up s (T1 ∖ T2)) (T1 ∖ T2) end. `````` Robbert Krebbers committed Feb 22, 2016 223 224 225 226 ``````Hint Extern 50 (equiv (A:=set _) _ _) => set_solver : sts. Hint Extern 50 (¬equiv (A:=set _) _ _) => set_solver : sts. Hint Extern 50 (_ ∈ _) => set_solver : sts. Hint Extern 50 (_ ⊆ _) => set_solver : sts. `````` Ralf Jung committed Feb 21, 2016 227 ``````Global Instance sts_equivalence: Equivalence ((≡) : relation (car sts)). `````` Robbert Krebbers committed Feb 16, 2016 228 229 ``````Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 230 231 `````` - by intros []; constructor. - by destruct 1; constructor. `````` Ralf Jung committed Feb 20, 2016 232 `````` - destruct 1; inversion_clear 1; constructor; etrans; eauto. `````` Robbert Krebbers committed Feb 16, 2016 233 234 ``````Qed. Global Instance sts_dra : DRA (car sts). `````` Robbert Krebbers committed Nov 11, 2015 235 236 ``````Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 237 238 239 240 241 242 `````` - apply _. - by do 2 destruct 1; constructor; setoid_subst. - by destruct 1; constructor; setoid_subst. - by destruct 1; simpl; intros ?; setoid_subst. - by intros ? [|]; destruct 1; inversion_clear 1; constructor; setoid_subst. - by do 2 destruct 1; constructor; setoid_subst. `````` Robbert Krebbers committed Mar 03, 2016 243 `````` - destruct 3; simpl in *; destruct_and?; eauto using closed_op; `````` Robbert Krebbers committed Feb 22, 2016 244 `````` match goal with H : closed _ _ |- _ => destruct H end; set_solver. `````` Robbert Krebbers committed Mar 03, 2016 245 `````` - intros []; simpl; intros; destruct_and?; split; `````` Robbert Krebbers committed Feb 22, 2016 246 `````` eauto using closed_up, up_non_empty, closed_up_set, up_set_empty with sts. `````` 247 `````` - intros ???? (z&Hy&?&Hxz); destruct Hxz; inversion Hy; clear Hy; `````` Robbert Krebbers committed Mar 03, 2016 248 `````` setoid_subst; destruct_and?; split_and?; `````` Robbert Krebbers committed Feb 22, 2016 249 250 `````` rewrite disjoint_union_difference //; eauto using up_set_non_empty, up_non_empty, closed_up, closed_disjoint; []. `````` Robbert Krebbers committed Feb 22, 2016 251 `````` eapply closed_up_set=> s ?; eapply closed_disjoint; eauto with sts. `````` Robbert Krebbers committed Feb 17, 2016 252 253 254 255 256 257 258 259 `````` - intros [] [] []; constructor; rewrite ?assoc; auto with sts. - destruct 4; inversion_clear 1; constructor; auto with sts. - destruct 4; inversion_clear 1; constructor; auto with sts. - destruct 1; constructor; auto with sts. - destruct 3; constructor; auto with sts. - intros [|S T]; constructor; auto using elem_of_up with sts. - intros [|S T]; constructor; auto with sts. - intros [s T|S T]; constructor; auto with sts. `````` Robbert Krebbers committed Jan 13, 2016 260 `````` + rewrite (up_closed (up _ _)); auto using closed_up with sts. `````` Robbert Krebbers committed Feb 24, 2016 261 `````` + rewrite (up_closed (up_set _ _)); eauto using closed_up_set with sts. `````` Ralf Jung committed Mar 08, 2016 262 `````` - intros x y ?? (z&Hy&?&Hxz); exists (core (x ⋅ y)); split_and?. `````` Robbert Krebbers committed Feb 24, 2016 263 `````` + destruct Hxz; inversion_clear Hy; constructor; unfold up_set; set_solver. `````` 264 `````` + destruct Hxz; inversion_clear Hy; simpl; split_and?; `````` Robbert Krebbers committed Feb 22, 2016 265 266 `````` auto using closed_up_set_empty, closed_up_empty, up_non_empty; []. apply up_set_non_empty. set_solver. `````` Robbert Krebbers committed Dec 08, 2015 267 268 `````` + destruct Hxz; inversion_clear Hy; constructor; repeat match goal with `````` 269 270 271 272 `````` | |- context [ up_set ?S ?T ] => unless (S ⊆ up_set S T) by done; pose proof (subseteq_up_set S T) | |- context [ up ?s ?T ] => unless (s ∈ up s T) by done; pose proof (elem_of_up s T) `````` Robbert Krebbers committed Dec 08, 2015 273 `````` end; auto with sts. `````` Robbert Krebbers committed Feb 17, 2016 274 `````` - intros x y ?? (z&Hy&_&Hxz); destruct Hxz; inversion_clear Hy; constructor; `````` Robbert Krebbers committed Dec 08, 2015 275 `````` repeat match goal with `````` 276 277 278 279 `````` | |- context [ up_set ?S ?T ] => unless (S ⊆ up_set S T) by done; pose proof (subseteq_up_set S T) | |- context [ up ?s ?T ] => unless (s ∈ up s T) by done; pose proof (elem_of_up s T) `````` Robbert Krebbers committed Dec 08, 2015 280 `````` end; auto with sts. `````` Robbert Krebbers committed Feb 17, 2016 281 `````` - intros x y ?? (z&Hy&?&Hxz); destruct Hxz as [S1 S2 T1 T2| |]; `````` Robbert Krebbers committed Nov 20, 2015 282 `````` inversion Hy; clear Hy; constructor; setoid_subst; `````` Robbert Krebbers committed Jan 13, 2016 283 `````` rewrite ?disjoint_union_difference; auto. `````` Robbert Krebbers committed Dec 08, 2015 284 `````` split; [|apply intersection_greatest; auto using subseteq_up_set with sts]. `````` Robbert Krebbers committed Nov 20, 2015 285 `````` apply intersection_greatest; [auto with sts|]. `````` Robbert Krebbers committed Mar 03, 2016 286 `````` intros s2; rewrite elem_of_intersection. destruct_and?. `````` Robbert Krebbers committed Nov 20, 2015 287 288 `````` unfold up_set; rewrite elem_of_bind; intros (?&s1&?&?&?). apply closed_steps with T2 s1; auto with sts. `````` Robbert Krebbers committed Feb 22, 2016 289 ``````Qed. `````` Robbert Krebbers committed Mar 01, 2016 290 ``````Canonical Structure R : cmraT := validityR (car sts). `````` Robbert Krebbers committed Feb 16, 2016 291 292 293 294 ``````End sts_dra. End sts_dra. (** * The STS Resource Algebra *) (** Finally, the general theory of STS that should be used by users *) `````` Robbert Krebbers committed Mar 01, 2016 295 ``````Notation stsR := (@sts_dra.R). `````` Robbert Krebbers committed Feb 16, 2016 296 297 298 `````` Section sts_definitions. Context {sts : stsT}. `````` Robbert Krebbers committed Mar 01, 2016 299 `````` Definition sts_auth (s : sts.state sts) (T : sts.tokens sts) : stsR sts := `````` Robbert Krebbers committed Feb 16, 2016 300 `````` to_validity (sts_dra.auth s T). `````` Robbert Krebbers committed Mar 01, 2016 301 `````` Definition sts_frag (S : sts.states sts) (T : sts.tokens sts) : stsR sts := `````` Robbert Krebbers committed Feb 16, 2016 302 `````` to_validity (sts_dra.frag S T). `````` Robbert Krebbers committed Mar 01, 2016 303 `````` Definition sts_frag_up (s : sts.state sts) (T : sts.tokens sts) : stsR sts := `````` Robbert Krebbers committed Feb 16, 2016 304 305 306 307 308 309 310 311 312 313 314 315 316 `````` sts_frag (sts.up s T) T. End sts_definitions. Instance: Params (@sts_auth) 2. Instance: Params (@sts_frag) 1. Instance: Params (@sts_frag_up) 2. Section stsRA. Import sts. Context {sts : stsT}. Implicit Types s : state sts. Implicit Types S : states sts. Implicit Types T : tokens sts. `````` Robbert Krebbers committed Mar 01, 2016 317 ``````Global Instance sts_cmra_discrete : CMRADiscrete (stsR sts). `````` Robbert Krebbers committed Feb 24, 2016 318 319 ``````Proof. apply validity_cmra_discrete. Qed. `````` Robbert Krebbers committed Feb 16, 2016 320 321 322 323 324 325 326 ``````(** Setoids *) Global Instance sts_auth_proper s : Proper ((≡) ==> (≡)) (sts_auth s). Proof. (* this proof is horrible *) intros T1 T2 HT. rewrite /sts_auth. by eapply to_validity_proper; try apply _; constructor. Qed. Global Instance sts_frag_proper : Proper ((≡) ==> (≡) ==> (≡)) (@sts_frag sts). `````` Robbert Krebbers committed Nov 11, 2015 327 ``````Proof. `````` Robbert Krebbers committed Feb 16, 2016 328 329 `````` intros S1 S2 ? T1 T2 HT; rewrite /sts_auth. by eapply to_validity_proper; try apply _; constructor. `````` Robbert Krebbers committed Nov 11, 2015 330 ``````Qed. `````` Robbert Krebbers committed Feb 16, 2016 331 332 ``````Global Instance sts_frag_up_proper s : Proper ((≡) ==> (≡)) (sts_frag_up s). Proof. intros T1 T2 HT. by rewrite /sts_frag_up HT. Qed. `````` Robbert Krebbers committed Nov 11, 2015 333 `````` `````` Robbert Krebbers committed Feb 16, 2016 334 335 ``````(** Validity *) Lemma sts_auth_valid s T : ✓ sts_auth s T ↔ tok s ∩ T ≡ ∅. `````` Robbert Krebbers committed Feb 24, 2016 336 ``````Proof. done. Qed. `````` 337 ``````Lemma sts_frag_valid S T : ✓ sts_frag S T ↔ closed S T ∧ S ≢ ∅. `````` Robbert Krebbers committed Feb 24, 2016 338 ``````Proof. done. Qed. `````` Robbert Krebbers committed Feb 16, 2016 339 ``````Lemma sts_frag_up_valid s T : tok s ∩ T ≡ ∅ → ✓ sts_frag_up s T. `````` Robbert Krebbers committed Feb 22, 2016 340 ``````Proof. intros. by apply sts_frag_valid; auto using closed_up, up_non_empty. Qed. `````` Robbert Krebbers committed Nov 11, 2015 341 `````` `````` Robbert Krebbers committed Feb 16, 2016 342 343 ``````Lemma sts_auth_frag_valid_inv s S T1 T2 : ✓ (sts_auth s T1 ⋅ sts_frag S T2) → s ∈ S. `````` Robbert Krebbers committed Feb 24, 2016 344 ``````Proof. by intros (?&?&Hdisj); inversion Hdisj. Qed. `````` Ralf Jung committed Feb 15, 2016 345 `````` `````` Robbert Krebbers committed Feb 16, 2016 346 347 348 349 ``````(** Op *) Lemma sts_op_auth_frag s S T : s ∈ S → closed S T → sts_auth s ∅ ⋅ sts_frag S T ≡ sts_auth s T. Proof. `````` Robbert Krebbers committed Feb 17, 2016 350 `````` intros; split; [split|constructor; set_solver]; simpl. `````` 351 `````` - intros (?&?&?); by apply closed_disjoint with S. `````` Robbert Krebbers committed Feb 24, 2016 352 `````` - intros; split_and?; last constructor; set_solver. `````` Robbert Krebbers committed Feb 16, 2016 353 354 ``````Qed. Lemma sts_op_auth_frag_up s T : `````` Ralf Jung committed Feb 20, 2016 355 356 357 `````` sts_auth s ∅ ⋅ sts_frag_up s T ≡ sts_auth s T. Proof. intros; split; [split|constructor; set_solver]; simpl. `````` Robbert Krebbers committed Feb 24, 2016 358 `````` - intros (?&[??]&?). by apply closed_disjoint with (up s T), elem_of_up. `````` Ralf Jung committed Feb 20, 2016 359 360 361 `````` - intros; split_and?. + set_solver+. + by apply closed_up. `````` Robbert Krebbers committed Feb 22, 2016 362 `````` + apply up_non_empty. `````` Ralf Jung committed Feb 20, 2016 363 364 `````` + constructor; last set_solver. apply elem_of_up. Qed. `````` Robbert Krebbers committed Feb 16, 2016 365 `````` `````` Ralf Jung committed Feb 17, 2016 366 ``````Lemma sts_op_frag S1 S2 T1 T2 : `````` 367 `````` T1 ∩ T2 ≡ ∅ → sts.closed S1 T1 → sts.closed S2 T2 → `````` Ralf Jung committed Feb 17, 2016 368 369 `````` sts_frag (S1 ∩ S2) (T1 ∪ T2) ≡ sts_frag S1 T1 ⋅ sts_frag S2 T2. Proof. `````` Ralf Jung committed Feb 17, 2016 370 371 `````` intros HT HS1 HS2. rewrite /sts_frag. (* FIXME why does rewrite not work?? *) `````` 372 373 374 `````` etrans; last eapply to_validity_op; first done; []. move=>/=[??]. split_and!; [auto; set_solver..|]. constructor; done. `````` Ralf Jung committed Feb 17, 2016 375 376 ``````Qed. `````` Robbert Krebbers committed Feb 16, 2016 377 378 ``````(** Frame preserving updates *) Lemma sts_update_auth s1 s2 T1 T2 : `````` Ralf Jung committed Feb 20, 2016 379 `````` steps (s1,T1) (s2,T2) → sts_auth s1 T1 ~~> sts_auth s2 T2. `````` Robbert Krebbers committed Nov 11, 2015 380 ``````Proof. `````` Robbert Krebbers committed Feb 22, 2016 381 `````` intros ?; apply validity_update. `````` Robbert Krebbers committed Mar 03, 2016 382 `````` inversion 3 as [|? S ? Tf|]; simplify_eq/=; destruct_and?. `````` Ralf Jung committed Feb 20, 2016 383 `````` destruct (steps_closed s1 s2 T1 T2 S Tf) as (?&?&?); auto; []. `````` Robbert Krebbers committed Nov 16, 2015 384 `````` repeat (done || constructor). `````` Robbert Krebbers committed Nov 11, 2015 385 ``````Qed. `````` Ralf Jung committed Feb 15, 2016 386 `````` `````` 387 388 ``````Lemma sts_update_frag S1 S2 T1 T2 : closed S2 T2 → S1 ⊆ S2 → T2 ⊆ T1 → sts_frag S1 T1 ~~> sts_frag S2 T2. `````` Ralf Jung committed Feb 15, 2016 389 ``````Proof. `````` 390 `````` rewrite /sts_frag=> ? HS HT. apply validity_update. `````` Robbert Krebbers committed Feb 17, 2016 391 `````` inversion 3 as [|? S ? Tf|]; simplify_eq/=. `````` 392 393 `````` - split_and!; first done; first set_solver. constructor; set_solver. - split_and!; first done; first set_solver. constructor; set_solver. `````` Ralf Jung committed Feb 15, 2016 394 395 ``````Qed. `````` 396 397 ``````Lemma sts_update_frag_up s1 S2 T1 T2 : closed S2 T2 → s1 ∈ S2 → T2 ⊆ T1 → sts_frag_up s1 T1 ~~> sts_frag S2 T2. `````` Ralf Jung committed Feb 15, 2016 398 ``````Proof. `````` 399 400 `````` intros ? ? HT; apply sts_update_frag; [intros; eauto using closed_steps..]. rewrite <-HT. eapply up_subseteq; done. `````` Robbert Krebbers committed Feb 16, 2016 401 402 ``````Qed. `````` Ralf Jung committed Feb 21, 2016 403 404 405 406 407 ``````Lemma up_set_intersection S1 Sf Tf : closed Sf Tf → S1 ∩ Sf ≡ S1 ∩ up_set (S1 ∩ Sf) Tf. Proof. intros Hclf. apply (anti_symm (⊆)). `````` Robbert Krebbers committed Feb 24, 2016 408 409 410 `````` + move=>s [HS1 HSf]. split. by apply HS1. by apply subseteq_up_set. + move=>s [HS1 [s' [/elem_of_mkSet Hsup Hs']]]. split; first done. eapply closed_steps, Hsup; first done. set_solver +Hs'. `````` Ralf Jung committed Feb 21, 2016 411 412 ``````Qed. `````` Robbert Krebbers committed Feb 16, 2016 413 ``````(** Inclusion *) `````` Ralf Jung committed Feb 21, 2016 414 415 416 ``````(* This is surprisingly different from to_validity_included. I am not sure whether this is because to_validity_included is non-canonical, or this one here is non-canonical - but I suspect both. *) `````` Robbert Krebbers committed Feb 16, 2016 417 ``````Lemma sts_frag_included S1 S2 T1 T2 : `````` Ralf Jung committed Feb 21, 2016 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 `````` closed S2 T2 → S2 ≢ ∅ → (sts_frag S1 T1 ≼ sts_frag S2 T2) ↔ (closed S1 T1 ∧ S1 ≢ ∅ ∧ ∃ Tf, T2 ≡ T1 ∪ Tf ∧ T1 ∩ Tf ≡ ∅ ∧ S2 ≡ S1 ∩ up_set S2 Tf). Proof. destruct (to_validity_included (sts_dra.car sts) (sts_dra.frag S1 T1) (sts_dra.frag S2 T2)) as [Hfincl Htoincl]. intros Hcl2 HS2ne. split. - intros Hincl. destruct Hfincl as ((Hcl1 & ?) & (z & EQ & Hval & Hdisj)). { split; last done. split; done. } clear Htoincl. split_and!; try done; []. destruct z as [sf Tf|Sf Tf]. { exfalso. inversion_clear EQ. } exists Tf. inversion_clear EQ as [|? ? ? ? HT2 HS2]. inversion_clear Hdisj as [? ? ? ? _ HTdisj | |]. split_and!; [done..|]. rewrite HS2. apply up_set_intersection. apply Hval. - intros (Hcl & Hne & (Tf & HT & HTdisj & HS)). destruct Htoincl as ((Hcl' & ?) & (z & EQ)); last first. { exists z. exact EQ. } clear Hfincl. split; first (split; done). exists (sts_dra.frag (up_set S2 Tf) Tf). split_and!. + constructor; done. + simpl. split. * apply closed_up_set. move=>s Hs2. move:(closed_disjoint _ _ Hcl2 _ Hs2). set_solver +HT. `````` Robbert Krebbers committed Feb 22, 2016 440 `````` * by apply up_set_non_empty. `````` Ralf Jung committed Feb 21, 2016 441 `````` + constructor; last done. by rewrite -HS. `````` 442 443 ``````Qed. `````` Robbert Krebbers committed Feb 16, 2016 444 ``````Lemma sts_frag_included' S1 S2 T : `````` Ralf Jung committed Feb 21, 2016 445 `````` closed S2 T → closed S1 T → S2 ≢ ∅ → S1 ≢ ∅ → S2 ≡ S1 ∩ up_set S2 ∅ → `````` Robbert Krebbers committed Feb 16, 2016 446 `````` sts_frag S1 T ≼ sts_frag S2 T. `````` 447 ``````Proof. `````` Robbert Krebbers committed Feb 19, 2016 448 449 `````` intros. apply sts_frag_included; split_and?; auto. exists ∅; split_and?; done || set_solver+. `````` 450 ``````Qed. `````` Ralf Jung committed Feb 21, 2016 451 `````` `````` Robbert Krebbers committed Feb 01, 2016 452 ``````End stsRA. `````` Ralf Jung committed Mar 07, 2016 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 `````` (** STSs without tokens: Some stuff is simpler *) Module sts_notok. Structure stsT := STS { state : Type; prim_step : relation state; }. Arguments STS {_} _. Arguments prim_step {_} _ _. Notation states sts := (set (state sts)). Canonical sts_notok (sts : stsT) : sts.stsT := sts.STS (token:=Empty_set) (@prim_step sts) (λ _, ∅). Section sts. Context {sts : stsT}. Implicit Types s : state sts. Implicit Types S : states sts. Notation prim_steps := (rtc prim_step). Lemma sts_step s1 s2 : prim_step s1 s2 → sts.step (s1, ∅) (s2, ∅). Proof. intros. split; set_solver. Qed. Lemma sts_steps s1 s2 : prim_steps s1 s2 → sts.steps (s1, ∅) (s2, ∅). Proof. induction 1; eauto using sts_step, rtc_refl, rtc_l. Qed. Lemma frame_prim_step T s1 s2 : sts.frame_step T s1 s2 → prim_step s1 s2. Proof. inversion 1 as [??? Hstep]. inversion_clear Hstep. done. Qed. Lemma prim_frame_step T s1 s2 : prim_step s1 s2 → sts.frame_step T s1 s2. Proof. intros Hstep. apply sts.Frame_step with ∅ ∅; first set_solver. by apply sts_step. Qed. Lemma mk_closed S : (∀ s1 s2, s1 ∈ S → prim_step s1 s2 → s2 ∈ S) → sts.closed S ∅. Proof. intros ?. constructor; first by set_solver. intros ????. eauto using frame_prim_step. Qed. End sts. Notation steps := (rtc prim_step). End sts_notok. Coercion sts_notok.sts_notok : sts_notok.stsT >-> sts.stsT. Notation sts_notokT := sts_notok.stsT. Notation STS_NoTok := sts_notok.STS. Section sts_notokRA. Import sts_notok. Context {sts : sts_notokT}. Implicit Types s : state sts. Implicit Types S : states sts. Lemma sts_notok_update_auth s1 s2 : rtc prim_step s1 s2 → sts_auth s1 ∅ ~~> sts_auth s2 ∅. Proof. intros. by apply sts_update_auth, sts_steps. Qed. End sts_notokRA. ``````