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)
+| 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).
-definition eq_stream_repl_back (A) (R:predicate …) (t1,t2) ≝
- t1 ≐⦋A⦌ t2 → R t1 → R 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.
+definition eq_stream_repl_fwd (A) (R:predicate …) ≝
+ ∀t1,t2. t2 ≐⦋A⦌ t1 → R t1 → R t2.
(* Basic inversion lemmas ***************************************************)
qed.
let corec eq_stream_refl: ∀A. reflexive … (eq_stream A) ≝ ?.
-#A * #b #t @eq_sec //
+#A * #b #t @eq_seq //
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/
+#t1 #t2 #b1 #b2 #Hb #Ht @eq_seq /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-.
+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 **********************************************************)
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/
+#Hb2 #Ht2 @eq_seq /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-.
+/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=4 by eq_stream_trans, eq_stream_repl_sym/ qed-.
+/3 width=3 by eq_stream_trans, eq_stream_sym/ qed-.