matita/matita/contribs/lambdadelta/basic_2/static/lfeq_lfeq.ma

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   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
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  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/relocation/lreq_lreq.ma".
  16 include "basic_2/static/frees_frees.ma".
  17 include "basic_2/static/lfxs_lfxs.ma".
  18 include "basic_2/static/lfeq_lreq.ma".
  19
  20 (* EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *******************)
  21
  22 (* Main properties **********************************************************)
  23
  24 theorem lfeq_bind: ∀p,I,L1,L2,V1,V2,T.
  25                    L1 ≡[V1] L2 → L1.ⓑ{I}V1 ≡[T] L2.ⓑ{I}V2 →
  26                    L1 ≡[ⓑ{p,I}V1.T] L2.
  27 /2 width=2 by lfxs_bind/ qed.
  28
  29 theorem lfeq_flat: ∀I,L1,L2,V,T. L1 ≡[V] L2 → L1 ≡[T] L2 →
  30                    L1 ≡[ⓕ{I}V.T] L2.
  31 /2 width=1 by lfxs_flat/ qed.
  32
  33 (* Note: /2 width=3 by lfeq_lfxs_trans/ *)
  34 theorem lfeq_trans: ∀T. Transitive … (lfeq T).
  35 #T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
  36 lapply (frees_lreq_conf … Hf1 … HL1) #H0
  37 lapply (frees_mono … Hf2 … H0) -Hf2 -H0
  38 /4 width=7 by lreq_trans, lexs_eq_repl_back, ex2_intro/
  39 qed-.
  40
  41 theorem lfeq_canc_sn: ∀T. left_cancellable … (lfeq T).
  42 /3 width=3 by lfeq_trans, lfeq_sym/ qed-.
  43
  44 theorem lfeq_canc_dx: ∀T. right_cancellable … (lfeq T).
  45 /3 width=3 by lfeq_trans, lfeq_sym/ qed-.
  46
  47 (* Advanced properies on negated lazy equivalence *****************************)
  48
  49 (* Note: for use in auto, works with /4 width=8/ so lfeq_canc_sn is preferred *)
  50 lemma lfeq_nlfeq_trans: ∀T,L1,L. L1 ≡[T] L →
  51                         ∀L2. (L ≡[T] L2 → ⊥) → (L1 ≡[T] L2 → ⊥).
  52 /3 width=3 by lfeq_canc_sn/ qed-.
  53
  54 lemma nlfeq_lfeq_div: ∀T,L2,L. L2 ≡[T] L →
  55                       ∀L1. (L1 ≡[T] L → ⊥) → (L1 ≡[T] L2 → ⊥).
  56 /3 width=3 by lfeq_trans/ qed-.