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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/substitution/ldrop.ma".
16 include "Basic_2/unfold/gr2_minus.ma".
17 include "Basic_2/unfold/lifts.ma".
19 (* GENERIC LOCAL ENVIRONMENT SLICING ****************************************)
21 inductive ldrops: list2 nat nat → relation lenv ≝
22 | ldrops_nil : ∀L. ldrops ⟠ L L
23 | ldrops_cons: ∀L1,L,L2,des,d,e.
24 ldrops des L1 L → ⇩[d,e] L ≡ L2 → ldrops ({d, e} :: des) L1 L2
27 interpretation "generic local environment slicing"
28 'RDropStar des T1 T2 = (ldrops des T1 T2).
30 (* Basic inversion lemmas ***************************************************)
32 fact ldrops_inv_nil_aux: ∀L1,L2,des. ⇩*[des] L1 ≡ L2 → des = ⟠ → L1 = L2.
33 #L1 #L2 #des * -L1 -L2 -des //
34 #L1 #L #L2 #d #e #des #_ #_ #H destruct
37 lemma ldrops_inv_nil: ∀L1,L2. ⇩*[⟠] L1 ≡ L2 → L1 = L2.
40 fact ldrops_inv_cons_aux: ∀L1,L2,des. ⇩*[des] L1 ≡ L2 →
41 ∀d,e,tl. des = {d, e} :: tl →
42 ∃∃L. ⇩*[tl] L1 ≡ L & ⇩[d, e] L ≡ L2.
43 #L1 #L2 #des * -L1 -L2 -des
44 [ #L #d #e #tl #H destruct
45 | #L1 #L #L2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct
49 lemma ldrops_inv_cons: ∀L1,L2,d,e,des. ⇩*[{d, e} :: des] L1 ≡ L2 →
50 ∃∃L. ⇩*[des] L1 ≡ L & ⇩[d, e] L ≡ L2.
53 lemma ldrops_inv_skip2: ∀I,des,i,des2. des ▭ i ≡ des2 →
54 ∀L1,K2,V2. ⇩*[des2] L1 ≡ K2. ⓑ{I} V2 →
55 ∃∃K1,V1,des1. des + 1 ▭ i + 1 ≡ des1 + 1 &
59 #I #des #i #des2 #H elim H -des -i -des2
61 >(ldrops_inv_nil … H) -L1 /2 width=7/
62 | #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H
63 elim (ldrops_inv_cons … H) -H #L #HL1 #H
64 elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ #K #V >minus_plus #HK2 #HV2 #H destruct
65 elim (IHdes2 … HL1) -IHdes2 -HL1 #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct
66 @(ex4_3_intro … K1 V1 … ) // [3,4: /2 width=7/ | skip ]
67 normalize >plus_minus // @minuss_lt // /2 width=1/ (**) (* explicit constructors, /3 width=1/ is a bit slow *)
68 | #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H
69 elim (IHdes2 … H) -IHdes2 -H #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct
74 (* Basic properties *********************************************************)
76 lemma ldrops_skip: ∀L1,L2,des. ⇩*[des] L1 ≡ L2 → ∀V1,V2. ⇧*[des] V2 ≡ V1 →
77 ∀I. ⇩*[des + 1] L1. ⓑ{I} V1 ≡ L2. ⓑ{I} V2.
78 #L1 #L2 #des #H elim H -L1 -L2 -des
80 >(lifts_inv_nil … HV12) -HV12 //
81 | #L1 #L #L2 #des #d #e #_ #HL2 #IHL #V1 #V2 #H #I
82 elim (lifts_inv_cons … H) -H /3 width=5/
86 (* Basic_1: removed theorems 1: drop1_getl_trans