%------------------------------------------------------------------------------
% Composition is Continuous
%
% All references are to BA Davey and HA Priestly "Introduction to Lattices and
% Orders", CUP, 1990
%
% Author: David Lester, Manchester University, NIA, Université Perpignan
%
% Version 1.0 25/12/07 Initial Version
%------------------------------------------------------------------------------
scott_composition_continuity[T1,T2,T3:TYPE, (IMPORTING orders@directed_orders)
le1:(directed_complete_partial_order?[T1]),
le2:(directed_complete_partial_order?[T2]),
le3:(directed_complete_partial_order?[T3])]: THEORY
BEGIN
IMPORTING scott_continuity[T1,T2,le1,le2],
scott_continuity[T2,T3,le2,le3],
scott_continuity[T1,T3,le1,le3],
topology@composition_continuity[T1,scott_open_sets[T1,le1],
T2,scott_open_sets[T2,le2],
T3,scott_open_sets[T3,le3]]
f: VAR scott_continuous[T2,T3,le2,le3]
g: VAR scott_continuous[T1,T2,le1,le2]
composition_scott_continuous: JUDGEMENT
o(f:(scott_continuous?[T2,T3,le2,le3]),
g:(scott_continuous?[T1,T2,le1,le2]))
HAS_TYPE scott_continuous[T1,T3,le1,le3]
END scott_composition_continuity
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