:: STRUCT_0 semantic presentation

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

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

definition
let S be 1-sorted ;
attr S 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 empty 1-sorted ;
existence
ex b1 being 1-sorted st b1 is empty
proof end;
end;

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 S -> non empty ;
coherence
not the carrier of S is empty
by Def1;
end;

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

registration
let S be non empty 1-sorted ;
cluster non empty Element of K10(the carrier of S);
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 S,X is Function of the carrier of S,X;
mode Function of X,S is Function of X,the carrier of S;
end;

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

definition
let T be 1-sorted ;
func {} T -> Subset of T equals :: STRUCT_0:def 2
{} ;
coherence
{} is Subset of T
proof end;
func [#] T -> Subset of T equals :: STRUCT_0:def 3
the carrier of T;
coherence
the carrier of T is Subset of T
proof end;
end;

:: deftheorem defines {} STRUCT_0:def 2 :
for T being 1-sorted holds {} T = {} ;

:: deftheorem defines [#] STRUCT_0:def 3 :
for T being 1-sorted holds [#] T = the carrier of T;

registration
let T be 1-sorted ;
cluster {} T -> empty ;
coherence
{} T is empty
;
end;

registration
let T be empty 1-sorted ;
cluster [#] T -> empty ;
coherence
[#] T is empty
by Def1;
end;

registration
let T be non empty 1-sorted ;
cluster [#] T -> non empty ;
coherence
not [#] T is empty
;
end;

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

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