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14
15 include "ground/arith/nat_plus.ma".
16 (* * it should not depend on pr_isi *)
17 include "ground/relocation/pr_isi_uni.ma".
18 include "ground/relocation/pr_after_isi.ma".
19
20 (* RELATIONAL COMPOSITION FOR PARTIAL RELOCATION MAPS ***********************)
21
22 (* Constructions with pr_uni ************************************************)
23
24 (*** after_uni *)
25 lemma pr_after_uni (h1) (h2): 𝐮❨h1❩ ⊚ 𝐮❨h2❩ ≘ 𝐮❨h2+h1❩.
26 #h1 @(nat_ind_succ … h1) -h1
27 /3 width=5 by pr_after_isi_sn, pr_after_next, eq_f/
28 qed.
29
30 (*** after_uni_sn_pushs *)
31 lemma pr_after_uni_sn_pushs (h):
32       ∀f. 𝐮❨h❩ ⊚ f ≘ ↑*[h]f.
33 #h @(nat_ind_succ … h) -h
34 /2 width=5 by pr_after_isi_sn, pr_after_next/
35 qed.
36
37 lemma pr_after_uni_isi_next (h1):
38       ∀f2. 𝐈❨f2❩ → 𝐮❨h1❩ ⊚ ↑f2 ≘ ↑𝐮❨h1❩.
39 #h1 @(nat_ind_succ … h1) -h1
40 /5 width=7 by pr_after_isi_dx, pr_after_eq_repl_back_sn, pr_after_next, pr_after_push, pr_isi_inv_eq_push/
41 qed.
42
43 lemma pr_after_uni_next_sn (h2):
44       ∀f1,f. ↑𝐮❨h2❩ ⊚ f1 ≘ f → 𝐮❨h2❩ ⊚ ↑f1 ≘ f.
45 #h2 @(nat_ind_succ … h2) -h2
46 [ #f1 #f #Hf
47   elim (pr_after_inv_next_sn … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
48   /4 width=7 by pr_after_isi_inv_sn, pr_after_isi_sn, pr_after_eq_repl_back, pr_eq_next/
49 | #h2 #IH #f1 #f #Hf
50   elim (pr_after_inv_next_sn … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
51   /3 width=5 by pr_after_next/
52 ]
53 qed.