matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_fqus.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 (* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *************)
  16
  17 include "basic_2/s_computation/fqus_fqup.ma".
  18 include "basic_2/rt_transition/cpx_drops.ma".
  19
  20 (* Properties on supclosure *************************************************)
  21
  22 lemma fqu_cpx_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
  23                      ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
  24                      ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
  25 #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
  26 /3 width=3 by fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, cpx_pair_sn, cpx_bind, cpx_flat, ex2_intro/
  27 [ #I #G #L2 #V2 #X2 #HVX2
  28   elim (lifts_total X2 (𝐔❴1❵))
  29   /3 width=3 by fqu_drop, cpx_delta, ex2_intro/
  30 | #I #G #L2 #V #T2 #X2 #HTX2 #U2 #HTU2
  31   elim (cpx_lifts … HTU2 (Ⓣ) … (L2.ⓑ{I}V) … HTX2)
  32   /3 width=3 by fqu_drop, drops_refl, drops_drop, ex2_intro/
  33 ]
  34 qed-.
  35
  36 lemma fquq_cpx_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
  37                       ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
  38                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
  39 #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H
  40 [ #HT12 #U2 #HTU2 elim (fqu_cpx_trans … HT12 … HTU2) /3 width=3 by fqu_fquq, ex2_intro/
  41 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
  42 ]
  43 qed-.
  44
  45 lemma fqup_cpx_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
  46                       ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
  47                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
  48 #h #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
  49 [ #G2 #L2 #T2 #H12 #U2 #HTU2 elim (fqu_cpx_trans … H12 … HTU2) -T2
  50   /3 width=3 by fqu_fqup, ex2_intro/
  51 | #G #G2 #L #L2 #T #T2 #_ #HT2 #IHT1 #U2 #HTU2
  52   elim (fqu_cpx_trans … HT2 … HTU2) -T2 #T2 #HT2 #HTU2
  53   elim (IHT1 … HT2) -T /3 width=7 by fqup_strap1, ex2_intro/
  54 ]
  55 qed-.
  56
  57 lemma fqus_cpx_trans: ∀h,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
  58                       ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h] U2 →
  59                       ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
  60 #h #G1 #G2 #L1 #L2 #T1 #T2 #H elim (fqus_inv_fqup … H) -H
  61 [ #HT12 #U2 #HTU2 elim (fqup_cpx_trans … HT12 … HTU2) /3 width=3 by fqup_fqus, ex2_intro/
  62 | * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
  63 ]
  64 qed-.
  65 (*
  66 lemma fqu_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
  67                          ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 → (T2 = U2 → ⊥) →
  68                          ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
  69 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
  70 [ #I #G #L #V1 #V2 #HV12 #_ elim (lift_total V2 0 1)
  71   #U2 #HVU2 @(ex3_intro … U2)
  72   [1,3: /3 width=7 by fqu_drop, cpx_delta, drop_pair, drop_drop/
  73   | #H destruct
  74     lapply (lift_inv_lref2_be … HVU2 ? ?) -HVU2 //
  75   ]
  76 | #I #G #L #V1 #T #V2 #HV12 #H @(ex3_intro … (②{I}V2.T))
  77   [1,3: /2 width=4 by fqu_pair_sn, cpx_pair_sn/
  78   | #H0 destruct /2 width=1 by/
  79   ]
  80 | #a #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓑ{a,I}V.T2))
  81   [1,3: /2 width=4 by fqu_bind_dx, cpx_bind/
  82   | #H0 destruct /2 width=1 by/
  83   ]
  84 | #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓕ{I}V.T2))
  85   [1,3: /2 width=4 by fqu_flat_dx, cpx_flat/
  86   | #H0 destruct /2 width=1 by/
  87   ]
  88 | #G #L #K #T1 #U1 #k #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (k+1))
  89   #U2 #HTU2 @(ex3_intro … U2)
  90   [1,3: /2 width=10 by cpx_lift, fqu_drop/
  91   | #H0 destruct /3 width=5 by lift_inj/
  92 ]
  93 qed-.
  94
  95 lemma fquq_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
  96                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 → (T2 = U2 → ⊥) →
  97                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
  98 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fquq_inv_gen … H12) -H12
  99 [ #H12 elim (fqu_cpx_trans_neq … H12 … HTU2 H) -T2
 100   /3 width=4 by fqu_fquq, ex3_intro/
 101 | * #HG #HL #HT destruct /3 width=4 by ex3_intro/
 102 ]
 103 qed-.
 104
 105 lemma fqup_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
 106                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 → (T2 = U2 → ⊥) →
 107                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
 108 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
 109 [ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpx_trans_neq … H12 … HTU2 H) -T2
 110   /3 width=4 by fqu_fqup, ex3_intro/
 111 | #G #G1 #L #L1 #T #T1 #H1 #_ #IH12 #U2 #HTU2 #H elim (IH12 … HTU2 H) -T2
 112   #U1 #HTU1 #H #H12 elim (fqu_cpx_trans_neq … H1 … HTU1 H) -T1
 113   /3 width=8 by fqup_strap2, ex3_intro/
 114 ]
 115 qed-.
 116
 117 lemma fqus_cpx_trans_neq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
 118                           ∀U2. ⦃G2, L2⦄ ⊢ T2 ⬈[h, o] U2 → (T2 = U2 → ⊥) →
 119                           ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h, o] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
 120 #h #o #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_gen … H12) -H12
 121 [ #H12 elim (fqup_cpx_trans_neq … H12 … HTU2 H) -T2
 122   /3 width=4 by fqup_fqus, ex3_intro/
 123 | * #HG #HL #HT destruct /3 width=4 by ex3_intro/
 124 ]
 125 qed-.
 126 *)