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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 "ground_2/notation/functions/liftstar_2.ma".
16 include "ground_2/relocation/rtmap_eq.ma".
18 (* RELOCATION MAP ***********************************************************)
20 rec definition pushs (f:rtmap) (n:nat) on n: rtmap ≝ match n with
21 [ O ⇒ f | S m ⇒ ↑(pushs f m) ].
23 interpretation "pushs (rtmap)" 'LiftStar n f = (pushs f n).
25 (* Basic_inversion lemmas *****************************************************)
27 lemma eq_inv_pushs_sn: ∀n,f1,g2. ↑*[n] f1 ≗ g2 →
28 ∃∃f2. f1 ≗ f2 & ↑*[n] f2 = g2.
29 #n elim n -n /2 width=3 by ex2_intro/
30 #n #IH #f1 #g2 #H elim (eq_inv_px … H) -H [|*: // ]
31 #f0 #Hf10 #H1 elim (IH … Hf10) -IH -Hf10 #f2 #Hf12 #H2 destruct
32 /2 width=3 by ex2_intro/
35 lemma eq_inv_pushs_dx: ∀n,f2,g1. g1 ≗ ↑*[n] f2 →
36 ∃∃f1. f1 ≗ f2 & ↑*[n] f1 = g1.
37 #n elim n -n /2 width=3 by ex2_intro/
38 #n #IH #f2 #g1 #H elim (eq_inv_xp … H) -H [|*: // ]
39 #f0 #Hf02 #H1 elim (IH … Hf02) -IH -Hf02 #f1 #Hf12 #H2 destruct
40 /2 width=3 by ex2_intro/
43 (* Basic properties *********************************************************)
45 lemma pushs_O: ∀f. f = ↑*[0] f.
48 lemma pushs_S: ∀f,n. ↑↑*[n] f = ↑*[⫯n] f.
51 lemma pushs_eq_repl: ∀n. eq_repl (λf1,f2. ↑*[n] f1 ≗ ↑*[n] f2).
52 #n elim n -n /3 width=5 by eq_push/
55 (* Advanced properties ******************************************************)
57 lemma pushs_xn: ∀n,f. ↑*[n] ↑f = ↑*[⫯n] f.