matita/matita/contribs/lambdadelta/basic_2/rt_transition/cnh_tdeq.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   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/rt_transition/cpr_tdeq.ma".
  16 include "basic_2/rt_transition/cnh_simple.ma".
  17 include "basic_2/rt_transition/cnh_drops.ma".
  18
  19 (* NORMAL TERMS FOR HEAD T-UNUNBOUND RT-TRANSITION **************************)
  20
  21 (* Properties with context-free sort-irrelevant equivalence for terms *******)
  22
  23 lemma cnh_dec_tdeq (h) (G) (L):
  24       ∀T1. ∨∨ ⦃G,L⦄ ⊢ ⥲[h] 𝐍⦃T1⦄
  25             | ∃∃n,T2. ⦃G,L⦄ ⊢ T1 ➡[n,h] T2 & (T1 ≛ T2 → ⊥).
  26 #h #G #L * *
  27 [ #s /3 width=5 by cnh_sort, or_introl/
  28 | #i elim (drops_F_uni L i)
  29   [ /3 width=7 by cnh_atom_drops, or_introl/
  30   | * * [ #I | * #V ] #K #HLK
  31     [ /3 width=8 by cnh_unit_drops, or_introl/
  32     | elim (lifts_total V 𝐔❴↑i❵) #W #HVW
  33       @or_intror @(ex2_2_intro … W) [1,2: /2 width=7 by cpm_delta_drops/ ] #H
  34       lapply (tdeq_inv_lref1 … H) -H #H destruct
  35       /2 width=5 by lifts_inv_lref2_uni_lt/
  36     | elim (lifts_total V 𝐔❴↑i❵) #W #HVW
  37       @or_intror @(ex2_2_intro … W) [1,2: /2 width=7 by cpm_ell_drops/ ] #H
  38       lapply (tdeq_inv_lref1 … H) -H #H destruct
  39       /2 width=5 by lifts_inv_lref2_uni_lt/
  40     ]
  41   ]
  42 | #l /3 width=5 by cnh_gref, or_introl/
  43 | #p * [ cases p ] #V1 #T1
  44   [ elim (cpr_subst h G (L.ⓓV1) T1 0 L V1) [| /2 width=1 by drops_refl/ ] #T2 #X2 #HT12 #HXT2
  45     elim (tdeq_dec T1 T2) [ -HT12 #HT12 | #HnT12 ]
  46     [ elim (tdeq_inv_lifts_dx … HT12 … HXT2) -T2 #X1 #HXT1 #_ -X2
  47       @or_intror @(ex2_2_intro … X1) [1,2: /2 width=4 by cpm_zeta/ ] #H
  48       /2 width=7 by tdeq_lifts_inv_pair_sn/
  49     | @or_intror @(ex2_2_intro … (+ⓓV1.T2)) [1,2: /2 width=2 by cpm_bind/ ] #H
  50       elim (tdeq_inv_pair … H) -H /2 width=1 by/
  51     ]
  52   | /3 width=5 by cnh_abbr_neg, or_introl/
  53   | /3 width=5 by cnh_abst, or_introl/
  54   ]
  55 | * #V1 #T1
  56   [ elim (simple_dec_ex T1) [ #HT1 | * #p * #W1 #U1 #H destruct ]
  57     [ /3 width=5 by cnh_appl_simple, or_introl/
  58     | elim (lifts_total V1 𝐔❴1❵) #X1 #HVX1
  59       @or_intror @(ex2_2_intro … (ⓓ{p}W1.ⓐX1.U1)) [1,2: /2 width=3 by cpm_theta/ ] #H
  60       elim (tdeq_inv_pair … H) -H #H destruct
  61     | @or_intror @(ex2_2_intro … (ⓓ{p}ⓝW1.V1.U1)) [1,2: /2 width=2 by cpm_beta/ ] #H
  62       elim (tdeq_inv_pair … H) -H #H destruct
  63     ]
  64   | @or_intror @(ex2_2_intro … T1) [1,2: /2 width=2 by cpm_eps/ ] #H
  65     /2 width=4 by tdeq_inv_pair_xy_y/
  66   ]
  67 ]
  68 qed-.