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14
15 include "ground/relocation/pr_tl_eq.ma".
16 include "ground/relocation/pr_sand.ma".
17
18 (* RELATIONAL INTERSECTION FOR PARTIAL RELOCATION MAPS **********************)
19
20 (* Constructions with pr_eq *************************************************)
21
22 (*** sand_eq_repl_back1 *)
23 corec lemma pr_sand_eq_repl_back_sn:
24             ∀f2,f. pr_eq_repl_back … (λf1. f1 ⋒ f2 ≘ f).
25 #f2 #f #f1 * -f1 -f2 -f
26 #f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x #Hx
27 try cases (pr_eq_inv_push_sn … Hx … H1) try cases (pr_eq_inv_next_sn … Hx … H1) -g1
28 /3 width=7 by pr_sand_push_bi, pr_sand_next_push, pr_sand_push_next, pr_sand_next_bi/
29 qed-.
30
31 (*** sand_eq_repl_fwd1 *)
32 lemma pr_sand_eq_repl_fwd_sn:
33       ∀f2,f. pr_eq_repl_fwd … (λf1. f1 ⋒ f2 ≘ f).
34 #f2 #f @pr_eq_repl_sym /2 width=3 by pr_sand_eq_repl_back_sn/
35 qed-.
36
37 (*** sand_eq_repl_back2 *)
38 corec lemma pr_sand_eq_repl_back_dx:
39             ∀f1,f. pr_eq_repl_back … (λf2. f1 ⋒ f2 ≘ f).
40 #f1 #f #f2 * -f1 -f2 -f
41 #f1 #f2 #f #g1 #g2 #g #Hf #H #H2 #H0 #x #Hx
42 try cases (pr_eq_inv_push_sn … Hx … H2) try cases (pr_eq_inv_next_sn … Hx … H2) -g2
43 /3 width=7 by pr_sand_push_bi, pr_sand_next_push, pr_sand_push_next, pr_sand_next_bi/
44 qed-.
45
46 (*** sand_eq_repl_fwd2 *)
47 lemma sand_eq_repl_fwd_dx:
48       ∀f1,f. pr_eq_repl_fwd … (λf2. f1 ⋒ f2 ≘ f).
49 #f1 #f @pr_eq_repl_sym /2 width=3 by pr_sand_eq_repl_back_dx/
50 qed-.
51
52 (*** sand_eq_repl_back3 *)
53 corec lemma pr_sand_eq_repl_back:
54             ∀f1,f2. pr_eq_repl_back … (λf. f1 ⋒ f2 ≘ f).
55 #f1 #f2 #f * -f1 -f2 -f
56 #f1 #f2 #f #g1 #g2 #g #Hf #H #H2 #H0 #x #Hx
57 try cases (pr_eq_inv_push_sn … Hx … H0) try cases (pr_eq_inv_next_sn … Hx … H0) -g
58 /3 width=7 by pr_sand_push_bi, pr_sand_next_push, pr_sand_push_next, pr_sand_next_bi/
59 qed-.
60
61 (*** sand_eq_repl_fwd3 *)
62 lemma pr_sand_eq_repl_fwd:
63       ∀f1,f2. pr_eq_repl_fwd … (λf. f1 ⋒ f2 ≘ f).
64 #f1 #f2 @pr_eq_repl_sym /2 width=3 by pr_sand_eq_repl_back/
65 qed-.