- ng_kernel: catched Invalid_argument "List.nth" in typeof of Rel i (raied when i...

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / lib / streams.ma

diff --git a/matita/matita/contribs/lambdadelta/ground_2/lib/streams.ma b/matita/matita/contribs/lambdadelta/ground_2/lib/streams.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ground_2/notation/constructors/cons_2.ma".
+include "ground_2/notation/relations/exteq_3.ma".
+include "ground_2/lib/star.ma".
+
+(* STREAMS ******************************************************************)
+
+coinductive stream (A:Type[0]): Type[0] ≝
+| seq: A → stream A → stream A
+.
+
+interpretation "cons (nstream)" 'Cons b t = (seq ? b t).
+
+coinductive eq_stream (A): relation (stream A) ≝
+| eq_sec: ∀t1,t2,b1,b2. b1 = b2 → eq_stream A t1 t2 → eq_stream A (b1@t1) (b2@t2)
+.
+
+interpretation "extensional equivalence (nstream)"
+   'ExtEq A t1 t2 = (eq_stream A t1 t2).
+
+definition eq_stream_repl_back (A) (R:predicate …) (t1,t2) ≝
+                               t1 ≐⦋A⦌ t2 → R t1 → R t2.
+
+definition eq_stream_repl_fwd (A) (R:predicate …) (t1,t2) ≝
+                              t2 ≐⦋A⦌ t1 → R t1 → R t2.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact eq_stream_inv_seq_aux: ∀A,t1,t2. t1 ≐⦋A⦌ t2 →
+                            ∀u1,u2,a1,a2. t1 = a1@u1 → t2 = a2@u2 →
+                            a1 = a2 ∧ u1 ≐ u2.
+#A #t1 #t2 * -t1 -t2
+#t1 #t2 #b1 #b2 #Hb #Ht #u1 #u2 #a1 #a2 #H1 #H2 destruct /2 width=1 by conj/
+qed-.
+
+lemma eq_stream_inv_seq: ∀A,t1,t2,b1,b2. b1@t1 ≐⦋A⦌ b2@t2 → b1 = b2 ∧ t1 ≐ t2.
+/2 width=5 by eq_stream_inv_seq_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma stream_expand (A) (t:stream A): t = match t with [ seq a u ⇒ a @ u ].
+#A * //
+qed.
+
+let corec eq_stream_refl: ∀A. reflexive … (eq_stream A) ≝ ?.
+#A * #b #t @eq_sec //
+qed.
+
+let corec eq_stream_sym: ∀A. symmetric … (eq_stream A) ≝ ?.
+#A #t1 #t2 * -t1 -t2
+#t1 #t2 #b1 #b2 #Hb #Ht @eq_sec /2 width=1 by/
+qed-.
+
+lemma eq_stream_repl_sym: ∀A,R,t1,t2. eq_stream_repl_back A R t1 t2 → eq_stream_repl_fwd A R t1 t2.
+/3 width=1 by eq_stream_sym/ qed-.
+
+(* Main properties **********************************************************)
+
+let corec eq_stream_trans: ∀A. Transitive … (eq_stream A) ≝ ?.
+#A #t1 #t * -t1 -t
+#t1 #t #b1 #b #Hb1 #Ht1 * #b2 #t2 #H cases (eq_stream_inv_seq A … H) -H
+#Hb2 #Ht2 @eq_sec /2 width=3 by/
+qed-.
+
+theorem eq_stream_canc_sn: ∀A,t,t1,t2. t ≐ t1 → t ≐ t2 → t1 ≐⦋A⦌ t2.
+/3 width=4 by eq_stream_trans, eq_stream_repl_sym/ qed-.
+
+theorem eq_stream_canc_dx: ∀A,t,t1,t2. t1 ≐ t → t2 ≐ t → t1 ≐⦋A⦌ t2.
+/3 width=4 by eq_stream_trans, eq_stream_repl_sym/ qed-.