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" 'RDrop L1 d e L2 = (drop L1 d e L2).
27 (* Basic properties *********************************************************)
29 lemma drop_drop_lt: ∀L1,L2,I,V,e.
30 ↓[0, e - 1] L1 ≡ L2 → 0 < e → ↓[0, e] L1. 𝕓{I} V ≡ L2.
31 #L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) /2/
34 (* Basic inversion lemmas ***************************************************)
36 lemma drop_inv_refl_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
37 #d #e #L1 #L2 #H elim H -H d e L1 L2
39 | #L1 #L2 #I #V #e #_ #_ #_ #H
40 elim (plus_S_eq_O_false … H)
41 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #_ #H
42 elim (plus_S_eq_O_false … H)
46 lemma drop_inv_refl: ∀L1,L2. ↓[0, 0] L1 ≡ L2 → L1 = L2.
49 lemma drop_inv_O1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 →
50 ∀K,I,V. L1 = K. 𝕓{I} V →
51 (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
52 (0 < e ∧ ↓[d, e - 1] K ≡ L2).
53 #d #e #L1 #L2 #H elim H -H d e L1 L2
55 | #L1 #L2 #I #V #e #HL12 #_ #_ #K #J #W #H destruct -L1 I V /3/
56 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #_ #H elim (plus_S_eq_O_false … H)
60 lemma drop_inv_O1: ∀e,K,I,V,L2. ↓[0, e] K. 𝕓{I} V ≡ L2 →
61 (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
62 (0 < e ∧ ↓[0, e - 1] K ≡ L2).
65 lemma drop_inv_drop1: ∀e,K,I,V,L2.
66 ↓[0, e] K. 𝕓{I} V ≡ L2 → 0 < e → ↓[0, e - 1] K ≡ L2.
67 #e #K #I #V #L2 #H #He
68 elim (drop_inv_O1 … H) -H * // #H destruct -e;
69 elim (lt_refl_false … He)
72 lemma drop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
73 ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
74 ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 &
77 #d #e #L1 #L2 #H elim H -H d e L1 L2
78 [ #L #H elim (lt_refl_false … H)
79 | #L1 #L2 #I #V #e #_ #_ #H elim (lt_refl_false … H)
80 | #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #_ #I #L2 #V2 #H destruct -X Y Z;
85 lemma drop_inv_skip2: ∀d,e,I,L1,K2,V2. ↓[d, e] L1 ≡ K2. 𝕓{I} V2 → 0 < d →
86 ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 & ↑[d - 1, e] V2 ≡ V1 &