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14
15 include "ground/relocation/pr_pat_lt.ma".
16 include "ground/relocation/pr_nat.ma".
17 include "ground/relocation/pr_ist.ma".
18 include "ground/relocation/pr_after_pat.ma".
19
20 (* RELATIONAL COMPOSITION FOR PARTIAL RELOCATION MAPS ***********************)
21
22 (* Destructions with pr_ist *************************************************)
23
24 (*** after_istot_fwd *)
25 lemma pr_after_ist_des:
26       โˆ€f2,f1,f. f2 โŠš f1 โ‰˜ f โ†’ ๐“โจf2โฉ โ†’ ๐“โจf1โฉ โ†’ ๐“โจfโฉ.
27 #f2 #f1 #f #Hf #Hf2 #Hf1 #i1 elim (Hf1 i1) -Hf1
28 #i2 #Hf1 elim (Hf2 i2) -Hf2
29 /3 width=7 by pr_after_des_pat, ex_intro/
30 qed-.
31
32 (*** after_fwd_istot_dx *)
33 lemma pr_after_des_ist_dx:
34       โˆ€f2,f1,f. f2 โŠš f1 โ‰˜ f โ†’ ๐“โจfโฉ โ†’ ๐“โจf1โฉ.
35 #f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
36 #i2 #Hf elim (pr_after_pat_des โ€ฆ Hf โ€ฆ H) -f /2 width=2 by ex_intro/
37 qed-.
38
39 (*** after_fwd_istot_sn *)
40 lemma pr_after_des_ist_sn:
41       โˆ€f2,f1,f. f2 โŠš f1 โ‰˜ f โ†’ ๐“โจfโฉ โ†’ ๐“โจf2โฉ.
42 #f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
43 #i #Hf elim (pr_after_pat_des โ€ฆ Hf โ€ฆ H) -f
44 #i2 #Hf1 #Hf2 lapply (pr_pat_increasing โ€ฆ Hf1) -f1
45 #Hi12 elim (pr_pat_le_ex โ€ฆ Hf2 โ€ฆ Hi12) -i2 /2 width=2 by ex_intro/
46 qed-.
47
48 (*** after_at1_fwd *)
49 lemma pr_after_des_ist_pat:
50       โˆ€f1,i1,i2. @โจi1, f1โฉ โ‰˜ i2 โ†’ โˆ€f2. ๐“โจf2โฉ โ†’ โˆ€f. f2 โŠš f1 โ‰˜ f โ†’
51       โˆƒโˆƒi. @โจi2, f2โฉ โ‰˜ i & @โจi1, fโฉ โ‰˜ i.
52 #f1 #i1 #i2 #Hf1 #f2 #Hf2 #f #Hf elim (Hf2 i2) -Hf2
53 /3 width=8 by pr_after_des_pat, ex2_intro/
54 qed-.
55
56 lemma pr_after_des_ist_nat:
57       โˆ€f1,l1,l2. @โ†‘โจl1, f1โฉ โ‰˜ l2 โ†’ โˆ€f2. ๐“โจf2โฉ โ†’ โˆ€f. f2 โŠš f1 โ‰˜ f โ†’
58       โˆƒโˆƒl. @โ†‘โจl2, f2โฉ โ‰˜ l & @โ†‘โจl1, fโฉ โ‰˜ l.
59 #f1 #l1 #l2 #H1 #f2 #H2 #f #Hf
60 elim (pr_after_des_ist_pat โ€ฆ H1 โ€ฆ H2 โ€ฆ Hf) -f1 -H2
61 /2 width=3 by ex2_intro/
62 qed-.
63
64 (* Inversions with pr_ist ***************************************************)
65
66 (*** after_inv_istot *)
67 lemma pr_after_inv_ist:
68       โˆ€f2,f1,f. f2 โŠš f1 โ‰˜ f โ†’ ๐“โจfโฉ โ†’ โˆงโˆง ๐“โจf2โฉ & ๐“โจf1โฉ.
69 /3 width=4 by pr_after_des_ist_sn, pr_after_des_ist_dx, conj/ qed-.