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 "ground/xoa/ex_3_2.ma".
16 include "ground/steps/rtc_max.ma".
17 include "ground/steps/rtc_isrt.ma".
19 (* RT-TRANSITION COUNTER ****************************************************)
21 (* Properties with test for constrained rt-transition counter ***************)
23 lemma isrt_max: โn1,n2,c1,c2. ๐๐โชn1,c1โซ โ ๐๐โชn2,c2โซ โ ๐๐โชn1โจn2,c1โจc2โซ.
24 #n1 #n2 #c1 #c2 * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct
25 /2 width=3 by ex1_2_intro/
28 lemma isrt_max_O1: โn,c1,c2. ๐๐โช0,c1โซ โ ๐๐โชn,c2โซ โ ๐๐โชn,c1โจc2โซ.
29 /2 width=1 by isrt_max/ qed.
31 lemma isrt_max_O2: โn,c1,c2. ๐๐โชn,c1โซ โ ๐๐โช0,c2โซ โ ๐๐โชn,c1โจc2โซ.
32 #n #c1 #c2 #H1 #H2 >(max_O2 n) /2 width=1 by isrt_max/
35 lemma isrt_max_idem1: โn,c1,c2. ๐๐โชn,c1โซ โ ๐๐โชn,c2โซ โ ๐๐โชn,c1โจc2โซ.
36 #n #c1 #c2 #H1 #H2 >(idempotent_max n) /2 width=1 by isrt_max/
39 (* Inversion properties with test for constrained rt-transition counter *****)
41 lemma isrt_inv_max: โn,c1,c2. ๐๐โชn,c1 โจ c2โซ โ
42 โโn1,n2. ๐๐โชn1,c1โซ & ๐๐โชn2,c2โซ & (n1 โจ n2) = n.
43 #n #c1 #c2 * #ri #rs #H
44 elim (max_inv_dx โฆ H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #_ #_ #H1 #H2 #H3 #H4
45 elim (max_inv_O3 โฆ H1) -H1 /3 width=5 by ex3_2_intro, ex1_2_intro/
48 lemma isrt_O_inv_max: โc1,c2. ๐๐โช0,c1 โจ c2โซ โ โงโง ๐๐โช0,c1โซ & ๐๐โช0,c2โซ.
50 elim (isrt_inv_max โฆ H) -H #n1 #n2 #Hn1 #Hn2 #H
51 elim (max_inv_O3 โฆ H) -H #H1 #H2 destruct
55 lemma isrt_inv_max_O_dx: โn,c1,c2. ๐๐โชn,c1 โจ c2โซ โ ๐๐โช0,c2โซ โ ๐๐โชn,c1โซ.
57 elim (isrt_inv_max โฆ H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
58 lapply (isrt_inj โฆ Hn2 H2) -c2 #H destruct //
61 lemma isrt_inv_max_eq_t: โn,c1,c2. ๐๐โชn,c1 โจ c2โซ โ rtc_eq_t c1 c2 โ
62 โงโง ๐๐โชn,c1โซ & ๐๐โชn,c2โซ.
64 elim (isrt_inv_max โฆ H) -H #n1 #n2 #Hc1 #Hc2 #H destruct
65 lapply (isrt_eq_t_trans โฆ Hc1 โฆ Hc12) -Hc12 #H
66 <(isrt_inj โฆ H โฆ Hc2) -Hc2
67 <idempotent_max /2 width=1 by conj/