1 (**************************************************************************)
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 *)
13 (**************************************************************************)
15 include "basic_2/relocation/lifts.ma".
17 (* GENERIC RELOCATION FOR TERMS *********************************************)
19 (* Main properties **********************************************************)
21 (* Basic_1: includes: lift_gen_lift *)
22 (* Basic_2A1: includes: lift_div_le lift_div_be *)
23 theorem lifts_div: ∀T,T2,f2. ⬆*[f2] T2 ≡ T → ∀T1,f. ⬆*[f] T1 ≡ T →
24 ∀f1. f2 ⊚ f1 ≡ f → ⬆*[f1] T1 ≡ T2.
25 #T #T2 #f2 #H elim H -T -T2 -f2
26 [ #s #f2 #T1 #f #H >(lifts_inv_sort2 … H) -T1 //
27 | #i2 #i #f2 #Hi2 #T1 #f #H #f1 #Ht21 elim (lifts_inv_lref2 … H) -H
28 #i1 #Hi1 #H destruct /3 width=6 by lifts_lref, after_fwd_at1/
29 | #l #f2 #T1 #f #H >(lifts_inv_gref2 … H) -T1 //
30 | #p #I #W2 #W #U2 #U #f2 #_ #_ #IHW #IHU #T1 #f #H
31 elim (lifts_inv_bind2 … H) -H #W1 #U1 #HW1 #HU1 #H destruct
32 /4 width=3 by lifts_bind, after_O2/
33 | #I #W2 #W #U2 #U #f2 #_ #_ #IHW #IHU #T1 #f #H
34 elim (lifts_inv_flat2 … H) -H #W1 #U1 #HW1 #HU1 #H destruct
35 /3 width=3 by lifts_flat/
39 (* Basic_1: was: lift1_lift1 (left to right) *)
40 (* Basic_1: includes: lift_free (left to right) lift_d lift1_xhg (right to left) lift1_free (right to left) *)
41 (* Basic_2A1: includes: lift_trans_be lift_trans_le lift_trans_ge lifts_lift_trans_le lifts_lift_trans *)
42 theorem lifts_trans: ∀T1,T,f1. ⬆*[f1] T1 ≡ T → ∀T2,f2. ⬆*[f2] T ≡ T2 →
43 ∀f. f2 ⊚ f1 ≡ f → ⬆*[f] T1 ≡ T2.
44 #T1 #T #f1 #H elim H -T1 -T -f1
45 [ #s #f1 #T2 #f2 #H >(lifts_inv_sort1 … H) -T2 //
46 | #i1 #i #f1 #Hi1 #T2 #f2 #H #f #Ht21 elim (lifts_inv_lref1 … H) -H
47 #i2 #Hi2 #H destruct /3 width=6 by lifts_lref, after_fwd_at/
48 | #l #f1 #T2 #f2 #H >(lifts_inv_gref1 … H) -T2 //
49 | #p #I #W1 #W #U1 #U #f1 #_ #_ #IHW #IHU #T2 #f2 #H
50 elim (lifts_inv_bind1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
51 /4 width=3 by lifts_bind, after_O2/
52 | #I #W1 #W #U1 #U #f1 #_ #_ #IHW #IHU #T2 #f2 #H
53 elim (lifts_inv_flat1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
54 /3 width=3 by lifts_flat/
58 (* Basic_2A1: includes: lift_conf_O1 lift_conf_be *)
59 theorem lifts_conf: ∀T,T1,f1. ⬆*[f1] T ≡ T1 → ∀T2,f. ⬆*[f] T ≡ T2 →
60 ∀f2. f2 ⊚ f1 ≡ f → ⬆*[f2] T1 ≡ T2.
61 #T #T1 #f1 #H elim H -T -T1 -f1
62 [ #s #f1 #T2 #f #H >(lifts_inv_sort1 … H) -T2 //
63 | #i #i1 #f1 #Hi1 #T2 #f #H #f2 #Ht21 elim (lifts_inv_lref1 … H) -H
64 #i2 #Hi2 #H destruct /3 width=6 by lifts_lref, after_fwd_at2/
65 | #l #f1 #T2 #f #H >(lifts_inv_gref1 … H) -T2 //
66 | #p #I #W #W1 #U #U1 #f1 #_ #_ #IHW #IHU #T2 #f #H
67 elim (lifts_inv_bind1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
68 /4 width=3 by lifts_bind, after_O2/
69 | #I #W #W1 #U #U1 #f1 #_ #_ #IHW #IHU #T2 #f #H
70 elim (lifts_inv_flat1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
71 /3 width=3 by lifts_flat/
75 (* Advanced proprerties *****************************************************)
77 (* Basic_2A1: includes: lift_inj *)
78 lemma lifts_inj: ∀T1,U,f. ⬆*[f] T1 ≡ U → ∀T2. ⬆*[f] T2 ≡ U → T1 = T2.
79 #T1 #U #f #H1 #T2 #H2 lapply (isid_after_dx 𝐈𝐝 … f)
80 /3 width=6 by lifts_div, lifts_fwd_isid/
83 (* Basic_2A1: includes: lift_mono *)
84 lemma lifts_mono: ∀T,U1,f. ⬆*[f] T ≡ U1 → ∀U2. ⬆*[f] T ≡ U2 → U1 = U2.
85 #T #U1 #f #H1 #U2 #H2 lapply (isid_after_sn 𝐈𝐝 … f)
86 /3 width=6 by lifts_conf, lifts_fwd_isid/