Quelle fun_preds_partial.pvs Sprache: PVS

fun_preds_partial[T1,T2:TYPE,le1:(partial_order?[T1]),
                             le2:(partial_order?[T2])]: THEORY

  BEGIN

  f,g,h: VAR [T1->T2]
  x,y:   VAR T1
  a,b:   VAR T2

  lt1(x,y)  : MACRO bool = le1(x,y) AND x /= y
  lt2(a,b)  : MACRO bool = le2(a,b) AND a /= b
  min(x,y):T1 = IF le1(x,y) THEN x ELSE y ENDIF
  max(x,y):T1 = IF le1(x,y) THEN x ELSE y ENDIF

  increasing?(f)        : bool = FORALL x,y : le1(x,y) IMPLIES le2(f(x),f(y))
  decreasing?(f)        : bool = FORALL x,y : le1(x,y) IMPLIES le2(f(y),f(x))
  strict_increasing?(f) : bool = FORALL x,y : lt1(x,y) IMPLIES lt2(f(x),f(y))
  strict_decreasing?(f) : bool = FORALL x,y : lt1(x,y) IMPLIES lt2(f(y),f(x))

  constant?(f)          : bool = FORALL x,y : f(x) = f(y)

  monotonic?(f)         : bool = increasing?(f) OR decreasing?(f)
  strict_monotonic?(f)  : bool = strict_increasing?(f) OR strict_decreasing?(f)

  min(f,g):[T1->T2] = LAMBDA x: IF le2(f(x),g(x)) THEN f(x) ELSE g(x) ENDIF
  max(f,g):[T1->T2] = LAMBDA x: IF le2(f(x),g(x)) THEN g(x) ELSE f(x) ENDIF



  mid(g,f,h):[T1->T2] = min(max(g,f),h)

  increasing_is_function: JUDGEMENT (increasing?) SUBTYPE_OF [T1->T2]
  decreasing_is_function: JUDGEMENT (decreasing?) SUBTYPE_OF [T1->T2]
  constant_is_function  : JUDGEMENT (constant?)   SUBTYPE_OF [T1->T2]
  monotonic_is_function : JUDGEMENT (monotonic?)  SUBTYPE_OF [T1->T2]

  strict_increasing_is_increasing:
                       JUDGEMENT (strict_increasing?) SUBTYPE_OF (increasing?)
  strict_decreasing_is_decreasing:
                       JUDGEMENT (strict_decreasing?) SUBTYPE_OF (decreasing?)
  strict_monotonic_is_monotonic:
                       JUDGEMENT (strict_monotonic?)  SUBTYPE_OF (monotonic?)

%  min_preserves_increasing: JUDGEMENT

  END fun_preds_partial

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Wurzel

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Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

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