matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_da.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "basic_2/static/da_da.ma".
  16 include "basic_2/unfold/lstas_lstas.ma".
  17
  18 (* NAT-ITERATED STATIC TYPE ASSIGNMENT FOR TERMS ****************************)
  19
  20 (* Properties on degree assignment for terms ********************************)
  21
  22 lemma da_lstas: ∀h,g,G,L,T,l1. ⦃G, L⦄ ⊢ T ▪[h, g] l1 → ∀l2.
  23                 ∃∃U. ⦃G, L⦄ ⊢ T •*[h, l2] U & ⦃G, L⦄ ⊢ U ▪[h, g] l1-l2.
  24 #h #g #G #L #T #l1 #H elim H -G -L -T -l1
  25 [ /4 width=3 by da_sort, deg_iter, ex2_intro/
  26 | #G #L #K #V #i #l1 #HLK #_ #IHV #l2
  27   elim (IHV l2) -IHV #W
  28   elim (lift_total W 0 (i+1))
  29   lapply (drop_fwd_drop2 … HLK)
  30   /3 width=10 by lstas_ldef, da_lift, ex2_intro/
  31 | #G #L #K #W #i #l1 #HLK #HW #IHW #l2 @(nat_ind_plus … l2) -l2
  32   [ elim (IHW 0) -IHW /3 width=6 by lstas_zero, da_ldec, ex2_intro/
  33   | #l #_ elim (IHW l) -IHW #V
  34     elim (lift_total V 0 (i+1))
  35     lapply (drop_fwd_drop2 … HLK)
  36     /3 width=10 by lstas_succ, da_lift, ex2_intro/
  37   ]
  38 | #a #I #G #L #V #T #l1 #_ #IHT #l2 elim (IHT … l2) -IHT
  39   /3 width=6 by lstas_bind, da_bind, ex2_intro/
  40 | * #G #L #V #T #l1 #_ #IHT #l2 elim (IHT … l2) -IHT
  41   /3 width=5 by lstas_appl, lstas_cast, da_flat, ex2_intro/
  42 ]
  43 qed-.
  44
  45 lemma lstas_da_conf: ∀h,g,G,L,T,U,l2. ⦃G, L⦄ ⊢ T •*[h, l2] U →
  46                      ∀l1. ⦃G, L⦄ ⊢ T ▪[h, g] l1 → ⦃G, L⦄ ⊢ U ▪[h, g] l1-l2.
  47 #h #g #G #L #T #U #l2 #HTU #l1 #HT
  48 elim (da_lstas … HT l2) -HT #X #HTX
  49 lapply (lstas_mono … HTX … HTU) -T //
  50 qed-.
  51
  52 (* inversion lemmas on degree assignment for terms **************************)
  53
  54 lemma lstas_inv_da: ∀h,g,G,L,T,U,l2. ⦃G, L⦄ ⊢ T •*[h, l2] U →
  55                     ∃∃l1. ⦃G, L⦄ ⊢ T ▪[h, g] l1 & ⦃G, L⦄ ⊢ U ▪[h, g] l1-l2.
  56 #h #g #G #L #T #U #l2 #H elim H -G -L -T -U -l2
  57 [ #G #L #l2 #k elim (deg_total h g k) /4 width=3 by da_sort, deg_iter, ex2_intro/
  58 | #G #L #K #V #W #U #i #l2 #HLK #_ #HWU *
  59   lapply (drop_fwd_drop2 … HLK) /3 width=10 by da_lift, da_ldef, ex2_intro/
  60 | #G #L #K #W #V #i #HLK #_ * /3 width=6 by da_ldec, ex2_intro/
  61 | #G #L #K #W #V #U #i #l2 #HLK #_ #HVU *
  62   lapply (drop_fwd_drop2 … HLK) /3 width=10 by da_lift, da_ldec, ex2_intro/
  63 | #a #I #G #L #V #T #U #l2 #_ * /3 width=3 by da_bind, ex2_intro/
  64 | #G #L #V #T #U #l2 #_ * /3 width=3 by da_flat, ex2_intro/
  65 | #G #L #W #T #U #l2 #_ * /3 width=3 by da_flat, ex2_intro/
  66 ]
  67 qed-.
  68
  69 lemma lstas_inv_da_ge: ∀h,G,L,T,U,l2,l. ⦃G, L⦄ ⊢ T •*[h, l2] U →
  70                        ∃∃g,l1. ⦃G, L⦄ ⊢ T ▪[h, g] l1 & ⦃G, L⦄ ⊢ U ▪[h, g] l1-l2 & l ≤ l1.
  71 #h #G #L #T #U #l2 #l #H elim H -G -L -T -U -l2
  72 [ /4 width=5 by da_sort, deg_iter, ex3_2_intro/
  73 | #G #L #K #V #W #U #i #l2 #HLK #_ #HWU *
  74   lapply (drop_fwd_drop2 … HLK) /3 width=10 by da_lift, da_ldef, ex3_2_intro/
  75 | #G #L #K #W #V #i #HLK #_ *
  76   #g #l1 #HW #HV #Hl1 /4 width=6 by da_ldec, lt_to_le, le_S_S, ex3_2_intro/
  77 | #G #L #K #W #V #U #i #l2 #HLK #_ #HVU *
  78   lapply (drop_fwd_drop2 … HLK)
  79   /4 width=10 by da_lift, da_ldec, lt_to_le, le_S_S, ex3_2_intro/
  80 | #a #I #G #L #V #T #U #l2 #_ * /3 width=5 by da_bind, ex3_2_intro/
  81 | #G #L #V #T #U #l2 #_ * /3 width=5 by da_flat, ex3_2_intro/
  82 | #G #L #W #T #U #l2 #_ * /3 width=5 by da_flat, ex3_2_intro/
  83 ]
  84 qed-.
  85
  86 (* Advanced inversion lemmas ************************************************)
  87
  88 lemma lstas_inv_refl_pos: ∀h,G,L,T,l. ⦃G, L⦄ ⊢ T •*[h, l+1] T → ⊥.
  89 #h #G #L #T #l2 #H elim (lstas_inv_da_ge … (l2+1) H) -H
  90 #g #l1 #HT1 #HT12 #Hl21 lapply (da_mono … HT1 … HT12) -h -G -L -T
  91 #H elim (discr_x_minus_xy … H) -H
  92 [ #H destruct /2 width=3 by le_plus_xSy_O_false/
  93 | -l1 <plus_n_Sm #H destruct
  94 ]
  95 qed-.