:: GROEB_1 semantic presentation
begin
definition
let n be Ordinal;
let L be non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr ;
let p be Polynomial of n,L;
:: original: {
redefine func{p} -> Subset of (Polynom-Ring (n,L));
coherence
{p} is Subset of (Polynom-Ring (n,L))
proof
now__::_thesis:_for_u_being_set_st_u_in_{p}_holds_
u_in_the_carrier_of_(Polynom-Ring_(n,L))
let u be set ; ::_thesis: ( u in {p} implies u in the carrier of (Polynom-Ring (n,L)) )
assume u in {p} ; ::_thesis: u in the carrier of (Polynom-Ring (n,L))
then u = p by TARSKI:def_1;
hence u in the carrier of (Polynom-Ring (n,L)) by POLYNOM1:def_10; ::_thesis: verum
end;
hence {p} is Subset of (Polynom-Ring (n,L)) by TARSKI:def_3; ::_thesis: verum
end;
end;
theorem Th1: :: GROEB_1:1
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, p, g being Polynomial of n,L st f reduces_to g,p,T holds
ex m being Monomial of n,L st g = f - (m *' p)
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, p, g being Polynomial of n,L st f reduces_to g,p,T holds
ex m being Monomial of n,L st g = f - (m *' p)
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, p, g being Polynomial of n,L st f reduces_to g,p,T holds
ex m being Monomial of n,L st g = f - (m *' p)
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, p, g being Polynomial of n,L st f reduces_to g,p,T holds
ex m being Monomial of n,L st g = f - (m *' p)
let f, p, g be Polynomial of n,L; ::_thesis: ( f reduces_to g,p,T implies ex m being Monomial of n,L st g = f - (m *' p) )
assume f reduces_to g,p,T ; ::_thesis: ex m being Monomial of n,L st g = f - (m *' p)
then consider b being bag of n such that
A1: f reduces_to g,p,b,T by POLYRED:def_6;
consider s being bag of n such that
s + (HT (p,T)) = b and
A2: g = f - (((f . b) / (HC (p,T))) * (s *' p)) by A1, POLYRED:def_5;
((f . b) / (HC (p,T))) * (s *' p) = (Monom (((f . b) / (HC (p,T))),s)) *' p by POLYRED:22;
hence ex m being Monomial of n,L st g = f - (m *' p) by A2; ::_thesis: verum
end;
theorem :: GROEB_1:2
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, p, g being Polynomial of n,L st f reduces_to g,p,T holds
ex m being Monomial of n,L st
( g = f - (m *' p) & not HT ((m *' p),T) in Support g & HT ((m *' p),T) <= HT (f,T),T )
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, p, g being Polynomial of n,L st f reduces_to g,p,T holds
ex m being Monomial of n,L st
( g = f - (m *' p) & not HT ((m *' p),T) in Support g & HT ((m *' p),T) <= HT (f,T),T )
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, p, g being Polynomial of n,L st f reduces_to g,p,T holds
ex m being Monomial of n,L st
( g = f - (m *' p) & not HT ((m *' p),T) in Support g & HT ((m *' p),T) <= HT (f,T),T )
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, p, g being Polynomial of n,L st f reduces_to g,p,T holds
ex m being Monomial of n,L st
( g = f - (m *' p) & not HT ((m *' p),T) in Support g & HT ((m *' p),T) <= HT (f,T),T )
let f, p, g be Polynomial of n,L; ::_thesis: ( f reduces_to g,p,T implies ex m being Monomial of n,L st
( g = f - (m *' p) & not HT ((m *' p),T) in Support g & HT ((m *' p),T) <= HT (f,T),T ) )
assume f reduces_to g,p,T ; ::_thesis: ex m being Monomial of n,L st
( g = f - (m *' p) & not HT ((m *' p),T) in Support g & HT ((m *' p),T) <= HT (f,T),T )
then consider b being bag of n such that
A1: f reduces_to g,p,b,T by POLYRED:def_6;
b in Support f by A1, POLYRED:def_5;
then A2: f . b <> 0. L by POLYNOM1:def_3;
p <> 0_ (n,L) by A1, POLYRED:def_5;
then reconsider p = p as non-zero Polynomial of n,L by POLYNOM7:def_1;
consider s being bag of n such that
A3: s + (HT (p,T)) = b and
A4: g = f - (((f . b) / (HC (p,T))) * (s *' p)) by A1, POLYRED:def_5;
set m = Monom (((f . b) / (HC (p,T))),s);
A5: (HC (p,T)) " <> 0. L by VECTSP_1:25;
(f . b) / (HC (p,T)) = (f . b) * ((HC (p,T)) ") by VECTSP_1:def_11;
then A6: (f . b) / (HC (p,T)) <> 0. L by A2, A5, VECTSP_2:def_1;
then A7: not (f . b) / (HC (p,T)) is zero by STRUCT_0:def_12;
coefficient (Monom (((f . b) / (HC (p,T))),s)) <> 0. L by A6, POLYNOM7:9;
then HC ((Monom (((f . b) / (HC (p,T))),s)),T) <> 0. L by TERMORD:23;
then Monom (((f . b) / (HC (p,T))),s) <> 0_ (n,L) by TERMORD:17;
then reconsider m = Monom (((f . b) / (HC (p,T))),s) as non-zero Monomial of n,L by POLYNOM7:def_1;
A8: HT ((m *' p),T) = (HT (m,T)) + (HT (p,T)) by TERMORD:31
.= (term m) + (HT (p,T)) by TERMORD:23
.= s + (HT (p,T)) by A7, POLYNOM7:10 ;
then HT ((m *' p),T) in Support f by A1, A3, POLYRED:def_5;
then ( ((f . b) / (HC (p,T))) * (s *' p) = (Monom (((f . b) / (HC (p,T))),s)) *' p & HT ((m *' p),T) <= HT (f,T),T ) by POLYRED:22, TERMORD:def_6;
hence ex m being Monomial of n,L st
( g = f - (m *' p) & not HT ((m *' p),T) in Support g & HT ((m *' p),T) <= HT (f,T),T ) by A1, A3, A4, A8, POLYRED:39; ::_thesis: verum
end;
Lm1: for L being non empty add-cancelable right_complementable associative well-unital distributive add-associative right_zeroed left_zeroed doubleLoopStr
for P being Subset of L
for p being Element of L st p in P holds
p in P -Ideal
proof
let L be non empty add-cancelable right_complementable associative well-unital distributive add-associative right_zeroed left_zeroed doubleLoopStr ; ::_thesis: for P being Subset of L
for p being Element of L st p in P holds
p in P -Ideal
let P be Subset of L; ::_thesis: for p being Element of L st p in P holds
p in P -Ideal
let p be Element of L; ::_thesis: ( p in P implies p in P -Ideal )
set f = <*p*>;
assume A1: p in P ; ::_thesis: p in P -Ideal
then reconsider P9 = P as non empty Subset of L ;
now__::_thesis:_for_i_being_set_st_i_in_dom_<*p*>_holds_
ex_u,_v_being_Element_of_L_ex_a_being_Element_of_P9_st_<*p*>_/._i_=_(u_*_a)_*_v
let i be set ; ::_thesis: ( i in dom <*p*> implies ex u, v being Element of L ex a being Element of P9 st <*p*> /. i = (u * a) * v )
assume A2: i in dom <*p*> ; ::_thesis: ex u, v being Element of L ex a being Element of P9 st <*p*> /. i = (u * a) * v
dom <*p*> = {1} by FINSEQ_1:2, FINSEQ_1:38;
then i = 1 by A2, TARSKI:def_1;
then <*p*> /. i = <*p*> . 1 by A2, PARTFUN1:def_6
.= p by FINSEQ_1:40
.= (1. L) * p by VECTSP_1:def_8
.= ((1. L) * p) * (1. L) by VECTSP_1:def_4 ;
hence ex u, v being Element of L ex a being Element of P9 st <*p*> /. i = (u * a) * v by A1; ::_thesis: verum
end;
then reconsider f = <*p*> as LinearCombination of P9 by IDEAL_1:def_8;
Sum f = p by RLVECT_1:44;
hence p in P -Ideal by IDEAL_1:60; ::_thesis: verum
end;
Lm2: for n being Ordinal
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for p, q being Polynomial of n,L
for f, g being Element of (Polynom-Ring (n,L)) st f = p & g = q holds
f - g = p - q
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for p, q being Polynomial of n,L
for f, g being Element of (Polynom-Ring (n,L)) st f = p & g = q holds
f - g = p - q
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for p, q being Polynomial of n,L
for f, g being Element of (Polynom-Ring (n,L)) st f = p & g = q holds
f - g = p - q
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for p, q being Polynomial of n,L
for f, g being Element of (Polynom-Ring (n,L)) st f = p & g = q holds
f - g = p - q
let p, q be Polynomial of n,L; ::_thesis: for f, g being Element of (Polynom-Ring (n,L)) st f = p & g = q holds
f - g = p - q
let f, g be Element of (Polynom-Ring (n,L)); ::_thesis: ( f = p & g = q implies f - g = p - q )
assume that
A1: f = p and
A2: g = q ; ::_thesis: f - g = p - q
reconsider x = - q as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider x = x as Element of (Polynom-Ring (n,L)) ;
x + g = (- q) + q by A2, POLYNOM1:def_10
.= 0_ (n,L) by POLYRED:3
.= 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10 ;
then A3: - q = - g by RLVECT_1:6;
thus p - q = p + (- q) by POLYNOM1:def_6
.= f + (- g) by A1, A3, POLYNOM1:def_10
.= f - g by RLVECT_1:def_11 ; ::_thesis: verum
end;
Lm3: for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f being Polynomial of n,L holds f is_irreducible_wrt 0_ (n,L),T
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f being Polynomial of n,L holds f is_irreducible_wrt 0_ (n,L),T
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f being Polynomial of n,L holds f is_irreducible_wrt 0_ (n,L),T
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for f being Polynomial of n,L holds f is_irreducible_wrt 0_ (n,L),T
let f be Polynomial of n,L; ::_thesis: f is_irreducible_wrt 0_ (n,L),T
assume f is_reducible_wrt 0_ (n,L),T ; ::_thesis: contradiction
then consider g being Polynomial of n,L such that
A1: f reduces_to g, 0_ (n,L),T by POLYRED:def_8;
ex b being bag of n st f reduces_to g, 0_ (n,L),b,T by A1, POLYRED:def_6;
hence contradiction by POLYRED:def_5; ::_thesis: verum
end;
theorem Th3: :: GROEB_1:3
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, g being Polynomial of n,L
for P, Q being Subset of (Polynom-Ring (n,L)) st P c= Q & f reduces_to g,P,T holds
f reduces_to g,Q,T
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, g being Polynomial of n,L
for P, Q being Subset of (Polynom-Ring (n,L)) st P c= Q & f reduces_to g,P,T holds
f reduces_to g,Q,T
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, g being Polynomial of n,L
for P, Q being Subset of (Polynom-Ring (n,L)) st P c= Q & f reduces_to g,P,T holds
f reduces_to g,Q,T
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, g being Polynomial of n,L
for P, Q being Subset of (Polynom-Ring (n,L)) st P c= Q & f reduces_to g,P,T holds
f reduces_to g,Q,T
let f, g be Polynomial of n,L; ::_thesis: for P, Q being Subset of (Polynom-Ring (n,L)) st P c= Q & f reduces_to g,P,T holds
f reduces_to g,Q,T
let P, Q be Subset of (Polynom-Ring (n,L)); ::_thesis: ( P c= Q & f reduces_to g,P,T implies f reduces_to g,Q,T )
assume A1: P c= Q ; ::_thesis: ( not f reduces_to g,P,T or f reduces_to g,Q,T )
assume f reduces_to g,P,T ; ::_thesis: f reduces_to g,Q,T
then ex p being Polynomial of n,L st
( p in P & f reduces_to g,p,T ) by POLYRED:def_7;
hence f reduces_to g,Q,T by A1, POLYRED:def_7; ::_thesis: verum
end;
theorem Th4: :: GROEB_1:4
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P, Q being Subset of (Polynom-Ring (n,L)) st P c= Q holds
PolyRedRel (P,T) c= PolyRedRel (Q,T)
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P, Q being Subset of (Polynom-Ring (n,L)) st P c= Q holds
PolyRedRel (P,T) c= PolyRedRel (Q,T)
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P, Q being Subset of (Polynom-Ring (n,L)) st P c= Q holds
PolyRedRel (P,T) c= PolyRedRel (Q,T)
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for P, Q being Subset of (Polynom-Ring (n,L)) st P c= Q holds
PolyRedRel (P,T) c= PolyRedRel (Q,T)
let P, Q be Subset of (Polynom-Ring (n,L)); ::_thesis: ( P c= Q implies PolyRedRel (P,T) c= PolyRedRel (Q,T) )
assume A1: P c= Q ; ::_thesis: PolyRedRel (P,T) c= PolyRedRel (Q,T)
now__::_thesis:_for_u_being_set_st_u_in_PolyRedRel_(P,T)_holds_
u_in_PolyRedRel_(Q,T)
let u be set ; ::_thesis: ( u in PolyRedRel (P,T) implies u in PolyRedRel (Q,T) )
assume A2: u in PolyRedRel (P,T) ; ::_thesis: u in PolyRedRel (Q,T)
then consider p, q being set such that
A3: p in NonZero (Polynom-Ring (n,L)) and
A4: q in the carrier of (Polynom-Ring (n,L)) and
A5: u = [p,q] by ZFMISC_1:def_2;
reconsider q = q as Polynomial of n,L by A4, POLYNOM1:def_10;
0_ (n,L) = 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10;
then not p in {(0_ (n,L))} by A3, XBOOLE_0:def_5;
then p <> 0_ (n,L) by TARSKI:def_1;
then reconsider p = p as non-zero Polynomial of n,L by A3, POLYNOM1:def_10, POLYNOM7:def_1;
p reduces_to q,P,T by A2, A5, POLYRED:def_13;
then p reduces_to q,Q,T by A1, Th3;
hence u in PolyRedRel (Q,T) by A5, POLYRED:def_13; ::_thesis: verum
end;
hence PolyRedRel (P,T) c= PolyRedRel (Q,T) by TARSKI:def_3; ::_thesis: verum
end;
theorem Th5: :: GROEB_1:5
for n being Ordinal
for L being non empty right_complementable add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L holds Support (- p) = Support p
proof
let n be Ordinal; ::_thesis: for L being non empty right_complementable add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L holds Support (- p) = Support p
let L be non empty right_complementable add-associative right_zeroed doubleLoopStr ; ::_thesis: for p being Polynomial of n,L holds Support (- p) = Support p
let p be Polynomial of n,L; ::_thesis: Support (- p) = Support p
A1: now__::_thesis:_for_u_being_set_st_u_in_Support_p_holds_
u_in_Support_(-_p)
let u be set ; ::_thesis: ( u in Support p implies u in Support (- p) )
assume A2: u in Support p ; ::_thesis: u in Support (- p)
then reconsider u9 = u as Element of Bags n ;
A3: p . u9 <> 0. L by A2, POLYNOM1:def_3;
now__::_thesis:_not_(-_p)_._u9_=_0._L
assume (- p) . u9 = 0. L ; ::_thesis: contradiction
then (- p) . u9 = - (- (0. L)) by RLVECT_1:17;
then - (p . u9) = - (- (0. L)) by POLYNOM1:17
.= 0. L by RLVECT_1:17 ;
then p . u9 = - (0. L) by RLVECT_1:17;
hence contradiction by A3, RLVECT_1:12; ::_thesis: verum
end;
hence u in Support (- p) by POLYNOM1:def_3; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_Support_(-_p)_holds_
u_in_Support_p
let u be set ; ::_thesis: ( u in Support (- p) implies u in Support p )
assume A4: u in Support (- p) ; ::_thesis: u in Support p
then reconsider u9 = u as Element of Bags n ;
A5: (- p) . u9 <> 0. L by A4, POLYNOM1:def_3;
now__::_thesis:_not_p_._u9_=_0._L
A6: (- p) . u9 = - (p . u9) by POLYNOM1:17;
assume p . u9 = 0. L ; ::_thesis: contradiction
hence contradiction by A5, A6, RLVECT_1:12; ::_thesis: verum
end;
hence u in Support p by POLYNOM1:def_3; ::_thesis: verum
end;
hence Support (- p) = Support p by A1, TARSKI:1; ::_thesis: verum
end;
theorem Th6: :: GROEB_1:6
for n being Ordinal
for T being connected TermOrder of n
for L being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L holds HT ((- p),T) = HT (p,T)
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L holds HT ((- p),T) = HT (p,T)
let T be connected TermOrder of n; ::_thesis: for L being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L holds HT ((- p),T) = HT (p,T)
let L be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for p being Polynomial of n,L holds HT ((- p),T) = HT (p,T)
let p be Polynomial of n,L; ::_thesis: HT ((- p),T) = HT (p,T)
percases ( p = 0_ (n,L) or p <> 0_ (n,L) ) ;
supposeA1: p = 0_ (n,L) ; ::_thesis: HT ((- p),T) = HT (p,T)
reconsider x = - p as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider x = x as Element of (Polynom-Ring (n,L)) ;
A2: - (0_ (n,L)) = (- (0_ (n,L))) + (0_ (n,L)) by POLYNOM1:23
.= 0_ (n,L) by POLYRED:3 ;
0. (Polynom-Ring (n,L)) = 0_ (n,L) by POLYNOM1:def_10;
then x + (0. (Polynom-Ring (n,L))) = (- p) + (0_ (n,L)) by POLYNOM1:def_10
.= 0_ (n,L) by A1, A2, POLYNOM1:23
.= 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10 ;
then - p = - (0. (Polynom-Ring (n,L))) by RLVECT_1:6
.= 0. (Polynom-Ring (n,L)) by RLVECT_1:12
.= p by A1, POLYNOM1:def_10 ;
hence HT ((- p),T) = HT (p,T) ; ::_thesis: verum
end;
suppose p <> 0_ (n,L) ; ::_thesis: HT ((- p),T) = HT (p,T)
then A3: Support p <> {} by POLYNOM7:1;
then Support (- p) <> {} by Th5;
then HT ((- p),T) in Support (- p) by TERMORD:def_6;
then HT ((- p),T) in Support p by Th5;
then A4: HT ((- p),T) <= HT (p,T),T by TERMORD:def_6;
HT (p,T) in Support p by A3, TERMORD:def_6;
then HT (p,T) in Support (- p) by Th5;
then HT (p,T) <= HT ((- p),T),T by TERMORD:def_6;
hence HT ((- p),T) = HT (p,T) by A4, TERMORD:7; ::_thesis: verum
end;
end;
end;
theorem Th7: :: GROEB_1:7
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for p, q being Polynomial of n,L holds HT ((p - q),T) <= max ((HT (p,T)),(HT (q,T)),T),T
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for p, q being Polynomial of n,L holds HT ((p - q),T) <= max ((HT (p,T)),(HT (q,T)),T),T
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for p, q being Polynomial of n,L holds HT ((p - q),T) <= max ((HT (p,T)),(HT (q,T)),T),T
let L be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for p, q being Polynomial of n,L holds HT ((p - q),T) <= max ((HT (p,T)),(HT (q,T)),T),T
let p, q be Polynomial of n,L; ::_thesis: HT ((p - q),T) <= max ((HT (p,T)),(HT (q,T)),T),T
HT ((p + (- q)),T) <= max ((HT (p,T)),(HT ((- q),T)),T),T by TERMORD:34;
then HT ((p - q),T) <= max ((HT (p,T)),(HT ((- q),T)),T),T by POLYNOM1:def_6;
hence HT ((p - q),T) <= max ((HT (p,T)),(HT (q,T)),T),T by Th6; ::_thesis: verum
end;
theorem Th8: :: GROEB_1:8
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for p, q being Polynomial of n,L st Support q c= Support p holds
q <= p,T
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for p, q being Polynomial of n,L st Support q c= Support p holds
q <= p,T
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for p, q being Polynomial of n,L st Support q c= Support p holds
q <= p,T
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for p, q being Polynomial of n,L st Support q c= Support p holds
q <= p,T
let p, q be Polynomial of n,L; ::_thesis: ( Support q c= Support p implies q <= p,T )
assume A1: Support q c= Support p ; ::_thesis: q <= p,T
defpred S1[ Element of NAT ] means for f, g being Polynomial of n,L st Support f c= Support g & card (Support f) = $1 holds
f <= g,T;
A2: now__::_thesis:_for_k_being_Element_of_NAT_st_S1[k]_holds_
S1[k_+_1]
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A3: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_for_f,_g_being_Polynomial_of_n,L_st_Support_f_c=_Support_g_&_card_(Support_f)_=_k_+_1_holds_
f_<=_g,T
set R = RelStr(# (Bags n),T #);
let f, g be Polynomial of n,L; ::_thesis: ( Support f c= Support g & card (Support f) = k + 1 implies f <= g,T )
assume that
A4: Support f c= Support g and
A5: card (Support f) = k + 1 ; ::_thesis: f <= g,T
set rf = Red (f,T);
set rg = Red (g,T);
A6: Support f <> {} by A5;
then A7: HT (f,T) in Support f by TERMORD:def_6;
f <> 0_ (n,L) by A6, POLYNOM7:1;
then A8: f is non-zero by POLYNOM7:def_1;
g <> 0_ (n,L) by A4, A7, POLYNOM7:1;
then A9: g is non-zero by POLYNOM7:def_1;
now__::_thesis:_(_(_HT_(f,T)_=_HT_(g,T)_&_f_<=_g,T_)_or_(_HT_(f,T)_<>_HT_(g,T)_&_f_<=_g,T_)_)
percases ( HT (f,T) = HT (g,T) or HT (f,T) <> HT (g,T) ) ;
caseA10: HT (f,T) = HT (g,T) ; ::_thesis: f <= g,T
A11: Support (Red (f,T)) = (Support f) \ {(HT (f,T))} by TERMORD:36;
A12: Support (Red (g,T)) = (Support g) \ {(HT (g,T))} by TERMORD:36;
now__::_thesis:_for_u_being_set_st_u_in_Support_(Red_(f,T))_holds_
u_in_Support_(Red_(g,T))
let u be set ; ::_thesis: ( u in Support (Red (f,T)) implies u in Support (Red (g,T)) )
assume u in Support (Red (f,T)) ; ::_thesis: u in Support (Red (g,T))
then ( u in Support f & not u in {(HT (f,T))} ) by A11, XBOOLE_0:def_5;
hence u in Support (Red (g,T)) by A4, A10, A12, XBOOLE_0:def_5; ::_thesis: verum
end;
then A13: Support (Red (f,T)) c= Support (Red (g,T)) by TARSKI:def_3;
for u being set st u in {(HT (f,T))} holds
u in Support f by A7, TARSKI:def_1;
then A14: {(HT (f,T))} c= Support f by TARSKI:def_3;
A15: ( Support (f,T) <> {} & Support (g,T) <> {} ) by A4, A7, POLYRED:def_4;
A16: Support ((Red (f,T)),T) = Support (Red (f,T)) by POLYRED:def_4;
HT (f,T) in {(HT (f,T))} by TARSKI:def_1;
then A17: not HT (f,T) in Support (Red (f,T)) by A11, XBOOLE_0:def_5;
A18: PosetMax (Support (f,T)) = HT (g,T) by A8, A10, POLYRED:24
.= PosetMax (Support (g,T)) by A9, POLYRED:24 ;
A19: Support ((Red (g,T)),T) = Support (Red (g,T)) by POLYRED:def_4;
A20: Support (g,T) = Support g by POLYRED:def_4;
then A21: (Support (g,T)) \ {(PosetMax (Support (g,T)))} = Support ((Red (g,T)),T) by A9, A12, A19, POLYRED:24;
(Support (Red (f,T))) \/ {(HT (f,T))} = (Support f) \/ {(HT (f,T))} by A11, XBOOLE_1:39
.= Support f by A14, XBOOLE_1:12 ;
then (card (Support (Red (f,T)))) + 1 = k + 1 by A5, A17, CARD_2:41;
then Red (f,T) <= Red (g,T),T by A3, A13;
then [(Support (Red (f,T))),(Support (Red (g,T)))] in FinOrd RelStr(# (Bags n),T #) by POLYRED:def_2;
then A22: [(Support ((Red (f,T)),T)),(Support ((Red (g,T)),T))] in union (rng (FinOrd-Approx RelStr(# (Bags n),T #))) by A16, A19, BAGORDER:def_15;
A23: Support (f,T) = Support f by POLYRED:def_4;
then (Support (f,T)) \ {(PosetMax (Support (f,T)))} = Support ((Red (f,T)),T) by A8, A11, A16, POLYRED:24;
then [(Support (f,T)),(Support (g,T))] in union (rng (FinOrd-Approx RelStr(# (Bags n),T #))) by A22, A15, A18, A21, BAGORDER:35;
then [(Support f),(Support g)] in FinOrd RelStr(# (Bags n),T #) by A23, A20, BAGORDER:def_15;
hence f <= g,T by POLYRED:def_2; ::_thesis: verum
end;
caseA24: HT (f,T) <> HT (g,T) ; ::_thesis: f <= g,T
now__::_thesis:_not_HT_(g,T)_<_HT_(f,T),T
assume HT (g,T) < HT (f,T),T ; ::_thesis: contradiction
then not HT (f,T) <= HT (g,T),T by TERMORD:5;
hence contradiction by A4, A7, TERMORD:def_6; ::_thesis: verum
end;
then HT (f,T) <= HT (g,T),T by TERMORD:5;
then HT (f,T) < HT (g,T),T by A24, TERMORD:def_3;
then f < g,T by POLYRED:32;
hence f <= g,T by POLYRED:def_3; ::_thesis: verum
end;
end;
end;
hence f <= g,T ; ::_thesis: verum
end;
hence S1[k + 1] ; ::_thesis: verum
end;
A25: ex k being Element of NAT st card (Support q) = k ;
A26: S1[ 0 ]
proof
let f, g be Polynomial of n,L; ::_thesis: ( Support f c= Support g & card (Support f) = 0 implies f <= g,T )
assume that
Support f c= Support g and
A27: card (Support f) = 0 ; ::_thesis: f <= g,T
Support f = {} by A27;
then f = 0_ (n,L) by POLYNOM7:1;
hence f <= g,T by POLYRED:30; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A26, A2);
hence q <= p,T by A1, A25; ::_thesis: verum
end;
theorem Th9: :: GROEB_1:9
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, p being non-zero Polynomial of n,L st f is_reducible_wrt p,T holds
HT (p,T) <= HT (f,T),T
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, p being non-zero Polynomial of n,L st f is_reducible_wrt p,T holds
HT (p,T) <= HT (f,T),T
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, p being non-zero Polynomial of n,L st f is_reducible_wrt p,T holds
HT (p,T) <= HT (f,T),T
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, p being non-zero Polynomial of n,L st f is_reducible_wrt p,T holds
HT (p,T) <= HT (f,T),T
let f, p be non-zero Polynomial of n,L; ::_thesis: ( f is_reducible_wrt p,T implies HT (p,T) <= HT (f,T),T )
assume f is_reducible_wrt p,T ; ::_thesis: HT (p,T) <= HT (f,T),T
then consider b being bag of n such that
A1: ( b in Support f & HT (p,T) divides b ) by POLYRED:36;
( b <= HT (f,T),T & HT (p,T) <= b,T ) by A1, TERMORD:10, TERMORD:def_6;
hence HT (p,T) <= HT (f,T),T by TERMORD:8; ::_thesis: verum
end;
begin
theorem Th10: :: GROEB_1:10
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L holds PolyRedRel ({p},T) is locally-confluent
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L holds PolyRedRel ({p},T) is locally-confluent
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L holds PolyRedRel ({p},T) is locally-confluent
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for p being Polynomial of n,L holds PolyRedRel ({p},T) is locally-confluent
let p be Polynomial of n,L; ::_thesis: PolyRedRel ({p},T) is locally-confluent
set R = PolyRedRel ({p},T);
A1: 0_ (n,L) = 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10;
percases ( p = 0_ (n,L) or p <> 0_ (n,L) ) ;
supposeA2: p = 0_ (n,L) ; ::_thesis: PolyRedRel ({p},T) is locally-confluent
now__::_thesis:_for_a,_b,_c_being_set_st_[a,b]_in_PolyRedRel_({p},T)_&_[a,c]_in_PolyRedRel_({p},T)_holds_
b,c_are_convergent_wrt_PolyRedRel_({p},T)
let a, b, c be set ; ::_thesis: ( [a,b] in PolyRedRel ({p},T) & [a,c] in PolyRedRel ({p},T) implies b,c are_convergent_wrt PolyRedRel ({p},T) )
assume that
A3: [a,b] in PolyRedRel ({p},T) and
[a,c] in PolyRedRel ({p},T) ; ::_thesis: b,c are_convergent_wrt PolyRedRel ({p},T)
consider p9, q being set such that
A4: p9 in NonZero (Polynom-Ring (n,L)) and
A5: q in the carrier of (Polynom-Ring (n,L)) and
A6: [a,b] = [p9,q] by A3, ZFMISC_1:def_2;
reconsider q = q as Polynomial of n,L by A5, POLYNOM1:def_10;
not p9 in {(0_ (n,L))} by A1, A4, XBOOLE_0:def_5;
then p9 <> 0_ (n,L) by TARSKI:def_1;
then reconsider p9 = p9 as non-zero Polynomial of n,L by A4, POLYNOM1:def_10, POLYNOM7:def_1;
p9 reduces_to q,{p},T by A3, A6, POLYRED:def_13;
then consider g being Polynomial of n,L such that
A7: g in {p} and
A8: p9 reduces_to q,g,T by POLYRED:def_7;
g = 0_ (n,L) by A2, A7, TARSKI:def_1;
then p9 is_reducible_wrt 0_ (n,L),T by A8, POLYRED:def_8;
hence b,c are_convergent_wrt PolyRedRel ({p},T) by Lm3; ::_thesis: verum
end;
hence PolyRedRel ({p},T) is locally-confluent by REWRITE1:def_24; ::_thesis: verum
end;
suppose p <> 0_ (n,L) ; ::_thesis: PolyRedRel ({p},T) is locally-confluent
then reconsider p = p as non-zero Polynomial of n,L by POLYNOM7:def_1;
now__::_thesis:_for_a,_b,_c_being_set_st_[a,b]_in_PolyRedRel_({p},T)_&_[a,c]_in_PolyRedRel_({p},T)_holds_
b,c_are_convergent_wrt_PolyRedRel_({p},T)
let a, b, c be set ; ::_thesis: ( [a,b] in PolyRedRel ({p},T) & [a,c] in PolyRedRel ({p},T) implies b,c are_convergent_wrt PolyRedRel ({p},T) )
assume that
A9: [a,b] in PolyRedRel ({p},T) and
A10: [a,c] in PolyRedRel ({p},T) ; ::_thesis: b,c are_convergent_wrt PolyRedRel ({p},T)
consider pa, pb being set such that
A11: pa in NonZero (Polynom-Ring (n,L)) and
A12: pb in the carrier of (Polynom-Ring (n,L)) and
A13: [a,b] = [pa,pb] by A9, ZFMISC_1:def_2;
not pa in {(0_ (n,L))} by A1, A11, XBOOLE_0:def_5;
then pa <> 0_ (n,L) by TARSKI:def_1;
then reconsider pa = pa as non-zero Polynomial of n,L by A11, POLYNOM1:def_10, POLYNOM7:def_1;
reconsider pb = pb as Polynomial of n,L by A12, POLYNOM1:def_10;
A14: pb = b by A13, XTUPLE_0:1;
A15: pa = a by A13, XTUPLE_0:1;
then pa reduces_to pb,{p},T by A9, A14, POLYRED:def_13;
then ex p9 being Polynomial of n,L st
( p9 in {p} & pa reduces_to pb,p9,T ) by POLYRED:def_7;
then A16: pa reduces_to pb,p,T by TARSKI:def_1;
consider pa9, pc being set such that
pa9 in NonZero (Polynom-Ring (n,L)) and
A17: pc in the carrier of (Polynom-Ring (n,L)) and
A18: [a,c] = [pa9,pc] by A10, ZFMISC_1:def_2;
reconsider pc = pc as Polynomial of n,L by A17, POLYNOM1:def_10;
A19: p in {p} by TARSKI:def_1;
A20: pc = c by A18, XTUPLE_0:1;
then pa reduces_to pc,{p},T by A10, A15, POLYRED:def_13;
then ex p9 being Polynomial of n,L st
( p9 in {p} & pa reduces_to pc,p9,T ) by POLYRED:def_7;
then A21: pa reduces_to pc,p,T by TARSKI:def_1;
now__::_thesis:_(_(_pb_=_0__(n,L)_&_ex_d_being_set_st_
(_PolyRedRel_({p},T)_reduces_b,d_&_PolyRedRel_({p},T)_reduces_c,d_)_)_or_(_pc_=_0__(n,L)_&_ex_d_being_set_st_
(_PolyRedRel_({p},T)_reduces_b,d_&_PolyRedRel_({p},T)_reduces_c,d_)_)_or_(_not_pb_=_0__(n,L)_&_not_pc_=_0__(n,L)_&_ex_d_being_set_st_
(_PolyRedRel_({p},T)_reduces_b,d_&_PolyRedRel_({p},T)_reduces_c,d_)_)_)
percases ( pb = 0_ (n,L) or pc = 0_ (n,L) or ( not pb = 0_ (n,L) & not pc = 0_ (n,L) ) ) ;
caseA22: pb = 0_ (n,L) ; ::_thesis: ex d being set st
( PolyRedRel ({p},T) reduces b,d & PolyRedRel ({p},T) reduces c,d )
then consider mb being Monomial of n,L such that
A23: 0_ (n,L) = pa - (mb *' p) by A16, Th1;
(0_ (n,L)) + (mb *' p) = (pa + (- (mb *' p))) + (mb *' p) by A23, POLYNOM1:def_6;
then mb *' p = (pa + (- (mb *' p))) + (mb *' p) by POLYRED:2;
then mb *' p = pa + ((- (mb *' p)) + (mb *' p)) by POLYNOM1:21;
then mb *' p = pa + (0_ (n,L)) by POLYRED:3;
then mb *' p = pa by POLYNOM1:23;
then consider mc being Monomial of n,L such that
A24: pc = (mb *' p) - (mc *' p) by A21, Th1;
pc = (mb *' p) + (- (mc *' p)) by A24, POLYNOM1:def_6;
then pc = (mb *' p) + ((- mc) *' p) by POLYRED:6;
then A25: pc = (mb + (- mc)) *' p by POLYNOM1:26;
then A26: pc = (mb - mc) *' p by POLYNOM1:def_6;
now__::_thesis:_(_(_mb_=_mc_&_ex_d_being_set_st_
(_PolyRedRel_({p},T)_reduces_b,d_&_PolyRedRel_({p},T)_reduces_c,d_)_)_or_(_mb_<>_mc_&_ex_d_being_set_st_
(_PolyRedRel_({p},T)_reduces_b,d_&_PolyRedRel_({p},T)_reduces_c,d_)_)_)
percases ( mb = mc or mb <> mc ) ;
case mb = mc ; ::_thesis: ex d being set st
( PolyRedRel ({p},T) reduces b,d & PolyRedRel ({p},T) reduces c,d )
then pc = (0_ (n,L)) *' p by A26, POLYNOM1:24;
then pc = 0_ (n,L) by POLYRED:5;
hence ex d being set st
( PolyRedRel ({p},T) reduces b,d & PolyRedRel ({p},T) reduces c,d ) by A14, A20, A22, REWRITE1:12; ::_thesis: verum
end;
case mb <> mc ; ::_thesis: ex d being set st
( PolyRedRel ({p},T) reduces b,d & PolyRedRel ({p},T) reduces c,d )
PolyRedRel ({p},T) reduces pb, 0_ (n,L) by A22, REWRITE1:12;
hence ex d being set st
( PolyRedRel ({p},T) reduces b,d & PolyRedRel ({p},T) reduces c,d ) by A14, A20, A19, A25, POLYRED:45; ::_thesis: verum
end;
end;
end;
hence ex d being set st
( PolyRedRel ({p},T) reduces b,d & PolyRedRel ({p},T) reduces c,d ) ; ::_thesis: verum
end;
caseA27: pc = 0_ (n,L) ; ::_thesis: ex d being set st
( PolyRedRel ({p},T) reduces b,d & PolyRedRel ({p},T) reduces c,d )
then consider mc being Monomial of n,L such that
A28: 0_ (n,L) = pa - (mc *' p) by A21, Th1;
(0_ (n,L)) + (mc *' p) = (pa + (- (mc *' p))) + (mc *' p) by A28, POLYNOM1:def_6;
then mc *' p = (pa + (- (mc *' p))) + (mc *' p) by POLYRED:2;
then mc *' p = pa + ((- (mc *' p)) + (mc *' p)) by POLYNOM1:21;
then mc *' p = pa + (0_ (n,L)) by POLYRED:3;
then mc *' p = pa by POLYNOM1:23;
then consider mb being Monomial of n,L such that
A29: pb = (mc *' p) - (mb *' p) by A16, Th1;
pb = (mc *' p) + (- (mb *' p)) by A29, POLYNOM1:def_6;
then pb = (mc *' p) + ((- mb) *' p) by POLYRED:6;
then A30: pb = (mc + (- mb)) *' p by POLYNOM1:26;
then A31: pb = (mc - mb) *' p by POLYNOM1:def_6;
now__::_thesis:_(_(_mb_=_mc_&_ex_d_being_set_st_
(_PolyRedRel_({p},T)_reduces_b,d_&_PolyRedRel_({p},T)_reduces_c,d_)_)_or_(_mb_<>_mc_&_ex_d_being_set_st_
(_PolyRedRel_({p},T)_reduces_b,d_&_PolyRedRel_({p},T)_reduces_c,d_)_)_)
percases ( mb = mc or mb <> mc ) ;
case mb = mc ; ::_thesis: ex d being set st
( PolyRedRel ({p},T) reduces b,d & PolyRedRel ({p},T) reduces c,d )
then pb = (0_ (n,L)) *' p by A31, POLYNOM1:24;
then pb = 0_ (n,L) by POLYRED:5;
hence ex d being set st
( PolyRedRel ({p},T) reduces b,d & PolyRedRel ({p},T) reduces c,d ) by A14, A20, A27, REWRITE1:12; ::_thesis: verum
end;
case mb <> mc ; ::_thesis: ex d being set st
( PolyRedRel ({p},T) reduces b,d & PolyRedRel ({p},T) reduces c,d )
PolyRedRel ({p},T) reduces pc, 0_ (n,L) by A27, REWRITE1:12;
hence ex d being set st
( PolyRedRel ({p},T) reduces b,d & PolyRedRel ({p},T) reduces c,d ) by A14, A20, A19, A30, POLYRED:45; ::_thesis: verum
end;
end;
end;
hence ex d being set st
( PolyRedRel ({p},T) reduces b,d & PolyRedRel ({p},T) reduces c,d ) ; ::_thesis: verum
end;
case ( not pb = 0_ (n,L) & not pc = 0_ (n,L) ) ; ::_thesis: ex d being set st
( PolyRedRel ({p},T) reduces b,d & PolyRedRel ({p},T) reduces c,d )
then reconsider pb = pb, pc = pc as non-zero Polynomial of n,L by POLYNOM7:def_1;
now__::_thesis:_(_(_pb_=_pc_&_ex_d_being_set_st_
(_PolyRedRel_({p},T)_reduces_b,d_&_PolyRedRel_({p},T)_reduces_c,d_)_)_or_(_pb_<>_pc_&_ex_d_being_set_st_
(_PolyRedRel_({p},T)_reduces_b,d_&_PolyRedRel_({p},T)_reduces_c,d_)_)_)
percases ( pb = pc or pb <> pc ) ;
case pb = pc ; ::_thesis: ex d being set st
( PolyRedRel ({p},T) reduces b,d & PolyRedRel ({p},T) reduces c,d )
hence ex d being set st
( PolyRedRel ({p},T) reduces b,d & PolyRedRel ({p},T) reduces c,d ) by A14, A20, REWRITE1:12; ::_thesis: verum
end;
caseA32: pb <> pc ; ::_thesis: ex d being set st
( PolyRedRel ({p},T) reduces b,d & PolyRedRel ({p},T) reduces c,d )
now__::_thesis:_not_pb_-_pc_=_0__(n,L)
assume pb - pc = 0_ (n,L) ; ::_thesis: contradiction
then (pb + (- pc)) + pc = (0_ (n,L)) + pc by POLYNOM1:def_6;
then pb + ((- pc) + pc) = (0_ (n,L)) + pc by POLYNOM1:21;
then pb + (0_ (n,L)) = (0_ (n,L)) + pc by POLYRED:3;
then pb + (0_ (n,L)) = pc by POLYRED:2;
hence contradiction by A32, POLYNOM1:23; ::_thesis: verum
end;
then reconsider h = pb - pc as non-zero Polynomial of n,L by POLYNOM7:def_1;
consider mb being Monomial of n,L such that
A33: pb = pa - (mb *' p) by A16, Th1;
consider mc being Monomial of n,L such that
A34: pc = pa - (mc *' p) by A21, Th1;
now__::_thesis:_not_(-_mb)_+_mc_=_0__(n,L)
assume (- mb) + mc = 0_ (n,L) ; ::_thesis: contradiction
then mc + ((- mb) + mb) = (0_ (n,L)) + mb by POLYNOM1:21;
then mc + (0_ (n,L)) = (0_ (n,L)) + mb by POLYRED:3;
then mc + (0_ (n,L)) = mb by POLYRED:2;
hence contradiction by A32, A33, A34, POLYNOM1:23; ::_thesis: verum
end;
then reconsider hh = (- mb) + mc as non-zero Polynomial of n,L by POLYNOM7:def_1;
A35: - (- (mc *' p)) = mc *' p by POLYNOM1:19;
h = (pa - (mb *' p)) + (- (pa - (mc *' p))) by A33, A34, POLYNOM1:def_6
.= (pa - (mb *' p)) + (- (pa + (- (mc *' p)))) by POLYNOM1:def_6
.= (pa - (mb *' p)) + ((- pa) + (- (- (mc *' p)))) by POLYRED:1
.= (pa + (- (mb *' p))) + ((- pa) + (- (- (mc *' p)))) by POLYNOM1:def_6
.= ((pa + (- (mb *' p))) + (- pa)) + (mc *' p) by A35, POLYNOM1:21
.= ((pa + (- pa)) + (- (mb *' p))) + (mc *' p) by POLYNOM1:21
.= ((0_ (n,L)) + (- (mb *' p))) + (mc *' p) by POLYRED:3
.= (- (mb *' p)) + (mc *' p) by POLYRED:2
.= ((- mb) *' p) + (mc *' p) by POLYRED:6
.= hh *' p by POLYNOM1:26 ;
then consider f1, g1 being Polynomial of n,L such that
A36: f1 - g1 = 0_ (n,L) and
A37: ( PolyRedRel ({p},T) reduces pb,f1 & PolyRedRel ({p},T) reduces pc,g1 ) by A19, POLYRED:45, POLYRED:49;
(f1 + (- g1)) + g1 = (0_ (n,L)) + g1 by A36, POLYNOM1:def_6;
then f1 + ((- g1) + g1) = (0_ (n,L)) + g1 by POLYNOM1:21;
then f1 + (0_ (n,L)) = (0_ (n,L)) + g1 by POLYRED:3;
then f1 + (0_ (n,L)) = g1 by POLYRED:2;
then f1 = g1 by POLYNOM1:23;
hence ex d being set st
( PolyRedRel ({p},T) reduces b,d & PolyRedRel ({p},T) reduces c,d ) by A14, A20, A37; ::_thesis: verum
end;
end;
end;
hence ex d being set st
( PolyRedRel ({p},T) reduces b,d & PolyRedRel ({p},T) reduces c,d ) ; ::_thesis: verum
end;
end;
end;
hence b,c are_convergent_wrt PolyRedRel ({p},T) by REWRITE1:def_7; ::_thesis: verum
end;
hence PolyRedRel ({p},T) is locally-confluent by REWRITE1:def_24; ::_thesis: verum
end;
end;
end;
theorem :: GROEB_1:11
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st ex p being Polynomial of n,L st
( p in P & P -Ideal = {p} -Ideal ) holds
PolyRedRel (P,T) is locally-confluent
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st ex p being Polynomial of n,L st
( p in P & P -Ideal = {p} -Ideal ) holds
PolyRedRel (P,T) is locally-confluent
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st ex p being Polynomial of n,L st
( p in P & P -Ideal = {p} -Ideal ) holds
PolyRedRel (P,T) is locally-confluent
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for P being Subset of (Polynom-Ring (n,L)) st ex p being Polynomial of n,L st
( p in P & P -Ideal = {p} -Ideal ) holds
PolyRedRel (P,T) is locally-confluent
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: ( ex p being Polynomial of n,L st
( p in P & P -Ideal = {p} -Ideal ) implies PolyRedRel (P,T) is locally-confluent )
set R = PolyRedRel (P,T);
assume ex p being Polynomial of n,L st
( p in P & P -Ideal = {p} -Ideal ) ; ::_thesis: PolyRedRel (P,T) is locally-confluent
then consider p being Polynomial of n,L such that
A1: p in P and
A2: P -Ideal = {p} -Ideal ;
now__::_thesis:_for_a,_b,_c_being_set_st_[a,b]_in_PolyRedRel_(P,T)_&_[a,c]_in_PolyRedRel_(P,T)_holds_
b,c_are_convergent_wrt_PolyRedRel_(P,T)
set Rp = PolyRedRel ({p},T);
reconsider Rp = PolyRedRel ({p},T) as strongly-normalizing locally-confluent Relation by Th10;
let a, b, c be set ; ::_thesis: ( [a,b] in PolyRedRel (P,T) & [a,c] in PolyRedRel (P,T) implies b,c are_convergent_wrt PolyRedRel (P,T) )
assume that
A3: [a,b] in PolyRedRel (P,T) and
A4: [a,c] in PolyRedRel (P,T) ; ::_thesis: b,c are_convergent_wrt PolyRedRel (P,T)
a,b are_convertible_wrt PolyRedRel (P,T) by A3, REWRITE1:29;
then A5: b,a are_convertible_wrt PolyRedRel (P,T) by REWRITE1:31;
consider pa, pb being set such that
pa in NonZero (Polynom-Ring (n,L)) and
A6: pb in the carrier of (Polynom-Ring (n,L)) and
A7: [a,b] = [pa,pb] by A3, ZFMISC_1:def_2;
reconsider pb = pb as Polynomial of n,L by A6, POLYNOM1:def_10;
A8: pb = b by A7, XTUPLE_0:1;
consider pa9, pc being set such that
pa9 in NonZero (Polynom-Ring (n,L)) and
A9: pc in the carrier of (Polynom-Ring (n,L)) and
A10: [a,c] = [pa9,pc] by A4, ZFMISC_1:def_2;
reconsider pc = pc as Polynomial of n,L by A9, POLYNOM1:def_10;
A11: pc = c by A10, XTUPLE_0:1;
reconsider pb9 = pb, pc9 = pc as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider pc9 = pc9, pb9 = pb9 as Element of (Polynom-Ring (n,L)) ;
a,c are_convertible_wrt PolyRedRel (P,T) by A4, REWRITE1:29;
then pb9,pc9 are_congruent_mod {p} -Ideal by A2, A8, A11, A5, POLYRED:57, REWRITE1:30;
then pb,pc are_convertible_wrt PolyRedRel ({p},T) by POLYRED:58;
then b,c are_convergent_wrt Rp by A8, A11, REWRITE1:def_23;
then consider d being set such that
A12: ( Rp reduces b,d & Rp reduces c,d ) by REWRITE1:def_7;
for u being set st u in {p} holds
u in P by A1, TARSKI:def_1;
then {p} c= P by TARSKI:def_3;
then ( PolyRedRel (P,T) reduces b,d & PolyRedRel (P,T) reduces c,d ) by A12, Th4, REWRITE1:22;
hence b,c are_convergent_wrt PolyRedRel (P,T) by REWRITE1:def_7; ::_thesis: verum
end;
hence PolyRedRel (P,T) is locally-confluent by REWRITE1:def_24; ::_thesis: verum
end;
definition
let n be Ordinal;
let T be connected TermOrder of n;
let L be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr ;
let P be Subset of (Polynom-Ring (n,L));
func HT (P,T) -> Subset of (Bags n) equals :: GROEB_1:def 1
{ (HT (p,T)) where p is Polynomial of n,L : ( p in P & p <> 0_ (n,L) ) } ;
coherence
{ (HT (p,T)) where p is Polynomial of n,L : ( p in P & p <> 0_ (n,L) ) } is Subset of (Bags n)
proof
set M = { (HT (p,T)) where p is Polynomial of n,L : ( p in P & p <> 0_ (n,L) ) } ;
now__::_thesis:_for_u_being_set_st_u_in__{__(HT_(p,T))_where_p_is_Polynomial_of_n,L_:_(_p_in_P_&_p_<>_0__(n,L)_)__}__holds_
u_in_Bags_n
let u be set ; ::_thesis: ( u in { (HT (p,T)) where p is Polynomial of n,L : ( p in P & p <> 0_ (n,L) ) } implies u in Bags n )
assume u in { (HT (p,T)) where p is Polynomial of n,L : ( p in P & p <> 0_ (n,L) ) } ; ::_thesis: u in Bags n
then ex p being Polynomial of n,L st
( u = HT (p,T) & p in P & p <> 0_ (n,L) ) ;
hence u in Bags n ; ::_thesis: verum
end;
hence { (HT (p,T)) where p is Polynomial of n,L : ( p in P & p <> 0_ (n,L) ) } is Subset of (Bags n) by TARSKI:def_3; ::_thesis: verum
end;
end;
:: deftheorem defines HT GROEB_1:def_1_:_
for n being Ordinal
for T being connected TermOrder of n
for L being non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) holds HT (P,T) = { (HT (p,T)) where p is Polynomial of n,L : ( p in P & p <> 0_ (n,L) ) } ;
definition
let n be Ordinal;
let S be Subset of (Bags n);
func multiples S -> Subset of (Bags n) equals :: GROEB_1:def 2
{ b where b is Element of Bags n : ex b9 being bag of n st
( b9 in S & b9 divides b ) } ;
coherence
{ b where b is Element of Bags n : ex b9 being bag of n st
( b9 in S & b9 divides b ) } is Subset of (Bags n)
proof
set M = { b where b is Element of Bags n : ex b9 being bag of n st
( b9 in S & b9 divides b ) } ;
now__::_thesis:_for_u_being_set_st_u_in__{__b_where_b_is_Element_of_Bags_n_:_ex_b9_being_bag_of_n_st_
(_b9_in_S_&_b9_divides_b_)__}__holds_
u_in_Bags_n
let u be set ; ::_thesis: ( u in { b where b is Element of Bags n : ex b9 being bag of n st
( b9 in S & b9 divides b ) } implies u in Bags n )
assume u in { b where b is Element of Bags n : ex b9 being bag of n st
( b9 in S & b9 divides b ) } ; ::_thesis: u in Bags n
then ex b being Element of Bags n st
( u = b & ex b9 being bag of n st
( b9 in S & b9 divides b ) ) ;
hence u in Bags n ; ::_thesis: verum
end;
hence { b where b is Element of Bags n : ex b9 being bag of n st
( b9 in S & b9 divides b ) } is Subset of (Bags n) by TARSKI:def_3; ::_thesis: verum
end;
end;
:: deftheorem defines multiples GROEB_1:def_2_:_
for n being Ordinal
for S being Subset of (Bags n) holds multiples S = { b where b is Element of Bags n : ex b9 being bag of n st
( b9 in S & b9 divides b ) } ;
theorem :: GROEB_1:12
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) is locally-confluent holds
PolyRedRel (P,T) is confluent ;
theorem :: GROEB_1:13
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) is confluent holds
PolyRedRel (P,T) is with_UN_property ;
theorem :: GROEB_1:14
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) is with_UN_property holds
PolyRedRel (P,T) is with_Church-Rosser_property ;
Lm4: for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f being Polynomial of n,L
for g being set
for P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds
g is Polynomial of n,L
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f being Polynomial of n,L
for g being set
for P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds
g is Polynomial of n,L
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f being Polynomial of n,L
for g being set
for P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds
g is Polynomial of n,L
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for f being Polynomial of n,L
for g being set
for P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds
g is Polynomial of n,L
let f be Polynomial of n,L; ::_thesis: for g being set
for P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds
g is Polynomial of n,L
let g be set ; ::_thesis: for P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds
g is Polynomial of n,L
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: ( PolyRedRel (P,T) reduces f,g implies g is Polynomial of n,L )
set R = PolyRedRel (P,T);
assume PolyRedRel (P,T) reduces f,g ; ::_thesis: g is Polynomial of n,L
then consider p being RedSequence of PolyRedRel (P,T) such that
A1: p . 1 = f and
A2: p . (len p) = g by REWRITE1:def_3;
reconsider l = (len p) - 1 as Element of NAT by INT_1:5, NAT_1:14;
A3: 1 <= len p by NAT_1:14;
set h = p . l;
1 <= l + 1 by NAT_1:12;
then l + 1 in Seg (len p) by FINSEQ_1:1;
then A4: l + 1 in dom p by FINSEQ_1:def_3;
percases ( len p = 1 or len p <> 1 ) ;
suppose len p = 1 ; ::_thesis: g is Polynomial of n,L
hence g is Polynomial of n,L by A1, A2; ::_thesis: verum
end;
suppose len p <> 1 ; ::_thesis: g is Polynomial of n,L
then 0 + 1 < l + 1 by A3, XXREAL_0:1;
then A5: 1 <= l by NAT_1:13;
l <= l + 1 by NAT_1:13;
then l in Seg (len p) by A5, FINSEQ_1:1;
then l in dom p by FINSEQ_1:def_3;
then [(p . l),g] in PolyRedRel (P,T) by A2, A4, REWRITE1:def_2;
then consider h9, g9 being set such that
A6: [(p . l),g] = [h9,g9] and
h9 in NonZero (Polynom-Ring (n,L)) and
A7: g9 in the carrier of (Polynom-Ring (n,L)) by RELSET_1:2;
g = g9 by A6, XTUPLE_0:1;
hence g is Polynomial of n,L by A7, POLYNOM1:def_10; ::_thesis: verum
end;
end;
end;
Lm5: for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed 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 & g <> f holds
ex h being Polynomial of n,L st
( f reduces_to h,P,T & PolyRedRel (P,T) reduces h,g )
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed 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 & g <> f holds
ex h being Polynomial of n,L st
( f reduces_to h,P,T & PolyRedRel (P,T) reduces h,g )
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed 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 & g <> f holds
ex h being Polynomial of n,L st
( f reduces_to h,P,T & PolyRedRel (P,T) reduces h,g )
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, g being Polynomial of n,L
for P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g & g <> f holds
ex h being Polynomial of n,L st
( f reduces_to h,P,T & PolyRedRel (P,T) reduces h,g )
let f, g be Polynomial of n,L; ::_thesis: for P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g & g <> f holds
ex h being Polynomial of n,L st
( f reduces_to h,P,T & PolyRedRel (P,T) reduces h,g )
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: ( PolyRedRel (P,T) reduces f,g & g <> f implies ex h being Polynomial of n,L st
( f reduces_to h,P,T & PolyRedRel (P,T) reduces h,g ) )
set R = PolyRedRel (P,T);
assume that
A1: PolyRedRel (P,T) reduces f,g and
A2: g <> f ; ::_thesis: ex h being Polynomial of n,L st
( f reduces_to h,P,T & PolyRedRel (P,T) reduces h,g )
consider p being RedSequence of PolyRedRel (P,T) such that
A3: p . 1 = f and
A4: p . (len p) = g by A1, REWRITE1:def_3;
set h = p . 2;
len p > 0 by REWRITE1:def_2;
then (len p) + 1 > 0 + 1 by XREAL_1:8;
then A5: 1 <= len p by NAT_1:13;
then 1 < len p by A2, A3, A4, XXREAL_0:1;
then A6: 1 + 1 <= len p by NAT_1:13;
then 1 + 1 in Seg (len p) by FINSEQ_1:1;
then A7: 1 + 1 in dom p by FINSEQ_1:def_3;
1 in Seg (len p) by A5, FINSEQ_1:1;
then 1 in dom p by FINSEQ_1:def_3;
then A8: [f,(p . 2)] in PolyRedRel (P,T) by A3, A7, REWRITE1:def_2;
then consider f9, h9 being set such that
A9: [f,(p . 2)] = [f9,h9] and
f9 in NonZero (Polynom-Ring (n,L)) and
A10: h9 in the carrier of (Polynom-Ring (n,L)) by RELSET_1:2;
A11: p . 2 = h9 by A9, XTUPLE_0:1;
len p in Seg (len p) by A5, FINSEQ_1:1;
then A12: len p in dom p by FINSEQ_1:def_3;
reconsider h = p . 2 as Polynomial of n,L by A10, A11, POLYNOM1:def_10;
f reduces_to h,P,T by A8, POLYRED:def_13;
hence ex h being Polynomial of n,L st
( f reduces_to h,P,T & PolyRedRel (P,T) reduces h,g ) by A4, A6, A7, A12, REWRITE1:17; ::_thesis: verum
end;
theorem Th15: :: GROEB_1:15
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being non empty Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) is with_Church-Rosser_property holds
for f being Polynomial of n,L st f in P -Ideal holds
PolyRedRel (P,T) reduces f, 0_ (n,L)
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being non empty Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) is with_Church-Rosser_property holds
for f being Polynomial of n,L st f in P -Ideal holds
PolyRedRel (P,T) reduces f, 0_ (n,L)
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being non empty Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) is with_Church-Rosser_property holds
for f being Polynomial of n,L st f in P -Ideal holds
PolyRedRel (P,T) reduces f, 0_ (n,L)
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for P being non empty Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) is with_Church-Rosser_property holds
for f being Polynomial of n,L st f in P -Ideal holds
PolyRedRel (P,T) reduces f, 0_ (n,L)
let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: ( PolyRedRel (P,T) is with_Church-Rosser_property implies for f being Polynomial of n,L st f in P -Ideal holds
PolyRedRel (P,T) reduces f, 0_ (n,L) )
set R = PolyRedRel (P,T);
assume A1: PolyRedRel (P,T) is with_Church-Rosser_property ; ::_thesis: for f being Polynomial of n,L st f in P -Ideal holds
PolyRedRel (P,T) reduces f, 0_ (n,L)
now__::_thesis:_for_f_being_Polynomial_of_n,L_st_f_in_P_-Ideal_holds_
PolyRedRel_(P,T)_reduces_f,_0__(n,L)
reconsider e = 0_ (n,L) as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
let f be Polynomial of n,L; ::_thesis: ( f in P -Ideal implies PolyRedRel (P,T) reduces f, 0_ (n,L) )
assume A2: f in P -Ideal ; ::_thesis: PolyRedRel (P,T) reduces f, 0_ (n,L)
reconsider e = e as Element of (Polynom-Ring (n,L)) ;
reconsider f9 = f as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider f9 = f9 as Element of (Polynom-Ring (n,L)) ;
f - (0_ (n,L)) = f9 - e by Lm2;
then f9 - e in P -Ideal by A2, POLYRED:4;
then f9,e are_congruent_mod P -Ideal by POLYRED:def_14;
then f9,e are_convertible_wrt PolyRedRel (P,T) by POLYRED:58;
then f9,e are_convergent_wrt PolyRedRel (P,T) by A1, REWRITE1:def_23;
then consider c being set such that
A3: PolyRedRel (P,T) reduces f,c and
A4: PolyRedRel (P,T) reduces 0_ (n,L),c by REWRITE1:def_7;
reconsider c9 = c as Polynomial of n,L by A3, Lm4;
now__::_thesis:_not_c9_<>_0__(n,L)
assume c9 <> 0_ (n,L) ; ::_thesis: contradiction
then consider h being Polynomial of n,L such that
A5: 0_ (n,L) reduces_to h,P,T and
PolyRedRel (P,T) reduces h,c9 by A4, Lm5;
consider pp being Polynomial of n,L such that
pp in P and
A6: 0_ (n,L) reduces_to h,pp,T by A5, POLYRED:def_7;
0_ (n,L) is_reducible_wrt pp,T by A6, POLYRED:def_8;
hence contradiction by POLYRED:37; ::_thesis: verum
end;
hence PolyRedRel (P,T) reduces f, 0_ (n,L) by A3; ::_thesis: verum
end;
hence for f being Polynomial of n,L st f in P -Ideal holds
PolyRedRel (P,T) reduces f, 0_ (n,L) ; ::_thesis: verum
end;
theorem Th16: :: GROEB_1:16
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st ( for f being Polynomial of n,L st f in P -Ideal holds
PolyRedRel (P,T) reduces f, 0_ (n,L) ) holds
for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_reducible_wrt P,T
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st ( for f being Polynomial of n,L st f in P -Ideal holds
PolyRedRel (P,T) reduces f, 0_ (n,L) ) holds
for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_reducible_wrt P,T
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st ( for f being Polynomial of n,L st f in P -Ideal holds
PolyRedRel (P,T) reduces f, 0_ (n,L) ) holds
for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_reducible_wrt P,T
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for P being Subset of (Polynom-Ring (n,L)) st ( for f being Polynomial of n,L st f in P -Ideal holds
PolyRedRel (P,T) reduces f, 0_ (n,L) ) holds
for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_reducible_wrt P,T
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: ( ( for f being Polynomial of n,L st f in P -Ideal holds
PolyRedRel (P,T) reduces f, 0_ (n,L) ) implies for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_reducible_wrt P,T )
assume A1: for f being Polynomial of n,L st f in P -Ideal holds
PolyRedRel (P,T) reduces f, 0_ (n,L) ; ::_thesis: for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_reducible_wrt P,T
now__::_thesis:_for_f_being_non-zero_Polynomial_of_n,L_st_f_in_P_-Ideal_holds_
f_is_reducible_wrt_P,T
let f be non-zero Polynomial of n,L; ::_thesis: ( f in P -Ideal implies f is_reducible_wrt P,T )
assume f in P -Ideal ; ::_thesis: f is_reducible_wrt P,T
then A2: PolyRedRel (P,T) reduces f, 0_ (n,L) by A1;
f <> 0_ (n,L) by POLYNOM7:def_1;
then ex g being Polynomial of n,L st
( f reduces_to g,P,T & PolyRedRel (P,T) reduces g, 0_ (n,L) ) by A2, Lm5;
hence f is_reducible_wrt P,T by POLYRED:def_9; ::_thesis: verum
end;
hence for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_reducible_wrt P,T ; ::_thesis: verum
end;
theorem Th17: :: GROEB_1:17
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st ( for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_reducible_wrt P,T ) holds
for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_top_reducible_wrt P,T
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st ( for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_reducible_wrt P,T ) holds
for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_top_reducible_wrt P,T
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st ( for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_reducible_wrt P,T ) holds
for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_top_reducible_wrt P,T
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for P being Subset of (Polynom-Ring (n,L)) st ( for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_reducible_wrt P,T ) holds
for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_top_reducible_wrt P,T
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: ( ( for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_reducible_wrt P,T ) implies for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_top_reducible_wrt P,T )
assume A1: for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_reducible_wrt P,T ; ::_thesis: for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_top_reducible_wrt P,T
thus for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_top_reducible_wrt P,T ::_thesis: verum
proof
set H = { g where g is non-zero Polynomial of n,L : ( g in P -Ideal & not g is_top_reducible_wrt P,T ) } ;
let f be non-zero Polynomial of n,L; ::_thesis: ( f in P -Ideal implies f is_top_reducible_wrt P,T )
assume A2: f in P -Ideal ; ::_thesis: f is_top_reducible_wrt P,T
assume not f is_top_reducible_wrt P,T ; ::_thesis: contradiction
then A3: f in { g where g is non-zero Polynomial of n,L : ( g in P -Ideal & not g is_top_reducible_wrt P,T ) } by A2;
now__::_thesis:_for_u_being_set_st_u_in__{__g_where_g_is_non-zero_Polynomial_of_n,L_:_(_g_in_P_-Ideal_&_not_g_is_top_reducible_wrt_P,T_)__}__holds_
u_in_the_carrier_of_(Polynom-Ring_(n,L))
let u be set ; ::_thesis: ( u in { g where g is non-zero Polynomial of n,L : ( g in P -Ideal & not g is_top_reducible_wrt P,T ) } implies u in the carrier of (Polynom-Ring (n,L)) )
assume u in { g where g is non-zero Polynomial of n,L : ( g in P -Ideal & not g is_top_reducible_wrt P,T ) } ; ::_thesis: u in the carrier of (Polynom-Ring (n,L))
then ex g9 being non-zero Polynomial of n,L st
( u = g9 & g9 in P -Ideal & not g9 is_top_reducible_wrt P,T ) ;
hence u in the carrier of (Polynom-Ring (n,L)) ; ::_thesis: verum
end;
then reconsider H = { g where g is non-zero Polynomial of n,L : ( g in P -Ideal & not g is_top_reducible_wrt P,T ) } as non empty Subset of (Polynom-Ring (n,L)) by A3, TARSKI:def_3;
consider p being Polynomial of n,L such that
A4: p in H and
A5: for q being Polynomial of n,L st q in H holds
p <= q,T by POLYRED:31;
A6: ex p9 being non-zero Polynomial of n,L st
( p9 = p & p9 in P -Ideal & not p9 is_top_reducible_wrt P,T ) by A4;
then reconsider p = p as non-zero Polynomial of n,L ;
p is_reducible_wrt P,T by A1, A6;
then consider q being Polynomial of n,L such that
A7: p reduces_to q,P,T by POLYRED:def_9;
consider u being Polynomial of n,L such that
A8: u in P and
A9: p reduces_to q,u,T by A7, POLYRED:def_7;
ex b being bag of n st p reduces_to q,u,b,T by A9, POLYRED:def_6;
then A10: u <> 0_ (n,L) by POLYRED:def_5;
then reconsider u = u as non-zero Polynomial of n,L by POLYNOM7:def_1;
consider b being bag of n such that
A11: p reduces_to q,u,b,T by A9, POLYRED:def_6;
A12: now__::_thesis:_not_b_=_HT_(p,T)
assume b = HT (p,T) ; ::_thesis: contradiction
then p top_reduces_to q,u,T by A11, POLYRED:def_10;
then p is_top_reducible_wrt u,T by POLYRED:def_11;
hence contradiction by A6, A8, POLYRED:def_12; ::_thesis: verum
end;
consider m being Monomial of n,L such that
A13: q = p - (m *' u) by A9, Th1;
reconsider uu = u, pp = p, mm = m as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider uu = uu, pp = pp, mm = mm as Element of (Polynom-Ring (n,L)) ;
uu in P -Ideal by A8, Lm1;
then mm * uu in P -Ideal by IDEAL_1:def_2;
then - (mm * uu) in P -Ideal by IDEAL_1:13;
then A14: pp + (- (mm * uu)) in P -Ideal by A6, IDEAL_1:def_1;
mm * uu = m *' u by POLYNOM1:def_10;
then p - (m *' u) = pp - (mm * uu) by Lm2;
then A15: q in P -Ideal by A13, A14, RLVECT_1:def_11;
A16: q < p,T by A9, POLYRED:43;
A17: p <> 0_ (n,L) by POLYNOM7:def_1;
then Support p <> {} by POLYNOM7:1;
then A18: HT (p,T) in Support p by TERMORD:def_6;
b in Support p by A11, POLYRED:def_5;
then b <= HT (p,T),T by TERMORD:def_6;
then b < HT (p,T),T by A12, TERMORD:def_3;
then A19: HT (p,T) in Support q by A18, A11, POLYRED:40;
now__::_thesis:_(_(_q_<>_0__(n,L)_&_contradiction_)_or_(_q_=_0__(n,L)_&_contradiction_)_)
percases ( q <> 0_ (n,L) or q = 0_ (n,L) ) ;
caseA20: q <> 0_ (n,L) ; ::_thesis: contradiction
then reconsider q = q as non-zero Polynomial of n,L by POLYNOM7:def_1;
Support q <> {} by A20, POLYNOM7:1;
then HT (q,T) in Support q by TERMORD:def_6;
then A21: HT (q,T) <= HT (p,T),T by A9, POLYRED:42;
HT (p,T) <= HT (q,T),T by A19, TERMORD:def_6;
then A22: HT (q,T) = HT (p,T) by A21, TERMORD:7;
now__::_thesis:_q_is_top_reducible_wrt_P,T
assume not q is_top_reducible_wrt P,T ; ::_thesis: contradiction
then q in H by A15;
then p <= q,T by A5;
hence contradiction by A16, POLYRED:29; ::_thesis: verum
end;
then consider u9 being Polynomial of n,L such that
A23: u9 in P and
A24: q is_top_reducible_wrt u9,T by POLYRED:def_12;
consider q9 being Polynomial of n,L such that
A25: q top_reduces_to q9,u9,T by A24, POLYRED:def_11;
A26: p <> 0_ (n,L) by POLYNOM7:def_1;
then Support p <> {} by POLYNOM7:1;
then A27: HT (p,T) in Support p by TERMORD:def_6;
A28: q reduces_to q9,u9, HT (q,T),T by A25, POLYRED:def_10;
then consider s being bag of n such that
A29: s + (HT (u9,T)) = HT (q,T) and
q9 = q - (((q . (HT (q,T))) / (HC (u9,T))) * (s *' u9)) by POLYRED:def_5;
set qq = p - (((p . (HT (p,T))) / (HC (u9,T))) * (s *' u9));
u9 <> 0_ (n,L) by A28, POLYRED:def_5;
then p reduces_to p - (((p . (HT (p,T))) / (HC (u9,T))) * (s *' u9)),u9, HT (p,T),T by A22, A29, A26, A27, POLYRED:def_5;
then p top_reduces_to p - (((p . (HT (p,T))) / (HC (u9,T))) * (s *' u9)),u9,T by POLYRED:def_10;
then p is_top_reducible_wrt u9,T by POLYRED:def_11;
hence contradiction by A6, A23, POLYRED:def_12; ::_thesis: verum
end;
case q = 0_ (n,L) ; ::_thesis: contradiction
then A30: m *' u = (p - (m *' u)) + (m *' u) by A13, POLYRED:2
.= (p + (- (m *' u))) + (m *' u) by POLYNOM1:def_6
.= p + ((- (m *' u)) + (m *' u)) by POLYNOM1:21
.= p + (0_ (n,L)) by POLYRED:3
.= p by POLYNOM1:23 ;
now__::_thesis:_not_m_=_0__(n,L)
A31: p <> 0_ (n,L) by POLYNOM7:def_1;
assume m = 0_ (n,L) ; ::_thesis: contradiction
hence contradiction by A30, A31, POLYRED:5; ::_thesis: verum
end;
then reconsider m = m as non-zero Polynomial of n,L by POLYNOM7:def_1;
set pp = p - (((p . (HT (p,T))) / (HC (u,T))) * ((HT (m,T)) *' u));
HT (p,T) = (HT (m,T)) + (HT (u,T)) by A30, TERMORD:31;
then p reduces_to p - (((p . (HT (p,T))) / (HC (u,T))) * ((HT (m,T)) *' u)),u, HT (p,T),T by A10, A17, A18, POLYRED:def_5;
then p top_reduces_to p - (((p . (HT (p,T))) / (HC (u,T))) * ((HT (m,T)) *' u)),u,T by POLYRED:def_10;
then p is_top_reducible_wrt u,T by POLYRED:def_11;
hence contradiction by A6, A8, POLYRED:def_12; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
theorem Th18: :: GROEB_1:18
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st ( for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_top_reducible_wrt P,T ) holds
for b being bag of n st b in HT ((P -Ideal),T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b )
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st ( for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_top_reducible_wrt P,T ) holds
for b being bag of n st b in HT ((P -Ideal),T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b )
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st ( for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_top_reducible_wrt P,T ) holds
for b being bag of n st b in HT ((P -Ideal),T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b )
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for P being Subset of (Polynom-Ring (n,L)) st ( for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_top_reducible_wrt P,T ) holds
for b being bag of n st b in HT ((P -Ideal),T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b )
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: ( ( for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_top_reducible_wrt P,T ) implies for b being bag of n st b in HT ((P -Ideal),T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) )
assume A1: for f being non-zero Polynomial of n,L st f in P -Ideal holds
f is_top_reducible_wrt P,T ; ::_thesis: for b being bag of n st b in HT ((P -Ideal),T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b )
now__::_thesis:_for_b_being_bag_of_n_st_b_in_HT_((P_-Ideal),T)_holds_
ex_b9_being_bag_of_n_st_
(_b9_in_HT_(P,T)_&_b9_divides_b_)
let b be bag of n; ::_thesis: ( b in HT ((P -Ideal),T) implies ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) )
assume b in HT ((P -Ideal),T) ; ::_thesis: ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b )
then consider p being Polynomial of n,L such that
A2: b = HT (p,T) and
A3: p in P -Ideal and
A4: p <> 0_ (n,L) ;
reconsider p = p as non-zero Polynomial of n,L by A4, POLYNOM7:def_1;
p is_top_reducible_wrt P,T by A1, A3;
then consider u being Polynomial of n,L such that
A5: u in P and
A6: p is_top_reducible_wrt u,T by POLYRED:def_12;
consider q being Polynomial of n,L such that
A7: p top_reduces_to q,u,T by A6, POLYRED:def_11;
A8: p reduces_to q,u, HT (p,T),T by A7, POLYRED:def_10;
then u <> 0_ (n,L) by POLYRED:def_5;
then A9: HT (u,T) in { (HT (r,T)) where r is Polynomial of n,L : ( r in P & r <> 0_ (n,L) ) } by A5;
ex s being bag of n st
( s + (HT (u,T)) = HT (p,T) & q = p - (((p . (HT (p,T))) / (HC (u,T))) * (s *' u)) ) by A8, POLYRED:def_5;
hence ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) by A2, A9, PRE_POLY:50; ::_thesis: verum
end;
hence for b being bag of n st b in HT ((P -Ideal),T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) ; ::_thesis: verum
end;
theorem :: GROEB_1:19
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st ( for b being bag of n st b in HT ((P -Ideal),T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) ) holds
HT ((P -Ideal),T) c= multiples (HT (P,T))
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st ( for b being bag of n st b in HT ((P -Ideal),T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) ) holds
HT ((P -Ideal),T) c= multiples (HT (P,T))
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st ( for b being bag of n st b in HT ((P -Ideal),T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) ) holds
HT ((P -Ideal),T) c= multiples (HT (P,T))
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for P being Subset of (Polynom-Ring (n,L)) st ( for b being bag of n st b in HT ((P -Ideal),T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) ) holds
HT ((P -Ideal),T) c= multiples (HT (P,T))
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: ( ( for b being bag of n st b in HT ((P -Ideal),T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) ) implies HT ((P -Ideal),T) c= multiples (HT (P,T)) )
assume A1: for b being bag of n st b in HT ((P -Ideal),T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) ; ::_thesis: HT ((P -Ideal),T) c= multiples (HT (P,T))
now__::_thesis:_for_u_being_set_st_u_in_HT_((P_-Ideal),T)_holds_
u_in_multiples_(HT_(P,T))
let u be set ; ::_thesis: ( u in HT ((P -Ideal),T) implies u in multiples (HT (P,T)) )
assume A2: u in HT ((P -Ideal),T) ; ::_thesis: u in multiples (HT (P,T))
then reconsider u9 = u as Element of Bags n ;
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides u9 ) by A1, A2;
hence u in multiples (HT (P,T)) ; ::_thesis: verum
end;
hence HT ((P -Ideal),T) c= multiples (HT (P,T)) by TARSKI:def_3; ::_thesis: verum
end;
theorem :: GROEB_1:20
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st HT ((P -Ideal),T) c= multiples (HT (P,T)) holds
PolyRedRel (P,T) is locally-confluent
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st HT ((P -Ideal),T) c= multiples (HT (P,T)) holds
PolyRedRel (P,T) is locally-confluent
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) st HT ((P -Ideal),T) c= multiples (HT (P,T)) holds
PolyRedRel (P,T) is locally-confluent
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for P being Subset of (Polynom-Ring (n,L)) st HT ((P -Ideal),T) c= multiples (HT (P,T)) holds
PolyRedRel (P,T) is locally-confluent
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: ( HT ((P -Ideal),T) c= multiples (HT (P,T)) implies PolyRedRel (P,T) is locally-confluent )
set R = PolyRedRel (P,T);
assume A1: HT ((P -Ideal),T) c= multiples (HT (P,T)) ; ::_thesis: PolyRedRel (P,T) is locally-confluent
A2: for f being Polynomial of n,L st f in P -Ideal & f <> 0_ (n,L) holds
f is_reducible_wrt P,T
proof
let f be Polynomial of n,L; ::_thesis: ( f in P -Ideal & f <> 0_ (n,L) implies f is_reducible_wrt P,T )
assume that
A3: f in P -Ideal and
A4: f <> 0_ (n,L) ; ::_thesis: f is_reducible_wrt P,T
HT (f,T) in { (HT (p,T)) where p is Polynomial of n,L : ( p in P -Ideal & p <> 0_ (n,L) ) } by A3, A4;
then HT (f,T) in multiples (HT (P,T)) by A1;
then ex b being Element of Bags n st
( b = HT (f,T) & ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) ) ;
then consider b9 being bag of n such that
A5: b9 in HT (P,T) and
A6: b9 divides HT (f,T) ;
consider p being Polynomial of n,L such that
A7: b9 = HT (p,T) and
A8: p in P and
A9: p <> 0_ (n,L) by A5;
consider s being bag of n such that
A10: b9 + s = HT (f,T) by A6, TERMORD:1;
set g = f - (((f . (HT (f,T))) / (HC (p,T))) * (s *' p));
Support f <> {} by A4, POLYNOM7:1;
then HT (f,T) in Support f by TERMORD:def_6;
then f reduces_to f - (((f . (HT (f,T))) / (HC (p,T))) * (s *' p)),p, HT (f,T),T by A4, A7, A9, A10, POLYRED:def_5;
then f reduces_to f - (((f . (HT (f,T))) / (HC (p,T))) * (s *' p)),p,T by POLYRED:def_6;
then f reduces_to f - (((f . (HT (f,T))) / (HC (p,T))) * (s *' p)),P,T by A8, POLYRED:def_7;
hence f is_reducible_wrt P,T by POLYRED:def_9; ::_thesis: verum
end;
A11: for f being Polynomial of n,L st f in P -Ideal holds
PolyRedRel (P,T) reduces f, 0_ (n,L)
proof
let f be Polynomial of n,L; ::_thesis: ( f in P -Ideal implies PolyRedRel (P,T) reduces f, 0_ (n,L) )
assume A12: f in P -Ideal ; ::_thesis: PolyRedRel (P,T) reduces f, 0_ (n,L)
percases ( f = 0_ (n,L) or f <> 0_ (n,L) ) ;
suppose f = 0_ (n,L) ; ::_thesis: PolyRedRel (P,T) reduces f, 0_ (n,L)
hence PolyRedRel (P,T) reduces f, 0_ (n,L) by REWRITE1:12; ::_thesis: verum
end;
suppose f <> 0_ (n,L) ; ::_thesis: PolyRedRel (P,T) reduces f, 0_ (n,L)
then f is_reducible_wrt P,T by A2, A12;
then consider v being Polynomial of n,L such that
A13: f reduces_to v,P,T by POLYRED:def_9;
[f,v] in PolyRedRel (P,T) by A13, POLYRED:def_13;
then f in field (PolyRedRel (P,T)) by RELAT_1:15;
then f has_a_normal_form_wrt PolyRedRel (P,T) by REWRITE1:def_14;
then consider g being set such that
A14: g is_a_normal_form_of f, PolyRedRel (P,T) by REWRITE1:def_11;
A15: PolyRedRel (P,T) reduces f,g by A14, REWRITE1:def_6;
then reconsider g9 = g as Polynomial of n,L by Lm4;
reconsider ff = f, gg = g9 as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider ff = ff, gg = gg as Element of (Polynom-Ring (n,L)) ;
f - g9 = ff - gg by Lm2;
then ff - gg in P -Ideal by A15, POLYRED:59;
then A16: (ff - gg) - ff in P -Ideal by A12, IDEAL_1:16;
(ff - gg) - ff = (ff + (- gg)) - ff by RLVECT_1:def_11
.= (ff + (- gg)) + (- ff) by RLVECT_1:def_11
.= (ff + (- ff)) + (- gg) by RLVECT_1:def_3
.= (0. (Polynom-Ring (n,L))) + (- gg) by RLVECT_1:5
.= - gg by ALGSTR_1:def_2 ;
then - (- gg) in P -Ideal by A16, IDEAL_1:14;
then A17: g in P -Ideal by RLVECT_1:17;
assume not PolyRedRel (P,T) reduces f, 0_ (n,L) ; ::_thesis: contradiction
then g <> 0_ (n,L) by A14, REWRITE1:def_6;
then g9 is_reducible_wrt P,T by A2, A17;
then consider u being Polynomial of n,L such that
A18: g9 reduces_to u,P,T by POLYRED:def_9;
A19: [g9,u] in PolyRedRel (P,T) by A18, POLYRED:def_13;
g is_a_normal_form_wrt PolyRedRel (P,T) by A14, REWRITE1:def_6;
hence contradiction by A19, REWRITE1:def_5; ::_thesis: verum
end;
end;
end;
now__::_thesis:_for_a,_b,_c_being_set_st_[a,b]_in_PolyRedRel_(P,T)_&_[a,c]_in_PolyRedRel_(P,T)_holds_
b,c_are_convergent_wrt_PolyRedRel_(P,T)
let a, b, c be set ; ::_thesis: ( [a,b] in PolyRedRel (P,T) & [a,c] in PolyRedRel (P,T) implies b,c are_convergent_wrt PolyRedRel (P,T) )
assume that
A20: [a,b] in PolyRedRel (P,T) and
A21: [a,c] in PolyRedRel (P,T) ; ::_thesis: b,c are_convergent_wrt PolyRedRel (P,T)
consider a9, b9 being set such that
a9 in NonZero (Polynom-Ring (n,L)) and
A22: b9 in the carrier of (Polynom-Ring (n,L)) and
A23: [a,b] = [a9,b9] by A20, ZFMISC_1:def_2;
A24: b9 = b by A23, XTUPLE_0:1;
a,b are_convertible_wrt PolyRedRel (P,T) by A20, REWRITE1:29;
then A25: b,a are_convertible_wrt PolyRedRel (P,T) by REWRITE1:31;
consider aa, c9 being set such that
aa in NonZero (Polynom-Ring (n,L)) and
A26: c9 in the carrier of (Polynom-Ring (n,L)) and
A27: [a,c] = [aa,c9] by A21, ZFMISC_1:def_2;
A28: c9 = c by A27, XTUPLE_0:1;
reconsider b9 = b9, c9 = c9 as Polynomial of n,L by A22, A26, POLYNOM1:def_10;
reconsider bb = b9, cc = c9 as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider bb = bb, cc = cc as Element of (Polynom-Ring (n,L)) ;
a,c are_convertible_wrt PolyRedRel (P,T) by A21, REWRITE1:29;
then bb,cc are_congruent_mod P -Ideal by A24, A28, A25, POLYRED:57, REWRITE1:30;
then A29: bb - cc in P -Ideal by POLYRED:def_14;
b9 - c9 = bb - cc by Lm2;
hence b,c are_convergent_wrt PolyRedRel (P,T) by A11, A24, A28, A29, POLYRED:50; ::_thesis: verum
end;
hence PolyRedRel (P,T) is locally-confluent by REWRITE1:def_24; ::_thesis: verum
end;
definition
let n be Ordinal;
let T be connected TermOrder of n;
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ;
let G be Subset of (Polynom-Ring (n,L));
predG is_Groebner_basis_wrt T means :Def3: :: GROEB_1:def 3
PolyRedRel (G,T) is locally-confluent ;
end;
:: deftheorem Def3 defines is_Groebner_basis_wrt GROEB_1:def_3_:_
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for G being Subset of (Polynom-Ring (n,L)) holds
( G is_Groebner_basis_wrt T iff PolyRedRel (G,T) is locally-confluent );
definition
let n be Ordinal;
let T be connected TermOrder of n;
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ;
let G, I be Subset of (Polynom-Ring (n,L));
predG is_Groebner_basis_of I,T means :Def4: :: GROEB_1:def 4
( G -Ideal = I & PolyRedRel (G,T) is locally-confluent );
end;
:: deftheorem Def4 defines is_Groebner_basis_of GROEB_1:def_4_:_
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for G, I being Subset of (Polynom-Ring (n,L)) holds
( G is_Groebner_basis_of I,T iff ( G -Ideal = I & PolyRedRel (G,T) is locally-confluent ) );
Lm6: for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for a, b being set st a <> b & PolyRedRel (P,T) reduces a,b holds
( a is Polynomial of n,L & b is Polynomial of n,L )
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for a, b being set st a <> b & PolyRedRel (P,T) reduces a,b holds
( a is Polynomial of n,L & b is Polynomial of n,L )
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for a, b being set st a <> b & PolyRedRel (P,T) reduces a,b holds
( a is Polynomial of n,L & b is Polynomial of n,L )
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for P being Subset of (Polynom-Ring (n,L))
for a, b being set st a <> b & PolyRedRel (P,T) reduces a,b holds
( a is Polynomial of n,L & b is Polynomial of n,L )
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: for a, b being set st a <> b & PolyRedRel (P,T) reduces a,b holds
( a is Polynomial of n,L & b is Polynomial of n,L )
let f, g be set ; ::_thesis: ( f <> g & PolyRedRel (P,T) reduces f,g implies ( f is Polynomial of n,L & g is Polynomial of n,L ) )
set R = PolyRedRel (P,T);
assume A1: f <> g ; ::_thesis: ( not PolyRedRel (P,T) reduces f,g or ( f is Polynomial of n,L & g is Polynomial of n,L ) )
assume PolyRedRel (P,T) reduces f,g ; ::_thesis: ( f is Polynomial of n,L & g is Polynomial of n,L )
then consider p being RedSequence of PolyRedRel (P,T) such that
A2: p . 1 = f and
A3: p . (len p) = g by REWRITE1:def_3;
reconsider l = (len p) - 1 as Element of NAT by INT_1:5, NAT_1:14;
set q = p . (1 + 1);
set h = p . l;
A4: 1 <= len p by NAT_1:14;
now__::_thesis:_(_(_len_p_=_1_&_f_is_Polynomial_of_n,L_)_or_(_len_p_<>_1_&_f_is_Polynomial_of_n,L_)_)
percases ( len p = 1 or len p <> 1 ) ;
case len p = 1 ; ::_thesis: f is Polynomial of n,L
hence f is Polynomial of n,L by A1, A2, A3; ::_thesis: verum
end;
case len p <> 1 ; ::_thesis: f is Polynomial of n,L
then 1 < len p by A4, XXREAL_0:1;
then 1 + 1 <= len p by NAT_1:13;
then 1 + 1 in Seg (len p) by FINSEQ_1:1;
then A5: 1 + 1 in dom p by FINSEQ_1:def_3;
1 in Seg (len p) by A4, FINSEQ_1:1;
then 1 in dom p by FINSEQ_1:def_3;
then [f,(p . (1 + 1))] in PolyRedRel (P,T) by A2, A5, REWRITE1:def_2;
then consider h9, g9 being set such that
A6: [f,(p . (1 + 1))] = [h9,g9] and
A7: h9 in NonZero (Polynom-Ring (n,L)) and
g9 in the carrier of (Polynom-Ring (n,L)) by RELSET_1:2;
f = h9 by A6, XTUPLE_0:1;
hence f is Polynomial of n,L by A7, POLYNOM1:def_10; ::_thesis: verum
end;
end;
end;
hence f is Polynomial of n,L ; ::_thesis: g is Polynomial of n,L
1 <= l + 1 by NAT_1:12;
then l + 1 in Seg (len p) by FINSEQ_1:1;
then A8: l + 1 in dom p by FINSEQ_1:def_3;
now__::_thesis:_(_(_len_p_=_1_&_g_is_Polynomial_of_n,L_)_or_(_len_p_<>_1_&_g_is_Polynomial_of_n,L_)_)
percases ( len p = 1 or len p <> 1 ) ;
case len p = 1 ; ::_thesis: g is Polynomial of n,L
hence g is Polynomial of n,L by A1, A2, A3; ::_thesis: verum
end;
case len p <> 1 ; ::_thesis: g is Polynomial of n,L
then 0 + 1 < l + 1 by A4, XXREAL_0:1;
then A9: 1 <= l by NAT_1:13;
l <= l + 1 by NAT_1:13;
then l in Seg (len p) by A9, FINSEQ_1:1;
then l in dom p by FINSEQ_1:def_3;
then [(p . l),g] in PolyRedRel (P,T) by A3, A8, REWRITE1:def_2;
then consider h9, g9 being set such that
A10: [(p . l),g] = [h9,g9] and
h9 in NonZero (Polynom-Ring (n,L)) and
A11: g9 in the carrier of (Polynom-Ring (n,L)) by RELSET_1:2;
g = g9 by A10, XTUPLE_0:1;
hence g is Polynomial of n,L by A11, POLYNOM1:def_10; ::_thesis: verum
end;
end;
end;
hence g is Polynomial of n,L ; ::_thesis: verum
end;
theorem :: GROEB_1:21
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for G, P being non empty Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_of P,T holds
PolyRedRel (G,T) is Completion of PolyRedRel (P,T)
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for G, P being non empty Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_of P,T holds
PolyRedRel (G,T) is Completion of PolyRedRel (P,T)
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for G, P being non empty Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_of P,T holds
PolyRedRel (G,T) is Completion of PolyRedRel (P,T)
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for G, P being non empty Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_of P,T holds
PolyRedRel (G,T) is Completion of PolyRedRel (P,T)
let G, P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: ( G is_Groebner_basis_of P,T implies PolyRedRel (G,T) is Completion of PolyRedRel (P,T) )
set R = PolyRedRel (P,T);
assume A1: G is_Groebner_basis_of P,T ; ::_thesis: PolyRedRel (G,T) is Completion of PolyRedRel (P,T)
then PolyRedRel (G,T) is locally-confluent by Def4;
then reconsider RG = PolyRedRel (G,T) as complete Relation ;
for a, b being set holds
( a,b are_convertible_wrt PolyRedRel (P,T) iff a,b are_convergent_wrt RG )
proof
let a, b be set ; ::_thesis: ( a,b are_convertible_wrt PolyRedRel (P,T) iff a,b are_convergent_wrt RG )
A2: G -Ideal = P by A1, Def4;
A3: now__::_thesis:_(_a,b_are_convertible_wrt_PolyRedRel_(P,T)_implies_a,b_are_convergent_wrt_RG_)
assume A4: a,b are_convertible_wrt PolyRedRel (P,T) ; ::_thesis: a,b are_convergent_wrt RG
now__::_thesis:_(_(_a_=_b_&_a,b_are_convergent_wrt_RG_)_or_(_a_<>_b_&_a,b_are_convergent_wrt_RG_)_)
percases ( a = b or a <> b ) ;
case a = b ; ::_thesis: a,b are_convergent_wrt RG
hence a,b are_convergent_wrt RG by REWRITE1:38; ::_thesis: verum
end;
caseA5: a <> b ; ::_thesis: a,b are_convergent_wrt RG
(PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) reduces a,b by A4, REWRITE1:def_4;
then consider p being RedSequence of (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) such that
A6: p . 1 = a and
A7: p . (len p) = b by REWRITE1:def_3;
reconsider l = (len p) - 1 as Element of NAT by INT_1:5, NAT_1:14;
A8: 1 <= len p by NAT_1:14;
set h = p . l;
set g = p . (1 + 1);
1 <= l + 1 by NAT_1:12;
then l + 1 in Seg (len p) by FINSEQ_1:1;
then A9: l + 1 in dom p by FINSEQ_1:def_3;
now__::_thesis:_(_(_len_p_=_1_&_a_is_Polynomial_of_n,L_&_b_is_Polynomial_of_n,L_)_or_(_len_p_<>_1_&_a_is_Polynomial_of_n,L_&_b_is_Polynomial_of_n,L_)_)
percases ( len p = 1 or len p <> 1 ) ;
case len p = 1 ; ::_thesis: ( a is Polynomial of n,L & b is Polynomial of n,L )
hence ( a is Polynomial of n,L & b is Polynomial of n,L ) by A5, A6, A7; ::_thesis: verum
end;
caseA10: len p <> 1 ; ::_thesis: ( a is Polynomial of n,L & b is Polynomial of n,L )
then 0 + 1 < l + 1 by A8, XXREAL_0:1;
then A11: 1 <= l by NAT_1:13;
l <= l + 1 by NAT_1:13;
then l in Seg (len p) by A11, FINSEQ_1:1;
then l in dom p by FINSEQ_1:def_3;
then A12: [(p . l),b] in (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) by A7, A9, REWRITE1:def_2;
A13: now__::_thesis:_(_(_[(p_._l),b]_in_PolyRedRel_(P,T)_&_b_is_Polynomial_of_n,L_)_or_(_[(p_._l),b]_in_(PolyRedRel_(P,T))_~_&_b_is_Polynomial_of_n,L_)_)
percases ( [(p . l),b] in PolyRedRel (P,T) or [(p . l),b] in (PolyRedRel (P,T)) ~ ) by A12, XBOOLE_0:def_3;
case [(p . l),b] in PolyRedRel (P,T) ; ::_thesis: b is Polynomial of n,L
then consider h9, b9 being set such that
A14: [(p . l),b] = [h9,b9] and
h9 in NonZero (Polynom-Ring (n,L)) and
A15: b9 in the carrier of (Polynom-Ring (n,L)) by RELSET_1:2;
b = b9 by A14, XTUPLE_0:1;
hence b is Polynomial of n,L by A15, POLYNOM1:def_10; ::_thesis: verum
end;
case [(p . l),b] in (PolyRedRel (P,T)) ~ ; ::_thesis: b is Polynomial of n,L
then [b,(p . l)] in PolyRedRel (P,T) by RELAT_1:def_7;
then consider h9, b9 being set such that
A16: [b,(p . l)] = [h9,b9] and
A17: h9 in NonZero (Polynom-Ring (n,L)) and
b9 in the carrier of (Polynom-Ring (n,L)) by RELSET_1:2;
b = h9 by A16, XTUPLE_0:1;
hence b is Polynomial of n,L by A17, POLYNOM1:def_10; ::_thesis: verum
end;
end;
end;
1 < len p by A8, A10, XXREAL_0:1;
then 1 + 1 <= len p by NAT_1:13;
then 1 + 1 in Seg (len p) by FINSEQ_1:1;
then A18: 1 + 1 in dom p by FINSEQ_1:def_3;
1 in Seg (len p) by A8, FINSEQ_1:1;
then 1 in dom p by FINSEQ_1:def_3;
then A19: [a,(p . (1 + 1))] in (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) by A6, A18, REWRITE1:def_2;
now__::_thesis:_(_(_[a,(p_._(1_+_1))]_in_PolyRedRel_(P,T)_&_a_is_Polynomial_of_n,L_)_or_(_[a,(p_._(1_+_1))]_in_(PolyRedRel_(P,T))_~_&_a_is_Polynomial_of_n,L_)_)
percases ( [a,(p . (1 + 1))] in PolyRedRel (P,T) or [a,(p . (1 + 1))] in (PolyRedRel (P,T)) ~ ) by A19, XBOOLE_0:def_3;
case [a,(p . (1 + 1))] in PolyRedRel (P,T) ; ::_thesis: a is Polynomial of n,L
then consider h9, b9 being set such that
A20: [a,(p . (1 + 1))] = [h9,b9] and
A21: h9 in NonZero (Polynom-Ring (n,L)) and
b9 in the carrier of (Polynom-Ring (n,L)) by RELSET_1:2;
a = h9 by A20, XTUPLE_0:1;
hence a is Polynomial of n,L by A21, POLYNOM1:def_10; ::_thesis: verum
end;
case [a,(p . (1 + 1))] in (PolyRedRel (P,T)) ~ ; ::_thesis: a is Polynomial of n,L
then [(p . (1 + 1)),a] in PolyRedRel (P,T) by RELAT_1:def_7;
then consider h9, b9 being set such that
A22: [(p . (1 + 1)),a] = [h9,b9] and
h9 in NonZero (Polynom-Ring (n,L)) and
A23: b9 in the carrier of (Polynom-Ring (n,L)) by RELSET_1:2;
a = b9 by A22, XTUPLE_0:1;
hence a is Polynomial of n,L by A23, POLYNOM1:def_10; ::_thesis: verum
end;
end;
end;
hence ( a is Polynomial of n,L & b is Polynomial of n,L ) by A13; ::_thesis: verum
end;
end;
end;
then reconsider a9 = a, b9 = b as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider a9 = a9, b9 = b9 as Element of (Polynom-Ring (n,L)) ;
G -Ideal = P -Ideal by A2, IDEAL_1:44;
then a9,b9 are_congruent_mod G -Ideal by A4, POLYRED:57;
then a9,b9 are_convertible_wrt RG by POLYRED:58;
hence a,b are_convergent_wrt RG by REWRITE1:def_23; ::_thesis: verum
end;
end;
end;
hence a,b are_convergent_wrt RG ; ::_thesis: verum
end;
now__::_thesis:_(_a,b_are_convergent_wrt_RG_implies_a,b_are_convertible_wrt_PolyRedRel_(P,T)_)
assume A24: a,b are_convergent_wrt RG ; ::_thesis: a,b are_convertible_wrt PolyRedRel (P,T)
now__::_thesis:_(_(_a_=_b_&_a,b_are_convertible_wrt_PolyRedRel_(P,T)_)_or_(_a_<>_b_&_a,b_are_convertible_wrt_PolyRedRel_(P,T)_)_)
percases ( a = b or a <> b ) ;
case a = b ; ::_thesis: a,b are_convertible_wrt PolyRedRel (P,T)
hence a,b are_convertible_wrt PolyRedRel (P,T) by REWRITE1:26; ::_thesis: verum
end;
caseA25: a <> b ; ::_thesis: a,b are_convertible_wrt PolyRedRel (P,T)
consider c being set such that
A26: RG reduces a,c and
A27: RG reduces b,c by A24, REWRITE1:def_7;
( a is Polynomial of n,L & b is Polynomial of n,L )
proof
now__::_thesis:_(_(_a_=_c_&_a_is_Polynomial_of_n,L_&_b_is_Polynomial_of_n,L_)_or_(_a_<>_c_&_a_is_Polynomial_of_n,L_&_b_is_Polynomial_of_n,L_)_)
percases ( a = c or a <> c ) ;
case a = c ; ::_thesis: ( a is Polynomial of n,L & b is Polynomial of n,L )
hence ( a is Polynomial of n,L & b is Polynomial of n,L ) by A25, A27, Lm6; ::_thesis: verum
end;
caseA28: a <> c ; ::_thesis: ( a is Polynomial of n,L & b is Polynomial of n,L )
now__::_thesis:_(_(_b_=_c_&_a_is_Polynomial_of_n,L_&_b_is_Polynomial_of_n,L_)_or_(_b_<>_c_&_b_is_Polynomial_of_n,L_)_)
percases ( b = c or b <> c ) ;
case b = c ; ::_thesis: ( a is Polynomial of n,L & b is Polynomial of n,L )
hence ( a is Polynomial of n,L & b is Polynomial of n,L ) by A25, A26, Lm6; ::_thesis: verum
end;
case b <> c ; ::_thesis: b is Polynomial of n,L
hence b is Polynomial of n,L by A27, Lm6; ::_thesis: verum
end;
end;
end;
hence ( a is Polynomial of n,L & b is Polynomial of n,L ) by A26, A28, Lm6; ::_thesis: verum
end;
end;
end;
hence ( a is Polynomial of n,L & b is Polynomial of n,L ) ; ::_thesis: verum
end;
then reconsider a9 = a, b9 = b as Element of the carrier of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider a9 = a9, b9 = b9 as Element of (Polynom-Ring (n,L)) ;
( G -Ideal = P -Ideal & a9,b9 are_convertible_wrt RG ) by A2, A24, IDEAL_1:44, REWRITE1:37;
then a9,b9 are_congruent_mod P -Ideal by POLYRED:57;
hence a,b are_convertible_wrt PolyRedRel (P,T) by POLYRED:58; ::_thesis: verum
end;
end;
end;
hence a,b are_convertible_wrt PolyRedRel (P,T) ; ::_thesis: verum
end;
hence ( a,b are_convertible_wrt PolyRedRel (P,T) iff a,b are_convergent_wrt RG ) by A3; ::_thesis: verum
end;
hence PolyRedRel (G,T) is Completion of PolyRedRel (P,T) by REWRITE1:def_28; ::_thesis: verum
end;
theorem :: GROEB_1:22
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p, q being Element of (Polynom-Ring (n,L))
for G being non empty Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_wrt T holds
( p,q are_congruent_mod G -Ideal iff nf (p,(PolyRedRel (G,T))) = nf (q,(PolyRedRel (G,T))) )
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p, q being Element of (Polynom-Ring (n,L))
for G being non empty Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_wrt T holds
( p,q are_congruent_mod G -Ideal iff nf (p,(PolyRedRel (G,T))) = nf (q,(PolyRedRel (G,T))) )
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p, q being Element of (Polynom-Ring (n,L))
for G being non empty Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_wrt T holds
( p,q are_congruent_mod G -Ideal iff nf (p,(PolyRedRel (G,T))) = nf (q,(PolyRedRel (G,T))) )
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for p, q being Element of (Polynom-Ring (n,L))
for G being non empty Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_wrt T holds
( p,q are_congruent_mod G -Ideal iff nf (p,(PolyRedRel (G,T))) = nf (q,(PolyRedRel (G,T))) )
let p, q be Element of (Polynom-Ring (n,L)); ::_thesis: for G being non empty Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_wrt T holds
( p,q are_congruent_mod G -Ideal iff nf (p,(PolyRedRel (G,T))) = nf (q,(PolyRedRel (G,T))) )
let G be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: ( G is_Groebner_basis_wrt T implies ( p,q are_congruent_mod G -Ideal iff nf (p,(PolyRedRel (G,T))) = nf (q,(PolyRedRel (G,T))) ) )
set R = PolyRedRel (G,T);
assume G is_Groebner_basis_wrt T ; ::_thesis: ( p,q are_congruent_mod G -Ideal iff nf (p,(PolyRedRel (G,T))) = nf (q,(PolyRedRel (G,T))) )
then A1: PolyRedRel (G,T) is locally-confluent by Def3;
now__::_thesis:_(_nf_(p,(PolyRedRel_(G,T)))_=_nf_(q,(PolyRedRel_(G,T)))_implies_p,q_are_congruent_mod_G_-Ideal_)
nf (q,(PolyRedRel (G,T))) is_a_normal_form_of q, PolyRedRel (G,T) by A1, REWRITE1:54;
then PolyRedRel (G,T) reduces q, nf (q,(PolyRedRel (G,T))) by REWRITE1:def_6;
then A2: nf (q,(PolyRedRel (G,T))),q are_convertible_wrt PolyRedRel (G,T) by REWRITE1:25;
nf (p,(PolyRedRel (G,T))) is_a_normal_form_of p, PolyRedRel (G,T) by A1, REWRITE1:54;
then PolyRedRel (G,T) reduces p, nf (p,(PolyRedRel (G,T))) by REWRITE1:def_6;
then A3: p, nf (p,(PolyRedRel (G,T))) are_convertible_wrt PolyRedRel (G,T) by REWRITE1:25;
assume nf (p,(PolyRedRel (G,T))) = nf (q,(PolyRedRel (G,T))) ; ::_thesis: p,q are_congruent_mod G -Ideal
hence p,q are_congruent_mod G -Ideal by A3, A2, POLYRED:57, REWRITE1:30; ::_thesis: verum
end;
hence ( p,q are_congruent_mod G -Ideal iff nf (p,(PolyRedRel (G,T))) = nf (q,(PolyRedRel (G,T))) ) by A1, POLYRED:58, REWRITE1:55; ::_thesis: verum
end;
theorem :: GROEB_1:23
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f being Polynomial of n,L
for P being non empty Subset of (Polynom-Ring (n,L)) st P is_Groebner_basis_wrt T holds
( f in P -Ideal iff PolyRedRel (P,T) reduces f, 0_ (n,L) )
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f being Polynomial of n,L
for P being non empty Subset of (Polynom-Ring (n,L)) st P is_Groebner_basis_wrt T holds
( f in P -Ideal iff PolyRedRel (P,T) reduces f, 0_ (n,L) )
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f being Polynomial of n,L
for P being non empty Subset of (Polynom-Ring (n,L)) st P is_Groebner_basis_wrt T holds
( f in P -Ideal iff PolyRedRel (P,T) reduces f, 0_ (n,L) )
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for f being Polynomial of n,L
for P being non empty Subset of (Polynom-Ring (n,L)) st P is_Groebner_basis_wrt T holds
( f in P -Ideal iff PolyRedRel (P,T) reduces f, 0_ (n,L) )
let f be Polynomial of n,L; ::_thesis: for P being non empty Subset of (Polynom-Ring (n,L)) st P is_Groebner_basis_wrt T holds
( f in P -Ideal iff PolyRedRel (P,T) reduces f, 0_ (n,L) )
let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: ( P is_Groebner_basis_wrt T implies ( f in P -Ideal iff PolyRedRel (P,T) reduces f, 0_ (n,L) ) )
assume P is_Groebner_basis_wrt T ; ::_thesis: ( f in P -Ideal iff PolyRedRel (P,T) reduces f, 0_ (n,L) )
then PolyRedRel (P,T) is locally-confluent by Def3;
hence ( f in P -Ideal iff PolyRedRel (P,T) reduces f, 0_ (n,L) ) by Th15, POLYRED:60; ::_thesis: verum
end;
Lm7: for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being LeftIdeal of (Polynom-Ring (n,L))
for G being non empty Subset of (Polynom-Ring (n,L)) st G c= I & ( for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L) ) holds
G -Ideal = I
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being LeftIdeal of (Polynom-Ring (n,L))
for G being non empty Subset of (Polynom-Ring (n,L)) st G c= I & ( for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L) ) holds
G -Ideal = I
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being LeftIdeal of (Polynom-Ring (n,L))
for G being non empty Subset of (Polynom-Ring (n,L)) st G c= I & ( for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L) ) holds
G -Ideal = I
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for I being LeftIdeal of (Polynom-Ring (n,L))
for G being non empty Subset of (Polynom-Ring (n,L)) st G c= I & ( for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L) ) holds
G -Ideal = I
let I be LeftIdeal of (Polynom-Ring (n,L)); ::_thesis: for G being non empty Subset of (Polynom-Ring (n,L)) st G c= I & ( for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L) ) holds
G -Ideal = I
let G be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: ( G c= I & ( for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L) ) implies G -Ideal = I )
assume A1: G c= I ; ::_thesis: ( ex f being Polynomial of n,L st
( f in I & not PolyRedRel (G,T) reduces f, 0_ (n,L) ) or G -Ideal = I )
A2: now__::_thesis:_for_u_being_set_st_u_in_G_-Ideal_holds_
u_in_I
let u be set ; ::_thesis: ( u in G -Ideal implies u in I )
assume A3: u in G -Ideal ; ::_thesis: u in I
G -Ideal c= I by A1, IDEAL_1:def_14;
hence u in I by A3; ::_thesis: verum
end;
assume A4: for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L) ; ::_thesis: G -Ideal = I
now__::_thesis:_for_u_being_set_st_u_in_I_holds_
u_in_G_-Ideal
let u be set ; ::_thesis: ( u in I implies u in G -Ideal )
assume A5: u in I ; ::_thesis: u in G -Ideal
then reconsider u9 = u as Element of (Polynom-Ring (n,L)) ;
reconsider u9 = u9 as Polynomial of n,L by POLYNOM1:def_10;
PolyRedRel (G,T) reduces u9, 0_ (n,L) by A4, A5;
hence u in G -Ideal by POLYRED:60; ::_thesis: verum
end;
hence G -Ideal = I by A2, TARSKI:1; ::_thesis: verum
end;
theorem Th24: :: GROEB_1:24
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being Subset of (Polynom-Ring (n,L))
for G being non empty Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_of I,T holds
for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L)
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being Subset of (Polynom-Ring (n,L))
for G being non empty Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_of I,T holds
for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L)
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being Subset of (Polynom-Ring (n,L))
for G being non empty Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_of I,T holds
for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L)
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for I being Subset of (Polynom-Ring (n,L))
for G being non empty Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_of I,T holds
for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L)
let I be Subset of (Polynom-Ring (n,L)); ::_thesis: for G being non empty Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_of I,T holds
for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L)
let G be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: ( G is_Groebner_basis_of I,T implies for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L) )
assume G is_Groebner_basis_of I,T ; ::_thesis: for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L)
then ( G -Ideal = I & PolyRedRel (G,T) is locally-confluent ) by Def4;
hence for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L) by Th15; ::_thesis: verum
end;
theorem Th25: :: GROEB_1:25
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for G, I being Subset of (Polynom-Ring (n,L)) st ( for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L) ) holds
for f being non-zero Polynomial of n,L st f in I holds
f is_reducible_wrt G,T
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for G, I being Subset of (Polynom-Ring (n,L)) st ( for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L) ) holds
for f being non-zero Polynomial of n,L st f in I holds
f is_reducible_wrt G,T
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for G, I being Subset of (Polynom-Ring (n,L)) st ( for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L) ) holds
for f being non-zero Polynomial of n,L st f in I holds
f is_reducible_wrt G,T
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for G, I being Subset of (Polynom-Ring (n,L)) st ( for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L) ) holds
for f being non-zero Polynomial of n,L st f in I holds
f is_reducible_wrt G,T
let G, I be Subset of (Polynom-Ring (n,L)); ::_thesis: ( ( for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L) ) implies for f being non-zero Polynomial of n,L st f in I holds
f is_reducible_wrt G,T )
assume A1: for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L) ; ::_thesis: for f being non-zero Polynomial of n,L st f in I holds
f is_reducible_wrt G,T
now__::_thesis:_for_f_being_non-zero_Polynomial_of_n,L_st_f_in_I_holds_
f_is_reducible_wrt_G,T
let f be non-zero Polynomial of n,L; ::_thesis: ( f in I implies f is_reducible_wrt G,T )
assume f in I ; ::_thesis: f is_reducible_wrt G,T
then A2: PolyRedRel (G,T) reduces f, 0_ (n,L) by A1;
f <> 0_ (n,L) by POLYNOM7:def_1;
then ex g being Polynomial of n,L st
( f reduces_to g,G,T & PolyRedRel (G,T) reduces g, 0_ (n,L) ) by A2, Lm5;
hence f is_reducible_wrt G,T by POLYRED:def_9; ::_thesis: verum
end;
hence for f being non-zero Polynomial of n,L st f in I holds
f is_reducible_wrt G,T ; ::_thesis: verum
end;
theorem Th26: :: GROEB_1:26
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being add-closed left-ideal Subset of (Polynom-Ring (n,L))
for G being Subset of (Polynom-Ring (n,L)) st G c= I & ( for f being non-zero Polynomial of n,L st f in I holds
f is_reducible_wrt G,T ) holds
for f being non-zero Polynomial of n,L st f in I holds
f is_top_reducible_wrt G,T
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being add-closed left-ideal Subset of (Polynom-Ring (n,L))
for G being Subset of (Polynom-Ring (n,L)) st G c= I & ( for f being non-zero Polynomial of n,L st f in I holds
f is_reducible_wrt G,T ) holds
for f being non-zero Polynomial of n,L st f in I holds
f is_top_reducible_wrt G,T
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being add-closed left-ideal Subset of (Polynom-Ring (n,L))
for G being Subset of (Polynom-Ring (n,L)) st G c= I & ( for f being non-zero Polynomial of n,L st f in I holds
f is_reducible_wrt G,T ) holds
for f being non-zero Polynomial of n,L st f in I holds
f is_top_reducible_wrt G,T
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for I being add-closed left-ideal Subset of (Polynom-Ring (n,L))
for G being Subset of (Polynom-Ring (n,L)) st G c= I & ( for f being non-zero Polynomial of n,L st f in I holds
f is_reducible_wrt G,T ) holds
for f being non-zero Polynomial of n,L st f in I holds
f is_top_reducible_wrt G,T
let I be add-closed left-ideal Subset of (Polynom-Ring (n,L)); ::_thesis: for G being Subset of (Polynom-Ring (n,L)) st G c= I & ( for f being non-zero Polynomial of n,L st f in I holds
f is_reducible_wrt G,T ) holds
for f being non-zero Polynomial of n,L st f in I holds
f is_top_reducible_wrt G,T
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: ( P c= I & ( for f being non-zero Polynomial of n,L st f in I holds
f is_reducible_wrt P,T ) implies for f being non-zero Polynomial of n,L st f in I holds
f is_top_reducible_wrt P,T )
assume A1: P c= I ; ::_thesis: ( ex f being non-zero Polynomial of n,L st
( f in I & not f is_reducible_wrt P,T ) or for f being non-zero Polynomial of n,L st f in I holds
f is_top_reducible_wrt P,T )
assume A2: for f being non-zero Polynomial of n,L st f in I holds
f is_reducible_wrt P,T ; ::_thesis: for f being non-zero Polynomial of n,L st f in I holds
f is_top_reducible_wrt P,T
thus for f being non-zero Polynomial of n,L st f in I holds
f is_top_reducible_wrt P,T ::_thesis: verum
proof
set H = { g where g is non-zero Polynomial of n,L : ( g in I & not g is_top_reducible_wrt P,T ) } ;
let f be non-zero Polynomial of n,L; ::_thesis: ( f in I implies f is_top_reducible_wrt P,T )
assume A3: f in I ; ::_thesis: f is_top_reducible_wrt P,T
assume not f is_top_reducible_wrt P,T ; ::_thesis: contradiction
then A4: f in { g where g is non-zero Polynomial of n,L : ( g in I & not g is_top_reducible_wrt P,T ) } by A3;
now__::_thesis:_for_u_being_set_st_u_in__{__g_where_g_is_non-zero_Polynomial_of_n,L_:_(_g_in_I_&_not_g_is_top_reducible_wrt_P,T_)__}__holds_
u_in_the_carrier_of_(Polynom-Ring_(n,L))
let u be set ; ::_thesis: ( u in { g where g is non-zero Polynomial of n,L : ( g in I & not g is_top_reducible_wrt P,T ) } implies u in the carrier of (Polynom-Ring (n,L)) )
assume u in { g where g is non-zero Polynomial of n,L : ( g in I & not g is_top_reducible_wrt P,T ) } ; ::_thesis: u in the carrier of (Polynom-Ring (n,L))
then ex g9 being non-zero Polynomial of n,L st
( u = g9 & g9 in I & not g9 is_top_reducible_wrt P,T ) ;
hence u in the carrier of (Polynom-Ring (n,L)) ; ::_thesis: verum
end;
then reconsider H = { g where g is non-zero Polynomial of n,L : ( g in I & not g is_top_reducible_wrt P,T ) } as non empty Subset of (Polynom-Ring (n,L)) by A4, TARSKI:def_3;
consider p being Polynomial of n,L such that
A5: p in H and
A6: for q being Polynomial of n,L st q in H holds
p <= q,T by POLYRED:31;
A7: ex p9 being non-zero Polynomial of n,L st
( p9 = p & p9 in I & not p9 is_top_reducible_wrt P,T ) by A5;
then reconsider p = p as non-zero Polynomial of n,L ;
p is_reducible_wrt P,T by A2, A7;
then consider q being Polynomial of n,L such that
A8: p reduces_to q,P,T by POLYRED:def_9;
consider u being Polynomial of n,L such that
A9: u in P and
A10: p reduces_to q,u,T by A8, POLYRED:def_7;
ex b being bag of n st p reduces_to q,u,b,T by A10, POLYRED:def_6;
then A11: u <> 0_ (n,L) by POLYRED:def_5;
then reconsider u = u as non-zero Polynomial of n,L by POLYNOM7:def_1;
consider b being bag of n such that
A12: p reduces_to q,u,b,T by A10, POLYRED:def_6;
A13: now__::_thesis:_not_b_=_HT_(p,T)
assume b = HT (p,T) ; ::_thesis: contradiction
then p top_reduces_to q,u,T by A12, POLYRED:def_10;
then p is_top_reducible_wrt u,T by POLYRED:def_11;
hence contradiction by A7, A9, POLYRED:def_12; ::_thesis: verum
end;
consider m being Monomial of n,L such that
A14: q = p - (m *' u) by A10, Th1;
reconsider uu = u, pp = p, mm = m as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider uu = uu, pp = pp, mm = mm as Element of (Polynom-Ring (n,L)) ;
mm * uu in I by A1, A9, IDEAL_1:def_2;
then - (mm * uu) in I by IDEAL_1:13;
then A15: pp + (- (mm * uu)) in I by A7, IDEAL_1:def_1;
mm * uu = m *' u by POLYNOM1:def_10;
then p - (m *' u) = pp - (mm * uu) by Lm2;
then A16: q in I by A14, A15, RLVECT_1:def_11;
A17: q < p,T by A10, POLYRED:43;
A18: p <> 0_ (n,L) by POLYNOM7:def_1;
then Support p <> {} by POLYNOM7:1;
then A19: HT (p,T) in Support p by TERMORD:def_6;
b in Support p by A12, POLYRED:def_5;
then b <= HT (p,T),T by TERMORD:def_6;
then b < HT (p,T),T by A13, TERMORD:def_3;
then A20: HT (p,T) in Support q by A19, A12, POLYRED:40;
now__::_thesis:_(_(_q_<>_0__(n,L)_&_contradiction_)_or_(_q_=_0__(n,L)_&_contradiction_)_)
percases ( q <> 0_ (n,L) or q = 0_ (n,L) ) ;
caseA21: q <> 0_ (n,L) ; ::_thesis: contradiction
then reconsider q = q as non-zero Polynomial of n,L by POLYNOM7:def_1;
Support q <> {} by A21, POLYNOM7:1;
then HT (q,T) in Support q by TERMORD:def_6;
then A22: HT (q,T) <= HT (p,T),T by A10, POLYRED:42;
HT (p,T) <= HT (q,T),T by A20, TERMORD:def_6;
then A23: HT (q,T) = HT (p,T) by A22, TERMORD:7;
now__::_thesis:_q_is_top_reducible_wrt_P,T
assume not q is_top_reducible_wrt P,T ; ::_thesis: contradiction
then q in H by A16;
then p <= q,T by A6;
hence contradiction by A17, POLYRED:29; ::_thesis: verum
end;
then consider u9 being Polynomial of n,L such that
A24: u9 in P and
A25: q is_top_reducible_wrt u9,T by POLYRED:def_12;
consider q9 being Polynomial of n,L such that
A26: q top_reduces_to q9,u9,T by A25, POLYRED:def_11;
A27: p <> 0_ (n,L) by POLYNOM7:def_1;
then Support p <> {} by POLYNOM7:1;
then A28: HT (p,T) in Support p by TERMORD:def_6;
A29: q reduces_to q9,u9, HT (q,T),T by A26, POLYRED:def_10;
then consider s being bag of n such that
A30: s + (HT (u9,T)) = HT (q,T) and
q9 = q - (((q . (HT (q,T))) / (HC (u9,T))) * (s *' u9)) by POLYRED:def_5;
set qq = p - (((p . (HT (p,T))) / (HC (u9,T))) * (s *' u9));
u9 <> 0_ (n,L) by A29, POLYRED:def_5;
then p reduces_to p - (((p . (HT (p,T))) / (HC (u9,T))) * (s *' u9)),u9, HT (p,T),T by A23, A30, A27, A28, POLYRED:def_5;
then p top_reduces_to p - (((p . (HT (p,T))) / (HC (u9,T))) * (s *' u9)),u9,T by POLYRED:def_10;
then p is_top_reducible_wrt u9,T by POLYRED:def_11;
hence contradiction by A7, A24, POLYRED:def_12; ::_thesis: verum
end;
case q = 0_ (n,L) ; ::_thesis: contradiction
then A31: m *' u = (p - (m *' u)) + (m *' u) by A14, POLYRED:2
.= (p + (- (m *' u))) + (m *' u) by POLYNOM1:def_6
.= p + ((- (m *' u)) + (m *' u)) by POLYNOM1:21
.= p + (0_ (n,L)) by POLYRED:3
.= p by POLYNOM1:23 ;
now__::_thesis:_not_m_=_0__(n,L)
A32: p <> 0_ (n,L) by POLYNOM7:def_1;
assume m = 0_ (n,L) ; ::_thesis: contradiction
hence contradiction by A31, A32, POLYRED:5; ::_thesis: verum
end;
then reconsider m = m as non-zero Polynomial of n,L by POLYNOM7:def_1;
set pp = p - (((p . (HT (p,T))) / (HC (u,T))) * ((HT (m,T)) *' u));
HT (p,T) = (HT (m,T)) + (HT (u,T)) by A31, TERMORD:31;
then p reduces_to p - (((p . (HT (p,T))) / (HC (u,T))) * ((HT (m,T)) *' u)),u, HT (p,T),T by A11, A18, A19, POLYRED:def_5;
then p top_reduces_to p - (((p . (HT (p,T))) / (HC (u,T))) * ((HT (m,T)) *' u)),u,T by POLYRED:def_10;
then p is_top_reducible_wrt u,T by POLYRED:def_11;
hence contradiction by A7, A9, POLYRED:def_12; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
theorem Th27: :: GROEB_1:27
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for G, I being Subset of (Polynom-Ring (n,L)) st ( for f being non-zero Polynomial of n,L st f in I holds
f is_top_reducible_wrt G,T ) holds
for b being bag of n st b in HT (I,T) holds
ex b9 being bag of n st
( b9 in HT (G,T) & b9 divides b )
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for G, I being Subset of (Polynom-Ring (n,L)) st ( for f being non-zero Polynomial of n,L st f in I holds
f is_top_reducible_wrt G,T ) holds
for b being bag of n st b in HT (I,T) holds
ex b9 being bag of n st
( b9 in HT (G,T) & b9 divides b )
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for G, I being Subset of (Polynom-Ring (n,L)) st ( for f being non-zero Polynomial of n,L st f in I holds
f is_top_reducible_wrt G,T ) holds
for b being bag of n st b in HT (I,T) holds
ex b9 being bag of n st
( b9 in HT (G,T) & b9 divides b )
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for G, I being Subset of (Polynom-Ring (n,L)) st ( for f being non-zero Polynomial of n,L st f in I holds
f is_top_reducible_wrt G,T ) holds
for b being bag of n st b in HT (I,T) holds
ex b9 being bag of n st
( b9 in HT (G,T) & b9 divides b )
let P, I be Subset of (Polynom-Ring (n,L)); ::_thesis: ( ( for f being non-zero Polynomial of n,L st f in I holds
f is_top_reducible_wrt P,T ) implies for b being bag of n st b in HT (I,T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) )
assume A1: for f being non-zero Polynomial of n,L st f in I holds
f is_top_reducible_wrt P,T ; ::_thesis: for b being bag of n st b in HT (I,T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b )
now__::_thesis:_for_b_being_bag_of_n_st_b_in_HT_(I,T)_holds_
ex_b9_being_bag_of_n_st_
(_b9_in_HT_(P,T)_&_b9_divides_b_)
let b be bag of n; ::_thesis: ( b in HT (I,T) implies ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) )
assume b in HT (I,T) ; ::_thesis: ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b )
then consider p being Polynomial of n,L such that
A2: b = HT (p,T) and
A3: p in I and
A4: p <> 0_ (n,L) ;
reconsider p = p as non-zero Polynomial of n,L by A4, POLYNOM7:def_1;
p is_top_reducible_wrt P,T by A1, A3;
then consider u being Polynomial of n,L such that
A5: u in P and
A6: p is_top_reducible_wrt u,T by POLYRED:def_12;
consider q being Polynomial of n,L such that
A7: p top_reduces_to q,u,T by A6, POLYRED:def_11;
A8: p reduces_to q,u, HT (p,T),T by A7, POLYRED:def_10;
then u <> 0_ (n,L) by POLYRED:def_5;
then A9: HT (u,T) in { (HT (r,T)) where r is Polynomial of n,L : ( r in P & r <> 0_ (n,L) ) } by A5;
ex s being bag of n st
( s + (HT (u,T)) = HT (p,T) & q = p - (((p . (HT (p,T))) / (HC (u,T))) * (s *' u)) ) by A8, POLYRED:def_5;
hence ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) by A2, A9, PRE_POLY:50; ::_thesis: verum
end;
hence for b being bag of n st b in HT (I,T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) ; ::_thesis: verum
end;
theorem Th28: :: GROEB_1:28
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for G, I being Subset of (Polynom-Ring (n,L)) st ( for b being bag of n st b in HT (I,T) holds
ex b9 being bag of n st
( b9 in HT (G,T) & b9 divides b ) ) holds
HT (I,T) c= multiples (HT (G,T))
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for G, I being Subset of (Polynom-Ring (n,L)) st ( for b being bag of n st b in HT (I,T) holds
ex b9 being bag of n st
( b9 in HT (G,T) & b9 divides b ) ) holds
HT (I,T) c= multiples (HT (G,T))
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for G, I being Subset of (Polynom-Ring (n,L)) st ( for b being bag of n st b in HT (I,T) holds
ex b9 being bag of n st
( b9 in HT (G,T) & b9 divides b ) ) holds
HT (I,T) c= multiples (HT (G,T))
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for G, I being Subset of (Polynom-Ring (n,L)) st ( for b being bag of n st b in HT (I,T) holds
ex b9 being bag of n st
( b9 in HT (G,T) & b9 divides b ) ) holds
HT (I,T) c= multiples (HT (G,T))
let P, I be Subset of (Polynom-Ring (n,L)); ::_thesis: ( ( for b being bag of n st b in HT (I,T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) ) implies HT (I,T) c= multiples (HT (P,T)) )
assume A1: for b being bag of n st b in HT (I,T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) ; ::_thesis: HT (I,T) c= multiples (HT (P,T))
now__::_thesis:_for_u_being_set_st_u_in_HT_(I,T)_holds_
u_in_multiples_(HT_(P,T))
let u be set ; ::_thesis: ( u in HT (I,T) implies u in multiples (HT (P,T)) )
assume A2: u in HT (I,T) ; ::_thesis: u in multiples (HT (P,T))
then reconsider u9 = u as Element of Bags n ;
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides u9 ) by A1, A2;
hence u in multiples (HT (P,T)) ; ::_thesis: verum
end;
hence HT (I,T) c= multiples (HT (P,T)) by TARSKI:def_3; ::_thesis: verum
end;
theorem Th29: :: GROEB_1:29
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L))
for G being non empty Subset of (Polynom-Ring (n,L)) st G c= I & HT (I,T) c= multiples (HT (G,T)) holds
G is_Groebner_basis_of I,T
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L))
for G being non empty Subset of (Polynom-Ring (n,L)) st G c= I & HT (I,T) c= multiples (HT (G,T)) holds
G is_Groebner_basis_of I,T
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L))
for G being non empty Subset of (Polynom-Ring (n,L)) st G c= I & HT (I,T) c= multiples (HT (G,T)) holds
G is_Groebner_basis_of I,T
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L))
for G being non empty Subset of (Polynom-Ring (n,L)) st G c= I & HT (I,T) c= multiples (HT (G,T)) holds
G is_Groebner_basis_of I,T
let I be non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)); ::_thesis: for G being non empty Subset of (Polynom-Ring (n,L)) st G c= I & HT (I,T) c= multiples (HT (G,T)) holds
G is_Groebner_basis_of I,T
let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: ( P c= I & HT (I,T) c= multiples (HT (P,T)) implies P is_Groebner_basis_of I,T )
assume A1: P c= I ; ::_thesis: ( not HT (I,T) c= multiples (HT (P,T)) or P is_Groebner_basis_of I,T )
set R = PolyRedRel (P,T);
assume A2: HT (I,T) c= multiples (HT (P,T)) ; ::_thesis: P is_Groebner_basis_of I,T
A3: for f being Polynomial of n,L st f in I & f <> 0_ (n,L) holds
f is_reducible_wrt P,T
proof
let f be Polynomial of n,L; ::_thesis: ( f in I & f <> 0_ (n,L) implies f is_reducible_wrt P,T )
assume that
A4: f in I and
A5: f <> 0_ (n,L) ; ::_thesis: f is_reducible_wrt P,T
HT (f,T) in { (HT (p,T)) where p is Polynomial of n,L : ( p in I & p <> 0_ (n,L) ) } by A4, A5;
then HT (f,T) in multiples (HT (P,T)) by A2;
then ex b being Element of Bags n st
( b = HT (f,T) & ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) ) ;
then consider b9 being bag of n such that
A6: b9 in HT (P,T) and
A7: b9 divides HT (f,T) ;
consider p being Polynomial of n,L such that
A8: b9 = HT (p,T) and
A9: p in P and
A10: p <> 0_ (n,L) by A6;
consider s being bag of n such that
A11: b9 + s = HT (f,T) by A7, TERMORD:1;
set g = f - (((f . (HT (f,T))) / (HC (p,T))) * (s *' p));
Support f <> {} by A5, POLYNOM7:1;
then HT (f,T) in Support f by TERMORD:def_6;
then f reduces_to f - (((f . (HT (f,T))) / (HC (p,T))) * (s *' p)),p, HT (f,T),T by A5, A8, A10, A11, POLYRED:def_5;
then f reduces_to f - (((f . (HT (f,T))) / (HC (p,T))) * (s *' p)),p,T by POLYRED:def_6;
then f reduces_to f - (((f . (HT (f,T))) / (HC (p,T))) * (s *' p)),P,T by A9, POLYRED:def_7;
hence f is_reducible_wrt P,T by POLYRED:def_9; ::_thesis: verum
end;
A12: PolyRedRel (P,T) c= PolyRedRel (I,T) by A1, Th4;
A13: for f being Polynomial of n,L st f in I holds
PolyRedRel (P,T) reduces f, 0_ (n,L)
proof
let f be Polynomial of n,L; ::_thesis: ( f in I implies PolyRedRel (P,T) reduces f, 0_ (n,L) )
assume A14: f in I ; ::_thesis: PolyRedRel (P,T) reduces f, 0_ (n,L)
percases ( f = 0_ (n,L) or f <> 0_ (n,L) ) ;
suppose f = 0_ (n,L) ; ::_thesis: PolyRedRel (P,T) reduces f, 0_ (n,L)
hence PolyRedRel (P,T) reduces f, 0_ (n,L) by REWRITE1:12; ::_thesis: verum
end;
suppose f <> 0_ (n,L) ; ::_thesis: PolyRedRel (P,T) reduces f, 0_ (n,L)
then f is_reducible_wrt P,T by A3, A14;
then consider v being Polynomial of n,L such that
A15: f reduces_to v,P,T by POLYRED:def_9;
[f,v] in PolyRedRel (P,T) by A15, POLYRED:def_13;
then f in field (PolyRedRel (P,T)) by RELAT_1:15;
then f has_a_normal_form_wrt PolyRedRel (P,T) by REWRITE1:def_14;
then consider g being set such that
A16: g is_a_normal_form_of f, PolyRedRel (P,T) by REWRITE1:def_11;
A17: PolyRedRel (P,T) reduces f,g by A16, REWRITE1:def_6;
then reconsider g9 = g as Polynomial of n,L by Lm4;
reconsider ff = f, gg = g9 as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider ff = ff, gg = gg as Element of (Polynom-Ring (n,L)) ;
f - g9 = ff - gg by Lm2;
then ff - gg in I -Ideal by A12, A17, POLYRED:59, REWRITE1:22;
then ff - gg in I by IDEAL_1:44;
then A18: (ff - gg) - ff in I by A14, IDEAL_1:16;
(ff - gg) - ff = (ff + (- gg)) - ff by RLVECT_1:def_11
.= (ff + (- gg)) + (- ff) by RLVECT_1:def_11
.= (ff + (- ff)) + (- gg) by RLVECT_1:def_3
.= (0. (Polynom-Ring (n,L))) + (- gg) by RLVECT_1:5
.= - gg by ALGSTR_1:def_2 ;
then - (- gg) in I by A18, IDEAL_1:14;
then A19: g in I by RLVECT_1:17;
assume not PolyRedRel (P,T) reduces f, 0_ (n,L) ; ::_thesis: contradiction
then g <> 0_ (n,L) by A16, REWRITE1:def_6;
then g9 is_reducible_wrt P,T by A3, A19;
then consider u being Polynomial of n,L such that
A20: g9 reduces_to u,P,T by POLYRED:def_9;
A21: [g9,u] in PolyRedRel (P,T) by A20, POLYRED:def_13;
g is_a_normal_form_wrt PolyRedRel (P,T) by A16, REWRITE1:def_6;
hence contradiction by A21, REWRITE1:def_5; ::_thesis: verum
end;
end;
end;
then A22: P -Ideal = I by A1, Lm7;
now__::_thesis:_for_a,_b,_c_being_set_st_[a,b]_in_PolyRedRel_(P,T)_&_[a,c]_in_PolyRedRel_(P,T)_holds_
b,c_are_convergent_wrt_PolyRedRel_(P,T)
let a, b, c be set ; ::_thesis: ( [a,b] in PolyRedRel (P,T) & [a,c] in PolyRedRel (P,T) implies b,c are_convergent_wrt PolyRedRel (P,T) )
assume that
A23: [a,b] in PolyRedRel (P,T) and
A24: [a,c] in PolyRedRel (P,T) ; ::_thesis: b,c are_convergent_wrt PolyRedRel (P,T)
consider a9, b9 being set such that
a9 in NonZero (Polynom-Ring (n,L)) and
A25: b9 in the carrier of (Polynom-Ring (n,L)) and
A26: [a,b] = [a9,b9] by A23, ZFMISC_1:def_2;
A27: b9 = b by A26, XTUPLE_0:1;
a,b are_convertible_wrt PolyRedRel (P,T) by A23, REWRITE1:29;
then A28: b,a are_convertible_wrt PolyRedRel (P,T) by REWRITE1:31;
consider aa, c9 being set such that
aa in NonZero (Polynom-Ring (n,L)) and
A29: c9 in the carrier of (Polynom-Ring (n,L)) and
A30: [a,c] = [aa,c9] by A24, ZFMISC_1:def_2;
A31: c9 = c by A30, XTUPLE_0:1;
reconsider b9 = b9, c9 = c9 as Polynomial of n,L by A25, A29, POLYNOM1:def_10;
reconsider bb = b9, cc = c9 as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider bb = bb, cc = cc as Element of (Polynom-Ring (n,L)) ;
a,c are_convertible_wrt PolyRedRel (P,T) by A24, REWRITE1:29;
then bb,cc are_congruent_mod I by A22, A27, A31, A28, POLYRED:57, REWRITE1:30;
then A32: bb - cc in I by POLYRED:def_14;
b9 - c9 = bb - cc by Lm2;
hence b,c are_convergent_wrt PolyRedRel (P,T) by A13, A27, A31, A32, POLYRED:50; ::_thesis: verum
end;
then PolyRedRel (P,T) is locally-confluent by REWRITE1:def_24;
hence P is_Groebner_basis_of I,T by A22, Def4; ::_thesis: verum
end;
begin
theorem Th30: :: GROEB_1:30
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds {(0_ (n,L))} is_Groebner_basis_of {(0_ (n,L))},T
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds {(0_ (n,L))} is_Groebner_basis_of {(0_ (n,L))},T
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr holds {(0_ (n,L))} is_Groebner_basis_of {(0_ (n,L))},T
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: {(0_ (n,L))} is_Groebner_basis_of {(0_ (n,L))},T
set I = {(0_ (n,L))};
set G = {(0_ (n,L))};
set R = PolyRedRel ({(0_ (n,L))},T);
A1: 0_ (n,L) = 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10;
now__::_thesis:_for_a,_b,_c_being_set_st_[a,b]_in_PolyRedRel_({(0__(n,L))},T)_&_[a,c]_in_PolyRedRel_({(0__(n,L))},T)_holds_
b,c_are_convergent_wrt_PolyRedRel_({(0__(n,L))},T)
let a, b, c be set ; ::_thesis: ( [a,b] in PolyRedRel ({(0_ (n,L))},T) & [a,c] in PolyRedRel ({(0_ (n,L))},T) implies b,c are_convergent_wrt PolyRedRel ({(0_ (n,L))},T) )
assume that
A2: [a,b] in PolyRedRel ({(0_ (n,L))},T) and
[a,c] in PolyRedRel ({(0_ (n,L))},T) ; ::_thesis: b,c are_convergent_wrt PolyRedRel ({(0_ (n,L))},T)
consider p, q being set such that
A3: p in NonZero (Polynom-Ring (n,L)) and
A4: q in the carrier of (Polynom-Ring (n,L)) and
A5: [a,b] = [p,q] by A2, ZFMISC_1:def_2;
reconsider q = q as Polynomial of n,L by A4, POLYNOM1:def_10;
not p in {(0_ (n,L))} by A1, A3, XBOOLE_0:def_5;
then p <> 0_ (n,L) by TARSKI:def_1;
then reconsider p = p as non-zero Polynomial of n,L by A3, POLYNOM1:def_10, POLYNOM7:def_1;
p reduces_to q,{(0_ (n,L))},T by A2, A5, POLYRED:def_13;
then consider g being Polynomial of n,L such that
A6: g in {(0_ (n,L))} and
A7: p reduces_to q,g,T by POLYRED:def_7;
g = 0_ (n,L) by A6, TARSKI:def_1;
then p is_reducible_wrt 0_ (n,L),T by A7, POLYRED:def_8;
hence b,c are_convergent_wrt PolyRedRel ({(0_ (n,L))},T) by Lm3; ::_thesis: verum
end;
then A8: PolyRedRel ({(0_ (n,L))},T) is locally-confluent by REWRITE1:def_24;
0_ (n,L) = 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10;
then {(0_ (n,L))} -Ideal = {(0_ (n,L))} by IDEAL_1:44;
hence {(0_ (n,L))} is_Groebner_basis_of {(0_ (n,L))},T by A8, Def4; ::_thesis: verum
end;
theorem :: GROEB_1:31
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L holds {p} is_Groebner_basis_of {p} -Ideal ,T
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L holds {p} is_Groebner_basis_of {p} -Ideal ,T
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L holds {p} is_Groebner_basis_of {p} -Ideal ,T
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for p being Polynomial of n,L holds {p} is_Groebner_basis_of {p} -Ideal ,T
let p be Polynomial of n,L; ::_thesis: {p} is_Groebner_basis_of {p} -Ideal ,T
percases ( p = 0_ (n,L) or p <> 0_ (n,L) ) ;
supposeA1: p = 0_ (n,L) ; ::_thesis: {p} is_Groebner_basis_of {p} -Ideal ,T
0_ (n,L) = 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10;
then {p} -Ideal = {(0_ (n,L))} by A1, IDEAL_1:44;
hence {p} is_Groebner_basis_of {p} -Ideal ,T by A1, Th30; ::_thesis: verum
end;
suppose p <> 0_ (n,L) ; ::_thesis: {p} is_Groebner_basis_of {p} -Ideal ,T
then reconsider p = p as non-zero Polynomial of n,L by POLYNOM7:def_1;
PolyRedRel ({p},T) is locally-confluent by Th10;
hence {p} is_Groebner_basis_of {p} -Ideal ,T by Def4; ::_thesis: verum
end;
end;
end;
theorem :: GROEB_1:32
for T being connected admissible TermOrder of {}
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring ({},L))
for P being non empty Subset of (Polynom-Ring ({},L)) st P c= I & P <> {(0_ ({},L))} holds
P is_Groebner_basis_of I,T
proof
now__::_thesis:_not__{__i_where_i_is_Element_of_NAT_:_i_<_0__}__<>_{}
set j = the Element of { i where i is Element of NAT : i < 0 } ;
assume { i where i is Element of NAT : i < 0 } <> {} ; ::_thesis: contradiction
then the Element of { i where i is Element of NAT : i < 0 } in { i where i is Element of NAT : i < 0 } ;
then ex i being Element of NAT st
( i = the Element of { i where i is Element of NAT : i < 0 } & i < 0 ) ;
hence contradiction by NAT_1:2; ::_thesis: verum
end;
then reconsider n = {} as Element of NAT ;
let T be connected admissible TermOrder of {}; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring ({},L))
for P being non empty Subset of (Polynom-Ring ({},L)) st P c= I & P <> {(0_ ({},L))} holds
P is_Groebner_basis_of I,T
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for I being non empty add-closed left-ideal Subset of (Polynom-Ring ({},L))
for P being non empty Subset of (Polynom-Ring ({},L)) st P c= I & P <> {(0_ ({},L))} holds
P is_Groebner_basis_of I,T
let I be non empty add-closed left-ideal Subset of (Polynom-Ring ({},L)); ::_thesis: for P being non empty Subset of (Polynom-Ring ({},L)) st P c= I & P <> {(0_ ({},L))} holds
P is_Groebner_basis_of I,T
let P be non empty Subset of (Polynom-Ring ({},L)); ::_thesis: ( P c= I & P <> {(0_ ({},L))} implies P is_Groebner_basis_of I,T )
assume that
A1: P c= I and
A2: P <> {(0_ ({},L))} ; ::_thesis: P is_Groebner_basis_of I,T
reconsider T = T as connected admissible TermOrder of n ;
reconsider P = P as non empty Subset of (Polynom-Ring (n,L)) ;
reconsider I = I as non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)) ;
A3: ex q being Element of P st q <> 0_ (n,L)
proof
assume A4: for q being Element of P holds not q <> 0_ (n,L) ; ::_thesis: contradiction
A5: now__::_thesis:_for_u_being_set_st_u_in_{(0__(n,L))}_holds_
u_in_P
let u be set ; ::_thesis: ( u in {(0_ (n,L))} implies u in P )
assume u in {(0_ (n,L))} ; ::_thesis: u in P
then A6: u = 0_ (n,L) by TARSKI:def_1;
now__::_thesis:_u_in_P
assume not u in P ; ::_thesis: contradiction
then for v being set holds not v in P by A4, A6;
hence contradiction by XBOOLE_0:def_1; ::_thesis: verum
end;
hence u in P ; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_P_holds_
u_in_{(0__(n,L))}
let u be set ; ::_thesis: ( u in P implies u in {(0_ (n,L))} )
assume u in P ; ::_thesis: u in {(0_ (n,L))}
then u = 0_ (n,L) by A4;
hence u in {(0_ (n,L))} by TARSKI:def_1; ::_thesis: verum
end;
hence contradiction by A2, A5, TARSKI:1; ::_thesis: verum
end;
now__::_thesis:_for_f_being_non-zero_Polynomial_of_n,L_st_f_in_I_holds_
f_is_reducible_wrt_P,T
consider p being Element of P such that
A7: p <> 0_ (n,L) by A3;
reconsider p = p as Polynomial of n,L by POLYNOM1:def_10;
reconsider p = p as non-zero Polynomial of n,L by A7, POLYNOM7:def_1;
let f be non-zero Polynomial of n,L; ::_thesis: ( f in I implies f is_reducible_wrt P,T )
assume f in I ; ::_thesis: f is_reducible_wrt P,T
f <> 0_ (n,L) by POLYNOM7:def_1;
then Support f <> {} by POLYNOM7:1;
then HT (f,T) in Support f by TERMORD:def_6;
then ( HT (p,T) = EmptyBag n & EmptyBag n in Support f ) ;
then f is_reducible_wrt p,T by POLYRED:36;
then consider g being Polynomial of n,L such that
A8: f reduces_to g,p,T by POLYRED:def_8;
f reduces_to g,P,T by A8, POLYRED:def_7;
hence f is_reducible_wrt P,T by POLYRED:def_9; ::_thesis: verum
end;
then for f being non-zero Polynomial of n,L st f in I holds
f is_top_reducible_wrt P,T by A1, Th26;
then for b being bag of n st b in HT (I,T) holds
ex b9 being bag of n st
( b9 in HT (P,T) & b9 divides b ) by Th27;
then HT (I,T) c= multiples (HT (P,T)) by Th28;
hence P is_Groebner_basis_of I,T by A1, Th29; ::_thesis: verum
end;
theorem :: GROEB_1:33
for n being non empty Ordinal
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr holds
not for P being Subset of (Polynom-Ring (n,L)) holds P is_Groebner_basis_wrt T
proof
let n be non empty Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr holds
not for P being Subset of (Polynom-Ring (n,L)) holds P is_Groebner_basis_wrt T
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr holds
not for P being Subset of (Polynom-Ring (n,L)) holds P is_Groebner_basis_wrt T
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: not for P being Subset of (Polynom-Ring (n,L)) holds P is_Groebner_basis_wrt T
set 1bag = (EmptyBag n) +* ({},1);
reconsider 1bag = (EmptyBag n) +* ({},1) as Element of Bags n by PRE_POLY:def_12;
set p = ((1. L) | (n,L)) +* (1bag,(1. L));
reconsider p = ((1. L) | (n,L)) +* (1bag,(1. L)) as Function of (Bags n),L ;
reconsider p = p as Series of n,L ;
A1: 1. L <> 0. L ;
set q = ((0. L) | (n,L)) +* (1bag,(1. L));
reconsider q = ((0. L) | (n,L)) +* (1bag,(1. L)) as Function of (Bags n),L ;
reconsider q = q as Series of n,L ;
A2: now__::_thesis:_for_u_being_bag_of_n_st_u_<>_EmptyBag_n_&_u_<>_1bag_holds_
p_._u_=_0._L
let u be bag of n; ::_thesis: ( u <> EmptyBag n & u <> 1bag implies p . u = 0. L )
assume that
A3: u <> EmptyBag n and
A4: u <> 1bag ; ::_thesis: p . u = 0. L
p . u = ((1. L) | (n,L)) . u by A4, FUNCT_7:32;
then p . u = (1_ (n,L)) . u by POLYNOM7:20;
hence p . u = 0. L by A3, POLYNOM1:25; ::_thesis: verum
end;
A5: now__::_thesis:_for_u9_being_set_st_u9_in_Support_p_holds_
u9_in_{(EmptyBag_n),1bag}
let u9 be set ; ::_thesis: ( u9 in Support p implies u9 in {(EmptyBag n),1bag} )
assume A6: u9 in Support p ; ::_thesis: u9 in {(EmptyBag n),1bag}
then reconsider u = u9 as Element of Bags n ;
A7: p . u <> 0. L by A6, POLYNOM1:def_3;
now__::_thesis:_u_in_{(EmptyBag_n),1bag}
assume not u in {(EmptyBag n),1bag} ; ::_thesis: contradiction
then ( u <> EmptyBag n & u <> 1bag ) by TARSKI:def_2;
hence contradiction by A2, A7; ::_thesis: verum
end;
hence u9 in {(EmptyBag n),1bag} ; ::_thesis: verum
end;
( {} in n & dom (EmptyBag n) = n ) by ORDINAL1:14, PARTFUN1:def_2;
then 1bag . {} = 1 by FUNCT_7:31;
then A8: EmptyBag n <> 1bag by PRE_POLY:52;
then A9: q . (EmptyBag n) = ((0. L) | (n,L)) . (EmptyBag n) by FUNCT_7:32
.= (0_ (n,L)) . (EmptyBag n) by POLYNOM7:19
.= 0. L by POLYNOM1:22 ;
A10: now__::_thesis:_for_u_being_bag_of_n_st_u_<>_1bag_holds_
q_._u_=_0._L
let u be bag of n; ::_thesis: ( u <> 1bag implies q . u = 0. L )
assume u <> 1bag ; ::_thesis: q . u = 0. L
then q . u = ((0. L) | (n,L)) . u by FUNCT_7:32;
then q . u = (0_ (n,L)) . u by POLYNOM7:19;
hence q . u = 0. L by POLYNOM1:22; ::_thesis: verum
end;
A11: now__::_thesis:_for_u9_being_set_st_u9_in_Support_q_holds_
u9_in_{1bag}
let u9 be set ; ::_thesis: ( u9 in Support q implies u9 in {1bag} )
assume A12: u9 in Support q ; ::_thesis: u9 in {1bag}
then reconsider u = u9 as Element of Bags n ;
A13: q . u <> 0. L by A12, POLYNOM1:def_3;
now__::_thesis:_u_in_{1bag}
assume not u in {1bag} ; ::_thesis: contradiction
then u <> 1bag by TARSKI:def_1;
hence contradiction by A10, A13; ::_thesis: verum
end;
hence u9 in {1bag} ; ::_thesis: verum
end;
dom ((0. L) | (n,L)) = Bags n by FUNCT_2:def_1;
then A14: q . 1bag = 1. L by FUNCT_7:31;
then A15: q <> 0_ (n,L) by POLYNOM1:22;
now__::_thesis:_for_u_being_set_st_u_in_{1bag}_holds_
u_in_Support_q
let u be set ; ::_thesis: ( u in {1bag} implies u in Support q )
assume u in {1bag} ; ::_thesis: u in Support q
then u = 1bag by TARSKI:def_1;
hence u in Support q by A14, POLYNOM1:def_3; ::_thesis: verum
end;
then A16: Support q = {1bag} by A11, TARSKI:1;
then reconsider q = q as Polynomial of n,L by POLYNOM1:def_4;
reconsider q = q as non-zero Polynomial of n,L by A15, POLYNOM7:def_1;
set q1 = q - (((q . (HT (q,T))) / (HC (q,T))) * ((EmptyBag n) *' q));
Support q <> {} by A15, POLYNOM7:1;
then A17: HT (q,T) in Support q by TERMORD:def_6;
(EmptyBag n) + (HT (q,T)) = HT (q,T) by PRE_POLY:53;
then q reduces_to q - (((q . (HT (q,T))) / (HC (q,T))) * ((EmptyBag n) *' q)),q, HT (q,T),T by A15, A17, POLYRED:def_5;
then A18: q reduces_to q - (((q . (HT (q,T))) / (HC (q,T))) * ((EmptyBag n) *' q)),q,T by POLYRED:def_6;
A19: q - (((q . (HT (q,T))) / (HC (q,T))) * ((EmptyBag n) *' q)) = q - (((HC (q,T)) / (HC (q,T))) * ((EmptyBag n) *' q)) by TERMORD:def_7
.= q - (((HC (q,T)) * ((HC (q,T)) ")) * ((EmptyBag n) *' q)) by VECTSP_1:def_11
.= q - ((1. L) * ((EmptyBag n) *' q)) by VECTSP_1:def_10
.= q - ((1. L) * q) by POLYRED:17
.= q - (((1. L) | (n,L)) *' q) by POLYNOM7:27
.= q - ((1_ (n,L)) *' q) by POLYNOM7:20
.= q - q by POLYNOM1:30
.= 0_ (n,L) by POLYNOM1:24 ;
A20: dom ((1. L) | (n,L)) = Bags n by FUNCT_2:def_1;
then A21: p . 1bag = 1. L by FUNCT_7:31;
then A22: p <> 0_ (n,L) by A1, POLYNOM1:22;
A23: p . (EmptyBag n) = ((1. L) | (n,L)) . (EmptyBag n) by A8, FUNCT_7:32
.= (1_ (n,L)) . (EmptyBag n) by POLYNOM7:20
.= 1. L by POLYNOM1:25 ;
now__::_thesis:_for_u_being_set_st_u_in_{(EmptyBag_n),1bag}_holds_
u_in_Support_p
let u be set ; ::_thesis: ( u in {(EmptyBag n),1bag} implies u in Support p )
assume A24: u in {(EmptyBag n),1bag} ; ::_thesis: u in Support p
now__::_thesis:_(_(_u_=_EmptyBag_n_&_u_in_Support_p_)_or_(_u_=_1bag_&_u_in_Support_p_)_)
percases ( u = EmptyBag n or u = 1bag ) by A24, TARSKI:def_2;
case u = EmptyBag n ; ::_thesis: u in Support p
hence u in Support p by A1, A23, POLYNOM1:def_3; ::_thesis: verum
end;
case u = 1bag ; ::_thesis: u in Support p
hence u in Support p by A1, A21, POLYNOM1:def_3; ::_thesis: verum
end;
end;
end;
hence u in Support p ; ::_thesis: verum
end;
then A25: Support p = {(EmptyBag n),1bag} by A5, TARSKI:1;
then reconsider p = p as Polynomial of n,L by POLYNOM1:def_4;
reconsider p = p as non-zero Polynomial of n,L by A22, POLYNOM7:def_1;
A26: (EmptyBag n) + (HT (p,T)) = HT (p,T) by PRE_POLY:53;
A27: now__::_thesis:_not_HT_(p,T)_=_EmptyBag_n
A28: EmptyBag n <= 1bag,T by TERMORD:9;
assume A29: HT (p,T) = EmptyBag n ; ::_thesis: contradiction
1bag in Support p by A25, TARSKI:def_2;
then 1bag <= EmptyBag n,T by A29, TERMORD:def_6;
hence contradiction by A8, A28, TERMORD:7; ::_thesis: verum
end;
set p1 = q - (((q . (HT (p,T))) / (HC (p,T))) * ((EmptyBag n) *' p));
Support p <> {} by A22, POLYNOM7:1;
then A30: HT (p,T) in Support p by TERMORD:def_6;
then A31: HT (p,T) = 1bag by A25, A27, TARSKI:def_2;
then HT (p,T) in Support q by A16, TARSKI:def_1;
then q reduces_to q - (((q . (HT (p,T))) / (HC (p,T))) * ((EmptyBag n) *' p)),p, HT (p,T),T by A22, A15, A26, POLYRED:def_5;
then A32: q reduces_to q - (((q . (HT (p,T))) / (HC (p,T))) * ((EmptyBag n) *' p)),p,T by POLYRED:def_6;
A33: now__::_thesis:_not_Support_q_=_Support_p
assume Support q = Support p ; ::_thesis: contradiction
then EmptyBag n in {1bag} by A25, A16, TARSKI:def_2;
hence contradiction by A8, TARSKI:def_1; ::_thesis: verum
end;
A34: now__::_thesis:_not_q_-_p_=_0__(n,L)
assume q - p = 0_ (n,L) ; ::_thesis: contradiction
then p = (q - p) + p by POLYRED:2
.= (q + (- p)) + p by POLYNOM1:def_6
.= q + ((- p) + p) by POLYNOM1:21
.= q + (0_ (n,L)) by POLYRED:3 ;
hence contradiction by A33, POLYNOM1:23; ::_thesis: verum
end;
set P = {p,q};
now__::_thesis:_for_u_being_set_st_u_in_{p,q}_holds_
u_in_the_carrier_of_(Polynom-Ring_(n,L))
let u be set ; ::_thesis: ( u in {p,q} implies u in the carrier of (Polynom-Ring (n,L)) )
assume u in {p,q} ; ::_thesis: u in the carrier of (Polynom-Ring (n,L))
then ( u = p or u = q ) by TARSKI:def_2;
hence u in the carrier of (Polynom-Ring (n,L)) by POLYNOM1:def_10; ::_thesis: verum
end;
then reconsider P = {p,q} as Subset of (Polynom-Ring (n,L)) by TARSKI:def_3;
reconsider P = P as Subset of (Polynom-Ring (n,L)) ;
set R = PolyRedRel (P,T);
take P ; ::_thesis: not P is_Groebner_basis_wrt T
A35: p in P by TARSKI:def_2;
q in P by TARSKI:def_2;
then q reduces_to 0_ (n,L),P,T by A18, A19, POLYRED:def_7;
then A36: [q,(0_ (n,L))] in PolyRedRel (P,T) by POLYRED:def_13;
q - (((q . (HT (p,T))) / (HC (p,T))) * ((EmptyBag n) *' p)) = q - (((1. L) / (p . 1bag)) * ((EmptyBag n) *' p)) by A14, A31, TERMORD:def_7
.= q - (((1. L) / (1. L)) * ((EmptyBag n) *' p)) by A20, FUNCT_7:31
.= q - (((1. L) * ((1. L) ")) * ((EmptyBag n) *' p)) by VECTSP_1:def_11
.= q - ((1. L) * ((EmptyBag n) *' p)) by VECTSP_1:def_10
.= q - ((1. L) * p) by POLYRED:17
.= q - (((1. L) | (n,L)) *' p) by POLYNOM7:27
.= q - ((1_ (n,L)) *' p) by POLYNOM7:20
.= q - p by POLYNOM1:30 ;
then q reduces_to q - p,P,T by A32, A35, POLYRED:def_7;
then A37: [q,(q - p)] in PolyRedRel (P,T) by POLYRED:def_13;
now__::_thesis:_not_PolyRedRel_(P,T)_is_locally-confluent
A38: now__::_thesis:_for_u_being_set_st_u_in_Support_(q_-_p)_holds_
u_in_{(EmptyBag_n)}
let u be set ; ::_thesis: ( u in Support (q - p) implies u in {(EmptyBag n)} )
now__::_thesis:_for_u_being_set_st_u_in_{1bag}_holds_
u_in_{(EmptyBag_n),1bag}
let u be set ; ::_thesis: ( u in {1bag} implies u in {(EmptyBag n),1bag} )
assume u in {1bag} ; ::_thesis: u in {(EmptyBag n),1bag}
then u = 1bag by TARSKI:def_1;
hence u in {(EmptyBag n),1bag} by TARSKI:def_2; ::_thesis: verum
end;
then {1bag} c= {(EmptyBag n),1bag} by TARSKI:def_3;
then A39: {1bag} \/ {(EmptyBag n),1bag} = {(EmptyBag n),1bag} by XBOOLE_1:12;
A40: (q - p) . 1bag = (q + (- p)) . 1bag by POLYNOM1:def_6
.= (q . 1bag) + ((- p) . 1bag) by POLYNOM1:15
.= (q . 1bag) + (- (p . 1bag)) by POLYNOM1:17
.= (1. L) + (- (1. L)) by A20, A14, FUNCT_7:31
.= 0. L by RLVECT_1:5 ;
Support (q - p) = Support (q + (- p)) by POLYNOM1:def_6;
then Support (q - p) c= (Support q) \/ (Support (- p)) by POLYNOM1:20;
then A41: Support (q - p) c= {1bag} \/ {(EmptyBag n),1bag} by A25, A16, Th5;
assume A42: u in Support (q - p) ; ::_thesis: u in {(EmptyBag n)}
then (q - p) . u <> 0. L by POLYNOM1:def_3;
then u = EmptyBag n by A42, A41, A39, A40, TARSKI:def_2;
hence u in {(EmptyBag n)} by TARSKI:def_1; ::_thesis: verum
end;
assume PolyRedRel (P,T) is locally-confluent ; ::_thesis: contradiction
then 0_ (n,L),q - p are_convergent_wrt PolyRedRel (P,T) by A37, A36, REWRITE1:def_24;
then consider c being set such that
A43: PolyRedRel (P,T) reduces 0_ (n,L),c and
A44: PolyRedRel (P,T) reduces q - p,c by REWRITE1:def_7;
reconsider c = c as Polynomial of n,L by A43, Lm4;
consider s being FinSequence such that
A45: len s > 0 and
A46: s . 1 = 0_ (n,L) and
A47: s . (len s) = c and
A48: for i being Element of NAT st i in dom s & i + 1 in dom s holds
[(s . i),(s . (i + 1))] in PolyRedRel (P,T) by A43, REWRITE1:11;
now__::_thesis:_not_len_s_<>_1
A49: 0 + 1 <= len s by A45, NAT_1:13;
A50: dom s = Seg (len s) by FINSEQ_1:def_3;
assume len s <> 1 ; ::_thesis: contradiction
then 1 < len s by A49, XXREAL_0:1;
then 1 + 1 <= len s by NAT_1:13;
then A51: 1 + 1 in dom s by A50, FINSEQ_1:1;
A52: 1 in dom s by A49, A50, FINSEQ_1:1;
then consider f9, h9 being set such that
A53: [(0_ (n,L)),(s . 2)] = [f9,h9] and
f9 in NonZero (Polynom-Ring (n,L)) and
A54: h9 in the carrier of (Polynom-Ring (n,L)) by A46, A48, A51, RELSET_1:2;
s . 2 = h9 by A53, XTUPLE_0:1;
then reconsider c9 = s . 2 as Polynomial of n,L by A54, POLYNOM1:def_10;
[(s . 1),(s . 2)] in PolyRedRel (P,T) by A48, A52, A51;
then 0_ (n,L) reduces_to c9,P,T by A46, POLYRED:def_13;
then consider g being Polynomial of n,L such that
g in P and
A55: 0_ (n,L) reduces_to c9,g,T by POLYRED:def_7;
0_ (n,L) is_reducible_wrt g,T by A55, POLYRED:def_8;
hence contradiction by POLYRED:37; ::_thesis: verum
end;
then consider s being FinSequence such that
A56: len s > 0 and
A57: s . 1 = q - p and
A58: s . (len s) = 0_ (n,L) and
A59: for i being Element of NAT st i in dom s & i + 1 in dom s holds
[(s . i),(s . (i + 1))] in PolyRedRel (P,T) by A44, A46, A47, REWRITE1:11;
A60: now__::_thesis:_not_-_(1._L)_=_0._L
assume - (1. L) = 0. L ; ::_thesis: contradiction
then - (- (1. L)) = 0. L by RLVECT_1:12;
hence contradiction by RLVECT_1:17; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_{(EmptyBag_n)}_holds_
u_in_Support_(q_-_p)
let u be set ; ::_thesis: ( u in {(EmptyBag n)} implies u in Support (q - p) )
assume A61: u in {(EmptyBag n)} ; ::_thesis: u in Support (q - p)
then reconsider u9 = u as Element of Bags n by TARSKI:def_1;
(q - p) . u9 = (q + (- p)) . u9 by POLYNOM1:def_6
.= (q . u9) + ((- p) . u9) by POLYNOM1:15
.= (q . u9) + (- (p . u9)) by POLYNOM1:17
.= (0. L) + (- (p . u9)) by A9, A61, TARSKI:def_1
.= (0. L) + (- (1. L)) by A23, A61, TARSKI:def_1
.= - (1. L) by ALGSTR_1:def_2 ;
hence u in Support (q - p) by A60, POLYNOM1:def_3; ::_thesis: verum
end;
then A62: Support (q - p) = {(EmptyBag n)} by A38, TARSKI:1;
A63: now__::_thesis:_not_q_-_p_is_reducible_wrt_P,T
assume q - p is_reducible_wrt P,T ; ::_thesis: contradiction
then consider g being Polynomial of n,L such that
A64: q - p reduces_to g,P,T by POLYRED:def_9;
consider h being Polynomial of n,L such that
A65: h in P and
A66: q - p reduces_to g,h,T by A64, POLYRED:def_7;
ex b being bag of n st q - p reduces_to g,h,b,T by A66, POLYRED:def_6;
then h <> 0_ (n,L) by POLYRED:def_5;
then reconsider h = h as non-zero Polynomial of n,L by POLYNOM7:def_1;
q - p is_reducible_wrt h,T by A66, POLYRED:def_8;
then consider b9 being bag of n such that
A67: b9 in Support (q - p) and
A68: HT (h,T) divides b9 by POLYRED:36;
A69: HT (h,T) = 1bag
proof
now__::_thesis:_(_(_h_=_p_&_HT_(h,T)_=_1bag_)_or_(_h_=_q_&_HT_(h,T)_=_1bag_)_)
percases ( h = p or h = q ) by A65, TARSKI:def_2;
case h = p ; ::_thesis: HT (h,T) = 1bag
hence HT (h,T) = 1bag by A25, A30, A27, TARSKI:def_2; ::_thesis: verum
end;
case h = q ; ::_thesis: HT (h,T) = 1bag
hence HT (h,T) = 1bag by A16, A17, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
hence HT (h,T) = 1bag ; ::_thesis: verum
end;
b9 = EmptyBag n by A62, A67, TARSKI:def_1;
hence contradiction by A8, A68, A69, PRE_POLY:58; ::_thesis: verum
end;
now__::_thesis:_not_len_s_<>_1
A70: 0 + 1 <= len s by A56, NAT_1:13;
A71: dom s = Seg (len s) by FINSEQ_1:def_3;
assume len s <> 1 ; ::_thesis: contradiction
then 1 < len s by A70, XXREAL_0:1;
then 1 + 1 <= len s by NAT_1:13;
then A72: 1 + 1 in dom s by A71, FINSEQ_1:1;
A73: 1 in dom s by A70, A71, FINSEQ_1:1;
then consider f9, h9 being set such that
A74: [(q - p),(s . 2)] = [f9,h9] and
f9 in NonZero (Polynom-Ring (n,L)) and
A75: h9 in the carrier of (Polynom-Ring (n,L)) by A57, A59, A72, RELSET_1:2;
s . 2 = h9 by A74, XTUPLE_0:1;
then reconsider c9 = s . 2 as Polynomial of n,L by A75, POLYNOM1:def_10;
[(q - p),(s . 2)] in PolyRedRel (P,T) by A57, A59, A73, A72;
then q - p reduces_to c9,P,T by POLYRED:def_13;
hence contradiction by A63, POLYRED:def_9; ::_thesis: verum
end;
hence contradiction by A34, A57, A58; ::_thesis: verum
end;
hence not P is_Groebner_basis_wrt T by Def3; ::_thesis: verum
end;
Lm8: for n being Ordinal
for b1, b2, b3 being bag of n st b1 divides b2 & b2 divides b3 holds
b1 divides b3
proof
let n be Ordinal; ::_thesis: for b1, b2, b3 being bag of n st b1 divides b2 & b2 divides b3 holds
b1 divides b3
let b1, b2, b3 be bag of n; ::_thesis: ( b1 divides b2 & b2 divides b3 implies b1 divides b3 )
assume A1: ( b1 divides b2 & b2 divides b3 ) ; ::_thesis: b1 divides b3
now__::_thesis:_for_k_being_set_holds_b1_._k_<=_b3_._k
let k be set ; ::_thesis: b1 . k <= b3 . k
( b1 . k <= b2 . k & b2 . k <= b3 . k ) by A1, PRE_POLY:def_11;
hence b1 . k <= b3 . k by XXREAL_0:2; ::_thesis: verum
end;
hence b1 divides b3 by PRE_POLY:def_11; ::_thesis: verum
end;
definition
let n be Ordinal;
func DivOrder n -> Order of (Bags n) means :Def5: :: GROEB_1:def 5
for b1, b2 being bag of n holds
( [b1,b2] in it iff b1 divides b2 );
existence
ex b1 being Order of (Bags n) st
for b1, b2 being bag of n holds
( [b1,b2] in b1 iff b1 divides b2 )
proof
defpred S1[ set , set ] means ex b1, b2 being Element of Bags n st
( $1 = b1 & $2 = b2 & b1 divides b2 );
consider BO being Relation of (Bags n),(Bags n) such that
A1: for x, y being set holds
( [x,y] in BO iff ( x in Bags n & y in Bags n & S1[x,y] ) ) from RELSET_1:sch_1();
A2: BO is_transitive_in Bags n
proof
let x, y, z be set ; :: according to RELAT_2:def_8 ::_thesis: ( not x in Bags n or not y in Bags n or not z in Bags n or not [x,y] in BO or not [y,z] in BO or [x,z] in BO )
assume that
x in Bags n and
y in Bags n and
z in Bags n and
A3: [x,y] in BO and
A4: [y,z] in BO ; ::_thesis: [x,z] in BO
consider b1, b2 being Element of Bags n such that
A5: x = b1 and
A6: ( y = b2 & b1 divides b2 ) by A1, A3;
consider b11, b22 being Element of Bags n such that
A7: y = b11 and
A8: z = b22 and
A9: b11 divides b22 by A1, A4;
reconsider B1 = b1, B29 = b22 as bag of n ;
B1 divides B29 by A6, A7, A9, Lm8;
hence [x,z] in BO by A1, A5, A8; ::_thesis: verum
end;
A10: BO is_reflexive_in Bags n
proof
let x be set ; :: according to RELAT_2:def_1 ::_thesis: ( not x in Bags n or [x,x] in BO )
assume x in Bags n ; ::_thesis: [x,x] in BO
hence [x,x] in BO by A1; ::_thesis: verum
end;
then A11: ( dom BO = Bags n & field BO = Bags n ) by ORDERS_1:13;
BO is_antisymmetric_in Bags n
proof
let x, y be set ; :: according to RELAT_2:def_4 ::_thesis: ( not x in Bags n or not y in Bags n or not [x,y] in BO or not [y,x] in BO or x = y )
assume that
x in Bags n and
y in Bags n and
A12: [x,y] in BO and
A13: [y,x] in BO ; ::_thesis: x = y
consider b19, b29 being Element of Bags n such that
A14: ( y = b19 & x = b29 ) and
A15: b19 divides b29 by A1, A13;
consider b11, b22 being Element of Bags n such that
A16: ( x = b11 & y = b22 ) and
A17: b11 divides b22 by A1, A12;
reconsider b11 = b11, b22 = b22 as bag of n ;
A18: now__::_thesis:_for_k_being_set_st_k_in_dom_b11_holds_
b11_._k_=_b22_._k
let k be set ; ::_thesis: ( k in dom b11 implies b11 . k = b22 . k )
assume k in dom b11 ; ::_thesis: b11 . k = b22 . k
( b11 . k <= b22 . k & b19 . k <= b29 . k ) by A17, A15, PRE_POLY:def_11;
hence b11 . k = b22 . k by A16, A14, XXREAL_0:1; ::_thesis: verum
end;
dom b11 = n by PARTFUN1:def_2
.= dom b22 by PARTFUN1:def_2 ;
hence x = y by A16, A18, FUNCT_1:2; ::_thesis: verum
end;
then reconsider BO = BO as Order of (Bags n) by A10, A2, A11, PARTFUN1:def_2, RELAT_2:def_9, RELAT_2:def_12, RELAT_2:def_16;
take BO ; ::_thesis: for b1, b2 being bag of n holds
( [b1,b2] in BO iff b1 divides b2 )
let p, q be bag of n; ::_thesis: ( [p,q] in BO iff p divides q )
hereby ::_thesis: ( p divides q implies [p,q] in BO )
assume [p,q] in BO ; ::_thesis: p divides q
then ex b19, b29 being Element of Bags n st
( p = b19 & q = b29 & b19 divides b29 ) by A1;
hence p divides q ; ::_thesis: verum
end;
( p in Bags n & q in Bags n ) by PRE_POLY:def_12;
hence ( p divides q implies [p,q] in BO ) by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being Order of (Bags n) st ( for b1, b2 being bag of n holds
( [b1,b2] in b1 iff b1 divides b2 ) ) & ( for b1, b2 being bag of n holds
( [b1,b2] in b2 iff b1 divides b2 ) ) holds
b1 = b2
proof
let B1, B2 be Order of (Bags n); ::_thesis: ( ( for b1, b2 being bag of n holds
( [b1,b2] in B1 iff b1 divides b2 ) ) & ( for b1, b2 being bag of n holds
( [b1,b2] in B2 iff b1 divides b2 ) ) implies B1 = B2 )
assume that
A19: for p, q being bag of n holds
( [p,q] in B1 iff p divides q ) and
A20: for p, q being bag of n holds
( [p,q] in B2 iff p divides q ) ; ::_thesis: B1 = B2
let a, b be set ; :: according to RELAT_1:def_2 ::_thesis: ( ( not [a,b] in B1 or [a,b] in B2 ) & ( not [a,b] in B2 or [a,b] in B1 ) )
hereby ::_thesis: ( not [a,b] in B2 or [a,b] in B1 )
assume A21: [a,b] in B1 ; ::_thesis: [a,b] in B2
then consider b1, b2 being set such that
A22: [a,b] = [b1,b2] and
A23: ( b1 in Bags n & b2 in Bags n ) by RELSET_1:2;
reconsider b1 = b1, b2 = b2 as bag of n by A23;
b1 divides b2 by A19, A21, A22;
hence [a,b] in B2 by A20, A22; ::_thesis: verum
end;
assume A24: [a,b] in B2 ; ::_thesis: [a,b] in B1
then consider b1, b2 being set such that
A25: [a,b] = [b1,b2] and
A26: ( b1 in Bags n & b2 in Bags n ) by RELSET_1:2;
reconsider b1 = b1, b2 = b2 as bag of n by A26;
b1 divides b2 by A20, A24, A25;
hence [a,b] in B1 by A19, A25; ::_thesis: verum
end;
end;
:: deftheorem Def5 defines DivOrder GROEB_1:def_5_:_
for n being Ordinal
for b2 being Order of (Bags n) holds
( b2 = DivOrder n iff for b1, b2 being bag of n holds
( [b1,b2] in b2 iff b1 divides b2 ) );
registration
let n be Element of NAT ;
cluster RelStr(# (Bags n),(DivOrder n) #) -> Dickson ;
coherence
RelStr(# (Bags n),(DivOrder n) #) is Dickson
proof
set R = RelStr(# (Bags n),(DivOrder n) #);
set S = product (Carrier (n --> OrderedNAT));
set SJ = Carrier (n --> OrderedNAT);
set P = product (n --> OrderedNAT);
set J = n --> OrderedNAT;
defpred S1[ set , set ] means ( n in product (Carrier (n --> OrderedNAT)) & ex b being bag of n st
( b = c2 & b = n ) );
A1: for x being set st x in product (Carrier (n --> OrderedNAT)) holds
for g being Function st x = g holds
( dom g = n & ( for y being set st y in dom g holds
g . y in NAT ) )
proof
let x be set ; ::_thesis: ( x in product (Carrier (n --> OrderedNAT)) implies for g being Function st x = g holds
( dom g = n & ( for y being set st y in dom g holds
g . y in NAT ) ) )
assume x in product (Carrier (n --> OrderedNAT)) ; ::_thesis: for g being Function st x = g holds
( dom g = n & ( for y being set st y in dom g holds
g . y in NAT ) )
then consider g being Function such that
A2: x = g and
A3: dom g = dom (Carrier (n --> OrderedNAT)) and
A4: for y being set st y in dom (Carrier (n --> OrderedNAT)) holds
g . y in (Carrier (n --> OrderedNAT)) . y by CARD_3:def_5;
let g9 be Function; ::_thesis: ( x = g9 implies ( dom g9 = n & ( for y being set st y in dom g9 holds
g9 . y in NAT ) ) )
assume A5: x = g9 ; ::_thesis: ( dom g9 = n & ( for y being set st y in dom g9 holds
g9 . y in NAT ) )
hence dom g9 = n by A2, A3, PARTFUN1:def_2; ::_thesis: for y being set st y in dom g9 holds
g9 . y in NAT
thus for y being set st y in dom g9 holds
g9 . y in NAT ::_thesis: verum
proof
let y be set ; ::_thesis: ( y in dom g9 implies g9 . y in NAT )
assume A6: y in dom g9 ; ::_thesis: g9 . y in NAT
then A7: y in n by A5, A2, A3;
then consider R being 1-sorted such that
A8: R = (n --> OrderedNAT) . y and
A9: (Carrier (n --> OrderedNAT)) . y = the carrier of R by PRALG_1:def_13;
g . y in the carrier of R by A5, A2, A3, A4, A6, A9;
hence g9 . y in NAT by A5, A2, A7, A8, DICKSON:def_15, FUNCOP_1:7; ::_thesis: verum
end;
end;
A10: for x being set st x in product (Carrier (n --> OrderedNAT)) holds
ex y being set st S1[x,y]
proof
let x be set ; ::_thesis: ( x in product (Carrier (n --> OrderedNAT)) implies ex y being set st S1[x,y] )
assume A11: x in product (Carrier (n --> OrderedNAT)) ; ::_thesis: ex y being set st S1[x,y]
then consider g being Function such that
A12: x = g and
dom g = dom (Carrier (n --> OrderedNAT)) and
for y being set st y in dom (Carrier (n --> OrderedNAT)) holds
g . y in (Carrier (n --> OrderedNAT)) . y by CARD_3:def_5;
defpred S2[ set , set ] means c2 = g . n;
A13: for x being set st x in n holds
ex y being set st S2[x,y] ;
consider b being Function such that
A14: ( dom b = n & ( for x being set st x in n holds
S2[x,b . x] ) ) from CLASSES1:sch_1(A13);
reconsider b = b as ManySortedSet of n by A14, PARTFUN1:def_2, RELAT_1:def_18;
A15: dom g = n by A1, A11, A12;
now__::_thesis:_for_u_being_set_st_u_in_rng_b_holds_
u_in_NAT
let u be set ; ::_thesis: ( u in rng b implies u in NAT )
assume u in rng b ; ::_thesis: u in NAT
then consider x being set such that
A16: ( x in dom b & u = b . x ) by FUNCT_1:def_3;
( u = g . x & x in dom g ) by A15, A14, A16;
hence u in NAT by A1, A11, A12; ::_thesis: verum
end;
then rng b c= NAT by TARSKI:def_3;
then reconsider b = b as bag of n by VALUED_0:def_6;
take b ; ::_thesis: S1[x,b]
thus x in product (Carrier (n --> OrderedNAT)) by A11; ::_thesis: ex b being bag of n st
( b = b & b = x )
take b ; ::_thesis: ( b = b & b = x )
thus b = b ; ::_thesis: b = x
thus b = x by A12, A15, A14, FUNCT_1:2; ::_thesis: verum
end;
consider i being Function such that
A17: ( dom i = product (Carrier (n --> OrderedNAT)) & ( for x being set st x in product (Carrier (n --> OrderedNAT)) holds
S1[x,i . x] ) ) from CLASSES1:sch_1(A10);
A18: for x being Element of RelStr(# (Bags n),(DivOrder n) #) ex u being Element of (product (n --> OrderedNAT)) st
( u in dom i & i . u = x )
proof
let x be Element of RelStr(# (Bags n),(DivOrder n) #); ::_thesis: ex u being Element of (product (n --> OrderedNAT)) st
( u in dom i & i . u = x )
reconsider g = x as bag of n ;
A19: now__::_thesis:_for_x_being_set_st_x_in_dom_(Carrier_(n_-->_OrderedNAT))_holds_
g_._x_in_(Carrier_(n_-->_OrderedNAT))_._x
let x be set ; ::_thesis: ( x in dom (Carrier (n --> OrderedNAT)) implies g . x in (Carrier (n --> OrderedNAT)) . x )
assume x in dom (Carrier (n --> OrderedNAT)) ; ::_thesis: g . x in (Carrier (n --> OrderedNAT)) . x
then A20: x in n ;
then consider L being 1-sorted such that
A21: L = (n --> OrderedNAT) . x and
A22: (Carrier (n --> OrderedNAT)) . x = the carrier of L by PRALG_1:def_13;
the carrier of L = NAT by A20, A21, DICKSON:def_15, FUNCOP_1:7;
hence g . x in (Carrier (n --> OrderedNAT)) . x by A22; ::_thesis: verum
end;
A23: dom g = n by PARTFUN1:def_2
.= dom (Carrier (n --> OrderedNAT)) by PARTFUN1:def_2 ;
then A24: g in product (Carrier (n --> OrderedNAT)) by A19, CARD_3:def_5;
then reconsider g = g as Element of (product (n --> OrderedNAT)) by YELLOW_1:def_4;
take g ; ::_thesis: ( g in dom i & i . g = x )
thus g in dom i by A17, A23, A19, CARD_3:def_5; ::_thesis: i . g = x
S1[g,i . g] by A17, A24;
hence i . g = x ; ::_thesis: verum
end;
A25: now__::_thesis:_for_v_being_set_st_v_in_rng_i_holds_
v_in_Bags_n
let v be set ; ::_thesis: ( v in rng i implies v in Bags n )
assume v in rng i ; ::_thesis: v in Bags n
then consider u being set such that
A26: u in dom i and
A27: v = i . u by FUNCT_1:def_3;
ex b being bag of n st
( b = i . u & b = u ) by A17, A26;
hence v in Bags n by A27, PRE_POLY:def_12; ::_thesis: verum
end;
now__::_thesis:_for_N_being_Subset_of_RelStr(#_(Bags_n),(DivOrder_n)_#)_ex_B_being_set_st_
(_B_is_Dickson-basis_of_N,_RelStr(#_(Bags_n),(DivOrder_n)_#)_&_B_is_finite_)
let N be Subset of RelStr(# (Bags n),(DivOrder n) #); ::_thesis: ex B being set st
( B is_Dickson-basis_of N, RelStr(# (Bags n),(DivOrder n) #) & B is finite )
set N9 = i " N;
A28: i " N c= product (Carrier (n --> OrderedNAT)) by A17, RELAT_1:132;
then reconsider N9 = i " N as Subset of (product (n --> OrderedNAT)) by YELLOW_1:def_4;
consider B9 being set such that
A29: B9 is_Dickson-basis_of N9, product (n --> OrderedNAT) and
A30: B9 is finite by DICKSON:def_10;
set B = i .: B9;
A31: B9 c= N9 by A29, DICKSON:def_9;
A32: for a, b being Element of (product (n --> OrderedNAT))
for ta, tb being Element of Bags n st a = ta & b = tb & a in product (Carrier (n --> OrderedNAT)) & a <= b holds
ta divides tb
proof
let a, b be Element of (product (n --> OrderedNAT)); ::_thesis: for ta, tb being Element of Bags n st a = ta & b = tb & a in product (Carrier (n --> OrderedNAT)) & a <= b holds
ta divides tb
let ta, tb be Element of Bags n; ::_thesis: ( a = ta & b = tb & a in product (Carrier (n --> OrderedNAT)) & a <= b implies ta divides tb )
assume that
A33: ( a = ta & b = tb ) and
A34: a in product (Carrier (n --> OrderedNAT)) ; ::_thesis: ( not a <= b or ta divides tb )
assume a <= b ; ::_thesis: ta divides tb
then consider f, g being Function such that
A35: ( f = a & g = b ) and
A36: for i being set st i in n holds
ex R being RelStr ex ai, bi being Element of R st
( R = (n --> OrderedNAT) . i & ai = f . i & bi = g . i & ai <= bi ) by A34, YELLOW_1:def_4;
now__::_thesis:_for_k_being_set_st_k_in_n_holds_
ta_._k_<=_tb_._k
let k be set ; ::_thesis: ( k in n implies ta . k <= tb . k )
assume A37: k in n ; ::_thesis: ta . k <= tb . k
then consider R being RelStr , ak, bk being Element of R such that
A38: R = (n --> OrderedNAT) . k and
A39: ( ak = f . k & bk = g . k ) and
A40: ak <= bk by A36;
(n --> OrderedNAT) . k = OrderedNAT by A37, FUNCOP_1:7;
then [ak,bk] in NATOrd by A38, A40, DICKSON:def_15, ORDERS_2:def_5;
then consider a9, b9 being Element of NAT such that
A41: [a9,b9] = [ak,bk] and
A42: a9 <= b9 by DICKSON:def_14;
A43: b9 = bk by A41, XTUPLE_0:1;
a9 = ak by A41, XTUPLE_0:1;
hence ta . k <= tb . k by A33, A35, A39, A42, A43; ::_thesis: verum
end;
hence ta divides tb by PRE_POLY:46; ::_thesis: verum
end;
A44: for a being Element of RelStr(# (Bags n),(DivOrder n) #) st a in N holds
ex b being Element of RelStr(# (Bags n),(DivOrder n) #) st
( b in i .: B9 & b <= a )
proof
let a be Element of RelStr(# (Bags n),(DivOrder n) #); ::_thesis: ( a in N implies ex b being Element of RelStr(# (Bags n),(DivOrder n) #) st
( b in i .: B9 & b <= a ) )
consider a9 being Element of (product (n --> OrderedNAT)) such that
A45: a9 in dom i and
A46: i . a9 = a by A18;
A47: ex b being bag of n st
( b = i . a9 & b = a9 ) by A17, A45;
assume a in N ; ::_thesis: ex b being Element of RelStr(# (Bags n),(DivOrder n) #) st
( b in i .: B9 & b <= a )
then a9 in N9 by A45, A46, FUNCT_1:def_7;
then consider b9 being Element of (product (n --> OrderedNAT)) such that
A48: b9 in B9 and
A49: b9 <= a9 by A29, DICKSON:def_9;
set b = i . b9;
A50: B9 c= product (Carrier (n --> OrderedNAT)) by A28, A31, XBOOLE_1:1;
then i . b9 in rng i by A17, A48, FUNCT_1:def_3;
then reconsider b = i . b9 as Element of Bags n by A25;
reconsider b = b as Element of RelStr(# (Bags n),(DivOrder n) #) ;
take b ; ::_thesis: ( b in i .: B9 & b <= a )
thus b in i .: B9 by A17, A48, A50, FUNCT_1:def_6; ::_thesis: b <= a
reconsider aa = a, bb = b as Element of Bags n ;
ex b1 being bag of n st
( b1 = i . b9 & b1 = b9 ) by A17, A48, A50;
then bb divides aa by A32, A46, A47, A48, A49, A50;
then [b,a] in DivOrder n by Def5;
hence b <= a by ORDERS_2:def_5; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_i_.:_B9_holds_
u_in_N
let u be set ; ::_thesis: ( u in i .: B9 implies u in N )
assume u in i .: B9 ; ::_thesis: u in N
then ex v being set st
( v in dom i & v in B9 & u = i . v ) by FUNCT_1:def_6;
hence u in N by A31, FUNCT_1:def_7; ::_thesis: verum
end;
then i .: B9 c= N by TARSKI:def_3;
then i .: B9 is_Dickson-basis_of N, RelStr(# (Bags n),(DivOrder n) #) by A44, DICKSON:def_9;
hence ex B being set st
( B is_Dickson-basis_of N, RelStr(# (Bags n),(DivOrder n) #) & B is finite ) by A30; ::_thesis: verum
end;
hence RelStr(# (Bags n),(DivOrder n) #) is Dickson by DICKSON:def_10; ::_thesis: verum
end;
end;
theorem Th34: :: GROEB_1:34
for n being Element of NAT
for N being Subset of RelStr(# (Bags n),(DivOrder n) #) ex B being finite Subset of (Bags n) st B is_Dickson-basis_of N, RelStr(# (Bags n),(DivOrder n) #)
proof
let n be Element of NAT ; ::_thesis: for N being Subset of RelStr(# (Bags n),(DivOrder n) #) ex B being finite Subset of (Bags n) st B is_Dickson-basis_of N, RelStr(# (Bags n),(DivOrder n) #)
let N be Subset of RelStr(# (Bags n),(DivOrder n) #); ::_thesis: ex B being finite Subset of (Bags n) st B is_Dickson-basis_of N, RelStr(# (Bags n),(DivOrder n) #)
consider B being set such that
A1: B is_Dickson-basis_of N, RelStr(# (Bags n),(DivOrder n) #) and
A2: B is finite by DICKSON:def_10;
now__::_thesis:_for_u_being_set_st_u_in_B_holds_
u_in_N
let u be set ; ::_thesis: ( u in B implies u in N )
assume A3: u in B ; ::_thesis: u in N
B c= N by A1, DICKSON:def_9;
hence u in N by A3; ::_thesis: verum
end;
then reconsider B = B as finite Subset of N by A2, TARSKI:def_3;
reconsider B = B as finite Subset of (Bags n) by XBOOLE_1:1;
take B ; ::_thesis: B is_Dickson-basis_of N, RelStr(# (Bags n),(DivOrder n) #)
thus B is_Dickson-basis_of N, RelStr(# (Bags n),(DivOrder n) #) by A1; ::_thesis: verum
end;
theorem Th35: :: GROEB_1:35
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)) ex G being finite Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_of I,T
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)) ex G being finite Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_of I,T
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)) ex G being finite Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_of I,T
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)) ex G being finite Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_of I,T
let I be non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)); ::_thesis: ex G being finite Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_of I,T
A1: 0_ (n,L) = 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10;
percases ( I = {(0_ (n,L))} or I <> {(0_ (n,L))} ) ;
supposeA2: I = {(0_ (n,L))} ; ::_thesis: ex G being finite Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_of I,T
set G = {(0_ (n,L))};
set R = PolyRedRel ({(0_ (n,L))},T);
take {(0_ (n,L))} ; ::_thesis: {(0_ (n,L))} is_Groebner_basis_of I,T
now__::_thesis:_for_a,_b,_c_being_set_st_[a,b]_in_PolyRedRel_({(0__(n,L))},T)_&_[a,c]_in_PolyRedRel_({(0__(n,L))},T)_holds_
b,c_are_convergent_wrt_PolyRedRel_({(0__(n,L))},T)
let a, b, c be set ; ::_thesis: ( [a,b] in PolyRedRel ({(0_ (n,L))},T) & [a,c] in PolyRedRel ({(0_ (n,L))},T) implies b,c are_convergent_wrt PolyRedRel ({(0_ (n,L))},T) )
assume that
A3: [a,b] in PolyRedRel ({(0_ (n,L))},T) and
[a,c] in PolyRedRel ({(0_ (n,L))},T) ; ::_thesis: b,c are_convergent_wrt PolyRedRel ({(0_ (n,L))},T)
consider p, q being set such that
A4: p in NonZero (Polynom-Ring (n,L)) and
A5: q in the carrier of (Polynom-Ring (n,L)) and
A6: [a,b] = [p,q] by A3, ZFMISC_1:def_2;
reconsider q = q as Polynomial of n,L by A5, POLYNOM1:def_10;
not p in {(0_ (n,L))} by A1, A4, XBOOLE_0:def_5;
then p <> 0_ (n,L) by TARSKI:def_1;
then reconsider p = p as non-zero Polynomial of n,L by A4, POLYNOM1:def_10, POLYNOM7:def_1;
p reduces_to q,{(0_ (n,L))},T by A3, A6, POLYRED:def_13;
then consider g being Polynomial of n,L such that
A7: g in {(0_ (n,L))} and
A8: p reduces_to q,g,T by POLYRED:def_7;
g = 0_ (n,L) by A7, TARSKI:def_1;
then p is_reducible_wrt 0_ (n,L),T by A8, POLYRED:def_8;
hence b,c are_convergent_wrt PolyRedRel ({(0_ (n,L))},T) by Lm3; ::_thesis: verum
end;
then A9: PolyRedRel ({(0_ (n,L))},T) is locally-confluent by REWRITE1:def_24;
{(0_ (n,L))} -Ideal = I by A2, IDEAL_1:44;
hence {(0_ (n,L))} is_Groebner_basis_of I,T by A9, Def4; ::_thesis: verum
end;
supposeA10: I <> {(0_ (n,L))} ; ::_thesis: ex G being finite Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_of I,T
ex q being Element of I st q <> 0_ (n,L)
proof
assume A11: for q being Element of I holds not q <> 0_ (n,L) ; ::_thesis: contradiction
A12: now__::_thesis:_for_u_being_set_st_u_in_{(0__(n,L))}_holds_
u_in_I
let u be set ; ::_thesis: ( u in {(0_ (n,L))} implies u in I )
assume u in {(0_ (n,L))} ; ::_thesis: u in I
then A13: u = 0_ (n,L) by TARSKI:def_1;
now__::_thesis:_u_in_I
assume not u in I ; ::_thesis: contradiction
then for v being set holds not v in I by A11, A13;
hence contradiction by XBOOLE_0:def_1; ::_thesis: verum
end;
hence u in I ; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_I_holds_
u_in_{(0__(n,L))}
let u be set ; ::_thesis: ( u in I implies u in {(0_ (n,L))} )
assume u in I ; ::_thesis: u in {(0_ (n,L))}
then u = 0_ (n,L) by A11;
hence u in {(0_ (n,L))} by TARSKI:def_1; ::_thesis: verum
end;
hence contradiction by A10, A12, TARSKI:1; ::_thesis: verum
end;
then consider q being Element of I such that
A14: q <> 0_ (n,L) ;
set R = RelStr(# (Bags n),(DivOrder n) #);
set hti = HT (I,T);
reconsider hti = HT (I,T) as Subset of RelStr(# (Bags n),(DivOrder n) #) ;
consider S being finite Subset of (Bags n) such that
A15: S is_Dickson-basis_of hti, RelStr(# (Bags n),(DivOrder n) #) by Th34;
set M = { { p where p is Polynomial of n,L : ( p in I & HT (p,T) = s & p <> 0_ (n,L) ) } where s is Element of Bags n : s in S } ;
set s = the Element of S;
reconsider q = q as Polynomial of n,L by POLYNOM1:def_10;
set hq = HT (q,T);
reconsider hq = HT (q,T) as Element of RelStr(# (Bags n),(DivOrder n) #) ;
hq in { (HT (p,T)) where p is Polynomial of n,L : ( p in I & p <> 0_ (n,L) ) } by A14;
then ex b being Element of RelStr(# (Bags n),(DivOrder n) #) st
( b in S & b <= hq ) by A15, DICKSON:def_9;
then the Element of S in S ;
then { r where r is Polynomial of n,L : ( r in I & HT (r,T) = the Element of S & r <> 0_ (n,L) ) } in { { p where p is Polynomial of n,L : ( p in I & HT (p,T) = s9 & p <> 0_ (n,L) ) } where s9 is Element of Bags n : s9 in S } ;
then reconsider M = { { p where p is Polynomial of n,L : ( p in I & HT (p,T) = s & p <> 0_ (n,L) ) } where s is Element of Bags n : s in S } as non empty set ;
A16: for x, y being set st x in M & y in M & x <> y holds
x misses y
proof
let x, y be set ; ::_thesis: ( x in M & y in M & x <> y implies x misses y )
assume that
A17: x in M and
A18: y in M and
A19: x <> y ; ::_thesis: x misses y
consider t being Element of Bags n such that
A20: y = { p where p is Polynomial of n,L : ( p in I & HT (p,T) = t & p <> 0_ (n,L) ) } and
t in S by A18;
consider s being Element of Bags n such that
A21: x = { p where p is Polynomial of n,L : ( p in I & HT (p,T) = s & p <> 0_ (n,L) ) } and
s in S by A17;
now__::_thesis:_not_x_/\_y_<>_{}
set u = the Element of x /\ y;
assume A22: x /\ y <> {} ; ::_thesis: contradiction
then the Element of x /\ y in y by XBOOLE_0:def_4;
then A23: ex v being Polynomial of n,L st
( the Element of x /\ y = v & v in I & HT (v,T) = t & v <> 0_ (n,L) ) by A20;
the Element of x /\ y in x by A22, XBOOLE_0:def_4;
then ex r being Polynomial of n,L st
( the Element of x /\ y = r & r in I & HT (r,T) = s & r <> 0_ (n,L) ) by A21;
hence contradiction by A19, A21, A20, A23; ::_thesis: verum
end;
hence x misses y by XBOOLE_0:def_7; ::_thesis: verum
end;
A24: S c= hti by A15, DICKSON:def_9;
for x being set st x in M holds
x <> {}
proof
let x be set ; ::_thesis: ( x in M implies x <> {} )
assume x in M ; ::_thesis: x <> {}
then consider s being Element of Bags n such that
A25: x = { p where p is Polynomial of n,L : ( p in I & HT (p,T) = s & p <> 0_ (n,L) ) } and
A26: s in S ;
s in hti by A24, A26;
then consider q being Polynomial of n,L such that
A27: ( s = HT (q,T) & q in I & q <> 0_ (n,L) ) ;
q in x by A25, A27;
hence x <> {} ; ::_thesis: verum
end;
then consider G9 being set such that
A28: for x being set st x in M holds
ex y being set st G9 /\ x = {y} by A16, WELLORD2:18;
set xx = the Element of M;
A29: M is finite
proof
defpred S1[ set , set ] means $2 = { p where p is Polynomial of n,L : ( p in I & HT (p,T) = $1 & p <> 0_ (n,L) ) } ;
A30: for x being set st x in S holds
ex y being set st S1[x,y] ;
consider f being Function such that
A31: ( dom f = S & ( for x being set st x in S holds
S1[x,f . x] ) ) from CLASSES1:sch_1(A30);
A32: now__::_thesis:_for_u_being_set_st_u_in_rng_f_holds_
u_in_M
let u be set ; ::_thesis: ( u in rng f implies u in M )
assume u in rng f ; ::_thesis: u in M
then consider s being set such that
A33: s in dom f and
A34: u = f . s by FUNCT_1:def_3;
u = { p where p is Polynomial of n,L : ( p in I & HT (p,T) = s & p <> 0_ (n,L) ) } by A31, A33, A34;
hence u in M by A31, A33; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_M_holds_
u_in_rng_f
let u be set ; ::_thesis: ( u in M implies u in rng f )
assume u in M ; ::_thesis: u in rng f
then consider s being Element of Bags n such that
A35: u = { p where p is Polynomial of n,L : ( p in I & HT (p,T) = s & p <> 0_ (n,L) ) } and
A36: s in S ;
f . s in rng f by A31, A36, FUNCT_1:3;
hence u in rng f by A31, A35, A36; ::_thesis: verum
end;
then rng f = M by A32, TARSKI:1;
hence M is finite by A31, FINSET_1:8; ::_thesis: verum
end;
A37: ex y being set st G9 /\ the Element of M = {y} by A28;
set xx = the Element of M;
reconsider G9 = G9 as non empty set by A37;
set G = { x where x is Element of G9 : ex y being set st
( y in M & G9 /\ y = {x} ) } ;
now__::_thesis:_for_u_being_set_st_u_in__{__x_where_x_is_Element_of_G9_:_ex_y_being_set_st_
(_y_in_M_&_G9_/\_y_=_{x}_)__}__holds_
u_in_the_carrier_of_(Polynom-Ring_(n,L))
let u be set ; ::_thesis: ( u in { x where x is Element of G9 : ex y being set st
( y in M & G9 /\ y = {x} ) } implies u in the carrier of (Polynom-Ring (n,L)) )
assume u in { x where x is Element of G9 : ex y being set st
( y in M & G9 /\ y = {x} ) } ; ::_thesis: u in the carrier of (Polynom-Ring (n,L))
then consider x being Element of G9 such that
A38: u = x and
A39: ex y being set st
( y in M & G9 /\ y = {x} ) ;
consider y being set such that
A40: y in M and
A41: G9 /\ y = {x} by A39;
consider s being Element of Bags n such that
A42: y = { p where p is Polynomial of n,L : ( p in I & HT (p,T) = s & p <> 0_ (n,L) ) } and
s in S by A40;
x in G9 /\ y by A41, TARSKI:def_1;
then x in y by XBOOLE_0:def_4;
then ex q being Polynomial of n,L st
( x = q & q in I & HT (q,T) = s & q <> 0_ (n,L) ) by A42;
hence u in the carrier of (Polynom-Ring (n,L)) by A38; ::_thesis: verum
end;
then reconsider G = { x where x is Element of G9 : ex y being set st
( y in M & G9 /\ y = {x} ) } as Subset of (Polynom-Ring (n,L)) by TARSKI:def_3;
defpred S1[ set , set ] means ( G9 /\ $1 = {$2} & $2 in G );
A43: for x being set st x in M holds
ex y being set st S1[x,y]
proof
let x be set ; ::_thesis: ( x in M implies ex y being set st S1[x,y] )
assume A44: x in M ; ::_thesis: ex y being set st S1[x,y]
then consider y being set such that
A45: G9 /\ x = {y} by A28;
y in G9 /\ x by A45, TARSKI:def_1;
then reconsider y = y as Element of G9 by XBOOLE_0:def_4;
y in G by A44, A45;
hence ex y being set st S1[x,y] by A45; ::_thesis: verum
end;
consider f being Function such that
A46: ( dom f = M & ( for x being set st x in M holds
S1[x,f . x] ) ) from CLASSES1:sch_1(A43);
A47: now__::_thesis:_for_u_being_set_st_u_in_G_holds_
u_in_rng_f
let u be set ; ::_thesis: ( u in G implies u in rng f )
assume u in G ; ::_thesis: u in rng f
then consider x being Element of G9 such that
A48: u = x and
A49: ex y being set st
( y in M & G9 /\ y = {x} ) ;
consider y being set such that
A50: y in M and
A51: G9 /\ y = {x} by A49;
G9 /\ y = {(f . y)} by A46, A50;
then A52: x in {(f . y)} by A51, TARSKI:def_1;
f . y in rng f by A46, A50, FUNCT_1:3;
hence u in rng f by A48, A52, TARSKI:def_1; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_rng_f_holds_
u_in_G
let u be set ; ::_thesis: ( u in rng f implies u in G )
assume u in rng f ; ::_thesis: u in G
then ex s being set st
( s in dom f & u = f . s ) by FUNCT_1:def_3;
hence u in G by A46; ::_thesis: verum
end;
then A53: rng f = G by A47, TARSKI:1;
ex y being set st G9 /\ the Element of M = {y} by A28;
then reconsider G = G as non empty finite Subset of (Polynom-Ring (n,L)) by A29, A46, A53, FINSET_1:8;
for b being bag of n st b in HT (I,T) holds
ex b9 being bag of n st
( b9 in HT (G,T) & b9 divides b )
proof
let b be bag of n; ::_thesis: ( b in HT (I,T) implies ex b9 being bag of n st
( b9 in HT (G,T) & b9 divides b ) )
reconsider bb = b as Element of RelStr(# (Bags n),(DivOrder n) #) by PRE_POLY:def_12;
assume b in HT (I,T) ; ::_thesis: ex b9 being bag of n st
( b9 in HT (G,T) & b9 divides b )
then consider bb9 being Element of RelStr(# (Bags n),(DivOrder n) #) such that
A54: bb9 in S and
A55: bb9 <= bb by A15, DICKSON:def_9;
set N = { p where p is Polynomial of n,L : ( p in I & HT (p,T) = bb9 & p <> 0_ (n,L) ) } ;
A56: { p where p is Polynomial of n,L : ( p in I & HT (p,T) = bb9 & p <> 0_ (n,L) ) } in M by A54;
then consider y being set such that
A57: G9 /\ { p where p is Polynomial of n,L : ( p in I & HT (p,T) = bb9 & p <> 0_ (n,L) ) } = {y} by A28;
reconsider b9 = bb9 as bag of n ;
take b9 ; ::_thesis: ( b9 in HT (G,T) & b9 divides b )
A58: [bb9,bb] in DivOrder n by A55, ORDERS_2:def_5;
A59: y in G9 /\ { p where p is Polynomial of n,L : ( p in I & HT (p,T) = bb9 & p <> 0_ (n,L) ) } by A57, TARSKI:def_1;
then reconsider y = y as Element of G9 by XBOOLE_0:def_4;
y in { p where p is Polynomial of n,L : ( p in I & HT (p,T) = bb9 & p <> 0_ (n,L) ) } by A59, XBOOLE_0:def_4;
then A60: ex r being Polynomial of n,L st
( y = r & r in I & HT (r,T) = bb9 & r <> 0_ (n,L) ) ;
y in G by A56, A57;
hence ( b9 in HT (G,T) & b9 divides b ) by A58, A60, Def5; ::_thesis: verum
end;
then A61: HT (I,T) c= multiples (HT (G,T)) by Th28;
take G ; ::_thesis: G is_Groebner_basis_of I,T
now__::_thesis:_for_u_being_set_st_u_in_G_holds_
u_in_I
let u be set ; ::_thesis: ( u in G implies u in I )
assume u in G ; ::_thesis: u in I
then consider x being Element of G9 such that
A62: u = x and
A63: ex y being set st
( y in M & G9 /\ y = {x} ) ;
consider y being set such that
A64: y in M and
A65: G9 /\ y = {x} by A63;
consider s being Element of Bags n such that
A66: y = { p where p is Polynomial of n,L : ( p in I & HT (p,T) = s & p <> 0_ (n,L) ) } and
s in S by A64;
x in G9 /\ y by A65, TARSKI:def_1;
then x in y by XBOOLE_0:def_4;
then ex q being Polynomial of n,L st
( x = q & q in I & HT (q,T) = s & q <> 0_ (n,L) ) by A66;
hence u in I by A62; ::_thesis: verum
end;
then G c= I by TARSKI:def_3;
hence G is_Groebner_basis_of I,T by A61, Th29; ::_thesis: verum
end;
end;
end;
Lm9: for L being non empty right_complementable associative well-unital distributive add-associative right_zeroed left_zeroed doubleLoopStr
for A, B being non empty Subset of L st B = A \ {(0. L)} holds
for f being LinearCombination of A
for u being set st u = Sum f holds
ex g being LinearCombination of B st u = Sum g
proof
let L be non empty right_complementable associative well-unital distributive add-associative right_zeroed left_zeroed doubleLoopStr ; ::_thesis: for A, B being non empty Subset of L st B = A \ {(0. L)} holds
for f being LinearCombination of A
for u being set st u = Sum f holds
ex g being LinearCombination of B st u = Sum g
let A, B be non empty Subset of L; ::_thesis: ( B = A \ {(0. L)} implies for f being LinearCombination of A
for u being set st u = Sum f holds
ex g being LinearCombination of B st u = Sum g )
defpred S1[ Element of NAT ] means for f being LinearCombination of A st len f = $1 holds
for u being set st u = Sum f holds
ex g being LinearCombination of B st u = Sum g;
assume A1: B = A \ {(0. L)} ; ::_thesis: for f being LinearCombination of A
for u being set st u = Sum f holds
ex g being LinearCombination of B st u = Sum g
A2: now__::_thesis:_for_k_being_Element_of_NAT_st_S1[k]_holds_
S1[k_+_1]
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A3: S1[k] ; ::_thesis: S1[k + 1]
for f being LinearCombination of A st len f = k + 1 holds
for u being set st u = Sum f holds
ex g being LinearCombination of B st u = Sum g
proof
let f be LinearCombination of A; ::_thesis: ( len f = k + 1 implies for u being set st u = Sum f holds
ex g being LinearCombination of B st u = Sum g )
set g = f | (Seg k);
reconsider g = f | (Seg k) as FinSequence by FINSEQ_1:15;
A4: rng f c= the carrier of L by FINSEQ_1:def_4;
set h = f /. (len f);
assume A5: len f = k + 1 ; ::_thesis: for u being set st u = Sum f holds
ex g being LinearCombination of B st u = Sum g
then 1 <= len f by NAT_1:12;
then len f in Seg (len f) by FINSEQ_1:1;
then A6: len f in dom f by FINSEQ_1:def_3;
then A7: f /. (len f) = f . (len f) by PARTFUN1:def_6;
A8: k <= k + 1 by NAT_1:12;
then A9: len g = k by A5, FINSEQ_1:17;
A10: dom g = Seg k by A5, A8, FINSEQ_1:17;
A11: dom f = Seg (k + 1) by A5, FINSEQ_1:def_3;
then A12: dom g c= dom f by A8, A10, FINSEQ_1:5;
now__::_thesis:_for_u_being_set_st_u_in_rng_g_holds_
u_in_the_carrier_of_L
let u be set ; ::_thesis: ( u in rng g implies u in the carrier of L )
assume u in rng g ; ::_thesis: u in the carrier of L
then consider x being set such that
A13: x in dom g and
A14: u = g . x by FUNCT_1:def_3;
g . x = f . x by A13, FUNCT_1:47;
then u in rng f by A12, A13, A14, FUNCT_1:def_3;
hence u in the carrier of L by A4; ::_thesis: verum
end;
then rng g c= the carrier of L by TARSKI:def_3;
then reconsider g = g as FinSequence of the carrier of L by FINSEQ_1:def_4;
for i being set st i in dom g holds
ex u, v being Element of L ex a being Element of A st g /. i = (u * a) * v
proof
let i be set ; ::_thesis: ( i in dom g implies ex u, v being Element of L ex a being Element of A st g /. i = (u * a) * v )
assume A15: i in dom g ; ::_thesis: ex u, v being Element of L ex a being Element of A st g /. i = (u * a) * v
then reconsider i = i as Element of NAT ;
i <= k by A10, A15, FINSEQ_1:1;
then A16: i <= k + 1 by NAT_1:12;
1 <= i by A10, A15, FINSEQ_1:1;
then A17: i in dom f by A11, A16, FINSEQ_1:1;
g /. i = g . i by A15, PARTFUN1:def_6
.= f . i by A15, FUNCT_1:47
.= f /. i by A17, PARTFUN1:def_6 ;
hence ex u, v being Element of L ex a being Element of A st g /. i = (u * a) * v by A17, IDEAL_1:def_8; ::_thesis: verum
end;
then reconsider g = g as LinearCombination of A by IDEAL_1:def_8;
consider g9 being LinearCombination of B such that
A18: Sum g = Sum g9 by A3, A9;
let u be set ; ::_thesis: ( u = Sum f implies ex g being LinearCombination of B st u = Sum g )
assume A19: u = Sum f ; ::_thesis: ex g being LinearCombination of B st u = Sum g
A20: len f = (len g) + 1 by A5, A8, FINSEQ_1:17;
then A21: Sum f = (Sum g) + (f /. (len f)) by A10, A7, RLVECT_1:38;
now__::_thesis:_(_(_f_/._(len_f)_=_0._L_&_Sum_f_=_Sum_g_)_or_(_f_/._(len_f)_<>_0._L_&_ex_g_being_LinearCombination_of_B_st_u_=_Sum_g_)_)
percases ( f /. (len f) = 0. L or f /. (len f) <> 0. L ) ;
case f /. (len f) = 0. L ; ::_thesis: Sum f = Sum g
hence Sum f = Sum g by A21, RLVECT_1:def_4; ::_thesis: verum
end;
caseA22: f /. (len f) <> 0. L ; ::_thesis: ex g being LinearCombination of B st u = Sum g
set l = g9 ^ <*(f /. (len f))*>;
for i being set st i in dom (g9 ^ <*(f /. (len f))*>) holds
ex u, v being Element of L ex a being Element of B st (g9 ^ <*(f /. (len f))*>) /. i = (u * a) * v
proof
let i be set ; ::_thesis: ( i in dom (g9 ^ <*(f /. (len f))*>) implies ex u, v being Element of L ex a being Element of B st (g9 ^ <*(f /. (len f))*>) /. i = (u * a) * v )
assume A23: i in dom (g9 ^ <*(f /. (len f))*>) ; ::_thesis: ex u, v being Element of L ex a being Element of B st (g9 ^ <*(f /. (len f))*>) /. i = (u * a) * v
then reconsider i = i as Element of NAT ;
A24: len (g9 ^ <*(f /. (len f))*>) = (len g9) + (len <*(f /. (len f))*>) by FINSEQ_1:22
.= (len g9) + 1 by FINSEQ_1:39 ;
now__::_thesis:_(_(_i_=_len_(g9_^_<*(f_/._(len_f))*>)_&_ex_u,_v_being_Element_of_L_ex_a_being_Element_of_B_st_(g9_^_<*(f_/._(len_f))*>)_/._i_=_(u_*_a)_*_v_)_or_(_i_<>_len_(g9_^_<*(f_/._(len_f))*>)_&_ex_u,_v_being_Element_of_L_ex_a_being_Element_of_B_st_(g9_^_<*(f_/._(len_f))*>)_/._i_=_(u_*_a)_*_v_)_)
percases ( i = len (g9 ^ <*(f /. (len f))*>) or i <> len (g9 ^ <*(f /. (len f))*>) ) ;
caseA25: i = len (g9 ^ <*(f /. (len f))*>) ; ::_thesis: ex u, v being Element of L ex a being Element of B st (g9 ^ <*(f /. (len f))*>) /. i = (u * a) * v
consider u, v being Element of L, a being Element of A such that
A26: f /. (len f) = (u * a) * v by A6, IDEAL_1:def_8;
A27: now__::_thesis:_a_in_B
assume not a in B ; ::_thesis: contradiction
then a in {(0. L)} by A1, XBOOLE_0:def_5;
then a = 0. L by TARSKI:def_1;
then (u * a) * v = (0. L) * v by VECTSP_1:6
.= 0. L by VECTSP_1:7 ;
hence contradiction by A22, A26; ::_thesis: verum
end;
(g9 ^ <*(f /. (len f))*>) /. i = (g9 ^ <*(f /. (len f))*>) . i by A23, PARTFUN1:def_6
.= f /. (len f) by A24, A25, FINSEQ_1:42 ;
hence ex u, v being Element of L ex a being Element of B st (g9 ^ <*(f /. (len f))*>) /. i = (u * a) * v by A26, A27; ::_thesis: verum
end;
caseA28: i <> len (g9 ^ <*(f /. (len f))*>) ; ::_thesis: ex u, v being Element of L ex a being Element of B st (g9 ^ <*(f /. (len f))*>) /. i = (u * a) * v
A29: i in Seg (len (g9 ^ <*(f /. (len f))*>)) by A23, FINSEQ_1:def_3;
then i <= len (g9 ^ <*(f /. (len f))*>) by FINSEQ_1:1;
then i < len (g9 ^ <*(f /. (len f))*>) by A28, XXREAL_0:1;
then A30: i <= len g9 by A24, NAT_1:13;
1 <= i by A29, FINSEQ_1:1;
then i in Seg (len g9) by A30, FINSEQ_1:1;
then A31: i in dom g9 by FINSEQ_1:def_3;
(g9 ^ <*(f /. (len f))*>) /. i = (g9 ^ <*(f /. (len f))*>) . i by A23, PARTFUN1:def_6
.= g9 . i by A31, FINSEQ_1:def_7
.= g9 /. i by A31, PARTFUN1:def_6 ;
hence ex u, v being Element of L ex a being Element of B st (g9 ^ <*(f /. (len f))*>) /. i = (u * a) * v by A31, IDEAL_1:def_8; ::_thesis: verum
end;
end;
end;
hence ex u, v being Element of L ex a being Element of B st (g9 ^ <*(f /. (len f))*>) /. i = (u * a) * v ; ::_thesis: verum
end;
then reconsider l = g9 ^ <*(f /. (len f))*> as LinearCombination of B by IDEAL_1:def_8;
Sum l = (Sum g9) + (Sum <*(f /. (len f))*>) by RLVECT_1:41
.= Sum f by A10, A18, A7, A20, RLVECT_1:38, RLVECT_1:44 ;
hence ex g being LinearCombination of B st u = Sum g by A19; ::_thesis: verum
end;
end;
end;
hence ex g being LinearCombination of B st u = Sum g by A19, A18; ::_thesis: verum
end;
hence S1[k + 1] ; ::_thesis: verum
end;
let f be LinearCombination of A; ::_thesis: for u being set st u = Sum f holds
ex g being LinearCombination of B st u = Sum g
let u be set ; ::_thesis: ( u = Sum f implies ex g being LinearCombination of B st u = Sum g )
assume A32: u = Sum f ; ::_thesis: ex g being LinearCombination of B st u = Sum g
A33: ex n being Element of NAT st len f = n ;
A34: S1[ 0 ]
proof
set g = <*> the carrier of L;
reconsider g = <*> the carrier of L as FinSequence of the carrier of L ;
for i being set st i in dom g holds
ex u, v being Element of L ex a being Element of B st g /. i = (u * a) * v ;
then reconsider g = g as LinearCombination of B by IDEAL_1:def_8;
let f be LinearCombination of A; ::_thesis: ( len f = 0 implies for u being set st u = Sum f holds
ex g being LinearCombination of B st u = Sum g )
A35: g = <*> the carrier of L ;
assume len f = 0 ; ::_thesis: for u being set st u = Sum f holds
ex g being LinearCombination of B st u = Sum g
then A36: f = <*> the carrier of L ;
let u be set ; ::_thesis: ( u = Sum f implies ex g being LinearCombination of B st u = Sum g )
assume u = Sum f ; ::_thesis: ex g being LinearCombination of B st u = Sum g
hence ex g being LinearCombination of B st u = Sum g by A36, A35; ::_thesis: verum
end;
for k being Element of NAT holds S1[k] from NAT_1:sch_1(A34, A2);
hence ex g being LinearCombination of B st u = Sum g by A32, A33; ::_thesis: verum
end;
theorem :: GROEB_1:36
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)) st I <> {(0_ (n,L))} holds
ex G being finite Subset of (Polynom-Ring (n,L)) st
( G is_Groebner_basis_of I,T & not 0_ (n,L) in G )
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)) st I <> {(0_ (n,L))} holds
ex G being finite Subset of (Polynom-Ring (n,L)) st
( G is_Groebner_basis_of I,T & not 0_ (n,L) in G )
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)) st I <> {(0_ (n,L))} holds
ex G being finite Subset of (Polynom-Ring (n,L)) st
( G is_Groebner_basis_of I,T & not 0_ (n,L) in G )
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)) st I <> {(0_ (n,L))} holds
ex G being finite Subset of (Polynom-Ring (n,L)) st
( G is_Groebner_basis_of I,T & not 0_ (n,L) in G )
let I be non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)); ::_thesis: ( I <> {(0_ (n,L))} implies ex G being finite Subset of (Polynom-Ring (n,L)) st
( G is_Groebner_basis_of I,T & not 0_ (n,L) in G ) )
assume A1: I <> {(0_ (n,L))} ; ::_thesis: ex G being finite Subset of (Polynom-Ring (n,L)) st
( G is_Groebner_basis_of I,T & not 0_ (n,L) in G )
A2: 0_ (n,L) = 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10;
consider G being finite Subset of (Polynom-Ring (n,L)) such that
A3: G is_Groebner_basis_of I,T by Th35;
set R = PolyRedRel (G,T);
A4: G -Ideal = I by A3, Def4;
A5: PolyRedRel (G,T) is locally-confluent by A3, Def4;
set G9 = G \ {(0_ (n,L))};
set R9 = PolyRedRel ((G \ {(0_ (n,L))}),T);
reconsider G9 = G \ {(0_ (n,L))} as finite Subset of (Polynom-Ring (n,L)) ;
A6: now__::_thesis:_(_(_G9_=_{}_&_G9_-Ideal_=_I_)_or_(_G9_<>_{}_&_G9_-Ideal_=_I_)_)
percases ( G9 = {} or G9 <> {} ) ;
caseA7: G9 = {} ; ::_thesis: G9 -Ideal = I
now__::_thesis:_(_(_G_=_{}_&_G9_-Ideal_=_I_)_or_(_G_<>_{}_&_G9_-Ideal_=_I_)_)
percases ( G = {} or G <> {} ) ;
case G = {} ; ::_thesis: G9 -Ideal = I
hence G9 -Ideal = I by A3, Def4; ::_thesis: verum
end;
caseA8: G <> {} ; ::_thesis: G9 -Ideal = I
A9: now__::_thesis:_for_u_being_set_st_u_in_{(0__(n,L))}_holds_
u_in_G
let u be set ; ::_thesis: ( u in {(0_ (n,L))} implies u in G )
assume u in {(0_ (n,L))} ; ::_thesis: u in G
then A10: u = 0_ (n,L) by TARSKI:def_1;
A11: G c= {(0_ (n,L))} by A7, XBOOLE_1:37;
now__::_thesis:_(_not_u_in_G_implies_G_=_{}_)
assume not u in G ; ::_thesis: G = {}
then for v being set holds not v in G by A10, A11, TARSKI:def_1;
hence G = {} by XBOOLE_0:def_1; ::_thesis: verum
end;
hence u in G by A8; ::_thesis: verum
end;
A12: 0_ (n,L) = 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10;
now__::_thesis:_for_u_being_set_st_u_in_G_holds_
u_in_{(0__(n,L))}
let u be set ; ::_thesis: ( u in G implies u in {(0_ (n,L))} )
assume A13: u in G ; ::_thesis: u in {(0_ (n,L))}
G c= {(0_ (n,L))} by A7, XBOOLE_1:37;
hence u in {(0_ (n,L))} by A13; ::_thesis: verum
end;
then G = {(0_ (n,L))} by A9, TARSKI:1;
hence G9 -Ideal = I by A1, A4, A12, IDEAL_1:44; ::_thesis: verum
end;
end;
end;
hence G9 -Ideal = I ; ::_thesis: verum
end;
case G9 <> {} ; ::_thesis: G9 -Ideal = I
then reconsider GG = G, GG9 = G9 as non empty Subset of (Polynom-Ring (n,L)) ;
A14: 0. (Polynom-Ring (n,L)) = 0_ (n,L) by POLYNOM1:def_10;
A15: now__::_thesis:_for_u_being_set_st_u_in_G_-Ideal_holds_
u_in_G9_-Ideal
let u be set ; ::_thesis: ( u in G -Ideal implies u in G9 -Ideal )
assume u in G -Ideal ; ::_thesis: u in G9 -Ideal
then ex f being LinearCombination of GG st u = Sum f by IDEAL_1:60;
then ex g being LinearCombination of GG9 st u = Sum g by A14, Lm9;
hence u in G9 -Ideal by IDEAL_1:60; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_G9_-Ideal_holds_
u_in_G_-Ideal
let u be set ; ::_thesis: ( u in G9 -Ideal implies u in G -Ideal )
A16: GG9 -Ideal c= GG -Ideal by IDEAL_1:57, XBOOLE_1:36;
assume u in G9 -Ideal ; ::_thesis: u in G -Ideal
hence u in G -Ideal by A16; ::_thesis: verum
end;
hence G9 -Ideal = I by A4, A15, TARSKI:1; ::_thesis: verum
end;
end;
end;
A17: now__::_thesis:_not_0__(n,L)_in_G9
assume 0_ (n,L) in G9 ; ::_thesis: contradiction
then not 0_ (n,L) in {(0_ (n,L))} by XBOOLE_0:def_5;
hence contradiction by TARSKI:def_1; ::_thesis: verum
end;
A18: for u being set st u in PolyRedRel (G,T) holds
u in PolyRedRel ((G \ {(0_ (n,L))}),T)
proof
let u be set ; ::_thesis: ( u in PolyRedRel (G,T) implies u in PolyRedRel ((G \ {(0_ (n,L))}),T) )
assume A19: u in PolyRedRel (G,T) ; ::_thesis: u in PolyRedRel ((G \ {(0_ (n,L))}),T)
then consider p, q being set such that
A20: p in NonZero (Polynom-Ring (n,L)) and
A21: q in the carrier of (Polynom-Ring (n,L)) and
A22: u = [p,q] by ZFMISC_1:def_2;
reconsider q = q as Polynomial of n,L by A21, POLYNOM1:def_10;
not p in {(0_ (n,L))} by A2, A20, XBOOLE_0:def_5;
then p <> 0_ (n,L) by TARSKI:def_1;
then reconsider p = p as non-zero Polynomial of n,L by A20, POLYNOM1:def_10, POLYNOM7:def_1;
p reduces_to q,G,T by A19, A22, POLYRED:def_13;
then consider f being Polynomial of n,L such that
A23: f in G and
A24: p reduces_to q,f,T by POLYRED:def_7;
ex b being bag of n st p reduces_to q,f,b,T by A24, POLYRED:def_6;
then f <> 0_ (n,L) by POLYRED:def_5;
then not f in {(0_ (n,L))} by TARSKI:def_1;
then f in G9 by A23, XBOOLE_0:def_5;
then p reduces_to q,G9,T by A24, POLYRED:def_7;
hence u in PolyRedRel ((G \ {(0_ (n,L))}),T) by A22, POLYRED:def_13; ::_thesis: verum
end;
PolyRedRel ((G \ {(0_ (n,L))}),T) c= PolyRedRel (G,T) by Th4, XBOOLE_1:36;
then for u being set st u in PolyRedRel ((G \ {(0_ (n,L))}),T) holds
u in PolyRedRel (G,T) ;
then PolyRedRel ((G \ {(0_ (n,L))}),T) is locally-confluent by A5, A18, TARSKI:1;
then G9 is_Groebner_basis_of I,T by A6, Def4;
hence ex G being finite Subset of (Polynom-Ring (n,L)) st
( G is_Groebner_basis_of I,T & not 0_ (n,L) in G ) by A17; ::_thesis: verum
end;
definition
let n be Ordinal;
let T be connected TermOrder of n;
let L be non empty multLoopStr_0 ;
let p be Polynomial of n,L;
predp is_monic_wrt T means :Def6: :: GROEB_1:def 6
HC (p,T) = 1. L;
end;
:: deftheorem Def6 defines is_monic_wrt GROEB_1:def_6_:_
for n being Ordinal
for T being connected TermOrder of n
for L being non empty multLoopStr_0
for p being Polynomial of n,L holds
( p is_monic_wrt T iff HC (p,T) = 1. L );
definition
let n be Ordinal;
let T be connected TermOrder of n;
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ;
let P be Subset of (Polynom-Ring (n,L));
predP is_reduced_wrt T means :Def7: :: GROEB_1:def 7
for p being Polynomial of n,L st p in P holds
( p is_monic_wrt T & p is_irreducible_wrt P \ {p},T );
end;
:: deftheorem Def7 defines is_reduced_wrt GROEB_1:def_7_:_
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L)) holds
( P is_reduced_wrt T iff for p being Polynomial of n,L st p in P holds
( p is_monic_wrt T & p is_irreducible_wrt P \ {p},T ) );
notation
let n be Ordinal;
let T be connected TermOrder of n;
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ;
let P be Subset of (Polynom-Ring (n,L));
synonym P is_autoreduced_wrt T for P is_reduced_wrt T;
end;
theorem Th37: :: GROEB_1:37
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being add-closed left-ideal Subset of (Polynom-Ring (n,L))
for m being Monomial of n,L
for f, g being Polynomial of n,L st f in I & g in I & HM (f,T) = m & HM (g,T) = m & ( for p being Polynomial of n,L holds
( not p in I or not p < f,T or not HM (p,T) = m ) ) & ( for p being Polynomial of n,L holds
( not p in I or not p < g,T or not HM (p,T) = m ) ) holds
f = g
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being add-closed left-ideal Subset of (Polynom-Ring (n,L))
for m being Monomial of n,L
for f, g being Polynomial of n,L st f in I & g in I & HM (f,T) = m & HM (g,T) = m & ( for p being Polynomial of n,L holds
( not p in I or not p < f,T or not HM (p,T) = m ) ) & ( for p being Polynomial of n,L holds
( not p in I or not p < g,T or not HM (p,T) = m ) ) holds
f = g
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being add-closed left-ideal Subset of (Polynom-Ring (n,L))
for m being Monomial of n,L
for f, g being Polynomial of n,L st f in I & g in I & HM (f,T) = m & HM (g,T) = m & ( for p being Polynomial of n,L holds
( not p in I or not p < f,T or not HM (p,T) = m ) ) & ( for p being Polynomial of n,L holds
( not p in I or not p < g,T or not HM (p,T) = m ) ) holds
f = g
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for I being add-closed left-ideal Subset of (Polynom-Ring (n,L))
for m being Monomial of n,L
for f, g being Polynomial of n,L st f in I & g in I & HM (f,T) = m & HM (g,T) = m & ( for p being Polynomial of n,L holds
( not p in I or not p < f,T or not HM (p,T) = m ) ) & ( for p being Polynomial of n,L holds
( not p in I or not p < g,T or not HM (p,T) = m ) ) holds
f = g
let I be add-closed left-ideal Subset of (Polynom-Ring (n,L)); ::_thesis: for m being Monomial of n,L
for f, g being Polynomial of n,L st f in I & g in I & HM (f,T) = m & HM (g,T) = m & ( for p being Polynomial of n,L holds
( not p in I or not p < f,T or not HM (p,T) = m ) ) & ( for p being Polynomial of n,L holds
( not p in I or not p < g,T or not HM (p,T) = m ) ) holds
f = g
let m be Monomial of n,L; ::_thesis: for f, g being Polynomial of n,L st f in I & g in I & HM (f,T) = m & HM (g,T) = m & ( for p being Polynomial of n,L holds
( not p in I or not p < f,T or not HM (p,T) = m ) ) & ( for p being Polynomial of n,L holds
( not p in I or not p < g,T or not HM (p,T) = m ) ) holds
f = g
let f, g be Polynomial of n,L; ::_thesis: ( f in I & g in I & HM (f,T) = m & HM (g,T) = m & ( for p being Polynomial of n,L holds
( not p in I or not p < f,T or not HM (p,T) = m ) ) & ( for p being Polynomial of n,L holds
( not p in I or not p < g,T or not HM (p,T) = m ) ) implies f = g )
assume that
A1: f in I and
A2: g in I and
A3: HM (f,T) = m and
A4: HM (g,T) = m ; ::_thesis: ( ex p being Polynomial of n,L st
( p in I & p < f,T & HM (p,T) = m ) or ex p being Polynomial of n,L st
( p in I & p < g,T & HM (p,T) = m ) or f = g )
A5: HT (f,T) = term (HM (f,T)) by TERMORD:22
.= HT (g,T) by A3, A4, TERMORD:22 ;
A6: HC (f,T) = f . (HT (f,T)) by TERMORD:def_7
.= (HM (f,T)) . (HT (f,T)) by TERMORD:18
.= g . (HT (g,T)) by A3, A4, A5, TERMORD:18
.= HC (g,T) by TERMORD:def_7 ;
assume that
A7: for p being Polynomial of n,L holds
( not p in I or not p < f,T or not HM (p,T) = m ) and
A8: for p being Polynomial of n,L holds
( not p in I or not p < g,T or not HM (p,T) = m ) ; ::_thesis: f = g
reconsider I = I as LeftIdeal of (Polynom-Ring (n,L)) by A1;
percases ( f - g = 0_ (n,L) or f - g <> 0_ (n,L) ) ;
suppose f - g = 0_ (n,L) ; ::_thesis: f = g
hence g = (f - g) + g by POLYRED:2
.= (f + (- g)) + g by POLYNOM1:def_6
.= f + ((- g) + g) by POLYNOM1:21
.= f + (0_ (n,L)) by POLYRED:3
.= f by POLYNOM1:23 ;
::_thesis: verum
end;
supposeA9: f - g <> 0_ (n,L) ; ::_thesis: f = g
now__::_thesis:_(_(_f_=_0__(n,L)_&_f_=_g_)_or_(_g_=_0__(n,L)_&_f_=_g_)_or_(_f_<>_0__(n,L)_&_g_<>_0__(n,L)_&_contradiction_)_)
percases ( f = 0_ (n,L) or g = 0_ (n,L) or ( f <> 0_ (n,L) & g <> 0_ (n,L) ) ) ;
caseA10: f = 0_ (n,L) ; ::_thesis: f = g
then HC (g,T) = 0. L by A6, TERMORD:17;
hence f = g by A10, TERMORD:17; ::_thesis: verum
end;
caseA11: g = 0_ (n,L) ; ::_thesis: f = g
then HC (f,T) = 0. L by A6, TERMORD:17;
hence f = g by A11, TERMORD:17; ::_thesis: verum
end;
caseA12: ( f <> 0_ (n,L) & g <> 0_ (n,L) ) ; ::_thesis: contradiction
set s = HT ((f - g),T);
set d = (f . (HT ((f - g),T))) - (g . (HT ((f - g),T)));
set c = (f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ");
set h = f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g));
A13: Support (f - g) <> {} by A9, POLYNOM7:1;
then A14: HT ((f - g),T) in Support (f - g) by TERMORD:def_6;
A15: now__::_thesis:_not_HT_((f_-_g),T)_=_HT_(f,T)
assume HT ((f - g),T) = HT (f,T) ; ::_thesis: contradiction
then (f - g) . (HT ((f - g),T)) = (f + (- g)) . (HT (f,T)) by POLYNOM1:def_6
.= (f . (HT (f,T))) + ((- g) . (HT (g,T))) by A5, POLYNOM1:15
.= (f . (HT (f,T))) + (- (g . (HT (g,T)))) by POLYNOM1:17
.= (HC (f,T)) + (- (g . (HT (g,T)))) by TERMORD:def_7
.= (HC (f,T)) + (- (HC (g,T))) by TERMORD:def_7
.= 0. L by A6, RLVECT_1:5 ;
hence contradiction by A14, POLYNOM1:def_3; ::_thesis: verum
end;
HT ((f - g),T) <= max ((HT (f,T)),(HT (f,T)),T),T by A5, Th7;
then HT ((f - g),T) <= HT (f,T),T by TERMORD:12;
then HT ((f - g),T) < HT (f,T),T by A15, TERMORD:def_3;
then not HT (f,T) <= HT ((f - g),T),T by TERMORD:5;
then not HT (f,T) in Support (f - g) by TERMORD:def_6;
then A16: (f - g) . (HT (f,T)) = 0. L by POLYNOM1:def_3;
A17: (f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))) . (HT (f,T)) = (f + (- (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g)))) . (HT (f,T)) by POLYNOM1:def_6
.= (f . (HT (f,T))) + ((- (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))) . (HT (f,T))) by POLYNOM1:15
.= (f . (HT (f,T))) + (- ((((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g)) . (HT (f,T)))) by POLYNOM1:17
.= (f . (HT (f,T))) + (- (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (0. L))) by A16, POLYNOM7:def_9
.= (f . (HT (f,T))) + (- (0. L)) by VECTSP_1:7
.= (f . (HT (f,T))) + (0. L) by RLVECT_1:12
.= f . (HT (f,T)) by RLVECT_1:def_4 ;
Support f <> {} by A12, POLYNOM7:1;
then HT (f,T) in Support f by TERMORD:def_6;
then (f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))) . (HT (f,T)) <> 0. L by A17, POLYNOM1:def_3;
then A18: HT (f,T) in Support (f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))) by POLYNOM1:def_3;
then A19: HT (f,T) <= HT ((f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))),T),T by TERMORD:def_6;
Support (f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))) = Support (f + (- (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g)))) by POLYNOM1:def_6;
then Support (f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))) c= (Support f) \/ (Support (- (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g)))) by POLYNOM1:20;
then A20: Support (f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))) c= (Support f) \/ (Support (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))) by Th5;
(Support f) \/ (Support (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))) c= (Support f) \/ (Support (f - g)) by POLYRED:19, XBOOLE_1:9;
then A21: Support (f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))) c= (Support f) \/ (Support (f - g)) by A20, XBOOLE_1:1;
not g < f,T by A2, A4, A7;
then A22: f <= g,T by POLYRED:29;
not f < g,T by A1, A3, A8;
then g <= f,T by POLYRED:29;
then A23: Support f = Support g by A22, POLYRED:26;
( Support (f - g) = Support (f + (- g)) & Support (f + (- g)) c= (Support f) \/ (Support (- g)) ) by POLYNOM1:20, POLYNOM1:def_6;
then A24: Support (f - g) c= (Support f) \/ (Support g) by Th5;
then A25: (Support f) \/ (Support (f - g)) c= (Support f) \/ (Support f) by A23, XBOOLE_1:9;
then A26: Support (f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))) c= Support f by A21, XBOOLE_1:1;
HT ((f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))),T) in Support (f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))) by A18, TERMORD:def_6;
then HT ((f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))),T) <= HT (f,T),T by A26, TERMORD:def_6;
then A27: HT ((f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))),T) = HT (f,T) by A19, TERMORD:7;
then HC ((f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))),T) = f . (HT (f,T)) by A17, TERMORD:def_7
.= HC (f,T) by TERMORD:def_7 ;
then A28: HM ((f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))),T) = Monom ((HC (f,T)),(HT (f,T))) by A27, TERMORD:def_8
.= m by A3, TERMORD:def_8 ;
reconsider cp = (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) | (n,L)) *' (f - g) as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider cc = ((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) | (n,L) as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider f9 = f, g9 = g as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
A29: (f - g) . (HT ((f - g),T)) = (f + (- g)) . (HT ((f - g),T)) by POLYNOM1:def_6
.= (f . (HT ((f - g),T))) + ((- g) . (HT ((f - g),T))) by POLYNOM1:15
.= (f . (HT ((f - g),T))) + (- (g . (HT ((f - g),T)))) by POLYNOM1:17
.= (f . (HT ((f - g),T))) - (g . (HT ((f - g),T))) by RLVECT_1:def_11 ;
A30: HT ((f - g),T) in Support (f - g) by A13, TERMORD:def_6;
A31: now__::_thesis:_not_Support_(f_-_(((f_._(HT_((f_-_g),T)))_*_(((f_._(HT_((f_-_g),T)))_-_(g_._(HT_((f_-_g),T))))_"))_*_(f_-_g)))_=_Support_f
A32: (f - g) . (HT ((f - g),T)) <> 0. L by A30, POLYNOM1:def_3;
A33: - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * ((f . (HT ((f - g),T))) - (g . (HT ((f - g),T))))) = - ((f . (HT ((f - g),T))) * ((((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ") * ((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))))) by GROUP_1:def_3
.= - ((f . (HT ((f - g),T))) * (1. L)) by A29, A32, VECTSP_1:def_10
.= - (f . (HT ((f - g),T))) by VECTSP_1:def_8 ;
assume A34: Support (f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))) = Support f ; ::_thesis: contradiction
(f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))) . (HT ((f - g),T)) = (f + (- (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g)))) . (HT ((f - g),T)) by POLYNOM1:def_6
.= (f . (HT ((f - g),T))) + ((- (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g))) . (HT ((f - g),T))) by POLYNOM1:15
.= (f . (HT ((f - g),T))) + (((- ((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) "))) * (f - g)) . (HT ((f - g),T))) by POLYRED:9
.= (f . (HT ((f - g),T))) + ((- ((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) "))) * ((f - g) . (HT ((f - g),T)))) by POLYNOM7:def_9
.= (f . (HT ((f - g),T))) + (- (f . (HT ((f - g),T)))) by A29, A33, VECTSP_1:9
.= 0. L by RLVECT_1:5 ;
hence contradiction by A23, A14, A24, A34, POLYNOM1:def_3; ::_thesis: verum
end;
f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g)) <= f,T by A21, A25, Th8, XBOOLE_1:1;
then A35: f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g)) < f,T by A31, POLYRED:def_3;
reconsider cp = cp as Element of (Polynom-Ring (n,L)) ;
reconsider cc = cc as Element of (Polynom-Ring (n,L)) ;
reconsider f9 = f9, g9 = g9 as Element of (Polynom-Ring (n,L)) ;
f - g = f9 - g9 by Lm2;
then A36: cp = cc * (f9 - g9) by POLYNOM1:def_10;
f9 - g9 in I by A1, A2, IDEAL_1:15;
then A37: cc * (f9 - g9) in I by IDEAL_1:def_2;
f9 - cp = f - ((((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) | (n,L)) *' (f - g)) by Lm2
.= f - (((f . (HT ((f - g),T))) * (((f . (HT ((f - g),T))) - (g . (HT ((f - g),T)))) ")) * (f - g)) by POLYNOM7:27 ;
hence contradiction by A1, A7, A28, A35, A37, A36, IDEAL_1:15; ::_thesis: verum
end;
end;
end;
hence f = g ; ::_thesis: verum
end;
end;
end;
Lm10: for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)) st p in I & p <> 0_ (n,L) holds
ex q being Polynomial of n,L st
( q in I & HM (q,T) = Monom ((1. L),(HT (p,T))) & q <> 0_ (n,L) )
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)) st p in I & p <> 0_ (n,L) holds
ex q being Polynomial of n,L st
( q in I & HM (q,T) = Monom ((1. L),(HT (p,T))) & q <> 0_ (n,L) )
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)) st p in I & p <> 0_ (n,L) holds
ex q being Polynomial of n,L st
( q in I & HM (q,T) = Monom ((1. L),(HT (p,T))) & q <> 0_ (n,L) )
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for p being Polynomial of n,L
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)) st p in I & p <> 0_ (n,L) holds
ex q being Polynomial of n,L st
( q in I & HM (q,T) = Monom ((1. L),(HT (p,T))) & q <> 0_ (n,L) )
let p be Polynomial of n,L; ::_thesis: for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)) st p in I & p <> 0_ (n,L) holds
ex q being Polynomial of n,L st
( q in I & HM (q,T) = Monom ((1. L),(HT (p,T))) & q <> 0_ (n,L) )
let I be non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)); ::_thesis: ( p in I & p <> 0_ (n,L) implies ex q being Polynomial of n,L st
( q in I & HM (q,T) = Monom ((1. L),(HT (p,T))) & q <> 0_ (n,L) ) )
assume that
A1: p in I and
A2: p <> 0_ (n,L) ; ::_thesis: ex q being Polynomial of n,L st
( q in I & HM (q,T) = Monom ((1. L),(HT (p,T))) & q <> 0_ (n,L) )
set c = (HC (p,T)) " ;
A3: HC (p,T) <> 0. L by A2, TERMORD:17;
now__::_thesis:_not_(HC_(p,T))_"_=_0._L
assume (HC (p,T)) " = 0. L ; ::_thesis: contradiction
then 0. L = ((HC (p,T)) ") * (HC (p,T)) by VECTSP_1:7
.= 1. L by A3, VECTSP_1:def_10 ;
hence contradiction ; ::_thesis: verum
end;
then reconsider c = (HC (p,T)) " as non zero Element of L by STRUCT_0:def_12;
set q = c * p;
take c * p ; ::_thesis: ( c * p in I & HM ((c * p),T) = Monom ((1. L),(HT (p,T))) & c * p <> 0_ (n,L) )
reconsider pp = p, cc = c | (n,L) as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider pp = pp, cc = cc as Element of (Polynom-Ring (n,L)) ;
c * p = (c | (n,L)) *' p by POLYNOM7:27
.= cc * pp by POLYNOM1:def_10 ;
hence c * p in I by A1, IDEAL_1:def_2; ::_thesis: ( HM ((c * p),T) = Monom ((1. L),(HT (p,T))) & c * p <> 0_ (n,L) )
A4: HT ((c * p),T) = HT (p,T) by POLYRED:21;
then HC ((c * p),T) = (c * p) . (HT (p,T)) by TERMORD:def_7
.= c * (p . (HT (p,T))) by POLYNOM7:def_9
.= (HC (p,T)) * ((HC (p,T)) ") by TERMORD:def_7
.= 1. L by A3, VECTSP_1:def_10 ;
hence HM ((c * p),T) = Monom ((1. L),(HT (p,T))) by A4, TERMORD:def_8; ::_thesis: c * p <> 0_ (n,L)
then 1. L = coefficient (HM ((c * p),T)) by POLYNOM7:9
.= HC ((c * p),T) by TERMORD:22 ;
then HC ((c * p),T) <> 0. L ;
hence c * p <> 0_ (n,L) by TERMORD:17; ::_thesis: verum
end;
theorem :: GROEB_1:38
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L))
for G being Subset of (Polynom-Ring (n,L))
for p being Polynomial of n,L
for q being non-zero Polynomial of n,L st p in G & q in G & p <> q & HT (q,T) divides HT (p,T) & G is_Groebner_basis_of I,T holds
G \ {p} is_Groebner_basis_of I,T
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L))
for G being Subset of (Polynom-Ring (n,L))
for p being Polynomial of n,L
for q being non-zero Polynomial of n,L st p in G & q in G & p <> q & HT (q,T) divides HT (p,T) & G is_Groebner_basis_of I,T holds
G \ {p} is_Groebner_basis_of I,T
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L))
for G being Subset of (Polynom-Ring (n,L))
for p being Polynomial of n,L
for q being non-zero Polynomial of n,L st p in G & q in G & p <> q & HT (q,T) divides HT (p,T) & G is_Groebner_basis_of I,T holds
G \ {p} is_Groebner_basis_of I,T
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L))
for G being Subset of (Polynom-Ring (n,L))
for p being Polynomial of n,L
for q being non-zero Polynomial of n,L st p in G & q in G & p <> q & HT (q,T) divides HT (p,T) & G is_Groebner_basis_of I,T holds
G \ {p} is_Groebner_basis_of I,T
let I be non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)); ::_thesis: for G being Subset of (Polynom-Ring (n,L))
for p being Polynomial of n,L
for q being non-zero Polynomial of n,L st p in G & q in G & p <> q & HT (q,T) divides HT (p,T) & G is_Groebner_basis_of I,T holds
G \ {p} is_Groebner_basis_of I,T
let G be Subset of (Polynom-Ring (n,L)); ::_thesis: for p being Polynomial of n,L
for q being non-zero Polynomial of n,L st p in G & q in G & p <> q & HT (q,T) divides HT (p,T) & G is_Groebner_basis_of I,T holds
G \ {p} is_Groebner_basis_of I,T
let p be Polynomial of n,L; ::_thesis: for q being non-zero Polynomial of n,L st p in G & q in G & p <> q & HT (q,T) divides HT (p,T) & G is_Groebner_basis_of I,T holds
G \ {p} is_Groebner_basis_of I,T
let q be non-zero Polynomial of n,L; ::_thesis: ( p in G & q in G & p <> q & HT (q,T) divides HT (p,T) & G is_Groebner_basis_of I,T implies G \ {p} is_Groebner_basis_of I,T )
assume that
A1: p in G and
A2: q in G and
A3: p <> q and
A4: HT (q,T) divides HT (p,T) ; ::_thesis: ( not G is_Groebner_basis_of I,T or G \ {p} is_Groebner_basis_of I,T )
reconsider GG = G as non empty Subset of (Polynom-Ring (n,L)) by A1;
assume A5: G is_Groebner_basis_of I,T ; ::_thesis: G \ {p} is_Groebner_basis_of I,T
set G9 = G \ {p};
A6: not q in {p} by A3, TARSKI:def_1;
then ( q <> 0_ (n,L) & q in G \ {p} ) by A2, POLYNOM7:def_1, XBOOLE_0:def_5;
then A7: HT (q,T) in { (HT (u,T)) where u is Polynomial of n,L : ( u in G \ {p} & u <> 0_ (n,L) ) } ;
GG c= GG -Ideal by IDEAL_1:def_14;
then A8: G c= I by A5, Def4;
for f being Polynomial of n,L st f in I holds
PolyRedRel (G,T) reduces f, 0_ (n,L) by A1, A5, Th24;
then for f being non-zero Polynomial of n,L st f in I holds
f is_reducible_wrt G,T by Th25;
then A9: for f being non-zero Polynomial of n,L st f in I holds
f is_top_reducible_wrt G,T by A8, Th26;
for b being bag of n st b in HT (I,T) holds
ex b9 being bag of n st
( b9 in HT ((G \ {p}),T) & b9 divides b )
proof
let b be bag of n; ::_thesis: ( b in HT (I,T) implies ex b9 being bag of n st
( b9 in HT ((G \ {p}),T) & b9 divides b ) )
assume b in HT (I,T) ; ::_thesis: ex b9 being bag of n st
( b9 in HT ((G \ {p}),T) & b9 divides b )
then consider bb being bag of n such that
A10: bb in HT (G,T) and
A11: bb divides b by A9, Th27;
consider r being Polynomial of n,L such that
A12: bb = HT (r,T) and
A13: r in G and
A14: r <> 0_ (n,L) by A10;
now__::_thesis:_(_(_r_=_p_&_ex_b9_being_bag_of_n_st_
(_b9_in_HT_((G_\_{p}),T)_&_b9_divides_b_)_)_or_(_r_<>_p_&_ex_b9_being_bag_of_n_st_
(_b9_in_HT_((G_\_{p}),T)_&_b9_divides_b_)_)_)
percases ( r = p or r <> p ) ;
case r = p ; ::_thesis: ex b9 being bag of n st
( b9 in HT ((G \ {p}),T) & b9 divides b )
hence ex b9 being bag of n st
( b9 in HT ((G \ {p}),T) & b9 divides b ) by A4, A7, A11, A12, Lm8; ::_thesis: verum
end;
case r <> p ; ::_thesis: ex b9 being bag of n st
( b9 in HT ((G \ {p}),T) & b9 divides b )
then not r in {p} by TARSKI:def_1;
then r in G \ {p} by A13, XBOOLE_0:def_5;
then bb in { (HT (u,T)) where u is Polynomial of n,L : ( u in G \ {p} & u <> 0_ (n,L) ) } by A12, A14;
hence ex b9 being bag of n st
( b9 in HT ((G \ {p}),T) & b9 divides b ) by A11; ::_thesis: verum
end;
end;
end;
hence ex b9 being bag of n st
( b9 in HT ((G \ {p}),T) & b9 divides b ) ; ::_thesis: verum
end;
then A15: HT (I,T) c= multiples (HT ((G \ {p}),T)) by Th28;
G \ {p} c= G by XBOOLE_1:36;
then A16: G \ {p} c= I by A8, XBOOLE_1:1;
G \ {p} <> {} by A2, A6, XBOOLE_0:def_5;
hence G \ {p} is_Groebner_basis_of I,T by A16, A15, Th29; ::_thesis: verum
end;
theorem :: GROEB_1:39
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)) st I <> {(0_ (n,L))} holds
ex G being finite Subset of (Polynom-Ring (n,L)) st
( G is_Groebner_basis_of I,T & G is_reduced_wrt T )
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)) st I <> {(0_ (n,L))} holds
ex G being finite Subset of (Polynom-Ring (n,L)) st
( G is_Groebner_basis_of I,T & G is_reduced_wrt T )
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)) st I <> {(0_ (n,L))} holds
ex G being finite Subset of (Polynom-Ring (n,L)) st
( G is_Groebner_basis_of I,T & G is_reduced_wrt T )
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)) st I <> {(0_ (n,L))} holds
ex G being finite Subset of (Polynom-Ring (n,L)) st
( G is_Groebner_basis_of I,T & G is_reduced_wrt T )
let I be non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)); ::_thesis: ( I <> {(0_ (n,L))} implies ex G being finite Subset of (Polynom-Ring (n,L)) st
( G is_Groebner_basis_of I,T & G is_reduced_wrt T ) )
set R = RelStr(# (Bags n),(DivOrder n) #);
assume A1: I <> {(0_ (n,L))} ; ::_thesis: ex G being finite Subset of (Polynom-Ring (n,L)) st
( G is_Groebner_basis_of I,T & G is_reduced_wrt T )
ex q being Element of I st q <> 0_ (n,L)
proof
assume A2: for q being Element of I holds not q <> 0_ (n,L) ; ::_thesis: contradiction
A3: now__::_thesis:_for_u_being_set_st_u_in_{(0__(n,L))}_holds_
u_in_I
let u be set ; ::_thesis: ( u in {(0_ (n,L))} implies u in I )
assume u in {(0_ (n,L))} ; ::_thesis: u in I
then A4: u = 0_ (n,L) by TARSKI:def_1;
now__::_thesis:_u_in_I
assume not u in I ; ::_thesis: contradiction
then for v being set holds not v in I by A2, A4;
hence contradiction by XBOOLE_0:def_1; ::_thesis: verum
end;
hence u in I ; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_I_holds_
u_in_{(0__(n,L))}
let u be set ; ::_thesis: ( u in I implies u in {(0_ (n,L))} )
assume u in I ; ::_thesis: u in {(0_ (n,L))}
then u = 0_ (n,L) by A2;
hence u in {(0_ (n,L))} by TARSKI:def_1; ::_thesis: verum
end;
hence contradiction by A1, A3, TARSKI:1; ::_thesis: verum
end;
then consider q being Element of I such that
A5: q <> 0_ (n,L) ;
reconsider q = q as Polynomial of n,L by POLYNOM1:def_10;
HT (q,T) in { (HT (p,T)) where p is Polynomial of n,L : ( p in I & p <> 0_ (n,L) ) } by A5;
then reconsider hti = HT (I,T) as non empty Subset of RelStr(# (Bags n),(DivOrder n) #) ;
set hq = HT (q,T);
consider S being set such that
A6: S is_Dickson-basis_of hti, RelStr(# (Bags n),(DivOrder n) #) and
A7: for C being set st C is_Dickson-basis_of hti, RelStr(# (Bags n),(DivOrder n) #) holds
S c= C by DICKSON:34;
reconsider hq = HT (q,T) as Element of RelStr(# (Bags n),(DivOrder n) #) ;
A8: now__::_thesis:_S_is_finite
A9: ex B being set st
( B is_Dickson-basis_of hti, RelStr(# (Bags n),(DivOrder n) #) & B is finite ) by DICKSON:def_10;
assume not S is finite ; ::_thesis: contradiction
hence contradiction by A7, A9, FINSET_1:1; ::_thesis: verum
end;
A10: S c= hti by A6, DICKSON:def_9;
now__::_thesis:_for_u_being_set_st_u_in_S_holds_
u_in_Bags_n
let u be set ; ::_thesis: ( u in S implies u in Bags n )
assume u in S ; ::_thesis: u in Bags n
then u in hti by A10;
hence u in Bags n ; ::_thesis: verum
end;
then reconsider S = S as finite Subset of (Bags n) by A8, TARSKI:def_3;
set M = { { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),s) ) ) ) } where s is Element of Bags n : s in S } ;
set s = the Element of S;
hq in { (HT (p,T)) where p is Polynomial of n,L : ( p in I & p <> 0_ (n,L) ) } by A5;
then A11: ex b being Element of RelStr(# (Bags n),(DivOrder n) #) st
( b in S & b <= hq ) by A6, DICKSON:def_9;
then the Element of S in S ;
then reconsider s = the Element of S as Element of Bags n ;
{ p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) & ( for r being Polynomial of n,L holds
( not r in I or not r < p,T or not r <> 0_ (n,L) or not HM (r,T) = Monom ((1. L),s) ) ) ) } in { { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),s) ) ) ) } where s is Element of Bags n : s in S } by A11;
then reconsider M = { { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),s) ) ) ) } where s is Element of Bags n : s in S } as non empty set ;
set G = { r where r is Polynomial of n,L : ex x being Element of M st x = {r} } ;
A12: for x being set st x in M holds
ex q being Polynomial of n,L st
( q in I & x = {q} & q <> 0_ (n,L) )
proof
let x be set ; ::_thesis: ( x in M implies ex q being Polynomial of n,L st
( q in I & x = {q} & q <> 0_ (n,L) ) )
assume x in M ; ::_thesis: ex q being Polynomial of n,L st
( q in I & x = {q} & q <> 0_ (n,L) )
then consider s being Element of Bags n such that
A13: x = { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),s) ) ) ) } and
A14: s in S ;
s in hti by A10, A14;
then consider q being Polynomial of n,L such that
A15: s = HT (q,T) and
A16: ( q in I & q <> 0_ (n,L) ) ;
consider qq being Polynomial of n,L such that
A17: ( qq in I & HM (qq,T) = Monom ((1. L),(HT (q,T))) & qq <> 0_ (n,L) ) by A16, Lm10;
set M9 = { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) ) } ;
A18: now__::_thesis:_for_u_being_set_st_u_in__{__p_where_p_is_Polynomial_of_n,L_:_(_p_in_I_&_HM_(p,T)_=_Monom_((1._L),s)_&_p_<>_0__(n,L)_)__}__holds_
u_in_the_carrier_of_(Polynom-Ring_(n,L))
let u be set ; ::_thesis: ( u in { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) ) } implies u in the carrier of (Polynom-Ring (n,L)) )
assume u in { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) ) } ; ::_thesis: u in the carrier of (Polynom-Ring (n,L))
then ex pp being Polynomial of n,L st
( u = pp & pp in I & HM (pp,T) = Monom ((1. L),s) & pp <> 0_ (n,L) ) ;
hence u in the carrier of (Polynom-Ring (n,L)) ; ::_thesis: verum
end;
qq in { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) ) } by A15, A17;
then reconsider M9 = { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) ) } as non empty Subset of (Polynom-Ring (n,L)) by A18, TARSKI:def_3;
reconsider M9 = M9 as non empty Subset of (Polynom-Ring (n,L)) ;
consider p being Polynomial of n,L such that
A19: p in M9 and
A20: for r being Polynomial of n,L st r in M9 holds
p <= r,T by POLYRED:31;
consider p9 being Polynomial of n,L such that
A21: p9 = p and
A22: p9 in I and
A23: HM (p9,T) = Monom ((1. L),s) and
A24: p9 <> 0_ (n,L) by A19;
A25: now__::_thesis:_for_q_being_Polynomial_of_n,L_holds_
(_not_q_in_I_or_not_q_<_p9,T_or_not_q_<>_0__(n,L)_or_not_HM_(q,T)_=_Monom_((1._L),s)_)
assume ex q being Polynomial of n,L st
( q in I & q < p9,T & q <> 0_ (n,L) & HM (q,T) = Monom ((1. L),s) ) ; ::_thesis: contradiction
then consider q being Polynomial of n,L such that
A26: q in I and
A27: q < p9,T and
A28: ( q <> 0_ (n,L) & HM (q,T) = Monom ((1. L),s) ) ;
q in M9 by A26, A28;
then p <= q,T by A20;
hence contradiction by A21, A27, POLYRED:29; ::_thesis: verum
end;
A29: now__::_thesis:_for_q_being_Polynomial_of_n,L_holds_
(_not_q_in_I_or_not_q_<_p9,T_or_not_HM_(q,T)_=_Monom_((1._L),s)_)
A30: 1. L <> 0. L ;
assume ex q being Polynomial of n,L st
( q in I & q < p9,T & HM (q,T) = Monom ((1. L),s) ) ; ::_thesis: contradiction
then consider q being Polynomial of n,L such that
A31: ( q in I & q < p9,T ) and
A32: HM (q,T) = Monom ((1. L),s) ;
HC (q,T) = coefficient (Monom ((1. L),s)) by A32, TERMORD:22
.= 1. L by POLYNOM7:9 ;
then q <> 0_ (n,L) by A30, TERMORD:17;
hence contradiction by A25, A31, A32; ::_thesis: verum
end;
A33: now__::_thesis:_for_u_being_set_st_u_in_x_holds_
u_in_{p9}
let u be set ; ::_thesis: ( u in x implies u in {p9} )
assume u in x ; ::_thesis: u in {p9}
then consider u9 being Polynomial of n,L such that
A34: u9 = u and
A35: ( u9 in I & HM (u9,T) = Monom ((1. L),s) ) and
u9 <> 0_ (n,L) and
A36: for q being Polynomial of n,L holds
( not q in I or not q < u9,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),s) ) by A13;
now__::_thesis:_for_q_being_Polynomial_of_n,L_holds_
(_not_q_in_I_or_not_q_<_u9,T_or_not_HM_(q,T)_=_Monom_((1._L),s)_)
A37: 1. L <> 0. L ;
assume ex q being Polynomial of n,L st
( q in I & q < u9,T & HM (q,T) = Monom ((1. L),s) ) ; ::_thesis: contradiction
then consider q being Polynomial of n,L such that
A38: ( q in I & q < u9,T ) and
A39: HM (q,T) = Monom ((1. L),s) ;
HC (q,T) = coefficient (Monom ((1. L),s)) by A39, TERMORD:22
.= 1. L by POLYNOM7:9 ;
then q <> 0_ (n,L) by A37, TERMORD:17;
hence contradiction by A36, A38, A39; ::_thesis: verum
end;
then u9 = p9 by A22, A23, A29, A35, Th37;
hence u in {p9} by A34, TARSKI:def_1; ::_thesis: verum
end;
take p9 ; ::_thesis: ( p9 in I & x = {p9} & p9 <> 0_ (n,L) )
p9 in x by A13, A22, A23, A24, A25;
then for u being set st u in {p9} holds
u in x by TARSKI:def_1;
hence ( p9 in I & x = {p9} & p9 <> 0_ (n,L) ) by A22, A24, A33, TARSKI:1; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in__{__r_where_r_is_Polynomial_of_n,L_:_ex_x_being_Element_of_M_st_x_=_{r}__}__holds_
u_in_I
let u be set ; ::_thesis: ( u in { r where r is Polynomial of n,L : ex x being Element of M st x = {r} } implies u in I )
assume u in { r where r is Polynomial of n,L : ex x being Element of M st x = {r} } ; ::_thesis: u in I
then consider r being Polynomial of n,L such that
A40: u = r and
A41: ex x being Element of M st x = {r} ;
consider x being Element of M such that
A42: x = {r} by A41;
consider r9 being Polynomial of n,L such that
A43: r9 in I and
A44: x = {r9} and
r9 <> 0_ (n,L) by A12;
r9 in {r} by A42, A44, TARSKI:def_1;
hence u in I by A40, A43, TARSKI:def_1; ::_thesis: verum
end;
then A45: { r where r is Polynomial of n,L : ex x being Element of M st x = {r} } c= I by TARSKI:def_3;
A46: M is finite
proof
defpred S1[ set , set ] means $2 = { p where p is Polynomial of n,L : ex b being bag of n st
( b = $1 & p in I & HM (p,T) = Monom ((1. L),b) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),b) ) ) ) } ;
A47: for x being set st x in S holds
ex y being set st S1[x,y] ;
consider f being Function such that
A48: ( dom f = S & ( for x being set st x in S holds
S1[x,f . x] ) ) from CLASSES1:sch_1(A47);
A49: now__::_thesis:_for_u_being_set_st_u_in_rng_f_holds_
u_in_M
let u be set ; ::_thesis: ( u in rng f implies u in M )
assume u in rng f ; ::_thesis: u in M
then consider s being set such that
A50: s in dom f and
A51: u = f . s by FUNCT_1:def_3;
reconsider s = s as Element of Bags n by A48, A50;
set V = { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),s) ) ) ) } ;
A52: ex b being bag of n st f . s = { p where p is Polynomial of n,L : ex b being bag of n st
( b = s & p in I & HM (p,T) = Monom ((1. L),b) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),b) ) ) ) } by A48, A50;
A53: now__::_thesis:_for_v_being_set_st_v_in__{__p_where_p_is_Polynomial_of_n,L_:_(_p_in_I_&_HM_(p,T)_=_Monom_((1._L),s)_&_p_<>_0__(n,L)_&_(_for_q_being_Polynomial_of_n,L_holds_
(_not_q_in_I_or_not_q_<_p,T_or_not_q_<>_0__(n,L)_or_not_HM_(q,T)_=_Monom_((1._L),s)_)_)_)__}__holds_
v_in_f_._s
let v be set ; ::_thesis: ( v in { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),s) ) ) ) } implies v in f . s )
assume v in { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),s) ) ) ) } ; ::_thesis: v in f . s
then ex p being Polynomial of n,L st
( v = p & p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),s) ) ) ) ;
hence v in f . s by A52; ::_thesis: verum
end;
now__::_thesis:_for_v_being_set_st_v_in_f_._s_holds_
v_in__{__p_where_p_is_Polynomial_of_n,L_:_(_p_in_I_&_HM_(p,T)_=_Monom_((1._L),s)_&_p_<>_0__(n,L)_&_(_for_q_being_Polynomial_of_n,L_holds_
(_not_q_in_I_or_not_q_<_p,T_or_not_q_<>_0__(n,L)_or_not_HM_(q,T)_=_Monom_((1._L),s)_)_)_)__}_
let v be set ; ::_thesis: ( v in f . s implies v in { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),s) ) ) ) } )
assume v in f . s ; ::_thesis: v in { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),s) ) ) ) }
then ex p being Polynomial of n,L st
( v = p & ex b being bag of n st
( b = s & p in I & HM (p,T) = Monom ((1. L),b) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),b) ) ) ) ) by A52;
hence v in { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),s) ) ) ) } ; ::_thesis: verum
end;
then u = { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),s) ) ) ) } by A51, A53, TARSKI:1;
hence u in M by A48, A50; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_M_holds_
u_in_rng_f
let u be set ; ::_thesis: ( u in M implies u in rng f )
assume u in M ; ::_thesis: u in rng f
then consider s being Element of Bags n such that
A54: u = { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),s) ) ) ) } and
A55: s in S ;
A56: ex b being bag of n st f . s = { p where p is Polynomial of n,L : ex b being bag of n st
( b = s & p in I & HM (p,T) = Monom ((1. L),b) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),b) ) ) ) } by A48, A55;
A57: now__::_thesis:_for_v_being_set_st_v_in_u_holds_
v_in_f_._s
let v be set ; ::_thesis: ( v in u implies v in f . s )
assume v in u ; ::_thesis: v in f . s
then ex p being Polynomial of n,L st
( v = p & p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),s) ) ) ) by A54;
hence v in f . s by A56; ::_thesis: verum
end;
A58: now__::_thesis:_for_v_being_set_st_v_in_f_._s_holds_
v_in_u
let v be set ; ::_thesis: ( v in f . s implies v in u )
assume v in f . s ; ::_thesis: v in u
then ex p being Polynomial of n,L st
( v = p & ex b being bag of n st
( b = s & p in I & HM (p,T) = Monom ((1. L),b) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),b) ) ) ) ) by A56;
hence v in u by A54; ::_thesis: verum
end;
f . s in rng f by A48, A55, FUNCT_1:3;
hence u in rng f by A57, A58, TARSKI:1; ::_thesis: verum
end;
then rng f = M by A49, TARSKI:1;
hence M is finite by A48, FINSET_1:8; ::_thesis: verum
end;
A59: { r where r is Polynomial of n,L : ex x being Element of M st x = {r} } is finite
proof
defpred S1[ set , set ] means ex p being Polynomial of n,L ex x being Element of M st
( $1 = x & $2 = p & x = {p} );
A60: for x being set st x in M holds
ex y being set st S1[x,y]
proof
let x be set ; ::_thesis: ( x in M implies ex y being set st S1[x,y] )
assume A61: x in M ; ::_thesis: ex y being set st S1[x,y]
then reconsider x9 = x as Element of M ;
consider q being Polynomial of n,L such that
q in I and
A62: x = {q} and
q <> 0_ (n,L) by A12, A61;
take q ; ::_thesis: S1[x,q]
take q ; ::_thesis: ex x being Element of M st
( x = x & q = q & x = {q} )
take x9 ; ::_thesis: ( x = x9 & q = q & x9 = {q} )
thus x = x9 ; ::_thesis: ( q = q & x9 = {q} )
thus q = q ; ::_thesis: x9 = {q}
thus x9 = {q} by A62; ::_thesis: verum
end;
consider f being Function such that
A63: ( dom f = M & ( for x being set st x in M holds
S1[x,f . x] ) ) from CLASSES1:sch_1(A60);
A64: now__::_thesis:_for_u_being_set_st_u_in__{__r_where_r_is_Polynomial_of_n,L_:_ex_x_being_Element_of_M_st_x_=_{r}__}__holds_
u_in_rng_f
let u be set ; ::_thesis: ( u in { r where r is Polynomial of n,L : ex x being Element of M st x = {r} } implies u in rng f )
assume u in { r where r is Polynomial of n,L : ex x being Element of M st x = {r} } ; ::_thesis: u in rng f
then consider r being Polynomial of n,L such that
A65: u = r and
A66: ex x being Element of M st x = {r} ;
consider x being Element of M such that
A67: x = {r} by A66;
S1[x,f . x] by A63;
then consider p9 being Polynomial of n,L, x9 being Element of M such that
x9 = x and
A68: p9 = f . x and
A69: x = {p9} ;
A70: f . x in rng f by A63, FUNCT_1:3;
p9 in {r} by A67, A69, TARSKI:def_1;
hence u in rng f by A65, A70, A68, TARSKI:def_1; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_rng_f_holds_
u_in__{__r_where_r_is_Polynomial_of_n,L_:_ex_x_being_Element_of_M_st_x_=_{r}__}_
let u be set ; ::_thesis: ( u in rng f implies u in { r where r is Polynomial of n,L : ex x being Element of M st x = {r} } )
assume u in rng f ; ::_thesis: u in { r where r is Polynomial of n,L : ex x being Element of M st x = {r} }
then consider s being set such that
A71: s in dom f and
A72: u = f . s by FUNCT_1:def_3;
reconsider s = s as Element of M by A63, A71;
ex p9 being Polynomial of n,L ex x9 being Element of M st
( x9 = s & p9 = f . s & x9 = {p9} ) by A63;
hence u in { r where r is Polynomial of n,L : ex x being Element of M st x = {r} } by A72; ::_thesis: verum
end;
then rng f = { r where r is Polynomial of n,L : ex x being Element of M st x = {r} } by A64, TARSKI:1;
hence { r where r is Polynomial of n,L : ex x being Element of M st x = {r} } is finite by A46, A63, FINSET_1:8; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in__{__r_where_r_is_Polynomial_of_n,L_:_ex_x_being_Element_of_M_st_x_=_{r}__}__holds_
u_in_the_carrier_of_(Polynom-Ring_(n,L))
let u be set ; ::_thesis: ( u in { r where r is Polynomial of n,L : ex x being Element of M st x = {r} } implies u in the carrier of (Polynom-Ring (n,L)) )
assume u in { r where r is Polynomial of n,L : ex x being Element of M st x = {r} } ; ::_thesis: u in the carrier of (Polynom-Ring (n,L))
then ex r being Polynomial of n,L st
( u = r & ex x being Element of M st x = {r} ) ;
hence u in the carrier of (Polynom-Ring (n,L)) by POLYNOM1:def_10; ::_thesis: verum
end;
then reconsider G = { r where r is Polynomial of n,L : ex x being Element of M st x = {r} } as Subset of (Polynom-Ring (n,L)) by TARSKI:def_3;
G <> {}
proof
set z = the Element of M;
consider r being Polynomial of n,L such that
r in I and
A73: the Element of M = {r} and
r <> 0_ (n,L) by A12;
r in G by A73;
hence G <> {} ; ::_thesis: verum
end;
then reconsider G = G as non empty finite Subset of (Polynom-Ring (n,L)) by A59;
take G ; ::_thesis: ( G is_Groebner_basis_of I,T & G is_reduced_wrt T )
for b being bag of n st b in HT (I,T) holds
ex b9 being bag of n st
( b9 in HT (G,T) & b9 divides b )
proof
let b be bag of n; ::_thesis: ( b in HT (I,T) implies ex b9 being bag of n st
( b9 in HT (G,T) & b9 divides b ) )
reconsider bb = b as Element of RelStr(# (Bags n),(DivOrder n) #) by PRE_POLY:def_12;
assume b in HT (I,T) ; ::_thesis: ex b9 being bag of n st
( b9 in HT (G,T) & b9 divides b )
then consider bb9 being Element of RelStr(# (Bags n),(DivOrder n) #) such that
A74: bb9 in S and
A75: bb9 <= bb by A6, DICKSON:def_9;
A76: [bb9,bb] in DivOrder n by A75, ORDERS_2:def_5;
reconsider b9 = bb9 as bag of n ;
set N = { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),b9) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),b9) ) ) ) } ;
{ p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),b9) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),b9) ) ) ) } in M by A74;
then reconsider N = { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),b9) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),b9) ) ) ) } as Element of M ;
take b9 ; ::_thesis: ( b9 in HT (G,T) & b9 divides b )
consider r being Polynomial of n,L such that
r in I and
A77: N = {r} and
r <> 0_ (n,L) by A12;
r in N by A77, TARSKI:def_1;
then consider r9 being Polynomial of n,L such that
A78: r = r9 and
r9 in I and
A79: HM (r9,T) = Monom ((1. L),b9) and
A80: r9 <> 0_ (n,L) and
for q being Polynomial of n,L holds
( not q in I or not q < r9,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),b9) ) ;
A81: r9 in G by A77, A78;
b9 = term (HM (r9,T)) by A79, POLYNOM7:10
.= HT (r9,T) by TERMORD:22 ;
hence ( b9 in HT (G,T) & b9 divides b ) by A76, A80, A81, Def5; ::_thesis: verum
end;
then HT (I,T) c= multiples (HT (G,T)) by Th28;
hence G is_Groebner_basis_of I,T by A45, Th29; ::_thesis: G is_reduced_wrt T
now__::_thesis:_for_q_being_Polynomial_of_n,L_st_q_in_G_holds_
(_q_is_monic_wrt_T_&_q_is_irreducible_wrt_G_\_{q},T_)
let q be Polynomial of n,L; ::_thesis: ( q in G implies ( q is_monic_wrt T & q is_irreducible_wrt G \ {q},T ) )
assume A82: q in G ; ::_thesis: ( q is_monic_wrt T & q is_irreducible_wrt G \ {q},T )
then consider rr being Polynomial of n,L such that
A83: q = rr and
A84: ex x being Element of M st x = {rr} ;
consider x being Element of M such that
A85: x = {rr} by A84;
x in M ;
then consider s being Element of Bags n such that
A86: x = { p where p is Polynomial of n,L : ( p in I & HM (p,T) = Monom ((1. L),s) & p <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),s) ) ) ) } and
A87: s in S ;
rr in x by A85, TARSKI:def_1;
then consider p being Polynomial of n,L such that
A88: rr = p and
p in I and
A89: HM (p,T) = Monom ((1. L),s) and
p <> 0_ (n,L) and
A90: for q being Polynomial of n,L holds
( not q in I or not q < p,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),s) ) by A86;
A91: 1. L = coefficient (HM (rr,T)) by A88, A89, POLYNOM7:9
.= HC (rr,T) by TERMORD:22 ;
hence q is_monic_wrt T by A83, Def6; ::_thesis: q is_irreducible_wrt G \ {q},T
A92: s = term (HM (rr,T)) by A88, A89, POLYNOM7:10
.= HT (q,T) by A83, TERMORD:22 ;
now__::_thesis:_(_q_is_reducible_wrt_G_\_{q},T_implies_q_is_irreducible_wrt_G_\_{q},T_)
reconsider htq = HT (q,T) as Element of RelStr(# (Bags n),(DivOrder n) #) ;
assume q is_reducible_wrt G \ {q},T ; ::_thesis: q is_irreducible_wrt G \ {q},T
then consider pp being Polynomial of n,L such that
A93: q reduces_to pp,G \ {q},T by POLYRED:def_9;
consider g being Polynomial of n,L such that
A94: g in G \ {q} and
A95: q reduces_to pp,g,T by A93, POLYRED:def_7;
A96: g in G by A94, XBOOLE_0:def_5;
A97: not g in {q} by A94, XBOOLE_0:def_5;
reconsider htg = HT (g,T) as Element of RelStr(# (Bags n),(DivOrder n) #) ;
consider b being bag of n such that
A98: q reduces_to pp,g,b,T by A95, POLYRED:def_6;
A99: b in Support q by A98, POLYRED:def_5;
A100: ex s being bag of n st
( s + (HT (g,T)) = b & pp = q - (((q . b) / (HC (g,T))) * (s *' g)) ) by A98, POLYRED:def_5;
now__::_thesis:_(_(_b_=_HT_(q,T)_&_q_is_irreducible_wrt_G_\_{q},T_)_or_(_b_<>_HT_(q,T)_&_contradiction_)_)
percases ( b = HT (q,T) or b <> HT (q,T) ) ;
caseA101: b = HT (q,T) ; ::_thesis: q is_irreducible_wrt G \ {q},T
set S9 = S \ {htq};
consider z being Polynomial of n,L such that
A102: g = z and
A103: ex x being Element of M st x = {z} by A96;
consider x1 being Element of M such that
A104: x1 = {z} by A103;
x1 in M ;
then consider t being Element of Bags n such that
A105: x1 = { u where u is Polynomial of n,L : ( u in I & HM (u,T) = Monom ((1. L),t) & u <> 0_ (n,L) & ( for q being Polynomial of n,L holds
( not q in I or not q < u,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),t) ) ) ) } and
A106: t in S ;
z in x1 by A104, TARSKI:def_1;
then consider p1 being Polynomial of n,L such that
A107: z = p1 and
p1 in I and
A108: HM (p1,T) = Monom ((1. L),t) and
p1 <> 0_ (n,L) and
for q being Polynomial of n,L holds
( not q in I or not q < p1,T or not q <> 0_ (n,L) or not HM (q,T) = Monom ((1. L),t) ) by A105;
A109: t = term (HM (p1,T)) by A108, POLYNOM7:10
.= htg by A102, A107, TERMORD:22 ;
now__::_thesis:_not_htg_in_{htq}
assume htg in {htq} ; ::_thesis: contradiction
then t = s by A92, A109, TARSKI:def_1;
hence contradiction by A83, A85, A86, A97, A102, A104, A105, TARSKI:def_1; ::_thesis: verum
end;
then A110: htg in S \ {htq} by A106, A109, XBOOLE_0:def_5;
HT (g,T) divides HT (q,T) by A100, A101, PRE_POLY:50;
then [htg,htq] in DivOrder n by Def5;
then A111: htg <= htq by ORDERS_2:def_5;
A112: now__::_thesis:_for_a_being_Element_of_RelStr(#_(Bags_n),(DivOrder_n)_#)_st_a_in_hti_holds_
ex_b_being_Element_of_RelStr(#_(Bags_n),(DivOrder_n)_#)_st_
(_b_in_S_\_{htq}_&_b_<=_a_)
let a be Element of RelStr(# (Bags n),(DivOrder n) #); ::_thesis: ( a in hti implies ex b being Element of RelStr(# (Bags n),(DivOrder n) #) st
( b in S \ {htq} & b <= a ) )
assume a in hti ; ::_thesis: ex b being Element of RelStr(# (Bags n),(DivOrder n) #) st
( b in S \ {htq} & b <= a )
then consider b being Element of RelStr(# (Bags n),(DivOrder n) #) such that
A113: b in S and
A114: b <= a by A6, DICKSON:def_9;
now__::_thesis:_(_(_b_in_S_\_{htq}_&_ex_b_being_Element_of_RelStr(#_(Bags_n),(DivOrder_n)_#)_st_
(_b_in_S_\_{htq}_&_b_<=_a_)_)_or_(_not_b_in_S_\_{htq}_&_ex_b_being_Element_of_RelStr(#_(Bags_n),(DivOrder_n)_#)_st_
(_b_in_S_\_{htq}_&_b_<=_a_)_)_)
percases ( b in S \ {htq} or not b in S \ {htq} ) ;
case b in S \ {htq} ; ::_thesis: ex b being Element of RelStr(# (Bags n),(DivOrder n) #) st
( b in S \ {htq} & b <= a )
hence ex b being Element of RelStr(# (Bags n),(DivOrder n) #) st
( b in S \ {htq} & b <= a ) by A114; ::_thesis: verum
end;
case not b in S \ {htq} ; ::_thesis: ex b being Element of RelStr(# (Bags n),(DivOrder n) #) st
( b in S \ {htq} & b <= a )
then b in {htq} by A113, XBOOLE_0:def_5;
then htg <= b by A111, TARSKI:def_1;
hence ex b being Element of RelStr(# (Bags n),(DivOrder n) #) st
( b in S \ {htq} & b <= a ) by A110, A114, ORDERS_2:3; ::_thesis: verum
end;
end;
end;
hence ex b being Element of RelStr(# (Bags n),(DivOrder n) #) st
( b in S \ {htq} & b <= a ) ; ::_thesis: verum
end;
A115: now__::_thesis:_not_htq_in_S_\_{htq}
assume htq in S \ {htq} ; ::_thesis: contradiction
then not htq in {htq} by XBOOLE_0:def_5;
hence contradiction by TARSKI:def_1; ::_thesis: verum
end;
S \ {htq} c= S by XBOOLE_1:36;
then S \ {htq} c= hti by A10, XBOOLE_1:1;
then S \ {htq} is_Dickson-basis_of hti, RelStr(# (Bags n),(DivOrder n) #) by A112, DICKSON:def_9;
then S c= S \ {htq} by A7;
hence q is_irreducible_wrt G \ {q},T by A87, A92, A115; ::_thesis: verum
end;
caseA116: b <> HT (q,T) ; ::_thesis: contradiction
b <= HT (q,T),T by A99, TERMORD:def_6;
then b < HT (q,T),T by A116, TERMORD:def_3;
then A117: pp . (HT (q,T)) = q . (HT (q,T)) by A98, POLYRED:41
.= 1. L by A83, A91, TERMORD:def_7 ;
1. L <> 0. L ;
then A118: HT (q,T) in Support pp by A117, POLYNOM1:def_3;
now__::_thesis:_not_HT_(q,T)_<_HT_(pp,T),T
A119: b <= HT (q,T),T by A99, TERMORD:def_6;
assume A120: HT (q,T) < HT (pp,T),T ; ::_thesis: contradiction
then HT (q,T) <= HT (pp,T),T by TERMORD:def_3;
then ( b <= HT (pp,T),T & b <> HT (pp,T) ) by A116, A119, TERMORD:7, TERMORD:8;
then b < HT (pp,T),T by TERMORD:def_3;
then ( HT (pp,T) in Support q iff HT (pp,T) in Support pp ) by A98, POLYRED:40;
then HT (pp,T) <= HT (q,T),T by A118, TERMORD:def_6;
hence contradiction by A120, TERMORD:5; ::_thesis: verum
end;
then A121: HT (pp,T) <= HT (q,T),T by TERMORD:5;
HT (q,T) <= HT (pp,T),T by A118, TERMORD:def_6;
then HT (pp,T) = HT (q,T) by A121, TERMORD:7;
then Monom ((HC (pp,T)),(HT (pp,T))) = Monom ((1. L),s) by A92, A117, TERMORD:def_7;
then A122: HM (pp,T) = HM (q,T) by A83, A88, A89, TERMORD:def_8;
A123: now__::_thesis:_not_pp_=_0__(n,L)
assume pp = 0_ (n,L) ; ::_thesis: contradiction
then 0. L = HC (pp,T) by TERMORD:17
.= coefficient (HM (pp,T)) by TERMORD:22
.= 1. L by A83, A91, A122, TERMORD:22 ;
hence contradiction ; ::_thesis: verum
end;
consider m being Monomial of n,L such that
A124: pp = q - (m *' g) by A95, Th1;
reconsider gg = g, qq = q, mm = m as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider gg = gg, qq = qq, mm = mm as Element of (Polynom-Ring (n,L)) ;
g in G by A94, XBOOLE_0:def_5;
then mm * gg in I by A45, IDEAL_1:def_2;
then - (mm * gg) in I by IDEAL_1:13;
then A125: qq + (- (mm * gg)) in I by A45, A82, IDEAL_1:def_1;
mm * gg = m *' g by POLYNOM1:def_10;
then q - (m *' g) = qq - (mm * gg) by Lm2;
then pp in I by A124, A125, RLVECT_1:def_11;
hence contradiction by A83, A88, A89, A90, A95, A122, A123, POLYRED:43; ::_thesis: verum
end;
end;
end;
hence q is_irreducible_wrt G \ {q},T ; ::_thesis: verum
end;
hence q is_irreducible_wrt G \ {q},T ; ::_thesis: verum
end;
hence G is_reduced_wrt T by Def7; ::_thesis: verum
end;
theorem :: GROEB_1:40
for n being Element of NAT
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L))
for G1, G2 being non empty finite Subset of (Polynom-Ring (n,L)) st G1 is_Groebner_basis_of I,T & G1 is_reduced_wrt T & G2 is_Groebner_basis_of I,T & G2 is_reduced_wrt T holds
G1 = G2
proof
let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L))
for G1, G2 being non empty finite Subset of (Polynom-Ring (n,L)) st G1 is_Groebner_basis_of I,T & G1 is_reduced_wrt T & G2 is_Groebner_basis_of I,T & G2 is_reduced_wrt T holds
G1 = G2
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L))
for G1, G2 being non empty finite Subset of (Polynom-Ring (n,L)) st G1 is_Groebner_basis_of I,T & G1 is_reduced_wrt T & G2 is_Groebner_basis_of I,T & G2 is_reduced_wrt T holds
G1 = G2
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for I being non empty add-closed left-ideal Subset of (Polynom-Ring (n,L))
for G1, G2 being non empty finite Subset of (Polynom-Ring (n,L)) st G1 is_Groebner_basis_of I,T & G1 is_reduced_wrt T & G2 is_Groebner_basis_of I,T & G2 is_reduced_wrt T holds
G1 = G2
let I be non empty add-closed left-ideal Subset of (Polynom-Ring (n,L)); ::_thesis: for G1, G2 being non empty finite Subset of (Polynom-Ring (n,L)) st G1 is_Groebner_basis_of I,T & G1 is_reduced_wrt T & G2 is_Groebner_basis_of I,T & G2 is_reduced_wrt T holds
G1 = G2
let G, H be non empty finite Subset of (Polynom-Ring (n,L)); ::_thesis: ( G is_Groebner_basis_of I,T & G is_reduced_wrt T & H is_Groebner_basis_of I,T & H is_reduced_wrt T implies G = H )
assume that
A1: G is_Groebner_basis_of I,T and
A2: G is_reduced_wrt T and
A3: H is_Groebner_basis_of I,T and
A4: H is_reduced_wrt T ; ::_thesis: G = H
A5: H -Ideal = I by A3, Def4;
set GH = (G \/ H) \ (G /\ H);
assume A6: G <> H ; ::_thesis: contradiction
now__::_thesis:_not_(G_\/_H)_\_(G_/\_H)_=_{}
assume (G \/ H) \ (G /\ H) = {} ; ::_thesis: contradiction
then A7: G \/ H c= G /\ H by XBOOLE_1:37;
A8: now__::_thesis:_for_u_being_set_st_u_in_H_holds_
u_in_G
let u be set ; ::_thesis: ( u in H implies u in G )
assume u in H ; ::_thesis: u in G
then u in G \/ H by XBOOLE_0:def_3;
hence u in G by A7, XBOOLE_0:def_4; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_G_holds_
u_in_H
let u be set ; ::_thesis: ( u in G implies u in H )
assume u in G ; ::_thesis: u in H
then u in G \/ H by XBOOLE_0:def_3;
hence u in H by A7, XBOOLE_0:def_4; ::_thesis: verum
end;
hence contradiction by A6, A8, TARSKI:1; ::_thesis: verum
end;
then reconsider GH = (G \/ H) \ (G /\ H) as non empty Subset of (Polynom-Ring (n,L)) ;
A9: now__::_thesis:_for_u_being_set_st_u_in_GH_holds_
u_in_(G_\_H)_\/_(H_\_G)
let u be set ; ::_thesis: ( u in GH implies u in (G \ H) \/ (H \ G) )
assume A10: u in GH ; ::_thesis: u in (G \ H) \/ (H \ G)
then A11: not u in G /\ H by XBOOLE_0:def_5;
A12: u in G \/ H by A10, XBOOLE_0:def_5;
( u in G \ H or u in H \ G )
proof
assume A13: not u in G \ H ; ::_thesis: u in H \ G
now__::_thesis:_(_(_not_u_in_G_&_not_u_in_G_&_u_in_H_)_or_(_u_in_H_&_u_in_H_&_not_u_in_G_)_)
percases ( not u in G or u in H ) by A13, XBOOLE_0:def_5;
case not u in G ; ::_thesis: ( not u in G & u in H )
hence ( not u in G & u in H ) by A12, XBOOLE_0:def_3; ::_thesis: verum
end;
case u in H ; ::_thesis: ( u in H & not u in G )
hence ( u in H & not u in G ) by A11, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
end;
hence u in H \ G by XBOOLE_0:def_5; ::_thesis: verum
end;
hence u in (G \ H) \/ (H \ G) by XBOOLE_0:def_3; ::_thesis: verum
end;
consider g being Polynomial of n,L such that
A14: g in GH and
A15: for q being Polynomial of n,L st q in GH holds
g <= q,T by POLYRED:31;
A16: G -Ideal = I by A1, Def4;
A17: now__::_thesis:_for_u_being_set_st_(_u_in_G_or_u_in_H_)_holds_
(_u_is_Polynomial_of_n,L_&_u_<>_0__(n,L)_)
let u be set ; ::_thesis: ( ( u in G or u in H ) implies ( u is Polynomial of n,L & u <> 0_ (n,L) ) )
assume A18: ( u in G or u in H ) ; ::_thesis: ( u is Polynomial of n,L & u <> 0_ (n,L) )
now__::_thesis:_(_(_u_in_G_&_u_is_Polynomial_of_n,L_&_u_<>_0__(n,L)_)_or_(_u_in_H_&_u_is_Polynomial_of_n,L_&_u_<>_0__(n,L)_)_)
percases ( u in G or u in H ) by A18;
caseA19: u in G ; ::_thesis: ( u is Polynomial of n,L & u <> 0_ (n,L) )
then reconsider u9 = u as Element of (Polynom-Ring (n,L)) ;
reconsider u9 = u9 as Polynomial of n,L by POLYNOM1:def_10;
u9 is_monic_wrt T by A2, A19, Def7;
then A20: HC (u9,T) = 1. L by Def6;
1. L <> 0. L ;
hence ( u is Polynomial of n,L & u <> 0_ (n,L) ) by A20, TERMORD:17; ::_thesis: verum
end;
caseA21: u in H ; ::_thesis: ( u is Polynomial of n,L & u <> 0_ (n,L) )
then reconsider u9 = u as Element of (Polynom-Ring (n,L)) ;
reconsider u9 = u9 as Polynomial of n,L by POLYNOM1:def_10;
u9 is_monic_wrt T by A4, A21, Def7;
then A22: HC (u9,T) = 1. L by Def6;
1. L <> 0. L ;
hence ( u is Polynomial of n,L & u <> 0_ (n,L) ) by A22, TERMORD:17; ::_thesis: verum
end;
end;
end;
hence ( u is Polynomial of n,L & u <> 0_ (n,L) ) ; ::_thesis: verum
end;
PolyRedRel (G,T) is locally-confluent by A1, Def4;
then for f being Polynomial of n,L st f in G -Ideal holds
PolyRedRel (G,T) reduces f, 0_ (n,L) by Th15;
then for f being non-zero Polynomial of n,L st f in G -Ideal holds
f is_reducible_wrt G,T by Th16;
then A23: for f being non-zero Polynomial of n,L st f in G -Ideal holds
f is_top_reducible_wrt G,T by Th17;
A24: now__::_thesis:_for_u_being_Polynomial_of_n,L_st_(_u_in_G_or_u_in_H_)_holds_
HC_(u,T)_=_1._L
let u be Polynomial of n,L; ::_thesis: ( ( u in G or u in H ) implies HC (u,T) = 1. L )
assume ( u in G or u in H ) ; ::_thesis: HC (u,T) = 1. L
then u is_monic_wrt T by A2, A4, Def7;
hence HC (u,T) = 1. L by Def6; ::_thesis: verum
end;
A25: now__::_thesis:_for_u_being_set_holds_
(_not_u_in_GH_or_u_in_G_\_H_or_u_in_H_\_G_)
let u be set ; ::_thesis: ( not u in GH or u in G \ H or u in H \ G )
assume A26: u in GH ; ::_thesis: ( u in G \ H or u in H \ G )
then not u in G /\ H by XBOOLE_0:def_5;
then A27: ( not u in G or not u in H ) by XBOOLE_0:def_4;
A28: u in G \/ H by A26, XBOOLE_0:def_5;
now__::_thesis:_(_(_u_in_G_&_u_in_G_\_H_)_or_(_u_in_H_&_u_in_H_\_G_)_)
percases ( u in G or u in H ) by A28, XBOOLE_0:def_3;
case u in G ; ::_thesis: u in G \ H
hence u in G \ H by A27, XBOOLE_0:def_5; ::_thesis: verum
end;
case u in H ; ::_thesis: u in H \ G
hence u in H \ G by A27, XBOOLE_0:def_5; ::_thesis: verum
end;
end;
end;
hence ( u in G \ H or u in H \ G ) ; ::_thesis: verum
end;
now__::_thesis:_for_u_being_set_st_u_in_(G_\_H)_\/_(H_\_G)_holds_
u_in_GH
let u be set ; ::_thesis: ( u in (G \ H) \/ (H \ G) implies u in GH )
assume A29: u in (G \ H) \/ (H \ G) ; ::_thesis: u in GH
now__::_thesis:_(_(_u_in_G_\_H_&_u_in_G_\/_H_&_not_u_in_G_/\_H_)_or_(_u_in_H_\_G_&_u_in_G_\/_H_&_not_u_in_G_/\_H_)_)
percases ( u in G \ H or u in H \ G ) by A29, XBOOLE_0:def_3;
case u in G \ H ; ::_thesis: ( u in G \/ H & not u in G /\ H )
then ( u in G & not u in H ) by XBOOLE_0:def_5;
hence ( u in G \/ H & not u in G /\ H ) by XBOOLE_0:def_3, XBOOLE_0:def_4; ::_thesis: verum
end;
case u in H \ G ; ::_thesis: ( u in G \/ H & not u in G /\ H )
then ( u in H & not u in G ) by XBOOLE_0:def_5;
hence ( u in G \/ H & not u in G /\ H ) by XBOOLE_0:def_3, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
end;
hence u in GH by XBOOLE_0:def_5; ::_thesis: verum
end;
then A30: GH = (G \ H) \/ (H \ G) by A9, TARSKI:1;
PolyRedRel (H,T) is locally-confluent by A3, Def4;
then for f being Polynomial of n,L st f in H -Ideal holds
PolyRedRel (H,T) reduces f, 0_ (n,L) by Th15;
then for f being non-zero Polynomial of n,L st f in H -Ideal holds
f is_reducible_wrt H,T by Th16;
then A31: for f being non-zero Polynomial of n,L st f in H -Ideal holds
f is_top_reducible_wrt H,T by Th17;
percases ( g in G \ H or g in H \ G ) by A25, A14;
supposeA32: g in G \ H ; ::_thesis: contradiction
then A33: g in G by XBOOLE_0:def_5;
then A34: g <> 0_ (n,L) by A17;
then reconsider g = g as non-zero Polynomial of n,L by POLYNOM7:def_1;
A35: G c= G -Ideal by IDEAL_1:def_14;
then HT (g,T) in HT ((H -Ideal),T) by A16, A5, A33, A34;
then consider b9 being bag of n such that
A36: b9 in HT (H,T) and
A37: b9 divides HT (g,T) by A31, Th18;
consider h being Polynomial of n,L such that
A38: b9 = HT (h,T) and
A39: h in H and
A40: h <> 0_ (n,L) by A36;
reconsider h = h as non-zero Polynomial of n,L by A40, POLYNOM7:def_1;
set f = g - h;
A41: h <> g by A32, A39, XBOOLE_0:def_5;
A42: now__::_thesis:_not_g_-_h_=_0__(n,L)
assume A43: g - h = 0_ (n,L) ; ::_thesis: contradiction
h = (0_ (n,L)) + h by POLYRED:2
.= (g + (- h)) + h by A43, POLYNOM1:def_6
.= g + ((- h) + h) by POLYNOM1:21
.= g + (0_ (n,L)) by POLYRED:3 ;
hence contradiction by A41, POLYNOM1:23; ::_thesis: verum
end;
Support g <> {} by A34, POLYNOM7:1;
then HT (g,T) in Support g by TERMORD:def_6;
then A44: g is_reducible_wrt h,T by A37, A38, POLYRED:36;
then A45: ex r being Polynomial of n,L st g reduces_to r,h,T by POLYRED:def_8;
now__::_thesis:_h_in_H_\_G
assume not h in H \ G ; ::_thesis: contradiction
then A46: h in G by A39, XBOOLE_0:def_5;
not h in {g} by A41, TARSKI:def_1;
then h in G \ {g} by A46, XBOOLE_0:def_5;
then consider r being Polynomial of n,L such that
A47: ( h in G \ {g} & g reduces_to r,h,T ) by A45;
A48: g reduces_to r,G \ {g},T by A47, POLYRED:def_7;
g is_irreducible_wrt G \ {g},T by A2, A33, Def7;
hence contradiction by A48, POLYRED:def_9; ::_thesis: verum
end;
then h in GH by A30, XBOOLE_0:def_3;
then g <= h,T by A15;
then not h < g,T by POLYRED:29;
then not HT (h,T) < HT (g,T),T by POLYRED:32;
then A49: HT (g,T) <= HT (h,T),T by TERMORD:5;
HT (h,T) <= HT (g,T),T by A44, Th9;
then A50: HT (h,T) = HT (g,T) by A49, TERMORD:7;
reconsider f = g - h as non-zero Polynomial of n,L by A42, POLYNOM7:def_1;
Support f <> {} by A42, POLYNOM7:1;
then A51: HT (f,T) in Support f by TERMORD:def_6;
f . (HT (g,T)) = (g + (- h)) . (HT (g,T)) by POLYNOM1:def_6
.= (g . (HT (g,T))) + ((- h) . (HT (g,T))) by POLYNOM1:15
.= (g . (HT (g,T))) + (- (h . (HT (h,T)))) by A50, POLYNOM1:17
.= (HC (g,T)) + (- (h . (HT (h,T)))) by TERMORD:def_7
.= (HC (g,T)) + (- (HC (h,T))) by TERMORD:def_7
.= (1. L) + (- (HC (h,T))) by A24, A33
.= (1. L) + (- (1. L)) by A24, A39
.= 0. L by RLVECT_1:5 ;
then A52: HT (f,T) <> HT (g,T) by A51, POLYNOM1:def_3;
HT (f,T) <= max ((HT (g,T)),(HT (h,T)),T),T by Th7;
then A53: HT (f,T) <= HT (g,T),T by A50, TERMORD:12;
reconsider g9 = g, h9 = h as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider g9 = g9, h9 = h9 as Element of (Polynom-Ring (n,L)) ;
H c= H -Ideal by IDEAL_1:def_14;
then g9 - h9 in I by A16, A5, A33, A35, A39, IDEAL_1:15;
then f in I by Lm2;
then A54: HT (f,T) in HT (I,T) by A42;
Support (g + (- h)) c= (Support g) \/ (Support (- h)) by POLYNOM1:20;
then Support f c= (Support g) \/ (Support (- h)) by POLYNOM1:def_6;
then A55: Support f c= (Support g) \/ (Support h) by Th5;
now__::_thesis:_(_(_HT_(f,T)_in_Support_g_&_contradiction_)_or_(_HT_(f,T)_in_Support_h_&_contradiction_)_)
percases ( HT (f,T) in Support g or HT (f,T) in Support h ) by A51, A55, XBOOLE_0:def_3;
caseA56: HT (f,T) in Support g ; ::_thesis: contradiction
consider b9 being bag of n such that
A57: b9 in HT (G,T) and
A58: b9 divides HT (f,T) by A16, A23, A54, Th18;
consider q being Polynomial of n,L such that
A59: b9 = HT (q,T) and
A60: q in G and
A61: q <> 0_ (n,L) by A57;
reconsider q = q as non-zero Polynomial of n,L by A61, POLYNOM7:def_1;
g is_reducible_wrt q,T by A56, A58, A59, POLYRED:36;
then consider r being Polynomial of n,L such that
A62: g reduces_to r,q,T by POLYRED:def_8;
HT (q,T) <= HT (f,T),T by A58, A59, TERMORD:10;
then q <> g by A53, A52, TERMORD:7;
then not q in {g} by TARSKI:def_1;
then q in G \ {g} by A60, XBOOLE_0:def_5;
then g reduces_to r,G \ {g},T by A62, POLYRED:def_7;
then g is_reducible_wrt G \ {g},T by POLYRED:def_9;
hence contradiction by A2, A33, Def7; ::_thesis: verum
end;
caseA63: HT (f,T) in Support h ; ::_thesis: contradiction
consider b9 being bag of n such that
A64: b9 in HT (H,T) and
A65: b9 divides HT (f,T) by A5, A31, A54, Th18;
consider q being Polynomial of n,L such that
A66: b9 = HT (q,T) and
A67: q in H and
A68: q <> 0_ (n,L) by A64;
reconsider q = q as non-zero Polynomial of n,L by A68, POLYNOM7:def_1;
h is_reducible_wrt q,T by A63, A65, A66, POLYRED:36;
then consider r being Polynomial of n,L such that
A69: h reduces_to r,q,T by POLYRED:def_8;
HT (q,T) <= HT (f,T),T by A65, A66, TERMORD:10;
then q <> h by A50, A53, A52, TERMORD:7;
then not q in {h} by TARSKI:def_1;
then q in H \ {h} by A67, XBOOLE_0:def_5;
then h reduces_to r,H \ {h},T by A69, POLYRED:def_7;
then h is_reducible_wrt H \ {h},T by POLYRED:def_9;
hence contradiction by A4, A39, Def7; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
supposeA70: g in H \ G ; ::_thesis: contradiction
then A71: not g in G by XBOOLE_0:def_5;
A72: g in H by A70, XBOOLE_0:def_5;
then A73: g <> 0_ (n,L) by A17;
then reconsider g = g as non-zero Polynomial of n,L by POLYNOM7:def_1;
A74: H c= H -Ideal by IDEAL_1:def_14;
then HT (g,T) in HT ((G -Ideal),T) by A16, A5, A72, A73;
then consider b9 being bag of n such that
A75: b9 in HT (G,T) and
A76: b9 divides HT (g,T) by A23, Th18;
consider h being Polynomial of n,L such that
A77: b9 = HT (h,T) and
A78: h in G and
A79: h <> 0_ (n,L) by A75;
reconsider h = h as non-zero Polynomial of n,L by A79, POLYNOM7:def_1;
set f = g - h;
A80: now__::_thesis:_not_g_-_h_=_0__(n,L)
assume A81: g - h = 0_ (n,L) ; ::_thesis: contradiction
h = (0_ (n,L)) + h by POLYRED:2
.= (g + (- h)) + h by A81, POLYNOM1:def_6
.= g + ((- h) + h) by POLYNOM1:21
.= g + (0_ (n,L)) by POLYRED:3 ;
hence contradiction by A71, A78, POLYNOM1:23; ::_thesis: verum
end;
Support g <> {} by A73, POLYNOM7:1;
then HT (g,T) in Support g by TERMORD:def_6;
then A82: g is_reducible_wrt h,T by A76, A77, POLYRED:36;
then A83: ex r being Polynomial of n,L st g reduces_to r,h,T by POLYRED:def_8;
now__::_thesis:_h_in_G_\_H
assume not h in G \ H ; ::_thesis: contradiction
then A84: h in H by A78, XBOOLE_0:def_5;
not h in {g} by A71, A78, TARSKI:def_1;
then h in H \ {g} by A84, XBOOLE_0:def_5;
then consider r being Polynomial of n,L such that
A85: ( h in H \ {g} & g reduces_to r,h,T ) by A83;
A86: g reduces_to r,H \ {g},T by A85, POLYRED:def_7;
g is_irreducible_wrt H \ {g},T by A4, A72, Def7;
hence contradiction by A86, POLYRED:def_9; ::_thesis: verum
end;
then h in GH by A30, XBOOLE_0:def_3;
then g <= h,T by A15;
then not h < g,T by POLYRED:29;
then not HT (h,T) < HT (g,T),T by POLYRED:32;
then A87: HT (g,T) <= HT (h,T),T by TERMORD:5;
HT (h,T) <= HT (g,T),T by A82, Th9;
then A88: HT (h,T) = HT (g,T) by A87, TERMORD:7;
reconsider f = g - h as non-zero Polynomial of n,L by A80, POLYNOM7:def_1;
Support f <> {} by A80, POLYNOM7:1;
then A89: HT (f,T) in Support f by TERMORD:def_6;
f . (HT (g,T)) = (g + (- h)) . (HT (g,T)) by POLYNOM1:def_6
.= (g . (HT (g,T))) + ((- h) . (HT (g,T))) by POLYNOM1:15
.= (g . (HT (g,T))) + (- (h . (HT (h,T)))) by A88, POLYNOM1:17
.= (HC (g,T)) + (- (h . (HT (h,T)))) by TERMORD:def_7
.= (HC (g,T)) + (- (HC (h,T))) by TERMORD:def_7
.= (1. L) + (- (HC (h,T))) by A24, A72
.= (1. L) + (- (1. L)) by A24, A78
.= 0. L by RLVECT_1:5 ;
then A90: HT (f,T) <> HT (g,T) by A89, POLYNOM1:def_3;
HT (f,T) <= max ((HT (g,T)),(HT (h,T)),T),T by Th7;
then A91: HT (f,T) <= HT (g,T),T by A88, TERMORD:12;
reconsider g9 = g, h9 = h as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider g9 = g9, h9 = h9 as Element of (Polynom-Ring (n,L)) ;
G c= G -Ideal by IDEAL_1:def_14;
then g9 - h9 in I by A16, A5, A72, A74, A78, IDEAL_1:15;
then f in I by Lm2;
then A92: HT (f,T) in HT (I,T) by A80;
Support (g + (- h)) c= (Support g) \/ (Support (- h)) by POLYNOM1:20;
then Support f c= (Support g) \/ (Support (- h)) by POLYNOM1:def_6;
then A93: Support f c= (Support g) \/ (Support h) by Th5;
now__::_thesis:_(_(_HT_(f,T)_in_Support_g_&_contradiction_)_or_(_HT_(f,T)_in_Support_h_&_contradiction_)_)
percases ( HT (f,T) in Support g or HT (f,T) in Support h ) by A89, A93, XBOOLE_0:def_3;
caseA94: HT (f,T) in Support g ; ::_thesis: contradiction
consider b9 being bag of n such that
A95: b9 in HT (H,T) and
A96: b9 divides HT (f,T) by A5, A31, A92, Th18;
consider q being Polynomial of n,L such that
A97: b9 = HT (q,T) and
A98: q in H and
A99: q <> 0_ (n,L) by A95;
reconsider q = q as non-zero Polynomial of n,L by A99, POLYNOM7:def_1;
g is_reducible_wrt q,T by A94, A96, A97, POLYRED:36;
then consider r being Polynomial of n,L such that
A100: g reduces_to r,q,T by POLYRED:def_8;
HT (q,T) <= HT (f,T),T by A96, A97, TERMORD:10;
then q <> g by A91, A90, TERMORD:7;
then not q in {g} by TARSKI:def_1;
then q in H \ {g} by A98, XBOOLE_0:def_5;
then g reduces_to r,H \ {g},T by A100, POLYRED:def_7;
then g is_reducible_wrt H \ {g},T by POLYRED:def_9;
hence contradiction by A4, A72, Def7; ::_thesis: verum
end;
caseA101: HT (f,T) in Support h ; ::_thesis: contradiction
consider b9 being bag of n such that
A102: b9 in HT (G,T) and
A103: b9 divides HT (f,T) by A16, A23, A92, Th18;
consider q being Polynomial of n,L such that
A104: b9 = HT (q,T) and
A105: q in G and
A106: q <> 0_ (n,L) by A102;
reconsider q = q as non-zero Polynomial of n,L by A106, POLYNOM7:def_1;
h is_reducible_wrt q,T by A101, A103, A104, POLYRED:36;
then consider r being Polynomial of n,L such that
A107: h reduces_to r,q,T by POLYRED:def_8;
HT (q,T) <= HT (f,T),T by A103, A104, TERMORD:10;
then HT (q,T) <> HT (h,T) by A88, A91, A90, TERMORD:7;
then not q in {h} by TARSKI:def_1;
then q in G \ {h} by A105, XBOOLE_0:def_5;
then h reduces_to r,G \ {h},T by A107, POLYRED:def_7;
then h is_reducible_wrt G \ {h},T by POLYRED:def_9;
hence contradiction by A2, A78, Def7; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
end;
end;