sts.v 18.4 KB
 Robbert Krebbers committed Jul 22, 2016 1 From iris.prelude Require Export set.  Robbert Krebbers committed Mar 10, 2016 2 3 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 :=  Ralf Jung committed Aug 16, 2016 33  (* Probably equivalent definition: (\mathcal{L}(s') ⊥ T) ∧ s \rightarrow s' *)  Robbert Krebbers committed Nov 11, 2015 34  | Frame_step T1 T2 :  Robbert Krebbers committed Mar 23, 2016 35  T1 ⊥ tok s1 ∪ T → step (s1,T1) (s2,T2) → frame_step T s1 s2.  Ralf Jung committed Aug 09, 2016 36 Notation frame_steps T := (rtc (frame_step T)).  Robbert Krebbers committed Feb 16, 2016 37 38 39  (** ** Closure under frame steps *) Record closed (S : states sts) (T : tokens sts) : Prop := Closed {  Robbert Krebbers committed Mar 23, 2016 40  closed_disjoint s : s ∈ S → tok s ⊥ T;  Robbert Krebbers committed Nov 11, 2015 41 42  closed_step s1 s2 : s1 ∈ S → frame_step T s1 s2 → s2 ∈ S }.  Robbert Krebbers committed Feb 16, 2016 43 Definition up (s : state sts) (T : tokens sts) : states sts :=  Ralf Jung committed Aug 09, 2016 44  {[ s' | frame_steps T s s' ]}.  Robbert Krebbers committed Feb 16, 2016 45 Definition up_set (S : states sts) (T : tokens sts) : states sts :=  Robbert Krebbers committed Feb 16, 2016 46  S ≫= λ s, up s T.  Robbert Krebbers committed Nov 11, 2015 47   Robbert Krebbers committed Feb 16, 2016 48 49 (** Tactic setup *) Hint Resolve Step.  50 51 52 53 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 54 Hint Extern 50 (_ ⊥ _) => set_solver : sts.  Robbert Krebbers committed Feb 16, 2016 55 56  (** ** Setoids *)  Ralf Jung committed Feb 17, 2016 57 58 59 Instance framestep_mono : Proper (flip (⊆) ==> (=) ==> (=) ==> impl) frame_step. Proof. intros ?? HT ?? <- ?? <-; destruct 1; econstructor;  Robbert Krebbers committed Feb 17, 2016 60  eauto with sts; set_solver.  Ralf Jung committed Feb 17, 2016 61 Qed.  Robbert Krebbers committed Feb 16, 2016 62 Global Instance framestep_proper : Proper ((≡) ==> (=) ==> (=) ==> iff) frame_step.  Robbert Krebbers committed Jul 22, 2016 63 Proof. move=> ?? /collection_equiv_spec [??]; split; by apply framestep_mono. Qed.  Robbert Krebbers committed Nov 16, 2015 64 Instance closed_proper' : Proper ((≡) ==> (≡) ==> impl) closed.  Robbert Krebbers committed Mar 23, 2016 65 Proof. destruct 3; constructor; intros until 0; setoid_subst; eauto. Qed.  Robbert Krebbers committed Feb 16, 2016 66 Global Instance closed_proper : Proper ((≡) ==> (≡) ==> iff) closed.  Robbert Krebbers committed Nov 16, 2015 67 Proof. by split; apply closed_proper'. Qed.  Robbert Krebbers committed Feb 16, 2016 68 Global Instance up_preserving : Proper ((=) ==> flip (⊆) ==> (⊆)) up.  Robbert Krebbers committed Nov 11, 2015 69 Proof.  70  intros s ? <- T T' HT ; apply elem_of_subseteq.  Robbert Krebbers committed Nov 11, 2015 71  induction 1 as [|s1 s2 s3 [T1 T2]]; [constructor|].  Robbert Krebbers committed Feb 24, 2016 72  eapply elem_of_mkSet, rtc_l; [eapply Frame_step with T1 T2|]; eauto with sts.  Robbert Krebbers committed Nov 11, 2015 73 Qed.  Robbert Krebbers committed Feb 16, 2016 74 Global Instance up_proper : Proper ((=) ==> (≡) ==> (≡)) up.  Robbert Krebbers committed Jul 22, 2016 75 76 77 Proof. by move=> ??? ?? /collection_equiv_spec [??]; split; apply up_preserving. Qed.  Robbert Krebbers committed Feb 16, 2016 78 Global Instance up_set_preserving : Proper ((⊆) ==> flip (⊆) ==> (⊆)) up_set.  Ralf Jung committed Feb 15, 2016 79 80 Proof. intros S1 S2 HS T1 T2 HT. rewrite /up_set.  Ralf Jung committed Feb 25, 2016 81  f_equiv; last done. move =>s1 s2 Hs. simpl in HT. by apply up_preserving.  Ralf Jung committed Feb 15, 2016 82 Qed.  Robbert Krebbers committed Feb 16, 2016 83 Global Instance up_set_proper : Proper ((≡) ==> (≡) ==> (≡)) up_set.  Robbert Krebbers committed Jul 22, 2016 84 85 86 87 Proof. move=> S1 S2 /collection_equiv_spec [??] T1 T2 /collection_equiv_spec [??]; split; by apply up_set_preserving. Qed.  Robbert Krebbers committed Feb 16, 2016 88 89 90  (** ** Properties of closure under frame steps *) Lemma closed_steps S T s1 s2 :  Ralf Jung committed Aug 09, 2016 91  closed S T → s1 ∈ S → frame_steps T s1 s2 → s2 ∈ S.  Robbert Krebbers committed Feb 16, 2016 92 93 Proof. induction 3; eauto using closed_step. Qed. Lemma closed_op T1 T2 S1 S2 :  94  closed S1 T1 → closed S2 T2 → closed (S1 ∩ S2) (T1 ∪ T2).  Robbert Krebbers committed Feb 16, 2016 95 Proof.  96  intros [? Hstep1] [? Hstep2]; split; [set_solver|].  Robbert Krebbers committed Feb 16, 2016 97  intros s3 s4; rewrite !elem_of_intersection; intros [??] [T3 T4 ?]; split.  Robbert Krebbers committed Feb 17, 2016 98 99  - 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 100 101 Qed. Lemma step_closed s1 s2 T1 T2 S Tf :  Robbert Krebbers committed Mar 23, 2016 102 103  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 104 Proof.  105  inversion_clear 1 as [???? HR Hs1 Hs2]; intros [? Hstep]??; split_and?; auto.  Robbert Krebbers committed Feb 17, 2016 106  - eapply Hstep with s1, Frame_step with T1 T2; auto with sts.  Robbert Krebbers committed Feb 17, 2016 107  - set_solver -Hstep Hs1 Hs2.  Robbert Krebbers committed Feb 16, 2016 108 Qed.  Ralf Jung committed Feb 20, 2016 109 Lemma steps_closed s1 s2 T1 T2 S Tf :  Robbert Krebbers committed Mar 23, 2016 110 111  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 112 Proof.  Robbert Krebbers committed Feb 22, 2016 113 114 115 116 117  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 118 Qed.  Robbert Krebbers committed Feb 16, 2016 119 120  (** ** Properties of the closure operators *)  121 Lemma elem_of_up s T : s ∈ up s T.  Robbert Krebbers committed Nov 11, 2015 122 Proof. constructor. Qed.  123 Lemma subseteq_up_set S T : S ⊆ up_set S T.  Robbert Krebbers committed Nov 11, 2015 124 Proof. intros s ?; apply elem_of_bind; eauto using elem_of_up. Qed.  Ralf Jung committed Feb 15, 2016 125 126 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 127 Lemma closed_up_set S T : (∀ s, s ∈ S → tok s ⊥ T) → closed (up_set S T) T.  Robbert Krebbers committed Nov 11, 2015 128 Proof.  129  intros HS; unfold up_set; split.  Robbert Krebbers committed Feb 17, 2016 130  - intros s; rewrite !elem_of_bind; intros (s'&Hstep&Hs').  131  specialize (HS s' Hs'); clear Hs' S.  Ralf Jung committed Feb 16, 2016 132  induction Hstep as [s|s1 s2 s3 [T1 T2 ? Hstep] ? IH]; first done.  Robbert Krebbers committed Nov 11, 2015 133  inversion_clear Hstep; apply IH; clear IH; auto with sts.  Robbert Krebbers committed Feb 24, 2016 134  - intros s1 s2; rewrite /up; set_unfold; intros (s&?&?) ?; exists s.  Robbert Krebbers committed Nov 11, 2015 135 136  split; [eapply rtc_r|]; eauto. Qed.  Robbert Krebbers committed Mar 23, 2016 137 Lemma closed_up s T : tok s ⊥ T → closed (up s T) T.  Robbert Krebbers committed Nov 11, 2015 138 Proof.  139  intros; rewrite -(collection_bind_singleton (λ s, up s T) s).  Robbert Krebbers committed Feb 17, 2016 140  apply closed_up_set; set_solver.  Robbert Krebbers committed Nov 11, 2015 141 Qed.  142 143 Lemma closed_up_set_empty S : closed (up_set S ∅) ∅. Proof. eauto using closed_up_set with sts. Qed.  144 Lemma closed_up_empty s : closed (up s ∅) ∅.  Robbert Krebbers committed Nov 11, 2015 145 Proof. eauto using closed_up with sts. Qed.  146 Lemma up_set_empty S T : up_set S T ≡ ∅ → S ≡ ∅.  Robbert Krebbers committed Feb 22, 2016 147 148 Proof. move:(subseteq_up_set S T). set_solver. Qed. Lemma up_set_non_empty S T : S ≢ ∅ → up_set S T ≢ ∅.  149 Proof. by move=>? /up_set_empty. Qed.  Robbert Krebbers committed Feb 22, 2016 150 151 Lemma up_non_empty s T : up s T ≢ ∅. Proof. eapply non_empty_inhabited, elem_of_up. Qed.  152 Lemma up_closed S T : closed S T → up_set S T ≡ S.  Robbert Krebbers committed Nov 11, 2015 153 Proof.  Robbert Krebbers committed Jul 22, 2016 154 155  intros ?; apply collection_equiv_spec; split; auto using subseteq_up_set. intros s; unfold up_set; rewrite elem_of_bind; intros (s'&Hstep&?).  Robbert Krebbers committed Nov 11, 2015 156 157  induction Hstep; eauto using closed_step. Qed.  Robbert Krebbers committed Feb 22, 2016 158 159 160 161 162 163 164 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).  Ralf Jung committed Aug 09, 2016 165 Notation frame_steps T := (rtc (frame_step T)).  Robbert Krebbers committed Feb 16, 2016 166 167 168 169 170 171 172 173  (* 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 {_} _ _.  Robbert Krebbers committed May 25, 2016 174 End sts.  Robbert Krebbers committed Feb 16, 2016 175   Robbert Krebbers committed May 25, 2016 176 177 178 179 Notation stsT := sts.stsT. Notation STS := sts.STS. (** * STSs form a disjoint RA *)  Robbert Krebbers committed Feb 16, 2016 180 Section sts_dra.  Robbert Krebbers committed May 25, 2016 181 182 Context (sts : stsT). Import sts.  Robbert Krebbers committed Feb 16, 2016 183 184 185 186 187 188 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.  Robbert Krebbers committed May 25, 2016 189 190 Existing Instance sts_equiv. Instance sts_valid : Valid (car sts) := λ x,  191  match x with  Robbert Krebbers committed Mar 23, 2016 192  | auth s T => tok s ⊥ T  Robbert Krebbers committed Feb 22, 2016 193 194  | frag S' T => closed S' T ∧ S' ≢ ∅ end.  Robbert Krebbers committed May 25, 2016 195 Instance sts_core : Core (car sts) := λ x,  Robbert Krebbers committed Feb 16, 2016 196 197 198 199 200 201  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 202 203 204  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.  Robbert Krebbers committed May 25, 2016 205 206 Existing Instance sts_disjoint. Instance sts_op : Op (car sts) := λ x1 x2,  Robbert Krebbers committed Feb 16, 2016 207 208 209 210 211 212 213  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 214 215 216 217 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 218 219 Hint Extern 50 (_ ⊥ _) => set_solver : sts.  Robbert Krebbers committed May 25, 2016 220 221 222 223 224 225 Global Instance auth_proper s : Proper ((≡) ==> (≡)) (@auth sts s). Proof. by constructor. Qed. Global Instance frag_proper : Proper ((≡) ==> (≡) ==> (≡)) (@frag sts). Proof. by constructor. Qed. Instance sts_equivalence: Equivalence ((≡) : relation (car sts)).  Robbert Krebbers committed Feb 16, 2016 226 227 Proof. split.  Robbert Krebbers committed Feb 17, 2016 228 229  - by intros []; constructor. - by destruct 1; constructor.  Ralf Jung committed Feb 20, 2016 230  - destruct 1; inversion_clear 1; constructor; etrans; eauto.  Robbert Krebbers committed Feb 16, 2016 231 Qed.  Robbert Krebbers committed May 25, 2016 232 Lemma sts_dra_mixin : DRAMixin (car sts).  Robbert Krebbers committed Nov 11, 2015 233 234 Proof. split.  Robbert Krebbers committed Feb 17, 2016 235 236 237 238 239  - 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 240  - destruct 3; simpl in *; destruct_and?; eauto using closed_op;  Robbert Krebbers committed Feb 22, 2016 241  match goal with H : closed _ _ |- _ => destruct H end; set_solver.  Robbert Krebbers committed Mar 03, 2016 242  - intros []; simpl; intros; destruct_and?; split;  Robbert Krebbers committed Feb 22, 2016 243  eauto using closed_up, up_non_empty, closed_up_set, up_set_empty with sts.  Robbert Krebbers committed Feb 17, 2016 244 245 246 247 248 249 250 251  - 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 252  + rewrite (up_closed (up _ _)); auto using closed_up with sts.  Robbert Krebbers committed Feb 24, 2016 253  + rewrite (up_closed (up_set _ _)); eauto using closed_up_set with sts.  Robbert Krebbers committed Mar 11, 2016 254 255 256  - 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 257 258  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 259  + destruct Hxy; constructor;  Robbert Krebbers committed Dec 08, 2015 260  repeat match goal with  261 262 263 264  | |- 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 265  end; auto with sts.  Robbert Krebbers committed Feb 22, 2016 266 Qed.  Robbert Krebbers committed May 25, 2016 267 268 Canonical Structure stsDR : draT := DRAT (car sts) sts_dra_mixin. End sts_dra.  Robbert Krebbers committed Feb 16, 2016 269 270 271  (** * The STS Resource Algebra *) (** Finally, the general theory of STS that should be used by users *)  Robbert Krebbers committed May 25, 2016 272 273 Notation stsC sts := (validityC (stsDR sts)). Notation stsR sts := (validityR (stsDR sts)).  Robbert Krebbers committed Feb 16, 2016 274 275 276  Section sts_definitions. Context {sts : stsT}.  Robbert Krebbers committed Mar 01, 2016 277  Definition sts_auth (s : sts.state sts) (T : sts.tokens sts) : stsR sts :=  Robbert Krebbers committed May 25, 2016 278  to_validity (sts.auth s T).  Robbert Krebbers committed Mar 01, 2016 279  Definition sts_frag (S : sts.states sts) (T : sts.tokens sts) : stsR sts :=  Robbert Krebbers committed May 25, 2016 280  to_validity (sts.frag S T).  Robbert Krebbers committed Mar 01, 2016 281  Definition sts_frag_up (s : sts.state sts) (T : sts.tokens sts) : stsR sts :=  Robbert Krebbers committed Feb 16, 2016 282 283 284 285 286 287 288 289 290 291 292 293  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 May 25, 2016 294 Arguments dra_valid _ !_/.  Robbert Krebbers committed Feb 24, 2016 295   Robbert Krebbers committed Feb 16, 2016 296 297 (** Setoids *) Global Instance sts_auth_proper s : Proper ((≡) ==> (≡)) (sts_auth s).  Robbert Krebbers committed May 25, 2016 298 Proof. solve_proper. Qed.  Robbert Krebbers committed Feb 16, 2016 299 Global Instance sts_frag_proper : Proper ((≡) ==> (≡) ==> (≡)) (@sts_frag sts).  Robbert Krebbers committed May 25, 2016 300 Proof. solve_proper. Qed.  Robbert Krebbers committed Feb 16, 2016 301 Global Instance sts_frag_up_proper s : Proper ((≡) ==> (≡)) (sts_frag_up s).  Robbert Krebbers committed May 25, 2016 302 Proof. solve_proper. Qed.  Robbert Krebbers committed Nov 11, 2015 303   Robbert Krebbers committed Feb 16, 2016 304 (** Validity *)  Robbert Krebbers committed Mar 23, 2016 305 Lemma sts_auth_valid s T : ✓ sts_auth s T ↔ tok s ⊥ T.  Robbert Krebbers committed Feb 24, 2016 306 Proof. done. Qed.  307 Lemma sts_frag_valid S T : ✓ sts_frag S T ↔ closed S T ∧ S ≢ ∅.  Robbert Krebbers committed Feb 24, 2016 308 Proof. done. Qed.  Robbert Krebbers committed Mar 23, 2016 309 Lemma sts_frag_up_valid s T : tok s ⊥ T → ✓ sts_frag_up s T.  Robbert Krebbers committed Feb 22, 2016 310 Proof. intros. by apply sts_frag_valid; auto using closed_up, up_non_empty. Qed.  Robbert Krebbers committed Nov 11, 2015 311   Robbert Krebbers committed Feb 16, 2016 312 313 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 314 Proof. by intros (?&?&Hdisj); inversion Hdisj. Qed.  Ralf Jung committed Feb 15, 2016 315   Robbert Krebbers committed Feb 16, 2016 316 317 318 319 (** 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 320  intros; split; [split|constructor; set_solver]; simpl.  321  - intros (?&?&?); by apply closed_disjoint with S.  Robbert Krebbers committed Feb 24, 2016 322  - intros; split_and?; last constructor; set_solver.  Robbert Krebbers committed Feb 16, 2016 323 324 Qed. Lemma sts_op_auth_frag_up s T :  Ralf Jung committed Feb 20, 2016 325 326 327  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 328  - intros (?&[??]&?). by apply closed_disjoint with (up s T), elem_of_up.  Ralf Jung committed Feb 20, 2016 329 330 331  - intros; split_and?. + set_solver+. + by apply closed_up.  Robbert Krebbers committed Feb 22, 2016 332  + apply up_non_empty.  Ralf Jung committed Feb 20, 2016 333 334  + constructor; last set_solver. apply elem_of_up. Qed.  Robbert Krebbers committed Feb 16, 2016 335   Ralf Jung committed Feb 17, 2016 336 Lemma sts_op_frag S1 S2 T1 T2 :  Robbert Krebbers committed Mar 23, 2016 337  T1 ⊥ T2 → sts.closed S1 T1 → sts.closed S2 T2 →  Ralf Jung committed Feb 17, 2016 338 339  sts_frag (S1 ∩ S2) (T1 ∪ T2) ≡ sts_frag S1 T1 ⋅ sts_frag S2 T2. Proof.  Robbert Krebbers committed May 25, 2016 340 341  intros HT HS1 HS2. rewrite /sts_frag -to_validity_op //. move=>/=[??]. split_and!; [auto; set_solver..|by constructor].  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.  Robbert Krebbers committed May 25, 2016 370 371 Lemma sts_up_set_intersection S1 Sf Tf : closed Sf Tf → S1 ∩ Sf ≡ S1 ∩ up_set (S1 ∩ Sf) Tf.  Ralf Jung committed Feb 21, 2016 372 373 Proof. intros Hclf. apply (anti_symm (⊆)).  Robbert Krebbers committed Jul 22, 2016 374 375  - 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.  Robbert Krebbers committed Feb 24, 2016 376  eapply closed_steps, Hsup; first done. set_solver +Hs'.  Ralf Jung committed Feb 21, 2016 377 378 Qed.  Janno committed Oct 10, 2016 379 380 381 382 383 Global Instance sts_frag_peristent S : Persistent (sts_frag S ∅). Proof. constructor; split=> //= [[??]]. by rewrite /dra.dra_core /= sts.up_closed. Qed.  Robbert Krebbers committed Feb 16, 2016 384 (** Inclusion *)  Ralf Jung committed Feb 21, 2016 385 386 387 (* 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 388 (* TODO: These have to be proven again. *)  Robbert Krebbers committed Mar 11, 2016 389 (*  Robbert Krebbers committed Feb 16, 2016 390 Lemma sts_frag_included S1 S2 T1 T2 :  Ralf Jung committed Feb 21, 2016 391 392  closed S2 T2 → S2 ≢ ∅ → (sts_frag S1 T1 ≼ sts_frag S2 T2) ↔  Robbert Krebbers committed Mar 23, 2016 393  (closed S1 T1 ∧ S1 ≢ ∅ ∧ ∃ Tf, T2 ≡ T1 ∪ Tf ∧ T1 ⊥ Tf ∧  Ralf Jung committed Feb 21, 2016 394 395  S2 ≡ S1 ∩ up_set S2 Tf). Proof.  Robbert Krebbers committed Mar 11, 2016 396  intros ??; split.  Robbert Krebbers committed Mar 11, 2016 397  - intros [[???] ?].  Ralf Jung committed Feb 21, 2016 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414  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 415  * by apply up_set_non_empty.  Ralf Jung committed Feb 21, 2016 416  + constructor; last done. by rewrite -HS.  417 418 Qed.  Robbert Krebbers committed Feb 16, 2016 419 Lemma sts_frag_included' S1 S2 T :  Ralf Jung committed Feb 21, 2016 420  closed S2 T → closed S1 T → S2 ≢ ∅ → S1 ≢ ∅ → S2 ≡ S1 ∩ up_set S2 ∅ →  Robbert Krebbers committed Feb 16, 2016 421  sts_frag S1 T ≼ sts_frag S2 T.  422 Proof.  Robbert Krebbers committed Feb 19, 2016 423 424  intros. apply sts_frag_included; split_and?; auto. exists ∅; split_and?; done || set_solver+.  Robbert Krebbers committed Mar 11, 2016 425 Qed. *)  Robbert Krebbers committed Feb 01, 2016 426 End stsRA.  Ralf Jung committed Mar 07, 2016 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500  (** 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.