(* *)
(**************************************************************************)
-include "redex_pointer_sequence.ma".
-include "labelled_sequential_reduction.ma".
+include "pointer_list.ma".
+include "parallel_computation.ma".
(* LABELLED SEQUENTIAL COMPUTATION (MULTISTEP) ******************************)
-(* Note: this comes from "star term lsred" *)
-inductive lsreds: rpseq → relation term ≝
-| lsreds_nil : ∀M. lsreds (◊) M M
-| lsreds_cons: ∀p,M1,M. M1 ⇀[p] M →
- ∀s,M2. lsreds s M M2 → lsreds (p::s) M1 M2
-.
+definition lsreds: ptrl → relation term ≝ lstar … lsred.
interpretation "labelled sequential computation"
'SeqRedStar M s N = (lsreds s M N).
-notation "hvbox( M break â\87\80* [ term 46 s ] break term 46 N )"
+notation "hvbox( M break â\86¦* [ term 46 s ] break term 46 N )"
non associative with precedence 45
for @{ 'SeqRedStar $M $s $N }.
-lemma lsred_lsreds: ∀p,M1,M2. M1 ⇀[p] M2 → M1 ⇀*[p::◊] M2.
+lemma lsreds_refl: reflexive … (lsreds (◊)).
+//
+qed.
+
+lemma lsreds_step_sn: ∀p,M1,M. M1 ↦[p] M → ∀s,M2. M ↦*[s] M2 → M1 ↦*[p::s] M2.
+/2 width=3/
+qed-.
+
+lemma lsreds_step_dx: ∀s,M1,M. M1 ↦*[s] M → ∀p,M2. M ↦[p] M2 → M1 ↦*[s@p::◊] M2.
/2 width=3/
+qed-.
+
+lemma lsreds_step_rc: ∀p,M1,M2. M1 ↦[p] M2 → M1 ↦*[p::◊] M2.
+/2 width=1/
qed.
-lemma lsreds_inv_nil: ∀s,M1,M2. M1 ⇀*[s] M2 → ◊ = s → M1 = M2.
-#s #M1 #M2 * -s -M1 -M2 //
-#p #M1 #M #_ #s #M2 #_ #H destruct
+lemma lsreds_inv_nil: ∀s,M1,M2. M1 ↦*[s] M2 → ◊ = s → M1 = M2.
+/2 width=5 by lstar_inv_nil/
qed-.
-lemma lsreds_inv_cons: ∀s,M1,M2. M1 ⇀*[s] M2 → ∀q,r. q::r = s →
- ∃∃M. M1 ⇀[q] M & M ⇀*[r] M2.
-#s #M1 #M2 * -s -M1 -M2
-[ #M #q #r #H destruct
-| #p #M1 #M #HM1 #s #M2 #HM2 #q #r #H destruct /2 width=3/
-]
+lemma lsreds_inv_cons: ∀s,M1,M2. M1 ↦*[s] M2 → ∀q,r. q::r = s →
+ ∃∃M. M1 ↦[q] M & M ↦*[r] M2.
+/2 width=3 by lstar_inv_cons/
qed-.
-lemma lsreds_inv_lsred: ∀p,M1,M2. M1 ⇀*[p::◊] M2 → M1 ⇀[p] M2.
-#p #M1 #M2 #H
-elim (lsreds_inv_cons … H ???) -H [4: // |2,3: skip ] #M #HM1 #H (**) (* simplify line *)
-<(lsreds_inv_nil … H ?) -H //
+lemma lsreds_inv_step_rc: ∀p,M1,M2. M1 ↦*[p::◊] M2 → M1 ↦[p] M2.
+/2 width=1 by lstar_inv_step/
qed-.
-(* Note: "|s|" should be unparetesized *)
-lemma lsreds_fwd_mult: ∀s,M1,M2. M1 ⇀*[s] M2 → #{M2} ≤ #{M1} ^ (2 ^ (|s|)).
-#s #M1 #M2 #H elim H -s -M1 -M2 normalize //
-#p #M1 #M #HM1 #s #M2 #_ #IHM2
-lapply (lsred_fwd_mult … HM1) -p #HM1
-@(transitive_le … IHM2) -M2
-/3 width=1 by le_exp1, lt_O_exp, lt_to_le/ (**) (* auto: slow without trace *)
+lemma lsreds_inv_pos: ∀s,M1,M2. M1 ↦*[s] M2 → 0 < |s| →
+ ∃∃p,r,M. p::r = s & M1 ↦[p] M & M ↦*[r] M2.
+/2 width=1 by lstar_inv_pos/
qed-.
+lemma lsred_compatible_rc: ho_compatible_rc lsreds.
+/3 width=1/
+qed.
+
+lemma lsreds_compatible_sn: ho_compatible_sn lsreds.
+/3 width=1/
+qed.
+
+lemma lsreds_compatible_dx: ho_compatible_dx lsreds.
+/3 width=1/
+qed.
+
lemma lsreds_lift: ∀s. liftable (lsreds s).
-#s #h #M1 #M2 #H elim H -s -M1 -M2 // /3 width=3/
+/2 width=1/
qed.
-lemma lsreds_inv_lift: ∀s. deliftable (lsreds s).
-#s #h #N1 #N2 #H elim H -s -N1 -N2 /2 width=3/
-#p #N1 #N #HN1 #s #N2 #_ #IHN2 #d #M1 #HMN1
-elim (lsred_inv_lift … HN1 … HMN1) -N1 #M #HM1 #HMN
-elim (IHN2 … HMN) -N /3 width=3/
+lemma lsreds_inv_lift: ∀s. deliftable_sn (lsreds s).
+/3 width=3 by lstar_deliftable_sn, lsred_inv_lift/
qed-.
lemma lsreds_dsubst: ∀s. dsubstable_dx (lsreds s).
-#s #D #M1 #M2 #H elim H -s -M1 -M2 // /3 width=3/
+/2 width=1/
qed.
theorem lsreds_mono: ∀s. singlevalued … (lsreds s).
-#s #M #N1 #H elim H -s -M -N1
-[ /2 width=3 by lsreds_inv_nil/
-| #p #M #M1 #HM1 #s #N1 #_ #IHMN1 #N2 #H
- elim (lsreds_inv_cons … H ???) -H [4: // |2,3: skip ] #M2 #HM2 #HMN2 (**) (* simplify line *)
- lapply (lsred_mono … HM1 … HM2) -M #H destruct /2 width=1/
-]
+/3 width=7 by lstar_singlevalued, lsred_mono/
+qed-.
+
+theorem lsreds_trans: ltransitive … lsreds.
+/2 width=3 by lstar_ltransitive/
+qed-.
+
+lemma lsreds_compatible_appl: ∀r,B1,B2. B1 ↦*[r] B2 → ∀s,A1,A2. A1 ↦*[s] A2 →
+ @B1.A1 ↦*[(sn:::r)@dx:::s] @B2.A2.
+#r #B1 #B2 #HB12 #s #A1 #A2 #HA12
+@(lsreds_trans … (@B2.A1)) /2 width=1/
+qed.
+
+lemma lsreds_compatible_beta: ∀r,B1,B2. B1 ↦*[r] B2 → ∀s,A1,A2. A1 ↦*[s] A2 →
+ @B1.𝛌.A1 ↦*[(sn:::r)@(dx:::sn:::s)@◊::◊] [↙B2] A2.
+#r #B1 #B2 #HB12 #s #A1 #A2 #HA12
+@(lsreds_trans … (@B2.𝛌.A1)) /2 width=1/ -r -B1
+@(lsreds_step_dx … (@B2.𝛌.A2)) // /3 width=1/
+qed.
+
+(* Note: "|s|" should be unparetesized *)
+lemma lsreds_fwd_mult: ∀s,M1,M2. M1 ↦*[s] M2 → #{M2} ≤ #{M1} ^ (2 ^ (|s|)).
+#s #M1 #M2 #H @(lstar_ind_l ????????? H) -s -M1 normalize //
+#p #s #M1 #M #HM1 #_ #IHM2
+lapply (lsred_fwd_mult … HM1) -p #HM1
+@(transitive_le … IHM2) -M2
+/3 width=1 by le_exp1, lt_O_exp, lt_to_le/ (**) (* auto: slow without trace *)
+qed-.
+
+theorem lsreds_preds: ∀s,M1,M2. M1 ↦*[s] M2 → M1 ⤇* M2.
+#s #M1 #M2 #H @(lstar_ind_l ????????? H) -s -M1 //
+#a #s #M1 #M #HM1 #_ #IHM2
+@(preds_step_sn … IHM2) -M2 /2 width=2/
+qed.
+
+lemma pred_lsreds: ∀M1,M2. M1 ⤇ M2 → ∃r. M1 ↦*[r] M2.
+#M1 #M2 #H elim H -M1 -M2 /2 width=2/
+[ #A1 #A2 #_ * /3 width=2/
+| #B1 #B2 #A1 #A2 #_ #_ * #r #HB12 * /3 width=2/
+| #B1 #B2 #A1 #A2 #_ #_ * #r #HB12 * /3 width=2/
+qed-.
+
+theorem preds_lsreds: ∀M1,M2. M1 ⤇* M2 → ∃r. M1 ↦*[r] M2.
+#M1 #M2 #H elim H -M2 /2 width=2/
+#M #M2 #_ #HM2 * #r #HM1
+elim (pred_lsreds … HM2) -HM2 #s #HM2
+lapply (lsreds_trans … HM1 … HM2) -M /2 width=2/
qed-.
-theorem lsreds_trans: ∀s1,M1,M. M1 ⇀*[s1] M →
- ∀s2,M2. M ⇀*[s2] M2 → M1 ⇀*[s1@s2] M2.
-#s1 #M1 #M #H elim H -s1 -M1 -M normalize // /3 width=3/
+theorem lsreds_conf: ∀s1,M0,M1. M0 ↦*[s1] M1 → ∀s2,M2. M0 ↦*[s2] M2 →
+ ∃∃r1,r2,M. M1 ↦*[r1] M & M2 ↦*[r2] M.
+#s1 #M0 #M1 #HM01 #s2 #M2 #HM02
+lapply (lsreds_preds … HM01) -s1 #HM01
+lapply (lsreds_preds … HM02) -s2 #HM02
+elim (preds_conf … HM01 … HM02) -M0 #M #HM1 #HM2
+elim (preds_lsreds … HM1) -HM1
+elim (preds_lsreds … HM2) -HM2 /2 width=5/
qed-.
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