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14
15 include "ground/relocation/pr_isi.ma".
16 include "ground/relocation/pr_after_after.ma".
17
18 (* RELATIONAL COMPOSITION FOR PARTIAL RELOCATION MAPS ***********************)
19
20 (* Constructions with pr_isi ************************************************)
21
22 (*** after_isid_sn *)
23 corec lemma pr_after_isi_sn:
24             âˆ€f1. đˆâ¨f1❊ â†’ âˆ€f2. f1 âŠš f2 â‰˜ f2.
25 #f1 * -f1
26 #f1 #g1 #Hf1 #H1 #f2 cases (pr_map_split_tl f2) #H2
27 /3 width=7 by pr_after_push, pr_after_refl/
28 qed.
29
30 (*** after_isid_dx *)
31 corec lemma pr_after_isi_dx:
32             âˆ€f2. đˆâ¨f2❊ â†’ âˆ€f1. f1 âŠš f2 â‰˜ f1.
33 #f2 * -f2
34 #f2 #g2 #Hf2 #H2 #f1 cases (pr_map_split_tl f1) #H1
35 [ /3 width=7 by pr_after_refl/
36 | @(pr_after_next â€Ś H1 H1) /3 width=3 by pr_isi_push/
37 ]
38 qed.
39
40 (* Destructions with pr_isi *************************************************)
41
42 (*** after_isid_inv_sn *)
43 lemma pr_after_isi_inv_sn:
44       âˆ€f1,f2,f. f1 âŠš f2 â‰˜ f â†’ đˆâ¨f1❊ â†’ f2 â‰Ą f.
45 /3 width=6 by pr_after_isi_sn, pr_after_mono/ qed-.
46
47 (*** after_isid_inv_dx *)
48 lemma pr_after_isi_inv_dx:
49       âˆ€f1,f2,f. f1 âŠš f2 â‰˜ f â†’ đˆâ¨f2❊ â†’ f1 â‰Ą f.
50 /3 width=6 by pr_after_isi_dx, pr_after_mono/ qed-.
51
52 (*** after_fwd_isid1 *)
53 corec lemma pr_after_des_isi_sn:
54             âˆ€f1,f2,f. f1 âŠš f2 â‰˜ f â†’ đˆâ¨f❊ â†’ đˆâ¨f1❊.
55 #f1 #f2 #f * -f1 -f2 -f
56 #f1 #f2 #f #g1 [1,2: #g2 ] #g #Hf #H1 [1,2: #H2 ] #H0 #H
57 [ /4 width=6 by pr_isi_inv_push, pr_isi_push/ ]
58 cases (pr_isi_inv_next â€Ś H â€Ś H0)
59 qed-.
60
61 (*** after_fwd_isid2 *)
62 corec lemma pr_after_des_isi_dx:
63             âˆ€f1,f2,f. f1 âŠš f2 â‰˜ f â†’ đˆâ¨f❊ â†’ đˆâ¨f2❊.
64 #f1 #f2 #f * -f1 -f2 -f
65 #f1 #f2 #f #g1 [1,2: #g2 ] #g #Hf #H1 [1,2: #H2 ] #H0 #H
66 [ /4 width=6 by pr_isi_inv_push, pr_isi_push/ ]
67 cases (pr_isi_inv_next â€Ś H â€Ś H0)
68 qed-.
69
70 (*** after_inv_isid3 *)
71 lemma pr_after_inv_isi:
72       âˆ€f1,f2,f. f1 âŠš f2 â‰˜ f â†’ đˆâ¨f❊ â†’ đˆâ¨f1❊ âˆ§ đˆâ¨f2❊.
73 /3 width=4 by pr_after_des_isi_dx, pr_after_des_isi_sn, conj/ qed-.