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14
15 include "ground/relocation/pr_pat_tls.ma".
16 include "ground/relocation/pr_coafter_nat_tls.ma".
17
18 (* RELATIONAL CO-COMPOSITION FOR PARTIAL RELOCATION MAPS ********************)
19
20 (* Constructions with pr_pat and pr_tls *************************************)
21
22 (* Note: this does not require ↑ first and second j *)
23 (*** coafter_tls_succ *)
24 lemma pr_coafter_tls_tl_tls:
25       ∀g2,g1,g. g2 ~⊚ g1 ≘ g →
26       ∀j. @❨𝟏, g2❩ ≘ j → ⫰*[j]g2 ~⊚ ⫰g1 ≘ ⫰*[j]g.
27 #g2 #g1 #g #Hg #j #Hg2
28 lapply (pr_nat_pred_bi … Hg2) -Hg2 #Hg2
29 lapply (pr_coafter_tls_bi_tls … Hg2 … Hg) -Hg #Hg
30 lapply (pr_pat_unit_succ_tls … Hg2) -Hg2 #H
31 elim (pr_pat_inv_unit_bi … H) -H [ |*: // ] #f2 #H2
32 elim (pr_coafter_inv_push_sn … Hg … H2) -Hg * #f1 #f #Hf #H1 #H0
33 >(npsucc_pred j) <pr_tls_succ <pr_tls_succ //
34 qed.
35
36 (* Note: parked for now
37 lemma coafter_fwd_xpx_pushs:
38       ∀g2,f1,g,i,j. @❨i, g2❩ ≘ j → g2 ~⊚ ⫯*[i]⫯f1 ≘ g →
39       ∃∃f.  ⫰*[↑j]g2 ~⊚ f1 ≘ f & ⫯*[j]⫯f = g.
40 #g2 #g1 #g #i #j #Hg2 <pushs_xn #Hg(coafter_fwd_pushs … Hg Hg2) #f #H0 destruct
41 lapply (coafter_tls … Hg2 Hg) -Hg <tls_pushs <tls_pushs #Hf
42 lapply (at_inv_tls … Hg2) -Hg2 #H
43 lapply (coafter_eq_repl_fwd2 … Hf … H) -H -Hf #Hf
44 elim (coafter_inv_ppx … Hf) [|*: // ] -Hf #g #Hg #H destruct
45 /2 width=3 by ex2_intro/
46 qed-.
47
48 lemma coafter_fwd_xnx_pushs:
49       ∀g2,f1,g,i,j. @❨i, g2❩ ≘ j → g2 ~⊚ ⫯*[i]↑f1 ≘ g →
50       ∃∃f. ⫰*[↑j]g2 ~⊚ f1 ≘ f & ⫯*[j] ↑f = g.
51 #g2 #g1 #g #i #j #Hg2 #Hg
52 elim (coafter_fwd_pushs … Hg Hg2) #f #H0 destruct
53 lapply (coafter_tls … Hg2 Hg) -Hg <tls_pushs <tls_pushs #Hf
54 lapply (at_inv_tls … Hg2) -Hg2 #H
55 lapply (coafter_eq_repl_fwd2 … Hf … H) -H -Hf #Hf
56 elim (coafter_inv_pnx … Hf) [|*: // ] -Hf #g #Hg #H destruct
57 /2 width=3 by ex2_intro/
58 qed-.
59 *)