From prelude Require Export sets. From algebra Require Export cmra. From algebra Require Import dra. Local Arguments valid _ _ !_ /. Local Arguments op _ _ !_ !_ /. Local Arguments unit _ _ !_ /. (** * Definition of STSs *) Module sts. Structure stsT := STS { state : Type; token : Type; prim_step : relation state; tok : state → set token; }. Arguments STS {_ _} _ _. Arguments prim_step {_} _ _. Arguments tok {_} _. Notation states sts := (set (state sts)). Notation tokens sts := (set (token sts)). (** * Theory and definitions *) Section sts. Context {sts : stsT}. (** ** Step relations *) Inductive step : relation (state sts * tokens sts) := | Step s1 s2 T1 T2 : (* TODO: This asks for ⊥ on sets: T1 ⊥ T2 := T1 ∩ T2 ⊆ ∅. *) prim_step s1 s2 → tok s1 ∩ T1 ≡ ∅ → tok s2 ∩ T2 ≡ ∅ → tok s1 ∪ T1 ≡ tok s2 ∪ T2 → step (s1,T1) (s2,T2). Inductive frame_step (T : tokens sts) (s1 s2 : state sts) : Prop := | Frame_step T1 T2 : T1 ∩ (tok s1 ∪ T) ≡ ∅ → step (s1,T1) (s2,T2) → frame_step T s1 s2. (** ** Closure under frame steps *) Record closed (S : states sts) (T : tokens sts) : Prop := Closed { closed_ne : S ≢ ∅; closed_disjoint s : s ∈ S → tok s ∩ T ⊆ ∅; closed_step s1 s2 : s1 ∈ S → frame_step T s1 s2 → s2 ∈ S }. Definition up (s : state sts) (T : tokens sts) : states sts := mkSet (rtc (frame_step T) s). Definition up_set (S : states sts) (T : tokens sts) : states sts := S ≫= λ s, up s T. (** Tactic setup *) Hint Resolve Step. 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. (** ** Setoids *) Instance framestep_mono : Proper (flip (⊆) ==> (=) ==> (=) ==> impl) frame_step. Proof. intros ?? HT ?? <- ?? <-; destruct 1; econstructor; eauto with sts; solve_elem_of. Qed. Global Instance framestep_proper : Proper ((≡) ==> (=) ==> (=) ==> iff) frame_step. Proof. by intros ?? [??] ??????; split; apply framestep_mono. Qed. Instance closed_proper' : Proper ((≡) ==> (≡) ==> impl) closed. Proof. intros ?? HT ?? HS; destruct 1; constructor; intros until 0; rewrite -?HS -?HT; eauto. Qed. Global Instance closed_proper : Proper ((≡) ==> (≡) ==> iff) closed. Proof. by split; apply closed_proper'. Qed. Global Instance up_preserving : Proper ((=) ==> flip (⊆) ==> (⊆)) up. Proof. intros s ? <- T T' HT ; apply elem_of_subseteq. induction 1 as [|s1 s2 s3 [T1 T2]]; [constructor|]. eapply rtc_l; [eapply Frame_step with T1 T2|]; eauto with sts. Qed. Global Instance up_proper : Proper ((=) ==> (≡) ==> (≡)) up. Proof. by intros ??? ?? [??]; split; apply up_preserving. Qed. Global Instance up_set_preserving : Proper ((⊆) ==> flip (⊆) ==> (⊆)) up_set. Proof. intros S1 S2 HS T1 T2 HT. rewrite /up_set. f_equiv; last done. move =>s1 s2 Hs. simpl in HT. by apply up_preserving. Qed. Global Instance up_set_proper : Proper ((≡) ==> (≡) ==> (≡)) up_set. Proof. by intros S1 S2 [??] T1 T2 [??]; split; apply up_set_preserving. Qed. (** ** Properties of closure under frame steps *) Lemma closed_disjoint' S T s : closed S T → s ∈ S → tok s ∩ T ≡ ∅. Proof. intros [_ ? _]; solve_elem_of. Qed. 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 : closed S1 T1 → closed S2 T2 → T1 ∩ T2 ≡ ∅ → S1 ∩ S2 ≢ ∅ → closed (S1 ∩ S2) (T1 ∪ T2). Proof. intros [_ ? Hstep1] [_ ? Hstep2] ?; split; [done|solve_elem_of|]. 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. 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. inversion_clear 1 as [???? HR Hs1 Hs2]; intros [?? Hstep]??; split_ands; auto. - eapply Hstep with s1, Frame_step with T1 T2; auto with sts. - solve_elem_of -Hstep Hs1 Hs2. Qed. (** ** Properties of the closure operators *) Lemma elem_of_up s T : s ∈ up s T. Proof. constructor. Qed. Lemma subseteq_up_set S T : S ⊆ up_set S T. Proof. intros s ?; apply elem_of_bind; eauto using elem_of_up. Qed. Lemma up_up_set s T : up s T ≡ up_set {[ s ]} T. Proof. by rewrite /up_set collection_bind_singleton. Qed. Lemma closed_up_set S T : (∀ s, s ∈ S → tok s ∩ T ⊆ ∅) → S ≢ ∅ → closed (up_set S T) T. Proof. intros HS Hne; unfold up_set; split. - assert (∀ s, s ∈ up s T) by eauto using elem_of_up. solve_elem_of. - intros s; rewrite !elem_of_bind; intros (s'&Hstep&Hs'). specialize (HS s' Hs'); clear Hs' Hne S. induction Hstep as [s|s1 s2 s3 [T1 T2 ? Hstep] ? IH]; first done. 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. Lemma closed_up_set_empty S : S ≢ ∅ → closed (up_set S ∅) ∅. Proof. eauto using closed_up_set with sts. Qed. Lemma closed_up s T : tok s ∩ T ≡ ∅ → closed (up s T) T. Proof. intros; rewrite -(collection_bind_singleton (λ s, up s T) s). apply closed_up_set; solve_elem_of. Qed. Lemma closed_up_empty s : closed (up s ∅) ∅. Proof. eauto using closed_up with sts. Qed. Lemma up_closed S T : closed S T → up_set S T ≡ S. Proof. intros; split; auto using subseteq_up_set; intros s. unfold up_set; rewrite elem_of_bind; intros (s'&Hstep&?). induction Hstep; eauto using closed_step. Qed. End sts. End sts. 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. Existing Instance sts_equiv. Instance sts_valid : Valid (car sts) := λ x, match x with auth s T => tok s ∩ T ≡ ∅ | frag S' T => closed S' T end. Instance sts_unit : Unit (car sts) := λ x, 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. Existing Instance sts_disjoint. Instance sts_op : Op (car sts) := λ x1 x2, 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. Instance sts_minus : Minus (car sts) := λ x1 x2, 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. 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. Instance sts_equivalence: Equivalence ((≡) : relation (car sts)). Proof. split. - by intros []; constructor. - by destruct 1; constructor. - destruct 1; inversion_clear 1; constructor; etransitivity; eauto. Qed. Global Instance sts_dra : DRA (car sts). Proof. split. - 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. - assert (∀ T T' S s, closed S T → s ∈ S → tok s ∩ T' ≡ ∅ → tok s ∩ (T ∪ T') ≡ ∅). { intros S T T' s [??]; solve_elem_of. } destruct 3; simpl in *; auto using closed_op with sts. - intros []; simpl; eauto using closed_up, closed_up_set, closed_ne with sts. - intros ???? (z&Hy&?&Hxz); destruct Hxz; inversion Hy;clear Hy; setoid_subst; rewrite ?disjoint_union_difference; auto using closed_up with sts. eapply closed_up_set; eauto 2 using closed_disjoint with sts. - 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. assert (S ⊆ up_set S ∅ ∧ S ≢ ∅) by eauto using subseteq_up_set, closed_ne. solve_elem_of. - intros [|S T]; constructor; auto with sts. assert (S ⊆ up_set S ∅); auto using subseteq_up_set with sts. - intros [s T|S T]; constructor; auto with sts. + rewrite (up_closed (up _ _)); auto using closed_up with sts. + rewrite (up_closed (up_set _ _)); eauto using closed_up_set, closed_ne with sts. - intros x y ?? (z&Hy&?&Hxz); exists (unit (x ⋅ y)); split_ands. + destruct Hxz;inversion_clear Hy;constructor;unfold up_set; solve_elem_of. + 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 | |- 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) end; auto with sts. - intros x y ?? (z&Hy&_&Hxz); destruct Hxz; inversion_clear Hy; constructor; repeat match goal with | |- 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) end; auto with sts. - intros x y ?? (z&Hy&?&Hxz); destruct Hxz as [S1 S2 T1 T2| |]; inversion Hy; clear Hy; constructor; setoid_subst; rewrite ?disjoint_union_difference; auto. split; [|apply intersection_greatest; auto using subseteq_up_set with sts]. 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. Qed. Canonical Structure RA : cmraT := validityRA (car sts). End sts_dra. End sts_dra. (** * The STS Resource Algebra *) (** Finally, the general theory of STS that should be used by users *) Notation stsRA := (@sts_dra.RA). Section sts_definitions. Context {sts : stsT}. Definition sts_auth (s : sts.state sts) (T : sts.tokens sts) : stsRA sts := to_validity (sts_dra.auth s T). Definition sts_frag (S : sts.states sts) (T : sts.tokens sts) : stsRA sts := to_validity (sts_dra.frag S T). Definition sts_frag_up (s : sts.state sts) (T : sts.tokens sts) : stsRA sts := 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. (** 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). Proof. intros S1 S2 ? T1 T2 HT; rewrite /sts_auth. by eapply to_validity_proper; try apply _; constructor. Qed. Global Instance sts_frag_up_proper s : Proper ((≡) ==> (≡)) (sts_frag_up s). Proof. intros T1 T2 HT. by rewrite /sts_frag_up HT. Qed. (** Validity *) Lemma sts_auth_valid s T : ✓ sts_auth s T ↔ tok s ∩ T ≡ ∅. Proof. split. by move=> /(_ 0). by intros ??. Qed. Lemma sts_frag_valid S T : ✓ sts_frag S T ↔ closed S T. Proof. split. by move=> /(_ 0). by intros ??. Qed. Lemma sts_frag_up_valid s T : tok s ∩ T ≡ ∅ → ✓ sts_frag_up s T. Proof. intros; by apply sts_frag_valid, closed_up. Qed. Lemma sts_auth_frag_valid_inv s S T1 T2 : ✓ (sts_auth s T1 ⋅ sts_frag S T2) → s ∈ S. Proof. by move=> /(_ 0) [? [? Hdisj]]; inversion Hdisj. Qed. (** 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. intros; split; [split|constructor; solve_elem_of]; simpl. - intros (?&?&?); by apply closed_disjoint' with S. - intros; split_ands. solve_elem_of+. done. constructor; solve_elem_of. Qed. Lemma sts_op_auth_frag_up s T : tok s ∩ T ≡ ∅ → sts_auth s ∅ ⋅ sts_frag_up s T ≡ sts_auth s T. Proof. intros; apply sts_op_auth_frag; auto using elem_of_up, closed_up. Qed. Lemma sts_op_frag S1 S2 T1 T2 : T1 ∪ T2 ⊆ ∅ → sts.closed S1 T1 → sts.closed S2 T2 → sts_frag (S1 ∩ S2) (T1 ∪ T2) ≡ sts_frag S1 T1 ⋅ sts_frag S2 T2. Proof. intros HT HS1 HS2. rewrite /sts_frag. (* FIXME why does rewrite not work?? *) etransitivity; last eapply to_validity_op; try done; []. intros Hval. constructor; last solve_elem_of. eapply closed_ne, Hval. Qed. (** Frame preserving updates *) Lemma sts_update_auth s1 s2 T1 T2 : step (s1,T1) (s2,T2) → sts_auth s1 T1 ~~> sts_auth s2 T2. Proof. intros ?; apply validity_update; inversion 3 as [|? S ? Tf|]; subst. destruct (step_closed s1 s2 T1 T2 S Tf) as (?&?&?); auto. repeat (done || constructor). Qed. Lemma sts_update_frag S1 S2 T : closed S2 T → S1 ⊆ S2 → sts_frag S1 T ~~> sts_frag S2 T. Proof. rewrite /sts_frag=> HS Hcl. apply validity_update. inversion 3 as [|? S ? Tf|]; simplify_equality'. - split; first done. constructor; [solve_elem_of|done]. - split; first done. constructor; solve_elem_of. Qed. Lemma sts_update_frag_up s1 S2 T : closed S2 T → s1 ∈ S2 → sts_frag_up s1 T ~~> sts_frag S2 T. Proof. intros; by apply sts_update_frag; [|intros ?; eauto using closed_steps]. Qed. (** Inclusion *) Lemma sts_frag_included S1 S2 T1 T2 : closed S2 T2 → sts_frag S1 T1 ≼ sts_frag S2 T2 ↔ (closed S1 T1 ∧ ∃ Tf, T2 ≡ T1 ∪ Tf ∧ T1 ∩ Tf ≡ ∅ ∧ S2 ≡ S1 ∩ up_set S2 Tf). Proof. (* This should use some general properties of DRAs. To be improved when we have RAs back *) move=>Hcl2. split. - intros [[[Sf Tf|Sf Tf] vf Hvf] EQ]. { exfalso. inversion_clear EQ as [Hv EQ']. apply EQ' in Hcl2. simpl in Hcl2. inversion Hcl2. } inversion_clear EQ as [Hv EQ']. move:(EQ' Hcl2)=>{EQ'} EQ. inversion_clear EQ as [|? ? ? ? HT HS]. destruct Hv as [Hv _]. move:(Hv Hcl2)=>{Hv} [/= Hcl1 [Hclf Hdisj]]. apply Hvf in Hclf. simpl in Hclf. clear Hvf. inversion_clear Hdisj. split; last (exists Tf; split_ands); [done..|]. apply (anti_symm (⊆)). + move=>s HS2. apply elem_of_intersection. split; first by apply HS. by apply subseteq_up_set. + move=>s /elem_of_intersection [HS1 Hscl]. apply HS. split; first done. destruct Hscl as [s' [Hsup Hs']]. eapply closed_steps; last (hnf in Hsup; eexact Hsup); first done. solve_elem_of +HS Hs'. - intros (Hcl1 & Tf & Htk & Hf & Hs). exists (sts_frag (up_set S2 Tf) Tf). split; first split; simpl;[|done|]. + intros _. split_ands; first done. * apply closed_up_set; last by eapply closed_ne. move=>s Hs2. move:(closed_disjoint _ _ Hcl2 _ Hs2). solve_elem_of +Htk. * constructor; last done. rewrite -Hs. by eapply closed_ne. + intros _. constructor; [ solve_elem_of +Htk | done]. Qed. Lemma sts_frag_included' S1 S2 T : closed S2 T → closed S1 T → S2 ≡ S1 ∩ up_set S2 ∅ → sts_frag S1 T ≼ sts_frag S2 T. Proof. intros. apply sts_frag_included; split_ands; auto. exists ∅; split_ands; done || solve_elem_of+. Qed. End stsRA.