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14
15 include "ground/xoa/ex_3_2.ma".
16 include "ground/counters/rtc_max.ma".
17 include "ground/counters/rtc_ist.ma".
18
19 (* T-BOUND RT-TRANSITION COUNTERS *******************************************)
20
21 (* Constructions with rtc_max ***********************************************)
22
23 lemma rtc_ist_max (n1) (n2) (c1) (c2): ๐“โจn1,c1โฉ โ†’ ๐“โจn2,c2โฉ โ†’ ๐“โจn1โˆจn2,c1โˆจc2โฉ.
24 #n1 #n2 #c1 #c2 #H1 #H2 destruct //
25 qed.
26
27 lemma rtc_ist_max_zero_sn (n) (c1) (c2): ๐“โจ๐ŸŽ,c1โฉ โ†’ ๐“โจn,c2โฉ โ†’ ๐“โจn,c1โˆจc2โฉ.
28 /2 width=1 by rtc_ist_max/ qed.
29
30 lemma rtc_ist_max_zero_dx (n) (c1) (c2): ๐“โจn,c1โฉ โ†’ ๐“โจ๐ŸŽ,c2โฉ โ†’ ๐“โจn,c1โˆจc2โฉ.
31 // qed.
32
33 lemma rtc_ist_max_idem_sn (n) (c1) (c2): ๐“โจn,c1โฉ โ†’ ๐“โจn,c2โฉ โ†’ ๐“โจn,c1โˆจc2โฉ.
34 #n #c1 #c2 #H1 #H2 >(nmax_idem n) /2 width=1 by rtc_ist_max/
35 qed.
36
37 (* Inversions with rtc_max **************************************************)
38
39 lemma rtc_ist_inv_max (n) (c1) (c2): ๐“โจn,c1 โˆจ c2โฉ โ†’
40       โˆƒโˆƒn1,n2. ๐“โจn1,c1โฉ & ๐“โจn2,c2โฉ & (n1 โˆจ n2) = n.
41 #n #c1 #c2 #H
42 elim (rtc_max_inv_dx โ€ฆ H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #H1 #H2 #H3 #H4 #H5 #H6 destruct
43 elim (eq_inv_nmax_zero โ€ฆ H1) -H1 #H11 #H12 destruct
44 elim (eq_inv_nmax_zero โ€ฆ H2) -H2 #H21 #H22 destruct
45 elim (eq_inv_nmax_zero โ€ฆ H3) -H3 #H31 #H32 destruct
46 /2 width=5 by ex3_2_intro/
47 qed-.
48
49 lemma rtc_ist_inv_zero_max (c1) (c2): ๐“โจ๐ŸŽ,c1 โˆจ c2โฉ โ†’ โˆงโˆง ๐“โจ๐ŸŽ,c1โฉ & ๐“โจ๐ŸŽ,c2โฉ.
50 #c1 #c2 #H
51 elim (rtc_ist_inv_max โ€ฆ H) -H #n1 #n2 #Hn1 #Hn2 #H
52 elim (eq_inv_nmax_zero โ€ฆ H) -H #H1 #H2 destruct
53 /2 width=1 by conj/
54 qed-.
55
56 lemma rtc_ist_inv_max_zero_dx (n) (c1) (c2): ๐“โจn,c1 โˆจ c2โฉ โ†’ ๐“โจ๐ŸŽ,c2โฉ โ†’ ๐“โจn,c1โฉ.
57 #n #c1 #c2 #H #H2
58 elim (rtc_ist_inv_max โ€ฆ H) -H #n1 #n2 #Hn1 #Hn2 #H destruct //
59 qed-.