:: TERMORD semantic presentation
:: deftheorem Def1 defines non-zero TERMORD:def 1 :
theorem Th1: :: TERMORD:1
for
X being
set for
b1,
b2 being
bag of
X holds
(
b1 divides b2 iff ex
b being
bag of
X st
b2 = b1 + b )
theorem Th2: :: TERMORD:2
theorem Th3: :: TERMORD:3
theorem Th4: :: TERMORD:4
Lemma98:
for n being Ordinal
for T being TermOrder of n
for b being set st b in field T holds
b is bag of n
:: deftheorem Def2 defines <= TERMORD:def 2 :
:: deftheorem Def3 defines < TERMORD:def 3 :
definition
let n be
Ordinal;
let T be
TermOrder of
n;
let b1 be
bag of
n,
b2 be
bag of
n;
func min c3,
c4,
c2 -> bag of
a1 equals :
Def4:
:: TERMORD:def 4
b1 if b1 <= b2,
T otherwise b2;
correctness
coherence
( ( b1 <= b2,T implies b1 is bag of n ) & ( not b1 <= b2,T implies b2 is bag of n ) );
consistency
for b1 being bag of n holds verum;
;
func max c3,
c4,
c2 -> bag of
a1 equals :
Def5:
:: TERMORD:def 5
b1 if b2 <= b1,
T otherwise b2;
correctness
coherence
( ( b2 <= b1,T implies b1 is bag of n ) & ( not b2 <= b1,T implies b2 is bag of n ) );
consistency
for b1 being bag of n holds verum;
;
end;
:: deftheorem Def4 defines min TERMORD:def 4 :
for
n being
Ordinal for
T being
TermOrder of
n for
b1,
b2 being
bag of
n holds
( (
b1 <= b2,
T implies
min b1,
b2,
T = b1 ) & ( not
b1 <= b2,
T implies
min b1,
b2,
T = b2 ) );
:: deftheorem Def5 defines max TERMORD:def 5 :
for
n being
Ordinal for
T being
TermOrder of
n for
b1,
b2 being
bag of
n holds
( (
b2 <= b1,
T implies
max b1,
b2,
T = b1 ) & ( not
b2 <= b1,
T implies
max b1,
b2,
T = b2 ) );
Lemma104:
for n being Ordinal
for T being TermOrder of n
for b being bag of n holds b <= b,T
Lemma105:
for n being Ordinal
for T being TermOrder of n
for b1, b2 being bag of n st b1 <= b2,T & b2 <= b1,T holds
b1 = b2
Lemma106:
for n being Ordinal
for T being TermOrder of n
for b being bag of n holds b in field T
theorem Th5: :: TERMORD:5
Lemma108:
for n being Ordinal
for T being connected TermOrder of n
for b1, b2 being bag of n holds
( b1 <= b2,T or b2 <= b1,T )
theorem Th6: :: TERMORD:6
theorem Th7: :: TERMORD:7
theorem Th8: :: TERMORD:8
theorem Th9: :: TERMORD:9
theorem Th10: :: TERMORD:10
theorem Th11: :: TERMORD:11
theorem Th12: :: TERMORD:12
Lemma114:
for n being Ordinal
for T being TermOrder of n
for b being bag of n holds
( min b,b,T = b & max b,b,T = b )
theorem Th13: :: TERMORD:13
theorem Th14: :: TERMORD:14
theorem Th15: :: TERMORD:15
theorem Th16: :: TERMORD:16
:: deftheorem Def6 defines HT TERMORD:def 6 :
:: deftheorem Def7 defines HC TERMORD:def 7 :
definition
let n be
Ordinal;
let T be
connected TermOrder of
n;
let L be non
empty ZeroStr ;
let p be
Polynomial of
n,
L;
func HM c4,
c2 -> Monomial of
a1,
a3 equals :: TERMORD:def 8
Monom (HC p,T),
(HT p,T);
correctness
coherence
Monom (HC p,T),(HT p,T) is Monomial of n,L;
;
end;
:: deftheorem Def8 defines HM TERMORD:def 8 :
Lemma122:
for n being Ordinal
for O being connected TermOrder of n
for L being non empty ZeroStr
for p being Polynomial of n,L holds
( HC p,O = 0. L iff p = 0_ n,L )
Lemma123:
for n being Ordinal
for O being connected TermOrder of n
for L being non trivial ZeroStr
for p being Polynomial of n,L holds (HM p,O) . (HT p,O) = p . (HT p,O)
Lemma124:
for n being Ordinal
for O being connected TermOrder of n
for L being non trivial ZeroStr
for p being Polynomial of n,L st HC p,O <> 0. L holds
HT p,O in Support (HM p,O)
Lemma125:
for n being Ordinal
for O being connected TermOrder of n
for L being non trivial ZeroStr
for p being Polynomial of n,L st HC p,O = 0. L holds
Support (HM p,O) = {}
Lemma126:
for n being Ordinal
for O being connected TermOrder of n
for L being non empty ZeroStr
for m being Monomial of n,L holds
( HT m,O = term m & HC m,O = coefficient m & HM m,O = m )
theorem Th17: :: TERMORD:17
theorem Th18: :: TERMORD:18
theorem Th19: :: TERMORD:19
Lemma128:
for n being Ordinal
for O being connected TermOrder of n
for L being non trivial ZeroStr
for p being Polynomial of n,L holds
( Support (HM p,O) = {} or Support (HM p,O) = {(HT p,O)} )
theorem Th20: :: TERMORD:20
theorem Th21: :: TERMORD:21
theorem Th22: :: TERMORD:22
theorem Th23: :: TERMORD:23
theorem Th24: :: TERMORD:24
theorem Th25: :: TERMORD:25
theorem Th26: :: TERMORD:26
theorem Th27: :: TERMORD:27
theorem Th28: :: TERMORD:28
Lemma131:
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
Lemma133:
for n being Ordinal
for O being connected admissible TermOrder of n
for L being add-associative right_zeroed right_complementable unital distributive left_zeroed non trivial doubleLoopStr
for p, q being non-zero Polynomial of n,L holds (p *' q) . ((HT p,O) + (HT q,O)) = (p . (HT p,O)) * (q . (HT q,O))
theorem Th29: :: TERMORD:29
theorem Th30: :: TERMORD:30
theorem Th31: :: TERMORD:31
theorem Th32: :: TERMORD:32
theorem Th33: :: TERMORD:33
theorem Th34: :: TERMORD:34
:: deftheorem Def9 defines Red TERMORD:def 9 :
Lemma145:
for n being Ordinal
for O being connected TermOrder of n
for L being add-associative right_zeroed right_complementable non trivial LoopStr
for p being Polynomial of n,L holds not HT p,O in Support (Red p,O)
Lemma146:
for n being Ordinal
for O 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 b in Support p & b <> HT p,O holds
b in Support (Red p,O)
Lemma147:
for n being Ordinal
for O being connected TermOrder of n
for L being add-associative right_zeroed right_complementable non trivial LoopStr
for p being Polynomial of n,L holds Support (Red p,O) = (Support p) \ {(HT p,O)}
theorem Th35: :: TERMORD:35
theorem Th36: :: TERMORD:36
Lemma148:
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 holds (Red p,T) . (HT p,T) = 0. L
Lemma149:
for n being Ordinal
for O 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 b <> HT p,O holds
(Red p,O) . b = p . b
theorem Th37: :: TERMORD:37
theorem Th38: :: TERMORD:38
theorem Th39: :: TERMORD:39
theorem Th40: :: TERMORD:40