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14
15 include "ground/relocation/pr_tl_eq.ma".
16 include "ground/relocation/pr_after.ma".
17
18 (* RELATIONAL COMPOSITION FOR PARTIAL RELOCATION MAPS ***********************)
19
20 (* Constructions with pr_eq *************************************************)
21
22 (*** after_eq_repl_back2 *)
23 corec lemma pr_after_eq_repl_back_sn:
24             ∀f1,f. pr_eq_repl_back (λf2. f2 ⊚ f1 ≘ f).
25 #f1 #f #f2 * -f2 -f1 -f
26 #f21 #f1 #f #g21 [1,2: #g1 ] #g #Hf #H21 [1,2: #H1 ] #H #g22 #H0
27 [ cases (pr_eq_inv_push_sn …  H0 …  H21) -g21 /3 width=7 by pr_after_refl/
28 | cases (pr_eq_inv_push_sn …  H0 …  H21) -g21 /3 width=7 by pr_after_push/
29 | cases (pr_eq_inv_next_sn …  H0 …  H21) -g21 /3 width=5 by pr_after_next/
30 ]
31 qed-.
32
33 (*** after_eq_repl_fwd2 *)
34 lemma pr_after_eq_repl_fwd_sn:
35       ∀f1,f. pr_eq_repl_fwd (λf2. f2 ⊚ f1 ≘ f).
36 #f1 #f @pr_eq_repl_sym /2 width=3 by pr_after_eq_repl_back_sn/
37 qed-.
38
39 (*** after_eq_repl_back1 *)
40 corec lemma pr_after_eq_repl_back_dx:
41             ∀f2,f. pr_eq_repl_back (λf1. f2 ⊚ f1 ≘ f).
42 #f2 #f #f1 * -f2 -f1 -f
43 #f2 #f11 #f #g2 [1,2: #g11 ] #g #Hf #H2 [1,2: #H11 ] #H #g2 #H0
44 [ cases (pr_eq_inv_push_sn …  H0 …  H11) -g11 /3 width=7 by pr_after_refl/
45 | cases (pr_eq_inv_next_sn …  H0 …  H11) -g11 /3 width=7 by pr_after_push/
46 | @(pr_after_next … H2 H) /2 width=5 by/
47 ]
48 qed-.
49
50 (*** after_eq_repl_fwd1 *)
51 lemma pr_after_eq_repl_fwd_dx:
52       ∀f2,f. pr_eq_repl_fwd (λf1. f2 ⊚ f1 ≘ f).
53 #f2 #f @pr_eq_repl_sym /2 width=3 by pr_after_eq_repl_back_dx/
54 qed-.
55
56 (*** after_eq_repl_back0 *)
57 corec lemma pr_after_eq_repl_back:
58             ∀f1,f2. pr_eq_repl_back (λf. f2 ⊚ f1 ≘ f).
59 #f2 #f1 #f * -f2 -f1 -f
60 #f2 #f1 #f01 #g2 [1,2: #g1 ] #g01 #Hf01 #H2 [1,2: #H1 ] #H01 #g02 #H0
61 [ cases (pr_eq_inv_push_sn …  H0 …  H01) -g01 /3 width=7 by pr_after_refl/
62 | cases (pr_eq_inv_next_sn …  H0 …  H01) -g01 /3 width=7 by pr_after_push/
63 | cases (pr_eq_inv_next_sn …  H0 …  H01) -g01 /3 width=5 by pr_after_next/
64 ]
65 qed-.
66
67 (*** after_eq_repl_fwd0 *)
68 lemma pr_after_eq_repl_fwd:
69       ∀f2,f1. pr_eq_repl_fwd (λf. f2 ⊚ f1 ≘ f).
70 #f2 #f1 @pr_eq_repl_sym /2 width=3 by pr_after_eq_repl_back/
71 qed-.