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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "basic_2/syntax/ext2.ma".
16 include "basic_2/relocation/lifts.ma".
18 (* GENERIC RELOCATION FOR BINDERS *******************************************)
20 definition liftsb: rtmap → relation bind ≝
23 interpretation "uniform relocation (binder for local environments)"
24 'RLiftStar i I1 I2 = (liftsb (uni i) I1 I2).
26 interpretation "generic relocation (binder for local environments)"
27 'RLiftStar f I1 I2 = (liftsb f I1 I2).
29 (* Basic_inversion lemmas **************************************************)
31 lemma liftsb_inv_unit_sn: ∀f,I,Z2. ⬆*[f] BUnit I ≘ Z2 → Z2 = BUnit I.
32 /2 width=2 by ext2_inv_unit_sn/ qed-.
34 lemma liftsb_inv_pair_sn: ∀f:rtmap. ∀Z2,I,V1. ⬆*[f] BPair I V1 ≘ Z2 →
35 ∃∃V2. ⬆*[f] V1 ≘ V2 & Z2 = BPair I V2.
36 /2 width=1 by ext2_inv_pair_sn/ qed-.
38 lemma liftsb_inv_unit_dx: ∀f,I,Z1. ⬆*[f] Z1 ≘ BUnit I → Z1 = BUnit I.
39 /2 width=2 by ext2_inv_unit_dx/ qed-.
41 lemma liftsb_inv_pair_dx: ∀f:rtmap. ∀Z1,I,V2. ⬆*[f] Z1 ≘ BPair I V2 →
42 ∃∃V1. ⬆*[f] V1 ≘ V2 & Z1 = BPair I V1.
43 /2 width=1 by ext2_inv_pair_dx/ qed-.
45 (* Basic properties *********************************************************)
47 lemma liftsb_eq_repl_back: ∀I1,I2. eq_repl_back … (λf. ⬆*[f] I1 ≘ I2).
48 #I1 #I2 #f1 * -I1 -I2 /3 width=3 by lifts_eq_repl_back, ext2_pair/
51 lemma liftsb_refl: ∀f. 𝐈⦃f⦄ → reflexive … (liftsb f).
52 /3 width=1 by lifts_refl, ext2_refl/ qed.
54 lemma liftsb_total: ∀I1,f. ∃I2. ⬆*[f] I1 ≘ I2.
55 * [2: #I #T1 #f elim (lifts_total T1 f) ]
56 /3 width=2 by ext2_unit, ext2_pair, ex_intro/
59 lemma liftsb_split_trans: ∀f,I1,I2. ⬆*[f] I1 ≘ I2 →
61 ∃∃I. ⬆*[f1] I1 ≘ I & ⬆*[f2] I ≘ I2.
62 #f #I1 #I2 * -I1 -I2 /2 width=3 by ext2_unit, ex2_intro/
63 #I #V1 #V2 #HV12 #f1 #f2 #Hf elim (lifts_split_trans … HV12 … Hf) -f
64 /3 width=3 by ext2_pair, ex2_intro/
67 (* Basic forward lemmas *****************************************************)
69 lemma liftsb_fwd_isid: ∀f,I1,I2. ⬆*[f] I1 ≘ I2 → 𝐈⦃f⦄ → I1 = I2.
70 #f #I1 #I2 * -I1 -I2 /3 width=3 by lifts_fwd_isid, eq_f2/