matita/matita/contribs/lambdadelta/basic_2/etc/cpys/cpys_cpys.etc

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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/grammar/lpx_sn_lpx_sn.ma".
  16 include "basic_2/substitution/fqup.ma".
  17 include "basic_2/substitution/lpys_ldrop.ma".
  18
  19 (* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
  20
  21 (* Main properties **********************************************************)
  22
  23 theorem cpys_antisym: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*× T2 → ⦃G, L⦄ ⊢ T2 ▶*× T1 → T1 = T2.
  24 #G #L #T1 #T2 #H elim H -G -L -T1 -T2 //
  25 [ #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #_ #HW2 lapply (ldrop_fwd_ldrop2 … HLK) -I -V1
  26   #HLK elim (cpys_inv_lift1 … HW2 … HLK … HVW2) -L -HVW2
  27   #X #H #_ elim (lift_inv_lref2_be … H) -G -K -V2 -W2 -X //
  28 | #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #H elim (cpys_inv_bind1 … H) -H
  29   #V #T #HV2 #HT2 #H destruct
  30   lapply (IHV12 HV2) #H destruct -IHV12 -HV2 /3 width=1 by eq_f2/
  31 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #H elim (cpys_inv_flat1 … H) -H
  32   #V #T #HV2 #HT2 #H destruct /3 width=1 by eq_f2/
  33 ]
  34 qed-.
  35
  36 theorem cpys_trans_lpys: ∀G. lpx_sn_transitive (cpys G) (cpys G).
  37 #G0 #L0 #T0 @(fqup_wf_ind_eq … G0 L0 T0) -G0 -L0 -T0 #G0 #L0 #T0 #IH #G1 #L1 * [|*]
  38 [ #I #HG #HL #HT #T #H1 #L2 #HL12 #T2 #HT2 destruct
  39   elim (cpys_inv_atom1 … H1) -H1
  40   [ #H destruct
  41     elim (cpys_inv_atom1 … HT2) -HT2
  42     [ #H destruct //
  43     | * #I2 #K2 #V #V2 #i #HLK2 #HV2 #HVT2 #H destruct
  44       elim (lpys_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
  45       elim (lpys_inv_pair2 … H) -H #K1 #V1 #HK12 #HV1 #H destruct
  46       lapply (fqup_lref … G1 … HLK1) /3 width=10 by cpys_delta/
  47     ]
  48   | * #I1 #K1 #V1 #V #i #HLK1 #HV1 #HVT #H destruct
  49     elim (lpys_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
  50     elim (lpys_inv_pair1 … H) -H #K2 #W2 #HK12 #_ #H destruct
  51     lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
  52     elim (cpys_inv_lift1 … HT2 … HLK2 … HVT) -L2 -T
  53     lapply (fqup_lref … G1 … HLK1) /3 width=10 by cpys_delta/
  54   ]
  55 | #a #I #V1 #T1 #HG #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
  56   elim (cpys_inv_bind1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
  57   elim (cpys_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
  58   /4 width=5 by cpys_bind, lpys_pair/
  59 | #I #V1 #T1 #HG #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
  60   elim (cpys_inv_flat1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
  61   elim (cpys_inv_flat1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
  62   /3 width=5 by cpys_flat/
  63 ]
  64 qed-.
  65
  66 theorem cpys_trans: ∀G,L. Transitive … (cpys G L).
  67 /2 width=5 by cpys_trans_lpys/ qed-.
  68
  69 (* Advanced properties ******************************************************)
  70
  71 lemma lpys_cpys_trans: ∀G. lsub_trans … (cpys G) (lpys G).
  72 /2 width=5 by cpys_trans_lpys/ qed-.