1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 include "pointer_list.ma".
16 include "parallel_computation.ma".
18 (* LABELLED SEQUENTIAL COMPUTATION (MULTISTEP) ******************************)
20 definition lsreds: ptrl → relation term ≝ lstar … lsred.
22 interpretation "labelled sequential computation"
23 'SeqRedStar M s N = (lsreds s M N).
25 notation "hvbox( M break ↦* [ term 46 s ] break term 46 N )"
26 non associative with precedence 45
27 for @{ 'SeqRedStar $M $s $N }.
29 lemma lsreds_refl: reflexive … (lsreds (◊)).
33 lemma lsreds_step_sn: ∀p,M1,M. M1 ↦[p] M → ∀s,M2. M ↦*[s] M2 → M1 ↦*[p::s] M2.
37 lemma lsreds_step_dx: ∀s,M1,M. M1 ↦*[s] M → ∀p,M2. M ↦[p] M2 → M1 ↦*[s@p::◊] M2.
41 lemma lsreds_step_rc: ∀p,M1,M2. M1 ↦[p] M2 → M1 ↦*[p::◊] M2.
45 lemma lsreds_inv_nil: ∀s,M1,M2. M1 ↦*[s] M2 → ◊ = s → M1 = M2.
46 /2 width=5 by lstar_inv_nil/
49 lemma lsreds_inv_cons: ∀s,M1,M2. M1 ↦*[s] M2 → ∀q,r. q::r = s →
50 ∃∃M. M1 ↦[q] M & M ↦*[r] M2.
51 /2 width=3 by lstar_inv_cons/
54 lemma lsreds_inv_step_rc: ∀p,M1,M2. M1 ↦*[p::◊] M2 → M1 ↦[p] M2.
55 /2 width=1 by lstar_inv_step/
58 lemma lsreds_inv_pos: ∀s,M1,M2. M1 ↦*[s] M2 → 0 < |s| →
59 ∃∃p,r,M. p::r = s & M1 ↦[p] M & M ↦*[r] M2.
60 /2 width=1 by lstar_inv_pos/
63 lemma lsred_compatible_rc: ho_compatible_rc lsreds.
67 lemma lsreds_compatible_sn: ho_compatible_sn lsreds.
71 lemma lsreds_compatible_dx: ho_compatible_dx lsreds.
75 lemma lsreds_lift: ∀s. liftable (lsreds s).
79 lemma lsreds_inv_lift: ∀s. deliftable_sn (lsreds s).
80 /3 width=3 by lstar_deliftable_sn, lsred_inv_lift/
83 lemma lsreds_dsubst: ∀s. dsubstable_dx (lsreds s).
87 theorem lsreds_mono: ∀s. singlevalued … (lsreds s).
88 /3 width=7 by lstar_singlevalued, lsred_mono/
91 theorem lsreds_trans: ltransitive … lsreds.
92 /2 width=3 by lstar_ltransitive/
95 lemma lsreds_compatible_appl: ∀r,B1,B2. B1 ↦*[r] B2 → ∀s,A1,A2. A1 ↦*[s] A2 →
96 @B1.A1 ↦*[(sn:::r)@dx:::s] @B2.A2.
97 #r #B1 #B2 #HB12 #s #A1 #A2 #HA12
98 @(lsreds_trans … (@B2.A1)) /2 width=1/
101 lemma lsreds_compatible_beta: ∀r,B1,B2. B1 ↦*[r] B2 → ∀s,A1,A2. A1 ↦*[s] A2 →
102 @B1.𝛌.A1 ↦*[(sn:::r)@(dx:::sn:::s)@◊::◊] [↙B2] A2.
103 #r #B1 #B2 #HB12 #s #A1 #A2 #HA12
104 @(lsreds_trans … (@B2.𝛌.A1)) /2 width=1/ -r -B1
105 @(lsreds_step_dx … (@B2.𝛌.A2)) // /3 width=1/
108 (* Note: "|s|" should be unparetesized *)
109 lemma lsreds_fwd_mult: ∀s,M1,M2. M1 ↦*[s] M2 → #{M2} ≤ #{M1} ^ (2 ^ (|s|)).
110 #s #M1 #M2 #H @(lstar_ind_l ????????? H) -s -M1 normalize //
111 #p #s #M1 #M #HM1 #_ #IHM2
112 lapply (lsred_fwd_mult … HM1) -p #HM1
113 @(transitive_le … IHM2) -M2
114 /3 width=1 by le_exp1, lt_O_exp, lt_to_le/ (**) (* auto: slow without trace *)
117 theorem lsreds_preds: ∀s,M1,M2. M1 ↦*[s] M2 → M1 ⤇* M2.
118 #s #M1 #M2 #H @(lstar_ind_l ????????? H) -s -M1 //
119 #a #s #M1 #M #HM1 #_ #IHM2
120 @(preds_step_sn … IHM2) -M2 /2 width=2/
123 lemma pred_lsreds: ∀M1,M2. M1 ⤇ M2 → ∃r. M1 ↦*[r] M2.
124 #M1 #M2 #H elim H -M1 -M2 /2 width=2/
125 [ #A1 #A2 #_ * /3 width=2/
126 | #B1 #B2 #A1 #A2 #_ #_ * #r #HB12 * /3 width=2/
127 | #B1 #B2 #A1 #A2 #_ #_ * #r #HB12 * /3 width=2/
130 theorem preds_lsreds: ∀M1,M2. M1 ⤇* M2 → ∃r. M1 ↦*[r] M2.
131 #M1 #M2 #H elim H -M2 /2 width=2/
132 #M #M2 #_ #HM2 * #r #HM1
133 elim (pred_lsreds … HM2) -HM2 #s #HM2
134 lapply (lsreds_trans … HM1 … HM2) -M /2 width=2/
137 theorem lsreds_conf: ∀s1,M0,M1. M0 ↦*[s1] M1 → ∀s2,M2. M0 ↦*[s2] M2 →
138 ∃∃r1,r2,M. M1 ↦*[r1] M & M2 ↦*[r2] M.
139 #s1 #M0 #M1 #HM01 #s2 #M2 #HM02
140 lapply (lsreds_preds … HM01) -s1 #HM01
141 lapply (lsreds_preds … HM02) -s2 #HM02
142 elim (preds_conf … HM01 … HM02) -M0 #M #HM1 #HM2
143 elim (preds_lsreds … HM1) -HM1
144 elim (preds_lsreds … HM2) -HM2 /2 width=5/