:: 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;