sts.v 9.92 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 6 7 ``````Local Arguments valid _ _ !_ /. Local Arguments op _ _ !_ !_ /. Local Arguments unit _ _ !_ /. `````` Robbert Krebbers committed Feb 01, 2016 8 9 10 ``````Inductive sts {A B} (R : relation A) (tok : A → set B) := | auth : A → set B → sts R tok | frag : set A → set B → sts R tok. `````` Robbert Krebbers committed Nov 16, 2015 11 12 ``````Arguments auth {_ _ _ _} _ _. Arguments frag {_ _ _ _} _ _. `````` Robbert Krebbers committed Nov 11, 2015 13 `````` `````` Robbert Krebbers committed Feb 01, 2016 14 ``````Module sts. `````` Robbert Krebbers committed Nov 11, 2015 15 ``````Section sts_core. `````` Robbert Krebbers committed Nov 20, 2015 16 17 ``````Context {A B : Type} (R : relation A) (tok : A → set B). Infix "≼" := dra_included. `````` Robbert Krebbers committed Nov 11, 2015 18 `````` `````` Robbert Krebbers committed Feb 01, 2016 19 ``````Inductive sts_equiv : Equiv (sts R tok) := `````` Robbert Krebbers committed Nov 16, 2015 20 21 `````` | 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 Nov 11, 2015 22 ``````Global Existing Instance sts_equiv. `````` Robbert Krebbers committed Nov 20, 2015 23 ``````Inductive step : relation (A * set B) := `````` Robbert Krebbers committed Nov 11, 2015 24 `````` | Step s1 s2 T1 T2 : `````` Robbert Krebbers committed Nov 16, 2015 25 `````` R s1 s2 → tok s1 ∩ T1 ≡ ∅ → tok s2 ∩ T2 ≡ ∅ → tok s1 ∪ T1 ≡ tok s2 ∪ T2 → `````` Robbert Krebbers committed Nov 11, 2015 26 27 `````` step (s1,T1) (s2,T2). Hint Resolve Step. `````` Robbert Krebbers committed Nov 20, 2015 28 ``````Inductive frame_step (T : set B) (s1 s2 : A) : Prop := `````` Robbert Krebbers committed Nov 11, 2015 29 `````` | Frame_step T1 T2 : `````` Robbert Krebbers committed Nov 16, 2015 30 `````` T1 ∩ (tok s1 ∪ T) ≡ ∅ → step (s1,T1) (s2,T2) → frame_step T s1 s2. `````` Robbert Krebbers committed Nov 11, 2015 31 ``````Hint Resolve Frame_step. `````` 32 ``````Record closed (S : set A) (T : set B) : Prop := Closed { `````` Robbert Krebbers committed Dec 08, 2015 33 `````` closed_ne : S ≢ ∅; `````` Robbert Krebbers committed Nov 16, 2015 34 `````` closed_disjoint s : s ∈ S → tok s ∩ T ≡ ∅; `````` Robbert Krebbers committed Nov 11, 2015 35 36 37 `````` closed_step s1 s2 : s1 ∈ S → frame_step T s1 s2 → s2 ∈ S }. Lemma closed_steps S T s1 s2 : `````` 38 `````` closed S T → s1 ∈ S → rtc (frame_step T) s1 s2 → s2 ∈ S. `````` Robbert Krebbers committed Nov 11, 2015 39 ``````Proof. induction 3; eauto using closed_step. Qed. `````` Robbert Krebbers committed Feb 01, 2016 40 ``````Global Instance sts_valid : Valid (sts R tok) := λ x, `````` 41 42 43 `````` match x with auth s T => tok s ∩ T ≡ ∅ | frag S' T => closed S' T end. Definition up (s : A) (T : set B) : set A := mkSet (rtc (frame_step T) s). Definition up_set (S : set A) (T : set B) : set A := S ≫= λ s, up s T. `````` Robbert Krebbers committed Feb 01, 2016 44 ``````Global Instance sts_unit : Unit (sts R tok) := λ x, `````` Robbert Krebbers committed Nov 11, 2015 45 `````` match x with `````` 46 `````` | frag S' _ => frag (up_set S' ∅ ) ∅ | auth s _ => frag (up s ∅) ∅ `````` Robbert Krebbers committed Nov 11, 2015 47 `````` end. `````` Robbert Krebbers committed Feb 01, 2016 48 ``````Inductive sts_disjoint : Disjoint (sts R tok) := `````` Robbert Krebbers committed Dec 08, 2015 49 50 `````` | frag_frag_disjoint S1 S2 T1 T2 : S1 ∩ S2 ≢ ∅ → T1 ∩ T2 ≡ ∅ → frag S1 T1 ⊥ frag S2 T2 `````` Robbert Krebbers committed Nov 16, 2015 51 52 `````` | 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 Nov 11, 2015 53 ``````Global Existing Instance sts_disjoint. `````` Robbert Krebbers committed Feb 01, 2016 54 ``````Global Instance sts_op : Op (sts R tok) := λ x1 x2, `````` Robbert Krebbers committed Nov 11, 2015 55 56 57 58 59 60 `````` 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 01, 2016 61 ``````Global Instance sts_minus : Minus (sts R tok) := λ x1 x2, `````` Robbert Krebbers committed Nov 11, 2015 62 `````` match x1, x2 with `````` 63 `````` | frag S1 T1, frag S2 T2 => frag (up_set S1 (T1 ∖ T2)) (T1 ∖ T2) `````` Robbert Krebbers committed Nov 11, 2015 64 65 `````` | auth s T1, frag _ T2 => auth s (T1 ∖ T2) | frag _ T2, auth s T1 => auth s (T1 ∖ T2) (* never happens *) `````` 66 `````` | auth s T1, auth _ T2 => frag (up s (T1 ∖ T2)) (T1 ∖ T2) `````` Robbert Krebbers committed Nov 11, 2015 67 68 `````` end. `````` Robbert Krebbers committed Jan 16, 2016 69 70 71 72 ``````Hint Extern 10 (equiv (A:=set _) _ _) => solve_elem_of : sts. Hint Extern 10 (¬(equiv (A:=set _) _ _)) => solve_elem_of : sts. Hint Extern 10 (_ ∈ _) => solve_elem_of : sts. Hint Extern 10 (_ ⊆ _) => solve_elem_of : sts. `````` Robbert Krebbers committed Feb 01, 2016 73 ``````Instance: Equivalence ((≡) : relation (sts R tok)). `````` Robbert Krebbers committed Nov 11, 2015 74 75 76 77 78 79 ``````Proof. split. * by intros []; constructor. * by destruct 1; constructor. * destruct 1; inversion_clear 1; constructor; etransitivity; eauto. Qed. `````` Robbert Krebbers committed Nov 16, 2015 80 81 82 ``````Instance framestep_proper : Proper ((≡) ==> (=) ==> (=) ==> impl) frame_step. Proof. intros ?? HT ?? <- ?? <-; destruct 1; econstructor; eauto with sts. Qed. Instance closed_proper' : Proper ((≡) ==> (≡) ==> impl) closed. `````` Robbert Krebbers committed Nov 11, 2015 83 ``````Proof. `````` Robbert Krebbers committed Nov 16, 2015 84 `````` intros ?? HT ?? HS; destruct 1; `````` Robbert Krebbers committed Jan 13, 2016 85 `````` constructor; intros until 0; rewrite -?HS -?HT; eauto. `````` Robbert Krebbers committed Nov 11, 2015 86 ``````Qed. `````` Robbert Krebbers committed Nov 16, 2015 87 88 ``````Instance closed_proper : Proper ((≡) ==> (≡) ==> iff) closed. Proof. by split; apply closed_proper'. Qed. `````` Robbert Krebbers committed Nov 11, 2015 89 ``````Lemma closed_op T1 T2 S1 S2 : `````` 90 91 `````` closed S1 T1 → closed S2 T2 → T1 ∩ T2 ≡ ∅ → S1 ∩ S2 ≢ ∅ → closed (S1 ∩ S2) (T1 ∪ T2). `````` Robbert Krebbers committed Nov 11, 2015 92 ``````Proof. `````` Robbert Krebbers committed Jan 16, 2016 93 `````` intros [_ ? Hstep1] [_ ? Hstep2] ?; split; [done|solve_elem_of|]. `````` Robbert Krebbers committed Dec 08, 2015 94 95 96 `````` intros s3 s4; rewrite !elem_of_intersection; intros [??] [T3 T4 ?]; split. * 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 Nov 11, 2015 97 ``````Qed. `````` 98 ``````Instance up_preserving : Proper ((=) ==> flip (⊆) ==> (⊆)) up. `````` Robbert Krebbers committed Nov 11, 2015 99 ``````Proof. `````` 100 `````` intros s ? <- T T' HT ; apply elem_of_subseteq. `````` Robbert Krebbers committed Nov 11, 2015 101 102 103 `````` induction 1 as [|s1 s2 s3 [T1 T2]]; [constructor|]. eapply rtc_l; [eapply Frame_step with T1 T2|]; eauto with sts. Qed. `````` 104 105 ``````Instance up_proper : Proper ((=) ==> (≡) ==> (≡)) up. Proof. by intros ??? ?? [??]; split; apply up_preserving. Qed. `````` Robbert Krebbers committed Nov 16, 2015 106 ``````Instance up_set_proper : Proper ((≡) ==> (≡) ==> (≡)) up_set. `````` 107 108 109 110 111 ``````Proof. intros S1 S2 HS T1 T2 HT. rewrite /up_set HS. f_equiv=>s1 s2 Hs. by rewrite Hs HT. Qed. Lemma elem_of_up s T : s ∈ up s T. `````` Robbert Krebbers committed Nov 11, 2015 112 ``````Proof. constructor. Qed. `````` 113 ``````Lemma subseteq_up_set S T : S ⊆ up_set S T. `````` Robbert Krebbers committed Nov 11, 2015 114 ``````Proof. intros s ?; apply elem_of_bind; eauto using elem_of_up. Qed. `````` Robbert Krebbers committed Dec 08, 2015 115 ``````Lemma closed_up_set S T : `````` 116 `````` (∀ s, s ∈ S → tok s ∩ T ≡ ∅) → S ≢ ∅ → closed (up_set S T) T. `````` Robbert Krebbers committed Nov 11, 2015 117 ``````Proof. `````` Robbert Krebbers committed Dec 08, 2015 118 `````` intros HS Hne; unfold up_set; split. `````` 119 `````` * assert (∀ s, s ∈ up s T) by eauto using elem_of_up. solve_elem_of. `````` Robbert Krebbers committed Nov 11, 2015 120 `````` * intros s; rewrite !elem_of_bind; intros (s'&Hstep&Hs'). `````` Robbert Krebbers committed Dec 08, 2015 121 `````` specialize (HS s' Hs'); clear Hs' Hne S. `````` Robbert Krebbers committed Nov 11, 2015 122 123 124 125 126 `````` induction Hstep as [s|s1 s2 s3 [T1 T2 ? Hstep] ? IH]; auto. inversion_clear Hstep; apply IH; clear IH; auto with sts. * intros s1 s2; rewrite !elem_of_bind; intros (s&?&?) ?; exists s. split; [eapply rtc_r|]; eauto. Qed. `````` 127 ``````Lemma closed_up_set_empty S : S ≢ ∅ → closed (up_set S ∅) ∅. `````` Robbert Krebbers committed Nov 11, 2015 128 ``````Proof. eauto using closed_up_set with sts. Qed. `````` 129 ``````Lemma closed_up s T : tok s ∩ T ≡ ∅ → closed (up s T) T. `````` Robbert Krebbers committed Nov 11, 2015 130 ``````Proof. `````` 131 `````` intros; rewrite -(collection_bind_singleton (λ s, up s T) s). `````` Robbert Krebbers committed Jan 16, 2016 132 `````` apply closed_up_set; solve_elem_of. `````` Robbert Krebbers committed Nov 11, 2015 133 ``````Qed. `````` 134 ``````Lemma closed_up_empty s : closed (up s ∅) ∅. `````` Robbert Krebbers committed Nov 11, 2015 135 ``````Proof. eauto using closed_up with sts. Qed. `````` 136 ``````Lemma up_closed S T : closed S T → up_set S T ≡ S. `````` Robbert Krebbers committed Nov 11, 2015 137 ``````Proof. `````` Robbert Krebbers committed Dec 08, 2015 138 `````` intros; split; auto using subseteq_up_set; intros s. `````` Robbert Krebbers committed Nov 11, 2015 139 140 141 `````` unfold up_set; rewrite elem_of_bind; intros (s'&Hstep&?). induction Hstep; eauto using closed_step. Qed. `````` Robbert Krebbers committed Feb 01, 2016 142 ``````Global Instance sts_dra : DRA (sts R tok). `````` Robbert Krebbers committed Nov 11, 2015 143 144 145 146 147 148 149 150 ``````Proof. split. * apply _. * by do 2 destruct 1; constructor; setoid_subst. * by destruct 1; constructor; setoid_subst. * by intros ? [|]; destruct 1; inversion_clear 1; constructor; setoid_subst. * by do 2 destruct 1; constructor; setoid_subst. * assert (∀ T T' S s, `````` 151 `````` closed S T → s ∈ S → tok s ∩ T' ≡ ∅ → tok s ∩ (T ∪ T') ≡ ∅). `````` Robbert Krebbers committed Jan 16, 2016 152 `````` { intros S T T' s [??]; solve_elem_of. } `````` Robbert Krebbers committed Nov 11, 2015 153 `````` destruct 3; simpl in *; auto using closed_op with sts. `````` Robbert Krebbers committed Dec 08, 2015 154 `````` * intros []; simpl; eauto using closed_up, closed_up_set, closed_ne with sts. `````` Robbert Krebbers committed Nov 20, 2015 155 156 `````` * intros ???? (z&Hy&?&Hxz); destruct Hxz; inversion Hy;clear Hy; setoid_subst; rewrite ?disjoint_union_difference; auto using closed_up with sts. `````` Robbert Krebbers committed Nov 11, 2015 157 `````` eapply closed_up_set; eauto 2 using closed_disjoint with sts. `````` Robbert Krebbers committed Feb 11, 2016 158 `````` * intros [] [] []; constructor; rewrite ?assoc; auto with sts. `````` Robbert Krebbers committed Nov 11, 2015 159 160 161 162 `````` * 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. `````` Robbert Krebbers committed Dec 08, 2015 163 `````` * intros [|S T]; constructor; auto using elem_of_up with sts. `````` 164 `````` assert (S ⊆ up_set S ∅ ∧ S ≢ ∅) by eauto using subseteq_up_set, closed_ne. `````` Robbert Krebbers committed Jan 16, 2016 165 `````` solve_elem_of. `````` Robbert Krebbers committed Nov 11, 2015 166 `````` * intros [|S T]; constructor; auto with sts. `````` 167 `````` assert (S ⊆ up_set S ∅); auto using subseteq_up_set with sts. `````` Robbert Krebbers committed Nov 11, 2015 168 `````` * intros [s T|S T]; constructor; auto with sts. `````` Robbert Krebbers committed Jan 13, 2016 169 170 171 `````` + rewrite (up_closed (up _ _)); auto using closed_up with sts. + rewrite (up_closed (up_set _ _)); eauto using closed_up_set, closed_ne with sts. `````` Robbert Krebbers committed Dec 08, 2015 172 `````` * intros x y ?? (z&Hy&?&Hxz); exists (unit (x ⋅ y)); split_ands. `````` Robbert Krebbers committed Jan 16, 2016 173 `````` + destruct Hxz;inversion_clear Hy;constructor;unfold up_set; solve_elem_of. `````` Robbert Krebbers committed Dec 08, 2015 174 175 176 177 `````` + destruct Hxz; inversion_clear Hy; simpl; auto using closed_up_set_empty, closed_up_empty with sts. + destruct Hxz; inversion_clear Hy; constructor; repeat match goal with `````` 178 179 180 181 `````` | |- 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 182 183 184 `````` end; auto with sts. * intros x y ?? (z&Hy&_&Hxz); destruct Hxz; inversion_clear Hy; constructor; repeat match goal with `````` 185 186 187 188 `````` | |- 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 189 `````` end; auto with sts. `````` Robbert Krebbers committed Nov 20, 2015 190 191 `````` * intros x y ?? (z&Hy&?&Hxz); destruct Hxz as [S1 S2 T1 T2| |]; inversion Hy; clear Hy; constructor; setoid_subst; `````` Robbert Krebbers committed Jan 13, 2016 192 `````` rewrite ?disjoint_union_difference; auto. `````` Robbert Krebbers committed Dec 08, 2015 193 `````` split; [|apply intersection_greatest; auto using subseteq_up_set with sts]. `````` Robbert Krebbers committed Nov 20, 2015 194 195 196 197 `````` apply intersection_greatest; [auto with sts|]. intros s2; rewrite elem_of_intersection. unfold up_set; rewrite elem_of_bind; intros (?&s1&?&?&?). apply closed_steps with T2 s1; auto with sts. `````` Robbert Krebbers committed Nov 11, 2015 198 199 ``````Qed. Lemma step_closed s1 s2 T1 T2 S Tf : `````` 200 `````` step (s1,T1) (s2,T2) → closed S Tf → s1 ∈ S → T1 ∩ Tf ≡ ∅ → `````` Robbert Krebbers committed Nov 16, 2015 201 `````` s2 ∈ S ∧ T2 ∩ Tf ≡ ∅ ∧ tok s2 ∩ T2 ≡ ∅. `````` Robbert Krebbers committed Nov 11, 2015 202 ``````Proof. `````` Robbert Krebbers committed Dec 08, 2015 203 `````` inversion_clear 1 as [???? HR Hs1 Hs2]; intros [?? Hstep]??; split_ands; auto. `````` Robbert Krebbers committed Nov 11, 2015 204 `````` * eapply Hstep with s1, Frame_step with T1 T2; auto with sts. `````` Robbert Krebbers committed Jan 16, 2016 205 `````` * solve_elem_of -Hstep Hs1 Hs2. `````` Robbert Krebbers committed Nov 11, 2015 206 207 208 209 ``````Qed. End sts_core. End sts. `````` Robbert Krebbers committed Feb 01, 2016 210 ``````Section stsRA. `````` Robbert Krebbers committed Nov 22, 2015 211 ``````Context {A B : Type} (R : relation A) (tok : A → set B). `````` Robbert Krebbers committed Nov 11, 2015 212 `````` `````` Robbert Krebbers committed Feb 01, 2016 213 214 215 ``````Canonical Structure stsRA := validityRA (sts R tok). Definition sts_auth (s : A) (T : set B) : stsRA := to_validity (auth s T). Definition sts_frag (S : set A) (T : set B) : stsRA := to_validity (frag S T). `````` Robbert Krebbers committed Nov 11, 2015 216 ``````Lemma sts_update s1 s2 T1 T2 : `````` Ralf Jung committed Feb 03, 2016 217 `````` sts.step R tok (s1,T1) (s2,T2) → sts_auth s1 T1 ~~> sts_auth s2 T2. `````` Robbert Krebbers committed Nov 11, 2015 218 ``````Proof. `````` Robbert Krebbers committed Nov 22, 2015 219 `````` intros ?; apply validity_update; inversion 3 as [|? S ? Tf|]; subst. `````` Robbert Krebbers committed Nov 11, 2015 220 `````` destruct (sts.step_closed R tok s1 s2 T1 T2 S Tf) as (?&?&?); auto. `````` Robbert Krebbers committed Nov 16, 2015 221 `````` repeat (done || constructor). `````` Robbert Krebbers committed Nov 11, 2015 222 ``````Qed. `````` Robbert Krebbers committed Feb 01, 2016 223 ``End stsRA.``