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-delta/syntax/lenv.ma".
13 include "lambda-delta/substitution/lift_defs.ma".
15 (* DROPPING *****************************************************************)
17 inductive drop: lenv → nat → nat → lenv → Prop ≝
18 | drop_refl: ∀L. drop L 0 0 L
19 | drop_drop: ∀L1,L2,I,V,e. drop L1 0 e L2 → drop (L1. 𝕓{I} V) 0 (e + 1) L2
20 | drop_skip: ∀L1,L2,I,V1,V2,d,e.
21 drop L1 d e L2 → ↑[d,e] V2 ≡ V1 →
22 drop (L1. 𝕓{I} V1) (d + 1) e (L2. 𝕓{I} V2)
25 interpretation "dropping" 'RLift L2 d e L1 = (drop L1 d e L2).
27 (* the basic inversion lemmas ***********************************************)
29 lemma drop_inv_drop1_aux: ∀d,e,L2,L1. ↑[d, e] L2 ≡ L1 → 0 < e → d = 0 →
30 ∀K,I,V. L1 = K. 𝕓{I} V → ↑[d, e - 1] L2 ≡ K.
31 #d #e #L2 #L1 #H elim H -H d e L2 L1
32 [ #L #H elim (lt_refl_false … H)
33 | #L1 #L2 #I #V #e #HL12 #_ #_ #_ #K #J #W #H destruct -L1 I V //
34 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #_ #_ #H elim (plus_S_eq_O_false … H)
38 lemma drop_inv_drop1: ∀e,L2,K,I,V. ↑[0, e] L2 ≡ K. 𝕓{I} V → 0 < e →
42 lemma drop_inv_skip2_aux: ∀d,e,L1,L2. ↑[d, e] L2 ≡ L1 → 0 < d →
43 ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
44 ∃∃K1,V1. ↑[d - 1, e] K2 ≡ K1 &
47 #d #e #L1 #L2 #H elim H -H d e L1 L2
48 [ #L #H elim (lt_refl_false … H)
49 | #L1 #L2 #I #V #e #_ #_ #H elim (lt_refl_false … H)
50 | #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #_ #I #L2 #V2 #H destruct -X Y Z;
55 lemma drop_inv_skip2: ∀d,e,I,L1,K2,V2. ↑[d, e] K2. 𝕓{I} V2 ≡ L1 → 0 < d →
56 ∃∃K1,V1. ↑[d - 1, e] K2 ≡ K1 & ↑[d - 1, e] V2 ≡ V1 &