ARYTM_2

:: Non negative real numbers. Part I
:: by Andrzej Trybulec
::
:: Received March 7, 1998
:: Copyright (c) 1998 Association of Mizar Users

:: ARYTM_2 semantic presentation

definition

func DEDEKIND_CUTS -> Subset-Family of RAT+ equals :: ARYTM_2:def 1
{ A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) } \ {RAT+ };
coherence

proof end;

end;

:: deftheorem defines DEDEKIND_CUTS ARYTM_2:def 1 :

registration

cluster DEDEKIND_CUTS -> non empty ;
coherence

proof end;

end;

definition

func REAL+ -> set equals :: ARYTM_2:def 2
(RAT+ \/ DEDEKIND_CUTS ) \ { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } ;
coherence ;

end;

:: deftheorem defines REAL+ ARYTM_2:def 2 :

set IR = { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) } ;

set RA = { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } ;

Lm1: for x, y being set holds not [x,y] in { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) }

proof end;

Lm2: RAT+ misses { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} }

proof end;

theorem Th1: :: ARYTM_2:1

RAT+ c= REAL+ by Lm2, XBOOLE_1:7, XBOOLE_1:86;

RAT+ \/ DEDEKIND_CUTS c= RAT+ \/ { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) }
by XBOOLE_1:9;

then Lm3: REAL+ c= RAT+ \/ { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) }
by XBOOLE_1:1;

REAL+ = RAT+ \/ (({ A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) } \ {RAT+ }) \ { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } )
by Lm2, XBOOLE_1:87;

then Lm4: DEDEKIND_CUTS \ { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } c= REAL+
by XBOOLE_1:7;

Lm5: omega c= ({ [c,d] where c, d is Element of omega : ( c,d are_relative_prime & d <> {} ) } \ { [k,1] where k is Element of omega : verum } ) \/ omega
by XBOOLE_1:7;

theorem :: ARYTM_2:2

omega c= REAL+ by Lm5, Th1, XBOOLE_1:1;

registration

cluster REAL+ -> non empty ;
coherence by Th1;

end;

definition

let x be Element of REAL+ ;
func DEDEKIND_CUT x -> Element of DEDEKIND_CUTS means :Def3: :: ARYTM_2:def 3
ex r being Element of RAT+ st
( x = r & it = { s where s is Element of RAT+ : s < r } ) if x in RAT+
otherwise it = x;
existence

proof end;

uniqueness ;
consistency ;

end;

:: deftheorem Def3 defines DEDEKIND_CUT ARYTM_2:def 3 :

theorem :: ARYTM_2:3

for y being set holds not [{} ,y] in REAL+

proof end;

Lm6: for s, t being Element of RAT+ st ( for r being Element of RAT+ holds
( r < s iff r < t ) ) holds
s = t

proof end;

definition

let x be Element of DEDEKIND_CUTS ;
func GLUED x -> Element of REAL+ means :Def4: :: ARYTM_2:def 4
ex r being Element of RAT+ st
( it = r & ( for s being Element of RAT+ holds
( s in x iff s < r ) ) ) if ex r being Element of RAT+ st
for s being Element of RAT+ holds
( s in x iff s < r )
otherwise it = x;
uniqueness

proof end;

existence

proof end;

consistency ;

end;

:: deftheorem Def4 defines GLUED ARYTM_2:def 4 :

Lm7: for r being Element of RAT+
for B being set st B in DEDEKIND_CUTS & r in B holds
ex s being Element of RAT+ st
( s in B & r < s )

proof end;

Lm8: {} in DEDEKIND_CUTS

proof end;

Lm9: DEDEKIND_CUTS /\ RAT+ = {{} }

proof end;

Lm10: for x being Element of REAL+ holds
( DEDEKIND_CUT x = {} iff x = {} )

proof end;

Lm11: for A being Element of DEDEKIND_CUTS holds
( GLUED A = {} iff A = {} )

proof end;

Lm12: for A being Element of DEDEKIND_CUTS holds DEDEKIND_CUT (GLUED A) = A

proof end;

Lm13: for x, y being set st x in { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) } & y in { A where A is Subset of RAT+ : for r being Element of RAT+ st r in A holds
( ( for s being Element of RAT+ st s <=' r holds
s in A ) & ex s being Element of RAT+ st
( s in A & r < s ) ) } & not x c= y holds
y c= x

proof end;

definition

let x, y be Element of REAL+ ;
pred x <=' y means :Def5: :: ARYTM_2:def 5
ex x', y' being Element of RAT+ st
( x = x' & y = y' & x' <=' y' ) if ( x in RAT+ & y in RAT+ )
x in y if ( x in RAT+ & not y in RAT+ )
not y in x if ( not x in RAT+ & y in RAT+ )
otherwise x c= y;
consistency ;
connectedness

proof end;

end;

:: deftheorem Def5 defines <=' ARYTM_2:def 5 :

notation

let x, y be Element of REAL+ ;
antonym y < x for x <=' y;

end;

Lm14: for x', y' being Element of RAT+
for x, y being Element of REAL+ st x = x' & y = y' holds
( x <=' y iff x' <=' y' )

proof end;

Lm15: for B being set st B in DEDEKIND_CUTS & B <> {} holds
ex r being Element of RAT+ st
( r in B & r <> {} )

proof end;

Lm16: for r, s being Element of RAT+
for B being set st B in DEDEKIND_CUTS & r in B & s <=' r holds
s in B

proof end;

Lm17: for B being set st B in DEDEKIND_CUTS & B <> {} holds
{} in B

proof end;

Lm18: for r being Element of RAT+
for B being set st B in DEDEKIND_CUTS \ { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } & not r in B & B <> {} holds
ex s being Element of RAT+ st
( not s in B & s < r )

proof end;

Lm19: for x being Element of REAL+ st DEDEKIND_CUT x in { { s where s is Element of RAT+ : s < t } where t is Element of RAT+ : t <> {} } holds
x in RAT+

proof end;

Lm20: for x, y being Element of REAL+ st DEDEKIND_CUT x c= DEDEKIND_CUT y holds
x <=' y

proof end;

Lm21: for x, y being Element of REAL+ st x <=' y & y <=' x holds
x = y

proof end;

Lm22: for x, y being Element of REAL+ st DEDEKIND_CUT x = DEDEKIND_CUT y holds
x = y

proof end;

Lm23: for x being Element of REAL+ holds GLUED (DEDEKIND_CUT x) = x

proof end;

definition

let A, B be Element of DEDEKIND_CUTS ;
func A + B -> Element of DEDEKIND_CUTS equals :Def6: :: ARYTM_2:def 6
{ (r + s) where r, s is Element of RAT+ : ( r in A & s in B ) } ;
coherence

proof end;

commutativity

proof end;

end;

:: deftheorem Def6 defines + ARYTM_2:def 6 :

Lm24: for B being set st B in DEDEKIND_CUTS holds
ex r being Element of RAT+ st not r in B

proof end;

definition

let A, B be Element of DEDEKIND_CUTS ;
func A *' B -> Element of DEDEKIND_CUTS equals :: ARYTM_2:def 7
{ (r *' s) where r, s is Element of RAT+ : ( r in A & s in B ) } ;
coherence

proof end;

commutativity

proof end;

end;

:: deftheorem defines *' ARYTM_2:def 7 :

definition

let x, y be Element of REAL+ ;
func x + y -> Element of REAL+ equals :Def8: :: ARYTM_2:def 8
x if y = {}
y if x = {}
otherwise GLUED ((DEDEKIND_CUT x) + (DEDEKIND_CUT y));
coherence ;
consistency ;
commutativity ;
func x *' y -> Element of REAL+ equals :: ARYTM_2:def 9
GLUED ((DEDEKIND_CUT x) *' (DEDEKIND_CUT y));
coherence ;
commutativity ;

end;

:: deftheorem Def8 defines + ARYTM_2:def 8 :

:: deftheorem defines *' ARYTM_2:def 9 :

theorem Th4: :: ARYTM_2:4

for x, y being Element of REAL+ st x = {} holds
x *' y = {}

proof end;

theorem :: ARYTM_2:5

canceled;

theorem Th6: :: ARYTM_2:6

for x, y being Element of REAL+ st x + y = {} holds
x = {}

proof end;

Lm25: for A, B, C being Element of DEDEKIND_CUTS holds A + (B + C) c= (A + B) + C

proof end;

Lm26: for A, B, C being Element of DEDEKIND_CUTS holds A + (B + C) = (A + B) + C

proof end;

Lm27: for A, B being Element of DEDEKIND_CUTS holds
( not A + B = {} or A = {} or B = {} )

proof end;

theorem Th7: :: ARYTM_2:7

for x, y, z being Element of REAL+ holds x + (y + z) = (x + y) + z

proof end;

theorem :: ARYTM_2:8

{ A where A is Subset of RAT+ : for