:: POLYNOM6 semantic presentation
theorem Th1: :: POLYNOM6:1
for b
1, b
2 being
Ordinalfor b
3 being
set holds
( ( for b
4 being
set holds
( b
4 in b
3 iff ex b
5 being
Ordinal st
( b
4 = b
1 +^ b
5 & b
5 in b
2 ) ) ) implies b
1 +^ b
2 = b
1 \/ b
3 )
theorem Th2: :: POLYNOM6:2
for b
1 being
Ordinalfor b
2, b
3 being
bag of b
1 holds
not ( b
2 < b
3 & ( for b
4 being
Ordinal holds
not ( b
4 in b
1 & b
2 . b
4 < b
3 . b
4 & ( for b
5 being
Ordinal holds
( b
5 in b
4 implies b
2 . b
5 = b
3 . b
5 ) ) ) ) )
:: deftheorem Def1 defines +^ POLYNOM6:def 1 :
theorem Th3: :: POLYNOM6:3
theorem Th4: :: POLYNOM6:4
theorem Th5: :: POLYNOM6:5
theorem Th6: :: POLYNOM6:6
theorem Th7: :: POLYNOM6:7
theorem Th8: :: POLYNOM6:8
definition
let c
1, c
2 be non
empty Ordinal;
let c
3 be non
empty add-associative right_zeroed right_complementable unital distributive non
trivial doubleLoopStr ;
let c
4 be
Polynomial of c
1,
(Polynom-Ring c2,c3);
func Compress c
4 -> Polynomial of
(a1 +^ a2),a
3 means :
Def2:
:: POLYNOM6:def 2
for b
1 being
Element of
Bags (a1 +^ a2) holds
ex b
2 being
Element of
Bags a
1ex b
3 being
Element of
Bags a
2ex b
4 being
Polynomial of a
2,a
3 st
( b
4 = a
4 . b
2 & b
1 = b
2 +^ b
3 & a
5 . b
1 = b
4 . b
3 );
existence
ex b1 being Polynomial of (c1 +^ c2),c3 st
for b2 being Element of Bags (c1 +^ c2) holds
ex b3 being Element of Bags c1ex b4 being Element of Bags c2ex b5 being Polynomial of c2,c3 st
( b5 = c4 . b3 & b2 = b3 +^ b4 & b1 . b2 = b5 . b4 )
uniqueness
for b1, b2 being Polynomial of (c1 +^ c2),c3 holds
( ( for b3 being Element of Bags (c1 +^ c2) holds
ex b4 being Element of Bags c1ex b5 being Element of Bags c2ex b6 being Polynomial of c2,c3 st
( b6 = c4 . b4 & b3 = b4 +^ b5 & b1 . b3 = b6 . b5 ) ) & ( for b3 being Element of Bags (c1 +^ c2) holds
ex b4 being Element of Bags c1ex b5 being Element of Bags c2ex b6 being Polynomial of c2,c3 st
( b6 = c4 . b4 & b3 = b4 +^ b5 & b2 . b3 = b6 . b5 ) ) implies b1 = b2 )
end;
:: deftheorem Def2 defines Compress POLYNOM6:def 2 :
theorem Th9: :: POLYNOM6:9
theorem Th10: :: POLYNOM6:10
theorem Th11: :: POLYNOM6:11
theorem Th12: :: POLYNOM6:12
theorem Th13: :: POLYNOM6:13
theorem Th14: :: POLYNOM6:14
theorem Th15: :: POLYNOM6:15