Quelle interm_value_thm.pvs Sprache: PVS

interm_value_thm: THEORY
%-----------------------------------------------------------------
% API for Intermediate Value Theorem
%-----------------------------------------------------------------
BEGIN

 IMPORTING continuity_interval, reals@intervals_real

 a,b,x: VAR real

 interm_value1 : LEMMA a real]):
 continuous?(f) AND f(a) < x AND x < f(b)
 IMPLIES EXISTS (c: closed_interval(a,b)): f(c) = x

 interm_value2 : LEMMA a real]):
 continuous?(f) AND f(a) <= x AND x <= f(b)
 IMPLIES EXISTS (c: closed_interval(a,b)): f(c) = x

 interm_value3 : LEMMA a real]):
 continuous?(f) AND f(b) < x AND x < f(a)
 IMPLIES EXISTS (c: closed_interval(a,b)): f(c) = x

 interm_value4 : LEMMA a real]):
 continuous?(f) AND f(b) <= x AND x <= f(a)
 IMPLIES EXISTS (c: closed_interval(a,b)): f(c) = x

 interm_value5 : LEMMA a real],p,q:closed_interval(a,b)):
 continuous?(f) AND f(p) <= x AND x <= f(q)
 IMPLIES EXISTS (c: closed_interval(a,b)):
 ((p<=c AND c<=q) OR (q<=c AND c<=p)) AND f(c) = x

 f: VAR [real -> real]

 cont_intv : LEMMA a < b AND continuous?(f)
 IMPLIES continuous?(LAMBDA (x: closed_interval(a, b)): f(x))

 cont_intv1: LEMMA a < b AND continuous?[closed_interval[real](a, b)](f)
 IMPLIES continuous?(LAMBDA (x: closed_interval(a, b)): f(x))

 interm_val1 : LEMMA a < b IMPLIES
 continuous?[closed_interval(a,b)](f) AND f(a) < x AND x < f(b)
 IMPLIES EXISTS (c: closed_interval(a,b)): f(c) = x

 interm_val2 : LEMMA a real]):
 continuous?[closed_interval(a,b)](f) AND f(a) <= x AND x <= f(b)
 IMPLIES EXISTS (c: closed_interval(a,b)): f(c) = x

 interm_val3 : LEMMA a real]):
 continuous?[closed_interval(a,b)](f) AND f(b) < x AND x < f(a)
 IMPLIES EXISTS (c: closed_interval(a,b)): f(c) = x

 interm_val4 : LEMMA a real]):
 continuous?[closed_interval(a,b)](f) AND f(b) <= x AND x <= f(a)
 IMPLIES EXISTS (c: closed_interval(a,b)): f(c) = x

 % ----------------------------------------------------------------------------------

 xl,xu: VAR real

 zeros_interm: LEMMA continuous?(f)
 AND f(xl) = 0 AND f(xu) = 0 AND
 (FORALL (x: open_interval(xl,xu)): f(x) /= 0)
 IMPLIES
 (FORALL (x: open_interval(xl,xu)): f(x) > 0) OR
 (FORALL (x: open_interval(xl,xu)): f(x) < 0)

 % ----------------------------------------------------------------------------------

 IMPORTING reals@connected_set

 ab : VAR Connected

 IntermediateValue : THEOREM
 FORALL (f:[real->real]):
 continuous?(f) AND
 (FORALL (c:(ab)) : f(c) /= 0) IMPLIES
 (FORALL (x,y:(ab)) : f(x)*f(y) > 0)

END interm_value_thm

quality94%

¤ Dauer der Verarbeitung: 0.10 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.