From stdpp Require Export set. From iris.algebra Require Export cmra. From iris.algebra Require Import dra. Set Default Proof Using "Type". Local Arguments valid _ _ !_ /. Local Arguments op _ _ !_ !_ /. Local Arguments core _ _ !_ /. (** * 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 : prim_step s1 s2 → tok s1 ⊥ T1 → tok s2 ⊥ T2 → tok s1 ∪ T1 ≡ tok s2 ∪ T2 → step (s1,T1) (s2,T2). Notation steps := (rtc step). Inductive frame_step (T : tokens sts) (s1 s2 : state sts) : Prop := (* Probably equivalent definition: (\mathcal{L}(s') ⊥ T) ∧ s \rightarrow s' *) | Frame_step T1 T2 : T1 ⊥ tok s1 ∪ T → step (s1,T1) (s2,T2) → frame_step T s1 s2. Notation frame_steps T := (rtc (frame_step T)). (** ** Closure under frame steps *) Record closed (S : states sts) (T : tokens sts) : Prop := Closed { 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 := {[ s' | frame_steps T s 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 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. Hint Extern 50 (_ ⊥ _) => set_solver : sts. (** ** Setoids *) Instance frame_step_mono : Proper (flip (⊆) ==> (=) ==> (=) ==> impl) frame_step. Proof. intros ?? HT ?? <- ?? <-; destruct 1; econstructor; eauto with sts; set_solver. Qed. Global Instance frame_step_proper : Proper ((≡) ==> (=) ==> (=) ==> iff) frame_step. Proof. move=> ?? /collection_equiv_spec [??]; split; by apply frame_step_mono. Qed. Instance closed_proper' : Proper ((≡) ==> (≡) ==> impl) closed. Proof. destruct 3; constructor; intros until 0; setoid_subst; 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 elem_of_mkSet, rtc_l; [eapply Frame_step with T1 T2|]; eauto with sts. Qed. Global Instance up_proper : Proper ((=) ==> (≡) ==> (≡)) up. Proof. by move=> ??? ?? /collection_equiv_spec [??]; 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=> // s1 s2 Hs. by apply up_preserving. Qed. Global Instance up_set_proper : Proper ((≡) ==> (≡) ==> (≡)) up_set. Proof. move=> S1 S2 /collection_equiv_spec [??] T1 T2 /collection_equiv_spec [??]; split; by apply up_set_preserving. Qed. (** ** Properties of closure under frame steps *) Lemma closed_steps S T s1 s2 : closed S T → s1 ∈ S → frame_steps 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 → closed (S1 ∩ S2) (T1 ∪ T2). Proof. intros [? Hstep1] [? Hstep2]; split; [set_solver|]. 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_and?; auto. - eapply Hstep with s1, Frame_step with T1 T2; auto with sts. - set_solver -Hstep Hs1 Hs2. Qed. 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. 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. 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 elem_of_up_set S T s : s ∈ S → s ∈ up_set S T. Proof. apply subseteq_up_set. 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) → closed (up_set S T) T. Proof. intros HS; unfold up_set; split. - intros s; rewrite !elem_of_bind; intros (s'&Hstep&Hs'). specialize (HS s' Hs'); clear Hs' 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 /up; set_unfold; intros (s&?&?) ?; exists s. split; [eapply rtc_r|]; eauto. 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; set_solver. Qed. Lemma closed_up_set_empty S : closed (up_set S ∅) ∅. Proof. eauto using closed_up_set with sts. 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 ?; apply collection_equiv_spec; 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. 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. Lemma up_op s T1 T2 : up s (T1 ∪ T2) ⊆ up s T1 ∩ up s T2. Proof. (* Notice that the other direction does not hold. *) intros x Hx. split; eapply elem_of_mkSet, rtc_subrel; try exact Hx. - intros; eapply frame_step_mono; last first; try done. set_solver+. - intros; eapply frame_step_mono; last first; try done. set_solver+. Qed. End sts. Notation steps := (rtc step). Notation frame_steps T := (rtc (frame_step T)). (* 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 {_} _ _. End sts. Notation stsT := sts.stsT. Notation Sts := sts.Sts. (** * STSs form a disjoint RA *) Section sts_dra. Context (sts : stsT). Import sts. 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 ∧ ∃ s, s ∈ S' end. Instance sts_core : Core (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 : (∃ s, s ∈ 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. Hint Extern 50 (equiv (A:=set _) _ _) => set_solver : sts. Hint Extern 50 (∃ s : state sts, _) => set_solver : sts. Hint Extern 50 (_ ∈ _) => set_solver : sts. Hint Extern 50 (_ ⊆ _) => set_solver : sts. Hint Extern 50 (_ ⊥ _) => set_solver : sts. 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)). Proof. split. - by intros []; constructor. - by destruct 1; constructor. - destruct 1; inversion_clear 1; constructor; etrans; eauto. Qed. Lemma sts_dra_mixin : DraMixin (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; econstructor; setoid_subst. - destruct 3; simpl in *; destruct_and?; eauto using closed_op; match goal with H : closed _ _ |- _ => destruct H end; set_solver. - intros []; naive_solver eauto using closed_up, closed_up_set, elem_of_up, elem_of_up_set 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 []; constructor; eauto with sts. - intros []; constructor; auto 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 with sts. - intros x y. exists (core (x ⋅ y))=> ?? Hxy; split_and?. + destruct Hxy; constructor; unfold up_set; set_solver. + destruct Hxy; simpl; eauto using closed_up_set_empty, closed_up_empty with sts. + destruct Hxy; econstructor; 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. Qed. Canonical Structure stsDR : draT := DraT (car sts) sts_dra_mixin. End sts_dra. (** * The STS Resource Algebra *) (** Finally, the general theory of STS that should be used by users *) Notation stsC sts := (validityC (stsDR sts)). Notation stsR sts := (validityR (stsDR sts)). Section sts_definitions. Context {sts : stsT}. Definition sts_auth (s : sts.state sts) (T : sts.tokens sts) : stsR sts := to_validity (sts.auth s T). Definition sts_frag (S : sts.states sts) (T : sts.tokens sts) : stsR sts := to_validity (sts.frag S T). Definition sts_frag_up (s : sts.state sts) (T : sts.tokens sts) : stsR 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. Arguments dra_valid _ !_/. (** Setoids *) Global Instance sts_auth_proper s : Proper ((≡) ==> (≡)) (sts_auth s). Proof. solve_proper. Qed. Global Instance sts_frag_proper : Proper ((≡) ==> (≡) ==> (≡)) (@sts_frag sts). Proof. solve_proper. Qed. Global Instance sts_frag_up_proper s : Proper ((≡) ==> (≡)) (sts_frag_up s). Proof. solve_proper. Qed. (** Validity *) Lemma sts_auth_valid s T : ✓ sts_auth s T ↔ tok s ⊥ T. Proof. done. Qed. Lemma sts_frag_valid S T : ✓ sts_frag S T ↔ closed S T ∧ ∃ s, s ∈ S. Proof. done. Qed. Lemma sts_frag_up_valid s T : ✓ sts_frag_up s T ↔ tok s ⊥ T. Proof. split. - move=>/sts_frag_valid [H _]. apply H, elem_of_up. - intros. apply sts_frag_valid; split. by apply closed_up. set_solver. Qed. Lemma sts_auth_frag_valid_inv s S T1 T2 : ✓ (sts_auth s T1 ⋅ sts_frag S T2) → s ∈ S. Proof. by intros (?&?&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; set_solver]; simpl. - intros (?&?&?); by apply closed_disjoint with S. - intros; split_and?; last constructor; set_solver. Qed. Lemma sts_op_auth_frag_up s T : sts_auth s ∅ ⋅ sts_frag_up s T ≡ sts_auth s T. Proof. intros; split; [split|constructor; set_solver]; simpl. - intros (?&[??]&?). by apply closed_disjoint with (up s T), elem_of_up. - intros; split_and?. + set_solver+. + by apply closed_up. + exists s. set_solver. + constructor; last set_solver. apply elem_of_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 -to_validity_op //. move=>/=[?[? ?]]. split_and!; [set_solver..|constructor; set_solver]. Qed. (* Notice that the following does *not* hold -- the composition of the two closures is weaker than the closure with the itnersected token set. Also see up_op. Lemma sts_op_frag_up s T1 T2 : T1 ⊥ T2 → sts_frag_up s (T1 ∪ T2) ≡ sts_frag_up s T1 ⋅ sts_frag_up s T2. *) (** Frame preserving updates *) Lemma sts_update_auth s1 s2 T1 T2 : steps (s1,T1) (s2,T2) → sts_auth s1 T1 ~~> sts_auth s2 T2. Proof. intros ?; apply validity_update. inversion 3 as [|? S ? Tf|]; simplify_eq/=; destruct_and?. destruct (steps_closed s1 s2 T1 T2 S Tf) as (?&?&?); auto; []. repeat (done || constructor). Qed. Lemma sts_update_frag S1 S2 T1 T2 : (closed S1 T1 → closed S2 T2 ∧ S1 ⊆ S2 ∧ T2 ⊆ T1) → sts_frag S1 T1 ~~> sts_frag S2 T2. Proof. rewrite /sts_frag=> HC HS HT. apply validity_update. inversion 3 as [|? S ? Tf|]; simplify_eq/=; (destruct HC as (? & ? & ?); first by destruct_and?). - split_and!. done. set_solver. constructor; set_solver. - split_and!. done. set_solver. constructor; set_solver. Qed. Lemma sts_update_frag_up s1 S2 T1 T2 : (tok s1 ⊥ T1 → closed S2 T2) → s1 ∈ S2 → T2 ⊆ T1 → sts_frag_up s1 T1 ~~> sts_frag S2 T2. Proof. intros HC ? HT; apply sts_update_frag. intros HC1; split; last split; eauto using closed_steps. - eapply HC, HC1, elem_of_up. - rewrite <-HT. eapply up_subseteq; last done. apply HC, HC1, elem_of_up. Qed. Lemma sts_up_set_intersection S1 Sf Tf : closed Sf Tf → S1 ∩ Sf ≡ S1 ∩ up_set (S1 ∩ Sf) Tf. Proof. intros Hclf. apply (anti_symm (⊆)). - 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'. Qed. Global Instance sts_frag_core_id S : CoreId (sts_frag S ∅). Proof. constructor; split=> //= [[??]]. by rewrite /dra.dra_core /= sts.up_closed. Qed. (** Inclusion *) (* 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. *) (* TODO: These have to be proven again. *) (* Lemma sts_frag_included S1 S2 T1 T2 : 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. intros ??; split. - intros [[???] ?]. 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. * by apply up_set_non_empty. + constructor; last done. by rewrite -HS. Qed. Lemma sts_frag_included' S1 S2 T : closed S2 T → closed S1 T → S2 ≢ ∅ → S1 ≢ ∅ → S2 ≡ S1 ∩ up_set S2 ∅ → sts_frag S1 T ≼ sts_frag S2 T. Proof. intros. apply sts_frag_included; split_and?; auto. exists ∅; split_and?; done || set_solver+. Qed. *) End stsRA. (** 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)). Definition stsT_token := Empty_set. Definition stsT_tok {sts : stsT} (_ : state sts) : set stsT_token := ∅. Canonical Structure sts_notok (sts : stsT) : sts.stsT := sts.Sts (@prim_step sts) stsT_tok. Coercion sts_notok.sts_notok : sts_notok.stsT >-> sts.stsT. 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]. by inversion_clear Hstep. 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; [by set_solver|eauto using frame_prim_step]. Qed. End sts. End sts_notok. Notation sts_notokT := sts_notok.stsT. Notation Sts_NoTok := sts_notok.Sts. Section sts_notokRA. Context {sts : sts_notokT}. Import sts_notok. 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.