:: ALTCAT_1 semantic presentation

registration

let A be functional set ;
cluster -> functional Element of bool A;
coherence

proof end;

end;

registration

let f be Function-yielding Function;
let C be set ;
cluster f | C -> Function-yielding ;
correctness by MSUHOM_1:3;

end;

registration

let f be Function;
cluster {f} -> functional ;
coherence

proof end;

end;

theorem :: ALTCAT_1:1

canceled;

theorem Th2: :: ALTCAT_1:2

for A being set holds id A in Funcs A,A

proof end;

theorem Th3: :: ALTCAT_1:3

Funcs {} ,{} = {(id {} )}

proof end;

theorem Th4: :: ALTCAT_1:4

for A, B, C being set
for f, g being Function st f in Funcs A,B & g in Funcs B,C holds
g * f in Funcs A,C

proof end;

theorem Th5: :: ALTCAT_1:5

for A, B, C being set st Funcs A,B <> {} & Funcs B,C <> {} holds
Funcs A,C <> {}

proof end;

theorem :: ALTCAT_1:6

canceled;

theorem :: ALTCAT_1:7

for A, B being set
for F being ManySortedSet of [:B,A:]
for C being Subset of A
for D being Subset of B
for x, y being set st x in C & y in D holds
F . y,x = (F | [:D,C:]) . y,x

proof end;

scheme :: ALTCAT_1:sch 1
MSSLambda2{ F₁() -> set , F₂() -> set , F₃( set , set ) -> set } :

ex M being ManySortedSet of [:F₁(),F₂():] st
for i, j being set st i in F₁() & j in F₂() holds
M . i,j = F₃(i,j)

proof end;

scheme :: ALTCAT_1:sch 2
MSSLambda2D{ F₁() -> non empty set , F₂() -> non empty set , F₃( set , set ) -> set } :

ex M being ManySortedSet of [:F₁(),F₂():] st
for i being Element of F₁()
for j being Element of F₂() holds M . i,j = F₃(i,j)

proof end;

scheme :: ALTCAT_1:sch 3
MSSLambda3{ F₁() -> set , F₂() -> set , F₃() -> set , F₄( set , set , set ) -> set } :

ex M being ManySortedSet of [:F₁(),F₂(),F₃():] st
for i, j, k being set st i in F₁() & j in F₂() & k in F₃() holds
M . i,j,k = F₄(i,j,k)

proof end;

scheme :: ALTCAT_1:sch 4
MSSLambda3D{ F₁() -> non empty set , F₂() -> non empty set , F₃() -> non empty set , F₄( set , set , set ) -> set } :

ex M being ManySortedSet of [:F₁(),F₂(),F₃():] st
for i being Element of F₁()
for j being Element of F₂()
for k being Element of F₃() holds M . i,j,k = F₄(i,j,k)

proof end;

theorem Th8: :: ALTCAT_1:8

for A, B being set
for N, M being ManySortedSet of [:A,B:] st ( for i, j being set st i in A & j in B holds
N . i,j = M . i,j ) holds
M = N

proof end;

theorem Th9: :: ALTCAT_1:9

for A, B being non empty set
for N, M being ManySortedSet of [:A,B:] st ( for i being Element of A
for j being Element of B holds N . i,j = M . i,j ) holds
M = N

proof end;

theorem Th10: :: ALTCAT_1:10

for A being set
for N, M being ManySortedSet of [:A,A,A:] st ( for i, j, k being set st i in A & j in A & k in A holds
N . i,j,k = M . i,j,k ) holds
M = N

proof end;

theorem Th11: :: ALTCAT_1:11

for i, j, k being set holds i,j :-> k = [i,j] .--> k

proof end;

theorem Th12: :: ALTCAT_1:12

for i, j, k being set holds (i,j :-> k) . i,j = k

proof end;

definition

attr c₁ is strict;
struct AltGraph -> 1-sorted ;
aggr AltGraph(# carrier, Arrows #) -> AltGraph ;
sel Arrows c₁ -> ManySortedSet of [:the carrier of c₁,the carrier of c₁:];

end;

definition

let G be AltGraph ;
mode object of G is Element of G;

end;

definition

let G be AltGraph ;
let o1, o2 be object of G;
canceled;
func <^o1,o2^> -> set equals :: ALTCAT_1:def 2
the Arrows of G . o1,o2;
correctness ;

end;

:: deftheorem ALTCAT_1:def 1 :

:: deftheorem defines <^ ALTCAT_1:def 2 :

definition

let G be AltGraph ;
let o1, o2 be object of G;
mode Morphism of o1,o2 is Element of <^o1,o2^>;

end;

definition

let G be AltGraph ;
canceled;
attr G is transitive means :Def4: :: ALTCAT_1:def 4
for o1, o2, o3 being object of G st <^o1,o2^> <> {} & <^o2,o3^> <> {} holds
<^o1,o3^> <> {} ;

end;

:: deftheorem ALTCAT_1:def 3 :

:: deftheorem Def4 defines transitive ALTCAT_1:def 4 :

definition

let I be set ;
let G be ManySortedSet of [:I,I:];
func {|G|} -> ManySortedSet of [:I,I,I:] means :Def5: :: ALTCAT_1:def 5
for i, j, k being set st i in I & j in I & k in I holds
it . i,j,k = G . i,k;
existence

proof end;

uniqueness

proof end;

let H be ManySortedSet of [:I,I:];
func {|G,H|} -> ManySortedSet of [:I,I,I:] means :Def6: :: ALTCAT_1:def 6
for i, j, k being set st i in I & j in I & k in I holds
it . i,j,k = [:(H . j,k),(G . i,j):];
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def5 defines {| ALTCAT_1:def 5 :

:: deftheorem Def6 defines {| ALTCAT_1:def 6 :

definition

let I be set ;
let G be ManySortedSet of [:I,I:];
mode BinComp of G is ManySortedFunction of {|G,G|},{|G|};

end;

definition

let I be non empty set ;
let G be ManySortedSet of [:I,I:];
let o be BinComp of G;
let i, j, k be Element of I;
:: original: .
redefine func o . i,j,k -> Function of [:(G . j,k),(G . i,j):],G . i,k;
coherence

proof end;

end;

definition

let I be non empty set ;
let G be ManySortedSet of [:I,I:];
let IT be BinComp of G;
attr IT is associative means :Def7: :: ALTCAT_1:def 7
for i, j, k, l being Element of I
for f, g, h being set st f in G . i,j & g in G . j,k & h in G . k,l holds
(IT . i,k,l) . h,((IT . i,j,k) . g,f) = (IT . i,j,l) . ((IT . j,k,l) . h,g),f;
attr IT is with_right_units means :Def8: :: ALTCAT_1:def 8
for i being Element of I ex e being set st
( e in G . i,i & ( for j being Element of I
for f being set st f in G . i,j holds
(IT . i,i,j) . f,e = f ) );
attr IT is with_left_units means :Def9: :: ALTCAT_1:def 9
for j being Element of I ex e being set st
( e in G . j,j & ( for i being Element of I
for f being set st f in G . i,j holds
(IT . i,j,j) . e,f = f ) );

end;

:: deftheorem Def7 defines associative ALTCAT_1:def 7 :

:: deftheorem Def8 defines with_right_units ALTCAT_1:def 8 :

:: deftheorem Def9 defines with_left_units ALTCAT_1:def 9 :

definition

attr c₁ is strict;
struct AltCatStr -> AltGraph ;
aggr AltCatStr(# carrier, Arrows, Comp #) -> AltCatStr ;
sel Comp c₁ -> BinComp of the Arrows of c₁;

end;

registration

cluster non empty strict AltCatStr ;
existence

proof end;

end;

definition

let C be non empty AltCatStr ;
let o1, o2, o3 be object of C;
assume A1: ( <^o1,o2^> <> {} & <^o2,o3^> <> {} ) ;
let f be Morphism of o1,o2;
let g be Morphism of o2,o3;
func g * f -> Morphism of o1,o3 equals :Def10: :: ALTCAT_1:def 10
(the Comp of C . o1,o2,o3) . g,f;
coherence

proof end;

correctness ;

end;

:: deftheorem Def10 defines * ALTCAT_1:def 10 :

definition

let IT be Function;
attr IT is compositional means :Def11: :: ALTCAT_1:def 11
for x being set st x in dom IT holds
ex f, g being Function st
( x = [g,f] & IT . x = g * f );

end;

:: deftheorem Def11 defines compositional ALTCAT_1:def 11 :

registration

let A, B be functional set ;
cluster compositional ManySortedSet of [:A,B:];
existence

proof end;

end;

theorem Th13: :: ALTCAT_1:13

for A, B being functional set
for F being compositional ManySortedSet of [:A,B:]
for g, f being Function st g in A & f in B holds
F . g,f = g * f

proof end;

definition

let A, B be functional set ;
func FuncComp A,B -> compositional ManySortedFunction of [:B,A:] means :Def12: :: ALTCAT_1:def 12
verum;
uniqueness

proof end;

correctness ;

end;

:: deftheorem Def12 defines FuncComp ALTCAT_1:def 12 :

theorem Th14: :: ALTCAT_1:14

for A, B, C being set holds rng (FuncComp (Funcs A,B),(Funcs B,C)) c= Funcs A,C

proof end;

theorem Th15: :: ALTCAT_1:15

for o being set holds FuncComp {(id o)},{(id o)} = (id o),(id o) :-> (id o)

proof end;

theorem Th16: :: ALTCAT_1:16

for A, B being functional set
for A1 being Subset of A
for B1 being Subset of B holds FuncComp A1,B1 = (FuncComp A,B) | [:B1,A1:]

proof end;

definition

let C be non empty AltCatStr ;
attr C is quasi-functional means :Def13: :: ALTCAT_1:def 13
for a1, a2 being object of C holds <^a1,a2^> c= Funcs a1,a2;
attr C is semi-functional means :Def14: :: ALTCAT_1:def 14
for a1, a2, a3 being object of C st <^a1,a2^> <> {} & <^a2,a3^> <> {} & <^a1,a3^> <> {} holds
for f being Morphism of a1,a2
for g being Morphism of a2,a3
for f', g' being Function st f = f' & g = g' holds
g * f = g' * f';
attr C is pseudo-functional means :Def15: :: ALTCAT_1:def 15
for o1, o2, o3 being object of C holds the Comp of C . o1,o2,o3 = (FuncComp (Funcs o1,o2),(Funcs o2,o3)) | [:<^o2,o3^>,<^o1,o2^>:];

end;

:: deftheorem Def13 defines quasi-functional ALTCAT_1:def 13 :

:: deftheorem Def14 defines semi-functional ALTCAT_1:def 14 :

:: deftheorem Def15 defines pseudo-functional ALTCAT_1:def 15 :

registration

let X be non empty set ;
let A be ManySortedSet of [:X,X:];
let C be BinComp of A;
cluster AltCatStr(# X,A,C #) -> non empty ;
coherence by STRUCT_0:def 1;

end;

registration

cluster non empty strict pseudo-functional AltCatStr ;
existence

proof end;

end;

theorem :: ALTCAT_1:17

for C being non empty AltCatStr
for a1, a2, a3 being object of C holds the Comp of C . a1,a2,a3 is Function of [:<^a2,a3^>,<^a1,a2^>:],<^a1,a3^> ;

theorem Th18: :: ALTCAT_1:18

for C being non empty pseudo-functional AltCatStr
for a1, a2, a3 being object of C st <^a1,a2^> <> {} & <^a2,a3^> <> {} & <^a1,a3^> <> {} holds
for f being Morphism of a1,a2
for g being Morphism of a2,a3
for f', g' being Function st f = f' & g = g' holds
g * f = g' * f'

proof end;

definition

let A be non empty set ;
func EnsCat A -> non empty strict pseudo-functional AltCatStr means :Def16: :: ALTCAT_1:def 16
( the carrier of it = A & ( for a1, a2 being object of it holds <^a1,a2^> = Funcs a1,a2 ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def16 defines EnsCat ALTCAT_1:def 16 :

definition

let C be non empty AltCatStr ;
attr C is associative means :Def17: :: ALTCAT_1:def 17
the Comp of C is associative;
attr C is with_units means :Def18: :: ALTCAT_1:def 18
( the Comp of C is with_left_units & the Comp of C is with_right_units );

end;

:: deftheorem Def17 defines associative ALTCAT_1:def 17 :

:: deftheorem Def18 defines with_units ALTCAT_1:def 18 :

Lm1: for A being non empty set holds
( EnsCat A is transitive & EnsCat A is associative & EnsCat A is with_units )

proof end;

registration

cluster non empty transitive strict associative with_units AltCatStr ;
existence

proof end;

end;

theorem :: ALTCAT_1:19

canceled;

theorem :: ALTCAT_1:20

for C being non empty transitive AltCatStr
for a1, a2, a3 being object of C holds
( dom (the Comp of C . a1,a2,a3) = [:<^a2,a3^>,<^a1,a2^>:] & rng (the Comp of C . a1,a2,a3) c= <^a1,a3^> )

proof end;

theorem Th21: :: ALTCAT_1:21

for C being non empty with_units AltCatStr
for o being object of C holds <^o,o^> <> {}

proof end;

registration

let A be non empty set ;
cluster EnsCat A -> non empty transitive strict pseudo-functional associative with_units ;
coherence by Lm1;

end;

registration

cluster non empty transitive quasi-functional semi-functional -> non empty pseudo-functional AltCatStr ;
coherence

proof end;

cluster non empty transitive pseudo-functional with_units -> non empty quasi-functional semi-functional AltCatStr ;
coherence

proof end;

end;

definition

mode category is non empty transitive associative with_units AltCatStr ;

end;

definition

let C be non empty with_units AltCatStr ;
let o be object of C;
func idm o -> Morphism of o,o means :Def19: :: ALTCAT_1:def 19
for o' being object of C st <^o,o'^> <> {} holds
for a being Morphism of o,o' holds a * it = a;
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def19 defines idm ALTCAT_1:def 19 :

theorem :: ALTCAT_1:22

canceled;

theorem Th23: :: ALTCAT_1:23

for C being non empty with_units AltCatStr
for o being object of C holds idm o in <^o,o^>

proof end;

theorem :: ALTCAT_1:24

for C being non empty with_units AltCatStr
for o1, o2 being object of C st <^o1,o2^> <> {} holds
for a being Morphism of o1,o2 holds (idm o2) * a = a

proof end;

theorem :: ALTCAT_1:25

for C being non empty transitive associative AltCatStr
for o1, o2, o3, o4 being object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} & <^o3,o4^> <> {} holds
for a being Morphism of o1,o2
for b being Morphism of o2,o3
for c being Morphism of o3,o4 holds c * (b * a) = (c * b) * a

proof end;

definition