:: WAYBEL27 semantic presentation
:: deftheorem Def1 defines uncurrying WAYBEL27:def 1 :
:: deftheorem Def2 defines currying WAYBEL27:def 2 :
:: deftheorem Def3 defines commuting WAYBEL27:def 3 :
theorem Th1: :: WAYBEL27:1
theorem Th2: :: WAYBEL27:2
registration
let c
1, c
2, c
3 be
set ;
cluster uncurrying ManySortedSet of
Funcs a
1,
(Funcs a2,a3);
existence
ex b1 being ManySortedSet of Funcs c1,(Funcs c2,c3) st b1 is uncurrying
cluster currying ManySortedSet of
Funcs [:a1,a2:],a
3;
existence
ex b1 being ManySortedSet of Funcs [:c1,c2:],c3 st b1 is currying
end;
theorem Th3: :: WAYBEL27:3
theorem Th4: :: WAYBEL27:4
theorem Th5: :: WAYBEL27:5
theorem Th6: :: WAYBEL27:6
theorem Th7: :: WAYBEL27:7
theorem Th8: :: WAYBEL27:8
theorem Th9: :: WAYBEL27:9
theorem Th10: :: WAYBEL27:10
theorem Th11: :: WAYBEL27:11
theorem Th12: :: WAYBEL27:12
theorem Th13: :: WAYBEL27:13
theorem Th14: :: WAYBEL27:14
theorem Th15: :: WAYBEL27:15
theorem Th16: :: WAYBEL27:16
theorem Th17: :: WAYBEL27:17
theorem Th18: :: WAYBEL27:18
theorem Th19: :: WAYBEL27:19
theorem Th20: :: WAYBEL27:20
theorem Th21: :: WAYBEL27:21
:: deftheorem Def4 defines UPS WAYBEL27:def 4 :
theorem Th22: :: WAYBEL27:22
theorem Th23: :: WAYBEL27:23
theorem Th24: :: WAYBEL27:24
theorem Th25: :: WAYBEL27:25
theorem Th26: :: WAYBEL27:26
theorem Th27: :: WAYBEL27:27
definition
let c
1, c
2, c
3, c
4 be non
empty reflexive antisymmetric RelStr ;
let c
5 be
Function of c
1,c
2;
assume E30:
c
5 is
directed-sups-preserving
;
let c
6 be
Function of c
3,c
4;
assume E31:
c
6 is
directed-sups-preserving
;
func UPS c
5,c
6 -> Function of
(UPS a2,a3),
(UPS a1,a4) means :
Def5:
:: WAYBEL27:def 5
for b
1 being
directed-sups-preserving Function of a
2,a
3 holds a
7 . b
1 = (a6 * b1) * a
5;
existence
ex b1 being Function of (UPS c2,c3),(UPS c1,c4) st
for b2 being directed-sups-preserving Function of c2,c3 holds b1 . b2 = (c6 * b2) * c5
uniqueness
for b1, b2 being Function of (UPS c2,c3),(UPS c1,c4) holds
( ( for b3 being directed-sups-preserving Function of c2,c3 holds b1 . b3 = (c6 * b3) * c5 ) & ( for b3 being directed-sups-preserving Function of c2,c3 holds b2 . b3 = (c6 * b3) * c5 ) implies b1 = b2 )
end;
:: deftheorem Def5 defines UPS WAYBEL27:def 5 :
for b
1, b
2, b
3, b
4 being non
empty reflexive antisymmetric RelStr for b
5 being
Function of b
1,b
2 holds
( b
5 is
directed-sups-preserving implies for b
6 being
Function of b
3,b
4 holds
( b
6 is
directed-sups-preserving implies for b
7 being
Function of
(UPS b2,b3),
(UPS b1,b4) holds
( b
7 = UPS b
5,b
6 iff for b
8 being
directed-sups-preserving Function of b
2,b
3 holds b
7 . b
8 = (b6 * b8) * b
5 ) ) );
theorem Th28: :: WAYBEL27:28
for b
1, b
2, b
3, b
4, b
5, b
6 being non
empty Posetfor b
7 being
directed-sups-preserving Function of b
2,b
3for b
8 being
directed-sups-preserving Function of b
1,b
2for b
9 being
directed-sups-preserving Function of b
4,b
5for b
10 being
directed-sups-preserving Function of b
5,b
6 holds
(UPS b8,b10) * (UPS b7,b9) = UPS (b7 * b8),
(b10 * b9)
theorem Th29: :: WAYBEL27:29
theorem Th30: :: WAYBEL27:30
theorem Th31: :: WAYBEL27:31
theorem Th32: :: WAYBEL27:32
theorem Th33: :: WAYBEL27:33
theorem Th34: :: WAYBEL27:34
theorem Th35: :: WAYBEL27:35
theorem Th36: :: WAYBEL27:36
theorem Th37: :: WAYBEL27:37
E40:
now
let c
1 be non
empty set ;
let c
2, c
3 be non
empty Poset;
let c
4 be
directed-sups-preserving Function of c
2,
(c3 |^ c1);
the
carrier of
(c3 |^ c1) = Funcs c
1,the
carrier of c
3
by YELLOW_1:28;
then
(
dom c
4 = the
carrier of c
2 &
rng c
4 c= Funcs c
1,the
carrier of c
3 )
by FUNCT_2:def 1;
hence
c
4 in Funcs the
carrier of c
2,
(Funcs c1,the carrier of c3)
by FUNCT_2:def 2;
then
commute c
4 in Funcs c
1,
(Funcs the carrier of c2,the carrier of c3)
by FUNCT_6:85;
then
ex b
1 being
Function st
(
commute c
4 = b
1 &
dom b
1 = c
1 &
rng b
1 c= Funcs the
carrier of c
2,the
carrier of c
3 )
by FUNCT_2:def 2;
hence
(
rng (commute c4) c= Funcs the
carrier of c
2,the
carrier of c
3 &
dom (commute c4) = c
1 )
;
end;
theorem Th38: :: WAYBEL27:38
theorem Th39: :: WAYBEL27:39
theorem Th40: :: WAYBEL27:40
theorem Th41: :: WAYBEL27:41
theorem Th42: :: WAYBEL27:42
theorem Th43: :: WAYBEL27:43
theorem Th44: :: WAYBEL27:44
theorem Th45: :: WAYBEL27:45
theorem Th46: :: WAYBEL27:46
theorem Th47: :: WAYBEL27:47