sts.v 18.2 KB
 Robbert Krebbers committed Mar 10, 2016 1 2 3 ``````From iris.prelude Require Export sets. From iris.algebra Require Export cmra. From iris.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 : `````` Robbert Krebbers committed Mar 23, 2016 29 `````` prim_step s1 s2 → tok s1 ⊥ T1 → tok s2 ⊥ T2 → `````` Ralf Jung committed Feb 15, 2016 30 `````` tok s1 ∪ T1 ≡ tok s2 ∪ T2 → step (s1,T1) (s2,T2). `````` Robbert Krebbers committed Feb 22, 2016 31 ``````Notation steps := (rtc step). `````` Robbert Krebbers committed Feb 16, 2016 32 ``````Inductive frame_step (T : tokens sts) (s1 s2 : state sts) : Prop := `````` Robbert Krebbers committed Nov 11, 2015 33 `````` | Frame_step T1 T2 : `````` Robbert Krebbers committed Mar 23, 2016 34 `````` T1 ⊥ tok s1 ∪ T → step (s1,T1) (s2,T2) → frame_step T s1 s2. `````` Robbert Krebbers committed Feb 16, 2016 35 36 37 `````` (** ** Closure under frame steps *) Record closed (S : states sts) (T : tokens sts) : Prop := Closed { `````` Robbert Krebbers committed Mar 23, 2016 38 `````` closed_disjoint s : s ∈ S → tok s ⊥ T; `````` Robbert Krebbers committed Nov 11, 2015 39 40 `````` closed_step s1 s2 : s1 ∈ S → frame_step T s1 s2 → s2 ∈ S }. `````` Robbert Krebbers committed Feb 16, 2016 41 ``````Definition up (s : state sts) (T : tokens sts) : states sts := `````` Robbert Krebbers committed Feb 24, 2016 42 `````` {[ s' | rtc (frame_step T) s s' ]}. `````` Robbert Krebbers committed Feb 16, 2016 43 ``````Definition up_set (S : states sts) (T : tokens sts) : states sts := `````` Robbert Krebbers committed Feb 16, 2016 44 `````` S ≫= λ s, up s T. `````` Robbert Krebbers committed Nov 11, 2015 45 `````` `````` Robbert Krebbers committed Feb 16, 2016 46 47 ``````(** Tactic setup *) Hint Resolve Step. `````` 48 49 50 51 ``````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 Mar 23, 2016 52 ``````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 Mar 23, 2016 63 ``````Proof. destruct 3; constructor; intros until 0; setoid_subst; eauto. Qed. `````` Robbert Krebbers committed Feb 16, 2016 64 ``````Global Instance closed_proper : Proper ((≡) ==> (≡) ==> iff) closed. `````` Robbert Krebbers committed Nov 16, 2015 65 ``````Proof. by split; apply closed_proper'. Qed. `````` Robbert Krebbers committed Feb 16, 2016 66 ``````Global Instance up_preserving : Proper ((=) ==> flip (⊆) ==> (⊆)) up. `````` Robbert Krebbers committed Nov 11, 2015 67 ``````Proof. `````` 68 `````` intros s ? <- T T' HT ; apply elem_of_subseteq. `````` Robbert Krebbers committed Nov 11, 2015 69 `````` induction 1 as [|s1 s2 s3 [T1 T2]]; [constructor|]. `````` Robbert Krebbers committed Feb 24, 2016 70 `````` eapply elem_of_mkSet, rtc_l; [eapply Frame_step with T1 T2|]; eauto with sts. `````` Robbert Krebbers committed Nov 11, 2015 71 ``````Qed. `````` Robbert Krebbers committed Feb 16, 2016 72 ``````Global Instance up_proper : Proper ((=) ==> (≡) ==> (≡)) up. `````` 73 ``````Proof. by intros ??? ?? [??]; split; apply up_preserving. Qed. `````` Robbert Krebbers committed Feb 16, 2016 74 ``````Global Instance up_set_preserving : Proper ((⊆) ==> flip (⊆) ==> (⊆)) up_set. `````` Ralf Jung committed Feb 15, 2016 75 76 ``````Proof. intros S1 S2 HS T1 T2 HT. rewrite /up_set. `````` Ralf Jung committed Feb 25, 2016 77 `````` f_equiv; last done. move =>s1 s2 Hs. simpl in HT. by apply up_preserving. `````` Ralf Jung committed Feb 15, 2016 78 ``````Qed. `````` Robbert Krebbers committed Feb 16, 2016 79 ``````Global Instance up_set_proper : Proper ((≡) ==> (≡) ==> (≡)) up_set. `````` Robbert Krebbers committed Feb 16, 2016 80 ``````Proof. by intros S1 S2 [??] T1 T2 [??]; split; apply up_set_preserving. Qed. `````` Robbert Krebbers committed Feb 16, 2016 81 82 83 84 85 86 `````` (** ** 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 : `````` 87 `````` closed S1 T1 → closed S2 T2 → closed (S1 ∩ S2) (T1 ∪ T2). `````` Robbert Krebbers committed Feb 16, 2016 88 ``````Proof. `````` 89 `````` intros [? Hstep1] [? Hstep2]; split; [set_solver|]. `````` Robbert Krebbers committed Feb 16, 2016 90 `````` intros s3 s4; rewrite !elem_of_intersection; intros [??] [T3 T4 ?]; split. `````` Robbert Krebbers committed Feb 17, 2016 91 92 `````` - 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 93 94 ``````Qed. Lemma step_closed s1 s2 T1 T2 S Tf : `````` Robbert Krebbers committed Mar 23, 2016 95 96 `````` step (s1,T1) (s2,T2) → closed S Tf → s1 ∈ S → T1 ⊥ Tf → s2 ∈ S ∧ T2 ⊥ Tf ∧ tok s2 ⊥ T2. `````` Robbert Krebbers committed Feb 16, 2016 97 ``````Proof. `````` 98 `````` inversion_clear 1 as [???? HR Hs1 Hs2]; intros [? Hstep]??; split_and?; auto. `````` Robbert Krebbers committed Feb 17, 2016 99 `````` - eapply Hstep with s1, Frame_step with T1 T2; auto with sts. `````` Robbert Krebbers committed Feb 17, 2016 100 `````` - set_solver -Hstep Hs1 Hs2. `````` Robbert Krebbers committed Feb 16, 2016 101 ``````Qed. `````` Ralf Jung committed Feb 20, 2016 102 ``````Lemma steps_closed s1 s2 T1 T2 S Tf : `````` Robbert Krebbers committed Mar 23, 2016 103 104 `````` steps (s1,T1) (s2,T2) → closed S Tf → s1 ∈ S → T1 ⊥ Tf → tok s1 ⊥ T1 → s2 ∈ S ∧ T2 ⊥ Tf ∧ tok s2 ⊥ T2. `````` Ralf Jung committed Feb 20, 2016 105 ``````Proof. `````` Robbert Krebbers committed Feb 22, 2016 106 107 108 109 110 `````` 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 111 ``````Qed. `````` Robbert Krebbers committed Feb 16, 2016 112 113 `````` (** ** Properties of the closure operators *) `````` 114 ``````Lemma elem_of_up s T : s ∈ up s T. `````` Robbert Krebbers committed Nov 11, 2015 115 ``````Proof. constructor. Qed. `````` 116 ``````Lemma subseteq_up_set S T : S ⊆ up_set S T. `````` Robbert Krebbers committed Nov 11, 2015 117 ``````Proof. intros s ?; apply elem_of_bind; eauto using elem_of_up. Qed. `````` Ralf Jung committed Feb 15, 2016 118 119 ``````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 Mar 23, 2016 120 ``````Lemma closed_up_set S T : (∀ s, s ∈ S → tok s ⊥ T) → closed (up_set S T) T. `````` Robbert Krebbers committed Nov 11, 2015 121 ``````Proof. `````` 122 `````` intros HS; unfold up_set; split. `````` Robbert Krebbers committed Feb 17, 2016 123 `````` - intros s; rewrite !elem_of_bind; intros (s'&Hstep&Hs'). `````` 124 `````` specialize (HS s' Hs'); clear Hs' S. `````` Ralf Jung committed Feb 16, 2016 125 `````` induction Hstep as [s|s1 s2 s3 [T1 T2 ? Hstep] ? IH]; first done. `````` Robbert Krebbers committed Nov 11, 2015 126 `````` inversion_clear Hstep; apply IH; clear IH; auto with sts. `````` Robbert Krebbers committed Feb 24, 2016 127 `````` - intros s1 s2; rewrite /up; set_unfold; intros (s&?&?) ?; exists s. `````` Robbert Krebbers committed Nov 11, 2015 128 129 `````` split; [eapply rtc_r|]; eauto. Qed. `````` Robbert Krebbers committed Mar 23, 2016 130 ``````Lemma closed_up s T : tok s ⊥ T → closed (up s T) T. `````` Robbert Krebbers committed Nov 11, 2015 131 ``````Proof. `````` 132 `````` intros; rewrite -(collection_bind_singleton (λ s, up s T) s). `````` Robbert Krebbers committed Feb 17, 2016 133 `````` apply closed_up_set; set_solver. `````` Robbert Krebbers committed Nov 11, 2015 134 ``````Qed. `````` 135 136 ``````Lemma closed_up_set_empty S : closed (up_set S ∅) ∅. Proof. eauto using closed_up_set with sts. Qed. `````` 137 ``````Lemma closed_up_empty s : closed (up s ∅) ∅. `````` Robbert Krebbers committed Nov 11, 2015 138 ``````Proof. eauto using closed_up with sts. Qed. `````` 139 ``````Lemma up_set_empty S T : up_set S T ≡ ∅ → S ≡ ∅. `````` Robbert Krebbers committed Feb 22, 2016 140 141 ``````Proof. move:(subseteq_up_set S T). set_solver. Qed. Lemma up_set_non_empty S T : S ≢ ∅ → up_set S T ≢ ∅. `````` 142 ``````Proof. by move=>? /up_set_empty. Qed. `````` Robbert Krebbers committed Feb 22, 2016 143 144 ``````Lemma up_non_empty s T : up s T ≢ ∅. Proof. eapply non_empty_inhabited, elem_of_up. Qed. `````` 145 ``````Lemma up_closed S T : closed S T → up_set S T ≡ S. `````` Robbert Krebbers committed Nov 11, 2015 146 ``````Proof. `````` Robbert Krebbers committed Dec 08, 2015 147 `````` intros; split; auto using subseteq_up_set; intros s. `````` Robbert Krebbers committed Nov 11, 2015 148 149 150 `````` unfold up_set; rewrite elem_of_bind; intros (s'&Hstep&?). induction Hstep; eauto using closed_step. Qed. `````` Robbert Krebbers committed Feb 22, 2016 151 152 153 154 155 156 157 158 ``````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 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 `````` 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}. 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 184 185 ``````Global Existing Instance sts_equiv. Global Instance sts_valid : Valid (car sts) := λ x, `````` 186 `````` match x with `````` Robbert Krebbers committed Mar 23, 2016 187 `````` | auth s T => tok s ⊥ T `````` Robbert Krebbers committed Feb 22, 2016 188 189 `````` | frag S' T => closed S' T ∧ S' ≢ ∅ end. `````` Ralf Jung committed Mar 08, 2016 190 ``````Global Instance sts_core : Core (car sts) := λ x, `````` Robbert Krebbers committed Feb 16, 2016 191 192 193 194 195 196 `````` 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 : `````` Robbert Krebbers committed Mar 23, 2016 197 198 199 `````` 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 200 201 ``````Global Existing Instance sts_disjoint. Global Instance sts_op : Op (car sts) := λ x1 x2, `````` Robbert Krebbers committed Feb 16, 2016 202 203 204 205 206 207 208 `````` 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. `````` Robbert Krebbers committed Feb 22, 2016 209 210 211 212 ``````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 Mar 23, 2016 213 214 ``````Hint Extern 50 (_ ⊥ _) => set_solver : sts. `````` Ralf Jung committed Feb 21, 2016 215 ``````Global Instance sts_equivalence: Equivalence ((≡) : relation (car sts)). `````` Robbert Krebbers committed Feb 16, 2016 216 217 ``````Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 218 219 `````` - by intros []; constructor. - by destruct 1; constructor. `````` Ralf Jung committed Feb 20, 2016 220 `````` - destruct 1; inversion_clear 1; constructor; etrans; eauto. `````` Robbert Krebbers committed Feb 16, 2016 221 222 ``````Qed. Global Instance sts_dra : DRA (car sts). `````` Robbert Krebbers committed Nov 11, 2015 223 224 ``````Proof. split. `````` Robbert Krebbers committed Feb 17, 2016 225 226 227 228 229 `````` - 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. `````` Robbert Krebbers committed Mar 03, 2016 230 `````` - destruct 3; simpl in *; destruct_and?; eauto using closed_op; `````` Robbert Krebbers committed Feb 22, 2016 231 `````` match goal with H : closed _ _ |- _ => destruct H end; set_solver. `````` Robbert Krebbers committed Mar 03, 2016 232 `````` - intros []; simpl; intros; destruct_and?; split; `````` Robbert Krebbers committed Feb 22, 2016 233 `````` eauto using closed_up, up_non_empty, closed_up_set, up_set_empty with sts. `````` Robbert Krebbers committed Feb 17, 2016 234 235 236 237 238 239 240 241 `````` - 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 242 `````` + rewrite (up_closed (up _ _)); auto using closed_up with sts. `````` Robbert Krebbers committed Feb 24, 2016 243 `````` + rewrite (up_closed (up_set _ _)); eauto using closed_up_set with sts. `````` Robbert Krebbers committed Mar 11, 2016 244 245 246 `````` - intros x y. exists (core (x ⋅ y))=> ?? Hxy; split_and?. + destruct Hxy; constructor; unfold up_set; set_solver. + destruct Hxy; simpl; split_and?; `````` Robbert Krebbers committed Feb 22, 2016 247 248 `````` auto using closed_up_set_empty, closed_up_empty, up_non_empty; []. apply up_set_non_empty. set_solver. `````` Robbert Krebbers committed Mar 11, 2016 249 `````` + destruct Hxy; constructor; `````` Robbert Krebbers committed Dec 08, 2015 250 `````` repeat match goal with `````` 251 252 253 254 `````` | |- 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 255 `````` end; auto with sts. `````` Robbert Krebbers committed Feb 22, 2016 256 ``````Qed. `````` Robbert Krebbers committed Mar 01, 2016 257 ``````Canonical Structure R : cmraT := validityR (car sts). `````` Robbert Krebbers committed Feb 16, 2016 258 259 260 261 ``````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 262 ``````Notation stsR := (@sts_dra.R). `````` Robbert Krebbers committed Feb 16, 2016 263 264 265 `````` Section sts_definitions. Context {sts : stsT}. `````` Robbert Krebbers committed Mar 01, 2016 266 `````` Definition sts_auth (s : sts.state sts) (T : sts.tokens sts) : stsR sts := `````` Robbert Krebbers committed Feb 16, 2016 267 `````` to_validity (sts_dra.auth s T). `````` Robbert Krebbers committed Mar 01, 2016 268 `````` Definition sts_frag (S : sts.states sts) (T : sts.tokens sts) : stsR sts := `````` Robbert Krebbers committed Feb 16, 2016 269 `````` to_validity (sts_dra.frag S T). `````` Robbert Krebbers committed Mar 01, 2016 270 `````` Definition sts_frag_up (s : sts.state sts) (T : sts.tokens sts) : stsR sts := `````` Robbert Krebbers committed Feb 16, 2016 271 272 273 274 275 276 277 278 279 280 281 282 283 `````` 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 284 ``````Global Instance sts_cmra_discrete : CMRADiscrete (stsR sts). `````` Robbert Krebbers committed Feb 24, 2016 285 286 ``````Proof. apply validity_cmra_discrete. Qed. `````` Robbert Krebbers committed Feb 16, 2016 287 288 289 290 291 292 293 ``````(** 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 294 ``````Proof. `````` Robbert Krebbers committed Feb 16, 2016 295 296 `````` intros S1 S2 ? T1 T2 HT; rewrite /sts_auth. by eapply to_validity_proper; try apply _; constructor. `````` Robbert Krebbers committed Nov 11, 2015 297 ``````Qed. `````` Robbert Krebbers committed Feb 16, 2016 298 299 ``````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 300 `````` `````` Robbert Krebbers committed Feb 16, 2016 301 ``````(** Validity *) `````` Robbert Krebbers committed Mar 23, 2016 302 ``````Lemma sts_auth_valid s T : ✓ sts_auth s T ↔ tok s ⊥ T. `````` Robbert Krebbers committed Feb 24, 2016 303 ``````Proof. done. Qed. `````` 304 ``````Lemma sts_frag_valid S T : ✓ sts_frag S T ↔ closed S T ∧ S ≢ ∅. `````` Robbert Krebbers committed Feb 24, 2016 305 ``````Proof. done. Qed. `````` Robbert Krebbers committed Mar 23, 2016 306 ``````Lemma sts_frag_up_valid s T : tok s ⊥ T → ✓ sts_frag_up s T. `````` Robbert Krebbers committed Feb 22, 2016 307 ``````Proof. intros. by apply sts_frag_valid; auto using closed_up, up_non_empty. Qed. `````` Robbert Krebbers committed Nov 11, 2015 308 `````` `````` Robbert Krebbers committed Feb 16, 2016 309 310 ``````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 311 ``````Proof. by intros (?&?&Hdisj); inversion Hdisj. Qed. `````` Ralf Jung committed Feb 15, 2016 312 `````` `````` Robbert Krebbers committed Feb 16, 2016 313 314 315 316 ``````(** 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 317 `````` intros; split; [split|constructor; set_solver]; simpl. `````` 318 `````` - intros (?&?&?); by apply closed_disjoint with S. `````` Robbert Krebbers committed Feb 24, 2016 319 `````` - intros; split_and?; last constructor; set_solver. `````` Robbert Krebbers committed Feb 16, 2016 320 321 ``````Qed. Lemma sts_op_auth_frag_up s T : `````` Ralf Jung committed Feb 20, 2016 322 323 324 `````` 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 325 `````` - intros (?&[??]&?). by apply closed_disjoint with (up s T), elem_of_up. `````` Ralf Jung committed Feb 20, 2016 326 327 328 `````` - intros; split_and?. + set_solver+. + by apply closed_up. `````` Robbert Krebbers committed Feb 22, 2016 329 `````` + apply up_non_empty. `````` Ralf Jung committed Feb 20, 2016 330 331 `````` + constructor; last set_solver. apply elem_of_up. Qed. `````` Robbert Krebbers committed Feb 16, 2016 332 `````` `````` Ralf Jung committed Feb 17, 2016 333 ``````Lemma sts_op_frag S1 S2 T1 T2 : `````` Robbert Krebbers committed Mar 23, 2016 334 `````` T1 ⊥ T2 → sts.closed S1 T1 → sts.closed S2 T2 → `````` Ralf Jung committed Feb 17, 2016 335 336 `````` sts_frag (S1 ∩ S2) (T1 ∪ T2) ≡ sts_frag S1 T1 ⋅ sts_frag S2 T2. Proof. `````` Ralf Jung committed Feb 17, 2016 337 338 `````` intros HT HS1 HS2. rewrite /sts_frag. (* FIXME why does rewrite not work?? *) `````` 339 340 341 `````` etrans; last eapply to_validity_op; first done; []. move=>/=[??]. split_and!; [auto; set_solver..|]. constructor; done. `````` Ralf Jung committed Feb 17, 2016 342 343 ``````Qed. `````` Robbert Krebbers committed Feb 16, 2016 344 345 ``````(** Frame preserving updates *) Lemma sts_update_auth s1 s2 T1 T2 : `````` Ralf Jung committed Feb 20, 2016 346 `````` steps (s1,T1) (s2,T2) → sts_auth s1 T1 ~~> sts_auth s2 T2. `````` Robbert Krebbers committed Nov 11, 2015 347 ``````Proof. `````` Robbert Krebbers committed Feb 22, 2016 348 `````` intros ?; apply validity_update. `````` Robbert Krebbers committed Mar 03, 2016 349 `````` inversion 3 as [|? S ? Tf|]; simplify_eq/=; destruct_and?. `````` Ralf Jung committed Feb 20, 2016 350 `````` destruct (steps_closed s1 s2 T1 T2 S Tf) as (?&?&?); auto; []. `````` Robbert Krebbers committed Nov 16, 2015 351 `````` repeat (done || constructor). `````` Robbert Krebbers committed Nov 11, 2015 352 ``````Qed. `````` Ralf Jung committed Feb 15, 2016 353 `````` `````` 354 355 ``````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 356 ``````Proof. `````` 357 `````` rewrite /sts_frag=> ? HS HT. apply validity_update. `````` Robbert Krebbers committed Feb 17, 2016 358 `````` inversion 3 as [|? S ? Tf|]; simplify_eq/=. `````` 359 360 `````` - 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 361 362 ``````Qed. `````` 363 364 ``````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 365 ``````Proof. `````` 366 367 `````` intros ? ? HT; apply sts_update_frag; [intros; eauto using closed_steps..]. rewrite <-HT. eapply up_subseteq; done. `````` Robbert Krebbers committed Feb 16, 2016 368 369 ``````Qed. `````` Ralf Jung committed Feb 21, 2016 370 371 372 373 374 ``````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 375 376 377 `````` + 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 378 379 ``````Qed. `````` Robbert Krebbers committed Feb 16, 2016 380 ``````(** Inclusion *) `````` Ralf Jung committed Feb 21, 2016 381 382 383 ``````(* 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. *) `````` Ralf Jung committed Mar 11, 2016 384 ``````(* TODO: These have to be proven again. *) `````` Robbert Krebbers committed Mar 11, 2016 385 ``````(* `````` Robbert Krebbers committed Feb 16, 2016 386 ``````Lemma sts_frag_included S1 S2 T1 T2 : `````` Ralf Jung committed Feb 21, 2016 387 388 `````` closed S2 T2 → S2 ≢ ∅ → (sts_frag S1 T1 ≼ sts_frag S2 T2) ↔ `````` Robbert Krebbers committed Mar 23, 2016 389 `````` (closed S1 T1 ∧ S1 ≢ ∅ ∧ ∃ Tf, T2 ≡ T1 ∪ Tf ∧ T1 ⊥ Tf ∧ `````` Ralf Jung committed Feb 21, 2016 390 391 `````` S2 ≡ S1 ∩ up_set S2 Tf). Proof. `````` Robbert Krebbers committed Mar 11, 2016 392 `````` intros ??; split. `````` Robbert Krebbers committed Mar 11, 2016 393 `````` - intros [[???] ?]. `````` Ralf Jung committed Feb 21, 2016 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 `````` 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 411 `````` * by apply up_set_non_empty. `````` Ralf Jung committed Feb 21, 2016 412 `````` + constructor; last done. by rewrite -HS. `````` 413 414 ``````Qed. `````` Robbert Krebbers committed Feb 16, 2016 415 ``````Lemma sts_frag_included' S1 S2 T : `````` Ralf Jung committed Feb 21, 2016 416 `````` closed S2 T → closed S1 T → S2 ≢ ∅ → S1 ≢ ∅ → S2 ≡ S1 ∩ up_set S2 ∅ → `````` Robbert Krebbers committed Feb 16, 2016 417 `````` sts_frag S1 T ≼ sts_frag S2 T. `````` 418 ``````Proof. `````` Robbert Krebbers committed Feb 19, 2016 419 420 `````` intros. apply sts_frag_included; split_and?; auto. exists ∅; split_and?; done || set_solver+. `````` Robbert Krebbers committed Mar 11, 2016 421 ``````Qed. *) `````` Robbert Krebbers committed Feb 01, 2016 422 ``````End stsRA. `````` Ralf Jung committed Mar 07, 2016 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 `````` (** 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.``````