:: STRUCT_0 semantic presentation

definition
attr a1 is strict;
struct 1-sorted -> ;
aggr 1-sorted(# carrier #) -> 1-sorted ;
sel carrier c1 -> set ;
end;

definition
attr a1 is strict;
struct ZeroStr -> 1-sorted ;
aggr ZeroStr(# carrier, Zero #) -> ZeroStr ;
sel Zero c1 -> Element of the carrier of a1;
end;

definition
let S be 1-sorted ;
attr a1 is empty means :Def1: :: STRUCT_0:def 1
the carrier of S is empty;
end;

:: deftheorem Def1 defines empty STRUCT_0:def 1 :
for S being 1-sorted holds
( S is empty iff the carrier of S is empty );

registration
cluster non empty 1-sorted ;
existence
not for b1 being 1-sorted holds b1 is empty
proof end;
end;

registration
cluster non empty ZeroStr ;
existence
not for b1 being ZeroStr holds b1 is empty
proof end;
end;

registration
let S be non empty 1-sorted ;
cluster the carrier of a1 -> non empty ;
coherence
not the carrier of S is empty
by ;
end;

definition
let S be 1-sorted ;
mode Element of a1 is Element of the carrier of a1;
mode Subset of a1 is Subset of the carrier of a1;
mode Subset-Family of a1 is Subset-Family of the carrier of a1;
end;

registration
let S be non empty 1-sorted ;
cluster non empty Element of K10(the carrier of a1);
existence
not for b1 being Subset of S holds b1 is empty
proof end;
end;

definition
let S be 1-sorted ;
let X be set ;
mode Function of a1,a2 is Function of the carrier of a1,a2;
mode Function of a2,a1 is Function of a2,the carrier of a1;
end;

definition
let S be 1-sorted , T be 1-sorted ;
mode Function of a1,a2 is Function of the carrier of a1,the carrier of a2;
end;

definition
let S be 1-sorted ;
mode FinSequence of a1 is FinSequence of the carrier of a1;
end;

definition
let S be 1-sorted ;
mode ManySortedSet of a1 is ManySortedSet of the carrier of a1;
end;