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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_ism.ma".
18
19 (* T-BOUND RT-TRANSITION COUNTERS *******************************************)
20
21 (* Constructions with rtc_max ***********************************************)
22
23 lemma rtc_ism_max (n1) (n2) (c1) (c2): ๐Œโจn1,c1โฉ โ†’ ๐Œโจn2,c2โฉ โ†’ ๐Œโจn1โˆจn2,c1โˆจc2โฉ.
24 #n1 #n2 #c1 #c2 * #ri1 #rs1 #H1 * #ri2 #rs2 #H2 destruct
25 /2 width=3 by ex1_2_intro/
26 qed.
27
28 lemma rtc_ism_max_zero_sn (n) (c1) (c2): ๐Œโจ๐ŸŽ,c1โฉ โ†’ ๐Œโจn,c2โฉ โ†’ ๐Œโจn,c1โˆจc2โฉ.
29 /2 width=1 by rtc_ism_max/ qed.
30
31 lemma rtc_ism_max_zero_dx (n) (c1) (c2): ๐Œโจn,c1โฉ โ†’ ๐Œโจ๐ŸŽ,c2โฉ โ†’ ๐Œโจn,c1โˆจc2โฉ.
32 #n #c1 #c2 #H1 #H2 >(nmax_zero_dx n) /2 width=1 by rtc_ism_max/
33 qed.
34
35 lemma rtc_ism_max_idem_sn (n) (c1) (c2): ๐Œโจn,c1โฉ โ†’ ๐Œโจn,c2โฉ โ†’ ๐Œโจn,c1โˆจc2โฉ.
36 #n #c1 #c2 #H1 #H2 >(nmax_idem n) /2 width=1 by rtc_ism_max/
37 qed.
38
39 (* Inversions with rtc_max **************************************************)
40
41 lemma rtc_ism_inv_max (n) (c1) (c2): ๐Œโจn,c1 โˆจ c2โฉ โ†’
42       โˆƒโˆƒn1,n2. ๐Œโจn1,c1โฉ & ๐Œโจn2,c2โฉ & (n1 โˆจ n2) = n.
43 #n #c1 #c2 * #ri #rs #H
44 elim (rtc_max_inv_dx โ€ฆ H) -H #ri1 #rs1 #ti1 #ts1 #ri2 #rs2 #ti2 #ts2 #_ #_ #H1 #H2 #H3 #H4
45 elim (eq_inv_nmax_zero โ€ฆ H1) -H1 /3 width=5 by ex3_2_intro, ex1_2_intro/
46 qed-.
47
48 lemma rtc_isr_inv_max (c1) (c2): ๐Œโจ๐ŸŽ,c1 โˆจ c2โฉ โ†’ โˆงโˆง ๐Œโจ๐ŸŽ,c1โฉ & ๐Œโจ๐ŸŽ,c2โฉ.
49 #c1 #c2 #H
50 elim (rtc_ism_inv_max โ€ฆ H) -H #n1 #n2 #Hn1 #Hn2 #H
51 elim (eq_inv_nmax_zero โ€ฆ H) -H #H1 #H2 destruct
52 /2 width=1 by conj/
53 qed-.
54
55 lemma rtc_ism_inv_max_zero_dx (n) (c1) (c2): ๐Œโจn,c1 โˆจ c2โฉ โ†’ ๐Œโจ๐ŸŽ,c2โฉ โ†’ ๐Œโจn,c1โฉ.
56 #n #c1 #c2 #H #H2
57 elim (rtc_ism_inv_max โ€ฆ H) -H #n1 #n2 #Hn1 #Hn2 #H destruct
58 lapply (rtc_ism_inj โ€ฆ Hn2 H2) -c2 #H destruct //
59 qed-.
60
61 lemma rtc_ism_inv_max_eq_t (n) (c1) (c2): ๐Œโจn,c1 โˆจ c2โฉ โ†’ rtc_eq_t c1 c2 โ†’
62       โˆงโˆง ๐Œโจn,c1โฉ & ๐Œโจn,c2โฉ.
63 #n #c1 #c2 #H #Hc12
64 elim (rtc_ism_inv_max โ€ฆ H) -H #n1 #n2 #Hc1 #Hc2 #H destruct
65 lapply (rtc_ism_eq_t_trans โ€ฆ Hc1 โ€ฆ Hc12) -Hc12 #H
66 <(rtc_ism_inj โ€ฆ H โ€ฆ Hc2) -Hc2
67 <nmax_idem /2 width=1 by conj/
68 qed-.