propagating the arithmetics library, partial commit

[helm.git] / matita / matita / contribs / lambdadelta / ground / lib / stream_eq.ma

diff --git a/matita/matita/contribs/lambdadelta/ground/lib/stream_eq.ma b/matita/matita/contribs/lambdadelta/ground/lib/stream_eq.ma
index 425f91e825759d5b191a5470d851d295ff1530c9..311930686057e113b7d20faecfa5cdb1513220e0 100644 (file)
--- a/matita/matita/contribs/lambdadelta/ground/lib/stream_eq.ma
+++ b/matita/matita/contribs/lambdadelta/ground/lib/stream_eq.ma
@@ -15,57 +15,58 @@
 include "ground/notation/relations/ringeq_3.ma".
 include "ground/lib/stream.ma".
 
-(* STREAMS ******************************************************************)
+(* EXTENSIONAL EQUIVALENCE FOR 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)
+coinductive stream_eq (A): relation (stream A) ≝
+| stream_eq_cons: ∀t1,t2,b1,b2. b1 = b2 → stream_eq A t1 t2 → stream_eq A (b1⨮t1) (b2⨮t2)
 .
 
-interpretation "extensional equivalence (stream)"
-   'RingEq A t1 t2 = (eq_stream A t1 t2).
+interpretation
+  "extensional equivalence (streams)"
+  'RingEq A t1 t2 = (stream_eq A t1 t2).
 
-definition eq_stream_repl (A) (R:relation …) ≝
-                          ∀t1,t2. t1 ≗{A} t2 → R t1 t2.
+definition stream_eq_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 stream_eq_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.
+definition stream_eq_repl_fwd (A) (R:predicate …) ≝
+           ∀t1. R t1 → ∀t2. t2 ≗{A} t1 → R t2.
 
-(* Basic inversion lemmas ***************************************************)
+(* Basic inversions *********************************************************)
 
-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.
+lemma stream_eq_inv_cons: ∀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 *********************************************************)
+(* Basic constructions ******************************************************)
 
-corec lemma eq_stream_refl: ∀A. reflexive … (eq_stream A).
-#A * #b #t @eq_seq //
+corec lemma stream_eq_refl: ∀A. reflexive … (stream_eq A).
+#A * #b #t @stream_eq_cons //
 qed.
 
-corec lemma eq_stream_sym: ∀A. symmetric … (eq_stream A).
+corec lemma stream_eq_sym: ∀A. symmetric … (stream_eq A).
 #A #t1 #t2 * -t1 -t2
-#t1 #t2 #b1 #b2 #Hb #Ht @eq_seq /2 width=1 by/
+#t1 #t2 #b1 #b2 #Hb #Ht @stream_eq_cons /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-.
+lemma stream_eq_repl_sym: ∀A,R. stream_eq_repl_back A R → stream_eq_repl_fwd A R.
+/3 width=3 by stream_eq_sym/ qed-.
 
-(* Main properties **********************************************************)
+(* Main constructions *******************************************************)
 
-corec theorem eq_stream_trans: ∀A. Transitive … (eq_stream A).
+corec theorem stream_eq_trans: ∀A. Transitive … (stream_eq 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/
+#t1 #t #b1 #b * #Ht1 * #b2 #t2 #H cases (stream_eq_inv_cons A … H) -H -b
+/3 width=7 by stream_eq_cons/
 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 stream_eq_canc_sn: ∀A,t,t1,t2. t ≗ t1 → t ≗ t2 → t1 ≗{A} t2.
+/3 width=3 by stream_eq_trans, stream_eq_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-.
+theorem stream_eq_canc_dx: ∀A,t,t1,t2. t1 ≗ t → t2 ≗ t → t1 ≗{A} t2.
+/3 width=3 by stream_eq_trans, stream_eq_sym/ qed-.