matita/matita/contribs/lambdadelta/basic_2/multiple/ldrops.ma

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  14
  15 include "basic_2/notation/relations/rdropstar_3.ma".
  16 include "basic_2/notation/relations/rdropstar_4.ma".
  17 include "basic_2/substitution/ldrop.ma".
  18 include "basic_2/multiple/gr2_minus.ma".
  19 include "basic_2/multiple/lifts.ma".
  20
  21 (* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
  22
  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
  27 .
  28
  29 interpretation "iterated slicing (local environment) abstract"
  30    'RDropStar s des T1 T2 = (ldrops s des T1 T2).
  31 (*
  32 interpretation "iterated slicing (local environment) general"
  33    'RDropStar des T1 T2 = (ldrops true des T1 T2).
  34 *)
  35
  36 (* Basic inversion lemmas ***************************************************)
  37
  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
  41 qed-.
  42
  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-.
  46
  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/
  54 ]
  55 qed-.
  56
  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-.
  61
  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 &
  65                                       ⇩*[s, des1] K1 ≡ K2 &
  66                                       ⇧*[des1] V2 ≡ V1 &
  67                                       L1 = K1. ⓑ{I} V1.
  68 #I #s #des #des2 #i #H elim H -des -des2 -i
  69 [ #i #L1 #K2 #V2 #H
  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/
  80 ]
  81 qed-.
  82
  83 (* Basic properties *********************************************************)
  84
  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
  89 [ #L #V1 #V2 #HV12 #I
  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/
  93 ].
  94 qed.
  95
  96 (* Basic_1: removed theorems 1: drop1_getl_trans *)