From algebra Require Export sts. From program_logic Require Import ghost_ownership. (** The STS describing the main barrier protocol. Every state has an index-set associated with it. These indices are actually [gname], because we use them with saved propositions. *) Inductive phase := Low | High. Record state := State { state_phase : phase; state_I : gset gname }. Add Printing Constructor state. Inductive token := Change (i : gname) | Send. Global Instance stateT_inhabited: Inhabited state := populate (State Low ∅). Global Instance Change_inj : Inj (=) (=) Change. Proof. by injection 1. Qed. Inductive prim_step : relation state := | ChangeI p I2 I1 : prim_step (State p I1) (State p I2) | ChangePhase I : prim_step (State Low I) (State High I). Definition tok (s : state) : set token := {[ t | ∃ i, t = Change i ∧ i ∉ state_I s ]} ∪ (if state_phase s is High then {[ Send ]} else ∅). Global Arguments tok !_ /. Canonical Structure sts := sts.STS prim_step tok. (* The set of states containing some particular i *) Definition i_states (i : gname) : set state := {[ s | i ∈ state_I s ]}. (* The set of low states *) Definition low_states : set state := {[ s | state_phase s = Low ]}. Lemma i_states_closed i : sts.closed (i_states i) {[ Change i ]}. Proof. split; first (intros [[] I]; set_solver). (* If we do the destruct of the states early, and then inversion on the proof of a transition, it doesn't work - we do not obtain the equalities we need. So we destruct the states late, because this means we can use "destruct" instead of "inversion". *) intros s1 s2 Hs1 [T1 T2 Hdisj Hstep']. inversion_clear Hstep' as [? ? ? ? Htrans _ _ Htok]. destruct Htrans as [[] ??|]; done || set_solver. Qed. Lemma low_states_closed : sts.closed low_states {[ Send ]}. Proof. split; first (intros [??]; set_solver). intros s1 s2 Hs1 [T1 T2 Hdisj Hstep']. inversion_clear Hstep' as [? ? ? ? Htrans _ _ Htok]. destruct Htrans as [[] ??|]; done || set_solver. Qed. (* Proof that we can take the steps we need. *) Lemma signal_step I : sts.steps (State Low I, {[Send]}) (State High I, ∅). Proof. apply rtc_once. constructor; first constructor; set_solver. Qed. Lemma wait_step i I : i ∈ I → sts.steps (State High I, {[ Change i ]}) (State High (I ∖ {[ i ]}), ∅). Proof. intros. apply rtc_once. constructor; first constructor; [set_solver..|]. apply elem_of_equiv=>-[j|]; last set_solver. destruct (decide (i = j)); set_solver. Qed. Lemma split_step p i i1 i2 I : i ∈ I → i1 ∉ I → i2 ∉ I → i1 ≠ i2 → sts.steps (State p I, {[ Change i ]}) (State p ({[i1]} ∪ ({[i2]} ∪ (I ∖ {[i]}))), {[ Change i1; Change i2 ]}). Proof. intros. apply rtc_once. constructor; first constructor. - destruct p; set_solver. - destruct p; set_solver. - apply elem_of_equiv=> /= -[j|]; last set_solver. set_unfold; rewrite !(inj_iff Change). assert (Change j ∈ match p with Low => ∅ | High => {[Send]} end ↔ False) as -> by (destruct p; set_solver). destruct (decide (i1 = j)) as [->|]; first naive_solver. destruct (decide (i2 = j)) as [->|]; first naive_solver. destruct (decide (i = j)) as [->|]; naive_solver. Qed.