:: POLYRED semantic presentation
Lemma29:
for X being set
for S being Subset of X
for R being Order of X st R is_linear-order holds
R linearly_orders S
Lemma32:
for n being Ordinal
for b1, b2, b3 being bag of n st b1 <=' b2 holds
b1 + b3 <=' b2 + b3
Lemma39:
for n being Ordinal
for b1, b2 being bag of n st b1 <=' b2 & b2 <=' b1 holds
b1 = b2
Lemma40:
for n being Ordinal
for b1, b2 being bag of n holds
( not b1 < b2 iff b2 <=' b1 )
Lemma41:
for n being Ordinal
for L being non trivial ZeroStr
for p being finite-Support non-zero Series of n,L ex b being bag of n st
( p . b <> 0. L & ( for b' being bag of n st b < b' holds
p . b' = 0. L ) )
Lemma72:
for L being non empty Abelian add-associative right_zeroed right_complementable LoopStr
for f, g being FinSequence of the carrier of L
for n being Nat st len f = n + 1 & g = f | (Seg n) holds
Sum f = (Sum g) + (f /. (len f))
theorem Th1: :: POLYRED:1
theorem Th2: :: POLYRED:2
theorem Th3: :: POLYRED:3
theorem Th4: :: POLYRED:4
theorem Th5: :: POLYRED:5
Lemma96:
for n being Ordinal
for L being Abelian add-associative right_zeroed right_complementable unital associative commutative distributive non trivial doubleLoopStr
for p being Polynomial of n,L
for q being Element of (Polynom-Ring n,L) st p = q holds
- p = - q
theorem Th6: :: POLYRED:6
Lemma99:
for n being Ordinal
for L being non empty add-associative right_zeroed right_complementable LoopStr
for p being Polynomial of n,L
for m being Monomial of n,L
for b being bag of n st b <> term m holds
m . b = 0. L
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 :
Lemma127:
for n being Ordinal
for b, b' being bag of n
for L being non empty ZeroStr
for p being Series of n,L holds (b *' p) . (b' + b) = p . b'
Lemma128:
for n being Ordinal
for L being non empty ZeroStr
for p being Polynomial of n,L
for b being bag of n holds Support (b *' p) c= { (b + b') where b' is Element of Bags n : b' in Support p }
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
Lemma148:
for n being Ordinal
for T being connected TermOrder of n
for L being non empty ZeroStr
for p being Polynomial of n,L holds Support p in Fin the carrier of RelStr(# (Bags n),T #)
:: 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
Lemma154:
for n being Ordinal
for T being connected TermOrder of n
for L being non empty ZeroStr
for p being Polynomial of n,L holds 0_ n,L <= p,T
theorem Th26: :: POLYRED:26
theorem Th27: :: POLYRED:27
theorem Th28: :: POLYRED:28
theorem Th29: :: POLYRED:29
Lemma161:
for n being Ordinal
for T being connected TermOrder of n
for L being add-associative right_zeroed right_complementable non trivial LoopStr
for p being Polynomial of n,L
for b being bag of n st [(HT p,T),b] in T & b <> HT p,T holds
p . b = 0. L
Lemma162:
for n being Ordinal
for T being connected admissible TermOrder of n
for L being add-associative right_zeroed right_complementable non trivial LoopStr
for p being Polynomial of n,L st HT p,T = EmptyBag n holds
Red p,T = 0_ n,L
Lemma164:
for n being Ordinal
for T being connected admissible TermOrder of n
for L being add-associative right_zeroed right_complementable non trivial LoopStr
for p, q being Polynomial of n,L holds
( p < q,T iff ( ( p = 0_ n,L & q <> 0_ n,L ) or HT p,T < HT q,T,T or ( HT p,T = HT q,T & Red p,T < Red q,T,T ) ) )
theorem Th30: :: POLYRED:30
Lemma180:
for n being Ordinal
for T being connected admissible TermOrder of n
for L being add-associative right_zeroed right_complementable non trivial LoopStr
for p, q being Polynomial of n,L st q <> 0_ n,L & HT p,T = HT q,T & Red p,T <= Red q,T,T holds
p <= q,T
theorem Th31: :: POLYRED:31
theorem Th32: :: POLYRED:32
for
n being
Ordinal for
T being
connected admissible TermOrder of
n for
L being
add-associative right_zeroed right_complementable non
trivial LoopStr for
p,
q being
Polynomial of
n,
L holds
(
p < q,
T iff ( (
p = 0_ n,
L &
q <> 0_ n,
L ) or
HT p,
T < HT q,
T,
T or (
HT p,
T = HT q,
T &
Red p,
T < Red q,
T,
T ) ) )
by ;
theorem Th33: :: POLYRED:33
theorem Th34: :: POLYRED:34
theorem Th35: :: POLYRED:35
definition
let n be
Ordinal;
let T be
connected TermOrder of
n;
let L be
add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non
trivial doubleLoopStr ;
let f be
Polynomial of
n,
L,
p be
Polynomial of
n,
L,
g be
Polynomial of
n,
L;
let b be
bag of
n;
pred c4 reduces_to c6,
c5,
c7,
c2 means :
Def5:
:: POLYRED:def 5
(
f <> 0_ n,
L &
p <> 0_ n,
L &
b in Support f & ex
s being
bag of
n st
(
s + (HT p,T) = b &
g = f - (((f . b) / (HC p,T)) * (s *' p)) ) );
end;
:: deftheorem Def5 defines reduces_to POLYRED:def 5 :
for
n being
Ordinal for
T being
connected TermOrder of
n for
L being
add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non
trivial doubleLoopStr for
f,
p,
g being
Polynomial of
n,
L for
b being
bag of
n holds
(
f reduces_to g,
p,
b,
T iff (
f <> 0_ n,
L &
p <> 0_ n,
L &
b in Support f & ex
s being
bag of
n st
(
s + (HT p,T) = b &
g = f - (((f . b) / (HC p,T)) * (s *' p)) ) ) );
definition
let n be
Ordinal;
let T be
connected TermOrder of
n;
let L be
add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non
trivial doubleLoopStr ;
let f be
Polynomial of
n,
L,
p be
Polynomial of
n,
L,
g be
Polynomial of
n,
L;
pred c4 reduces_to c6,
c5,
c2 means :
Def6:
:: POLYRED:def 6
ex
b being
bag of
n st
f reduces_to g,
p,
b,
T;
end;
:: 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 :
definition
let n be
Ordinal;
let T be
connected TermOrder of
n;
let L be
add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non
trivial doubleLoopStr ;
let f be
Polynomial of
n,
L,
p be
Polynomial of
n,
L,
g be
Polynomial of
n,
L;
pred c4 top_reduces_to c6,
c5,
c2 means :: POLYRED:def 10
f reduces_to g,
p,
HT f,
T,
T;
end;
:: 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
Lemma193:
for n being Ordinal
for T being connected TermOrder of n
for L being add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non trivial doubleLoopStr
for f, p, g being Polynomial of n,L
for b being bag of n st f reduces_to g,p,b,T holds
not b in Support g
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 n be
Ordinal;
let T be
connected TermOrder of
n;
let L be
add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non
trivial doubleLoopStr ;
let P be
Subset of
(Polynom-Ring n,L);
func PolyRedRel c4,
c2 -> 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
p,
q being
Polynomial of
n,
L holds
(
[p,q] in it iff
p reduces_to q,
P,
T );
existence
ex b1 being Relation of the carrier of (Polynom-Ring n,L) \ {(0_ n,L)},the carrier of (Polynom-Ring n,L) st
for p, q being Polynomial of n,L holds
( [p,q] in b1 iff p reduces_to q,P,T )
uniqueness
for b1, b2 being Relation of the carrier of (Polynom-Ring n,L) \ {(0_ n,L)},the carrier of (Polynom-Ring n,L) st ( for p, q being Polynomial of n,L holds
( [p,q] in b1 iff p reduces_to q,P,T ) ) & ( for p, q being Polynomial of n,L holds
( [p,q] in b2 iff p reduces_to q,P,T ) ) holds
b1 = b2
end;
:: deftheorem Def13 defines PolyRedRel POLYRED:def 13 :
for
n being
Ordinal for
T being
connected TermOrder of
n for
L being
add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non
trivial doubleLoopStr for
P being
Subset of
(Polynom-Ring n,L) for
b5 being
Relation of the
carrier of
(Polynom-Ring n,L) \ {(0_ n,L)},the
carrier of
(Polynom-Ring n,L) holds
(
b5 = PolyRedRel P,
T iff for
p,
q being
Polynomial of
n,
L holds
(
[p,q] in b5 iff
p reduces_to q,
P,
T ) );
Lemma204:
for n being Ordinal
for T being connected TermOrder of n
for L being add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non trivial doubleLoopStr
for f, g, p being Polynomial of n,L st f reduces_to g,p,T holds
( f <> 0_ n,L & p <> 0_ n,L )
theorem Th44: :: POLYRED:44
for
n being
Ordinal for
T being
connected admissible TermOrder of
n for
L being non
empty Abelian add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non
degenerated doubleLoopStr for
f,
g being
Polynomial of
n,
L for
P being
Subset of
(Polynom-Ring n,L) st
PolyRedRel P,
T reduces f,
g holds
(
g <= f,
T & (
g = 0_ n,
L or
HT g,
T <= HT f,
T,
T ) )
theorem Th45: :: POLYRED:45
theorem Th46: :: POLYRED:46
theorem Th47: :: POLYRED:47
theorem Th48: :: POLYRED:48
theorem Th49: :: POLYRED:49
for
n being
Ordinal for
T being
connected TermOrder of
n for
L being
Abelian add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non
trivial doubleLoopStr for
P being
Subset of
(Polynom-Ring n,L) for
f,
g,
h,
h1 being
Polynomial of
n,
L st
f - g = h &
PolyRedRel P,
T reduces h,
h1 holds
ex
f1,
g1 being
Polynomial of
n,
L st
(
f1 - g1 = h1 &
PolyRedRel P,
T reduces f,
f1 &
PolyRedRel P,
T reduces g,
g1 )
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
Lemma230:
for n being Ordinal
for T being connected TermOrder of n
for L being Abelian add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non trivial doubleLoopStr
for P being Subset of (Polynom-Ring n,L)
for f being non-zero Polynomial of n,L
for g being Polynomial of n,L
for f', g' being Element of (Polynom-Ring n,L) st f = f' & g = g' & f reduces_to g,P,T holds
f',g' are_congruent_mod P -Ideal
theorem Th57: :: POLYRED:57
Lemma234:
for n being Nat
for T being connected admissible TermOrder of n
for L being non empty Abelian add-associative right_zeroed right_complementable unital associative commutative distributive Field-like non degenerated doubleLoopStr
for P being non empty Subset of (Polynom-Ring n,L)
for f, p, h being Element of (Polynom-Ring n,L) st p in P & p <> 0_ n,L & h <> 0_ n,L holds
f,f + (h * p) are_convertible_wrt PolyRedRel P,T
theorem Th58: :: POLYRED:58
theorem Th59: :: POLYRED:59
theorem Th60: :: POLYRED:60