matita/matita/contribs/lambda_delta/basic_2/reducibility/ltpr_aaa.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/aaa_ltpss_dx.ma".
  16 include "basic_2/static/lsuba_aaa.ma".
  17 include "basic_2/reducibility/ltpr_ldrop.ma".
  18
  19 (* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
  20
  21 (* Properties about atomic arity assignment on terms ************************)
  22
  23 fact aaa_ltpr_tpr_conf_aux: ∀L,T,L1,T1,A. L1 ⊢ T1 ⁝ A → L = L1 → T = T1 →
  24                             ∀L2. L1 ➡ L2 → ∀T2. T1 ➡ T2 → L2 ⊢ T2 ⁝ A.
  25 #L #T @(fw_ind … L T) -L -T #L #T #IH
  26 #L1 #T1 #A * -L1 -T1 -A
  27 [ -IH #L1 #k #H1 #H2 #L2 #_ #T2 #H destruct
  28   >(tpr_inv_atom1 … H) -H //
  29 | #I #L1 #K1 #V1 #B #i #HLK1 #HK1 #H1 #H2 #L2 #HL12 #T2 #H destruct
  30   >(tpr_inv_atom1 … H) -T2
  31   lapply (ldrop_pair2_fwd_fw … HLK1 (#i)) #HKV1
  32   elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
  33   elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
  34   lapply (IH … HKV1 … HK1 … HK12 … HV12) // -L1 -K1 -V1 /2 width=5/
  35 | #a #L1 #V1 #T1 #B #A #HB #HA #H1 #H2 #L2 #HL12 #X #H destruct
  36   elim (tpr_inv_abbr1 … H) -H *
  37   [ #V2 #T #T2 #HV12 #HT1 #HT2 #H destruct
  38     lapply (tps_lsubs_trans … HT2 (L2.ⓓV2) ?) -HT2 /2 width=1/ #HT2
  39     lapply (IH … HB … HL12 … HV12) -HB /width=5/ #HB
  40     lapply (IH … HA … (L2.ⓓV2) … HT1) -IH -HA -HT1 /width=5/ -T1 /2 width=1/ -L1 -V1 /3 width=5/
  41   | -B #T #HT1 #HXT #H destruct
  42     lapply (IH … HA … (L2.ⓓV1) … HT1) /width=5/ -T1 /2 width=1/ -L1 #HA
  43     @(aaa_inv_lift … HA … HXT) /2 width=1/
  44   ]
  45 | #a #L1 #V1 #T1 #B #A #HB #HA #H1 #H2 #L2 #HL12 #X #H destruct
  46   elim (tpr_inv_abst1 … H) -H #V2 #T2 #HV12 #HT12 #H destruct
  47   lapply (IH … HB … HL12 … HV12) -HB /width=5/ #HB
  48   lapply (IH … HA … (L2.ⓛV2) … HT12) -IH -HA -HT12 /width=5/ -T1 /2 width=1/
  49 | #L1 #V1 #T1 #B #A #HV1 #HT1 #H1 #H2 #L2 #HL12 #X #H destruct
  50   elim (tpr_inv_appl1 … H) -H *
  51   [ #V2 #T2 #HV12 #HT12 #H destruct
  52     lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HB
  53     lapply (IH … HT1 … HL12 … HT12) -IH -HT1 -HL12 -HT12 /width=5/ /2 width=3/
  54   | #a #V2 #W2 #T0 #T2 #HV12 #HT02 #H1 #H2 destruct
  55     elim (aaa_inv_abst … HT1) -HT1 #B0 #A0 #HB0 #HA0 #H destruct
  56     lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HB
  57     lapply (IH … HB0  … HL12 W2 ?) -HB0 /width=5/ #HB0
  58     lapply (IH … HA0 … (L2.ⓛW2) … HT02) -IH -HA0 -HT02 [2,4: // |3,5: skip ] /2 width=1/ -T0 -L1 -V1 /4 width=7/
  59   | #a #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HW02 #HT02 #HV02 #H1 #H2 destruct
  60     elim (aaa_inv_abbr … HT1) -HT1 #B0 #HW0 #HT0
  61     lapply (IH … HW0  … HL12 … HW02) -HW0 /width=5/ #HW2
  62     lapply (IH … HV1 … HL12 … HV10) -HV1 -HV10 /width=5/ #HV0
  63     lapply (IH … HT0 … (L2.ⓓW2) … HT02) -IH -HT0 -HT02 [2,4: // |3,5: skip ] /2 width=1/ -V1 -T0 -L1 -W0 #HT2
  64     @(aaa_abbr … HW2) -HW2
  65     @(aaa_appl … HT2) -HT2 /3 width=7/ (**) (* explict constructors, /5 width=7/ is too slow *)
  66   ]
  67 | #L1 #V1 #T1 #A #HV1 #HT1 #H1 #H2 #L2 #HL12 #X #H destruct
  68   elim (tpr_inv_cast1 … H) -H
  69   [ * #V2 #T2 #HV12 #HT12 #H destruct
  70     lapply (IH … HV1 … HL12 … HV12) -HV1 -HV12 /width=5/ #HV2
  71     lapply (IH … HT1 … HL12 … HT12) -IH -HT1 -HL12 -HT12 /width=5/ -L1 -V1 -T1 /2 width=1/
  72   | -HV1 #HT1X
  73      lapply (IH … HT1 … HL12 … HT1X) -IH -HT1 -HL12 -HT1X /width=5/
  74   ]
  75 ]
  76 qed.
  77
  78 lemma aaa_ltpr_tpr_conf: ∀L1,T1,A. L1 ⊢ T1 ⁝ A → ∀L2. L1 ➡ L2 →
  79                          ∀T2. T1 ➡ T2 → L2 ⊢ T2 ⁝ A.
  80 /2 width=9/ qed.
  81
  82 lemma aaa_ltpr_conf: ∀L1,T,A. L1 ⊢ T ⁝ A → ∀L2. L1 ➡ L2 → L2 ⊢ T ⁝ A.
  83 /2 width=5/ qed.
  84
  85 lemma aaa_tpr_conf: ∀L,T1,A. L ⊢ T1 ⁝ A → ∀T2. T1 ➡ T2 → L ⊢ T2 ⁝ A.
  86 /2 width=5/ qed.