matita/matita/contribs/lambdadelta/basic_2/static/ffdeq_ffdeq.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                  *)
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  13 (**************************************************************************)
  14
  15 include "basic_2/static/lfdeq_lfdeq.ma".
  16 include "basic_2/static/ffdeq.ma".
  17
  18 (* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************)
  19
  20 (* Advanced properties ******************************************************)
  21
  22 lemma ffdeq_sym: ∀h,o. tri_symmetric … (ffdeq h o).
  23 #h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -L1 -T1
  24 /3 width=1 by ffdeq_intro_dx, lfdeq_sym, tdeq_sym/
  25 qed-.
  26
  27 (* Main properties **********************************************************)
  28
  29 theorem ffdeq_trans: ∀h,o. tri_transitive … (ffdeq h o).
  30 #h #o #G1 #G #L1 #L #T1 #T * -G -L -T
  31 #L #T #HL1 #HT1 #G2 #L2 #T2 * -G2 -L2 -T2
  32 /4 width=5 by ffdeq_intro_sn, lfdeq_trans, tdeq_lfdeq_div, tdeq_trans/
  33 qed-.
  34
  35 theorem ffdeq_canc_sn: ∀h,o,G,G1,G2,L,L1,L2,T,T1,T2.
  36                        ⦃G, L, T⦄ ≛[h, o] ⦃G1, L1, T1⦄→ ⦃G, L, T⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄.
  37 /3 width=5 by ffdeq_trans, ffdeq_sym/ qed-.
  38
  39 theorem ffdeq_canc_dx: ∀h,o,G1,G2,G,L1,L2,L,T1,T2,T.
  40                        ⦃G1, L1, T1⦄ ≛[h, o] ⦃G, L, T⦄ → ⦃G2, L2, T2⦄ ≛[h, o] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄.
  41 /3 width=5 by ffdeq_trans, ffdeq_sym/ qed-.