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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 *)
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11 (* v GNU General Public License Version 2 *)
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15 include "static_2/syntax/ext2.ma".
16 include "static_2/relocation/lifts.ma".
18 (* GENERIC RELOCATION FOR BINDERS *******************************************)
20 definition liftsb: pr_map → relation bind ≝
23 interpretation "generic relocation (binder for local environments)"
24 'RLiftStar f I1 I2 = (liftsb f I1 I2).
26 interpretation "uniform relocation (binder for local environments)"
27 'RLift i I1 I2 = (liftsb (pr_uni i) I1 I2).
29 (* Basic_inversion lemmas **************************************************)
31 lemma liftsb_inv_unit_sn (f):
32 ∀I,Z2. ⇧*[f] BUnit I ≘ Z2 → Z2 = BUnit I.
33 /2 width=2 by ext2_inv_unit_sn/ qed-.
35 lemma liftsb_inv_pair_sn (f):
36 ∀Z2,I,V1. ⇧*[f] BPair I V1 ≘ Z2 →
37 ∃∃V2. ⇧*[f] V1 ≘ V2 & Z2 = BPair I V2.
38 /2 width=1 by ext2_inv_pair_sn/ qed-.
40 lemma liftsb_inv_unit_dx (f):
41 ∀I,Z1. ⇧*[f] Z1 ≘ BUnit I → Z1 = BUnit I.
42 /2 width=2 by ext2_inv_unit_dx/ qed-.
44 lemma liftsb_inv_pair_dx (f):
45 ∀Z1,I,V2. ⇧*[f] Z1 ≘ BPair I V2 →
46 ∃∃V1. ⇧*[f] V1 ≘ V2 & Z1 = BPair I V1.
47 /2 width=1 by ext2_inv_pair_dx/ qed-.
49 (* Basic properties *********************************************************)
51 lemma liftsb_eq_repl_back: ∀I1,I2. pr_eq_repl_back … (λf. ⇧*[f] I1 ≘ I2).
52 #I1 #I2 #f1 * -I1 -I2 /3 width=3 by lifts_eq_repl_back, ext2_pair/
55 lemma liftsb_refl (f): 𝐈❨f❩ → reflexive … (liftsb f).
56 /3 width=1 by lifts_refl, ext2_refl/ qed.
58 lemma liftsb_total: ∀I1,f. ∃I2. ⇧*[f] I1 ≘ I2.
59 * [2: #I #T1 #f elim (lifts_total T1 f) ]
60 /3 width=2 by ext2_unit, ext2_pair, ex_intro/
63 lemma liftsb_split_trans (f):
64 ∀I1,I2. ⇧*[f] I1 ≘ I2 → ∀f1,f2. f2 ⊚ f1 ≘ f →
65 ∃∃I. ⇧*[f1] I1 ≘ I & ⇧*[f2] I ≘ I2.
66 #f #I1 #I2 * -I1 -I2 /2 width=3 by ext2_unit, ex2_intro/
67 #I #V1 #V2 #HV12 #f1 #f2 #Hf elim (lifts_split_trans … HV12 … Hf) -f
68 /3 width=3 by ext2_pair, ex2_intro/
71 (* Basic forward lemmas *****************************************************)
73 lemma liftsb_fwd_isid (f):
74 ∀I1,I2. ⇧*[f] I1 ≘ I2 → 𝐈❨f❩ → I1 = I2.
75 #f #I1 #I2 * -I1 -I2 /3 width=3 by lifts_fwd_isid, eq_f2/