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14
15 include "ground/relocation/pr_tl_eq.ma".
16 include "ground/relocation/pr_pat_lt.ma".
17
18 (* POSITIVE APPLICATION FOR PARTIAL RELOCATION MAPS *************************)
19
20 (* Constructions with pr_eq *************************************************)
21
22 (*** at_eq_repl_back *)
23 corec lemma pr_pat_eq_repl_back (i1) (i2):
24             pr_eq_repl_back (λf. @❨i1,f❩ ≘ i2).
25 #f1 * -f1 -i1 -i2
26 [ #f1 #g1 #j1 #j2 #H #H1 #H2 #f2 #H12
27   cases (pr_eq_inv_push_sn … H12 … H) -g1 /2 width=2 by pr_pat_refl/
28 | #f1 #i1 #i2 #Hf1 #g1 #j1 #j2 #H #H1 #H2 #f2 #H12
29   cases (pr_eq_inv_push_sn … H12 … H) -g1 /3 width=7 by pr_pat_push/
30 | #f1 #i1 #i2 #Hf1 #g1 #j2 #H #H2 #f2 #H12
31   cases (pr_eq_inv_next_sn … H12 … H) -g1 /3 width=5 by pr_pat_next/
32 ]
33 qed-.
34
35 (*** at_eq_repl_fwd *)
36 lemma pr_pat_eq_repl_fwd (i1) (i2):
37       pr_eq_repl_fwd (λf. @❨i1,f❩ ≘ i2).
38 #i1 #i2 @pr_eq_repl_sym /2 width=3 by pr_pat_eq_repl_back/
39 qed-.
40
41 lemma pr_pat_eq (f): ⫯f ≡ f → ∀i. @❨i,f❩ ≘ i.
42 #f #Hf #i elim i -i
43 [ /3 width=3 by pr_pat_eq_repl_back, pr_pat_refl/
44 | /3 width=7 by pr_pat_eq_repl_back, pr_pat_push/
45 ]
46 qed.
47
48 (* Inversions with pr_eq ****************************************************)
49
50 corec lemma pr_pat_inv_eq (f):
51             (∀i. @❨i,f❩ ≘ i) → ⫯f ≡ f.
52 #Hf
53 lapply (Hf (𝟏)) #H
54 lapply (pr_pat_des_id … H) -H #H
55 cases H in Hf; -H #Hf
56 @pr_eq_push [3:|*: // ]
57 /3 width=7 by pr_pat_inv_succ_push_succ/
58 qed-.