:: DICKSON semantic presentation
begin
theorem Th1: :: DICKSON:1
for g being Function
for x being set st dom g = {x} holds
g = x .--> (g . x)
proof
let g be Function; ::_thesis: for x being set st dom g = {x} holds
g = x .--> (g . x)
let x be set ; ::_thesis: ( dom g = {x} implies g = x .--> (g . x) )
assume A1: dom g = {x} ; ::_thesis: g = x .--> (g . x)
now__::_thesis:_for_a,_b_being_set_holds_
(_(_[a,b]_in_g_implies_[a,b]_in_x_.-->_(g_._x)_)_&_(_[a,b]_in_x_.-->_(g_._x)_implies_[a,b]_in_g_)_)
let a, b be set ; ::_thesis: ( ( [a,b] in g implies [a,b] in x .--> (g . x) ) & ( [a,b] in x .--> (g . x) implies [a,b] in g ) )
A2: dom (x .--> (g . x)) = {x} by FUNCOP_1:13;
hereby ::_thesis: ( [a,b] in x .--> (g . x) implies [a,b] in g )
assume A3: [a,b] in g ; ::_thesis: [a,b] in x .--> (g . x)
then A4: a in {x} by A1, FUNCT_1:1;
then A5: a = x by TARSKI:def_1;
then b = g . x by A3, FUNCT_1:1;
then (x .--> (g . x)) . a = b by A5, FUNCOP_1:72;
hence [a,b] in x .--> (g . x) by A2, A4, FUNCT_1:1; ::_thesis: verum
end;
assume A6: [a,b] in x .--> (g . x) ; ::_thesis: [a,b] in g
then A7: a in {x} by A2, FUNCT_1:1;
then A8: a = x by TARSKI:def_1;
b = (x .--> (g . x)) . a by A6, FUNCT_1:1
.= g . a by A8, FUNCOP_1:72 ;
hence [a,b] in g by A1, A7, FUNCT_1:1; ::_thesis: verum
end;
hence g = x .--> (g . x) by RELAT_1:def_2; ::_thesis: verum
end;
theorem Th2: :: DICKSON:2
for n being Nat holds n c= n + 1
proof
let n be Nat; ::_thesis: n c= n + 1
n + 1 = n \/ {n} by AFINSQ_1:2;
hence n c= n + 1 by XBOOLE_1:7; ::_thesis: verum
end;
scheme :: DICKSON:sch 1
FinSegRng2{ F1() -> Element of NAT , F2() -> Element of NAT , F3( set ) -> set , P1[ set ] } :
{ F3(i) where i is Element of NAT : ( F1() < i & i <= F2() & P1[i] ) } is finite
proof
set F1 = { F3(i) where i is Element of NAT : ( F1() <= i & i <= F2() & P1[i] ) } ;
set F2 = { F3(i) where i is Element of NAT : ( F1() < i & i <= F2() & P1[i] ) } ;
A1: { F3(i) where i is Element of NAT : ( F1() <= i & i <= F2() & P1[i] ) } is finite from FINSEQ_1:sch_6();
{ F3(i) where i is Element of NAT : ( F1() < i & i <= F2() & P1[i] ) } c= { F3(i) where i is Element of NAT : ( F1() <= i & i <= F2() & P1[i] ) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { F3(i) where i is Element of NAT : ( F1() < i & i <= F2() & P1[i] ) } or x in { F3(i) where i is Element of NAT : ( F1() <= i & i <= F2() & P1[i] ) } )
assume x in { F3(i) where i is Element of NAT : ( F1() < i & i <= F2() & P1[i] ) } ; ::_thesis: x in { F3(i) where i is Element of NAT : ( F1() <= i & i <= F2() & P1[i] ) }
then ex i being Element of NAT st
( F3(i) = x & F1() < i & i <= F2() & P1[i] ) ;
hence x in { F3(i) where i is Element of NAT : ( F1() <= i & i <= F2() & P1[i] ) } ; ::_thesis: verum
end;
hence { F3(i) where i is Element of NAT : ( F1() < i & i <= F2() & P1[i] ) } is finite by A1; ::_thesis: verum
end;
theorem Th3: :: DICKSON:3
for X being infinite set ex f being Function of NAT,X st f is one-to-one
proof
let X be infinite set ; ::_thesis: ex f being Function of NAT,X st f is one-to-one
card NAT c= card X by CARD_1:47, CARD_3:85;
then consider f being Function such that
A1: f is one-to-one and
A2: dom f = NAT and
A3: rng f c= X by CARD_1:10;
now__::_thesis:_for_x_being_set_st_x_in_NAT_holds_
f_._x_in_X
let x be set ; ::_thesis: ( x in NAT implies f . x in X )
assume x in NAT ; ::_thesis: f . x in X
then f . x in rng f by A2, FUNCT_1:3;
hence f . x in X by A3; ::_thesis: verum
end;
then reconsider f = f as Function of NAT,X by A2, FUNCT_2:3;
take f ; ::_thesis: f is one-to-one
thus f is one-to-one by A1; ::_thesis: verum
end;
definition
let R be RelStr ;
let f be sequence of R;
attrf is ascending means :: DICKSON:def 1
for n being Element of NAT holds
( f . (n + 1) <> f . n & [(f . n),(f . (n + 1))] in the InternalRel of R );
end;
:: deftheorem defines ascending DICKSON:def_1_:_
for R being RelStr
for f being sequence of R holds
( f is ascending iff for n being Element of NAT holds
( f . (n + 1) <> f . n & [(f . n),(f . (n + 1))] in the InternalRel of R ) );
definition
let R be RelStr ;
let f be sequence of R;
attrf is weakly-ascending means :Def2: :: DICKSON:def 2
for n being Element of NAT holds [(f . n),(f . (n + 1))] in the InternalRel of R;
end;
:: deftheorem Def2 defines weakly-ascending DICKSON:def_2_:_
for R being RelStr
for f being sequence of R holds
( f is weakly-ascending iff for n being Element of NAT holds [(f . n),(f . (n + 1))] in the InternalRel of R );
theorem Th4: :: DICKSON:4
for R being non empty transitive RelStr
for f being sequence of R st f is weakly-ascending holds
for i, j being Element of NAT st i < j holds
f . i <= f . j
proof
let R be non empty transitive RelStr ; ::_thesis: for f being sequence of R st f is weakly-ascending holds
for i, j being Element of NAT st i < j holds
f . i <= f . j
let f be sequence of R; ::_thesis: ( f is weakly-ascending implies for i, j being Element of NAT st i < j holds
f . i <= f . j )
assume A1: f is weakly-ascending ; ::_thesis: for i, j being Element of NAT st i < j holds
f . i <= f . j
let i be Element of NAT ; ::_thesis: for j being Element of NAT st i < j holds
f . i <= f . j
defpred S1[ Element of NAT ] means ( i < $1 implies f . i <= f . $1 );
A2: S1[ 0 ] by NAT_1:2;
A3: for j being Element of NAT st S1[j] holds
S1[j + 1]
proof
let j be Element of NAT ; ::_thesis: ( S1[j] implies S1[j + 1] )
assume that
A4: S1[j] and
A5: i < j + 1 ; ::_thesis: f . i <= f . (j + 1)
reconsider fj1 = f . (j + 1) as Element of R ;
A6: [(f . j),(f . (j + 1))] in the InternalRel of R by A1, Def2;
then A7: f . j <= fj1 by ORDERS_2:def_5;
A8: i <= j by A5, NAT_1:13;
percases ( i < j or i = j ) by A8, XXREAL_0:1;
suppose i < j ; ::_thesis: f . i <= f . (j + 1)
hence f . i <= f . (j + 1) by A4, A7, ORDERS_2:3; ::_thesis: verum
end;
suppose i = j ; ::_thesis: f . i <= f . (j + 1)
hence f . i <= f . (j + 1) by A6, ORDERS_2:def_5; ::_thesis: verum
end;
end;
end;
thus for j being Element of NAT holds S1[j] from NAT_1:sch_1(A2, A3); ::_thesis: verum
end;
theorem Th5: :: DICKSON:5
for R being non empty RelStr holds
( R is connected iff the InternalRel of R is_strongly_connected_in the carrier of R )
proof
let R be non empty RelStr ; ::_thesis: ( R is connected iff the InternalRel of R is_strongly_connected_in the carrier of R )
set IR = the InternalRel of R;
set CR = the carrier of R;
hereby ::_thesis: ( the InternalRel of R is_strongly_connected_in the carrier of R implies R is connected )
assume A1: R is connected ; ::_thesis: the InternalRel of R is_strongly_connected_in the carrier of R
now__::_thesis:_for_x,_y_being_set_st_x_in_the_carrier_of_R_&_y_in_the_carrier_of_R_&_not_[x,y]_in_the_InternalRel_of_R_holds_
[y,x]_in_the_InternalRel_of_R
let x, y be set ; ::_thesis: ( x in the carrier of R & y in the carrier of R & not [x,y] in the InternalRel of R implies [y,x] in the InternalRel of R )
assume that
A2: x in the carrier of R and
A3: y in the carrier of R ; ::_thesis: ( [x,y] in the InternalRel of R or [y,x] in the InternalRel of R )
reconsider x9 = x, y9 = y as Element of R by A2, A3;
( x9 <= y9 or y9 <= x9 ) by A1, WAYBEL_0:def_29;
hence ( [x,y] in the InternalRel of R or [y,x] in the InternalRel of R ) by ORDERS_2:def_5; ::_thesis: verum
end;
hence the InternalRel of R is_strongly_connected_in the carrier of R by RELAT_2:def_7; ::_thesis: verum
end;
assume A4: the InternalRel of R is_strongly_connected_in the carrier of R ; ::_thesis: R is connected
now__::_thesis:_for_x,_y_being_Element_of_R_holds_
(_x_<=_y_or_y_<=_x_)
let x, y be Element of R; ::_thesis: ( x <= y or y <= x )
( [x,y] in the InternalRel of R or [y,x] in the InternalRel of R ) by A4, RELAT_2:def_7;
hence ( x <= y or y <= x ) by ORDERS_2:def_5; ::_thesis: verum
end;
hence R is connected by WAYBEL_0:def_29; ::_thesis: verum
end;
theorem Th6: :: DICKSON:6
for L being RelStr
for Y, a being set holds
( ( the InternalRel of L -Seg a misses Y & a in Y ) iff a is_minimal_wrt Y, the InternalRel of L )
proof
let L be RelStr ; ::_thesis: for Y, a being set holds
( ( the InternalRel of L -Seg a misses Y & a in Y ) iff a is_minimal_wrt Y, the InternalRel of L )
let Y, a be set ; ::_thesis: ( ( the InternalRel of L -Seg a misses Y & a in Y ) iff a is_minimal_wrt Y, the InternalRel of L )
set IR = the InternalRel of L;
hereby ::_thesis: ( a is_minimal_wrt Y, the InternalRel of L implies ( the InternalRel of L -Seg a misses Y & a in Y ) )
assume that
A1: the InternalRel of L -Seg a misses Y and
A2: a in Y ; ::_thesis: a is_minimal_wrt Y, the InternalRel of L
A3: ( the InternalRel of L -Seg a) /\ Y = {} by A1, XBOOLE_0:def_7;
now__::_thesis:_for_y_being_set_holds_
(_not_y_in_Y_or_not_y_<>_a_or_not_[y,a]_in_the_InternalRel_of_L_)
assume ex y being set st
( y in Y & y <> a & [y,a] in the InternalRel of L ) ; ::_thesis: contradiction
then consider y being set such that
A4: y in Y and
A5: y <> a and
A6: [y,a] in the InternalRel of L ;
y in the InternalRel of L -Seg a by A5, A6, WELLORD1:1;
hence contradiction by A3, A4, XBOOLE_0:def_4; ::_thesis: verum
end;
hence a is_minimal_wrt Y, the InternalRel of L by A2, WAYBEL_4:def_25; ::_thesis: verum
end;
assume A7: a is_minimal_wrt Y, the InternalRel of L ; ::_thesis: ( the InternalRel of L -Seg a misses Y & a in Y )
now__::_thesis:_the_InternalRel_of_L_-Seg_a_misses_Y
assume not the InternalRel of L -Seg a misses Y ; ::_thesis: contradiction
then ( the InternalRel of L -Seg a) /\ Y <> {} by XBOOLE_0:def_7;
then consider y being set such that
A8: y in ( the InternalRel of L -Seg a) /\ Y by XBOOLE_0:def_1;
A9: y in the InternalRel of L -Seg a by A8, XBOOLE_0:def_4;
A10: y in Y by A8, XBOOLE_0:def_4;
A11: y <> a by A9, WELLORD1:1;
[y,a] in the InternalRel of L by A9, WELLORD1:1;
hence contradiction by A7, A10, A11, WAYBEL_4:def_25; ::_thesis: verum
end;
hence the InternalRel of L -Seg a misses Y ; ::_thesis: a in Y
thus a in Y by A7, WAYBEL_4:def_25; ::_thesis: verum
end;
theorem Th7: :: DICKSON:7
for L being non empty transitive antisymmetric RelStr
for x being Element of L
for a, N being set st a is_minimal_wrt ( the InternalRel of L -Seg x) /\ N, the InternalRel of L holds
a is_minimal_wrt N, the InternalRel of L
proof
let L be non empty transitive antisymmetric RelStr ; ::_thesis: for x being Element of L
for a, N being set st a is_minimal_wrt ( the InternalRel of L -Seg x) /\ N, the InternalRel of L holds
a is_minimal_wrt N, the InternalRel of L
let x be Element of L; ::_thesis: for a, N being set st a is_minimal_wrt ( the InternalRel of L -Seg x) /\ N, the InternalRel of L holds
a is_minimal_wrt N, the InternalRel of L
let a, N be set ; ::_thesis: ( a is_minimal_wrt ( the InternalRel of L -Seg x) /\ N, the InternalRel of L implies a is_minimal_wrt N, the InternalRel of L )
assume A1: a is_minimal_wrt ( the InternalRel of L -Seg x) /\ N, the InternalRel of L ; ::_thesis: a is_minimal_wrt N, the InternalRel of L
set IR = the InternalRel of L;
set CR = the carrier of L;
A2: the InternalRel of L is_transitive_in the carrier of L by ORDERS_2:def_3;
now__::_thesis:_(_a_in_N_&_(_for_y_being_set_holds_
(_not_y_in_N_or_not_y_<>_a_or_not_[y,a]_in_the_InternalRel_of_L_)_)_)
A3: a in ( the InternalRel of L -Seg x) /\ N by A1, WAYBEL_4:def_25;
then A4: a in the InternalRel of L -Seg x by XBOOLE_0:def_4;
then A5: a <> x by WELLORD1:1;
A6: [a,x] in the InternalRel of L by A4, WELLORD1:1;
then reconsider a9 = a as Element of L by ZFMISC_1:87;
thus a in N by A3, XBOOLE_0:def_4; ::_thesis: for y being set holds
( not y in N or not y <> a or not [y,a] in the InternalRel of L )
now__::_thesis:_for_y_being_set_holds_
(_not_y_in_N_or_not_y_<>_a_or_not_[y,a]_in_the_InternalRel_of_L_)
assume ex y being set st
( y in N & y <> a & [y,a] in the InternalRel of L ) ; ::_thesis: contradiction
then consider y being set such that
A7: y in N and
A8: y <> a and
A9: [y,a] in the InternalRel of L ;
A10: a in the carrier of L by A9, ZFMISC_1:87;
y in the carrier of L by A9, ZFMISC_1:87;
then A11: [y,x] in the InternalRel of L by A2, A6, A9, A10, RELAT_2:def_8;
percases ( x = y or x <> y ) ;
suppose x = y ; ::_thesis: contradiction
then A12: x <= a9 by A9, ORDERS_2:def_5;
a9 <= x by A6, ORDERS_2:def_5;
hence contradiction by A5, A12, ORDERS_2:2; ::_thesis: verum
end;
suppose x <> y ; ::_thesis: contradiction
then y in the InternalRel of L -Seg x by A11, WELLORD1:1;
then y in ( the InternalRel of L -Seg x) /\ N by A7, XBOOLE_0:def_4;
hence contradiction by A1, A8, A9, WAYBEL_4:def_25; ::_thesis: verum
end;
end;
end;
hence for y being set holds
( not y in N or not y <> a or not [y,a] in the InternalRel of L ) ; ::_thesis: verum
end;
hence a is_minimal_wrt N, the InternalRel of L by WAYBEL_4:def_25; ::_thesis: verum
end;
begin
definition
let R be RelStr ;
attrR is quasi_ordered means :Def3: :: DICKSON:def 3
( R is reflexive & R is transitive );
end;
:: deftheorem Def3 defines quasi_ordered DICKSON:def_3_:_
for R being RelStr holds
( R is quasi_ordered iff ( R is reflexive & R is transitive ) );
definition
let R be RelStr ;
assume B1: R is quasi_ordered ;
func EqRel R -> Equivalence_Relation of the carrier of R equals :Def4: :: DICKSON:def 4
the InternalRel of R /\ ( the InternalRel of R ~);
coherence
the InternalRel of R /\ ( the InternalRel of R ~) is Equivalence_Relation of the carrier of R
proof
set IR = the InternalRel of R;
set CR = the carrier of R;
R is reflexive by B1, Def3;
then A1: the InternalRel of R is_reflexive_in the carrier of R by ORDERS_2:def_2;
R is transitive by B1, Def3;
then A2: the InternalRel of R is_transitive_in the carrier of R by ORDERS_2:def_3;
then A3: the InternalRel of R quasi_orders the carrier of R by A1, ORDERS_1:def_6;
A4: the InternalRel of R /\ ( the InternalRel of R ~) is_reflexive_in the carrier of R
proof
let x be set ; :: according to RELAT_2:def_1 ::_thesis: ( not x in the carrier of R or [x,x] in the InternalRel of R /\ ( the InternalRel of R ~) )
assume x in the carrier of R ; ::_thesis: [x,x] in the InternalRel of R /\ ( the InternalRel of R ~)
then A5: [x,x] in the InternalRel of R by A1, RELAT_2:def_1;
then [x,x] in the InternalRel of R ~ by RELAT_1:def_7;
hence [x,x] in the InternalRel of R /\ ( the InternalRel of R ~) by A5, XBOOLE_0:def_4; ::_thesis: verum
end;
then A6: dom ( the InternalRel of R /\ ( the InternalRel of R ~)) = the carrier of R by ORDERS_1:13;
A7: field ( the InternalRel of R /\ ( the InternalRel of R ~)) = the carrier of R by A4, ORDERS_1:13;
A8: the InternalRel of R /\ ( the InternalRel of R ~) is_symmetric_in the carrier of R
proof
let x, y be set ; :: according to RELAT_2:def_3 ::_thesis: ( not x in the carrier of R or not y in the carrier of R or not [x,y] in the InternalRel of R /\ ( the InternalRel of R ~) or [y,x] in the InternalRel of R /\ ( the InternalRel of R ~) )
assume that
x in the carrier of R and
y in the carrier of R and
A9: [x,y] in the InternalRel of R /\ ( the InternalRel of R ~) ; ::_thesis: [y,x] in the InternalRel of R /\ ( the InternalRel of R ~)
A10: [x,y] in the InternalRel of R by A9, XBOOLE_0:def_4;
A11: [x,y] in the InternalRel of R ~ by A9, XBOOLE_0:def_4;
A12: [y,x] in the InternalRel of R ~ by A10, RELAT_1:def_7;
[y,x] in the InternalRel of R by A11, RELAT_1:def_7;
hence [y,x] in the InternalRel of R /\ ( the InternalRel of R ~) by A12, XBOOLE_0:def_4; ::_thesis: verum
end;
the InternalRel of R /\ ( the InternalRel of R ~) is_transitive_in the carrier of R
proof
let x, y, z be set ; :: according to RELAT_2:def_8 ::_thesis: ( not x in the carrier of R or not y in the carrier of R or not z in the carrier of R or not [x,y] in the InternalRel of R /\ ( the InternalRel of R ~) or not [y,z] in the InternalRel of R /\ ( the InternalRel of R ~) or [x,z] in the InternalRel of R /\ ( the InternalRel of R ~) )
assume that
A13: x in the carrier of R and
A14: y in the carrier of R and
A15: z in the carrier of R and
A16: [x,y] in the InternalRel of R /\ ( the InternalRel of R ~) and
A17: [y,z] in the InternalRel of R /\ ( the InternalRel of R ~) ; ::_thesis: [x,z] in the InternalRel of R /\ ( the InternalRel of R ~)
A18: [x,y] in the InternalRel of R by A16, XBOOLE_0:def_4;
A19: [x,y] in the InternalRel of R ~ by A16, XBOOLE_0:def_4;
A20: [y,z] in the InternalRel of R by A17, XBOOLE_0:def_4;
A21: [y,z] in the InternalRel of R ~ by A17, XBOOLE_0:def_4;
A22: [x,z] in the InternalRel of R by A2, A13, A14, A15, A18, A20, RELAT_2:def_8;
the InternalRel of R ~ quasi_orders the carrier of R by A3, ORDERS_1:40;
then the InternalRel of R ~ is_transitive_in the carrier of R by ORDERS_1:def_6;
then [x,z] in the InternalRel of R ~ by A13, A14, A15, A19, A21, RELAT_2:def_8;
hence [x,z] in the InternalRel of R /\ ( the InternalRel of R ~) by A22, XBOOLE_0:def_4; ::_thesis: verum
end;
hence the InternalRel of R /\ ( the InternalRel of R ~) is Equivalence_Relation of the carrier of R by A6, A7, A8, PARTFUN1:def_2, RELAT_2:def_11, RELAT_2:def_16; ::_thesis: verum
end;
end;
:: deftheorem Def4 defines EqRel DICKSON:def_4_:_
for R being RelStr st R is quasi_ordered holds
EqRel R = the InternalRel of R /\ ( the InternalRel of R ~);
theorem Th8: :: DICKSON:8
for R being RelStr
for x, y being Element of R st R is quasi_ordered holds
( x in Class ((EqRel R),y) iff ( x <= y & y <= x ) )
proof
let R be RelStr ; ::_thesis: for x, y being Element of R st R is quasi_ordered holds
( x in Class ((EqRel R),y) iff ( x <= y & y <= x ) )
let x, y be Element of R; ::_thesis: ( R is quasi_ordered implies ( x in Class ((EqRel R),y) iff ( x <= y & y <= x ) ) )
assume A1: R is quasi_ordered ; ::_thesis: ( x in Class ((EqRel R),y) iff ( x <= y & y <= x ) )
set IR = the InternalRel of R;
hereby ::_thesis: ( x <= y & y <= x implies x in Class ((EqRel R),y) )
assume x in Class ((EqRel R),y) ; ::_thesis: ( x <= y & y <= x )
then [x,y] in EqRel R by EQREL_1:19;
then A2: [x,y] in the InternalRel of R /\ ( the InternalRel of R ~) by A1, Def4;
then A3: [x,y] in the InternalRel of R by XBOOLE_0:def_4;
A4: [x,y] in the InternalRel of R ~ by A2, XBOOLE_0:def_4;
thus x <= y by A3, ORDERS_2:def_5; ::_thesis: y <= x
[y,x] in the InternalRel of R by A4, RELAT_1:def_7;
hence y <= x by ORDERS_2:def_5; ::_thesis: verum
end;
assume that
A5: x <= y and
A6: y <= x ; ::_thesis: x in Class ((EqRel R),y)
A7: [y,x] in the InternalRel of R by A6, ORDERS_2:def_5;
A8: [x,y] in the InternalRel of R by A5, ORDERS_2:def_5;
[x,y] in the InternalRel of R ~ by A7, RELAT_1:def_7;
then [x,y] in the InternalRel of R /\ ( the InternalRel of R ~) by A8, XBOOLE_0:def_4;
then [x,y] in EqRel R by A1, Def4;
hence x in Class ((EqRel R),y) by EQREL_1:19; ::_thesis: verum
end;
definition
let R be RelStr ;
func <=E R -> Relation of (Class (EqRel R)) means :Def5: :: DICKSON:def 5
for A, B being set holds
( [A,B] in it iff ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) );
existence
ex b1 being Relation of (Class (EqRel R)) st
for A, B being set holds
( [A,B] in b1 iff ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) )
proof
set IR = the InternalRel of R;
set CR = the carrier of R;
percases ( the carrier of R = {} or not the carrier of R is empty ) ;
supposeA1: the carrier of R = {} ; ::_thesis: ex b1 being Relation of (Class (EqRel R)) st
for A, B being set holds
( [A,B] in b1 iff ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) )
reconsider IT = {} as Relation of (Class (EqRel R)) by RELSET_1:12;
take IT ; ::_thesis: for A, B being set holds
( [A,B] in IT iff ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) )
let A, B be set ; ::_thesis: ( [A,B] in IT iff ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) )
thus ( [A,B] in IT implies ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) ) ; ::_thesis: ( ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) implies [A,B] in IT )
given a, b being Element of R such that A = Class ((EqRel R),a) and
B = Class ((EqRel R),b) and
A2: a <= b ; ::_thesis: [A,B] in IT
the InternalRel of R = {} by A1;
hence [A,B] in IT by A2, ORDERS_2:def_5; ::_thesis: verum
end;
suppose not the carrier of R is empty ; ::_thesis: ex b1 being Relation of (Class (EqRel R)) st
for A, B being set holds
( [A,B] in b1 iff ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) )
then reconsider R9 = R as non empty RelStr ;
set IT = { [(Class ((EqRel R),a)),(Class ((EqRel R),b))] where a, b is Element of R9 : a <= b } ;
set Y = Class (EqRel R);
{ [(Class ((EqRel R),a)),(Class ((EqRel R),b))] where a, b is Element of R9 : a <= b } c= [:(Class (EqRel R)),(Class (EqRel R)):]
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { [(Class ((EqRel R),a)),(Class ((EqRel R),b))] where a, b is Element of R9 : a <= b } or x in [:(Class (EqRel R)),(Class (EqRel R)):] )
assume x in { [(Class ((EqRel R),a)),(Class ((EqRel R),b))] where a, b is Element of R9 : a <= b } ; ::_thesis: x in [:(Class (EqRel R)),(Class (EqRel R)):]
then consider a, b being Element of R9 such that
A3: x = [(Class ((EqRel R),a)),(Class ((EqRel R),b))] and
a <= b ;
A4: Class ((EqRel R),a) in Class (EqRel R) by EQREL_1:def_3;
Class ((EqRel R),b) in Class (EqRel R) by EQREL_1:def_3;
hence x in [:(Class (EqRel R)),(Class (EqRel R)):] by A3, A4, ZFMISC_1:def_2; ::_thesis: verum
end;
then reconsider IT = { [(Class ((EqRel R),a)),(Class ((EqRel R),b))] where a, b is Element of R9 : a <= b } as Relation of (Class (EqRel R)) ;
take IT ; ::_thesis: for A, B being set holds
( [A,B] in IT iff ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) )
let A, B be set ; ::_thesis: ( [A,B] in IT iff ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) )
hereby ::_thesis: ( ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) implies [A,B] in IT )
assume [A,B] in IT ; ::_thesis: ex a9, b9 being Element of R st
( A = Class ((EqRel R),a9) & B = Class ((EqRel R),b9) & a9 <= b9 )
then consider a, b being Element of R such that
A5: [A,B] = [(Class ((EqRel R),a)),(Class ((EqRel R),b))] and
A6: a <= b ;
reconsider a9 = a, b9 = b as Element of R ;
take a9 = a9; ::_thesis: ex b9 being Element of R st
( A = Class ((EqRel R),a9) & B = Class ((EqRel R),b9) & a9 <= b9 )
take b9 = b9; ::_thesis: ( A = Class ((EqRel R),a9) & B = Class ((EqRel R),b9) & a9 <= b9 )
thus ( A = Class ((EqRel R),a9) & B = Class ((EqRel R),b9) & a9 <= b9 ) by A5, A6, XTUPLE_0:1; ::_thesis: verum
end;
given a, b being Element of R such that A7: A = Class ((EqRel R),a) and
A8: B = Class ((EqRel R),b) and
A9: a <= b ; ::_thesis: [A,B] in IT
thus [A,B] in IT by A7, A8, A9; ::_thesis: verum
end;
end;
end;
uniqueness
for b1, b2 being Relation of (Class (EqRel R)) st ( for A, B being set holds
( [A,B] in b1 iff ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) ) ) & ( for A, B being set holds
( [A,B] in b2 iff ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) ) ) holds
b1 = b2
proof
let IT1, IT2 be Relation of (Class (EqRel R)); ::_thesis: ( ( for A, B being set holds
( [A,B] in IT1 iff ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) ) ) & ( for A, B being set holds
( [A,B] in IT2 iff ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) ) ) implies IT1 = IT2 )
assume that
A10: for A, B being set holds
( [A,B] in IT1 iff ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) ) and
A11: for A, B being set holds
( [A,B] in IT2 iff ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) ) ; ::_thesis: IT1 = IT2
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_IT1_implies_x_in_IT2_)_&_(_x_in_IT2_implies_x_in_IT1_)_)
let x be set ; ::_thesis: ( ( x in IT1 implies x in IT2 ) & ( x in IT2 implies x in IT1 ) )
hereby ::_thesis: ( x in IT2 implies x in IT1 )
assume A12: x in IT1 ; ::_thesis: x in IT2
set Y = Class (EqRel R);
consider A, B being set such that
A in Class (EqRel R) and
B in Class (EqRel R) and
A13: x = [A,B] by A12, ZFMISC_1:def_2;
ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) by A10, A12, A13;
hence x in IT2 by A11, A13; ::_thesis: verum
end;
assume A14: x in IT2 ; ::_thesis: x in IT1
set Y = Class (EqRel R);
consider A, B being set such that
A in Class (EqRel R) and
B in Class (EqRel R) and
A15: x = [A,B] by A14, ZFMISC_1:def_2;
ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) by A11, A14, A15;
hence x in IT1 by A10, A15; ::_thesis: verum
end;
hence IT1 = IT2 by TARSKI:1; ::_thesis: verum
end;
end;
:: deftheorem Def5 defines <=E DICKSON:def_5_:_
for R being RelStr
for b2 being Relation of (Class (EqRel R)) holds
( b2 = <=E R iff for A, B being set holds
( [A,B] in b2 iff ex a, b being Element of R st
( A = Class ((EqRel R),a) & B = Class ((EqRel R),b) & a <= b ) ) );
theorem Th9: :: DICKSON:9
for R being RelStr st R is quasi_ordered holds
<=E R partially_orders Class (EqRel R)
proof
let R be RelStr ; ::_thesis: ( R is quasi_ordered implies <=E R partially_orders Class (EqRel R) )
set CR = the carrier of R;
set IR = the InternalRel of R;
assume A1: R is quasi_ordered ; ::_thesis: <=E R partially_orders Class (EqRel R)
then R is transitive by Def3;
then A2: the InternalRel of R is_transitive_in the carrier of R by ORDERS_2:def_3;
thus <=E R is_reflexive_in Class (EqRel R) :: according to ORDERS_1:def_7 ::_thesis: ( <=E R is_transitive_in Class (EqRel R) & <=E R is_antisymmetric_in Class (EqRel R) )
proof
let x be set ; :: according to RELAT_2:def_1 ::_thesis: ( not x in Class (EqRel R) or [x,x] in <=E R )
assume x in Class (EqRel R) ; ::_thesis: [x,x] in <=E R
then consider a being set such that
A3: a in the carrier of R and
A4: x = Class ((EqRel R),a) by EQREL_1:def_3;
R is reflexive by A1, Def3;
then the InternalRel of R is_reflexive_in the carrier of R by ORDERS_2:def_2;
then A5: [a,a] in the InternalRel of R by A3, RELAT_2:def_1;
reconsider a9 = a as Element of R by A3;
a9 <= a9 by A5, ORDERS_2:def_5;
hence [x,x] in <=E R by A4, Def5; ::_thesis: verum
end;
thus <=E R is_transitive_in Class (EqRel R) ::_thesis: <=E R is_antisymmetric_in Class (EqRel R)
proof
let x, y, z be set ; :: according to RELAT_2:def_8 ::_thesis: ( not x in Class (EqRel R) or not y in Class (EqRel R) or not z in Class (EqRel R) or not [x,y] in <=E R or not [y,z] in <=E R or [x,z] in <=E R )
assume that
A6: x in Class (EqRel R) and
y in Class (EqRel R) and
z in Class (EqRel R) and
A7: [x,y] in <=E R and
A8: [y,z] in <=E R ; ::_thesis: [x,z] in <=E R
consider a, b being Element of R such that
A9: x = Class ((EqRel R),a) and
A10: y = Class ((EqRel R),b) and
A11: a <= b by A7, Def5;
consider c, d being Element of R such that
A12: y = Class ((EqRel R),c) and
A13: z = Class ((EqRel R),d) and
A14: c <= d by A8, Def5;
A15: [a,b] in the InternalRel of R by A11, ORDERS_2:def_5;
A16: [c,d] in the InternalRel of R by A14, ORDERS_2:def_5;
A17: ex x1 being set st
( x1 in the carrier of R & x = Class ((EqRel R),x1) ) by A6, EQREL_1:def_3;
then b in Class ((EqRel R),c) by A10, A12, EQREL_1:23;
then [b,c] in EqRel R by EQREL_1:19;
then [b,c] in the InternalRel of R /\ ( the InternalRel of R ~) by A1, Def4;
then [b,c] in the InternalRel of R by XBOOLE_0:def_4;
then [a,c] in the InternalRel of R by A2, A15, A17, RELAT_2:def_8;
then [a,d] in the InternalRel of R by A2, A16, A17, RELAT_2:def_8;
then a <= d by ORDERS_2:def_5;
hence [x,z] in <=E R by A9, A13, Def5; ::_thesis: verum
end;
thus <=E R is_antisymmetric_in Class (EqRel R) ::_thesis: verum
proof
let x, y be set ; :: according to RELAT_2:def_4 ::_thesis: ( not x in Class (EqRel R) or not y in Class (EqRel R) or not [x,y] in <=E R or not [y,x] in <=E R or x = y )
assume that
A18: x in Class (EqRel R) and
y in Class (EqRel R) and
A19: [x,y] in <=E R and
A20: [y,x] in <=E R ; ::_thesis: x = y
consider a, b being Element of R such that
A21: x = Class ((EqRel R),a) and
A22: y = Class ((EqRel R),b) and
A23: a <= b by A19, Def5;
consider c, d being Element of R such that
A24: y = Class ((EqRel R),c) and
A25: x = Class ((EqRel R),d) and
A26: c <= d by A20, Def5;
A27: [a,b] in the InternalRel of R by A23, ORDERS_2:def_5;
A28: [c,d] in the InternalRel of R by A26, ORDERS_2:def_5;
A29: ex x1 being set st
( x1 in the carrier of R & x = Class ((EqRel R),x1) ) by A18, EQREL_1:def_3;
then A30: d in Class ((EqRel R),a) by A21, A25, EQREL_1:23;
a in Class ((EqRel R),a) by A29, EQREL_1:20;
then A31: [a,d] in EqRel R by A30, EQREL_1:22;
A32: c in Class ((EqRel R),b) by A22, A24, A29, EQREL_1:23;
b in Class ((EqRel R),b) by A29, EQREL_1:20;
then A33: [b,c] in EqRel R by A32, EQREL_1:22;
[a,d] in the InternalRel of R /\ ( the InternalRel of R ~) by A1, A31, Def4;
then [a,d] in the InternalRel of R ~ by XBOOLE_0:def_4;
then A34: [d,a] in the InternalRel of R by RELAT_1:def_7;
[b,c] in the InternalRel of R /\ ( the InternalRel of R ~) by A1, A33, Def4;
then [b,c] in the InternalRel of R by XBOOLE_0:def_4;
then [b,d] in the InternalRel of R by A2, A28, A29, RELAT_2:def_8;
then A35: [b,a] in the InternalRel of R by A2, A29, A34, RELAT_2:def_8;
[b,a] in the InternalRel of R ~ by A27, RELAT_1:def_7;
then [b,a] in the InternalRel of R /\ ( the InternalRel of R ~) by A35, XBOOLE_0:def_4;
then [b,a] in EqRel R by A1, Def4;
then b in Class ((EqRel R),a) by EQREL_1:19;
hence x = y by A21, A22, EQREL_1:23; ::_thesis: verum
end;
end;
theorem :: DICKSON:10
for R being non empty RelStr st R is quasi_ordered & R is connected holds
<=E R linearly_orders Class (EqRel R)
proof
let R be non empty RelStr ; ::_thesis: ( R is quasi_ordered & R is connected implies <=E R linearly_orders Class (EqRel R) )
assume that
A1: R is quasi_ordered and
A2: R is connected ; ::_thesis: <=E R linearly_orders Class (EqRel R)
A3: <=E R partially_orders Class (EqRel R) by A1, Th9;
hence <=E R is_reflexive_in Class (EqRel R) by ORDERS_1:def_7; :: according to ORDERS_1:def_8 ::_thesis: ( <=E R is_transitive_in Class (EqRel R) & <=E R is_antisymmetric_in Class (EqRel R) & <=E R is_connected_in Class (EqRel R) )
thus <=E R is_transitive_in Class (EqRel R) by A3, ORDERS_1:def_7; ::_thesis: ( <=E R is_antisymmetric_in Class (EqRel R) & <=E R is_connected_in Class (EqRel R) )
thus <=E R is_antisymmetric_in Class (EqRel R) by A3, ORDERS_1:def_7; ::_thesis: <=E R is_connected_in Class (EqRel R)
thus <=E R is_connected_in Class (EqRel R) ::_thesis: verum
proof
set CR = the carrier of R;
let x, y be set ; :: according to RELAT_2:def_6 ::_thesis: ( not x in Class (EqRel R) or not y in Class (EqRel R) or x = y or [x,y] in <=E R or [y,x] in <=E R )
assume that
A4: x in Class (EqRel R) and
A5: y in Class (EqRel R) and
x <> y and
A6: not [x,y] in <=E R ; ::_thesis: [y,x] in <=E R
consider a being set such that
A7: a in the carrier of R and
A8: x = Class ((EqRel R),a) by A4, EQREL_1:def_3;
consider b being set such that
A9: b in the carrier of R and
A10: y = Class ((EqRel R),b) by A5, EQREL_1:def_3;
reconsider a9 = a, b9 = b as Element of R by A7, A9;
not a9 <= b9 by A6, A8, A10, Def5;
then b9 <= a9 by A2, WAYBEL_0:def_29;
hence [y,x] in <=E R by A8, A10, Def5; ::_thesis: verum
end;
end;
definition
let R be Relation;
funcR \~ -> Relation equals :: DICKSON:def 6
R \ (R ~);
correctness
coherence
R \ (R ~) is Relation;
;
end;
:: deftheorem defines \~ DICKSON:def_6_:_
for R being Relation holds R \~ = R \ (R ~);
registration
let R be Relation;
clusterR \~ -> asymmetric ;
coherence
R \~ is asymmetric
proof
now__::_thesis:_for_x,_y_being_set_st_x_in_field_(R_\~)_&_y_in_field_(R_\~)_&_[x,y]_in_R_\~_holds_
not_[y,x]_in_R_\~
let x, y be set ; ::_thesis: ( x in field (R \~) & y in field (R \~) & [x,y] in R \~ implies not [y,x] in R \~ )
assume that
x in field (R \~) and
y in field (R \~) and
A1: [x,y] in R \~ ; ::_thesis: not [y,x] in R \~
not [x,y] in R ~ by A1, XBOOLE_0:def_5;
hence not [y,x] in R \~ by RELAT_1:def_7; ::_thesis: verum
end;
then R \~ is_asymmetric_in field (R \~) by RELAT_2:def_5;
hence R \~ is asymmetric by RELAT_2:def_13; ::_thesis: verum
end;
end;
definition
let X be set ;
let R be Relation of X;
:: original: \~
redefine funcR \~ -> Relation of X;
coherence
R \~ is Relation of X
proof
R \~ = R \ (R ~) ;
hence R \~ is Relation of X ; ::_thesis: verum
end;
end;
definition
let R be RelStr ;
funcR \~ -> strict RelStr equals :: DICKSON:def 7
RelStr(# the carrier of R,( the InternalRel of R \~) #);
correctness
coherence
RelStr(# the carrier of R,( the InternalRel of R \~) #) is strict RelStr ;
;
end;
:: deftheorem defines \~ DICKSON:def_7_:_
for R being RelStr holds R \~ = RelStr(# the carrier of R,( the InternalRel of R \~) #);
registration
let R be non empty RelStr ;
clusterR \~ -> non empty strict ;
coherence
not R \~ is empty ;
end;
registration
let R be transitive RelStr ;
clusterR \~ -> strict transitive ;
coherence
R \~ is transitive
proof
set IR = the InternalRel of R;
set CR = the carrier of R;
set IR9 = the InternalRel of (R \~);
set CR9 = the carrier of (R \~);
A1: the InternalRel of R is_transitive_in the carrier of R by ORDERS_2:def_3;
now__::_thesis:_for_x,_y,_z_being_set_st_x_in_the_carrier_of_(R_\~)_&_y_in_the_carrier_of_(R_\~)_&_z_in_the_carrier_of_(R_\~)_&_[x,y]_in_the_InternalRel_of_(R_\~)_&_[y,z]_in_the_InternalRel_of_(R_\~)_holds_
[x,z]_in_the_InternalRel_of_(R_\~)
let x, y, z be set ; ::_thesis: ( x in the carrier of (R \~) & y in the carrier of (R \~) & z in the carrier of (R \~) & [x,y] in the InternalRel of (R \~) & [y,z] in the InternalRel of (R \~) implies [x,z] in the InternalRel of (R \~) )
assume that
A2: x in the carrier of (R \~) and
A3: y in the carrier of (R \~) and
A4: z in the carrier of (R \~) and
A5: [x,y] in the InternalRel of (R \~) and
A6: [y,z] in the InternalRel of (R \~) ; ::_thesis: [x,z] in the InternalRel of (R \~)
A7: not [x,y] in the InternalRel of R ~ by A5, XBOOLE_0:def_5;
A8: [x,z] in the InternalRel of R by A1, A2, A3, A4, A5, A6, RELAT_2:def_8;
now__::_thesis:_not_[x,z]_in_the_InternalRel_of_R_~
assume [x,z] in the InternalRel of R ~ ; ::_thesis: contradiction
then [z,x] in the InternalRel of R by RELAT_1:def_7;
then [y,x] in the InternalRel of R by A1, A2, A3, A4, A6, RELAT_2:def_8;
hence contradiction by A7, RELAT_1:def_7; ::_thesis: verum
end;
hence [x,z] in the InternalRel of (R \~) by A8, XBOOLE_0:def_5; ::_thesis: verum
end;
then the InternalRel of (R \~) is_transitive_in the carrier of (R \~) by RELAT_2:def_8;
hence R \~ is transitive by ORDERS_2:def_3; ::_thesis: verum
end;
end;
registration
let R be RelStr ;
clusterR \~ -> strict antisymmetric ;
coherence
R \~ is antisymmetric
proof
set IR = the InternalRel of R;
set IR9 = the InternalRel of (R \~);
set CR9 = the carrier of (R \~);
now__::_thesis:_for_x,_y_being_set_st_x_in_the_carrier_of_(R_\~)_&_y_in_the_carrier_of_(R_\~)_&_[x,y]_in_the_InternalRel_of_(R_\~)_&_[y,x]_in_the_InternalRel_of_(R_\~)_holds_
x_=_y
let x, y be set ; ::_thesis: ( x in the carrier of (R \~) & y in the carrier of (R \~) & [x,y] in the InternalRel of (R \~) & [y,x] in the InternalRel of (R \~) implies x = y )
assume that
x in the carrier of (R \~) and
y in the carrier of (R \~) and
A1: [x,y] in the InternalRel of (R \~) and
A2: [y,x] in the InternalRel of (R \~) ; ::_thesis: x = y
not [x,y] in the InternalRel of R ~ by A1, XBOOLE_0:def_5;
hence x = y by A2, RELAT_1:def_7; ::_thesis: verum
end;
then the InternalRel of (R \~) is_antisymmetric_in the carrier of (R \~) by RELAT_2:def_4;
hence R \~ is antisymmetric by ORDERS_2:def_4; ::_thesis: verum
end;
end;
theorem :: DICKSON:11
for R being non empty Poset
for x being Element of R holds Class ((EqRel R),x) = {x}
proof
let R be non empty Poset; ::_thesis: for x being Element of R holds Class ((EqRel R),x) = {x}
set IR = the InternalRel of R;
set CR = the carrier of R;
let x be Element of the carrier of R; ::_thesis: Class ((EqRel R),x) = {x}
A1: R is quasi_ordered by Def3;
A2: the InternalRel of R is_antisymmetric_in the carrier of R by ORDERS_2:def_4;
now__::_thesis:_for_z_being_set_holds_
(_(_z_in_Class_((EqRel_R),x)_implies_z_in_{x}_)_&_(_z_in_{x}_implies_z_in_Class_((EqRel_R),x)_)_)
let z be set ; ::_thesis: ( ( z in Class ((EqRel R),x) implies z in {x} ) & ( z in {x} implies z in Class ((EqRel R),x) ) )
hereby ::_thesis: ( z in {x} implies z in Class ((EqRel R),x) )
assume z in Class ((EqRel R),x) ; ::_thesis: z in {x}
then [z,x] in EqRel R by EQREL_1:19;
then A3: [z,x] in the InternalRel of R /\ ( the InternalRel of R ~) by A1, Def4;
then A4: [z,x] in the InternalRel of R by XBOOLE_0:def_4;
[z,x] in the InternalRel of R ~ by A3, XBOOLE_0:def_4;
then A5: [x,z] in the InternalRel of R by RELAT_1:def_7;
z in dom the InternalRel of R by A4, XTUPLE_0:def_12;
then z = x by A2, A4, A5, RELAT_2:def_4;
hence z in {x} by TARSKI:def_1; ::_thesis: verum
end;
assume z in {x} ; ::_thesis: z in Class ((EqRel R),x)
then z = x by TARSKI:def_1;
hence z in Class ((EqRel R),x) by EQREL_1:20; ::_thesis: verum
end;
hence Class ((EqRel R),x) = {x} by TARSKI:1; ::_thesis: verum
end;
theorem :: DICKSON:12
for R being Relation holds
( R = R \~ iff R is asymmetric )
proof
let R be Relation; ::_thesis: ( R = R \~ iff R is asymmetric )
thus ( R = R \~ implies R is asymmetric ) ; ::_thesis: ( R is asymmetric implies R = R \~ )
assume R is asymmetric ; ::_thesis: R = R \~
then A1: R is_asymmetric_in field R by RELAT_2:def_13;
now__::_thesis:_for_a,_b_being_set_holds_
(_(_[a,b]_in_R_implies_[a,b]_in_R_\~_)_&_(_[a,b]_in_R_\~_implies_[a,b]_in_R_)_)
let a, b be set ; ::_thesis: ( ( [a,b] in R implies [a,b] in R \~ ) & ( [a,b] in R \~ implies [a,b] in R ) )
hereby ::_thesis: ( [a,b] in R \~ implies [a,b] in R )
assume A2: [a,b] in R ; ::_thesis: [a,b] in R \~
then A3: a in field R by RELAT_1:15;
b in field R by A2, RELAT_1:15;
then not [b,a] in R by A1, A2, A3, RELAT_2:def_5;
then not [a,b] in R ~ by RELAT_1:def_7;
hence [a,b] in R \~ by A2, XBOOLE_0:def_5; ::_thesis: verum
end;
assume [a,b] in R \~ ; ::_thesis: [a,b] in R
hence [a,b] in R ; ::_thesis: verum
end;
hence R = R \~ by RELAT_1:def_2; ::_thesis: verum
end;
theorem :: DICKSON:13
for R being Relation st R is transitive holds
R \~ is transitive
proof
let R be Relation; ::_thesis: ( R is transitive implies R \~ is transitive )
assume R is transitive ; ::_thesis: R \~ is transitive
then A1: R is_transitive_in field R by RELAT_2:def_16;
now__::_thesis:_for_x,_y,_z_being_set_st_x_in_field_(R_\~)_&_y_in_field_(R_\~)_&_z_in_field_(R_\~)_&_[x,y]_in_R_\~_&_[y,z]_in_R_\~_holds_
[x,z]_in_R_\~
let x, y, z be set ; ::_thesis: ( x in field (R \~) & y in field (R \~) & z in field (R \~) & [x,y] in R \~ & [y,z] in R \~ implies [x,z] in R \~ )
assume that
x in field (R \~) and
y in field (R \~) and
z in field (R \~) and
A2: [x,y] in R \~ and
A3: [y,z] in R \~ ; ::_thesis: [x,z] in R \~
A4: x in field R by A2, RELAT_1:15;
A5: y in field R by A2, RELAT_1:15;
A6: z in field R by A3, RELAT_1:15;
then A7: [x,z] in R by A1, A2, A3, A4, A5, RELAT_2:def_8;
not [x,y] in R ~ by A2, XBOOLE_0:def_5;
then not [y,x] in R by RELAT_1:def_7;
then not [z,x] in R by A1, A3, A4, A5, A6, RELAT_2:def_8;
then not [x,z] in R ~ by RELAT_1:def_7;
hence [x,z] in R \~ by A7, XBOOLE_0:def_5; ::_thesis: verum
end;
then R \~ is_transitive_in field (R \~) by RELAT_2:def_8;
hence R \~ is transitive by RELAT_2:def_16; ::_thesis: verum
end;
theorem :: DICKSON:14
for R being Relation
for a, b being set st R is antisymmetric holds
( [a,b] in R \~ iff ( [a,b] in R & a <> b ) )
proof
let R be Relation; ::_thesis: for a, b being set st R is antisymmetric holds
( [a,b] in R \~ iff ( [a,b] in R & a <> b ) )
let a, b be set ; ::_thesis: ( R is antisymmetric implies ( [a,b] in R \~ iff ( [a,b] in R & a <> b ) ) )
assume R is antisymmetric ; ::_thesis: ( [a,b] in R \~ iff ( [a,b] in R & a <> b ) )
then A1: R is_antisymmetric_in field R by RELAT_2:def_12;
A2: R \~ is_asymmetric_in field (R \~) by RELAT_2:def_13;
hereby ::_thesis: ( [a,b] in R & a <> b implies [a,b] in R \~ )
assume A3: [a,b] in R \~ ; ::_thesis: ( [a,b] in R & a <> b )
hence [a,b] in R ; ::_thesis: a <> b
now__::_thesis:_not_a_=_b
assume A4: a = b ; ::_thesis: contradiction
a in field (R \~) by A3, RELAT_1:15;
hence contradiction by A2, A3, A4, RELAT_2:def_5; ::_thesis: verum
end;
hence a <> b ; ::_thesis: verum
end;
assume that
A5: [a,b] in R and
A6: a <> b ; ::_thesis: [a,b] in R \~
A7: a in field R by A5, RELAT_1:15;
b in field R by A5, RELAT_1:15;
then not [b,a] in R by A1, A5, A6, A7, RELAT_2:def_4;
then not [a,b] in R ~ by RELAT_1:def_7;
hence [a,b] in R \~ by A5, XBOOLE_0:def_5; ::_thesis: verum
end;
theorem Th15: :: DICKSON:15
for R being RelStr st R is well_founded holds
R \~ is well_founded
proof
let R be RelStr ; ::_thesis: ( R is well_founded implies R \~ is well_founded )
assume A1: R is well_founded ; ::_thesis: R \~ is well_founded
set IR = the InternalRel of R;
set CR = the carrier of R;
set IR9 = the InternalRel of (R \~);
set CR9 = the carrier of (R \~);
A2: the InternalRel of R is_well_founded_in the carrier of R by A1, WELLFND1:def_2;
now__::_thesis:_for_Y_being_set_st_Y_c=_the_carrier_of_(R_\~)_&_Y_<>_{}_holds_
ex_a_being_set_st_
(_a_in_Y_&_the_InternalRel_of_(R_\~)_-Seg_a_misses_Y_)
let Y be set ; ::_thesis: ( Y c= the carrier of (R \~) & Y <> {} implies ex a being set st
( a in Y & the InternalRel of (R \~) -Seg a misses Y ) )
assume that
A3: Y c= the carrier of (R \~) and
A4: Y <> {} ; ::_thesis: ex a being set st
( a in Y & the InternalRel of (R \~) -Seg a misses Y )
consider a being set such that
A5: a in Y and
A6: the InternalRel of R -Seg a misses Y by A2, A3, A4, WELLORD1:def_3;
A7: ( the InternalRel of R -Seg a) /\ Y = {} by A6, XBOOLE_0:def_7;
take a = a; ::_thesis: ( a in Y & the InternalRel of (R \~) -Seg a misses Y )
thus a in Y by A5; ::_thesis: the InternalRel of (R \~) -Seg a misses Y
now__::_thesis:_for_z_being_set_holds_not_z_in_(_the_InternalRel_of_(R_\~)_-Seg_a)_/\_Y
assume ex z being set st z in ( the InternalRel of (R \~) -Seg a) /\ Y ; ::_thesis: contradiction
then consider z being set such that
A8: z in ( the InternalRel of (R \~) -Seg a) /\ Y ;
A9: z in the InternalRel of (R \~) -Seg a by A8, XBOOLE_0:def_4;
A10: z in Y by A8, XBOOLE_0:def_4;
A11: z <> a by A9, WELLORD1:1;
[z,a] in the InternalRel of (R \~) by A9, WELLORD1:1;
then z in the InternalRel of R -Seg a by A11, WELLORD1:1;
hence contradiction by A7, A10, XBOOLE_0:def_4; ::_thesis: verum
end;
then ( the InternalRel of (R \~) -Seg a) /\ Y = {} by XBOOLE_0:def_1;
hence the InternalRel of (R \~) -Seg a misses Y by XBOOLE_0:def_7; ::_thesis: verum
end;
then the InternalRel of (R \~) is_well_founded_in the carrier of (R \~) by WELLORD1:def_3;
hence R \~ is well_founded by WELLFND1:def_2; ::_thesis: verum
end;
theorem Th16: :: DICKSON:16
for R being RelStr st R \~ is well_founded & R is antisymmetric holds
R is well_founded
proof
let R be RelStr ; ::_thesis: ( R \~ is well_founded & R is antisymmetric implies R is well_founded )
assume that
A1: R \~ is well_founded and
A2: R is antisymmetric ; ::_thesis: R is well_founded
set IR = the InternalRel of R;
set CR = the carrier of R;
set IR9 = the InternalRel of (R \~);
A3: the InternalRel of R is_antisymmetric_in the carrier of R by A2, ORDERS_2:def_4;
A4: the InternalRel of (R \~) is_well_founded_in the carrier of R by A1, WELLFND1:def_2;
now__::_thesis:_for_Y_being_set_st_Y_c=_the_carrier_of_R_&_Y_<>_{}_holds_
ex_a_being_set_st_
(_a_in_Y_&_the_InternalRel_of_R_-Seg_a_misses_Y_)
let Y be set ; ::_thesis: ( Y c= the carrier of R & Y <> {} implies ex a being set st
( a in Y & the InternalRel of R -Seg a misses Y ) )
assume that
A5: Y c= the carrier of R and
A6: Y <> {} ; ::_thesis: ex a being set st
( a in Y & the InternalRel of R -Seg a misses Y )
consider a being set such that
A7: a in Y and
A8: the InternalRel of (R \~) -Seg a misses Y by A4, A5, A6, WELLORD1:def_3;
A9: ( the InternalRel of (R \~) -Seg a) /\ Y = {} by A8, XBOOLE_0:def_7;
take a = a; ::_thesis: ( a in Y & the InternalRel of R -Seg a misses Y )
thus a in Y by A7; ::_thesis: the InternalRel of R -Seg a misses Y
now__::_thesis:_not_(_the_InternalRel_of_R_-Seg_a)_/\_Y_<>_{}
assume ( the InternalRel of R -Seg a) /\ Y <> {} ; ::_thesis: contradiction
then consider z being set such that
A10: z in ( the InternalRel of R -Seg a) /\ Y by XBOOLE_0:def_1;
A11: z in the InternalRel of R -Seg a by A10, XBOOLE_0:def_4;
A12: z in Y by A10, XBOOLE_0:def_4;
A13: z <> a by A11, WELLORD1:1;
A14: [z,a] in the InternalRel of R by A11, WELLORD1:1;
then not [a,z] in the InternalRel of R by A3, A5, A7, A12, A13, RELAT_2:def_4;
then not [z,a] in the InternalRel of R ~ by RELAT_1:def_7;
then [z,a] in the InternalRel of R \ ( the InternalRel of R ~) by A14, XBOOLE_0:def_5;
then z in the InternalRel of (R \~) -Seg a by A13, WELLORD1:1;
hence contradiction by A9, A12, XBOOLE_0:def_4; ::_thesis: verum
end;
hence the InternalRel of R -Seg a misses Y by XBOOLE_0:def_7; ::_thesis: verum
end;
then the InternalRel of R is_well_founded_in the carrier of R by WELLORD1:def_3;
hence R is well_founded by WELLFND1:def_2; ::_thesis: verum
end;
begin
theorem Th17: :: DICKSON:17
for L being RelStr
for N being set
for x being Element of (L \~) holds
( x is_minimal_wrt N, the InternalRel of (L \~) iff ( x in N & ( for y being Element of L st y in N & [y,x] in the InternalRel of L holds
[x,y] in the InternalRel of L ) ) )
proof
let L be RelStr ; ::_thesis: for N being set
for x being Element of (L \~) holds
( x is_minimal_wrt N, the InternalRel of (L \~) iff ( x in N & ( for y being Element of L st y in N & [y,x] in the InternalRel of L holds
[x,y] in the InternalRel of L ) ) )
let N be set ; ::_thesis: for x being Element of (L \~) holds
( x is_minimal_wrt N, the InternalRel of (L \~) iff ( x in N & ( for y being Element of L st y in N & [y,x] in the InternalRel of L holds
[x,y] in the InternalRel of L ) ) )
let x be Element of (L \~); ::_thesis: ( x is_minimal_wrt N, the InternalRel of (L \~) iff ( x in N & ( for y being Element of L st y in N & [y,x] in the InternalRel of L holds
[x,y] in the InternalRel of L ) ) )
set IR = the InternalRel of L;
set IR9 = the InternalRel of (L \~);
hereby ::_thesis: ( x in N & ( for y being Element of L st y in N & [y,x] in the InternalRel of L holds
[x,y] in the InternalRel of L ) implies x is_minimal_wrt N, the InternalRel of (L \~) )
assume A1: x is_minimal_wrt N, the InternalRel of (L \~) ; ::_thesis: ( x in N & ( for y being Element of L st y in N & [y,x] in the InternalRel of L holds
[x,y] in the InternalRel of L ) )
hence x in N by WAYBEL_4:def_25; ::_thesis: for y being Element of L st y in N & [y,x] in the InternalRel of L holds
[x,y] in the InternalRel of L
let y be Element of L; ::_thesis: ( y in N & [y,x] in the InternalRel of L implies [x,y] in the InternalRel of L )
assume A2: y in N ; ::_thesis: ( [y,x] in the InternalRel of L implies [x,y] in the InternalRel of L )
assume A3: [y,x] in the InternalRel of L ; ::_thesis: [x,y] in the InternalRel of L
now__::_thesis:_[x,y]_in_the_InternalRel_of_L
percases ( x = y or x <> y ) ;
suppose x = y ; ::_thesis: [x,y] in the InternalRel of L
hence [x,y] in the InternalRel of L by A3; ::_thesis: verum
end;
suppose x <> y ; ::_thesis: [x,y] in the InternalRel of L
then not [y,x] in the InternalRel of (L \~) by A1, A2, WAYBEL_4:def_25;
then [y,x] in the InternalRel of L ~ by A3, XBOOLE_0:def_5;
hence [x,y] in the InternalRel of L by RELAT_1:def_7; ::_thesis: verum
end;
end;
end;
hence [x,y] in the InternalRel of L ; ::_thesis: verum
end;
assume that
A4: x in N and
A5: for y being Element of L st y in N & [y,x] in the InternalRel of L holds
[x,y] in the InternalRel of L ; ::_thesis: x is_minimal_wrt N, the InternalRel of (L \~)
now__::_thesis:_for_y_being_set_holds_
(_not_y_in_N_or_not_y_<>_x_or_not_[y,x]_in_the_InternalRel_of_(L_\~)_)
assume ex y being set st
( y in N & y <> x & [y,x] in the InternalRel of (L \~) ) ; ::_thesis: contradiction
then consider y being set such that
A6: y in N and
y <> x and
A7: [y,x] in the InternalRel of (L \~) ;
reconsider y9 = y as Element of L by A7, ZFMISC_1:87;
A8: not [y,x] in the InternalRel of L ~ by A7, XBOOLE_0:def_5;
( [y9,x] in the InternalRel of L implies [x,y9] in the InternalRel of L ) by A5, A6;
hence contradiction by A7, A8, RELAT_1:def_7; ::_thesis: verum
end;
hence x is_minimal_wrt N, the InternalRel of (L \~) by A4, WAYBEL_4:def_25; ::_thesis: verum
end;
theorem :: DICKSON:18
for R, S being non empty RelStr
for m being Function of R,S st R is quasi_ordered & S is antisymmetric & S \~ is well_founded & ( for a, b being Element of R holds
( ( a <= b implies m . a <= m . b ) & ( m . a = m . b implies [a,b] in EqRel R ) ) ) holds
R \~ is well_founded
proof
let R, S be non empty RelStr ; ::_thesis: for m being Function of R,S st R is quasi_ordered & S is antisymmetric & S \~ is well_founded & ( for a, b being Element of R holds
( ( a <= b implies m . a <= m . b ) & ( m . a = m . b implies [a,b] in EqRel R ) ) ) holds
R \~ is well_founded
let m be Function of R,S; ::_thesis: ( R is quasi_ordered & S is antisymmetric & S \~ is well_founded & ( for a, b being Element of R holds
( ( a <= b implies m . a <= m . b ) & ( m . a = m . b implies [a,b] in EqRel R ) ) ) implies R \~ is well_founded )
assume that
A1: R is quasi_ordered and
A2: S is antisymmetric and
A3: S \~ is well_founded and
A4: for a, b being Element of R holds
( ( a <= b implies m . a <= m . b ) & ( m . a = m . b implies [a,b] in EqRel R ) ) ; ::_thesis: R \~ is well_founded
set IR = the InternalRel of R;
set IS = the InternalRel of S;
A5: the InternalRel of S is_antisymmetric_in the carrier of S by A2, ORDERS_2:def_4;
now__::_thesis:_for_f_being_sequence_of_(R_\~)_holds_not_f_is_descending
assume ex f being sequence of (R \~) st f is descending ; ::_thesis: contradiction
then consider f being sequence of (R \~) such that
A6: f is descending ;
reconsider f9 = f as sequence of R ;
deffunc H1( Element of NAT ) -> Element of the carrier of S = m . (f9 . $1);
consider g9 being Function of NAT, the carrier of S such that
A7: for k being Element of NAT holds g9 . k = H1(k) from FUNCT_2:sch_4();
reconsider g = g9 as sequence of (S \~) ;
now__::_thesis:_for_n_being_Nat_holds_
(_g_._(n_+_1)_<>_g_._n_&_[(g_._(n_+_1)),(g_._n)]_in_the_InternalRel_of_(S_\~)_)
let n be Nat; ::_thesis: ( g . (n + 1) <> g . n & [(g . (n + 1)),(g . n)] in the InternalRel of (S \~) )
reconsider n1 = n as Element of NAT by ORDINAL1:def_12;
A8: [(f . (n + 1)),(f . n)] in the InternalRel of (R \~) by A6, WELLFND1:def_6;
A9: [(f . (n + 1)),(f . n)] in the InternalRel of R \ ( the InternalRel of R ~) by A6, WELLFND1:def_6;
A10: not [(f . (n + 1)),(f . n)] in the InternalRel of R ~ by A8, XBOOLE_0:def_5;
A11: g . n1 = m . (f . n1) by A7;
A12: now__::_thesis:_not_g_._(n_+_1)_=_g_._n
assume g . (n + 1) = g . n ; ::_thesis: contradiction
then m . (f . (n + 1)) = m . (f . n) by A7, A11;
then [(f9 . (n + 1)),(f9 . n)] in EqRel R by A4;
then [(f . (n + 1)),(f . n)] in the InternalRel of R /\ ( the InternalRel of R ~) by A1, Def4;
hence contradiction by A10, XBOOLE_0:def_4; ::_thesis: verum
end;
hence g . (n + 1) <> g . n ; ::_thesis: [(g . (n + 1)),(g . n)] in the InternalRel of (S \~)
reconsider fn1 = f . (n + 1) as Element of R ;
reconsider fn = f . n as Element of R ;
A13: fn1 <= fn by A9, ORDERS_2:def_5;
A14: g9 . (n + 1) = m . fn1 by A7;
g9 . n1 = m . fn by A7;
then g9 . (n + 1) <= g9 . n by A4, A13, A14;
then A15: [(g . (n + 1)),(g . n)] in the InternalRel of S by ORDERS_2:def_5;
then not [(g . n),(g . (n + 1))] in the InternalRel of S by A5, A12, RELAT_2:def_4;
then not [(g . (n + 1)),(g . n)] in the InternalRel of S ~ by RELAT_1:def_7;
hence [(g . (n + 1)),(g . n)] in the InternalRel of (S \~) by A15, XBOOLE_0:def_5; ::_thesis: verum
end;
then g is descending by WELLFND1:def_6;
hence contradiction by A3, WELLFND1:14; ::_thesis: verum
end;
hence R \~ is well_founded by WELLFND1:14; ::_thesis: verum
end;
definition
let R be non empty RelStr ;
let N be Subset of R;
func min-classes N -> Subset-Family of R means :Def8: :: DICKSON:def 8
for x being set holds
( x in it iff ex y being Element of (R \~) st
( y is_minimal_wrt N, the InternalRel of (R \~) & x = (Class ((EqRel R),y)) /\ N ) );
existence
ex b1 being Subset-Family of R st
for x being set holds
( x in b1 iff ex y being Element of (R \~) st
( y is_minimal_wrt N, the InternalRel of (R \~) & x = (Class ((EqRel R),y)) /\ N ) )
proof
set IR9 = the InternalRel of (R \~);
set C = { (x /\ N) where x is Element of Class (EqRel R) : ex y being Element of (R \~) st
( x = Class ((EqRel R),y) & y is_minimal_wrt N, the InternalRel of (R \~) ) } ;
now__::_thesis:_for_x_being_set_st_x_in__{__(x_/\_N)_where_x_is_Element_of_Class_(EqRel_R)_:_ex_y_being_Element_of_(R_\~)_st_
(_x_=_Class_((EqRel_R),y)_&_y_is_minimal_wrt_N,_the_InternalRel_of_(R_\~)_)__}__holds_
x_in_bool_the_carrier_of_R
let x be set ; ::_thesis: ( x in { (x /\ N) where x is Element of Class (EqRel R) : ex y being Element of (R \~) st
( x = Class ((EqRel R),y) & y is_minimal_wrt N, the InternalRel of (R \~) ) } implies x in bool the carrier of R )
assume x in { (x /\ N) where x is Element of Class (EqRel R) : ex y being Element of (R \~) st
( x = Class ((EqRel R),y) & y is_minimal_wrt N, the InternalRel of (R \~) ) } ; ::_thesis: x in bool the carrier of R
then ex xx being Element of Class (EqRel R) st
( x = xx /\ N & ex y being Element of (R \~) st
( xx = Class ((EqRel R),y) & y is_minimal_wrt N, the InternalRel of (R \~) ) ) ;
hence x in bool the carrier of R ; ::_thesis: verum
end;
then reconsider C = { (x /\ N) where x is Element of Class (EqRel R) : ex y being Element of (R \~) st
( x = Class ((EqRel R),y) & y is_minimal_wrt N, the InternalRel of (R \~) ) } as Subset-Family of R by TARSKI:def_3;
take C ; ::_thesis: for x being set holds
( x in C iff ex y being Element of (R \~) st
( y is_minimal_wrt N, the InternalRel of (R \~) & x = (Class ((EqRel R),y)) /\ N ) )
let x be set ; ::_thesis: ( x in C iff ex y being Element of (R \~) st
( y is_minimal_wrt N, the InternalRel of (R \~) & x = (Class ((EqRel R),y)) /\ N ) )
hereby ::_thesis: ( ex y being Element of (R \~) st
( y is_minimal_wrt N, the InternalRel of (R \~) & x = (Class ((EqRel R),y)) /\ N ) implies x in C )
assume x in C ; ::_thesis: ex y being Element of (R \~) st
( y is_minimal_wrt N, the InternalRel of (R \~) & x = (Class ((EqRel R),y)) /\ N )
then ex xx being Element of Class (EqRel R) st
( x = xx /\ N & ex y being Element of (R \~) st
( xx = Class ((EqRel R),y) & y is_minimal_wrt N, the InternalRel of (R \~) ) ) ;
hence ex y being Element of (R \~) st
( y is_minimal_wrt N, the InternalRel of (R \~) & x = (Class ((EqRel R),y)) /\ N ) ; ::_thesis: verum
end;
given y being Element of (R \~) such that A1: y is_minimal_wrt N, the InternalRel of (R \~) and
A2: x = (Class ((EqRel R),y)) /\ N ; ::_thesis: x in C
reconsider y9 = y as Element of R ;
EqClass ((EqRel R),y9) in Class (EqRel R) ;
hence x in C by A1, A2; ::_thesis: verum
end;
uniqueness
for b1, b2 being Subset-Family of R st ( for x being set holds
( x in b1 iff ex y being Element of (R \~) st
( y is_minimal_wrt N, the InternalRel of (R \~) & x = (Class ((EqRel R),y)) /\ N ) ) ) & ( for x being set holds
( x in b2 iff ex y being Element of (R \~) st
( y is_minimal_wrt N, the InternalRel of (R \~) & x = (Class ((EqRel R),y)) /\ N ) ) ) holds
b1 = b2
proof
set IR = the InternalRel of (R \~);
let IT1, IT2 be Subset-Family of R; ::_thesis: ( ( for x being set holds
( x in IT1 iff ex y being Element of (R \~) st
( y is_minimal_wrt N, the InternalRel of (R \~) & x = (Class ((EqRel R),y)) /\ N ) ) ) & ( for x being set holds
( x in IT2 iff ex y being Element of (R \~) st
( y is_minimal_wrt N, the InternalRel of (R \~) & x = (Class ((EqRel R),y)) /\ N ) ) ) implies IT1 = IT2 )
assume that
A3: for x being set holds
( x in IT1 iff ex y being Element of (R \~) st
( y is_minimal_wrt N, the InternalRel of (R \~) & x = (Class ((EqRel R),y)) /\ N ) ) and
A4: for x being set holds
( x in IT2 iff ex y being Element of (R \~) st
( y is_minimal_wrt N, the InternalRel of (R \~) & x = (Class ((EqRel R),y)) /\ N ) ) ; ::_thesis: IT1 = IT2
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_IT1_implies_x_in_IT2_)_&_(_x_in_IT2_implies_x_in_IT1_)_)
let x be set ; ::_thesis: ( ( x in IT1 implies x in IT2 ) & ( x in IT2 implies x in IT1 ) )
hereby ::_thesis: ( x in IT2 implies x in IT1 )
assume x in IT1 ; ::_thesis: x in IT2
then ex y being Element of (R \~) st
( y is_minimal_wrt N, the InternalRel of (R \~) & x = (Class ((EqRel R),y)) /\ N ) by A3;
hence x in IT2 by A4; ::_thesis: verum
end;
assume x in IT2 ; ::_thesis: x in IT1
then ex y being Element of (R \~) st
( y is_minimal_wrt N, the InternalRel of (R \~) & x = (Class ((EqRel R),y)) /\ N ) by A4;
hence x in IT1 by A3; ::_thesis: verum
end;
hence IT1 = IT2 by TARSKI:1; ::_thesis: verum
end;
end;
:: deftheorem Def8 defines min-classes DICKSON:def_8_:_
for R being non empty RelStr
for N being Subset of R
for b3 being Subset-Family of R holds
( b3 = min-classes N iff for x being set holds
( x in b3 iff ex y being Element of (R \~) st
( y is_minimal_wrt N, the InternalRel of (R \~) & x = (Class ((EqRel R),y)) /\ N ) ) );
theorem Th19: :: DICKSON:19
for R being non empty RelStr
for N being Subset of R
for x being set st R is quasi_ordered & x in min-classes N holds
for y being Element of (R \~) st y in x holds
y is_minimal_wrt N, the InternalRel of (R \~)
proof
let R be non empty RelStr ; ::_thesis: for N being Subset of R
for x being set st R is quasi_ordered & x in min-classes N holds
for y being Element of (R \~) st y in x holds
y is_minimal_wrt N, the InternalRel of (R \~)
let N be Subset of R; ::_thesis: for x being set st R is quasi_ordered & x in min-classes N holds
for y being Element of (R \~) st y in x holds
y is_minimal_wrt N, the InternalRel of (R \~)
let x be set ; ::_thesis: ( R is quasi_ordered & x in min-classes N implies for y being Element of (R \~) st y in x holds
y is_minimal_wrt N, the InternalRel of (R \~) )
assume that
A1: R is quasi_ordered and
A2: x in min-classes N ; ::_thesis: for y being Element of (R \~) st y in x holds
y is_minimal_wrt N, the InternalRel of (R \~)
set IR = the InternalRel of R;
set CR = the carrier of R;
set IR9 = the InternalRel of (R \~);
let c be Element of (R \~); ::_thesis: ( c in x implies c is_minimal_wrt N, the InternalRel of (R \~) )
assume A3: c in x ; ::_thesis: c is_minimal_wrt N, the InternalRel of (R \~)
consider b being Element of (R \~) such that
A4: b is_minimal_wrt N, the InternalRel of (R \~) and
A5: x = (Class ((EqRel R),b)) /\ N by A2, Def8;
c in Class ((EqRel R),b) by A3, A5, XBOOLE_0:def_4;
then [c,b] in EqRel R by EQREL_1:19;
then [c,b] in the InternalRel of R /\ ( the InternalRel of R ~) by A1, Def4;
then A6: [c,b] in the InternalRel of R by XBOOLE_0:def_4;
A7: now__::_thesis:_for_d_being_set_holds_
(_not_d_in_N_or_not_d_<>_c_or_not_[d,c]_in_the_InternalRel_of_(R_\~)_)
assume ex d being set st
( d in N & d <> c & [d,c] in the InternalRel of (R \~) ) ; ::_thesis: contradiction
then consider d being set such that
A8: d in N and
d <> c and
A9: [d,c] in the InternalRel of (R \~) ;
A10: not [d,c] in the InternalRel of R ~ by A9, XBOOLE_0:def_5;
R is transitive by A1, Def3;
then A11: the InternalRel of R is_transitive_in the carrier of R by ORDERS_2:def_3;
then A12: [d,b] in the InternalRel of R by A6, A8, A9, RELAT_2:def_8;
now__::_thesis:_not_[d,b]_in_the_InternalRel_of_R_~
assume [d,b] in the InternalRel of R ~ ; ::_thesis: contradiction
then [b,d] in the InternalRel of R by RELAT_1:def_7;
then [c,d] in the InternalRel of R by A6, A8, A11, RELAT_2:def_8;
hence contradiction by A10, RELAT_1:def_7; ::_thesis: verum
end;
then A13: [d,b] in the InternalRel of (R \~) by A12, XBOOLE_0:def_5;
b <> d by A6, A10, RELAT_1:def_7;
hence contradiction by A4, A8, A13, WAYBEL_4:def_25; ::_thesis: verum
end;
c in N by A3, A5, XBOOLE_0:def_4;
hence c is_minimal_wrt N, the InternalRel of (R \~) by A7, WAYBEL_4:def_25; ::_thesis: verum
end;
theorem Th20: :: DICKSON:20
for R being non empty RelStr holds
( R \~ is well_founded iff for N being Subset of R st N <> {} holds
ex x being set st x in min-classes N )
proof
let R be non empty RelStr ; ::_thesis: ( R \~ is well_founded iff for N being Subset of R st N <> {} holds
ex x being set st x in min-classes N )
set CR = the carrier of R;
set IR9 = the InternalRel of (R \~);
set CR9 = the carrier of (R \~);
hereby ::_thesis: ( ( for N being Subset of R st N <> {} holds
ex x being set st x in min-classes N ) implies R \~ is well_founded )
assume R \~ is well_founded ; ::_thesis: for N being Subset of the carrier of R st N <> {} holds
ex x being set st x in min-classes N
then A1: the InternalRel of (R \~) is_well_founded_in the carrier of (R \~) by WELLFND1:def_2;
let N be Subset of the carrier of R; ::_thesis: ( N <> {} implies ex x being set st x in min-classes N )
assume A2: N <> {} ; ::_thesis: ex x being set st x in min-classes N
reconsider N9 = N as Subset of the carrier of (R \~) ;
consider y being set such that
A3: y in N9 and
A4: the InternalRel of (R \~) -Seg y misses N9 by A1, A2, WELLORD1:def_3;
A5: ( the InternalRel of (R \~) -Seg y) /\ N9 = {} by A4, XBOOLE_0:def_7;
reconsider y = y as Element of (R \~) by A3;
set x = (Class ((EqRel R),y)) /\ N;
now__::_thesis:_for_z_being_set_holds_
(_not_z_in_N_or_not_z_<>_y_or_not_[z,y]_in_the_InternalRel_of_(R_\~)_)
assume ex z being set st
( z in N & z <> y & [z,y] in the InternalRel of (R \~) ) ; ::_thesis: contradiction
then consider z being set such that
A6: z in N and
A7: z <> y and
A8: [z,y] in the InternalRel of (R \~) ;
z in the InternalRel of (R \~) -Seg y by A7, A8, WELLORD1:1;
hence contradiction by A5, A6, XBOOLE_0:def_4; ::_thesis: verum
end;
then y is_minimal_wrt N, the InternalRel of (R \~) by A3, WAYBEL_4:def_25;
then (Class ((EqRel R),y)) /\ N in min-classes N by Def8;
hence ex x being set st x in min-classes N ; ::_thesis: verum
end;
assume A9: for N being Subset of R st N <> {} holds
ex x being set st x in min-classes N ; ::_thesis: R \~ is well_founded
now__::_thesis:_for_N_being_set_st_N_c=_the_carrier_of_(R_\~)_&_N_<>_{}_holds_
ex_a9_being_set_st_
(_a9_in_N_&_the_InternalRel_of_(R_\~)_-Seg_a9_misses_N_)
let N be set ; ::_thesis: ( N c= the carrier of (R \~) & N <> {} implies ex a9 being set st
( a9 in N & the InternalRel of (R \~) -Seg a9 misses N ) )
assume that
A10: N c= the carrier of (R \~) and
A11: N <> {} ; ::_thesis: ex a9 being set st
( a9 in N & the InternalRel of (R \~) -Seg a9 misses N )
reconsider N9 = N as Subset of R by A10;
consider x being set such that
A12: x in min-classes N9 by A9, A11;
consider a being Element of (R \~) such that
A13: a is_minimal_wrt N9, the InternalRel of (R \~) and
x = (Class ((EqRel R),a)) /\ N9 by A12, Def8;
reconsider a9 = a as set ;
take a9 = a9; ::_thesis: ( a9 in N & the InternalRel of (R \~) -Seg a9 misses N )
thus a9 in N by A13, WAYBEL_4:def_25; ::_thesis: the InternalRel of (R \~) -Seg a9 misses N
now__::_thesis:_not_(_the_InternalRel_of_(R_\~)_-Seg_a9)_/\_N_<>_{}
assume ( the InternalRel of (R \~) -Seg a9) /\ N <> {} ; ::_thesis: contradiction
then consider z being set such that
A14: z in ( the InternalRel of (R \~) -Seg a9) /\ N by XBOOLE_0:def_1;
A15: z in the InternalRel of (R \~) -Seg a9 by A14, XBOOLE_0:def_4;
A16: z in N by A14, XBOOLE_0:def_4;
then reconsider z = z as Element of (R \~) by A10;
A17: z <> a9 by A15, WELLORD1:1;
[z,a] in the InternalRel of (R \~) by A15, WELLORD1:1;
hence contradiction by A13, A16, A17, WAYBEL_4:def_25; ::_thesis: verum
end;
hence the InternalRel of (R \~) -Seg a9 misses N by XBOOLE_0:def_7; ::_thesis: verum
end;
then the InternalRel of (R \~) is_well_founded_in the carrier of (R \~) by WELLORD1:def_3;
hence R \~ is well_founded by WELLFND1:def_2; ::_thesis: verum
end;
theorem Th21: :: DICKSON:21
for R being non empty RelStr
for N being Subset of R
for y being Element of (R \~) st y is_minimal_wrt N, the InternalRel of (R \~) holds
not min-classes N is empty
proof
let R be non empty RelStr ; ::_thesis: for N being Subset of R
for y being Element of (R \~) st y is_minimal_wrt N, the InternalRel of (R \~) holds
not min-classes N is empty
let N be Subset of R; ::_thesis: for y being Element of (R \~) st y is_minimal_wrt N, the InternalRel of (R \~) holds
not min-classes N is empty
let y be Element of (R \~); ::_thesis: ( y is_minimal_wrt N, the InternalRel of (R \~) implies not min-classes N is empty )
assume A1: y is_minimal_wrt N, the InternalRel of (R \~) ; ::_thesis: not min-classes N is empty
ex x being set st x = (Class ((EqRel R),y)) /\ N ;
hence not min-classes N is empty by A1, Def8; ::_thesis: verum
end;
theorem Th22: :: DICKSON:22
for R being non empty RelStr
for N being Subset of R
for x being set st x in min-classes N holds
not x is empty
proof
let R be non empty RelStr ; ::_thesis: for N being Subset of R
for x being set st x in min-classes N holds
not x is empty
let N be Subset of R; ::_thesis: for x being set st x in min-classes N holds
not x is empty
let x be set ; ::_thesis: ( x in min-classes N implies not x is empty )
assume x in min-classes N ; ::_thesis: not x is empty
then consider y being Element of (R \~) such that
A1: y is_minimal_wrt N, the InternalRel of (R \~) and
A2: x = (Class ((EqRel R),y)) /\ N by Def8;
A3: y in N by A1, WAYBEL_4:def_25;
y in Class ((EqRel R),y) by EQREL_1:20;
hence not x is empty by A2, A3, XBOOLE_0:def_4; ::_thesis: verum
end;
theorem Th23: :: DICKSON:23
for R being non empty RelStr st R is quasi_ordered holds
( ( R is connected & R \~ is well_founded ) iff for N being non empty Subset of R holds card (min-classes N) = 1 )
proof
let R be non empty RelStr ; ::_thesis: ( R is quasi_ordered implies ( ( R is connected & R \~ is well_founded ) iff for N being non empty Subset of R holds card (min-classes N) = 1 ) )
assume A1: R is quasi_ordered ; ::_thesis: ( ( R is connected & R \~ is well_founded ) iff for N being non empty Subset of R holds card (min-classes N) = 1 )
set IR = the InternalRel of R;
set CR = the carrier of R;
set IR9 = the InternalRel of (R \~);
hereby ::_thesis: ( ( for N being non empty Subset of R holds card (min-classes N) = 1 ) implies ( R is connected & R \~ is well_founded ) )
assume that
A2: R is connected and
A3: R \~ is well_founded ; ::_thesis: for N being non empty Subset of the carrier of R holds card (min-classes N) = 1
let N be non empty Subset of the carrier of R; ::_thesis: card (min-classes N) = 1
consider x being set such that
A4: x in min-classes N by A3, Th20;
consider a being Element of (R \~) such that
A5: a is_minimal_wrt N, the InternalRel of (R \~) and
A6: x = (Class ((EqRel R),a)) /\ N by A4, Def8;
A7: a in N by A5, WAYBEL_4:def_25;
now__::_thesis:_for_y_being_set_holds_
(_(_y_in_min-classes_N_implies_y_in_{x}_)_&_(_y_in_{x}_implies_y_in_min-classes_N_)_)
let y be set ; ::_thesis: ( ( y in min-classes N implies y in {x} ) & ( y in {x} implies y in min-classes N ) )
hereby ::_thesis: ( y in {x} implies y in min-classes N )
assume y in min-classes N ; ::_thesis: y in {x}
then consider b being Element of (R \~) such that
A8: b is_minimal_wrt N, the InternalRel of (R \~) and
A9: y = (Class ((EqRel R),b)) /\ N by Def8;
A10: b in N by A8, WAYBEL_4:def_25;
reconsider a9 = a as Element of R ;
reconsider b9 = b as Element of R ;
A11: now__::_thesis:_(_[a,b]_in_the_InternalRel_of_R_&_[b,a]_in_the_InternalRel_of_R_)
percases ( a9 <= b9 or b9 <= a9 ) by A2, WAYBEL_0:def_29;
supposeA12: a9 <= b9 ; ::_thesis: ( [a,b] in the InternalRel of R & [b,a] in the InternalRel of R )
then A13: [a,b] in the InternalRel of R by ORDERS_2:def_5;
now__::_thesis:_[b,a]_in_the_InternalRel_of_R
percases ( a = b or a <> b ) ;
suppose a = b ; ::_thesis: [b,a] in the InternalRel of R
hence [b,a] in the InternalRel of R by A12, ORDERS_2:def_5; ::_thesis: verum
end;
supposeA14: a <> b ; ::_thesis: [b,a] in the InternalRel of R
now__::_thesis:_[b,a]_in_the_InternalRel_of_R
assume not [b,a] in the InternalRel of R ; ::_thesis: contradiction
then not [a,b] in the InternalRel of R ~ by RELAT_1:def_7;
then [a,b] in the InternalRel of R \ ( the InternalRel of R ~) by A13, XBOOLE_0:def_5;
hence contradiction by A7, A8, A14, WAYBEL_4:def_25; ::_thesis: verum
end;
hence [b,a] in the InternalRel of R ; ::_thesis: verum
end;
end;
end;
hence ( [a,b] in the InternalRel of R & [b,a] in the InternalRel of R ) by A12, ORDERS_2:def_5; ::_thesis: verum
end;
supposeA15: b9 <= a9 ; ::_thesis: ( [a,b] in the InternalRel of R & [b,a] in the InternalRel of R )
then A16: [b,a] in the InternalRel of R by ORDERS_2:def_5;
now__::_thesis:_[a,b]_in_the_InternalRel_of_R
percases ( a = b or a <> b ) ;
suppose a = b ; ::_thesis: [a,b] in the InternalRel of R
hence [a,b] in the InternalRel of R by A15, ORDERS_2:def_5; ::_thesis: verum
end;
supposeA17: a <> b ; ::_thesis: [a,b] in the InternalRel of R
now__::_thesis:_[a,b]_in_the_InternalRel_of_R
assume not [a,b] in the InternalRel of R ; ::_thesis: contradiction
then not [b,a] in the InternalRel of R ~ by RELAT_1:def_7;
then [b,a] in the InternalRel of R \ ( the InternalRel of R ~) by A16, XBOOLE_0:def_5;
hence contradiction by A5, A10, A17, WAYBEL_4:def_25; ::_thesis: verum
end;
hence [a,b] in the InternalRel of R ; ::_thesis: verum
end;
end;
end;
hence ( [a,b] in the InternalRel of R & [b,a] in the InternalRel of R ) by A15, ORDERS_2:def_5; ::_thesis: verum
end;
end;
end;
then [b,a] in the InternalRel of R ~ by RELAT_1:def_7;
then [b,a] in the InternalRel of R /\ ( the InternalRel of R ~) by A11, XBOOLE_0:def_4;
then [b,a] in EqRel R by A1, Def4;
then b in Class ((EqRel R),a) by EQREL_1:19;
then Class ((EqRel R),b) = Class ((EqRel R),a) by EQREL_1:23;
hence y in {x} by A6, A9, TARSKI:def_1; ::_thesis: verum
end;
assume y in {x} ; ::_thesis: y in min-classes N
then y = (Class ((EqRel R),a)) /\ N by A6, TARSKI:def_1;
hence y in min-classes N by A5, Def8; ::_thesis: verum
end;
then min-classes N = {x} by TARSKI:1;
hence card (min-classes N) = 1 by CARD_1:30; ::_thesis: verum
end;
assume A18: for N being non empty Subset of R holds card (min-classes N) = 1 ; ::_thesis: ( R is connected & R \~ is well_founded )
now__::_thesis:_for_a,_b_being_Element_of_R_st_not_a_<=_b_holds_
a_>=_b
let a, b be Element of R; ::_thesis: ( not a <= b implies a >= b )
assume that
A19: not a <= b and
A20: not a >= b ; ::_thesis: contradiction
A21: not [a,b] in the InternalRel of R by A19, ORDERS_2:def_5;
then A22: not [b,a] in the InternalRel of R ~ by RELAT_1:def_7;
not [b,a] in the InternalRel of R by A20, ORDERS_2:def_5;
then A23: not [a,b] in the InternalRel of R ~ by RELAT_1:def_7;
set N9 = {a,b};
set MCN = {{a},{b}};
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_min-classes_{a,b}_implies_x_in_{{a},{b}}_)_&_(_x_in_{{a},{b}}_implies_x_in_min-classes_{a,b}_)_)
let x be set ; ::_thesis: ( ( x in min-classes {a,b} implies x in {{a},{b}} ) & ( x in {{a},{b}} implies b1 in min-classes {a,b} ) )
hereby ::_thesis: ( x in {{a},{b}} implies b1 in min-classes {a,b} )
assume x in min-classes {a,b} ; ::_thesis: x in {{a},{b}}
then consider y being Element of (R \~) such that
A24: y is_minimal_wrt {a,b}, the InternalRel of (R \~) and
A25: x = (Class ((EqRel R),y)) /\ {a,b} by Def8;
A26: y in {a,b} by A24, WAYBEL_4:def_25;
percases ( y = a or y = b ) by A26, TARSKI:def_2;
supposeA27: y = a ; ::_thesis: x in {{a},{b}}
now__::_thesis:_for_z_being_set_holds_
(_(_z_in_x_implies_z_in_{a}_)_&_(_z_in_{a}_implies_z_in_x_)_)
let z be set ; ::_thesis: ( ( z in x implies z in {a} ) & ( z in {a} implies z in x ) )
hereby ::_thesis: ( z in {a} implies z in x )
assume A28: z in x ; ::_thesis: z in {a}
then A29: z in Class ((EqRel R),a) by A25, A27, XBOOLE_0:def_4;
A30: z in {a,b} by A25, A28, XBOOLE_0:def_4;
now__::_thesis:_z_in_{a}
percases ( z = a or z = b ) by A30, TARSKI:def_2;
suppose z = a ; ::_thesis: z in {a}
hence z in {a} by TARSKI:def_1; ::_thesis: verum
end;
suppose z = b ; ::_thesis: z in {a}
then [b,a] in EqRel R by A29, EQREL_1:19;
then [b,a] in the InternalRel of R /\ ( the InternalRel of R ~) by A1, Def4;
hence z in {a} by A22, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
end;
hence z in {a} ; ::_thesis: verum
end;
assume z in {a} ; ::_thesis: z in x
then A31: z = a by TARSKI:def_1;
then A32: z in {a,b} by TARSKI:def_2;
z in Class ((EqRel R),a) by A31, EQREL_1:20;
hence z in x by A25, A27, A32, XBOOLE_0:def_4; ::_thesis: verum
end;
then x = {a} by TARSKI:1;
hence x in {{a},{b}} by TARSKI:def_2; ::_thesis: verum
end;
supposeA33: y = b ; ::_thesis: x in {{a},{b}}
now__::_thesis:_for_z_being_set_holds_
(_(_z_in_x_implies_z_in_{b}_)_&_(_z_in_{b}_implies_z_in_x_)_)
let z be set ; ::_thesis: ( ( z in x implies z in {b} ) & ( z in {b} implies z in x ) )
hereby ::_thesis: ( z in {b} implies z in x )
assume A34: z in x ; ::_thesis: z in {b}
then A35: z in Class ((EqRel R),b) by A25, A33, XBOOLE_0:def_4;
A36: z in {a,b} by A25, A34, XBOOLE_0:def_4;
now__::_thesis:_z_in_{b}
percases ( z = b or z = a ) by A36, TARSKI:def_2;
suppose z = b ; ::_thesis: z in {b}
hence z in {b} by TARSKI:def_1; ::_thesis: verum
end;
suppose z = a ; ::_thesis: z in {b}
then [a,b] in EqRel R by A35, EQREL_1:19;
then [a,b] in the InternalRel of R /\ ( the InternalRel of R ~) by A1, Def4;
hence z in {b} by A21, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
end;
hence z in {b} ; ::_thesis: verum
end;
assume z in {b} ; ::_thesis: z in x
then A37: z = b by TARSKI:def_1;
then A38: z in {a,b} by TARSKI:def_2;
z in Class ((EqRel R),b) by A37, EQREL_1:20;
hence z in x by A25, A33, A38, XBOOLE_0:def_4; ::_thesis: verum
end;
then x = {b} by TARSKI:1;
hence x in {{a},{b}} by TARSKI:def_2; ::_thesis: verum
end;
end;
end;
assume A39: x in {{a},{b}} ; ::_thesis: b1 in min-classes {a,b}
percases ( x = {a} or x = {b} ) by A39, TARSKI:def_2;
supposeA40: x = {a} ; ::_thesis: b1 in min-classes {a,b}
now__::_thesis:_ex_a9_being_Element_of_(R_\~)_st_
(_a9_is_minimal_wrt_{a,b},_the_InternalRel_of_(R_\~)_&_x_=_(Class_((EqRel_R),a9))_/\_{a,b}_)
reconsider a9 = a as Element of (R \~) ;
take a9 = a9; ::_thesis: ( a9 is_minimal_wrt {a,b}, the InternalRel of (R \~) & x = (Class ((EqRel R),a9)) /\ {a,b} )
A41: a9 in {a,b} by TARSKI:def_2;
now__::_thesis:_for_y_being_set_holds_
(_not_y_in_{a,b}_or_not_y_<>_a9_or_not_[y,a9]_in_the_InternalRel_of_(R_\~)_)
assume ex y being set st
( y in {a,b} & y <> a9 & [y,a9] in the InternalRel of (R \~) ) ; ::_thesis: contradiction
then consider y being set such that
A42: y in {a,b} and
A43: y <> a9 and
A44: [y,a9] in the InternalRel of (R \~) ;
y = b by A42, A43, TARSKI:def_2;
hence contradiction by A20, A44, ORDERS_2:def_5; ::_thesis: verum
end;
hence a9 is_minimal_wrt {a,b}, the InternalRel of (R \~) by A41, WAYBEL_4:def_25; ::_thesis: x = (Class ((EqRel R),a9)) /\ {a,b}
now__::_thesis:_for_z_being_set_holds_
(_(_z_in_x_implies_z_in_(Class_((EqRel_R),a))_/\_{a,b}_)_&_(_z_in_(Class_((EqRel_R),a))_/\_{a,b}_implies_z_in_x_)_)
let z be set ; ::_thesis: ( ( z in x implies z in (Class ((EqRel R),a)) /\ {a,b} ) & ( z in (Class ((EqRel R),a)) /\ {a,b} implies b1 in x ) )
hereby ::_thesis: ( z in (Class ((EqRel R),a)) /\ {a,b} implies b1 in x )
assume z in x ; ::_thesis: z in (Class ((EqRel R),a)) /\ {a,b}
then A45: z = a by A40, TARSKI:def_1;
then z in Class ((EqRel R),a) by EQREL_1:20;
hence z in (Class ((EqRel R),a)) /\ {a,b} by A41, A45, XBOOLE_0:def_4; ::_thesis: verum
end;
assume A46: z in (Class ((EqRel R),a)) /\ {a,b} ; ::_thesis: b1 in x
then A47: z in Class ((EqRel R),a) by XBOOLE_0:def_4;
A48: z in {a,b} by A46, XBOOLE_0:def_4;
percases ( z = a or z = b ) by A48, TARSKI:def_2;
suppose z = a ; ::_thesis: b1 in x
hence z in x by A40, TARSKI:def_1; ::_thesis: verum
end;
suppose z = b ; ::_thesis: b1 in x
then [b,a] in EqRel R by A47, EQREL_1:19;
then [b,a] in the InternalRel of R /\ ( the InternalRel of R ~) by A1, Def4;
hence z in x by A22, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
end;
hence x = (Class ((EqRel R),a9)) /\ {a,b} by TARSKI:1; ::_thesis: verum
end;
hence x in min-classes {a,b} by Def8; ::_thesis: verum
end;
supposeA49: x = {b} ; ::_thesis: b1 in min-classes {a,b}
now__::_thesis:_ex_b9_being_Element_of_(R_\~)_st_
(_b9_is_minimal_wrt_{a,b},_the_InternalRel_of_(R_\~)_&_x_=_(Class_((EqRel_R),b9))_/\_{a,b}_)
reconsider b9 = b as Element of (R \~) ;
take b9 = b9; ::_thesis: ( b9 is_minimal_wrt {a,b}, the InternalRel of (R \~) & x = (Class ((EqRel R),b9)) /\ {a,b} )
A50: b9 in {a,b} by TARSKI:def_2;
now__::_thesis:_for_y_being_set_holds_
(_not_y_in_{a,b}_or_not_y_<>_b9_or_not_[y,b9]_in_the_InternalRel_of_(R_\~)_)
assume ex y being set st
( y in {a,b} & y <> b9 & [y,b9] in the InternalRel of (R \~) ) ; ::_thesis: contradiction
then consider y being set such that
A51: y in {a,b} and
A52: y <> b9 and
A53: [y,b9] in the InternalRel of (R \~) ;
y = a by A51, A52, TARSKI:def_2;
hence contradiction by A19, A53, ORDERS_2:def_5; ::_thesis: verum
end;
hence b9 is_minimal_wrt {a,b}, the InternalRel of (R \~) by A50, WAYBEL_4:def_25; ::_thesis: x = (Class ((EqRel R),b9)) /\ {a,b}
now__::_thesis:_for_z_being_set_holds_
(_(_z_in_x_implies_z_in_(Class_((EqRel_R),b))_/\_{a,b}_)_&_(_z_in_(Class_((EqRel_R),b))_/\_{a,b}_implies_z_in_x_)_)
let z be set ; ::_thesis: ( ( z in x implies z in (Class ((EqRel R),b)) /\ {a,b} ) & ( z in (Class ((EqRel R),b)) /\ {a,b} implies b1 in x ) )
hereby ::_thesis: ( z in (Class ((EqRel R),b)) /\ {a,b} implies b1 in x )
assume z in x ; ::_thesis: z in (Class ((EqRel R),b)) /\ {a,b}
then A54: z = b by A49, TARSKI:def_1;
then z in Class ((EqRel R),b) by EQREL_1:20;
hence z in (Class ((EqRel R),b)) /\ {a,b} by A50, A54, XBOOLE_0:def_4; ::_thesis: verum
end;
assume A55: z in (Class ((EqRel R),b)) /\ {a,b} ; ::_thesis: b1 in x
then A56: z in Class ((EqRel R),b) by XBOOLE_0:def_4;
A57: z in {a,b} by A55, XBOOLE_0:def_4;
percases ( z = b or z = a ) by A57, TARSKI:def_2;
suppose z = b ; ::_thesis: b1 in x
hence z in x by A49, TARSKI:def_1; ::_thesis: verum
end;
suppose z = a ; ::_thesis: b1 in x
then [a,b] in EqRel R by A56, EQREL_1:19;
then [a,b] in the InternalRel of R /\ ( the InternalRel of R ~) by A1, Def4;
hence z in x by A23, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
end;
hence x = (Class ((EqRel R),b9)) /\ {a,b} by TARSKI:1; ::_thesis: verum
end;
hence x in min-classes {a,b} by Def8; ::_thesis: verum
end;
end;
end;
then min-classes {a,b} = {{a},{b}} by TARSKI:1;
then A58: card {{a},{b}} = 1 by A18;
now__::_thesis:_not_{a}_=_{b}
assume {a} = {b} ; ::_thesis: contradiction
then A59: a = b by ZFMISC_1:3;
R is reflexive by A1, Def3;
then the InternalRel of R is_reflexive_in the carrier of R by ORDERS_2:def_2;
hence contradiction by A21, A59, RELAT_2:def_1; ::_thesis: verum
end;
hence contradiction by A58, CARD_2:57; ::_thesis: verum
end;
hence R is connected by WAYBEL_0:def_29; ::_thesis: R \~ is well_founded
now__::_thesis:_for_N_being_Subset_of_R_st_N_<>_{}_holds_
ex_x_being_set_st_x_in_min-classes_N
let N be Subset of R; ::_thesis: ( N <> {} implies ex x being set st x in min-classes N )
assume N <> {} ; ::_thesis: ex x being set st x in min-classes N
then card (min-classes N) = 1 by A18;
then min-classes N <> {} ;
hence ex x being set st x in min-classes N by XBOOLE_0:def_1; ::_thesis: verum
end;
hence R \~ is well_founded by Th20; ::_thesis: verum
end;
theorem :: DICKSON:24
for R being non empty Poset holds
( the InternalRel of R well_orders the carrier of R iff for N being non empty Subset of R holds card (min-classes N) = 1 )
proof
let R be non empty Poset; ::_thesis: ( the InternalRel of R well_orders the carrier of R iff for N being non empty Subset of R holds card (min-classes N) = 1 )
set IR = the InternalRel of R;
set CR = the carrier of R;
A1: R is quasi_ordered by Def3;
hereby ::_thesis: ( ( for N being non empty Subset of R holds card (min-classes N) = 1 ) implies the InternalRel of R well_orders the carrier of R )
assume A2: the InternalRel of R well_orders the carrier of R ; ::_thesis: for N being non empty Subset of R holds card (min-classes N) = 1
then A3: the InternalRel of R is_reflexive_in the carrier of R by WELLORD1:def_5;
A4: the InternalRel of R is_connected_in the carrier of R by A2, WELLORD1:def_5;
A5: the InternalRel of R is_well_founded_in the carrier of R by A2, WELLORD1:def_5;
the InternalRel of R is_strongly_connected_in the carrier of R by A3, A4, ORDERS_1:7;
then A6: R is connected by Th5;
R is well_founded by A5, WELLFND1:def_2;
then R \~ is well_founded by Th15;
hence for N being non empty Subset of R holds card (min-classes N) = 1 by A1, A6, Th23; ::_thesis: verum
end;
assume A7: for N being non empty Subset of R holds card (min-classes N) = 1 ; ::_thesis: the InternalRel of R well_orders the carrier of R
then A8: R is connected by A1, Th23;
A9: R \~ is well_founded by A1, A7, Th23;
A10: the InternalRel of R is_strongly_connected_in the carrier of R by A8, Th5;
A11: R is well_founded by A9, Th16;
A12: the InternalRel of R is_reflexive_in the carrier of R by ORDERS_2:def_2;
A13: the InternalRel of R is_transitive_in the carrier of R by ORDERS_2:def_3;
A14: the InternalRel of R is_antisymmetric_in the carrier of R by ORDERS_2:def_4;
A15: the InternalRel of R is_connected_in the carrier of R by A10, ORDERS_1:7;
the InternalRel of R is_well_founded_in the carrier of R by A11, WELLFND1:def_2;
hence the InternalRel of R well_orders the carrier of R by A12, A13, A14, A15, WELLORD1:def_5; ::_thesis: verum
end;
definition
let R be RelStr ;
let N be Subset of R;
let B be set ;
predB is_Dickson-basis_of N,R means :Def9: :: DICKSON:def 9
( B c= N & ( for a being Element of R st a in N holds
ex b being Element of R st
( b in B & b <= a ) ) );
end;
:: deftheorem Def9 defines is_Dickson-basis_of DICKSON:def_9_:_
for R being RelStr
for N being Subset of R
for B being set holds
( B is_Dickson-basis_of N,R iff ( B c= N & ( for a being Element of R st a in N holds
ex b being Element of R st
( b in B & b <= a ) ) ) );
theorem Th25: :: DICKSON:25
for R being RelStr holds {} is_Dickson-basis_of {} the carrier of R,R
proof
let R be RelStr ; ::_thesis: {} is_Dickson-basis_of {} the carrier of R,R
set N = {} the carrier of R;
thus {} c= {} the carrier of R ; :: according to DICKSON:def_9 ::_thesis: for a being Element of R st a in {} the carrier of R holds
ex b being Element of R st
( b in {} & b <= a )
thus for a being Element of R st a in {} the carrier of R holds
ex b being Element of R st
( b in {} & b <= a ) ; ::_thesis: verum
end;
theorem Th26: :: DICKSON:26
for R being non empty RelStr
for N being non empty Subset of R
for B being set st B is_Dickson-basis_of N,R holds
not B is empty
proof
let R be non empty RelStr ; ::_thesis: for N being non empty Subset of R
for B being set st B is_Dickson-basis_of N,R holds
not B is empty
let N be non empty Subset of R; ::_thesis: for B being set st B is_Dickson-basis_of N,R holds
not B is empty
let B be set ; ::_thesis: ( B is_Dickson-basis_of N,R implies not B is empty )
assume A1: B is_Dickson-basis_of N,R ; ::_thesis: not B is empty
set a = the Element of N;
ex b being Element of R st
( b in B & b <= the Element of N ) by A1, Def9;
hence not B is empty ; ::_thesis: verum
end;
definition
let R be RelStr ;
attrR is Dickson means :Def10: :: DICKSON:def 10
for N being Subset of R ex B being set st
( B is_Dickson-basis_of N,R & B is finite );
end;
:: deftheorem Def10 defines Dickson DICKSON:def_10_:_
for R being RelStr holds
( R is Dickson iff for N being Subset of R ex B being set st
( B is_Dickson-basis_of N,R & B is finite ) );
theorem Th27: :: DICKSON:27
for R being non empty RelStr st R \~ is well_founded & R is connected holds
R is Dickson
proof
let R be non empty RelStr ; ::_thesis: ( R \~ is well_founded & R is connected implies R is Dickson )
assume that
A1: R \~ is well_founded and
A2: R is connected ; ::_thesis: R is Dickson
set IR9 = the InternalRel of (R \~);
set CR9 = the carrier of (R \~);
set IR = the InternalRel of R;
set CR = the carrier of R;
let N be Subset of the carrier of R; :: according to DICKSON:def_10 ::_thesis: ex B being set st
( B is_Dickson-basis_of N,R & B is finite )
percases ( N = {} the carrier of R or N <> {} the carrier of R ) ;
supposeA3: N = {} the carrier of R ; ::_thesis: ex B being set st
( B is_Dickson-basis_of N,R & B is finite )
take B = {} ; ::_thesis: ( B is_Dickson-basis_of N,R & B is finite )
thus B is_Dickson-basis_of N,R by A3, Th25; ::_thesis: B is finite
thus B is finite ; ::_thesis: verum
end;
supposeA4: N <> {} the carrier of R ; ::_thesis: ex B being set st
( B is_Dickson-basis_of N,R & B is finite )
the InternalRel of (R \~) is_well_founded_in the carrier of (R \~) by A1, WELLFND1:def_2;
then consider b being set such that
A5: b in N and
A6: the InternalRel of (R \~) -Seg b misses N by A4, WELLORD1:def_3;
A7: ( the InternalRel of (R \~) -Seg b) /\ N = {} by A6, XBOOLE_0:def_7;
take B = {b}; ::_thesis: ( B is_Dickson-basis_of N,R & B is finite )
reconsider b = b as Element of N by A5;
thus B is_Dickson-basis_of N,R ::_thesis: B is finite
proof
{b} is Subset of N by A4, SUBSET_1:33;
hence B c= N ; :: according to DICKSON:def_9 ::_thesis: for a being Element of R st a in N holds
ex b being Element of R st
( b in B & b <= a )
let a be Element of R; ::_thesis: ( a in N implies ex b being Element of R st
( b in B & b <= a ) )
assume A8: a in N ; ::_thesis: ex b being Element of R st
( b in B & b <= a )
reconsider b = b as Element of R by A5;
take b ; ::_thesis: ( b in B & b <= a )
thus b in B by TARSKI:def_1; ::_thesis: b <= a
percases ( b <= a or a <= b ) by A2, WAYBEL_0:def_29;
suppose b <= a ; ::_thesis: b <= a
hence b <= a ; ::_thesis: verum
end;
supposeA9: a <= b ; ::_thesis: b <= a
then A10: [a,b] in the InternalRel of R by ORDERS_2:def_5;
now__::_thesis:_b_<=_a
percases ( a = b or not a = b ) ;
suppose a = b ; ::_thesis: b <= a
hence b <= a by A9; ::_thesis: verum
end;
supposeA11: not a = b ; ::_thesis: b <= a
now__::_thesis:_b_<=_a
percases ( [a,b] in the InternalRel of (R \~) or not [a,b] in the InternalRel of (R \~) ) ;
suppose [a,b] in the InternalRel of (R \~) ; ::_thesis: b <= a
then a in the InternalRel of (R \~) -Seg b by A11, WELLORD1:1;
hence b <= a by A7, A8, XBOOLE_0:def_4; ::_thesis: verum
end;
suppose not [a,b] in the InternalRel of (R \~) ; ::_thesis: b <= a
then [a,b] in the InternalRel of R ~ by A10, XBOOLE_0:def_5;
then [b,a] in the InternalRel of R by RELAT_1:def_7;
hence b <= a by ORDERS_2:def_5; ::_thesis: verum
end;
end;
end;
hence b <= a ; ::_thesis: verum
end;
end;
end;
hence b <= a ; ::_thesis: verum
end;
end;
end;
thus B is finite ; ::_thesis: verum
end;
end;
end;
theorem Th28: :: DICKSON:28
for R, S being RelStr st the InternalRel of R c= the InternalRel of S & R is Dickson & the carrier of R = the carrier of S holds
S is Dickson
proof
let r, s be RelStr ; ::_thesis: ( the InternalRel of r c= the InternalRel of s & r is Dickson & the carrier of r = the carrier of s implies s is Dickson )
assume that
A1: the InternalRel of r c= the InternalRel of s and
A2: r is Dickson and
A3: the carrier of r = the carrier of s ; ::_thesis: s is Dickson
let N be Subset of s; :: according to DICKSON:def_10 ::_thesis: ex B being set st
( B is_Dickson-basis_of N,s & B is finite )
reconsider N9 = N as Subset of r by A3;
consider B being set such that
A4: B is_Dickson-basis_of N9,r and
A5: B is finite by A2, Def10;
take B ; ::_thesis: ( B is_Dickson-basis_of N,s & B is finite )
thus B c= N by A4, Def9; :: according to DICKSON:def_9 ::_thesis: ( ( for a being Element of s st a in N holds
ex b being Element of s st
( b in B & b <= a ) ) & B is finite )
hereby ::_thesis: B is finite
let a be Element of s; ::_thesis: ( a in N implies ex b9 being Element of s st
( b9 in B & b9 <= a ) )
assume A6: a in N ; ::_thesis: ex b9 being Element of s st
( b9 in B & b9 <= a )
reconsider a9 = a as Element of r by A3;
consider b being Element of r such that
A7: b in B and
A8: b <= a9 by A4, A6, Def9;
reconsider b9 = b as Element of s by A3;
take b9 = b9; ::_thesis: ( b9 in B & b9 <= a )
[b,a9] in the InternalRel of r by A8, ORDERS_2:def_5;
hence ( b9 in B & b9 <= a ) by A1, A7, ORDERS_2:def_5; ::_thesis: verum
end;
thus B is finite by A5; ::_thesis: verum
end;
definition
let f be Function;
let b be set ;
assume that
B1: dom f = NAT and
B2: b in rng f ;
funcf mindex b -> Element of NAT means :Def11: :: DICKSON:def 11
( f . it = b & ( for i being Element of NAT st f . i = b holds
it <= i ) );
existence
ex b1 being Element of NAT st
( f . b1 = b & ( for i being Element of NAT st f . i = b holds
b1 <= i ) )
proof
set N = { i where i is Element of NAT : f . i = b } ;
consider x being set such that
A1: x in NAT and
A2: f . x = b by B1, B2, FUNCT_1:def_3;
reconsider x = x as Element of NAT by A1;
A3: x in { i where i is Element of NAT : f . i = b } by A2;
now__::_thesis:_for_x_being_set_st_x_in__{__i_where_i_is_Element_of_NAT_:_f_._i_=_b__}__holds_
x_in_NAT
let x be set ; ::_thesis: ( x in { i where i is Element of NAT : f . i = b } implies x in NAT )
assume x in { i where i is Element of NAT : f . i = b } ; ::_thesis: x in NAT
then ex i being Element of NAT st
( x = i & f . i = b ) ;
hence x in NAT ; ::_thesis: verum
end;
then reconsider N = { i where i is Element of NAT : f . i = b } as non empty Subset of NAT by A3, TARSKI:def_3;
take I = min N; ::_thesis: ( f . I = b & ( for i being Element of NAT st f . i = b holds
I <= i ) )
I in N by XXREAL_2:def_7;
then ex II being Element of NAT st
( II = I & f . II = b ) ;
hence f . I = b ; ::_thesis: for i being Element of NAT st f . i = b holds
I <= i
let i be Element of NAT ; ::_thesis: ( f . i = b implies I <= i )
assume f . i = b ; ::_thesis: I <= i
then i in N ;
hence I <= i by XXREAL_2:def_7; ::_thesis: verum
end;
uniqueness
for b1, b2 being Element of NAT st f . b1 = b & ( for i being Element of NAT st f . i = b holds
b1 <= i ) & f . b2 = b & ( for i being Element of NAT st f . i = b holds
b2 <= i ) holds
b1 = b2
proof
let IT1, IT2 be Element of NAT ; ::_thesis: ( f . IT1 = b & ( for i being Element of NAT st f . i = b holds
IT1 <= i ) & f . IT2 = b & ( for i being Element of NAT st f . i = b holds
IT2 <= i ) implies IT1 = IT2 )
assume that
A4: f . IT1 = b and
A5: for i being Element of NAT st f . i = b holds
IT1 <= i and
A6: f . IT2 = b and
A7: for i being Element of NAT st f . i = b holds
IT2 <= i ; ::_thesis: IT1 = IT2
assume A8: IT1 <> IT2 ; ::_thesis: contradiction
percases ( IT1 < IT2 or IT1 > IT2 ) by A8, XXREAL_0:1;
suppose IT1 < IT2 ; ::_thesis: contradiction
hence contradiction by A4, A7; ::_thesis: verum
end;
suppose IT1 > IT2 ; ::_thesis: contradiction
hence contradiction by A5, A6; ::_thesis: verum
end;
end;
end;
end;
:: deftheorem Def11 defines mindex DICKSON:def_11_:_
for f being Function
for b being set st dom f = NAT & b in rng f holds
for b3 being Element of NAT holds
( b3 = f mindex b iff ( f . b3 = b & ( for i being Element of NAT st f . i = b holds
b3 <= i ) ) );
definition
let R be non empty 1-sorted ;
let f be sequence of R;
let b be set ;
let m be Element of NAT ;
assume A1: ex j being Element of NAT st
( m < j & f . j = b ) ;
funcf mindex (b,m) -> Element of NAT means :Def12: :: DICKSON:def 12
( f . it = b & m < it & ( for i being Element of NAT st m < i & f . i = b holds
it <= i ) );
existence
ex b1 being Element of NAT st
( f . b1 = b & m < b1 & ( for i being Element of NAT st m < i & f . i = b holds
b1 <= i ) )
proof
set N = { i where i is Element of NAT : ( m < i & f . i = b ) } ;
consider j being Element of NAT such that
A2: m < j and
A3: f . j = b by A1;
A4: j in { i where i is Element of NAT : ( m < i & f . i = b ) } by A2, A3;
now__::_thesis:_for_x_being_set_st_x_in__{__i_where_i_is_Element_of_NAT_:_(_m_<_i_&_f_._i_=_b_)__}__holds_
x_in_NAT
let x be set ; ::_thesis: ( x in { i where i is Element of NAT : ( m < i & f . i = b ) } implies x in NAT )
assume x in { i where i is Element of NAT : ( m < i & f . i = b ) } ; ::_thesis: x in NAT
then ex i being Element of NAT st
( x = i & m < i & f . i = b ) ;
hence x in NAT ; ::_thesis: verum
end;
then reconsider N = { i where i is Element of NAT : ( m < i & f . i = b ) } as non empty Subset of NAT by A4, TARSKI:def_3;
take I = min N; ::_thesis: ( f . I = b & m < I & ( for i being Element of NAT st m < i & f . i = b holds
I <= i ) )
I in N by XXREAL_2:def_7;
then ex II being Element of NAT st
( II = I & m < II & f . II = b ) ;
hence ( f . I = b & m < I ) ; ::_thesis: for i being Element of NAT st m < i & f . i = b holds
I <= i
let i be Element of NAT ; ::_thesis: ( m < i & f . i = b implies I <= i )
assume that
A5: m < i and
A6: f . i = b ; ::_thesis: I <= i
i in N by A5, A6;
hence I <= i by XXREAL_2:def_7; ::_thesis: verum
end;
uniqueness
for b1, b2 being Element of NAT st f . b1 = b & m < b1 & ( for i being Element of NAT st m < i & f . i = b holds
b1 <= i ) & f . b2 = b & m < b2 & ( for i being Element of NAT st m < i & f . i = b holds
b2 <= i ) holds
b1 = b2
proof
let IT1, IT2 be Element of NAT ; ::_thesis: ( f . IT1 = b & m < IT1 & ( for i being Element of NAT st m < i & f . i = b holds
IT1 <= i ) & f . IT2 = b & m < IT2 & ( for i being Element of NAT st m < i & f . i = b holds
IT2 <= i ) implies IT1 = IT2 )
assume that
A7: f . IT1 = b and
A8: m < IT1 and
A9: for i being Element of NAT st m < i & f . i = b holds
IT1 <= i and
A10: f . IT2 = b and
A11: m < IT2 and
A12: for i being Element of NAT st m < i & f . i = b holds
IT2 <= i ; ::_thesis: IT1 = IT2
assume A13: IT1 <> IT2 ; ::_thesis: contradiction
percases ( IT1 < IT2 or IT1 > IT2 ) by A13, XXREAL_0:1;
suppose IT1 < IT2 ; ::_thesis: contradiction
hence contradiction by A7, A8, A12; ::_thesis: verum
end;
suppose IT1 > IT2 ; ::_thesis: contradiction
hence contradiction by A9, A10, A11; ::_thesis: verum
end;
end;
end;
end;
:: deftheorem Def12 defines mindex DICKSON:def_12_:_
for R being non empty 1-sorted
for f being sequence of R
for b being set
for m being Element of NAT st ex j being Element of NAT st
( m < j & f . j = b ) holds
for b5 being Element of NAT holds
( b5 = f mindex (b,m) iff ( f . b5 = b & m < b5 & ( for i being Element of NAT st m < i & f . i = b holds
b5 <= i ) ) );
theorem Th29: :: DICKSON:29
for R being non empty RelStr st R is Dickson holds
for f being sequence of R ex i, j being Element of NAT st
( i < j & f . i <= f . j )
proof
let R be non empty RelStr ; ::_thesis: ( R is Dickson implies for f being sequence of R ex i, j being Element of NAT st
( i < j & f . i <= f . j ) )
assume A1: R is Dickson ; ::_thesis: for f being sequence of R ex i, j being Element of NAT st
( i < j & f . i <= f . j )
let f be sequence of R; ::_thesis: ex i, j being Element of NAT st
( i < j & f . i <= f . j )
set N = rng f;
A2: dom f = NAT by FUNCT_2:def_1;
consider B being set such that
A3: B is_Dickson-basis_of rng f,R and
A4: B is finite by A1, Def10;
reconsider B = B as non empty finite set by A3, A4, Th26;
defpred S1[ set ] means verum;
deffunc H1( set ) -> Element of NAT = f mindex $1;
set BI = { H1(b) where b is Element of B : S1[b] } ;
A5: { H1(b) where b is Element of B : S1[b] } is finite from PRE_CIRC:sch_1();
set b = the Element of B;
A6: f mindex the Element of B in { H1(b) where b is Element of B : S1[b] } ;
now__::_thesis:_for_x_being_set_st_x_in__{__H1(b)_where_b_is_Element_of_B_:_S1[b]__}__holds_
x_in_NAT
let x be set ; ::_thesis: ( x in { H1(b) where b is Element of B : S1[b] } implies x in NAT )
assume x in { H1(b) where b is Element of B : S1[b] } ; ::_thesis: x in NAT
then ex b being Element of B st x = f mindex b ;
hence x in NAT ; ::_thesis: verum
end;
then reconsider BI = { H1(b) where b is Element of B : S1[b] } as non empty finite Subset of NAT by A5, A6, TARSKI:def_3;
reconsider mB = max BI as Element of NAT by ORDINAL1:def_12;
set j = mB + 1;
reconsider fj = f . (mB + 1) as Element of R ;
fj in rng f by A2, FUNCT_1:3;
then consider b being Element of R such that
A7: b in B and
A8: b <= fj by A3, Def9;
A9: B c= rng f by A3, Def9;
take i = f mindex b; ::_thesis: ex j being Element of NAT st
( i < j & f . i <= f . j )
take mB + 1 ; ::_thesis: ( i < mB + 1 & f . i <= f . (mB + 1) )
i in BI by A7;
then i <= max BI by XXREAL_2:def_8;
hence i < mB + 1 by NAT_1:13; ::_thesis: f . i <= f . (mB + 1)
dom f = NAT by NORMSP_1:12;
hence f . i <= f . (mB + 1) by A7, A8, A9, Def11; ::_thesis: verum
end;
theorem Th30: :: DICKSON:30
for R being RelStr
for N being Subset of R
for x being Element of (R \~) st R is quasi_ordered & x in N & ( the InternalRel of R -Seg x) /\ N c= Class ((EqRel R),x) holds
x is_minimal_wrt N, the InternalRel of (R \~)
proof
let R be RelStr ; ::_thesis: for N being Subset of R
for x being Element of (R \~) st R is quasi_ordered & x in N & ( the InternalRel of R -Seg x) /\ N c= Class ((EqRel R),x) holds
x is_minimal_wrt N, the InternalRel of (R \~)
let N be Subset of R; ::_thesis: for x being Element of (R \~) st R is quasi_ordered & x in N & ( the InternalRel of R -Seg x) /\ N c= Class ((EqRel R),x) holds
x is_minimal_wrt N, the InternalRel of (R \~)
let x be Element of (R \~); ::_thesis: ( R is quasi_ordered & x in N & ( the InternalRel of R -Seg x) /\ N c= Class ((EqRel R),x) implies x is_minimal_wrt N, the InternalRel of (R \~) )
assume that
A1: R is quasi_ordered and
A2: x in N and
A3: ( the InternalRel of R -Seg x) /\ N c= Class ((EqRel R),x) ; ::_thesis: x is_minimal_wrt N, the InternalRel of (R \~)
set IR = the InternalRel of R;
set IR9 = the InternalRel of (R \~);
now__::_thesis:_for_y_being_set_holds_
(_not_y_in_N_or_not_y_<>_x_or_not_[y,x]_in_the_InternalRel_of_(R_\~)_)
assume ex y being set st
( y in N & y <> x & [y,x] in the InternalRel of (R \~) ) ; ::_thesis: contradiction
then consider y being set such that
A4: y in N and
A5: y <> x and
A6: [y,x] in the InternalRel of (R \~) ;
A7: not [y,x] in the InternalRel of R ~ by A6, XBOOLE_0:def_5;
y in the InternalRel of R -Seg x by A5, A6, WELLORD1:1;
then y in ( the InternalRel of R -Seg x) /\ N by A4, XBOOLE_0:def_4;
then [y,x] in EqRel R by A3, EQREL_1:19;
then [y,x] in the InternalRel of R /\ ( the InternalRel of R ~) by A1, Def4;
hence contradiction by A7, XBOOLE_0:def_4; ::_thesis: verum
end;
hence x is_minimal_wrt N, the InternalRel of (R \~) by A2, WAYBEL_4:def_25; ::_thesis: verum
end;
theorem Th31: :: DICKSON:31
for R being non empty RelStr st R is quasi_ordered & ( for f being sequence of R ex i, j being Element of NAT st
( i < j & f . i <= f . j ) ) holds
for N being non empty Subset of R holds
( min-classes N is finite & not min-classes N is empty )
proof
let R be non empty RelStr ; ::_thesis: ( R is quasi_ordered & ( for f being sequence of R ex i, j being Element of NAT st
( i < j & f . i <= f . j ) ) implies for N being non empty Subset of R holds
( min-classes N is finite & not min-classes N is empty ) )
assume that
A1: R is quasi_ordered and
A2: for f being sequence of R ex i, j being Element of NAT st
( i < j & f . i <= f . j ) ; ::_thesis: for N being non empty Subset of R holds
( min-classes N is finite & not min-classes N is empty )
set IR = the InternalRel of R;
set IR9 = the InternalRel of (R \~);
A3: R is transitive by A1, Def3;
let N be non empty Subset of R; ::_thesis: ( min-classes N is finite & not min-classes N is empty )
assume A4: ( not min-classes N is finite or min-classes N is empty ) ; ::_thesis: contradiction
percases ( min-classes N is infinite or min-classes N is empty ) by A4;
supposeA5: min-classes N is infinite ; ::_thesis: contradiction
then reconsider MCN = min-classes N as infinite set ;
consider f being Function of NAT,(min-classes N) such that
A6: f is one-to-one by A5, Th3;
deffunc H1( set ) -> Element of f . $1 = choose (f . $1);
A7: now__::_thesis:_for_x_being_set_st_x_in_NAT_holds_
H1(x)_in_the_carrier_of_R
let x be set ; ::_thesis: ( x in NAT implies H1(x) in the carrier of R )
assume x in NAT ; ::_thesis: H1(x) in the carrier of R
then reconsider x9 = x as Element of NAT ;
f . x9 is Element of min-classes N ;
then A8: f . x in MCN ;
then not f . x is empty by Th22;
then choose (f . x9) in f . x ;
hence H1(x) in the carrier of R by A8; ::_thesis: verum
end;
consider g being Function of NAT, the carrier of R such that
A9: for x being set st x in NAT holds
g . x = H1(x) from FUNCT_2:sch_2(A7);
reconsider g = g as sequence of R ;
consider i, j being Element of NAT such that
A10: i < j and
A11: g . i <= g . j by A2;
reconsider gi = g . i, gj = g . j as Element of (R \~) ;
A12: [gi,gj] in the InternalRel of R by A11, ORDERS_2:def_5;
A13: f . i in MCN ;
then A14: not f . i is empty by Th22;
A15: f . j in MCN ;
then A16: not f . j is empty by Th22;
A17: g . j = choose (f . j) by A9;
A18: g . i = choose (f . i) by A9;
A19: gj is_minimal_wrt N, the InternalRel of (R \~) by A1, A15, A16, A17, Th19;
gi is_minimal_wrt N, the InternalRel of (R \~) by A1, A13, A14, A18, Th19;
then A20: gi in N by WAYBEL_4:def_25;
A21: now__::_thesis:_[gi,gj]_in_the_InternalRel_of_R_~
percases ( gi = gj or gi <> gj ) ;
suppose gi = gj ; ::_thesis: [gi,gj] in the InternalRel of R ~
hence [gi,gj] in the InternalRel of R ~ by A12, RELAT_1:def_7; ::_thesis: verum
end;
supposeA22: gi <> gj ; ::_thesis: [gi,gj] in the InternalRel of R ~
now__::_thesis:_[gi,gj]_in_the_InternalRel_of_R_~
assume not [gi,gj] in the InternalRel of R ~ ; ::_thesis: contradiction
then [gi,gj] in the InternalRel of R \ ( the InternalRel of R ~) by A12, XBOOLE_0:def_5;
hence contradiction by A19, A20, A22, WAYBEL_4:def_25; ::_thesis: verum
end;
hence [gi,gj] in the InternalRel of R ~ ; ::_thesis: verum
end;
end;
end;
[gi,gj] in the InternalRel of R by A11, ORDERS_2:def_5;
then [gi,gj] in the InternalRel of R /\ ( the InternalRel of R ~) by A21, XBOOLE_0:def_4;
then [gi,gj] in EqRel R by A1, Def4;
then gi in Class ((EqRel R),gj) by EQREL_1:19;
then A23: Class ((EqRel R),gj) = Class ((EqRel R),gi) by EQREL_1:23;
consider mj being Element of (R \~) such that
mj is_minimal_wrt N, the InternalRel of (R \~) and
A24: f . j = (Class ((EqRel R),mj)) /\ N by A15, Def8;
consider mi being Element of (R \~) such that
mi is_minimal_wrt N, the InternalRel of (R \~) and
A25: f . i = (Class ((EqRel R),mi)) /\ N by A13, Def8;
gj in Class ((EqRel R),mj) by A16, A17, A24, XBOOLE_0:def_4;
then A26: Class ((EqRel R),gj) = Class ((EqRel R),mj) by EQREL_1:23;
gi in Class ((EqRel R),mi) by A14, A18, A25, XBOOLE_0:def_4;
then f . i = f . j by A23, A24, A25, A26, EQREL_1:23;
hence contradiction by A5, A6, A10, FUNCT_2:19; ::_thesis: verum
end;
supposeA27: min-classes N is empty ; ::_thesis: contradiction
deffunc H1( set , set ) -> Element of (( the InternalRel of R -Seg $2) /\ N) \ (Class ((EqRel R),$2)) = choose ((( the InternalRel of R -Seg $2) /\ N) \ (Class ((EqRel R),$2)));
consider f being Function such that
A28: dom f = NAT and
A29: f . 0 = choose N and
A30: for n being Nat holds f . (n + 1) = H1(n,f . n) from NAT_1:sch_11();
defpred S1[ Nat] means f . $1 in N;
A31: S1[ 0 ] by A29;
A32: now__::_thesis:_for_i_being_Element_of_NAT_st_S1[i]_holds_
S1[i_+_1]
let i be Element of NAT ; ::_thesis: ( S1[i] implies S1[i + 1] )
assume A33: S1[i] ; ::_thesis: S1[i + 1]
reconsider fi = f . i as Element of (R \~) by A33;
set IC = (( the InternalRel of R -Seg fi) /\ N) \ (Class ((EqRel R),fi));
A34: f . (i + 1) = choose ((( the InternalRel of R -Seg (f . i)) /\ N) \ (Class ((EqRel R),(f . i)))) by A30;
now__::_thesis:_not_((_the_InternalRel_of_R_-Seg_fi)_/\_N)_\_(Class_((EqRel_R),fi))_is_empty
assume (( the InternalRel of R -Seg fi) /\ N) \ (Class ((EqRel R),fi)) is empty ; ::_thesis: contradiction
then ( the InternalRel of R -Seg fi) /\ N c= Class ((EqRel R),fi) by XBOOLE_1:37;
hence contradiction by A1, A27, A33, Th21, Th30; ::_thesis: verum
end;
then f . (i + 1) in ( the InternalRel of R -Seg (f . i)) /\ N by A34, XBOOLE_0:def_5;
hence S1[i + 1] by XBOOLE_0:def_4; ::_thesis: verum
end;
A35: for i being Element of NAT holds S1[i] from NAT_1:sch_1(A31, A32);
now__::_thesis:_for_x_being_set_st_x_in_NAT_holds_
f_._x_in_the_carrier_of_R
let x be set ; ::_thesis: ( x in NAT implies f . x in the carrier of R )
assume x in NAT ; ::_thesis: f . x in the carrier of R
then f . x in N by A35;
hence f . x in the carrier of R ; ::_thesis: verum
end;
then reconsider f = f as sequence of R by A28, FUNCT_2:3;
A36: now__::_thesis:_for_i,_j_being_Element_of_NAT_holds_S2[j]
let i be Element of NAT ; ::_thesis: for j being Element of NAT holds S2[j]
defpred S2[ Element of NAT ] means ( i < $1 implies f . i >= f . $1 );
A37: S2[ 0 ] by NAT_1:2;
A38: for j being Element of NAT st S2[j] holds
S2[j + 1]
proof
let j be Element of NAT ; ::_thesis: ( S2[j] implies S2[j + 1] )
assume that
A39: ( i < j implies f . i >= f . j ) and
A40: i < j + 1 ; ::_thesis: f . i >= f . (j + 1)
A41: i <= j by A40, NAT_1:13;
reconsider fj = f . j, fj1 = f . (j + 1) as Element of (R \~) ;
set IC = (( the InternalRel of R -Seg fj) /\ N) \ (Class ((EqRel R),fj));
A42: fj in N by A35;
A43: fj1 = choose ((( the InternalRel of R -Seg fj) /\ N) \ (Class ((EqRel R),fj))) by A30;
now__::_thesis:_not_((_the_InternalRel_of_R_-Seg_fj)_/\_N)_\_(Class_((EqRel_R),fj))_is_empty
assume (( the InternalRel of R -Seg fj) /\ N) \ (Class ((EqRel R),fj)) is empty ; ::_thesis: contradiction
then ( the InternalRel of R -Seg fj) /\ N c= Class ((EqRel R),fj) by XBOOLE_1:37;
hence contradiction by A1, A27, A42, Th21, Th30; ::_thesis: verum
end;
then fj1 in ( the InternalRel of R -Seg fj) /\ N by A43, XBOOLE_0:def_5;
then fj1 in the InternalRel of R -Seg fj by XBOOLE_0:def_4;
then A44: [fj1,fj] in the InternalRel of R by WELLORD1:1;
then A45: f . j >= f . (j + 1) by ORDERS_2:def_5;
percases ( i < j or i = j ) by A41, XXREAL_0:1;
suppose i < j ; ::_thesis: f . i >= f . (j + 1)
hence f . i >= f . (j + 1) by A3, A39, A45, ORDERS_2:3; ::_thesis: verum
end;
suppose i = j ; ::_thesis: f . i >= f . (j + 1)
hence f . i >= f . (j + 1) by A44, ORDERS_2:def_5; ::_thesis: verum
end;
end;
end;
thus for j being Element of NAT holds S2[j] from NAT_1:sch_1(A37, A38); ::_thesis: verum
end;
now__::_thesis:_for_i,_j_being_Element_of_NAT_holds_S2[j]
let i be Element of NAT ; ::_thesis: for j being Element of NAT holds S2[j]
defpred S2[ Element of NAT ] means ( i < $1 implies not f . i <= f . $1 );
A46: S2[ 0 ] by NAT_1:2;
A47: for j being Element of NAT st S2[j] holds
S2[j + 1]
proof
let j be Element of NAT ; ::_thesis: ( S2[j] implies S2[j + 1] )
assume that
( i < j implies not f . i <= f . j ) and
A48: i < j + 1 ; ::_thesis: not f . i <= f . (j + 1)
A49: i <= j by A48, NAT_1:13;
reconsider fj = f . j, fj1 = f . (j + 1) as Element of (R \~) ;
A50: fj in N by A35;
percases ( i < j or i = j ) by A49, XXREAL_0:1;
supposeA51: i < j ; ::_thesis: not f . i <= f . (j + 1)
assume A52: f . i <= f . (j + 1) ; ::_thesis: contradiction
j < j + 1 by NAT_1:13;
then A53: f . j >= f . (j + 1) by A36;
f . i >= f . j by A36, A51;
then f . j <= f . (j + 1) by A3, A52, ORDERS_2:3;
then A54: fj1 in Class ((EqRel R),fj) by A1, A53, Th8;
set IC = (( the InternalRel of R -Seg fj) /\ N) \ (Class ((EqRel R),fj));
A55: fj1 = choose ((( the InternalRel of R -Seg fj) /\ N) \ (Class ((EqRel R),fj))) by A30;
now__::_thesis:_not_((_the_InternalRel_of_R_-Seg_fj)_/\_N)_\_(Class_((EqRel_R),fj))_is_empty
assume (( the InternalRel of R -Seg fj) /\ N) \ (Class ((EqRel R),fj)) is empty ; ::_thesis: contradiction
then ( the InternalRel of R -Seg fj) /\ N c= Class ((EqRel R),fj) by XBOOLE_1:37;
hence contradiction by A1, A27, A50, Th21, Th30; ::_thesis: verum
end;
hence contradiction by A54, A55, XBOOLE_0:def_5; ::_thesis: verum
end;
supposeA56: i = j ; ::_thesis: not f . i <= f . (j + 1)
assume A57: f . i <= f . (j + 1) ; ::_thesis: contradiction
j < j + 1 by NAT_1:13;
then f . (j + 1) <= f . j by A36;
then A58: fj1 in Class ((EqRel R),fj) by A1, A56, A57, Th8;
set IC = (( the InternalRel of R -Seg fj) /\ N) \ (Class ((EqRel R),fj));
A59: fj1 = choose ((( the InternalRel of R -Seg fj) /\ N) \ (Class ((EqRel R),fj))) by A30;
now__::_thesis:_not_((_the_InternalRel_of_R_-Seg_fj)_/\_N)_\_(Class_((EqRel_R),fj))_is_empty
assume (( the InternalRel of R -Seg fj) /\ N) \ (Class ((EqRel R),fj)) is empty ; ::_thesis: contradiction
then ( the InternalRel of R -Seg fj) /\ N c= Class ((EqRel R),fj) by XBOOLE_1:37;
hence contradiction by A1, A27, A50, Th21, Th30; ::_thesis: verum
end;
hence contradiction by A58, A59, XBOOLE_0:def_5; ::_thesis: verum
end;
end;
end;
thus for j being Element of NAT holds S2[j] from NAT_1:sch_1(A46, A47); ::_thesis: verum
end;
hence contradiction by A2; ::_thesis: verum
end;
end;
end;
theorem Th32: :: DICKSON:32
for R being non empty RelStr st R is quasi_ordered & ( for N being non empty Subset of R holds
( min-classes N is finite & not min-classes N is empty ) ) holds
R is Dickson
proof
let R be non empty RelStr ; ::_thesis: ( R is quasi_ordered & ( for N being non empty Subset of R holds
( min-classes N is finite & not min-classes N is empty ) ) implies R is Dickson )
assume that
A1: R is quasi_ordered and
A2: for N being non empty Subset of R holds
( min-classes N is finite & not min-classes N is empty ) ; ::_thesis: R is Dickson
A3: R is transitive by A1, Def3;
A4: R is reflexive by A1, Def3;
set IR = the InternalRel of R;
set CR = the carrier of R;
set IR9 = the InternalRel of (R \~);
let N be Subset of the carrier of R; :: according to DICKSON:def_10 ::_thesis: ex B being set st
( B is_Dickson-basis_of N,R & B is finite )
percases ( N = {} the carrier of R or not N is empty ) ;
supposeA5: N = {} the carrier of R ; ::_thesis: ex B being set st
( B is_Dickson-basis_of N,R & B is finite )
take B = {} ; ::_thesis: ( B is_Dickson-basis_of N,R & B is finite )
thus B is_Dickson-basis_of N,R by A5, Th25; ::_thesis: B is finite
thus B is finite ; ::_thesis: verum
end;
suppose not N is empty ; ::_thesis: ex B being set st
( B is_Dickson-basis_of N,R & B is finite )
then reconsider N9 = N as non empty Subset of the carrier of R ;
reconsider MCN9 = min-classes N9 as non empty finite Subset-Family of the carrier of R by A2;
take B = { (choose x) where x is Element of MCN9 : verum } ; ::_thesis: ( B is_Dickson-basis_of N,R & B is finite )
thus B is_Dickson-basis_of N,R ::_thesis: B is finite
proof
now__::_thesis:_for_x_being_set_st_x_in_B_holds_
x_in_N
let x be set ; ::_thesis: ( x in B implies x in N )
assume x in B ; ::_thesis: x in N
then consider y being Element of MCN9 such that
A6: x = choose y ;
A7: ex z being Element of (R \~) st
( z is_minimal_wrt N, the InternalRel of (R \~) & y = (Class ((EqRel R),z)) /\ N ) by Def8;
not y is empty by Th22;
hence x in N by A6, A7, XBOOLE_0:def_4; ::_thesis: verum
end;
hence B c= N by TARSKI:def_3; :: according to DICKSON:def_9 ::_thesis: for a being Element of R st a in N holds
ex b being Element of R st
( b in B & b <= a )
let a be Element of R; ::_thesis: ( a in N implies ex b being Element of R st
( b in B & b <= a ) )
assume A8: a in N ; ::_thesis: ex b being Element of R st
( b in B & b <= a )
defpred S1[ Element of R] means $1 <= a;
set NN9 = { d where d is Element of N9 : S1[d] } ;
A9: { d where d is Element of N9 : S1[d] } is Subset of N9 from DOMAIN_1:sch_7();
a <= a by A4, ORDERS_2:1;
then a in { d where d is Element of N9 : S1[d] } by A8;
then reconsider NN9 = { d where d is Element of N9 : S1[d] } as non empty Subset of the carrier of R by A9, XBOOLE_1:1;
set m = the Element of min-classes NN9;
A10: not min-classes NN9 is empty by A2;
then reconsider m9 = the Element of min-classes NN9 as non empty set by Th22;
set c = the Element of m9;
consider y being Element of (R \~) such that
y is_minimal_wrt NN9, the InternalRel of (R \~) and
A11: m9 = (Class ((EqRel R),y)) /\ NN9 by A10, Def8;
A12: the Element of m9 in Class ((EqRel R),y) by A11, XBOOLE_0:def_4;
A13: the Element of m9 in NN9 by A11, XBOOLE_0:def_4;
reconsider c = the Element of m9 as Element of (R \~) by A12;
reconsider c9 = c as Element of R ;
A14: ex d being Element of N9 st
( c = d & d <= a ) by A13;
A15: c is_minimal_wrt NN9, the InternalRel of (R \~) by A1, A10, Th19;
now__::_thesis:_c_is_minimal_wrt_N,_the_InternalRel_of_(R_\~)
assume not c is_minimal_wrt N, the InternalRel of (R \~) ; ::_thesis: contradiction
then consider w being set such that
A16: w in N and
A17: w <> c and
A18: [w,c] in the InternalRel of (R \~) by A14, WAYBEL_4:def_25;
reconsider w9 = w as Element of R by A18, ZFMISC_1:87;
w9 <= c9 by A18, ORDERS_2:def_5;
then w9 <= a by A3, A14, ORDERS_2:3;
then w9 in NN9 by A16;
hence contradiction by A15, A17, A18, WAYBEL_4:def_25; ::_thesis: verum
end;
then A19: (Class ((EqRel R),c)) /\ N in min-classes N by Def8;
then A20: not (Class ((EqRel R),c)) /\ N is empty by Th22;
set t = choose ((Class ((EqRel R),c)) /\ N);
choose ((Class ((EqRel R),c)) /\ N) in N by A20, XBOOLE_0:def_4;
then reconsider t = choose ((Class ((EqRel R),c)) /\ N) as Element of R ;
take t ; ::_thesis: ( t in B & t <= a )
thus t in B by A19; ::_thesis: t <= a
t in Class ((EqRel R),c) by A20, XBOOLE_0:def_4;
then [t,c] in EqRel R by EQREL_1:19;
then [t,c] in the InternalRel of R /\ ( the InternalRel of R ~) by A1, Def4;
then [t,c] in the InternalRel of R by XBOOLE_0:def_4;
then t <= c9 by ORDERS_2:def_5;
hence t <= a by A3, A14, ORDERS_2:3; ::_thesis: verum
end;
deffunc H1( set ) -> Element of $1 = choose $1;
defpred S1[ set ] means verum;
{ H1(x) where x is Element of MCN9 : S1[x] } is finite from PRE_CIRC:sch_1();
hence B is finite ; ::_thesis: verum
end;
end;
end;
theorem Th33: :: DICKSON:33
for R being non empty RelStr st R is quasi_ordered & R is Dickson holds
R \~ is well_founded
proof
let R be non empty RelStr ; ::_thesis: ( R is quasi_ordered & R is Dickson implies R \~ is well_founded )
assume that
A1: R is quasi_ordered and
A2: R is Dickson ; ::_thesis: R \~ is well_founded
A3: for f being sequence of R ex i, j being Element of NAT st
( i < j & f . i <= f . j ) by A2, Th29;
now__::_thesis:_for_N_being_Subset_of_R_st_N_<>_{}_holds_
ex_x_being_set_st_x_in_min-classes_N
let N be Subset of R; ::_thesis: ( N <> {} implies ex x being set st x in min-classes N )
assume N <> {} ; ::_thesis: ex x being set st x in min-classes N
then not min-classes N is empty by A1, A3, Th31;
hence ex x being set st x in min-classes N by XBOOLE_0:def_1; ::_thesis: verum
end;
hence R \~ is well_founded by Th20; ::_thesis: verum
end;
theorem :: DICKSON:34
for R being non empty Poset
for N being non empty Subset of R st R is Dickson holds
ex B being set st
( B is_Dickson-basis_of N,R & ( for C being set st C is_Dickson-basis_of N,R holds
B c= C ) )
proof
let R be non empty Poset; ::_thesis: for N being non empty Subset of R st R is Dickson holds
ex B being set st
( B is_Dickson-basis_of N,R & ( for C being set st C is_Dickson-basis_of N,R holds
B c= C ) )
let N be non empty Subset of R; ::_thesis: ( R is Dickson implies ex B being set st
( B is_Dickson-basis_of N,R & ( for C being set st C is_Dickson-basis_of N,R holds
B c= C ) ) )
assume A1: R is Dickson ; ::_thesis: ex B being set st
( B is_Dickson-basis_of N,R & ( for C being set st C is_Dickson-basis_of N,R holds
B c= C ) )
set IR = the InternalRel of R;
set CR = the carrier of R;
set IR9 = the InternalRel of (R \~);
set B = { b where b is Element of (R \~) : b is_minimal_wrt N, the InternalRel of (R \~) } ;
A2: R is quasi_ordered by Def3;
for f being sequence of R ex i, j being Element of NAT st
( i < j & f . i <= f . j ) by A1, Th29;
then not min-classes N is empty by A2, Th31;
then consider x being set such that
A3: x in min-classes N by XBOOLE_0:def_1;
consider y being Element of (R \~) such that
A4: y is_minimal_wrt N, the InternalRel of (R \~) and
x = (Class ((EqRel R),y)) /\ N by A3, Def8;
y in { b where b is Element of (R \~) : b is_minimal_wrt N, the InternalRel of (R \~) } by A4;
then reconsider B = { b where b is Element of (R \~) : b is_minimal_wrt N, the InternalRel of (R \~) } as non empty set ;
take B ; ::_thesis: ( B is_Dickson-basis_of N,R & ( for C being set st C is_Dickson-basis_of N,R holds
B c= C ) )
A5: now__::_thesis:_for_x_being_set_st_x_in_B_holds_
x_in_N
let x be set ; ::_thesis: ( x in B implies x in N )
assume x in B ; ::_thesis: x in N
then ex b being Element of (R \~) st
( x = b & b is_minimal_wrt N, the InternalRel of (R \~) ) ;
hence x in N by WAYBEL_4:def_25; ::_thesis: verum
end;
then A6: B c= N by TARSKI:def_3;
thus B is_Dickson-basis_of N,R ::_thesis: for C being set st C is_Dickson-basis_of N,R holds
B c= C
proof
thus B c= N by A5, TARSKI:def_3; :: according to DICKSON:def_9 ::_thesis: for a being Element of R st a in N holds
ex b being Element of R st
( b in B & b <= a )
let a be Element of R; ::_thesis: ( a in N implies ex b being Element of R st
( b in B & b <= a ) )
assume A7: a in N ; ::_thesis: ex b being Element of R st
( b in B & b <= a )
reconsider a9 = a as Element of (R \~) ;
now__::_thesis:_ex_b_being_Element_of_R_st_
(_b_in_B_&_b_<=_a_)
assume A8: for b being Element of R holds
( not b in B or not b <= a ) ; ::_thesis: contradiction
percases ( ( the InternalRel of (R \~) -Seg a) /\ N = {} or ( the InternalRel of (R \~) -Seg a) /\ N <> {} ) ;
suppose ( the InternalRel of (R \~) -Seg a) /\ N = {} ; ::_thesis: contradiction
then the InternalRel of (R \~) -Seg a misses N by XBOOLE_0:def_7;
then a9 is_minimal_wrt N, the InternalRel of (R \~) by A7, Th6;
then a in B ;
hence contradiction by A8; ::_thesis: verum
end;
supposeA9: ( the InternalRel of (R \~) -Seg a) /\ N <> {} ; ::_thesis: contradiction
R \~ is well_founded by A1, A2, Th33;
then the InternalRel of (R \~) is_well_founded_in the carrier of R by WELLFND1:def_2;
then consider z being set such that
A10: z in ( the InternalRel of (R \~) -Seg a) /\ N and
A11: the InternalRel of (R \~) -Seg z misses ( the InternalRel of (R \~) -Seg a) /\ N by A9, WELLORD1:def_3;
A12: z in the InternalRel of (R \~) -Seg a by A10, XBOOLE_0:def_4;
then [z,a] in the InternalRel of (R \~) by WELLORD1:1;
then z in dom the InternalRel of (R \~) by XTUPLE_0:def_12;
then reconsider z = z as Element of (R \~) ;
reconsider z9 = z as Element of R ;
z is_minimal_wrt ( the InternalRel of (R \~) -Seg a9) /\ N, the InternalRel of (R \~) by A10, A11, Th6;
then z is_minimal_wrt N, the InternalRel of (R \~) by Th7;
then A13: z in B ;
[z,a] in the InternalRel of R \ ( the InternalRel of R ~) by A12, WELLORD1:1;
then z9 <= a by ORDERS_2:def_5;
hence contradiction by A8, A13; ::_thesis: verum
end;
end;
end;
hence ex b being Element of R st
( b in B & b <= a ) ; ::_thesis: verum
end;
let C be set ; ::_thesis: ( C is_Dickson-basis_of N,R implies B c= C )
assume A14: C is_Dickson-basis_of N,R ; ::_thesis: B c= C
A15: C c= N by A14, Def9;
now__::_thesis:_for_b_being_set_st_b_in_B_holds_
b_in_C
let b be set ; ::_thesis: ( b in B implies b in C )
assume A16: b in B ; ::_thesis: b in C
b in N by A5, A16;
then reconsider b9 = b as Element of R ;
consider c being Element of R such that
A17: c in C and
A18: c <= b9 by A6, A14, A16, Def9;
A19: ex b99 being Element of (R \~) st
( b99 = b & b99 is_minimal_wrt N, the InternalRel of (R \~) ) by A16;
A20: [c,b] in the InternalRel of R by A18, ORDERS_2:def_5;
A21: the InternalRel of R is_antisymmetric_in the carrier of R by ORDERS_2:def_4;
[b,c] in the InternalRel of R by A15, A17, A19, A20, Th17;
hence b in C by A17, A18, A20, A21, RELAT_2:def_4; ::_thesis: verum
end;
hence B c= C by TARSKI:def_3; ::_thesis: verum
end;
definition
let R be non empty RelStr ;
let N be Subset of R;
assume B1: R is Dickson ;
func Dickson-bases (N,R) -> non empty Subset-Family of R means :Def13: :: DICKSON:def 13
for B being set holds
( B in it iff B is_Dickson-basis_of N,R );
existence
ex b1 being non empty Subset-Family of R st
for B being set holds
( B in b1 iff B is_Dickson-basis_of N,R )
proof
set BB = { b where b is Subset of N : b is_Dickson-basis_of N,R } ;
set CR = the carrier of R;
consider bp being set such that
A1: bp is_Dickson-basis_of N,R and
bp is finite by B1, Def10;
bp c= N by A1, Def9;
then A2: bp in { b where b is Subset of N : b is_Dickson-basis_of N,R } by A1;
now__::_thesis:_for_x_being_set_st_x_in__{__b_where_b_is_Subset_of_N_:_b_is_Dickson-basis_of_N,R__}__holds_
x_in_bool_the_carrier_of_R
let x be set ; ::_thesis: ( x in { b where b is Subset of N : b is_Dickson-basis_of N,R } implies x in bool the carrier of R )
assume x in { b where b is Subset of N : b is_Dickson-basis_of N,R } ; ::_thesis: x in bool the carrier of R
then consider b being Subset of N such that
A3: x = b and
b is_Dickson-basis_of N,R ;
b c= the carrier of R by XBOOLE_1:1;
hence x in bool the carrier of R by A3; ::_thesis: verum
end;
then reconsider BB = { b where b is Subset of N : b is_Dickson-basis_of N,R } as non empty Subset-Family of the carrier of R by A2, TARSKI:def_3;
take BB ; ::_thesis: for B being set holds
( B in BB iff B is_Dickson-basis_of N,R )
let B be set ; ::_thesis: ( B in BB iff B is_Dickson-basis_of N,R )
hereby ::_thesis: ( B is_Dickson-basis_of N,R implies B in BB )
assume B in BB ; ::_thesis: B is_Dickson-basis_of N,R
then ex b being Subset of N st
( b = B & b is_Dickson-basis_of N,R ) ;
hence B is_Dickson-basis_of N,R ; ::_thesis: verum
end;
assume A4: B is_Dickson-basis_of N,R ; ::_thesis: B in BB
then B c= N by Def9;
hence B in BB by A4; ::_thesis: verum
end;
uniqueness
for b1, b2 being non empty Subset-Family of R st ( for B being set holds
( B in b1 iff B is_Dickson-basis_of N,R ) ) & ( for B being set holds
( B in b2 iff B is_Dickson-basis_of N,R ) ) holds
b1 = b2
proof
let IT1, IT2 be non empty Subset-Family of R; ::_thesis: ( ( for B being set holds
( B in IT1 iff B is_Dickson-basis_of N,R ) ) & ( for B being set holds
( B in IT2 iff B is_Dickson-basis_of N,R ) ) implies IT1 = IT2 )
assume that
A5: for B being set holds
( B in IT1 iff B is_Dickson-basis_of N,R ) and
A6: for B being set holds
( B in IT2 iff B is_Dickson-basis_of N,R ) ; ::_thesis: IT1 = IT2
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_IT1_implies_x_in_IT2_)_&_(_x_in_IT2_implies_x_in_IT1_)_)
let x be set ; ::_thesis: ( ( x in IT1 implies x in IT2 ) & ( x in IT2 implies x in IT1 ) )
hereby ::_thesis: ( x in IT2 implies x in IT1 )
assume x in IT1 ; ::_thesis: x in IT2
then x is_Dickson-basis_of N,R by A5;
hence x in IT2 by A6; ::_thesis: verum
end;
assume x in IT2 ; ::_thesis: x in IT1
then x is_Dickson-basis_of N,R by A6;
hence x in IT1 by A5; ::_thesis: verum
end;
hence IT1 = IT2 by TARSKI:1; ::_thesis: verum
end;
end;
:: deftheorem Def13 defines Dickson-bases DICKSON:def_13_:_
for R being non empty RelStr
for N being Subset of R st R is Dickson holds
for b3 being non empty Subset-Family of R holds
( b3 = Dickson-bases (N,R) iff for B being set holds
( B in b3 iff B is_Dickson-basis_of N,R ) );
theorem Th35: :: DICKSON:35
for R being non empty RelStr
for s being sequence of R st R is Dickson holds
ex t being sequence of R st
( t is subsequence of s & t is weakly-ascending )
proof
let R be non empty RelStr ; ::_thesis: for s being sequence of R st R is Dickson holds
ex t being sequence of R st
( t is subsequence of s & t is weakly-ascending )
let s be sequence of R; ::_thesis: ( R is Dickson implies ex t being sequence of R st
( t is subsequence of s & t is weakly-ascending ) )
assume A1: R is Dickson ; ::_thesis: ex t being sequence of R st
( t is subsequence of s & t is weakly-ascending )
set CR = the carrier of R;
deffunc H1( Element of rng s, Element of NAT ) -> set = { n where n is Element of NAT : ( $1 <= s . n & $2 < n ) } ;
deffunc H2( Element of rng s) -> set = { n where n is Element of NAT : $1 <= s . n } ;
defpred S1[ set , Element of NAT , set ] means ex N being Subset of the carrier of R ex B being non empty Subset of the carrier of R st
( N = { (s . w) where w is Element of NAT : ( s . $2 <= s . w & $2 < w ) } & { w where w is Element of NAT : ( s . $2 <= s . w & $2 < w ) } is infinite & B = choose { BB where BB is Element of Dickson-bases (N,R) : BB is finite } & $3 = s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,$2) is infinite ) } ),$2) );
defpred S2[ set , Element of NAT , set ] means ( { w where w is Element of NAT : ( s . $2 <= s . w & $2 < w ) } is infinite implies S1[$1,$2,$3] );
A2: for n, x being Element of NAT ex y being Element of NAT st S2[n,x,y]
proof
let n, x be Element of NAT ; ::_thesis: ex y being Element of NAT st S2[n,x,y]
set N = { (s . w) where w is Element of NAT : ( s . x <= s . w & x < w ) } ;
now__::_thesis:_for_y_being_set_st_y_in__{__(s_._w)_where_w_is_Element_of_NAT_:_(_s_._x_<=_s_._w_&_x_<_w_)__}__holds_
y_in_the_carrier_of_R
let y be set ; ::_thesis: ( y in { (s . w) where w is Element of NAT : ( s . x <= s . w & x < w ) } implies y in the carrier of R )
assume y in { (s . w) where w is Element of NAT : ( s . x <= s . w & x < w ) } ; ::_thesis: y in the carrier of R
then ex w being Element of NAT st
( y = s . w & s . x <= s . w & x < w ) ;
hence y in the carrier of R ; ::_thesis: verum
end;
then reconsider N = { (s . w) where w is Element of NAT : ( s . x <= s . w & x < w ) } as Subset of the carrier of R by TARSKI:def_3;
set W = { w where w is Element of NAT : ( s . x <= s . w & x < w ) } ;
percases ( N is empty or not N is empty ) ;
supposeA3: N is empty ; ::_thesis: ex y being Element of NAT st S2[n,x,y]
take 1 ; ::_thesis: S2[n,x,1]
assume { w where w is Element of NAT : ( s . x <= s . w & x < w ) } is infinite ; ::_thesis: S1[n,x,1]
then consider ww being set such that
A4: ww in { w where w is Element of NAT : ( s . x <= s . w & x < w ) } by XBOOLE_0:def_1;
consider www being Element of NAT such that
www = ww and
A5: s . x <= s . www and
A6: x < www by A4;
s . www in N by A5, A6;
hence S1[n,x,1] by A3; ::_thesis: verum
end;
supposeA7: not N is empty ; ::_thesis: ex y being Element of NAT st S2[n,x,y]
set BBX = { BB where BB is Element of Dickson-bases (N,R) : BB is finite } ;
set B = choose { BB where BB is Element of Dickson-bases (N,R) : BB is finite } ;
consider BD1 being set such that
A8: BD1 is_Dickson-basis_of N,R and
A9: BD1 is finite by A1, Def10;
BD1 in Dickson-bases (N,R) by A1, A8, Def13;
then BD1 in { BB where BB is Element of Dickson-bases (N,R) : BB is finite } by A9;
then choose { BB where BB is Element of Dickson-bases (N,R) : BB is finite } in { BB where BB is Element of Dickson-bases (N,R) : BB is finite } ;
then ex BBB being Element of Dickson-bases (N,R) st
( choose { BB where BB is Element of Dickson-bases (N,R) : BB is finite } = BBB & BBB is finite ) ;
then A10: choose { BB where BB is Element of Dickson-bases (N,R) : BB is finite } is_Dickson-basis_of N,R by A1, Def13;
then choose { BB where BB is Element of Dickson-bases (N,R) : BB is finite } c= N by Def9;
then reconsider B = choose { BB where BB is Element of Dickson-bases (N,R) : BB is finite } as non empty Subset of the carrier of R by A7, A10, Th26, XBOOLE_1:1;
set y = s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,x) is infinite ) } ),x);
take s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,x) is infinite ) } ),x) ; ::_thesis: S2[n,x,s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,x) is infinite ) } ),x)]
set W = { w where w is Element of NAT : ( s . x <= s . w & x < w ) } ;
assume A11: { w where w is Element of NAT : ( s . x <= s . w & x < w ) } is infinite ; ::_thesis: S1[n,x,s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,x) is infinite ) } ),x)]
take N ; ::_thesis: ex B being non empty Subset of the carrier of R st
( N = { (s . w) where w is Element of NAT : ( s . x <= s . w & x < w ) } & { w where w is Element of NAT : ( s . x <= s . w & x < w ) } is infinite & B = choose { BB where BB is Element of Dickson-bases (N,R) : BB is finite } & s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,x) is infinite ) } ),x) = s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,x) is infinite ) } ),x) )
reconsider B = B as non empty Subset of the carrier of R ;
take B ; ::_thesis: ( N = { (s . w) where w is Element of NAT : ( s . x <= s . w & x < w ) } & { w where w is Element of NAT : ( s . x <= s . w & x < w ) } is infinite & B = choose { BB where BB is Element of Dickson-bases (N,R) : BB is finite } & s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,x) is infinite ) } ),x) = s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,x) is infinite ) } ),x) )
thus N = { (s . w) where w is Element of NAT : ( s . x <= s . w & x < w ) } ; ::_thesis: ( { w where w is Element of NAT : ( s . x <= s . w & x < w ) } is infinite & B = choose { BB where BB is Element of Dickson-bases (N,R) : BB is finite } & s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,x) is infinite ) } ),x) = s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,x) is infinite ) } ),x) )
thus { w where w is Element of NAT : ( s . x <= s . w & x < w ) } is infinite by A11; ::_thesis: ( B = choose { BB where BB is Element of Dickson-bases (N,R) : BB is finite } & s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,x) is infinite ) } ),x) = s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,x) is infinite ) } ),x) )
thus B = choose { BB where BB is Element of Dickson-bases (N,R) : BB is finite } ; ::_thesis: s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,x) is infinite ) } ),x) = s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,x) is infinite ) } ),x)
thus s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,x) is infinite ) } ),x) = s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,x) is infinite ) } ),x) ; ::_thesis: verum
end;
end;
end;
consider B being set such that
A12: B is_Dickson-basis_of rng s,R and
A13: B is finite by A1, Def10;
reconsider B = B as non empty set by A12, Th26;
set BALL = { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H2(b9) is infinite ) } ;
set b1 = choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H2(b9) is infinite ) } ;
consider f being Function of NAT,NAT such that
A14: f . 0 = s mindex (choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H2(b9) is infinite ) } ) and
A15: for n being Element of NAT holds S2[n,f . n,f . (n + 1)] from RECDEF_1:sch_2(A2);
A16: dom f = NAT by FUNCT_2:def_1;
A17: rng f c= NAT ;
now__::_thesis:_ex_b_being_Element_of_rng_s_st_
(_b_in_B_&_not_H2(b)_is_finite_)
A18: B is finite by A13;
assume A19: for b being Element of rng s st b in B holds
H2(b) is finite ; ::_thesis: contradiction
set Ball = { H2(b) where b is Element of rng s : b in B } ;
A20: { H2(b) where b is Element of rng s : b in B } is finite from FRAENKEL:sch_21(A18);
now__::_thesis:_for_X_being_set_st_X_in__{__H2(b)_where_b_is_Element_of_rng_s_:_b_in_B__}__holds_
X_is_finite
let X be set ; ::_thesis: ( X in { H2(b) where b is Element of rng s : b in B } implies X is finite )
assume X in { H2(b) where b is Element of rng s : b in B } ; ::_thesis: X is finite
then ex b being Element of rng s st
( X = H2(b) & b in B ) ;
hence X is finite by A19; ::_thesis: verum
end;
then A21: union { H2(b) where b is Element of rng s : b in B } is finite by A20, FINSET_1:7;
now__::_thesis:_for_x_being_set_st_x_in_NAT_holds_
x_in_union__{__H2(b)_where_b_is_Element_of_rng_s_:_b_in_B__}_
let x be set ; ::_thesis: ( x in NAT implies x in union { H2(b) where b is Element of rng s : b in B } )
assume x in NAT ; ::_thesis: x in union { H2(b) where b is Element of rng s : b in B }
then reconsider x9 = x as Element of NAT ;
dom s = NAT by FUNCT_2:def_1;
then x9 in dom s ;
then A22: s . x in rng s by FUNCT_1:3;
then reconsider sx = s . x as Element of R ;
consider b being Element of R such that
A23: b in B and
A24: b <= sx by A12, A22, Def9;
B c= rng s by A12, Def9;
then reconsider b = b as Element of rng s by A23;
A25: x9 in H2(b) by A24;
H2(b) in { H2(b) where b is Element of rng s : b in B } by A23;
hence x in union { H2(b) where b is Element of rng s : b in B } by A25, TARSKI:def_4; ::_thesis: verum
end;
then NAT c= union { H2(b) where b is Element of rng s : b in B } by TARSKI:def_3;
hence contradiction by A21; ::_thesis: verum
end;
then consider tb being Element of rng s such that
A26: tb in B and
A27: H2(tb) is infinite ;
A28: tb in { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H2(b9) is infinite ) } by A26, A27;
then choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H2(b9) is infinite ) } in { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H2(b9) is infinite ) } ;
then A29: ex bt being Element of B st
( choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H2(b9) is infinite ) } = bt & ex b9 being Element of rng s st
( b9 = bt & H2(b9) is infinite ) ) ;
dom s = NAT by NORMSP_1:12;
then A30: s . (f . 0) = choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H2(b9) is infinite ) } by A14, A29, Def11;
choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H2(b9) is infinite ) } in { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H2(b9) is infinite ) } by A28;
then A31: ex eb being Element of B st
( choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H2(b9) is infinite ) } = eb & ex eb9 being Element of rng s st
( eb9 = eb & H2(eb9) is infinite ) ) ;
deffunc H3( set ) -> set = $1;
defpred S3[ Element of NAT ] means s . (f . 0) <= s . $1;
set W1 = { w where w is Element of NAT : s . (f . 0) <= s . w } ;
set W2 = { w where w is Element of NAT : ( s . (f . 0) <= s . w & f . 0 < w ) } ;
set W3 = { H3(w) where w is Element of NAT : ( 0 <= w & w <= f . 0 & S3[w] ) } ;
A32: { H3(w) where w is Element of NAT : ( 0 <= w & w <= f . 0 & S3[w] ) } is finite from FINSEQ_1:sch_6();
now__::_thesis:_for_x_being_set_holds_
(_(_x_in__{__w_where_w_is_Element_of_NAT_:_s_._(f_._0)_<=_s_._w__}__implies_x_in__{__w_where_w_is_Element_of_NAT_:_(_s_._(f_._0)_<=_s_._w_&_f_._0_<_w_)__}__\/__{__H3(w)_where_w_is_Element_of_NAT_:_(_0_<=_w_&_w_<=_f_._0_&_S3[w]_)__}__)_&_(_x_in__{__w_where_w_is_Element_of_NAT_:_(_s_._(f_._0)_<=_s_._w_&_f_._0_<_w_)__}__\/__{__H3(w)_where_w_is_Element_of_NAT_:_(_0_<=_w_&_w_<=_f_._0_&_S3[w]_)__}__implies_x_in__{__w_where_w_is_Element_of_NAT_:_s_._(f_._0)_<=_s_._w__}__)_)
let x be set ; ::_thesis: ( ( x in { w where w is Element of NAT : s . (f . 0) <= s . w } implies x in { w where w is Element of NAT : ( s . (f . 0) <= s . w & f . 0 < w ) } \/ { H3(w) where w is Element of NAT : ( 0 <= w & w <= f . 0 & S3[w] ) } ) & ( x in { w where w is Element of NAT : ( s . (f . 0) <= s . w & f . 0 < w ) } \/ { H3(w) where w is Element of NAT : ( 0 <= w & w <= f . 0 & S3[w] ) } implies b1 in { b2 where w is Element of NAT : s . (f . 0) <= s . b2 } ) )
hereby ::_thesis: ( x in { w where w is Element of NAT : ( s . (f . 0) <= s . w & f . 0 < w ) } \/ { H3(w) where w is Element of NAT : ( 0 <= w & w <= f . 0 & S3[w] ) } implies b1 in { b2 where w is Element of NAT : s . (f . 0) <= s . b2 } )
assume x in { w where w is Element of NAT : s . (f . 0) <= s . w } ; ::_thesis: x in { w where w is Element of NAT : ( s . (f . 0) <= s . w & f . 0 < w ) } \/ { H3(w) where w is Element of NAT : ( 0 <= w & w <= f . 0 & S3[w] ) }
then consider w being Element of NAT such that
A33: x = w and
A34: s . (f . 0) <= s . w ;
A35: 0 <= w by NAT_1:2;
( w <= f . 0 or w > f . 0 ) ;
then ( x in { w where w is Element of NAT : ( s . (f . 0) <= s . w & f . 0 < w ) } or x in { H3(w) where w is Element of NAT : ( 0 <= w & w <= f . 0 & S3[w] ) } ) by A33, A34, A35;
hence x in { w where w is Element of NAT : ( s . (f . 0) <= s . w & f . 0 < w ) } \/ { H3(w) where w is Element of NAT : ( 0 <= w & w <= f . 0 & S3[w] ) } by XBOOLE_0:def_3; ::_thesis: verum
end;
assume A36: x in { w where w is Element of NAT : ( s . (f . 0) <= s . w & f . 0 < w ) } \/ { H3(w) where w is Element of NAT : ( 0 <= w & w <= f . 0 & S3[w] ) } ; ::_thesis: b1 in { b2 where w is Element of NAT : s . (f . 0) <= s . b2 }
percases ( x in { w where w is Element of NAT : ( s . (f . 0) <= s . w & f . 0 < w ) } or x in { H3(w) where w is Element of NAT : ( 0 <= w & w <= f . 0 & S3[w] ) } ) by A36, XBOOLE_0:def_3;
suppose x in { w where w is Element of NAT : ( s . (f . 0) <= s . w & f . 0 < w ) } ; ::_thesis: b1 in { b2 where w is Element of NAT : s . (f . 0) <= s . b2 }
then ex w being Element of NAT st
( x = w & s . (f . 0) <= s . w & f . 0 < w ) ;
hence x in { w where w is Element of NAT : s . (f . 0) <= s . w } ; ::_thesis: verum
end;
suppose x in { H3(w) where w is Element of NAT : ( 0 <= w & w <= f . 0 & S3[w] ) } ; ::_thesis: b1 in { b2 where w is Element of NAT : s . (f . 0) <= s . b2 }
then ex w being Element of NAT st
( x = w & 0 <= w & w <= f . 0 & s . (f . 0) <= s . w ) ;
hence x in { w where w is Element of NAT : s . (f . 0) <= s . w } ; ::_thesis: verum
end;
end;
end;
then A37: { w where w is Element of NAT : ( s . (f . 0) <= s . w & f . 0 < w ) } is infinite by A30, A31, A32, TARSKI:1;
defpred S4[ Element of NAT ] means S1[$1,f . $1,f . ($1 + 1)];
A38: S4[ 0 ] by A15, A37;
A39: now__::_thesis:_for_k_being_Element_of_NAT_st_S4[k]_holds_
S4[k_+_1]
let k be Element of NAT ; ::_thesis: ( S4[k] implies S4[k + 1] )
assume S4[k] ; ::_thesis: S4[k + 1]
then consider N being Subset of the carrier of R, B being non empty Subset of the carrier of R such that
A40: N = { (s . w) where w is Element of NAT : ( s . (f . k) <= s . w & f . k < w ) } and
A41: { w where w is Element of NAT : ( s . (f . k) <= s . w & f . k < w ) } is infinite and
A42: B = choose { BB where BB is Element of Dickson-bases (N,R) : BB is finite } and
A43: f . (k + 1) = s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,f . k) is infinite ) } ),(f . k)) ;
set BALL = { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,f . k) is infinite ) } ;
set BBX = { BB where BB is Element of Dickson-bases (N,R) : BB is finite } ;
set iN = { w where w is Element of NAT : ( s . (f . k) <= s . w & f . k < w ) } ;
consider BD being set such that
A44: BD is_Dickson-basis_of N,R and
A45: BD is finite by A1, Def10;
BD in Dickson-bases (N,R) by A1, A44, Def13;
then BD in { BB where BB is Element of Dickson-bases (N,R) : BB is finite } by A45;
then B in { BB where BB is Element of Dickson-bases (N,R) : BB is finite } by A42;
then A46: ex BB being Element of Dickson-bases (N,R) st
( B = BB & BB is finite ) ;
then A47: B is_Dickson-basis_of N,R by A1, Def13;
now__::_thesis:_ex_b_being_Element_of_rng_s_st_
(_b_in_B_&_not_H1(b,f_._k)_is_finite_)
deffunc H4( Element of rng s) -> set = H1($1,f . k);
A48: B is finite by A46;
assume A49: for b being Element of rng s st b in B holds
H1(b,f . k) is finite ; ::_thesis: contradiction
set Ball = { H4(b) where b is Element of rng s : b in B } ;
A50: { H4(b) where b is Element of rng s : b in B } is finite from FRAENKEL:sch_21(A48);
now__::_thesis:_for_X_being_set_st_X_in__{__H4(b)_where_b_is_Element_of_rng_s_:_b_in_B__}__holds_
X_is_finite
let X be set ; ::_thesis: ( X in { H4(b) where b is Element of rng s : b in B } implies X is finite )
assume X in { H4(b) where b is Element of rng s : b in B } ; ::_thesis: X is finite
then ex b being Element of rng s st
( X = H1(b,f . k) & b in B ) ;
hence X is finite by A49; ::_thesis: verum
end;
then A51: union { H4(b) where b is Element of rng s : b in B } is finite by A50, FINSET_1:7;
{ w where w is Element of NAT : ( s . (f . k) <= s . w & f . k < w ) } c= union { H4(b) where b is Element of rng s : b in B }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { w where w is Element of NAT : ( s . (f . k) <= s . w & f . k < w ) } or x in union { H4(b) where b is Element of rng s : b in B } )
assume x in { w where w is Element of NAT : ( s . (f . k) <= s . w & f . k < w ) } ; ::_thesis: x in union { H4(b) where b is Element of rng s : b in B }
then consider w being Element of NAT such that
A52: x = w and
A53: s . (f . k) <= s . w and
A54: f . k < w ;
A55: s . w in N by A40, A53, A54;
reconsider sw = s . w as Element of R ;
consider b being Element of R such that
A56: b in B and
A57: b <= sw by A47, A55, Def9;
A58: B c= N by A47, Def9;
N c= rng s
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in N or x in rng s )
assume x in N ; ::_thesis: x in rng s
then A59: ex u being Element of NAT st
( x = s . u & s . (f . k) <= s . u & f . k < u ) by A40;
dom s = NAT by FUNCT_2:def_1;
hence x in rng s by A59, FUNCT_1:3; ::_thesis: verum
end;
then B c= rng s by A58, XBOOLE_1:1;
then reconsider b = b as Element of rng s by A56;
A60: w in H1(b,f . k) by A54, A57;
H1(b,f . k) in { H4(b) where b is Element of rng s : b in B } by A56;
hence x in union { H4(b) where b is Element of rng s : b in B } by A52, A60, TARSKI:def_4; ::_thesis: verum
end;
hence contradiction by A41, A51; ::_thesis: verum
end;
then consider b being Element of rng s such that
A61: b in B and
A62: H1(b,f . k) is infinite ;
b in { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,f . k) is infinite ) } by A61, A62;
then choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,f . k) is infinite ) } in { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,f . k) is infinite ) } ;
then consider b being Element of B such that
A63: b = choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,f . k) is infinite ) } and
A64: ex b9 being Element of rng s st
( b9 = b & H1(b9,f . k) is infinite ) ;
A65: b in B ;
B c= N by A47, Def9;
then b in N by A65;
then A66: ex w being Element of NAT st
( b = s . w & s . (f . k) <= s . w & f . k < w ) by A40;
then A67: s . (f . (k + 1)) = choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,f . k) is infinite ) } by A43, A63, Def12;
A68: f . k < f . (k + 1) by A43, A63, A66, Def12;
deffunc H4( set ) -> set = $1;
defpred S5[ Element of NAT ] means s . (f . (k + 1)) <= s . $1;
set W1 = { w1 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . w1 & f . k < w1 ) } ;
set W2 = { w1 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . w1 & f . (k + 1) < w1 ) } ;
set W3 = { H4(w1) where w1 is Element of NAT : ( f . k < w1 & w1 <= f . (k + 1) & S5[w1] ) } ;
A69: { H4(w1) where w1 is Element of NAT : ( f . k < w1 & w1 <= f . (k + 1) & S5[w1] ) } is finite from DICKSON:sch_1();
now__::_thesis:_for_x_being_set_holds_
(_(_x_in__{__w1_where_w1_is_Element_of_NAT_:_(_s_._(f_._(k_+_1))_<=_s_._w1_&_f_._k_<_w1_)__}__implies_x_in__{__w1_where_w1_is_Element_of_NAT_:_(_s_._(f_._(k_+_1))_<=_s_._w1_&_f_._(k_+_1)_<_w1_)__}__\/__{__H4(w1)_where_w1_is_Element_of_NAT_:_(_f_._k_<_w1_&_w1_<=_f_._(k_+_1)_&_S5[w1]_)__}__)_&_(_x_in__{__w1_where_w1_is_Element_of_NAT_:_(_s_._(f_._(k_+_1))_<=_s_._w1_&_f_._(k_+_1)_<_w1_)__}__\/__{__H4(w1)_where_w1_is_Element_of_NAT_:_(_f_._k_<_w1_&_w1_<=_f_._(k_+_1)_&_S5[w1]_)__}__implies_x_in__{__w1_where_w1_is_Element_of_NAT_:_(_s_._(f_._(k_+_1))_<=_s_._w1_&_f_._k_<_w1_)__}__)_)
let x be set ; ::_thesis: ( ( x in { w1 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . w1 & f . k < w1 ) } implies x in { w1 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . w1 & f . (k + 1) < w1 ) } \/ { H4(w1) where w1 is Element of NAT : ( f . k < w1 & w1 <= f . (k + 1) & S5[w1] ) } ) & ( x in { w1 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . w1 & f . (k + 1) < w1 ) } \/ { H4(w1) where w1 is Element of NAT : ( f . k < w1 & w1 <= f . (k + 1) & S5[w1] ) } implies b1 in { b2 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . b2 & f . k < b2 ) } ) )
hereby ::_thesis: ( x in { w1 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . w1 & f . (k + 1) < w1 ) } \/ { H4(w1) where w1 is Element of NAT : ( f . k < w1 & w1 <= f . (k + 1) & S5[w1] ) } implies b1 in { b2 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . b2 & f . k < b2 ) } )
assume x in { w1 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . w1 & f . k < w1 ) } ; ::_thesis: x in { w1 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . w1 & f . (k + 1) < w1 ) } \/ { H4(w1) where w1 is Element of NAT : ( f . k < w1 & w1 <= f . (k + 1) & S5[w1] ) }
then consider w being Element of NAT such that
A70: x = w and
A71: s . (f . (k + 1)) <= s . w and
A72: f . k < w ;
( w <= f . (k + 1) or w > f . (k + 1) ) ;
then ( x in { w1 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . w1 & f . (k + 1) < w1 ) } or x in { H4(w1) where w1 is Element of NAT : ( f . k < w1 & w1 <= f . (k + 1) & S5[w1] ) } ) by A70, A71, A72;
hence x in { w1 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . w1 & f . (k + 1) < w1 ) } \/ { H4(w1) where w1 is Element of NAT : ( f . k < w1 & w1 <= f . (k + 1) & S5[w1] ) } by XBOOLE_0:def_3; ::_thesis: verum
end;
assume A73: x in { w1 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . w1 & f . (k + 1) < w1 ) } \/ { H4(w1) where w1 is Element of NAT : ( f . k < w1 & w1 <= f . (k + 1) & S5[w1] ) } ; ::_thesis: b1 in { b2 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . b2 & f . k < b2 ) }
percases ( x in { w1 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . w1 & f . (k + 1) < w1 ) } or x in { H4(w1) where w1 is Element of NAT : ( f . k < w1 & w1 <= f . (k + 1) & S5[w1] ) } ) by A73, XBOOLE_0:def_3;
suppose x in { w1 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . w1 & f . (k + 1) < w1 ) } ; ::_thesis: b1 in { b2 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . b2 & f . k < b2 ) }
then consider w being Element of NAT such that
A74: x = w and
A75: s . (f . (k + 1)) <= s . w and
A76: f . (k + 1) < w ;
f . k < w by A68, A76, XXREAL_0:2;
hence x in { w1 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . w1 & f . k < w1 ) } by A74, A75; ::_thesis: verum
end;
suppose x in { H4(w1) where w1 is Element of NAT : ( f . k < w1 & w1 <= f . (k + 1) & S5[w1] ) } ; ::_thesis: b1 in { b2 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . b2 & f . k < b2 ) }
then ex w being Element of NAT st
( x = w & f . k < w & w <= f . (k + 1) & s . (f . (k + 1)) <= s . w ) ;
hence x in { w1 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . w1 & f . k < w1 ) } ; ::_thesis: verum
end;
end;
end;
then { w1 where w1 is Element of NAT : ( s . (f . (k + 1)) <= s . w1 & f . (k + 1) < w1 ) } is infinite by A63, A64, A67, A69, TARSKI:1;
hence S4[k + 1] by A15; ::_thesis: verum
end;
A77: for n being Element of NAT holds S4[n] from NAT_1:sch_1(A38, A39);
set t = s * f;
take s * f ; ::_thesis: ( s * f is subsequence of s & s * f is weakly-ascending )
reconsider f = f as Function of NAT,REAL by FUNCT_2:7;
now__::_thesis:_(_f_is_increasing_&_(_for_n_being_Element_of_NAT_holds_f_._n_is_Element_of_NAT_)_)
now__::_thesis:_for_n_being_Element_of_NAT_holds_f_._n_<_f_._(n_+_1)
let n be Element of NAT ; ::_thesis: f . n < f . (n + 1)
f . n in rng f by A16, FUNCT_1:def_3;
then reconsider fn = f . n as Element of NAT by A17;
consider N being Subset of the carrier of R, B being non empty Subset of the carrier of R such that
A78: N = { (s . w) where w is Element of NAT : ( s . fn <= s . w & fn < w ) } and
A79: { w where w is Element of NAT : ( s . fn <= s . w & fn < w ) } is infinite and
A80: B = choose { BB where BB is Element of Dickson-bases (N,R) : BB is finite } and
A81: f . (n + 1) = s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,fn) is infinite ) } ),fn) by A77;
set BBX = { BB where BB is Element of Dickson-bases (N,R) : BB is finite } ;
set BJ = { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,fn) is infinite ) } ;
set BC = choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,fn) is infinite ) } ;
consider BD being set such that
A82: BD is_Dickson-basis_of N,R and
A83: BD is finite by A1, Def10;
BD in Dickson-bases (N,R) by A1, A82, Def13;
then BD in { BB where BB is Element of Dickson-bases (N,R) : BB is finite } by A83;
then B in { BB where BB is Element of Dickson-bases (N,R) : BB is finite } by A80;
then A84: ex BB being Element of Dickson-bases (N,R) st
( B = BB & BB is finite ) ;
then A85: B is_Dickson-basis_of N,R by A1, Def13;
then A86: B c= N by Def9;
now__::_thesis:_ex_b_being_Element_of_rng_s_st_
(_b_in_B_&_not_H1(b,fn)_is_finite_)
A87: B is finite by A84;
assume A88: for b being Element of rng s st b in B holds
H1(b,fn) is finite ; ::_thesis: contradiction
deffunc H4( Element of rng s) -> set = H1($1,fn);
set Ball = { H4(b) where b is Element of rng s : b in B } ;
set iN = { w where w is Element of NAT : ( s . fn <= s . w & fn < w ) } ;
A89: { H4(b) where b is Element of rng s : b in B } is finite from FRAENKEL:sch_21(A87);
now__::_thesis:_for_X_being_set_st_X_in__{__H4(b)_where_b_is_Element_of_rng_s_:_b_in_B__}__holds_
X_is_finite
let X be set ; ::_thesis: ( X in { H4(b) where b is Element of rng s : b in B } implies X is finite )
assume X in { H4(b) where b is Element of rng s : b in B } ; ::_thesis: X is finite
then ex b being Element of rng s st
( X = H1(b,fn) & b in B ) ;
hence X is finite by A88; ::_thesis: verum
end;
then A90: union { H4(b) where b is Element of rng s : b in B } is finite by A89, FINSET_1:7;
{ w where w is Element of NAT : ( s . fn <= s . w & fn < w ) } c= union { H4(b) where b is Element of rng s : b in B }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { w where w is Element of NAT : ( s . fn <= s . w & fn < w ) } or x in union { H4(b) where b is Element of rng s : b in B } )
assume x in { w where w is Element of NAT : ( s . fn <= s . w & fn < w ) } ; ::_thesis: x in union { H4(b) where b is Element of rng s : b in B }
then consider w being Element of NAT such that
A91: x = w and
A92: s . fn <= s . w and
A93: f . n < w ;
A94: s . w in N by A78, A92, A93;
reconsider sw = s . w as Element of R ;
consider b being Element of R such that
A95: b in B and
A96: b <= sw by A85, A94, Def9;
A97: B c= N by A85, Def9;
N c= rng s
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in N or x in rng s )
assume x in N ; ::_thesis: x in rng s
then A98: ex u being Element of NAT st
( x = s . u & s . fn <= s . u & fn < u ) by A78;
dom s = NAT by FUNCT_2:def_1;
hence x in rng s by A98, FUNCT_1:3; ::_thesis: verum
end;
then B c= rng s by A97, XBOOLE_1:1;
then reconsider b = b as Element of rng s by A95;
A99: w in H1(b,fn) by A93, A96;
H1(b,fn) in { H4(b) where b is Element of rng s : b in B } by A95;
hence x in union { H4(b) where b is Element of rng s : b in B } by A91, A99, TARSKI:def_4; ::_thesis: verum
end;
hence contradiction by A79, A90; ::_thesis: verum
end;
then consider b being Element of rng s such that
A100: b in B and
A101: H1(b,fn) is infinite ;
b in { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,fn) is infinite ) } by A100, A101;
then choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,fn) is infinite ) } in { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,fn) is infinite ) } ;
then ex b being Element of B st
( choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,fn) is infinite ) } = b & ex b9 being Element of rng s st
( b9 = b & H1(b9,fn) is infinite ) ) ;
then choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,fn) is infinite ) } in N by A86, TARSKI:def_3;
then ex j being Element of NAT st
( choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,fn) is infinite ) } = s . j & s . fn <= s . j & fn < j ) by A78;
hence f . n < f . (n + 1) by A81, Def12; ::_thesis: verum
end;
hence f is increasing by SEQM_3:def_6; ::_thesis: for n being Element of NAT holds f . n is Element of NAT
let n be Element of NAT ; ::_thesis: f . n is Element of NAT
f . n in rng f by A16, FUNCT_1:def_3;
hence f . n is Element of NAT by A17; ::_thesis: verum
end;
then reconsider f = f as increasing sequence of NAT ;
s * f = s * f ;
hence s * f is subsequence of s ; ::_thesis: s * f is weakly-ascending
let n be Element of NAT ; :: according to DICKSON:def_2 ::_thesis: [((s * f) . n),((s * f) . (n + 1))] in the InternalRel of R
A102: (s * f) . n = s . (f . n) by A16, FUNCT_1:13;
A103: (s * f) . (n + 1) = s . (f . (n + 1)) by A16, FUNCT_1:13;
consider N being Subset of the carrier of R, B being non empty Subset of the carrier of R such that
A104: N = { (s . w) where w is Element of NAT : ( s . (f . n) <= s . w & f . n < w ) } and
A105: { w where w is Element of NAT : ( s . (f . n) <= s . w & f . n < w ) } is infinite and
A106: B = choose { BB where BB is Element of Dickson-bases (N,R) : BB is finite } and
A107: f . (n + 1) = s mindex ((choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,f . n) is infinite ) } ),(f . n)) by A77;
set BX = { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,f . n) is infinite ) } ;
set sfn1 = choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,f . n) is infinite ) } ;
set BBX = { BB where BB is Element of Dickson-bases (N,R) : BB is finite } ;
consider BD being set such that
A108: BD is_Dickson-basis_of N,R and
A109: BD is finite by A1, Def10;
BD in Dickson-bases (N,R) by A1, A108, Def13;
then BD in { BB where BB is Element of Dickson-bases (N,R) : BB is finite } by A109;
then B in { BB where BB is Element of Dickson-bases (N,R) : BB is finite } by A106;
then A110: ex BB being Element of Dickson-bases (N,R) st
( BB = B & BB is finite ) ;
then A111: B is_Dickson-basis_of N,R by A1, Def13;
now__::_thesis:_ex_b_being_Element_of_rng_s_st_
(_b_in_B_&_not_H1(b,f_._n)_is_finite_)
A112: B is finite by A110;
assume A113: for b being Element of rng s st b in B holds
H1(b,f . n) is finite ; ::_thesis: contradiction
deffunc H4( Element of rng s) -> set = H1($1,f . n);
set Ball = { H4(b) where b is Element of rng s : b in B } ;
set iN = { w where w is Element of NAT : ( s . (f . n) <= s . w & f . n < w ) } ;
A114: { H4(b) where b is Element of rng s : b in B } is finite from FRAENKEL:sch_21(A112);
now__::_thesis:_for_X_being_set_st_X_in__{__H4(b)_where_b_is_Element_of_rng_s_:_b_in_B__}__holds_
X_is_finite
let X be set ; ::_thesis: ( X in { H4(b) where b is Element of rng s : b in B } implies X is finite )
assume X in { H4(b) where b is Element of rng s : b in B } ; ::_thesis: X is finite
then ex b being Element of rng s st
( X = H1(b,f . n) & b in B ) ;
hence X is finite by A113; ::_thesis: verum
end;
then A115: union { H4(b) where b is Element of rng s : b in B } is finite by A114, FINSET_1:7;
{ w where w is Element of NAT : ( s . (f . n) <= s . w & f . n < w ) } c= union { H4(b) where b is Element of rng s : b in B }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { w where w is Element of NAT : ( s . (f . n) <= s . w & f . n < w ) } or x in union { H4(b) where b is Element of rng s : b in B } )
assume x in { w where w is Element of NAT : ( s . (f . n) <= s . w & f . n < w ) } ; ::_thesis: x in union { H4(b) where b is Element of rng s : b in B }
then consider w being Element of NAT such that
A116: x = w and
A117: s . (f . n) <= s . w and
A118: f . n < w ;
A119: s . w in N by A104, A117, A118;
reconsider sw = s . w as Element of R ;
consider b being Element of R such that
A120: b in B and
A121: b <= sw by A111, A119, Def9;
A122: B c= N by A111, Def9;
N c= rng s
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in N or x in rng s )
assume x in N ; ::_thesis: x in rng s
then A123: ex u being Element of NAT st
( x = s . u & s . (f . n) <= s . u & f . n < u ) by A104;
dom s = NAT by FUNCT_2:def_1;
hence x in rng s by A123, FUNCT_1:3; ::_thesis: verum
end;
then B c= rng s by A122, XBOOLE_1:1;
then reconsider b = b as Element of rng s by A120;
A124: w in H1(b,f . n) by A118, A121;
H1(b,f . n) in { H4(b) where b is Element of rng s : b in B } by A120;
hence x in union { H4(b) where b is Element of rng s : b in B } by A116, A124, TARSKI:def_4; ::_thesis: verum
end;
hence contradiction by A105, A115; ::_thesis: verum
end;
then consider b being Element of rng s such that
A125: b in B and
A126: H1(b,f . n) is infinite ;
b in { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,f . n) is infinite ) } by A125, A126;
then choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,f . n) is infinite ) } in { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,f . n) is infinite ) } ;
then ex b being Element of B st
( b = choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,f . n) is infinite ) } & ex b9 being Element of rng s st
( b9 = b & H1(b9,f . n) is infinite ) ) ;
then A127: choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,f . n) is infinite ) } in B ;
B c= N by A111, Def9;
then choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,f . n) is infinite ) } in N by A127;
then ex w being Element of NAT st
( choose { b where b is Element of B : ex b9 being Element of rng s st
( b9 = b & H1(b9,f . n) is infinite ) } = s . w & s . (f . n) <= s . w & f . n < w ) by A104;
then (s * f) . n <= (s * f) . (n + 1) by A102, A103, A107, Def12;
hence [((s * f) . n),((s * f) . (n + 1))] in the InternalRel of R by ORDERS_2:def_5; ::_thesis: verum
end;
theorem Th36: :: DICKSON:36
for R being RelStr st R is empty holds
R is Dickson
proof
let R be RelStr ; ::_thesis: ( R is empty implies R is Dickson )
assume A1: R is empty ; ::_thesis: R is Dickson
now__::_thesis:_for_N_being_Subset_of_R_ex_B_being_set_st_
(_B_is_Dickson-basis_of_N,R_&_B_is_finite_)
let N be Subset of R; ::_thesis: ex B being set st
( B is_Dickson-basis_of N,R & B is finite )
take B = {} ; ::_thesis: ( B is_Dickson-basis_of N,R & B is finite )
N = {} the carrier of R by A1;
hence B is_Dickson-basis_of N,R by Th25; ::_thesis: B is finite
thus B is finite ; ::_thesis: verum
end;
hence R is Dickson by Def10; ::_thesis: verum
end;
theorem Th37: :: DICKSON:37
for M, N being RelStr st M is Dickson & N is Dickson & M is quasi_ordered & N is quasi_ordered holds
( [:M,N:] is quasi_ordered & [:M,N:] is Dickson )
proof
let M, N be RelStr ; ::_thesis: ( M is Dickson & N is Dickson & M is quasi_ordered & N is quasi_ordered implies ( [:M,N:] is quasi_ordered & [:M,N:] is Dickson ) )
assume that
A1: M is Dickson and
A2: N is Dickson and
A3: M is quasi_ordered and
A4: N is quasi_ordered ; ::_thesis: ( [:M,N:] is quasi_ordered & [:M,N:] is Dickson )
reconsider M9 = M as reflexive transitive RelStr by A3, Def3;
reconsider N9 = N as reflexive transitive RelStr by A4, Def3;
[:M9,N9:] is reflexive ;
hence A5: [:M,N:] is quasi_ordered by Def3; ::_thesis: [:M,N:] is Dickson
percases ( ( not M is empty & not N is empty ) or M is empty or N is empty ) ;
suppose ( not M is empty & not N is empty ) ; ::_thesis: [:M,N:] is Dickson
then reconsider Me = M, Ne = N as non empty RelStr ;
set CPMN = [:Me,Ne:];
for f being sequence of [:Me,Ne:] ex i, j being Element of NAT st
( i < j & f . i <= f . j )
proof
let f be sequence of [:Me,Ne:]; ::_thesis: ex i, j being Element of NAT st
( i < j & f . i <= f . j )
deffunc H1( Element of NAT ) -> Element of the carrier of Me = (f . $1) `1 ;
consider a being Function of NAT, the carrier of Me such that
A6: for x being Element of NAT holds a . x = H1(x) from FUNCT_2:sch_4();
reconsider a = a as sequence of Me ;
consider sa being sequence of Me such that
A7: sa is subsequence of a and
A8: sa is weakly-ascending by A1, Th35;
consider NS being increasing sequence of NAT such that
A9: sa = a * NS by A7, VALUED_0:def_17;
deffunc H2( Element of NAT ) -> Element of the carrier of Ne = (f . (NS . $1)) `2 ;
consider b being Function of NAT, the carrier of Ne such that
A10: for x being Element of NAT holds b . x = H2(x) from FUNCT_2:sch_4();
reconsider b = b as sequence of Ne ;
consider i, j being Element of NAT such that
A11: i < j and
A12: b . i <= b . j by A2, Th29;
take NS . i ; ::_thesis: ex j being Element of NAT st
( NS . i < j & f . (NS . i) <= f . j )
take NS . j ; ::_thesis: ( NS . i < NS . j & f . (NS . i) <= f . (NS . j) )
dom NS = NAT by FUNCT_2:def_1;
hence NS . i < NS . j by A11, VALUED_0:def_13; ::_thesis: f . (NS . i) <= f . (NS . j)
reconsider x = f . (NS . i), y = f . (NS . j) as Element of [:Me,Ne:] ;
A13: dom sa = NAT by FUNCT_2:def_1;
then A14: sa . i = a . (NS . i) by A9, FUNCT_1:12
.= (f . (NS . i)) `1 by A6 ;
A15: sa . j = a . (NS . j) by A9, A13, FUNCT_1:12
.= (f . (NS . j)) `1 by A6 ;
M is transitive by A3, Def3;
then A16: x `1 <= y `1 by A8, A11, A14, A15, Th4;
A17: b . i = x `2 by A10;
b . j = y `2 by A10;
hence f . (NS . i) <= f . (NS . j) by A12, A16, A17, YELLOW_3:12; ::_thesis: verum
end;
then for N being non empty Subset of [:Me,Ne:] holds
( min-classes N is finite & not min-classes N is empty ) by A5, Th31;
hence [:M,N:] is Dickson by A5, Th32; ::_thesis: verum
end;
supposeA18: ( M is empty or N is empty ) ; ::_thesis: [:M,N:] is Dickson
now__::_thesis:_[:_the_carrier_of_M,_the_carrier_of_N:]_is_empty
percases ( M is empty or N is empty ) by A18;
suppose M is empty ; ::_thesis: [: the carrier of M, the carrier of N:] is empty
then reconsider M2 = the carrier of M as empty set ;
[:M2, the carrier of N:] is empty ;
hence [: the carrier of M, the carrier of N:] is empty ; ::_thesis: verum
end;
suppose N is empty ; ::_thesis: [: the carrier of M, the carrier of N:] is empty
then reconsider N2 = the carrier of N as empty set ;
[: the carrier of M,N2:] is empty ;
hence [: the carrier of M, the carrier of N:] is empty ; ::_thesis: verum
end;
end;
end;
then [:M,N:] is empty by YELLOW_3:def_2;
hence [:M,N:] is Dickson by Th36; ::_thesis: verum
end;
end;
end;
theorem Th38: :: DICKSON:38
for R, S being RelStr st R,S are_isomorphic & R is Dickson & R is quasi_ordered holds
( S is quasi_ordered & S is Dickson )
proof
let R, S be RelStr ; ::_thesis: ( R,S are_isomorphic & R is Dickson & R is quasi_ordered implies ( S is quasi_ordered & S is Dickson ) )
assume that
A1: R,S are_isomorphic and
A2: R is Dickson and
A3: R is quasi_ordered ; ::_thesis: ( S is quasi_ordered & S is Dickson )
set CRS = the carrier of S;
set IRS = the InternalRel of S;
percases ( ( not R is empty & not S is empty ) or R is empty or S is empty ) ;
suppose ( not R is empty & not S is empty ) ; ::_thesis: ( S is quasi_ordered & S is Dickson )
then reconsider Re = R, Se = S as non empty RelStr ;
consider f being Function of Re,Se such that
A4: f is isomorphic by A1, WAYBEL_1:def_8;
A5: f is V7() by A4, WAYBEL_0:66;
A6: rng f = the carrier of Se by A4, WAYBEL_0:66;
A7: Re is reflexive by A3, Def3;
A8: Re is transitive by A3, Def3;
A9: Se is reflexive by A1, A7, WAYBEL20:15;
Se is transitive by A1, A8, WAYBEL20:16;
hence A10: S is quasi_ordered by A9, Def3; ::_thesis: S is Dickson
now__::_thesis:_for_t_being_sequence_of_Se_ex_i,_j_being_Element_of_NAT_st_
(_i_<_j_&_t_._i_<=_t_._j_)
let t be sequence of Se; ::_thesis: ex i, j being Element of NAT st
( i < j & t . i <= t . j )
reconsider fi = f " as Function of the carrier of Se, the carrier of Re by A5, A6, FUNCT_2:25;
deffunc H1( Element of NAT ) -> Element of the carrier of Re = fi . (t . $1);
consider sr being Function of NAT, the carrier of Re such that
A11: for x being Element of NAT holds sr . x = H1(x) from FUNCT_2:sch_4();
reconsider sr = sr as sequence of Re ;
consider i, j being Element of NAT such that
A12: i < j and
A13: sr . i <= sr . j by A2, Th29;
take i = i; ::_thesis: ex j being Element of NAT st
( i < j & t . i <= t . j )
take j = j; ::_thesis: ( i < j & t . i <= t . j )
thus i < j by A12; ::_thesis: t . i <= t . j
A14: f . (sr . i) = f . ((f ") . (t . i)) by A11
.= t . i by A5, A6, FUNCT_1:35 ;
f . (sr . j) = f . ((f ") . (t . j)) by A11
.= t . j by A5, A6, FUNCT_1:35 ;
hence t . i <= t . j by A4, A13, A14, WAYBEL_0:66; ::_thesis: verum
end;
then for N being non empty Subset of Se holds
( min-classes N is finite & not min-classes N is empty ) by A10, Th31;
hence S is Dickson by A10, Th32; ::_thesis: verum
end;
supposeA15: ( R is empty or S is empty ) ; ::_thesis: ( S is quasi_ordered & S is Dickson )
A16: now__::_thesis:_S_is_empty
percases ( R is empty or S is empty ) by A15;
supposeA17: R is empty ; ::_thesis: S is empty
ex f being Function of R,S st f is isomorphic by A1, WAYBEL_1:def_8;
hence S is empty by A17, WAYBEL_0:def_38; ::_thesis: verum
end;
suppose S is empty ; ::_thesis: S is empty
hence S is empty ; ::_thesis: verum
end;
end;
end;
then for x being set st x in the carrier of S holds
[x,x] in the InternalRel of S ;
then A18: the InternalRel of S is_reflexive_in the carrier of S by RELAT_2:def_1;
for x, y, z being set st x in the carrier of S & y in the carrier of S & z in the carrier of S & [x,y] in the InternalRel of S & [y,z] in the InternalRel of S holds
[x,z] in the InternalRel of S by A16;
then A19: the InternalRel of S is_transitive_in the carrier of S by RELAT_2:def_8;
A20: S is reflexive by A18, ORDERS_2:def_2;
S is transitive by A19, ORDERS_2:def_3;
hence S is quasi_ordered by A20, Def3; ::_thesis: S is Dickson
thus S is Dickson by A16, Th36; ::_thesis: verum
end;
end;
end;
theorem Th39: :: DICKSON:39
for p being RelStr-yielding ManySortedSet of 1
for z being Element of 1 holds p . z, product p are_isomorphic
proof
let p be RelStr-yielding ManySortedSet of 1; ::_thesis: for z being Element of 1 holds p . z, product p are_isomorphic
let z be Element of 1; ::_thesis: p . z, product p are_isomorphic
deffunc H1( set ) -> set = 0 .--> $1;
consider f being Function such that
A1: dom f = the carrier of (p . z) and
A2: for x being set st x in the carrier of (p . z) holds
f . x = H1(x) from FUNCT_1:sch_3();
A3: z = 0 by CARD_1:49, TARSKI:def_1;
A4: 0 in 1 by CARD_1:49, TARSKI:def_1;
now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_(p_._z)_holds_
f_._x_in_the_carrier_of_(product_p)
let x be set ; ::_thesis: ( x in the carrier of (p . z) implies f . x in the carrier of (product p) )
assume A5: x in the carrier of (p . z) ; ::_thesis: f . x in the carrier of (product p)
then A6: f . x = 0 .--> x by A2;
set g = 0 .--> x;
A7: dom (0 .--> x) = {0} by FUNCOP_1:13
.= dom (Carrier p) by CARD_1:49, PARTFUN1:def_2 ;
now__::_thesis:_for_y_being_set_st_y_in_dom_(Carrier_p)_holds_
(0_.-->_x)_._y_in_(Carrier_p)_._y
let y be set ; ::_thesis: ( y in dom (Carrier p) implies (0 .--> x) . y in (Carrier p) . y )
assume y in dom (Carrier p) ; ::_thesis: (0 .--> x) . y in (Carrier p) . y
then A8: y in 1 ;
then A9: y = 0 by CARD_1:49, TARSKI:def_1;
ex R being 1-sorted st
( R = p . y & (Carrier p) . y = the carrier of R ) by A8, PRALG_1:def_13;
hence (0 .--> x) . y in (Carrier p) . y by A3, A5, A9, FUNCOP_1:72; ::_thesis: verum
end;
then f . x in product (Carrier p) by A6, A7, CARD_3:def_5;
hence f . x in the carrier of (product p) by YELLOW_1:def_4; ::_thesis: verum
end;
then reconsider f = f as Function of (p . z),(product p) by A1, FUNCT_2:3;
now__::_thesis:_f_is_isomorphic
percases ( p . z is empty or not p . z is empty ) ;
supposeA10: p . z is empty ; ::_thesis: f is isomorphic
now__::_thesis:_the_carrier_of_(product_p)_is_empty
assume not the carrier of (product p) is empty ; ::_thesis: contradiction
then A11: not product (Carrier p) is empty by YELLOW_1:def_4;
set x = the Element of product (Carrier p);
A12: ex g being Function st
( the Element of product (Carrier p) = g & dom g = dom (Carrier p) & ( for y being set st y in dom (Carrier p) holds
g . y in (Carrier p) . y ) ) by A11, CARD_3:def_5;
A13: 0 in dom (Carrier p) by A4, PARTFUN1:def_2;
consider R being 1-sorted such that
A14: R = p . 0 and
A15: (Carrier p) . 0 = the carrier of R by A4, PRALG_1:def_13;
R is empty by A10, A14, CARD_1:49, TARSKI:def_1;
hence contradiction by A12, A13, A15; ::_thesis: verum
end;
then product p is empty ;
hence f is isomorphic by A10, WAYBEL_0:def_38; ::_thesis: verum
end;
supposeA16: not p . z is empty ; ::_thesis: f is isomorphic
then reconsider pz = p . z as non empty RelStr ;
now__::_thesis:_for_i_being_set_st_i_in_rng_p_holds_
i_is_non_empty_1-sorted
let i be set ; ::_thesis: ( i in rng p implies i is non empty 1-sorted )
assume i in rng p ; ::_thesis: i is non empty 1-sorted
then consider x being set such that
A17: x in dom p and
A18: i = p . x by FUNCT_1:def_3;
x in 1 by A17;
then i = p . 0 by A18, CARD_1:49, TARSKI:def_1;
hence i is non empty 1-sorted by A16, CARD_1:49, TARSKI:def_1; ::_thesis: verum
end;
then reconsider np = p as RelStr-yielding yielding_non-empty_carriers ManySortedSet of 1 by YELLOW_6:def_2;
not product np is empty ;
then reconsider pp = product p as non empty RelStr ;
now__::_thesis:_for_x1,_x2_being_set_st_x1_in_dom_f_&_x2_in_dom_f_&_f_._x1_=_f_._x2_holds_
x1_=_x2
let x1, x2 be set ; ::_thesis: ( x1 in dom f & x2 in dom f & f . x1 = f . x2 implies x1 = x2 )
assume that
A19: x1 in dom f and
A20: x2 in dom f and
A21: f . x1 = f . x2 ; ::_thesis: x1 = x2
A22: f . x1 = 0 .--> x1 by A2, A19
.= [:{0},{x1}:] ;
A23: f . x2 = 0 .--> x2 by A2, A20
.= [:{0},{x2}:] ;
A24: 0 in {0} by TARSKI:def_1;
x1 in {x1} by TARSKI:def_1;
then [0,x1] in f . x2 by A21, A22, A24, ZFMISC_1:def_2;
then ex xa, ya being set st
( xa in {0} & ya in {x2} & [0,x1] = [xa,ya] ) by A23, ZFMISC_1:def_2;
then x1 in {x2} by XTUPLE_0:1;
hence x1 = x2 by TARSKI:def_1; ::_thesis: verum
end;
then A25: f is one-to-one by FUNCT_1:def_4;
now__::_thesis:_for_y_being_set_holds_
(_(_y_in_rng_f_implies_y_in_the_carrier_of_(product_p)_)_&_(_y_in_the_carrier_of_(product_p)_implies_y_in_rng_f_)_)
let y be set ; ::_thesis: ( ( y in rng f implies y in the carrier of (product p) ) & ( y in the carrier of (product p) implies y in rng f ) )
thus ( y in rng f implies y in the carrier of (product p) ) ; ::_thesis: ( y in the carrier of (product p) implies y in rng f )
assume y in the carrier of (product p) ; ::_thesis: y in rng f
then y in product (Carrier p) by YELLOW_1:def_4;
then consider g being Function such that
A26: y = g and
A27: dom g = dom (Carrier p) and
A28: for x being set st x in dom (Carrier p) holds
g . x in (Carrier p) . x by CARD_3:def_5;
A29: dom g = {0} by A27, CARD_1:49, PARTFUN1:def_2;
A30: 0 in dom (Carrier p) by A4, PARTFUN1:def_2;
A31: ex R being 1-sorted st
( R = p . 0 & (Carrier p) . 0 = the carrier of R ) by A4, PRALG_1:def_13;
A32: g = 0 .--> (g . 0) by A29, Th1;
A33: f . (g . 0) = 0 .--> (g . 0) by A2, A3, A28, A30, A31;
g . 0 in dom f by A1, A3, A28, A30, A31;
hence y in rng f by A26, A32, A33, FUNCT_1:def_3; ::_thesis: verum
end;
then A34: rng f = the carrier of (product p) by TARSKI:1;
reconsider f9 = f as Function of pz,pp ;
now__::_thesis:_for_x,_y_being_Element_of_pz_holds_
(_(_x_<=_y_implies_f9_._x_<=_f9_._y_)_&_(_f9_._x_<=_f9_._y_implies_x_<=_y_)_)
let x, y be Element of pz; ::_thesis: ( ( x <= y implies f9 . x <= f9 . y ) & ( f9 . x <= f9 . y implies x <= y ) )
not product np is empty ;
then A35: not product (Carrier p) is empty by YELLOW_1:def_4;
A36: f9 . x is Element of product (Carrier p) by YELLOW_1:def_4;
hereby ::_thesis: ( f9 . x <= f9 . y implies x <= y )
assume A37: x <= y ; ::_thesis: f9 . x <= f9 . y
ex f1, g1 being Function st
( f1 = f9 . x & g1 = f9 . y & ( for i being set st i in 1 holds
ex R being RelStr ex xi, yi being Element of R st
( R = p . i & xi = f1 . i & yi = g1 . i & xi <= yi ) ) )
proof
set f1 = 0 .--> x;
set g1 = 0 .--> y;
reconsider f1 = 0 .--> x as Function ;
reconsider g1 = 0 .--> y as Function ;
take f1 ; ::_thesis: ex g1 being Function st
( f1 = f9 . x & g1 = f9 . y & ( for i being set st i in 1 holds
ex R being RelStr ex xi, yi being Element of R st
( R = p . i & xi = f1 . i & yi = g1 . i & xi <= yi ) ) )
take g1 ; ::_thesis: ( f1 = f9 . x & g1 = f9 . y & ( for i being set st i in 1 holds
ex R being RelStr ex xi, yi being Element of R st
( R = p . i & xi = f1 . i & yi = g1 . i & xi <= yi ) ) )
thus f1 = f9 . x by A2; ::_thesis: ( g1 = f9 . y & ( for i being set st i in 1 holds
ex R being RelStr ex xi, yi being Element of R st
( R = p . i & xi = f1 . i & yi = g1 . i & xi <= yi ) ) )
thus g1 = f9 . y by A2; ::_thesis: for i being set st i in 1 holds
ex R being RelStr ex xi, yi being Element of R st
( R = p . i & xi = f1 . i & yi = g1 . i & xi <= yi )
let i be set ; ::_thesis: ( i in 1 implies ex R being RelStr ex xi, yi being Element of R st
( R = p . i & xi = f1 . i & yi = g1 . i & xi <= yi ) )
assume A38: i in 1 ; ::_thesis: ex R being RelStr ex xi, yi being Element of R st
( R = p . i & xi = f1 . i & yi = g1 . i & xi <= yi )
A39: i = 0 by A38, CARD_1:49, TARSKI:def_1;
0 in dom p by A4, PARTFUN1:def_2;
then p . 0 in rng p by FUNCT_1:def_3;
then reconsider p0 = p . 0 as RelStr by YELLOW_1:def_3;
set R = p0;
reconsider x9 = x as Element of p0 by CARD_1:49, TARSKI:def_1;
reconsider y9 = y as Element of p0 by CARD_1:49, TARSKI:def_1;
take p0 ; ::_thesis: ex xi, yi being Element of p0 st
( p0 = p . i & xi = f1 . i & yi = g1 . i & xi <= yi )
take x9 ; ::_thesis: ex yi being Element of p0 st
( p0 = p . i & x9 = f1 . i & yi = g1 . i & x9 <= yi )
take y9 ; ::_thesis: ( p0 = p . i & x9 = f1 . i & y9 = g1 . i & x9 <= y9 )
thus p0 = p . i by A38, CARD_1:49, TARSKI:def_1; ::_thesis: ( x9 = f1 . i & y9 = g1 . i & x9 <= y9 )
thus x9 = f1 . i by A39, FUNCOP_1:72; ::_thesis: ( y9 = g1 . i & x9 <= y9 )
thus y9 = g1 . i by A39, FUNCOP_1:72; ::_thesis: x9 <= y9
thus x9 <= y9 by A37, CARD_1:49, TARSKI:def_1; ::_thesis: verum
end;
hence f9 . x <= f9 . y by A35, A36, YELLOW_1:def_4; ::_thesis: verum
end;
assume f9 . x <= f9 . y ; ::_thesis: x <= y
then consider f1, g1 being Function such that
A40: f1 = f9 . x and
A41: g1 = f9 . y and
A42: for i being set st i in 1 holds
ex R being RelStr ex xi, yi being Element of R st
( R = p . i & xi = f1 . i & yi = g1 . i & xi <= yi ) by A35, A36, YELLOW_1:def_4;
consider R being RelStr , xi, yi being Element of R such that
A43: R = p . 0 and
A44: xi = f1 . 0 and
A45: yi = g1 . 0 and
A46: xi <= yi by A4, A42;
f1 = 0 .--> x by A2, A40;
then A47: xi = x by A44, FUNCOP_1:72;
A48: g1 = 0 .--> y by A2, A41;
R = pz by A43, CARD_1:49, TARSKI:def_1;
hence x <= y by A45, A46, A47, A48, FUNCOP_1:72; ::_thesis: verum
end;
hence f is isomorphic by A25, A34, WAYBEL_0:66; ::_thesis: verum
end;
end;
end;
hence p . z, product p are_isomorphic by WAYBEL_1:def_8; ::_thesis: verum
end;
registration
let X be set ;
let p be RelStr-yielding ManySortedSet of X;
let Y be Subset of X;
clusterp | Y -> RelStr-yielding ;
coherence
p | Y is RelStr-yielding
proof
now__::_thesis:_for_v_being_set_st_v_in_rng_(p_|_Y)_holds_
v_is_RelStr
let v be set ; ::_thesis: ( v in rng (p | Y) implies v is RelStr )
assume v in rng (p | Y) ; ::_thesis: v is RelStr
then consider x being set such that
A1: x in dom (p | Y) and
A2: v = (p | Y) . x by FUNCT_1:def_3;
A3: x in Y by A1;
then A4: (p | Y) . x = p . x by FUNCT_1:49;
x in X by A3;
then x in dom p by PARTFUN1:def_2;
then p . x in rng p by FUNCT_1:3;
hence v is RelStr by A2, A4, YELLOW_1:def_3; ::_thesis: verum
end;
hence p | Y is RelStr-yielding by YELLOW_1:def_3; ::_thesis: verum
end;
end;
theorem Th40: :: DICKSON:40
for n being non empty Nat
for p being RelStr-yielding ManySortedSet of n holds
( not product p is empty iff p is non-Empty )
proof
let n be non empty Nat; ::_thesis: for p being RelStr-yielding ManySortedSet of n holds
( not product p is empty iff p is non-Empty )
let p be RelStr-yielding ManySortedSet of n; ::_thesis: ( not product p is empty iff p is non-Empty )
hereby ::_thesis: ( p is non-Empty implies not product p is empty )
assume not product p is empty ; ::_thesis: p is non-Empty
then not product (Carrier p) is empty by YELLOW_1:def_4;
then consider z being set such that
A1: z in product (Carrier p) by XBOOLE_0:def_1;
A2: ex g being Function st
( z = g & dom g = dom (Carrier p) & ( for i being set st i in dom (Carrier p) holds
g . i in (Carrier p) . i ) ) by A1, CARD_3:def_5;
now__::_thesis:_for_S_being_1-sorted_st_S_in_rng_p_holds_
not_S_is_empty
let S be 1-sorted ; ::_thesis: ( S in rng p implies not S is empty )
assume S in rng p ; ::_thesis: not S is empty
then consider x being set such that
A3: x in dom p and
A4: S = p . x by FUNCT_1:def_3;
A5: x in n by A3;
then A6: x in dom (Carrier p) by PARTFUN1:def_2;
ex R being 1-sorted st
( R = p . x & (Carrier p) . x = the carrier of R ) by A5, PRALG_1:def_13;
hence not S is empty by A2, A4, A6; ::_thesis: verum
end;
hence p is non-Empty by WAYBEL_3:def_7; ::_thesis: verum
end;
assume A7: p is non-Empty ; ::_thesis: not product p is empty
A8: dom (Carrier p) = n by PARTFUN1:def_2;
deffunc H1( set ) -> Element of (Carrier p) . $1 = choose ((Carrier p) . $1);
consider g being Function such that
A9: dom g = dom (Carrier p) and
A10: for i being set st i in dom (Carrier p) holds
g . i = H1(i) from FUNCT_1:sch_3();
set x = g;
now__::_thesis:_ex_g_being_Function_st_
(_g_=_g_&_dom_g_=_dom_(Carrier_p)_&_(_for_i_being_set_st_i_in_dom_(Carrier_p)_holds_
g_._i_in_(Carrier_p)_._i_)_)
take g = g; ::_thesis: ( g = g & dom g = dom (Carrier p) & ( for i being set st i in dom (Carrier p) holds
g . i in (Carrier p) . i ) )
thus g = g ; ::_thesis: ( dom g = dom (Carrier p) & ( for i being set st i in dom (Carrier p) holds
g . i in (Carrier p) . i ) )
thus dom g = dom (Carrier p) by A9; ::_thesis: for i being set st i in dom (Carrier p) holds
g . i in (Carrier p) . i
thus for i being set st i in dom (Carrier p) holds
g . i in (Carrier p) . i ::_thesis: verum
proof
let i be set ; ::_thesis: ( i in dom (Carrier p) implies g . i in (Carrier p) . i )
assume A11: i in dom (Carrier p) ; ::_thesis: g . i in (Carrier p) . i
i in dom p by A8, A11, PARTFUN1:def_2;
then A12: p . i in rng p by FUNCT_1:def_3;
then reconsider pi1 = p . i as RelStr by YELLOW_1:def_3;
not pi1 is empty by A7, A12, WAYBEL_3:def_7;
then A13: not the carrier of pi1 is empty ;
i in n by A11;
then A14: ex R being 1-sorted st
( R = p . i & (Carrier p) . i = the carrier of R ) by PRALG_1:def_13;
g . i = choose ((Carrier p) . i) by A10, A11;
hence g . i in (Carrier p) . i by A13, A14; ::_thesis: verum
end;
end;
then not product (Carrier p) is empty by CARD_3:def_5;
hence not product p is empty by YELLOW_1:def_4; ::_thesis: verum
end;
theorem Th41: :: DICKSON:41
for n being non empty Nat
for p being RelStr-yielding ManySortedSet of n + 1
for ns being Subset of (n + 1)
for ne being Element of n + 1 st ns = n & ne = n holds
[:(product (p | ns)),(p . ne):], product p are_isomorphic
proof
let n be non empty Nat; ::_thesis: for p being RelStr-yielding ManySortedSet of n + 1
for ns being Subset of (n + 1)
for ne being Element of n + 1 st ns = n & ne = n holds
[:(product (p | ns)),(p . ne):], product p are_isomorphic
let p be RelStr-yielding ManySortedSet of n + 1; ::_thesis: for ns being Subset of (n + 1)
for ne being Element of n + 1 st ns = n & ne = n holds
[:(product (p | ns)),(p . ne):], product p are_isomorphic
let ns be Subset of (n + 1); ::_thesis: for ne being Element of n + 1 st ns = n & ne = n holds
[:(product (p | ns)),(p . ne):], product p are_isomorphic
let ne be Element of n + 1; ::_thesis: ( ns = n & ne = n implies [:(product (p | ns)),(p . ne):], product p are_isomorphic )
assume that
A1: ns = n and
A2: ne = n ; ::_thesis: [:(product (p | ns)),(p . ne):], product p are_isomorphic
set S = [:(product (p | ns)),(p . ne):];
set T = product p;
set CP = [: the carrier of (product (p | ns)), the carrier of (p . ne):];
set CPP = the carrier of (product p);
A3: dom (Carrier (p | ns)) = n by A1, PARTFUN1:def_2;
percases ( the carrier of (product p) is empty or not the carrier of (product p) is empty ) ;
supposeA4: the carrier of (product p) is empty ; ::_thesis: [:(product (p | ns)),(p . ne):], product p are_isomorphic
then A5: product p is empty ;
not p is non-Empty by A4;
then consider r1 being 1-sorted such that
A6: r1 in rng p and
A7: r1 is empty by WAYBEL_3:def_7;
consider x being set such that
A8: x in dom p and
A9: r1 = p . x by A6, FUNCT_1:def_3;
x in n + 1 by A8;
then A10: x in n \/ {n} by AFINSQ_1:2;
A11: [:(product (p | ns)),(p . ne):] is empty
proof
percases ( x in n or x in {n} ) by A10, XBOOLE_0:def_3;
supposeA12: x in n ; ::_thesis: [:(product (p | ns)),(p . ne):] is empty
then A13: (p | ns) . x = p . x by A1, FUNCT_1:49;
x in dom (p | ns) by A1, A12, PARTFUN1:def_2;
then p . x in rng (p | ns) by A13, FUNCT_1:def_3;
then not p | ns is non-Empty by A7, A9, WAYBEL_3:def_7;
then reconsider ptemp = the carrier of (product (p | ns)) as empty set by A1, Th40;
[:ptemp, the carrier of (p . ne):] is empty ;
hence [:(product (p | ns)),(p . ne):] is empty by YELLOW_3:def_2; ::_thesis: verum
end;
suppose x in {n} ; ::_thesis: [:(product (p | ns)),(p . ne):] is empty
then p . ne is empty by A2, A7, A9, TARSKI:def_1;
then reconsider pne = the carrier of (p . ne) as empty set ;
reconsider ptemp = the carrier of (product (p | ns)) as set ;
[:ptemp,pne:] is empty ;
hence [:(product (p | ns)),(p . ne):] is empty by YELLOW_3:def_2; ::_thesis: verum
end;
end;
end;
then A14: dom {} = the carrier of [:(product (p | ns)),(p . ne):] ;
for x being set st x in {} holds
{} . x in the carrier of (product p) ;
then reconsider f = {} as Function of [:(product (p | ns)),(p . ne):],(product p) by A14, FUNCT_2:3;
f is isomorphic by A5, A11, WAYBEL_0:def_38;
hence [:(product (p | ns)),(p . ne):], product p are_isomorphic by WAYBEL_1:def_8; ::_thesis: verum
end;
supposeA15: not the carrier of (product p) is empty ; ::_thesis: [:(product (p | ns)),(p . ne):], product p are_isomorphic
then reconsider CPP = the carrier of (product p) as non empty set ;
reconsider T = product p as non empty RelStr by A15;
A16: p is non-Empty by A15, Th40;
reconsider p9 = p as RelStr-yielding non-Empty ManySortedSet of n + 1 by A15, Th40;
not p9 . ne is empty ;
then reconsider pne2 = the carrier of (p . ne) as non empty set ;
now__::_thesis:_for_S_being_1-sorted_st_S_in_rng_(p_|_ns)_holds_
not_S_is_empty
let S be 1-sorted ; ::_thesis: ( S in rng (p | ns) implies not S is empty )
assume S in rng (p | ns) ; ::_thesis: not S is empty
then consider x being set such that
A17: x in dom (p | ns) and
A18: S = (p | ns) . x by FUNCT_1:def_3;
x in ns by A17;
then x in n + 1 ;
then A19: x in dom p by PARTFUN1:def_2;
S = p . x by A17, A18, FUNCT_1:47;
then S in rng p by A19, FUNCT_1:def_3;
hence not S is empty by A16, WAYBEL_3:def_7; ::_thesis: verum
end;
then A20: p | ns is non-Empty by WAYBEL_3:def_7;
then A21: not product (p | ns) is empty ;
reconsider ppns2 = the carrier of (product (p | ns)) as non empty set by A20;
[: the carrier of (product (p | ns)), the carrier of (p . ne):] = [:ppns2,pne2:] ;
then reconsider S = [:(product (p | ns)),(p . ne):] as non empty RelStr by YELLOW_3:def_2;
[: the carrier of (product (p | ns)), the carrier of (p . ne):] = the carrier of S by YELLOW_3:def_2;
then reconsider CP9 = [: the carrier of (product (p | ns)), the carrier of (p . ne):] as non empty set ;
defpred S1[ set , set ] means ex a being Function ex b being Element of pne2 st
( a in ppns2 & $1 = [a,b] & $2 = a +* (n .--> b) );
A22: for x being Element of CP9 ex y being Element of CPP st S1[x,y]
proof
let x be Element of CP9; ::_thesis: ex y being Element of CPP st S1[x,y]
reconsider xx = x as Element of [:ppns2,pne2:] ;
set a = xx `1 ;
set b = xx `2 ;
reconsider a = xx `1 as Element of ppns2 by MCART_1:10;
reconsider b = xx `2 as Element of pne2 by MCART_1:10;
A23: not product (Carrier (p | ns)) is empty by A21, YELLOW_1:def_4;
A24: a is Element of product (Carrier (p | ns)) by YELLOW_1:def_4;
then A25: ex g being Function st
( a = g & dom g = dom (Carrier (p | ns)) & ( for i being set st i in dom (Carrier (p | ns)) holds
g . i in (Carrier (p | ns)) . i ) ) by A23, CARD_3:def_5;
reconsider a = a as Function by A24;
set y = a +* (n .--> b);
now__::_thesis:_ex_g1_being_Function_st_
(_a_+*_(n_.-->_b)_=_g1_&_dom_g1_=_dom_(Carrier_p)_&_(_for_x_being_set_st_x_in_dom_(Carrier_p)_holds_
g1_._x_in_(Carrier_p)_._x_)_)
set g1 = a +* (n .--> b);
reconsider g1 = a +* (n .--> b) as Function ;
take g1 = g1; ::_thesis: ( a +* (n .--> b) = g1 & dom g1 = dom (Carrier p) & ( for x being set st x in dom (Carrier p) holds
b2 . b3 in (Carrier p) . b3 ) )
thus a +* (n .--> b) = g1 ; ::_thesis: ( dom g1 = dom (Carrier p) & ( for x being set st x in dom (Carrier p) holds
b2 . b3 in (Carrier p) . b3 ) )
A26: dom a = ns by A25, PARTFUN1:def_2;
thus dom g1 = (dom a) \/ (dom (n .--> b)) by FUNCT_4:def_1
.= n \/ {n} by A1, A26, FUNCOP_1:13
.= n + 1 by AFINSQ_1:2
.= dom (Carrier p) by PARTFUN1:def_2 ; ::_thesis: for x being set st x in dom (Carrier p) holds
b2 . b3 in (Carrier p) . b3
let x be set ; ::_thesis: ( x in dom (Carrier p) implies b1 . b2 in (Carrier p) . b2 )
assume x in dom (Carrier p) ; ::_thesis: b1 . b2 in (Carrier p) . b2
then A27: x in n + 1 ;
then A28: x in n \/ {n} by AFINSQ_1:2;
percases ( x in n or x in {n} ) by A28, XBOOLE_0:def_3;
supposeA29: x in n ; ::_thesis: b1 . b2 in (Carrier p) . b2
then x <> n ;
then not x in dom (n .--> b) by TARSKI:def_1;
then A30: g1 . x = a . x by FUNCT_4:11;
A31: x in dom (Carrier (p | ns)) by A1, A29, PARTFUN1:def_2;
A32: ex R being 1-sorted st
( R = (p | ns) . x & (Carrier (p | ns)) . x = the carrier of R ) by A1, A29, PRALG_1:def_13;
ex R2 being 1-sorted st
( R2 = p . x & (Carrier p) . x = the carrier of R2 ) by A27, PRALG_1:def_13;
then (Carrier (p | ns)) . x = (Carrier p) . x by A1, A29, A32, FUNCT_1:49;
hence g1 . x in (Carrier p) . x by A25, A30, A31; ::_thesis: verum
end;
supposeA33: x in {n} ; ::_thesis: b1 . b2 in (Carrier p) . b2
then A34: x = n by TARSKI:def_1;
x in dom (n .--> b) by A33, FUNCOP_1:13;
then A35: g1 . x = (n .--> b) . n by A34, FUNCT_4:13
.= b by FUNCOP_1:72 ;
ex R being 1-sorted st
( R = p . ne & (Carrier p) . ne = the carrier of R ) by PRALG_1:def_13;
hence g1 . x in (Carrier p) . x by A2, A34, A35; ::_thesis: verum
end;
end;
end;
then a +* (n .--> b) in product (Carrier p) by CARD_3:def_5;
then reconsider y = a +* (n .--> b) as Element of CPP by YELLOW_1:def_4;
take y ; ::_thesis: S1[x,y]
take a ; ::_thesis: ex b being Element of pne2 st
( a in ppns2 & x = [a,b] & y = a +* (n .--> b) )
take b ; ::_thesis: ( a in ppns2 & x = [a,b] & y = a +* (n .--> b) )
thus a in ppns2 ; ::_thesis: ( x = [a,b] & y = a +* (n .--> b) )
thus x = [a,b] by MCART_1:21; ::_thesis: y = a +* (n .--> b)
thus y = a +* (n .--> b) ; ::_thesis: verum
end;
consider f being Function of CP9,CPP such that
A36: for x being Element of CP9 holds S1[x,f . x] from FUNCT_2:sch_3(A22);
now__::_thesis:_ex_f_being_Function_of_[:(product_(p_|_ns)),(p_._ne):],(product_p)_st_f_is_isomorphic
the carrier of [:(product (p | ns)),(p . ne):] = [: the carrier of (product (p | ns)), the carrier of (p . ne):] by YELLOW_3:def_2;
then reconsider f = f as Function of [:(product (p | ns)),(p . ne):],(product p) ;
reconsider f9 = f as Function of S,T ;
take f = f; ::_thesis: f is isomorphic
now__::_thesis:_for_x1,_x2_being_set_st_x1_in_dom_f_&_x2_in_dom_f_&_f_._x1_=_f_._x2_holds_
x1_=_x2
let x1, x2 be set ; ::_thesis: ( x1 in dom f & x2 in dom f & f . x1 = f . x2 implies x1 = x2 )
assume that
A37: x1 in dom f and
A38: x2 in dom f and
A39: f . x1 = f . x2 ; ::_thesis: x1 = x2
x1 is Element of [: the carrier of (product (p | ns)), the carrier of (p . ne):] by A37, YELLOW_3:def_2;
then consider a1 being Function, b1 being Element of pne2 such that
A40: a1 in ppns2 and
A41: x1 = [a1,b1] and
A42: f . x1 = a1 +* (n .--> b1) by A36;
x2 is Element of [: the carrier of (product (p | ns)), the carrier of (p . ne):] by A38, YELLOW_3:def_2;
then consider a2 being Function, b2 being Element of pne2 such that
A43: a2 in ppns2 and
A44: x2 = [a2,b2] and
A45: f . x2 = a2 +* (n .--> b2) by A36;
a1 in product (Carrier (p | ns)) by A40, YELLOW_1:def_4;
then A46: ex g1 being Function st
( a1 = g1 & dom g1 = dom (Carrier (p | ns)) & ( for x being set st x in dom (Carrier (p | ns)) holds
g1 . x in (Carrier (p | ns)) . x ) ) by CARD_3:def_5;
a2 in product (Carrier (p | ns)) by A43, YELLOW_1:def_4;
then A47: ex g2 being Function st
( a2 = g2 & dom g2 = dom (Carrier (p | ns)) & ( for x being set st x in dom (Carrier (p | ns)) holds
g2 . x in (Carrier (p | ns)) . x ) ) by CARD_3:def_5;
A48: dom (n .--> b1) = {n} by FUNCOP_1:13;
then A49: dom (n .--> b1) = dom (n .--> b2) by FUNCOP_1:13;
A50: dom a1 = n by A1, A46, PARTFUN1:def_2;
A51: now__::_thesis:_n_misses_{n}
assume not n misses {n} ; ::_thesis: contradiction
then n /\ {n} <> {} by XBOOLE_0:def_7;
then consider x being set such that
A52: x in n /\ {n} by XBOOLE_0:def_1;
A53: x in n by A52, XBOOLE_0:def_4;
x in {n} by A52, XBOOLE_0:def_4;
then x = n by TARSKI:def_1;
hence contradiction by A53; ::_thesis: verum
end;
then A54: dom a1 misses dom (n .--> b1) by A50, FUNCOP_1:13;
A55: dom a2 misses dom (n .--> b2) by A46, A47, A50, A51, FUNCOP_1:13;
A56: now__::_thesis:_for_a,_b_being_set_holds_
(_(_[a,b]_in_a1_implies_[a,b]_in_a2_)_&_(_[a,b]_in_a2_implies_[a,b]_in_a1_)_)
let a, b be set ; ::_thesis: ( ( [a,b] in a1 implies [a,b] in a2 ) & ( [a,b] in a2 implies [a,b] in a1 ) )
hereby ::_thesis: ( [a,b] in a2 implies [a,b] in a1 )
assume A57: [a,b] in a1 ; ::_thesis: [a,b] in a2
then A58: a in dom a1 by FUNCT_1:1;
A59: b = a1 . a by A57, FUNCT_1:1;
A60: a in (dom a1) \/ (dom (n .--> b1)) by A58, XBOOLE_0:def_3;
then not a in dom (n .--> b1) by A54, A58, XBOOLE_0:5;
then A61: (a2 +* (n .--> b2)) . a = a1 . a by A39, A42, A45, A60, FUNCT_4:def_1;
A62: a in (dom a2) \/ (dom (n .--> b2)) by A46, A47, A58, XBOOLE_0:def_3;
A63: a in dom a2 by A46, A47, A57, FUNCT_1:1;
then not a in dom (n .--> b2) by A55, A62, XBOOLE_0:5;
then b = a2 . a by A59, A61, A62, FUNCT_4:def_1;
hence [a,b] in a2 by A63, FUNCT_1:1; ::_thesis: verum
end;
assume A64: [a,b] in a2 ; ::_thesis: [a,b] in a1
then A65: a in dom a2 by FUNCT_1:1;
A66: b = a2 . a by A64, FUNCT_1:1;
A67: a in (dom a2) \/ (dom (n .--> b2)) by A65, XBOOLE_0:def_3;
then not a in dom (n .--> b2) by A55, A65, XBOOLE_0:5;
then A68: (a1 +* (n .--> b1)) . a = a2 . a by A39, A42, A45, A67, FUNCT_4:def_1;
A69: a in (dom a1) \/ (dom (n .--> b1)) by A46, A47, A65, XBOOLE_0:def_3;
A70: a in dom a1 by A46, A47, A64, FUNCT_1:1;
then not a in dom (n .--> b1) by A54, A69, XBOOLE_0:5;
then b = a1 . a by A66, A68, A69, FUNCT_4:def_1;
hence [a,b] in a1 by A70, FUNCT_1:1; ::_thesis: verum
end;
A71: n in dom (n .--> b1) by A48, TARSKI:def_1;
then A72: n in (dom a1) \/ (dom (n .--> b1)) by XBOOLE_0:def_3;
A73: n in dom (n .--> b2) by A48, A49, TARSKI:def_1;
then n in (dom a2) \/ (dom (n .--> b2)) by XBOOLE_0:def_3;
then (a1 +* (n .--> b1)) . n = (n .--> b2) . n by A39, A42, A45, A73, FUNCT_4:def_1
.= b2 by FUNCOP_1:72 ;
then b2 = (n .--> b1) . n by A71, A72, FUNCT_4:def_1
.= b1 by FUNCOP_1:72 ;
hence x1 = x2 by A41, A44, A56, RELAT_1:def_2; ::_thesis: verum
end;
then A74: f is one-to-one by FUNCT_1:def_4;
now__::_thesis:_for_y_being_set_holds_
(_(_y_in_rng_f_implies_y_in_the_carrier_of_(product_p)_)_&_(_y_in_the_carrier_of_(product_p)_implies_y_in_rng_f_)_)
let y be set ; ::_thesis: ( ( y in rng f implies y in the carrier of (product p) ) & ( y in the carrier of (product p) implies y in rng f ) )
thus ( y in rng f implies y in the carrier of (product p) ) ; ::_thesis: ( y in the carrier of (product p) implies y in rng f )
assume y in the carrier of (product p) ; ::_thesis: y in rng f
then y in product (Carrier p) by YELLOW_1:def_4;
then consider g being Function such that
A75: y = g and
A76: dom g = dom (Carrier p) and
A77: for x being set st x in dom (Carrier p) holds
g . x in (Carrier p) . x by CARD_3:def_5;
A78: dom (Carrier p) = n + 1 by PARTFUN1:def_2;
A79: n + 1 = { i where i is Element of NAT : i < n + 1 } by AXIOMS:4;
A80: n in NAT by ORDINAL1:def_12;
n < n + 1 by NAT_1:13;
then A81: n in n + 1 by A79, A80;
set a = g | n;
set b = g . n;
set x = [(g | n),(g . n)];
A82: dom (Carrier (p | ns)) = ns by PARTFUN1:def_2;
A83: dom (g | n) = (dom g) /\ n by RELAT_1:61
.= (n + 1) /\ n by A76, PARTFUN1:def_2 ;
then A84: dom (g | n) = n by Th2, XBOOLE_1:28;
A85: dom (g | n) = dom (Carrier (p | ns)) by A1, A82, A83, XBOOLE_1:28;
now__::_thesis:_for_x_being_set_st_x_in_dom_(Carrier_(p_|_ns))_holds_
(g_|_n)_._x_in_(Carrier_(p_|_ns))_._x
let x be set ; ::_thesis: ( x in dom (Carrier (p | ns)) implies (g | n) . x in (Carrier (p | ns)) . x )
assume x in dom (Carrier (p | ns)) ; ::_thesis: (g | n) . x in (Carrier (p | ns)) . x
then A86: x in n by A1;
A87: n c= n + 1 by Th2;
A88: (g | n) . x = g . x by A86, FUNCT_1:49;
A89: g . x in (Carrier p) . x by A77, A78, A86, A87;
A90: ex R1 being 1-sorted st
( R1 = p . x & (Carrier p) . x = the carrier of R1 ) by A86, A87, PRALG_1:def_13;
ex R2 being 1-sorted st
( R2 = (p | ns) . x & (Carrier (p | ns)) . x = the carrier of R2 ) by A1, A86, PRALG_1:def_13;
hence (g | n) . x in (Carrier (p | ns)) . x by A1, A86, A88, A89, A90, FUNCT_1:49; ::_thesis: verum
end;
then g | n in product (Carrier (p | ns)) by A85, CARD_3:9;
then A91: g | n in the carrier of (product (p | ns)) by YELLOW_1:def_4;
ex R1 being 1-sorted st
( R1 = p . n & (Carrier p) . n = the carrier of R1 ) by A81, PRALG_1:def_13;
then A92: g . n in the carrier of (p . ne) by A2, A77, A78;
then [(g | n),(g . n)] in [: the carrier of (product (p | ns)), the carrier of (p . ne):] by A91, ZFMISC_1:87;
then A93: [(g | n),(g . n)] in dom f by FUNCT_2:def_1;
[(g | n),(g . n)] is Element of CP9 by A91, A92, ZFMISC_1:87;
then consider ta being Function, tb being Element of pne2 such that
ta in ppns2 and
A94: [(g | n),(g . n)] = [ta,tb] and
A95: f . [(g | n),(g . n)] = ta +* (n .--> tb) by A36;
A96: ta = g | n by A94, XTUPLE_0:1;
A97: tb = g . n by A94, XTUPLE_0:1;
now__::_thesis:_for_i,_j_being_set_holds_
(_(_[i,j]_in_(g_|_n)_+*_(n_.-->_(g_._n))_implies_[i,j]_in_g_)_&_(_[i,j]_in_g_implies_[i,j]_in_(g_|_n)_+*_(n_.-->_(g_._n))_)_)
let i, j be set ; ::_thesis: ( ( [i,j] in (g | n) +* (n .--> (g . n)) implies [i,j] in g ) & ( [i,j] in g implies [b1,b2] in (g | n) +* (n .--> (g . n)) ) )
hereby ::_thesis: ( [i,j] in g implies [b1,b2] in (g | n) +* (n .--> (g . n)) )
assume A98: [i,j] in (g | n) +* (n .--> (g . n)) ; ::_thesis: [i,j] in g
then i in dom ((g | n) +* (n .--> (g . n))) by FUNCT_1:1;
then A99: i in (dom (g | n)) \/ (dom (n .--> (g . n))) by FUNCT_4:def_1;
then A100: i in n \/ {n} by A84, FUNCOP_1:13;
A101: ((g | n) +* (n .--> (g . n))) . i = j by A98, FUNCT_1:1;
percases ( i in dom (n .--> (g . n)) or not i in dom (n .--> (g . n)) ) ;
supposeA102: i in dom (n .--> (g . n)) ; ::_thesis: [i,j] in g
then i in {n} ;
then A103: i = n by TARSKI:def_1;
((g | n) +* (n .--> (g . n))) . i = (n .--> (g . n)) . i by A99, A102, FUNCT_4:def_1
.= g . n by A103, FUNCOP_1:72 ;
then A104: g . i = j by A98, A103, FUNCT_1:1;
i in dom g by A76, A78, A100, AFINSQ_1:2;
hence [i,j] in g by A104, FUNCT_1:1; ::_thesis: verum
end;
supposeA105: not i in dom (n .--> (g . n)) ; ::_thesis: [i,j] in g
then not i in {n} by FUNCOP_1:13;
then A106: i in n by A100, XBOOLE_0:def_3;
((g | n) +* (n .--> (g . n))) . i = (g | n) . i by A99, A105, FUNCT_4:def_1;
then A107: g . i = j by A101, A106, FUNCT_1:49;
i in dom g by A76, A78, A100, AFINSQ_1:2;
hence [i,j] in g by A107, FUNCT_1:1; ::_thesis: verum
end;
end;
end;
assume A108: [i,j] in g ; ::_thesis: [b1,b2] in (g | n) +* (n .--> (g . n))
then A109: i in n + 1 by A76, A78, FUNCT_1:1;
then A110: i in n \/ {n} by AFINSQ_1:2;
dom ((g | n) +* (n .--> (g . n))) = (dom (g | n)) \/ (dom (n .--> (g . n))) by FUNCT_4:def_1
.= n \/ {n} by A84, FUNCOP_1:13 ;
then A111: i in dom ((g | n) +* (n .--> (g . n))) by A109, AFINSQ_1:2;
then A112: i in (dom (g | n)) \/ (dom (n .--> (g . n))) by FUNCT_4:def_1;
percases ( i in dom (n .--> (g . n)) or not i in dom (n .--> (g . n)) ) ;
supposeA113: i in dom (n .--> (g . n)) ; ::_thesis: [b1,b2] in (g | n) +* (n .--> (g . n))
then i in {n} ;
then A114: i = n by TARSKI:def_1;
((g | n) +* (n .--> (g . n))) . i = (n .--> (g . n)) . i by A112, A113, FUNCT_4:def_1
.= g . n by A114, FUNCOP_1:72
.= j by A108, A114, FUNCT_1:1 ;
hence [i,j] in (g | n) +* (n .--> (g . n)) by A111, FUNCT_1:1; ::_thesis: verum
end;
supposeA115: not i in dom (n .--> (g . n)) ; ::_thesis: [b1,b2] in (g | n) +* (n .--> (g . n))
then not i in {n} by FUNCOP_1:13;
then A116: i in n by A110, XBOOLE_0:def_3;
((g | n) +* (n .--> (g . n))) . i = (g | n) . i by A112, A115, FUNCT_4:def_1
.= g . i by A116, FUNCT_1:49
.= j by A108, FUNCT_1:1 ;
hence [i,j] in (g | n) +* (n .--> (g . n)) by A111, FUNCT_1:1; ::_thesis: verum
end;
end;
end;
then f . [(g | n),(g . n)] = y by A75, A95, A96, A97, RELAT_1:def_2;
hence y in rng f by A93, FUNCT_1:def_3; ::_thesis: verum
end;
then A117: rng f = the carrier of (product p) by TARSKI:1;
now__::_thesis:_for_x,_y_being_Element_of_S_holds_
(_(_x_<=_y_implies_f9_._x_<=_f9_._y_)_&_(_f9_._x_<=_f9_._y_implies_x_<=_y_)_)
let x, y be Element of S; ::_thesis: ( ( x <= y implies f9 . x <= f9 . y ) & ( f9 . x <= f9 . y implies x <= y ) )
A118: x is Element of [: the carrier of (product (p | ns)), the carrier of (p . ne):] by YELLOW_3:def_2;
then consider xa being Function, xb being Element of pne2 such that
A119: xa in ppns2 and
A120: x = [xa,xb] and
A121: f . x = xa +* (n .--> xb) by A36;
dom f = CP9 by FUNCT_2:def_1;
then f . x in the carrier of (product p) by A117, A118, FUNCT_1:def_3;
then A122: f9 . x in product (Carrier p) by YELLOW_1:def_4;
y is Element of [: the carrier of (product (p | ns)), the carrier of (p . ne):] by YELLOW_3:def_2;
then consider ya being Function, yb being Element of pne2 such that
A123: ya in ppns2 and
A124: y = [ya,yb] and
A125: f . y = ya +* (n .--> yb) by A36;
A126: [xa,xb] `1 = xa ;
A127: [xa,xb] `2 = xb ;
A128: xa in product (Carrier (p | ns)) by A119, YELLOW_1:def_4;
then A129: ex gx being Function st
( xa = gx & dom gx = dom (Carrier (p | ns)) & ( for x being set st x in dom (Carrier (p | ns)) holds
gx . x in (Carrier (p | ns)) . x ) ) by CARD_3:def_5;
then A130: dom xa = n by A1, PARTFUN1:def_2;
then A131: (dom xa) \/ (dom (n .--> xb)) = n \/ {n} by FUNCOP_1:13;
ya in product (Carrier (p | ns)) by A123, YELLOW_1:def_4;
then A132: ex gy being Function st
( ya = gy & dom gy = dom (Carrier (p | ns)) & ( for x being set st x in dom (Carrier (p | ns)) holds
gy . x in (Carrier (p | ns)) . x ) ) by CARD_3:def_5;
then A133: dom ya = n by A1, PARTFUN1:def_2;
then A134: (dom ya) \/ (dom (n .--> yb)) = n \/ {n} by FUNCOP_1:13;
reconsider xa9 = xa as Element of (product (p | ns)) by A119;
reconsider ya9 = ya as Element of (product (p | ns)) by A123;
hereby ::_thesis: ( f9 . x <= f9 . y implies x <= y )
assume x <= y ; ::_thesis: f9 . x <= f9 . y
then [x,y] in the InternalRel of S by ORDERS_2:def_5;
then A135: [x,y] in [" the InternalRel of (product (p | ns)), the InternalRel of (p . ne)"] by YELLOW_3:def_2;
then A136: [(([x,y] `1) `1),(([x,y] `2) `1)] in the InternalRel of (product (p | ns)) by YELLOW_3:10;
A137: [(([x,y] `1) `2),(([x,y] `2) `2)] in the InternalRel of (p . ne) by A135, YELLOW_3:10;
[ya,yb] `1 = ya ;
then A138: [xa,ya] in the InternalRel of (product (p | ns)) by A124, A126, A136, A120;
A139: xa in product (Carrier (p | ns)) by A119, YELLOW_1:def_4;
xa9 <= ya9 by A138, ORDERS_2:def_5;
then consider ffx, ggx being Function such that
A140: ffx = xa and
A141: ggx = ya and
A142: for i being set st i in n holds
ex RR being RelStr ex xxi, yyi being Element of RR st
( RR = (p | ns) . i & xxi = ffx . i & yyi = ggx . i & xxi <= yyi ) by A1, A139, YELLOW_1:def_4;
[ya,yb] `2 = yb ;
then A143: [xb,yb] in the InternalRel of (p . ne) by A124, A127, A137, A120;
reconsider xb9 = xb as Element of (p . ne) ;
reconsider yb9 = yb as Element of (p . ne) ;
A144: xb9 <= yb9 by A143, ORDERS_2:def_5;
ex f1, g1 being Function st
( f1 = f . x & g1 = f . y & ( for i being set st i in n + 1 holds
ex R being RelStr ex xi, yi being Element of R st
( R = p . i & xi = f1 . i & yi = g1 . i & xi <= yi ) ) )
proof
set f1 = xa +* (n .--> xb);
set g1 = ya +* (n .--> yb);
take xa +* (n .--> xb) ; ::_thesis: ex g1 being Function st
( xa +* (n .--> xb) = f . x & g1 = f . y & ( for i being set st i in n + 1 holds
ex R being RelStr ex xi, yi being Element of R st
( R = p . i & xi = (xa +* (n .--> xb)) . i & yi = g1 . i & xi <= yi ) ) )
take ya +* (n .--> yb) ; ::_thesis: ( xa +* (n .--> xb) = f . x & ya +* (n .--> yb) = f . y & ( for i being set st i in n + 1 holds
ex R being RelStr ex xi, yi being Element of R st
( R = p . i & xi = (xa +* (n .--> xb)) . i & yi = (ya +* (n .--> yb)) . i & xi <= yi ) ) )
thus xa +* (n .--> xb) = f . x by A121; ::_thesis: ( ya +* (n .--> yb) = f . y & ( for i being set st i in n + 1 holds
ex R being RelStr ex xi, yi being Element of R st
( R = p . i & xi = (xa +* (n .--> xb)) . i & yi = (ya +* (n .--> yb)) . i & xi <= yi ) ) )
thus ya +* (n .--> yb) = f . y by A125; ::_thesis: for i being set st i in n + 1 holds
ex R being RelStr ex xi, yi being Element of R st
( R = p . i & xi = (xa +* (n .--> xb)) . i & yi = (ya +* (n .--> yb)) . i & xi <= yi )
let i be set ; ::_thesis: ( i in n + 1 implies ex R being RelStr ex xi, yi being Element of R st
( R = p . i & xi = (xa +* (n .--> xb)) . i & yi = (ya +* (n .--> yb)) . i & xi <= yi ) )
assume A145: i in n + 1 ; ::_thesis: ex R being RelStr ex xi, yi being Element of R st
( R = p . i & xi = (xa +* (n .--> xb)) . i & yi = (ya +* (n .--> yb)) . i & xi <= yi )
A146: i in n \/ {n} by A145, AFINSQ_1:2;
set R = p . i;
set xi = (xa +* (n .--> xb)) . i;
set yi = (ya +* (n .--> yb)) . i;
i in dom p by A145, PARTFUN1:def_2;
then p . i in rng p by FUNCT_1:def_3;
then reconsider R = p . i as RelStr by YELLOW_1:def_3;
A147: i in (dom xa) \/ (dom (n .--> xb)) by A131, A145, AFINSQ_1:2;
now__::_thesis:_(xa_+*_(n_.-->_xb))_._i_is_Element_of_R
percases ( i in dom (n .--> xb) or not i in dom (n .--> xb) ) ;
supposeA148: i in dom (n .--> xb) ; ::_thesis: (xa +* (n .--> xb)) . i is Element of R
then i in {n} ;
then A149: i = n by TARSKI:def_1;
(xa +* (n .--> xb)) . i = (n .--> xb) . i by A147, A148, FUNCT_4:def_1;
hence (xa +* (n .--> xb)) . i is Element of R by A2, A149, FUNCOP_1:72; ::_thesis: verum
end;
supposeA150: not i in dom (n .--> xb) ; ::_thesis: (xa +* (n .--> xb)) . i is Element of R
then A151: not i in {n} by FUNCOP_1:13;
then A152: i in n by A146, XBOOLE_0:def_3;
A153: i in dom (Carrier (p | ns)) by A3, A146, A151, XBOOLE_0:def_3;
(xa +* (n .--> xb)) . i = xa . i by A147, A150, FUNCT_4:def_1;
then A154: (xa +* (n .--> xb)) . i in (Carrier (p | ns)) . i by A129, A153;
ex R2 being 1-sorted st
( R2 = (p | ns) . i & (Carrier (p | ns)) . i = the carrier of R2 ) by A1, A152, PRALG_1:def_13;
hence (xa +* (n .--> xb)) . i is Element of R by A1, A152, A154, FUNCT_1:49; ::_thesis: verum
end;
end;
end;
then reconsider xi = (xa +* (n .--> xb)) . i as Element of R ;
A155: i in (dom ya) \/ (dom (n .--> yb)) by A134, A145, AFINSQ_1:2;
now__::_thesis:_(ya_+*_(n_.-->_yb))_._i_is_Element_of_R
percases ( i in dom (n .--> yb) or not i in dom (n .--> yb) ) ;
supposeA156: i in dom (n .--> yb) ; ::_thesis: (ya +* (n .--> yb)) . i is Element of R
then i in {n} ;
then A157: i = n by TARSKI:def_1;
(ya +* (n .--> yb)) . i = (n .--> yb) . i by A155, A156, FUNCT_4:def_1;
hence (ya +* (n .--> yb)) . i is Element of R by A2, A157, FUNCOP_1:72; ::_thesis: verum
end;
supposeA158: not i in dom (n .--> yb) ; ::_thesis: (ya +* (n .--> yb)) . i is Element of R
then A159: not i in {n} by FUNCOP_1:13;
then A160: i in n by A146, XBOOLE_0:def_3;
A161: i in dom (Carrier (p | ns)) by A3, A146, A159, XBOOLE_0:def_3;
(ya +* (n .--> yb)) . i = ya . i by A155, A158, FUNCT_4:def_1;
then A162: (ya +* (n .--> yb)) . i in (Carrier (p | ns)) . i by A132, A161;
ex R2 being 1-sorted st
( R2 = (p | ns) . i & (Carrier (p | ns)) . i = the carrier of R2 ) by A1, A160, PRALG_1:def_13;
hence (ya +* (n .--> yb)) . i is Element of R by A1, A160, A162, FUNCT_1:49; ::_thesis: verum
end;
end;
end;
then reconsider yi = (ya +* (n .--> yb)) . i as Element of R ;
take R ; ::_thesis: ex xi, yi being Element of R st
( R = p . i & xi = (xa +* (n .--> xb)) . i & yi = (ya +* (n .--> yb)) . i & xi <= yi )
take xi ; ::_thesis: ex yi being Element of R st
( R = p . i & xi = (xa +* (n .--> xb)) . i & yi = (ya +* (n .--> yb)) . i & xi <= yi )
take yi ; ::_thesis: ( R = p . i & xi = (xa +* (n .--> xb)) . i & yi = (ya +* (n .--> yb)) . i & xi <= yi )
thus R = p . i ; ::_thesis: ( xi = (xa +* (n .--> xb)) . i & yi = (ya +* (n .--> yb)) . i & xi <= yi )
thus xi = (xa +* (n .--> xb)) . i ; ::_thesis: ( yi = (ya +* (n .--> yb)) . i & xi <= yi )
thus yi = (ya +* (n .--> yb)) . i ; ::_thesis: xi <= yi
percases ( i in n or i in {n} ) by A146, XBOOLE_0:def_3;
supposeA163: i in n ; ::_thesis: xi <= yi
A164: not i in {n}
proof
assume i in {n} ; ::_thesis: contradiction
then n in n by A163, TARSKI:def_1;
hence contradiction ; ::_thesis: verum
end;
then A165: not i in dom (n .--> xb) ;
not i in dom (n .--> yb) by A164;
then A166: yi = ya . i by A155, FUNCT_4:def_1;
consider RR being RelStr , xxi, yyi being Element of RR such that
A167: RR = (p | ns) . i and
A168: xxi = ffx . i and
A169: yyi = ggx . i and
A170: xxi <= yyi by A142, A163;
RR = R by A1, A163, A167, FUNCT_1:49;
hence xi <= yi by A140, A141, A147, A165, A166, A168, A169, A170, FUNCT_4:def_1; ::_thesis: verum
end;
supposeA171: i in {n} ; ::_thesis: xi <= yi
then A172: i = n by TARSKI:def_1;
A173: i in dom (n .--> xb) by A171, FUNCOP_1:13;
A174: i in dom (n .--> yb) by A171, FUNCOP_1:13;
A175: xi = (n .--> xb) . i by A147, A173, FUNCT_4:def_1
.= xb by A172, FUNCOP_1:72 ;
yi = (n .--> yb) . i by A155, A174, FUNCT_4:def_1
.= yb by A172, FUNCOP_1:72 ;
hence xi <= yi by A2, A144, A171, A175, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
hence f9 . x <= f9 . y by A122, YELLOW_1:def_4; ::_thesis: verum
end;
assume f9 . x <= f9 . y ; ::_thesis: x <= y
then consider f1, g1 being Function such that
A176: f1 = f . x and
A177: g1 = f . y and
A178: for i being set st i in n + 1 holds
ex R being RelStr ex xi, yi being Element of R st
( R = p . i & xi = f1 . i & yi = g1 . i & xi <= yi ) by A122, YELLOW_1:def_4;
now__::_thesis:_ex_f2,_g2_being_Function_st_
(_f2_=_xa9_&_g2_=_ya9_&_(_for_i_being_set_st_i_in_ns_holds_
ex_R_being_RelStr_ex_xi,_yi_being_Element_of_R_st_
(_R_=_(p_|_ns)_._i_&_xi_=_f2_._i_&_yi_=_g2_._i_&_xi_<=_yi_)_)_)
set f2 = xa9;
set g2 = ya9;
reconsider f2 = xa9, g2 = ya9 as Function ;
take f2 = f2; ::_thesis: ex g2 being Function st
( f2 = xa9 & g2 = ya9 & ( for i being set st i in ns holds
ex R being RelStr ex xi, yi being Element of R st
( R = (p | ns) . i & xi = f2 . i & yi = g2 . i & xi <= yi ) ) )
take g2 = g2; ::_thesis: ( f2 = xa9 & g2 = ya9 & ( for i being set st i in ns holds
ex R being RelStr ex xi, yi being Element of R st
( R = (p | ns) . i & xi = f2 . i & yi = g2 . i & xi <= yi ) ) )
thus ( f2 = xa9 & g2 = ya9 ) ; ::_thesis: for i being set st i in ns holds
ex R being RelStr ex xi, yi being Element of R st
( R = (p | ns) . i & xi = f2 . i & yi = g2 . i & xi <= yi )
let i be set ; ::_thesis: ( i in ns implies ex R being RelStr ex xi, yi being Element of R st
( R = (p | ns) . i & xi = f2 . i & yi = g2 . i & xi <= yi ) )
assume A179: i in ns ; ::_thesis: ex R being RelStr ex xi, yi being Element of R st
( R = (p | ns) . i & xi = f2 . i & yi = g2 . i & xi <= yi )
A180: not i in {n}
proof
assume i in {n} ; ::_thesis: contradiction
then i = n by TARSKI:def_1;
hence contradiction by A1, A179; ::_thesis: verum
end;
then A181: not i in dom (n .--> yb) ;
A182: not i in dom (n .--> xb) by A180;
set R = (p | ns) . i;
i in dom (p | ns) by A179, PARTFUN1:def_2;
then (p | ns) . i in rng (p | ns) by FUNCT_1:def_3;
then reconsider R = (p | ns) . i as RelStr by YELLOW_1:def_3;
take R = R; ::_thesis: ex xi, yi being Element of R st
( R = (p | ns) . i & xi = f2 . i & yi = g2 . i & xi <= yi )
set xi = xa . i;
set yi = ya . i;
A183: i in (dom xa) \/ (dom (n .--> xb)) by A1, A130, A179, XBOOLE_0:def_3;
A184: i in dom (Carrier (p | ns)) by A179, PARTFUN1:def_2;
ex R2 being 1-sorted st
( R2 = (p | ns) . i & (Carrier (p | ns)) . i = the carrier of R2 ) by A179, PRALG_1:def_13;
then reconsider xi = xa . i as Element of R by A129, A184;
A185: i in (dom ya) \/ (dom (n .--> yb)) by A1, A133, A179, XBOOLE_0:def_3;
ex R2 being 1-sorted st
( R2 = (p | ns) . i & (Carrier (p | ns)) . i = the carrier of R2 ) by A179, PRALG_1:def_13;
then reconsider yi = ya . i as Element of R by A132, A184;
take xi = xi; ::_thesis: ex yi being Element of R st
( R = (p | ns) . i & xi = f2 . i & yi = g2 . i & xi <= yi )
take yi = yi; ::_thesis: ( R = (p | ns) . i & xi = f2 . i & yi = g2 . i & xi <= yi )
thus ( R = (p | ns) . i & xi = f2 . i & yi = g2 . i ) ; ::_thesis: xi <= yi
consider R2 being RelStr , xi2, yi2 being Element of R2 such that
A186: R2 = p . i and
A187: xi2 = f1 . i and
A188: yi2 = g1 . i and
A189: xi2 <= yi2 by A178, A179;
A190: R2 = R by A179, A186, FUNCT_1:49;
(xa +* (n .--> xb)) . i = xi by A182, A183, FUNCT_4:def_1;
hence xi <= yi by A121, A125, A176, A177, A181, A185, A187, A188, A189, A190, FUNCT_4:def_1; ::_thesis: verum
end;
then xa9 <= ya9 by A128, YELLOW_1:def_4;
then A191: [xa,ya] in the InternalRel of (product (p | ns)) by ORDERS_2:def_5;
[ya,yb] `1 = ya ;
then A192: [(([x,y] `1) `1),(([x,y] `2) `1)] in the InternalRel of (product (p | ns)) by A124, A126, A120, A191;
consider Rn being RelStr , xni, yni being Element of Rn such that
A193: Rn = p . ne and
A194: xni = f1 . n and
A195: yni = g1 . n and
A196: xni <= yni by A2, A178;
A197: [xni,yni] in the InternalRel of (p . ne) by A193, A196, ORDERS_2:def_5;
dom (n .--> xb) = {n} by FUNCOP_1:13;
then A198: n in dom (n .--> xb) by TARSKI:def_1;
then n in (dom xa) \/ (dom (n .--> xb)) by XBOOLE_0:def_3;
then A199: (xa +* (n .--> xb)) . n = (n .--> xb) . n by A198, FUNCT_4:def_1
.= xb by FUNCOP_1:72 ;
dom (n .--> yb) = {n} by FUNCOP_1:13;
then A200: n in dom (n .--> yb) by TARSKI:def_1;
then n in (dom ya) \/ (dom (n .--> yb)) by XBOOLE_0:def_3;
then A201: (ya +* (n .--> yb)) . n = (n .--> yb) . n by A200, FUNCT_4:def_1
.= yb by FUNCOP_1:72 ;
[ya,yb] `2 = yb ;
then A202: [(([x,y] `1) `2),(([x,y] `2) `2)] in the InternalRel of (p . ne) by A121, A124, A125, A127, A176, A177, A194, A195, A197, A199, A201, A120;
A203: [x,y] `1 = [xa,xb] by A120;
[x,y] `2 = [ya,yb] by A124;
then [x,y] in [" the InternalRel of (product (p | ns)), the InternalRel of (p . ne)"] by A192, A202, A203, YELLOW_3:10;
then [x,y] in the InternalRel of S by YELLOW_3:def_2;
hence x <= y by ORDERS_2:def_5; ::_thesis: verum
end;
hence f is isomorphic by A74, A117, WAYBEL_0:66; ::_thesis: verum
end;
hence [:(product (p | ns)),(p . ne):], product p are_isomorphic by WAYBEL_1:def_8; ::_thesis: verum
end;
end;
end;
theorem Th42: :: DICKSON:42
for n being non empty Nat
for p being RelStr-yielding ManySortedSet of n st ( for i being Element of n holds
( p . i is Dickson & p . i is quasi_ordered ) ) holds
( product p is quasi_ordered & product p is Dickson )
proof
defpred S1[ non empty Nat] means for p being RelStr-yielding ManySortedSet of $1 st ( for i being Element of $1 holds
( p . i is Dickson & p . i is quasi_ordered ) ) holds
( product p is quasi_ordered & product p is Dickson );
A1: S1[1]
proof
let p be RelStr-yielding ManySortedSet of 1; ::_thesis: ( ( for i being Element of 1 holds
( p . i is Dickson & p . i is quasi_ordered ) ) implies ( product p is quasi_ordered & product p is Dickson ) )
assume A2: for i being Element of 1 holds
( p . i is Dickson & p . i is quasi_ordered ) ; ::_thesis: ( product p is quasi_ordered & product p is Dickson )
reconsider z = 0 as Element of 1 by CARD_1:49, TARSKI:def_1;
A3: p . z is Dickson by A2;
A4: p . z is quasi_ordered by A2;
p . z, product p are_isomorphic by Th39;
hence ( product p is quasi_ordered & product p is Dickson ) by A3, A4, Th38; ::_thesis: verum
end;
A5: now__::_thesis:_for_n_being_non_empty_Nat_st_S1[n]_holds_
S1[n_+_1]
let n be non empty Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A6: S1[n] ; ::_thesis: S1[n + 1]
thus S1[n + 1] ::_thesis: verum
proof
let p be RelStr-yielding ManySortedSet of n + 1; ::_thesis: ( ( for i being Element of n + 1 holds
( p . i is Dickson & p . i is quasi_ordered ) ) implies ( product p is quasi_ordered & product p is Dickson ) )
assume A7: for i being Element of n + 1 holds
( p . i is Dickson & p . i is quasi_ordered ) ; ::_thesis: ( product p is quasi_ordered & product p is Dickson )
n <= n + 1 by NAT_1:11;
then reconsider ns = n as Subset of (n + 1) by NAT_1:39;
A8: n + 1 = { k where k is Element of NAT : k < n + 1 } by AXIOMS:4;
A9: n in NAT by ORDINAL1:def_12;
n < n + 1 by NAT_1:13;
then n in n + 1 by A8, A9;
then reconsider ne = n as Element of n + 1 ;
reconsider pns = p | ns as RelStr-yielding ManySortedSet of n ;
A10: for i being Element of n holds
( pns . i is Dickson & pns . i is quasi_ordered )
proof
let i be Element of n; ::_thesis: ( pns . i is Dickson & pns . i is quasi_ordered )
A11: pns . i = p . i by FUNCT_1:49;
A12: n = { k where k is Element of NAT : k < n } by AXIOMS:4;
i in n ;
then consider k being Element of NAT such that
A13: k = i and
A14: k < n by A12;
k < n + 1 by A14, NAT_1:13;
then i in n + 1 by A8, A13;
then reconsider i9 = i as Element of n + 1 ;
i9 = i ;
hence ( pns . i is Dickson & pns . i is quasi_ordered ) by A7, A11; ::_thesis: verum
end;
then A15: product pns is Dickson by A6;
A16: product pns is quasi_ordered by A6, A10;
A17: p . ne is Dickson by A7;
A18: p . ne is quasi_ordered by A7;
then A19: [:(product (p | ns)),(p . ne):] is Dickson by A15, A16, A17, Th37;
A20: [:(product (p | ns)),(p . ne):] is quasi_ordered by A15, A16, A17, A18, Th37;
[:(product (p | ns)),(p . ne):], product p are_isomorphic by Th41;
hence ( product p is quasi_ordered & product p is Dickson ) by A19, A20, Th38; ::_thesis: verum
end;
end;
thus for n being non empty Nat holds S1[n] from NAT_1:sch_10(A1, A5); ::_thesis: verum
end;
Lm1: for p being RelStr-yielding ManySortedSet of {} holds
( not product p is empty & product p is quasi_ordered & product p is Dickson & product p is antisymmetric )
proof
let p be RelStr-yielding ManySortedSet of {} ; ::_thesis: ( not product p is empty & product p is quasi_ordered & product p is Dickson & product p is antisymmetric )
A1: product p = RelStr(# {{}},(id {{}}) #) by YELLOW_1:26;
set pp = product p;
set cpp = the carrier of (product p);
set ipp = the InternalRel of (product p);
A2: the InternalRel of (product p) = id {{}} by YELLOW_1:26;
thus not product p is empty by YELLOW_1:26; ::_thesis: ( product p is quasi_ordered & product p is Dickson & product p is antisymmetric )
thus product p is quasi_ordered ::_thesis: ( product p is Dickson & product p is antisymmetric )
proof
thus product p is reflexive by YELLOW_1:26; :: according to DICKSON:def_3 ::_thesis: product p is transitive
let x, y, z be set ; :: according to RELAT_2:def_8,ORDERS_2:def_3 ::_thesis: ( not x in the carrier of (product p) or not y in the carrier of (product p) or not z in the carrier of (product p) or not [x,y] in the InternalRel of (product p) or not [y,z] in the InternalRel of (product p) or [x,z] in the InternalRel of (product p) )
assume that
x in the carrier of (product p) and
y in the carrier of (product p) and
z in the carrier of (product p) and
A3: [x,y] in the InternalRel of (product p) and
A4: [y,z] in the InternalRel of (product p) ; ::_thesis: [x,z] in the InternalRel of (product p)
thus [x,z] in the InternalRel of (product p) by A2, A3, A4, RELAT_1:def_10; ::_thesis: verum
end;
thus product p is Dickson ::_thesis: product p is antisymmetric
proof
let N be Subset of the carrier of (product p); :: according to DICKSON:def_10 ::_thesis: ex B being set st
( B is_Dickson-basis_of N, product p & B is finite )
percases ( N = {} or N = {{}} ) by A1, ZFMISC_1:33;
supposeA5: N = {} ; ::_thesis: ex B being set st
( B is_Dickson-basis_of N, product p & B is finite )
take B = {} ; ::_thesis: ( B is_Dickson-basis_of N, product p & B is finite )
N = {} the carrier of (product p) by A5;
hence B is_Dickson-basis_of N, product p by Th25; ::_thesis: B is finite
thus B is finite ; ::_thesis: verum
end;
supposeA6: N = {{}} ; ::_thesis: ex B being set st
( B is_Dickson-basis_of N, product p & B is finite )
take B = {{}}; ::_thesis: ( B is_Dickson-basis_of N, product p & B is finite )
thus B is_Dickson-basis_of N, product p ::_thesis: B is finite
proof
thus B c= N by A6; :: according to DICKSON:def_9 ::_thesis: for a being Element of (product p) st a in N holds
ex b being Element of (product p) st
( b in B & b <= a )
let a be Element of (product p); ::_thesis: ( a in N implies ex b being Element of (product p) st
( b in B & b <= a ) )
assume A7: a in N ; ::_thesis: ex b being Element of (product p) st
( b in B & b <= a )
take b = a; ::_thesis: ( b in B & b <= a )
thus b in B by A6, A7; ::_thesis: b <= a
[b,a] in id {{}} by A6, A7, RELAT_1:def_10;
hence b <= a by A2, ORDERS_2:def_5; ::_thesis: verum
end;
thus B is finite ; ::_thesis: verum
end;
end;
end;
thus product p is antisymmetric by YELLOW_1:26; ::_thesis: verum
end;
registration
let p be RelStr-yielding ManySortedSet of {} ;
cluster product p -> non empty ;
coherence
not product p is empty by Lm1;
cluster product p -> antisymmetric ;
coherence
product p is antisymmetric by Lm1;
cluster product p -> quasi_ordered ;
coherence
product p is quasi_ordered by Lm1;
cluster product p -> Dickson ;
coherence
product p is Dickson by Lm1;
end;
definition
func NATOrd -> Relation of NAT equals :: DICKSON:def 14
{ [x,y] where x, y is Element of NAT : x <= y } ;
correctness
coherence
{ [x,y] where x, y is Element of NAT : x <= y } is Relation of NAT;
proof
set NO = { [x,y] where x, y is Element of NAT : x <= y } ;
now__::_thesis:_for_z_being_set_st_z_in__{__[x,y]_where_x,_y_is_Element_of_NAT_:_x_<=_y__}__holds_
z_in_[:NAT,NAT:]
let z be set ; ::_thesis: ( z in { [x,y] where x, y is Element of NAT : x <= y } implies z in [:NAT,NAT:] )
assume z in { [x,y] where x, y is Element of NAT : x <= y } ; ::_thesis: z in [:NAT,NAT:]
then ex x, y being Element of NAT st
( z = [x,y] & x <= y ) ;
hence z in [:NAT,NAT:] ; ::_thesis: verum
end;
hence { [x,y] where x, y is Element of NAT : x <= y } is Relation of NAT by TARSKI:def_3; ::_thesis: verum
end;
end;
:: deftheorem defines NATOrd DICKSON:def_14_:_
NATOrd = { [x,y] where x, y is Element of NAT : x <= y } ;
theorem Th43: :: DICKSON:43
NATOrd is_reflexive_in NAT
proof
let x be set ; :: according to RELAT_2:def_1 ::_thesis: ( not x in NAT or [x,x] in NATOrd )
assume x in NAT ; ::_thesis: [x,x] in NATOrd
then reconsider x9 = x as Element of NAT ;
x9 <= x9 ;
hence [x,x] in NATOrd ; ::_thesis: verum
end;
theorem Th44: :: DICKSON:44
NATOrd is_antisymmetric_in NAT
proof
let x, y be set ; :: according to RELAT_2:def_4 ::_thesis: ( not x in NAT or not y in NAT or not [x,y] in NATOrd or not [y,x] in NATOrd or x = y )
assume that
x in NAT and
y in NAT and
A1: [x,y] in NATOrd and
A2: [y,x] in NATOrd ; ::_thesis: x = y
consider x9, y9 being Element of NAT such that
A3: [x,y] = [x9,y9] and
A4: x9 <= y9 by A1;
A5: x = x9 by A3, XTUPLE_0:1;
A6: y = y9 by A3, XTUPLE_0:1;
consider y2, x2 being Element of NAT such that
A7: [y,x] = [y2,x2] and
A8: y2 <= x2 by A2;
A9: y2 = y9 by A6, A7, XTUPLE_0:1;
x2 = x9 by A5, A7, XTUPLE_0:1;
hence x = y by A4, A5, A6, A8, A9, XXREAL_0:1; ::_thesis: verum
end;
theorem Th45: :: DICKSON:45
NATOrd is_strongly_connected_in NAT
proof
now__::_thesis:_for_x,_y_being_set_st_x_in_NAT_&_y_in_NAT_&_not_[x,y]_in_NATOrd_holds_
[y,x]_in_NATOrd
let x, y be set ; ::_thesis: ( x in NAT & y in NAT & not [x,y] in NATOrd implies [y,x] in NATOrd )
assume that
A1: x in NAT and
A2: y in NAT ; ::_thesis: ( [x,y] in NATOrd or [y,x] in NATOrd )
thus ( [x,y] in NATOrd or [y,x] in NATOrd ) ::_thesis: verum
proof
assume that
A3: not [x,y] in NATOrd and
A4: not [y,x] in NATOrd ; ::_thesis: contradiction
reconsider x = x, y = y as Element of NAT by A1, A2;
percases ( x <= y or y <= x ) ;
suppose x <= y ; ::_thesis: contradiction
hence contradiction by A3; ::_thesis: verum
end;
suppose y <= x ; ::_thesis: contradiction
hence contradiction by A4; ::_thesis: verum
end;
end;
end;
end;
hence NATOrd is_strongly_connected_in NAT by RELAT_2:def_7; ::_thesis: verum
end;
theorem Th46: :: DICKSON:46
NATOrd is_transitive_in NAT
proof
let x, y, z be set ; :: according to RELAT_2:def_8 ::_thesis: ( not x in NAT or not y in NAT or not z in NAT or not [x,y] in NATOrd or not [y,z] in NATOrd or [x,z] in NATOrd )
assume that
x in NAT and
y in NAT and
z in NAT and
A1: [x,y] in NATOrd and
A2: [y,z] in NATOrd ; ::_thesis: [x,z] in NATOrd
consider x1, y1 being Element of NAT such that
A3: [x,y] = [x1,y1] and
A4: x1 <= y1 by A1;
A5: x = x1 by A3, XTUPLE_0:1;
A6: y = y1 by A3, XTUPLE_0:1;
consider y2, z2 being Element of NAT such that
A7: [y,z] = [y2,z2] and
A8: y2 <= z2 by A2;
A9: y = y2 by A7, XTUPLE_0:1;
A10: z = z2 by A7, XTUPLE_0:1;
x1 <= z2 by A4, A6, A8, A9, XXREAL_0:2;
hence [x,z] in NATOrd by A5, A10; ::_thesis: verum
end;
definition
func OrderedNAT -> non empty RelStr equals :: DICKSON:def 15
RelStr(# NAT,NATOrd #);
correctness
coherence
RelStr(# NAT,NATOrd #) is non empty RelStr ;
;
end;
:: deftheorem defines OrderedNAT DICKSON:def_15_:_
OrderedNAT = RelStr(# NAT,NATOrd #);
Lm2: now__::_thesis:_OrderedNAT_is_connected
now__::_thesis:_for_x,_y_being_Element_of_OrderedNAT_st_not_x_<=_y_holds_
y_<=_x
let x, y be Element of OrderedNAT; ::_thesis: ( not x <= y implies y <= x )
assume not x <= y ; ::_thesis: y <= x
then not [x,y] in NATOrd by ORDERS_2:def_5;
then [y,x] in NATOrd by Th45, RELAT_2:def_7;
hence y <= x by ORDERS_2:def_5; ::_thesis: verum
end;
hence OrderedNAT is connected by WAYBEL_0:def_29; ::_thesis: verum
end;
registration
cluster OrderedNAT -> non empty connected ;
coherence
OrderedNAT is connected by Lm2;
cluster OrderedNAT -> non empty Dickson ;
coherence
OrderedNAT is Dickson
proof
set IR9 = the InternalRel of (OrderedNAT \~);
set CR9 = the carrier of (OrderedNAT \~);
now__::_thesis:_for_N_being_set_st_N_c=_the_carrier_of_(OrderedNAT_\~)_&_N_<>_{}_holds_
ex_m_being_set_st_
(_m_in_N_&_the_InternalRel_of_(OrderedNAT_\~)_-Seg_m_misses_N_)
let N be set ; ::_thesis: ( N c= the carrier of (OrderedNAT \~) & N <> {} implies ex m being set st
( m in N & the InternalRel of (OrderedNAT \~) -Seg m misses N ) )
assume that
A1: N c= the carrier of (OrderedNAT \~) and
A2: N <> {} ; ::_thesis: ex m being set st
( m in N & the InternalRel of (OrderedNAT \~) -Seg m misses N )
A3: ex k being set st k in N by A2, XBOOLE_0:def_1;
defpred S1[ Nat] means c1 in N;
A4: ex k being Nat st S1[k] by A1, A3;
consider m being Nat such that
A5: S1[m] and
A6: for n being Nat st S1[n] holds
m <= n from NAT_1:sch_5(A4);
reconsider m = m as Element of OrderedNAT by ORDINAL1:def_12;
thus ex m being set st
( m in N & the InternalRel of (OrderedNAT \~) -Seg m misses N ) ::_thesis: verum
proof
take m ; ::_thesis: ( m in N & the InternalRel of (OrderedNAT \~) -Seg m misses N )
thus m in N by A5; ::_thesis: the InternalRel of (OrderedNAT \~) -Seg m misses N
now__::_thesis:_not_(_the_InternalRel_of_(OrderedNAT_\~)_-Seg_m)_/\_N_<>_{}
assume ( the InternalRel of (OrderedNAT \~) -Seg m) /\ N <> {} ; ::_thesis: contradiction
then consider z being set such that
A7: z in ( the InternalRel of (OrderedNAT \~) -Seg m) /\ N by XBOOLE_0:def_1;
A8: z in the InternalRel of (OrderedNAT \~) -Seg m by A7, XBOOLE_0:def_4;
A9: z in N by A7, XBOOLE_0:def_4;
A10: z <> m by A8, WELLORD1:1;
A11: [z,m] in the InternalRel of (OrderedNAT \~) by A8, WELLORD1:1;
then [z,m] in NATOrd ;
then consider x, y being Element of NAT such that
A12: [z,m] = [x,y] and
x <= y ;
A13: z = x by A12, XTUPLE_0:1;
A14: m = y by A12, XTUPLE_0:1;
then y <= x by A6, A9, A13;
then [y,x] in NATOrd ;
hence contradiction by A10, A11, A13, A14, Th44, RELAT_2:def_4; ::_thesis: verum
end;
hence the InternalRel of (OrderedNAT \~) -Seg m misses N by XBOOLE_0:def_7; ::_thesis: verum
end;
end;
then the InternalRel of (OrderedNAT \~) is_well_founded_in the carrier of (OrderedNAT \~) by WELLORD1:def_3;
then OrderedNAT \~ is well_founded by WELLFND1:def_2;
hence OrderedNAT is Dickson by Th27; ::_thesis: verum
end;
cluster OrderedNAT -> non empty quasi_ordered ;
coherence
OrderedNAT is quasi_ordered
proof
A15: OrderedNAT is reflexive by Th43, ORDERS_2:def_2;
OrderedNAT is transitive by Th46, ORDERS_2:def_3;
hence OrderedNAT is quasi_ordered by A15, Def3; ::_thesis: verum
end;
cluster OrderedNAT -> non empty antisymmetric ;
coherence
OrderedNAT is antisymmetric by Th44, ORDERS_2:def_4;
cluster OrderedNAT -> non empty transitive ;
coherence
OrderedNAT is transitive by Th46, ORDERS_2:def_3;
cluster OrderedNAT -> non empty well_founded ;
coherence
OrderedNAT is well_founded
proof
set ir = the InternalRel of OrderedNAT;
now__::_thesis:_for_f_being_sequence_of_OrderedNAT_holds_not_f_is_descending
given f being sequence of OrderedNAT such that A16: f is descending ; ::_thesis: contradiction
A17: dom f = NAT by FUNCT_2:def_1;
reconsider rf = rng f as non empty Subset of NAT ;
set m = min rf;
min rf in rng f by XXREAL_2:def_7;
then consider d being set such that
A18: d in dom f and
A19: min rf = f . d by FUNCT_1:def_3;
reconsider d = d as Element of NAT by A18;
A20: f . (d + 1) <> f . d by A16, WELLFND1:def_6;
[(f . (d + 1)),(f . d)] in the InternalRel of OrderedNAT by A16, WELLFND1:def_6;
then consider x, y being Element of NAT such that
A21: [(f . (d + 1)),(f . d)] = [x,y] and
A22: x <= y ;
reconsider fd1 = f . (d + 1), fd = f . d as Element of NAT ;
A23: f . (d + 1) = x by A21, XTUPLE_0:1;
f . d = y by A21, XTUPLE_0:1;
then A24: fd1 < fd by A20, A22, A23, XXREAL_0:1;
f . (d + 1) in rf by A17, FUNCT_1:3;
hence contradiction by A19, A24, XXREAL_2:def_7; ::_thesis: verum
end;
hence OrderedNAT is well_founded by WELLFND1:14; ::_thesis: verum
end;
end;
Lm3: now__::_thesis:_for_n_being_Element_of_NAT_holds_
(_not_product_(n_-->_OrderedNAT)_is_empty_&_product_(n_-->_OrderedNAT)_is_Dickson_&_product_(n_-->_OrderedNAT)_is_quasi_ordered_&_product_(n_-->_OrderedNAT)_is_antisymmetric_)
let n be Element of NAT ; ::_thesis: ( not product (b1 --> OrderedNAT) is empty & product (b1 --> OrderedNAT) is Dickson & product (b1 --> OrderedNAT) is quasi_ordered & product (b1 --> OrderedNAT) is antisymmetric )
set pp = product (n --> OrderedNAT);
percases ( n is empty or not n is empty ) ;
suppose n is empty ; ::_thesis: ( not product (b1 --> OrderedNAT) is empty & product (b1 --> OrderedNAT) is Dickson & product (b1 --> OrderedNAT) is quasi_ordered & product (b1 --> OrderedNAT) is antisymmetric )
hence ( not product (n --> OrderedNAT) is empty & product (n --> OrderedNAT) is Dickson & product (n --> OrderedNAT) is quasi_ordered & product (n --> OrderedNAT) is antisymmetric ) ; ::_thesis: verum
end;
suppose not n is empty ; ::_thesis: ( not product (b1 --> OrderedNAT) is empty & product (b1 --> OrderedNAT) is Dickson & product (b1 --> OrderedNAT) is quasi_ordered & product (b1 --> OrderedNAT) is antisymmetric )
then reconsider n9 = n as non empty Element of NAT ;
set p = n9 --> OrderedNAT;
thus not product (n --> OrderedNAT) is empty ; ::_thesis: ( product (n --> OrderedNAT) is Dickson & product (n --> OrderedNAT) is quasi_ordered & product (n --> OrderedNAT) is antisymmetric )
for i being Element of n9 holds
( (n9 --> OrderedNAT) . i is Dickson & (n9 --> OrderedNAT) . i is quasi_ordered ) by FUNCOP_1:7;
hence ( product (n --> OrderedNAT) is Dickson & product (n --> OrderedNAT) is quasi_ordered ) by Th42; ::_thesis: product (n --> OrderedNAT) is antisymmetric
product (n9 --> OrderedNAT) is antisymmetric ;
hence product (n --> OrderedNAT) is antisymmetric ; ::_thesis: verum
end;
end;
end;
registration
let n be Element of NAT ;
cluster product (n --> OrderedNAT) -> non empty ;
coherence
not product (n --> OrderedNAT) is empty ;
cluster product (n --> OrderedNAT) -> Dickson ;
coherence
product (n --> OrderedNAT) is Dickson by Lm3;
cluster product (n --> OrderedNAT) -> quasi_ordered ;
coherence
product (n --> OrderedNAT) is quasi_ordered by Lm3;
cluster product (n --> OrderedNAT) -> antisymmetric ;
coherence
product (n --> OrderedNAT) is antisymmetric by Lm3;
end;
theorem :: DICKSON:47
for M being RelStr st M is Dickson & M is quasi_ordered holds
( [:M,OrderedNAT:] is quasi_ordered & [:M,OrderedNAT:] is Dickson ) by Th37;
theorem Th48: :: DICKSON:48
for R, S being non empty RelStr st R is Dickson & S is quasi_ordered & the InternalRel of R c= the InternalRel of S & the carrier of R = the carrier of S holds
S \~ is well_founded
proof
let R, S be non empty RelStr ; ::_thesis: ( R is Dickson & S is quasi_ordered & the InternalRel of R c= the InternalRel of S & the carrier of R = the carrier of S implies S \~ is well_founded )
assume that
A1: R is Dickson and
A2: S is quasi_ordered and
A3: the InternalRel of R c= the InternalRel of S and
A4: the carrier of R = the carrier of S ; ::_thesis: S \~ is well_founded
S is Dickson by A1, A3, A4, Th28;
hence S \~ is well_founded by A2, Th33; ::_thesis: verum
end;
theorem :: DICKSON:49
for R being non empty RelStr st R is quasi_ordered holds
( R is Dickson iff for S being non empty RelStr st S is quasi_ordered & the InternalRel of R c= the InternalRel of S & the carrier of R = the carrier of S holds
S \~ is well_founded )
proof
let R be non empty RelStr ; ::_thesis: ( R is quasi_ordered implies ( R is Dickson iff for S being non empty RelStr st S is quasi_ordered & the InternalRel of R c= the InternalRel of S & the carrier of R = the carrier of S holds
S \~ is well_founded ) )
assume A1: R is quasi_ordered ; ::_thesis: ( R is Dickson iff for S being non empty RelStr st S is quasi_ordered & the InternalRel of R c= the InternalRel of S & the carrier of R = the carrier of S holds
S \~ is well_founded )
A2: R is reflexive by A1, Def3;
A3: R is transitive by A1, Def3;
set IR = the InternalRel of R;
set CR = the carrier of R;
thus ( R is Dickson implies for S being non empty RelStr st S is quasi_ordered & the InternalRel of R c= the InternalRel of S & the carrier of R = the carrier of S holds
S \~ is well_founded ) by Th48; ::_thesis: ( ( for S being non empty RelStr st S is quasi_ordered & the InternalRel of R c= the InternalRel of S & the carrier of R = the carrier of S holds
S \~ is well_founded ) implies R is Dickson )
assume A4: for S being non empty RelStr st S is quasi_ordered & the InternalRel of R c= the InternalRel of S & the carrier of R = the carrier of S holds
S \~ is well_founded ; ::_thesis: R is Dickson
now__::_thesis:_R_is_Dickson
assume not R is Dickson ; ::_thesis: contradiction
then not for N being non empty Subset of R holds
( min-classes N is finite & not min-classes N is empty ) by A1, Th32;
then consider f being sequence of R such that
A5: for i, j being Element of NAT st i < j holds
not f . i <= f . j by A1, Th31;
defpred S1[ set , set ] means ( [$1,$2] in the InternalRel of R or ex i, j being Element of NAT st
( i < j & [$1,(f . j)] in the InternalRel of R & [(f . i),$2] in the InternalRel of R ) );
consider QOE being Relation of the carrier of R, the carrier of R such that
A6: for x, y being set holds
( [x,y] in QOE iff ( x in the carrier of R & y in the carrier of R & S1[x,y] ) ) from RELSET_1:sch_1();
set S = RelStr(# the carrier of R,QOE #);
now__::_thesis:_for_x,_y_being_set_st_[x,y]_in_the_InternalRel_of_R_holds_
[x,y]_in_QOE
let x, y be set ; ::_thesis: ( [x,y] in the InternalRel of R implies [x,y] in QOE )
assume A7: [x,y] in the InternalRel of R ; ::_thesis: [x,y] in QOE
A8: x in dom the InternalRel of R by A7, XTUPLE_0:def_12;
y in rng the InternalRel of R by A7, XTUPLE_0:def_13;
hence [x,y] in QOE by A6, A7, A8; ::_thesis: verum
end;
then A9: the InternalRel of R c= the InternalRel of RelStr(# the carrier of R,QOE #) by RELAT_1:def_3;
A10: the InternalRel of R is_reflexive_in the carrier of R by A2, ORDERS_2:def_2;
now__::_thesis:_for_x_being_set_st_x_in_the_carrier_of_R_holds_
[x,x]_in_QOE
let x be set ; ::_thesis: ( x in the carrier of R implies [x,x] in QOE )
assume A11: x in the carrier of R ; ::_thesis: [x,x] in QOE
[x,x] in the InternalRel of R by A10, A11, RELAT_2:def_1;
hence [x,x] in QOE by A6, A11; ::_thesis: verum
end;
then QOE is_reflexive_in the carrier of R by RELAT_2:def_1;
then A12: RelStr(# the carrier of R,QOE #) is reflexive by ORDERS_2:def_2;
A13: the InternalRel of R is_transitive_in the carrier of R by A3, ORDERS_2:def_3;
now__::_thesis:_for_x,_y,_z_being_set_st_x_in_the_carrier_of_R_&_y_in_the_carrier_of_R_&_z_in_the_carrier_of_R_&_[x,y]_in_QOE_&_[y,z]_in_QOE_holds_
[x,z]_in_QOE
let x, y, z be set ; ::_thesis: ( x in the carrier of R & y in the carrier of R & z in the carrier of R & [x,y] in QOE & [y,z] in QOE implies [x,z] in QOE )
assume that
A14: x in the carrier of R and
A15: y in the carrier of R and
A16: z in the carrier of R and
A17: [x,y] in QOE and
A18: [y,z] in QOE ; ::_thesis: [x,z] in QOE
now__::_thesis:_[x,z]_in_QOE
percases ( [x,y] in the InternalRel of R or ex i, j being Element of NAT st
( i < j & [x,(f . j)] in the InternalRel of R & [(f . i),y] in the InternalRel of R ) ) by A6, A17;
supposeA19: [x,y] in the InternalRel of R ; ::_thesis: [x,z] in QOE
now__::_thesis:_[x,z]_in_QOE
percases ( [y,z] in the InternalRel of R or ex i, j being Element of NAT st
( i < j & [y,(f . j)] in the InternalRel of R & [(f . i),z] in the InternalRel of R ) ) by A6, A18;
suppose [y,z] in the InternalRel of R ; ::_thesis: [x,z] in QOE
then [x,z] in the InternalRel of R by A13, A14, A15, A16, A19, RELAT_2:def_8;
hence [x,z] in QOE by A6, A14, A16; ::_thesis: verum
end;
suppose ex i, j being Element of NAT st
( i < j & [y,(f . j)] in the InternalRel of R & [(f . i),z] in the InternalRel of R ) ; ::_thesis: [x,z] in QOE
then consider i, j being Element of NAT such that
A20: i < j and
A21: [y,(f . j)] in the InternalRel of R and
A22: [(f . i),z] in the InternalRel of R ;
[x,(f . j)] in the InternalRel of R by A13, A14, A15, A19, A21, RELAT_2:def_8;
hence [x,z] in QOE by A6, A14, A16, A20, A22; ::_thesis: verum
end;
end;
end;
hence [x,z] in QOE ; ::_thesis: verum
end;
suppose ex i, j being Element of NAT st
( i < j & [x,(f . j)] in the InternalRel of R & [(f . i),y] in the InternalRel of R ) ; ::_thesis: [x,z] in QOE
then consider i, j being Element of NAT such that
A23: i < j and
A24: [x,(f . j)] in the InternalRel of R and
A25: [(f . i),y] in the InternalRel of R ;
now__::_thesis:_[x,z]_in_QOE
percases ( [y,z] in the InternalRel of R or ex a, b being Element of NAT st
( a < b & [y,(f . b)] in the InternalRel of R & [(f . a),z] in the InternalRel of R ) ) by A6, A18;
suppose [y,z] in the InternalRel of R ; ::_thesis: [x,z] in QOE
then [(f . i),z] in the InternalRel of R by A13, A15, A16, A25, RELAT_2:def_8;
hence [x,z] in QOE by A6, A14, A16, A23, A24; ::_thesis: verum
end;
suppose ex a, b being Element of NAT st
( a < b & [y,(f . b)] in the InternalRel of R & [(f . a),z] in the InternalRel of R ) ; ::_thesis: [x,z] in QOE
then consider a, b being Element of NAT such that
A26: a < b and
A27: [y,(f . b)] in the InternalRel of R and
A28: [(f . a),z] in the InternalRel of R ;
[(f . i),(f . b)] in the InternalRel of R by A13, A15, A25, A27, RELAT_2:def_8;
then f . i <= f . b by ORDERS_2:def_5;
then not i < b by A5;
then a <= i by A26, XXREAL_0:2;
then a < j by A23, XXREAL_0:2;
hence [x,z] in QOE by A6, A14, A16, A24, A28; ::_thesis: verum
end;
end;
end;
hence [x,z] in QOE ; ::_thesis: verum
end;
end;
end;
hence [x,z] in QOE ; ::_thesis: verum
end;
then QOE is_transitive_in the carrier of R by RELAT_2:def_8;
then RelStr(# the carrier of R,QOE #) is transitive by ORDERS_2:def_3;
then RelStr(# the carrier of R,QOE #) is quasi_ordered by A12, Def3;
then A29: RelStr(# the carrier of R,QOE #) \~ is well_founded by A4, A9;
reconsider f9 = f as sequence of (RelStr(# the carrier of R,QOE #) \~) ;
now__::_thesis:_for_n_being_Nat_holds_
(_f9_._(n_+_1)_<>_f9_._n_&_[(f9_._(n_+_1)),(f9_._n)]_in_the_InternalRel_of_(RelStr(#_the_carrier_of_R,QOE_#)_\~)_)
let n be Nat; ::_thesis: ( f9 . (n + 1) <> f9 . n & [(f9 . (n + 1)),(f9 . n)] in the InternalRel of (RelStr(# the carrier of R,QOE #) \~) )
reconsider n1 = n as Element of NAT by ORDINAL1:def_12;
A30: n < n + 1 by NAT_1:13;
then not f . n1 <= f . (n1 + 1) by A5;
then A31: not [(f . n),(f . (n + 1))] in the InternalRel of R by ORDERS_2:def_5;
hence f9 . (n + 1) <> f9 . n by A10, RELAT_2:def_1; ::_thesis: [(f9 . (n + 1)),(f9 . n)] in the InternalRel of (RelStr(# the carrier of R,QOE #) \~)
A32: [(f9 . (n + 1)),(f9 . (n + 1))] in the InternalRel of R by A10, RELAT_2:def_1;
A33: [(f9 . n),(f9 . n)] in the InternalRel of R by A10, RELAT_2:def_1;
A34: now__::_thesis:_not_[(f9_._n),(f9_._(n_+_1))]_in_QOE
assume [(f9 . n),(f9 . (n + 1))] in QOE ; ::_thesis: contradiction
then ex i, j being Element of NAT st
( i < j & [(f9 . n),(f . j)] in the InternalRel of R & [(f . i),(f9 . (n + 1))] in the InternalRel of R ) by A6, A31;
then consider l, k being Element of NAT such that
A35: k < l and
A36: [(f9 . n),(f . l)] in the InternalRel of R and
A37: [(f . k),(f9 . (n + 1))] in the InternalRel of R ;
A38: f . n <= f . l by A36, ORDERS_2:def_5;
A39: f . k <= f . (n + 1) by A37, ORDERS_2:def_5;
A40: l <= n1 by A5, A38;
A41: n + 1 <= k by A5, A39;
l < n + 1 by A40, NAT_1:13;
hence contradiction by A35, A41, XXREAL_0:2; ::_thesis: verum
end;
A42: [(f9 . (n1 + 1)),(f9 . n1)] in QOE by A6, A30, A32, A33;
not [(f9 . (n + 1)),(f9 . n)] in QOE ~ by A34, RELAT_1:def_7;
hence [(f9 . (n + 1)),(f9 . n)] in the InternalRel of (RelStr(# the carrier of R,QOE #) \~) by A42, XBOOLE_0:def_5; ::_thesis: verum
end;
then f9 is descending by WELLFND1:def_6;
hence contradiction by A29, WELLFND1:14; ::_thesis: verum
end;
hence R is Dickson ; ::_thesis: verum
end;