### Simplify barrier protocol proofs.

parent 2d9c5f33
Pipeline #156 passed with stage
 ... ... @@ -17,12 +17,9 @@ 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 change_tok (I : gset gname) : set token := {[ t | match t with Change i => i ∉ I | Send => False end ]}. Definition send_tok (p : phase) : set token := match p with Low => ∅ | High => {[ Send ]} end. Definition tok (s : state) : set token := change_tok (state_I s) ∪ send_tok (state_phase s). {[ 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. ... ... @@ -35,30 +32,22 @@ Definition low_states : set state := {[ s | state_phase s = Low ]}. Lemma i_states_closed i : sts.closed (i_states i) {[ Change i ]}. Proof. split. - intros [p I]. rewrite /= /i_states /change_tok. destruct p; 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; simpl in *; last done. move: Hs1 Hdisj Htok. rewrite elem_of_equiv_empty elem_of_equiv. move=> ? /(_ (Change i)) Hdisj /(_ (Change i)); move: Hdisj. rewrite elem_of_intersection elem_of_union !elem_of_mkSet. intros; apply dec_stable. destruct p; set_solver. 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. - intros [p I]. rewrite /low_states. set_solver. - intros s1 s2 Hs1 [T1 T2 Hdisj Hstep']. inversion_clear Hstep' as [? ? ? ? Htrans _ _ Htok]. destruct Htrans; simpl in *; first by destruct p. exfalso; apply dec_stable; set_solver. 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. *) ... ... @@ -70,12 +59,8 @@ Lemma wait_step i I : sts.steps (State High I, {[ Change i ]}) (State High (I ∖ {[ i ]}), ∅). Proof. intros. apply rtc_once. constructor; first constructor; rewrite /= /change_tok; [set_solver by eauto..|]. (* TODO this proof is rather annoying. *) apply elem_of_equiv=>t. rewrite !elem_of_union. rewrite !elem_of_mkSet /change_tok /=. destruct t as [j|]; last set_solver. rewrite elem_of_difference elem_of_singleton. constructor; first constructor; [set_solver..|]. apply elem_of_equiv=>-[j|]; last set_solver. destruct (decide (i = j)); set_solver. Qed. ... ... @@ -85,17 +70,14 @@ Lemma split_step p i i1 i2 I : (State p I, {[ Change i ]}) (State p ({[i1]} ∪ ({[i2]} ∪ (I ∖ {[i]}))), {[ Change i1; Change i2 ]}). Proof. intros. apply rtc_once. constructor; first constructor; simpl. intros. apply rtc_once. constructor; first constructor. - destruct p; set_solver. - destruct p; set_solver. (* This gets annoying... and I think I can see a pattern with all these proofs. Automatable? *) - apply elem_of_equiv=>t. destruct t; last set_solver. rewrite !elem_of_mkSet !not_elem_of_union !not_elem_of_singleton not_elem_of_difference elem_of_singleton !(inj_iff Change). destruct p; naive_solver. - apply elem_of_equiv=>t. destruct t as [j|]; last set_solver. rewrite !elem_of_mkSet !not_elem_of_union !not_elem_of_singleton not_elem_of_difference elem_of_singleton !(inj_iff Change). destruct (decide (i1 = j)) as [->|]; first tauto. destruct (decide (i2 = j)) as [->|]; intuition. - 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.
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