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14
15 include "ground/relocation/pr_tls.ma".
16 include "ground/relocation/pr_nat.ma".
17 include "ground/relocation/pr_coafter.ma".
18
19 (* RELATIONAL CO-COMPOSITION FOR PARTIAL RELOCATION MAPS ********************)
20
21 (* Constructions with pr_nat and pr_tls *************************************)
22
23 (*** coafter_tls *)
24 lemma pr_coafter_tls_bi_tls (n2) (n1):
25       ∀f1,f2,f. @↑❨n1, f1❩ ≘ n2 →
26       f1 ~⊚ f2 ≘ f → ⫰*[n2]f1 ~⊚ ⫰*[n1]f2 ≘ ⫰*[n2]f.
27 #n2 @(nat_ind_succ … n2) -n2 [ #n1 | #n2 #IH * [| #n1 ] ] #f1 #f2 #f #Hf1 #Hf
28 [ elim (pr_nat_inv_zero_dx … Hf1) -Hf1 [ |*: // ] #g1 #Hg1 #H1 destruct //
29 | elim (pr_nat_inv_zero_succ … Hf1) -Hf1 [ |*: // ] #g1 #Hg1 #H1
30   elim (pr_coafter_inv_next_sn … Hf … H1) -Hf #g #Hg #H0 destruct
31   lapply (IH … Hg1 Hg) -IH -Hg1 -Hg //
32 | elim (pr_nat_inv_succ_dx … Hf1) -Hf1 [1,3: * |*: // ] #g1 [ #n1 ] #Hg1 [ #H ] #H1
33   [ elim (pr_coafter_inv_push_sn … Hf … H1) -Hf * #g2 #g #Hg #H2 #H0 destruct
34     lapply (IH … Hg1 Hg) -IH -Hg1 -Hg #H //
35   | elim (pr_coafter_inv_next_sn … Hf … H1) -Hf #g #Hg #H0 destruct
36     lapply (IH … Hg1 Hg) -IH -Hg1 -Hg #H //
37   ]
38 ]
39 qed.
40
41 (*** coafter_tls_O *)
42 lemma pr_coafter_tls_sn_tls:
43       ∀n,f1,f2,f. @↑❨𝟎, f1❩ ≘ n →
44       f1 ~⊚ f2 ≘ f → ⫰*[n]f1 ~⊚ f2 ≘ ⫰*[n]f.
45 /2 width=1 by pr_coafter_tls_bi_tls/ qed.