:: POLYRED semantic presentation
Lemma1:
for b1 being set
for b2 being Subset of b1
for b3 being Order of b1 holds
( b3 is_linear-order implies b3 linearly_orders b2 )
Lemma2:
for b1 being Ordinal
for b2, b3, b4 being bag of b1 holds
( b2 <=' b3 implies b2 + b4 <=' b3 + b4 )
Lemma3:
for b1 being Ordinal
for b2, b3 being bag of b1 holds
( b2 <=' b3 & b3 <=' b2 implies b2 = b3 )
Lemma4:
for b1 being Ordinal
for b2, b3 being bag of b1 holds
( not b2 < b3 iff b3 <=' b2 )
Lemma5:
for b1 being Ordinal
for b2 being non trivial ZeroStr
for b3 being finite-Support non-zero Series of b1,b2 holds
ex b4 being bag of b1 st
( b3 . b4 <> 0. b2 & ( for b5 being bag of b1 holds
( b4 < b5 implies b3 . b5 = 0. b2 ) ) )
Lemma6:
for b1 being non empty Abelian add-associative right_zeroed right_complementable LoopStr
for b2, b3 being FinSequence of the carrier of b1
for b4 being Nat holds
( len b2 = b4 + 1 & b3 = b2 | (Seg b4) implies Sum b2 = (Sum b3) + (b2 /. (len b2)) )
theorem Th1: :: POLYRED:1
theorem Th2: :: POLYRED:2
theorem Th3: :: POLYRED:3
theorem Th4: :: POLYRED:4
theorem Th5: :: POLYRED:5
Lemma12:
for b1 being Ordinal
for b2 being Abelian add-associative right_zeroed right_complementable unital associative commutative distributive non trivial doubleLoopStr
for b3 being Polynomial of b1,b2
for b4 being Element of (Polynom-Ring b1,b2) holds
( b3 = b4 implies - b3 = - b4 )
theorem Th6: :: POLYRED:6
Lemma14:
for b1 being Ordinal
for b2 being non empty add-associative right_zeroed right_complementable LoopStr
for b3 being Polynomial of b1,b2
for b4 being Monomial of b1,b2
for b5 being bag of b1 holds
( b5 <> term b4 implies b4 . b5 = 0. b2 )
theorem Th7: :: POLYRED:7
theorem Th8: :: POLYRED:8
theorem Th9: :: POLYRED:9
theorem Th10: :: POLYRED:10
theorem Th11: :: POLYRED:11
theorem Th12: :: POLYRED:12
:: deftheorem Def1 defines *' POLYRED:def 1 :
for b
1 being
Ordinalfor b
2 being
bag of b
1for b
3 being non
empty ZeroStr for b
4, b
5 being
Series of b
1,b
3 holds
( b
5 = b
2 *' b
4 iff for b
6 being
bag of b
1 holds
( b
2 divides b
6 implies ( b
5 . b
6 = b
4 . (b6 -' b2) & ( for b
7 being
bag of b
1 holds
( not b
2 divides b
7 implies b
5 . b
7 = 0. b
3 ) ) ) ) );
Lemma22:
for b1 being Ordinal
for b2, b3 being bag of b1
for b4 being non empty ZeroStr
for b5 being Series of b1,b4 holds (b2 *' b5) . (b3 + b2) = b5 . b3
Lemma23:
for b1 being Ordinal
for b2 being non empty ZeroStr
for b3 being Polynomial of b1,b2
for b4 being bag of b1 holds Support (b4 *' b3) c= { (b4 + b5) where B is Element of Bags b1 : b5 in Support b3 }
theorem Th13: :: POLYRED:13
theorem Th14: :: POLYRED:14
theorem Th15: :: POLYRED:15
theorem Th16: :: POLYRED:16
theorem Th17: :: POLYRED:17
theorem Th18: :: POLYRED:18
theorem Th19: :: POLYRED:19
theorem Th20: :: POLYRED:20
theorem Th21: :: POLYRED:21
theorem Th22: :: POLYRED:22
theorem Th23: :: POLYRED:23
Lemma30:
for b1 being Ordinal
for b2 being connected TermOrder of b1
for b3 being non empty ZeroStr
for b4 being Polynomial of b1,b3 holds Support b4 in Fin the carrier of RelStr(# (Bags b1),b2 #)
:: deftheorem Def2 defines <= POLYRED:def 2 :
:: deftheorem Def3 defines < POLYRED:def 3 :
:: deftheorem Def4 defines Support POLYRED:def 4 :
theorem Th24: :: POLYRED:24
theorem Th25: :: POLYRED:25
Lemma35:
for b1 being Ordinal
for b2 being connected TermOrder of b1
for b3 being non empty ZeroStr
for b4 being Polynomial of b1,b3 holds 0_ b1,b3 <= b4,b2
theorem Th26: :: POLYRED:26
theorem Th27: :: POLYRED:27
theorem Th28: :: POLYRED:28
theorem Th29: :: POLYRED:29
Lemma40:
for b1 being Ordinal
for b2 being connected TermOrder of b1
for b3 being add-associative right_zeroed right_complementable non trivial LoopStr
for b4 being Polynomial of b1,b3
for b5 being bag of b1 holds
( [(HT b4,b2),b5] in b2 & b5 <> HT b4,b2 implies b4 . b5 = 0. b3 )
Lemma41:
for b1 being Ordinal
for b2 being connected admissible TermOrder of b1
for b3 being add-associative right_zeroed right_complementable non trivial LoopStr
for b4 being Polynomial of b1,b3 holds
( HT b4,b2 = EmptyBag b1 implies Red b4,b2 = 0_ b1,b3 )
Lemma42:
for b1 being Ordinal
for b2 being connected admissible TermOrder of b1
for b3 being add-associative right_zeroed right_complementable non trivial LoopStr
for b4, b5 being Polynomial of b1,b3 holds
( b4 < b5,b2 iff not ( not ( b4 = 0_ b1,b3 & b5 <> 0_ b1,b3 ) & not HT b4,b2 < HT b5,b2,b2 & not ( HT b4,b2 = HT b5,b2 & Red b4,b2 < Red b5,b2,b2 ) ) )
theorem Th30: :: POLYRED:30
Lemma43:
for b1 being Ordinal
for b2 being connected admissible TermOrder of b1
for b3 being add-associative right_zeroed right_complementable non trivial LoopStr
for b4, b5 being Polynomial of b1,b3 holds
( b5 <> 0_ b1,b3 & HT b4,b2 = HT b5,b2 & Red b4,b2 <= Red b5,b2,b2 implies b4 <= b5,b2 )
theorem Th31: :: POLYRED:31
theorem Th32: :: POLYRED:32
for b
1 being
Ordinalfor b
2 being
connected admissible TermOrder of b
1for b
3 being
add-associative right_zeroed right_complementable non
trivial LoopStr for b
4, b
5 being
Polynomial of b
1,b
3 holds
( b
4 < b
5,b
2 iff not ( not ( b
4 = 0_ b
1,b
3 & b
5 <> 0_ b
1,b
3 ) & not
HT b
4,b
2 < HT b
5,b
2,b
2 & not (
HT b
4,b
2 = HT b
5,b
2 &
Red b
4,b
2 < Red b
5,b
2,b
2 ) ) )
by Lemma42;
theorem Th33: :: POLYRED:33
theorem Th34: :: POLYRED:34
theorem Th35: :: POLYRED:35
definition
let c
1 be
Ordinal;
let c
2 be
connected TermOrder of c
1;
let c
3 be
add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non
trivial doubleLoopStr ;
let c
4, c
5, c
6 be
Polynomial of c
1,c
3;
let c
7 be
bag of c
1;
pred c
4 reduces_to c
6,c
5,c
7,c
2 means :
Def5:
:: POLYRED:def 5
( a
4 <> 0_ a
1,a
3 & a
5 <> 0_ a
1,a
3 & a
7 in Support a
4 & ex b
1 being
bag of a
1 st
( b
1 + (HT a5,a2) = a
7 & a
6 = a
4 - (((a4 . a7) / (HC a5,a2)) * (b1 *' a5)) ) );
end;
:: deftheorem Def5 defines reduces_to POLYRED:def 5 :
for b
1 being
Ordinalfor b
2 being
connected TermOrder of b
1for b
3 being
add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non
trivial doubleLoopStr for b
4, b
5, b
6 being
Polynomial of b
1,b
3for b
7 being
bag of b
1 holds
( b
4 reduces_to b
6,b
5,b
7,b
2 iff ( b
4 <> 0_ b
1,b
3 & b
5 <> 0_ b
1,b
3 & b
7 in Support b
4 & ex b
8 being
bag of b
1 st
( b
8 + (HT b5,b2) = b
7 & b
6 = b
4 - (((b4 . b7) / (HC b5,b2)) * (b8 *' b5)) ) ) );
:: deftheorem Def6 defines reduces_to POLYRED:def 6 :
:: deftheorem Def7 defines reduces_to POLYRED:def 7 :
:: deftheorem Def8 defines is_reducible_wrt POLYRED:def 8 :
:: deftheorem Def9 defines is_reducible_wrt POLYRED:def 9 :
:: deftheorem Def10 defines top_reduces_to POLYRED:def 10 :
:: deftheorem Def11 defines is_top_reducible_wrt POLYRED:def 11 :
:: deftheorem Def12 defines is_top_reducible_wrt POLYRED:def 12 :
theorem Th36: :: POLYRED:36
Lemma52:
for b1 being Ordinal
for b2 being connected TermOrder of b1
for b3 being add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non trivial doubleLoopStr
for b4, b5, b6 being Polynomial of b1,b3
for b7 being bag of b1 holds
not ( b4 reduces_to b6,b5,b7,b2 & b7 in Support b6 )
theorem Th37: :: POLYRED:37
theorem Th38: :: POLYRED:38
theorem Th39: :: POLYRED:39
theorem Th40: :: POLYRED:40
theorem Th41: :: POLYRED:41
theorem Th42: :: POLYRED:42
theorem Th43: :: POLYRED:43
definition
let c
1 be
Ordinal;
let c
2 be
connected TermOrder of c
1;
let c
3 be
add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non
trivial doubleLoopStr ;
let c
4 be
Subset of
(Polynom-Ring c1,c3);
func PolyRedRel c
4,c
2 -> Relation of the
carrier of
(Polynom-Ring a1,a3) \ {(0_ a1,a3)},the
carrier of
(Polynom-Ring a1,a3) means :
Def13:
:: POLYRED:def 13
for b
1, b
2 being
Polynomial of a
1,a
3 holds
(
[b1,b2] in a
5 iff b
1 reduces_to b
2,a
4,a
2 );
existence
ex b1 being Relation of the carrier of (Polynom-Ring c1,c3) \ {(0_ c1,c3)},the carrier of (Polynom-Ring c1,c3) st
for b2, b3 being Polynomial of c1,c3 holds
( [b2,b3] in b1 iff b2 reduces_to b3,c4,c2 )
uniqueness
for b1, b2 being Relation of the carrier of (Polynom-Ring c1,c3) \ {(0_ c1,c3)},the carrier of (Polynom-Ring c1,c3) holds
( ( for b3, b4 being Polynomial of c1,c3 holds
( [b3,b4] in b1 iff b3 reduces_to b4,c4,c2 ) ) & ( for b3, b4 being Polynomial of c1,c3 holds
( [b3,b4] in b2 iff b3 reduces_to b4,c4,c2 ) ) implies b1 = b2 )
end;
:: deftheorem Def13 defines PolyRedRel POLYRED:def 13 :
for b
1 being
Ordinalfor b
2 being
connected TermOrder of b
1for b
3 being
add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non
trivial doubleLoopStr for b
4 being
Subset of
(Polynom-Ring b1,b3)for b
5 being
Relation of the
carrier of
(Polynom-Ring b1,b3) \ {(0_ b1,b3)},the
carrier of
(Polynom-Ring b1,b3) holds
( b
5 = PolyRedRel b
4,b
2 iff for b
6, b
7 being
Polynomial of b
1,b
3 holds
(
[b6,b7] in b
5 iff b
6 reduces_to b
7,b
4,b
2 ) );
Lemma59:
for b1 being Ordinal
for b2 being connected TermOrder of b1
for b3 being add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non trivial doubleLoopStr
for b4, b5, b6 being Polynomial of b1,b3 holds
( b4 reduces_to b5,b6,b2 implies ( b4 <> 0_ b1,b3 & b6 <> 0_ b1,b3 ) )
theorem Th44: :: POLYRED:44
for b
1 being
Ordinalfor b
2 being
connected admissible TermOrder of b
1for b
3 being non
empty Abelian add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non
degenerated doubleLoopStr for b
4, b
5 being
Polynomial of b
1,b
3for b
6 being
Subset of
(Polynom-Ring b1,b3) holds
(
PolyRedRel b
6,b
2 reduces b
4,b
5 implies ( b
5 <= b
4,b
2 & ( b
5 = 0_ b
1,b
3 or
HT b
5,b
2 <= HT b
4,b
2,b
2 ) ) )
theorem Th45: :: POLYRED:45
theorem Th46: :: POLYRED:46
theorem Th47: :: POLYRED:47
theorem Th48: :: POLYRED:48
theorem Th49: :: POLYRED:49
for b
1 being
Ordinalfor b
2 being
connected TermOrder of b
1for b
3 being
Abelian add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non
trivial doubleLoopStr for b
4 being
Subset of
(Polynom-Ring b1,b3)for b
5, b
6, b
7, b
8 being
Polynomial of b
1,b
3 holds
not ( b
5 - b
6 = b
7 &
PolyRedRel b
4,b
2 reduces b
7,b
8 & ( for b
9, b
10 being
Polynomial of b
1,b
3 holds
not ( b
9 - b
10 = b
8 &
PolyRedRel b
4,b
2 reduces b
5,b
9 &
PolyRedRel b
4,b
2 reduces b
6,b
10 ) ) )
theorem Th50: :: POLYRED:50
theorem Th51: :: POLYRED:51
:: deftheorem Def14 defines are_congruent_mod POLYRED:def 14 :
theorem Th52: :: POLYRED:52
theorem Th53: :: POLYRED:53
theorem Th54: :: POLYRED:54
theorem Th55: :: POLYRED:55
theorem Th56: :: POLYRED:56
Lemma69:
for b1 being Ordinal
for b2 being connected TermOrder of b1
for b3 being Abelian add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non trivial doubleLoopStr
for b4 being Subset of (Polynom-Ring b1,b3)
for b5 being non-zero Polynomial of b1,b3
for b6 being Polynomial of b1,b3
for b7, b8 being Element of (Polynom-Ring b1,b3) holds
( b5 = b7 & b6 = b8 & b5 reduces_to b6,b4,b2 implies b7,b8 are_congruent_mod b4 -Ideal )
theorem Th57: :: POLYRED:57
Lemma71:
for b1 being Nat
for b2 being connected admissible TermOrder of b1
for b3 being non empty Abelian add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non degenerated doubleLoopStr
for b4 being non empty Subset of (Polynom-Ring b1,b3)
for b5, b6, b7 being Element of (Polynom-Ring b1,b3) holds
( b6 in b4 & b6 <> 0_ b1,b3 & b7 <> 0_ b1,b3 implies b5,b5 + (b7 * b6) are_convertible_wrt PolyRedRel b4,b2 )
theorem Th58: :: POLYRED:58
theorem Th59: :: POLYRED:59
theorem Th60: :: POLYRED:60