:: GROEB_2 semantic presentation begin theorem :: GROEB_2:1 for L being non empty right_complementable add-associative right_zeroed addLoopStr for p being FinSequence of L for n being Element of NAT st ( for k being Element of NAT st k in dom p & k > n holds p . k = 0. L ) holds Sum p = Sum (p | n) proof let L be non empty right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for p being FinSequence of L for n being Element of NAT st ( for k being Element of NAT st k in dom p & k > n holds p . k = 0. L ) holds Sum p = Sum (p | n) let p be FinSequence of L; ::_thesis: for n being Element of NAT st ( for k being Element of NAT st k in dom p & k > n holds p . k = 0. L ) holds Sum p = Sum (p | n) let n be Element of NAT ; ::_thesis: ( ( for k being Element of NAT st k in dom p & k > n holds p . k = 0. L ) implies Sum p = Sum (p | n) ) defpred S1[ Element of NAT ] means for p being FinSequence of L for n being Element of NAT st len p = $1 & ( for k being Element of NAT st k in dom p & k > n holds p . k = 0. L ) holds Sum p = Sum (p | n); A1: 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 A2: S1[k] ; ::_thesis: S1[k + 1] now__::_thesis:_for_p_being_FinSequence_of_L for_n_being_Element_of_NAT_st_len_p_=_k_+_1_&_(_for_l_being_Element_of_NAT_st_l_in_dom_p_&_l_>_n_holds_ p_._l_=_0._L_)_holds_ Sum_p_=_Sum_(p_|_n) let p be FinSequence of L; ::_thesis: for n being Element of NAT st len p = k + 1 & ( for l being Element of NAT st l in dom p & l > n holds p . l = 0. L ) holds Sum p = Sum (p | n) let n be Element of NAT ; ::_thesis: ( len p = k + 1 & ( for l being Element of NAT st l in dom p & l > n holds p . l = 0. L ) implies Sum p = Sum (p | n) ) assume that A3: len p = k + 1 and A4: for l being Element of NAT st l in dom p & l > n holds p . l = 0. L ; ::_thesis: Sum p = Sum (p | n) A5: dom p = Seg (k + 1) by A3, FINSEQ_1:def_3; set q = p | (Seg k); reconsider q = p | (Seg k) as FinSequence of L by FINSEQ_1:18; A6: k <= len p by A3, NAT_1:11; then A7: len q = k by FINSEQ_1:17; ( k <= k + 1 & dom q = Seg k ) by A6, FINSEQ_1:17, NAT_1:11; then A8: dom q c= dom p by A5, FINSEQ_1:5; A9: q = p | k by FINSEQ_1:def_15; A10: q ^ <*(p /. (k + 1))*> = q ^ <*(p . (k + 1))*> by A5, FINSEQ_1:4, PARTFUN1:def_6 .= p by A3, FINSEQ_3:55 ; now__::_thesis:_(_(_k_<_n_&_Sum_(p_|_n)_=_Sum_p_)_or_(_n_<=_k_&_Sum_p_=_Sum_(p_|_n)_)_) percases ( k < n or n <= k ) ; caseA11: k < n ; ::_thesis: Sum (p | n) = Sum p A12: dom (p | n) = dom (p | (Seg n)) by FINSEQ_1:def_15; A13: k + 1 <= n by A11, NAT_1:13; A14: now__::_thesis:_for_u_being_set_st_u_in_dom_p_holds_ u_in_dom_(p_|_n) let u be set ; ::_thesis: ( u in dom p implies u in dom (p | n) ) assume A15: u in dom p ; ::_thesis: u in dom (p | n) then reconsider u9 = u as Element of NAT ; A16: u in Seg (k + 1) by A3, A15, FINSEQ_1:def_3; then u9 <= k + 1 by FINSEQ_1:1; then A17: u9 <= n by A13, XXREAL_0:2; 1 <= u9 by A16, FINSEQ_1:1; then u9 in Seg n by A17, FINSEQ_1:1; then u9 in (dom p) /\ (Seg n) by A15, XBOOLE_0:def_4; hence u in dom (p | n) by A12, RELAT_1:61; ::_thesis: verum end; A18: for x being set st x in dom (p | (Seg n)) holds (p | (Seg n)) . x = p . x by FUNCT_1:47; now__::_thesis:_for_u_being_set_st_u_in_dom_(p_|_n)_holds_ u_in_dom_p let u be set ; ::_thesis: ( u in dom (p | n) implies u in dom p ) assume u in dom (p | n) ; ::_thesis: u in dom p then A19: u in dom (p | (Seg n)) by FINSEQ_1:def_15; dom (p | (Seg n)) c= dom p by RELAT_1:60; hence u in dom p by A19; ::_thesis: verum end; then dom (p | n) = dom p by A14, TARSKI:1; then p | (Seg n) = p by A12, A18, FUNCT_1:2; hence Sum (p | n) = Sum p by FINSEQ_1:def_15; ::_thesis: verum end; caseA20: n <= k ; ::_thesis: Sum p = Sum (p | n) A21: now__::_thesis:_for_l_being_Element_of_NAT_st_l_in_dom_q_&_l_>_n_holds_ q_._l_=_0._L let l be Element of NAT ; ::_thesis: ( l in dom q & l > n implies q . l = 0. L ) assume that A22: l in dom q and A23: l > n ; ::_thesis: q . l = 0. L A24: p . l = 0. L by A4, A8, A22, A23; thus q . l = q /. l by A22, PARTFUN1:def_6 .= p /. l by A9, A22, FINSEQ_4:70 .= 0. L by A8, A22, A24, PARTFUN1:def_6 ; ::_thesis: verum end; k + 1 > n by A20, NAT_1:13; then A25: 0. L = p . (k + 1) by A4, A5, FINSEQ_1:4 .= p /. (k + 1) by A5, FINSEQ_1:4, PARTFUN1:def_6 ; thus Sum p = (Sum q) + (Sum <*(p /. (k + 1))*>) by A10, RLVECT_1:41 .= (Sum q) + (p /. (k + 1)) by RLVECT_1:44 .= Sum q by A25, RLVECT_1:def_4 .= Sum (q | n) by A2, A7, A21 .= Sum (p | n) by A9, A20, FINSEQ_1:82 ; ::_thesis: verum end; end; end; hence Sum p = Sum (p | n) ; ::_thesis: verum end; hence S1[k + 1] ; ::_thesis: verum end; A26: S1[ 0 ] by FINSEQ_1:58; A27: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A26, A1); A28: ex k being Element of NAT st len p = k ; assume for k being Element of NAT st k in dom p & k > n holds p . k = 0. L ; ::_thesis: Sum p = Sum (p | n) hence Sum p = Sum (p | n) by A27, A28; ::_thesis: verum end; theorem :: GROEB_2:2 for L being non empty Abelian add-associative right_zeroed addLoopStr for f being FinSequence of L for i, j being Element of NAT holds Sum (Swap (f,i,j)) = Sum f proof let L be non empty Abelian add-associative right_zeroed addLoopStr ; ::_thesis: for f being FinSequence of L for i, j being Element of NAT holds Sum (Swap (f,i,j)) = Sum f let f be FinSequence of L; ::_thesis: for i, j being Element of NAT holds Sum (Swap (f,i,j)) = Sum f let i, j be Element of NAT ; ::_thesis: Sum (Swap (f,i,j)) = Sum f percases ( not 1 <= i or not i <= len f or not 1 <= j or not j <= len f or ( 1 <= i & i <= len f & 1 <= j & j <= len f ) ) ; suppose ( not 1 <= i or not i <= len f or not 1 <= j or not j <= len f ) ; ::_thesis: Sum (Swap (f,i,j)) = Sum f hence Sum (Swap (f,i,j)) = Sum f by FINSEQ_7:def_2; ::_thesis: verum end; supposeA1: ( 1 <= i & i <= len f & 1 <= j & j <= len f ) ; ::_thesis: Sum (Swap (f,i,j)) = Sum f then j in Seg (len f) by FINSEQ_1:1; then A2: j in dom f by FINSEQ_1:def_3; i in Seg (len f) by A1, FINSEQ_1:1; then A3: i in dom f by FINSEQ_1:def_3; now__::_thesis:_(_(_i_=_j_&_Sum_(Swap_(f,i,j))_=_Sum_f_)_or_(_i_<_j_&_Sum_(Swap_(f,i,j))_=_Sum_f_)_or_(_i_>_j_&_Sum_(Swap_(f,i,j))_=_Sum_f_)_) percases ( i = j or i < j or i > j ) by XXREAL_0:1; case i = j ; ::_thesis: Sum (Swap (f,i,j)) = Sum f hence Sum (Swap (f,i,j)) = Sum f by FINSEQ_7:19; ::_thesis: verum end; caseA4: i < j ; ::_thesis: Sum (Swap (f,i,j)) = Sum f then Swap (f,i,j) = ((((f | (i -' 1)) ^ <*(f /. j)*>) ^ ((f /^ i) | ((j -' i) -' 1))) ^ <*(f /. i)*>) ^ (f /^ j) by A1, FINSEQ_7:27; then A5: Sum (Swap (f,i,j)) = (Sum ((((f | (i -' 1)) ^ <*(f /. j)*>) ^ ((f /^ i) | ((j -' i) -' 1))) ^ <*(f /. i)*>)) + (Sum (f /^ j)) by RLVECT_1:41 .= ((Sum (((f | (i -' 1)) ^ <*(f /. j)*>) ^ ((f /^ i) | ((j -' i) -' 1)))) + (Sum <*(f /. i)*>)) + (Sum (f /^ j)) by RLVECT_1:41 .= (((Sum ((f | (i -' 1)) ^ <*(f /. j)*>)) + (Sum ((f /^ i) | ((j -' i) -' 1)))) + (Sum <*(f /. i)*>)) + (Sum (f /^ j)) by RLVECT_1:41 .= ((((Sum (f | (i -' 1))) + (Sum <*(f /. j)*>)) + (Sum ((f /^ i) | ((j -' i) -' 1)))) + (Sum <*(f /. i)*>)) + (Sum (f /^ j)) by RLVECT_1:41 .= ((((Sum (f | (i -' 1))) + (Sum <*(f /. j)*>)) + (Sum <*(f /. i)*>)) + (Sum ((f /^ i) | ((j -' i) -' 1)))) + (Sum (f /^ j)) by RLVECT_1:def_3 .= ((((Sum (f | (i -' 1))) + (Sum <*(f /. i)*>)) + (Sum <*(f /. j)*>)) + (Sum ((f /^ i) | ((j -' i) -' 1)))) + (Sum (f /^ j)) by RLVECT_1:def_3 .= (((Sum ((f | (i -' 1)) ^ <*(f /. i)*>)) + (Sum <*(f /. j)*>)) + (Sum ((f /^ i) | ((j -' i) -' 1)))) + (Sum (f /^ j)) by RLVECT_1:41 .= (((Sum ((f | (i -' 1)) ^ <*(f /. i)*>)) + (Sum ((f /^ i) | ((j -' i) -' 1)))) + (Sum <*(f /. j)*>)) + (Sum (f /^ j)) by RLVECT_1:def_3 .= ((Sum (((f | (i -' 1)) ^ <*(f /. i)*>) ^ ((f /^ i) | ((j -' i) -' 1)))) + (Sum <*(f /. j)*>)) + (Sum (f /^ j)) by RLVECT_1:41 .= (Sum ((((f | (i -' 1)) ^ <*(f /. i)*>) ^ ((f /^ i) | ((j -' i) -' 1))) ^ <*(f /. j)*>)) + (Sum (f /^ j)) by RLVECT_1:41 .= Sum (((((f | (i -' 1)) ^ <*(f /. i)*>) ^ ((f /^ i) | ((j -' i) -' 1))) ^ <*(f /. j)*>) ^ (f /^ j)) by RLVECT_1:41 ; ((((f | (i -' 1)) ^ <*(f /. i)*>) ^ ((f /^ i) | ((j -' i) -' 1))) ^ <*(f /. j)*>) ^ (f /^ j) = ((((f | (i -' 1)) ^ <*(f . i)*>) ^ ((f /^ i) | ((j -' i) -' 1))) ^ <*(f /. j)*>) ^ (f /^ j) by A3, PARTFUN1:def_6 .= ((((f | (i -' 1)) ^ <*(f . i)*>) ^ ((f /^ i) | ((j -' i) -' 1))) ^ <*(f . j)*>) ^ (f /^ j) by A2, PARTFUN1:def_6 .= f by A1, A4, FINSEQ_7:1 ; hence Sum (Swap (f,i,j)) = Sum f by A5; ::_thesis: verum end; caseA6: i > j ; ::_thesis: Sum (Swap (f,i,j)) = Sum f then Swap (f,j,i) = ((((f | (j -' 1)) ^ <*(f /. i)*>) ^ ((f /^ j) | ((i -' j) -' 1))) ^ <*(f /. j)*>) ^ (f /^ i) by A1, FINSEQ_7:27; then A7: Sum (Swap (f,j,i)) = (Sum ((((f | (j -' 1)) ^ <*(f /. i)*>) ^ ((f /^ j) | ((i -' j) -' 1))) ^ <*(f /. j)*>)) + (Sum (f /^ i)) by RLVECT_1:41 .= ((Sum (((f | (j -' 1)) ^ <*(f /. i)*>) ^ ((f /^ j) | ((i -' j) -' 1)))) + (Sum <*(f /. j)*>)) + (Sum (f /^ i)) by RLVECT_1:41 .= (((Sum ((f | (j -' 1)) ^ <*(f /. i)*>)) + (Sum ((f /^ j) | ((i -' j) -' 1)))) + (Sum <*(f /. j)*>)) + (Sum (f /^ i)) by RLVECT_1:41 .= ((((Sum (f | (j -' 1))) + (Sum <*(f /. i)*>)) + (Sum ((f /^ j) | ((i -' j) -' 1)))) + (Sum <*(f /. j)*>)) + (Sum (f /^ i)) by RLVECT_1:41 .= ((((Sum (f | (j -' 1))) + (Sum <*(f /. i)*>)) + (Sum <*(f /. j)*>)) + (Sum ((f /^ j) | ((i -' j) -' 1)))) + (Sum (f /^ i)) by RLVECT_1:def_3 .= ((((Sum (f | (j -' 1))) + (Sum <*(f /. j)*>)) + (Sum <*(f /. i)*>)) + (Sum ((f /^ j) | ((i -' j) -' 1)))) + (Sum (f /^ i)) by RLVECT_1:def_3 .= (((Sum ((f | (j -' 1)) ^ <*(f /. j)*>)) + (Sum <*(f /. i)*>)) + (Sum ((f /^ j) | ((i -' j) -' 1)))) + (Sum (f /^ i)) by RLVECT_1:41 .= (((Sum ((f | (j -' 1)) ^ <*(f /. j)*>)) + (Sum ((f /^ j) | ((i -' j) -' 1)))) + (Sum <*(f /. i)*>)) + (Sum (f /^ i)) by RLVECT_1:def_3 .= ((Sum (((f | (j -' 1)) ^ <*(f /. j)*>) ^ ((f /^ j) | ((i -' j) -' 1)))) + (Sum <*(f /. i)*>)) + (Sum (f /^ i)) by RLVECT_1:41 .= (Sum ((((f | (j -' 1)) ^ <*(f /. j)*>) ^ ((f /^ j) | ((i -' j) -' 1))) ^ <*(f /. i)*>)) + (Sum (f /^ i)) by RLVECT_1:41 .= Sum (((((f | (j -' 1)) ^ <*(f /. j)*>) ^ ((f /^ j) | ((i -' j) -' 1))) ^ <*(f /. i)*>) ^ (f /^ i)) by RLVECT_1:41 ; ((((f | (j -' 1)) ^ <*(f /. j)*>) ^ ((f /^ j) | ((i -' j) -' 1))) ^ <*(f /. i)*>) ^ (f /^ i) = ((((f | (j -' 1)) ^ <*(f . j)*>) ^ ((f /^ j) | ((i -' j) -' 1))) ^ <*(f /. i)*>) ^ (f /^ i) by A2, PARTFUN1:def_6 .= ((((f | (j -' 1)) ^ <*(f . j)*>) ^ ((f /^ j) | ((i -' j) -' 1))) ^ <*(f . i)*>) ^ (f /^ i) by A3, PARTFUN1:def_6 .= f by A1, A6, FINSEQ_7:1 ; hence Sum (Swap (f,i,j)) = Sum f by A7, FINSEQ_7:21; ::_thesis: verum end; end; end; hence Sum (Swap (f,i,j)) = Sum f ; ::_thesis: verum end; end; end; definition let X be set ; let b1, b2 be bag of X; assume A1: b2 divides b1 ; funcb1 / b2 -> bag of X means :Def1: :: GROEB_2:def 1 b2 + it = b1; existence ex b1 being bag of X st b2 + b1 = b1 by A1, TERMORD:1; uniqueness for b1, b2 being bag of X st b2 + b1 = b1 & b2 + b2 = b1 holds b1 = b2 proof let b3, b4 be bag of X; ::_thesis: ( b2 + b3 = b1 & b2 + b4 = b1 implies b3 = b4 ) assume A2: b2 + b3 = b1 ; ::_thesis: ( not b2 + b4 = b1 or b3 = b4 ) assume A3: b2 + b4 = b1 ; ::_thesis: b3 = b4 A4: now__::_thesis:_for_x_being_set_st_x_in_dom_b3_holds_ b3_._x_=_b4_._x let x be set ; ::_thesis: ( x in dom b3 implies b3 . x = b4 . x ) assume x in dom b3 ; ::_thesis: b3 . x = b4 . x thus b3 . x = ((b2 . x) + (b3 . x)) -' (b2 . x) by NAT_D:34 .= (b1 . x) -' (b2 . x) by A2, PRE_POLY:def_5 .= ((b2 . x) + (b4 . x)) -' (b2 . x) by A3, PRE_POLY:def_5 .= b4 . x by NAT_D:34 ; ::_thesis: verum end; dom b3 = X by PARTFUN1:def_2 .= dom b4 by PARTFUN1:def_2 ; hence b3 = b4 by A4, FUNCT_1:2; ::_thesis: verum end; end; :: deftheorem Def1 defines / GROEB_2:def_1_:_ for X being set for b1, b2 being bag of X st b2 divides b1 holds for b4 being bag of X holds ( b4 = b1 / b2 iff b2 + b4 = b1 ); definition let X be set ; let b1, b2 be bag of X; func lcm (b1,b2) -> bag of X means :Def2: :: GROEB_2:def 2 for k being set holds it . k = max ((b1 . k),(b2 . k)); existence ex b1 being bag of X st for k being set holds b1 . k = max ((b1 . k),(b2 . k)) proof defpred S1[ set , set ] means $2 = max ((b1 . $1),(b2 . $1)); A1: for x being set st x in X holds ex y being set st S1[x,y] ; consider b being Function such that A2: ( dom b = X & ( for x being set st x in X holds S1[x,b . x] ) ) from CLASSES1:sch_1(A1); reconsider b = b as ManySortedSet of X by A2, PARTFUN1:def_2, RELAT_1:def_18; 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 A3: ( x in dom b & u = b . x ) by FUNCT_1:def_3; u = max ((b1 . x),(b2 . x)) by A2, A3; hence u in NAT ; ::_thesis: verum end; then A4: rng b c= NAT by TARSKI:def_3; now__::_thesis:_for_u_being_set_st_u_in_support_b_holds_ u_in_(support_b1)_\/_(support_b2) let u be set ; ::_thesis: ( u in support b implies u in (support b1) \/ (support b2) ) A5: support b c= dom b by PRE_POLY:37; assume A6: u in support b ; ::_thesis: u in (support b1) \/ (support b2) then A7: b . u <> 0 by PRE_POLY:def_7; now__::_thesis:_u_in_(support_b1)_\/_(support_b2) assume A8: not u in (support b1) \/ (support b2) ; ::_thesis: contradiction then not u in support b2 by XBOOLE_0:def_3; then A9: b2 . u = 0 by PRE_POLY:def_7; not u in support b1 by A8, XBOOLE_0:def_3; then b1 . u = 0 by PRE_POLY:def_7; then max ((b1 . u),(b2 . u)) = 0 by A9; hence contradiction by A2, A6, A7, A5; ::_thesis: verum end; hence u in (support b1) \/ (support b2) ; ::_thesis: verum end; then support b c= (support b1) \/ (support b2) by TARSKI:def_3; then reconsider b = b as bag of X by A4, PRE_POLY:def_8, VALUED_0:def_6; A10: dom b = X by PARTFUN1:def_2 .= dom b2 by PARTFUN1:def_2 ; take b ; ::_thesis: for k being set holds b . k = max ((b1 . k),(b2 . k)) A11: dom b = X by PARTFUN1:def_2 .= dom b1 by PARTFUN1:def_2 ; now__::_thesis:_for_k_being_set_holds_b_._k_=_max_((b1_._k),(b2_._k)) let k be set ; ::_thesis: b . k = max ((b1 . k),(b2 . k)) now__::_thesis:_(_(_k_in_dom_b_&_b_._k_=_max_((b1_._k),(b2_._k))_)_or_(_not_k_in_dom_b_&_b_._k_=_max_((b1_._k),(b2_._k))_)_) percases ( k in dom b or not k in dom b ) ; case k in dom b ; ::_thesis: b . k = max ((b1 . k),(b2 . k)) hence b . k = max ((b1 . k),(b2 . k)) by A2; ::_thesis: verum end; caseA12: not k in dom b ; ::_thesis: b . k = max ((b1 . k),(b2 . k)) then ( b1 . k = 0 & b2 . k = 0 ) by A11, A10, FUNCT_1:def_2; hence b . k = max ((b1 . k),(b2 . k)) by A12, FUNCT_1:def_2; ::_thesis: verum end; end; end; hence b . k = max ((b1 . k),(b2 . k)) ; ::_thesis: verum end; hence for k being set holds b . k = max ((b1 . k),(b2 . k)) ; ::_thesis: verum end; uniqueness for b1, b2 being bag of X st ( for k being set holds b1 . k = max ((b1 . k),(b2 . k)) ) & ( for k being set holds b2 . k = max ((b1 . k),(b2 . k)) ) holds b1 = b2 proof let b3, b4 be bag of X; ::_thesis: ( ( for k being set holds b3 . k = max ((b1 . k),(b2 . k)) ) & ( for k being set holds b4 . k = max ((b1 . k),(b2 . k)) ) implies b3 = b4 ) assume A13: for k being set holds b3 . k = max ((b1 . k),(b2 . k)) ; ::_thesis: ( ex k being set st not b4 . k = max ((b1 . k),(b2 . k)) or b3 = b4 ) assume A14: for k being set holds b4 . k = max ((b1 . k),(b2 . k)) ; ::_thesis: b3 = b4 A15: now__::_thesis:_for_u_being_set_st_u_in_dom_b3_holds_ b3_._u_=_b4_._u let u be set ; ::_thesis: ( u in dom b3 implies b3 . u = b4 . u ) assume u in dom b3 ; ::_thesis: b3 . u = b4 . u thus b3 . u = max ((b1 . u),(b2 . u)) by A13 .= b4 . u by A14 ; ::_thesis: verum end; dom b3 = X by PARTFUN1:def_2 .= dom b4 by PARTFUN1:def_2 ; hence b3 = b4 by A15, FUNCT_1:2; ::_thesis: verum end; commutativity for b1, b1, b2 being bag of X st ( for k being set holds b1 . k = max ((b1 . k),(b2 . k)) ) holds for k being set holds b1 . k = max ((b2 . k),(b1 . k)) ; idempotence for b1 being bag of X for k being set holds b1 . k = max ((b1 . k),(b1 . k)) ; end; :: deftheorem Def2 defines lcm GROEB_2:def_2_:_ for X being set for b1, b2, b4 being bag of X holds ( b4 = lcm (b1,b2) iff for k being set holds b4 . k = max ((b1 . k),(b2 . k)) ); notation let X be set ; let b1, b2 be bag of X; synonym b1 lcm b2 for lcm (b1,b2); end; definition let X be set ; let b1, b2 be bag of X; predb1,b2 are_disjoint means :Def3: :: GROEB_2:def 3 for i being set holds ( b1 . i = 0 or b2 . i = 0 ); end; :: deftheorem Def3 defines are_disjoint GROEB_2:def_3_:_ for X being set for b1, b2 being bag of X holds ( b1,b2 are_disjoint iff for i being set holds ( b1 . i = 0 or b2 . i = 0 ) ); notation let X be set ; let b1, b2 be bag of X; antonym b1,b2 are_non_disjoint for b1,b2 are_disjoint ; end; theorem Th3: :: GROEB_2:3 for X being set for b1, b2 being bag of X holds ( b1 divides lcm (b1,b2) & b2 divides lcm (b1,b2) ) proof let X be set ; ::_thesis: for b1, b2 being bag of X holds ( b1 divides lcm (b1,b2) & b2 divides lcm (b1,b2) ) let b1, b2 be bag of X; ::_thesis: ( b1 divides lcm (b1,b2) & b2 divides lcm (b1,b2) ) set bb = lcm (b1,b2); now__::_thesis:_for_k_being_set_holds_b1_._k_<=_(lcm_(b1,b2))_._k let k be set ; ::_thesis: b1 . k <= (lcm (b1,b2)) . k b1 . k <= max ((b1 . k),(b2 . k)) by XXREAL_0:25; hence b1 . k <= (lcm (b1,b2)) . k by Def2; ::_thesis: verum end; hence b1 divides lcm (b1,b2) by PRE_POLY:def_11; ::_thesis: b2 divides lcm (b1,b2) now__::_thesis:_for_k_being_set_holds_b2_._k_<=_(lcm_(b1,b2))_._k let k be set ; ::_thesis: b2 . k <= (lcm (b1,b2)) . k b2 . k <= max ((b1 . k),(b2 . k)) by XXREAL_0:25; hence b2 . k <= (lcm (b1,b2)) . k by Def2; ::_thesis: verum end; hence b2 divides lcm (b1,b2) by PRE_POLY:def_11; ::_thesis: verum end; theorem Th4: :: GROEB_2:4 for X being set for b1, b2, b3 being bag of X st b1 divides b3 & b2 divides b3 holds lcm (b1,b2) divides b3 proof let X be set ; ::_thesis: for b1, b2, b3 being bag of X st b1 divides b3 & b2 divides b3 holds lcm (b1,b2) divides b3 let b1, b2, b3 be bag of X; ::_thesis: ( b1 divides b3 & b2 divides b3 implies lcm (b1,b2) divides b3 ) assume that A1: b1 divides b3 and A2: b2 divides b3 ; ::_thesis: lcm (b1,b2) divides b3 now__::_thesis:_for_k_being_set_st_k_in_X_holds_ (lcm_(b1,b2))_._k_<=_b3_._k let k be set ; ::_thesis: ( k in X implies (lcm (b1,b2)) . k <= b3 . k ) assume k in X ; ::_thesis: (lcm (b1,b2)) . k <= b3 . k now__::_thesis:_(_(_max_((b1_._k),(b2_._k))_=_b1_._k_&_max_((b1_._k),(b2_._k))_<=_b3_._k_)_or_(_max_((b1_._k),(b2_._k))_=_b2_._k_&_max_((b1_._k),(b2_._k))_<=_b3_._k_)_) percases ( max ((b1 . k),(b2 . k)) = b1 . k or max ((b1 . k),(b2 . k)) = b2 . k ) by XXREAL_0:16; case max ((b1 . k),(b2 . k)) = b1 . k ; ::_thesis: max ((b1 . k),(b2 . k)) <= b3 . k hence max ((b1 . k),(b2 . k)) <= b3 . k by A1, PRE_POLY:def_11; ::_thesis: verum end; case max ((b1 . k),(b2 . k)) = b2 . k ; ::_thesis: max ((b1 . k),(b2 . k)) <= b3 . k hence max ((b1 . k),(b2 . k)) <= b3 . k by A2, PRE_POLY:def_11; ::_thesis: verum end; end; end; hence (lcm (b1,b2)) . k <= b3 . k by Def2; ::_thesis: verum end; hence lcm (b1,b2) divides b3 by PRE_POLY:46; ::_thesis: verum end; theorem :: GROEB_2:5 for X being set for b1, b2 being bag of X holds ( b1,b2 are_disjoint iff lcm (b1,b2) = b1 + b2 ) proof let X be set ; ::_thesis: for b1, b2 being bag of X holds ( b1,b2 are_disjoint iff lcm (b1,b2) = b1 + b2 ) let b1, b2 be bag of X; ::_thesis: ( b1,b2 are_disjoint iff lcm (b1,b2) = b1 + b2 ) A1: now__::_thesis:_(_lcm_(b1,b2)_=_b1_+_b2_implies_b1,b2_are_disjoint_) assume A2: lcm (b1,b2) = b1 + b2 ; ::_thesis: b1,b2 are_disjoint now__::_thesis:_for_k_being_set_holds_ (_b1_._k_=_0_or_b2_._k_=_0_) let k be set ; ::_thesis: ( b1 . k = 0 or b2 . k = 0 ) A3: (lcm (b1,b2)) . k = max ((b1 . k),(b2 . k)) by Def2; now__::_thesis:_(_(_(b1_+_b2)_._k_=_b1_._k_&_b2_._k_=_0_)_or_(_(b1_+_b2)_._k_=_b2_._k_&_b1_._k_=_0_)_) percases ( (b1 + b2) . k = b1 . k or (b1 + b2) . k = b2 . k ) by A2, A3, XXREAL_0:16; case (b1 + b2) . k = b1 . k ; ::_thesis: b2 . k = 0 then (b1 . k) + (b2 . k) = (b1 . k) + 0 by PRE_POLY:def_5; hence b2 . k = 0 ; ::_thesis: verum end; case (b1 + b2) . k = b2 . k ; ::_thesis: b1 . k = 0 then (b1 . k) + (b2 . k) = 0 + (b2 . k) by PRE_POLY:def_5; hence b1 . k = 0 ; ::_thesis: verum end; end; end; hence ( b1 . k = 0 or b2 . k = 0 ) ; ::_thesis: verum end; hence b1,b2 are_disjoint by Def3; ::_thesis: verum end; now__::_thesis:_(_b1,b2_are_disjoint_implies_lcm_(b1,b2)_=_b1_+_b2_) assume A4: b1,b2 are_disjoint ; ::_thesis: lcm (b1,b2) = b1 + b2 now__::_thesis:_for_k_being_set_holds_(b1_+_b2)_._k_=_max_((b1_._k),(b2_._k)) let k be set ; ::_thesis: (b1 + b2) . k = max ((b1 . k),(b2 . k)) now__::_thesis:_(_(_b1_._k_=_0_&_(b1_+_b2)_._k_=_max_((b1_._k),(b2_._k))_)_or_(_b2_._k_=_0_&_(b1_+_b2)_._k_=_max_((b1_._k),(b2_._k))_)_) percases ( b1 . k = 0 or b2 . k = 0 ) by A4, Def3; caseA5: b1 . k = 0 ; ::_thesis: (b1 + b2) . k = max ((b1 . k),(b2 . k)) hence (b1 + b2) . k = 0 + (b2 . k) by PRE_POLY:def_5 .= max ((b1 . k),(b2 . k)) by A5, XXREAL_0:def_10 ; ::_thesis: verum end; caseA6: b2 . k = 0 ; ::_thesis: (b1 + b2) . k = max ((b1 . k),(b2 . k)) hence (b1 + b2) . k = (b1 . k) + 0 by PRE_POLY:def_5 .= max ((b1 . k),(b2 . k)) by A6, XXREAL_0:def_10 ; ::_thesis: verum end; end; end; hence (b1 + b2) . k = max ((b1 . k),(b2 . k)) ; ::_thesis: verum end; hence lcm (b1,b2) = b1 + b2 by Def2; ::_thesis: verum end; hence ( b1,b2 are_disjoint iff lcm (b1,b2) = b1 + b2 ) by A1; ::_thesis: verum end; theorem Th6: :: GROEB_2:6 for X being set for b being bag of X holds b / b = EmptyBag X proof let X be set ; ::_thesis: for b being bag of X holds b / b = EmptyBag X let b be bag of X; ::_thesis: b / b = EmptyBag X b + (EmptyBag X) = b by PRE_POLY:53; hence b / b = EmptyBag X by Def1; ::_thesis: verum end; theorem Th7: :: GROEB_2:7 for X being set for b1, b2 being bag of X holds ( b2 divides b1 iff lcm (b1,b2) = b1 ) proof let X be set ; ::_thesis: for b1, b2 being bag of X holds ( b2 divides b1 iff lcm (b1,b2) = b1 ) let b1, b2 be bag of X; ::_thesis: ( b2 divides b1 iff lcm (b1,b2) = b1 ) now__::_thesis:_(_b2_divides_b1_implies_lcm_(b1,b2)_=_b1_) assume A1: b2 divides b1 ; ::_thesis: lcm (b1,b2) = b1 now__::_thesis:_for_k_being_set_holds_b1_._k_=_max_((b1_._k),(b2_._k)) let k be set ; ::_thesis: b1 . k = max ((b1 . k),(b2 . k)) b2 . k <= b1 . k by A1, PRE_POLY:def_11; hence b1 . k = max ((b1 . k),(b2 . k)) by XXREAL_0:def_10; ::_thesis: verum end; hence lcm (b1,b2) = b1 by Def2; ::_thesis: verum end; hence ( b2 divides b1 iff lcm (b1,b2) = b1 ) by Th3; ::_thesis: verum end; theorem :: GROEB_2:8 for X being set for b1, b2, b3 being bag of X st b1 divides lcm (b2,b3) holds lcm (b2,b1) divides lcm (b2,b3) proof let X be set ; ::_thesis: for b1, b2, b3 being bag of X st b1 divides lcm (b2,b3) holds lcm (b2,b1) divides lcm (b2,b3) let b1, b2, b3 be bag of X; ::_thesis: ( b1 divides lcm (b2,b3) implies lcm (b2,b1) divides lcm (b2,b3) ) assume A1: b1 divides lcm (b2,b3) ; ::_thesis: lcm (b2,b1) divides lcm (b2,b3) for k being set st k in X holds (lcm (b2,b1)) . k <= (lcm (b2,b3)) . k proof let k be set ; ::_thesis: ( k in X implies (lcm (b2,b1)) . k <= (lcm (b2,b3)) . k ) assume k in X ; ::_thesis: (lcm (b2,b1)) . k <= (lcm (b2,b3)) . k b1 . k <= (lcm (b2,b3)) . k by A1, PRE_POLY:def_11; then A2: b1 . k <= max ((b2 . k),(b3 . k)) by Def2; b2 . k <= max ((b2 . k),(b3 . k)) by XXREAL_0:25; then max ((b2 . k),(b1 . k)) <= max ((b2 . k),(b3 . k)) by A2, XXREAL_0:28; then max ((b2 . k),(b1 . k)) <= (lcm (b2,b3)) . k by Def2; hence (lcm (b2,b1)) . k <= (lcm (b2,b3)) . k by Def2; ::_thesis: verum end; hence lcm (b2,b1) divides lcm (b2,b3) by PRE_POLY:46; ::_thesis: verum end; theorem :: GROEB_2:9 for X being set for b1, b2, b3 being bag of X st lcm (b2,b1) divides lcm (b2,b3) holds lcm (b1,b3) divides lcm (b2,b3) proof let X be set ; ::_thesis: for b1, b2, b3 being bag of X st lcm (b2,b1) divides lcm (b2,b3) holds lcm (b1,b3) divides lcm (b2,b3) let b1, b2, b3 be bag of X; ::_thesis: ( lcm (b2,b1) divides lcm (b2,b3) implies lcm (b1,b3) divides lcm (b2,b3) ) assume A1: lcm (b2,b1) divides lcm (b2,b3) ; ::_thesis: lcm (b1,b3) divides lcm (b2,b3) for k being set st k in X holds (lcm (b1,b3)) . k <= (lcm (b2,b3)) . k proof let k be set ; ::_thesis: ( k in X implies (lcm (b1,b3)) . k <= (lcm (b2,b3)) . k ) assume k in X ; ::_thesis: (lcm (b1,b3)) . k <= (lcm (b2,b3)) . k A2: b3 . k <= max ((b2 . k),(b3 . k)) by XXREAL_0:25; (lcm (b2,b1)) . k <= (lcm (b2,b3)) . k by A1, PRE_POLY:def_11; then max ((b2 . k),(b1 . k)) <= (lcm (b2,b3)) . k by Def2; then A3: max ((b2 . k),(b1 . k)) <= max ((b2 . k),(b3 . k)) by Def2; b1 . k <= max ((b2 . k),(b1 . k)) by XXREAL_0:25; then b1 . k <= max ((b2 . k),(b3 . k)) by A3, XXREAL_0:2; then max ((b1 . k),(b3 . k)) <= max ((b2 . k),(b3 . k)) by A2, XXREAL_0:28; then max ((b1 . k),(b3 . k)) <= (lcm (b2,b3)) . k by Def2; hence (lcm (b1,b3)) . k <= (lcm (b2,b3)) . k by Def2; ::_thesis: verum end; hence lcm (b1,b3) divides lcm (b2,b3) by PRE_POLY:46; ::_thesis: verum end; theorem :: GROEB_2:10 for n being set for b1, b2, b3 being bag of n st lcm (b1,b3) divides lcm (b2,b3) holds b1 divides lcm (b2,b3) proof let X be set ; ::_thesis: for b1, b2, b3 being bag of X st lcm (b1,b3) divides lcm (b2,b3) holds b1 divides lcm (b2,b3) let b1, b2, b3 be bag of X; ::_thesis: ( lcm (b1,b3) divides lcm (b2,b3) implies b1 divides lcm (b2,b3) ) assume A1: lcm (b1,b3) divides lcm (b2,b3) ; ::_thesis: b1 divides lcm (b2,b3) for k being set st k in X holds b1 . k <= (lcm (b2,b3)) . k proof let k be set ; ::_thesis: ( k in X implies b1 . k <= (lcm (b2,b3)) . k ) assume k in X ; ::_thesis: b1 . k <= (lcm (b2,b3)) . k (lcm (b1,b3)) . k <= (lcm (b2,b3)) . k by A1, PRE_POLY:def_11; then max ((b1 . k),(b3 . k)) <= (lcm (b2,b3)) . k by Def2; then A2: max ((b1 . k),(b3 . k)) <= max ((b2 . k),(b3 . k)) by Def2; b1 . k <= max ((b1 . k),(b3 . k)) by XXREAL_0:25; then b1 . k <= max ((b2 . k),(b3 . k)) by A2, XXREAL_0:2; hence b1 . k <= (lcm (b2,b3)) . k by Def2; ::_thesis: verum end; hence b1 divides lcm (b2,b3) by PRE_POLY:46; ::_thesis: verum end; theorem :: GROEB_2:11 for n being Element of NAT for T being connected admissible TermOrder of n for P being non empty Subset of (Bags n) ex b being bag of n st ( b in P & ( for b9 being bag of n st b9 in P holds b <= b9,T ) ) proof let n be Element of NAT ; ::_thesis: for T being connected admissible TermOrder of n for P being non empty Subset of (Bags n) ex b being bag of n st ( b in P & ( for b9 being bag of n st b9 in P holds b <= b9,T ) ) let T be connected admissible TermOrder of n; ::_thesis: for P being non empty Subset of (Bags n) ex b being bag of n st ( b in P & ( for b9 being bag of n st b9 in P holds b <= b9,T ) ) let P be non empty Subset of (Bags n); ::_thesis: ex b being bag of n st ( b in P & ( for b9 being bag of n st b9 in P holds b <= b9,T ) ) set R = RelStr(# (Bags n),T #); set m = MinElement (P,RelStr(# (Bags n),T #)); A1: T is_connected_in field T by RELAT_2:def_14; reconsider b = MinElement (P,RelStr(# (Bags n),T #)) as bag of n ; A2: MinElement (P,RelStr(# (Bags n),T #)) is_minimal_wrt P, the InternalRel of RelStr(# (Bags n),T #) by BAGORDER:def_17; A3: now__::_thesis:_for_b9_being_bag_of_n_st_b9_in_P_holds_ b_<=_b9,T let b9 be bag of n; ::_thesis: ( b9 in P implies b <= b9,T ) b <= b,T by TERMORD:6; then [b,b] in T by TERMORD:def_2; then A4: b in field T by RELAT_1:15; b9 <= b9,T by TERMORD:6; then [b9,b9] in T by TERMORD:def_2; then A5: b9 in field T by RELAT_1:15; assume A6: b9 in P ; ::_thesis: b <= b9,T now__::_thesis:_(_(_b9_=_b_&_b_<=_b9,T_)_or_(_b9_<>_b_&_b_<=_b9,T_)_) percases ( b9 = b or b9 <> b ) ; case b9 = b ; ::_thesis: b <= b9,T hence b <= b9,T by TERMORD:6; ::_thesis: verum end; caseA7: b9 <> b ; ::_thesis: b <= b9,T then not [b9,b] in T by A2, A6, WAYBEL_4:def_25; then [b,b9] in T by A1, A4, A5, A7, RELAT_2:def_6; hence b <= b9,T by TERMORD:def_2; ::_thesis: verum end; end; end; hence b <= b9,T ; ::_thesis: verum end; MinElement (P,RelStr(# (Bags n),T #)) in P by BAGORDER:def_17; hence ex b being bag of n st ( b in P & ( for b9 being bag of n st b9 in P holds b <= b9,T ) ) by A3; ::_thesis: verum end; registration let L be non trivial right_complementable add-associative right_zeroed addLoopStr ; let a be non zero Element of L; cluster - a -> non zero ; coherence not - a is zero proof now__::_thesis:_not_-_a_=_0._L assume - a = 0. L ; ::_thesis: contradiction then - (- a) = 0. L by RLVECT_1:12; hence contradiction by RLVECT_1:17; ::_thesis: verum end; hence not - a is zero by STRUCT_0:def_12; ::_thesis: verum end; end; registration let X be set ; let L be non empty add-cancelable distributive right_zeroed left_zeroed doubleLoopStr ; let m be Monomial of X,L; let a be Element of L; clustera * m -> monomial-like ; coherence a * m is monomial-like proof set p = a * m; now__::_thesis:_(_(_Support_m_=_{}_&_a_*_m_is_monomial-like_)_or_(_ex_b_being_bag_of_X_st_Support_m_=_{b}_&_a_*_m_is_monomial-like_)_) percases ( Support m = {} or ex b being bag of X st Support m = {b} ) by POLYNOM7:6; caseA1: Support m = {} ; ::_thesis: a * m is monomial-like now__::_thesis:_not_Support_(a_*_m)_<>_{} set b = the Element of Support (a * m); assume A2: Support (a * m) <> {} ; ::_thesis: contradiction then the Element of Support (a * m) in Support (a * m) ; then reconsider b = the Element of Support (a * m) as Element of Bags X ; (a * m) . b = a * (m . b) by POLYNOM7:def_9 .= a * (0. L) by A1, POLYNOM1:def_3 .= 0. L by BINOM:2 ; hence contradiction by A2, POLYNOM1:def_3; ::_thesis: verum end; hence a * m is monomial-like by POLYNOM7:6; ::_thesis: verum end; case ex b being bag of X st Support m = {b} ; ::_thesis: a * m is monomial-like then consider b being bag of X such that A3: Support m = {b} ; reconsider b = b as Element of Bags X by PRE_POLY:def_12; now__::_thesis:_(_(_a_=_0._L_&_Support_(a_*_m)_=_{}_)_or_(_a_<>_0._L_&_a_*_m_is_monomial-like_)_) percases ( a = 0. L or a <> 0. L ) ; caseA4: a = 0. L ; ::_thesis: Support (a * m) = {} now__::_thesis:_not_Support_(a_*_m)_<>_{} set b = the Element of Support (a * m); assume A5: Support (a * m) <> {} ; ::_thesis: contradiction then the Element of Support (a * m) in Support (a * m) ; then reconsider b = the Element of Support (a * m) as Element of Bags X ; (a * m) . b = a * (m . b) by POLYNOM7:def_9 .= 0. L by A4, BINOM:1 ; hence contradiction by A5, POLYNOM1:def_3; ::_thesis: verum end; hence Support (a * m) = {} ; ::_thesis: verum end; case a <> 0. L ; ::_thesis: a * m is monomial-like A6: now__::_thesis:_for_b9_being_Element_of_Bags_X_st_b9_<>_b_holds_ (a_*_m)_._b9_=_0._L let b9 be Element of Bags X; ::_thesis: ( b9 <> b implies (a * m) . b9 = 0. L ) assume b9 <> b ; ::_thesis: (a * m) . b9 = 0. L then not b9 in Support m by A3, TARSKI:def_1; then A7: m . b9 = 0. L by POLYNOM1:def_3; (a * m) . b9 = a * (m . b9) by POLYNOM7:def_9; hence (a * m) . b9 = 0. L by A7, BINOM:2; ::_thesis: verum end; now__::_thesis:_(_(_a_*_(m_._b)_=_0._L_&_Support_(a_*_m)_=_{}_)_or_(_a_*_(m_._b)_<>_0._L_&_Support_(a_*_m)_=_{b}_)_) percases ( a * (m . b) = 0. L or a * (m . b) <> 0. L ) ; caseA8: a * (m . b) = 0. L ; ::_thesis: Support (a * m) = {} now__::_thesis:_not_Support_(a_*_m)_<>_{} set b9 = the Element of Support (a * m); assume A9: Support (a * m) <> {} ; ::_thesis: contradiction then the Element of Support (a * m) in Support (a * m) ; then reconsider b9 = the Element of Support (a * m) as Element of Bags X ; A10: (a * m) . b9 <> 0. L by A9, POLYNOM1:def_3; then b9 = b by A6; hence contradiction by A8, A10, POLYNOM7:def_9; ::_thesis: verum end; hence Support (a * m) = {} ; ::_thesis: verum end; caseA11: a * (m . b) <> 0. L ; ::_thesis: Support (a * m) = {b} A12: now__::_thesis:_for_u_being_set_st_u_in_Support_(a_*_m)_holds_ u_in_{b} let u be set ; ::_thesis: ( u in Support (a * m) implies u in {b} ) assume A13: u in Support (a * m) ; ::_thesis: u in {b} then reconsider u9 = u as Element of Bags X ; (a * m) . u9 <> 0. L by A13, POLYNOM1:def_3; then u9 = b by A6; hence u in {b} by TARSKI:def_1; ::_thesis: verum end; now__::_thesis:_for_u_being_set_st_u_in_{b}_holds_ u_in_Support_(a_*_m) let u be set ; ::_thesis: ( u in {b} implies u in Support (a * m) ) assume u in {b} ; ::_thesis: u in Support (a * m) then A14: u = b by TARSKI:def_1; (a * m) . b <> 0. L by A11, POLYNOM7:def_9; hence u in Support (a * m) by A14, POLYNOM1:def_3; ::_thesis: verum end; hence Support (a * m) = {b} by A12, TARSKI:1; ::_thesis: verum end; end; end; hence a * m is monomial-like by POLYNOM7:6; ::_thesis: verum end; end; end; hence a * m is monomial-like by POLYNOM7:6; ::_thesis: verum end; end; end; hence a * m is monomial-like ; ::_thesis: verum end; end; registration let n be Ordinal; let L be non trivial add-cancelable distributive right_zeroed domRing-like left_zeroed doubleLoopStr ; let p be non-zero Polynomial of n,L; let a be non zero Element of L; clustera * p -> non-zero ; coherence a * p is non-zero proof set b = the Element of Support p; set ap = a * p; p <> 0_ (n,L) by POLYNOM7:def_1; then A1: Support p <> {} by POLYNOM7:1; then the Element of Support p in Support p ; then reconsider b = the Element of Support p as Element of Bags n ; p . b <> 0. L by A1, POLYNOM1:def_3; then a * (p . b) <> 0. L by VECTSP_2:def_1; then (a * p) . b <> 0. L by POLYNOM7:def_9; then b in Support (a * p) by POLYNOM1:def_3; then a * p <> 0_ (n,L) by POLYNOM7:1; hence a * p is non-zero by POLYNOM7:def_1; ::_thesis: verum end; end; theorem Th12: :: GROEB_2:12 for n being Ordinal for L being non empty right-distributive right_zeroed doubleLoopStr for p, q being Series of n,L for b being bag of n holds b *' (p + q) = (b *' p) + (b *' q) proof let n be Ordinal; ::_thesis: for L being non empty right-distributive right_zeroed doubleLoopStr for p, q being Series of n,L for b being bag of n holds b *' (p + q) = (b *' p) + (b *' q) let L be non empty right-distributive right_zeroed doubleLoopStr ; ::_thesis: for p, q being Series of n,L for b being bag of n holds b *' (p + q) = (b *' p) + (b *' q) let p1, p2 be Series of n,L; ::_thesis: for b being bag of n holds b *' (p1 + p2) = (b *' p1) + (b *' p2) let b be bag of n; ::_thesis: b *' (p1 + p2) = (b *' p1) + (b *' p2) set q1 = b *' (p1 + p2); set q2 = (b *' p1) + (b *' p2); A1: now__::_thesis:_for_x_being_set_st_x_in_dom_(b_*'_(p1_+_p2))_holds_ (b_*'_(p1_+_p2))_._x_=_((b_*'_p1)_+_(b_*'_p2))_._x let x be set ; ::_thesis: ( x in dom (b *' (p1 + p2)) implies (b *' (p1 + p2)) . x = ((b *' p1) + (b *' p2)) . x ) assume x in dom (b *' (p1 + p2)) ; ::_thesis: (b *' (p1 + p2)) . x = ((b *' p1) + (b *' p2)) . x then reconsider u = x as bag of n ; now__::_thesis:_(_(_b_divides_u_&_(b_*'_(p1_+_p2))_._u_=_((b_*'_p1)_+_(b_*'_p2))_._u_)_or_(_not_b_divides_u_&_(b_*'_(p1_+_p2))_._u_=_((b_*'_p1)_+_(b_*'_p2))_._u_)_) percases ( b divides u or not b divides u ) ; caseA2: b divides u ; ::_thesis: (b *' (p1 + p2)) . u = ((b *' p1) + (b *' p2)) . u hence (b *' (p1 + p2)) . u = (p1 + p2) . (u -' b) by POLYRED:def_1 .= (p1 . (u -' b)) + (p2 . (u -' b)) by POLYNOM1:15 .= ((b *' p1) . u) + (p2 . (u -' b)) by A2, POLYRED:def_1 .= ((b *' p1) . u) + ((b *' p2) . u) by A2, POLYRED:def_1 .= ((b *' p1) + (b *' p2)) . u by POLYNOM1:15 ; ::_thesis: verum end; caseA3: not b divides u ; ::_thesis: (b *' (p1 + p2)) . u = ((b *' p1) + (b *' p2)) . u hence (b *' (p1 + p2)) . u = 0. L by POLYRED:def_1 .= (0. L) + (0. L) by RLVECT_1:def_4 .= ((b *' p1) . u) + (0. L) by A3, POLYRED:def_1 .= ((b *' p1) . u) + ((b *' p2) . u) by A3, POLYRED:def_1 .= ((b *' p1) + (b *' p2)) . u by POLYNOM1:15 ; ::_thesis: verum end; end; end; hence (b *' (p1 + p2)) . x = ((b *' p1) + (b *' p2)) . x ; ::_thesis: verum end; dom (b *' (p1 + p2)) = Bags n by FUNCT_2:def_1 .= dom ((b *' p1) + (b *' p2)) by FUNCT_2:def_1 ; hence b *' (p1 + p2) = (b *' p1) + (b *' p2) by A1, FUNCT_1:2; ::_thesis: verum end; theorem Th13: :: GROEB_2:13 for n being Ordinal for L being non empty right_complementable add-associative right_zeroed addLoopStr for p being Series of n,L for b being bag of n holds b *' (- p) = - (b *' p) proof let n be Ordinal; ::_thesis: for L being non empty right_complementable add-associative right_zeroed addLoopStr for p being Series of n,L for b being bag of n holds b *' (- p) = - (b *' p) let L be non empty right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for p being Series of n,L for b being bag of n holds b *' (- p) = - (b *' p) let p be Series of n,L; ::_thesis: for b being bag of n holds b *' (- p) = - (b *' p) let b be bag of n; ::_thesis: b *' (- p) = - (b *' p) set q1 = b *' (- p); set q2 = - (b *' p); A1: now__::_thesis:_for_x_being_set_st_x_in_dom_(b_*'_(-_p))_holds_ (b_*'_(-_p))_._x_=_(-_(b_*'_p))_._x let x be set ; ::_thesis: ( x in dom (b *' (- p)) implies (b *' (- p)) . x = (- (b *' p)) . x ) assume x in dom (b *' (- p)) ; ::_thesis: (b *' (- p)) . x = (- (b *' p)) . x then reconsider u = x as bag of n ; now__::_thesis:_(_(_b_divides_u_&_(b_*'_(-_p))_._u_=_(-_(b_*'_p))_._u_)_or_(_not_b_divides_u_&_(b_*'_(-_p))_._u_=_(-_(b_*'_p))_._u_)_) percases ( b divides u or not b divides u ) ; caseA2: b divides u ; ::_thesis: (b *' (- p)) . u = (- (b *' p)) . u then A3: (b *' p) . u = p . (u -' b) by POLYRED:def_1; thus (b *' (- p)) . u = (- p) . (u -' b) by A2, POLYRED:def_1 .= - (p . (u -' b)) by POLYNOM1:17 .= (- (b *' p)) . u by A3, POLYNOM1:17 ; ::_thesis: verum end; caseA4: not b divides u ; ::_thesis: (b *' (- p)) . u = (- (b *' p)) . u then A5: (b *' p) . u = 0. L by POLYRED:def_1; thus (b *' (- p)) . u = 0. L by A4, POLYRED:def_1 .= - (0. L) by RLVECT_1:12 .= (- (b *' p)) . u by A5, POLYNOM1:17 ; ::_thesis: verum end; end; end; hence (b *' (- p)) . x = (- (b *' p)) . x ; ::_thesis: verum end; dom (b *' (- p)) = Bags n by FUNCT_2:def_1 .= dom (- (b *' p)) by FUNCT_2:def_1 ; hence b *' (- p) = - (b *' p) by A1, FUNCT_1:2; ::_thesis: verum end; theorem Th14: :: GROEB_2:14 for n being Ordinal for L being non empty right_add-cancelable right-distributive left_zeroed doubleLoopStr for p being Series of n,L for b being bag of n for a being Element of L holds b *' (a * p) = a * (b *' p) proof let n be Ordinal; ::_thesis: for L being non empty right_add-cancelable right-distributive left_zeroed doubleLoopStr for p being Series of n,L for b being bag of n for a being Element of L holds b *' (a * p) = a * (b *' p) let L be non empty right_add-cancelable right-distributive left_zeroed doubleLoopStr ; ::_thesis: for p being Series of n,L for b being bag of n for a being Element of L holds b *' (a * p) = a * (b *' p) let p be Series of n,L; ::_thesis: for b being bag of n for a being Element of L holds b *' (a * p) = a * (b *' p) let b be bag of n; ::_thesis: for a being Element of L holds b *' (a * p) = a * (b *' p) let a be Element of L; ::_thesis: b *' (a * p) = a * (b *' p) set q1 = b *' (a * p); set q2 = a * (b *' p); A1: now__::_thesis:_for_x_being_set_st_x_in_dom_(b_*'_(a_*_p))_holds_ (b_*'_(a_*_p))_._x_=_(a_*_(b_*'_p))_._x let x be set ; ::_thesis: ( x in dom (b *' (a * p)) implies (b *' (a * p)) . x = (a * (b *' p)) . x ) assume x in dom (b *' (a * p)) ; ::_thesis: (b *' (a * p)) . x = (a * (b *' p)) . x then reconsider u = x as bag of n ; now__::_thesis:_(_(_b_divides_u_&_(b_*'_(a_*_p))_._u_=_(a_*_(b_*'_p))_._u_)_or_(_not_b_divides_u_&_(b_*'_(a_*_p))_._u_=_(a_*_(b_*'_p))_._u_)_) percases ( b divides u or not b divides u ) ; caseA2: b divides u ; ::_thesis: (b *' (a * p)) . u = (a * (b *' p)) . u hence (b *' (a * p)) . u = (a * p) . (u -' b) by POLYRED:def_1 .= a * (p . (u -' b)) by POLYNOM7:def_9 .= a * ((b *' p) . u) by A2, POLYRED:def_1 .= (a * (b *' p)) . u by POLYNOM7:def_9 ; ::_thesis: verum end; caseA3: not b divides u ; ::_thesis: (b *' (a * p)) . u = (a * (b *' p)) . u hence (b *' (a * p)) . u = 0. L by POLYRED:def_1 .= a * (0. L) by BINOM:2 .= a * ((b *' p) . u) by A3, POLYRED:def_1 .= (a * (b *' p)) . u by POLYNOM7:def_9 ; ::_thesis: verum end; end; end; hence (b *' (a * p)) . x = (a * (b *' p)) . x ; ::_thesis: verum end; dom (b *' (a * p)) = Bags n by FUNCT_2:def_1 .= dom (a * (b *' p)) by FUNCT_2:def_1 ; hence b *' (a * p) = a * (b *' p) by A1, FUNCT_1:2; ::_thesis: verum end; theorem Th15: :: GROEB_2:15 for n being Ordinal for L being non empty right-distributive doubleLoopStr for p, q being Series of n,L for a being Element of L holds a * (p + q) = (a * p) + (a * q) proof let n be Ordinal; ::_thesis: for L being non empty right-distributive doubleLoopStr for p, q being Series of n,L for a being Element of L holds a * (p + q) = (a * p) + (a * q) let L be non empty right-distributive doubleLoopStr ; ::_thesis: for p, q being Series of n,L for a being Element of L holds a * (p + q) = (a * p) + (a * q) let p1, p2 be Series of n,L; ::_thesis: for a being Element of L holds a * (p1 + p2) = (a * p1) + (a * p2) let b be Element of L; ::_thesis: b * (p1 + p2) = (b * p1) + (b * p2) set q1 = b * (p1 + p2); set q2 = (b * p1) + (b * p2); A1: now__::_thesis:_for_x_being_set_st_x_in_dom_(b_*_(p1_+_p2))_holds_ (b_*_(p1_+_p2))_._x_=_((b_*_p1)_+_(b_*_p2))_._x let x be set ; ::_thesis: ( x in dom (b * (p1 + p2)) implies (b * (p1 + p2)) . x = ((b * p1) + (b * p2)) . x ) assume x in dom (b * (p1 + p2)) ; ::_thesis: (b * (p1 + p2)) . x = ((b * p1) + (b * p2)) . x then reconsider u = x as bag of n ; (b * (p1 + p2)) . u = b * ((p1 + p2) . u) by POLYNOM7:def_9 .= b * ((p1 . u) + (p2 . u)) by POLYNOM1:15 .= (b * (p1 . u)) + (b * (p2 . u)) by VECTSP_1:def_2 .= ((b * p1) . u) + (b * (p2 . u)) by POLYNOM7:def_9 .= ((b * p1) . u) + ((b * p2) . u) by POLYNOM7:def_9 .= ((b * p1) + (b * p2)) . u by POLYNOM1:15 ; hence (b * (p1 + p2)) . x = ((b * p1) + (b * p2)) . x ; ::_thesis: verum end; dom (b * (p1 + p2)) = Bags n by FUNCT_2:def_1 .= dom ((b * p1) + (b * p2)) by FUNCT_2:def_1 ; hence b * (p1 + p2) = (b * p1) + (b * p2) by A1, FUNCT_1:2; ::_thesis: verum end; theorem Th16: :: GROEB_2:16 for X being set for L being non empty right_complementable add-associative right_zeroed doubleLoopStr for a being Element of L holds - (a | (X,L)) = (- a) | (X,L) proof let n be set ; ::_thesis: for L being non empty right_complementable add-associative right_zeroed doubleLoopStr for a being Element of L holds - (a | (n,L)) = (- a) | (n,L) let L be non empty right_complementable add-associative right_zeroed doubleLoopStr ; ::_thesis: for a being Element of L holds - (a | (n,L)) = (- a) | (n,L) let a be Element of L; ::_thesis: - (a | (n,L)) = (- a) | (n,L) A1: now__::_thesis:_for_u_being_set_st_u_in_dom_((-_a)_|_(n,L))_holds_ ((-_a)_|_(n,L))_._u_=_(-_(a_|_(n,L)))_._u let u be set ; ::_thesis: ( u in dom ((- a) | (n,L)) implies ((- a) | (n,L)) . u = (- (a | (n,L))) . u ) assume u in dom ((- a) | (n,L)) ; ::_thesis: ((- a) | (n,L)) . u = (- (a | (n,L))) . u then reconsider u9 = u as Element of Bags n ; now__::_thesis:_(_(_u9_=_EmptyBag_n_&_-_((a_|_(n,L))_._u9)_=_((-_a)_|_(n,L))_._u9_)_or_(_u9_<>_EmptyBag_n_&_-_((a_|_(n,L))_._u9)_=_((-_a)_|_(n,L))_._u9_)_) percases ( u9 = EmptyBag n or u9 <> EmptyBag n ) ; caseA2: u9 = EmptyBag n ; ::_thesis: - ((a | (n,L)) . u9) = ((- a) | (n,L)) . u9 hence - ((a | (n,L)) . u9) = - a by POLYNOM7:18 .= ((- a) | (n,L)) . u9 by A2, POLYNOM7:18 ; ::_thesis: verum end; caseA3: u9 <> EmptyBag n ; ::_thesis: - ((a | (n,L)) . u9) = ((- a) | (n,L)) . u9 - (0. L) = 0. L by RLVECT_1:12; hence - ((a | (n,L)) . u9) = 0. L by A3, POLYNOM7:18 .= ((- a) | (n,L)) . u9 by A3, POLYNOM7:18 ; ::_thesis: verum end; end; end; hence ((- a) | (n,L)) . u = (- (a | (n,L))) . u by POLYNOM1:17; ::_thesis: verum end; dom (- (a | (n,L))) = Bags n by FUNCT_2:def_1 .= dom ((- a) | (n,L)) by FUNCT_2:def_1 ; hence - (a | (n,L)) = (- a) | (n,L) by A1, FUNCT_1:2; ::_thesis: verum end; Lm1: 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; A3: 1 <= len p by NAT_1:14; reconsider l = (len p) - 1 as Element of NAT by INT_1:5, 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; begin theorem Th17: :: GROEB_2:17 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 p1, p2 being Polynomial of n,L st p1 <> p2 & p1 in P & p2 in P holds for m1, m2 being Monomial of n,L st HM ((m1 *' p1),T) = HM ((m2 *' p2),T) holds PolyRedRel (P,T) reduces (m1 *' p1) - (m2 *' p2), 0_ (n,L) ) holds P is_Groebner_basis_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 P being Subset of (Polynom-Ring (n,L)) st ( for p1, p2 being Polynomial of n,L st p1 <> p2 & p1 in P & p2 in P holds for m1, m2 being Monomial of n,L st HM ((m1 *' p1),T) = HM ((m2 *' p2),T) holds PolyRedRel (P,T) reduces (m1 *' p1) - (m2 *' p2), 0_ (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 Abelian add-associative right_zeroed doubleLoopStr for P being Subset of (Polynom-Ring (n,L)) st ( for p1, p2 being Polynomial of n,L st p1 <> p2 & p1 in P & p2 in P holds for m1, m2 being Monomial of n,L st HM ((m1 *' p1),T) = HM ((m2 *' p2),T) holds PolyRedRel (P,T) reduces (m1 *' p1) - (m2 *' p2), 0_ (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 Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for P being Subset of (Polynom-Ring (n,L)) st ( for p1, p2 being Polynomial of n,L st p1 <> p2 & p1 in P & p2 in P holds for m1, m2 being Monomial of n,L st HM ((m1 *' p1),T) = HM ((m2 *' p2),T) holds PolyRedRel (P,T) reduces (m1 *' p1) - (m2 *' p2), 0_ (n,L) ) holds P is_Groebner_basis_wrt T let P be Subset of (Polynom-Ring (n,L)); ::_thesis: ( ( for p1, p2 being Polynomial of n,L st p1 <> p2 & p1 in P & p2 in P holds for m1, m2 being Monomial of n,L st HM ((m1 *' p1),T) = HM ((m2 *' p2),T) holds PolyRedRel (P,T) reduces (m1 *' p1) - (m2 *' p2), 0_ (n,L) ) implies P is_Groebner_basis_wrt T ) assume A1: for p1, p2 being Polynomial of n,L st p1 <> p2 & p1 in P & p2 in P holds for m1, m2 being Monomial of n,L st HM ((m1 *' p1),T) = HM ((m2 *' p2),T) holds PolyRedRel (P,T) reduces (m1 *' p1) - (m2 *' p2), 0_ (n,L) ; ::_thesis: P is_Groebner_basis_wrt T set R = PolyRedRel (P,T); A2: 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_(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 A4: [a,c] in PolyRedRel (P,T) ; ::_thesis: b,c are_convergent_wrt PolyRedRel (P,T) consider f, f1 being set such that A5: f in NonZero (Polynom-Ring (n,L)) and A6: f1 in the carrier of (Polynom-Ring (n,L)) and A7: [a,b] = [f,f1] by A3, ZFMISC_1:def_2; A8: not f in {(0_ (n,L))} by A2, A5, XBOOLE_0:def_5; reconsider f = f as Polynomial of n,L by A5, POLYNOM1:def_10; f <> 0_ (n,L) by A8, TARSKI:def_1; then reconsider f = f as non-zero Polynomial of n,L by POLYNOM7:def_1; reconsider f1 = f1 as Polynomial of n,L by A6, POLYNOM1:def_10; f reduces_to f1,P,T by A3, A7, POLYRED:def_13; then consider g1 being Polynomial of n,L such that A9: g1 in P and A10: f reduces_to f1,g1,T by POLYRED:def_7; ex b1 being bag of n st f reduces_to f1,g1,b1,T by A10, POLYRED:def_6; then A11: g1 <> 0_ (n,L) by POLYRED:def_5; consider f9, f2 being set such that f9 in NonZero (Polynom-Ring (n,L)) and A12: f2 in the carrier of (Polynom-Ring (n,L)) and A13: [a,c] = [f9,f2] by A4, ZFMISC_1:def_2; reconsider f2 = f2 as Polynomial of n,L by A12, POLYNOM1:def_10; A14: f2 = c by A13, XTUPLE_0:1; reconsider g1 = g1 as non-zero Polynomial of n,L by A11, POLYNOM7:def_1; consider m1 being Monomial of n,L such that A15: f1 = f - (m1 *' g1) and A16: not HT ((m1 *' g1),T) in Support f1 and HT ((m1 *' g1),T) <= HT (f,T),T by A10, GROEB_1:2; A17: f9 = a by A13, XTUPLE_0:1; A18: f9 = a by A13, XTUPLE_0:1 .= f by A7, XTUPLE_0:1 ; then f reduces_to f2,P,T by A4, A13, POLYRED:def_13; then consider g2 being Polynomial of n,L such that A19: g2 in P and A20: f reduces_to f2,g2,T by POLYRED:def_7; ex b2 being bag of n st f reduces_to f2,g2,b2,T by A20, POLYRED:def_6; then A21: g2 <> 0_ (n,L) by POLYRED:def_5; then reconsider g2 = g2 as non-zero Polynomial of n,L by POLYNOM7:def_1; consider m2 being Monomial of n,L such that A22: f2 = f - (m2 *' g2) and A23: not HT ((m2 *' g2),T) in Support f2 and HT ((m2 *' g2),T) <= HT (f,T),T by A20, GROEB_1:2; set mg1 = m1 *' g1; set mg2 = m2 *' g2; A24: f1 = b by A7, XTUPLE_0:1; now__::_thesis:_(_(_m1_=_0__(n,L)_&_b,c_are_convergent_wrt_PolyRedRel_(P,T)_)_or_(_m2_=_0__(n,L)_&_b,c_are_convergent_wrt_PolyRedRel_(P,T)_)_or_(_m1_<>_0__(n,L)_&_m2_<>_0__(n,L)_&_b,c_are_convergent_wrt_PolyRedRel_(P,T)_)_) percases ( m1 = 0_ (n,L) or m2 = 0_ (n,L) or ( m1 <> 0_ (n,L) & m2 <> 0_ (n,L) ) ) ; case m1 = 0_ (n,L) ; ::_thesis: b,c are_convergent_wrt PolyRedRel (P,T) then f1 = f - (0_ (n,L)) by A15, POLYRED:5 .= f by POLYRED:4 ; then A25: PolyRedRel (P,T) reduces b,c by A4, A18, A24, A17, REWRITE1:15; PolyRedRel (P,T) reduces c,c by REWRITE1:12; hence b,c are_convergent_wrt PolyRedRel (P,T) by A25, REWRITE1:def_7; ::_thesis: verum end; case m2 = 0_ (n,L) ; ::_thesis: b,c are_convergent_wrt PolyRedRel (P,T) then f2 = f - (0_ (n,L)) by A22, POLYRED:5 .= f by POLYRED:4 ; then A26: PolyRedRel (P,T) reduces c,b by A3, A18, A14, A17, REWRITE1:15; PolyRedRel (P,T) reduces b,b by REWRITE1:12; hence b,c are_convergent_wrt PolyRedRel (P,T) by A26, REWRITE1:def_7; ::_thesis: verum end; case ( m1 <> 0_ (n,L) & m2 <> 0_ (n,L) ) ; ::_thesis: b,c are_convergent_wrt PolyRedRel (P,T) then reconsider m1 = m1, m2 = m2 as non-zero Monomial of n,L by POLYNOM7:def_1; (HT (m1,T)) + (HT (g1,T)) in Support (m1 *' g1) by TERMORD:29; then A27: m1 *' g1 <> 0_ (n,L) by POLYNOM7:1; (HT (m2,T)) + (HT (g2,T)) in Support (m2 *' g2) by TERMORD:29; then A28: m2 *' g2 <> 0_ (n,L) by POLYNOM7:1; A29: HC (g2,T) <> 0. L ; A30: - (- (m1 *' g1)) = m1 *' g1 by POLYNOM1:19; A31: f2 - f1 = (f + (- (m2 *' g2))) - (f - (m1 *' g1)) by A15, A22, POLYNOM1:def_6 .= (f + (- (m2 *' g2))) - (f + (- (m1 *' g1))) by POLYNOM1:def_6 .= (f + (- (m2 *' g2))) + (- (f + (- (m1 *' g1)))) by POLYNOM1:def_6 .= (f + (- (m2 *' g2))) + ((- f) + (- (- (m1 *' g1)))) by POLYRED:1 .= f + ((- (m2 *' g2)) + ((- f) + (m1 *' g1))) by A30, POLYNOM1:21 .= f + ((- f) + ((- (m2 *' g2)) + (m1 *' g1))) by POLYNOM1:21 .= (f + (- f)) + ((- (m2 *' g2)) + (m1 *' g1)) by POLYNOM1:21 .= (0_ (n,L)) + ((- (m2 *' g2)) + (m1 *' g1)) by POLYRED:3 .= (m1 *' g1) + (- (m2 *' g2)) by POLYRED:2 .= (m1 *' g1) - (m2 *' g2) by POLYNOM1:def_6 ; A32: HC (g1,T) <> 0. L ; A33: - (- (m1 *' g1)) = m1 *' g1 by POLYNOM1:19; PolyRedRel (P,T) reduces f2 - f1, 0_ (n,L) proof now__::_thesis:_(_(_(m1_*'_g1)_-_(m2_*'_g2)_=_0__(n,L)_&_PolyRedRel_(P,T)_reduces_f2_-_f1,_0__(n,L)_)_or_(_(m1_*'_g1)_-_(m2_*'_g2)_<>_0__(n,L)_&_PolyRedRel_(P,T)_reduces_f2_-_f1,_0__(n,L)_)_) percases ( (m1 *' g1) - (m2 *' g2) = 0_ (n,L) or (m1 *' g1) - (m2 *' g2) <> 0_ (n,L) ) ; case (m1 *' g1) - (m2 *' g2) = 0_ (n,L) ; ::_thesis: PolyRedRel (P,T) reduces f2 - f1, 0_ (n,L) hence PolyRedRel (P,T) reduces f2 - f1, 0_ (n,L) by A31, REWRITE1:12; ::_thesis: verum end; caseA34: (m1 *' g1) - (m2 *' g2) <> 0_ (n,L) ; ::_thesis: PolyRedRel (P,T) reduces f2 - f1, 0_ (n,L) now__::_thesis:_(_(_g1_=_g2_&_PolyRedRel_(P,T)_reduces_f2_-_f1,_0__(n,L)_)_or_(_g1_<>_g2_&_PolyRedRel_(P,T)_reduces_f2_-_f1,_0__(n,L)_)_) percases ( g1 = g2 or g1 <> g2 ) ; case g1 = g2 ; ::_thesis: PolyRedRel (P,T) reduces f2 - f1, 0_ (n,L) then f2 - f1 = (m1 *' g1) + (- (m2 *' g1)) by A31, POLYNOM1:def_6 .= (g1 *' m1) + ((- m2) *' g1) by POLYRED:6 .= (m1 + (- m2)) *' g1 by POLYNOM1:26 ; hence PolyRedRel (P,T) reduces f2 - f1, 0_ (n,L) by A9, POLYRED:45; ::_thesis: verum end; caseA35: g1 <> g2 ; ::_thesis: PolyRedRel (P,T) reduces f2 - f1, 0_ (n,L) now__::_thesis:_(_(_HT_((m1_*'_g1),T)_<>_HT_((m2_*'_g2),T)_&_PolyRedRel_(P,T)_reduces_f2_-_f1,_0__(n,L)_)_or_(_HT_((m1_*'_g1),T)_=_HT_((m2_*'_g2),T)_&_PolyRedRel_(P,T)_reduces_f2_-_f1,_0__(n,L)_)_) percases ( HT ((m1 *' g1),T) <> HT ((m2 *' g2),T) or HT ((m1 *' g1),T) = HT ((m2 *' g2),T) ) ; caseA36: HT ((m1 *' g1),T) <> HT ((m2 *' g2),T) ; ::_thesis: PolyRedRel (P,T) reduces f2 - f1, 0_ (n,L) now__::_thesis:_(_(_HT_((m2_*'_g2),T)_<_HT_((m1_*'_g1),T),T_&_PolyRedRel_(P,T)_reduces_f2_-_f1,_0__(n,L)_)_or_(_not_HT_((m2_*'_g2),T)_<_HT_((m1_*'_g1),T),T_&_PolyRedRel_(P,T)_reduces_f2_-_f1,_0__(n,L)_)_) percases ( HT ((m2 *' g2),T) < HT ((m1 *' g1),T),T or not HT ((m2 *' g2),T) < HT ((m1 *' g1),T),T ) ; case HT ((m2 *' g2),T) < HT ((m1 *' g1),T),T ; ::_thesis: PolyRedRel (P,T) reduces f2 - f1, 0_ (n,L) then not HT ((m1 *' g1),T) <= HT ((m2 *' g2),T),T by TERMORD:5; then not HT ((m1 *' g1),T) in Support (m2 *' g2) by TERMORD:def_6; then A37: (m2 *' g2) . (HT ((m1 *' g1),T)) = 0. L by POLYNOM1:def_3; A38: ((m1 *' g1) - (m2 *' g2)) . (HT ((m1 *' g1),T)) = ((m1 *' g1) + (- (m2 *' g2))) . (HT ((m1 *' g1),T)) by POLYNOM1:def_6 .= ((m1 *' g1) . (HT ((m1 *' g1),T))) + ((- (m2 *' g2)) . (HT ((m1 *' g1),T))) by POLYNOM1:15 .= ((m1 *' g1) . (HT ((m1 *' g1),T))) + (- ((m2 *' g2) . (HT ((m1 *' g1),T)))) by POLYNOM1:17 .= ((m1 *' g1) . (HT ((m1 *' g1),T))) + (0. L) by A37, RLVECT_1:12 .= (m1 *' g1) . (HT ((m1 *' g1),T)) by RLVECT_1:def_4 .= HC ((m1 *' g1),T) by TERMORD:def_7 ; then ((m1 *' g1) - (m2 *' g2)) . (HT ((m1 *' g1),T)) <> 0. L by A27, TERMORD:17; then A39: HT ((m1 *' g1),T) in Support ((m1 *' g1) - (m2 *' g2)) by POLYNOM1:def_3; A40: (HT (m1,T)) + (HT (g1,T)) = HT ((m1 *' g1),T) by TERMORD:31; ((m1 *' g1) - (m2 *' g2)) - (((((m1 *' g1) - (m2 *' g2)) . (HT ((m1 *' g1),T))) / (HC (g1,T))) * ((HT (m1,T)) *' g1)) = ((m1 *' g1) - (m2 *' g2)) - ((((HC (m1,T)) * (HC (g1,T))) / (HC (g1,T))) * ((HT (m1,T)) *' g1)) by A38, TERMORD:32 .= ((m1 *' g1) - (m2 *' g2)) - ((((HC (m1,T)) * (HC (g1,T))) * ((HC (g1,T)) ")) * ((HT (m1,T)) *' g1)) by VECTSP_1:def_11 .= ((m1 *' g1) - (m2 *' g2)) - (((HC (m1,T)) * ((HC (g1,T)) * ((HC (g1,T)) "))) * ((HT (m1,T)) *' g1)) by GROUP_1:def_3 .= ((m1 *' g1) - (m2 *' g2)) - (((HC (m1,T)) * (1. L)) * ((HT (m1,T)) *' g1)) by A32, VECTSP_1:def_10 .= ((m1 *' g1) - (m2 *' g2)) - ((HC (m1,T)) * ((HT (m1,T)) *' g1)) by VECTSP_1:def_8 .= ((m1 *' g1) - (m2 *' g2)) - ((Monom ((HC (m1,T)),(HT (m1,T)))) *' g1) by POLYRED:22 .= ((m1 *' g1) - (m2 *' g2)) - ((Monom ((coefficient m1),(HT (m1,T)))) *' g1) by TERMORD:23 .= ((m1 *' g1) - (m2 *' g2)) - ((Monom ((coefficient m1),(term m1))) *' g1) by TERMORD:23 .= ((m1 *' g1) - (m2 *' g2)) - (m1 *' g1) by POLYNOM7:11 .= ((m1 *' g1) + (- (m2 *' g2))) - (m1 *' g1) by POLYNOM1:def_6 .= ((m1 *' g1) + (- (m2 *' g2))) + (- (m1 *' g1)) by POLYNOM1:def_6 .= ((m1 *' g1) + (- (m1 *' g1))) + (- (m2 *' g2)) by POLYNOM1:21 .= (0_ (n,L)) + (- (m2 *' g2)) by POLYRED:3 .= - (m2 *' g2) by POLYRED:2 ; then (m1 *' g1) - (m2 *' g2) reduces_to - (m2 *' g2),g1, HT ((m1 *' g1),T),T by A11, A34, A39, A40, POLYRED:def_5; then (m1 *' g1) - (m2 *' g2) reduces_to - (m2 *' g2),g1,T by POLYRED:def_6; then (m1 *' g1) - (m2 *' g2) reduces_to - (m2 *' g2),P,T by A9, POLYRED:def_7; then [((m1 *' g1) - (m2 *' g2)),(- (m2 *' g2))] in PolyRedRel (P,T) by POLYRED:def_13; then A41: PolyRedRel (P,T) reduces (m1 *' g1) - (m2 *' g2), - (m2 *' g2) by REWRITE1:15; PolyRedRel (P,T) reduces (- m2) *' g2, 0_ (n,L) by A19, POLYRED:45; then PolyRedRel (P,T) reduces - (m2 *' g2), 0_ (n,L) by POLYRED:6; hence PolyRedRel (P,T) reduces f2 - f1, 0_ (n,L) by A31, A41, REWRITE1:16; ::_thesis: verum end; case not HT ((m2 *' g2),T) < HT ((m1 *' g1),T),T ; ::_thesis: PolyRedRel (P,T) reduces f2 - f1, 0_ (n,L) then HT ((m1 *' g1),T) <= HT ((m2 *' g2),T),T by TERMORD:5; then HT ((m1 *' g1),T) < HT ((m2 *' g2),T),T by A36, TERMORD:def_3; then not HT ((m2 *' g2),T) <= HT ((m1 *' g1),T),T by TERMORD:5; then not HT ((m2 *' g2),T) in Support (m1 *' g1) by TERMORD:def_6; then A42: (m1 *' g1) . (HT ((m2 *' g2),T)) = 0. L by POLYNOM1:def_3; A43: ((m2 *' g2) - (m1 *' g1)) . (HT ((m2 *' g2),T)) = ((m2 *' g2) + (- (m1 *' g1))) . (HT ((m2 *' g2),T)) by POLYNOM1:def_6 .= ((m2 *' g2) . (HT ((m2 *' g2),T))) + ((- (m1 *' g1)) . (HT ((m2 *' g2),T))) by POLYNOM1:15 .= ((m2 *' g2) . (HT ((m2 *' g2),T))) + (- ((m1 *' g1) . (HT ((m2 *' g2),T)))) by POLYNOM1:17 .= ((m2 *' g2) . (HT ((m2 *' g2),T))) + (0. L) by A42, RLVECT_1:12 .= (m2 *' g2) . (HT ((m2 *' g2),T)) by RLVECT_1:def_4 .= HC ((m2 *' g2),T) by TERMORD:def_7 ; then ((m2 *' g2) - (m1 *' g1)) . (HT ((m2 *' g2),T)) <> 0. L by A28, TERMORD:17; then A44: HT ((m2 *' g2),T) in Support ((m2 *' g2) - (m1 *' g1)) by POLYNOM1:def_3; reconsider x = - (0_ (n,L)) as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10; A45: (HT (m2,T)) + (HT (g2,T)) = HT ((m2 *' g2),T) by TERMORD:31; reconsider x = x as Element of (Polynom-Ring (n,L)) ; 0. (Polynom-Ring (n,L)) = 0_ (n,L) by POLYNOM1:def_10; then A46: x + (0. (Polynom-Ring (n,L))) = (- (0_ (n,L))) + (0_ (n,L)) by POLYNOM1:def_10 .= 0_ (n,L) by POLYRED:3 .= 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10 ; A47: now__::_thesis:_not_(m2_*'_g2)_-_(m1_*'_g1)_=_0__(n,L) assume (m2 *' g2) - (m1 *' g1) = 0_ (n,L) ; ::_thesis: contradiction then - ((m2 *' g2) + (- (m1 *' g1))) = - (0_ (n,L)) by POLYNOM1:def_6; then (- (m2 *' g2)) + (- (- (m1 *' g1))) = - (0_ (n,L)) by POLYRED:1; then (m1 *' g1) + (- (m2 *' g2)) = - (0. (Polynom-Ring (n,L))) by A33, A46, RLVECT_1:6 .= 0. (Polynom-Ring (n,L)) by RLVECT_1:12 .= 0_ (n,L) by POLYNOM1:def_10 ; hence contradiction by A34, POLYNOM1:def_6; ::_thesis: verum end; ((m2 *' g2) - (m1 *' g1)) - (((((m2 *' g2) - (m1 *' g1)) . (HT ((m2 *' g2),T))) / (HC (g2,T))) * ((HT (m2,T)) *' g2)) = ((m2 *' g2) - (m1 *' g1)) - ((((HC (m2,T)) * (HC (g2,T))) / (HC (g2,T))) * ((HT (m2,T)) *' g2)) by A43, TERMORD:32 .= ((m2 *' g2) - (m1 *' g1)) - ((((HC (m2,T)) * (HC (g2,T))) * ((HC (g2,T)) ")) * ((HT (m2,T)) *' g2)) by VECTSP_1:def_11 .= ((m2 *' g2) - (m1 *' g1)) - (((HC (m2,T)) * ((HC (g2,T)) * ((HC (g2,T)) "))) * ((HT (m2,T)) *' g2)) by GROUP_1:def_3 .= ((m2 *' g2) - (m1 *' g1)) - (((HC (m2,T)) * (1. L)) * ((HT (m2,T)) *' g2)) by A29, VECTSP_1:def_10 .= ((m2 *' g2) - (m1 *' g1)) - ((HC (m2,T)) * ((HT (m2,T)) *' g2)) by VECTSP_1:def_8 .= ((m2 *' g2) - (m1 *' g1)) - ((Monom ((HC (m2,T)),(HT (m2,T)))) *' g2) by POLYRED:22 .= ((m2 *' g2) - (m1 *' g1)) - ((Monom ((coefficient m2),(HT (m2,T)))) *' g2) by TERMORD:23 .= ((m2 *' g2) - (m1 *' g1)) - ((Monom ((coefficient m2),(term m2))) *' g2) by TERMORD:23 .= ((m2 *' g2) - (m1 *' g1)) - (m2 *' g2) by POLYNOM7:11 .= ((m2 *' g2) + (- (m1 *' g1))) - (m2 *' g2) by POLYNOM1:def_6 .= ((m2 *' g2) + (- (m1 *' g1))) + (- (m2 *' g2)) by POLYNOM1:def_6 .= ((m2 *' g2) + (- (m2 *' g2))) + (- (m1 *' g1)) by POLYNOM1:21 .= (0_ (n,L)) + (- (m1 *' g1)) by POLYRED:3 .= - (m1 *' g1) by POLYRED:2 ; then (m2 *' g2) - (m1 *' g1) reduces_to - (m1 *' g1),g2, HT ((m2 *' g2),T),T by A21, A44, A45, A47, POLYRED:def_5; then (m2 *' g2) - (m1 *' g1) reduces_to - (m1 *' g1),g2,T by POLYRED:def_6; then (m2 *' g2) - (m1 *' g1) reduces_to - (m1 *' g1),P,T by A19, POLYRED:def_7; then [((m2 *' g2) - (m1 *' g1)),(- (m1 *' g1))] in PolyRedRel (P,T) by POLYRED:def_13; then A48: PolyRedRel (P,T) reduces (m2 *' g2) - (m1 *' g1), - (m1 *' g1) by REWRITE1:15; A49: - (1_ (n,L)) = - ((1. L) | (n,L)) by POLYNOM7:20 .= (- (1. L)) | (n,L) by Th16 ; PolyRedRel (P,T) reduces (- m1) *' g1, 0_ (n,L) by A9, POLYRED:45; then PolyRedRel (P,T) reduces - (m1 *' g1), 0_ (n,L) by POLYRED:6; then PolyRedRel (P,T) reduces (m2 *' g2) - (m1 *' g1), 0_ (n,L) by A48, REWRITE1:16; then A50: PolyRedRel (P,T) reduces (- (1_ (n,L))) *' ((m2 *' g2) - (m1 *' g1)),(- (1_ (n,L))) *' (0_ (n,L)) by A49, POLYRED:47; (- (1_ (n,L))) *' ((m2 *' g2) - (m1 *' g1)) = (- (1_ (n,L))) *' ((m2 *' g2) + (- (m1 *' g1))) by POLYNOM1:def_6 .= ((- (1_ (n,L))) *' (m2 *' g2)) + ((- (1_ (n,L))) *' (- (m1 *' g1))) by POLYNOM1:26 .= (- ((1_ (n,L)) *' (m2 *' g2))) + ((- (1_ (n,L))) *' (- (m1 *' g1))) by POLYRED:6 .= ((1_ (n,L)) *' (- (m2 *' g2))) + ((- (1_ (n,L))) *' (- (m1 *' g1))) by POLYRED:6 .= ((1_ (n,L)) *' (- (m2 *' g2))) + (- ((1_ (n,L)) *' (- (m1 *' g1)))) by POLYRED:6 .= ((1_ (n,L)) *' (- (m2 *' g2))) + ((1_ (n,L)) *' (- (- (m1 *' g1)))) by POLYRED:6 .= (- (m2 *' g2)) + ((1_ (n,L)) *' (m1 *' g1)) by A33, POLYNOM1:30 .= (m1 *' g1) + (- (m2 *' g2)) by POLYNOM1:30 .= (m1 *' g1) - (m2 *' g2) by POLYNOM1:def_6 ; hence PolyRedRel (P,T) reduces f2 - f1, 0_ (n,L) by A31, A50, POLYNOM1:28; ::_thesis: verum end; end; end; hence PolyRedRel (P,T) reduces f2 - f1, 0_ (n,L) ; ::_thesis: verum end; caseA51: HT ((m1 *' g1),T) = HT ((m2 *' g2),T) ; ::_thesis: PolyRedRel (P,T) reduces f2 - f1, 0_ (n,L) (f - (m2 *' g2)) . (HT ((m2 *' g2),T)) = 0. L by A22, A23, POLYNOM1:def_3; then (f + (- (m2 *' g2))) . (HT ((m2 *' g2),T)) = 0. L by POLYNOM1:def_6; then (f . (HT ((m2 *' g2),T))) + ((- (m2 *' g2)) . (HT ((m2 *' g2),T))) = 0. L by POLYNOM1:15; then (f . (HT ((m2 *' g2),T))) + (- ((m2 *' g2) . (HT ((m2 *' g2),T)))) = 0. L by POLYNOM1:17; then A52: f . (HT ((m2 *' g2),T)) = - (- ((m2 *' g2) . (HT ((m2 *' g2),T)))) by RLVECT_1:6; (f - (m1 *' g1)) . (HT ((m1 *' g1),T)) = 0. L by A15, A16, POLYNOM1:def_3; then (f + (- (m1 *' g1))) . (HT ((m1 *' g1),T)) = 0. L by POLYNOM1:def_6; then (f . (HT ((m1 *' g1),T))) + ((- (m1 *' g1)) . (HT ((m1 *' g1),T))) = 0. L by POLYNOM1:15; then (f . (HT ((m1 *' g1),T))) + (- ((m1 *' g1) . (HT ((m1 *' g1),T)))) = 0. L by POLYNOM1:17; then A53: f . (HT ((m1 *' g1),T)) = - (- ((m1 *' g1) . (HT ((m1 *' g1),T)))) by RLVECT_1:6; HC ((m1 *' g1),T) = (m1 *' g1) . (HT ((m1 *' g1),T)) by TERMORD:def_7 .= f . (HT ((m1 *' g1),T)) by A53, RLVECT_1:17 .= (m2 *' g2) . (HT ((m2 *' g2),T)) by A51, A52, RLVECT_1:17 .= HC ((m2 *' g2),T) by TERMORD:def_7 ; then HM ((m1 *' g1),T) = Monom ((HC ((m2 *' g2),T)),(HT ((m2 *' g2),T))) by A51, TERMORD:def_8 .= HM ((m2 *' g2),T) by TERMORD:def_8 ; hence PolyRedRel (P,T) reduces f2 - f1, 0_ (n,L) by A1, A9, A19, A31, A35; ::_thesis: verum end; end; end; hence PolyRedRel (P,T) reduces f2 - f1, 0_ (n,L) ; ::_thesis: verum end; end; end; hence PolyRedRel (P,T) reduces f2 - f1, 0_ (n,L) ; ::_thesis: verum end; end; end; hence PolyRedRel (P,T) reduces f2 - f1, 0_ (n,L) ; ::_thesis: verum end; hence b,c are_convergent_wrt PolyRedRel (P,T) by A24, A14, POLYRED:50, REWRITE1:40; ::_thesis: verum end; end; end; hence b,c are_convergent_wrt PolyRedRel (P,T) ; ::_thesis: verum end; then PolyRedRel (P,T) is locally-confluent by REWRITE1:def_24; hence P is_Groebner_basis_wrt T by GROEB_1:def_3; ::_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 p1, p2 be Polynomial of n,L; func S-Poly (p1,p2,T) -> Polynomial of n,L equals :: GROEB_2:def 4 ((HC (p2,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))) *' p1)) - ((HC (p1,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))) *' p2)); coherence ((HC (p2,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))) *' p1)) - ((HC (p1,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))) *' p2)) is Polynomial of n,L ; end; :: deftheorem defines S-Poly GROEB_2: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 p1, p2 being Polynomial of n,L holds S-Poly (p1,p2,T) = ((HC (p2,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))) *' p1)) - ((HC (p1,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))) *' p2)); Lm2: 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; Lm3: for n being Ordinal for T being connected 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 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 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 for f, g being Element of (Polynom-Ring (n,L)) st f = p & g = q holds f - g = p - q let T be connected 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 for f, g being Element of (Polynom-Ring (n,L)) st f = p & g = q holds f - g = p - q let L be non trivial right_complementable 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; theorem Th18: :: GROEB_2:18 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 Abelian add-associative right_zeroed doubleLoopStr for P being Subset of (Polynom-Ring (n,L)) for p1, p2 being Polynomial of n,L st p1 in P & p2 in P holds S-Poly (p1,p2,T) in P -Ideal 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 P being Subset of (Polynom-Ring (n,L)) for p1, p2 being Polynomial of n,L st p1 in P & p2 in P holds S-Poly (p1,p2,T) in P -Ideal 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 P being Subset of (Polynom-Ring (n,L)) for p1, p2 being Polynomial of n,L st p1 in P & p2 in P holds S-Poly (p1,p2,T) in P -Ideal 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 Subset of (Polynom-Ring (n,L)) for p1, p2 being Polynomial of n,L st p1 in P & p2 in P holds S-Poly (p1,p2,T) in P -Ideal let P be Subset of (Polynom-Ring (n,L)); ::_thesis: for p1, p2 being Polynomial of n,L st p1 in P & p2 in P holds S-Poly (p1,p2,T) in P -Ideal let p1, p2 be Polynomial of n,L; ::_thesis: ( p1 in P & p2 in P implies S-Poly (p1,p2,T) in P -Ideal ) assume that A1: p1 in P and A2: p2 in P ; ::_thesis: S-Poly (p1,p2,T) in P -Ideal set q1 = Monom ((HC (p2,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T)))); set q2 = Monom ((HC (p1,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T)))); reconsider p19 = p1, p29 = p2 as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10; reconsider p19 = p19, p29 = p29 as Element of (Polynom-Ring (n,L)) ; reconsider q19 = Monom ((HC (p2,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T)))), q29 = Monom ((HC (p1,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T)))) as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10; reconsider q19 = q19, q29 = q29 as Element of (Polynom-Ring (n,L)) ; p29 in P -Ideal by A2, Lm2; then A3: q29 * p29 in P -Ideal by IDEAL_1:def_2; p19 in P -Ideal by A1, Lm2; then q19 * p19 in P -Ideal by IDEAL_1:def_2; then A4: (q19 * p19) - (q29 * p29) in P -Ideal by A3, IDEAL_1:16; set q = S-Poly (p1,p2,T); A5: ( (Monom ((HC (p2,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))))) *' p1 = q19 * p19 & (Monom ((HC (p1,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))))) *' p2 = q29 * p29 ) by POLYNOM1:def_10; S-Poly (p1,p2,T) = ((Monom ((HC (p2,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))))) *' p1) - ((HC (p1,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))) *' p2)) by POLYRED:22 .= ((Monom ((HC (p2,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))))) *' p1) - ((Monom ((HC (p1,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))))) *' p2) by POLYRED:22 ; hence S-Poly (p1,p2,T) in P -Ideal by A4, A5, Lm3; ::_thesis: verum end; theorem Th19: :: GROEB_2:19 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 m1, m2 being Monomial of n,L holds S-Poly (m1,m2,T) = 0_ (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 m1, m2 being Monomial of n,L holds S-Poly (m1,m2,T) = 0_ (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 m1, m2 being Monomial of n,L holds S-Poly (m1,m2,T) = 0_ (n,L) let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for m1, m2 being Monomial of n,L holds S-Poly (m1,m2,T) = 0_ (n,L) let m1, m2 be Monomial of n,L; ::_thesis: S-Poly (m1,m2,T) = 0_ (n,L) percases ( m1 = 0_ (n,L) or m2 = 0_ (n,L) or ( m1 <> 0_ (n,L) & m2 <> 0_ (n,L) ) ) ; supposeA1: m1 = 0_ (n,L) ; ::_thesis: S-Poly (m1,m2,T) = 0_ (n,L) A2: HC ((Monom ((HC ((0_ (n,L)),T)),((lcm ((HT (m1,T)),(HT (m2,T)))) / (HT (m2,T))))),T) = coefficient (Monom ((HC ((0_ (n,L)),T)),((lcm ((HT (m1,T)),(HT (m2,T)))) / (HT (m2,T))))) by TERMORD:23 .= HC ((0_ (n,L)),T) by POLYNOM7:9 .= 0. L by TERMORD:17 ; thus S-Poly (m1,m2,T) = ((Monom ((HC (m2,T)),((lcm ((HT (m1,T)),(HT (m2,T)))) / (HT (m1,T))))) *' (0_ (n,L))) - ((HC ((0_ (n,L)),T)) * (((lcm ((HT (m1,T)),(HT (m2,T)))) / (HT (m2,T))) *' m2)) by A1, POLYRED:22 .= (0_ (n,L)) - ((HC ((0_ (n,L)),T)) * (((lcm ((HT (m1,T)),(HT (m2,T)))) / (HT (m2,T))) *' m2)) by POLYNOM1:28 .= (0_ (n,L)) - ((Monom ((HC ((0_ (n,L)),T)),((lcm ((HT (m1,T)),(HT (m2,T)))) / (HT (m2,T))))) *' m2) by POLYRED:22 .= (0_ (n,L)) - ((0_ (n,L)) *' m2) by A2, TERMORD:17 .= (0_ (n,L)) - (0_ (n,L)) by POLYRED:5 .= 0_ (n,L) by POLYNOM1:24 ; ::_thesis: verum end; supposeA3: m2 = 0_ (n,L) ; ::_thesis: S-Poly (m1,m2,T) = 0_ (n,L) A4: HC ((Monom ((HC ((0_ (n,L)),T)),((lcm ((HT (m1,T)),(HT (m2,T)))) / (HT (m1,T))))),T) = coefficient (Monom ((HC ((0_ (n,L)),T)),((lcm ((HT (m1,T)),(HT (m2,T)))) / (HT (m1,T))))) by TERMORD:23 .= HC ((0_ (n,L)),T) by POLYNOM7:9 .= 0. L by TERMORD:17 ; thus S-Poly (m1,m2,T) = ((HC ((0_ (n,L)),T)) * (((lcm ((HT (m1,T)),(HT (m2,T)))) / (HT (m1,T))) *' m1)) - ((Monom ((HC (m1,T)),((lcm ((HT (m1,T)),(HT (m2,T)))) / (HT (m2,T))))) *' (0_ (n,L))) by A3, POLYRED:22 .= ((HC ((0_ (n,L)),T)) * (((lcm ((HT (m1,T)),(HT (m2,T)))) / (HT (m1,T))) *' m1)) - (0_ (n,L)) by POLYNOM1:28 .= ((Monom ((HC ((0_ (n,L)),T)),((lcm ((HT (m1,T)),(HT (m2,T)))) / (HT (m1,T))))) *' m1) - (0_ (n,L)) by POLYRED:22 .= ((0_ (n,L)) *' m1) - (0_ (n,L)) by A4, TERMORD:17 .= (0_ (n,L)) - (0_ (n,L)) by POLYRED:5 .= 0_ (n,L) by POLYNOM1:24 ; ::_thesis: verum end; supposeA5: ( m1 <> 0_ (n,L) & m2 <> 0_ (n,L) ) ; ::_thesis: S-Poly (m1,m2,T) = 0_ (n,L) then HC (m2,T) <> 0. L by TERMORD:17; then A6: not HC (m2,T) is zero by STRUCT_0:def_12; HC (m1,T) <> 0. L by A5, TERMORD:17; then A7: not HC (m1,T) is zero by STRUCT_0:def_12; A8: HT (m2,T) divides lcm ((HT (m1,T)),(HT (m2,T))) by Th3; A9: m2 = Monom ((coefficient m2),(term m2)) by POLYNOM7:11 .= Monom ((HC (m2,T)),(term m2)) by TERMORD:23 .= Monom ((HC (m2,T)),(HT (m2,T))) by TERMORD:23 ; A10: HT (m1,T) divides lcm ((HT (m1,T)),(HT (m2,T))) by Th3; A11: m1 = Monom ((coefficient m1),(term m1)) by POLYNOM7:11 .= Monom ((HC (m1,T)),(term m1)) by TERMORD:23 .= Monom ((HC (m1,T)),(HT (m1,T))) by TERMORD:23 ; A12: (HC (m1,T)) * (((lcm ((HT (m1,T)),(HT (m2,T)))) / (HT (m2,T))) *' m2) = (Monom ((HC (m1,T)),((lcm ((HT (m1,T)),(HT (m2,T)))) / (HT (m2,T))))) *' m2 by POLYRED:22 .= Monom (((HC (m2,T)) * (HC (m1,T))),(((lcm ((HT (m1,T)),(HT (m2,T)))) / (HT (m2,T))) + (HT (m2,T)))) by A7, A6, A9, TERMORD:3 .= Monom (((HC (m2,T)) * (HC (m1,T))),(lcm ((HT (m1,T)),(HT (m2,T))))) by A8, Def1 ; (HC (m2,T)) * (((lcm ((HT (m1,T)),(HT (m2,T)))) / (HT (m1,T))) *' m1) = (Monom ((HC (m2,T)),((lcm ((HT (m1,T)),(HT (m2,T)))) / (HT (m1,T))))) *' m1 by POLYRED:22 .= Monom (((HC (m2,T)) * (HC (m1,T))),(((lcm ((HT (m1,T)),(HT (m2,T)))) / (HT (m1,T))) + (HT (m1,T)))) by A7, A6, A11, TERMORD:3 .= Monom (((HC (m2,T)) * (HC (m1,T))),(lcm ((HT (m1,T)),(HT (m2,T))))) by A10, Def1 ; hence S-Poly (m1,m2,T) = 0_ (n,L) by A12, POLYNOM1:24; ::_thesis: verum end; end; end; theorem Th20: :: GROEB_2:20 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 Polynomial of n,L holds ( S-Poly (p,(0_ (n,L)),T) = 0_ (n,L) & S-Poly ((0_ (n,L)),p,T) = 0_ (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 Polynomial of n,L holds ( S-Poly (p,(0_ (n,L)),T) = 0_ (n,L) & S-Poly ((0_ (n,L)),p,T) = 0_ (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 Polynomial of n,L holds ( S-Poly (p,(0_ (n,L)),T) = 0_ (n,L) & S-Poly ((0_ (n,L)),p,T) = 0_ (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 Polynomial of n,L holds ( S-Poly (p,(0_ (n,L)),T) = 0_ (n,L) & S-Poly ((0_ (n,L)),p,T) = 0_ (n,L) ) let p be Polynomial of n,L; ::_thesis: ( S-Poly (p,(0_ (n,L)),T) = 0_ (n,L) & S-Poly ((0_ (n,L)),p,T) = 0_ (n,L) ) set p2 = 0_ (n,L); thus S-Poly (p,(0_ (n,L)),T) = ((HC ((0_ (n,L)),T)) * (((lcm ((HT (p,T)),(HT ((0_ (n,L)),T)))) / (HT (p,T))) *' p)) - ((Monom ((HC (p,T)),((lcm ((HT (p,T)),(HT ((0_ (n,L)),T)))) / (HT ((0_ (n,L)),T))))) *' (0_ (n,L))) by POLYRED:22 .= ((HC ((0_ (n,L)),T)) * (((lcm ((HT (p,T)),(HT ((0_ (n,L)),T)))) / (HT (p,T))) *' p)) - (0_ (n,L)) by POLYNOM1:28 .= ((0. L) * (((lcm ((HT (p,T)),(HT ((0_ (n,L)),T)))) / (HT (p,T))) *' p)) - (0_ (n,L)) by TERMORD:17 .= (((0. L) | (n,L)) *' (((lcm ((HT (p,T)),(HT ((0_ (n,L)),T)))) / (HT (p,T))) *' p)) - (0_ (n,L)) by POLYNOM7:27 .= ((0_ (n,L)) *' (((lcm ((HT (p,T)),(HT ((0_ (n,L)),T)))) / (HT (p,T))) *' p)) - (0_ (n,L)) by POLYNOM7:19 .= (0_ (n,L)) - (0_ (n,L)) by POLYRED:5 .= 0_ (n,L) by POLYRED:4 ; ::_thesis: S-Poly ((0_ (n,L)),p,T) = 0_ (n,L) thus S-Poly ((0_ (n,L)),p,T) = ((Monom ((HC (p,T)),((lcm ((HT ((0_ (n,L)),T)),(HT (p,T)))) / (HT ((0_ (n,L)),T))))) *' (0_ (n,L))) - ((HC ((0_ (n,L)),T)) * (((lcm ((HT ((0_ (n,L)),T)),(HT (p,T)))) / (HT (p,T))) *' p)) by POLYRED:22 .= (0_ (n,L)) - ((HC ((0_ (n,L)),T)) * (((lcm ((HT ((0_ (n,L)),T)),(HT (p,T)))) / (HT (p,T))) *' p)) by POLYNOM1:28 .= (0_ (n,L)) - ((0. L) * (((lcm ((HT ((0_ (n,L)),T)),(HT (p,T)))) / (HT (p,T))) *' p)) by TERMORD:17 .= (0_ (n,L)) - (((0. L) | (n,L)) *' (((lcm ((HT ((0_ (n,L)),T)),(HT (p,T)))) / (HT (p,T))) *' p)) by POLYNOM7:27 .= (0_ (n,L)) - ((0_ (n,L)) *' (((lcm ((HT ((0_ (n,L)),T)),(HT (p,T)))) / (HT (p,T))) *' p)) by POLYNOM7:19 .= (0_ (n,L)) - (0_ (n,L)) by POLYRED:5 .= 0_ (n,L) by POLYRED:4 ; ::_thesis: verum end; theorem :: GROEB_2:21 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 p1, p2 being Polynomial of n,L holds ( S-Poly (p1,p2,T) = 0_ (n,L) or HT ((S-Poly (p1,p2,T)),T) < lcm ((HT (p1,T)),(HT (p2,T))),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 p1, p2 being Polynomial of n,L holds ( S-Poly (p1,p2,T) = 0_ (n,L) or HT ((S-Poly (p1,p2,T)),T) < lcm ((HT (p1,T)),(HT (p2,T))),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 p1, p2 being Polynomial of n,L holds ( S-Poly (p1,p2,T) = 0_ (n,L) or HT ((S-Poly (p1,p2,T)),T) < lcm ((HT (p1,T)),(HT (p2,T))),T ) let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for p1, p2 being Polynomial of n,L holds ( S-Poly (p1,p2,T) = 0_ (n,L) or HT ((S-Poly (p1,p2,T)),T) < lcm ((HT (p1,T)),(HT (p2,T))),T ) let p1, p2 be Polynomial of n,L; ::_thesis: ( S-Poly (p1,p2,T) = 0_ (n,L) or HT ((S-Poly (p1,p2,T)),T) < lcm ((HT (p1,T)),(HT (p2,T))),T ) assume A1: S-Poly (p1,p2,T) <> 0_ (n,L) ; ::_thesis: HT ((S-Poly (p1,p2,T)),T) < lcm ((HT (p1,T)),(HT (p2,T))),T set sp = S-Poly (p1,p2,T); set g1 = (HC (p2,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))) *' p1); set g2 = (HC (p1,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))) *' p2); percases ( p1 = 0_ (n,L) or p2 = 0_ (n,L) or ( p1 <> 0_ (n,L) & p2 <> 0_ (n,L) ) ) ; suppose ( p1 = 0_ (n,L) or p2 = 0_ (n,L) ) ; ::_thesis: HT ((S-Poly (p1,p2,T)),T) < lcm ((HT (p1,T)),(HT (p2,T))),T hence HT ((S-Poly (p1,p2,T)),T) < lcm ((HT (p1,T)),(HT (p2,T))),T by A1, Th20; ::_thesis: verum end; supposeA2: ( p1 <> 0_ (n,L) & p2 <> 0_ (n,L) ) ; ::_thesis: HT ((S-Poly (p1,p2,T)),T) < lcm ((HT (p1,T)),(HT (p2,T))),T then A3: HC (p2,T) <> 0. L by TERMORD:17; then A4: not HC (p2,T) is zero by STRUCT_0:def_12; A5: HT ((Monom ((HC (p2,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))))),T) = term (Monom ((HC (p2,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))))) by TERMORD:23 .= (lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T)) by A4, POLYNOM7:10 ; A6: p1 is non-zero by A2, POLYNOM7:def_1; HC ((Monom ((HC (p2,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))))),T) = coefficient (Monom ((HC (p2,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))))) by TERMORD:23 .= HC (p2,T) by POLYNOM7:9 ; then Monom ((HC (p2,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T)))) <> 0_ (n,L) by A3, TERMORD:17; then A7: Monom ((HC (p2,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T)))) is non-zero by POLYNOM7:def_1; A8: HC (((HC (p2,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))) *' p1)),T) = HC (((Monom ((HC (p2,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))))) *' p1),T) by POLYRED:22 .= (HC ((Monom ((HC (p2,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))))),T)) * (HC (p1,T)) by A6, A7, TERMORD:32 .= (coefficient (Monom ((HC (p2,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T)))))) * (HC (p1,T)) by TERMORD:23 .= (HC (p1,T)) * (HC (p2,T)) by POLYNOM7:9 ; A9: HT (p1,T) divides lcm ((HT (p1,T)),(HT (p2,T))) by Th3; A10: HT (((HC (p2,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))) *' p1)),T) = HT (((Monom ((HC (p2,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))))) *' p1),T) by POLYRED:22 .= (HT ((Monom ((HC (p2,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))))),T)) + (HT (p1,T)) by A6, A7, TERMORD:31 .= lcm ((HT (p1,T)),(HT (p2,T))) by A9, A5, Def1 ; A11: HC (p1,T) <> 0. L by A2, TERMORD:17; then A12: not HC (p1,T) is zero by STRUCT_0:def_12; A13: HT ((Monom ((HC (p1,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))))),T) = term (Monom ((HC (p1,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))))) by TERMORD:23 .= (lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T)) by A12, POLYNOM7:10 ; A14: p2 is non-zero by A2, POLYNOM7:def_1; HC ((Monom ((HC (p1,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))))),T) = coefficient (Monom ((HC (p1,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))))) by TERMORD:23 .= HC (p1,T) by POLYNOM7:9 ; then Monom ((HC (p1,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T)))) <> 0_ (n,L) by A11, TERMORD:17; then A15: Monom ((HC (p1,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T)))) is non-zero by POLYNOM7:def_1; Support (S-Poly (p1,p2,T)) <> {} by A1, POLYNOM7:1; then A16: HT ((S-Poly (p1,p2,T)),T) in Support (S-Poly (p1,p2,T)) by TERMORD:def_6; A17: HT (p2,T) divides lcm ((HT (p1,T)),(HT (p2,T))) by Th3; A18: HC (((HC (p1,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))) *' p2)),T) = HC (((Monom ((HC (p1,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))))) *' p2),T) by POLYRED:22 .= (HC ((Monom ((HC (p1,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))))),T)) * (HC (p2,T)) by A14, A15, TERMORD:32 .= (coefficient (Monom ((HC (p1,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T)))))) * (HC (p2,T)) by TERMORD:23 .= (HC (p1,T)) * (HC (p2,T)) by POLYNOM7:9 ; A19: HT (((HC (p1,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))) *' p2)),T) = HT (((Monom ((HC (p1,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))))) *' p2),T) by POLYRED:22 .= (HT ((Monom ((HC (p1,T)),((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))))),T)) + (HT (p2,T)) by A14, A15, TERMORD:31 .= lcm ((HT (p1,T)),(HT (p2,T))) by A17, A13, Def1 ; then (S-Poly (p1,p2,T)) . (lcm ((HT (p1,T)),(HT (p2,T)))) = (((HC (p2,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))) *' p1)) + (- ((HC (p1,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))) *' p2)))) . (HT (((HC (p1,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))) *' p2)),T)) by POLYNOM1:def_6 .= (((HC (p2,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))) *' p1)) . (HT (((HC (p1,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))) *' p2)),T))) + ((- ((HC (p1,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))) *' p2))) . (HT (((HC (p1,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))) *' p2)),T))) by POLYNOM1:15 .= (((HC (p2,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))) *' p1)) . (HT (((HC (p1,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))) *' p2)),T))) + (- (((HC (p1,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))) *' p2)) . (HT (((HC (p1,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))) *' p2)),T)))) by POLYNOM1:17 .= (HC (((HC (p2,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))) *' p1)),T)) + (- (((HC (p1,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))) *' p2)) . (HT (((HC (p1,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))) *' p2)),T)))) by A10, A19, TERMORD:def_7 .= (HC (((HC (p2,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))) *' p1)),T)) + (- (HC (((HC (p1,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))) *' p2)),T))) by TERMORD:def_7 .= 0. L by A8, A18, RLVECT_1:5 ; then A20: not lcm ((HT (p1,T)),(HT (p2,T))) in Support (S-Poly (p1,p2,T)) by POLYNOM1:def_3; HT ((S-Poly (p1,p2,T)),T) <= max ((HT (((HC (p2,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p1,T))) *' p1)),T)),(HT (((HC (p1,T)) * (((lcm ((HT (p1,T)),(HT (p2,T)))) / (HT (p2,T))) *' p2)),T)),T),T by GROEB_1:7; then HT ((S-Poly (p1,p2,T)),T) <= lcm ((HT (p1,T)),(HT (p2,T))),T by A10, A19, TERMORD:12; hence HT ((S-Poly (p1,p2,T)),T) < lcm ((HT (p1,T)),(HT (p2,T))),T by A16, A20, TERMORD:def_3; ::_thesis: verum end; end; end; theorem :: GROEB_2:22 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 p1, p2 being non-zero Polynomial of n,L st HT (p2,T) divides HT (p1,T) holds (HC (p2,T)) * p1 top_reduces_to S-Poly (p1,p2,T),p2,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 p1, p2 being non-zero Polynomial of n,L st HT (p2,T) divides HT (p1,T) holds (HC (p2,T)) * p1 top_reduces_to S-Poly (p1,p2,T),p2,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 p1, p2 being non-zero Polynomial of n,L st HT (p2,T) divides HT (p1,T) holds (HC (p2,T)) * p1 top_reduces_to S-Poly (p1,p2,T),p2,T let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for p1, p2 being non-zero Polynomial of n,L st HT (p2,T) divides HT (p1,T) holds (HC (p2,T)) * p1 top_reduces_to S-Poly (p1,p2,T),p2,T let p1, p2 be non-zero Polynomial of n,L; ::_thesis: ( HT (p2,T) divides HT (p1,T) implies (HC (p2,T)) * p1 top_reduces_to S-Poly (p1,p2,T),p2,T ) set hcp2 = HC (p2,T); assume A1: HT (p2,T) divides HT (p1,T) ; ::_thesis: (HC (p2,T)) * p1 top_reduces_to S-Poly (p1,p2,T),p2,T then consider b being bag of n such that A2: HT (p1,T) = (HT (p2,T)) + b by TERMORD:1; set g = ((HC (p2,T)) * p1) - (((((HC (p2,T)) * p1) . (HT (p1,T))) / (HC (p2,T))) * (b *' p2)); A3: p2 <> 0_ (n,L) by POLYNOM7:def_1; A4: HC (p2,T) <> 0. L ; p1 <> 0_ (n,L) by POLYNOM7:def_1; then Support p1 <> {} by POLYNOM7:1; then A5: HT (p1,T) in Support p1 by TERMORD:def_6; A6: Support p1 c= Support ((HC (p2,T)) * p1) by POLYRED:20; then (HC (p2,T)) * p1 <> 0_ (n,L) by A5, POLYNOM7:1; then A7: ( HT (((HC (p2,T)) * p1),T) = HT (p1,T) & (HC (p2,T)) * p1 reduces_to ((HC (p2,T)) * p1) - (((((HC (p2,T)) * p1) . (HT (p1,T))) / (HC (p2,T))) * (b *' p2)),p2, HT (p1,T),T ) by A3, A5, A2, A6, POLYRED:21, POLYRED:def_5; A8: lcm ((HT (p1,T)),(HT (p2,T))) = HT (p1,T) by A1, Th7; ((HC (p2,T)) * p1) - (((((HC (p2,T)) * p1) . (HT (p1,T))) / (HC (p2,T))) * (b *' p2)) = ((HC (p2,T)) * p1) - ((((HC (p2,T)) * (p1 . (HT (p1,T)))) / (HC (p2,T))) * (b *' p2)) by POLYNOM7:def_9 .= ((HC (p2,T)) * p1) - ((((HC (p2,T)) * (HC (p1,T))) / (HC (p2,T))) * (b *' p2)) by TERMORD:def_7 .= ((HC (p2,T)) * p1) - ((((HC (p2,T)) * (HC (p1,T))) * ((HC (p2,T)) ")) * (b *' p2)) by VECTSP_1:def_11 .= ((HC (p2,T)) * p1) - (((HC (p1,T)) * ((HC (p2,T)) * ((HC (p2,T)) "))) * (b *' p2)) by GROUP_1:def_3 .= ((HC (p2,T)) * p1) - (((HC (p1,T)) * (1. L)) * (b *' p2)) by A4, VECTSP_1:def_10 .= ((HC (p2,T)) * p1) - ((HC (p1,T)) * (b *' p2)) by VECTSP_1:def_4 .= ((HC (p2,T)) * ((EmptyBag n) *' p1)) - ((HC (p1,T)) * (b *' p2)) by POLYRED:17 .= ((HC (p2,T)) * (((HT (p1,T)) / (HT (p1,T))) *' p1)) - ((HC (p1,T)) * (b *' p2)) by Th6 .= S-Poly (p1,p2,T) by A1, A2, A8, Def1 ; hence (HC (p2,T)) * p1 top_reduces_to S-Poly (p1,p2,T),p2,T by A7, POLYRED:def_10; ::_thesis: verum end; theorem :: GROEB_2:23 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 G being Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_wrt T holds for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds h = 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 G being Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_wrt T holds for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds h = 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 G being Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_wrt T holds for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds h = 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 G being Subset of (Polynom-Ring (n,L)) st G is_Groebner_basis_wrt T holds for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds h = 0_ (n,L) let G be Subset of (Polynom-Ring (n,L)); ::_thesis: ( G is_Groebner_basis_wrt T implies for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds h = 0_ (n,L) ) assume A1: G is_Groebner_basis_wrt T ; ::_thesis: for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds h = 0_ (n,L) set R = PolyRedRel (G,T); A2: 0_ (n,L) = 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10; percases ( G = {} or G <> {} ) ; suppose G = {} ; ::_thesis: for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds h = 0_ (n,L) hence for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds h = 0_ (n,L) ; ::_thesis: verum end; suppose G <> {} ; ::_thesis: for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds h = 0_ (n,L) then reconsider G = G as non empty Subset of (Polynom-Ring (n,L)) ; A3: PolyRedRel (G,T) is locally-confluent by A1, GROEB_1:def_3; now__::_thesis:_for_g1,_g2,_h_being_Polynomial_of_n,L_st_g1_in_G_&_g2_in_G_&_h_is_a_normal_form_of_S-Poly_(g1,g2,T),_PolyRedRel_(G,T)_holds_ h_=_0__(n,L) A4: now__::_thesis:_0__(n,L)_is_a_normal_form_wrt_PolyRedRel_(G,T) assume not 0_ (n,L) is_a_normal_form_wrt PolyRedRel (G,T) ; ::_thesis: contradiction then consider b being set such that A5: [(0_ (n,L)),b] in PolyRedRel (G,T) by REWRITE1:def_5; consider f1, f2 being set such that A6: f1 in NonZero (Polynom-Ring (n,L)) and f2 in the carrier of (Polynom-Ring (n,L)) and A7: [(0_ (n,L)),b] = [f1,f2] by A5, ZFMISC_1:def_2; A8: f1 = 0_ (n,L) by A7, XTUPLE_0:1; not f1 in {(0_ (n,L))} by A2, A6, XBOOLE_0:def_5; hence contradiction by A8, TARSKI:def_1; ::_thesis: verum end; let g1, g2, h be Polynomial of n,L; ::_thesis: ( g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) implies h = 0_ (n,L) ) assume that A9: ( g1 in G & g2 in G ) and A10: h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) ; ::_thesis: h = 0_ (n,L) S-Poly (g1,g2,T) in G -Ideal by A9, Th18; then PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) by A3, GROEB_1:15; then A11: S-Poly (g1,g2,T), 0_ (n,L) are_convertible_wrt PolyRedRel (G,T) by REWRITE1:25; PolyRedRel (G,T) reduces S-Poly (g1,g2,T),h by A10, REWRITE1:def_6; then h, S-Poly (g1,g2,T) are_convertible_wrt PolyRedRel (G,T) by REWRITE1:25; then A12: h, 0_ (n,L) are_convertible_wrt PolyRedRel (G,T) by A11, REWRITE1:30; h is_a_normal_form_wrt PolyRedRel (G,T) by A10, REWRITE1:def_6; hence h = 0_ (n,L) by A3, A4, A12, REWRITE1:def_19; ::_thesis: verum end; hence for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds h = 0_ (n,L) ; ::_thesis: verum end; end; end; theorem :: GROEB_2:24 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 G being Subset of (Polynom-Ring (n,L)) st ( for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds h = 0_ (n,L) ) holds for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 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 G being Subset of (Polynom-Ring (n,L)) st ( for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds h = 0_ (n,L) ) holds for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 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 G being Subset of (Polynom-Ring (n,L)) st ( for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds h = 0_ (n,L) ) holds for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 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 G being Subset of (Polynom-Ring (n,L)) st ( for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds h = 0_ (n,L) ) holds for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) let G be Subset of (Polynom-Ring (n,L)); ::_thesis: ( ( for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds h = 0_ (n,L) ) implies for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ) set R = PolyRedRel (G,T); assume A1: for g1, g2, h being Polynomial of n,L st g1 in G & g2 in G & h is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) holds h = 0_ (n,L) ; ::_thesis: for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) now__::_thesis:_for_g1,_g2_being_Polynomial_of_n,L_st_g1_in_G_&_g2_in_G_holds_ PolyRedRel_(G,T)_reduces_S-Poly_(g1,g2,T),_0__(n,L) let g1, g2 be Polynomial of n,L; ::_thesis: ( g1 in G & g2 in G implies PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ) now__::_thesis:_(_(_S-Poly_(g1,g2,T)_in_field_(PolyRedRel_(G,T))_&_S-Poly_(g1,g2,T)_has_a_normal_form_wrt_PolyRedRel_(G,T)_)_or_(_not_S-Poly_(g1,g2,T)_in_field_(PolyRedRel_(G,T))_&_S-Poly_(g1,g2,T)_has_a_normal_form_wrt_PolyRedRel_(G,T)_)_) percases ( S-Poly (g1,g2,T) in field (PolyRedRel (G,T)) or not S-Poly (g1,g2,T) in field (PolyRedRel (G,T)) ) ; case S-Poly (g1,g2,T) in field (PolyRedRel (G,T)) ; ::_thesis: S-Poly (g1,g2,T) has_a_normal_form_wrt PolyRedRel (G,T) hence S-Poly (g1,g2,T) has_a_normal_form_wrt PolyRedRel (G,T) by REWRITE1:def_14; ::_thesis: verum end; case not S-Poly (g1,g2,T) in field (PolyRedRel (G,T)) ; ::_thesis: S-Poly (g1,g2,T) has_a_normal_form_wrt PolyRedRel (G,T) hence S-Poly (g1,g2,T) has_a_normal_form_wrt PolyRedRel (G,T) by REWRITE1:46; ::_thesis: verum end; end; end; then consider q being set such that A2: q is_a_normal_form_of S-Poly (g1,g2,T), PolyRedRel (G,T) by REWRITE1:def_11; PolyRedRel (G,T) reduces S-Poly (g1,g2,T),q by A2, REWRITE1:def_6; then reconsider q = q as Polynomial of n,L by Lm1; assume ( g1 in G & g2 in G ) ; ::_thesis: PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) then q = 0_ (n,L) by A1, A2; hence PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) by A2, REWRITE1:def_6; ::_thesis: verum end; hence for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ; ::_thesis: verum end; theorem Th25: :: GROEB_2:25 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 being Subset of (Polynom-Ring (n,L)) st not 0_ (n,L) in G & ( for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ) holds G is_Groebner_basis_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 G being Subset of (Polynom-Ring (n,L)) st not 0_ (n,L) in G & ( for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ) holds G 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 Abelian add-associative right_zeroed doubleLoopStr for G being Subset of (Polynom-Ring (n,L)) st not 0_ (n,L) in G & ( for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ) holds G is_Groebner_basis_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 G being Subset of (Polynom-Ring (n,L)) st not 0_ (n,L) in G & ( for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ) holds G is_Groebner_basis_wrt T let G be Subset of (Polynom-Ring (n,L)); ::_thesis: ( not 0_ (n,L) in G & ( for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ) implies G is_Groebner_basis_wrt T ) assume A1: not 0_ (n,L) in G ; ::_thesis: ( ex g1, g2 being Polynomial of n,L st ( g1 in G & g2 in G & not PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ) or G is_Groebner_basis_wrt T ) assume A2: for g1, g2 being Polynomial of n,L st g1 in G & g2 in G holds PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ; ::_thesis: G is_Groebner_basis_wrt T now__::_thesis:_for_g1,_g2_being_Polynomial_of_n,L_st_g1_<>_g2_&_g1_in_G_&_g2_in_G_holds_ for_m1,_m2_being_Monomial_of_n,L_st_HM_((m1_*'_g1),T)_=_HM_((m2_*'_g2),T)_holds_ PolyRedRel_(G,T)_reduces_(m1_*'_g1)_-_(m2_*'_g2),_0__(n,L) let g1, g2 be Polynomial of n,L; ::_thesis: ( g1 <> g2 & g1 in G & g2 in G implies for m1, m2 being Monomial of n,L st HM ((m1 *' g1),T) = HM ((m2 *' g2),T) holds PolyRedRel (G,T) reduces (m1 *' g1) - (m2 *' g2), 0_ (n,L) ) assume that g1 <> g2 and A3: g1 in G and A4: g2 in G ; ::_thesis: for m1, m2 being Monomial of n,L st HM ((m1 *' g1),T) = HM ((m2 *' g2),T) holds PolyRedRel (G,T) reduces (m1 *' g1) - (m2 *' g2), 0_ (n,L) thus for m1, m2 being Monomial of n,L st HM ((m1 *' g1),T) = HM ((m2 *' g2),T) holds PolyRedRel (G,T) reduces (m1 *' g1) - (m2 *' g2), 0_ (n,L) ::_thesis: verum proof set a1 = HC (g1,T); set a2 = HC (g2,T); set t1 = HT (g1,T); set t2 = HT (g2,T); let m1, m2 be Monomial of n,L; ::_thesis: ( HM ((m1 *' g1),T) = HM ((m2 *' g2),T) implies PolyRedRel (G,T) reduces (m1 *' g1) - (m2 *' g2), 0_ (n,L) ) assume A5: HM ((m1 *' g1),T) = HM ((m2 *' g2),T) ; ::_thesis: PolyRedRel (G,T) reduces (m1 *' g1) - (m2 *' g2), 0_ (n,L) A6: HC (g2,T) <> 0. L by A1, A4, TERMORD:17; reconsider g1 = g1, g2 = g2 as non-zero Polynomial of n,L by A1, A3, A4, POLYNOM7:def_1; set b1 = coefficient m1; set b2 = coefficient m2; set u1 = term m1; set u2 = term m2; A7: HC (g1,T) <> 0. L by A1, A3, TERMORD:17; then reconsider a1 = HC (g1,T), a2 = HC (g2,T) as non zero Element of L by A6, STRUCT_0:def_12; A8: HC ((m1 *' g1),T) = coefficient (HM ((m1 *' g1),T)) by TERMORD:22 .= HC ((m2 *' g2),T) by A5, TERMORD:22 ; now__::_thesis:_(_(_(_coefficient_m1_=_0._L_or_coefficient_m2_=_0._L_)_&_PolyRedRel_(G,T)_reduces_(m1_*'_g1)_-_(m2_*'_g2),_0__(n,L)_)_or_(_coefficient_m1_<>_0._L_&_coefficient_m2_<>_0._L_&_PolyRedRel_(G,T)_reduces_(m1_*'_g1)_-_(m2_*'_g2),_0__(n,L)_)_) percases ( coefficient m1 = 0. L or coefficient m2 = 0. L or ( coefficient m1 <> 0. L & coefficient m2 <> 0. L ) ) ; caseA9: ( coefficient m1 = 0. L or coefficient m2 = 0. L ) ; ::_thesis: PolyRedRel (G,T) reduces (m1 *' g1) - (m2 *' g2), 0_ (n,L) now__::_thesis:_(_(_coefficient_m1_=_0._L_&_PolyRedRel_(G,T)_reduces_(m1_*'_g1)_-_(m2_*'_g2),_0__(n,L)_)_or_(_coefficient_m2_=_0._L_&_PolyRedRel_(G,T)_reduces_(m1_*'_g1)_-_(m2_*'_g2),_0__(n,L)_)_) percases ( coefficient m1 = 0. L or coefficient m2 = 0. L ) by A9; case coefficient m1 = 0. L ; ::_thesis: PolyRedRel (G,T) reduces (m1 *' g1) - (m2 *' g2), 0_ (n,L) then HC (m1,T) = 0. L by TERMORD:23; then m1 = 0_ (n,L) by TERMORD:17; then A10: m1 *' g1 = 0_ (n,L) by POLYRED:5; then HC ((m2 *' g2),T) = 0. L by A8, TERMORD:17; then m2 *' g2 = 0_ (n,L) by TERMORD:17; then (m1 *' g1) - (m2 *' g2) = 0_ (n,L) by A10, POLYRED:4; hence PolyRedRel (G,T) reduces (m1 *' g1) - (m2 *' g2), 0_ (n,L) by REWRITE1:12; ::_thesis: verum end; case coefficient m2 = 0. L ; ::_thesis: PolyRedRel (G,T) reduces (m1 *' g1) - (m2 *' g2), 0_ (n,L) then HC (m2,T) = 0. L by TERMORD:23; then m2 = 0_ (n,L) by TERMORD:17; then A11: m2 *' g2 = 0_ (n,L) by POLYRED:5; then HC ((m1 *' g1),T) = 0. L by A8, TERMORD:17; then m1 *' g1 = 0_ (n,L) by TERMORD:17; then (m1 *' g1) - (m2 *' g2) = 0_ (n,L) by A11, POLYRED:4; hence PolyRedRel (G,T) reduces (m1 *' g1) - (m2 *' g2), 0_ (n,L) by REWRITE1:12; ::_thesis: verum end; end; end; hence PolyRedRel (G,T) reduces (m1 *' g1) - (m2 *' g2), 0_ (n,L) ; ::_thesis: verum end; caseA12: ( coefficient m1 <> 0. L & coefficient m2 <> 0. L ) ; ::_thesis: PolyRedRel (G,T) reduces (m1 *' g1) - (m2 *' g2), 0_ (n,L) then reconsider b1 = coefficient m1, b2 = coefficient m2 as non zero Element of L by STRUCT_0:def_12; b2 * a2 <> 0. L by VECTSP_2:def_1; then A13: not b2 * a2 is zero by STRUCT_0:def_12; HT (g1,T) divides lcm ((HT (g1,T)),(HT (g2,T))) by Th3; then consider s1 being bag of n such that A14: (HT (g1,T)) + s1 = lcm ((HT (g1,T)),(HT (g2,T))) by TERMORD:1; HC (m2,T) <> 0. L by A12, TERMORD:23; then A15: m2 <> 0_ (n,L) by TERMORD:17; HC (m1,T) <> 0. L by A12, TERMORD:23; then m1 <> 0_ (n,L) by TERMORD:17; then reconsider m1 = m1, m2 = m2 as non-zero Monomial of n,L by A15, POLYNOM7:def_1; A16: Monom ((b1 * a1),((term m1) + (HT (g1,T)))) = (Monom (b1,(term m1))) *' (Monom (a1,(HT (g1,T)))) by TERMORD:3 .= m1 *' (Monom (a1,(HT (g1,T)))) by POLYNOM7:11 .= (HM (m1,T)) *' (Monom (a1,(HT (g1,T)))) by TERMORD:23 .= (HM (m1,T)) *' (HM (g1,T)) by TERMORD:def_8 .= HM ((m2 *' g2),T) by A5, TERMORD:33 .= (HM (m2,T)) *' (HM (g2,T)) by TERMORD:33 .= (HM (m2,T)) *' (Monom (a2,(HT (g2,T)))) by TERMORD:def_8 .= m2 *' (Monom (a2,(HT (g2,T)))) by TERMORD:23 .= (Monom (b2,(term m2))) *' (Monom (a2,(HT (g2,T)))) by POLYNOM7:11 .= Monom ((b2 * a2),((term m2) + (HT (g2,T)))) by TERMORD:3 ; then b1 * a1 = coefficient (Monom ((b2 * a2),((term m2) + (HT (g2,T))))) by POLYNOM7:9 .= b2 * a2 by POLYNOM7:9 ; then (b1 * a1) / a2 = (b2 * a2) * (a2 ") by VECTSP_1:def_11 .= b2 * (a2 * (a2 ")) by GROUP_1:def_3 .= b2 * (1. L) by A6, VECTSP_1:def_10 ; then A17: b2 / a1 = ((b1 * a1) / a2) / a1 by VECTSP_1:def_4 .= ((b1 * a1) * (a2 ")) / a1 by VECTSP_1:def_11 .= ((b1 * a1) * (a2 ")) * (a1 ") by VECTSP_1:def_11 .= ((b1 * (a2 ")) * a1) * (a1 ") by GROUP_1:def_3 .= (b1 * (a2 ")) * (a1 * (a1 ")) by GROUP_1:def_3 .= (b1 * (a2 ")) * (1. L) by A7, VECTSP_1:def_10 .= b1 * (a2 ") by VECTSP_1:def_4 .= b1 / a2 by VECTSP_1:def_11 ; b1 * a1 <> 0. L by VECTSP_2:def_1; then not b1 * a1 is zero by STRUCT_0:def_12; then A18: (term m1) + (HT (g1,T)) = term (Monom ((b2 * a2),((term m2) + (HT (g2,T))))) by A16, POLYNOM7:10 .= (term m2) + (HT (g2,T)) by A13, POLYNOM7:10 ; then ( HT (g1,T) divides (term m1) + (HT (g1,T)) & HT (g2,T) divides (term m1) + (HT (g1,T)) ) by TERMORD:1; then lcm ((HT (g1,T)),(HT (g2,T))) divides (term m1) + (HT (g1,T)) by Th4; then consider v being bag of n such that A19: (term m1) + (HT (g1,T)) = (lcm ((HT (g1,T)),(HT (g2,T)))) + v by TERMORD:1; (term m1) + (HT (g1,T)) = (v + s1) + (HT (g1,T)) by A14, A19, PRE_POLY:35; then A20: term m1 = ((v + s1) + (HT (g1,T))) -' (HT (g1,T)) by PRE_POLY:48 .= v + s1 by PRE_POLY:48 ; HT (g2,T) divides lcm ((HT (g1,T)),(HT (g2,T))) by Th3; then consider s2 being bag of n such that A21: (HT (g2,T)) + s2 = lcm ((HT (g1,T)),(HT (g2,T))) by TERMORD:1; (term m2) + (HT (g2,T)) = (v + s2) + (HT (g2,T)) by A18, A21, A19, PRE_POLY:35; then A22: term m2 = ((v + s2) + (HT (g2,T))) -' (HT (g2,T)) by PRE_POLY:48 .= v + s2 by PRE_POLY:48 ; HT (g2,T) divides lcm ((HT (g1,T)),(HT (g2,T))) by Th3; then A23: s2 = (lcm ((HT (g1,T)),(HT (g2,T)))) / (HT (g2,T)) by A21, Def1; A24: (b2 / a1) * a1 = (b2 * (a1 ")) * a1 by VECTSP_1:def_11 .= b2 * ((a1 ") * a1) by GROUP_1:def_3 .= b2 * (1. L) by A7, VECTSP_1:def_10 .= b2 by VECTSP_1:def_4 ; HT (g1,T) divides lcm ((HT (g1,T)),(HT (g2,T))) by Th3; then A25: s1 = (lcm ((HT (g1,T)),(HT (g2,T)))) / (HT (g1,T)) by A14, Def1; A26: (b1 / a2) * a2 = (b1 * (a2 ")) * a2 by VECTSP_1:def_11 .= b1 * ((a2 ") * a2) by GROUP_1:def_3 .= b1 * (1. L) by A6, VECTSP_1:def_10 ; (m1 *' g1) - (m2 *' g2) = ((Monom (b1,(term m1))) *' g1) - (m2 *' g2) by POLYNOM7:11 .= ((Monom (b1,(term m1))) *' g1) - ((Monom (b2,(term m2))) *' g2) by POLYNOM7:11 .= (b1 * ((v + s1) *' g1)) - ((Monom (b2,(v + s2))) *' g2) by A20, A22, POLYRED:22 .= (b1 * (v *' (s1 *' g1))) - ((Monom (b2,(v + s2))) *' g2) by POLYRED:18 .= (b1 * (v *' (s1 *' g1))) - (b2 * ((v + s2) *' g2)) by POLYRED:22 .= (b1 * (v *' (s1 *' g1))) - (b2 * (v *' (s2 *' g2))) by POLYRED:18 .= (b1 * (v *' (s1 *' g1))) + (- (b2 * (v *' (s2 *' g2)))) by POLYNOM1:def_6 .= (b1 * (v *' (s1 *' g1))) + (b2 * (- (v *' (s2 *' g2)))) by POLYRED:9 .= (((b1 / a2) * a2) * (v *' (s1 *' g1))) + (((b2 / a1) * a1) * (- (v *' (s2 *' g2)))) by A26, A24, VECTSP_1:def_4 .= (((b1 / a2) * a2) * (v *' (s1 *' g1))) + (((b2 / a1) * a1) * (v *' (- (s2 *' g2)))) by Th13 .= (((b1 / a2) * a2) * (v *' (s1 *' g1))) + ((b1 / a2) * (a1 * (v *' (- (s2 *' g2))))) by A17, POLYRED:11 .= ((b1 / a2) * (a2 * (v *' (s1 *' g1)))) + ((b1 / a2) * (a1 * (v *' (- (s2 *' g2))))) by POLYRED:11 .= (b1 / a2) * ((a2 * (v *' (s1 *' g1))) + (a1 * (v *' (- (s2 *' g2))))) by Th15 .= (b1 / a2) * ((a2 * (v *' (s1 *' g1))) + (v *' (a1 * (- (s2 *' g2))))) by Th14 .= (b1 / a2) * ((a2 * (v *' (s1 *' g1))) + (v *' (- (a1 * (s2 *' g2))))) by POLYRED:9 .= (b1 / a2) * ((v *' (a2 * (s1 *' g1))) + (v *' (- (a1 * (s2 *' g2))))) by Th14 .= (b1 / a2) * (v *' ((a2 * (s1 *' g1)) + (- (a1 * (s2 *' g2))))) by Th12 .= (b1 / a2) * (v *' (S-Poly (g1,g2,T))) by A25, A23, POLYNOM1:def_6 .= (Monom ((b1 / a2),v)) *' (S-Poly (g1,g2,T)) by POLYRED:22 ; hence PolyRedRel (G,T) reduces (m1 *' g1) - (m2 *' g2), 0_ (n,L) by A2, A3, A4, POLYRED:48; ::_thesis: verum end; end; end; hence PolyRedRel (G,T) reduces (m1 *' g1) - (m2 *' g2), 0_ (n,L) ; ::_thesis: verum end; end; hence G is_Groebner_basis_wrt T by Th17; ::_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 P be Subset of (Polynom-Ring (n,L)); func S-Poly (P,T) -> Subset of (Polynom-Ring (n,L)) equals :: GROEB_2:def 5 { (S-Poly (p1,p2,T)) where p1, p2 is Polynomial of n,L : ( p1 in P & p2 in P ) } ; coherence { (S-Poly (p1,p2,T)) where p1, p2 is Polynomial of n,L : ( p1 in P & p2 in P ) } is Subset of (Polynom-Ring (n,L)) proof set M = { (S-Poly (p1,p2,T)) where p1, p2 is Polynomial of n,L : ( p1 in P & p2 in P ) } ; now__::_thesis:_for_u_being_set_st_u_in__{__(S-Poly_(p1,p2,T))_where_p1,_p2_is_Polynomial_of_n,L_:_(_p1_in_P_&_p2_in_P_)__}__holds_ u_in_the_carrier_of_(Polynom-Ring_(n,L)) let u be set ; ::_thesis: ( u in { (S-Poly (p1,p2,T)) where p1, p2 is Polynomial of n,L : ( p1 in P & p2 in P ) } implies u in the carrier of (Polynom-Ring (n,L)) ) assume u in { (S-Poly (p1,p2,T)) where p1, p2 is Polynomial of n,L : ( p1 in P & p2 in P ) } ; ::_thesis: u in the carrier of (Polynom-Ring (n,L)) then ex p1, p2 being Polynomial of n,L st ( u = S-Poly (p1,p2,T) & p1 in P & p2 in P ) ; hence u in the carrier of (Polynom-Ring (n,L)) by POLYNOM1:def_10; ::_thesis: verum end; hence { (S-Poly (p1,p2,T)) where p1, p2 is Polynomial of n,L : ( p1 in P & p2 in P ) } is Subset of (Polynom-Ring (n,L)) by TARSKI:def_3; ::_thesis: verum end; end; :: deftheorem defines S-Poly GROEB_2:def_5_:_ 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 S-Poly (P,T) = { (S-Poly (p1,p2,T)) where p1, p2 is Polynomial of n,L : ( p1 in P & p2 in P ) } ; registration 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 finite Subset of (Polynom-Ring (n,L)); cluster S-Poly (P,T) -> finite ; coherence S-Poly (P,T) is finite proof defpred S1[ set , set ] means ex p1, p2 being Polynomial of n,L st ( p1 = n `1 & n `2 = p2 & p1 in P & p2 in P & T = S-Poly (p1,p2,T) ); A1: for x being set st x in [:P,P:] holds ex y being set st S1[x,y] proof let x be set ; ::_thesis: ( x in [:P,P:] implies ex y being set st S1[x,y] ) assume x in [:P,P:] ; ::_thesis: ex y being set st S1[x,y] then consider p1, p2 being set such that A2: ( p1 in P & p2 in P ) and A3: [p1,p2] = x by ZFMISC_1:def_2; reconsider p1 = p1, p2 = p2 as Polynomial of n,L by A2, POLYNOM1:def_10; take S-Poly (p1,p2,T) ; ::_thesis: S1[x, S-Poly (p1,p2,T)] take p1 ; ::_thesis: ex p2 being Polynomial of n,L st ( p1 = x `1 & x `2 = p2 & p1 in P & p2 in P & S-Poly (p1,p2,T) = S-Poly (p1,p2,T) ) take p2 ; ::_thesis: ( p1 = x `1 & x `2 = p2 & p1 in P & p2 in P & S-Poly (p1,p2,T) = S-Poly (p1,p2,T) ) [p1,p2] `1 = p1 ; hence x `1 = p1 by A3; ::_thesis: ( x `2 = p2 & p1 in P & p2 in P & S-Poly (p1,p2,T) = S-Poly (p1,p2,T) ) [p1,p2] `2 = p2 ; hence x `2 = p2 by A3; ::_thesis: ( p1 in P & p2 in P & S-Poly (p1,p2,T) = S-Poly (p1,p2,T) ) thus ( p1 in P & p2 in P & S-Poly (p1,p2,T) = S-Poly (p1,p2,T) ) by A2; ::_thesis: verum end; consider f being Function such that A4: ( dom f = [:P,P:] & ( for x being set st x in [:P,P:] holds S1[x,f . x] ) ) from CLASSES1:sch_1(A1); A5: now__::_thesis:_for_v_being_set_st_v_in_S-Poly_(P,T)_holds_ v_in_rng_f let v be set ; ::_thesis: ( v in S-Poly (P,T) implies v in rng f ) assume v in S-Poly (P,T) ; ::_thesis: v in rng f then consider p1, p2 being Polynomial of n,L such that A6: v = S-Poly (p1,p2,T) and A7: ( p1 in P & p2 in P ) ; A8: [p1,p2] in dom f by A4, A7, ZFMISC_1:def_2; then consider p19, p29 being Polynomial of n,L such that A9: ( [p1,p2] `1 = p19 & [p1,p2] `2 = p29 ) and p19 in P and p29 in P and A10: f . [p1,p2] = S-Poly (p19,p29,T) by A4; ( p1 = p19 & p2 = p29 ) by A9; hence v in rng f by A6, A8, A10, FUNCT_1:def_3; ::_thesis: verum end; now__::_thesis:_for_v_being_set_st_v_in_rng_f_holds_ v_in_S-Poly_(P,T) let v be set ; ::_thesis: ( v in rng f implies v in S-Poly (P,T) ) assume v in rng f ; ::_thesis: v in S-Poly (P,T) then consider u being set such that A11: u in dom f and A12: v = f . u by FUNCT_1:def_3; ex p1, p2 being Polynomial of n,L st ( p1 = u `1 & u `2 = p2 & p1 in P & p2 in P & f . u = S-Poly (p1,p2,T) ) by A4, A11; hence v in S-Poly (P,T) by A12; ::_thesis: verum end; then rng f = S-Poly (P,T) by A5, TARSKI:1; hence S-Poly (P,T) is finite by A4, FINSET_1:8; ::_thesis: verum end; end; theorem :: GROEB_2:26 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 G being Subset of (Polynom-Ring (n,L)) st not 0_ (n,L) in G & ( for g being Polynomial of n,L st g in G holds g is Monomial of n,L ) holds G is_Groebner_basis_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 G being Subset of (Polynom-Ring (n,L)) st not 0_ (n,L) in G & ( for g being Polynomial of n,L st g in G holds g is Monomial of n,L ) holds G 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 Abelian add-associative right_zeroed doubleLoopStr for G being Subset of (Polynom-Ring (n,L)) st not 0_ (n,L) in G & ( for g being Polynomial of n,L st g in G holds g is Monomial of n,L ) holds G is_Groebner_basis_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 G being Subset of (Polynom-Ring (n,L)) st not 0_ (n,L) in G & ( for g being Polynomial of n,L st g in G holds g is Monomial of n,L ) holds G is_Groebner_basis_wrt T let G be Subset of (Polynom-Ring (n,L)); ::_thesis: ( not 0_ (n,L) in G & ( for g being Polynomial of n,L st g in G holds g is Monomial of n,L ) implies G is_Groebner_basis_wrt T ) assume that A1: not 0_ (n,L) in G and A2: for g being Polynomial of n,L st g in G holds g is Monomial of n,L ; ::_thesis: G is_Groebner_basis_wrt T now__::_thesis:_for_g1,_g2_being_Polynomial_of_n,L_st_g1_in_G_&_g2_in_G_holds_ PolyRedRel_(G,T)_reduces_S-Poly_(g1,g2,T),_0__(n,L) let g1, g2 be Polynomial of n,L; ::_thesis: ( g1 in G & g2 in G implies PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) ) assume ( g1 in G & g2 in G ) ; ::_thesis: PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) then ( g1 is Monomial of n,L & g2 is Monomial of n,L ) by A2; then S-Poly (g1,g2,T) = 0_ (n,L) by Th19; hence PolyRedRel (G,T) reduces S-Poly (g1,g2,T), 0_ (n,L) by REWRITE1:12; ::_thesis: verum end; hence G is_Groebner_basis_wrt T by A1, Th25; ::_thesis: verum end; begin theorem :: GROEB_2:27 for L being non empty multLoopStr for P being non empty Subset of L for A being LeftLinearCombination of P for i being Element of NAT holds A | i is LeftLinearCombination of P proof let L be non empty multLoopStr ; ::_thesis: for P being non empty Subset of L for A being LeftLinearCombination of P for i being Element of NAT holds A | i is LeftLinearCombination of P let P be non empty Subset of L; ::_thesis: for A being LeftLinearCombination of P for i being Element of NAT holds A | i is LeftLinearCombination of P let A be LeftLinearCombination of P; ::_thesis: for i being Element of NAT holds A | i is LeftLinearCombination of P let j be Element of NAT ; ::_thesis: A | j is LeftLinearCombination of P set C = A | (Seg j); reconsider C = A | (Seg j) as FinSequence of the carrier of L by FINSEQ_1:18; now__::_thesis:_for_i_being_set_st_i_in_dom_C_holds_ ex_u_being_Element_of_L_ex_a_being_Element_of_P_st_C_/._i_=_u_*_a let i be set ; ::_thesis: ( i in dom C implies ex u being Element of L ex a being Element of P st C /. i = u * a ) A1: dom C c= dom A by RELAT_1:60; assume A2: i in dom C ; ::_thesis: ex u being Element of L ex a being Element of P st C /. i = u * a then C . i = A . i by FUNCT_1:47; then C /. i = A . i by A2, PARTFUN1:def_6 .= A /. i by A2, A1, PARTFUN1:def_6 ; hence ex u being Element of L ex a being Element of P st C /. i = u * a by A2, A1, IDEAL_1:def_9; ::_thesis: verum end; then C is LeftLinearCombination of P by IDEAL_1:def_9; hence A | j is LeftLinearCombination of P by FINSEQ_1:def_15; ::_thesis: verum end; theorem :: GROEB_2:28 for L being non empty multLoopStr for P being non empty Subset of L for A being LeftLinearCombination of P for i being Element of NAT holds A /^ i is LeftLinearCombination of P proof let L be non empty multLoopStr ; ::_thesis: for P being non empty Subset of L for A being LeftLinearCombination of P for i being Element of NAT holds A /^ i is LeftLinearCombination of P let P be non empty Subset of L; ::_thesis: for A being LeftLinearCombination of P for i being Element of NAT holds A /^ i is LeftLinearCombination of P let A be LeftLinearCombination of P; ::_thesis: for i being Element of NAT holds A /^ i is LeftLinearCombination of P let j be Element of NAT ; ::_thesis: A /^ j is LeftLinearCombination of P set C = A /^ j; reconsider C = A /^ j as FinSequence of the carrier of L ; now__::_thesis:_(_(_j_<=_len_A_&_A_/^_j_is_LeftLinearCombination_of_P_)_or_(_not_j_<=_len_A_&_A_/^_j_is_LeftLinearCombination_of_P_)_) percases ( j <= len A or not j <= len A ) ; caseA1: j <= len A ; ::_thesis: A /^ j is LeftLinearCombination of P then reconsider m = (len A) - j as Element of NAT by INT_1:5; now__::_thesis:_for_i_being_set_st_i_in_dom_C_holds_ ex_u_being_Element_of_L_ex_a_being_Element_of_P_st_C_/._i_=_u_*_a let i be set ; ::_thesis: ( i in dom C implies ex u being Element of L ex a being Element of P st C /. i = u * a ) assume A2: i in dom C ; ::_thesis: ex u being Element of L ex a being Element of P st C /. i = u * a then reconsider k = i as Element of NAT ; A3: dom C = Seg (len C) by FINSEQ_1:def_3 .= Seg m by A1, RFINSEQ:def_1 ; then k <= (len A) - j by A2, FINSEQ_1:1; then A4: k + j <= ((len A) + (- j)) + j by XREAL_1:6; A5: k <= k + j by NAT_1:11; 1 <= k by A2, A3, FINSEQ_1:1; then 1 <= k + j by A5, XXREAL_0:2; then j + k in Seg (len A) by A4, FINSEQ_1:1; then j + k in dom A by FINSEQ_1:def_3; then ex u being Element of L ex a being Element of P st A /. (j + k) = u * a by IDEAL_1:def_9; hence ex u being Element of L ex a being Element of P st C /. i = u * a by A2, FINSEQ_5:27; ::_thesis: verum end; hence A /^ j is LeftLinearCombination of P by IDEAL_1:def_9; ::_thesis: verum end; case not j <= len A ; ::_thesis: A /^ j is LeftLinearCombination of P then C = <*> the carrier of L by RFINSEQ:def_1; then for i being set st i in dom C holds ex u being Element of L ex a being Element of P st C /. i = u * a ; hence A /^ j is LeftLinearCombination of P by IDEAL_1:def_9; ::_thesis: verum end; end; end; hence A /^ j is LeftLinearCombination of P ; ::_thesis: verum end; theorem :: GROEB_2:29 for L being non empty multLoopStr for P, Q being non empty Subset of L for A being LeftLinearCombination of P st P c= Q holds A is LeftLinearCombination of Q proof let L be non empty multLoopStr ; ::_thesis: for P, Q being non empty Subset of L for A being LeftLinearCombination of P st P c= Q holds A is LeftLinearCombination of Q let P, Q be non empty Subset of L; ::_thesis: for A being LeftLinearCombination of P st P c= Q holds A is LeftLinearCombination of Q let A be LeftLinearCombination of P; ::_thesis: ( P c= Q implies A is LeftLinearCombination of Q ) assume A1: P c= Q ; ::_thesis: A is LeftLinearCombination of Q now__::_thesis:_for_i_being_set_st_i_in_dom_A_holds_ ex_u_being_Element_of_L_ex_a_being_Element_of_Q_st_A_/._i_=_u_*_a let i be set ; ::_thesis: ( i in dom A implies ex u being Element of L ex a being Element of Q st A /. i = u * a ) assume i in dom A ; ::_thesis: ex u being Element of L ex a being Element of Q st A /. i = u * a then consider u being Element of L, a being Element of P such that A2: A /. i = u * a by IDEAL_1:def_9; a in P ; hence ex u being Element of L ex a being Element of Q st A /. i = u * a by A1, A2; ::_thesis: verum end; hence A is LeftLinearCombination of Q by IDEAL_1:def_9; ::_thesis: verum end; definition let n be Ordinal; let L be non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr ; let P be non empty Subset of (Polynom-Ring (n,L)); let A, B be LeftLinearCombination of P; :: original: ^ redefine funcA ^ B -> LeftLinearCombination of P; coherence A ^ B is LeftLinearCombination of P proof A ^ B is LeftLinearCombination of P \/ P ; hence A ^ B is LeftLinearCombination of P ; ::_thesis: verum end; end; definition let n be Ordinal; let L be non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr ; let f be Polynomial of n,L; let P be non empty Subset of (Polynom-Ring (n,L)); let A be LeftLinearCombination of P; predA is_MonomialRepresentation_of f means :Def6: :: GROEB_2:def 6 ( Sum A = f & ( for i being Element of NAT st i in dom A holds ex m being Monomial of n,L ex p being Polynomial of n,L st ( p in P & A /. i = m *' p ) ) ); end; :: deftheorem Def6 defines is_MonomialRepresentation_of GROEB_2:def_6_:_ for n being Ordinal for L being non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr for f being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P holds ( A is_MonomialRepresentation_of f iff ( Sum A = f & ( for i being Element of NAT st i in dom A holds ex m being Monomial of n,L ex p being Polynomial of n,L st ( p in P & A /. i = m *' p ) ) ) ); theorem :: GROEB_2:30 for n being Ordinal for L being non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr for f being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds Support f c= union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } proof let n be Ordinal; ::_thesis: for L being non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr for f being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds Support f c= union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } let L be non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for f being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds Support f c= union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } let f be Polynomial of n,L; ::_thesis: for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds Support f c= union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds Support f c= union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } let A be LeftLinearCombination of P; ::_thesis: ( A is_MonomialRepresentation_of f implies Support f c= union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } ) assume A1: A is_MonomialRepresentation_of f ; ::_thesis: Support f c= union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } defpred S1[ Element of NAT ] means for f being Polynomial of n,L for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f & len A = $1 holds Support f c= union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } ; A2: ex n being Element of NAT st len A = n ; A3: 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 A4: S1[k] ; ::_thesis: S1[k + 1] for f being Polynomial of n,L for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f & len A = k + 1 holds Support f c= union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } proof A5: k <= k + 1 by NAT_1:11; let f be Polynomial of n,L; ::_thesis: for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f & len A = k + 1 holds Support f c= union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } let A be LeftLinearCombination of P; ::_thesis: ( A is_MonomialRepresentation_of f & len A = k + 1 implies Support f c= union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } ) assume that A6: A is_MonomialRepresentation_of f and A7: len A = k + 1 ; ::_thesis: Support f c= union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } A8: A <> <*> the carrier of (Polynom-Ring (n,L)) by A7; A9: Sum A = f by A6, Def6; reconsider A = A as non empty LeftLinearCombination of P by A8; consider A9 being LeftLinearCombination of P, e being Element of (Polynom-Ring (n,L)) such that A10: A = A9 ^ <*e*> and <*e*> is LeftLinearCombination of P by IDEAL_1:33; A11: dom A = Seg (k + 1) by A7, FINSEQ_1:def_3; reconsider ep = Sum <*e*> as Polynomial of n,L by POLYNOM1:def_10; reconsider g = Sum A9 as Polynomial of n,L by POLYNOM1:def_10; f = (Sum A9) + (Sum <*e*>) by A9, A10, RLVECT_1:41 .= g + ep by POLYNOM1:def_10 ; then A12: Support f c= (Support g) \/ (Support ep) by POLYNOM1:20; A13: len A = (len A9) + (len <*e*>) by A10, FINSEQ_1:22 .= (len A9) + 1 by FINSEQ_1:39 ; then dom A9 = Seg k by A7, FINSEQ_1:def_3; then A14: dom A9 c= dom A by A11, A5, FINSEQ_1:5; now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_A9_holds_ ex_m_being_Monomial_of_n,L_ex_p_being_Polynomial_of_n,L_st_ (_p_in_P_&_A9_/._i_=_m_*'_p_) let i be Element of NAT ; ::_thesis: ( i in dom A9 implies ex m being Monomial of n,L ex p being Polynomial of n,L st ( p in P & A9 /. i = m *' p ) ) assume A15: i in dom A9 ; ::_thesis: ex m being Monomial of n,L ex p being Polynomial of n,L st ( p in P & A9 /. i = m *' p ) then A /. i = A . i by A14, PARTFUN1:def_6 .= A9 . i by A10, A15, FINSEQ_1:def_7 .= A9 /. i by A15, PARTFUN1:def_6 ; hence ex m being Monomial of n,L ex p being Polynomial of n,L st ( p in P & A9 /. i = m *' p ) by A6, A14, A15, Def6; ::_thesis: verum end; then A9 is_MonomialRepresentation_of g by Def6; then A16: Support g c= union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A9 & A9 /. i = m *' p ) } by A4, A7, A13; now__::_thesis:_for_x_being_set_st_x_in_Support_f_holds_ x_in_union__{__(Support_(m_*'_p))_where_m_is_Monomial_of_n,L,_p_is_Polynomial_of_n,L_:_ex_i_being_Element_of_NAT_st_ (_i_in_dom_A_&_A_/._i_=_m_*'_p_)__}_ let x be set ; ::_thesis: ( x in Support f implies x in union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } ) assume A17: x in Support f ; ::_thesis: x in union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } then reconsider u = x as Element of Bags n ; now__::_thesis:_(_(_u_in_Support_g_&_u_in_union__{__(Support_(m_*'_p))_where_m_is_Monomial_of_n,L,_p_is_Polynomial_of_n,L_:_ex_i_being_Element_of_NAT_st_ (_i_in_dom_A_&_A_/._i_=_m_*'_p_)__}__)_or_(_u_in_Support_ep_&_u_in_union__{__(Support_(m_*'_p))_where_m_is_Monomial_of_n,L,_p_is_Polynomial_of_n,L_:_ex_i_being_Element_of_NAT_st_ (_i_in_dom_A_&_A_/._i_=_m_*'_p_)__}__)_) percases ( u in Support g or u in Support ep ) by A12, A17, XBOOLE_0:def_3; case u in Support g ; ::_thesis: u in union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } then consider Y being set such that A18: u in Y and A19: Y in { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A9 & A9 /. i = m *' p ) } by A16, TARSKI:def_4; consider m9 being Monomial of n,L, p9 being Polynomial of n,L such that A20: Y = Support (m9 *' p9) and A21: ex i being Element of NAT st ( i in dom A9 & A9 /. i = m9 *' p9 ) by A19; consider i being Element of NAT such that A22: i in dom A9 and A23: A9 /. i = m9 *' p9 by A21; A /. i = A . i by A14, A22, PARTFUN1:def_6 .= A9 . i by A10, A22, FINSEQ_1:def_7 .= A9 /. i by A22, PARTFUN1:def_6 ; then Y in { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } by A14, A20, A22, A23; hence u in union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } by A18, TARSKI:def_4; ::_thesis: verum end; caseA24: u in Support ep ; ::_thesis: u in union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } 1 <= len A by A7, NAT_1:11; then A25: len A in Seg (len A) by FINSEQ_1:1; dom A = Seg (len A) by FINSEQ_1:def_3; then A26: ex m9 being Monomial of n,L ex p9 being Polynomial of n,L st ( p9 in P & A /. (len A) = m9 *' p9 ) by A6, A25, Def6; A27: ( A . (len A) = e & e = Sum <*e*> ) by A10, A13, FINSEQ_1:42, RLVECT_1:44; A28: len A in dom A by A25, FINSEQ_1:def_3; then A /. (len A) = A . (len A) by PARTFUN1:def_6; then Support ep in { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } by A28, A26, A27; hence u in union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } by A24, TARSKI:def_4; ::_thesis: verum end; end; end; hence x in union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } ; ::_thesis: verum end; hence Support f c= union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } by TARSKI:def_3; ::_thesis: verum end; hence S1[k + 1] ; ::_thesis: verum end; A29: S1[ 0 ] proof let f be Polynomial of n,L; ::_thesis: for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f & len A = 0 holds Support f c= union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } let A be LeftLinearCombination of P; ::_thesis: ( A is_MonomialRepresentation_of f & len A = 0 implies Support f c= union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } ) assume that A30: A is_MonomialRepresentation_of f and A31: len A = 0 ; ::_thesis: Support f c= union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } A = <*> the carrier of (Polynom-Ring (n,L)) by A31; then Sum A = 0. (Polynom-Ring (n,L)) by RLVECT_1:43; then f = 0. (Polynom-Ring (n,L)) by A30, Def6; then f = 0_ (n,L) by POLYNOM1:def_10; then Support f = {} by POLYNOM7:1; hence Support f c= union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } by XBOOLE_1:2; ::_thesis: verum end; for k being Element of NAT holds S1[k] from NAT_1:sch_1(A29, A3); hence Support f c= union { (Support (m *' p)) where m is Monomial of n,L, p is Polynomial of n,L : ex i being Element of NAT st ( i in dom A & A /. i = m *' p ) } by A1, A2; ::_thesis: verum end; theorem :: GROEB_2:31 for n being Ordinal for L being non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr for f, g being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A, B being LeftLinearCombination of P st A is_MonomialRepresentation_of f & B is_MonomialRepresentation_of g holds A ^ B is_MonomialRepresentation_of f + g proof let n be Ordinal; ::_thesis: for L being non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr for f, g being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A, B being LeftLinearCombination of P st A is_MonomialRepresentation_of f & B is_MonomialRepresentation_of g holds A ^ B is_MonomialRepresentation_of f + g let L be non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, g being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A, B being LeftLinearCombination of P st A is_MonomialRepresentation_of f & B is_MonomialRepresentation_of g holds A ^ B is_MonomialRepresentation_of f + g let f, g be Polynomial of n,L; ::_thesis: for P being non empty Subset of (Polynom-Ring (n,L)) for A, B being LeftLinearCombination of P st A is_MonomialRepresentation_of f & B is_MonomialRepresentation_of g holds A ^ B is_MonomialRepresentation_of f + g let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: for A, B being LeftLinearCombination of P st A is_MonomialRepresentation_of f & B is_MonomialRepresentation_of g holds A ^ B is_MonomialRepresentation_of f + g let A, B be LeftLinearCombination of P; ::_thesis: ( A is_MonomialRepresentation_of f & B is_MonomialRepresentation_of g implies A ^ B is_MonomialRepresentation_of f + g ) assume that A1: A is_MonomialRepresentation_of f and A2: B is_MonomialRepresentation_of g ; ::_thesis: A ^ B is_MonomialRepresentation_of f + g A3: now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_(A_^_B)_holds_ ex_m_being_Monomial_of_n,L_ex_p_being_Polynomial_of_n,L_st_ (_p_in_P_&_(A_^_B)_/._i_=_m_*'_p_) let i be Element of NAT ; ::_thesis: ( i in dom (A ^ B) implies ex m being Monomial of n,L ex p being Polynomial of n,L st ( p in P & (A ^ B) /. i = m *' p ) ) assume A4: i in dom (A ^ B) ; ::_thesis: ex m being Monomial of n,L ex p being Polynomial of n,L st ( p in P & (A ^ B) /. i = m *' p ) now__::_thesis:_(_(_i_in_dom_A_&_ex_m_being_Monomial_of_n,L_ex_p_being_Polynomial_of_n,L_st_ (_p_in_P_&_(A_^_B)_/._i_=_m_*'_p_)_)_or_(_ex_k_being_Nat_st_ (_k_in_dom_B_&_i_=_(len_A)_+_k_)_&_ex_m_being_Monomial_of_n,L_ex_p_being_Polynomial_of_n,L_st_ (_p_in_P_&_(A_^_B)_/._i_=_m_*'_p_)_)_) percases ( i in dom A or ex k being Nat st ( k in dom B & i = (len A) + k ) ) by A4, FINSEQ_1:25; caseA5: i in dom A ; ::_thesis: ex m being Monomial of n,L ex p being Polynomial of n,L st ( p in P & (A ^ B) /. i = m *' p ) dom A c= dom (A ^ B) by FINSEQ_1:26; then (A ^ B) /. i = (A ^ B) . i by A5, PARTFUN1:def_6 .= A . i by A5, FINSEQ_1:def_7 .= A /. i by A5, PARTFUN1:def_6 ; hence ex m being Monomial of n,L ex p being Polynomial of n,L st ( p in P & (A ^ B) /. i = m *' p ) by A1, A5, Def6; ::_thesis: verum end; case ex k being Nat st ( k in dom B & i = (len A) + k ) ; ::_thesis: ex m being Monomial of n,L ex p being Polynomial of n,L st ( p in P & (A ^ B) /. i = m *' p ) then consider k being Nat such that A6: k in dom B and A7: i = (len A) + k ; i in dom (A ^ B) by A6, A7, FINSEQ_1:28; then (A ^ B) /. i = (A ^ B) . i by PARTFUN1:def_6 .= B . k by A6, A7, FINSEQ_1:def_7 .= B /. k by A6, PARTFUN1:def_6 ; hence ex m being Monomial of n,L ex p being Polynomial of n,L st ( p in P & (A ^ B) /. i = m *' p ) by A2, A6, Def6; ::_thesis: verum end; end; end; hence ex m being Monomial of n,L ex p being Polynomial of n,L st ( p in P & (A ^ B) /. i = m *' p ) ; ::_thesis: verum end; reconsider f9 = f, g9 = g as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10; reconsider f9 = f9, g9 = g9 as Element of (Polynom-Ring (n,L)) ; Sum (A ^ B) = (Sum A) + (Sum B) by RLVECT_1:41 .= f9 + (Sum B) by A1, Def6 .= f9 + g9 by A2, Def6 .= f + g by POLYNOM1:def_10 ; hence A ^ B is_MonomialRepresentation_of f + g by A3, Def6; ::_thesis: verum end; Lm4: 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 f being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds for b being bag of n st ( for i being Element of NAT st i in dom A holds for m being Monomial of n,L for p being Polynomial of n,L st A . i = m *' p & p in P holds (m *' p) . b = 0. L ) holds f . b = 0. L 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 f being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds for b being bag of n st ( for i being Element of NAT st i in dom A holds for m being Monomial of n,L for p being Polynomial of n,L st A . i = m *' p & p in P holds (m *' p) . b = 0. L ) holds f . b = 0. L 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 f being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds for b being bag of n st ( for i being Element of NAT st i in dom A holds for m being Monomial of n,L for p being Polynomial of n,L st A . i = m *' p & p in P holds (m *' p) . b = 0. L ) holds f . b = 0. L let L be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for f being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds for b being bag of n st ( for i being Element of NAT st i in dom A holds for m being Monomial of n,L for p being Polynomial of n,L st A . i = m *' p & p in P holds (m *' p) . b = 0. L ) holds f . b = 0. L let f be Polynomial of n,L; ::_thesis: for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds for b being bag of n st ( for i being Element of NAT st i in dom A holds for m being Monomial of n,L for p being Polynomial of n,L st A . i = m *' p & p in P holds (m *' p) . b = 0. L ) holds f . b = 0. L let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds for b being bag of n st ( for i being Element of NAT st i in dom A holds for m being Monomial of n,L for p being Polynomial of n,L st A . i = m *' p & p in P holds (m *' p) . b = 0. L ) holds f . b = 0. L let A be LeftLinearCombination of P; ::_thesis: ( A is_MonomialRepresentation_of f implies for b being bag of n st ( for i being Element of NAT st i in dom A holds for m being Monomial of n,L for p being Polynomial of n,L st A . i = m *' p & p in P holds (m *' p) . b = 0. L ) holds f . b = 0. L ) assume A1: A is_MonomialRepresentation_of f ; ::_thesis: for b being bag of n st ( for i being Element of NAT st i in dom A holds for m being Monomial of n,L for p being Polynomial of n,L st A . i = m *' p & p in P holds (m *' p) . b = 0. L ) holds f . b = 0. L let b be bag of n; ::_thesis: ( ( for i being Element of NAT st i in dom A holds for m being Monomial of n,L for p being Polynomial of n,L st A . i = m *' p & p in P holds (m *' p) . b = 0. L ) implies f . b = 0. L ) assume A2: for i being Element of NAT st i in dom A holds for m being Monomial of n,L for p being Polynomial of n,L st A . i = m *' p & p in P holds (m *' p) . b = 0. L ; ::_thesis: f . b = 0. L defpred S1[ Element of NAT ] means for f being Polynomial of n,L for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f & f = Sum A & len A = $1 & ( for i being Element of NAT st i in dom A holds for m being Monomial of n,L for p being Polynomial of n,L st A . i = m *' p & p in P holds (m *' p) . b = 0. L ) holds f . b = 0. L; A3: 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 A4: S1[k] ; ::_thesis: S1[k + 1] for f being Polynomial of n,L for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f & f = Sum A & len A = k + 1 & ( for i being Element of NAT st i in dom A holds for m being Monomial of n,L for p being Polynomial of n,L st A . i = m *' p & p in P holds (m *' p) . b = 0. L ) holds f . b = 0. L proof let f be Polynomial of n,L; ::_thesis: for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f & f = Sum A & len A = k + 1 & ( for i being Element of NAT st i in dom A holds for m being Monomial of n,L for p being Polynomial of n,L st A . i = m *' p & p in P holds (m *' p) . b = 0. L ) holds f . b = 0. L let A be LeftLinearCombination of P; ::_thesis: ( A is_MonomialRepresentation_of f & f = Sum A & len A = k + 1 & ( for i being Element of NAT st i in dom A holds for m being Monomial of n,L for p being Polynomial of n,L st A . i = m *' p & p in P holds (m *' p) . b = 0. L ) implies f . b = 0. L ) assume that A5: A is_MonomialRepresentation_of f and A6: f = Sum A and A7: len A = k + 1 ; ::_thesis: ( ex i being Element of NAT st ( i in dom A & ex m being Monomial of n,L ex p being Polynomial of n,L st ( A . i = m *' p & p in P & not (m *' p) . b = 0. L ) ) or f . b = 0. L ) set B = A | (Seg k); reconsider B = A | (Seg k) as FinSequence of (Polynom-Ring (n,L)) by FINSEQ_1:18; reconsider B = B as LeftLinearCombination of P by IDEAL_1:42; set g = Sum B; reconsider g = Sum B as Polynomial of n,L by POLYNOM1:def_10; A8: dom A = Seg (k + 1) by A7, FINSEQ_1:def_3; then k + 1 in dom A by FINSEQ_1:4; then A9: ex m being Monomial of n,L ex p being Polynomial of n,L st ( p in P & A /. (k + 1) = m *' p ) by A5, Def6; A10: k <= len A by A7, NAT_1:11; then ( k <= k + 1 & dom B = Seg k ) by FINSEQ_1:17, NAT_1:11; then A11: dom B c= dom A by A8, FINSEQ_1:5; now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_B_holds_ ex_m_being_Monomial_of_n,L_ex_p_being_Polynomial_of_n,L_st_ (_p_in_P_&_B_/._i_=_m_*'_p_) let i be Element of NAT ; ::_thesis: ( i in dom B implies ex m being Monomial of n,L ex p being Polynomial of n,L st ( p in P & B /. i = m *' p ) ) assume A12: i in dom B ; ::_thesis: ex m being Monomial of n,L ex p being Polynomial of n,L st ( p in P & B /. i = m *' p ) then B /. i = B . i by PARTFUN1:def_6 .= A . i by A12, FUNCT_1:47 .= A /. i by A11, A12, PARTFUN1:def_6 ; hence ex m being Monomial of n,L ex p being Polynomial of n,L st ( p in P & B /. i = m *' p ) by A5, A11, A12, Def6; ::_thesis: verum end; then A13: B is_MonomialRepresentation_of g by Def6; assume A14: for i being Element of NAT st i in dom A holds for m being Monomial of n,L for p being Polynomial of n,L st A . i = m *' p & p in P holds (m *' p) . b = 0. L ; ::_thesis: f . b = 0. L A15: now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_B_holds_ for_m_being_Monomial_of_n,L for_p_being_Polynomial_of_n,L_st_B_._i_=_m_*'_p_&_p_in_P_holds_ (m_*'_p)_._b_=_0._L let i be Element of NAT ; ::_thesis: ( i in dom B implies for m being Monomial of n,L for p being Polynomial of n,L st B . i = m *' p & p in P holds (m *' p) . b = 0. L ) assume A16: i in dom B ; ::_thesis: for m being Monomial of n,L for p being Polynomial of n,L st B . i = m *' p & p in P holds (m *' p) . b = 0. L now__::_thesis:_for_m_being_Monomial_of_n,L for_p_being_Polynomial_of_n,L_st_B_._i_=_m_*'_p_&_p_in_P_holds_ (m_*'_p)_._b_=_0._L let m be Monomial of n,L; ::_thesis: for p being Polynomial of n,L st B . i = m *' p & p in P holds (m *' p) . b = 0. L let p be Polynomial of n,L; ::_thesis: ( B . i = m *' p & p in P implies (m *' p) . b = 0. L ) assume that A17: B . i = m *' p and A18: p in P ; ::_thesis: (m *' p) . b = 0. L A . i = m *' p by A16, A17, FUNCT_1:47; hence (m *' p) . b = 0. L by A14, A11, A16, A18; ::_thesis: verum end; hence for m being Monomial of n,L for p being Polynomial of n,L st B . i = m *' p & p in P holds (m *' p) . b = 0. L ; ::_thesis: verum end; reconsider h = A /. (k + 1) as Polynomial of n,L by POLYNOM1:def_10; B ^ <*(A /. (k + 1))*> = B ^ <*(A . (k + 1))*> by A8, FINSEQ_1:4, PARTFUN1:def_6 .= A by A7, FINSEQ_3:55 ; then f = (Sum B) + (Sum <*(A /. (k + 1))*>) by A6, RLVECT_1:41 .= (Sum B) + (A /. (k + 1)) by RLVECT_1:44 .= g + h by POLYNOM1:def_10 ; then A19: f . b = (g . b) + (h . b) by POLYNOM1:15; A /. (k + 1) = A . (k + 1) by A8, FINSEQ_1:4, PARTFUN1:def_6; then A20: 0. L = h . b by A14, A8, A9, FINSEQ_1:4; len B = k by A10, FINSEQ_1:17; then g . b = 0. L by A4, A13, A15; hence f . b = 0. L by A19, A20, RLVECT_1:def_4; ::_thesis: verum end; hence S1[k + 1] ; ::_thesis: verum end; A21: S1[ 0 ] proof let f be Polynomial of n,L; ::_thesis: for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f & f = Sum A & len A = 0 & ( for i being Element of NAT st i in dom A holds for m being Monomial of n,L for p being Polynomial of n,L st A . i = m *' p & p in P holds (m *' p) . b = 0. L ) holds f . b = 0. L let A be LeftLinearCombination of P; ::_thesis: ( A is_MonomialRepresentation_of f & f = Sum A & len A = 0 & ( for i being Element of NAT st i in dom A holds for m being Monomial of n,L for p being Polynomial of n,L st A . i = m *' p & p in P holds (m *' p) . b = 0. L ) implies f . b = 0. L ) assume that A is_MonomialRepresentation_of f and A22: ( f = Sum A & len A = 0 ) ; ::_thesis: ( ex i being Element of NAT st ( i in dom A & ex m being Monomial of n,L ex p being Polynomial of n,L st ( A . i = m *' p & p in P & not (m *' p) . b = 0. L ) ) or f . b = 0. L ) assume for i being Element of NAT st i in dom A holds for m being Monomial of n,L for p being Polynomial of n,L st A . i = m *' p & p in P holds (m *' p) . b = 0. L ; ::_thesis: f . b = 0. L A = <*> the carrier of (Polynom-Ring (n,L)) by A22; then f = Sum (<*> the carrier of (Polynom-Ring (n,L))) by A22 .= 0. (Polynom-Ring (n,L)) by RLVECT_1:43 .= 0_ (n,L) by POLYNOM1:def_10 ; hence f . b = 0. L by POLYNOM1:22; ::_thesis: verum end; A23: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A21, A3); A24: ex n being Element of NAT st n = len A ; Sum A = f by A1, Def6; hence f . b = 0. L by A1, A2, A23, A24; ::_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 f be Polynomial of n,L; let P be non empty Subset of (Polynom-Ring (n,L)); let A be LeftLinearCombination of P; let b be bag of n; predA is_Standard_Representation_of f,P,b,T means :Def7: :: GROEB_2:def 7 ( Sum A = f & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A /. i = m *' p & HT ((m *' p),T) <= b,T ) ) ); end; :: deftheorem Def7 defines is_Standard_Representation_of GROEB_2:def_7_:_ 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 f being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P for b being bag of n holds ( A is_Standard_Representation_of f,P,b,T iff ( Sum A = f & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A /. i = m *' p & HT ((m *' p),T) <= b,T ) ) ) ); 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 f be Polynomial of n,L; let P be non empty Subset of (Polynom-Ring (n,L)); let A be LeftLinearCombination of P; predA is_Standard_Representation_of f,P,T means :Def8: :: GROEB_2:def 8 A is_Standard_Representation_of f,P, HT (f,T),T; end; :: deftheorem Def8 defines is_Standard_Representation_of GROEB_2:def_8_:_ 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 f being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P holds ( A is_Standard_Representation_of f,P,T iff A is_Standard_Representation_of f,P, HT (f,T),T ); 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 f be Polynomial of n,L; let P be non empty Subset of (Polynom-Ring (n,L)); let b be bag of n; predf has_a_Standard_Representation_of P,b,T means :: GROEB_2:def 9 ex A being LeftLinearCombination of P st A is_Standard_Representation_of f,P,b,T; end; :: deftheorem defines has_a_Standard_Representation_of GROEB_2:def_9_:_ 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 f being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for b being bag of n holds ( f has_a_Standard_Representation_of P,b,T iff ex A being LeftLinearCombination of P st A is_Standard_Representation_of f,P,b,T ); 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 f be Polynomial of n,L; let P be non empty Subset of (Polynom-Ring (n,L)); predf has_a_Standard_Representation_of P,T means :Def10: :: GROEB_2:def 10 ex A being LeftLinearCombination of P st A is_Standard_Representation_of f,P,T; end; :: deftheorem Def10 defines has_a_Standard_Representation_of GROEB_2:def_10_:_ 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 f being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) holds ( f has_a_Standard_Representation_of P,T iff ex A being LeftLinearCombination of P st A is_Standard_Representation_of f,P,T ); theorem Th32: :: GROEB_2:32 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 f being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P for b being bag of n st A is_Standard_Representation_of f,P,b,T holds A is_MonomialRepresentation_of f 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 f being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P for b being bag of n st A is_Standard_Representation_of f,P,b,T holds A is_MonomialRepresentation_of f 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 f being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P for b being bag of n st A is_Standard_Representation_of f,P,b,T holds A is_MonomialRepresentation_of f let L be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for f being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P for b being bag of n st A is_Standard_Representation_of f,P,b,T holds A is_MonomialRepresentation_of f let f be Polynomial of n,L; ::_thesis: for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P for b being bag of n st A is_Standard_Representation_of f,P,b,T holds A is_MonomialRepresentation_of f let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: for A being LeftLinearCombination of P for b being bag of n st A is_Standard_Representation_of f,P,b,T holds A is_MonomialRepresentation_of f let A be LeftLinearCombination of P; ::_thesis: for b being bag of n st A is_Standard_Representation_of f,P,b,T holds A is_MonomialRepresentation_of f let b be bag of n; ::_thesis: ( A is_Standard_Representation_of f,P,b,T implies A is_MonomialRepresentation_of f ) assume A1: A is_Standard_Representation_of f,P,b,T ; ::_thesis: A is_MonomialRepresentation_of f A2: now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_A_holds_ ex_m_being_Monomial_of_n,L_ex_p_being_Polynomial_of_n,L_st_ (_p_in_P_&_A_/._i_=_m_*'_p_) let i be Element of NAT ; ::_thesis: ( i in dom A implies ex m being Monomial of n,L ex p being Polynomial of n,L st ( p in P & A /. i = m *' p ) ) assume i in dom A ; ::_thesis: ex m being Monomial of n,L ex p being Polynomial of n,L st ( p in P & A /. i = m *' p ) then ex m9 being non-zero Monomial of n,L ex p9 being non-zero Polynomial of n,L st ( p9 in P & A /. i = m9 *' p9 & HT ((m9 *' p9),T) <= b,T ) by A1, Def7; hence ex m being Monomial of n,L ex p being Polynomial of n,L st ( p in P & A /. i = m *' p ) ; ::_thesis: verum end; Sum A = f by A1, Def7; hence A is_MonomialRepresentation_of f by A2, Def6; ::_thesis: verum end; Lm5: 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, g being Polynomial of n,L for f9, g9 being Element of (Polynom-Ring (n,L)) st f = f9 & g = g9 holds for P being non empty Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & 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, g being Polynomial of n,L for f9, g9 being Element of (Polynom-Ring (n,L)) st f = f9 & g = g9 holds for P being non empty Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & 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, g being Polynomial of n,L for f9, g9 being Element of (Polynom-Ring (n,L)) st f = f9 & g = g9 holds for P being non empty Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & 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, g being Polynomial of n,L for f9, g9 being Element of (Polynom-Ring (n,L)) st f = f9 & g = g9 holds for P being non empty Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) let f, g be Polynomial of n,L; ::_thesis: for f9, g9 being Element of (Polynom-Ring (n,L)) st f = f9 & g = g9 holds for P being non empty Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) let f9, g9 be Element of (Polynom-Ring (n,L)); ::_thesis: ( f = f9 & g = g9 implies for P being non empty Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) ) assume A1: ( f = f9 & g = g9 ) ; ::_thesis: for P being non empty Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) defpred S1[ Nat] means for f, g being Polynomial of n,L for f9, g9 being Element of (Polynom-Ring (n,L)) st f = f9 & g = g9 holds for P being non empty Subset of (Polynom-Ring (n,L)) for R being RedSequence of PolyRedRel (P,T) st R . 1 = f & R . (len R) = g & len R = $1 holds ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ); let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: ( PolyRedRel (P,T) reduces f,g implies ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) ) assume PolyRedRel (P,T) reduces f,g ; ::_thesis: ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) then consider R being RedSequence of PolyRedRel (P,T) such that A2: ( R . 1 = f & R . (len R) = g ) by REWRITE1:def_3; A3: 0_ (n,L) = 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10; A4: now__::_thesis:_for_k_being_Nat_st_1_<=_k_&_S1[k]_holds_ S1[k_+_1] let k be Nat; ::_thesis: ( 1 <= k & S1[k] implies S1[k + 1] ) assume A5: 1 <= k ; ::_thesis: ( S1[k] implies S1[k + 1] ) thus ( S1[k] implies S1[k + 1] ) ::_thesis: verum proof assume A6: S1[k] ; ::_thesis: S1[k + 1] for f, g being Polynomial of n,L for f9, g9 being Element of (Polynom-Ring (n,L)) st f = f9 & g = g9 holds for P being non empty Subset of (Polynom-Ring (n,L)) for R being RedSequence of PolyRedRel (P,T) st R . 1 = f & R . (len R) = g & len R = k + 1 holds ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) proof let f, g be Polynomial of n,L; ::_thesis: for f9, g9 being Element of (Polynom-Ring (n,L)) st f = f9 & g = g9 holds for P being non empty Subset of (Polynom-Ring (n,L)) for R being RedSequence of PolyRedRel (P,T) st R . 1 = f & R . (len R) = g & len R = k + 1 holds ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) let f9, g9 be Element of (Polynom-Ring (n,L)); ::_thesis: ( f = f9 & g = g9 implies for P being non empty Subset of (Polynom-Ring (n,L)) for R being RedSequence of PolyRedRel (P,T) st R . 1 = f & R . (len R) = g & len R = k + 1 holds ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) ) assume that A7: f = f9 and A8: g = g9 ; ::_thesis: for P being non empty Subset of (Polynom-Ring (n,L)) for R being RedSequence of PolyRedRel (P,T) st R . 1 = f & R . (len R) = g & len R = k + 1 holds ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: for R being RedSequence of PolyRedRel (P,T) st R . 1 = f & R . (len R) = g & len R = k + 1 holds ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) let R be RedSequence of PolyRedRel (P,T); ::_thesis: ( R . 1 = f & R . (len R) = g & len R = k + 1 implies ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) ) assume that A9: R . 1 = f and A10: R . (len R) = g and A11: len R = k + 1 ; ::_thesis: ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) set Q = R | (Seg k); reconsider Q = R | (Seg k) as FinSequence by FINSEQ_1:15; A12: k <= k + 1 by NAT_1:11; then A13: dom Q = Seg k by A11, FINSEQ_1:17; A14: dom R = Seg (k + 1) by A11, FINSEQ_1:def_3; A15: now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_Q_&_i_+_1_in_dom_Q_holds_ [(Q_._i),(Q_._(i_+_1))]_in_PolyRedRel_(P,T) let i be Element of NAT ; ::_thesis: ( i in dom Q & i + 1 in dom Q implies [(Q . i),(Q . (i + 1))] in PolyRedRel (P,T) ) assume that A16: i in dom Q and A17: i + 1 in dom Q ; ::_thesis: [(Q . i),(Q . (i + 1))] in PolyRedRel (P,T) i + 1 <= k by A13, A17, FINSEQ_1:1; then A18: i + 1 <= k + 1 by A12, XXREAL_0:2; i <= k by A13, A16, FINSEQ_1:1; then A19: i <= k + 1 by A12, XXREAL_0:2; 1 <= i + 1 by A13, A17, FINSEQ_1:1; then A20: i + 1 in dom R by A14, A18, FINSEQ_1:1; 1 <= i by A13, A16, FINSEQ_1:1; then i in dom R by A14, A19, FINSEQ_1:1; then A21: [(R . i),(R . (i + 1))] in PolyRedRel (P,T) by A20, REWRITE1:def_2; R . i = Q . i by A16, FUNCT_1:47; hence [(Q . i),(Q . (i + 1))] in PolyRedRel (P,T) by A17, A21, FUNCT_1:47; ::_thesis: verum end; len Q = k by A11, A12, FINSEQ_1:17; then reconsider Q = Q as RedSequence of PolyRedRel (P,T) by A5, A15, REWRITE1:def_2; A22: 1 in dom Q by A5, A13, FINSEQ_1:1; then A23: Q . 1 = f by A9, FUNCT_1:47; 1 <= k + 1 by NAT_1:11; then A24: k + 1 in dom R by A14, FINSEQ_1:1; k in dom R by A5, A14, A12, FINSEQ_1:1; then A25: [(R . k),(R . (k + 1))] in PolyRedRel (P,T) by A24, REWRITE1:def_2; then consider h9, g2 being set such that A26: [(R . k),(R . (k + 1))] = [h9,g2] and A27: h9 in NonZero (Polynom-Ring (n,L)) and A28: g2 in the carrier of (Polynom-Ring (n,L)) by RELSET_1:2; A29: R . k = h9 by A26, XTUPLE_0:1; reconsider g2 = g2 as Polynomial of n,L by A28, POLYNOM1:def_10; not h9 in {(0_ (n,L))} by A3, A27, XBOOLE_0:def_5; then h9 <> 0_ (n,L) by TARSKI:def_1; then reconsider h9 = h9 as non-zero Polynomial of n,L by A27, POLYNOM1:def_10, POLYNOM7:def_1; A30: Q . k = h9 by A5, A13, A29, FINSEQ_1:3, FUNCT_1:47; then reconsider u9 = Q . k as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10; reconsider u = u9 as Polynomial of n,L by POLYNOM1:def_10; Q . (len Q) = u by A11, A12, FINSEQ_1:17; then consider A9 being LeftLinearCombination of P such that A31: f9 = u9 + (Sum A9) and A32: for i being Element of NAT st i in dom A9 holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A9 . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) by A6, A7, A11, A12, A23, FINSEQ_1:17; A33: k in dom Q by A5, A13, FINSEQ_1:3; now__::_thesis:_(_(_u9_=_g9_&_ex_A_being_LeftLinearCombination_of_P_st_ (_f9_=_g9_+_(Sum_A)_&_(_for_i_being_Element_of_NAT_st_i_in_dom_A_holds_ ex_m_being_non-zero_Monomial_of_n,L_ex_p_being_non-zero_Polynomial_of_n,L_st_ (_p_in_P_&_A_._i_=_m_*'_p_&_HT_((m_*'_p),T)_<=_HT_(f,T),T_)_)_)_)_or_(_u9_<>_g9_&_ex_B,_A_being_LeftLinearCombination_of_P_st_ (_f9_=_g9_+_(Sum_A)_&_(_for_i_being_Element_of_NAT_st_i_in_dom_A_holds_ ex_m_being_non-zero_Monomial_of_n,L_ex_p_being_non-zero_Polynomial_of_n,L_st_ (_p_in_P_&_A_._i_=_m_*'_p_&_HT_((m_*'_p),T)_<=_HT_(f,T),T_)_)_)_)_) percases ( u9 = g9 or u9 <> g9 ) ; case u9 = g9 ; ::_thesis: ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) hence ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) by A31, A32; ::_thesis: verum end; caseA34: u9 <> g9 ; ::_thesis: ex B, A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) reconsider hh = h9 as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10; A35: PolyRedRel (P,T) reduces f,h9 by A5, A33, A30, A22, A23, REWRITE1:17; A36: R . (k + 1) = g2 by A26, XTUPLE_0:1; then reconsider gg = g2 as Element of (Polynom-Ring (n,L)) by A8, A10, A11; h9 reduces_to g2,P,T by A25, A26, POLYRED:def_13; then consider p9 being Polynomial of n,L such that A37: p9 in P and A38: h9 reduces_to g2,p9,T by POLYRED:def_7; consider m9 being Monomial of n,L such that A39: g2 = h9 - (m9 *' p9) and not HT ((m9 *' p9),T) in Support g2 and A40: HT ((m9 *' p9),T) <= HT (h9,T),T by A38, GROEB_1:2; A41: now__::_thesis:_not_p9_=_0__(n,L) assume p9 = 0_ (n,L) ; ::_thesis: contradiction then g2 = h9 - (0_ (n,L)) by A39, POLYRED:5 .= Q . k by A30, POLYRED:4 ; hence contradiction by A8, A10, A11, A34, A36; ::_thesis: verum end; A42: now__::_thesis:_not_m9_=_0__(n,L) assume m9 = 0_ (n,L) ; ::_thesis: contradiction then g2 = h9 - (0_ (n,L)) by A39, POLYRED:5 .= Q . k by A30, POLYRED:4 ; hence contradiction by A8, A10, A11, A34, A36; ::_thesis: verum end; reconsider mp = m9 *' p9 as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10; reconsider pp = p9 as Element of P by A37; set B = A9 ^ <*mp*>; reconsider mm = m9 as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10; A43: gg = hh - mp by A39, Lm3; reconsider m9 = m9 as non-zero Monomial of n,L by A42, POLYNOM7:def_1; reconsider p9 = p9 as non-zero Polynomial of n,L by A41, POLYNOM7:def_1; len (A9 ^ <*mp*>) = (len A9) + (len <*(m9 *' p9)*>) by FINSEQ_1:22 .= (len A9) + 1 by FINSEQ_1:40 ; then A44: dom (A9 ^ <*mp*>) = Seg ((len A9) + 1) by FINSEQ_1:def_3; A45: mp = mm * pp by POLYNOM1:def_10; now__::_thesis:_for_i_being_set_st_i_in_dom_(A9_^_<*mp*>)_holds_ ex_u_being_Element_of_(Polynom-Ring_(n,L))_ex_a_being_Element_of_P_st_(A9_^_<*mp*>)_/._i_=_u_*_a let i be set ; ::_thesis: ( i in dom (A9 ^ <*mp*>) implies ex u being Element of (Polynom-Ring (n,L)) ex a being Element of P st (A9 ^ <*mp*>) /. i = u * a ) assume A46: i in dom (A9 ^ <*mp*>) ; ::_thesis: ex u being Element of (Polynom-Ring (n,L)) ex a being Element of P st (A9 ^ <*mp*>) /. i = u * a then reconsider j = i as Element of NAT ; A47: j <= (len A9) + 1 by A44, A46, FINSEQ_1:1; A48: 1 <= j by A44, A46, FINSEQ_1:1; now__::_thesis:_(_(_j_=_(len_A9)_+_1_&_ex_u_being_Element_of_(Polynom-Ring_(n,L))_ex_a_being_Element_of_P_st_(A9_^_<*mp*>)_/._i_=_u_*_a_)_or_(_j_<>_(len_A9)_+_1_&_ex_u_being_Element_of_(Polynom-Ring_(n,L))_ex_a_being_Element_of_P_st_(A9_^_<*mp*>)_/._i_=_u_*_a_)_) percases ( j = (len A9) + 1 or j <> (len A9) + 1 ) ; case j = (len A9) + 1 ; ::_thesis: ex u being Element of (Polynom-Ring (n,L)) ex a being Element of P st (A9 ^ <*mp*>) /. i = u * a then mp = (A9 ^ <*mp*>) . j by FINSEQ_1:42 .= (A9 ^ <*mp*>) /. j by A46, PARTFUN1:def_6 ; hence ex u being Element of (Polynom-Ring (n,L)) ex a being Element of P st (A9 ^ <*mp*>) /. i = u * a by A45; ::_thesis: verum end; case j <> (len A9) + 1 ; ::_thesis: ex u being Element of (Polynom-Ring (n,L)) ex a being Element of P st (A9 ^ <*mp*>) /. i = u * a then j < (len A9) + 1 by A47, XXREAL_0:1; then j <= len A9 by NAT_1:13; then j in Seg (len A9) by A48, FINSEQ_1:1; then A49: j in dom A9 by FINSEQ_1:def_3; then consider m being non-zero Monomial of n,L, p being non-zero Polynomial of n,L such that A50: p in P and A51: A9 . j = m *' p and HT ((m *' p),T) <= HT (f,T),T by A32; reconsider a9 = p as Element of P by A50; reconsider u9 = m as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10; A52: (A9 ^ <*mp*>) . j = (A9 ^ <*mp*>) /. j by A46, PARTFUN1:def_6; u9 * a9 = m *' p by POLYNOM1:def_10 .= (A9 ^ <*mp*>) /. j by A49, A51, A52, FINSEQ_1:def_7 ; hence ex u being Element of (Polynom-Ring (n,L)) ex a being Element of P st (A9 ^ <*mp*>) /. i = u * a ; ::_thesis: verum end; end; end; hence ex u being Element of (Polynom-Ring (n,L)) ex a being Element of P st (A9 ^ <*mp*>) /. i = u * a ; ::_thesis: verum end; then reconsider B = A9 ^ <*mp*> as LeftLinearCombination of P by IDEAL_1:def_9; h9 is_reducible_wrt p9,T by A38, POLYRED:def_8; then h9 <> 0_ (n,L) by POLYRED:37; then HT (h9,T) <= HT (f,T),T by A35, POLYRED:44; then A53: HT ((m9 *' p9),T) <= HT (f,T),T by A40, TERMORD:8; A54: now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_B_holds_ ex_m_being_non-zero_Monomial_of_n,L_ex_p_being_non-zero_Polynomial_of_n,L_st_ (_p_in_P_&_B_._i_=_m_*'_p_&_HT_((m_*'_p),T)_<=_HT_(f,T),T_) let i be Element of NAT ; ::_thesis: ( i in dom B implies ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & B . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) assume A55: i in dom B ; ::_thesis: ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & B . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) then A56: i <= (len A9) + 1 by A44, FINSEQ_1:1; A57: 1 <= i by A44, A55, FINSEQ_1:1; now__::_thesis:_(_(_i_=_(len_A9)_+_1_&_ex_m_being_non-zero_Monomial_of_n,L_ex_p_being_non-zero_Polynomial_of_n,L_st_ (_p_in_P_&_B_._i_=_m_*'_p_&_HT_((m_*'_p),T)_<=_HT_(f,T),T_)_)_or_(_i_<>_(len_A9)_+_1_&_ex_m_being_non-zero_Monomial_of_n,L_ex_p_being_non-zero_Polynomial_of_n,L_st_ (_p_in_P_&_B_._i_=_m_*'_p_&_HT_((m_*'_p),T)_<=_HT_(f,T),T_)_)_) percases ( i = (len A9) + 1 or i <> (len A9) + 1 ) ; case i = (len A9) + 1 ; ::_thesis: ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & B . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) hence ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & B . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) by A37, A53, FINSEQ_1:42; ::_thesis: verum end; case i <> (len A9) + 1 ; ::_thesis: ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & B . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) then i < (len A9) + 1 by A56, XXREAL_0:1; then i <= len A9 by NAT_1:13; then i in Seg (len A9) by A57, FINSEQ_1:1; then A58: i in dom A9 by FINSEQ_1:def_3; then consider m being non-zero Monomial of n,L, p being non-zero Polynomial of n,L such that A59: p in P and A60: A9 . i = m *' p and A61: HT ((m *' p),T) <= HT (f,T),T by A32; B . i = m *' p by A58, A60, FINSEQ_1:def_7; hence ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & B . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) by A59, A61; ::_thesis: verum end; end; end; hence ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & B . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ; ::_thesis: verum end; take B = B; ::_thesis: ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) gg + (Sum B) = gg + ((Sum A9) + (Sum <*mp*>)) by RLVECT_1:41 .= gg + ((Sum A9) + mp) by RLVECT_1:44 .= (hh + (- mp)) + ((Sum A9) + mp) by A43, RLVECT_1:def_11 .= hh + ((- mp) + ((Sum A9) + mp)) by RLVECT_1:def_3 .= hh + ((Sum A9) + ((- mp) + mp)) by RLVECT_1:def_3 .= hh + ((Sum A9) + (0. (Polynom-Ring (n,L)))) by RLVECT_1:5 .= hh + (Sum A9) by RLVECT_1:4 .= f9 by A5, A13, A29, A31, FINSEQ_1:3, FUNCT_1:47 ; hence ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) by A8, A10, A11, A36, A54; ::_thesis: verum end; end; end; hence ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) ; ::_thesis: verum end; hence S1[k + 1] ; ::_thesis: verum end; end; A62: S1[1] proof set A = <*> the carrier of (Polynom-Ring (n,L)); let f, g be Polynomial of n,L; ::_thesis: for f9, g9 being Element of (Polynom-Ring (n,L)) st f = f9 & g = g9 holds for P being non empty Subset of (Polynom-Ring (n,L)) for R being RedSequence of PolyRedRel (P,T) st R . 1 = f & R . (len R) = g & len R = 1 holds ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) let f9, g9 be Element of (Polynom-Ring (n,L)); ::_thesis: ( f = f9 & g = g9 implies for P being non empty Subset of (Polynom-Ring (n,L)) for R being RedSequence of PolyRedRel (P,T) st R . 1 = f & R . (len R) = g & len R = 1 holds ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) ) assume A63: ( f = f9 & g = g9 ) ; ::_thesis: for P being non empty Subset of (Polynom-Ring (n,L)) for R being RedSequence of PolyRedRel (P,T) st R . 1 = f & R . (len R) = g & len R = 1 holds ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: for R being RedSequence of PolyRedRel (P,T) st R . 1 = f & R . (len R) = g & len R = 1 holds ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) let R be RedSequence of PolyRedRel (P,T); ::_thesis: ( R . 1 = f & R . (len R) = g & len R = 1 implies ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) ) for i being set st i in dom (<*> the carrier of (Polynom-Ring (n,L))) holds ex u being Element of (Polynom-Ring (n,L)) ex a being Element of P st (<*> the carrier of (Polynom-Ring (n,L))) /. i = u * a ; then reconsider A = <*> the carrier of (Polynom-Ring (n,L)) as LeftLinearCombination of P by IDEAL_1:def_9; assume A64: ( R . 1 = f & R . (len R) = g & len R = 1 ) ; ::_thesis: ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) take A ; ::_thesis: ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) f9 = g9 + (0. (Polynom-Ring (n,L))) by A63, A64, RLVECT_1:def_4 .= g9 + (Sum A) by RLVECT_1:43 ; hence ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) ; ::_thesis: verum end; A65: for k being Nat st 1 <= k holds S1[k] from NAT_1:sch_8(A62, A4); 1 <= len R by NAT_1:14; hence ex A being LeftLinearCombination of P st ( f9 = g9 + (Sum A) & ( for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) ) by A1, A65, A2; ::_thesis: verum end; theorem :: GROEB_2:33 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 f, g being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A, B being LeftLinearCombination of P for b being bag of n st A is_Standard_Representation_of f,P,b,T & B is_Standard_Representation_of g,P,b,T holds A ^ B is_Standard_Representation_of f + g,P,b,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 f, g being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A, B being LeftLinearCombination of P for b being bag of n st A is_Standard_Representation_of f,P,b,T & B is_Standard_Representation_of g,P,b,T holds A ^ B is_Standard_Representation_of f + g,P,b,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 f, g being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A, B being LeftLinearCombination of P for b being bag of n st A is_Standard_Representation_of f,P,b,T & B is_Standard_Representation_of g,P,b,T holds A ^ B is_Standard_Representation_of f + g,P,b,T let L be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, g being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A, B being LeftLinearCombination of P for b being bag of n st A is_Standard_Representation_of f,P,b,T & B is_Standard_Representation_of g,P,b,T holds A ^ B is_Standard_Representation_of f + g,P,b,T let f, g be Polynomial of n,L; ::_thesis: for P being non empty Subset of (Polynom-Ring (n,L)) for A, B being LeftLinearCombination of P for b being bag of n st A is_Standard_Representation_of f,P,b,T & B is_Standard_Representation_of g,P,b,T holds A ^ B is_Standard_Representation_of f + g,P,b,T let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: for A, B being LeftLinearCombination of P for b being bag of n st A is_Standard_Representation_of f,P,b,T & B is_Standard_Representation_of g,P,b,T holds A ^ B is_Standard_Representation_of f + g,P,b,T let A, B be LeftLinearCombination of P; ::_thesis: for b being bag of n st A is_Standard_Representation_of f,P,b,T & B is_Standard_Representation_of g,P,b,T holds A ^ B is_Standard_Representation_of f + g,P,b,T let b be bag of n; ::_thesis: ( A is_Standard_Representation_of f,P,b,T & B is_Standard_Representation_of g,P,b,T implies A ^ B is_Standard_Representation_of f + g,P,b,T ) assume that A1: A is_Standard_Representation_of f,P,b,T and A2: B is_Standard_Representation_of g,P,b,T ; ::_thesis: A ^ B is_Standard_Representation_of f + g,P,b,T A3: now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_(A_^_B)_holds_ ex_m_being_non-zero_Monomial_of_n,L_ex_p_being_non-zero_Polynomial_of_n,L_st_ (_p_in_P_&_(A_^_B)_/._i_=_m_*'_p_&_HT_((m_*'_p),T)_<=_b,T_) let i be Element of NAT ; ::_thesis: ( i in dom (A ^ B) implies ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & (A ^ B) /. i = m *' p & HT ((m *' p),T) <= b,T ) ) assume A4: i in dom (A ^ B) ; ::_thesis: ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & (A ^ B) /. i = m *' p & HT ((m *' p),T) <= b,T ) now__::_thesis:_(_(_i_in_dom_A_&_ex_m_being_non-zero_Monomial_of_n,L_ex_p_being_non-zero_Polynomial_of_n,L_st_ (_p_in_P_&_(A_^_B)_/._i_=_m_*'_p_&_HT_((m_*'_p),T)_<=_b,T_)_)_or_(_ex_k_being_Nat_st_ (_k_in_dom_B_&_i_=_(len_A)_+_k_)_&_ex_m_being_non-zero_Monomial_of_n,L_ex_p_being_non-zero_Polynomial_of_n,L_st_ (_p_in_P_&_(A_^_B)_/._i_=_m_*'_p_&_HT_((m_*'_p),T)_<=_b,T_)_)_) percases ( i in dom A or ex k being Nat st ( k in dom B & i = (len A) + k ) ) by A4, FINSEQ_1:25; caseA5: i in dom A ; ::_thesis: ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & (A ^ B) /. i = m *' p & HT ((m *' p),T) <= b,T ) (A ^ B) /. i = (A ^ B) . i by A4, PARTFUN1:def_6 .= A . i by A5, FINSEQ_1:def_7 .= A /. i by A5, PARTFUN1:def_6 ; hence ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & (A ^ B) /. i = m *' p & HT ((m *' p),T) <= b,T ) by A1, A5, Def7; ::_thesis: verum end; case ex k being Nat st ( k in dom B & i = (len A) + k ) ; ::_thesis: ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & (A ^ B) /. i = m *' p & HT ((m *' p),T) <= b,T ) then consider k being Nat such that A6: k in dom B and A7: i = (len A) + k ; (A ^ B) /. i = (A ^ B) . i by A4, PARTFUN1:def_6 .= B . k by A6, A7, FINSEQ_1:def_7 .= B /. k by A6, PARTFUN1:def_6 ; hence ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & (A ^ B) /. i = m *' p & HT ((m *' p),T) <= b,T ) by A2, A6, Def7; ::_thesis: verum end; end; end; hence ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & (A ^ B) /. i = m *' p & HT ((m *' p),T) <= b,T ) ; ::_thesis: verum end; ( f = Sum A & g = Sum B ) by A1, A2, Def7; then f + g = (Sum A) + (Sum B) by POLYNOM1:def_10 .= Sum (A ^ B) by RLVECT_1:41 ; hence A ^ B is_Standard_Representation_of f + g,P,b,T by A3, Def7; ::_thesis: verum end; theorem :: GROEB_2:34 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 f, g being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A, B being LeftLinearCombination of P for b being bag of n for i being Element of NAT st A is_Standard_Representation_of f,P,b,T & B = A | i & g = Sum (A /^ i) holds B is_Standard_Representation_of f - g,P,b,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 f, g being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A, B being LeftLinearCombination of P for b being bag of n for i being Element of NAT st A is_Standard_Representation_of f,P,b,T & B = A | i & g = Sum (A /^ i) holds B is_Standard_Representation_of f - g,P,b,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 f, g being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A, B being LeftLinearCombination of P for b being bag of n for i being Element of NAT st A is_Standard_Representation_of f,P,b,T & B = A | i & g = Sum (A /^ i) holds B is_Standard_Representation_of f - g,P,b,T let L be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, g being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A, B being LeftLinearCombination of P for b being bag of n for i being Element of NAT st A is_Standard_Representation_of f,P,b,T & B = A | i & g = Sum (A /^ i) holds B is_Standard_Representation_of f - g,P,b,T let f, g be Polynomial of n,L; ::_thesis: for P being non empty Subset of (Polynom-Ring (n,L)) for A, B being LeftLinearCombination of P for b being bag of n for i being Element of NAT st A is_Standard_Representation_of f,P,b,T & B = A | i & g = Sum (A /^ i) holds B is_Standard_Representation_of f - g,P,b,T let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: for A, B being LeftLinearCombination of P for b being bag of n for i being Element of NAT st A is_Standard_Representation_of f,P,b,T & B = A | i & g = Sum (A /^ i) holds B is_Standard_Representation_of f - g,P,b,T let A, B be LeftLinearCombination of P; ::_thesis: for b being bag of n for i being Element of NAT st A is_Standard_Representation_of f,P,b,T & B = A | i & g = Sum (A /^ i) holds B is_Standard_Representation_of f - g,P,b,T let b be bag of n; ::_thesis: for i being Element of NAT st A is_Standard_Representation_of f,P,b,T & B = A | i & g = Sum (A /^ i) holds B is_Standard_Representation_of f - g,P,b,T let i be Element of NAT ; ::_thesis: ( A is_Standard_Representation_of f,P,b,T & B = A | i & g = Sum (A /^ i) implies B is_Standard_Representation_of f - g,P,b,T ) assume that A1: A is_Standard_Representation_of f,P,b,T and A2: B = A | i and A3: g = Sum (A /^ i) ; ::_thesis: B is_Standard_Representation_of f - g,P,b,T A4: Sum A = f by A1, Def7; dom (A | (Seg i)) c= dom A by RELAT_1:60; then A5: dom B c= dom A by A2, FINSEQ_1:def_15; A6: now__::_thesis:_for_j_being_Element_of_NAT_st_j_in_dom_B_holds_ ex_m_being_non-zero_Monomial_of_n,L_ex_p_being_non-zero_Polynomial_of_n,L_st_ (_p_in_P_&_B_/._j_=_m_*'_p_&_HT_((m_*'_p),T)_<=_b,T_) let j be Element of NAT ; ::_thesis: ( j in dom B implies ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & B /. j = m *' p & HT ((m *' p),T) <= b,T ) ) assume A7: j in dom B ; ::_thesis: ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & B /. j = m *' p & HT ((m *' p),T) <= b,T ) then A8: j in dom (A | (Seg i)) by A2, FINSEQ_1:def_15; A /. j = A . j by A5, A7, PARTFUN1:def_6 .= (A | (Seg i)) . j by A8, FUNCT_1:47 .= B . j by A2, FINSEQ_1:def_15 .= B /. j by A7, PARTFUN1:def_6 ; hence ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & B /. j = m *' p & HT ((m *' p),T) <= b,T ) by A1, A5, A7, Def7; ::_thesis: verum end; A = B ^ (A /^ i) by A2, RFINSEQ:8; then Sum A = (Sum B) + (Sum (A /^ i)) by RLVECT_1:41; then (Sum A) + (- (Sum (A /^ i))) = (Sum B) + ((Sum (A /^ i)) + (- (Sum (A /^ i)))) by RLVECT_1:def_3 .= (Sum B) + (0. (Polynom-Ring (n,L))) by RLVECT_1:5 .= Sum B by RLVECT_1:def_4 ; then Sum B = (Sum A) - (Sum (A /^ i)) by RLVECT_1:def_11 .= f - g by A3, A4, Lm3 ; hence B is_Standard_Representation_of f - g,P,b,T by A6, Def7; ::_thesis: verum end; theorem :: GROEB_2:35 for n being Ordinal for T being connected TermOrder of n for L being non empty non trivial right_complementable well-unital distributive Abelian add-associative right_zeroed doubleLoopStr for f, g being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A, B being LeftLinearCombination of P for b being bag of n for i being Element of NAT st A is_Standard_Representation_of f,P,b,T & B = A /^ i & g = Sum (A | i) & i <= len A holds B is_Standard_Representation_of f - g,P,b,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 Abelian add-associative right_zeroed doubleLoopStr for f, g being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A, B being LeftLinearCombination of P for b being bag of n for i being Element of NAT st A is_Standard_Representation_of f,P,b,T & B = A /^ i & g = Sum (A | i) & i <= len A holds B is_Standard_Representation_of f - g,P,b,T let T be connected TermOrder of n; ::_thesis: for L being non empty non trivial right_complementable well-unital distributive Abelian add-associative right_zeroed doubleLoopStr for f, g being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A, B being LeftLinearCombination of P for b being bag of n for i being Element of NAT st A is_Standard_Representation_of f,P,b,T & B = A /^ i & g = Sum (A | i) & i <= len A holds B is_Standard_Representation_of f - g,P,b,T let L be non empty non trivial right_complementable well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, g being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A, B being LeftLinearCombination of P for b being bag of n for i being Element of NAT st A is_Standard_Representation_of f,P,b,T & B = A /^ i & g = Sum (A | i) & i <= len A holds B is_Standard_Representation_of f - g,P,b,T let f, g be Polynomial of n,L; ::_thesis: for P being non empty Subset of (Polynom-Ring (n,L)) for A, B being LeftLinearCombination of P for b being bag of n for i being Element of NAT st A is_Standard_Representation_of f,P,b,T & B = A /^ i & g = Sum (A | i) & i <= len A holds B is_Standard_Representation_of f - g,P,b,T let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: for A, B being LeftLinearCombination of P for b being bag of n for i being Element of NAT st A is_Standard_Representation_of f,P,b,T & B = A /^ i & g = Sum (A | i) & i <= len A holds B is_Standard_Representation_of f - g,P,b,T let A, B be LeftLinearCombination of P; ::_thesis: for b being bag of n for i being Element of NAT st A is_Standard_Representation_of f,P,b,T & B = A /^ i & g = Sum (A | i) & i <= len A holds B is_Standard_Representation_of f - g,P,b,T let b be bag of n; ::_thesis: for i being Element of NAT st A is_Standard_Representation_of f,P,b,T & B = A /^ i & g = Sum (A | i) & i <= len A holds B is_Standard_Representation_of f - g,P,b,T let i be Element of NAT ; ::_thesis: ( A is_Standard_Representation_of f,P,b,T & B = A /^ i & g = Sum (A | i) & i <= len A implies B is_Standard_Representation_of f - g,P,b,T ) assume that A1: A is_Standard_Representation_of f,P,b,T and A2: B = A /^ i and A3: g = Sum (A | i) and A4: i <= len A ; ::_thesis: B is_Standard_Representation_of f - g,P,b,T A5: Sum A = f by A1, Def7; A6: now__::_thesis:_for_j_being_Element_of_NAT_st_j_in_dom_B_holds_ ex_m_being_non-zero_Monomial_of_n,L_ex_p_being_non-zero_Polynomial_of_n,L_st_ (_p_in_P_&_B_/._j_=_m_*'_p_&_HT_((m_*'_p),T)_<=_b,T_) let j be Element of NAT ; ::_thesis: ( j in dom B implies ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & B /. j = m *' p & HT ((m *' p),T) <= b,T ) ) assume A7: j in dom B ; ::_thesis: ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & B /. j = m *' p & HT ((m *' p),T) <= b,T ) then A8: j + i in dom A by A2, FINSEQ_5:26; B /. j = B . j by A7, PARTFUN1:def_6 .= A . (j + i) by A2, A4, A7, RFINSEQ:def_1 .= A /. (j + i) by A8, PARTFUN1:def_6 ; hence ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & B /. j = m *' p & HT ((m *' p),T) <= b,T ) by A1, A8, Def7; ::_thesis: verum end; A = (A | i) ^ B by A2, RFINSEQ:8; then Sum A = (Sum (A | i)) + (Sum B) by RLVECT_1:41; then (Sum A) + (- (Sum (A | i))) = ((Sum (A | i)) + (- (Sum (A | i)))) + (Sum B) by RLVECT_1:def_3 .= (0. (Polynom-Ring (n,L))) + (Sum B) by RLVECT_1:5 .= Sum B by ALGSTR_1:def_2 ; then Sum B = (Sum A) - (Sum (A | i)) by RLVECT_1:def_11 .= f - g by A3, A5, Lm3 ; hence B is_Standard_Representation_of f - g,P,b,T by A6, Def7; ::_thesis: verum end; theorem Th36: :: GROEB_2:36 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 f being non-zero Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds ex i being Element of NAT ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( i in dom A & p in P & A . i = m *' p & HT (f,T) <= HT ((m *' p),T),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 f being non-zero Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds ex i being Element of NAT ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( i in dom A & p in P & A . i = m *' p & HT (f,T) <= HT ((m *' p),T),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 f being non-zero Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds ex i being Element of NAT ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( i in dom A & p in P & A . i = m *' p & HT (f,T) <= HT ((m *' p),T),T ) let L be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for f being non-zero Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds ex i being Element of NAT ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( i in dom A & p in P & A . i = m *' p & HT (f,T) <= HT ((m *' p),T),T ) let f be non-zero Polynomial of n,L; ::_thesis: for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds ex i being Element of NAT ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( i in dom A & p in P & A . i = m *' p & HT (f,T) <= HT ((m *' p),T),T ) let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: for A being LeftLinearCombination of P st A is_MonomialRepresentation_of f holds ex i being Element of NAT ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( i in dom A & p in P & A . i = m *' p & HT (f,T) <= HT ((m *' p),T),T ) let A be LeftLinearCombination of P; ::_thesis: ( A is_MonomialRepresentation_of f implies ex i being Element of NAT ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( i in dom A & p in P & A . i = m *' p & HT (f,T) <= HT ((m *' p),T),T ) ) HC (f,T) <> 0. L ; then A1: f . (HT (f,T)) <> 0. L by TERMORD:def_7; assume A is_MonomialRepresentation_of f ; ::_thesis: ex i being Element of NAT ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( i in dom A & p in P & A . i = m *' p & HT (f,T) <= HT ((m *' p),T),T ) then consider i being Element of NAT such that A2: i in dom A and A3: ex m being Monomial of n,L ex p being Polynomial of n,L st ( A . i = m *' p & p in P & (m *' p) . (HT (f,T)) <> 0. L ) by A1, Lm4; consider m being Monomial of n,L, p being Polynomial of n,L such that A4: A . i = m *' p and A5: (m *' p) . (HT (f,T)) <> 0. L and A6: p in P by A3; A7: m *' p <> 0_ (n,L) by A5, POLYNOM1:22; then A8: m <> 0_ (n,L) by POLYRED:5; p <> 0_ (n,L) by A7, POLYNOM1:28; then reconsider p = p as non-zero Polynomial of n,L by POLYNOM7:def_1; reconsider m = m as non-zero Monomial of n,L by A8, POLYNOM7:def_1; HT (f,T) in Support (m *' p) by A5, POLYNOM1:def_3; then HT (f,T) <= HT ((m *' p),T),T by TERMORD:def_6; hence ex i being Element of NAT ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( i in dom A & p in P & A . i = m *' p & HT (f,T) <= HT ((m *' p),T),T ) by A2, A4, A6; ::_thesis: verum end; theorem Th37: :: GROEB_2:37 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 f being non-zero Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P st A is_Standard_Representation_of f,P,T holds ex i being Element of NAT ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & i in dom A & A /. i = m *' p & HT (f,T) = HT ((m *' 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 f being non-zero Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P st A is_Standard_Representation_of f,P,T holds ex i being Element of NAT ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & i in dom A & A /. i = m *' p & HT (f,T) = HT ((m *' 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 f being non-zero Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P st A is_Standard_Representation_of f,P,T holds ex i being Element of NAT ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & i in dom A & A /. i = m *' p & HT (f,T) = HT ((m *' p),T) ) let L be non empty non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for f being non-zero Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P st A is_Standard_Representation_of f,P,T holds ex i being Element of NAT ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & i in dom A & A /. i = m *' p & HT (f,T) = HT ((m *' p),T) ) let f be non-zero Polynomial of n,L; ::_thesis: for P being non empty Subset of (Polynom-Ring (n,L)) for A being LeftLinearCombination of P st A is_Standard_Representation_of f,P,T holds ex i being Element of NAT ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & i in dom A & A /. i = m *' p & HT (f,T) = HT ((m *' p),T) ) let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: for A being LeftLinearCombination of P st A is_Standard_Representation_of f,P,T holds ex i being Element of NAT ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & i in dom A & A /. i = m *' p & HT (f,T) = HT ((m *' p),T) ) let A be LeftLinearCombination of P; ::_thesis: ( A is_Standard_Representation_of f,P,T implies ex i being Element of NAT ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & i in dom A & A /. i = m *' p & HT (f,T) = HT ((m *' p),T) ) ) assume A is_Standard_Representation_of f,P,T ; ::_thesis: ex i being Element of NAT ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & i in dom A & A /. i = m *' p & HT (f,T) = HT ((m *' p),T) ) then A1: A is_Standard_Representation_of f,P, HT (f,T),T by Def8; then consider i being Element of NAT , m being non-zero Monomial of n,L, p being non-zero Polynomial of n,L such that A2: i in dom A and p in P and A3: A . i = m *' p and A4: HT (f,T) <= HT ((m *' p),T),T by Th32, Th36; consider m9 being non-zero Monomial of n,L, p9 being non-zero Polynomial of n,L such that A5: p9 in P and A6: A /. i = m9 *' p9 and A7: HT ((m9 *' p9),T) <= HT (f,T),T by A1, A2, Def7; take i ; ::_thesis: ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & i in dom A & A /. i = m *' p & HT (f,T) = HT ((m *' p),T) ) take m9 ; ::_thesis: ex p being non-zero Polynomial of n,L st ( p in P & i in dom A & A /. i = m9 *' p & HT (f,T) = HT ((m9 *' p),T) ) take p9 ; ::_thesis: ( p9 in P & i in dom A & A /. i = m9 *' p9 & HT (f,T) = HT ((m9 *' p9),T) ) m *' p = m9 *' p9 by A2, A3, A6, PARTFUN1:def_6; hence ( p9 in P & i in dom A & A /. i = m9 *' p9 & HT (f,T) = HT ((m9 *' p9),T) ) by A2, A4, A5, A6, A7, TERMORD:7; ::_thesis: verum end; theorem Th38: :: GROEB_2:38 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 being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f, 0_ (n,L) holds f has_a_Standard_Representation_of P,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 being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f, 0_ (n,L) holds f has_a_Standard_Representation_of 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 f being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f, 0_ (n,L) holds f has_a_Standard_Representation_of 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 f being Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f, 0_ (n,L) holds f has_a_Standard_Representation_of P,T let f be Polynomial of n,L; ::_thesis: for P being non empty Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f, 0_ (n,L) holds f has_a_Standard_Representation_of P,T let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: ( PolyRedRel (P,T) reduces f, 0_ (n,L) implies f has_a_Standard_Representation_of P,T ) reconsider f9 = f as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10; A1: 0_ (n,L) = 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10; assume PolyRedRel (P,T) reduces f, 0_ (n,L) ; ::_thesis: f has_a_Standard_Representation_of P,T then consider A being LeftLinearCombination of P such that A2: f9 = (0. (Polynom-Ring (n,L))) + (Sum A) and A3: for i being Element of NAT st i in dom A holds ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) by A1, Lm5; A4: now__::_thesis:_for_i_being_Element_of_NAT_st_i_in_dom_A_holds_ ex_m_being_non-zero_Monomial_of_n,L_ex_p_being_non-zero_Polynomial_of_n,L_st_ (_p_in_P_&_A_/._i_=_m_*'_p_&_HT_((m_*'_p),T)_<=_HT_(f,T),T_) let i be Element of NAT ; ::_thesis: ( i in dom A implies ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A /. i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) ) assume A5: i in dom A ; ::_thesis: ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A /. i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) then ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A . i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) by A3; hence ex m being non-zero Monomial of n,L ex p being non-zero Polynomial of n,L st ( p in P & A /. i = m *' p & HT ((m *' p),T) <= HT (f,T),T ) by A5, PARTFUN1:def_6; ::_thesis: verum end; f = Sum A by A2, RLVECT_1:def_4; then A is_Standard_Representation_of f,P, HT (f,T),T by A4, Def7; then A is_Standard_Representation_of f,P,T by Def8; hence f has_a_Standard_Representation_of P,T by Def10; ::_thesis: verum end; theorem Th39: :: GROEB_2:39 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 f being non-zero Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) st f has_a_Standard_Representation_of P,T holds f is_top_reducible_wrt 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 f being non-zero Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) st f has_a_Standard_Representation_of P,T holds f is_top_reducible_wrt 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 f being non-zero Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) st f has_a_Standard_Representation_of P,T holds f is_top_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 f being non-zero Polynomial of n,L for P being non empty Subset of (Polynom-Ring (n,L)) st f has_a_Standard_Representation_of P,T holds f is_top_reducible_wrt P,T let f be non-zero Polynomial of n,L; ::_thesis: for P being non empty Subset of (Polynom-Ring (n,L)) st f has_a_Standard_Representation_of P,T holds f is_top_reducible_wrt P,T let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: ( f has_a_Standard_Representation_of P,T implies f is_top_reducible_wrt P,T ) assume f has_a_Standard_Representation_of P,T ; ::_thesis: f is_top_reducible_wrt P,T then consider A being LeftLinearCombination of P such that A1: A is_Standard_Representation_of f,P,T by Def10; consider i being Element of NAT , m being non-zero Monomial of n,L, p being non-zero Polynomial of n,L such that A2: p in P and i in dom A and A /. i = m *' p and A3: HT (f,T) = HT ((m *' p),T) by A1, Th37; set s = HT (m,T); A4: HT (f,T) = (HT (m,T)) + (HT (p,T)) by A3, TERMORD:31; set g = f - (((f . (HT (f,T))) / (HC (p,T))) * ((HT (m,T)) *' p)); A5: f <> 0_ (n,L) by POLYNOM7:def_1; then Support f <> {} by POLYNOM7:1; then ( p <> 0_ (n,L) & HT (f,T) in Support f ) by POLYNOM7:def_1, TERMORD:def_6; then f reduces_to f - (((f . (HT (f,T))) / (HC (p,T))) * ((HT (m,T)) *' p)),p, HT (f,T),T by A5, A4, POLYRED:def_5; then f top_reduces_to f - (((f . (HT (f,T))) / (HC (p,T))) * ((HT (m,T)) *' p)),p,T by POLYRED:def_10; then f is_top_reducible_wrt p,T by POLYRED:def_11; hence f is_top_reducible_wrt P,T by A2, POLYRED:def_12; ::_thesis: verum end; theorem :: GROEB_2:40 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 G being non empty Subset of (Polynom-Ring (n,L)) holds ( G is_Groebner_basis_wrt T iff for f being non-zero Polynomial of n,L st f in G -Ideal holds f has_a_Standard_Representation_of 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 G being non empty Subset of (Polynom-Ring (n,L)) holds ( G is_Groebner_basis_wrt T iff for f being non-zero Polynomial of n,L st f in G -Ideal holds f has_a_Standard_Representation_of 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 G being non empty Subset of (Polynom-Ring (n,L)) holds ( G is_Groebner_basis_wrt T iff for f being non-zero Polynomial of n,L st f in G -Ideal holds f has_a_Standard_Representation_of 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 G being non empty Subset of (Polynom-Ring (n,L)) holds ( G is_Groebner_basis_wrt T iff for f being non-zero Polynomial of n,L st f in G -Ideal holds f has_a_Standard_Representation_of G,T ) let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: ( P is_Groebner_basis_wrt T iff for f being non-zero Polynomial of n,L st f in P -Ideal holds f has_a_Standard_Representation_of P,T ) A1: now__::_thesis:_(_(_for_f_being_non-zero_Polynomial_of_n,L_st_f_in_P_-Ideal_holds_ f_has_a_Standard_Representation_of_P,T_)_implies_P_is_Groebner_basis_wrt_T_) assume for f being non-zero Polynomial of n,L st f in P -Ideal holds f has_a_Standard_Representation_of P,T ; ::_thesis: P is_Groebner_basis_wrt T then for f being non-zero Polynomial of n,L st f in P -Ideal holds f is_top_reducible_wrt P,T by Th39; then 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 ) by GROEB_1:18; then HT ((P -Ideal),T) c= multiples (HT (P,T)) by GROEB_1:19; then PolyRedRel (P,T) is locally-confluent by GROEB_1:20; hence P is_Groebner_basis_wrt T by GROEB_1:def_3; ::_thesis: verum end; A2: 0_ (n,L) = 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10; now__::_thesis:_(_P_is_Groebner_basis_wrt_T_implies_for_f_being_non-zero_Polynomial_of_n,L_st_f_in_P_-Ideal_holds_ f_has_a_Standard_Representation_of_P,T_) assume P is_Groebner_basis_wrt T ; ::_thesis: for f being non-zero Polynomial of n,L st f in P -Ideal holds f has_a_Standard_Representation_of P,T then PolyRedRel (P,T) is locally-confluent by GROEB_1:def_3; hence for f being non-zero Polynomial of n,L st f in P -Ideal holds f has_a_Standard_Representation_of P,T by A2, Th38, GROEB_1:15; ::_thesis: verum end; hence ( P is_Groebner_basis_wrt T iff for f being non-zero Polynomial of n,L st f in P -Ideal holds f has_a_Standard_Representation_of P,T ) by A1; ::_thesis: verum end;