matita/matita/contribs/lambdadelta/basic_2/relocation/fsup.ma

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
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  13 (**************************************************************************)
  14
  15 include "basic_2/notation/relations/supterm_6.ma".
  16 include "basic_2/grammar/cl_weight.ma".
  17 include "basic_2/relocation/ldrop.ma".
  18
  19 (* SUPCLOSURE ***************************************************************)
  20
  21 (* activate genv *)
  22 inductive fsup: tri_relation genv lenv term ≝
  23 | fsup_lref_O  : ∀I,G,L,V. fsup G (L.ⓑ{I}V) (#0) G L V
  24 | fsup_pair_sn : ∀I,G,L,V,T. fsup G L (②{I}V.T) G L V
  25 | fsup_bind_dx : ∀a,I,G,L,V,T. fsup G L (ⓑ{a,I}V.T) G (L.ⓑ{I}V) T
  26 | fsup_flat_dx : ∀I,G,L,V,T. fsup G L (ⓕ{I}V.T) G L T
  27 | fsup_ldrop_lt: ∀G,L,K,T,U,d,e.
  28                  ⇩[d, e] L ≡ K → ⇧[d, e] T ≡ U → 0 < e → fsup G L U G K T
  29 | fsup_ldrop   : ∀G1,G2,L1,K1,K2,T1,T2,U1,d,e.
  30                  ⇩[d, e] L1 ≡ K1 → ⇧[d, e] T1 ≡ U1 →
  31                  fsup G1 K1 T1 G2 K2 T2 → fsup G1 L1 U1 G2 K2 T2
  32 .
  33
  34 interpretation
  35    "structural successor (closure)"
  36    'SupTerm G1 L1 T1 G2 L2 T2 = (fsup G1 L1 T1 G2 L2 T2).
  37
  38 (* Basic properties *********************************************************)
  39
  40 lemma fsup_lref_S_lt: ∀I,G1,G2,L,K,V,T,i. 0 < i → ⦃G1, L, #(i-1)⦄ ⊃ ⦃G2, K, T⦄ → ⦃G1, L.ⓑ{I}V, #i⦄ ⊃ ⦃G2, K, T⦄.
  41 #I #G1 #G2 #L #K #V #T #i #Hi #H /3 width=7 by fsup_ldrop, ldrop_ldrop, lift_lref_ge_minus/ (**) (* auto too slow without trace *)
  42 qed.
  43
  44 lemma fsup_lref: ∀I,G,K,V,i,L. ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊃ ⦃G, K, V⦄.
  45 #I #G #K #V #i @(nat_elim1 i) -i #i #IH #L #H
  46 elim (ldrop_inv_O1_pair2 … H) -H *
  47 [ #H1 #H2 destruct //
  48 | #I1 #K1 #V1 #HK1 #H #Hi destruct
  49   lapply (IH … HK1) /2 width=1/
  50 ]
  51 qed.
  52
  53 (* Basic forward lemmas *****************************************************)
  54
  55 lemma fsup_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} < ♯{G1, L1, T1}.
  56 #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
  57 [ #G #L #K #T #U #d #e #HLK #HTU #HKL
  58   lapply (ldrop_fwd_lw_lt … HLK HKL) -HKL -HLK #HKL
  59   lapply (lift_fwd_tw … HTU) -d -e #H
  60   normalize in ⊢ (?%%); /2 width=1/
  61 | #G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12
  62   lapply (ldrop_fwd_lw … HLK1) -HLK1 #HLK1
  63   lapply (lift_fwd_tw … HTU1) -HTU1 #HTU1
  64   @(lt_to_le_to_lt … IHT12) -IHT12 /2 width=1/
  65 ]
  66 qed-.
  67
  68 fact fsup_fwd_length_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
  69                                 ∀i. T1 = #i → |L2| < |L1|.
  70 #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
  71 [1: normalize //
  72 |3: #a
  73 |5: /2 width=4 by ldrop_fwd_length_lt4/
  74 |6: #G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 #d #e #HLK1 #HTU1 #_ #IHT12 #i #H destruct
  75     lapply (ldrop_fwd_length_le4 … HLK1) -HLK1 #HLK1
  76     elim (lift_inv_lref2 … HTU1) -HTU1 * #Hdei #H destruct
  77     @(lt_to_le_to_lt … HLK1) /2 width=2/
  78 ] #I #G #L #V #T #j #H destruct
  79 qed-.
  80
  81 lemma fsup_fwd_length_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊃ ⦃G2, L2, T2⦄ → |L2| < |L1|.
  82 /2 width=7 by fsup_fwd_length_lref1_aux/
  83 qed-.