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14
15 include "ground/xoa/ex_3_2.ma".
16 include "ground/counters/rtc_plus.ma".
17 include "ground/counters/rtc_ist.ma".
18
19 (* T-BOUND RT-TRANSITION COUNTERS *******************************************)
20
21 (* Constructions with rtc_plus **********************************************)
22
23 lemma rtc_ist_plus (n1) (n2) (c1) (c2): š“āØn1,c1ā© ā†’ š“āØn2,c2ā© ā†’ š“āØn1+n2,c1+c2ā©.
24 #n1 #n2 #c1 #c2 #H1 #H2 destruct //
25 qed.
26
27 lemma rtc_ist_plus_zero_sn (n) (c1) (c2): š“āØšŸŽ,c1ā© ā†’ š“āØn,c2ā© ā†’ š“āØn,c1+c2ā©.
28 #n #c1 #c2 #H1 #H2 >(nplus_zero_sn n)
29 /2 width=1 by rtc_ist_plus/
30 qed.
31
32 lemma rtc_ist_plus_zero_dx (n) (c1) (c2): š“āØn,c1ā© ā†’ š“āØšŸŽ,c2ā© ā†’ š“āØn,c1+c2ā©.
33 /2 width=1 by rtc_ist_plus/ qed.
34
35 lemma rtc_ist_succ (n) (c): š“āØn,cā© ā†’ š“āØā†‘n,c+šŸ˜šŸ™ā©.
36 #n #c #H >nplus_unit_dx
37 /2 width=1 by rtc_ist_plus/
38 qed.
39
40 (* Inversions with rtc_plus *************************************************)
41
42 lemma rtc_ist_inv_plus (n) (c1) (c2): š“āØn,c1 + c2ā© ā†’
43       āˆƒāˆƒn1,n2. š“āØn1,c1ā© & š“āØn2,c2ā© & n1 + n2 = n.
44 #n #c1 #c2 #H
45 elim (rtc_plus_inv_dx ā€¦ H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #H1 #H2 #H3 #H4 #H5 #H6 destruct
46 elim (eq_inv_nplus_zero ā€¦ H1) -H1 #H11 #H12 destruct
47 elim (eq_inv_nplus_zero ā€¦ H2) -H2 #H21 #H22 destruct
48 elim (eq_inv_nplus_zero ā€¦ H3) -H3 #H31 #H32 destruct
49 /3 width=5 by ex3_2_intro/
50 qed-.
51
52 lemma rtc_ist_inv_plus_zero_dx (n) (c1) (c2): š“āØn,c1 + c2ā© ā†’ š“āØšŸŽ,c2ā© ā†’ š“āØn,c1ā©.
53 #n #c1 #c2 #H #H2
54 elim (rtc_ist_inv_plus ā€¦ H) -H #n1 #n2 #Hn1 #Hn2 #H destruct //
55 qed-.
56
57 lemma rtc_ist_inv_plus_unit_dx:
58       āˆ€n,c1,c2. š“āØn,c1 + c2ā© ā†’ š“āØšŸ,c2ā© ā†’
59       āˆƒāˆƒm. š“āØm,c1ā© & n = ā†‘m.
60 #n #c1 #c2 #H #H2 destruct
61 elim (rtc_ist_inv_plus ā€¦ H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
62 /2 width=3 by ex2_intro/
63 qed-.
64
65 lemma rtc_ist_inv_plus_zu_dx (n) (c): š“āØn,c+šŸ™šŸ˜ā© ā†’ āŠ„.
66 #n #c #H
67 elim (rtc_ist_inv_plus ā€¦ H) -H #n1 #n2 #_ #H #_
68 /2 width=2 by rtc_ist_inv_uz/
69 qed-.