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/substitution/drop.ma".
18 include "basic_2/multiple/mr2_minus.ma".
19 include "basic_2/multiple/lifts_vector.ma".
21 (* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
23 inductive drops (s:bool): list2 nat nat → relation lenv ≝
24 | drops_nil : ∀L. drops s (◊) L L
25 | drops_cons: ∀L1,L,L2,des,d,e.
26 drops s des L1 L → ⇩[s, d, e] L ≡ L2 → drops s ({d, e} @ des) L1 L2
29 interpretation "iterated slicing (local environment) abstract"
30 'RDropStar s des T1 T2 = (drops s des T1 T2).
32 interpretation "iterated slicing (local environment) general"
33 'RDropStar des T1 T2 = (drops true des T1 T2).
36 definition l_liftable1: relation2 lenv term → predicate bool ≝
37 λR,s. ∀K,T. R K T → ∀L,d,e. ⇩[s, d, e] L ≡ K →
38 ∀U. ⇧[d, e] T ≡ U → R L U.
40 definition l_liftables1: relation2 lenv term → predicate bool ≝
41 λR,s. ∀L,K,des. ⇩*[s, des] L ≡ K →
42 ∀T,U. ⇧*[des] T ≡ U → R K T → R L U.
44 definition l_liftables1_all: relation2 lenv term → predicate bool ≝
45 λR,s. ∀L,K,des. ⇩*[s, des] L ≡ K →
46 ∀Ts,Us. ⇧*[des] Ts ≡ Us →
47 all … (R K) Ts → all … (R L) Us.
49 (* Basic inversion lemmas ***************************************************)
51 fact drops_inv_nil_aux: ∀L1,L2,s,des. ⇩*[s, des] L1 ≡ L2 → des = ◊ → L1 = L2.
52 #L1 #L2 #s #des * -L1 -L2 -des //
53 #L1 #L #L2 #d #e #des #_ #_ #H destruct
56 (* Basic_1: was: drop1_gen_pnil *)
57 lemma drops_inv_nil: ∀L1,L2,s. ⇩*[s, ◊] L1 ≡ L2 → L1 = L2.
58 /2 width=4 by drops_inv_nil_aux/ qed-.
60 fact drops_inv_cons_aux: ∀L1,L2,s,des. ⇩*[s, des] L1 ≡ L2 →
61 ∀d,e,tl. des = {d, e} @ tl →
62 ∃∃L. ⇩*[s, tl] L1 ≡ L & ⇩[s, d, e] L ≡ L2.
63 #L1 #L2 #s #des * -L1 -L2 -des
64 [ #L #d #e #tl #H destruct
65 | #L1 #L #L2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct
66 /2 width=3 by ex2_intro/
70 (* Basic_1: was: drop1_gen_pcons *)
71 lemma drops_inv_cons: ∀L1,L2,s,d,e,des. ⇩*[s, {d, e} @ des] L1 ≡ L2 →
72 ∃∃L. ⇩*[s, des] L1 ≡ L & ⇩[s, d, e] L ≡ L2.
73 /2 width=3 by drops_inv_cons_aux/ qed-.
75 lemma drops_inv_skip2: ∀I,s,des,des2,i. des ▭ i ≡ des2 →
76 ∀L1,K2,V2. ⇩*[s, des2] L1 ≡ K2. ⓑ{I} V2 →
77 ∃∃K1,V1,des1. des + 1 ▭ i + 1 ≡ des1 + 1 &
81 #I #s #des #des2 #i #H elim H -des -des2 -i
83 >(drops_inv_nil … H) -L1 /2 width=7 by lifts_nil, minuss_nil, ex4_3_intro, drops_nil/
84 | #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H
85 elim (drops_inv_cons … H) -H #L #HL1 #H
86 elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ #K #V >minus_plus #HK2 #HV2 #H destruct
87 elim (IHdes2 … HL1) -IHdes2 -HL1 #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct
88 @(ex4_3_intro … K1 V1 … ) // [3,4: /2 width=7 by lifts_cons, drops_cons/ | skip ]
89 normalize >plus_minus /3 width=1 by minuss_lt, lt_minus_to_plus/ (**) (* explicit constructors *)
90 | #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H
91 elim (IHdes2 … H) -IHdes2 -H #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct
92 /4 width=7 by minuss_ge, ex4_3_intro, le_S_S/
96 (* Basic properties *********************************************************)
98 (* Basic_1: was: drop1_skip_bind *)
99 lemma drops_skip: ∀L1,L2,s,des. ⇩*[s, des] L1 ≡ L2 → ∀V1,V2. ⇧*[des] V2 ≡ V1 →
100 ∀I. ⇩*[s, des + 1] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
101 #L1 #L2 #s #des #H elim H -L1 -L2 -des
102 [ #L #V1 #V2 #HV12 #I
103 >(lifts_inv_nil … HV12) -HV12 //
104 | #L1 #L #L2 #des #d #e #_ #HL2 #IHL #V1 #V2 #H #I
105 elim (lifts_inv_cons … H) -H /3 width=5 by drop_skip, drops_cons/
109 lemma l1_liftable_liftables: ∀R,s. l_liftable1 R s → l_liftables1 R s.
110 #R #s #HR #L #K #des #H elim H -L -K -des
111 [ #L #T #U #H #HT <(lifts_inv_nil … H) -H //
112 | #L1 #L #L2 #des #d #e #_ #HL2 #IHL #T2 #T1 #H #HLT2
113 elim (lifts_inv_cons … H) -H /3 width=10 by/
117 lemma l1_liftables_liftables_all: ∀R,s. l_liftables1 R s → l_liftables1_all R s.
118 #R #s #HR #L #K #des #HLK #Ts #Us #H elim H -Ts -Us normalize //
119 #Ts #Us #T #U #HTU #_ #IHTUs * /3 width=7 by conj/
122 (* Basic_1: removed theorems 1: drop1_getl_trans *)