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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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11 (* v GNU General Public License Version 2 *)
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15 include "ground/xoa/ex_4_3.ma".
16 include "ground/relocation/mr2_minus.ma".
17 include "basic_2A/notation/relations/rdropstar_3.ma".
18 include "basic_2A/notation/relations/rdropstar_4.ma".
19 include "basic_2A/substitution/drop.ma".
20 include "basic_2A/multiple/lifts_vector.ma".
22 (* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
24 inductive drops (s:bool): mr2 → relation lenv ≝
25 | drops_nil : ∀L. drops s (◊) L L
26 | drops_cons: ∀L1,L,L2,cs,l,m.
27 drops s cs L1 L → ⬇[s, l, m] L ≡ L2 → drops s (❨l, m❩; cs) L1 L2
30 interpretation "iterated slicing (local environment) abstract"
31 'RDropStar s cs T1 T2 = (drops s cs T1 T2).
33 interpretation "iterated slicing (local environment) general"
34 'RDropStar des T1 T2 = (drops true des T1 T2).
37 definition d_liftable1: relation2 lenv term → predicate bool ≝
38 λR,s. ∀K,T. R K T → ∀L,l,m. ⬇[s, l, m] L ≡ K →
39 ∀U. ⬆[l, m] T ≡ U → R L U.
41 definition d_liftables1: relation2 lenv term → predicate bool ≝
42 λR,s. ∀L,K,cs. ⬇*[s, cs] L ≡ K →
43 ∀T,U. ⬆*[cs] T ≡ U → R K T → R L U.
45 definition d_liftables1_all: relation2 lenv term → predicate bool ≝
46 λR,s. ∀L,K,cs. ⬇*[s, cs] L ≡ K →
47 ∀Ts,Us. ⬆*[cs] Ts ≡ Us →
48 all … (R K) Ts → all … (R L) Us.
50 (* Basic inversion lemmas ***************************************************)
52 fact drops_inv_nil_aux: ∀L1,L2,s,cs. ⬇*[s, cs] L1 ≡ L2 → cs = ◊ → L1 = L2.
53 #L1 #L2 #s #cs * -L1 -L2 -cs //
54 #L1 #L #L2 #l #m #cs #_ #_ #H destruct
57 (* Basic_1: was: drop1_gen_pnil *)
58 lemma drops_inv_nil: ∀L1,L2,s. ⬇*[s, ◊] L1 ≡ L2 → L1 = L2.
59 /2 width=4 by drops_inv_nil_aux/ qed-.
61 fact drops_inv_cons_aux: ∀L1,L2,s,cs. ⬇*[s, cs] L1 ≡ L2 →
62 ∀l,m,tl. cs = ❨l, m❩; tl →
63 ∃∃L. ⬇*[s, tl] L1 ≡ L & ⬇[s, l, m] L ≡ L2.
64 #L1 #L2 #s #cs * -L1 -L2 -cs
65 [ #L #l #m #tl #H destruct
66 | #L1 #L #L2 #cs #l #m #HT1 #HT2 #l0 #m0 #tl #H destruct
67 /2 width=3 by ex2_intro/
71 (* Basic_1: was: drop1_gen_pcons *)
72 lemma drops_inv_cons: ∀L1,L2,s,l,m,cs. ⬇*[s, ❨l, m❩; cs] L1 ≡ L2 →
73 ∃∃L. ⬇*[s, cs] L1 ≡ L & ⬇[s, l, m] L ≡ L2.
74 /2 width=3 by drops_inv_cons_aux/ qed-.
76 lemma drops_inv_skip2: ∀I,s,cs,cs2,i. cs ▭ i ≘ cs2 →
77 ∀L1,K2,V2. ⬇*[s, cs2] L1 ≡ K2. ⓑ{I} V2 →
78 ∃∃K1,V1,cs1. cs + 1 ▭ i + 1 ≘ cs1 + 1 &
82 #I #s #cs #cs2 #i #H elim H -cs -cs2 -i
84 >(drops_inv_nil … H) -L1 /2 width=7 by lifts_nil, minuss_nil, ex4_3_intro, drops_nil/
85 | #cs #cs2 #l #m #i #Hil #_ #IHcs2 #L1 #K2 #V2 #H
86 elim (drops_inv_cons … H) -H #L #HL1 #H
87 elim (drop_inv_skip2 … H) -H /2 width=1 by lt_plus_to_minus_r/ #K #V >minus_plus #HK2 #HV2 #H destruct
88 elim (IHcs2 … HL1) -IHcs2 -HL1 #K1 #V1 #cs1 #Hcs1 #HK1 #HV1 #X destruct
89 @(ex4_3_intro … K1 V1 … ) // [3,4: /2 width=7 by lifts_cons, drops_cons/ | skip ]
90 normalize >plus_minus /3 width=1 by minuss_lt, lt_minus_to_plus/ (**) (* explicit constructors *)
91 | #cs #cs2 #l #m #i #Hil #_ #IHcs2 #L1 #K2 #V2 #H
92 elim (IHcs2 … H) -IHcs2 -H #K1 #V1 #cs1 #Hcs1 #HK1 #HV1 #X destruct
93 /4 width=7 by minuss_ge, ex4_3_intro, le_S_S/
97 (* Basic properties *********************************************************)
99 (* Basic_1: was: drop1_skip_bind *)
100 lemma drops_skip: ∀L1,L2,s,cs. ⬇*[s, cs] L1 ≡ L2 → ∀V1,V2. ⬆*[cs] V2 ≡ V1 →
101 ∀I. ⬇*[s, cs + 1] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
102 #L1 #L2 #s #cs #H elim H -L1 -L2 -cs
103 [ #L #V1 #V2 #HV12 #I
104 >(lifts_inv_nil … HV12) -HV12 //
105 | #L1 #L #L2 #cs #l #m #_ #HL2 #IHL #V1 #V2 #H #I
106 elim (lifts_inv_cons … H) -H /3 width=5 by drop_skip, drops_cons/
110 lemma d1_liftable_liftables: ∀R,s. d_liftable1 R s → d_liftables1 R s.
111 #R #s #HR #L #K #cs #H elim H -L -K -cs
112 [ #L #T #U #H #HT <(lifts_inv_nil … H) -H //
113 | #L1 #L #L2 #cs #l #m #_ #HL2 #IHL #T2 #T1 #H #HLT2
114 elim (lifts_inv_cons … H) -H /3 width=10 by/
118 lemma d1_liftables_liftables_all: ∀R,s. d_liftables1 R s → d_liftables1_all R s.
119 #R #s #HR #L #K #cs #HLK #Ts #Us #H elim H -Ts -Us normalize //
120 #Ts #Us #T #U #HTU #_ #IHTUs * /3 width=7 by conj/
123 (* Basic_1: removed theorems 1: drop1_getl_trans *)