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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 *)
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15 include "ground/xoa/ex_3_2.ma".
16 include "ground/steps/rtc_plus.ma".
17 include "ground/steps/rtc_ist.ma".
19 (* RT-TRANSITION COUNTER ****************************************************)
21 (* Properties with test for t-transition counter ****************************)
23 lemma ist_plus: ān1,n2,c1,c2. šāŖn1,c1ā« ā šāŖn2,c2ā« ā šāŖn1+n2,c1+c2ā«.
24 #n1 #n2 #c1 #c2 #H1 #H2 destruct //
27 lemma ist_plus_O1: ān,c1,c2. šāŖ0,c1ā« ā šāŖn,c2ā« ā šāŖn,c1+c2ā«.
28 /2 width=1 by ist_plus/ qed.
30 lemma ist_plus_O2: ān,c1,c2. šāŖn,c1ā« ā šāŖ0,c2ā« ā šāŖn,c1+c2ā«.
31 #n #c1 #c2 #H1 #H2 >(plus_n_O n) /2 width=1 by ist_plus/
34 lemma ist_succ: ān,c. šāŖn,cā« ā šāŖān,c+ššā«.
35 /2 width=1 by ist_plus/ qed.
37 (* Inversion properties with test for constrained rt-transition counter *****)
40 ān,c1,c2. šāŖn,c1 + c2ā« ā
41 āān1,n2. šāŖn1,c1ā« & šāŖn2,c2ā« & n1 + n2 = n.
43 elim (plus_inv_dx ā¦ H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #H1 #H2 #H3 #H4 #H5 #H6 destruct
44 elim (plus_inv_O3 ā¦ H1) -H1 #H11 #H12 destruct
45 elim (plus_inv_O3 ā¦ H2) -H2 #H21 #H22 destruct
46 elim (plus_inv_O3 ā¦ H3) -H3 #H31 #H32 destruct
47 /3 width=5 by ex3_2_intro/
50 lemma ist_inv_plus_O_dx: ān,c1,c2. šāŖn,c1 + c2ā« ā šāŖ0,c2ā« ā šāŖn,c1ā«.
52 elim (ist_inv_plus ā¦ H) -H #n1 #n2 #Hn1 #Hn2 #H destruct //
55 lemma ist_inv_plus_SO_dx:
56 ān,c1,c2. šāŖn,c1 + c2ā« ā šāŖ1,c2ā« ā
57 āām. šāŖm,c1ā« & n = ām.
58 #n #c1 #c2 #H #H2 destruct
59 elim (ist_inv_plus ā¦ H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
60 /2 width=3 by ex2_intro/
63 lemma ist_inv_plus_10_dx: ān,c. šāŖn,c+ššā« ā ā„.
65 elim (ist_inv_plus ā¦ H) -H #n1 #n2 #_ #H #_
66 /2 width=2 by ist_inv_10/