-(**************************************************************************)
-(* ___ *)
-(* ||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/relations/ringeq_3.ma".
-include "ground_2/lib/streams.ma".
-
-(* STREAMS ******************************************************************)
-
-coinductive eq_stream (A): relation (stream A) ≝
-| eq_seq: ∀t1,t2,b1,b2. b1 = b2 → eq_stream A t1 t2 → eq_stream A (b1⨮t1) (b2⨮t2)
-.
-
-interpretation "extensional equivalence (nstream)"
- 'RingEq A t1 t2 = (eq_stream A t1 t2).
-
-definition eq_stream_repl (A) (R:relation …) ≝
- ∀t1,t2. t1 ≗{A} t2 → R t1 t2.
-
-definition eq_stream_repl_back (A) (R:predicate …) ≝
- ∀t1. R t1 → ∀t2. t1 ≗{A} t2 → R t2.
-
-definition eq_stream_repl_fwd (A) (R:predicate …) ≝
- ∀t1. R t1 → ∀t2. t2 ≗{A} t1 → R t2.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma eq_stream_inv_seq: ∀A,t1,t2. t1 ≗{A} t2 →
- ∀u1,u2,a1,a2. a1⨮u1 = t1 → a2⨮u2 = t2 →
- u1 ≗ u2 ∧ a1 = a2.
-#A #t1 #t2 * -t1 -t2
-#t1 #t2 #b1 #b2 #Hb #Ht #u1 #u2 #a1 #a2 #H1 #H2 destruct /2 width=1 by conj/
-qed-.
-
-(* Basic properties *********************************************************)
-
-corec lemma eq_stream_refl: ∀A. reflexive … (eq_stream A).
-#A * #b #t @eq_seq //
-qed.
-
-corec lemma eq_stream_sym: ∀A. symmetric … (eq_stream A).
-#A #t1 #t2 * -t1 -t2
-#t1 #t2 #b1 #b2 #Hb #Ht @eq_seq /2 width=1 by/
-qed-.
-
-lemma eq_stream_repl_sym: ∀A,R. eq_stream_repl_back A R → eq_stream_repl_fwd A R.
-/3 width=3 by eq_stream_sym/ qed-.
-
-(* Main properties **********************************************************)
-
-corec theorem eq_stream_trans: ∀A. Transitive … (eq_stream A).
-#A #t1 #t * -t1 -t
-#t1 #t #b1 #b * #Ht1 * #b2 #t2 #H cases (eq_stream_inv_seq A … H) -H -b
-/3 width=7 by eq_seq/
-qed-.
-
-theorem eq_stream_canc_sn: ∀A,t,t1,t2. t ≗ t1 → t ≗ t2 → t1 ≗{A} t2.
-/3 width=3 by eq_stream_trans, eq_stream_sym/ qed-.
-
-theorem eq_stream_canc_dx: ∀A,t,t1,t2. t1 ≗ t → t2 ≗ t → t1 ≗{A} t2.
-/3 width=3 by eq_stream_trans, eq_stream_sym/ qed-.