:: POLYRED semantic presentation
begin
registration
let n be Ordinal;
let R be non trivial ZeroStr ;
cluster non zero Relation-like Bags n -defined the carrier of R -valued Function-like total V46( Bags n, the carrier of R) non-zero monomial-like finite-Support for Element of bool [:(Bags n), the carrier of R:];
existence
ex b1 being Monomial of n,R st b1 is non-zero
proof
set a = the Element of NonZero R;
reconsider a = the Element of NonZero R as Element of R ;
take q = a | (n,R); ::_thesis: q is non-zero
A1: ( a <> 0. R & 0_ (n,R) = (0. R) | (n,R) ) by POLYNOM7:19, ZFMISC_1:56;
assume q = 0_ (n,R) ; :: according to POLYNOM7:def_1 ::_thesis: contradiction
hence contradiction by A1, POLYNOM7:21; ::_thesis: verum
end;
end;
registration
cluster non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible V102() associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like left_zeroed add-left-invertible add-right-invertible Loop-like for doubleLoopStr ;
existence
not for b1 being Field holds b1 is trivial
proof
set F = the Field;
take the Field ; ::_thesis: not the Field is trivial
thus not the Field is trivial ; ::_thesis: verum
end;
end;
Lm1: for X being set
for S being Subset of X
for R being Order of X st R is being_linear-order holds
R linearly_orders S
proof
let X be set ; ::_thesis: for S being Subset of X
for R being Order of X st R is being_linear-order holds
R linearly_orders S
let S be Subset of X; ::_thesis: for R being Order of X st R is being_linear-order holds
R linearly_orders S
let R be Order of X; ::_thesis: ( R is being_linear-order implies R linearly_orders S )
S c= X ;
then A1: S c= field R by ORDERS_1:15;
assume R is being_linear-order ; ::_thesis: R linearly_orders S
hence R linearly_orders S by A1, ORDERS_1:37, ORDERS_1:38; ::_thesis: verum
end;
Lm2: for n being Ordinal
for b1, b2, b3 being bag of n st b1 <=' b2 holds
b1 + b3 <=' b2 + b3
proof
let n be Ordinal; ::_thesis: for b1, b2, b3 being bag of n st b1 <=' b2 holds
b1 + b3 <=' b2 + b3
let b1, b2, b3 be bag of n; ::_thesis: ( b1 <=' b2 implies b1 + b3 <=' b2 + b3 )
assume A1: b1 <=' b2 ; ::_thesis: b1 + b3 <=' b2 + b3
percases ( b1 = b2 or b1 <> b2 ) ;
suppose b1 = b2 ; ::_thesis: b1 + b3 <=' b2 + b3
hence b1 + b3 <=' b2 + b3 ; ::_thesis: verum
end;
suppose b1 <> b2 ; ::_thesis: b1 + b3 <=' b2 + b3
then b1 < b2 by A1, PRE_POLY:def_10;
then consider k being Ordinal such that
A2: b1 . k < b2 . k and
A3: for l being Ordinal st l in k holds
b1 . l = b2 . l by PRE_POLY:def_9;
A4: now__::_thesis:_for_l_being_Ordinal_st_l_in_k_holds_
(b1_+_b3)_._l_=_(b2_+_b3)_._l
let l be Ordinal; ::_thesis: ( l in k implies (b1 + b3) . l = (b2 + b3) . l )
assume A5: l in k ; ::_thesis: (b1 + b3) . l = (b2 + b3) . l
thus (b1 + b3) . l = (b1 . l) + (b3 . l) by PRE_POLY:def_5
.= (b2 . l) + (b3 . l) by A3, A5
.= (b2 + b3) . l by PRE_POLY:def_5 ; ::_thesis: verum
end;
( (b1 + b3) . k = (b1 . k) + (b3 . k) & (b2 + b3) . k = (b2 . k) + (b3 . k) ) by PRE_POLY:def_5;
then (b1 + b3) . k < (b2 + b3) . k by A2, XREAL_1:8;
then b1 + b3 < b2 + b3 by A4, PRE_POLY:def_9;
hence b1 + b3 <=' b2 + b3 by PRE_POLY:def_10; ::_thesis: verum
end;
end;
end;
Lm3: for n being Ordinal
for b1, b2 being bag of n st b1 <=' b2 & b2 <=' b1 holds
b1 = b2
proof
let n be Ordinal; ::_thesis: for b1, b2 being bag of n st b1 <=' b2 & b2 <=' b1 holds
b1 = b2
let b1, b2 be bag of n; ::_thesis: ( b1 <=' b2 & b2 <=' b1 implies b1 = b2 )
assume that
A1: b1 <=' b2 and
A2: b2 <=' b1 ; ::_thesis: b1 = b2
now__::_thesis:_not_b1_<>_b2
assume A3: b1 <> b2 ; ::_thesis: contradiction
then b1 < b2 by A1, PRE_POLY:def_10;
hence contradiction by A2, A3, PRE_POLY:def_10; ::_thesis: verum
end;
hence b1 = b2 ; ::_thesis: verum
end;
Lm4: for n being Ordinal
for b1, b2 being bag of n holds
( not b1 < b2 iff b2 <=' b1 )
proof
let n be Ordinal; ::_thesis: for b1, b2 being bag of n holds
( not b1 < b2 iff b2 <=' b1 )
let b1, b2 be bag of n; ::_thesis: ( not b1 < b2 iff b2 <=' b1 )
A1: now__::_thesis:_(_not_b1_<_b2_implies_b2_<='_b1_)
assume A2: not b1 < b2 ; ::_thesis: b2 <=' b1
now__::_thesis:_(_(_b1_=_b2_&_b2_<='_b1_)_or_(_b1_<>_b2_&_b2_<='_b1_)_)
percases ( b1 = b2 or b1 <> b2 ) ;
case b1 = b2 ; ::_thesis: b2 <=' b1
hence b2 <=' b1 ; ::_thesis: verum
end;
case b1 <> b2 ; ::_thesis: b2 <=' b1
then not b1 <=' b2 by A2, PRE_POLY:def_10;
hence b2 <=' b1 by PRE_POLY:45; ::_thesis: verum
end;
end;
end;
hence b2 <=' b1 ; ::_thesis: verum
end;
now__::_thesis:_(_b2_<='_b1_implies_not_b1_<_b2_)
assume A3: b2 <=' b1 ; ::_thesis: not b1 < b2
now__::_thesis:_(_(_b2_<>_b1_&_not_b1_<_b2_)_or_(_b1_=_b2_&_(_for_k_being_Ordinal_holds_
(_not_b1_._k_<_b2_._k_or_ex_l_being_Ordinal_st_
(_l_in_k_&_not_b1_._l_=_b2_._l_)_)_)_)_)
percases ( b2 <> b1 or b1 = b2 ) ;
case b2 <> b1 ; ::_thesis: not b1 < b2
hence not b1 < b2 by A3, PRE_POLY:def_10; ::_thesis: verum
end;
case b1 = b2 ; ::_thesis: for k being Ordinal holds
( not b1 . k < b2 . k or ex l being Ordinal st
( l in k & not b1 . l = b2 . l ) )
hence for k being Ordinal holds
( not b1 . k < b2 . k or ex l being Ordinal st
( l in k & not b1 . l = b2 . l ) ) ; ::_thesis: verum
end;
end;
end;
hence not b1 < b2 ; ::_thesis: verum
end;
hence ( not b1 < b2 iff b2 <=' b1 ) by A1; ::_thesis: verum
end;
Lm5: for n being Ordinal
for L being non trivial ZeroStr
for p being non-zero finite-Support Series of n,L ex b being bag of n st
( p . b <> 0. L & ( for b9 being bag of n st b < b9 holds
p . b9 = 0. L ) )
proof
let n be Ordinal; ::_thesis: for L being non trivial ZeroStr
for p being non-zero finite-Support Series of n,L ex b being bag of n st
( p . b <> 0. L & ( for b9 being bag of n st b < b9 holds
p . b9 = 0. L ) )
let L be non trivial ZeroStr ; ::_thesis: for p being non-zero finite-Support Series of n,L ex b being bag of n st
( p . b <> 0. L & ( for b9 being bag of n st b < b9 holds
p . b9 = 0. L ) )
let p be non-zero finite-Support Series of n,L; ::_thesis: ex b being bag of n st
( p . b <> 0. L & ( for b9 being bag of n st b < b9 holds
p . b9 = 0. L ) )
defpred S1[ Nat] means for B being finite Subset of (Bags n) st card B = $1 holds
ex b being bag of n st
( b in B & ( for b9 being bag of n st b9 in B holds
b9 <=' b ) );
A1: for k being Nat st 1 <= k & S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( 1 <= k & S1[k] implies S1[k + 1] )
assume A2: 1 <= k ; ::_thesis: ( not S1[k] or S1[k + 1] )
thus ( S1[k] implies S1[k + 1] ) ::_thesis: verum
proof
assume A3: S1[k] ; ::_thesis: S1[k + 1]
thus S1[k + 1] ::_thesis: verum
proof
let B be finite Subset of (Bags n); ::_thesis: ( card B = k + 1 implies ex b being bag of n st
( b in B & ( for b9 being bag of n st b9 in B holds
b9 <=' b ) ) )
assume A4: card B = k + 1 ; ::_thesis: ex b being bag of n st
( b in B & ( for b9 being bag of n st b9 in B holds
b9 <=' b ) )
then reconsider B = B as non empty finite Subset of (Bags n) ;
set x = the Element of B;
reconsider x = the Element of B as Element of Bags n ;
reconsider x = x as bag of n ;
set X = B \ {x};
now__::_thesis:_for_u_being_set_st_u_in_{x}_holds_
u_in_B
let u be set ; ::_thesis: ( u in {x} implies u in B )
assume u in {x} ; ::_thesis: u in B
then u = x by TARSKI:def_1;
hence u in B ; ::_thesis: verum
end;
then {x} c= B by TARSKI:def_3;
then A5: B = {x} \/ B by XBOOLE_1:12
.= {x} \/ (B \ {x}) by XBOOLE_1:39 ;
( x in B \ {x} iff ( x in B & not x in {x} ) ) by XBOOLE_0:def_5;
then A6: (card (B \ {x})) + 1 = k + 1 by A4, A5, CARD_2:41, TARSKI:def_1;
then reconsider X = B \ {x} as non empty set by A2, XCMPLX_1:2;
reconsider X = X as non empty finite Subset of (Bags n) ;
consider b being bag of n such that
A7: b in X and
A8: for b9 being bag of n st b9 in X holds
b9 <=' b by A3, A6, XCMPLX_1:2;
A9: now__::_thesis:_(_(_x_<='_b_&_(_for_b9_being_bag_of_n_st_b9_in_B_holds_
b9_<='_b_)_)_or_(_b_<='_x_&_(_for_b9_being_bag_of_n_st_b9_in_B_holds_
b9_<='_x_)_)_)
percases ( x <=' b or b <=' x ) by PRE_POLY:45;
caseA10: x <=' b ; ::_thesis: for b9 being bag of n st b9 in B holds
b9 <=' b
now__::_thesis:_for_b9_being_bag_of_n_st_b9_in_B_holds_
b9_<='_b
let b9 be bag of n; ::_thesis: ( b9 in B implies b9 <=' b )
assume A11: b9 in B ; ::_thesis: b9 <=' b
now__::_thesis:_(_(_b9_in_X_&_b9_<='_b_)_or_(_not_b9_in_X_&_b9_<='_b_)_)
percases ( b9 in X or not b9 in X ) ;
case b9 in X ; ::_thesis: b9 <=' b
hence b9 <=' b by A8; ::_thesis: verum
end;
case not b9 in X ; ::_thesis: b9 <=' b
then b9 in {x} by A5, A11, XBOOLE_0:def_3;
hence b9 <=' b by A10, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
hence b9 <=' b ; ::_thesis: verum
end;
hence for b9 being bag of n st b9 in B holds
b9 <=' b ; ::_thesis: verum
end;
caseA12: b <=' x ; ::_thesis: for b9 being bag of n st b9 in B holds
b9 <=' x
now__::_thesis:_for_b9_being_bag_of_n_st_b9_in_B_holds_
b9_<='_x
let b9 be bag of n; ::_thesis: ( b9 in B implies b9 <=' x )
assume A13: b9 in B ; ::_thesis: b9 <=' x
now__::_thesis:_(_(_b9_in_X_&_b9_<='_x_)_or_(_not_b9_in_X_&_b9_<='_x_)_)
percases ( b9 in X or not b9 in X ) ;
case b9 in X ; ::_thesis: b9 <=' x
then b9 <=' b by A8;
hence b9 <=' x by A12, PRE_POLY:42; ::_thesis: verum
end;
case not b9 in X ; ::_thesis: b9 <=' x
then b9 in {x} by A5, A13, XBOOLE_0:def_3;
hence b9 <=' x by TARSKI:def_1; ::_thesis: verum
end;
end;
end;
hence b9 <=' x ; ::_thesis: verum
end;
hence for b9 being bag of n st b9 in B holds
b9 <=' x ; ::_thesis: verum
end;
end;
end;
b in B by A5, A7, XBOOLE_0:def_3;
hence ex b being bag of n st
( b in B & ( for b9 being bag of n st b9 in B holds
b9 <=' b ) ) by A9; ::_thesis: verum
end;
end;
end;
reconsider sp = Support p as finite set by POLYNOM1:def_4;
A14: Support p is finite Subset of (Bags n) by POLYNOM1:def_4;
card sp is finite ;
then consider m being Nat such that
A15: card (Support p) = card m by CARD_1:48;
p <> 0_ (n,L) by POLYNOM7:def_1;
then Support p <> {} by POLYNOM7:1;
then m <> 0 by A15;
then A16: 1 <= m by NAT_1:14;
A17: card (Support p) = m by A15, CARD_1:def_2;
A18: S1[1]
proof
let B be finite Subset of (Bags n); ::_thesis: ( card B = 1 implies ex b being bag of n st
( b in B & ( for b9 being bag of n st b9 in B holds
b9 <=' b ) ) )
assume card B = 1 ; ::_thesis: ex b being bag of n st
( b in B & ( for b9 being bag of n st b9 in B holds
b9 <=' b ) )
then card {{}} = card B by CARD_1:30;
then consider b being set such that
A19: B = {b} by CARD_1:29;
A20: b in B by A19, TARSKI:def_1;
then reconsider b = b as Element of Bags n ;
reconsider b = b as bag of n ;
for b9 being bag of n st b9 in B holds
b9 <=' b by A19, TARSKI:def_1;
hence ex b being bag of n st
( b in B & ( for b9 being bag of n st b9 in B holds
b9 <=' b ) ) by A20; ::_thesis: verum
end;
for k being Nat st 1 <= k holds
S1[k] from NAT_1:sch_8(A18, A1);
then consider b being bag of n such that
A21: b in Support p and
A22: for b9 being bag of n st b9 in Support p holds
b9 <=' b by A17, A16, A14;
A23: now__::_thesis:_for_b9_being_bag_of_n_st_b_<_b9_holds_
p_._b9_=_0._L
let b9 be bag of n; ::_thesis: ( b < b9 implies p . b9 = 0. L )
assume b < b9 ; ::_thesis: p . b9 = 0. L
then not b9 <=' b by Lm4;
then A24: not b9 in Support p by A22;
b9 is Element of Bags n by PRE_POLY:def_12;
hence p . b9 = 0. L by A24, POLYNOM1:def_3; ::_thesis: verum
end;
p . b <> 0. L by A21, POLYNOM1:def_3;
hence ex b being bag of n st
( p . b <> 0. L & ( for b9 being bag of n st b < b9 holds
p . b9 = 0. L ) ) by A23; ::_thesis: verum
end;
Lm6: for L being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for f, g being FinSequence of the carrier of L
for n being Nat st len f = n + 1 & g = f | (Seg n) holds
Sum f = (Sum g) + (f /. (len f))
proof
let L be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ; ::_thesis: for f, g being FinSequence of the carrier of L
for n being Nat st len f = n + 1 & g = f | (Seg n) holds
Sum f = (Sum g) + (f /. (len f))
let f, g be FinSequence of the carrier of L; ::_thesis: for n being Nat st len f = n + 1 & g = f | (Seg n) holds
Sum f = (Sum g) + (f /. (len f))
let n be Nat; ::_thesis: ( len f = n + 1 & g = f | (Seg n) implies Sum f = (Sum g) + (f /. (len f)) )
assume that
A1: len f = n + 1 and
A2: g = f | (Seg n) ; ::_thesis: Sum f = (Sum g) + (f /. (len f))
A3: n <= len f by A1, NAT_1:11;
set q = <*(f /. (len f))*>;
set p = g ^ <*(f /. (len f))*>;
A4: len <*(f /. (len f))*> = 1 by FINSEQ_1:39;
A5: dom f = Seg (n + 1) by A1, FINSEQ_1:def_3;
A6: now__::_thesis:_for_u_being_set_st_u_in_dom_f_holds_
f_._u_=_(g_^_<*(f_/._(len_f))*>)_._u
let u be set ; ::_thesis: ( u in dom f implies f . u = (g ^ <*(f /. (len f))*>) . u )
assume A7: u in dom f ; ::_thesis: f . u = (g ^ <*(f /. (len f))*>) . u
then u in { k where k is Element of NAT : ( 1 <= k & k <= n + 1 ) } by A5, FINSEQ_1:def_1;
then consider i being Element of NAT such that
A8: u = i and
A9: 1 <= i and
A10: i <= n + 1 ;
now__::_thesis:_(_(_i_=_n_+_1_&_(g_^_<*(f_/._(len_f))*>)_._i_=_f_._i_)_or_(_i_<>_n_+_1_&_(g_^_<*(f_/._(len_f))*>)_._i_=_f_._i_)_)
percases ( i = n + 1 or i <> n + 1 ) ;
caseA11: i = n + 1 ; ::_thesis: (g ^ <*(f /. (len f))*>) . i = f . i
then ( (len g) + 1 <= i & i <= (len g) + (len <*(f /. (len f))*>) ) by A2, A3, A4, FINSEQ_1:17;
hence (g ^ <*(f /. (len f))*>) . i = <*(f /. (len f))*> . (i - (len g)) by FINSEQ_1:23
.= <*(f /. (len f))*> . ((n + 1) - n) by A2, A3, A11, FINSEQ_1:17
.= <*(f /. (len f))*> . 1 by XCMPLX_1:26
.= f /. (n + 1) by A1, FINSEQ_1:40
.= f . i by A7, A8, A11, PARTFUN1:def_6 ;
::_thesis: verum
end;
case i <> n + 1 ; ::_thesis: (g ^ <*(f /. (len f))*>) . i = f . i
then i < n + 1 by A10, XXREAL_0:1;
then i <= n by NAT_1:13;
then i in { k where k is Element of NAT : ( 1 <= k & k <= n ) } by A9;
then i in Seg n by FINSEQ_1:def_1;
then A12: i in dom g by A2, A3, FINSEQ_1:17;
then (g ^ <*(f /. (len f))*>) . i = g . i by FINSEQ_1:def_7;
hence (g ^ <*(f /. (len f))*>) . i = f . i by A2, A12, FUNCT_1:47; ::_thesis: verum
end;
end;
end;
hence f . u = (g ^ <*(f /. (len f))*>) . u by A8; ::_thesis: verum
end;
len (g ^ <*(f /. (len f))*>) = (len g) + (len <*(f /. (len f))*>) by FINSEQ_1:22
.= (len g) + 1 by FINSEQ_1:40
.= len f by A1, A2, A3, FINSEQ_1:17 ;
then dom f = Seg (len (g ^ <*(f /. (len f))*>)) by FINSEQ_1:def_3
.= dom (g ^ <*(f /. (len f))*>) by FINSEQ_1:def_3 ;
then f = g ^ <*(f /. (len f))*> by A6, FUNCT_1:2;
hence Sum f = (Sum g) + (Sum <*(f /. (len f))*>) by RLVECT_1:41
.= (Sum g) + (f /. (len f)) by RLVECT_1:44 ;
::_thesis: verum
end;
registration
let n be Ordinal;
let L be non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like left_zeroed doubleLoopStr ;
let p, q be non-zero finite-Support Series of n,L;
clusterp *' q -> non-zero ;
coherence
p *' q is non-zero
proof
consider b22 being bag of n such that
A1: q . b22 <> 0. L and
A2: for b9 being bag of n st b22 < b9 holds
q . b9 = 0. L by Lm5;
consider b11 being bag of n such that
A3: p . b11 <> 0. L and
A4: for b9 being bag of n st b11 < b9 holds
p . b9 = 0. L by Lm5;
set b = b11 + b22;
consider s being FinSequence of the carrier of L such that
A5: (p *' q) . (b11 + b22) = Sum s and
A6: len s = len (decomp (b11 + b22)) and
A7: for k being Element of NAT st k in dom s holds
ex b1, b2 being bag of n st
( (decomp (b11 + b22)) /. k = <*b1,b2*> & s /. k = (p . b1) * (q . b2) ) by POLYNOM1:def_9;
A8: ( b11 + b22 is Element of Bags n & (p . b11) * (q . b22) <> 0. L ) by A3, A1, PRE_POLY:def_12, VECTSP_2:def_1;
consider S being non empty finite Subset of (Bags n) such that
A9: divisors (b11 + b22) = SgmX ((BagOrder n),S) and
A10: for p being bag of n holds
( p in S iff p divides b11 + b22 ) by PRE_POLY:def_16;
set sgm = SgmX ((BagOrder n),S);
A11: BagOrder n linearly_orders S by Lm1;
b11 divides b11 + b22 by PRE_POLY:50;
then b11 in S by A10;
then b11 in rng (SgmX ((BagOrder n),S)) by A11, PRE_POLY:def_2;
then consider i being set such that
A12: i in dom (SgmX ((BagOrder n),S)) and
A13: (SgmX ((BagOrder n),S)) . i = b11 by FUNCT_1:def_3;
A14: i in dom (decomp (b11 + b22)) by A9, A12, PRE_POLY:def_17;
(divisors (b11 + b22)) /. i = b11 by A9, A12, A13, PARTFUN1:def_6;
then A15: (decomp (b11 + b22)) /. i = <*b11,((b11 + b22) -' b11)*> by A14, PRE_POLY:def_17;
then A16: (decomp (b11 + b22)) /. i = <*b11,b22*> by PRE_POLY:48;
A17: dom s = Seg (len (decomp (b11 + b22))) by A6, FINSEQ_1:def_3
.= dom (decomp (b11 + b22)) by FINSEQ_1:def_3 ;
then A18: i in dom s by A9, A12, PRE_POLY:def_17;
reconsider i = i as Element of NAT by A12;
consider b1, b2 being bag of n such that
A19: (decomp (b11 + b22)) /. i = <*b1,b2*> and
A20: s /. i = (p . b1) * (q . b2) by A7, A18;
A21: b2 = <*b11,b22*> . 2 by A16, A19, FINSEQ_1:44
.= b22 by FINSEQ_1:44 ;
A22: now__::_thesis:_for_k_being_Element_of_NAT_st_k_in_dom_s_&_k_<>_i_holds_
s_/._k_=_0._L
let k be Element of NAT ; ::_thesis: ( k in dom s & k <> i implies s /. k = 0. L )
assume that
A23: k in dom s and
A24: k <> i ; ::_thesis: s /. k = 0. L
consider b1, b2 being bag of n such that
A25: (decomp (b11 + b22)) /. k = <*b1,b2*> and
A26: s /. k = (p . b1) * (q . b2) by A7, A23;
consider b19, b29 being bag of n such that
A27: (decomp (b11 + b22)) /. k = <*b19,b29*> and
A28: b11 + b22 = b19 + b29 by A17, A23, PRE_POLY:68;
A29: b2 = <*b19,b29*> . 2 by A27, A25, FINSEQ_1:44
.= b29 by FINSEQ_1:44 ;
A30: b1 = <*b19,b29*> . 1 by A27, A25, FINSEQ_1:44
.= b19 by FINSEQ_1:44 ;
A31: now__::_thesis:_(_p_._b1_<>_0._L_implies_not_q_._b2_<>_0._L_)
assume that
A32: p . b1 <> 0. L and
A33: q . b2 <> 0. L ; ::_thesis: contradiction
not b11 < b1 by A4, A32;
then A34: b1 <=' b11 by Lm4;
not b22 < b2 by A2, A33;
then A35: b2 <=' b22 by Lm4;
A36: now__::_thesis:_(_b1_=_b11_implies_not_b2_=_b22_)
assume ( b1 = b11 & b2 = b22 ) ; ::_thesis: contradiction
then (decomp (b11 + b22)) . k = <*b11,b22*> by A17, A23, A25, PARTFUN1:def_6
.= (decomp (b11 + b22)) /. i by A15, PRE_POLY:48
.= (decomp (b11 + b22)) . i by A14, PARTFUN1:def_6 ;
hence contradiction by A14, A17, A23, A24, FUNCT_1:def_4; ::_thesis: verum
end;
now__::_thesis:_(_(_b1_<>_b11_&_contradiction_)_or_(_b2_<>_b22_&_contradiction_)_)
percases ( b1 <> b11 or b2 <> b22 ) by A36;
caseA37: b1 <> b11 ; ::_thesis: contradiction
A38: now__::_thesis:_not_b1_+_b2_=_b11_+_b2
assume b1 + b2 = b11 + b2 ; ::_thesis: contradiction
then b1 = (b11 + b2) -' b2 by PRE_POLY:48;
hence contradiction by A37, PRE_POLY:48; ::_thesis: verum
end;
( b11 + b22 <=' b11 + b2 & b11 + b2 <=' b11 + b22 ) by A28, A30, A29, A34, A35, Lm2;
hence contradiction by A28, A30, A29, A38, Lm3; ::_thesis: verum
end;
caseA39: b2 <> b22 ; ::_thesis: contradiction
A40: now__::_thesis:_not_b2_+_b1_=_b22_+_b1
assume b2 + b1 = b22 + b1 ; ::_thesis: contradiction
then b2 = (b22 + b1) -' b1 by PRE_POLY:48;
hence contradiction by A39, PRE_POLY:48; ::_thesis: verum
end;
( b11 + b22 <=' b22 + b1 & b22 + b1 <=' b11 + b22 ) by A28, A30, A29, A34, A35, Lm2;
hence contradiction by A28, A30, A29, A40, Lm3; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
now__::_thesis:_(_(_p_._b1_=_0._L_&_(p_._b1)_*_(q_._b2)_=_0._L_)_or_(_q_._b2_=_0._L_&_(p_._b1)_*_(q_._b2)_=_0._L_)_)
percases ( p . b1 = 0. L or q . b2 = 0. L ) by A31;
case p . b1 = 0. L ; ::_thesis: (p . b1) * (q . b2) = 0. L
hence (p . b1) * (q . b2) = 0. L by BINOM:1; ::_thesis: verum
end;
case q . b2 = 0. L ; ::_thesis: (p . b1) * (q . b2) = 0. L
hence (p . b1) * (q . b2) = 0. L by BINOM:2; ::_thesis: verum
end;
end;
end;
hence s /. k = 0. L by A26; ::_thesis: verum
end;
b1 = <*b11,b22*> . 1 by A16, A19, FINSEQ_1:44
.= b11 by FINSEQ_1:44 ;
then (p *' q) . (b11 + b22) = (p . b11) * (q . b22) by A5, A18, A22, A20, A21, POLYNOM2:3;
then b11 + b22 in Support (p *' q) by A8, POLYNOM1:def_3;
then p *' q <> 0_ (n,L) by POLYNOM7:1;
hence p *' q is non-zero by POLYNOM7:def_1; ::_thesis: verum
end;
end;
begin
theorem Th1: :: POLYRED:1
for X being set
for L being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for p, q being Series of X,L holds - (p + q) = (- p) + (- q)
proof
let n be set ; ::_thesis: for L being non empty right_complementable Abelian add-associative right_zeroed addLoopStr
for p, q being Series of n,L holds - (p + q) = (- p) + (- q)
let L be non empty right_complementable Abelian add-associative right_zeroed addLoopStr ; ::_thesis: for p, q being Series of n,L holds - (p + q) = (- p) + (- q)
let p, q be Series of n,L; ::_thesis: - (p + q) = (- p) + (- q)
A1: now__::_thesis:_for_x_being_set_st_x_in_dom_(-_(p_+_q))_holds_
(-_(p_+_q))_._x_=_((-_p)_+_(-_q))_._x
let x be set ; ::_thesis: ( x in dom (- (p + q)) implies (- (p + q)) . x = ((- p) + (- q)) . x )
assume x in dom (- (p + q)) ; ::_thesis: (- (p + q)) . x = ((- p) + (- q)) . x
then reconsider b = x as bag of n ;
((- p) + (- q)) . b = ((- p) . b) + ((- q) . b) by POLYNOM1:15
.= (- (p . b)) + ((- q) . b) by POLYNOM1:17
.= (- (p . b)) + (- (q . b)) by POLYNOM1:17
.= - ((q . b) + (p . b)) by RLVECT_1:31
.= - ((p + q) . b) by POLYNOM1:15
.= (- (p + q)) . b by POLYNOM1:17 ;
hence (- (p + q)) . x = ((- p) + (- q)) . x ; ::_thesis: verum
end;
dom (- (p + q)) = Bags n by FUNCT_2:def_1
.= dom ((- p) + (- q)) by FUNCT_2:def_1 ;
hence - (p + q) = (- p) + (- q) by A1, FUNCT_1:2; ::_thesis: verum
end;
theorem Th2: :: POLYRED:2
for X being set
for L being non empty left_zeroed addLoopStr
for p being Series of X,L holds (0_ (X,L)) + p = p
proof
let n be set ; ::_thesis: for L being non empty left_zeroed addLoopStr
for p being Series of n,L holds (0_ (n,L)) + p = p
let L be non empty left_zeroed addLoopStr ; ::_thesis: for p being Series of n,L holds (0_ (n,L)) + p = p
let p be Series of n,L; ::_thesis: (0_ (n,L)) + p = p
reconsider ls = (0_ (n,L)) + p, p9 = p as Function of (Bags n), the carrier of L ;
now__::_thesis:_for_b_being_Element_of_Bags_n_holds_ls_._b_=_p9_._b
let b be Element of Bags n; ::_thesis: ls . b = p9 . b
thus ls . b = ((0_ (n,L)) . b) + (p . b) by POLYNOM1:15
.= (0. L) + (p9 . b) by POLYNOM1:22
.= p9 . b by ALGSTR_1:def_2 ; ::_thesis: verum
end;
hence (0_ (n,L)) + p = p by FUNCT_2:63; ::_thesis: verum
end;
theorem Th3: :: POLYRED:3
for X being set
for L being non empty right_complementable add-associative right_zeroed addLoopStr
for p being Series of X,L holds
( (- p) + p = 0_ (X,L) & p + (- p) = 0_ (X,L) )
proof
let n be set ; ::_thesis: for L being non empty right_complementable add-associative right_zeroed addLoopStr
for p being Series of n,L holds
( (- p) + p = 0_ (n,L) & p + (- p) = 0_ (n,L) )
let L be non empty right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for p being Series of n,L holds
( (- p) + p = 0_ (n,L) & p + (- p) = 0_ (n,L) )
let p be Series of n,L; ::_thesis: ( (- p) + p = 0_ (n,L) & p + (- p) = 0_ (n,L) )
set q = (- p) + p;
now__::_thesis:_for_b_being_Element_of_Bags_n_holds_((-_p)_+_p)_._b_=_(0__(n,L))_._b
let b be Element of Bags n; ::_thesis: ((- p) + p) . b = (0_ (n,L)) . b
thus ((- p) + p) . b = ((- p) . b) + (p . b) by POLYNOM1:15
.= (- (p . b)) + (p . b) by POLYNOM1:17
.= 0. L by RLVECT_1:5
.= (0_ (n,L)) . b by POLYNOM1:22 ; ::_thesis: verum
end;
hence (- p) + p = 0_ (n,L) by FUNCT_2:63; ::_thesis: p + (- p) = 0_ (n,L)
set q = p + (- p);
now__::_thesis:_for_b_being_Element_of_Bags_n_holds_(p_+_(-_p))_._b_=_(0__(n,L))_._b
let b be Element of Bags n; ::_thesis: (p + (- p)) . b = (0_ (n,L)) . b
thus (p + (- p)) . b = (p . b) + ((- p) . b) by POLYNOM1:15
.= (p . b) + (- (p . b)) by POLYNOM1:17
.= 0. L by RLVECT_1:5
.= (0_ (n,L)) . b by POLYNOM1:22 ; ::_thesis: verum
end;
hence p + (- p) = 0_ (n,L) by FUNCT_2:63; ::_thesis: verum
end;
theorem Th4: :: POLYRED:4
for n being set
for L being non empty right_complementable add-associative right_zeroed addLoopStr
for p being Series of n,L holds p - (0_ (n,L)) = p
proof
let n be set ; ::_thesis: for L being non empty right_complementable add-associative right_zeroed addLoopStr
for p being Series of n,L holds p - (0_ (n,L)) = p
let L be non empty right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for p being Series of n,L holds p - (0_ (n,L)) = p
let p be Series of n,L; ::_thesis: p - (0_ (n,L)) = p
reconsider pp = p - (0_ (n,L)) as Function of (Bags n), the carrier of L ;
now__::_thesis:_for_b_being_Element_of_Bags_n_holds_pp_._b_=_p_._b
let b be Element of Bags n; ::_thesis: pp . b = p . b
thus pp . b = (p + (- (0_ (n,L)))) . b by POLYNOM1:def_6
.= (p . b) + ((- (0_ (n,L))) . b) by POLYNOM1:15
.= (p . b) + (- ((0_ (n,L)) . b)) by POLYNOM1:17
.= (p . b) + (- (0. L)) by POLYNOM1:22
.= (p . b) - (0. L) by RLVECT_1:def_11
.= p . b by RLVECT_1:13 ; ::_thesis: verum
end;
hence p - (0_ (n,L)) = p by FUNCT_2:63; ::_thesis: verum
end;
theorem Th5: :: POLYRED:5
for n being Ordinal
for L being non empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed doubleLoopStr
for p being Series of n,L holds (0_ (n,L)) *' p = 0_ (n,L)
proof
let n be Ordinal; ::_thesis: for L being non empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed doubleLoopStr
for p being Series of n,L holds (0_ (n,L)) *' p = 0_ (n,L)
let L be non empty left_add-cancelable right_complementable left-distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for p being Series of n,L holds (0_ (n,L)) *' p = 0_ (n,L)
let p be Series of n,L; ::_thesis: (0_ (n,L)) *' p = 0_ (n,L)
set Z = 0_ (n,L);
now__::_thesis:_for_b_being_Element_of_Bags_n_holds_((0__(n,L))_*'_p)_._b_=_(0__(n,L))_._b
let b be Element of Bags n; ::_thesis: ((0_ (n,L)) *' p) . b = (0_ (n,L)) . b
consider s being FinSequence of the carrier of L such that
A1: ((0_ (n,L)) *' p) . b = Sum s and
len s = len (decomp b) and
A2: for k being Element of NAT st k in dom s holds
ex b1, b2 being bag of n st
( (decomp b) /. k = <*b1,b2*> & s /. k = ((0_ (n,L)) . b1) * (p . b2) ) by POLYNOM1:def_9;
now__::_thesis:_for_k_being_Nat_st_k_in_dom_s_holds_
s_/._k_=_0._L
let k be Nat; ::_thesis: ( k in dom s implies s /. k = 0. L )
assume k in dom s ; ::_thesis: s /. k = 0. L
then consider b1, b2 being bag of n such that
(decomp b) /. k = <*b1,b2*> and
A3: s /. k = ((0_ (n,L)) . b1) * (p . b2) by A2;
thus s /. k = (0. L) * (p . b2) by A3, POLYNOM1:22
.= 0. L by BINOM:1 ; ::_thesis: verum
end;
then Sum s = 0. L by MATRLIN:11;
hence ((0_ (n,L)) *' p) . b = (0_ (n,L)) . b by A1, POLYNOM1:22; ::_thesis: verum
end;
hence (0_ (n,L)) *' p = 0_ (n,L) by FUNCT_2:63; ::_thesis: verum
end;
Lm7: for n being Ordinal
for L being non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L
for q being Element of (Polynom-Ring (n,L)) st p = q holds
- p = - q
proof
let n be Ordinal; ::_thesis: for L being non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L
for q being Element of (Polynom-Ring (n,L)) st p = q holds
- p = - q
let L be non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for p being Polynomial of n,L
for q being Element of (Polynom-Ring (n,L)) st p = q holds
- p = - q
let p be Polynomial of n,L; ::_thesis: for q being Element of (Polynom-Ring (n,L)) st p = q holds
- p = - q
let q be Element of (Polynom-Ring (n,L)); ::_thesis: ( p = q implies - p = - q )
set R = Polynom-Ring (n,L);
reconsider x = - p as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider x = x as Element of (Polynom-Ring (n,L)) ;
assume p = q ; ::_thesis: - p = - q
then x + q = (- p) + p by POLYNOM1:def_10
.= 0_ (n,L) by Th3
.= 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10 ;
hence - p = - q by RLVECT_1:6; ::_thesis: verum
end;
theorem Th6: :: POLYRED:6
for n being Ordinal
for L being non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p, q being Polynomial of n,L holds
( - (p *' q) = (- p) *' q & - (p *' q) = p *' (- q) )
proof
let n be Ordinal; ::_thesis: for L being non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for p, q being Polynomial of n,L holds
( - (p *' q) = (- p) *' q & - (p *' q) = p *' (- q) )
let L be non trivial right_complementable associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for p, q being Polynomial of n,L holds
( - (p *' q) = (- p) *' q & - (p *' q) = p *' (- q) )
let p, q be Polynomial of n,L; ::_thesis: ( - (p *' q) = (- p) *' q & - (p *' q) = p *' (- q) )
reconsider p9 = p, q9 = q as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider p9 = p9, q9 = q9 as Element of (Polynom-Ring (n,L)) ;
A1: p9 * q9 = p *' q by POLYNOM1:def_10;
- p = - p9 by Lm7;
then A2: (- p9) * q9 = (- p) *' q by POLYNOM1:def_10;
- q = - q9 by Lm7;
then A3: p9 * (- q9) = p *' (- q) by POLYNOM1:def_10;
( - (p9 * q9) = (- p9) * q9 & - (p9 * q9) = p9 * (- q9) ) by VECTSP_1:9;
hence ( - (p *' q) = (- p) *' q & - (p *' q) = p *' (- q) ) by A2, A3, A1, Lm7; ::_thesis: verum
end;
Lm8: for n being Ordinal
for L being non empty right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L
for m being Monomial of n,L
for b being bag of n st b <> term m holds
m . b = 0. L
proof
let n be Ordinal; ::_thesis: for L being non empty right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L
for m being Monomial of n,L
for b being bag of n st b <> term m holds
m . b = 0. L
let L be non empty right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for p being Polynomial of n,L
for m being Monomial of n,L
for b being bag of n st b <> term m holds
m . b = 0. L
let p be Polynomial of n,L; ::_thesis: for m being Monomial of n,L
for b being bag of n st b <> term m holds
m . b = 0. L
let m be Monomial of n,L; ::_thesis: for b being bag of n st b <> term m holds
m . b = 0. L
let b be bag of n; ::_thesis: ( b <> term m implies m . b = 0. L )
assume A1: b <> term m ; ::_thesis: m . b = 0. L
percases ( Support m = {} or Support m = {(term m)} ) by POLYNOM7:7;
suppose Support m = {} ; ::_thesis: m . b = 0. L
then m = 0_ (n,L) by POLYNOM7:1;
hence m . b = 0. L by POLYNOM1:22; ::_thesis: verum
end;
supposeA2: Support m = {(term m)} ; ::_thesis: m . b = 0. L
A3: b is Element of Bags n by PRE_POLY:def_12;
not b in Support m by A1, A2, TARSKI:def_1;
hence m . b = 0. L by A3, POLYNOM1:def_3; ::_thesis: verum
end;
end;
end;
theorem Th7: :: POLYRED:7
for n being Ordinal
for L being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L
for m being Monomial of n,L
for b being bag of n holds (m *' p) . ((term m) + b) = (m . (term m)) * (p . b)
proof
let n be Ordinal; ::_thesis: for L being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L
for m being Monomial of n,L
for b being bag of n holds (m *' p) . ((term m) + b) = (m . (term m)) * (p . b)
let L be non empty right_complementable distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for p being Polynomial of n,L
for m being Monomial of n,L
for b being bag of n holds (m *' p) . ((term m) + b) = (m . (term m)) * (p . b)
let p be Polynomial of n,L; ::_thesis: for m being Monomial of n,L
for b being bag of n holds (m *' p) . ((term m) + b) = (m . (term m)) * (p . b)
let m be Monomial of n,L; ::_thesis: for b being bag of n holds (m *' p) . ((term m) + b) = (m . (term m)) * (p . b)
let b2 be bag of n; ::_thesis: (m *' p) . ((term m) + b2) = (m . (term m)) * (p . b2)
set q = m *' p;
set b = (term m) + b2;
consider s being FinSequence of the carrier of L such that
A1: (m *' p) . ((term m) + b2) = Sum s and
A2: len s = len (decomp ((term m) + b2)) and
A3: for k being Element of NAT st k in dom s holds
ex b1, b2 being bag of n st
( (decomp ((term m) + b2)) /. k = <*b1,b2*> & s /. k = (m . b1) * (p . b2) ) by POLYNOM1:def_9;
consider k being Element of NAT such that
A4: k in dom (decomp ((term m) + b2)) and
A5: (decomp ((term m) + b2)) /. k = <*(term m),b2*> by PRE_POLY:69;
A6: dom s = Seg (len s) by FINSEQ_1:def_3
.= dom (decomp ((term m) + b2)) by A2, FINSEQ_1:def_3 ;
then consider b19, b29 being bag of n such that
A7: (decomp ((term m) + b2)) /. k = <*b19,b29*> and
A8: s /. k = (m . b19) * (p . b29) by A3, A4;
A9: b2 = <*(term m),b2*> . 2 by FINSEQ_1:44
.= b29 by A5, A7, FINSEQ_1:44 ;
A10: for k9 being Element of NAT st k9 in dom s & k9 <> k holds
s /. k9 = 0. L
proof
let k9 be Element of NAT ; ::_thesis: ( k9 in dom s & k9 <> k implies s /. k9 = 0. L )
assume that
A11: k9 in dom s and
A12: k9 <> k ; ::_thesis: s /. k9 = 0. L
consider b19, b29 being bag of n such that
A13: (decomp ((term m) + b2)) /. k9 = <*b19,b29*> and
A14: s /. k9 = (m . b19) * (p . b29) by A3, A11;
A15: b19 = (divisors ((term m) + b2)) /. k9 by A6, A11, A13, PRE_POLY:70;
A16: ((term m) + b2) -' b19 = <*b19,(((term m) + b2) -' b19)*> . 2 by FINSEQ_1:44
.= <*b19,b29*> . 2 by A6, A11, A13, A15, PRE_POLY:def_17
.= b29 by FINSEQ_1:44 ;
percases ( ( b19 = term m & b29 = b2 ) or b19 <> term m or b29 <> b2 ) ;
supposeA17: ( b19 = term m & b29 = b2 ) ; ::_thesis: s /. k9 = 0. L
(decomp ((term m) + b2)) . k9 = (decomp ((term m) + b2)) /. k9 by A6, A11, PARTFUN1:def_6
.= (decomp ((term m) + b2)) . k by A4, A5, A13, A17, PARTFUN1:def_6 ;
hence s /. k9 = 0. L by A6, A4, A11, A12, FUNCT_1:def_4; ::_thesis: verum
end;
suppose b19 <> term m ; ::_thesis: s /. k9 = 0. L
then m . b19 = 0. L by Lm8;
hence s /. k9 = 0. L by A14, VECTSP_1:7; ::_thesis: verum
end;
suppose b29 <> b2 ; ::_thesis: s /. k9 = 0. L
then b19 <> term m by A16, PRE_POLY:48;
then m . b19 = 0. L by Lm8;
hence s /. k9 = 0. L by A14, VECTSP_1:7; ::_thesis: verum
end;
end;
end;
term m = <*b19,b29*> . 1 by A5, A7, FINSEQ_1:44
.= b19 by FINSEQ_1:44 ;
hence (m *' p) . ((term m) + b2) = (m . (term m)) * (p . b2) by A1, A6, A4, A8, A9, A10, POLYNOM2:3; ::_thesis: verum
end;
theorem Th8: :: POLYRED:8
for X being set
for L being non empty left_add-cancelable left-distributive right_zeroed doubleLoopStr
for p being Series of X,L holds (0. L) * p = 0_ (X,L)
proof
let n be set ; ::_thesis: for L being non empty left_add-cancelable left-distributive right_zeroed doubleLoopStr
for p being Series of n,L holds (0. L) * p = 0_ (n,L)
let L be non empty left_add-cancelable left-distributive right_zeroed doubleLoopStr ; ::_thesis: for p being Series of n,L holds (0. L) * p = 0_ (n,L)
let p be Series of n,L; ::_thesis: (0. L) * p = 0_ (n,L)
set op = (0. L) * p;
A1: now__::_thesis:_for_u_being_set_st_u_in_dom_((0._L)_*_p)_holds_
((0._L)_*_p)_._u_=_(0__(n,L))_._u
let u be set ; ::_thesis: ( u in dom ((0. L) * p) implies ((0. L) * p) . u = (0_ (n,L)) . u )
assume u in dom ((0. L) * p) ; ::_thesis: ((0. L) * p) . u = (0_ (n,L)) . u
then reconsider u9 = u as bag of n ;
((0. L) * p) . u9 = (0. L) * (p . u9) by POLYNOM7:def_9
.= 0. L by BINOM:1 ;
hence ((0. L) * p) . u = (0_ (n,L)) . u by POLYNOM1:22; ::_thesis: verum
end;
dom ((0. L) * p) = Bags n by FUNCT_2:def_1
.= dom (0_ (n,L)) by FUNCT_2:def_1 ;
hence (0. L) * p = 0_ (n,L) by A1, FUNCT_1:2; ::_thesis: verum
end;
theorem Th9: :: POLYRED:9
for X being set
for L being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for p being Series of X,L
for a being Element of L holds
( - (a * p) = (- a) * p & - (a * p) = a * (- p) )
proof
let n be set ; ::_thesis: for L being non empty right_complementable distributive add-associative right_zeroed doubleLoopStr
for p being Series of n,L
for a being Element of L holds
( - (a * p) = (- a) * p & - (a * p) = a * (- p) )
let L be non empty right_complementable distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for p being Series of n,L
for a being Element of L holds
( - (a * p) = (- a) * p & - (a * p) = a * (- p) )
let p be Series of n,L; ::_thesis: for a being Element of L holds
( - (a * p) = (- a) * p & - (a * p) = a * (- p) )
let a be Element of L; ::_thesis: ( - (a * p) = (- a) * p & - (a * p) = a * (- p) )
set ap = a * p;
A1: now__::_thesis:_for_u_being_set_st_u_in_dom_(-_(a_*_p))_holds_
(-_(a_*_p))_._u_=_((-_a)_*_p)_._u
let u be set ; ::_thesis: ( u in dom (- (a * p)) implies (- (a * p)) . u = ((- a) * p) . u )
assume u in dom (- (a * p)) ; ::_thesis: (- (a * p)) . u = ((- a) * p) . u
then reconsider u9 = u as bag of n ;
(- (a * p)) . u9 = - ((a * p) . u9) by POLYNOM1:17
.= - (a * (p . u9)) by POLYNOM7:def_9
.= (- a) * (p . u9) by VECTSP_1:9
.= ((- a) * p) . u9 by POLYNOM7:def_9 ;
hence (- (a * p)) . u = ((- a) * p) . u ; ::_thesis: verum
end;
dom (- (a * p)) = Bags n by FUNCT_2:def_1
.= dom ((- a) * p) by FUNCT_2:def_1 ;
hence - (a * p) = (- a) * p by A1, FUNCT_1:2; ::_thesis: - (a * p) = a * (- p)
A2: now__::_thesis:_for_u_being_set_st_u_in_dom_(-_(a_*_p))_holds_
(-_(a_*_p))_._u_=_(a_*_(-_p))_._u
let u be set ; ::_thesis: ( u in dom (- (a * p)) implies (- (a * p)) . u = (a * (- p)) . u )
assume u in dom (- (a * p)) ; ::_thesis: (- (a * p)) . u = (a * (- p)) . u
then reconsider u9 = u as bag of n ;
(- (a * p)) . u9 = - ((a * p) . u9) by POLYNOM1:17
.= - (a * (p . u9)) by POLYNOM7:def_9
.= a * (- (p . u9)) by VECTSP_1:8
.= a * ((- p) . u9) by POLYNOM1:17
.= (a * (- p)) . u9 by POLYNOM7:def_9 ;
hence (- (a * p)) . u = (a * (- p)) . u ; ::_thesis: verum
end;
dom (- (a * p)) = Bags n by FUNCT_2:def_1
.= dom (a * (- p)) by FUNCT_2:def_1 ;
hence - (a * p) = a * (- p) by A2, FUNCT_1:2; ::_thesis: verum
end;
theorem Th10: :: POLYRED:10
for X being set
for L being non empty left-distributive doubleLoopStr
for p being Series of X,L
for a, a9 being Element of L holds (a * p) + (a9 * p) = (a + a9) * p
proof
let n be set ; ::_thesis: for L being non empty left-distributive doubleLoopStr
for p being Series of n,L
for a, a9 being Element of L holds (a * p) + (a9 * p) = (a + a9) * p
let L be non empty left-distributive doubleLoopStr ; ::_thesis: for p being Series of n,L
for a, a9 being Element of L holds (a * p) + (a9 * p) = (a + a9) * p
let p be Series of n,L; ::_thesis: for a, a9 being Element of L holds (a * p) + (a9 * p) = (a + a9) * p
let a, a9 be Element of L; ::_thesis: (a * p) + (a9 * p) = (a + a9) * p
set p1 = (a * p) + (a9 * p);
set p2 = (a + a9) * p;
A1: now__::_thesis:_for_u_being_set_st_u_in_dom_((a_*_p)_+_(a9_*_p))_holds_
((a_*_p)_+_(a9_*_p))_._u_=_((a_+_a9)_*_p)_._u
let u be set ; ::_thesis: ( u in dom ((a * p) + (a9 * p)) implies ((a * p) + (a9 * p)) . u = ((a + a9) * p) . u )
assume u in dom ((a * p) + (a9 * p)) ; ::_thesis: ((a * p) + (a9 * p)) . u = ((a + a9) * p) . u
then reconsider u9 = u as bag of n ;
((a * p) + (a9 * p)) . u9 = ((a * p) . u9) + ((a9 * p) . u9) by POLYNOM1:15
.= (a * (p . u9)) + ((a9 * p) . u9) by POLYNOM7:def_9
.= (a * (p . u9)) + (a9 * (p . u9)) by POLYNOM7:def_9
.= (a + a9) * (p . u9) by VECTSP_1:def_3
.= ((a + a9) * p) . u9 by POLYNOM7:def_9 ;
hence ((a * p) + (a9 * p)) . u = ((a + a9) * p) . u ; ::_thesis: verum
end;
dom ((a * p) + (a9 * p)) = Bags n by FUNCT_2:def_1
.= dom ((a + a9) * p) by FUNCT_2:def_1 ;
hence (a * p) + (a9 * p) = (a + a9) * p by A1, FUNCT_1:2; ::_thesis: verum
end;
theorem Th11: :: POLYRED:11
for X being set
for L being non empty associative multLoopStr_0
for p being Series of X,L
for a, a9 being Element of L holds (a * a9) * p = a * (a9 * p)
proof
let n be set ; ::_thesis: for L being non empty associative multLoopStr_0
for p being Series of n,L
for a, a9 being Element of L holds (a * a9) * p = a * (a9 * p)
let L be non empty associative multLoopStr_0 ; ::_thesis: for p being Series of n,L
for a, a9 being Element of L holds (a * a9) * p = a * (a9 * p)
let p be Series of n,L; ::_thesis: for a, a9 being Element of L holds (a * a9) * p = a * (a9 * p)
let a, a9 be Element of L; ::_thesis: (a * a9) * p = a * (a9 * p)
set q = (a * a9) * p;
set r = a * (a9 * p);
A1: now__::_thesis:_for_u_being_set_st_u_in_dom_((a_*_a9)_*_p)_holds_
((a_*_a9)_*_p)_._u_=_(a_*_(a9_*_p))_._u
let u be set ; ::_thesis: ( u in dom ((a * a9) * p) implies ((a * a9) * p) . u = (a * (a9 * p)) . u )
assume u in dom ((a * a9) * p) ; ::_thesis: ((a * a9) * p) . u = (a * (a9 * p)) . u
then reconsider b = u as bag of n ;
((a * a9) * p) . b = (a * a9) * (p . b) by POLYNOM7:def_9
.= a * (a9 * (p . b)) by GROUP_1:def_3
.= a * ((a9 * p) . b) by POLYNOM7:def_9
.= (a * (a9 * p)) . b by POLYNOM7:def_9 ;
hence ((a * a9) * p) . u = (a * (a9 * p)) . u ; ::_thesis: verum
end;
dom ((a * a9) * p) = Bags n by FUNCT_2:def_1
.= dom (a * (a9 * p)) by FUNCT_2:def_1 ;
hence (a * a9) * p = a * (a9 * p) by A1, FUNCT_1:2; ::_thesis: verum
end;
theorem Th12: :: POLYRED:12
for n being Ordinal
for L being non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for p, p9 being Series of n,L
for a being Element of L holds a * (p *' p9) = p *' (a * p9)
proof
let n be Ordinal; ::_thesis: for L being non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for p, p9 being Series of n,L
for a being Element of L holds a * (p *' p9) = p *' (a * p9)
let L be non empty right_complementable associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for p, p9 being Series of n,L
for a being Element of L holds a * (p *' p9) = p *' (a * p9)
let p, p9 be Series of n,L; ::_thesis: for a being Element of L holds a * (p *' p9) = p *' (a * p9)
let a be Element of L; ::_thesis: a * (p *' p9) = p *' (a * p9)
set app = a * (p *' p9);
set pap = p *' (a * p9);
set pp = p *' p9;
A1: now__::_thesis:_for_u_being_set_st_u_in_dom_(a_*_(p_*'_p9))_holds_
(a_*_(p_*'_p9))_._u_=_(p_*'_(a_*_p9))_._u
let u be set ; ::_thesis: ( u in dom (a * (p *' p9)) implies (a * (p *' p9)) . u = (p *' (a * p9)) . u )
assume u in dom (a * (p *' p9)) ; ::_thesis: (a * (p *' p9)) . u = (p *' (a * p9)) . u
then reconsider b = u as bag of n ;
consider s being FinSequence of the carrier of L such that
A2: (p *' (a * p9)) . b = Sum s and
A3: len s = len (decomp b) and
A4: for k being Element of NAT st k in dom s holds
ex b1, b2 being bag of n st
( (decomp b) /. k = <*b1,b2*> & s /. k = (p . b1) * ((a * p9) . b2) ) by POLYNOM1:def_9;
consider t being FinSequence of the carrier of L such that
A5: (p *' p9) . b = Sum t and
A6: len t = len (decomp b) and
A7: for k being Element of NAT st k in dom t holds
ex b1, b2 being bag of n st
( (decomp b) /. k = <*b1,b2*> & t /. k = (p . b1) * (p9 . b2) ) by POLYNOM1:def_9;
A8: dom t = Seg (len s) by A3, A6, FINSEQ_1:def_3
.= dom s by FINSEQ_1:def_3 ;
now__::_thesis:_for_i_being_set_st_i_in_dom_t_holds_
s_/._i_=_a_*_(t_/._i)
let i be set ; ::_thesis: ( i in dom t implies s /. i = a * (t /. i) )
assume A9: i in dom t ; ::_thesis: s /. i = a * (t /. i)
then reconsider k = i as Element of NAT ;
consider b1, b2 being bag of n such that
A10: (decomp b) /. k = <*b1,b2*> and
A11: t /. k = (p . b1) * (p9 . b2) by A7, A9;
consider a1, a2 being bag of n such that
A12: (decomp b) /. k = <*a1,a2*> and
A13: s /. k = (p . a1) * ((a * p9) . a2) by A4, A8, A9;
A14: b2 = <*a1,a2*> . 2 by A10, A12, FINSEQ_1:44
.= a2 by FINSEQ_1:44 ;
b1 = <*a1,a2*> . 1 by A10, A12, FINSEQ_1:44
.= a1 by FINSEQ_1:44 ;
hence s /. i = (p . b1) * (a * (p9 . b2)) by A13, A14, POLYNOM7:def_9
.= a * (t /. i) by A11, GROUP_1:def_3 ;
::_thesis: verum
end;
then s = a * t by A8, POLYNOM1:def_1;
then (p *' (a * p9)) . b = a * (Sum t) by A2, POLYNOM1:12
.= (a * (p *' p9)) . b by A5, POLYNOM7:def_9 ;
hence (a * (p *' p9)) . u = (p *' (a * p9)) . u ; ::_thesis: verum
end;
dom (a * (p *' p9)) = Bags n by FUNCT_2:def_1
.= dom (p *' (a * p9)) by FUNCT_2:def_1 ;
hence a * (p *' p9) = p *' (a * p9) by A1, FUNCT_1:2; ::_thesis: verum
end;
begin
definition
let n be Ordinal;
let b be bag of n;
let L be non empty ZeroStr ;
let p be Series of n,L;
funcb *' p -> Series of n,L means :Def1: :: POLYRED:def 1
for b9 being bag of n st b divides b9 holds
( it . b9 = p . (b9 -' b) & ( for b9 being bag of n st not b divides b9 holds
it . b9 = 0. L ) );
existence
ex b1 being Series of n,L st
for b9 being bag of n st b divides b9 holds
( b1 . b9 = p . (b9 -' b) & ( for b9 being bag of n st not b divides b9 holds
b1 . b9 = 0. L ) )
proof
set M1 = { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } ;
set M2 = { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } ;
set M = { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } \/ { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } ;
now__::_thesis:_for_u_being_set_st_u_in__{__[b9,(p_._(b9_-'_b))]_where_b9_is_Element_of_Bags_n_:_b_divides_b9__}__\/__{__[b9,(0._L)]_where_b9_is_Element_of_Bags_n_:_not_b_divides_b9__}__holds_
u_in_[:(Bags_n),_the_carrier_of_L:]
let u be set ; ::_thesis: ( u in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } \/ { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } implies u in [:(Bags n), the carrier of L:] )
assume A1: u in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } \/ { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } ; ::_thesis: u in [:(Bags n), the carrier of L:]
now__::_thesis:_(_(_u_in__{__[b9,(p_._(b9_-'_b))]_where_b9_is_Element_of_Bags_n_:_b_divides_b9__}__&_u_in_[:(Bags_n),_the_carrier_of_L:]_)_or_(_u_in__{__[b9,(0._L)]_where_b9_is_Element_of_Bags_n_:_not_b_divides_b9__}__&_u_in_[:(Bags_n),_the_carrier_of_L:]_)_)
percases ( u in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } or u in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } ) by A1, XBOOLE_0:def_3;
case u in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } ; ::_thesis: u in [:(Bags n), the carrier of L:]
then ex b9 being Element of Bags n st
( u = [b9,(p . (b9 -' b))] & b divides b9 ) ;
hence u in [:(Bags n), the carrier of L:] by ZFMISC_1:def_2; ::_thesis: verum
end;
case u in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } ; ::_thesis: u in [:(Bags n), the carrier of L:]
then ex b9 being Element of Bags n st
( u = [b9,(0. L)] & not b divides b9 ) ;
hence u in [:(Bags n), the carrier of L:] by ZFMISC_1:def_2; ::_thesis: verum
end;
end;
end;
hence u in [:(Bags n), the carrier of L:] ; ::_thesis: verum
end;
then reconsider M = { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } \/ { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } as Relation of (Bags n), the carrier of L by TARSKI:def_3;
A2: now__::_thesis:_for_u_being_set_st_u_in_Bags_n_holds_
u_in_dom_M
let u be set ; ::_thesis: ( u in Bags n implies u in dom M )
assume u in Bags n ; ::_thesis: u in dom M
then reconsider u9 = u as bag of n ;
A3: u9 is Element of Bags n by PRE_POLY:def_12;
now__::_thesis:_(_(_not_b_divides_u9_&_u_in_dom_M_)_or_(_b_divides_u9_&_u_in_dom_M_)_)
percases ( not b divides u9 or b divides u9 ) ;
case not b divides u9 ; ::_thesis: u in dom M
then [u9,(0. L)] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } by A3;
then [u9,(0. L)] in M by XBOOLE_0:def_3;
hence u in dom M by XTUPLE_0:def_12; ::_thesis: verum
end;
case b divides u9 ; ::_thesis: u in dom M
then [u9,(p . (u9 -' b))] in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } by A3;
then [u9,(p . (u9 -' b))] in M by XBOOLE_0:def_3;
hence u in dom M by XTUPLE_0:def_12; ::_thesis: verum
end;
end;
end;
hence u in dom M ; ::_thesis: verum
end;
for u being set st u in dom M holds
u in Bags n ;
then A4: dom M = Bags n by A2, TARSKI:1;
A5: now__::_thesis:_for_x,_y1,_y2_being_set_st_[x,y1]_in_M_&_[x,y2]_in_M_&_not_(_[x,y1]_in__{__[b9,(p_._(b9_-'_b))]_where_b9_is_Element_of_Bags_n_:_b_divides_b9__}__&_[x,y2]_in__{__[b9,(p_._(b9_-'_b))]_where_b9_is_Element_of_Bags_n_:_b_divides_b9__}__)_holds_
(_[x,y1]_in__{__[b9,(0._L)]_where_b9_is_Element_of_Bags_n_:_not_b_divides_b9__}__&_[x,y2]_in__{__[b9,(0._L)]_where_b9_is_Element_of_Bags_n_:_not_b_divides_b9__}__)
let x, y1, y2 be set ; ::_thesis: ( [x,y1] in M & [x,y2] in M & not ( [x,y1] in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } & [x,y2] in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } ) implies ( [x,y1] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } & [x,y2] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } ) )
assume A6: ( [x,y1] in M & [x,y2] in M ) ; ::_thesis: ( ( [x,y1] in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } & [x,y2] in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } ) or ( [x,y1] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } & [x,y2] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } ) )
A7: now__::_thesis:_(_[x,y1]_in__{__[b9,(0._L)]_where_b9_is_Element_of_Bags_n_:_not_b_divides_b9__}__implies_not_[x,y2]_in__{__[b9,(p_._(b9_-'_b))]_where_b9_is_Element_of_Bags_n_:_b_divides_b9__}__)
assume that
A8: [x,y1] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } and
A9: [x,y2] in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } ; ::_thesis: contradiction
consider v being Element of Bags n such that
A10: [v,(0. L)] = [x,y1] and
A11: not b divides v by A8;
consider u being Element of Bags n such that
A12: [u,(p . (u -' b))] = [x,y2] and
A13: b divides u by A9;
u = x by A12, XTUPLE_0:1
.= v by A10, XTUPLE_0:1 ;
hence contradiction by A13, A11; ::_thesis: verum
end;
A14: now__::_thesis:_(_[x,y1]_in__{__[b9,(p_._(b9_-'_b))]_where_b9_is_Element_of_Bags_n_:_b_divides_b9__}__implies_not_[x,y2]_in__{__[b9,(0._L)]_where_b9_is_Element_of_Bags_n_:_not_b_divides_b9__}__)
assume that
A15: [x,y1] in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } and
A16: [x,y2] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } ; ::_thesis: contradiction
consider v being Element of Bags n such that
A17: [v,(0. L)] = [x,y2] and
A18: not b divides v by A16;
consider u being Element of Bags n such that
A19: [u,(p . (u -' b))] = [x,y1] and
A20: b divides u by A15;
u = x by A19, XTUPLE_0:1
.= v by A17, XTUPLE_0:1 ;
hence contradiction by A20, A18; ::_thesis: verum
end;
thus ( ( [x,y1] in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } & [x,y2] in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } ) or ( [x,y1] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } & [x,y2] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } ) ) ::_thesis: verum
proof
assume A21: ( not [x,y1] in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } or not [x,y2] in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } ) ; ::_thesis: ( [x,y1] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } & [x,y2] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } )
now__::_thesis:_(_(_not_[x,y1]_in__{__[b9,(p_._(b9_-'_b))]_where_b9_is_Element_of_Bags_n_:_b_divides_b9__}__&_[x,y1]_in__{__[b9,(0._L)]_where_b9_is_Element_of_Bags_n_:_not_b_divides_b9__}__&_[x,y2]_in__{__[b9,(0._L)]_where_b9_is_Element_of_Bags_n_:_not_b_divides_b9__}__)_or_(_not_[x,y2]_in__{__[b9,(p_._(b9_-'_b))]_where_b9_is_Element_of_Bags_n_:_b_divides_b9__}__&_[x,y1]_in__{__[b9,(0._L)]_where_b9_is_Element_of_Bags_n_:_not_b_divides_b9__}__&_[x,y2]_in__{__[b9,(0._L)]_where_b9_is_Element_of_Bags_n_:_not_b_divides_b9__}__)_)
percases ( not [x,y1] in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } or not [x,y2] in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } ) by A21;
case not [x,y1] in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } ; ::_thesis: ( [x,y1] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } & [x,y2] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } )
hence ( [x,y1] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } & [x,y2] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } ) by A6, A7, XBOOLE_0:def_3; ::_thesis: verum
end;
case not [x,y2] in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } ; ::_thesis: ( [x,y1] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } & [x,y2] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } )
hence ( [x,y1] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } & [x,y2] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } ) by A6, A14, XBOOLE_0:def_3; ::_thesis: verum
end;
end;
end;
hence ( [x,y1] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } & [x,y2] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } ) ; ::_thesis: verum
end;
end;
now__::_thesis:_for_x,_y1,_y2_being_set_st_[x,y1]_in_M_&_[x,y2]_in_M_holds_
y1_=_y2
let x, y1, y2 be set ; ::_thesis: ( [x,y1] in M & [x,y2] in M implies y1 = y2 )
assume A22: ( [x,y1] in M & [x,y2] in M ) ; ::_thesis: y1 = y2
now__::_thesis:_(_(_[x,y1]_in__{__[b9,(p_._(b9_-'_b))]_where_b9_is_Element_of_Bags_n_:_b_divides_b9__}__&_[x,y2]_in__{__[b9,(p_._(b9_-'_b))]_where_b9_is_Element_of_Bags_n_:_b_divides_b9__}__&_y1_=_y2_)_or_(_[x,y1]_in__{__[b9,(0._L)]_where_b9_is_Element_of_Bags_n_:_not_b_divides_b9__}__&_[x,y2]_in__{__[b9,(0._L)]_where_b9_is_Element_of_Bags_n_:_not_b_divides_b9__}__&_y1_=_y2_)_)
percases ( ( [x,y1] in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } & [x,y2] in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } ) or ( [x,y1] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } & [x,y2] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } ) ) by A5, A22;
caseA23: ( [x,y1] in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } & [x,y2] in { [b9,(p . (b9 -' b))] where b9 is Element of Bags n : b divides b9 } ) ; ::_thesis: y1 = y2
then consider v being Element of Bags n such that
A24: [v,(p . (v -' b))] = [x,y2] and
b divides v ;
consider u being Element of Bags n such that
A25: [u,(p . (u -' b))] = [x,y1] and
b divides u by A23;
u = x by A25, XTUPLE_0:1
.= v by A24, XTUPLE_0:1 ;
hence y1 = p . (v -' b) by A25, XTUPLE_0:1
.= y2 by A24, XTUPLE_0:1 ;
::_thesis: verum
end;
caseA26: ( [x,y1] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } & [x,y2] in { [b9,(0. L)] where b9 is Element of Bags n : not b divides b9 } ) ; ::_thesis: y1 = y2
then A27: ex v being Element of Bags n st
( [v,(0. L)] = [x,y2] & not b divides v ) ;
A28: ex u being Element of Bags n st
( [u,(0. L)] = [x,y1] & not b divides u ) by A26;
thus y1 = 0. L by A28, XTUPLE_0:1
.= y2 by A27, XTUPLE_0:1 ; ::_thesis: verum
end;
end;
end;
hence y1 = y2 ; ::_thesis: verum
end;
then reconsider M = M as Function of (Bags n), the carrier of L by A4, FUNCT_1:def_1, FUNCT_2:def_1;
reconsider M = M as Function of (Bags n),L ;
take M ; ::_thesis: for b9 being bag of n st b divides b9 holds
( M . b9 = p . (b9 -' b) & ( for b9 being bag of n st not b divides b9 holds
M . b9 = 0. L ) )
A29: now__::_thesis:_for_b9_being_bag_of_n_st_not_b_divides_b9_holds_
M_._b9_=_0._L
let b9 be bag of n; ::_thesis: ( not b divides b9 implies M . b9 = 0. L )
A30: b9 is Element of Bags n by PRE_POLY:def_12;
assume not b divides b9 ; ::_thesis: M . b9 = 0. L
then [b9,(0. L)] in { [u,(0. L)] where u is Element of Bags n : not b divides u } by A30;
then [b9,(0. L)] in M by XBOOLE_0:def_3;
hence M . b9 = 0. L by FUNCT_1:1; ::_thesis: verum
end;
now__::_thesis:_for_b9_being_bag_of_n_st_b_divides_b9_holds_
M_._b9_=_p_._(b9_-'_b)
let b9 be bag of n; ::_thesis: ( b divides b9 implies M . b9 = p . (b9 -' b) )
A31: b9 is Element of Bags n by PRE_POLY:def_12;
assume b divides b9 ; ::_thesis: M . b9 = p . (b9 -' b)
then [b9,(p . (b9 -' b))] in { [u,(p . (u -' b))] where u is Element of Bags n : b divides u } by A31;
then [b9,(p . (b9 -' b))] in M by XBOOLE_0:def_3;
hence M . b9 = p . (b9 -' b) by FUNCT_1:1; ::_thesis: verum
end;
hence for b9 being bag of n st b divides b9 holds
( M . b9 = p . (b9 -' b) & ( for b9 being bag of n st not b divides b9 holds
M . b9 = 0. L ) ) by A29; ::_thesis: verum
end;
uniqueness
for b1, b2 being Series of n,L st ( for b9 being bag of n st b divides b9 holds
( b1 . b9 = p . (b9 -' b) & ( for b9 being bag of n st not b divides b9 holds
b1 . b9 = 0. L ) ) ) & ( for b9 being bag of n st b divides b9 holds
( b2 . b9 = p . (b9 -' b) & ( for b9 being bag of n st not b divides b9 holds
b2 . b9 = 0. L ) ) ) holds
b1 = b2
proof
let p1, p2 be Series of n,L; ::_thesis: ( ( for b9 being bag of n st b divides b9 holds
( p1 . b9 = p . (b9 -' b) & ( for b9 being bag of n st not b divides b9 holds
p1 . b9 = 0. L ) ) ) & ( for b9 being bag of n st b divides b9 holds
( p2 . b9 = p . (b9 -' b) & ( for b9 being bag of n st not b divides b9 holds
p2 . b9 = 0. L ) ) ) implies p1 = p2 )
assume that
A32: for b9 being bag of n st b divides b9 holds
( p1 . b9 = p . (b9 -' b) & ( for b9 being bag of n st not b divides b9 holds
p1 . b9 = 0. L ) ) and
A33: for b9 being bag of n st b divides b9 holds
( p2 . b9 = p . (b9 -' b) & ( for b9 being bag of n st not b divides b9 holds
p2 . b9 = 0. L ) ) ; ::_thesis: p1 = p2
now__::_thesis:_for_x_being_Element_of_Bags_n_holds_p1_._x_=_p2_._x
let x be Element of Bags n; ::_thesis: p1 . x = p2 . x
now__::_thesis:_(_(_b_divides_x_&_p1_._x_=_p2_._x_)_or_(_not_b_divides_x_&_p1_._x_=_p2_._x_)_)
percases ( b divides x or not b divides x ) ;
caseA34: b divides x ; ::_thesis: p1 . x = p2 . x
hence p1 . x = p . (x -' b) by A32
.= p2 . x by A33, A34 ;
::_thesis: verum
end;
caseA35: not b divides x ; ::_thesis: p1 . x = p2 . x
hence p1 . x = 0. L by A32
.= p2 . x by A33, A35 ;
::_thesis: verum
end;
end;
end;
hence p1 . x = p2 . x ; ::_thesis: verum
end;
hence p1 = p2 by FUNCT_2:63; ::_thesis: verum
end;
end;
:: deftheorem Def1 defines *' POLYRED:def_1_:_
for n being Ordinal
for b being bag of n
for L being non empty ZeroStr
for p, b5 being Series of n,L holds
( b5 = b *' p iff for b9 being bag of n st b divides b9 holds
( b5 . b9 = p . (b9 -' b) & ( for b9 being bag of n st not b divides b9 holds
b5 . b9 = 0. L ) ) );
Lm9: for n being Ordinal
for b, b9 being bag of n
for L being non empty ZeroStr
for p being Series of n,L holds (b *' p) . (b9 + b) = p . b9
proof
let n be Ordinal; ::_thesis: for b, b9 being bag of n
for L being non empty ZeroStr
for p being Series of n,L holds (b *' p) . (b9 + b) = p . b9
let b, b9 be bag of n; ::_thesis: for L being non empty ZeroStr
for p being Series of n,L holds (b *' p) . (b9 + b) = p . b9
let L be non empty ZeroStr ; ::_thesis: for p being Series of n,L holds (b *' p) . (b9 + b) = p . b9
let p be Series of n,L; ::_thesis: (b *' p) . (b9 + b) = p . b9
b divides b9 + b by PRE_POLY:50;
hence (b *' p) . (b9 + b) = p . ((b9 + b) -' b) by Def1
.= p . b9 by PRE_POLY:48 ;
::_thesis: verum
end;
Lm10: for n being Ordinal
for L being non empty ZeroStr
for p being Polynomial of n,L
for b being bag of n holds Support (b *' p) c= { (b + b9) where b9 is Element of Bags n : b9 in Support p }
proof
let n be Ordinal; ::_thesis: for L being non empty ZeroStr
for p being Polynomial of n,L
for b being bag of n holds Support (b *' p) c= { (b + b9) where b9 is Element of Bags n : b9 in Support p }
let L be non empty ZeroStr ; ::_thesis: for p being Polynomial of n,L
for b being bag of n holds Support (b *' p) c= { (b + b9) where b9 is Element of Bags n : b9 in Support p }
let p be Polynomial of n,L; ::_thesis: for b being bag of n holds Support (b *' p) c= { (b + b9) where b9 is Element of Bags n : b9 in Support p }
let b be bag of n; ::_thesis: Support (b *' p) c= { (b + b9) where b9 is Element of Bags n : b9 in Support p }
now__::_thesis:_for_u_being_set_st_u_in_Support_(b_*'_p)_holds_
u_in__{__(b_+_b9)_where_b9_is_Element_of_Bags_n_:_b9_in_Support_p__}_
let u be set ; ::_thesis: ( u in Support (b *' p) implies u in { (b + b9) where b9 is Element of Bags n : b9 in Support p } )
assume A1: u in Support (b *' p) ; ::_thesis: u in { (b + b9) where b9 is Element of Bags n : b9 in Support p }
then reconsider u9 = u as Element of Bags n ;
A2: (b *' p) . u9 <> 0. L by A1, POLYNOM1:def_3;
then b divides u9 by Def1;
then consider s being bag of n such that
A3: u9 = b + s by TERMORD:1;
( s is Element of Bags n & p . s <> 0. L ) by A2, A3, Lm9, PRE_POLY:def_12;
then s in Support p by POLYNOM1:def_3;
hence u in { (b + b9) where b9 is Element of Bags n : b9 in Support p } by A3; ::_thesis: verum
end;
hence Support (b *' p) c= { (b + b9) where b9 is Element of Bags n : b9 in Support p } by TARSKI:def_3; ::_thesis: verum
end;
registration
let n be Ordinal;
let b be bag of n;
let L be non empty ZeroStr ;
let p be finite-Support Series of n,L;
clusterb *' p -> finite-Support ;
coherence
b *' p is finite-Support
proof
set f = { [b9,(b + b9)] where b9 is Element of Bags n : b9 in Support p } ;
A1: now__::_thesis:_for_u_being_set_st_u_in__{__[b9,(b_+_b9)]_where_b9_is_Element_of_Bags_n_:_b9_in_Support_p__}__holds_
ex_y,_z_being_set_st_u_=_[y,z]
let u be set ; ::_thesis: ( u in { [b9,(b + b9)] where b9 is Element of Bags n : b9 in Support p } implies ex y, z being set st u = [y,z] )
assume u in { [b9,(b + b9)] where b9 is Element of Bags n : b9 in Support p } ; ::_thesis: ex y, z being set st u = [y,z]
then ex b9 being Element of Bags n st
( u = [b9,(b + b9)] & b9 in Support p ) ;
hence ex y, z being set st u = [y,z] ; ::_thesis: verum
end;
now__::_thesis:_for_x,_y1,_y2_being_set_st_[x,y1]_in__{__[b9,(b_+_b9)]_where_b9_is_Element_of_Bags_n_:_b9_in_Support_p__}__&_[x,y2]_in__{__[b9,(b_+_b9)]_where_b9_is_Element_of_Bags_n_:_b9_in_Support_p__}__holds_
y1_=_y2
let x, y1, y2 be set ; ::_thesis: ( [x,y1] in { [b9,(b + b9)] where b9 is Element of Bags n : b9 in Support p } & [x,y2] in { [b9,(b + b9)] where b9 is Element of Bags n : b9 in Support p } implies y1 = y2 )
assume that
A2: [x,y1] in { [b9,(b + b9)] where b9 is Element of Bags n : b9 in Support p } and
A3: [x,y2] in { [b9,(b + b9)] where b9 is Element of Bags n : b9 in Support p } ; ::_thesis: y1 = y2
consider b2 being Element of Bags n such that
A4: [x,y2] = [b2,(b + b2)] and
b2 in Support p by A3;
consider b1 being Element of Bags n such that
A5: [x,y1] = [b1,(b + b1)] and
b1 in Support p by A2;
b1 = x by A5, XTUPLE_0:1
.= b2 by A4, XTUPLE_0:1 ;
hence y1 = y2 by A5, A4, XTUPLE_0:1; ::_thesis: verum
end;
then reconsider f = { [b9,(b + b9)] where b9 is Element of Bags n : b9 in Support p } as Function by A1, FUNCT_1:def_1, RELAT_1:def_1;
A6: now__::_thesis:_for_u_being_set_st_u_in_dom_f_holds_
u_in_Support_p
let u be set ; ::_thesis: ( u in dom f implies u in Support p )
assume u in dom f ; ::_thesis: u in Support p
then consider v being set such that
A7: [u,v] in f by XTUPLE_0:def_12;
consider b9 being Element of Bags n such that
A8: [u,v] = [b9,(b + b9)] and
A9: b9 in Support p by A7;
u = b9 by A8, XTUPLE_0:1;
hence u in Support p by A9; ::_thesis: verum
end;
A10: Support (b *' p) c= { (b + b9) where b9 is Element of Bags n : b9 in Support p } by Lm10;
now__::_thesis:_for_u_being_set_st_u_in_Support_(b_*'_p)_holds_
u_in_rng_f
let u be set ; ::_thesis: ( u in Support (b *' p) implies u in rng f )
assume u in Support (b *' p) ; ::_thesis: u in rng f
then u in { (b + b9) where b9 is Element of Bags n : b9 in Support p } by A10;
then consider b9 being Element of Bags n such that
A11: ( u = b + b9 & b9 in Support p ) ;
[b9,u] in f by A11;
hence u in rng f by XTUPLE_0:def_13; ::_thesis: verum
end;
then A12: Support (b *' p) c= rng f by TARSKI:def_3;
now__::_thesis:_for_u_being_set_st_u_in_Support_p_holds_
u_in_dom_f
let u be set ; ::_thesis: ( u in Support p implies u in dom f )
assume A13: u in Support p ; ::_thesis: u in dom f
then reconsider u9 = u as Element of Bags n ;
[u9,(b + u9)] in { [b9,(b + b9)] where b9 is Element of Bags n : b9 in Support p } by A13;
hence u in dom f by XTUPLE_0:def_12; ::_thesis: verum
end;
then dom f = Support p by A6, TARSKI:1;
then dom f is finite by POLYNOM1:def_4;
then rng f is finite by FINSET_1:8;
hence b *' p is finite-Support by A12, POLYNOM1:def_4; ::_thesis: verum
end;
end;
theorem :: POLYRED:13
for n being Ordinal
for b, b9 being bag of n
for L being non empty ZeroStr
for p being Series of n,L holds (b *' p) . (b9 + b) = p . b9 by Lm9;
theorem :: POLYRED:14
for n being Ordinal
for L being non empty ZeroStr
for p being Polynomial of n,L
for b being bag of n holds Support (b *' p) c= { (b + b9) where b9 is Element of Bags n : b9 in Support p } by Lm10;
theorem Th15: :: POLYRED:15
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial ZeroStr
for p being non-zero Polynomial of n,L
for b being bag of n holds HT ((b *' p),T) = b + (HT (p,T))
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial ZeroStr
for p being non-zero Polynomial of n,L
for b being bag of n holds HT ((b *' p),T) = b + (HT (p,T))
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial ZeroStr
for p being non-zero Polynomial of n,L
for b being bag of n holds HT ((b *' p),T) = b + (HT (p,T))
let L be non trivial ZeroStr ; ::_thesis: for p being non-zero Polynomial of n,L
for b being bag of n holds HT ((b *' p),T) = b + (HT (p,T))
let p be non-zero Polynomial of n,L; ::_thesis: for b being bag of n holds HT ((b *' p),T) = b + (HT (p,T))
let b be bag of n; ::_thesis: HT ((b *' p),T) = b + (HT (p,T))
set htp = HT (p,T);
percases ( Support (b *' p) = {} or Support (b *' p) <> {} ) ;
supposeA1: Support (b *' p) = {} ; ::_thesis: HT ((b *' p),T) = b + (HT (p,T))
now__::_thesis:_not_Support_p_<>_{}
assume Support p <> {} ; ::_thesis: contradiction
then reconsider sp = Support p as non empty set ;
set u = the Element of sp;
the Element of sp in Support p ;
then reconsider u9 = the Element of sp as Element of Bags n ;
b divides b + u9 by PRE_POLY:50;
then (b *' p) . (b + u9) = p . ((b + u9) -' b) by Def1
.= p . u9 by PRE_POLY:48 ;
then A2: (b *' p) . (b + u9) <> 0. L by POLYNOM1:def_3;
b + u9 is Element of Bags n by PRE_POLY:def_12;
hence contradiction by A1, A2, POLYNOM1:def_3; ::_thesis: verum
end;
then p = 0_ (n,L) by POLYNOM7:1;
hence HT ((b *' p),T) = b + (HT (p,T)) by POLYNOM7:def_1; ::_thesis: verum
end;
supposeA3: Support (b *' p) <> {} ; ::_thesis: HT ((b *' p),T) = b + (HT (p,T))
now__::_thesis:_not_Support_p_=_{}
reconsider sp = Support (b *' p) as non empty set by A3;
set u = the Element of sp;
the Element of sp in Support (b *' p) ;
then reconsider u9 = the Element of sp as Element of Bags n ;
A4: u9 -' b is Element of Bags n by PRE_POLY:def_12;
A5: (b *' p) . u9 <> 0. L by POLYNOM1:def_3;
then b divides u9 by Def1;
then A6: p . (u9 -' b) <> 0. L by A5, Def1;
assume Support p = {} ; ::_thesis: contradiction
hence contradiction by A6, A4, POLYNOM1:def_3; ::_thesis: verum
end;
then HT (p,T) in Support p by TERMORD:def_6;
then A7: p . (HT (p,T)) <> 0. L by POLYNOM1:def_3;
A8: now__::_thesis:_for_b9_being_bag_of_n_st_b9_in_Support_(b_*'_p)_holds_
b9_<=_b_+_(HT_(p,T)),T
let b9 be bag of n; ::_thesis: ( b9 in Support (b *' p) implies b9 <= b + (HT (p,T)),T )
assume b9 in Support (b *' p) ; ::_thesis: b9 <= b + (HT (p,T)),T
then A9: (b *' p) . b9 <> 0. L by POLYNOM1:def_3;
then b divides b9 by Def1;
then consider b3 being bag of n such that
A10: b + b3 = b9 by TERMORD:1;
A11: b3 is Element of Bags n by PRE_POLY:def_12;
(b *' p) . b9 = p . b3 by A10, Lm9;
then b3 in Support p by A9, A11, POLYNOM1:def_3;
then b3 <= HT (p,T),T by TERMORD:def_6;
then [b3,(HT (p,T))] in T by TERMORD:def_2;
then [b9,(b + (HT (p,T)))] in T by A10, BAGORDER:def_5;
hence b9 <= b + (HT (p,T)),T by TERMORD:def_2; ::_thesis: verum
end;
( (b *' p) . (b + (HT (p,T))) = p . (HT (p,T)) & b + (HT (p,T)) is Element of Bags n ) by Lm9, PRE_POLY:def_12;
then b + (HT (p,T)) in Support (b *' p) by A7, POLYNOM1:def_3;
hence HT ((b *' p),T) = b + (HT (p,T)) by A8, TERMORD:def_6; ::_thesis: verum
end;
end;
end;
theorem Th16: :: POLYRED:16
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non empty ZeroStr
for p being Polynomial of n,L
for b, b9 being bag of n st b9 in Support (b *' p) holds
b9 <= b + (HT (p,T)),T
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty ZeroStr
for p being Polynomial of n,L
for b, b9 being bag of n st b9 in Support (b *' p) holds
b9 <= b + (HT (p,T)),T
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty ZeroStr
for p being Polynomial of n,L
for b, b9 being bag of n st b9 in Support (b *' p) holds
b9 <= b + (HT (p,T)),T
let L be non empty ZeroStr ; ::_thesis: for p being Polynomial of n,L
for b, b9 being bag of n st b9 in Support (b *' p) holds
b9 <= b + (HT (p,T)),T
let p be Polynomial of n,L; ::_thesis: for b, b9 being bag of n st b9 in Support (b *' p) holds
b9 <= b + (HT (p,T)),T
let b, b9 be bag of n; ::_thesis: ( b9 in Support (b *' p) implies b9 <= b + (HT (p,T)),T )
assume A1: b9 in Support (b *' p) ; ::_thesis: b9 <= b + (HT (p,T)),T
Support (b *' p) c= { (b + b2) where b2 is Element of Bags n : b2 in Support p } by Lm10;
then b9 in { (b + b2) where b2 is Element of Bags n : b2 in Support p } by A1;
then consider s being Element of Bags n such that
A2: b9 = b + s and
A3: s in Support p ;
s <= HT (p,T),T by A3, TERMORD:def_6;
then [s,(HT (p,T))] in T by TERMORD:def_2;
then [(b + s),(b + (HT (p,T)))] in T by BAGORDER:def_5;
hence b9 <= b + (HT (p,T)),T by A2, TERMORD:def_2; ::_thesis: verum
end;
theorem :: POLYRED:17
for n being Ordinal
for T being connected TermOrder of n
for L being non empty ZeroStr
for p being Series of n,L holds (EmptyBag n) *' p = p
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non empty ZeroStr
for p being Series of n,L holds (EmptyBag n) *' p = p
let T be connected TermOrder of n; ::_thesis: for L being non empty ZeroStr
for p being Series of n,L holds (EmptyBag n) *' p = p
let L be non empty ZeroStr ; ::_thesis: for p being Series of n,L holds (EmptyBag n) *' p = p
let p be Series of n,L; ::_thesis: (EmptyBag n) *' p = p
set e = EmptyBag n;
A1: now__::_thesis:_for_u_being_set_st_u_in_dom_p_holds_
((EmptyBag_n)_*'_p)_._u_=_p_._u
let u be set ; ::_thesis: ( u in dom p implies ((EmptyBag n) *' p) . u = p . u )
assume u in dom p ; ::_thesis: ((EmptyBag n) *' p) . u = p . u
then reconsider u9 = u as Element of Bags n ;
EmptyBag n divides u9 by PRE_POLY:59;
then ((EmptyBag n) *' p) . u9 = p . (u9 -' (EmptyBag n)) by Def1
.= p . u9 by PRE_POLY:54 ;
hence ((EmptyBag n) *' p) . u = p . u ; ::_thesis: verum
end;
dom ((EmptyBag n) *' p) = Bags n by FUNCT_2:def_1
.= dom p by FUNCT_2:def_1 ;
hence (EmptyBag n) *' p = p by A1, FUNCT_1:2; ::_thesis: verum
end;
theorem Th18: :: POLYRED:18
for n being Ordinal
for T being connected TermOrder of n
for L being non empty ZeroStr
for p being Series of n,L
for b1, b2 being bag of n holds (b1 + b2) *' p = b1 *' (b2 *' p)
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non empty ZeroStr
for p being Series of n,L
for b1, b2 being bag of n holds (b1 + b2) *' p = b1 *' (b2 *' p)
let T be connected TermOrder of n; ::_thesis: for L being non empty ZeroStr
for p being Series of n,L
for b1, b2 being bag of n holds (b1 + b2) *' p = b1 *' (b2 *' p)
let L be non empty ZeroStr ; ::_thesis: for p being Series of n,L
for b1, b2 being bag of n holds (b1 + b2) *' p = b1 *' (b2 *' p)
let p be Series of n,L; ::_thesis: for b1, b2 being bag of n holds (b1 + b2) *' p = b1 *' (b2 *' p)
let b1, b2 be bag of n; ::_thesis: (b1 + b2) *' p = b1 *' (b2 *' p)
set q = (b1 + b2) *' p;
set r = b1 *' (b2 *' p);
A1: now__::_thesis:_for_u_being_set_st_u_in_dom_((b1_+_b2)_*'_p)_holds_
((b1_+_b2)_*'_p)_._u_=_(b1_*'_(b2_*'_p))_._u
let u be set ; ::_thesis: ( u in dom ((b1 + b2) *' p) implies ((b1 + b2) *' p) . u = (b1 *' (b2 *' p)) . u )
assume u in dom ((b1 + b2) *' p) ; ::_thesis: ((b1 + b2) *' p) . u = (b1 *' (b2 *' p)) . u
then reconsider b = u as bag of n ;
now__::_thesis:_(_(_b1_+_b2_divides_b_&_(b1_*'_(b2_*'_p))_._b_=_((b1_+_b2)_*'_p)_._b_)_or_(_not_b1_+_b2_divides_b_&_((b1_+_b2)_*'_p)_._b_=_(b1_*'_(b2_*'_p))_._b_)_)
percases ( b1 + b2 divides b or not b1 + b2 divides b ) ;
caseA2: b1 + b2 divides b ; ::_thesis: (b1 *' (b2 *' p)) . b = ((b1 + b2) *' p) . b
then consider b3 being bag of n such that
A3: (b1 + b2) + b3 = b by TERMORD:1;
A4: b1 + (b2 + b3) = b by A3, PRE_POLY:35;
then b2 + b3 = b -' b1 by PRE_POLY:48;
then A5: b2 divides b -' b1 by TERMORD:1;
b1 divides b by A4, TERMORD:1;
then (b1 *' (b2 *' p)) . b = (b2 *' p) . (b -' b1) by Def1;
hence (b1 *' (b2 *' p)) . b = p . ((b -' b1) -' b2) by A5, Def1
.= p . (b -' (b1 + b2)) by PRE_POLY:36
.= ((b1 + b2) *' p) . b by A2, Def1 ;
::_thesis: verum
end;
caseA6: not b1 + b2 divides b ; ::_thesis: ((b1 + b2) *' p) . b = (b1 *' (b2 *' p)) . b
then A7: ((b1 + b2) *' p) . b = 0. L by Def1;
now__::_thesis:_(_(_b1_divides_b_&_((b1_+_b2)_*'_p)_._b_=_(b1_*'_(b2_*'_p))_._b_)_or_(_not_b1_divides_b_&_((b1_+_b2)_*'_p)_._b_=_(b1_*'_(b2_*'_p))_._b_)_)
percases ( b1 divides b or not b1 divides b ) ;
caseA8: b1 divides b ; ::_thesis: ((b1 + b2) *' p) . b = (b1 *' (b2 *' p)) . b
A9: now__::_thesis:_not_b2_divides_b_-'_b1
assume b2 divides b -' b1 ; ::_thesis: contradiction
then ((b -' b1) -' b2) + b2 = b -' b1 by PRE_POLY:47;
then (((b -' b1) -' b2) + b2) + b1 = b by A8, PRE_POLY:47;
then ((b -' b1) -' b2) + (b2 + b1) = b by PRE_POLY:35;
hence contradiction by A6, TERMORD:1; ::_thesis: verum
end;
(b1 *' (b2 *' p)) . b = (b2 *' p) . (b -' b1) by A8, Def1;
hence ((b1 + b2) *' p) . b = (b1 *' (b2 *' p)) . b by A7, A9, Def1; ::_thesis: verum
end;
case not b1 divides b ; ::_thesis: ((b1 + b2) *' p) . b = (b1 *' (b2 *' p)) . b
hence ((b1 + b2) *' p) . b = (b1 *' (b2 *' p)) . b by A7, Def1; ::_thesis: verum
end;
end;
end;
hence ((b1 + b2) *' p) . b = (b1 *' (b2 *' p)) . b ; ::_thesis: verum
end;
end;
end;
hence ((b1 + b2) *' p) . u = (b1 *' (b2 *' p)) . u ; ::_thesis: verum
end;
dom ((b1 + b2) *' p) = Bags n by FUNCT_2:def_1
.= dom (b1 *' (b2 *' p)) by FUNCT_2:def_1 ;
hence (b1 + b2) *' p = b1 *' (b2 *' p) by A1, FUNCT_1:2; ::_thesis: verum
end;
theorem Th19: :: POLYRED:19
for n being Ordinal
for L being non trivial right_complementable distributive add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L
for a being Element of L holds Support (a * p) c= Support p
proof
let n be Ordinal; ::_thesis: for L being non trivial right_complementable distributive add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L
for a being Element of L holds Support (a * p) c= Support p
let L be non trivial right_complementable distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for p being Polynomial of n,L
for a being Element of L holds Support (a * p) c= Support p
let p be Polynomial of n,L; ::_thesis: for a being Element of L holds Support (a * p) c= Support p
let a9 be Element of L; ::_thesis: Support (a9 * p) c= Support p
A1: dom (0_ (n,L)) = Bags n by FUNCT_2:def_1
.= dom (a9 * p) by FUNCT_2:def_1 ;
percases ( a9 = 0. L or a9 <> 0. L ) ;
supposeA2: a9 = 0. L ; ::_thesis: Support (a9 * p) c= Support p
now__::_thesis:_for_u_being_set_st_u_in_dom_(a9_*_p)_holds_
(a9_*_p)_._u_=_(0__(n,L))_._u
let u be set ; ::_thesis: ( u in dom (a9 * p) implies (a9 * p) . u = (0_ (n,L)) . u )
assume u in dom (a9 * p) ; ::_thesis: (a9 * p) . u = (0_ (n,L)) . u
then reconsider u9 = u as Element of Bags n ;
(a9 * p) . u9 = a9 * (p . u9) by POLYNOM7:def_9
.= 0. L by A2, VECTSP_1:7
.= (0_ (n,L)) . u9 by POLYNOM1:22 ;
hence (a9 * p) . u = (0_ (n,L)) . u ; ::_thesis: verum
end;
then a9 * p = 0_ (n,L) by A1, FUNCT_1:2;
then for u being set st u in Support (a9 * p) holds
u in Support p by POLYNOM7:1;
hence Support (a9 * p) c= Support p by TARSKI:def_3; ::_thesis: verum
end;
suppose a9 <> 0. L ; ::_thesis: Support (a9 * p) c= Support p
then reconsider a = a9 as non zero Element of L by STRUCT_0:def_12;
now__::_thesis:_for_u_being_set_st_u_in_Support_(a_*_p)_holds_
u_in_Support_p
let u be set ; ::_thesis: ( u in Support (a * p) implies u in Support p )
assume A3: u in Support (a * p) ; ::_thesis: u in Support p
then reconsider u9 = u as Element of Bags n ;
A4: (a * p) . u9 = a * (p . u9) by POLYNOM7:def_9;
(a * p) . u9 <> 0. L by A3, POLYNOM1:def_3;
then p . u9 <> 0. L by A4, VECTSP_1:6;
hence u in Support p by POLYNOM1:def_3; ::_thesis: verum
end;
hence Support (a9 * p) c= Support p by TARSKI:def_3; ::_thesis: verum
end;
end;
end;
theorem :: POLYRED:20
for n being Ordinal
for L being non trivial domRing-like doubleLoopStr
for p being Polynomial of n,L
for a being non zero Element of L holds Support p c= Support (a * p)
proof
let n be Ordinal; ::_thesis: for L being non trivial domRing-like doubleLoopStr
for p being Polynomial of n,L
for a being non zero Element of L holds Support p c= Support (a * p)
let L be non trivial domRing-like doubleLoopStr ; ::_thesis: for p being Polynomial of n,L
for a being non zero Element of L holds Support p c= Support (a * p)
let p be Polynomial of n,L; ::_thesis: for a being non zero Element of L holds Support p c= Support (a * p)
let a be non zero Element of L; ::_thesis: Support p c= Support (a * p)
now__::_thesis:_for_u_being_set_st_u_in_Support_p_holds_
u_in_Support_(a_*_p)
let u be set ; ::_thesis: ( u in Support p implies u in Support (a * p) )
assume A1: u in Support p ; ::_thesis: u in Support (a * p)
then reconsider u9 = u as Element of Bags n ;
A2: ( (a * p) . u9 = a * (p . u9) & a <> 0. L ) by POLYNOM7:def_9;
p . u9 <> 0. L by A1, POLYNOM1:def_3;
then (a * p) . u9 <> 0. L by A2, VECTSP_2:def_1;
hence u in Support (a * p) by POLYNOM1:def_3; ::_thesis: verum
end;
hence Support p c= Support (a * p) by TARSKI:def_3; ::_thesis: verum
end;
theorem Th21: :: POLYRED:21
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable distributive add-associative right_zeroed domRing-like doubleLoopStr
for p being Polynomial of n,L
for a being non zero Element of L holds HT ((a * p),T) = HT (p,T)
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable distributive add-associative right_zeroed domRing-like doubleLoopStr
for p being Polynomial of n,L
for a being non zero Element of L holds HT ((a * p),T) = HT (p,T)
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable distributive add-associative right_zeroed domRing-like doubleLoopStr
for p being Polynomial of n,L
for a being non zero Element of L holds HT ((a * p),T) = HT (p,T)
let L be non trivial right_complementable distributive add-associative right_zeroed domRing-like doubleLoopStr ; ::_thesis: for p being Polynomial of n,L
for a being non zero Element of L holds HT ((a * p),T) = HT (p,T)
let p be Polynomial of n,L; ::_thesis: for a being non zero Element of L holds HT ((a * p),T) = HT (p,T)
let a be non zero Element of L; ::_thesis: HT ((a * p),T) = HT (p,T)
set ht = HT ((a * p),T);
set htp = HT (p,T);
percases ( Support (a * p) = {} or Support (a * p) <> {} ) ;
supposeA1: Support (a * p) = {} ; ::_thesis: HT ((a * p),T) = HT (p,T)
now__::_thesis:_not_Support_p_<>_{}
assume Support p <> {} ; ::_thesis: contradiction
then reconsider sp = Support p as non empty set ;
set u = the Element of sp;
the Element of sp in Support p ;
then reconsider u9 = the Element of sp as Element of Bags n ;
A2: (a * p) . u9 = a * (p . u9) by POLYNOM7:def_9;
( p . u9 <> 0. L & a <> 0. L ) by POLYNOM1:def_3;
then (a * p) . u9 <> 0. L by A2, VECTSP_2:def_1;
hence contradiction by A1, POLYNOM1:def_3; ::_thesis: verum
end;
hence HT (p,T) = EmptyBag n by TERMORD:def_6
.= HT ((a * p),T) by A1, TERMORD:def_6 ;
::_thesis: verum
end;
supposeA3: Support (a * p) <> {} ; ::_thesis: HT ((a * p),T) = HT (p,T)
now__::_thesis:_not_Support_p_=_{}
reconsider sp = Support (a * p) as non empty set by A3;
set u = the Element of sp;
the Element of sp in Support (a * p) ;
then reconsider u9 = the Element of sp as Element of Bags n ;
( (a * p) . u9 <> 0. L & (a * p) . u9 = a * (p . u9) ) by POLYNOM1:def_3, POLYNOM7:def_9;
then A4: p . u9 <> 0. L by VECTSP_1:6;
assume Support p = {} ; ::_thesis: contradiction
hence contradiction by A4, POLYNOM1:def_3; ::_thesis: verum
end;
then HT (p,T) in Support p by TERMORD:def_6;
then A5: p . (HT (p,T)) <> 0. L by POLYNOM1:def_3;
A6: now__::_thesis:_for_b_being_bag_of_n_st_b_in_Support_(a_*_p)_holds_
b_<=_HT_(p,T),T
let b be bag of n; ::_thesis: ( b in Support (a * p) implies b <= HT (p,T),T )
assume A7: b in Support (a * p) ; ::_thesis: b <= HT (p,T),T
Support (a * p) c= Support p by Th19;
hence b <= HT (p,T),T by A7, TERMORD:def_6; ::_thesis: verum
end;
(a * p) . (HT (p,T)) = a * (p . (HT (p,T))) by POLYNOM7:def_9;
then (a * p) . (HT (p,T)) <> 0. L by A5, VECTSP_2:def_1;
then HT (p,T) in Support (a * p) by POLYNOM1:def_3;
hence HT ((a * p),T) = HT (p,T) by A6, TERMORD:def_6; ::_thesis: verum
end;
end;
end;
theorem Th22: :: POLYRED:22
for n being Ordinal
for L being non trivial right_complementable distributive add-associative right_zeroed doubleLoopStr
for p being Series of n,L
for b being bag of n
for a being Element of L holds a * (b *' p) = (Monom (a,b)) *' p
proof
let n be Ordinal; ::_thesis: for L being non trivial right_complementable distributive add-associative right_zeroed doubleLoopStr
for p being Series of n,L
for b being bag of n
for a being Element of L holds a * (b *' p) = (Monom (a,b)) *' p
let L be non trivial right_complementable distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for p being Series of n,L
for b being bag of n
for a being Element of L holds a * (b *' p) = (Monom (a,b)) *' p
let p be Series of n,L; ::_thesis: for b being bag of n
for a being Element of L holds a * (b *' p) = (Monom (a,b)) *' p
let b be bag of n; ::_thesis: for a being Element of L holds a * (b *' p) = (Monom (a,b)) *' p
let a be Element of L; ::_thesis: a * (b *' p) = (Monom (a,b)) *' p
set q = a * (b *' p);
set q9 = (Monom (a,b)) *' p;
set m = Monom (a,b);
percases ( a <> 0. L or a = 0. L ) ;
suppose a <> 0. L ; ::_thesis: a * (b *' p) = (Monom (a,b)) *' p
then reconsider a = a as non zero Element of L by STRUCT_0:def_12;
A1: now__::_thesis:_for_u_being_set_st_u_in_dom_(a_*_(b_*'_p))_holds_
(a_*_(b_*'_p))_._u_=_((Monom_(a,b))_*'_p)_._u
let u be set ; ::_thesis: ( u in dom (a * (b *' p)) implies (a * (b *' p)) . u = ((Monom (a,b)) *' p) . u )
assume u in dom (a * (b *' p)) ; ::_thesis: (a * (b *' p)) . u = ((Monom (a,b)) *' p) . u
then reconsider s = u as bag of n ;
consider t being FinSequence of the carrier of L such that
A2: ((Monom (a,b)) *' p) . s = Sum t and
A3: len t = len (decomp s) and
A4: for k being Element of NAT st k in dom t holds
ex b1, b2 being bag of n st
( (decomp s) /. k = <*b1,b2*> & t /. k = ((Monom (a,b)) . b1) * (p . b2) ) by POLYNOM1:def_9;
A5: dom t = Seg (len (decomp s)) by A3, FINSEQ_1:def_3
.= dom (decomp s) by FINSEQ_1:def_3 ;
A6: term (Monom (a,b)) = b by POLYNOM7:10;
now__::_thesis:_(_(_b_divides_s_&_(a_*_(b_*'_p))_._s_=_((Monom_(a,b))_*'_p)_._s_)_or_(_not_b_divides_s_&_(a_*_(b_*'_p))_._u_=_((Monom_(a,b))_*'_p)_._u_)_)
percases ( b divides s or not b divides s ) ;
caseA7: b divides s ; ::_thesis: (a * (b *' p)) . s = ((Monom (a,b)) *' p) . s
A8: (a * (b *' p)) . s = a * ((b *' p) . s) by POLYNOM7:def_9
.= a * (p . (s -' b)) by A7, Def1 ;
consider s9 being bag of n such that
A9: b + s9 = s by A7, TERMORD:1;
consider i being Element of NAT such that
A10: i in dom (decomp s) and
A11: (decomp s) /. i = <*b,s9*> by A9, PRE_POLY:69;
consider b1, b2 being bag of n such that
A12: (decomp s) /. i = <*b1,b2*> and
A13: t /. i = ((Monom (a,b)) . b1) * (p . b2) by A4, A5, A10;
A14: b2 = <*b,s9*> . 2 by A11, A12, FINSEQ_1:44
.= s9 by FINSEQ_1:44 ;
A15: s -' b = s9 by A9, PRE_POLY:48;
A16: now__::_thesis:_for_i9_being_Element_of_NAT_st_i9_in_dom_t_&_i9_<>_i_holds_
t_/._i9_=_0._L
let i9 be Element of NAT ; ::_thesis: ( i9 in dom t & i9 <> i implies t /. i9 = 0. L )
assume that
A17: i9 in dom t and
A18: i9 <> i ; ::_thesis: t /. i9 = 0. L
consider b19, b29 being bag of n such that
A19: (decomp s) /. i9 = <*b19,b29*> and
A20: t /. i9 = ((Monom (a,b)) . b19) * (p . b29) by A4, A17;
consider h1, h2 being bag of n such that
A21: (decomp s) /. i9 = <*h1,h2*> and
A22: s = h1 + h2 by A5, A17, PRE_POLY:68;
A23: s -' h1 = h2 by A22, PRE_POLY:48;
A24: h1 = <*b19,b29*> . 1 by A19, A21, FINSEQ_1:44
.= b19 by FINSEQ_1:44 ;
now__::_thesis:_not_(Monom_(a,b))_._b19_<>_0._L
assume (Monom (a,b)) . b19 <> 0. L ; ::_thesis: contradiction
then b19 = b by A6, POLYNOM7:def_5;
then (decomp s) . i9 = (decomp s) /. i by A5, A11, A15, A17, A21, A24, A23, PARTFUN1:def_6
.= (decomp s) . i by A10, PARTFUN1:def_6 ;
hence contradiction by A5, A10, A17, A18, FUNCT_1:def_4; ::_thesis: verum
end;
hence t /. i9 = 0. L by A20, BINOM:1; ::_thesis: verum
end;
b1 = <*b,s9*> . 1 by A11, A12, FINSEQ_1:44
.= b by FINSEQ_1:44 ;
then Sum t = (coefficient (Monom (a,b))) * (p . (s -' b)) by A6, A5, A10, A15, A13, A14, A16, POLYNOM2:3
.= a * (p . (s -' b)) by POLYNOM7:9 ;
hence (a * (b *' p)) . s = ((Monom (a,b)) *' p) . s by A2, A8; ::_thesis: verum
end;
caseA25: not b divides s ; ::_thesis: (a * (b *' p)) . u = ((Monom (a,b)) *' p) . u
consider t being FinSequence of the carrier of L such that
A26: ((Monom (a,b)) *' p) . s = Sum t and
A27: len t = len (decomp s) and
A28: for k being Element of NAT st k in dom t holds
ex b1, b2 being bag of n st
( (decomp s) /. k = <*b1,b2*> & t /. k = ((Monom (a,b)) . b1) * (p . b2) ) by POLYNOM1:def_9;
A29: now__::_thesis:_for_k_being_Nat_st_k_in_dom_t_holds_
t_/._k_=_0._L
let k be Nat; ::_thesis: ( k in dom t implies t /. k = 0. L )
assume A30: k in dom t ; ::_thesis: t /. k = 0. L
then consider b19, b29 being bag of n such that
A31: (decomp s) /. k = <*b19,b29*> and
A32: t /. k = ((Monom (a,b)) . b19) * (p . b29) by A28;
A33: dom t = Seg (len (decomp s)) by A27, FINSEQ_1:def_3
.= dom (decomp s) by FINSEQ_1:def_3 ;
now__::_thesis:_(_(_b19_=_term_(Monom_(a,b))_&_contradiction_)_or_(_b19_<>_term_(Monom_(a,b))_&_(Monom_(a,b))_._b19_=_0._L_)_)
percases ( b19 = term (Monom (a,b)) or b19 <> term (Monom (a,b)) ) ;
caseA34: b19 = term (Monom (a,b)) ; ::_thesis: contradiction
consider h1, h2 being bag of n such that
A35: (decomp s) /. k = <*h1,h2*> and
A36: s = h1 + h2 by A30, A33, PRE_POLY:68;
h1 = <*b19,b29*> . 1 by A31, A35, FINSEQ_1:44
.= b19 by FINSEQ_1:44 ;
hence contradiction by A6, A25, A34, A36, TERMORD:1; ::_thesis: verum
end;
case b19 <> term (Monom (a,b)) ; ::_thesis: (Monom (a,b)) . b19 = 0. L
hence (Monom (a,b)) . b19 = 0. L by Lm8; ::_thesis: verum
end;
end;
end;
hence t /. k = 0. L by A32, BINOM:1; ::_thesis: verum
end;
(a * (b *' p)) . s = a * ((b *' p) . s) by POLYNOM7:def_9
.= a * (0. L) by A25, Def1
.= 0. L by BINOM:2 ;
hence (a * (b *' p)) . u = ((Monom (a,b)) *' p) . u by A26, A29, MATRLIN:11; ::_thesis: verum
end;
end;
end;
hence (a * (b *' p)) . u = ((Monom (a,b)) *' p) . u ; ::_thesis: verum
end;
dom (a * (b *' p)) = Bags n by FUNCT_2:def_1
.= dom ((Monom (a,b)) *' p) by FUNCT_2:def_1 ;
hence a * (b *' p) = (Monom (a,b)) *' p by A1, FUNCT_1:2; ::_thesis: verum
end;
supposeA37: a = 0. L ; ::_thesis: a * (b *' p) = (Monom (a,b)) *' p
A38: now__::_thesis:_for_u_being_set_st_u_in_dom_(Monom_(a,b))_holds_
(Monom_(a,b))_._u_=_(0__(n,L))_._u
let u be set ; ::_thesis: ( u in dom (Monom (a,b)) implies (Monom (a,b)) . u = (0_ (n,L)) . u )
assume u in dom (Monom (a,b)) ; ::_thesis: (Monom (a,b)) . u = (0_ (n,L)) . u
then reconsider u9 = u as Element of Bags n ;
now__::_thesis:_(_(_u9_=_term_(Monom_(a,b))_&_(Monom_(a,b))_._u_=_(0__(n,L))_._u_)_or_(_u9_<>_term_(Monom_(a,b))_&_(Monom_(a,b))_._u_=_(0__(n,L))_._u_)_)
percases ( u9 = term (Monom (a,b)) or u9 <> term (Monom (a,b)) ) ;
caseA39: u9 = term (Monom (a,b)) ; ::_thesis: (Monom (a,b)) . u = (0_ (n,L)) . u
coefficient (Monom (a,b)) = 0. L by A37, POLYNOM7:8;
hence (Monom (a,b)) . u = (0_ (n,L)) . u by A39, POLYNOM1:22; ::_thesis: verum
end;
case u9 <> term (Monom (a,b)) ; ::_thesis: (Monom (a,b)) . u = (0_ (n,L)) . u
then (Monom (a,b)) . u9 = 0. L by Lm8
.= (0_ (n,L)) . u9 by POLYNOM1:22 ;
hence (Monom (a,b)) . u = (0_ (n,L)) . u ; ::_thesis: verum
end;
end;
end;
hence (Monom (a,b)) . u = (0_ (n,L)) . u ; ::_thesis: verum
end;
dom (Monom (a,b)) = Bags n by FUNCT_2:def_1
.= dom (0_ (n,L)) by FUNCT_2:def_1 ;
then A40: Monom (a,b) = 0_ (n,L) by A38, FUNCT_1:2;
A41: now__::_thesis:_for_u_being_set_st_u_in_dom_(a_*_(b_*'_p))_holds_
(a_*_(b_*'_p))_._u_=_(0__(n,L))_._u
let u be set ; ::_thesis: ( u in dom (a * (b *' p)) implies (a * (b *' p)) . u = (0_ (n,L)) . u )
assume u in dom (a * (b *' p)) ; ::_thesis: (a * (b *' p)) . u = (0_ (n,L)) . u
then reconsider u9 = u as bag of n ;
(a * (b *' p)) . u9 = (0. L) * ((b *' p) . u9) by A37, POLYNOM7:def_9
.= 0. L by BINOM:1
.= (0_ (n,L)) . u9 by POLYNOM1:22 ;
hence (a * (b *' p)) . u = (0_ (n,L)) . u ; ::_thesis: verum
end;
dom (a * (b *' p)) = Bags n by FUNCT_2:def_1
.= dom (0_ (n,L)) by FUNCT_2:def_1 ;
then a * (b *' p) = 0_ (n,L) by A41, FUNCT_1:2;
hence a * (b *' p) = (Monom (a,b)) *' p by A40, Th5; ::_thesis: verum
end;
end;
end;
theorem :: POLYRED:23
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for p being non-zero Polynomial of n,L
for q being Polynomial of n,L
for m being non-zero Monomial of n,L st HT (p,T) in Support q holds
HT ((m *' p),T) in Support (m *' q)
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for p being non-zero Polynomial of n,L
for q being Polynomial of n,L
for m being non-zero Monomial of n,L st HT (p,T) in Support q holds
HT ((m *' p),T) in Support (m *' q)
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr
for p being non-zero Polynomial of n,L
for q being Polynomial of n,L
for m being non-zero Monomial of n,L st HT (p,T) in Support q holds
HT ((m *' p),T) in Support (m *' q)
let L be non trivial right_complementable well-unital distributive add-associative right_zeroed domRing-like doubleLoopStr ; ::_thesis: for p being non-zero Polynomial of n,L
for q being Polynomial of n,L
for m being non-zero Monomial of n,L st HT (p,T) in Support q holds
HT ((m *' p),T) in Support (m *' q)
let p be non-zero Polynomial of n,L; ::_thesis: for q being Polynomial of n,L
for m being non-zero Monomial of n,L st HT (p,T) in Support q holds
HT ((m *' p),T) in Support (m *' q)
let q be Polynomial of n,L; ::_thesis: for m being non-zero Monomial of n,L st HT (p,T) in Support q holds
HT ((m *' p),T) in Support (m *' q)
let m be non-zero Monomial of n,L; ::_thesis: ( HT (p,T) in Support q implies HT ((m *' p),T) in Support (m *' q) )
set a = coefficient m;
set b = term m;
assume HT (p,T) in Support q ; ::_thesis: HT ((m *' p),T) in Support (m *' q)
then A1: q . (HT (p,T)) <> 0. L by POLYNOM1:def_3;
A2: HC (m,T) <> 0. L ;
then reconsider a = coefficient m as non zero Element of L by TERMORD:23;
m = Monom (a,(term m)) by POLYNOM7:11;
then m *' p = a * ((term m) *' p) by Th22;
then HT ((m *' p),T) = HT (((term m) *' p),T) by Th21
.= (term m) + (HT (p,T)) by Th15 ;
then A3: (m *' q) . (HT ((m *' p),T)) = (m . (term m)) * (q . (HT (p,T))) by Th7;
m . (HT (m,T)) <> 0. L by A2, TERMORD:def_7;
then m . (term m) <> 0. L by POLYNOM7:def_5;
then (m *' q) . (HT ((m *' p),T)) <> 0. L by A1, A3, VECTSP_2:def_1;
hence HT ((m *' p),T) in Support (m *' q) by POLYNOM1:def_3; ::_thesis: verum
end;
begin
registration
let n be Ordinal;
let T be connected TermOrder of n;
cluster RelStr(# (Bags n),T #) -> connected ;
coherence
RelStr(# (Bags n),T #) is connected
proof
set L = RelStr(# (Bags n),T #);
now__::_thesis:_for_x,_y_being_Element_of_RelStr(#_(Bags_n),T_#)_holds_
(_x_<=_y_or_y_<=_x_)
let x, y be Element of RelStr(# (Bags n),T #); ::_thesis: ( x <= y or y <= x )
reconsider x9 = x as bag of n ;
reconsider y9 = y as bag of n ;
y9 <= y9,T by TERMORD:6;
then [y9,y9] in T by TERMORD:def_2;
then A1: y in field T by RELAT_1:15;
x9 <= x9,T by TERMORD:6;
then [x9,x9] in T by TERMORD:def_2;
then A2: ( T is_connected_in field T & x in field T ) by RELAT_1:15, RELAT_2:def_14;
now__::_thesis:_(_(_x_<>_y_&_(_x_<=_y_or_y_<=_x_)_)_or_(_x_=_y_&_(_x_<=_y_or_y_<=_x_)_)_)
percases ( x <> y or x = y ) ;
case x <> y ; ::_thesis: ( x <= y or y <= x )
then ( [x,y] in the InternalRel of RelStr(# (Bags n),T #) or [y,x] in the InternalRel of RelStr(# (Bags n),T #) ) by A2, A1, RELAT_2:def_6;
hence ( x <= y or y <= x ) by ORDERS_2:def_5; ::_thesis: verum
end;
case x = y ; ::_thesis: ( x <= y or y <= x )
then x9 <= y9,T by TERMORD:6;
then [x9,y9] in the InternalRel of RelStr(# (Bags n),T #) by TERMORD:def_2;
hence ( x <= y or y <= x ) by ORDERS_2:def_5; ::_thesis: verum
end;
end;
end;
hence ( x <= y or y <= x ) ; ::_thesis: verum
end;
hence RelStr(# (Bags n),T #) is connected by WAYBEL_0:def_29; ::_thesis: verum
end;
end;
registration
let n be Nat;
let T be admissible TermOrder of n;
cluster RelStr(# (Bags n),T #) -> well_founded ;
coherence
RelStr(# (Bags n),T #) is well_founded
proof
set R = RelStr(# (Bags n),T #);
set X = the carrier of RelStr(# (Bags n),T #);
now__::_thesis:_for_Y_being_set_st_Y_c=_the_carrier_of_RelStr(#_(Bags_n),T_#)_&_Y_<>_{}_holds_
ex_a_being_set_st_
(_a_in_Y_&_T_-Seg_a_misses_Y_)
let Y be set ; ::_thesis: ( Y c= the carrier of RelStr(# (Bags n),T #) & Y <> {} implies ex a being set st
( a in Y & T -Seg a misses Y ) )
assume that
A1: Y c= the carrier of RelStr(# (Bags n),T #) and
A2: Y <> {} ; ::_thesis: ex a being set st
( a in Y & T -Seg a misses Y )
now__::_thesis:_for_u_being_set_st_u_in_Y_holds_
u_in_field_T
let u be set ; ::_thesis: ( u in Y implies u in field T )
assume u in Y ; ::_thesis: u in field T
then reconsider u9 = u as bag of n by A1;
u9 <= u9,T by TERMORD:6;
then [u9,u9] in T by TERMORD:def_2;
hence u in field T by RELAT_1:15; ::_thesis: verum
end;
then Y c= field T by TARSKI:def_3;
hence ex a being set st
( a in Y & T -Seg a misses Y ) by A2, WELLORD1:def_2; ::_thesis: verum
end;
then T is_well_founded_in the carrier of RelStr(# (Bags n),T #) by WELLORD1:def_3;
hence RelStr(# (Bags n),T #) is well_founded by WELLFND1:def_2; ::_thesis: verum
end;
end;
Lm11: for n being Ordinal
for T being connected TermOrder of n
for L being non empty ZeroStr
for p being Polynomial of n,L holds Support p in Fin the carrier of RelStr(# (Bags n),T #)
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non empty ZeroStr
for p being Polynomial of n,L holds Support p in Fin the carrier of RelStr(# (Bags n),T #)
let T be connected TermOrder of n; ::_thesis: for L being non empty ZeroStr
for p being Polynomial of n,L holds Support p in Fin the carrier of RelStr(# (Bags n),T #)
let L be non empty ZeroStr ; ::_thesis: for p being Polynomial of n,L holds Support p in Fin the carrier of RelStr(# (Bags n),T #)
let p be Polynomial of n,L; ::_thesis: Support p in Fin the carrier of RelStr(# (Bags n),T #)
set sp = Support p;
Support p is finite by POLYNOM1:def_4;
hence Support p in Fin the carrier of RelStr(# (Bags n),T #) by FINSUB_1:def_5; ::_thesis: verum
end;
definition
let n be Ordinal;
let T be connected TermOrder of n;
let L be non empty ZeroStr ;
let p, q be Polynomial of n,L;
predp <= q,T means :Def2: :: POLYRED:def 2
[(Support p),(Support q)] in FinOrd RelStr(# (Bags n),T #);
end;
:: deftheorem Def2 defines <= POLYRED:def_2_:_
for n being Ordinal
for T being connected TermOrder of n
for L being non empty ZeroStr
for p, q being Polynomial of n,L holds
( p <= q,T iff [(Support p),(Support q)] in FinOrd RelStr(# (Bags n),T #) );
definition
let n be Ordinal;
let T be connected TermOrder of n;
let L be non empty ZeroStr ;
let p, q be Polynomial of n,L;
predp < q,T means :Def3: :: POLYRED:def 3
( p <= q,T & Support p <> Support q );
end;
:: deftheorem Def3 defines < POLYRED:def_3_:_
for n being Ordinal
for T being connected TermOrder of n
for L being non empty ZeroStr
for p, q being Polynomial of n,L holds
( p < q,T iff ( p <= q,T & Support p <> Support q ) );
definition
let n be Ordinal;
let T be connected TermOrder of n;
let L be non empty ZeroStr ;
let p be Polynomial of n,L;
func Support (p,T) -> Element of Fin the carrier of RelStr(# (Bags n),T #) equals :: POLYRED:def 4
Support p;
coherence
Support p is Element of Fin the carrier of RelStr(# (Bags n),T #) by Lm11;
end;
:: deftheorem defines Support POLYRED:def_4_:_
for n being Ordinal
for T being connected TermOrder of n
for L being non empty ZeroStr
for p being Polynomial of n,L holds Support (p,T) = Support p;
theorem Th24: :: POLYRED:24
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial ZeroStr
for p being non-zero Polynomial of n,L holds PosetMax (Support (p,T)) = HT (p,T)
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial ZeroStr
for p being non-zero Polynomial of n,L holds PosetMax (Support (p,T)) = HT (p,T)
let T be connected TermOrder of n; ::_thesis: for L being non trivial ZeroStr
for p being non-zero Polynomial of n,L holds PosetMax (Support (p,T)) = HT (p,T)
let L be non trivial ZeroStr ; ::_thesis: for p being non-zero Polynomial of n,L holds PosetMax (Support (p,T)) = HT (p,T)
let p be non-zero Polynomial of n,L; ::_thesis: PosetMax (Support (p,T)) = HT (p,T)
set htp = HT (p,T);
set R = RelStr(# (Bags n),T #);
set sp = Support (p,T);
p <> 0_ (n,L) by POLYNOM7:def_1;
then Support p <> {} by POLYNOM7:1;
then A1: HT (p,T) in Support p by TERMORD:def_6;
now__::_thesis:_for_y_being_set_holds_
(_not_y_in_Support_(p,T)_or_not_y_<>_HT_(p,T)_or_not_[(HT_(p,T)),y]_in_the_InternalRel_of_RelStr(#_(Bags_n),T_#)_)
assume ex y being set st
( y in Support (p,T) & y <> HT (p,T) & [(HT (p,T)),y] in the InternalRel of RelStr(# (Bags n),T #) ) ; ::_thesis: contradiction
then consider y being set such that
A2: y in Support (p,T) and
A3: y <> HT (p,T) and
A4: [(HT (p,T)),y] in the InternalRel of RelStr(# (Bags n),T #) ;
y is Element of Bags n by A2;
then reconsider y9 = y as bag of n ;
( y9 <= HT (p,T),T & HT (p,T) <= y9,T ) by A2, A4, TERMORD:def_2, TERMORD:def_6;
hence contradiction by A3, TERMORD:7; ::_thesis: verum
end;
then HT (p,T) is_maximal_wrt Support (p,T), the InternalRel of RelStr(# (Bags n),T #) by A1, WAYBEL_4:def_23;
hence PosetMax (Support (p,T)) = HT (p,T) by A1, BAGORDER:def_13; ::_thesis: verum
end;
theorem Th25: :: POLYRED:25
for n being Ordinal
for T being connected TermOrder of n
for L being non empty addLoopStr
for p being Polynomial of n,L holds p <= p,T
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non empty addLoopStr
for p being Polynomial of n,L holds p <= p,T
let T be connected TermOrder of n; ::_thesis: for L being non empty addLoopStr
for p being Polynomial of n,L holds p <= p,T
let L be non empty addLoopStr ; ::_thesis: for p being Polynomial of n,L holds p <= p,T
let p be Polynomial of n,L; ::_thesis: p <= p,T
set O = FinOrd RelStr(# (Bags n),T #);
[(Support p),(Support p)] in FinOrd RelStr(# (Bags n),T #) by Lm11, ORDERS_1:3;
hence p <= p,T by Def2; ::_thesis: verum
end;
Lm12: for n being Ordinal
for T being connected TermOrder of n
for L being non empty ZeroStr
for p being Polynomial of n,L holds 0_ (n,L) <= p,T
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non empty ZeroStr
for p being Polynomial of n,L holds 0_ (n,L) <= p,T
let T be connected TermOrder of n; ::_thesis: for L being non empty ZeroStr
for p being Polynomial of n,L holds 0_ (n,L) <= p,T
let L be non empty ZeroStr ; ::_thesis: for p being Polynomial of n,L holds 0_ (n,L) <= p,T
let p be Polynomial of n,L; ::_thesis: 0_ (n,L) <= p,T
set sp0 = Support (0_ (n,L));
set sp = Support p;
set R = RelStr(# (Bags n),T #);
A1: Support p is Element of Fin the carrier of RelStr(# (Bags n),T #) by Lm11;
( Support (0_ (n,L)) = {} & Support (0_ (n,L)) is Element of Fin the carrier of RelStr(# (Bags n),T #) ) by Lm11, POLYNOM7:1;
then [(Support (0_ (n,L))),(Support p)] in { [x,y] where x, y is Element of Fin the carrier of RelStr(# (Bags n),T #) : ( x = {} or ( x <> {} & y <> {} & PosetMax x <> PosetMax y & [(PosetMax x),(PosetMax y)] in the InternalRel of RelStr(# (Bags n),T #) ) ) } by A1;
then A2: [(Support (0_ (n,L))),(Support p)] in (FinOrd-Approx RelStr(# (Bags n),T #)) . 0 by BAGORDER:def_14;
dom (FinOrd-Approx RelStr(# (Bags n),T #)) = NAT by BAGORDER:def_14;
then (FinOrd-Approx RelStr(# (Bags n),T #)) . 0 in rng (FinOrd-Approx RelStr(# (Bags n),T #)) by FUNCT_1:3;
then [(Support (0_ (n,L))),(Support p)] in union (rng (FinOrd-Approx RelStr(# (Bags n),T #))) by A2, TARSKI:def_4;
then [(Support (0_ (n,L))),(Support p)] in FinOrd RelStr(# (Bags n),T #) by BAGORDER:def_15;
hence 0_ (n,L) <= p,T by Def2; ::_thesis: verum
end;
theorem Th26: :: POLYRED:26
for n being Ordinal
for T being connected TermOrder of n
for L being non empty addLoopStr
for p, q being Polynomial of n,L holds
( ( p <= q,T & q <= p,T ) iff Support p = Support q )
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non empty addLoopStr
for p, q being Polynomial of n,L holds
( ( p <= q,T & q <= p,T ) iff Support p = Support q )
let T be connected TermOrder of n; ::_thesis: for L being non empty addLoopStr
for p, q being Polynomial of n,L holds
( ( p <= q,T & q <= p,T ) iff Support p = Support q )
let L be non empty addLoopStr ; ::_thesis: for p, q being Polynomial of n,L holds
( ( p <= q,T & q <= p,T ) iff Support p = Support q )
let p, q be Polynomial of n,L; ::_thesis: ( ( p <= q,T & q <= p,T ) iff Support p = Support q )
set O = FinOrd RelStr(# (Bags n),T #);
A1: now__::_thesis:_(_p_<=_q,T_&_q_<=_p,T_implies_Support_p_=_Support_q_)
assume ( p <= q,T & q <= p,T ) ; ::_thesis: Support p = Support q
then A2: ( [(Support p),(Support q)] in FinOrd RelStr(# (Bags n),T #) & [(Support q),(Support p)] in FinOrd RelStr(# (Bags n),T #) ) by Def2;
( Support p in Fin the carrier of RelStr(# (Bags n),T #) & Support q in Fin the carrier of RelStr(# (Bags n),T #) ) by Lm11;
hence Support p = Support q by A2, ORDERS_1:4; ::_thesis: verum
end;
now__::_thesis:_(_Support_p_=_Support_q_implies_(_p_<=_q,T_&_q_<=_p,T_)_)
assume A3: Support p = Support q ; ::_thesis: ( p <= q,T & q <= p,T )
then [(Support p),(Support q)] in FinOrd RelStr(# (Bags n),T #) by Lm11, ORDERS_1:3;
hence ( p <= q,T & q <= p,T ) by A3, Def2; ::_thesis: verum
end;
hence ( ( p <= q,T & q <= p,T ) iff Support p = Support q ) by A1; ::_thesis: verum
end;
theorem Th27: :: POLYRED:27
for n being Ordinal
for T being connected TermOrder of n
for L being non empty addLoopStr
for p, q, r being Polynomial of n,L st p <= q,T & q <= r,T holds
p <= r,T
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non empty addLoopStr
for p, q, r being Polynomial of n,L st p <= q,T & q <= r,T holds
p <= r,T
let T be connected TermOrder of n; ::_thesis: for L being non empty addLoopStr
for p, q, r being Polynomial of n,L st p <= q,T & q <= r,T holds
p <= r,T
let L be non empty addLoopStr ; ::_thesis: for p, q, r being Polynomial of n,L st p <= q,T & q <= r,T holds
p <= r,T
let p, q, r be Polynomial of n,L; ::_thesis: ( p <= q,T & q <= r,T implies p <= r,T )
set O = FinOrd RelStr(# (Bags n),T #);
A1: Support r in Fin the carrier of RelStr(# (Bags n),T #) by Lm11;
assume ( p <= q,T & q <= r,T ) ; ::_thesis: p <= r,T
then A2: ( [(Support p),(Support q)] in FinOrd RelStr(# (Bags n),T #) & [(Support q),(Support r)] in FinOrd RelStr(# (Bags n),T #) ) by Def2;
( Support p in Fin the carrier of RelStr(# (Bags n),T #) & Support q in Fin the carrier of RelStr(# (Bags n),T #) ) by Lm11;
then [(Support p),(Support r)] in FinOrd RelStr(# (Bags n),T #) by A2, A1, ORDERS_1:5;
hence p <= r,T by Def2; ::_thesis: verum
end;
theorem Th28: :: POLYRED:28
for n being Ordinal
for T being connected TermOrder of n
for L being non empty addLoopStr
for p, q being Polynomial of n,L holds
( p <= q,T or q <= p,T )
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non empty addLoopStr
for p, q being Polynomial of n,L holds
( p <= q,T or q <= p,T )
let T be connected TermOrder of n; ::_thesis: for L being non empty addLoopStr
for p, q being Polynomial of n,L holds
( p <= q,T or q <= p,T )
let L be non empty addLoopStr ; ::_thesis: for p, q being Polynomial of n,L holds
( p <= q,T or q <= p,T )
let p, q be Polynomial of n,L; ::_thesis: ( p <= q,T or q <= p,T )
set R = RelStr(# (Bags n),T #);
set O = RelStr(# (Fin the carrier of RelStr(# (Bags n),T #)),(FinOrd RelStr(# (Bags n),T #)) #);
reconsider sp = Support p, sq = Support q as Element of RelStr(# (Fin the carrier of RelStr(# (Bags n),T #)),(FinOrd RelStr(# (Bags n),T #)) #) by Lm11;
FinPoset RelStr(# (Bags n),T #) is connected ;
then RelStr(# (Fin the carrier of RelStr(# (Bags n),T #)),(FinOrd RelStr(# (Bags n),T #)) #) is connected by BAGORDER:def_16;
then ( sp <= sq or sq <= sp ) by WAYBEL_0:def_29;
then ( [(Support p),(Support q)] in FinOrd RelStr(# (Bags n),T #) or [(Support q),(Support p)] in FinOrd RelStr(# (Bags n),T #) ) by ORDERS_2:def_5;
hence ( p <= q,T or q <= p,T ) by Def2; ::_thesis: verum
end;
theorem Th29: :: POLYRED:29
for n being Ordinal
for T being connected TermOrder of n
for L being non empty addLoopStr
for p, q being Polynomial of n,L holds
( p <= q,T iff not q < p,T )
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non empty addLoopStr
for p, q being Polynomial of n,L holds
( p <= q,T iff not q < p,T )
let T be connected TermOrder of n; ::_thesis: for L being non empty addLoopStr
for p, q being Polynomial of n,L holds
( p <= q,T iff not q < p,T )
let L be non empty addLoopStr ; ::_thesis: for p, q being Polynomial of n,L holds
( p <= q,T iff not q < p,T )
let p, q be Polynomial of n,L; ::_thesis: ( p <= q,T iff not q < p,T )
A1: ( not q < p,T implies p <= q,T )
proof
assume A2: not q < p,T ; ::_thesis: p <= q,T
now__::_thesis:_(_(_not_Support_q_<>_Support_p_&_p_<=_q,T_)_or_(_not_q_<=_p,T_&_p_<=_q,T_)_)
percases ( not Support q <> Support p or not q <= p,T ) by A2, Def3;
case not Support q <> Support p ; ::_thesis: p <= q,T
hence p <= q,T by Th26; ::_thesis: verum
end;
case not q <= p,T ; ::_thesis: p <= q,T
hence p <= q,T by Th28; ::_thesis: verum
end;
end;
end;
hence p <= q,T ; ::_thesis: verum
end;
( p <= q,T implies not q < p,T )
proof
assume A3: p <= q,T ; ::_thesis: not q < p,T
assume A4: q < p,T ; ::_thesis: contradiction
then q <= p,T by Def3;
then Support q = Support p by A3, Th26;
hence contradiction by A4, Def3; ::_thesis: verum
end;
hence ( p <= q,T iff not q < p,T ) by A1; ::_thesis: verum
end;
Lm13: for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L
for b being bag of n st [(HT (p,T)),b] in T & b <> HT (p,T) holds
p . b = 0. L
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L
for b being bag of n st [(HT (p,T)),b] in T & b <> HT (p,T) holds
p . b = 0. L
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L
for b being bag of n st [(HT (p,T)),b] in T & b <> HT (p,T) holds
p . b = 0. L
let L be non trivial right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for p being Polynomial of n,L
for b being bag of n st [(HT (p,T)),b] in T & b <> HT (p,T) holds
p . b = 0. L
let p be Polynomial of n,L; ::_thesis: for b being bag of n st [(HT (p,T)),b] in T & b <> HT (p,T) holds
p . b = 0. L
let b be bag of n; ::_thesis: ( [(HT (p,T)),b] in T & b <> HT (p,T) implies p . b = 0. L )
A1: b is Element of Bags n by PRE_POLY:def_12;
assume A2: ( [(HT (p,T)),b] in T & b <> HT (p,T) ) ; ::_thesis: p . b = 0. L
now__::_thesis:_not_b_in_Support_p
assume b in Support p ; ::_thesis: contradiction
then b <= HT (p,T),T by TERMORD:def_6;
then [b,(HT (p,T))] in T by TERMORD:def_2;
hence contradiction by A2, A1, ORDERS_1:4; ::_thesis: verum
end;
hence p . b = 0. L by A1, POLYNOM1:def_3; ::_thesis: verum
end;
Lm14: for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L st HT (p,T) = EmptyBag n holds
Red (p,T) = 0_ (n,L)
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L st HT (p,T) = EmptyBag n holds
Red (p,T) = 0_ (n,L)
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L st HT (p,T) = EmptyBag n holds
Red (p,T) = 0_ (n,L)
let L be non trivial right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for p being Polynomial of n,L st HT (p,T) = EmptyBag n holds
Red (p,T) = 0_ (n,L)
let p be Polynomial of n,L; ::_thesis: ( HT (p,T) = EmptyBag n implies Red (p,T) = 0_ (n,L) )
set red = Red (p,T);
set e = 0_ (n,L);
assume A1: HT (p,T) = EmptyBag n ; ::_thesis: Red (p,T) = 0_ (n,L)
A2: now__::_thesis:_for_b_being_bag_of_n_st_b_<>_EmptyBag_n_holds_
p_._b_=_0._L
let b be bag of n; ::_thesis: ( b <> EmptyBag n implies p . b = 0. L )
A3: [(EmptyBag n),b] in T by BAGORDER:def_5;
assume b <> EmptyBag n ; ::_thesis: p . b = 0. L
hence p . b = 0. L by A1, A3, Lm13; ::_thesis: verum
end;
A4: now__::_thesis:_for_b_being_set_st_b_in_dom_(Red_(p,T))_holds_
(Red_(p,T))_._b_=_(0__(n,L))_._b
let b be set ; ::_thesis: ( b in dom (Red (p,T)) implies (Red (p,T)) . b = (0_ (n,L)) . b )
assume b in dom (Red (p,T)) ; ::_thesis: (Red (p,T)) . b = (0_ (n,L)) . b
then reconsider b9 = b as Element of Bags n ;
A5: (Red (p,T)) . b9 = (p - (HM (p,T))) . b9 by TERMORD:def_9
.= (p + (- (HM (p,T)))) . b9 by POLYNOM1:def_6
.= (p . b9) + ((- (HM (p,T))) . b9) by POLYNOM1:15
.= (p . b9) + (- ((HM (p,T)) . b9)) by POLYNOM1:17 ;
now__::_thesis:_(_(_b9_=_EmptyBag_n_&_(Red_(p,T))_._b9_=_(0__(n,L))_._b9_)_or_(_b9_<>_EmptyBag_n_&_(Red_(p,T))_._b9_=_(0__(n,L))_._b9_)_)
percases ( b9 = EmptyBag n or b9 <> EmptyBag n ) ;
case b9 = EmptyBag n ; ::_thesis: (Red (p,T)) . b9 = (0_ (n,L)) . b9
hence (Red (p,T)) . b9 = (p . (HT (p,T))) + (- (p . (HT (p,T)))) by A1, A5, TERMORD:18
.= 0. L by RLVECT_1:5
.= (0_ (n,L)) . b9 by POLYNOM1:22 ;
::_thesis: verum
end;
caseA6: b9 <> EmptyBag n ; ::_thesis: (Red (p,T)) . b9 = (0_ (n,L)) . b9
hence (Red (p,T)) . b9 = (0. L) + (- ((HM (p,T)) . b9)) by A2, A5
.= (0. L) + (- (0. L)) by A1, A6, TERMORD:19
.= 0. L by RLVECT_1:5
.= (0_ (n,L)) . b9 by POLYNOM1:22 ;
::_thesis: verum
end;
end;
end;
hence (Red (p,T)) . b = (0_ (n,L)) . b ; ::_thesis: verum
end;
dom (Red (p,T)) = Bags n by FUNCT_2:def_1
.= dom (0_ (n,L)) by FUNCT_2:def_1 ;
hence Red (p,T) = 0_ (n,L) by A4, FUNCT_1:2; ::_thesis: verum
end;
Lm15: for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p, q being Polynomial of n,L holds
( p < q,T iff ( ( p = 0_ (n,L) & q <> 0_ (n,L) ) or HT (p,T) < HT (q,T),T or ( HT (p,T) = HT (q,T) & Red (p,T) < Red (q,T),T ) ) )
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p, q being Polynomial of n,L holds
( p < q,T iff ( ( p = 0_ (n,L) & q <> 0_ (n,L) ) or HT (p,T) < HT (q,T),T or ( HT (p,T) = HT (q,T) & Red (p,T) < Red (q,T),T ) ) )
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p, q being Polynomial of n,L holds
( p < q,T iff ( ( p = 0_ (n,L) & q <> 0_ (n,L) ) or HT (p,T) < HT (q,T),T or ( HT (p,T) = HT (q,T) & Red (p,T) < Red (q,T),T ) ) )
let L be non trivial right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for p, q being Polynomial of n,L holds
( p < q,T iff ( ( p = 0_ (n,L) & q <> 0_ (n,L) ) or HT (p,T) < HT (q,T),T or ( HT (p,T) = HT (q,T) & Red (p,T) < Red (q,T),T ) ) )
let p, q be Polynomial of n,L; ::_thesis: ( p < q,T iff ( ( p = 0_ (n,L) & q <> 0_ (n,L) ) or HT (p,T) < HT (q,T),T or ( HT (p,T) = HT (q,T) & Red (p,T) < Red (q,T),T ) ) )
set R = RelStr(# (Bags n),T #);
set sp = Support p;
set sq = Support q;
set x = Support (p,T);
set y = Support (q,T);
A1: now__::_thesis:_(_(_(_p_=_0__(n,L)_&_q_<>_0__(n,L)_)_or_HT_(p,T)_<_HT_(q,T),T_or_(_HT_(p,T)_=_HT_(q,T)_&_Red_(p,T)_<_Red_(q,T),T_)_)_implies_p_<_q,T_)
assume A2: ( ( p = 0_ (n,L) & q <> 0_ (n,L) ) or HT (p,T) < HT (q,T),T or ( HT (p,T) = HT (q,T) & Red (p,T) < Red (q,T),T ) ) ; ::_thesis: p < q,T
now__::_thesis:_(_(_p_=_0__(n,L)_&_q_<>_0__(n,L)_&_p_<_q,T_)_or_(_HT_(p,T)_<_HT_(q,T),T_&_p_<_q,T_)_or_(_HT_(p,T)_=_HT_(q,T)_&_Red_(p,T)_<_Red_(q,T),T_&_p_<_q,T_)_)
percases ( ( p = 0_ (n,L) & q <> 0_ (n,L) ) or HT (p,T) < HT (q,T),T or ( HT (p,T) = HT (q,T) & Red (p,T) < Red (q,T),T ) ) by A2;
caseA3: ( p = 0_ (n,L) & q <> 0_ (n,L) ) ; ::_thesis: p < q,T
then Support p = {} by POLYNOM7:1;
then A4: Support p <> Support q by A3, POLYNOM7:1;
p <= q,T by A3, Lm12;
hence p < q,T by A4, Def3; ::_thesis: verum
end;
caseA5: HT (p,T) < HT (q,T),T ; ::_thesis: p < q,T
then A6: HT (p,T) <> HT (q,T) by TERMORD:def_3;
A7: HT (p,T) <= HT (q,T),T by A5, TERMORD:def_3;
then A8: [(HT (p,T)),(HT (q,T))] in T by TERMORD:def_2;
now__::_thesis:_(_(_Support_p_=_{}_&_p_<_q,T_)_or_(_Support_p_<>_{}_&_p_<_q,T_)_)
percases ( Support p = {} or Support p <> {} ) ;
caseA9: Support p = {} ; ::_thesis: p < q,T
then A10: p = 0_ (n,L) by POLYNOM7:1;
A11: now__::_thesis:_not_Support_p_=_Support_q
assume Support p = Support q ; ::_thesis: contradiction
then HT (p,T) = HT (q,T) by A9, A10, POLYNOM7:1;
hence contradiction by A5, TERMORD:def_3; ::_thesis: verum
end;
p <= q,T by A10, Lm12;
hence p < q,T by A11, Def3; ::_thesis: verum
end;
caseA12: Support p <> {} ; ::_thesis: p < q,T
A13: now__::_thesis:_not_Support_q_=_{}
assume Support q = {} ; ::_thesis: contradiction
then HT (q,T) = EmptyBag n by TERMORD:def_6;
then [(HT (q,T)),(HT (p,T))] in T by BAGORDER:def_5;
then HT (q,T) <= HT (p,T),T by TERMORD:def_2;
hence contradiction by A7, A6, TERMORD:7; ::_thesis: verum
end;
A14: now__::_thesis:_not_Support_p_=_Support_q
assume A15: Support p = Support q ; ::_thesis: contradiction
HT (q,T) in Support q by A13, TERMORD:def_6;
then A16: HT (q,T) <= HT (p,T),T by A15, TERMORD:def_6;
HT (p,T) in Support p by A12, TERMORD:def_6;
then HT (p,T) <= HT (q,T),T by A15, TERMORD:def_6;
hence contradiction by A6, A16, TERMORD:7; ::_thesis: verum
end;
p <> 0_ (n,L) by A12, POLYNOM7:1;
then p is non-zero by POLYNOM7:def_1;
then A17: PosetMax (Support (p,T)) = HT (p,T) by Th24;
q <> 0_ (n,L) by A13, POLYNOM7:1;
then q is non-zero by POLYNOM7:def_1;
then PosetMax (Support (q,T)) = HT (q,T) by Th24;
then [(Support (p,T)),(Support (q,T))] in union (rng (FinOrd-Approx RelStr(# (Bags n),T #))) by A6, A8, A12, A13, A17, BAGORDER:35;
then [(Support p),(Support q)] in FinOrd RelStr(# (Bags n),T #) by BAGORDER:def_15;
then p <= q,T by Def2;
hence p < q,T by A14, Def3; ::_thesis: verum
end;
end;
end;
hence p < q,T ; ::_thesis: verum
end;
caseA18: ( HT (p,T) = HT (q,T) & Red (p,T) < Red (q,T),T ) ; ::_thesis: p < q,T
then Red (p,T) <= Red (q,T),T by Def3;
then A19: [(Support (Red (p,T))),(Support (Red (q,T)))] in FinOrd RelStr(# (Bags n),T #) by Def2;
now__::_thesis:_(_(_Support_p_=_{}_&_contradiction_)_or_(_Support_p_<>_{}_&_p_<_q,T_)_)
percases ( Support p = {} or Support p <> {} ) ;
case Support p = {} ; ::_thesis: contradiction
then A20: HT (p,T) = EmptyBag n by TERMORD:def_6;
then Red (p,T) = 0_ (n,L) by Lm14;
then Support (Red (q,T)) = Support (Red (p,T)) by A18, A20, Lm14;
hence contradiction by A18, Def3; ::_thesis: verum
end;
caseA21: Support p <> {} ; ::_thesis: p < q,T
now__::_thesis:_(_(_Support_q_=_{}_&_contradiction_)_or_(_Support_q_<>_{}_&_p_<_q,T_)_)
percases ( Support q = {} or Support q <> {} ) ;
case Support q = {} ; ::_thesis: contradiction
then A22: HT (q,T) = EmptyBag n by TERMORD:def_6;
then Red (q,T) = 0_ (n,L) by Lm14;
then Support (Red (p,T)) = Support (Red (q,T)) by A18, A22, Lm14;
hence contradiction by A18, Def3; ::_thesis: verum
end;
caseA23: Support q <> {} ; ::_thesis: p < q,T
A24: now__::_thesis:_not_Support_p_=_Support_q
assume Support p = Support q ; ::_thesis: contradiction
then Support (Red (p,T)) = (Support q) \ {(HT (q,T))} by A18, TERMORD:36
.= Support (Red (q,T)) by TERMORD:36 ;
hence contradiction by A18, Def3; ::_thesis: verum
end;
set rp = Red (p,T);
set rq = Red (q,T);
p <> 0_ (n,L) by A21, POLYNOM7:1;
then A25: p is non-zero by POLYNOM7:def_1;
q <> 0_ (n,L) by A23, POLYNOM7:1;
then A26: q is non-zero by POLYNOM7:def_1;
then A27: PosetMax (Support (q,T)) = HT (q,T) by Th24;
A28: Support (Red (q,T)) = (Support q) \ {(HT (q,T))} by TERMORD:36
.= (Support (q,T)) \ {(PosetMax (Support (q,T)))} by A26, Th24 ;
Support (Red (p,T)) = (Support p) \ {(HT (p,T))} by TERMORD:36
.= (Support (p,T)) \ {(PosetMax (Support (p,T)))} by A25, Th24 ;
then A29: [((Support (p,T)) \ {(PosetMax (Support (p,T)))}),((Support (q,T)) \ {(PosetMax (Support (q,T)))})] in union (rng (FinOrd-Approx RelStr(# (Bags n),T #))) by A19, A28, BAGORDER:def_15;
PosetMax (Support (p,T)) = HT (p,T) by A25, Th24;
then [(Support (p,T)),(Support (q,T))] in union (rng (FinOrd-Approx RelStr(# (Bags n),T #))) by A18, A21, A23, A27, A29, BAGORDER:35;
then [(Support p),(Support q)] in FinOrd RelStr(# (Bags n),T #) by BAGORDER:def_15;
then p <= q,T by Def2;
hence p < q,T by A24, Def3; ::_thesis: verum
end;
end;
end;
hence p < q,T ; ::_thesis: verum
end;
end;
end;
hence p < q,T ; ::_thesis: verum
end;
end;
end;
hence p < q,T ; ::_thesis: verum
end;
now__::_thesis:_(_not_p_<_q,T_or_(_p_=_0__(n,L)_&_q_<>_0__(n,L)_)_or_HT_(p,T)_<_HT_(q,T),T_or_(_HT_(p,T)_=_HT_(q,T)_&_Red_(p,T)_<_Red_(q,T),T_)_)
assume A30: p < q,T ; ::_thesis: ( ( p = 0_ (n,L) & q <> 0_ (n,L) ) or HT (p,T) < HT (q,T),T or ( HT (p,T) = HT (q,T) & Red (p,T) < Red (q,T),T ) )
then p <= q,T by Def3;
then [(Support p),(Support q)] in FinOrd RelStr(# (Bags n),T #) by Def2;
then A31: [(Support p),(Support q)] in union (rng (FinOrd-Approx RelStr(# (Bags n),T #))) by BAGORDER:def_15;
A32: Support p <> Support q by A30, Def3;
now__::_thesis:_(_(_Support_(p,T)_=_{}_&_p_=_0__(n,L)_&_q_<>_0__(n,L)_)_or_(_Support_(p,T)_<>_{}_&_Support_(q,T)_<>_{}_&_PosetMax_(Support_(p,T))_<>_PosetMax_(Support_(q,T))_&_[(PosetMax_(Support_(p,T))),(PosetMax_(Support_(q,T)))]_in_the_InternalRel_of_RelStr(#_(Bags_n),T_#)_&_HT_(p,T)_<_HT_(q,T),T_)_or_(_Support_(p,T)_<>_{}_&_Support_(q,T)_<>_{}_&_PosetMax_(Support_(p,T))_=_PosetMax_(Support_(q,T))_&_[((Support_(p,T))_\_{(PosetMax_(Support_(p,T)))}),((Support_(q,T))_\_{(PosetMax_(Support_(q,T)))})]_in_union_(rng_(FinOrd-Approx_RelStr(#_(Bags_n),T_#)))_&_HT_(p,T)_=_HT_(q,T)_&_Red_(p,T)_<_Red_(q,T),T_)_)
percases ( Support (p,T) = {} or ( Support (p,T) <> {} & Support (q,T) <> {} & PosetMax (Support (p,T)) <> PosetMax (Support (q,T)) & [(PosetMax (Support (p,T))),(PosetMax (Support (q,T)))] in the InternalRel of RelStr(# (Bags n),T #) ) or ( Support (p,T) <> {} & Support (q,T) <> {} & PosetMax (Support (p,T)) = PosetMax (Support (q,T)) & [((Support (p,T)) \ {(PosetMax (Support (p,T)))}),((Support (q,T)) \ {(PosetMax (Support (q,T)))})] in union (rng (FinOrd-Approx RelStr(# (Bags n),T #))) ) ) by A31, BAGORDER:35;
case Support (p,T) = {} ; ::_thesis: ( p = 0_ (n,L) & q <> 0_ (n,L) )
hence ( p = 0_ (n,L) & q <> 0_ (n,L) ) by A32, POLYNOM7:1; ::_thesis: verum
end;
caseA33: ( Support (p,T) <> {} & Support (q,T) <> {} & PosetMax (Support (p,T)) <> PosetMax (Support (q,T)) & [(PosetMax (Support (p,T))),(PosetMax (Support (q,T)))] in the InternalRel of RelStr(# (Bags n),T #) ) ; ::_thesis: HT (p,T) < HT (q,T),T
then q <> 0_ (n,L) by POLYNOM7:1;
then q is non-zero by POLYNOM7:def_1;
then A34: PosetMax (Support (q,T)) = HT (q,T) by Th24;
p <> 0_ (n,L) by A33, POLYNOM7:1;
then p is non-zero by POLYNOM7:def_1;
then A35: PosetMax (Support (p,T)) = HT (p,T) by Th24;
then HT (p,T) <= HT (q,T),T by A33, A34, TERMORD:def_2;
hence HT (p,T) < HT (q,T),T by A33, A35, A34, TERMORD:def_3; ::_thesis: verum
end;
caseA36: ( Support (p,T) <> {} & Support (q,T) <> {} & PosetMax (Support (p,T)) = PosetMax (Support (q,T)) & [((Support (p,T)) \ {(PosetMax (Support (p,T)))}),((Support (q,T)) \ {(PosetMax (Support (q,T)))})] in union (rng (FinOrd-Approx RelStr(# (Bags n),T #))) ) ; ::_thesis: ( HT (p,T) = HT (q,T) & Red (p,T) < Red (q,T),T )
set rp = Red (p,T);
set rq = Red (q,T);
p <> 0_ (n,L) by A36, POLYNOM7:1;
then A37: p is non-zero by POLYNOM7:def_1;
then A38: PosetMax (Support (p,T)) = HT (p,T) by Th24;
q <> 0_ (n,L) by A36, POLYNOM7:1;
then A39: q is non-zero by POLYNOM7:def_1;
then A40: PosetMax (Support (q,T)) = HT (q,T) by Th24;
A41: now__::_thesis:_not_Support_(Red_(p,T))_=_Support_(Red_(q,T))
HT (q,T) in Support q by A36, TERMORD:def_6;
then for u being set st u in {(HT (q,T))} holds
u in Support q by TARSKI:def_1;
then A42: {(HT (q,T))} c= Support q by TARSKI:def_3;
Support (Red (q,T)) = (Support q) \ {(HT (q,T))} by TERMORD:36;
then A43: (Support (Red (q,T))) \/ {(HT (q,T))} = (Support q) \/ {(HT (q,T))} by XBOOLE_1:39
.= Support q by A42, XBOOLE_1:12 ;
HT (p,T) in Support p by A36, TERMORD:def_6;
then for u being set st u in {(HT (p,T))} holds
u in Support p by TARSKI:def_1;
then A44: {(HT (p,T))} c= Support p by TARSKI:def_3;
Support (Red (p,T)) = (Support p) \ {(HT (p,T))} by TERMORD:36;
then A45: (Support (Red (p,T))) \/ {(HT (p,T))} = (Support p) \/ {(HT (p,T))} by XBOOLE_1:39
.= Support p by A44, XBOOLE_1:12 ;
assume Support (Red (p,T)) = Support (Red (q,T)) ; ::_thesis: contradiction
hence contradiction by A30, A36, A38, A40, A45, A43, Def3; ::_thesis: verum
end;
A46: Support ((Red (p,T)),T) = (Support p) \ {(HT (p,T))} by TERMORD:36
.= (Support (p,T)) \ {(PosetMax (Support (p,T)))} by A37, Th24 ;
Support ((Red (q,T)),T) = (Support q) \ {(HT (q,T))} by TERMORD:36
.= (Support (q,T)) \ {(PosetMax (Support (q,T)))} by A39, Th24 ;
then [(Support ((Red (p,T)),T)),(Support ((Red (q,T)),T))] in FinOrd RelStr(# (Bags n),T #) by A36, A46, BAGORDER:def_15;
then Red (p,T) <= Red (q,T),T by Def2;
hence ( HT (p,T) = HT (q,T) & Red (p,T) < Red (q,T),T ) by A36, A39, A38, A41, Def3, Th24; ::_thesis: verum
end;
end;
end;
hence ( ( p = 0_ (n,L) & q <> 0_ (n,L) ) or HT (p,T) < HT (q,T),T or ( HT (p,T) = HT (q,T) & Red (p,T) < Red (q,T),T ) ) ; ::_thesis: verum
end;
hence ( p < q,T iff ( ( p = 0_ (n,L) & q <> 0_ (n,L) ) or HT (p,T) < HT (q,T),T or ( HT (p,T) = HT (q,T) & Red (p,T) < Red (q,T),T ) ) ) by A1; ::_thesis: verum
end;
theorem :: POLYRED:30
for n being Ordinal
for T being connected TermOrder of n
for L being non empty ZeroStr
for p being Polynomial of n,L holds 0_ (n,L) <= p,T by Lm12;
Lm16: for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p, q being Polynomial of n,L st q <> 0_ (n,L) & HT (p,T) = HT (q,T) & Red (p,T) <= Red (q,T),T holds
p <= q,T
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p, q being Polynomial of n,L st q <> 0_ (n,L) & HT (p,T) = HT (q,T) & Red (p,T) <= Red (q,T),T holds
p <= q,T
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p, q being Polynomial of n,L st q <> 0_ (n,L) & HT (p,T) = HT (q,T) & Red (p,T) <= Red (q,T),T holds
p <= q,T
let L be non trivial right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for p, q being Polynomial of n,L st q <> 0_ (n,L) & HT (p,T) = HT (q,T) & Red (p,T) <= Red (q,T),T holds
p <= q,T
let p, q be Polynomial of n,L; ::_thesis: ( q <> 0_ (n,L) & HT (p,T) = HT (q,T) & Red (p,T) <= Red (q,T),T implies p <= q,T )
assume A1: q <> 0_ (n,L) ; ::_thesis: ( not HT (p,T) = HT (q,T) or not Red (p,T) <= Red (q,T),T or p <= q,T )
set x = Support (p,T);
set y = Support (q,T);
set rp = Red (p,T);
set rq = Red (q,T);
set R = RelStr(# (Bags n),T #);
assume that
A2: HT (p,T) = HT (q,T) and
A3: Red (p,T) <= Red (q,T),T ; ::_thesis: p <= q,T
[(Support (Red (p,T))),(Support (Red (q,T)))] in FinOrd RelStr(# (Bags n),T #) by A3, Def2;
then A4: [(Support (Red (p,T))),(Support (Red (q,T)))] in union (rng (FinOrd-Approx RelStr(# (Bags n),T #))) by BAGORDER:def_15;
now__::_thesis:_(_(_p_=_0__(n,L)_&_p_<=_q,T_)_or_(_p_<>_0__(n,L)_&_p_<=_q,T_)_)
percases ( p = 0_ (n,L) or p <> 0_ (n,L) ) ;
case p = 0_ (n,L) ; ::_thesis: p <= q,T
hence p <= q,T by Lm12; ::_thesis: verum
end;
caseA5: p <> 0_ (n,L) ; ::_thesis: p <= q,T
then A6: Support (p,T) <> {} by POLYNOM7:1;
A7: q is non-zero by A1, POLYNOM7:def_1;
A8: p is non-zero by A5, POLYNOM7:def_1;
A9: Support (Red (p,T)) = (Support p) \ {(HT (p,T))} by TERMORD:36
.= (Support (p,T)) \ {(PosetMax (Support (p,T)))} by A8, Th24 ;
A10: Support (q,T) <> {} by A1, POLYNOM7:1;
A11: Support (Red (q,T)) = (Support q) \ {(HT (q,T))} by TERMORD:36
.= (Support (q,T)) \ {(PosetMax (Support (q,T)))} by A7, Th24 ;
PosetMax (Support (p,T)) = HT (q,T) by A2, A8, Th24
.= PosetMax (Support (q,T)) by A7, Th24 ;
then [(Support (p,T)),(Support (q,T))] in union (rng (FinOrd-Approx RelStr(# (Bags n),T #))) by A4, A6, A10, A9, A11, BAGORDER:35;
then [(Support (p,T)),(Support (q,T))] in FinOrd RelStr(# (Bags n),T #) by BAGORDER:def_15;
hence p <= q,T by Def2; ::_thesis: verum
end;
end;
end;
hence p <= q,T ; ::_thesis: verum
end;
theorem Th31: :: POLYRED:31
for n being Nat
for T being connected admissible TermOrder of n
for L being non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for P being non empty Subset of (Polynom-Ring (n,L)) ex p being Polynomial of n,L st
( p in P & ( for q being Polynomial of n,L st q in P holds
p <= q,T ) )
proof
let n be Nat; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for P being non empty Subset of (Polynom-Ring (n,L)) ex p being Polynomial of n,L st
( p in P & ( for q being Polynomial of n,L st q in P holds
p <= q,T ) )
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr
for P being non empty Subset of (Polynom-Ring (n,L)) ex p being Polynomial of n,L st
( p in P & ( for q being Polynomial of n,L st q in P holds
p <= q,T ) )
let L be non trivial right_complementable well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for P being non empty Subset of (Polynom-Ring (n,L)) ex p being Polynomial of n,L st
( p in P & ( for q being Polynomial of n,L st q in P holds
p <= q,T ) )
let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: ex p being Polynomial of n,L st
( p in P & ( for q being Polynomial of n,L st q in P holds
p <= q,T ) )
set P9 = { (Support (p,T)) where p is Polynomial of n,L : p in P } ;
set p = the Element of P;
reconsider p = the Element of P as Polynomial of n,L by POLYNOM1:def_10;
Support (p,T) in { (Support (p,T)) where p is Polynomial of n,L : p in P } ;
then reconsider P9 = { (Support (p,T)) where p is Polynomial of n,L : p in P } as non empty set ;
set R = RelStr(# (Bags n),T #);
set FR = FinPoset RelStr(# (Bags n),T #);
set m = MinElement (P9,(FinPoset RelStr(# (Bags n),T #)));
A1: FinPoset RelStr(# (Bags n),T #) = RelStr(# (Fin the carrier of RelStr(# (Bags n),T #)),(FinOrd RelStr(# (Bags n),T #)) #) by BAGORDER:def_16;
now__::_thesis:_for_u_being_set_st_u_in_P9_holds_
u_in_the_carrier_of_(FinPoset_RelStr(#_(Bags_n),T_#))
let u be set ; ::_thesis: ( u in P9 implies u in the carrier of (FinPoset RelStr(# (Bags n),T #)) )
assume u in P9 ; ::_thesis: u in the carrier of (FinPoset RelStr(# (Bags n),T #))
then ex p being Polynomial of n,L st
( u = Support (p,T) & p in P ) ;
hence u in the carrier of (FinPoset RelStr(# (Bags n),T #)) by A1; ::_thesis: verum
end;
then A2: P9 c= the carrier of (FinPoset RelStr(# (Bags n),T #)) by TARSKI:def_3;
A3: FinPoset RelStr(# (Bags n),T #) is well_founded by BAGORDER:41;
then MinElement (P9,(FinPoset RelStr(# (Bags n),T #))) in P9 by A2, BAGORDER:def_17;
then consider p being Polynomial of n,L such that
A4: Support (p,T) = MinElement (P9,(FinPoset RelStr(# (Bags n),T #))) and
A5: p in P ;
take p ; ::_thesis: ( p in P & ( for q being Polynomial of n,L st q in P holds
p <= q,T ) )
A6: MinElement (P9,(FinPoset RelStr(# (Bags n),T #))) is_minimal_wrt P9, the InternalRel of (FinPoset RelStr(# (Bags n),T #)) by A2, A3, BAGORDER:def_17;
now__::_thesis:_for_q_being_Polynomial_of_n,L_st_q_in_P_holds_
p_<=_q,T
let q be Polynomial of n,L; ::_thesis: ( q in P implies p <= q,T )
set sq = Support (q,T);
assume q in P ; ::_thesis: p <= q,T
then A7: Support (q,T) in P9 ;
now__::_thesis:_(_(_Support_p_=_Support_q_&_p_<=_q,T_)_or_(_Support_p_<>_Support_q_&_p_<=_q,T_)_)
percases ( Support p = Support q or Support p <> Support q ) ;
case Support p = Support q ; ::_thesis: p <= q,T
hence p <= q,T by Th26; ::_thesis: verum
end;
case Support p <> Support q ; ::_thesis: p <= q,T
then not [(Support (q,T)),(MinElement (P9,(FinPoset RelStr(# (Bags n),T #))))] in the InternalRel of (FinPoset RelStr(# (Bags n),T #)) by A6, A4, A7, WAYBEL_4:def_25;
then not q <= p,T by A1, A4, Def2;
hence p <= q,T by Th28; ::_thesis: verum
end;
end;
end;
hence p <= q,T ; ::_thesis: verum
end;
hence ( p in P & ( for q being Polynomial of n,L st q in P holds
p <= q,T ) ) by A5; ::_thesis: verum
end;
theorem :: POLYRED:32
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p, q being Polynomial of n,L holds
( p < q,T iff ( ( p = 0_ (n,L) & q <> 0_ (n,L) ) or HT (p,T) < HT (q,T),T or ( HT (p,T) = HT (q,T) & Red (p,T) < Red (q,T),T ) ) ) by Lm15;
theorem Th33: :: POLYRED:33
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being non-zero Polynomial of n,L holds Red (p,T) < HM (p,T),T
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being non-zero Polynomial of n,L holds Red (p,T) < HM (p,T),T
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being non-zero Polynomial of n,L holds Red (p,T) < HM (p,T),T
let L be non trivial right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for p being non-zero Polynomial of n,L holds Red (p,T) < HM (p,T),T
let p be non-zero Polynomial of n,L; ::_thesis: Red (p,T) < HM (p,T),T
set red = Red (p,T);
set htp = HT (p,T);
set sred = Support (Red (p,T));
set sp = Support (HM (p,T));
set R = RelStr(# (Bags n),T #);
p <> 0_ (n,L) by POLYNOM7:def_1;
then A1: Support p <> {} by POLYNOM7:1;
percases ( Red (p,T) = 0_ (n,L) or Red (p,T) <> 0_ (n,L) ) ;
suppose Red (p,T) = 0_ (n,L) ; ::_thesis: Red (p,T) < HM (p,T),T
then A2: Support (Red (p,T)) = {} by POLYNOM7:1;
HT (p,T) in Support p by A1, TERMORD:def_6;
then p . (HT (p,T)) <> 0. L by POLYNOM1:def_3;
then (HM (p,T)) . (HT (p,T)) <> 0. L by TERMORD:18;
then A3: HT (p,T) in Support (HM (p,T)) by POLYNOM1:def_3;
dom (FinOrd-Approx RelStr(# (Bags n),T #)) = NAT by BAGORDER:def_14;
then A4: (FinOrd-Approx RelStr(# (Bags n),T #)) . 0 in rng (FinOrd-Approx RelStr(# (Bags n),T #)) by FUNCT_1:3;
( Support (Red (p,T)) is Element of Fin the carrier of RelStr(# (Bags n),T #) & Support (HM (p,T)) is Element of Fin the carrier of RelStr(# (Bags n),T #) ) by Lm11;
then [(Support (Red (p,T))),(Support (HM (p,T)))] in { [x,y] where x, y is Element of Fin the carrier of RelStr(# (Bags n),T #) : ( x = {} or ( x <> {} & y <> {} & PosetMax x <> PosetMax y & [(PosetMax x),(PosetMax y)] in the InternalRel of RelStr(# (Bags n),T #) ) ) } by A2;
then [(Support (Red (p,T))),(Support (HM (p,T)))] in (FinOrd-Approx RelStr(# (Bags n),T #)) . 0 by BAGORDER:def_14;
then [(Support (Red (p,T))),(Support (HM (p,T)))] in union (rng (FinOrd-Approx RelStr(# (Bags n),T #))) by A4, TARSKI:def_4;
then [(Support (Red (p,T))),(Support (HM (p,T)))] in FinOrd RelStr(# (Bags n),T #) by BAGORDER:def_15;
then Red (p,T) <= HM (p,T),T by Def2;
hence Red (p,T) < HM (p,T),T by A2, A3, Def3; ::_thesis: verum
end;
suppose Red (p,T) <> 0_ (n,L) ; ::_thesis: Red (p,T) < HM (p,T),T
then Support (Red (p,T)) <> {} by POLYNOM7:1;
then A5: HT ((Red (p,T)),T) in Support (Red (p,T)) by TERMORD:def_6;
A6: now__::_thesis:_not_HT_((Red_(p,T)),T)_=_HT_(p,T)
assume HT ((Red (p,T)),T) = HT (p,T) ; ::_thesis: contradiction
then (Red (p,T)) . (HT ((Red (p,T)),T)) = 0. L by TERMORD:39;
hence contradiction by A5, POLYNOM1:def_3; ::_thesis: verum
end;
Support (Red (p,T)) c= Support p by TERMORD:35;
then HT ((Red (p,T)),T) <= HT (p,T),T by A5, TERMORD:def_6;
then HT ((Red (p,T)),T) < HT (p,T),T by A6, TERMORD:def_3;
then HT ((Red (p,T)),T) < HT ((HM (p,T)),T),T by TERMORD:26;
hence Red (p,T) < HM (p,T),T by Lm15; ::_thesis: verum
end;
end;
end;
theorem Th34: :: POLYRED:34
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L holds HM (p,T) <= p,T
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L holds HM (p,T) <= p,T
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being Polynomial of n,L holds HM (p,T) <= p,T
let L be non trivial right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for p being Polynomial of n,L holds HM (p,T) <= p,T
let p9 be Polynomial of n,L; ::_thesis: HM (p9,T) <= p9,T
percases ( p9 = 0_ (n,L) or p9 <> 0_ (n,L) ) ;
supposeA1: p9 = 0_ (n,L) ; ::_thesis: HM (p9,T) <= p9,T
now__::_thesis:_not_Support_(HM_(p9,T))_<>_{}
assume Support (HM (p9,T)) <> {} ; ::_thesis: contradiction
then consider u being bag of n such that
A2: Support (HM (p9,T)) = {u} by POLYNOM7:6;
A3: u in Support (HM (p9,T)) by A2, TARSKI:def_1;
now__::_thesis:_for_v_being_bag_of_n_st_v_in_Support_(HM_(p9,T))_holds_
v_<=_u,T
let v be bag of n; ::_thesis: ( v in Support (HM (p9,T)) implies v <= u,T )
assume v in Support (HM (p9,T)) ; ::_thesis: v <= u,T
then u = v by A2, TARSKI:def_1;
hence v <= u,T by TERMORD:6; ::_thesis: verum
end;
then A4: HT ((HM (p9,T)),T) = u by A3, TERMORD:def_6;
0. L = p9 . (HT (p9,T)) by A1, POLYNOM1:22
.= HC (p9,T) by TERMORD:def_7
.= HC ((HM (p9,T)),T) by TERMORD:27
.= (HM (p9,T)) . u by A4, TERMORD:def_7 ;
hence contradiction by A3, POLYNOM1:def_3; ::_thesis: verum
end;
then HM (p9,T) = 0_ (n,L) by POLYNOM7:1;
hence HM (p9,T) <= p9,T by A1, Th25; ::_thesis: verum
end;
suppose p9 <> 0_ (n,L) ; ::_thesis: HM (p9,T) <= p9,T
then reconsider p = p9 as non-zero Polynomial of n,L by POLYNOM7:def_1;
set hmp = HM (p,T);
set R = RelStr(# (Bags n),T #);
set x = Support ((HM (p,T)),T);
set y = Support (p,T);
A5: ( (Support ((HM (p,T)),T)) \ {(PosetMax (Support ((HM (p,T)),T)))} is Element of Fin the carrier of RelStr(# (Bags n),T #) & (Support (p,T)) \ {(PosetMax (Support (p,T)))} is Element of Fin the carrier of RelStr(# (Bags n),T #) ) by BAGORDER:37;
A6: PosetMax (Support ((HM (p,T)),T)) = HT ((HM (p,T)),T) by Th24
.= HT (p,T) by TERMORD:26 ;
p <> 0_ (n,L) by POLYNOM7:def_1;
then A7: Support p <> {} by POLYNOM7:1;
(HM (p,T)) . (HT (p,T)) = p . (HT (p,T)) by TERMORD:18
.= HC (p,T) by TERMORD:def_7 ;
then A8: (HM (p,T)) . (HT (p,T)) <> 0. L ;
then A9: Support ((HM (p,T)),T) <> {} by POLYNOM1:def_3;
HT (p,T) in Support (HM (p,T)) by A8, POLYNOM1:def_3;
then Support (HM (p,T)) = {(HT (p,T))} by TERMORD:21;
then (Support ((HM (p,T)),T)) \ {(PosetMax (Support ((HM (p,T)),T)))} = {} by A6, XBOOLE_1:37;
then A10: [((Support ((HM (p,T)),T)) \ {(PosetMax (Support ((HM (p,T)),T)))}),((Support (p,T)) \ {(PosetMax (Support (p,T)))})] in union (rng (FinOrd-Approx RelStr(# (Bags n),T #))) by A5, BAGORDER:35;
PosetMax (Support ((HM (p,T)),T)) = PosetMax (Support (p,T)) by A6, Th24;
then [(Support ((HM (p,T)),T)),(Support (p,T))] in union (rng (FinOrd-Approx RelStr(# (Bags n),T #))) by A7, A9, A10, BAGORDER:35;
then [(Support ((HM (p,T)),T)),(Support (p,T))] in FinOrd RelStr(# (Bags n),T #) by BAGORDER:def_15;
hence HM (p9,T) <= p9,T by Def2; ::_thesis: verum
end;
end;
end;
theorem Th35: :: POLYRED:35
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being non-zero Polynomial of n,L holds Red (p,T) < p,T
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being non-zero Polynomial of n,L holds Red (p,T) < p,T
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable add-associative right_zeroed addLoopStr
for p being non-zero Polynomial of n,L holds Red (p,T) < p,T
let L be non trivial right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for p being non-zero Polynomial of n,L holds Red (p,T) < p,T
let p be non-zero Polynomial of n,L; ::_thesis: Red (p,T) < p,T
(Red (p,T)) . (HT (p,T)) = 0. L by TERMORD:39;
then A1: not HT (p,T) in Support (Red (p,T)) by POLYNOM1:def_3;
p <> 0_ (n,L) by POLYNOM7:def_1;
then Support p <> {} by POLYNOM7:1;
then A2: HT (p,T) in Support p by TERMORD:def_6;
Red (p,T) < HM (p,T),T by Th33;
then A3: Red (p,T) <= HM (p,T),T by Def3;
HM (p,T) <= p,T by Th34;
then Red (p,T) <= p,T by A3, Th27;
hence Red (p,T) < p,T by A2, A1, Def3; ::_thesis: verum
end;
begin
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 f, p, g be Polynomial of n,L;
let b be bag of n;
predf reduces_to g,p,b,T means :Def5: :: POLYRED:def 5
( f <> 0_ (n,L) & p <> 0_ (n,L) & b in Support f & ex s being bag of n st
( s + (HT (p,T)) = b & g = f - (((f . b) / (HC (p,T))) * (s *' p)) ) );
end;
:: deftheorem Def5 defines reduces_to POLYRED: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 f, p, g being Polynomial of n,L
for b being bag of n holds
( f reduces_to g,p,b,T iff ( f <> 0_ (n,L) & p <> 0_ (n,L) & b in Support f & ex s being bag of n st
( s + (HT (p,T)) = b & g = f - (((f . b) / (HC (p,T))) * (s *' p)) ) ) );
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 f, p, g be Polynomial of n,L;
predf reduces_to g,p,T means :Def6: :: POLYRED:def 6
ex b being bag of n st f reduces_to g,p,b,T;
end;
:: deftheorem Def6 defines reduces_to POLYRED:def_6_:_
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, p, g being Polynomial of n,L holds
( f reduces_to g,p,T iff ex b being bag of n st f reduces_to g,p,b,T );
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 f, g be Polynomial of n,L;
let P be Subset of (Polynom-Ring (n,L));
predf reduces_to g,P,T means :Def7: :: POLYRED:def 7
ex p being Polynomial of n,L st
( p in P & f reduces_to g,p,T );
end;
:: deftheorem Def7 defines reduces_to POLYRED:def_7_:_
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, g being Polynomial of n,L
for P being Subset of (Polynom-Ring (n,L)) holds
( f reduces_to g,P,T iff ex p being Polynomial of n,L st
( p in P & f reduces_to g,p,T ) );
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 f, p be Polynomial of n,L;
predf is_reducible_wrt p,T means :Def8: :: POLYRED:def 8
ex g being Polynomial of n,L st f reduces_to g,p,T;
end;
:: deftheorem Def8 defines is_reducible_wrt POLYRED:def_8_:_
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, p being Polynomial of n,L holds
( f is_reducible_wrt p,T iff ex g being Polynomial of n,L st f reduces_to g,p,T );
notation
let n be Ordinal;
let T be connected TermOrder of n;
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ;
let f, p be Polynomial of n,L;
antonym f is_irreducible_wrt p,T for f is_reducible_wrt p,T;
antonym f is_in_normalform_wrt p,T for f is_reducible_wrt p,T;
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 f be Polynomial of n,L;
let P be Subset of (Polynom-Ring (n,L));
predf is_reducible_wrt P,T means :: POLYRED:def 9
ex g being Polynomial of n,L st f reduces_to g,P,T;
end;
:: deftheorem defines is_reducible_wrt POLYRED:def_9_:_
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 P being Subset of (Polynom-Ring (n,L)) holds
( f is_reducible_wrt P,T iff ex g being Polynomial of n,L st f reduces_to g,P,T );
notation
let n be Ordinal;
let T be connected TermOrder of n;
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ;
let f be Polynomial of n,L;
let P be Subset of (Polynom-Ring (n,L));
antonym f is_irreducible_wrt P,T for f is_reducible_wrt P,T;
antonym f is_in_normalform_wrt P,T for f is_reducible_wrt P,T;
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 f, p, g be Polynomial of n,L;
predf top_reduces_to g,p,T means :: POLYRED:def 10
f reduces_to g,p, HT (f,T),T;
end;
:: deftheorem defines top_reduces_to POLYRED:def_10_:_
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, p, g being Polynomial of n,L holds
( f top_reduces_to g,p,T iff f reduces_to g,p, HT (f,T),T );
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 f, p be Polynomial of n,L;
predf is_top_reducible_wrt p,T means :: POLYRED:def 11
ex g being Polynomial of n,L st f top_reduces_to g,p,T;
end;
:: deftheorem defines is_top_reducible_wrt POLYRED:def_11_:_
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, p being Polynomial of n,L holds
( f is_top_reducible_wrt p,T iff ex g being Polynomial of n,L st f top_reduces_to g,p,T );
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 f be Polynomial of n,L;
let P be Subset of (Polynom-Ring (n,L));
predf is_top_reducible_wrt P,T means :: POLYRED:def 12
ex p being Polynomial of n,L st
( p in P & f is_top_reducible_wrt p,T );
end;
:: deftheorem defines is_top_reducible_wrt POLYRED:def_12_:_
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 P being Subset of (Polynom-Ring (n,L)) holds
( f is_top_reducible_wrt P,T iff ex p being Polynomial of n,L st
( p in P & f is_top_reducible_wrt p,T ) );
theorem :: POLYRED:36
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 p being non-zero Polynomial of n,L holds
( f is_reducible_wrt p,T iff ex b being bag of n st
( b in Support f & HT (p,T) divides b ) )
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f being Polynomial of n,L
for p being non-zero Polynomial of n,L holds
( f is_reducible_wrt p,T iff ex b being bag of n st
( b in Support f & HT (p,T) divides b ) )
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f being Polynomial of n,L
for p being non-zero Polynomial of n,L holds
( f is_reducible_wrt p,T iff ex b being bag of n st
( b in Support f & HT (p,T) divides b ) )
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 p being non-zero Polynomial of n,L holds
( f is_reducible_wrt p,T iff ex b being bag of n st
( b in Support f & HT (p,T) divides b ) )
let f be Polynomial of n,L; ::_thesis: for p being non-zero Polynomial of n,L holds
( f is_reducible_wrt p,T iff ex b being bag of n st
( b in Support f & HT (p,T) divides b ) )
let p be non-zero Polynomial of n,L; ::_thesis: ( f is_reducible_wrt p,T iff ex b being bag of n st
( b in Support f & HT (p,T) divides b ) )
A1: now__::_thesis:_(_ex_b_being_bag_of_n_st_
(_b_in_Support_f_&_HT_(p,T)_divides_b_)_implies_f_is_reducible_wrt_p,T_)
A2: p <> 0_ (n,L) by POLYNOM7:def_1;
assume ex b being bag of n st
( b in Support f & HT (p,T) divides b ) ; ::_thesis: f is_reducible_wrt p,T
then consider b being bag of n such that
A3: b in Support f and
A4: HT (p,T) divides b ;
consider s being bag of n such that
A5: b = (HT (p,T)) + s by A4, TERMORD:1;
set g = f - (((f . b) / (HC (p,T))) * (s *' p));
f <> 0_ (n,L) by A3, POLYNOM7:1;
then f reduces_to f - (((f . b) / (HC (p,T))) * (s *' p)),p,b,T by A3, A5, A2, Def5;
then f reduces_to f - (((f . b) / (HC (p,T))) * (s *' p)),p,T by Def6;
hence f is_reducible_wrt p,T by Def8; ::_thesis: verum
end;
now__::_thesis:_(_f_is_reducible_wrt_p,T_implies_ex_b_being_bag_of_n_st_
(_b_in_Support_f_&_HT_(p,T)_divides_b_)_)
assume f is_reducible_wrt p,T ; ::_thesis: ex b being bag of n st
( b in Support f & HT (p,T) divides b )
then consider g being Polynomial of n,L such that
A6: f reduces_to g,p,T by Def8;
consider b being bag of n such that
A7: f reduces_to g,p,b,T by A6, Def6;
ex s being bag of n st
( s + (HT (p,T)) = b & g = f - (((f . b) / (HC (p,T))) * (s *' p)) ) by A7, Def5;
then A8: HT (p,T) divides b by TERMORD:1;
b in Support f by A7, Def5;
hence ex b being bag of n st
( b in Support f & HT (p,T) divides b ) by A8; ::_thesis: verum
end;
hence ( f is_reducible_wrt p,T iff ex b being bag of n st
( b in Support f & HT (p,T) divides b ) ) by A1; ::_thesis: verum
end;
Lm17: for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, p, g being Polynomial of n,L
for b being bag of n st f reduces_to g,p,b,T holds
not b in Support g
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, p, g being Polynomial of n,L
for b being bag of n st f reduces_to g,p,b,T holds
not b in Support g
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, p, g being Polynomial of n,L
for b being bag of n st f reduces_to g,p,b,T holds
not b in Support g
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, p, g being Polynomial of n,L
for b being bag of n st f reduces_to g,p,b,T holds
not b in Support g
let f, p, g be Polynomial of n,L; ::_thesis: for b being bag of n st f reduces_to g,p,b,T holds
not b in Support g
let b be bag of n; ::_thesis: ( f reduces_to g,p,b,T implies not b in Support g )
assume A1: f reduces_to g,p,b,T ; ::_thesis: not b in Support g
then ( b in Support f & ex s being bag of n st
( s + (HT (p,T)) = b & g = f - (((f . b) / (HC (p,T))) * (s *' p)) ) ) by Def5;
then consider s being bag of n such that
b in Support f and
A2: s + (HT (p,T)) = b and
A3: g = f - (((f . b) / (HC (p,T))) * (s *' p)) ;
p <> 0_ (n,L) by A1, Def5;
then A4: HC (p,T) <> 0. L by TERMORD:17;
set q = ((f . b) / (HC (p,T))) * (s *' p);
A5: (((f . b) / (HC (p,T))) * (s *' p)) . b = ((f . b) / (HC (p,T))) * ((s *' p) . b) by POLYNOM7:def_9
.= ((f . b) / (HC (p,T))) * (p . (HT (p,T))) by A2, Lm9
.= ((f . b) / (HC (p,T))) * (HC (p,T)) by TERMORD:def_7
.= ((f . b) * ((HC (p,T)) ")) * (HC (p,T)) by VECTSP_1:def_11
.= (f . b) * (((HC (p,T)) ") * (HC (p,T))) by GROUP_1:def_3
.= (f . b) * (1. L) by A4, VECTSP_1:def_10
.= f . b by VECTSP_1:def_4 ;
g = f + (- (((f . b) / (HC (p,T))) * (s *' p))) by A3, POLYNOM1:def_6;
then g . b = (f . b) + ((- (((f . b) / (HC (p,T))) * (s *' p))) . b) by POLYNOM1:15
.= (f . b) + (- ((((f . b) / (HC (p,T))) * (s *' p)) . b)) by POLYNOM1:17
.= 0. L by A5, RLVECT_1:5 ;
hence not b in Support g by POLYNOM1:def_3; ::_thesis: verum
end;
theorem Th37: :: POLYRED:37
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 0_ (n,L) is_irreducible_wrt p,T
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L holds 0_ (n,L) is_irreducible_wrt p,T
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for p being Polynomial of n,L holds 0_ (n,L) is_irreducible_wrt p,T
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for p being Polynomial of n,L holds 0_ (n,L) is_irreducible_wrt p,T
let p be Polynomial of n,L; ::_thesis: 0_ (n,L) is_irreducible_wrt p,T
assume 0_ (n,L) is_reducible_wrt p,T ; ::_thesis: contradiction
then consider g being Polynomial of n,L such that
A1: 0_ (n,L) reduces_to g,p,T by Def8;
ex b being bag of n st 0_ (n,L) reduces_to g,p,b,T by A1, Def6;
hence contradiction by Def5; ::_thesis: verum
end;
theorem :: POLYRED: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, p being Polynomial of n,L
for m being non-zero Monomial of n,L st f reduces_to f - (m *' p),p,T holds
HT ((m *' p),T) in Support f
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, p being Polynomial of n,L
for m being non-zero Monomial of n,L st f reduces_to f - (m *' p),p,T holds
HT ((m *' p),T) in Support f
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, p being Polynomial of n,L
for m being non-zero Monomial of n,L st f reduces_to f - (m *' p),p,T holds
HT ((m *' p),T) in Support f
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, p being Polynomial of n,L
for m being non-zero Monomial of n,L st f reduces_to f - (m *' p),p,T holds
HT ((m *' p),T) in Support f
let f, p be Polynomial of n,L; ::_thesis: for m being non-zero Monomial of n,L st f reduces_to f - (m *' p),p,T holds
HT ((m *' p),T) in Support f
let m be non-zero Monomial of n,L; ::_thesis: ( f reduces_to f - (m *' p),p,T implies HT ((m *' p),T) in Support f )
assume f reduces_to f - (m *' p),p,T ; ::_thesis: HT ((m *' p),T) in Support f
then consider b being bag of n such that
A1: f reduces_to f - (m *' p),p,b,T by Def6;
A2: p <> 0_ (n,L) by A1, Def5;
then A3: p is non-zero by POLYNOM7:def_1;
A4: HC (p,T) <> 0. L by A2, TERMORD:17;
A5: now__::_thesis:_not_(HC_(p,T))_"_=_0._L
assume (HC (p,T)) " = 0. L ; ::_thesis: contradiction
then 0. L = (HC (p,T)) * ((HC (p,T)) ") by VECTSP_1:7
.= 1. L by A4, VECTSP_1:def_10 ;
hence contradiction ; ::_thesis: verum
end;
b in Support f by A1, Def5;
then f . b <> 0. L by POLYNOM1:def_3;
then (f . b) * ((HC (p,T)) ") <> 0. L by A5, VECTSP_2:def_1;
then (f . b) / (HC (p,T)) <> 0. L by VECTSP_1:def_11;
then A6: not (f . b) / (HC (p,T)) is zero by STRUCT_0:def_12;
consider s being bag of n such that
A7: s + (HT (p,T)) = b and
A8: f - (m *' p) = f - (((f . b) / (HC (p,T))) * (s *' p)) by A1, Def5;
A9: ((f . b) / (HC (p,T))) * (s *' p) = - (- (((f . b) / (HC (p,T))) * (s *' p))) by POLYNOM1:19;
f = f + (0_ (n,L)) by POLYNOM1:23
.= f + ((m *' p) - (m *' p)) by POLYNOM1:24
.= f + ((m *' p) + (- (m *' p))) by POLYNOM1:def_6
.= (f + (- (m *' p))) + (m *' p) by POLYNOM1:21
.= (m *' p) + (f - (((f . b) / (HC (p,T))) * (s *' p))) by A8, POLYNOM1:def_6 ;
then 0_ (n,L) = f - ((m *' p) + (f - (((f . b) / (HC (p,T))) * (s *' p)))) by POLYNOM1:24
.= f + (- ((m *' p) + (f - (((f . b) / (HC (p,T))) * (s *' p))))) by POLYNOM1:def_6
.= f + ((- (m *' p)) + (- (f - (((f . b) / (HC (p,T))) * (s *' p))))) by Th1
.= f + ((- (m *' p)) + (- (f + (- (((f . b) / (HC (p,T))) * (s *' p)))))) by POLYNOM1:def_6
.= f + ((- (m *' p)) + ((- f) + (- (- (((f . b) / (HC (p,T))) * (s *' p)))))) by Th1
.= f + ((- f) + ((- (m *' p)) + (((f . b) / (HC (p,T))) * (s *' p)))) by A9, POLYNOM1:21
.= (f + (- f)) + ((- (m *' p)) + (((f . b) / (HC (p,T))) * (s *' p))) by POLYNOM1:21
.= (f - f) + ((- (m *' p)) + (((f . b) / (HC (p,T))) * (s *' p))) by POLYNOM1:def_6
.= (0_ (n,L)) + ((- (m *' p)) + (((f . b) / (HC (p,T))) * (s *' p))) by POLYNOM1:24
.= (- (m *' p)) + (((f . b) / (HC (p,T))) * (s *' p)) by Th2 ;
then m *' p = (m *' p) + ((- (m *' p)) + (((f . b) / (HC (p,T))) * (s *' p))) by POLYNOM1:23
.= ((m *' p) + (- (m *' p))) + (((f . b) / (HC (p,T))) * (s *' p)) by POLYNOM1:21
.= ((m *' p) - (m *' p)) + (((f . b) / (HC (p,T))) * (s *' p)) by POLYNOM1:def_6
.= (0_ (n,L)) + (((f . b) / (HC (p,T))) * (s *' p)) by POLYNOM1:24
.= ((f . b) / (HC (p,T))) * (s *' p) by Th2 ;
then HT ((m *' p),T) = HT ((s *' p),T) by A6, Th21
.= b by A7, A3, Th15 ;
hence HT ((m *' p),T) in Support f by A1, Def5; ::_thesis: verum
end;
theorem :: POLYRED:39
for n being Ordinal
for T being connected TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, p, g being Polynomial of n,L
for b being bag of n st f reduces_to g,p,b,T holds
not b in Support g by Lm17;
theorem Th40: :: POLYRED:40
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, p, g being Polynomial of n,L
for b, b9 being bag of n st b < b9,T & f reduces_to g,p,b,T holds
( b9 in Support g iff b9 in Support f )
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, p, g being Polynomial of n,L
for b, b9 being bag of n st b < b9,T & f reduces_to g,p,b,T holds
( b9 in Support g iff b9 in Support f )
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, p, g being Polynomial of n,L
for b, b9 being bag of n st b < b9,T & f reduces_to g,p,b,T holds
( b9 in Support g iff b9 in Support f )
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, p, g being Polynomial of n,L
for b, b9 being bag of n st b < b9,T & f reduces_to g,p,b,T holds
( b9 in Support g iff b9 in Support f )
let f, p, g be Polynomial of n,L; ::_thesis: for b, b9 being bag of n st b < b9,T & f reduces_to g,p,b,T holds
( b9 in Support g iff b9 in Support f )
let b, b9 be bag of n; ::_thesis: ( b < b9,T & f reduces_to g,p,b,T implies ( b9 in Support g iff b9 in Support f ) )
assume A1: b < b9,T ; ::_thesis: ( not f reduces_to g,p,b,T or ( b9 in Support g iff b9 in Support f ) )
assume f reduces_to g,p,b,T ; ::_thesis: ( b9 in Support g iff b9 in Support f )
then consider s being bag of n such that
A2: s + (HT (p,T)) = b and
A3: g = f - (((f . b) / (HC (p,T))) * (s *' p)) by Def5;
A4: b9 is Element of Bags n by PRE_POLY:def_12;
A5: now__::_thesis:_not_b9_in_Support_(s_*'_p)
assume b9 in Support (s *' p) ; ::_thesis: contradiction
then A6: b9 <= b,T by A2, Th16;
b <= b9,T by A1, TERMORD:def_3;
then b = b9 by A6, TERMORD:7;
hence contradiction by A1, TERMORD:def_3; ::_thesis: verum
end;
A7: now__::_thesis:_(_b9_in_Support_f_implies_b9_in_Support_g_)
A8: (((f . b) / (HC (p,T))) * (s *' p)) . b9 = ((f . b) / (HC (p,T))) * ((s *' p) . b9) by POLYNOM7:def_9
.= ((f . b) / (HC (p,T))) * (0. L) by A4, A5, POLYNOM1:def_3
.= 0. L by VECTSP_1:7 ;
assume A9: b9 in Support f ; ::_thesis: b9 in Support g
(f - (((f . b) / (HC (p,T))) * (s *' p))) . b9 = (f + (- (((f . b) / (HC (p,T))) * (s *' p)))) . b9 by POLYNOM1:def_6
.= (f . b9) + ((- (((f . b) / (HC (p,T))) * (s *' p))) . b9) by POLYNOM1:15
.= (f . b9) + (- (0. L)) by A8, POLYNOM1:17
.= (f . b9) + (0. L) by RLVECT_1:12
.= f . b9 by RLVECT_1:def_4 ;
then g . b9 <> 0. L by A3, A9, POLYNOM1:def_3;
hence b9 in Support g by A4, POLYNOM1:def_3; ::_thesis: verum
end;
now__::_thesis:_(_b9_in_Support_g_implies_b9_in_Support_f_)
A10: (((f . b) / (HC (p,T))) * (s *' p)) . b9 = ((f . b) / (HC (p,T))) * ((s *' p) . b9) by POLYNOM7:def_9
.= ((f . b) / (HC (p,T))) * (0. L) by A4, A5, POLYNOM1:def_3
.= 0. L by VECTSP_1:7 ;
assume A11: b9 in Support g ; ::_thesis: b9 in Support f
(f - (((f . b) / (HC (p,T))) * (s *' p))) . b9 = (f + (- (((f . b) / (HC (p,T))) * (s *' p)))) . b9 by POLYNOM1:def_6
.= (f . b9) + ((- (((f . b) / (HC (p,T))) * (s *' p))) . b9) by POLYNOM1:15
.= (f . b9) + (- (0. L)) by A10, POLYNOM1:17
.= (f . b9) + (0. L) by RLVECT_1:12
.= f . b9 by RLVECT_1:def_4 ;
then f . b9 <> 0. L by A3, A11, POLYNOM1:def_3;
hence b9 in Support f by A4, POLYNOM1:def_3; ::_thesis: verum
end;
hence ( b9 in Support g iff b9 in Support f ) by A7; ::_thesis: verum
end;
theorem Th41: :: POLYRED:41
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, p, g being Polynomial of n,L
for b, b9 being bag of n st b < b9,T & f reduces_to g,p,b,T holds
f . b9 = g . b9
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, p, g being Polynomial of n,L
for b, b9 being bag of n st b < b9,T & f reduces_to g,p,b,T holds
f . b9 = g . b9
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, p, g being Polynomial of n,L
for b, b9 being bag of n st b < b9,T & f reduces_to g,p,b,T holds
f . b9 = g . b9
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, p, g being Polynomial of n,L
for b, b9 being bag of n st b < b9,T & f reduces_to g,p,b,T holds
f . b9 = g . b9
let f, p, g be Polynomial of n,L; ::_thesis: for b, b9 being bag of n st b < b9,T & f reduces_to g,p,b,T holds
f . b9 = g . b9
let b, b9 be bag of n; ::_thesis: ( b < b9,T & f reduces_to g,p,b,T implies f . b9 = g . b9 )
assume A1: b < b9,T ; ::_thesis: ( not f reduces_to g,p,b,T or f . b9 = g . b9 )
assume f reduces_to g,p,b,T ; ::_thesis: f . b9 = g . b9
then consider s being bag of n such that
A2: s + (HT (p,T)) = b and
A3: g = f - (((f . b) / (HC (p,T))) * (s *' p)) by Def5;
A4: now__::_thesis:_not_b9_in_Support_(s_*'_p)
assume b9 in Support (s *' p) ; ::_thesis: contradiction
then A5: b9 <= b,T by A2, Th16;
b <= b9,T by A1, TERMORD:def_3;
then b = b9 by A5, TERMORD:7;
hence contradiction by A1, TERMORD:def_3; ::_thesis: verum
end;
A6: b9 is Element of Bags n by PRE_POLY:def_12;
A7: (((f . b) / (HC (p,T))) * (s *' p)) . b9 = ((f . b) / (HC (p,T))) * ((s *' p) . b9) by POLYNOM7:def_9
.= ((f . b) / (HC (p,T))) * (0. L) by A6, A4, POLYNOM1:def_3
.= 0. L by VECTSP_1:7 ;
(f - (((f . b) / (HC (p,T))) * (s *' p))) . b9 = (f + (- (((f . b) / (HC (p,T))) * (s *' p)))) . b9 by POLYNOM1:def_6
.= (f . b9) + ((- (((f . b) / (HC (p,T))) * (s *' p))) . b9) by POLYNOM1:15
.= (f . b9) + (- (0. L)) by A7, POLYNOM1:17
.= (f . b9) + (0. L) by RLVECT_1:12
.= f . b9 by RLVECT_1:def_4 ;
hence f . b9 = g . b9 by A3; ::_thesis: verum
end;
theorem Th42: :: POLYRED:42
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, p, g being Polynomial of n,L st f reduces_to g,p,T holds
for b being bag of n st b in Support g holds
b <= HT (f,T),T
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, p, g being Polynomial of n,L st f reduces_to g,p,T holds
for b being bag of n st b in Support g holds
b <= HT (f,T),T
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, p, g being Polynomial of n,L st f reduces_to g,p,T holds
for b being bag of n st b in Support g holds
b <= HT (f,T),T
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, p, g being Polynomial of n,L st f reduces_to g,p,T holds
for b being bag of n st b in Support g holds
b <= HT (f,T),T
let f, p, g be Polynomial of n,L; ::_thesis: ( f reduces_to g,p,T implies for b being bag of n st b in Support g holds
b <= HT (f,T),T )
A1: T is_connected_in field T by RELAT_2:def_14;
assume f reduces_to g,p,T ; ::_thesis: for b being bag of n st b in Support g holds
b <= HT (f,T),T
then consider b being bag of n such that
A2: f reduces_to g,p,b,T by Def6;
b in Support f by A2, Def5;
then A3: b <= HT (f,T),T by TERMORD:def_6;
now__::_thesis:_for_u_being_bag_of_n_st_u_in_Support_g_holds_
u_<=_HT_(f,T),T
let u be bag of n; ::_thesis: ( u in Support g implies u <= HT (f,T),T )
assume A4: u in Support g ; ::_thesis: u <= HT (f,T),T
now__::_thesis:_(_(_u_=_b_&_contradiction_)_or_(_u_<>_b_&_u_<=_HT_(f,T),T_)_)
percases ( u = b or u <> b ) ;
case u = b ; ::_thesis: contradiction
hence contradiction by A2, A4, Lm17; ::_thesis: verum
end;
caseA5: u <> b ; ::_thesis: u <= HT (f,T),T
b <= b,T by TERMORD:6;
then [b,b] in T by TERMORD:def_2;
then A6: b in field T by RELAT_1:15;
u <= u,T by TERMORD:6;
then [u,u] in T by TERMORD:def_2;
then u in field T by RELAT_1:15;
then A7: ( [u,b] in T or [b,u] in T ) by A1, A5, A6, RELAT_2:def_6;
now__::_thesis:_(_(_u_<=_b,T_&_u_<=_HT_(f,T),T_)_or_(_b_<=_u,T_&_u_<=_HT_(f,T),T_)_)
percases ( u <= b,T or b <= u,T ) by A7, TERMORD:def_2;
case u <= b,T ; ::_thesis: u <= HT (f,T),T
hence u <= HT (f,T),T by A3, TERMORD:8; ::_thesis: verum
end;
case b <= u,T ; ::_thesis: u <= HT (f,T),T
then b < u,T by A5, TERMORD:def_3;
then ( u in Support f iff u in Support g ) by A2, Th40;
hence u <= HT (f,T),T by A4, TERMORD:def_6; ::_thesis: verum
end;
end;
end;
hence u <= HT (f,T),T ; ::_thesis: verum
end;
end;
end;
hence u <= HT (f,T),T ; ::_thesis: verum
end;
hence for b being bag of n st b in Support g holds
b <= HT (f,T),T ; ::_thesis: verum
end;
theorem Th43: :: POLYRED:43
for n being Ordinal
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, p, g being Polynomial of n,L st f reduces_to g,p,T holds
g < f,T
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, p, g being Polynomial of n,L st f reduces_to g,p,T holds
g < f,T
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for f, p, g being Polynomial of n,L st f reduces_to g,p,T holds
g < f,T
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for f, p, g being Polynomial of n,L st f reduces_to g,p,T holds
g < f,T
let f, p, g be Polynomial of n,L; ::_thesis: ( f reduces_to g,p,T implies g < f,T )
set R = RelStr(# (Bags n),T #);
defpred S1[ Nat] means for f, p, g being Polynomial of n,L st card (Support f) = $1 & f reduces_to g,p,T holds
[(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #);
A1: ex k being Nat st card (Support f) = k ;
assume A2: f reduces_to g,p,T ; ::_thesis: g < f,T
then consider b being bag of n such that
A3: f reduces_to g,p,b,T by Def6;
A4: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A5: for f, p, g being Polynomial of n,L st card (Support f) = k & f reduces_to g,p,T holds
[(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #) ; ::_thesis: S1[k + 1]
now__::_thesis:_for_f,_p,_g_being_Polynomial_of_n,L_st_card_(Support_f)_=_k_+_1_&_f_reduces_to_g,p,T_holds_
[(Support_g),(Support_f)]_in_FinOrd_RelStr(#_(Bags_n),T_#)
A6: T is_connected_in field T by RELAT_2:def_14;
let f, p, g be Polynomial of n,L; ::_thesis: ( card (Support f) = k + 1 & f reduces_to g,p,T implies [(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #) )
assume that
A7: card (Support f) = k + 1 and
A8: f reduces_to g,p,T ; ::_thesis: [(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #)
consider b being bag of n such that
A9: f reduces_to g,p,b,T by A8, Def6;
consider s being bag of n such that
A10: s + (HT (p,T)) = b and
A11: g = f - (((f . b) / (HC (p,T))) * (s *' p)) by A9, Def5;
A12: b in Support f by A9, Def5;
A13: f <> 0_ (n,L) by A9, Def5;
now__::_thesis:_(_(_HT_(f,T)_<>_HT_(g,T)_&_[(Support_g),(Support_f)]_in_FinOrd_RelStr(#_(Bags_n),T_#)_)_or_(_HT_(g,T)_=_HT_(f,T)_&_[(Support_g),(Support_f)]_in_FinOrd_RelStr(#_(Bags_n),T_#)_)_)
percases ( HT (f,T) <> HT (g,T) or HT (g,T) = HT (f,T) ) ;
caseA14: HT (f,T) <> HT (g,T) ; ::_thesis: [(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #)
HT (g,T) <= HT (g,T),T by TERMORD:6;
then [(HT (g,T)),(HT (g,T))] in T by TERMORD:def_2;
then A15: HT (g,T) in field T by RELAT_1:15;
HT (f,T) <= HT (f,T),T by TERMORD:6;
then [(HT (f,T)),(HT (f,T))] in T by TERMORD:def_2;
then HT (f,T) in field T by RELAT_1:15;
then A16: ( [(HT (f,T)),(HT (g,T))] in T or [(HT (g,T)),(HT (f,T))] in T ) by A6, A14, A15, RELAT_2:def_6;
now__::_thesis:_(_(_HT_(f,T)_<=_HT_(g,T),T_&_[(Support_g),(Support_f)]_in_FinOrd_RelStr(#_(Bags_n),T_#)_)_or_(_HT_(g,T)_<=_HT_(f,T),T_&_[(Support_g),(Support_f)]_in_FinOrd_RelStr(#_(Bags_n),T_#)_)_)
percases ( HT (f,T) <= HT (g,T),T or HT (g,T) <= HT (f,T),T ) by A16, TERMORD:def_2;
caseA17: HT (f,T) <= HT (g,T),T ; ::_thesis: [(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #)
now__::_thesis:_HT_(g,T)_in_Support_g
assume not HT (g,T) in Support g ; ::_thesis: contradiction
then HT (g,T) = EmptyBag n by TERMORD:def_6;
then [(HT (g,T)),(HT (f,T))] in T by BAGORDER:def_5;
then HT (g,T) <= HT (f,T),T by TERMORD:def_2;
hence contradiction by A14, A17, TERMORD:7; ::_thesis: verum
end;
then HT (g,T) <= HT (f,T),T by A8, Th42;
hence [(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #) by A14, A17, TERMORD:7; ::_thesis: verum
end;
case HT (g,T) <= HT (f,T),T ; ::_thesis: [(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #)
then HT (g,T) < HT (f,T),T by A14, TERMORD:def_3;
then g < f,T by Lm15;
then g <= f,T by Def3;
hence [(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #) by Def2; ::_thesis: verum
end;
end;
end;
hence [(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #) ; ::_thesis: verum
end;
caseA18: HT (g,T) = HT (f,T) ; ::_thesis: [(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #)
now__::_thesis:_(_(_b_=_HT_(f,T)_&_[(Support_g),(Support_f)]_in_FinOrd_RelStr(#_(Bags_n),T_#)_)_or_(_b_<>_HT_(f,T)_&_[(Support_g),(Support_f)]_in_FinOrd_RelStr(#_(Bags_n),T_#)_)_)
percases ( b = HT (f,T) or b <> HT (f,T) ) ;
case b = HT (f,T) ; ::_thesis: [(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #)
then not HT (g,T) in Support g by A9, A18, Lm17;
then Support g = {} by TERMORD:def_6;
then g = 0_ (n,L) by POLYNOM7:1;
then g <= f,T by Lm12;
hence [(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #) by Def2; ::_thesis: verum
end;
caseA19: b <> HT (f,T) ; ::_thesis: [(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #)
HT (f,T) in Support f by A12, TERMORD:def_6;
then for u being set st u in {(HT (f,T))} holds
u in Support f by TARSKI:def_1;
then A20: {(HT (f,T))} c= Support f by TARSKI:def_3;
A21: Support (Red (f,T)) = (Support f) \ {(HT (f,T))} by TERMORD:36;
not b in {(HT (f,T))} by A19, TARSKI:def_1;
then A22: b in Support (Red (f,T)) by A12, A21, XBOOLE_0:def_5;
then Red (f,T) <> 0_ (n,L) by POLYNOM7:1;
then reconsider rf = Red (f,T) as non-zero Polynomial of n,L by POLYNOM7:def_1;
A23: ( rf <> 0_ (n,L) & p <> 0_ (n,L) ) by A9, Def5, POLYNOM7:def_1;
b <= HT (f,T),T by A12, TERMORD:def_6;
then b < HT (f,T),T by A19, TERMORD:def_3;
then f . (HT (f,T)) = g . (HT (g,T)) by A9, A18, Th41;
then HC (f,T) = g . (HT (g,T)) by TERMORD:def_7
.= HC (g,T) by TERMORD:def_7 ;
then HM (f,T) = Monom ((HC (g,T)),(HT (g,T))) by A18, TERMORD:def_8
.= HM (g,T) by TERMORD:def_8 ;
then Red (g,T) = (f - (((f . b) / (HC (p,T))) * (s *' p))) - (HM (f,T)) by A11, TERMORD:def_9
.= (((HM (f,T)) + rf) - (((f . b) / (HC (p,T))) * (s *' p))) - (HM (f,T)) by TERMORD:38
.= (((HM (f,T)) + rf) - (((rf . b) / (HC (p,T))) * (s *' p))) - (HM (f,T)) by A12, A19, TERMORD:40
.= (((HM (f,T)) + rf) + (- (((rf . b) / (HC (p,T))) * (s *' p)))) - (HM (f,T)) by POLYNOM1:def_6
.= ((HM (f,T)) + (rf + (- (((rf . b) / (HC (p,T))) * (s *' p))))) - (HM (f,T)) by POLYNOM1:21
.= ((HM (f,T)) + (rf + (- (((rf . b) / (HC (p,T))) * (s *' p))))) + (- (HM (f,T))) by POLYNOM1:def_6
.= (rf + (- (((rf . b) / (HC (p,T))) * (s *' p)))) + ((HM (f,T)) + (- (HM (f,T)))) by POLYNOM1:21
.= (rf - (((rf . b) / (HC (p,T))) * (s *' p))) + ((HM (f,T)) + (- (HM (f,T)))) by POLYNOM1:def_6
.= (rf - (((rf . b) / (HC (p,T))) * (s *' p))) + ((HM (f,T)) - (HM (f,T))) by POLYNOM1:def_6
.= (rf - (((rf . b) / (HC (p,T))) * (s *' p))) + (0_ (n,L)) by POLYNOM1:24
.= rf - (((rf . b) / (HC (p,T))) * (s *' p)) by POLYNOM1:23 ;
then rf reduces_to Red (g,T),p,b,T by A10, A22, A23, Def5;
then A24: rf reduces_to Red (g,T),p,T by Def6;
HT (f,T) in {(HT (f,T))} by TARSKI:def_1;
then A25: not HT (f,T) in Support (Red (f,T)) by A21, XBOOLE_0:def_5;
(Support (Red (f,T))) \/ {(HT (f,T))} = (Support f) \/ {(HT (f,T))} by A21, XBOOLE_1:39
.= Support f by A20, XBOOLE_1:12 ;
then (card (Support (Red (f,T)))) + 1 = k + 1 by A7, A25, CARD_2:41;
then [(Support (Red (g,T))),(Support rf)] in FinOrd RelStr(# (Bags n),T #) by A5, A24, XCMPLX_1:2;
then Red (g,T) <= Red (f,T),T by Def2;
then g <= f,T by A13, A18, Lm16;
hence [(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #) by Def2; ::_thesis: verum
end;
end;
end;
hence [(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #) ; ::_thesis: verum
end;
end;
end;
hence [(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #) ; ::_thesis: verum
end;
hence S1[k + 1] ; ::_thesis: verum
end;
A26: S1[ 0 ]
proof
let f, p, g be Polynomial of n,L; ::_thesis: ( card (Support f) = 0 & f reduces_to g,p,T implies [(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #) )
assume that
A27: card (Support f) = 0 and
A28: f reduces_to g,p,T ; ::_thesis: [(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #)
Support f = {} by A27;
then f = 0_ (n,L) by POLYNOM7:1;
then f is_irreducible_wrt p,T by Th37;
hence [(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #) by A28, Def8; ::_thesis: verum
end;
for k being Nat holds S1[k] from NAT_1:sch_2(A26, A4);
then [(Support g),(Support f)] in FinOrd RelStr(# (Bags n),T #) by A2, A1;
then A29: g <= f,T by Def2;
b in Support f by A3, Def5;
then Support f <> Support g by A3, Lm17;
hence g < f,T by A29, Def3; ::_thesis: verum
end;
begin
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 PolyRedRel (P,T) -> Relation of (NonZero (Polynom-Ring (n,L))), the carrier of (Polynom-Ring (n,L)) means :Def13: :: POLYRED:def 13
for p, q being Polynomial of n,L holds
( [p,q] in it iff p reduces_to q,P,T );
existence
ex b1 being Relation of (NonZero (Polynom-Ring (n,L))), the carrier of (Polynom-Ring (n,L)) st
for p, q being Polynomial of n,L holds
( [p,q] in b1 iff p reduces_to q,P,T )
proof
defpred S1[ set , set ] means ex p, q being Polynomial of n,L st
( p = $1 & q = $2 & p reduces_to q,P,T );
set A = NonZero (Polynom-Ring (n,L));
set B = the carrier of (Polynom-Ring (n,L));
consider R being Relation of (NonZero (Polynom-Ring (n,L))), the carrier of (Polynom-Ring (n,L)) such that
A1: for x, y being set holds
( [x,y] in R iff ( x in NonZero (Polynom-Ring (n,L)) & y in the carrier of (Polynom-Ring (n,L)) & S1[x,y] ) ) from RELSET_1:sch_1();
take R ; ::_thesis: for p, q being Polynomial of n,L holds
( [p,q] in R iff p reduces_to q,P,T )
now__::_thesis:_for_p,_q_being_Polynomial_of_n,L_holds_
(_[p,q]_in_R_iff_p_reduces_to_q,P,T_)
let p, q be Polynomial of n,L; ::_thesis: ( [p,q] in R iff p reduces_to q,P,T )
A2: now__::_thesis:_(_p_reduces_to_q,P,T_implies_[p,q]_in_R_)
assume A3: p reduces_to q,P,T ; ::_thesis: [p,q] in R
then consider pp being Polynomial of n,L such that
pp in P and
A4: p reduces_to q,pp,T by Def7;
ex b being bag of n st p reduces_to q,pp,b,T by A4, Def6;
then p <> 0_ (n,L) by Def5;
then A5: not p in {(0_ (n,L))} by TARSKI:def_1;
A6: q in the carrier of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
( 0_ (n,L) = 0. (Polynom-Ring (n,L)) & p in the carrier of (Polynom-Ring (n,L)) ) by POLYNOM1:def_10;
then p in NonZero (Polynom-Ring (n,L)) by A5, XBOOLE_0:def_5;
hence [p,q] in R by A1, A3, A6; ::_thesis: verum
end;
now__::_thesis:_(_[p,q]_in_R_implies_p_reduces_to_q,P,T_)
assume [p,q] in R ; ::_thesis: p reduces_to q,P,T
then S1[p,q] by A1;
hence p reduces_to q,P,T ; ::_thesis: verum
end;
hence ( [p,q] in R iff p reduces_to q,P,T ) by A2; ::_thesis: verum
end;
hence for p, q being Polynomial of n,L holds
( [p,q] in R iff p reduces_to q,P,T ) ; ::_thesis: verum
end;
uniqueness
for b1, b2 being Relation of (NonZero (Polynom-Ring (n,L))), the carrier of (Polynom-Ring (n,L)) st ( for p, q being Polynomial of n,L holds
( [p,q] in b1 iff p reduces_to q,P,T ) ) & ( for p, q being Polynomial of n,L holds
( [p,q] in b2 iff p reduces_to q,P,T ) ) holds
b1 = b2
proof
set A = NonZero (Polynom-Ring (n,L));
set B = the carrier of (Polynom-Ring (n,L));
let R1, R2 be Relation of (NonZero (Polynom-Ring (n,L))), the carrier of (Polynom-Ring (n,L)); ::_thesis: ( ( for p, q being Polynomial of n,L holds
( [p,q] in R1 iff p reduces_to q,P,T ) ) & ( for p, q being Polynomial of n,L holds
( [p,q] in R2 iff p reduces_to q,P,T ) ) implies R1 = R2 )
assume that
A7: for p, q being Polynomial of n,L holds
( [p,q] in R1 iff p reduces_to q,P,T ) and
A8: for p, q being Polynomial of n,L holds
( [p,q] in R2 iff p reduces_to q,P,T ) ; ::_thesis: R1 = R2
for p, q being set holds
( [p,q] in R1 iff [p,q] in R2 )
proof
let p, q be set ; ::_thesis: ( [p,q] in R1 iff [p,q] in R2 )
A9: 0_ (n,L) = 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10;
A10: now__::_thesis:_(_[p,q]_in_R2_implies_[p,q]_in_R1_)
assume A11: [p,q] in R2 ; ::_thesis: [p,q] in R1
then consider p9, q9 being set such that
A12: [p,q] = [p9,q9] and
A13: p9 in NonZero (Polynom-Ring (n,L)) and
A14: q9 in the carrier of (Polynom-Ring (n,L)) by RELSET_1:2;
reconsider p9 = p9, q9 = q9 as Polynomial of n,L by A13, A14, POLYNOM1:def_10;
not p9 in {(0_ (n,L))} by A9, A13, XBOOLE_0:def_5;
then p9 <> 0_ (n,L) by TARSKI:def_1;
then reconsider p9 = p9 as non-zero Polynomial of n,L by POLYNOM7:def_1;
A15: p = p9 by A12, XTUPLE_0:1;
A16: q = q9 by A12, XTUPLE_0:1;
p9 reduces_to q9,P,T by A8, A11, A12;
hence [p,q] in R1 by A7, A15, A16; ::_thesis: verum
end;
now__::_thesis:_(_[p,q]_in_R1_implies_[p,q]_in_R2_)
assume A17: [p,q] in R1 ; ::_thesis: [p,q] in R2
then consider p9, q9 being set such that
A18: [p,q] = [p9,q9] and
A19: p9 in NonZero (Polynom-Ring (n,L)) and
A20: q9 in the carrier of (Polynom-Ring (n,L)) by RELSET_1:2;
reconsider p9 = p9, q9 = q9 as Polynomial of n,L by A19, A20, POLYNOM1:def_10;
not p9 in {(0_ (n,L))} by A9, A19, XBOOLE_0:def_5;
then p9 <> 0_ (n,L) by TARSKI:def_1;
then reconsider p9 = p9 as non-zero Polynomial of n,L by POLYNOM7:def_1;
A21: p = p9 by A18, XTUPLE_0:1;
A22: q = q9 by A18, XTUPLE_0:1;
p9 reduces_to q9,P,T by A7, A17, A18;
hence [p,q] in R2 by A8, A21, A22; ::_thesis: verum
end;
hence ( [p,q] in R1 iff [p,q] in R2 ) by A10; ::_thesis: verum
end;
hence R1 = R2 by RELAT_1:def_2; ::_thesis: verum
end;
end;
:: deftheorem Def13 defines PolyRedRel POLYRED:def_13_:_
for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for b5 being Relation of (NonZero (Polynom-Ring (n,L))), the carrier of (Polynom-Ring (n,L)) holds
( b5 = PolyRedRel (P,T) iff for p, q being Polynomial of n,L holds
( [p,q] in b5 iff p reduces_to q,P,T ) );
Lm18: for n being Ordinal
for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive add-associative right_zeroed doubleLoopStr
for f, g, p being Polynomial of n,L st f reduces_to g,p,T holds
( f <> 0_ (n,L) & p <> 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 f, g, p being Polynomial of n,L st f reduces_to g,p,T holds
( f <> 0_ (n,L) & p <> 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 f, g, p being Polynomial of n,L st f reduces_to g,p,T holds
( f <> 0_ (n,L) & p <> 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 f, g, p being Polynomial of n,L st f reduces_to g,p,T holds
( f <> 0_ (n,L) & p <> 0_ (n,L) )
let f, g, p be Polynomial of n,L; ::_thesis: ( f reduces_to g,p,T implies ( f <> 0_ (n,L) & p <> 0_ (n,L) ) )
assume f reduces_to g,p,T ; ::_thesis: ( f <> 0_ (n,L) & p <> 0_ (n,L) )
then ex b being bag of n st f reduces_to g,p,b,T by Def6;
hence ( f <> 0_ (n,L) & p <> 0_ (n,L) ) by Def5; ::_thesis: verum
end;
theorem :: POLYRED:44
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 P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds
( g <= f,T & ( g = 0_ (n,L) or HT (g,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 P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds
( g <= f,T & ( g = 0_ (n,L) or HT (g,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 P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds
( g <= f,T & ( g = 0_ (n,L) or HT (g,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 P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds
( g <= f,T & ( g = 0_ (n,L) or HT (g,T) <= HT (f,T),T ) )
let f, g be Polynomial of n,L; ::_thesis: for P being Subset of (Polynom-Ring (n,L)) st PolyRedRel (P,T) reduces f,g holds
( g <= f,T & ( g = 0_ (n,L) or HT (g,T) <= HT (f,T),T ) )
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: ( PolyRedRel (P,T) reduces f,g implies ( g <= f,T & ( g = 0_ (n,L) or HT (g,T) <= HT (f,T),T ) ) )
set R = PolyRedRel (P,T);
defpred S1[ Nat] means for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f,g holds
for p being RedSequence of PolyRedRel (P,T) st p . 1 = f & p . (len p) = g & len p = $1 holds
g <= f,T;
assume A1: PolyRedRel (P,T) reduces f,g ; ::_thesis: ( g <= f,T & ( g = 0_ (n,L) or HT (g,T) <= HT (f,T),T ) )
then consider p being RedSequence of PolyRedRel (P,T) such that
A2: ( p . 1 = f & p . (len p) = g ) by REWRITE1:def_3;
A3: 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 A4: 1 <= k ; ::_thesis: ( S1[k] implies S1[k + 1] )
thus ( S1[k] implies S1[k + 1] ) ::_thesis: verum
proof
assume A5: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_for_f,_g_being_Polynomial_of_n,L_st_PolyRedRel_(P,T)_reduces_f,g_holds_
for_p_being_RedSequence_of_PolyRedRel_(P,T)_st_p_._1_=_f_&_p_._(len_p)_=_g_&_len_p_=_k_+_1_holds_
g_<=_f,T
let f, g be Polynomial of n,L; ::_thesis: ( PolyRedRel (P,T) reduces f,g implies for p being RedSequence of PolyRedRel (P,T) st p . 1 = f & p . (len p) = g & len p = k + 1 holds
g <= f,T )
assume PolyRedRel (P,T) reduces f,g ; ::_thesis: for p being RedSequence of PolyRedRel (P,T) st p . 1 = f & p . (len p) = g & len p = k + 1 holds
g <= f,T
let p be RedSequence of PolyRedRel (P,T); ::_thesis: ( p . 1 = f & p . (len p) = g & len p = k + 1 implies g <= f,T )
assume that
A6: p . 1 = f and
A7: p . (len p) = g and
A8: len p = k + 1 ; ::_thesis: g <= f,T
A9: dom p = Seg (k + 1) by A8, FINSEQ_1:def_3;
then A10: k + 1 in dom p by FINSEQ_1:4;
set q = p | (Seg k);
reconsider q = p | (Seg k) as FinSequence by FINSEQ_1:15;
A11: k <= k + 1 by NAT_1:11;
then A12: dom q = Seg k by A8, FINSEQ_1:17;
then A13: k in dom q by A4, FINSEQ_1:1;
set h = q . (len q);
A14: len q = k by A8, A11, FINSEQ_1:17;
k in dom p by A4, A9, A11, FINSEQ_1:1;
then [(p . k),(p . (k + 1))] in PolyRedRel (P,T) by A10, REWRITE1:def_2;
then A15: [(q . (len q)),g] in PolyRedRel (P,T) by A7, A8, A14, A13, FUNCT_1:47;
then consider h9, g9 being set such that
A16: [(q . (len q)),g] = [h9,g9] and
A17: h9 in NonZero (Polynom-Ring (n,L)) and
g9 in the carrier of (Polynom-Ring (n,L)) by RELSET_1:2;
A18: q . (len q) = h9 by A16, XTUPLE_0:1;
A19: 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
A20: i in dom q and
A21: i + 1 in dom q ; ::_thesis: [(q . i),(q . (i + 1))] in PolyRedRel (P,T)
i + 1 <= k by A12, A21, FINSEQ_1:1;
then A22: i + 1 <= k + 1 by A11, XXREAL_0:2;
i <= k by A12, A20, FINSEQ_1:1;
then A23: i <= k + 1 by A11, XXREAL_0:2;
1 <= i + 1 by A12, A21, FINSEQ_1:1;
then A24: i + 1 in dom p by A9, A22, FINSEQ_1:1;
1 <= i by A12, A20, FINSEQ_1:1;
then i in dom p by A9, A23, FINSEQ_1:1;
then A25: [(p . i),(p . (i + 1))] in PolyRedRel (P,T) by A24, REWRITE1:def_2;
p . i = q . i by A20, FUNCT_1:47;
hence [(q . i),(q . (i + 1))] in PolyRedRel (P,T) by A21, A25, FUNCT_1:47; ::_thesis: verum
end;
0_ (n,L) = 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10;
then not h9 in {(0_ (n,L))} by A17, XBOOLE_0:def_5;
then h9 <> 0_ (n,L) by TARSKI:def_1;
then reconsider h = q . (len q) as non-zero Polynomial of n,L by A17, A18, POLYNOM1:def_10, POLYNOM7:def_1;
reconsider q = q as RedSequence of PolyRedRel (P,T) by A4, A14, A19, REWRITE1:def_2;
1 in dom q by A4, A12, FINSEQ_1:1;
then A26: q . 1 = f by A6, FUNCT_1:47;
then PolyRedRel (P,T) reduces f,h by REWRITE1:def_3;
then A27: h <= f,T by A5, A8, A11, A26, FINSEQ_1:17;
h reduces_to g,P,T by A15, Def13;
then A28: ex r being Polynomial of n,L st
( r in P & h reduces_to g,r,T ) by Def7;
reconsider h = h as non-zero Polynomial of n,L ;
g < h,T by A28, Th43;
then g <= h,T by Def3;
hence g <= f,T by A27, Th27; ::_thesis: verum
end;
hence S1[k + 1] ; ::_thesis: verum
end;
end;
A29: S1[1] by Th25;
A30: for k being Nat st 1 <= k holds
S1[k] from NAT_1:sch_8(A29, A3);
consider k being Nat such that
A31: len p = k ;
1 <= k by A31, NAT_1:14;
hence A32: g <= f,T by A1, A30, A2, A31; ::_thesis: ( g = 0_ (n,L) or HT (g,T) <= HT (f,T),T )
now__::_thesis:_(_g_<>_0__(n,L)_implies_HT_(g,T)_<=_HT_(f,T),T_)
assume g <> 0_ (n,L) ; ::_thesis: HT (g,T) <= HT (f,T),T
then Support g <> {} by POLYNOM7:1;
then A33: HT (g,T) in Support g by TERMORD:def_6;
assume A34: not HT (g,T) <= HT (f,T),T ; ::_thesis: contradiction
now__::_thesis:_(_(_HT_(f,T)_=_HT_(g,T)_&_contradiction_)_or_(_HT_(f,T)_<>_HT_(g,T)_&_contradiction_)_)
percases ( HT (f,T) = HT (g,T) or HT (f,T) <> HT (g,T) ) ;
case HT (f,T) = HT (g,T) ; ::_thesis: contradiction
hence contradiction by A34, TERMORD:6; ::_thesis: verum
end;
caseA35: HT (f,T) <> HT (g,T) ; ::_thesis: contradiction
HT (g,T) <= HT (g,T),T by TERMORD:6;
then [(HT (g,T)),(HT (g,T))] in T by TERMORD:def_2;
then A36: HT (g,T) in field T by RELAT_1:15;
HT (f,T) <= HT (f,T),T by TERMORD:6;
then [(HT (f,T)),(HT (f,T))] in T by TERMORD:def_2;
then ( T is_connected_in field T & HT (f,T) in field T ) by RELAT_1:15, RELAT_2:def_14;
then ( [(HT (f,T)),(HT (g,T))] in T or [(HT (g,T)),(HT (f,T))] in T ) by A35, A36, RELAT_2:def_6;
then HT (f,T) <= HT (g,T),T by A34, TERMORD:def_2;
then A37: HT (f,T) < HT (g,T),T by A35, TERMORD:def_3;
then f < g,T by Lm15;
then f <= g,T by Def3;
then Support f = Support g by A32, Th26;
then HT (g,T) <= HT (f,T),T by A33, TERMORD:def_6;
hence contradiction by A37, TERMORD:5; ::_thesis: verum
end;
end;
end;
hence contradiction ; ::_thesis: verum
end;
hence ( g = 0_ (n,L) or HT (g,T) <= HT (f,T),T ) ; ::_thesis: verum
end;
registration
let n be Nat;
let T be connected admissible TermOrder of n;
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ;
let P be Subset of (Polynom-Ring (n,L));
cluster PolyRedRel (P,T) -> strongly-normalizing ;
coherence
PolyRedRel (P,T) is strongly-normalizing
proof
set BT = RelStr(# (Bags n),T #);
set X = the InternalRel of (FinPoset RelStr(# (Bags n),T #));
set R = PolyRedRel (P,T);
A1: 0_ (n,L) = 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10;
FinPoset RelStr(# (Bags n),T #) is well_founded by BAGORDER:41;
then A2: the InternalRel of (FinPoset RelStr(# (Bags n),T #)) is_well_founded_in the carrier of (FinPoset RelStr(# (Bags n),T #)) by WELLFND1:def_2;
now__::_thesis:_for_Y_being_set_st_Y_c=_field_((PolyRedRel_(P,T))_~)_&_Y_<>_{}_holds_
ex_p_being_set_st_
(_p_in_Y_&_((PolyRedRel_(P,T))_~)_-Seg_p_misses_Y_)
let Y be set ; ::_thesis: ( Y c= field ((PolyRedRel (P,T)) ~) & Y <> {} implies ex p being set st
( p in Y & ((PolyRedRel (P,T)) ~) -Seg p misses Y ) )
assume that
A3: Y c= field ((PolyRedRel (P,T)) ~) and
A4: Y <> {} ; ::_thesis: ex p being set st
( p in Y & ((PolyRedRel (P,T)) ~) -Seg p misses Y )
set z = the Element of Y;
the Element of Y in Y by A4;
then the Element of Y in field ((PolyRedRel (P,T)) ~) by A3;
then A5: the Element of Y in (dom ((PolyRedRel (P,T)) ~)) \/ (rng ((PolyRedRel (P,T)) ~)) by RELAT_1:def_6;
now__::_thesis:_(_(_the_Element_of_Y_in_dom_((PolyRedRel_(P,T))_~)_&_the_Element_of_Y_in_the_carrier_of_(Polynom-Ring_(n,L))_)_or_(_the_Element_of_Y_in_rng_((PolyRedRel_(P,T))_~)_&_the_Element_of_Y_in_the_carrier_of_(Polynom-Ring_(n,L))_)_)
percases ( the Element of Y in dom ((PolyRedRel (P,T)) ~) or the Element of Y in rng ((PolyRedRel (P,T)) ~) ) by A5, XBOOLE_0:def_3;
case the Element of Y in dom ((PolyRedRel (P,T)) ~) ; ::_thesis: the Element of Y in the carrier of (Polynom-Ring (n,L))
hence the Element of Y in the carrier of (Polynom-Ring (n,L)) ; ::_thesis: verum
end;
case the Element of Y in rng ((PolyRedRel (P,T)) ~) ; ::_thesis: the Element of Y in the carrier of (Polynom-Ring (n,L))
hence the Element of Y in the carrier of (Polynom-Ring (n,L)) by XBOOLE_0:def_5; ::_thesis: verum
end;
end;
end;
then reconsider z9 = the Element of Y as Polynomial of n,L by POLYNOM1:def_10;
set M = { (Support y9) where y9 is Polynomial of n,L : ex y being set st
( y in Y & y9 = y ) } ;
Support z9 in { (Support y9) where y9 is Polynomial of n,L : ex y being set st
( y in Y & y9 = y ) } by A4;
then reconsider M = { (Support y9) where y9 is Polynomial of n,L : ex y being set st
( y in Y & y9 = y ) } as non empty set ;
now__::_thesis:_for_u_being_set_st_u_in_M_holds_
u_in_the_carrier_of_(FinPoset_RelStr(#_(Bags_n),T_#))
let u be set ; ::_thesis: ( u in M implies u in the carrier of (FinPoset RelStr(# (Bags n),T #)) )
assume u in M ; ::_thesis: u in the carrier of (FinPoset RelStr(# (Bags n),T #))
then A6: ex p being Polynomial of n,L st
( Support p = u & ex y being set st
( y in Y & p = y ) ) ;
FinPoset RelStr(# (Bags n),T #) = RelStr(# (Fin the carrier of RelStr(# (Bags n),T #)),(FinOrd RelStr(# (Bags n),T #)) #) by BAGORDER:def_16;
hence u in the carrier of (FinPoset RelStr(# (Bags n),T #)) by A6, FINSUB_1:def_5; ::_thesis: verum
end;
then M c= the carrier of (FinPoset RelStr(# (Bags n),T #)) by TARSKI:def_3;
then consider a being set such that
A7: a in M and
A8: the InternalRel of (FinPoset RelStr(# (Bags n),T #)) -Seg a misses M by A2, WELLORD1:def_3;
consider p being Polynomial of n,L such that
A9: Support p = a and
A10: ex y being set st
( y in Y & p = y ) by A7;
(((PolyRedRel (P,T)) ~) -Seg p) /\ Y = {}
proof
set b = the Element of (((PolyRedRel (P,T)) ~) -Seg p) /\ Y;
A11: FinPoset RelStr(# (Bags n),T #) = RelStr(# (Fin the carrier of RelStr(# (Bags n),T #)),(FinOrd RelStr(# (Bags n),T #)) #) by BAGORDER:def_16;
assume A12: (((PolyRedRel (P,T)) ~) -Seg p) /\ Y <> {} ; ::_thesis: contradiction
then the Element of (((PolyRedRel (P,T)) ~) -Seg p) /\ Y in ((PolyRedRel (P,T)) ~) -Seg p by XBOOLE_0:def_4;
then [ the Element of (((PolyRedRel (P,T)) ~) -Seg p) /\ Y,p] in (PolyRedRel (P,T)) ~ by WELLORD1:1;
then A13: [p, the Element of (((PolyRedRel (P,T)) ~) -Seg p) /\ Y] in PolyRedRel (P,T) by RELAT_1:def_7;
then consider h9, g9 being set such that
A14: [p, the Element of (((PolyRedRel (P,T)) ~) -Seg p) /\ Y] = [h9,g9] and
A15: h9 in NonZero (Polynom-Ring (n,L)) and
A16: g9 in the carrier of (Polynom-Ring (n,L)) by RELSET_1:2;
the Element of (((PolyRedRel (P,T)) ~) -Seg p) /\ Y = g9 by A14, XTUPLE_0:1;
then reconsider b9 = the Element of (((PolyRedRel (P,T)) ~) -Seg p) /\ Y as Polynomial of n,L by A16, POLYNOM1:def_10;
A17: p = h9 by A14, XTUPLE_0:1;
not h9 in {(0_ (n,L))} by A1, A15, XBOOLE_0:def_5;
then h9 <> 0_ (n,L) by TARSKI:def_1;
then reconsider p = p as non-zero Polynomial of n,L by A17, POLYNOM7:def_1;
p reduces_to b9,P,T by A13, Def13;
then A18: ex u being Polynomial of n,L st
( u in P & p reduces_to b9,u,T ) by Def7;
reconsider p = p as non-zero Polynomial of n,L ;
A19: b9 < p,T by A18, Th43;
then A20: Support b9 <> Support p by Def3;
the Element of (((PolyRedRel (P,T)) ~) -Seg p) /\ Y in Y by A12, XBOOLE_0:def_4;
then A21: Support b9 in M ;
b9 <= p,T by A19, Def3;
then [(Support b9),(Support p)] in the InternalRel of (FinPoset RelStr(# (Bags n),T #)) by A11, Def2;
then Support b9 in the InternalRel of (FinPoset RelStr(# (Bags n),T #)) -Seg (Support p) by A20, WELLORD1:1;
then Support b9 in ( the InternalRel of (FinPoset RelStr(# (Bags n),T #)) -Seg a) /\ M by A9, A21, XBOOLE_0:def_4;
hence contradiction by A8, XBOOLE_0:def_7; ::_thesis: verum
end;
then ((PolyRedRel (P,T)) ~) -Seg p misses Y by XBOOLE_0:def_7;
hence ex p being set st
( p in Y & ((PolyRedRel (P,T)) ~) -Seg p misses Y ) by A10; ::_thesis: verum
end;
then (PolyRedRel (P,T)) ~ is well_founded by WELLORD1:def_2;
then A22: PolyRedRel (P,T) is co-well_founded by REWRITE1:def_13;
now__::_thesis:_for_x_being_set_st_x_in_field_(PolyRedRel_(P,T))_holds_
not_[x,x]_in_PolyRedRel_(P,T)
set A = the carrier of (Polynom-Ring (n,L)) \ {(0_ (n,L))};
set B = the carrier of (Polynom-Ring (n,L));
let x be set ; ::_thesis: ( x in field (PolyRedRel (P,T)) implies not [x,x] in PolyRedRel (P,T) )
assume x in field (PolyRedRel (P,T)) ; ::_thesis: not [x,x] in PolyRedRel (P,T)
then A23: x in (dom (PolyRedRel (P,T))) \/ (rng (PolyRedRel (P,T))) by RELAT_1:def_6;
now__::_thesis:_(_(_x_in_dom_(PolyRedRel_(P,T))_&_x_is_Polynomial_of_n,L_)_or_(_x_in_rng_(PolyRedRel_(P,T))_&_x_is_Polynomial_of_n,L_)_)
percases ( x in dom (PolyRedRel (P,T)) or x in rng (PolyRedRel (P,T)) ) by A23, XBOOLE_0:def_3;
case x in dom (PolyRedRel (P,T)) ; ::_thesis: x is Polynomial of n,L
then x in the carrier of (Polynom-Ring (n,L)) by XBOOLE_0:def_5;
hence x is Polynomial of n,L by POLYNOM1:def_10; ::_thesis: verum
end;
case x in rng (PolyRedRel (P,T)) ; ::_thesis: x is Polynomial of n,L
hence x is Polynomial of n,L by POLYNOM1:def_10; ::_thesis: verum
end;
end;
end;
then reconsider x9 = x as Polynomial of n,L ;
now__::_thesis:_not_[x,x]_in_PolyRedRel_(P,T)
assume A24: [x,x] in PolyRedRel (P,T) ; ::_thesis: contradiction
then consider x1, y1 being set such that
A25: [x,x] = [x1,y1] and
A26: x1 in the carrier of (Polynom-Ring (n,L)) \ {(0_ (n,L))} and
y1 in the carrier of (Polynom-Ring (n,L)) by A1, RELSET_1:2;
x = x1 by A25, XTUPLE_0:1;
then not x9 in {(0_ (n,L))} by A26, XBOOLE_0:def_5;
then x9 <> 0_ (n,L) by TARSKI:def_1;
then reconsider x9 = x9 as non-zero Polynomial of n,L by POLYNOM7:def_1;
x9 reduces_to x9,P,T by A24, Def13;
then ex p being Polynomial of n,L st
( p in P & x9 reduces_to x9,p,T ) by Def7;
then x9 < x9,T by Th43;
then Support x9 <> Support x9 by Def3;
hence contradiction ; ::_thesis: verum
end;
hence not [x,x] in PolyRedRel (P,T) ; ::_thesis: verum
end;
then PolyRedRel (P,T) is_irreflexive_in field (PolyRedRel (P,T)) by RELAT_2:def_2;
then PolyRedRel (P,T) is irreflexive by RELAT_2:def_10;
hence PolyRedRel (P,T) is strongly-normalizing by A22; ::_thesis: verum
end;
end;
theorem Th45: :: POLYRED:45
for n being Nat
for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for f, h being Polynomial of n,L st f in P holds
PolyRedRel (P,T) reduces h *' f, 0_ (n,L)
proof
let n be Nat; ::_thesis: for T being connected admissible TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for f, h being Polynomial of n,L st f in P holds
PolyRedRel (P,T) reduces h *' f, 0_ (n,L)
let T be connected admissible TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for f, h being Polynomial of n,L st f in P holds
PolyRedRel (P,T) reduces h *' f, 0_ (n,L)
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed left_zeroed doubleLoopStr ; ::_thesis: for P being Subset of (Polynom-Ring (n,L))
for f, h being Polynomial of n,L st f in P holds
PolyRedRel (P,T) reduces h *' f, 0_ (n,L)
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: for f, h being Polynomial of n,L st f in P holds
PolyRedRel (P,T) reduces h *' f, 0_ (n,L)
let f9, h9 be Polynomial of n,L; ::_thesis: ( f9 in P implies PolyRedRel (P,T) reduces h9 *' f9, 0_ (n,L) )
assume A1: f9 in P ; ::_thesis: PolyRedRel (P,T) reduces h9 *' f9, 0_ (n,L)
percases ( h9 = 0_ (n,L) or h9 <> 0_ (n,L) ) ;
suppose h9 = 0_ (n,L) ; ::_thesis: PolyRedRel (P,T) reduces h9 *' f9, 0_ (n,L)
then h9 *' f9 = 0_ (n,L) by Th5;
hence PolyRedRel (P,T) reduces h9 *' f9, 0_ (n,L) by REWRITE1:12; ::_thesis: verum
end;
suppose h9 <> 0_ (n,L) ; ::_thesis: PolyRedRel (P,T) reduces h9 *' f9, 0_ (n,L)
then reconsider h = h9 as non-zero Polynomial of n,L by POLYNOM7:def_1;
set H = { g where g is Polynomial of n,L : not PolyRedRel (P,T) reduces g *' f9, 0_ (n,L) } ;
now__::_thesis:_(_(_f9_=_0__(n,L)_&_PolyRedRel_(P,T)_reduces_h9_*'_f9,_0__(n,L)_)_or_(_f9_<>_0__(n,L)_&_(_not_PolyRedRel_(P,T)_reduces_h9_*'_f9,_0__(n,L)_implies_PolyRedRel_(P,T)_reduces_h9_*'_f9,_0__(n,L)_)_)_)
percases ( f9 = 0_ (n,L) or f9 <> 0_ (n,L) ) ;
case f9 = 0_ (n,L) ; ::_thesis: PolyRedRel (P,T) reduces h9 *' f9, 0_ (n,L)
then h9 *' f9 = 0_ (n,L) by POLYNOM1:28;
hence PolyRedRel (P,T) reduces h9 *' f9, 0_ (n,L) by REWRITE1:12; ::_thesis: verum
end;
case f9 <> 0_ (n,L) ; ::_thesis: ( not PolyRedRel (P,T) reduces h9 *' f9, 0_ (n,L) implies PolyRedRel (P,T) reduces h9 *' f9, 0_ (n,L) )
then reconsider f = f9 as non-zero Polynomial of n,L by POLYNOM7:def_1;
A2: now__::_thesis:_for_u_being_set_st_u_in__{__g_where_g_is_Polynomial_of_n,L_:_not_PolyRedRel_(P,T)_reduces_g_*'_f9,_0__(n,L)__}__holds_
u_in_the_carrier_of_(Polynom-Ring_(n,L))
let u be set ; ::_thesis: ( u in { g where g is Polynomial of n,L : not PolyRedRel (P,T) reduces g *' f9, 0_ (n,L) } implies u in the carrier of (Polynom-Ring (n,L)) )
assume u in { g where g is Polynomial of n,L : not PolyRedRel (P,T) reduces g *' f9, 0_ (n,L) } ; ::_thesis: u in the carrier of (Polynom-Ring (n,L))
then ex g9 being Polynomial of n,L st
( u = g9 & not PolyRedRel (P,T) reduces g9 *' f, 0_ (n,L) ) ;
hence u in the carrier of (Polynom-Ring (n,L)) by POLYNOM1:def_10; ::_thesis: verum
end;
assume not PolyRedRel (P,T) reduces h9 *' f9, 0_ (n,L) ; ::_thesis: PolyRedRel (P,T) reduces h9 *' f9, 0_ (n,L)
then h in { g where g is Polynomial of n,L : not PolyRedRel (P,T) reduces g *' f9, 0_ (n,L) } ;
then reconsider H = { g where g is Polynomial of n,L : not PolyRedRel (P,T) reduces g *' f9, 0_ (n,L) } as non empty Subset of (Polynom-Ring (n,L)) by A2, TARSKI:def_3;
now__::_thesis:_not_H_<>_{}
assume H <> {} ; ::_thesis: contradiction
reconsider H = H as non empty set ;
consider m being Polynomial of n,L such that
A3: m in H and
A4: for m9 being Polynomial of n,L st m9 in H holds
m <= m9,T by Th31;
m <> 0_ (n,L)
proof
assume m = 0_ (n,L) ; ::_thesis: contradiction
then A5: m *' f = 0_ (n,L) by Th5;
ex g9 being Polynomial of n,L st
( m = g9 & not PolyRedRel (P,T) reduces g9 *' f, 0_ (n,L) ) by A3;
hence contradiction by A5, REWRITE1:12; ::_thesis: verum
end;
then reconsider m = m as non-zero Polynomial of n,L by POLYNOM7:def_1;
Red (m,T) < m,T by Th35;
then not m <= Red (m,T),T by Th29;
then not Red (m,T) in H by A4;
then A6: PolyRedRel (P,T) reduces (Red (m,T)) *' f, 0_ (n,L) ;
set g = (m *' f) - ((((m *' f) . (HT ((m *' f),T))) / (HC (f,T))) * ((HT (m,T)) *' f));
A7: m *' f <> 0_ (n,L) by POLYNOM7:def_1;
A8: HC (f,T) <> 0. L ;
m *' f <> 0_ (n,L) by POLYNOM7:def_1;
then Support (m *' f) <> {} by POLYNOM7:1;
then A9: HT ((m *' f),T) in Support (m *' f) by TERMORD:def_6;
(((m *' f) . (HT ((m *' f),T))) / (HC (f,T))) * ((HT (m,T)) *' f) = ((HC ((m *' f),T)) / (HC (f,T))) * ((HT (m,T)) *' f) by TERMORD:def_7
.= (((HC (m,T)) * (HC (f,T))) / (HC (f,T))) * ((HT (m,T)) *' f) by TERMORD:32
.= (((HC (m,T)) * (HC (f,T))) * ((HC (f,T)) ")) * ((HT (m,T)) *' f) by VECTSP_1:def_11
.= ((HC (m,T)) * ((HC (f,T)) * ((HC (f,T)) "))) * ((HT (m,T)) *' f) by GROUP_1:def_3
.= ((HC (m,T)) * (1. L)) * ((HT (m,T)) *' f) by A8, VECTSP_1:def_10
.= (HC (m,T)) * ((HT (m,T)) *' f) by VECTSP_1:def_4
.= (Monom ((HC (m,T)),(HT (m,T)))) *' f by Th22
.= (HM (m,T)) *' f by TERMORD:def_8 ;
then A10: (m *' f) - ((((m *' f) . (HT ((m *' f),T))) / (HC (f,T))) * ((HT (m,T)) *' f)) = (m *' f) + (- ((HM (m,T)) *' f)) by POLYNOM1:def_6
.= (f *' m) + (f *' (- (HM (m,T)))) by Th6
.= (m + (- (HM (m,T)))) *' f by POLYNOM1:26
.= (m - (HM (m,T))) *' f by POLYNOM1:def_6
.= (Red (m,T)) *' f by TERMORD:def_9 ;
( HT ((m *' f),T) = (HT (m,T)) + (HT (f,T)) & f <> 0_ (n,L) ) by POLYNOM7:def_1, TERMORD:31;
then m *' f reduces_to (m *' f) - ((((m *' f) . (HT ((m *' f),T))) / (HC (f,T))) * ((HT (m,T)) *' f)),f, HT ((m *' f),T),T by A9, A7, Def5;
then m *' f reduces_to (Red (m,T)) *' f,f,T by A10, Def6;
then m *' f reduces_to (Red (m,T)) *' f,P,T by A1, Def7;
then [(m *' f),((Red (m,T)) *' f)] in PolyRedRel (P,T) by Def13;
then A11: PolyRedRel (P,T) reduces m *' f,(Red (m,T)) *' f by REWRITE1:15;
ex u being Polynomial of n,L st
( m = u & not PolyRedRel (P,T) reduces u *' f, 0_ (n,L) ) by A3;
hence contradiction by A11, A6, REWRITE1:16; ::_thesis: verum
end;
hence PolyRedRel (P,T) reduces h9 *' f9, 0_ (n,L) ; ::_thesis: verum
end;
end;
end;
hence PolyRedRel (P,T) reduces h9 *' f9, 0_ (n,L) ; ::_thesis: verum
end;
end;
end;
theorem Th46: :: POLYRED:46
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 P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L
for m being non-zero Monomial of n,L st f reduces_to g,P,T holds
m *' f reduces_to m *' g,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 P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L
for m being non-zero Monomial of n,L st f reduces_to g,P,T holds
m *' f reduces_to m *' g,P,T
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L
for m being non-zero Monomial of n,L st f reduces_to g,P,T holds
m *' f reduces_to m *' g,P,T
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L
for m being non-zero Monomial of n,L st f reduces_to g,P,T holds
m *' f reduces_to m *' g,P,T
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: for f, g being Polynomial of n,L
for m being non-zero Monomial of n,L st f reduces_to g,P,T holds
m *' f reduces_to m *' g,P,T
let f, g be Polynomial of n,L; ::_thesis: for m being non-zero Monomial of n,L st f reduces_to g,P,T holds
m *' f reduces_to m *' g,P,T
let m be non-zero Monomial of n,L; ::_thesis: ( f reduces_to g,P,T implies m *' f reduces_to m *' g,P,T )
assume f reduces_to g,P,T ; ::_thesis: m *' f reduces_to m *' g,P,T
then consider p being Polynomial of n,L such that
A1: p in P and
A2: f reduces_to g,p,T by Def7;
consider b being bag of n such that
A3: f reduces_to g,p,b,T by A2, Def6;
set b9 = b + (HT (m,T));
A4: b in Support f by A3, Def5;
A5: now__::_thesis:_not_(m_*'_f)_._(b_+_(HT_(m,T)))_=_0._L
m <> 0_ (n,L) by POLYNOM7:def_1;
then Support m <> {} by POLYNOM7:1;
then A6: m . (term m) <> 0. L by POLYNOM7:def_5;
assume A7: (m *' f) . (b + (HT (m,T))) = 0. L ; ::_thesis: contradiction
(m *' f) . (b + (HT (m,T))) = (m *' f) . ((term m) + b) by TERMORD:23
.= (m . (term m)) * (f . b) by Th7 ;
then f . b = 0. L by A7, A6, VECTSP_2:def_1;
hence contradiction by A4, POLYNOM1:def_3; ::_thesis: verum
end;
b + (HT (m,T)) is Element of Bags n by PRE_POLY:def_12;
then A8: b + (HT (m,T)) in Support (m *' f) by A5, POLYNOM1:def_3;
A9: p <> 0_ (n,L) by A2, Lm18;
consider s being bag of n such that
A10: s + (HT (p,T)) = b and
A11: g = f - (((f . b) / (HC (p,T))) * (s *' p)) by A3, Def5;
reconsider p = p as non-zero Polynomial of n,L by A9, POLYNOM7:def_1;
A12: (s + (HT (m,T))) + (HT (p,T)) = b + (HT (m,T)) by A10, PRE_POLY:35;
set t = s + (HT (m,T));
set h = (m *' f) - ((((m *' f) . (b + (HT (m,T)))) / (HC (p,T))) * ((s + (HT (m,T))) *' p));
f <> 0_ (n,L) by A3, Def5;
then reconsider f = f as non-zero Polynomial of n,L by POLYNOM7:def_1;
( m *' f <> 0_ (n,L) & p <> 0_ (n,L) ) by POLYNOM7:def_1;
then m *' f reduces_to (m *' f) - ((((m *' f) . (b + (HT (m,T)))) / (HC (p,T))) * ((s + (HT (m,T))) *' p)),p,b + (HT (m,T)),T by A8, A12, Def5;
then A13: m *' f reduces_to (m *' f) - ((((m *' f) . (b + (HT (m,T)))) / (HC (p,T))) * ((s + (HT (m,T))) *' p)),p,T by Def6;
A14: (m . (term m)) * ((f . b) / (HC (p,T))) = (m . (term m)) * ((f . b) * ((HC (p,T)) ")) by VECTSP_1:def_11
.= ((m . (term m)) * (f . b)) * ((HC (p,T)) ") by GROUP_1:def_3
.= ((m . (term m)) * (f . b)) / (HC (p,T)) by VECTSP_1:def_11 ;
(m *' f) . (b + (HT (m,T))) = (m *' f) . ((term m) + b) by TERMORD:23
.= (m . (term m)) * (f . b) by Th7 ;
then (m *' f) - ((((m *' f) . (b + (HT (m,T)))) / (HC (p,T))) * ((s + (HT (m,T))) *' p)) = (m *' f) - (((m . (term m)) * ((f . b) / (HC (p,T)))) * ((HT (m,T)) *' (s *' p))) by A14, Th18
.= (m *' f) - (((f . b) / (HC (p,T))) * ((m . (term m)) * ((HT (m,T)) *' (s *' p)))) by Th11
.= (m *' f) - (((f . b) / (HC (p,T))) * ((Monom ((m . (term m)),(HT (m,T)))) *' (s *' p))) by Th22
.= (m *' f) - (((f . b) / (HC (p,T))) * ((Monom ((coefficient m),(term m))) *' (s *' p))) by TERMORD:23
.= (m *' f) - (((f . b) / (HC (p,T))) * (m *' (s *' p))) by POLYNOM7:11
.= (m *' f) - (m *' (((f . b) / (HC (p,T))) * (s *' p))) by Th12
.= (m *' f) + (- (m *' (((f . b) / (HC (p,T))) * (s *' p)))) by POLYNOM1:def_6
.= (m *' f) + (m *' (- (((f . b) / (HC (p,T))) * (s *' p)))) by Th6
.= m *' (f + (- (((f . b) / (HC (p,T))) * (s *' p)))) by POLYNOM1:26
.= m *' (f - (((f . b) / (HC (p,T))) * (s *' p))) by POLYNOM1:def_6 ;
hence m *' f reduces_to m *' g,P,T by A1, A11, A13, Def7; ::_thesis: verum
end;
theorem Th47: :: POLYRED:47
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 P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L
for m being Monomial of n,L st PolyRedRel (P,T) reduces f,g holds
PolyRedRel (P,T) reduces m *' f,m *' g
proof
let n be Ordinal; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L
for m being Monomial of n,L st PolyRedRel (P,T) reduces f,g holds
PolyRedRel (P,T) reduces m *' f,m *' g
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L
for m being Monomial of n,L st PolyRedRel (P,T) reduces f,g holds
PolyRedRel (P,T) reduces m *' f,m *' g
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L
for m being Monomial of n,L st PolyRedRel (P,T) reduces f,g holds
PolyRedRel (P,T) reduces m *' f,m *' g
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: for f, g being Polynomial of n,L
for m being Monomial of n,L st PolyRedRel (P,T) reduces f,g holds
PolyRedRel (P,T) reduces m *' f,m *' g
let f, g be Polynomial of n,L; ::_thesis: for m being Monomial of n,L st PolyRedRel (P,T) reduces f,g holds
PolyRedRel (P,T) reduces m *' f,m *' g
let m be Monomial of n,L; ::_thesis: ( PolyRedRel (P,T) reduces f,g implies PolyRedRel (P,T) reduces m *' f,m *' g )
assume A1: PolyRedRel (P,T) reduces f,g ; ::_thesis: PolyRedRel (P,T) reduces m *' f,m *' g
set R = PolyRedRel (P,T);
percases ( m = 0_ (n,L) or m <> 0_ (n,L) ) ;
supposeA2: m = 0_ (n,L) ; ::_thesis: PolyRedRel (P,T) reduces m *' f,m *' g
then m *' f = 0_ (n,L) by Th5
.= m *' g by A2, Th5 ;
hence PolyRedRel (P,T) reduces m *' f,m *' g by REWRITE1:12; ::_thesis: verum
end;
suppose m <> 0_ (n,L) ; ::_thesis: PolyRedRel (P,T) reduces m *' f,m *' g
then reconsider m = m as non-zero Monomial of n,L by POLYNOM7:def_1;
defpred S1[ Nat] means for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f,g holds
for p being RedSequence of PolyRedRel (P,T) st p . 1 = f & p . (len p) = g & len p = $1 holds
PolyRedRel (P,T) reduces m *' f,m *' g;
consider p being RedSequence of PolyRedRel (P,T) such that
A3: ( p . 1 = f & p . (len p) = g ) by A1, REWRITE1:def_3;
consider k being Nat such that
A4: len p = k ;
A5: 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 A6: 1 <= k ; ::_thesis: ( S1[k] implies S1[k + 1] )
thus ( S1[k] implies S1[k + 1] ) ::_thesis: verum
proof
assume A7: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_for_f,_g_being_Polynomial_of_n,L_st_PolyRedRel_(P,T)_reduces_f,g_holds_
for_p_being_RedSequence_of_PolyRedRel_(P,T)_st_p_._1_=_f_&_p_._(len_p)_=_g_&_len_p_=_k_+_1_holds_
PolyRedRel_(P,T)_reduces_m_*'_f,m_*'_g
let f, g be Polynomial of n,L; ::_thesis: ( PolyRedRel (P,T) reduces f,g implies for p being RedSequence of PolyRedRel (P,T) st p . 1 = f & p . (len p) = g & len p = k + 1 holds
PolyRedRel (P,T) reduces m *' f,m *' g )
assume PolyRedRel (P,T) reduces f,g ; ::_thesis: for p being RedSequence of PolyRedRel (P,T) st p . 1 = f & p . (len p) = g & len p = k + 1 holds
PolyRedRel (P,T) reduces m *' f,m *' g
let p be RedSequence of PolyRedRel (P,T); ::_thesis: ( p . 1 = f & p . (len p) = g & len p = k + 1 implies PolyRedRel (P,T) reduces m *' f,m *' g )
assume that
A8: p . 1 = f and
A9: p . (len p) = g and
A10: len p = k + 1 ; ::_thesis: PolyRedRel (P,T) reduces m *' f,m *' g
A11: dom p = Seg (k + 1) by A10, FINSEQ_1:def_3;
then A12: k + 1 in dom p by FINSEQ_1:4;
set q = p | (Seg k);
reconsider q = p | (Seg k) as FinSequence by FINSEQ_1:15;
A13: k <= k + 1 by NAT_1:11;
then A14: dom q = Seg k by A10, FINSEQ_1:17;
then A15: k in dom q by A6, FINSEQ_1:1;
set h = q . (len q);
A16: len q = k by A10, A13, FINSEQ_1:17;
k in dom p by A6, A11, A13, FINSEQ_1:1;
then [(p . k),(p . (k + 1))] in PolyRedRel (P,T) by A12, REWRITE1:def_2;
then A17: [(q . (len q)),g] in PolyRedRel (P,T) by A9, A10, A16, A15, FUNCT_1:47;
then consider h9, g9 being set such that
A18: [(q . (len q)),g] = [h9,g9] and
A19: h9 in NonZero (Polynom-Ring (n,L)) and
g9 in the carrier of (Polynom-Ring (n,L)) by RELSET_1:2;
A20: q . (len q) = h9 by A18, XTUPLE_0:1;
A21: 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
A22: i in dom q and
A23: i + 1 in dom q ; ::_thesis: [(q . i),(q . (i + 1))] in PolyRedRel (P,T)
i + 1 <= k by A14, A23, FINSEQ_1:1;
then A24: i + 1 <= k + 1 by A13, XXREAL_0:2;
i <= k by A14, A22, FINSEQ_1:1;
then A25: i <= k + 1 by A13, XXREAL_0:2;
1 <= i + 1 by A14, A23, FINSEQ_1:1;
then A26: i + 1 in dom p by A11, A24, FINSEQ_1:1;
1 <= i by A14, A22, FINSEQ_1:1;
then i in dom p by A11, A25, FINSEQ_1:1;
then A27: [(p . i),(p . (i + 1))] in PolyRedRel (P,T) by A26, REWRITE1:def_2;
p . i = q . i by A22, FUNCT_1:47;
hence [(q . i),(q . (i + 1))] in PolyRedRel (P,T) by A23, A27, FUNCT_1:47; ::_thesis: verum
end;
0_ (n,L) = 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10;
then not h9 in {(0_ (n,L))} by A19, XBOOLE_0:def_5;
then h9 <> 0_ (n,L) by TARSKI:def_1;
then reconsider h = q . (len q) as non-zero Polynomial of n,L by A19, A20, POLYNOM1:def_10, POLYNOM7:def_1;
h reduces_to g,P,T by A17, Def13;
then m *' h reduces_to m *' g,P,T by Th46;
then [(m *' h),(m *' g)] in PolyRedRel (P,T) by Def13;
then A28: PolyRedRel (P,T) reduces m *' h,m *' g by REWRITE1:15;
reconsider q = q as RedSequence of PolyRedRel (P,T) by A6, A16, A21, REWRITE1:def_2;
1 in dom q by A6, A14, FINSEQ_1:1;
then A29: q . 1 = f by A8, FUNCT_1:47;
then PolyRedRel (P,T) reduces f,h by REWRITE1:def_3;
then PolyRedRel (P,T) reduces m *' f,m *' h by A7, A10, A13, A29, FINSEQ_1:17;
hence PolyRedRel (P,T) reduces m *' f,m *' g by A28, REWRITE1:16; ::_thesis: verum
end;
hence S1[k + 1] ; ::_thesis: verum
end;
end;
A30: S1[1] by REWRITE1:12;
A31: for k being Nat st 1 <= k holds
S1[k] from NAT_1:sch_8(A30, A5);
1 <= k by A4, NAT_1:14;
hence PolyRedRel (P,T) reduces m *' f,m *' g by A1, A31, A3, A4; ::_thesis: verum
end;
end;
end;
theorem :: POLYRED:48
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 P being Subset of (Polynom-Ring (n,L))
for f being Polynomial of n,L
for m being Monomial of n,L st PolyRedRel (P,T) reduces f, 0_ (n,L) holds
PolyRedRel (P,T) reduces m *' f, 0_ (n,L)
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 P being Subset of (Polynom-Ring (n,L))
for f being Polynomial of n,L
for m being Monomial of n,L st PolyRedRel (P,T) reduces f, 0_ (n,L) holds
PolyRedRel (P,T) reduces m *' f, 0_ (n,L)
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for f being Polynomial of n,L
for m being Monomial of n,L st PolyRedRel (P,T) reduces f, 0_ (n,L) holds
PolyRedRel (P,T) reduces m *' f, 0_ (n,L)
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for P being Subset of (Polynom-Ring (n,L))
for f being Polynomial of n,L
for m being Monomial of n,L st PolyRedRel (P,T) reduces f, 0_ (n,L) holds
PolyRedRel (P,T) reduces m *' f, 0_ (n,L)
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: for f being Polynomial of n,L
for m being Monomial of n,L st PolyRedRel (P,T) reduces f, 0_ (n,L) holds
PolyRedRel (P,T) reduces m *' f, 0_ (n,L)
let f be Polynomial of n,L; ::_thesis: for m being Monomial of n,L st PolyRedRel (P,T) reduces f, 0_ (n,L) holds
PolyRedRel (P,T) reduces m *' f, 0_ (n,L)
let m be Monomial of n,L; ::_thesis: ( PolyRedRel (P,T) reduces f, 0_ (n,L) implies PolyRedRel (P,T) reduces m *' f, 0_ (n,L) )
assume PolyRedRel (P,T) reduces f, 0_ (n,L) ; ::_thesis: PolyRedRel (P,T) reduces m *' f, 0_ (n,L)
then PolyRedRel (P,T) reduces m *' f,m *' (0_ (n,L)) by Th47;
hence PolyRedRel (P,T) reduces m *' f, 0_ (n,L) by POLYNOM1:28; ::_thesis: verum
end;
theorem Th49: :: POLYRED:49
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 f, g, h, h1 being Polynomial of n,L st f - g = h & PolyRedRel (P,T) reduces h,h1 holds
ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 )
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 f, g, h, h1 being Polynomial of n,L st f - g = h & PolyRedRel (P,T) reduces h,h1 holds
ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 )
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 f, g, h, h1 being Polynomial of n,L st f - g = h & PolyRedRel (P,T) reduces h,h1 holds
ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 )
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 f, g, h, h1 being Polynomial of n,L st f - g = h & PolyRedRel (P,T) reduces h,h1 holds
ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 )
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: for f, g, h, h1 being Polynomial of n,L st f - g = h & PolyRedRel (P,T) reduces h,h1 holds
ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 )
let f, g, h, h1 be Polynomial of n,L; ::_thesis: ( f - g = h & PolyRedRel (P,T) reduces h,h1 implies ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 ) )
assume that
A1: f - g = h and
A2: PolyRedRel (P,T) reduces h,h1 ; ::_thesis: ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 )
consider p being RedSequence of PolyRedRel (P,T) such that
A3: ( p . 1 = h & p . (len p) = h1 ) by A2, REWRITE1:def_3;
defpred S1[ Nat] means for f, g, h being Polynomial of n,L st f - g = h holds
for h1 being Polynomial of n,L
for p being RedSequence of PolyRedRel (P,T) st p . 1 = h & p . (len p) = h1 & len p = $1 holds
ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 );
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] )
assume A6: S1[k] ; ::_thesis: S1[k + 1]
thus S1[k + 1] ::_thesis: verum
proof
let f, g, h be Polynomial of n,L; ::_thesis: ( f - g = h implies for h1 being Polynomial of n,L
for p being RedSequence of PolyRedRel (P,T) st p . 1 = h & p . (len p) = h1 & len p = k + 1 holds
ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 ) )
assume A7: f - g = h ; ::_thesis: for h1 being Polynomial of n,L
for p being RedSequence of PolyRedRel (P,T) st p . 1 = h & p . (len p) = h1 & len p = k + 1 holds
ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 )
let h1 be Polynomial of n,L; ::_thesis: for p being RedSequence of PolyRedRel (P,T) st p . 1 = h & p . (len p) = h1 & len p = k + 1 holds
ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 )
let r be RedSequence of PolyRedRel (P,T); ::_thesis: ( r . 1 = h & r . (len r) = h1 & len r = k + 1 implies ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 ) )
assume that
A8: r . 1 = h and
A9: r . (len r) = h1 and
A10: len r = k + 1 ; ::_thesis: ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 )
set h2 = r . k;
A11: dom r = Seg (k + 1) by A10, FINSEQ_1:def_3;
1 <= k + 1 by NAT_1:11;
then A12: k + 1 in dom r by A11, FINSEQ_1:1;
k <= k + 1 by NAT_1:11;
then k in dom r by A5, A11, FINSEQ_1:1;
then A13: [(r . k),(r . (k + 1))] in PolyRedRel (P,T) by A12, REWRITE1:def_2;
then consider x, y being set such that
A14: x in NonZero (Polynom-Ring (n,L)) and
y in the carrier of (Polynom-Ring (n,L)) and
A15: [(r . k),(r . (k + 1))] = [x,y] by ZFMISC_1:def_2;
A16: r . k = x by A15, XTUPLE_0:1;
then reconsider h2 = r . k as Polynomial of n,L by A14, POLYNOM1:def_10;
0_ (n,L) = 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10;
then not r . k in {(0_ (n,L))} by A14, A16, XBOOLE_0:def_5;
then r . k <> 0_ (n,L) by TARSKI:def_1;
then reconsider h2 = h2 as non-zero Polynomial of n,L by POLYNOM7:def_1;
h2 reduces_to h1,P,T by A9, A10, A13, Def13;
then consider p being Polynomial of n,L such that
A17: p in P and
A18: h2 reduces_to h1,p,T by Def7;
consider b being bag of n such that
A19: h2 reduces_to h1,p,b,T by A18, Def6;
set q = r | (Seg k);
reconsider q = r | (Seg k) as FinSequence by FINSEQ_1:15;
A20: k <= k + 1 by NAT_1:11;
then A21: dom q = Seg k by A10, FINSEQ_1:17;
A22: dom r = Seg (k + 1) by A10, FINSEQ_1:def_3;
A23: 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
A24: i in dom q and
A25: i + 1 in dom q ; ::_thesis: [(q . i),(q . (i + 1))] in PolyRedRel (P,T)
i + 1 <= k by A21, A25, FINSEQ_1:1;
then A26: i + 1 <= k + 1 by A20, XXREAL_0:2;
i <= k by A21, A24, FINSEQ_1:1;
then A27: i <= k + 1 by A20, XXREAL_0:2;
1 <= i + 1 by A21, A25, FINSEQ_1:1;
then A28: i + 1 in dom r by A22, A26, FINSEQ_1:1;
1 <= i by A21, A24, FINSEQ_1:1;
then i in dom r by A22, A27, FINSEQ_1:1;
then A29: [(r . i),(r . (i + 1))] in PolyRedRel (P,T) by A28, REWRITE1:def_2;
r . i = q . i by A24, FUNCT_1:47;
hence [(q . i),(q . (i + 1))] in PolyRedRel (P,T) by A25, A29, FUNCT_1:47; ::_thesis: verum
end;
len q = k by A10, A20, FINSEQ_1:17;
then reconsider q = q as RedSequence of PolyRedRel (P,T) by A5, A23, REWRITE1:def_2;
A30: k in dom q by A5, A21, FINSEQ_1:1;
1 in dom q by A5, A21, FINSEQ_1:1;
then A31: q . 1 = h by A8, FUNCT_1:47;
q . (len q) = q . k by A10, A20, FINSEQ_1:17
.= h2 by A30, FUNCT_1:47 ;
then consider f2, g2 being Polynomial of n,L such that
A32: f2 - g2 = h2 and
A33: PolyRedRel (P,T) reduces f,f2 and
A34: PolyRedRel (P,T) reduces g,g2 by A6, A7, A10, A20, A31, FINSEQ_1:17;
consider s being bag of n such that
A35: s + (HT (p,T)) = b and
A36: h1 = h2 - (((h2 . b) / (HC (p,T))) * (s *' p)) by A19, Def5;
set f1 = f2 - (((f2 . b) / (HC (p,T))) * (s *' p));
set g1 = g2 - (((g2 . b) / (HC (p,T))) * (s *' p));
A37: ((f2 . b) / (HC (p,T))) + (- ((g2 . b) / (HC (p,T)))) = ((f2 . b) * ((HC (p,T)) ")) + (- ((g2 . b) / (HC (p,T)))) by VECTSP_1:def_11
.= ((f2 . b) * ((HC (p,T)) ")) + (- ((g2 . b) * ((HC (p,T)) "))) by VECTSP_1:def_11
.= ((f2 . b) * ((HC (p,T)) ")) + ((- (g2 . b)) * ((HC (p,T)) ")) by VECTSP_1:9
.= ((f2 . b) + (- (g2 . b))) * ((HC (p,T)) ") by VECTSP_1:def_7
.= ((f2 . b) + ((- g2) . b)) * ((HC (p,T)) ") by POLYNOM1:17
.= ((f2 + (- g2)) . b) * ((HC (p,T)) ") by POLYNOM1:15
.= ((f2 - g2) . b) * ((HC (p,T)) ") by POLYNOM1:def_6
.= ((f2 - g2) . b) / (HC (p,T)) by VECTSP_1:def_11 ;
A38: now__::_thesis:_(_(_not_b_in_Support_g2_&_PolyRedRel_(P,T)_reduces_g,g2_-_(((g2_._b)_/_(HC_(p,T)))_*_(s_*'_p))_)_or_(_b_in_Support_g2_&_PolyRedRel_(P,T)_reduces_g,g2_-_(((g2_._b)_/_(HC_(p,T)))_*_(s_*'_p))_)_)
percases ( not b in Support g2 or b in Support g2 ) ;
caseA39: not b in Support g2 ; ::_thesis: PolyRedRel (P,T) reduces g,g2 - (((g2 . b) / (HC (p,T))) * (s *' p))
b is Element of Bags n by PRE_POLY:def_12;
then g2 . b = 0. L by A39, POLYNOM1:def_3;
then g2 - (((g2 . b) / (HC (p,T))) * (s *' p)) = g2 - (((0. L) * ((HC (p,T)) ")) * (s *' p)) by VECTSP_1:def_11
.= g2 - ((0. L) * (s *' p)) by BINOM:1
.= g2 - (0_ (n,L)) by Th8
.= g2 by Th4 ;
hence PolyRedRel (P,T) reduces g,g2 - (((g2 . b) / (HC (p,T))) * (s *' p)) by A34; ::_thesis: verum
end;
caseA40: b in Support g2 ; ::_thesis: PolyRedRel (P,T) reduces g,g2 - (((g2 . b) / (HC (p,T))) * (s *' p))
then g2 <> 0_ (n,L) by POLYNOM7:1;
then reconsider g2 = g2 as non-zero Polynomial of n,L by POLYNOM7:def_1;
( g2 <> 0_ (n,L) & p <> 0_ (n,L) ) by A18, Lm18, POLYNOM7:def_1;
then g2 reduces_to g2 - (((g2 . b) / (HC (p,T))) * (s *' p)),p,b,T by A35, A40, Def5;
then g2 reduces_to g2 - (((g2 . b) / (HC (p,T))) * (s *' p)),p,T by Def6;
then g2 reduces_to g2 - (((g2 . b) / (HC (p,T))) * (s *' p)),P,T by A17, Def7;
then [g2,(g2 - (((g2 . b) / (HC (p,T))) * (s *' p)))] in PolyRedRel (P,T) by Def13;
then PolyRedRel (P,T) reduces g2,g2 - (((g2 . b) / (HC (p,T))) * (s *' p)) by REWRITE1:15;
hence PolyRedRel (P,T) reduces g,g2 - (((g2 . b) / (HC (p,T))) * (s *' p)) by A34, REWRITE1:16; ::_thesis: verum
end;
end;
end;
A41: now__::_thesis:_(_(_not_b_in_Support_f2_&_PolyRedRel_(P,T)_reduces_f,f2_-_(((f2_._b)_/_(HC_(p,T)))_*_(s_*'_p))_)_or_(_b_in_Support_f2_&_PolyRedRel_(P,T)_reduces_f,f2_-_(((f2_._b)_/_(HC_(p,T)))_*_(s_*'_p))_)_)
percases ( not b in Support f2 or b in Support f2 ) ;
caseA42: not b in Support f2 ; ::_thesis: PolyRedRel (P,T) reduces f,f2 - (((f2 . b) / (HC (p,T))) * (s *' p))
b is Element of Bags n by PRE_POLY:def_12;
then f2 . b = 0. L by A42, POLYNOM1:def_3;
then f2 - (((f2 . b) / (HC (p,T))) * (s *' p)) = f2 - (((0. L) * ((HC (p,T)) ")) * (s *' p)) by VECTSP_1:def_11
.= f2 - ((0. L) * (s *' p)) by BINOM:1
.= f2 - (0_ (n,L)) by Th8
.= f2 by Th4 ;
hence PolyRedRel (P,T) reduces f,f2 - (((f2 . b) / (HC (p,T))) * (s *' p)) by A33; ::_thesis: verum
end;
caseA43: b in Support f2 ; ::_thesis: PolyRedRel (P,T) reduces f,f2 - (((f2 . b) / (HC (p,T))) * (s *' p))
then f2 <> 0_ (n,L) by POLYNOM7:1;
then reconsider f2 = f2 as non-zero Polynomial of n,L by POLYNOM7:def_1;
( f2 <> 0_ (n,L) & p <> 0_ (n,L) ) by A18, Lm18, POLYNOM7:def_1;
then f2 reduces_to f2 - (((f2 . b) / (HC (p,T))) * (s *' p)),p,b,T by A35, A43, Def5;
then f2 reduces_to f2 - (((f2 . b) / (HC (p,T))) * (s *' p)),p,T by Def6;
then f2 reduces_to f2 - (((f2 . b) / (HC (p,T))) * (s *' p)),P,T by A17, Def7;
then [f2,(f2 - (((f2 . b) / (HC (p,T))) * (s *' p)))] in PolyRedRel (P,T) by Def13;
then PolyRedRel (P,T) reduces f2,f2 - (((f2 . b) / (HC (p,T))) * (s *' p)) by REWRITE1:15;
hence PolyRedRel (P,T) reduces f,f2 - (((f2 . b) / (HC (p,T))) * (s *' p)) by A33, REWRITE1:16; ::_thesis: verum
end;
end;
end;
A44: - (- (((g2 . b) / (HC (p,T))) * (s *' p))) = ((g2 . b) / (HC (p,T))) * (s *' p) by POLYNOM1:19;
A45: ((- ((f2 . b) / (HC (p,T)))) * (s *' p)) + (((g2 . b) / (HC (p,T))) * (s *' p)) = ((- ((f2 . b) / (HC (p,T)))) + ((g2 . b) / (HC (p,T)))) * (s *' p) by Th10
.= - (- (((- ((f2 . b) / (HC (p,T)))) + ((g2 . b) / (HC (p,T)))) * (s *' p))) by POLYNOM1:19 ;
(f2 - (((f2 . b) / (HC (p,T))) * (s *' p))) - (g2 - (((g2 . b) / (HC (p,T))) * (s *' p))) = (f2 - (((f2 . b) / (HC (p,T))) * (s *' p))) + (- (g2 - (((g2 . b) / (HC (p,T))) * (s *' p)))) by POLYNOM1:def_6
.= (f2 + (- (((f2 . b) / (HC (p,T))) * (s *' p)))) + (- (g2 - (((g2 . b) / (HC (p,T))) * (s *' p)))) by POLYNOM1:def_6
.= (f2 + (- (((f2 . b) / (HC (p,T))) * (s *' p)))) + (- (g2 + (- (((g2 . b) / (HC (p,T))) * (s *' p))))) by POLYNOM1:def_6
.= (f2 + (- (((f2 . b) / (HC (p,T))) * (s *' p)))) + ((- g2) + (- (- (((g2 . b) / (HC (p,T))) * (s *' p))))) by Th1
.= ((f2 + (- (((f2 . b) / (HC (p,T))) * (s *' p)))) + (- g2)) + (((g2 . b) / (HC (p,T))) * (s *' p)) by A44, POLYNOM1:21
.= ((- (((f2 . b) / (HC (p,T))) * (s *' p))) + (f2 + (- g2))) + (((g2 . b) / (HC (p,T))) * (s *' p)) by POLYNOM1:21
.= (f2 + (- g2)) + ((- (((f2 . b) / (HC (p,T))) * (s *' p))) + (((g2 . b) / (HC (p,T))) * (s *' p))) by POLYNOM1:21
.= (f2 - g2) + ((- (((f2 . b) / (HC (p,T))) * (s *' p))) + (((g2 . b) / (HC (p,T))) * (s *' p))) by POLYNOM1:def_6
.= (f2 - g2) + (((- ((f2 . b) / (HC (p,T)))) * (s *' p)) + (((g2 . b) / (HC (p,T))) * (s *' p))) by Th9
.= (f2 - g2) + (- ((- ((- ((f2 . b) / (HC (p,T)))) + ((g2 . b) / (HC (p,T))))) * (s *' p))) by A45, Th9
.= (f2 - g2) - ((- ((- ((f2 . b) / (HC (p,T)))) + ((g2 . b) / (HC (p,T))))) * (s *' p)) by POLYNOM1:def_6
.= (f2 - g2) - (((- (- ((f2 . b) / (HC (p,T))))) + (- ((g2 . b) / (HC (p,T))))) * (s *' p)) by RLVECT_1:31
.= h1 by A32, A36, A37, RLVECT_1:17 ;
hence ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 ) by A41, A38; ::_thesis: verum
end;
end;
A46: S1[1]
proof
let f, g, h be Polynomial of n,L; ::_thesis: ( f - g = h implies for h1 being Polynomial of n,L
for p being RedSequence of PolyRedRel (P,T) st p . 1 = h & p . (len p) = h1 & len p = 1 holds
ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 ) )
assume A47: f - g = h ; ::_thesis: for h1 being Polynomial of n,L
for p being RedSequence of PolyRedRel (P,T) st p . 1 = h & p . (len p) = h1 & len p = 1 holds
ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 )
let h1 be Polynomial of n,L; ::_thesis: for p being RedSequence of PolyRedRel (P,T) st p . 1 = h & p . (len p) = h1 & len p = 1 holds
ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 )
let p be RedSequence of PolyRedRel (P,T); ::_thesis: ( p . 1 = h & p . (len p) = h1 & len p = 1 implies ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 ) )
assume A48: ( p . 1 = h & p . (len p) = h1 & len p = 1 ) ; ::_thesis: ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 )
take f ; ::_thesis: ex g1 being Polynomial of n,L st
( f - g1 = h1 & PolyRedRel (P,T) reduces f,f & PolyRedRel (P,T) reduces g,g1 )
take g ; ::_thesis: ( f - g = h1 & PolyRedRel (P,T) reduces f,f & PolyRedRel (P,T) reduces g,g )
thus ( f - g = h1 & PolyRedRel (P,T) reduces f,f & PolyRedRel (P,T) reduces g,g ) by A47, A48, REWRITE1:12; ::_thesis: verum
end;
A49: for k being Nat st 1 <= k holds
S1[k] from NAT_1:sch_8(A46, A4);
consider k being Nat such that
A50: len p = k ;
1 <= k by A50, NAT_1:14;
hence ex f1, g1 being Polynomial of n,L st
( f1 - g1 = h1 & PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 ) by A1, A49, A3, A50; ::_thesis: verum
end;
theorem Th50: :: POLYRED:50
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 f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f - g, 0_ (n,L) holds
f,g are_convergent_wrt PolyRedRel (P,T)
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f - g, 0_ (n,L) holds
f,g are_convergent_wrt PolyRedRel (P,T)
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f - g, 0_ (n,L) holds
f,g are_convergent_wrt PolyRedRel (P,T)
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f - g, 0_ (n,L) holds
f,g are_convergent_wrt PolyRedRel (P,T)
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f - g, 0_ (n,L) holds
f,g are_convergent_wrt PolyRedRel (P,T)
let f, g be Polynomial of n,L; ::_thesis: ( PolyRedRel (P,T) reduces f - g, 0_ (n,L) implies f,g are_convergent_wrt PolyRedRel (P,T) )
assume PolyRedRel (P,T) reduces f - g, 0_ (n,L) ; ::_thesis: f,g are_convergent_wrt PolyRedRel (P,T)
then consider f1, g1 being Polynomial of n,L such that
A1: f1 - g1 = 0_ (n,L) and
A2: ( PolyRedRel (P,T) reduces f,f1 & PolyRedRel (P,T) reduces g,g1 ) by Th49;
g1 = (f1 - g1) + g1 by A1, Th2
.= (f1 + (- g1)) + g1 by POLYNOM1:def_6
.= f1 + ((- g1) + g1) by POLYNOM1:21
.= f1 + (0_ (n,L)) by Th3
.= f1 by POLYNOM1:23 ;
hence f,g are_convergent_wrt PolyRedRel (P,T) by A2, REWRITE1:def_7; ::_thesis: verum
end;
theorem Th51: :: POLYRED:51
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 f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f - g, 0_ (n,L) holds
f,g are_convertible_wrt PolyRedRel (P,T)
proof
let n be Ordinal; ::_thesis: for T being connected TermOrder of n
for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f - g, 0_ (n,L) holds
f,g are_convertible_wrt PolyRedRel (P,T)
let T be connected TermOrder of n; ::_thesis: for L being non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f - g, 0_ (n,L) holds
f,g are_convertible_wrt PolyRedRel (P,T)
let L be non trivial right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for P being Subset of (Polynom-Ring (n,L))
for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f - g, 0_ (n,L) holds
f,g are_convertible_wrt PolyRedRel (P,T)
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f - g, 0_ (n,L) holds
f,g are_convertible_wrt PolyRedRel (P,T)
let f, g be Polynomial of n,L; ::_thesis: ( PolyRedRel (P,T) reduces f - g, 0_ (n,L) implies f,g are_convertible_wrt PolyRedRel (P,T) )
set R = PolyRedRel (P,T);
assume PolyRedRel (P,T) reduces f - g, 0_ (n,L) ; ::_thesis: f,g are_convertible_wrt PolyRedRel (P,T)
then f,g are_convergent_wrt PolyRedRel (P,T) by Th50;
then consider h being set such that
A1: PolyRedRel (P,T) reduces f,h and
A2: PolyRedRel (P,T) reduces g,h by REWRITE1:def_7;
(PolyRedRel (P,T)) ~ reduces h,g by A2, REWRITE1:24;
then A3: (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) reduces h,g by REWRITE1:22, XBOOLE_1:7;
(PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) reduces f,h by A1, REWRITE1:22, XBOOLE_1:7;
then (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) reduces f,g by A3, REWRITE1:16;
hence f,g are_convertible_wrt PolyRedRel (P,T) by REWRITE1:def_4; ::_thesis: verum
end;
definition
let R be non empty addLoopStr ;
let I be Subset of R;
let a, b be Element of R;
preda,b are_congruent_mod I means :Def14: :: POLYRED:def 14
a - b in I;
end;
:: deftheorem Def14 defines are_congruent_mod POLYRED:def_14_:_
for R being non empty addLoopStr
for I being Subset of R
for a, b being Element of R holds
( a,b are_congruent_mod I iff a - b in I );
theorem :: POLYRED:52
for R being non empty right_complementable right-distributive add-associative right_zeroed left_zeroed doubleLoopStr
for I being non empty right-ideal Subset of R
for a being Element of R holds a,a are_congruent_mod I
proof
let R be non empty right_complementable right-distributive add-associative right_zeroed left_zeroed doubleLoopStr ; ::_thesis: for I being non empty right-ideal Subset of R
for a being Element of R holds a,a are_congruent_mod I
let I be non empty right-ideal Subset of R; ::_thesis: for a being Element of R holds a,a are_congruent_mod I
let a be Element of R; ::_thesis: a,a are_congruent_mod I
( a - a = 0. R & 0. R in I ) by IDEAL_1:3, RLVECT_1:15;
hence a,a are_congruent_mod I by Def14; ::_thesis: verum
end;
theorem Th53: :: POLYRED:53
for R being non empty right_complementable right-distributive well-unital add-associative right_zeroed doubleLoopStr
for I being non empty right-ideal Subset of R
for a, b being Element of R st a,b are_congruent_mod I holds
b,a are_congruent_mod I
proof
let R be non empty right_complementable right-distributive well-unital add-associative right_zeroed doubleLoopStr ; ::_thesis: for I being non empty right-ideal Subset of R
for a, b being Element of R st a,b are_congruent_mod I holds
b,a are_congruent_mod I
let I be non empty right-ideal Subset of R; ::_thesis: for a, b being Element of R st a,b are_congruent_mod I holds
b,a are_congruent_mod I
let a, b be Element of R; ::_thesis: ( a,b are_congruent_mod I implies b,a are_congruent_mod I )
assume a,b are_congruent_mod I ; ::_thesis: b,a are_congruent_mod I
then a - b in I by Def14;
then A1: - (a - b) in I by IDEAL_1:14;
(b - a) - (- (a - b)) = (b - a) + (- (- (a - b))) by RLVECT_1:def_11
.= (b - a) + (a - b) by RLVECT_1:17
.= (b + (- a)) + (a - b) by RLVECT_1:def_11
.= b + ((- a) + (a - b)) by RLVECT_1:def_3
.= b + ((- a) + (a + (- b))) by RLVECT_1:def_11
.= b + (((- a) + a) + (- b)) by RLVECT_1:def_3
.= b + ((0. R) + (- b)) by RLVECT_1:5
.= b + (- b) by ALGSTR_1:def_2
.= 0. R by RLVECT_1:5 ;
then b - a = - (a - b) by RLVECT_1:21;
hence b,a are_congruent_mod I by A1, Def14; ::_thesis: verum
end;
theorem Th54: :: POLYRED:54
for R being non empty right_complementable add-associative right_zeroed addLoopStr
for I being non empty add-closed Subset of R
for a, b, c being Element of R st a,b are_congruent_mod I & b,c are_congruent_mod I holds
a,c are_congruent_mod I
proof
let R be non empty right_complementable add-associative right_zeroed addLoopStr ; ::_thesis: for I being non empty add-closed Subset of R
for a, b, c being Element of R st a,b are_congruent_mod I & b,c are_congruent_mod I holds
a,c are_congruent_mod I
let I be non empty add-closed Subset of R; ::_thesis: for a, b, c being Element of R st a,b are_congruent_mod I & b,c are_congruent_mod I holds
a,c are_congruent_mod I
let a, b, c be Element of R; ::_thesis: ( a,b are_congruent_mod I & b,c are_congruent_mod I implies a,c are_congruent_mod I )
assume ( a,b are_congruent_mod I & b,c are_congruent_mod I ) ; ::_thesis: a,c are_congruent_mod I
then ( a - b in I & b - c in I ) by Def14;
then A1: (a - b) + (b - c) in I by IDEAL_1:def_1;
(a - b) + (b - c) = (a + (- b)) + (b - c) by RLVECT_1:def_11
.= a + ((- b) + (b - c)) by RLVECT_1:def_3
.= a + ((- b) + (b + (- c))) by RLVECT_1:def_11
.= a + (((- b) + b) + (- c)) by RLVECT_1:def_3
.= a + ((0. R) + (- c)) by RLVECT_1:5
.= a + (- c) by ALGSTR_1:def_2
.= a - c by RLVECT_1:def_11 ;
hence a,c are_congruent_mod I by A1, Def14; ::_thesis: verum
end;
theorem :: POLYRED:55
for R being non trivial right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for I being non empty add-closed Subset of R
for a, b, c, d being Element of R st a,b are_congruent_mod I & c,d are_congruent_mod I holds
a + c,b + d are_congruent_mod I
proof
let R be non trivial right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for I being non empty add-closed Subset of R
for a, b, c, d being Element of R st a,b are_congruent_mod I & c,d are_congruent_mod I holds
a + c,b + d are_congruent_mod I
let I be non empty add-closed Subset of R; ::_thesis: for a, b, c, d being Element of R st a,b are_congruent_mod I & c,d are_congruent_mod I holds
a + c,b + d are_congruent_mod I
let a, b, c, d be Element of R; ::_thesis: ( a,b are_congruent_mod I & c,d are_congruent_mod I implies a + c,b + d are_congruent_mod I )
assume ( a,b are_congruent_mod I & c,d are_congruent_mod I ) ; ::_thesis: a + c,b + d are_congruent_mod I
then ( a - b in I & c - d in I ) by Def14;
then A1: (a - b) + (c - d) in I by IDEAL_1:def_1;
(a + c) - (b + d) = (a + c) + (- (b + d)) by RLVECT_1:def_11
.= (a + c) + ((- d) + (- b)) by RLVECT_1:31
.= a + (c + ((- d) + (- b))) by RLVECT_1:def_3
.= a + ((c + (- d)) + (- b)) by RLVECT_1:def_3
.= (a + (- b)) + (c + (- d)) by RLVECT_1:def_3
.= (a - b) + (c + (- d)) by RLVECT_1:def_11
.= (a - b) + (c - d) by RLVECT_1:def_11 ;
hence a + c,b + d are_congruent_mod I by A1, Def14; ::_thesis: verum
end;
theorem :: POLYRED:56
for R being non empty right_complementable commutative distributive add-associative right_zeroed doubleLoopStr
for I being non empty add-closed right-ideal Subset of R
for a, b, c, d being Element of R st a,b are_congruent_mod I & c,d are_congruent_mod I holds
a * c,b * d are_congruent_mod I
proof
let R be non empty right_complementable commutative distributive add-associative right_zeroed doubleLoopStr ; ::_thesis: for I being non empty add-closed right-ideal Subset of R
for a, b, c, d being Element of R st a,b are_congruent_mod I & c,d are_congruent_mod I holds
a * c,b * d are_congruent_mod I
let I be non empty add-closed right-ideal Subset of R; ::_thesis: for a, b, c, d being Element of R st a,b are_congruent_mod I & c,d are_congruent_mod I holds
a * c,b * d are_congruent_mod I
let a, b, c, d be Element of R; ::_thesis: ( a,b are_congruent_mod I & c,d are_congruent_mod I implies a * c,b * d are_congruent_mod I )
assume that
A1: a,b are_congruent_mod I and
A2: c,d are_congruent_mod I ; ::_thesis: a * c,b * d are_congruent_mod I
c - d in I by A2, Def14;
then A3: (c - d) * b in I by IDEAL_1:def_3;
A4: (c - d) * b = (c + (- d)) * b by RLVECT_1:def_11
.= (c * b) + ((- d) * b) by VECTSP_1:def_3 ;
(a - b) * c = (a + (- b)) * c by RLVECT_1:def_11
.= (a * c) + ((- b) * c) by VECTSP_1:def_3 ;
then A5: ((a - b) * c) + ((c - d) * b) = (a * c) + (((- b) * c) + ((c * b) + ((- d) * b))) by A4, RLVECT_1:def_3
.= (a * c) + ((((- b) * c) + (c * b)) + ((- d) * b)) by RLVECT_1:def_3
.= (a * c) + (((- (b * c)) + (c * b)) + ((- d) * b)) by VECTSP_1:9
.= (a * c) + ((0. R) + ((- d) * b)) by RLVECT_1:5
.= (a * c) + ((- d) * b) by ALGSTR_1:def_2
.= (a * c) + (- (d * b)) by VECTSP_1:9
.= (a * c) - (b * d) by RLVECT_1:def_11 ;
a - b in I by A1, Def14;
then (a - b) * c in I by IDEAL_1:def_3;
then ((a - b) * c) + ((c - d) * b) in I by A3, IDEAL_1:def_1;
hence a * c,b * d are_congruent_mod I by A5, Def14; ::_thesis: verum
end;
Lm19: 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 f being non-zero Polynomial of n,L
for g being Polynomial of n,L
for f9, g9 being Element of (Polynom-Ring (n,L)) st f = f9 & g = g9 & f reduces_to g,P,T holds
f9,g9 are_congruent_mod 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 f being non-zero Polynomial of n,L
for g being Polynomial of n,L
for f9, g9 being Element of (Polynom-Ring (n,L)) st f = f9 & g = g9 & f reduces_to g,P,T holds
f9,g9 are_congruent_mod 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 f being non-zero Polynomial of n,L
for g being Polynomial of n,L
for f9, g9 being Element of (Polynom-Ring (n,L)) st f = f9 & g = g9 & f reduces_to g,P,T holds
f9,g9 are_congruent_mod 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 f being non-zero Polynomial of n,L
for g being Polynomial of n,L
for f9, g9 being Element of (Polynom-Ring (n,L)) st f = f9 & g = g9 & f reduces_to g,P,T holds
f9,g9 are_congruent_mod P -Ideal
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: for f being non-zero Polynomial of n,L
for g being Polynomial of n,L
for f9, g9 being Element of (Polynom-Ring (n,L)) st f = f9 & g = g9 & f reduces_to g,P,T holds
f9,g9 are_congruent_mod P -Ideal
let f be non-zero Polynomial of n,L; ::_thesis: for g being Polynomial of n,L
for f9, g9 being Element of (Polynom-Ring (n,L)) st f = f9 & g = g9 & f reduces_to g,P,T holds
f9,g9 are_congruent_mod P -Ideal
let g be Polynomial of n,L; ::_thesis: for f9, g9 being Element of (Polynom-Ring (n,L)) st f = f9 & g = g9 & f reduces_to g,P,T holds
f9,g9 are_congruent_mod P -Ideal
let f9, g9 be Element of (Polynom-Ring (n,L)); ::_thesis: ( f = f9 & g = g9 & f reduces_to g,P,T implies f9,g9 are_congruent_mod P -Ideal )
assume that
A1: f = f9 and
A2: g = g9 ; ::_thesis: ( not f reduces_to g,P,T or f9,g9 are_congruent_mod P -Ideal )
set R = Polynom-Ring (n,L);
reconsider x = - g as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider x = x as Element of (Polynom-Ring (n,L)) ;
x + g9 = (- g) + g by A2, POLYNOM1:def_10
.= 0_ (n,L) by Th3
.= 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10 ;
then A3: - g9 = - g by RLVECT_1:6;
assume f reduces_to g,P,T ; ::_thesis: f9,g9 are_congruent_mod P -Ideal
then consider p being Polynomial of n,L such that
A4: p in P and
A5: f reduces_to g,p,T by Def7;
consider b being bag of n such that
A6: f reduces_to g,p,b,T by A5, Def6;
consider s being bag of n such that
s + (HT (p,T)) = b and
A7: g = f - (((f . b) / (HC (p,T))) * (s *' p)) by A6, Def5;
reconsider P = P as non empty Subset of (Polynom-Ring (n,L)) by A4;
set q = ((f . b) / (HC (p,T))) * (s *' p);
set q9 = (Monom (((f . b) / (HC (p,T))),s)) *' p;
set r = <*((Monom (((f . b) / (HC (p,T))),s)) *' p)*>;
now__::_thesis:_for_u_being_set_st_u_in_rng_<*((Monom_(((f_._b)_/_(HC_(p,T))),s))_*'_p)*>_holds_
u_in_the_carrier_of_(Polynom-Ring_(n,L))
let u be set ; ::_thesis: ( u in rng <*((Monom (((f . b) / (HC (p,T))),s)) *' p)*> implies u in the carrier of (Polynom-Ring (n,L)) )
A8: rng <*((Monom (((f . b) / (HC (p,T))),s)) *' p)*> = {((Monom (((f . b) / (HC (p,T))),s)) *' p)} by FINSEQ_1:39;
assume u in rng <*((Monom (((f . b) / (HC (p,T))),s)) *' p)*> ; ::_thesis: u in the carrier of (Polynom-Ring (n,L))
then u = (Monom (((f . b) / (HC (p,T))),s)) *' p by A8, TARSKI:def_1;
hence u in the carrier of (Polynom-Ring (n,L)) by POLYNOM1:def_10; ::_thesis: verum
end;
then rng <*((Monom (((f . b) / (HC (p,T))),s)) *' p)*> c= the carrier of (Polynom-Ring (n,L)) by TARSKI:def_3;
then reconsider r = <*((Monom (((f . b) / (HC (p,T))),s)) *' p)*> as FinSequence of the carrier of (Polynom-Ring (n,L)) by FINSEQ_1:def_4;
now__::_thesis:_for_i_being_set_st_i_in_dom_r_holds_
ex_u,_v_being_Element_of_(Polynom-Ring_(n,L))_ex_a_being_Element_of_P_st_r_/._i_=_(u_*_a)_*_v
reconsider p9 = p as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider m = Monom (((f . b) / (HC (p,T))),s) as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
let i be set ; ::_thesis: ( i in dom r implies ex u, v being Element of (Polynom-Ring (n,L)) ex a being Element of P st r /. i = (u * a) * v )
assume A9: i in dom r ; ::_thesis: ex u, v being Element of (Polynom-Ring (n,L)) ex a being Element of P st r /. i = (u * a) * v
reconsider p9 = p9 as Element of (Polynom-Ring (n,L)) ;
reconsider m = m as Element of (Polynom-Ring (n,L)) ;
A10: (m * p9) * (1. (Polynom-Ring (n,L))) = m * p9 by VECTSP_1:def_4
.= (Monom (((f . b) / (HC (p,T))),s)) *' p by POLYNOM1:def_10 ;
dom r = Seg 1 by FINSEQ_1:38;
then i = 1 by A9, FINSEQ_1:2, TARSKI:def_1;
then r . i = (Monom (((f . b) / (HC (p,T))),s)) *' p by FINSEQ_1:40;
hence ex u, v being Element of (Polynom-Ring (n,L)) ex a being Element of P st r /. i = (u * a) * v by A4, A9, A10, PARTFUN1:def_6; ::_thesis: verum
end;
then reconsider r = r as LinearCombination of P by IDEAL_1:def_8;
(Monom (((f . b) / (HC (p,T))),s)) *' p is Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
then A11: Sum r = (Monom (((f . b) / (HC (p,T))),s)) *' p by BINOM:3;
(Monom (((f . b) / (HC (p,T))),s)) *' p = ((f . b) / (HC (p,T))) * (s *' p) by Th22;
then A12: ((f . b) / (HC (p,T))) * (s *' p) in P -Ideal by A11, IDEAL_1:60;
A13: f - g = f + (- g) by POLYNOM1:def_6
.= f9 + (- g9) by A1, A3, POLYNOM1:def_10
.= f9 - g9 by RLVECT_1:def_11 ;
A14: - (- (((f . b) / (HC (p,T))) * (s *' p))) = ((f . b) / (HC (p,T))) * (s *' p) by POLYNOM1:19;
f - g = f + (- (f - (((f . b) / (HC (p,T))) * (s *' p)))) by A7, POLYNOM1:def_6
.= f + (- (f + (- (((f . b) / (HC (p,T))) * (s *' p))))) by POLYNOM1:def_6
.= f + ((- f) + (- (- (((f . b) / (HC (p,T))) * (s *' p))))) by Th1
.= (f + (- f)) + (((f . b) / (HC (p,T))) * (s *' p)) by A14, POLYNOM1:21
.= (0_ (n,L)) + (((f . b) / (HC (p,T))) * (s *' p)) by Th3
.= ((f . b) / (HC (p,T))) * (s *' p) by Th2 ;
hence f9,g9 are_congruent_mod P -Ideal by A12, A13, Def14; ::_thesis: verum
end;
theorem Th57: :: POLYRED:57
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 f, g being Element of (Polynom-Ring (n,L)) st f,g are_convertible_wrt PolyRedRel (P,T) holds
f,g are_congruent_mod 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 f, g being Element of (Polynom-Ring (n,L)) st f,g are_convertible_wrt PolyRedRel (P,T) holds
f,g are_congruent_mod 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 f, g being Element of (Polynom-Ring (n,L)) st f,g are_convertible_wrt PolyRedRel (P,T) holds
f,g are_congruent_mod 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 f, g being Element of (Polynom-Ring (n,L)) st f,g are_convertible_wrt PolyRedRel (P,T) holds
f,g are_congruent_mod P -Ideal
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: for f, g being Element of (Polynom-Ring (n,L)) st f,g are_convertible_wrt PolyRedRel (P,T) holds
f,g are_congruent_mod P -Ideal
let f, g be Element of (Polynom-Ring (n,L)); ::_thesis: ( f,g are_convertible_wrt PolyRedRel (P,T) implies f,g are_congruent_mod P -Ideal )
set R = PolyRedRel (P,T);
set PR = Polynom-Ring (n,L);
defpred S1[ Nat] means for f, g being Element of (Polynom-Ring (n,L)) st (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) reduces f,g holds
for p being RedSequence of (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) st p . 1 = f & p . (len p) = g & len p = $1 holds
f,g are_congruent_mod P -Ideal ;
assume f,g are_convertible_wrt PolyRedRel (P,T) ; ::_thesis: f,g are_congruent_mod P -Ideal
then A1: (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) reduces f,g by REWRITE1:def_4;
then consider p being RedSequence of (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) such that
A2: ( p . 1 = f & p . (len p) = 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]
now__::_thesis:_for_f,_g_being_Element_of_(Polynom-Ring_(n,L))_st_(PolyRedRel_(P,T))_\/_((PolyRedRel_(P,T))_~)_reduces_f,g_holds_
for_p_being_RedSequence_of_(PolyRedRel_(P,T))_\/_((PolyRedRel_(P,T))_~)_st_p_._1_=_f_&_p_._(len_p)_=_g_&_len_p_=_k_+_1_holds_
f,g_are_congruent_mod_P_-Ideal
let f, g be Element of (Polynom-Ring (n,L)); ::_thesis: ( (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) reduces f,g implies for p being RedSequence of (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) st p . 1 = f & p . (len p) = g & len p = k + 1 holds
f,g are_congruent_mod P -Ideal )
assume (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) reduces f,g ; ::_thesis: for p being RedSequence of (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) st p . 1 = f & p . (len p) = g & len p = k + 1 holds
f,g are_congruent_mod P -Ideal
let p be RedSequence of (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~); ::_thesis: ( p . 1 = f & p . (len p) = g & len p = k + 1 implies f,g are_congruent_mod P -Ideal )
assume that
A7: p . 1 = f and
A8: p . (len p) = g and
A9: len p = k + 1 ; ::_thesis: f,g are_congruent_mod P -Ideal
A10: dom p = Seg (k + 1) by A9, FINSEQ_1:def_3;
then A11: k + 1 in dom p by FINSEQ_1:4;
set q = p | (Seg k);
reconsider q = p | (Seg k) as FinSequence by FINSEQ_1:15;
A12: k <= k + 1 by NAT_1:11;
then A13: dom q = Seg k by A9, FINSEQ_1:17;
then A14: k in dom q by A5, FINSEQ_1:1;
set h = q . (len q);
A15: len q = k by A9, A12, FINSEQ_1:17;
k in dom p by A5, A10, A12, FINSEQ_1:1;
then [(p . k),(p . (k + 1))] in (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) by A11, REWRITE1:def_2;
then [(q . (len q)),g] in (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) by A8, A9, A15, A14, FUNCT_1:47;
then A16: ( [(q . (len q)),g] in PolyRedRel (P,T) or [(q . (len q)),g] in (PolyRedRel (P,T)) ~ ) by XBOOLE_0:def_3;
A17: now__::_thesis:_(_(_[(q_._(len_q)),g]_in_PolyRedRel_(P,T)_&_q_._(len_q)_in_the_carrier_of_(Polynom-Ring_(n,L))_\_{(0__(n,L))}_&_q_._(len_q)_in_the_carrier_of_(Polynom-Ring_(n,L))_&_g_in_the_carrier_of_(Polynom-Ring_(n,L))_)_or_(_[g,(q_._(len_q))]_in_PolyRedRel_(P,T)_&_g_in_the_carrier_of_(Polynom-Ring_(n,L))_\_{(0__(n,L))}_&_g_in_the_carrier_of_(Polynom-Ring_(n,L))_&_q_._(len_q)_in_the_carrier_of_(Polynom-Ring_(n,L))_)_)
percases ( [(q . (len q)),g] in PolyRedRel (P,T) or [g,(q . (len q))] in PolyRedRel (P,T) ) by A16, RELAT_1:def_7;
case [(q . (len q)),g] in PolyRedRel (P,T) ; ::_thesis: ( q . (len q) in the carrier of (Polynom-Ring (n,L)) \ {(0_ (n,L))} & q . (len q) in the carrier of (Polynom-Ring (n,L)) & g in the carrier of (Polynom-Ring (n,L)) )
then consider h9, g9 being set such that
A18: [(q . (len q)),g] = [h9,g9] and
A19: h9 in NonZero (Polynom-Ring (n,L)) and
g9 in the carrier of (Polynom-Ring (n,L)) by RELSET_1:2;
q . (len q) = h9 by A18, XTUPLE_0:1;
hence ( q . (len q) in the carrier of (Polynom-Ring (n,L)) \ {(0_ (n,L))} & q . (len q) in the carrier of (Polynom-Ring (n,L)) & g in the carrier of (Polynom-Ring (n,L)) ) by A19, POLYNOM1:def_10; ::_thesis: verum
end;
case [g,(q . (len q))] in PolyRedRel (P,T) ; ::_thesis: ( g in the carrier of (Polynom-Ring (n,L)) \ {(0_ (n,L))} & g in the carrier of (Polynom-Ring (n,L)) & q . (len q) in the carrier of (Polynom-Ring (n,L)) )
then consider h9, g9 being set such that
A20: [g,(q . (len q))] = [h9,g9] and
A21: ( h9 in NonZero (Polynom-Ring (n,L)) & g9 in the carrier of (Polynom-Ring (n,L)) ) by RELSET_1:2;
A22: q . (len q) = g9 by A20, XTUPLE_0:1;
g = h9 by A20, XTUPLE_0:1;
hence ( g in the carrier of (Polynom-Ring (n,L)) \ {(0_ (n,L))} & g in the carrier of (Polynom-Ring (n,L)) & q . (len q) in the carrier of (Polynom-Ring (n,L)) ) by A21, A22, POLYNOM1:def_10; ::_thesis: verum
end;
end;
end;
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))_\/_((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)) \/ ((PolyRedRel (P,T)) ~) )
assume that
A23: i in dom q and
A24: i + 1 in dom q ; ::_thesis: [(q . i),(q . (i + 1))] in (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~)
i + 1 <= k by A13, A24, FINSEQ_1:1;
then A25: i + 1 <= k + 1 by A12, XXREAL_0:2;
i <= k by A13, A23, FINSEQ_1:1;
then A26: i <= k + 1 by A12, XXREAL_0:2;
1 <= i + 1 by A13, A24, FINSEQ_1:1;
then A27: i + 1 in dom p by A10, A25, FINSEQ_1:1;
1 <= i by A13, A23, FINSEQ_1:1;
then i in dom p by A10, A26, FINSEQ_1:1;
then A28: [(p . i),(p . (i + 1))] in (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) by A27, REWRITE1:def_2;
p . i = q . i by A23, FUNCT_1:47;
hence [(q . i),(q . (i + 1))] in (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) by A24, A28, FUNCT_1:47; ::_thesis: verum
end;
then reconsider q = q as RedSequence of (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) by A5, A15, REWRITE1:def_2;
reconsider h = q . (len q) as Polynomial of n,L by A17, POLYNOM1:def_10;
reconsider h9 = h as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider h9 = h9 as Element of (Polynom-Ring (n,L)) ;
1 in dom q by A5, A13, FINSEQ_1:1;
then A29: q . 1 = f by A7, FUNCT_1:47;
then (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) reduces f,h by REWRITE1:def_3;
then A30: f,h9 are_congruent_mod P -Ideal by A6, A9, A12, A29, FINSEQ_1:17;
now__::_thesis:_(_(_[h,g]_in_PolyRedRel_(P,T)_&_f_-_g_in_P_-Ideal_)_or_(_[g,h]_in_PolyRedRel_(P,T)_&_f_-_g_in_P_-Ideal_)_)
percases ( [h,g] in PolyRedRel (P,T) or [g,h] in PolyRedRel (P,T) ) by A16, RELAT_1:def_7;
caseA31: [h,g] in PolyRedRel (P,T) ; ::_thesis: f - g in P -Ideal
then consider h9, g9 being set such that
A32: [h,g] = [h9,g9] and
A33: h9 in NonZero (Polynom-Ring (n,L)) and
A34: g9 in the carrier of (Polynom-Ring (n,L)) by RELSET_1:2;
A35: h = h9 by A32, XTUPLE_0:1;
not h9 in {(0_ (n,L))} by A3, A33, XBOOLE_0:def_5;
then h9 <> 0_ (n,L) by TARSKI:def_1;
then reconsider h = h as non-zero Polynomial of n,L by A35, POLYNOM7:def_1;
A36: g = g9 by A32, XTUPLE_0:1;
reconsider g9 = g9 as Polynomial of n,L by A34, POLYNOM1:def_10;
reconsider h9 = h as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider h9 = h9 as Element of (Polynom-Ring (n,L)) ;
h reduces_to g9,P,T by A31, A36, Def13;
then h9,g are_congruent_mod P -Ideal by A36, Lm19;
then f,g are_congruent_mod P -Ideal by A30, Th54;
hence f - g in P -Ideal by Def14; ::_thesis: verum
end;
caseA37: [g,h] in PolyRedRel (P,T) ; ::_thesis: f - g in P -Ideal
then consider g9, h9 being set such that
A38: [g,h] = [g9,h9] and
A39: g9 in NonZero (Polynom-Ring (n,L)) and
A40: h9 in the carrier of (Polynom-Ring (n,L)) by RELSET_1:2;
A41: g = g9 by A38, XTUPLE_0:1;
not g9 in {(0_ (n,L))} by A3, A39, XBOOLE_0:def_5;
then A42: g9 <> 0_ (n,L) by TARSKI:def_1;
A43: h = h9 by A38, XTUPLE_0:1;
then reconsider h = h as Element of (Polynom-Ring (n,L)) by A40;
reconsider h9 = h9 as Polynomial of n,L by A43;
reconsider g9 = g as non-zero Polynomial of n,L by A41, A42, POLYNOM1:def_10, POLYNOM7:def_1;
reconsider gg = g9 as Element of (Polynom-Ring (n,L)) ;
reconsider gg = gg as Element of (Polynom-Ring (n,L)) ;
reconsider h = h as Element of (Polynom-Ring (n,L)) ;
g9 reduces_to h9,P,T by A37, A43, Def13;
then h,gg are_congruent_mod P -Ideal by A43, Lm19, Th53;
then f,gg are_congruent_mod P -Ideal by A30, Th54;
hence f - g in P -Ideal by Def14; ::_thesis: verum
end;
end;
end;
hence f,g are_congruent_mod P -Ideal by Def14; ::_thesis: verum
end;
hence S1[k + 1] ; ::_thesis: verum
end;
end;
A44: S1[1]
proof
let f, g be Element of (Polynom-Ring (n,L)); ::_thesis: ( (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) reduces f,g implies for p being RedSequence of (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) st p . 1 = f & p . (len p) = g & len p = 1 holds
f,g are_congruent_mod P -Ideal )
assume (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) reduces f,g ; ::_thesis: for p being RedSequence of (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~) st p . 1 = f & p . (len p) = g & len p = 1 holds
f,g are_congruent_mod P -Ideal
let p be RedSequence of (PolyRedRel (P,T)) \/ ((PolyRedRel (P,T)) ~); ::_thesis: ( p . 1 = f & p . (len p) = g & len p = 1 implies f,g are_congruent_mod P -Ideal )
assume ( p . 1 = f & p . (len p) = g & len p = 1 ) ; ::_thesis: f,g are_congruent_mod P -Ideal
then A45: f - g = 0. (Polynom-Ring (n,L)) by RLVECT_1:15;
0. (Polynom-Ring (n,L)) in P -Ideal by IDEAL_1:3;
hence f,g are_congruent_mod P -Ideal by A45, Def14; ::_thesis: verum
end;
A46: for k being Nat st 1 <= k holds
S1[k] from NAT_1:sch_8(A44, A4);
consider k being Nat such that
A47: len p = k ;
1 <= k by A47, NAT_1:14;
hence f,g are_congruent_mod P -Ideal by A46, A1, A2, A47; ::_thesis: verum
end;
Lm20: for n being Nat
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being non empty Subset of (Polynom-Ring (n,L))
for f, p, h being Element of (Polynom-Ring (n,L)) st p in P & p <> 0_ (n,L) & h <> 0_ (n,L) holds
f,f + (h * p) are_convertible_wrt PolyRedRel (P,T)
proof
let n be Nat; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being non empty Subset of (Polynom-Ring (n,L))
for f, p, h being Element of (Polynom-Ring (n,L)) st p in P & p <> 0_ (n,L) & h <> 0_ (n,L) holds
f,f + (h * p) are_convertible_wrt PolyRedRel (P,T)
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being non empty Subset of (Polynom-Ring (n,L))
for f, p, h being Element of (Polynom-Ring (n,L)) st p in P & p <> 0_ (n,L) & h <> 0_ (n,L) holds
f,f + (h * p) are_convertible_wrt PolyRedRel (P,T)
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for P being non empty Subset of (Polynom-Ring (n,L))
for f, p, h being Element of (Polynom-Ring (n,L)) st p in P & p <> 0_ (n,L) & h <> 0_ (n,L) holds
f,f + (h * p) are_convertible_wrt PolyRedRel (P,T)
let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: for f, p, h being Element of (Polynom-Ring (n,L)) st p in P & p <> 0_ (n,L) & h <> 0_ (n,L) holds
f,f + (h * p) are_convertible_wrt PolyRedRel (P,T)
let f, p, h be Element of (Polynom-Ring (n,L)); ::_thesis: ( p in P & p <> 0_ (n,L) & h <> 0_ (n,L) implies f,f + (h * p) are_convertible_wrt PolyRedRel (P,T) )
assume that
A1: p in P and
A2: p <> 0_ (n,L) and
A3: h <> 0_ (n,L) ; ::_thesis: f,f + (h * p) are_convertible_wrt PolyRedRel (P,T)
set PR = Polynom-Ring (n,L);
reconsider f9 = f, h9 = h, p9 = p as Element of (Polynom-Ring (n,L)) ;
reconsider f9 = f9, p9 = p9, h9 = h9 as Polynomial of n,L by POLYNOM1:def_10;
reconsider h9 = h9, p9 = p9 as non-zero Polynomial of n,L by A2, A3, POLYNOM7:def_1;
A4: PolyRedRel (P,T) reduces h9 *' p9, 0_ (n,L) by A1, Th45;
now__::_thesis:_(_(_f9_=_0__(n,L)_&_f,f_+_(h_*_p)_are_convertible_wrt_PolyRedRel_(P,T)_)_or_(_f9_<>_0__(n,L)_&_f,f_+_(h_*_p)_are_convertible_wrt_PolyRedRel_(P,T)_)_)
percases ( f9 = 0_ (n,L) or f9 <> 0_ (n,L) ) ;
case f9 = 0_ (n,L) ; ::_thesis: f,f + (h * p) are_convertible_wrt PolyRedRel (P,T)
then PolyRedRel (P,T) reduces f9 + (h9 *' p9),f9 by A4, Th2;
then A5: f9,f9 + (h9 *' p9) are_convertible_wrt PolyRedRel (P,T) by REWRITE1:25;
h9 *' p9 = h * p by POLYNOM1:def_10;
hence f,f + (h * p) are_convertible_wrt PolyRedRel (P,T) by A5, POLYNOM1:def_10; ::_thesis: verum
end;
case f9 <> 0_ (n,L) ; ::_thesis: f,f + (h * p) are_convertible_wrt PolyRedRel (P,T)
then reconsider f9 = f9 as non-zero Polynomial of n,L by POLYNOM7:def_1;
(f9 + (h9 *' p9)) - f9 = (f9 + (h9 *' p9)) + (- f9) by POLYNOM1:def_6
.= (h9 *' p9) + (f9 + (- f9)) by POLYNOM1:21
.= (0_ (n,L)) + (h9 *' p9) by Th3
.= h9 *' p9 by Th2 ;
then A6: PolyRedRel (P,T) reduces (f9 + (h9 *' p9)) - f9, 0_ (n,L) by A1, Th45;
now__::_thesis:_(_(_f9_+_(h9_*'_p9)_<>_0__(n,L)_&_f,f_+_(h_*_p)_are_convertible_wrt_PolyRedRel_(P,T)_)_or_(_f9_+_(h9_*'_p9)_=_0__(n,L)_&_f,f_+_(h_*_p)_are_convertible_wrt_PolyRedRel_(P,T)_)_)
percases ( f9 + (h9 *' p9) <> 0_ (n,L) or f9 + (h9 *' p9) = 0_ (n,L) ) ;
case f9 + (h9 *' p9) <> 0_ (n,L) ; ::_thesis: f,f + (h * p) are_convertible_wrt PolyRedRel (P,T)
then reconsider g9 = f9 + (h9 *' p9) as non-zero Polynomial of n,L by POLYNOM7:def_1;
h9 *' p9 = h * p by POLYNOM1:def_10;
then g9 = f + (h * p) by POLYNOM1:def_10;
hence f,f + (h * p) are_convertible_wrt PolyRedRel (P,T) by A6, Th51, REWRITE1:31; ::_thesis: verum
end;
caseA7: f9 + (h9 *' p9) = 0_ (n,L) ; ::_thesis: f,f + (h * p) are_convertible_wrt PolyRedRel (P,T)
now__::_thesis:_not_-_h9_=_0__(n,L)
assume A8: - h9 = 0_ (n,L) ; ::_thesis: contradiction
A9: now__::_thesis:_for_u_being_set_st_u_in_dom_h9_holds_
h9_._u_=_(0__(n,L))_._u
let u be set ; ::_thesis: ( u in dom h9 implies h9 . u = (0_ (n,L)) . u )
assume u in dom h9 ; ::_thesis: h9 . u = (0_ (n,L)) . u
then reconsider u9 = u as bag of n ;
- (h9 . u9) = (- h9) . u9 by POLYNOM1:17
.= 0. L by A8, POLYNOM1:22 ;
then h9 . u9 = - (0. L) by RLVECT_1:17
.= 0. L by RLVECT_1:12
.= (0_ (n,L)) . u9 by POLYNOM1:22 ;
hence h9 . u = (0_ (n,L)) . u ; ::_thesis: verum
end;
dom h9 = Bags n by FUNCT_2:def_1
.= dom (0_ (n,L)) by FUNCT_2:def_1 ;
hence contradiction by A3, A9, FUNCT_1:2; ::_thesis: verum
end;
then reconsider mh9 = - h9 as non-zero Polynomial of n,L by POLYNOM7:def_1;
reconsider x = mh9 as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider x = x as Element of (Polynom-Ring (n,L)) ;
h + x = mh9 + h9 by POLYNOM1:def_10
.= 0_ (n,L) by Th3
.= 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10 ;
then - h = mh9 by RLVECT_1:6;
then A10: (- h) * p = mh9 *' p9 by POLYNOM1:def_10;
PolyRedRel (P,T) reduces mh9 *' p9, 0_ (n,L) by A1, Th45;
then A11: mh9 *' p9, 0_ (n,L) are_convertible_wrt PolyRedRel (P,T) by REWRITE1:25;
h9 *' p9 = h * p by POLYNOM1:def_10;
then A12: f + (h * p) = 0_ (n,L) by A7, POLYNOM1:def_10
.= 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10 ;
then f = - (h * p) by RLVECT_1:6
.= (- h) * p by VECTSP_1:9 ;
hence f,f + (h * p) are_convertible_wrt PolyRedRel (P,T) by A12, A11, A10, POLYNOM1:def_10; ::_thesis: verum
end;
end;
end;
hence f,f + (h * p) are_convertible_wrt PolyRedRel (P,T) ; ::_thesis: verum
end;
end;
end;
hence f,f + (h * p) are_convertible_wrt PolyRedRel (P,T) ; ::_thesis: verum
end;
theorem :: POLYRED:58
for n being Nat
for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being non empty Subset of (Polynom-Ring (n,L))
for f, g being Element of (Polynom-Ring (n,L)) st f,g are_congruent_mod P -Ideal holds
f,g are_convertible_wrt PolyRedRel (P,T)
proof
let n be Nat; ::_thesis: for T being connected admissible TermOrder of n
for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being non empty Subset of (Polynom-Ring (n,L))
for f, g being Element of (Polynom-Ring (n,L)) st f,g are_congruent_mod P -Ideal holds
f,g are_convertible_wrt PolyRedRel (P,T)
let T be connected admissible TermOrder of n; ::_thesis: for L being non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr
for P being non empty Subset of (Polynom-Ring (n,L))
for f, g being Element of (Polynom-Ring (n,L)) st f,g are_congruent_mod P -Ideal holds
f,g are_convertible_wrt PolyRedRel (P,T)
let L be non empty non degenerated right_complementable almost_left_invertible associative commutative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; ::_thesis: for P being non empty Subset of (Polynom-Ring (n,L))
for f, g being Element of (Polynom-Ring (n,L)) st f,g are_congruent_mod P -Ideal holds
f,g are_convertible_wrt PolyRedRel (P,T)
let P be non empty Subset of (Polynom-Ring (n,L)); ::_thesis: for f, g being Element of (Polynom-Ring (n,L)) st f,g are_congruent_mod P -Ideal holds
f,g are_convertible_wrt PolyRedRel (P,T)
let f, g be Element of (Polynom-Ring (n,L)); ::_thesis: ( f,g are_congruent_mod P -Ideal implies f,g are_convertible_wrt PolyRedRel (P,T) )
set PR = Polynom-Ring (n,L);
defpred S1[ Nat] means for f, g being Element of (Polynom-Ring (n,L))
for p being LeftLinearCombination of P st Sum p = g - f & len p = $1 holds
f,g are_convertible_wrt PolyRedRel (P,T);
now__::_thesis:_for_k_being_Nat_st_S1[k]_holds_
S1[k_+_1]
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A1: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_for_f,_g_being_Element_of_(Polynom-Ring_(n,L))
for_p_being_LeftLinearCombination_of_P_st_Sum_p_=_g_-_f_&_len_p_=_k_+_1_holds_
f,g_are_convertible_wrt_PolyRedRel_(P,T)
let f, g be Element of (Polynom-Ring (n,L)); ::_thesis: for p being LeftLinearCombination of P st Sum p = g - f & len p = k + 1 holds
f,g are_convertible_wrt PolyRedRel (P,T)
let p be LeftLinearCombination of P; ::_thesis: ( Sum p = g - f & len p = k + 1 implies f,g are_convertible_wrt PolyRedRel (P,T) )
assume that
A2: Sum p = g - f and
A3: len p = k + 1 ; ::_thesis: f,g are_convertible_wrt PolyRedRel (P,T)
now__::_thesis:_f,g_are_convertible_wrt_PolyRedRel_(P,T)
set h = f + (p /. (k + 1));
set q = p | (Seg k);
reconsider q = p | (Seg k) as FinSequence by FINSEQ_1:15;
dom p = Seg (k + 1) by A3, FINSEQ_1:def_3;
then consider u being Element of (Polynom-Ring (n,L)), a being Element of P such that
A4: p /. (k + 1) = u * a by FINSEQ_1:4, IDEAL_1:def_9;
reconsider u9 = u, a9 = a as Element of (Polynom-Ring (n,L)) ;
reconsider u9 = u9, a9 = a9 as Polynomial of n,L by POLYNOM1:def_10;
A5: p /. (k + 1) = u9 *' a9 by A4, POLYNOM1:def_10;
k <= k + 1 by NAT_1:11;
then A6: len q = k by A3, FINSEQ_1:17;
k in NAT by ORDINAL1:def_12;
then reconsider q = q as LeftLinearCombination of P by IDEAL_1:42;
A7: Sum p = (Sum q) + (p /. (k + 1)) by A3, Lm6;
then (Sum p) - (p /. (k + 1)) = ((Sum q) + (p /. (k + 1))) + (- (p /. (k + 1))) by RLVECT_1:def_11
.= (Sum q) + ((p /. (k + 1)) + (- (p /. (k + 1)))) by RLVECT_1:def_3
.= (Sum q) + (0. (Polynom-Ring (n,L))) by RLVECT_1:5
.= Sum q by RLVECT_1:4 ;
then Sum q = (g - f) + (- (p /. (k + 1))) by A2, RLVECT_1:def_11
.= (g + (- f)) + (- (p /. (k + 1))) by RLVECT_1:def_11
.= g + ((- f) + (- (p /. (k + 1)))) by RLVECT_1:def_3
.= g + (- (f + (p /. (k + 1)))) by RLVECT_1:31
.= g - (f + (p /. (k + 1))) by RLVECT_1:def_11 ;
then A8: f + (p /. (k + 1)),g are_convertible_wrt PolyRedRel (P,T) by A1, A6;
now__::_thesis:_(_(_a_<>_0__(n,L)_&_u_<>_0__(n,L)_&_f,g_are_convertible_wrt_PolyRedRel_(P,T)_)_or_(_(_a_=_0__(n,L)_or_u_=_0__(n,L)_)_&_f,g_are_convertible_wrt_PolyRedRel_(P,T)_)_)
percases ( ( a <> 0_ (n,L) & u <> 0_ (n,L) ) or a = 0_ (n,L) or u = 0_ (n,L) ) ;
case ( a <> 0_ (n,L) & u <> 0_ (n,L) ) ; ::_thesis: f,g are_convertible_wrt PolyRedRel (P,T)
then f,f + (p /. (k + 1)) are_convertible_wrt PolyRedRel (P,T) by A4, Lm20;
hence f,g are_convertible_wrt PolyRedRel (P,T) by A8, REWRITE1:30; ::_thesis: verum
end;
caseA9: ( a = 0_ (n,L) or u = 0_ (n,L) ) ; ::_thesis: f,g are_convertible_wrt PolyRedRel (P,T)
reconsider sumq = Sum q as Polynomial of n,L by POLYNOM1:def_10;
now__::_thesis:_(_(_a_=_0__(n,L)_&_p_/._(k_+_1)_=_0__(n,L)_)_or_(_u_=_0__(n,L)_&_p_/._(k_+_1)_=_0__(n,L)_)_)
percases ( a = 0_ (n,L) or u = 0_ (n,L) ) by A9;
case a = 0_ (n,L) ; ::_thesis: p /. (k + 1) = 0_ (n,L)
hence p /. (k + 1) = 0_ (n,L) by A5, POLYNOM1:28; ::_thesis: verum
end;
case u = 0_ (n,L) ; ::_thesis: p /. (k + 1) = 0_ (n,L)
hence p /. (k + 1) = 0_ (n,L) by A5, Th5; ::_thesis: verum
end;
end;
end;
then Sum p = sumq + (0_ (n,L)) by A7, POLYNOM1:def_10
.= Sum q by POLYNOM1:23 ;
hence f,g are_convertible_wrt PolyRedRel (P,T) by A1, A2, A6; ::_thesis: verum
end;
end;
end;
hence f,g are_convertible_wrt PolyRedRel (P,T) ; ::_thesis: verum
end;
hence f,g are_convertible_wrt PolyRedRel (P,T) ; ::_thesis: verum
end;
hence S1[k + 1] ; ::_thesis: verum
end;
then A10: for k being Nat st S1[k] holds
S1[k + 1] ;
A11: S1[ 0 ]
proof
let f, g be Element of (Polynom-Ring (n,L)); ::_thesis: for p being LeftLinearCombination of P st Sum p = g - f & len p = 0 holds
f,g are_convertible_wrt PolyRedRel (P,T)
let p be LeftLinearCombination of P; ::_thesis: ( Sum p = g - f & len p = 0 implies f,g are_convertible_wrt PolyRedRel (P,T) )
assume that
A12: Sum p = g - f and
A13: len p = 0 ; ::_thesis: f,g are_convertible_wrt PolyRedRel (P,T)
p = <*> the carrier of (Polynom-Ring (n,L)) by A13;
then Sum p = 0. (Polynom-Ring (n,L)) by RLVECT_1:43;
then f = g by A12, RLVECT_1:21;
hence f,g are_convertible_wrt PolyRedRel (P,T) by REWRITE1:26; ::_thesis: verum
end;
A14: for k being Nat holds S1[k] from NAT_1:sch_2(A11, A10);
assume f,g are_congruent_mod P -Ideal ; ::_thesis: f,g are_convertible_wrt PolyRedRel (P,T)
then g,f are_congruent_mod P -Ideal by Th53;
then g - f in P -Ideal by Def14;
then g - f in P -LeftIdeal by IDEAL_1:63;
then consider p being LeftLinearCombination of P such that
A15: Sum p = g - f by IDEAL_1:61;
ex k being Nat st len p = k ;
hence f,g are_convertible_wrt PolyRedRel (P,T) by A14, A15; ::_thesis: verum
end;
theorem Th59: :: POLYRED:59
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 f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f,g holds
f - g 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 f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f,g holds
f - g 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 f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f,g holds
f - g 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 f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f,g holds
f - g in P -Ideal
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: for f, g being Polynomial of n,L st PolyRedRel (P,T) reduces f,g holds
f - g in P -Ideal
let f, g be Polynomial of n,L; ::_thesis: ( PolyRedRel (P,T) reduces f,g implies f - g in P -Ideal )
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)) ;
set R = Polynom-Ring (n,L);
reconsider x = - g as Element of (Polynom-Ring (n,L)) by POLYNOM1:def_10;
reconsider x = x as Element of (Polynom-Ring (n,L)) ;
x + g9 = (- g) + g by POLYNOM1:def_10
.= 0_ (n,L) by Th3
.= 0. (Polynom-Ring (n,L)) by POLYNOM1:def_10 ;
then A1: - g9 = - g by RLVECT_1:6;
assume PolyRedRel (P,T) reduces f,g ; ::_thesis: f - g in P -Ideal
then f,g are_convertible_wrt PolyRedRel (P,T) by REWRITE1:25;
then A2: f9,g9 are_congruent_mod P -Ideal by Th57;
f - g = f + (- g) by POLYNOM1:def_6
.= f9 + (- g9) by A1, POLYNOM1:def_10
.= f9 - g9 by RLVECT_1:def_11 ;
hence f - g in P -Ideal by A2, Def14; ::_thesis: verum
end;
theorem :: POLYRED:60
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 f being Polynomial of n,L st PolyRedRel (P,T) reduces f, 0_ (n,L) holds
f 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 f being Polynomial of n,L st PolyRedRel (P,T) reduces f, 0_ (n,L) holds
f 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 f being Polynomial of n,L st PolyRedRel (P,T) reduces f, 0_ (n,L) holds
f 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 f being Polynomial of n,L st PolyRedRel (P,T) reduces f, 0_ (n,L) holds
f in P -Ideal
let P be Subset of (Polynom-Ring (n,L)); ::_thesis: for f being Polynomial of n,L st PolyRedRel (P,T) reduces f, 0_ (n,L) holds
f in P -Ideal
let f be Polynomial of n,L; ::_thesis: ( PolyRedRel (P,T) reduces f, 0_ (n,L) implies f in P -Ideal )
assume PolyRedRel (P,T) reduces f, 0_ (n,L) ; ::_thesis: f in P -Ideal
then f - (0_ (n,L)) in P -Ideal by Th59;
hence f in P -Ideal by Th4; ::_thesis: verum
end;