(* *)
(**************************************************************************)
-include "ground_2/notation/relations/exteq_3.ma".
+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)
+| eq_seq: ∀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).
+ 'RingEq A t1 t2 = (eq_stream A t1 t2).
definition eq_stream_repl (A) (R:relation …) ≝
- â\88\80t1,t2. t1 â\89\90â¦\8bAâ¦\8c t2 → R t1 t2.
+ â\88\80t1,t2. t1 â\89\97{A} t2 → R t1 t2.
definition eq_stream_repl_back (A) (R:predicate …) ≝
- â\88\80t1. R t1 â\86\92 â\88\80t2. t1 â\89\90â¦\8bAâ¦\8c t2 → R t2.
+ â\88\80t1. R t1 â\86\92 â\88\80t2. t1 â\89\97{A} t2 → R t2.
definition eq_stream_repl_fwd (A) (R:predicate …) ≝
- â\88\80t1. R t1 â\86\92 â\88\80t2. t2 â\89\90â¦\8bAâ¦\8c t1 → R t2.
+ â\88\80t1. R t1 â\86\92 â\88\80t2. t2 â\89\97{A} t1 → R t2.
(* Basic inversion lemmas ***************************************************)
-lemma eq_stream_inv_seq: â\88\80A,t1,t2. t1 â\89\90â¦\8bAâ¦\8c t2 →
- ∀u1,u2,a1,a2. a1@u1 = t1 → a2@u2 = t2 →
- u1 â\89\90 u2 ∧ a1 = a2.
+lemma eq_stream_inv_seq: â\88\80A,t1,t2. t1 â\89\97{A} t2 →
+ ∀u1,u2,a1,a2. a1⨮u1 = t1 → a2⨮u2 = t2 →
+ u1 â\89\97 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 *********************************************************)
-let corec eq_stream_refl: ∀A. reflexive … (eq_stream A) ≝ ?.
+corec lemma eq_stream_refl: ∀A. reflexive … (eq_stream A).
#A * #b #t @eq_seq //
qed.
-let corec eq_stream_sym: ∀A. symmetric … (eq_stream A) ≝ ?.
+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-.
(* Main properties **********************************************************)
-let corec eq_stream_trans: ∀A. Transitive … (eq_stream A) ≝ ?.
+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: â\88\80A,t,t1,t2. t â\89\90 t1 â\86\92 t â\89\90 t2 â\86\92 t1 â\89\90â¦\8bAâ¦\8c t2.
+theorem eq_stream_canc_sn: â\88\80A,t,t1,t2. t â\89\97 t1 â\86\92 t â\89\97 t2 â\86\92 t1 â\89\97{A} t2.
/3 width=3 by eq_stream_trans, eq_stream_sym/ qed-.
-theorem eq_stream_canc_dx: â\88\80A,t,t1,t2. t1 â\89\90 t â\86\92 t2 â\89\90 t â\86\92 t1 â\89\90â¦\8bAâ¦\8c t2.
+theorem eq_stream_canc_dx: â\88\80A,t,t1,t2. t1 â\89\97 t â\86\92 t2 â\89\97 t â\86\92 t1 â\89\97{A} t2.
/3 width=3 by eq_stream_trans, eq_stream_sym/ qed-.