:: 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;