matita/matita/contribs/lambdadelta/ground/lib/stream_eq.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 include "ground/notation/relations/ringeq_3.ma".
  16 include "ground/lib/stream.ma".
  17
  18 (* EXTENSIONAL EQUIVALENCE FOR STREAMS **************************************)
  19
  20 coinductive stream_eq (A): relation (stream A) ≝
  21 | stream_eq_cons (a1) (a2) (t1) (t2):
  22   a1 = a2 → stream_eq A t1 t2 → stream_eq A (a1⨮t1) (a2⨮t2)
  23 .
  24
  25 interpretation
  26   "extensional equivalence (streams)"
  27   'RingEq A t1 t2 = (stream_eq A t1 t2).
  28
  29 definition stream_eq_repl (A) (R:relation …) ≝
  30            ∀t1,t2. t1 ≗{A} t2 → R t1 t2.
  31
  32 definition stream_eq_repl_back (A) (R:predicate …) ≝
  33            ∀t1. R t1 → ∀t2. t1 ≗{A} t2 → R t2.
  34
  35 definition stream_eq_repl_fwd (A) (R:predicate …) ≝
  36            ∀t1. R t1 → ∀t2. t2 ≗{A} t1 → R t2.
  37
  38 (* Basic constructions ******************************************************)
  39
  40 corec lemma stream_eq_refl (A:?):
  41             reflexive … (stream_eq A).
  42 * #a #t @stream_eq_cons //
  43 qed.
  44
  45 corec lemma stream_eq_sym (A):
  46             symmetric … (stream_eq A).
  47 #t1 #t2 * -t1 -t2
  48 #a1 #a2 #t1 #t2 #Ha #Ht
  49 @stream_eq_cons /2 width=1 by/
  50 qed-.
  51
  52 lemma stream_eq_repl_sym (A) (R):
  53       stream_eq_repl_back A R → stream_eq_repl_fwd A R.
  54 /3 width=3 by stream_eq_sym/ qed-.
  55
  56 (* Basic inversions *********************************************************)
  57
  58 lemma stream_eq_inv_cons_bi (A):
  59       ∀t1,t2. t1 ≗{A} t2 →
  60       ∀u1,u2,b1,b2. b1⨮u1 = t1 → b2⨮u2 = t2 →
  61       ∧∧ b1 = b2 & u1 ≗ u2.
  62 #A #t1 #t2 * -t1 -t2
  63 #a1 #a2 #t1 #t2 #Ha #Ht #u1 #u2 #b1 #b2 #H1 #H2 destruct /2 width=1 by conj/
  64 qed-.