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/notation/relations/rdropstar_3.ma".
16 include "basic_2/notation/relations/rdropstar_4.ma".
17 include "basic_2/relocation/ldrop.ma".
18 include "basic_2/substitution/gr2_minus.ma".
19 include "basic_2/substitution/lifts.ma".
21 (* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
23 inductive ldrops (s:bool): list2 nat nat → relation lenv ≝
24 | ldrops_nil : ∀L. ldrops s (⟠) L L
25 | ldrops_cons: ∀L1,L,L2,des,d,e.
26 ldrops s des L1 L → ⇩[s, d, e] L ≡ L2 → ldrops s ({d, e} @ des) L1 L2
29 interpretation "iterated slicing (local environment) abstract"
30 'RDropStar s des T1 T2 = (ldrops s des T1 T2).
32 interpretation "iterated slicing (local environment) general"
33 'RDropStar des T1 T2 = (ldrops true des T1 T2).
36 (* Basic inversion lemmas ***************************************************)
38 fact ldrops_inv_nil_aux: ∀L1,L2,s,des. ⇩*[s, des] L1 ≡ L2 → des = ⟠ → L1 = L2.
39 #L1 #L2 #s #des * -L1 -L2 -des //
40 #L1 #L #L2 #d #e #des #_ #_ #H destruct
43 (* Basic_1: was: drop1_gen_pnil *)
44 lemma ldrops_inv_nil: ∀L1,L2,s. ⇩*[s, ⟠] L1 ≡ L2 → L1 = L2.
45 /2 width=4 by ldrops_inv_nil_aux/ qed-.
47 fact ldrops_inv_cons_aux: ∀L1,L2,s,des. ⇩*[s, des] L1 ≡ L2 →
48 ∀d,e,tl. des = {d, e} @ tl →
49 ∃∃L. ⇩*[s, tl] L1 ≡ L & ⇩[s, d, e] L ≡ L2.
50 #L1 #L2 #s #des * -L1 -L2 -des
51 [ #L #d #e #tl #H destruct
52 | #L1 #L #L2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct
53 /2 width=3 by ex2_intro/
57 (* Basic_1: was: drop1_gen_pcons *)
58 lemma ldrops_inv_cons: ∀L1,L2,s,d,e,des. ⇩*[s, {d, e} @ des] L1 ≡ L2 →
59 ∃∃L. ⇩*[s, des] L1 ≡ L & ⇩[s, d, e] L ≡ L2.
60 /2 width=3 by ldrops_inv_cons_aux/ qed-.
62 lemma ldrops_inv_skip2: ∀I,s,des,des2,i. des ▭ i ≡ des2 →
63 ∀L1,K2,V2. ⇩*[s, des2] L1 ≡ K2. ⓑ{I} V2 →
64 ∃∃K1,V1,des1. des + 1 ▭ i + 1 ≡ des1 + 1 &
68 #I #s #des #des2 #i #H elim H -des -des2 -i
70 >(ldrops_inv_nil … H) -L1 /2 width=7 by lifts_nil, minuss_nil, ex4_3_intro, ldrops_nil/
71 | #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H
72 elim (ldrops_inv_cons … H) -H #L #HL1 #H
73 elim (ldrop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ #K #V >minus_plus #HK2 #HV2 #H destruct
74 elim (IHdes2 … HL1) -IHdes2 -HL1 #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct
75 @(ex4_3_intro … K1 V1 … ) // [3,4: /2 width=7 by lifts_cons, ldrops_cons/ | skip ]
76 normalize >plus_minus /3 width=1 by minuss_lt, lt_minus_to_plus/ (**) (* explicit constructors *)
77 | #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H
78 elim (IHdes2 … H) -IHdes2 -H #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct
79 /4 width=7 by minuss_ge, ex4_3_intro, le_S_S/
83 (* Basic properties *********************************************************)
85 (* Basic_1: was: drop1_skip_bind *)
86 lemma ldrops_skip: ∀L1,L2,s,des. ⇩*[s, des] L1 ≡ L2 → ∀V1,V2. ⇧*[des] V2 ≡ V1 →
87 ∀I. ⇩*[s, des + 1] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
88 #L1 #L2 #s #des #H elim H -L1 -L2 -des
90 >(lifts_inv_nil … HV12) -HV12 //
91 | #L1 #L #L2 #des #d #e #_ #HL2 #IHL #V1 #V2 #H #I
92 elim (lifts_inv_cons … H) -H /3 width=5 by ldrop_skip, ldrops_cons/
96 (* Basic_1: removed theorems 1: drop1_getl_trans *)