:: ALTCAT_1 semantic presentation
theorem Th1: :: ALTCAT_1:1
canceled;
theorem Th2: :: ALTCAT_1:2
theorem Th3: :: ALTCAT_1:3
theorem Th4: :: ALTCAT_1:4
theorem Th5: :: ALTCAT_1:5
theorem Th6: :: ALTCAT_1:6
canceled;
theorem Th7: :: ALTCAT_1:7
scheme :: ALTCAT_1:sch 3
s3{ F
1()
-> set , F
2()
-> set , F
3()
-> set , F
4(
set ,
set ,
set )
-> set } :
ex b
1 being
ManySortedSet of
[:F1(),F2(),F3():] st
for b
2, b
3, b
4 being
set holds
( b
2 in F
1() & b
3 in F
2() & b
4 in F
3() implies b
1 . b
2,b
3,b
4 = F
4(b
2,b
3,b
4) )
theorem Th8: :: ALTCAT_1:8
for b
1, b
2 being
set for b
3, b
4 being
ManySortedSet of
[:b1,b2:] holds
( ( for b
5, b
6 being
set holds
( b
5 in b
1 & b
6 in b
2 implies b
3 . b
5,b
6 = b
4 . b
5,b
6 ) ) implies b
4 = b
3 )
theorem Th9: :: ALTCAT_1:9
theorem Th10: :: ALTCAT_1:10
for b
1 being
set for b
2, b
3 being
ManySortedSet of
[:b1,b1,b1:] holds
( ( for b
4, b
5, b
6 being
set holds
( b
4 in b
1 & b
5 in b
1 & b
6 in b
1 implies b
2 . b
4,b
5,b
6 = b
3 . b
4,b
5,b
6 ) ) implies b
3 = b
2 )
theorem Th11: :: ALTCAT_1:11
theorem Th12: :: ALTCAT_1:12
for b
1, b
2, b
3 being
set holds
(b1,b2 :-> b3) . b
1,b
2 = b
3
:: deftheorem Def1 ALTCAT_1:def 1 :
canceled;
:: deftheorem Def2 defines <^ ALTCAT_1:def 2 :
:: deftheorem Def3 ALTCAT_1:def 3 :
canceled;
:: deftheorem Def4 defines transitive ALTCAT_1:def 4 :
definition
let c
1 be
set ;
let c
2 be
ManySortedSet of
[:c1,c1:];
func {|c2|} -> ManySortedSet of
[:a1,a1,a1:] means :
Def5:
:: ALTCAT_1:def 5
for b
1, b
2, b
3 being
set holds
( b
1 in a
1 & b
2 in a
1 & b
3 in a
1 implies a
3 . b
1,b
2,b
3 = a
2 . b
1,b
3 );
existence
ex b1 being ManySortedSet of [:c1,c1,c1:] st
for b2, b3, b4 being set holds
( b2 in c1 & b3 in c1 & b4 in c1 implies b1 . b2,b3,b4 = c2 . b2,b4 )
uniqueness
for b1, b2 being ManySortedSet of [:c1,c1,c1:] holds
( ( for b3, b4, b5 being set holds
( b3 in c1 & b4 in c1 & b5 in c1 implies b1 . b3,b4,b5 = c2 . b3,b5 ) ) & ( for b3, b4, b5 being set holds
( b3 in c1 & b4 in c1 & b5 in c1 implies b2 . b3,b4,b5 = c2 . b3,b5 ) ) implies b1 = b2 )
let c
3 be
ManySortedSet of
[:c1,c1:];
func {|c2,c3|} -> ManySortedSet of
[:a1,a1,a1:] means :
Def6:
:: ALTCAT_1:def 6
for b
1, b
2, b
3 being
set holds
( b
1 in a
1 & b
2 in a
1 & b
3 in a
1 implies a
4 . b
1,b
2,b
3 = [:(a3 . b2,b3),(a2 . b1,b2):] );
existence
ex b1 being ManySortedSet of [:c1,c1,c1:] st
for b2, b3, b4 being set holds
( b2 in c1 & b3 in c1 & b4 in c1 implies b1 . b2,b3,b4 = [:(c3 . b3,b4),(c2 . b2,b3):] )
uniqueness
for b1, b2 being ManySortedSet of [:c1,c1,c1:] holds
( ( for b3, b4, b5 being set holds
( b3 in c1 & b4 in c1 & b5 in c1 implies b1 . b3,b4,b5 = [:(c3 . b4,b5),(c2 . b3,b4):] ) ) & ( for b3, b4, b5 being set holds
( b3 in c1 & b4 in c1 & b5 in c1 implies b2 . b3,b4,b5 = [:(c3 . b4,b5),(c2 . b3,b4):] ) ) implies b1 = b2 )
end;
:: deftheorem Def5 defines {| ALTCAT_1:def 5 :
:: deftheorem Def6 defines {| ALTCAT_1:def 6 :
for b
1 being
set for b
2, b
3 being
ManySortedSet of
[:b1,b1:]for b
4 being
ManySortedSet of
[:b1,b1,b1:] holds
( b
4 = {|b2,b3|} iff for b
5, b
6, b
7 being
set holds
( b
5 in b
1 & b
6 in b
1 & b
7 in b
1 implies b
4 . b
5,b
6,b
7 = [:(b3 . b6,b7),(b2 . b5,b6):] ) );
definition
let c
1 be non
empty set ;
let c
2 be
ManySortedSet of
[:c1,c1:];
let c
3 be
BinComp of c
2;
let c
4, c
5, c
6 be
Element of c
1;
redefine func . as c
3 . c
4,c
5,c
6 -> Function of
[:(a2 . a5,a6),(a2 . a4,a5):],a
2 . a
4,a
6;
coherence
c3 . c4,c5,c6 is Function of [:(c2 . c5,c6),(c2 . c4,c5):],c2 . c4,c6
end;
definition
let c
1 be non
empty set ;
let c
2 be
ManySortedSet of
[:c1,c1:];
let c
3 be
BinComp of c
2;
attr a
3 is
associative means :
Def7:
:: ALTCAT_1:def 7
for b
1, b
2, b
3, b
4 being
Element of a
1for b
5, b
6, b
7 being
set holds
( b
5 in a
2 . b
1,b
2 & b
6 in a
2 . b
2,b
3 & b
7 in a
2 . b
3,b
4 implies
(a3 . b1,b3,b4) . b
7,
((a3 . b1,b2,b3) . b6,b5) = (a3 . b1,b2,b4) . ((a3 . b2,b3,b4) . b7,b6),b
5 );
attr a
3 is
with_right_units means :
Def8:
:: ALTCAT_1:def 8
for b
1 being
Element of a
1 holds
ex b
2 being
set st
( b
2 in a
2 . b
1,b
1 & ( for b
3 being
Element of a
1for b
4 being
set holds
( b
4 in a
2 . b
1,b
3 implies
(a3 . b1,b1,b3) . b
4,b
2 = b
4 ) ) );
attr a
3 is
with_left_units means :
Def9:
:: ALTCAT_1:def 9
for b
1 being
Element of a
1 holds
ex b
2 being
set st
( b
2 in a
2 . b
1,b
1 & ( for b
3 being
Element of a
1for b
4 being
set holds
( b
4 in a
2 . b
3,b
1 implies
(a3 . b3,b1,b1) . b
2,b
4 = b
4 ) ) );
end;
:: deftheorem Def7 defines associative ALTCAT_1:def 7 :
for b
1 being non
empty set for b
2 being
ManySortedSet of
[:b1,b1:]for b
3 being
BinComp of b
2 holds
( b
3 is
associative iff for b
4, b
5, b
6, b
7 being
Element of b
1for b
8, b
9, b
10 being
set holds
( b
8 in b
2 . b
4,b
5 & b
9 in b
2 . b
5,b
6 & b
10 in b
2 . b
6,b
7 implies
(b3 . b4,b6,b7) . b
10,
((b3 . b4,b5,b6) . b9,b8) = (b3 . b4,b5,b7) . ((b3 . b5,b6,b7) . b10,b9),b
8 ) );
:: deftheorem Def8 defines with_right_units ALTCAT_1:def 8 :
:: deftheorem Def9 defines with_left_units ALTCAT_1:def 9 :
definition
let c
1 be non
empty AltCatStr ;
let c
2, c
3, c
4 be
object of c
1;
assume E16:
(
<^c2,c3^> <> {} &
<^c3,c4^> <> {} )
;
let c
5 be
Morphism of c
2,c
3;
let c
6 be
Morphism of c
3,c
4;
func c
6 * c
5 -> Morphism of a
2,a
4 equals :
Def10:
:: ALTCAT_1:def 10
(the Comp of a1 . a2,a3,a4) . a
6,a
5;
coherence
(the Comp of c1 . c2,c3,c4) . c6,c5 is Morphism of c2,c4
correctness
;
end;
:: deftheorem Def10 defines * ALTCAT_1:def 10 :
:: deftheorem Def11 defines compositional ALTCAT_1:def 11 :
theorem Th13: :: ALTCAT_1:13
:: deftheorem Def12 defines FuncComp ALTCAT_1:def 12 :
theorem Th14: :: ALTCAT_1:14
theorem Th15: :: ALTCAT_1:15
theorem Th16: :: ALTCAT_1:16
definition
let c
1 be non
empty AltCatStr ;
attr a
1 is
quasi-functional means :
Def13:
:: ALTCAT_1:def 13
for b
1, b
2 being
object of a
1 holds
<^b1,b2^> c= Funcs b
1,b
2;
attr a
1 is
semi-functional means :
Def14:
:: ALTCAT_1:def 14
for b
1, b
2, b
3 being
object of a
1 holds
(
<^b1,b2^> <> {} &
<^b2,b3^> <> {} &
<^b1,b3^> <> {} implies for b
4 being
Morphism of b
1,b
2for b
5 being
Morphism of b
2,b
3for b
6, b
7 being
Function holds
( b
4 = b
6 & b
5 = b
7 implies b
5 * b
4 = b
7 * b
6 ) );
attr a
1 is
pseudo-functional means :
Def15:
:: ALTCAT_1:def 15
for b
1, b
2, b
3 being
object of a
1 holds the
Comp of a
1 . b
1,b
2,b
3 = (FuncComp (Funcs b1,b2),(Funcs b2,b3)) | [:<^b2,b3^>,<^b1,b2^>:];
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 :
for b
1 being non
empty AltCatStr holds
( b
1 is
pseudo-functional iff for b
2, b
3, b
4 being
object of b
1 holds the
Comp of b
1 . b
2,b
3,b
4 = (FuncComp (Funcs b2,b3),(Funcs b3,b4)) | [:<^b3,b4^>,<^b2,b3^>:] );
theorem Th17: :: ALTCAT_1:17
theorem Th18: :: ALTCAT_1:18
definition
let c
1 be non
empty set ;
func EnsCat c
1 -> non
empty strict pseudo-functional AltCatStr means :
Def16:
:: ALTCAT_1:def 16
( the
carrier of a
2 = a
1 & ( for b
1, b
2 being
object of a
2 holds
<^b1,b2^> = Funcs b
1,b
2 ) );
existence
ex b1 being non empty strict pseudo-functional AltCatStr st
( the carrier of b1 = c1 & ( for b2, b3 being object of b1 holds <^b2,b3^> = Funcs b2,b3 ) )
uniqueness
for b1, b2 being non empty strict pseudo-functional AltCatStr holds
( the carrier of b1 = c1 & ( for b3, b4 being object of b1 holds <^b3,b4^> = Funcs b3,b4 ) & the carrier of b2 = c1 & ( for b3, b4 being object of b2 holds <^b3,b4^> = Funcs b3,b4 ) implies b1 = b2 )
end;
:: deftheorem Def16 defines EnsCat ALTCAT_1:def 16 :
:: deftheorem Def17 defines associative ALTCAT_1:def 17 :
:: deftheorem Def18 defines with_units ALTCAT_1:def 18 :
Lemma30:
for b1 being non empty set holds
( EnsCat b1 is transitive & EnsCat b1 is associative & EnsCat b1 is with_units )
theorem Th19: :: ALTCAT_1:19
canceled;
theorem Th20: :: ALTCAT_1:20
for b
1 being non
empty transitive AltCatStr for b
2, b
3, b
4 being
object of b
1 holds
(
dom (the Comp of b1 . b2,b3,b4) = [:<^b3,b4^>,<^b2,b3^>:] &
rng (the Comp of b1 . b2,b3,b4) c= <^b2,b4^> )
theorem Th21: :: ALTCAT_1:21
definition
let c
1 be non
empty with_units AltCatStr ;
let c
2 be
object of c
1;
func idm c
2 -> Morphism of a
2,a
2 means :
Def19:
:: ALTCAT_1:def 19
for b
1 being
object of a
1 holds
(
<^a2,b1^> <> {} implies for b
2 being
Morphism of a
2,b
1 holds b
2 * a
3 = b
2 );
existence
ex b1 being Morphism of c2,c2 st
for b2 being object of c1 holds
( <^c2,b2^> <> {} implies for b3 being Morphism of c2,b2 holds b3 * b1 = b3 )
uniqueness
for b1, b2 being Morphism of c2,c2 holds
( ( for b3 being object of c1 holds
( <^c2,b3^> <> {} implies for b4 being Morphism of c2,b3 holds b4 * b1 = b4 ) ) & ( for b3 being object of c1 holds
( <^c2,b3^> <> {} implies for b4 being Morphism of c2,b3 holds b4 * b2 = b4 ) ) implies b1 = b2 )
end;
:: deftheorem Def19 defines idm ALTCAT_1:def 19 :
theorem Th22: :: ALTCAT_1:22
canceled;
theorem Th23: :: ALTCAT_1:23
theorem Th24: :: ALTCAT_1:24
theorem Th25: :: ALTCAT_1:25
:: deftheorem Def20 defines quasi-discrete ALTCAT_1:def 20 :
:: deftheorem Def21 defines pseudo-discrete ALTCAT_1:def 21 :
theorem Th26: :: ALTCAT_1:26
theorem Th27: :: ALTCAT_1:27
:: deftheorem Def22 defines DiscrCat ALTCAT_1:def 22 :
theorem Th28: :: ALTCAT_1:28
theorem Th29: :: ALTCAT_1:29