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4 (* ||A|| A project by Andrea Asperti *)
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7 (* ||T|| The HELM team. *)
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15 include "basic_2/grammar/leq_leq.ma".
16 include "basic_2/substitution/drop.ma".
18 (* BASIC SLICING FOR LOCAL ENVIRONMENTS *************************************)
20 definition dedropable_sn: predicate (relation lenv) ≝
21 λR. ∀L1,K1,s,d,e. ⇩[s, d, e] L1 ≡ K1 → ∀K2. R K1 K2 →
22 ∃∃L2. R L1 L2 & ⇩[s, d, e] L2 ≡ K2 & L1 ⩬[d, e] L2.
24 (* Properties on equivalence ************************************************)
26 lemma leq_drop_trans_be: ∀L1,L2,d,e. L1 ⩬[d, e] L2 →
27 ∀I,K2,W,s,i. ⇩[s, 0, i] L2 ≡ K2.ⓑ{I}W →
29 ∃∃K1. K1 ⩬[0, ⫰(d+e-i)] K2 & ⇩[s, 0, i] L1 ≡ K1.ⓑ{I}W.
30 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e
31 [ #d #e #J #K2 #W #s #i #H
32 elim (drop_inv_atom1 … H) -H #H destruct
33 | #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J #K2 #W #s #i #_ #_ #H
34 elim (ylt_yle_false … H) //
35 | #I #L1 #L2 #V #e #HL12 #IHL12 #J #K2 #W #s #i #H #_ >yplus_O1
36 elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
37 [ #_ destruct >ypred_succ
38 /2 width=3 by drop_pair, ex2_intro/
39 | lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/
40 #H <H -H #H lapply (ylt_inv_succ … H) -H
41 #Hie elim (IHL12 … HLK1) -IHL12 -HLK1 // -Hie
42 >yminus_succ <yminus_inj /3 width=3 by drop_drop_lt, ex2_intro/
44 | #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #J #K2 #W #s #i #HLK2 #Hdi
45 elim (yle_inv_succ1 … Hdi) -Hdi
46 #Hdi #Hi <Hi >yplus_succ1 #H lapply (ylt_inv_succ … H) -H
47 #Hide lapply (drop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/
48 #HLK1 elim (IHL12 … HLK1) -IHL12 -HLK1 <yminus_inj >yminus_SO2
49 /4 width=3 by ylt_O, drop_drop_lt, ex2_intro/
53 lemma leq_drop_conf_be: ∀L1,L2,d,e. L1 ⩬[d, e] L2 →
54 ∀I,K1,W,s,i. ⇩[s, 0, i] L1 ≡ K1.ⓑ{I}W →
56 ∃∃K2. K1 ⩬[0, ⫰(d+e-i)] K2 & ⇩[s, 0, i] L2 ≡ K2.ⓑ{I}W.
57 #L1 #L2 #d #e #HL12 #I #K1 #W #s #i #HLK1 #Hdi #Hide
58 elim (leq_drop_trans_be … (leq_sym … HL12) … HLK1) // -L1 -Hdi -Hide
59 /3 width=3 by leq_sym, ex2_intro/
62 lemma drop_O1_ex: ∀K2,i,L1. |L1| = |K2| + i →
63 ∃∃L2. L1 ⩬[0, i] L2 & ⇩[i] L2 ≡ K2.
64 #K2 #i @(nat_ind_plus … i) -i
65 [ /3 width=3 by leq_O2, ex2_intro/
66 | #i #IHi #Y #Hi elim (drop_O1_lt (Ⓕ) Y 0) //
67 #I #L1 #V #H lapply (drop_inv_O2 … H) -H #H destruct
68 normalize in Hi; elim (IHi L1) -IHi
69 /3 width=5 by drop_drop, leq_pair, injective_plus_l, ex2_intro/
73 lemma dedropable_sn_TC: ∀R. dedropable_sn R → dedropable_sn (TC … R).
74 #R #HR #L1 #K1 #s #d #e #HLK1 #K2 #H elim H -K2
75 [ #K2 #HK12 elim (HR … HLK1 … HK12) -HR -K1
76 /3 width=4 by inj, ex3_intro/
77 | #K #K2 #_ #HK2 * #L #H1L1 #HLK #H2L1 elim (HR … HLK … HK2) -HR -K
78 /3 width=6 by leq_trans, step, ex3_intro/
82 (* Inversion lemmas on equivalence ******************************************)
84 lemma drop_O1_inj: ∀i,L1,L2,K. ⇩[i] L1 ≡ K → ⇩[i] L2 ≡ K → L1 ⩬[i, ∞] L2.
85 #i @(nat_ind_plus … i) -i
86 [ #L1 #L2 #K #H <(drop_inv_O2 … H) -K #H <(drop_inv_O2 … H) -L1 //
87 | #i #IHi * [2: #L1 #I1 #V1 ] * [2,4: #L2 #I2 #V2 ] #K #HLK1 #HLK2 //
88 lapply (drop_fwd_length … HLK1)
89 <(drop_fwd_length … HLK2) [ /4 width=5 by drop_inv_drop1, leq_succ/ ]
90 normalize <plus_n_Sm #H destruct