2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
7 ||A|| This file is distributed under the terms of the
8 \ / GNU General Public License Version 2
10 V_______________________________________________________________ *)
12 include "lambda/terms.ma".
14 (* arguments: k is the depth (starts from 0), p is the height (starts from 0) *)
18 | Rel n ⇒ if_then_else T (leb (S n) k) (Rel n) (Rel (n+p))
19 | App m n ⇒ App (lift m k p) (lift n k p)
20 | Lambda m n ⇒ Lambda (lift m k p) (lift n (k+1) p)
21 | Prod m n ⇒ Prod (lift m k p) (lift n (k+1) p)
22 | D n ⇒ D (lift n k p)
26 ndefinition lift ≝ λt.λp.lift_aux t 0 p.
28 notation "↑ ^ n ( M )" non associative with precedence 40 for @{'Lift O $M}.
29 notation "↑ _ k ^ n ( M )" non associative with precedence 40 for @{'Lift $n $k $M}.
31 (* interpretation "Lift" 'Lift n M = (lift M n). *)
32 interpretation "Lift" 'Lift n k M = (lift M k n).
34 (*** properties of lift ***)
36 lemma lift_0: ∀t:T.∀k. lift t k 0 = t.
37 #t (elim t) normalize // #n #k cases (leb (S n) k) normalize //
40 (* nlemma lift_0: ∀t:T. lift t 0 = t.
41 #t; nelim t; nnormalize; //; nqed. *)
43 lemma lift_sort: ∀i,k,n. lift (Sort i) k n = Sort i.
46 lemma lift_rel: ∀i,n. lift (Rel i) 0 n = Rel (i+n).
49 lemma lift_rel1: ∀i.lift (Rel i) 0 1 = Rel (S i).
50 #i (change with (lift (Rel i) 0 1 = Rel (1 + i))) //
53 lemma lift_rel_lt : ∀n,k,i. i < k → lift (Rel i) k n = Rel i.
54 #n #k #i #ltik change with
55 (if_then_else ? (leb (S i) k) (Rel i) (Rel (i+n)) = Rel i)
56 >(le_to_leb_true … ltik) //
59 lemma lift_rel_ge : ∀n,k,i. k ≤ i → lift (Rel i) k n = Rel (i+n).
60 #n #k #i #leki change with
61 (if_then_else ? (leb (S i) k) (Rel i) (Rel (i+n)) = Rel (i+n))
62 >lt_to_leb_false // @le_S_S //
65 lemma lift_lift: ∀t.∀m,j.j ≤ m → ∀n,k.
66 lift (lift t k m) (j+k) n = lift t k (m+n).
67 #t #i #j #h (elim t) normalize // #n #h #k
68 @(leb_elim (S n) k) #Hnk normalize
69 [>(le_to_leb_true (S n) (j+k) ?) normalize /2/
70 |>(lt_to_leb_false (S n+i) (j+k) ?)
71 normalize // @le_S_S >(commutative_plus j k)
72 @le_plus // @not_lt_to_le /2/
76 lemma lift_lift_up: ∀n,m,t,k,i.
77 lift (lift t i m) (m+k+i) n = lift (lift t (k+i) n) i m.
79 [1,3,4,5,6: normalize //
80 |#p #k #i @(leb_elim i p);
81 [#leip >lift_rel_ge // @(leb_elim (k+i) p);
83 [>lift_rel_ge // >lift_rel_ge // @(transitive_le … leip) //
84 |>associative_plus >commutative_plus @monotonic_le_plus_l //
86 |#lefalse (cut (p < k+i)) [@not_le_to_lt //] #ltpki
87 >lift_rel_lt; [|>associative_plus >commutative_plus @monotonic_lt_plus_r //]
88 >lift_rel_lt // >lift_rel_ge //
90 |#lefalse (cut (p < i)) [@not_le_to_lt //] #ltpi
91 >lift_rel_lt // >lift_rel_lt; [|@(lt_to_le_to_lt … ltpi) //]
92 >lift_rel_lt; [|@(lt_to_le_to_lt … ltpi) //]
98 lemma lift_lift_up_sym: ∀n,m,t,k,i.
99 lift (lift t i m) (m+i+k) n = lift (lift t (i+k) n) i m.
102 lemma lift_lift_up_01: ∀t,k,p. (lift (lift t k p) 0 1 = lift (lift t 0 1) (k+1) p).
103 #t #k #p <(lift_lift_up_sym ? ? ? ? 0) //
106 lemma lift_lift1: ∀t.∀i,j,k.
107 lift(lift t k j) k i = lift t k (j+i).
110 lemma lift_lift2: ∀t.∀i,j,k.
111 lift (lift t k j) (j+k) i = lift t k (j+i).
115 nlemma lift_lift: ∀t.∀i,j. lift (lift t j) i = lift t (j+i).
116 nnormalize; //; nqed. *)
118 (********************* context lifting ********************)
120 let rec Lift G p ≝ match G with
122 | cons t F ⇒ cons … (lift t (|F|) p) (Lift F p)
125 interpretation "Lift (context)" 'Lift p G = (Lift G p).
127 lemma Lift_cons: ∀k,Gk. k = |Gk| →
128 ∀p,t. Lift (t::Gk) p = lift t k p :: Lift Gk p.
132 lemma Lift_length: ∀p,G. |Lift G p| = |G|.
133 #p #G elim G -G; normalize //