Quelle nat_indexed_sets.pvs Sprache: PVS

%------------------------------------------------------------------------------
% Indexed sets over nat
%
%     Author: David Lester, Manchester University
%
%     Version 1.0            14/06/09  Initial Version
%
% A new operation (along with the recurrence relations) is introduced:
%
%   IUnion(n,A) = A_0 u A_1 u ... u A_n
%
% We also show how to turn an indexed set into an increasing one
% with the same IUnion, or a decreasing one, with the same IIntersection.
%
% Finally, we show that there's a disjoint Union available.
%------------------------------------------------------------------------------
nat_indexed_sets[T: TYPE]: THEORY

BEGIN

  IMPORTING structures@fun_preds_partial[nat,set[T],reals.<=,subset?[T]],
            indexed_sets_aux[nat,T]

  n,i: VAR nat
  A,B: VAR sequence[set[T]]

  IUnion(n,A):set[T] = Union(image[nat,set[T]](A,{i | i <= n}))

  IUnion_0_def: LEMMA IUnion(0,A)   = A(0)
  IUnion_n_def: LEMMA IUnion(n+1,A) = union(IUnion(n,A),A(n+1))

  increasing_indexed?(A):bool = IUnion(A) = fullset[T] AND increasing?(A)

  increasing_indexed: TYPE+ = (increasing_indexed?) CONTAINING
                                                         (lambda n: fullset[T])

  increasing_IUnion: LEMMA
     EXISTS B: increasing?(B) AND
               B(0) = A(0) AND
               (FORALL n: B(n+1) = union(B(n),A(n+1))) AND
               IUnion(A) = IUnion(B)

  decreasing_IIntersection: LEMMA
     EXISTS B: decreasing?(B) AND
               B(0) = A(0) AND
               (FORALL n: B(n+1) = intersection(B(n),A(n+1))) AND
               IIntersection(A) = IIntersection(B)

  disjoint_IUnion: LEMMA
     EXISTS B: disjoint?(B) AND
               (FORALL n: IUnion(n,B) = IUnion(n,A)) AND
               B(0) = A(0) AND
               (FORALL n: B(n+1) = difference(A(n+1),IUnion(n,A))) AND
               IUnion(A) = IUnion(B)

END nat_indexed_sets

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