:: The Lattice of Real Numbers. The Lattice of Real Functions
:: by Marek Chmur
::
:: Received May 22, 1990
:: Copyright (c) 1990-2011 Association of Mizar Users

begin

definition

func minreal -> BinOp of REAL means :Def1: :: REAL_LAT:def 1
for x, y being Real holds it . (x,y) = min (x,y);
existence
ex b₁ being BinOp of REAL st
for x, y being Real holds b₁ . (x,y) = min (x,y)

proof end;

uniqueness
for b₁, b₂ being BinOp of REAL st ( for x, y being Real holds b₁ . (x,y) = min (x,y) ) & ( for x, y being Real holds b₂ . (x,y) = min (x,y) ) holds
b₁ = b₂

proof end;

func maxreal -> BinOp of REAL means :Def2: :: REAL_LAT:def 2
for x, y being Real holds it . (x,y) = max (x,y);
existence
ex b₁ being BinOp of REAL st
for x, y being Real holds b₁ . (x,y) = max (x,y)

proof end;

uniqueness
for b₁, b₂ being BinOp of REAL st ( for x, y being Real holds b₁ . (x,y) = max (x,y) ) & ( for x, y being Real holds b₂ . (x,y) = max (x,y) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines minreal REAL_LAT:def 1 :
for b₁ being BinOp of REAL holds
( b₁ = minreal iff for x, y being Real holds b₁ . (x,y) = min (x,y) );

:: deftheorem Def2 defines maxreal REAL_LAT:def 2 :
for b₁ being BinOp of REAL holds
( b₁ = maxreal iff for x, y being Real holds b₁ . (x,y) = max (x,y) );

definition

canceled;
func Real_Lattice -> strict LattStr equals :: REAL_LAT:def 4
LattStr(# REAL,maxreal,minreal #);
coherence
LattStr(# REAL,maxreal,minreal #) is strict LattStr ;

end;

:: deftheorem REAL_LAT:def 3 :
canceled;

:: deftheorem defines Real_Lattice REAL_LAT:def 4 :
Real_Lattice = LattStr(# REAL,maxreal,minreal #);

registration

cluster the carrier of Real_Lattice -> real-membered ;
coherence
the carrier of Real_Lattice is real-membered ;

end;

registration

cluster Real_Lattice -> non empty strict ;
coherence
not Real_Lattice is empty ;

end;

registration

let a, b be Element of Real_Lattice;
identify a "\/" b with max (a,b);
compatibility
a "\/" b = max (a,b) by Def2;
identify a "/\" b with min (a,b);
compatibility
a "/\" b = min (a,b) by Def1;

end;

Lm1: for a, b being Element of Real_Lattice holds a "\/" b = b "\/" a
;

Lm2: for a, b, c being Element of Real_Lattice holds a "\/" (b "\/" c) = (a "\/" b) "\/" c
by XXREAL_0:34;

Lm3: for a, b being Element of Real_Lattice holds (a "/\" b) "\/" b = b
by XXREAL_0:36;

Lm4: for a, b being Element of Real_Lattice holds a "/\" b = b "/\" a
;

Lm5: for a, b, c being Element of Real_Lattice holds a "/\" (b "/\" c) = (a "/\" b) "/\" c
by XXREAL_0:33;

Lm6: for a, b being Element of Real_Lattice holds a "/\" (a "\/" b) = a
by XXREAL_0:35;

registration

cluster Real_Lattice -> strict Lattice-like ;
coherence
Real_Lattice is Lattice-like

proof end;

end;

Lm7: for a, b, c being Element of Real_Lattice holds a "/\" (b "\/" c) = (a "/\" b) "\/" (a "/\" c)
by XXREAL_0:38;

theorem :: REAL_LAT:1

canceled;

theorem :: REAL_LAT:2

canceled;

theorem :: REAL_LAT:3

canceled;

theorem :: REAL_LAT:4

canceled;

theorem :: REAL_LAT:5

canceled;

theorem :: REAL_LAT:6

canceled;

theorem :: REAL_LAT:7

canceled;

theorem Th8: :: REAL_LAT:8

for p, q being Element of Real_Lattice holds maxreal . (p,q) = maxreal . (q,p)

proof end;

theorem Th9: :: REAL_LAT:9

for p, q being Element of Real_Lattice holds minreal . (p,q) = minreal . (q,p)

proof end;

theorem Th10: :: REAL_LAT:10

for p, q, r being Element of Real_Lattice holds
( maxreal . (p,(maxreal . (q,r))) = maxreal . ((maxreal . (q,r)),p) & maxreal . (p,(maxreal . (q,r))) = maxreal . ((maxreal . (p,q)),r) & maxreal . (p,(maxreal . (q,r))) = maxreal . ((maxreal . (q,p)),r) & maxreal . (p,(maxreal . (q,r))) = maxreal . ((maxreal . (r,p)),q) & maxreal . (p,(maxreal . (q,r))) = maxreal . ((maxreal . (r,q)),p) & maxreal . (p,(maxreal . (q,r))) = maxreal . ((maxreal . (p,r)),q) )

proof end;

theorem Th11: :: REAL_LAT:11

for p, q, r being Element of Real_Lattice holds
( minreal . (p,(minreal . (q,r))) = minreal . ((minreal . (q,r)),p) & minreal . (p,(minreal . (q,r))) = minreal . ((minreal . (p,q)),r) & minreal . (p,(minreal . (q,r))) = minreal . ((minreal . (q,p)),r) & minreal . (p,(minreal . (q,r))) = minreal . ((minreal . (r,p)),q) & minreal . (p,(minreal . (q,r))) = minreal . ((minreal . (r,q)),p) & minreal . (p,(minreal . (q,r))) = minreal . ((minreal . (p,r)),q) )

proof end;

theorem Th12: :: REAL_LAT:12

for p, q being Element of Real_Lattice holds
( maxreal . ((minreal . (p,q)),q) = q & maxreal . (q,(minreal . (p,q))) = q & maxreal . (q,(minreal . (q,p))) = q & maxreal . ((minreal . (q,p)),q) = q )

proof end;

theorem Th13: :: REAL_LAT:13

for q, p being Element of Real_Lattice holds
( minreal . (q,(maxreal . (q,p))) = q & minreal . ((maxreal . (p,q)),q) = q & minreal . (q,(maxreal . (p,q))) = q & minreal . ((maxreal . (q,p)),q) = q )

proof end;

theorem Th14: :: REAL_LAT:14

for q, p, r being Element of Real_Lattice holds minreal . (q,(maxreal . (p,r))) = maxreal . ((minreal . (q,p)),(minreal . (q,r)))

proof end;

registration

cluster Real_Lattice -> strict distributive ;
coherence
Real_Lattice is distributive by Lm7, LATTICES:def 11;

end;

definition

let A be non empty set ;
func maxfuncreal A -> BinOp of (Funcs (A,REAL)) means :Def5: :: REAL_LAT:def 5
for f, g being Element of Funcs (A,REAL) holds it . (f,g) = maxreal .: (f,g);
existence
ex b₁ being BinOp of (Funcs (A,REAL)) st
for f, g being Element of Funcs (A,REAL) holds b₁ . (f,g) = maxreal .: (f,g)

proof end;

uniqueness
for b₁, b₂ being BinOp of (Funcs (A,REAL)) st ( for f, g being Element of Funcs (A,REAL) holds b₁ . (f,g) = maxreal .: (f,g) ) & ( for f, g being Element of Funcs (A,REAL) holds b₂ . (f,g) = maxreal .: (f,g) ) holds
b₁ = b₂

proof end;

func minfuncreal A -> BinOp of (Funcs (A,REAL)) means :Def6: :: REAL_LAT:def 6
for f, g being Element of Funcs (A,REAL) holds it . (f,g) = minreal .: (f,g);
existence
ex b₁ being BinOp of (Funcs (A,REAL)) st
for f, g being Element of Funcs (A,REAL) holds b₁ . (f,g) = minreal .: (f,g)

proof end;

uniqueness
for b₁, b₂ being BinOp of (Funcs (A,REAL)) st ( for f, g being Element of Funcs (A,REAL) holds b₁ . (f,g) = minreal .: (f,g) ) & ( for f, g being Element of Funcs (A,REAL) holds b₂ . (f,g) = minreal .: (f,g) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def5 defines maxfuncreal REAL_LAT:def 5 :
for A being non empty set
for b₂ being BinOp of (Funcs (A,REAL)) holds
( b₂ = maxfuncreal A iff for f, g being Element of Funcs (A,REAL) holds b₂ . (f,g) = maxreal .: (f,g) );

:: deftheorem Def6 defines minfuncreal REAL_LAT:def 6 :
for A being non empty set
for b₂ being BinOp of (Funcs (A,REAL)) holds
( b₂ = minfuncreal A iff for f, g being Element of Funcs (A,REAL) holds b₂ . (f,g) = minreal .: (f,g) );

Lm8: for A, B being non empty set
for x being Element of A
for f being Function of A,B holds x in dom f

proof end;

theorem :: REAL_LAT:15

canceled;

theorem :: REAL_LAT:16

canceled;

theorem :: REAL_LAT:17

canceled;

theorem :: REAL_LAT:18

canceled;

theorem :: REAL_LAT:19

canceled;

theorem Th20: :: REAL_LAT:20

for A being non empty set
for f, g being Element of Funcs (A,REAL) holds (maxfuncreal A) . (f,g) = (maxfuncreal A) . (g,f)

proof end;

theorem Th21: :: REAL_LAT:21

for A being non empty set
for f, g being Element of Funcs (A,REAL) holds (minfuncreal A) . (f,g) = (minfuncreal A) . (g,f)

proof end;

theorem Th22: :: REAL_LAT:22

for A being non empty set
for f, g, h being Element of Funcs (A,REAL) holds (maxfuncreal A) . (((maxfuncreal A) . (f,g)),h) = (maxfuncreal A) . (f,((maxfuncreal A) . (g,h)))

proof end;

theorem Th23: :: REAL_LAT:23

for A being non empty set
for f, g, h being Element of Funcs (A,REAL) holds (minfuncreal A) . (((minfuncreal A) . (f,g)),h) = (minfuncreal A) . (f,((minfuncreal A) . (g,h)))

proof end;

theorem Th24: :: REAL_LAT:24

for A being non empty set
for f, g being Element of Funcs (A,REAL) holds (maxfuncreal A) . (f,((minfuncreal A) . (f,g))) = f

proof end;

theorem Th25: :: REAL_LAT:25

for A being non empty set
for f, g being Element of Funcs (A,REAL) holds (maxfuncreal A) . (((minfuncreal A) . (f,g)),f) = f

proof end;

theorem Th26: :: REAL_LAT:26

for A being non empty set
for g, f being Element of Funcs (A,REAL) holds (maxfuncreal A) . (((minfuncreal A) . (g,f)),f) = f

proof end;

theorem :: REAL_LAT:27

for A being non empty set
for f, g being Element of Funcs (A,REAL) holds (maxfuncreal A) . (f,((minfuncreal A) . (g,f))) = f

proof end;

theorem Th28: :: REAL_LAT:28

for A being non empty set
for f, g being Element of Funcs (A,REAL) holds (minfuncreal A) . (f,((maxfuncreal A) . (f,g))) = f

proof end;

theorem Th29: :: REAL_LAT:29

for A being non empty set
for f, g being Element of Funcs (A,REAL) holds (minfuncreal A) . (f,((maxfuncreal A) . (g,f))) = f

proof end;

theorem Th30: :: REAL_LAT:30

for A being non empty set
for g, f being Element of Funcs (A,REAL) holds (minfuncreal A) . (((maxfuncreal A) . (g,f)),f) = f

proof end;

theorem :: REAL_LAT:31

for A being non empty set
for f, g being Element of Funcs (A,REAL) holds (minfuncreal A) . (((maxfuncreal A) . (f,g)),f) = f

proof end;

theorem Th32: :: REAL_LAT:32

for A being non empty set
for f, g, h being Element of Funcs (A,REAL) holds (minfuncreal A) . (f,((maxfuncreal A) . (g,h))) = (maxfuncreal A) . (((minfuncreal A) . (f,g)),((minfuncreal A) . (f,h)))

proof end;

definition

let A be non empty set ;
let x be Element of LattStr(# (Funcs (A,REAL)),(maxfuncreal A),(minfuncreal A) #);
canceled;
canceled;
func @ x -> Element of Funcs (A,REAL) equals :: REAL_LAT:def 9
x;
coherence
x is Element of Funcs (A,REAL) ;

end;

:: deftheorem REAL_LAT:def 7 :
canceled;

:: deftheorem REAL_LAT:def 8 :
canceled;

:: deftheorem defines @ REAL_LAT:def 9 :
for A being non empty set
for x being Element of LattStr(# (Funcs (A,REAL)),(maxfuncreal A),(minfuncreal A) #) holds @ x = x;

Lm9: for A being non empty set
for a, b, c being Element of LattStr(# (Funcs (A,REAL)),(maxfuncreal A),(minfuncreal A) #) holds a "\/" (b "\/" c) = (a "\/" b) "\/" c
by Th22;

Lm10: for A being non empty set
for a, b, c being Element of LattStr(# (Funcs (A,REAL)),(maxfuncreal A),(minfuncreal A) #) holds a "/\" (b "/\" c) = (a "/\" b) "/\" c
by Th23;

definition

let A be non empty set ;
func RealFunc_Lattice A -> strict Lattice equals :: REAL_LAT:def 10
LattStr(# (Funcs (A,REAL)),(maxfuncreal A),(minfuncreal A) #);
coherence
LattStr(# (Funcs (A,REAL)),(maxfuncreal A),(minfuncreal A) #) is strict Lattice

proof end;

end;

:: deftheorem defines RealFunc_Lattice REAL_LAT:def 10 :
for A being non empty set holds RealFunc_Lattice A = LattStr(# (Funcs (A,REAL)),(maxfuncreal A),(minfuncreal A) #);

theorem :: REAL_LAT:33

canceled;

theorem :: REAL_LAT:34

canceled;

theorem :: REAL_LAT:35

canceled;

theorem :: REAL_LAT:36

canceled;

theorem :: REAL_LAT:37

canceled;

theorem :: REAL_LAT:38

canceled;

theorem :: REAL_LAT:39

canceled;

theorem Th40: :: REAL_LAT:40

for A being non empty set
for p, q being Element of (RealFunc_Lattice A) holds (maxfuncreal A) . (p,q) = (maxfuncreal A) . (q,p)

proof end;

theorem Th41: :: REAL_LAT:41

for A being non empty set
for p, q being Element of (RealFunc_Lattice A) holds (minfuncreal A) . (p,q) = (minfuncreal A) . (q,p)

proof end;

theorem :: REAL_LAT:42

for A being non empty set
for p, q, r being Element of (RealFunc_Lattice A) holds
( (maxfuncreal A) . (p,((maxfuncreal A) . (q,r))) = (maxfuncreal A) . (((maxfuncreal A) . (q,r)),p) & (maxfuncreal A) . (p,((maxfuncreal A) . (q,r))) = (maxfuncreal A) . (((maxfuncreal A) . (p,q)),r) & (maxfuncreal A) . (p,((maxfuncreal A) . (q,r))) = (maxfuncreal A) . (((maxfuncreal A) . (q,p)),r) & (maxfuncreal A) . (p,((maxfuncreal A) . (q,r))) = (maxfuncreal A) . (((maxfuncreal A) . (r,p)),q) & (maxfuncreal A) . (p,((maxfuncreal A) . (q,r))) = (maxfuncreal A) . (((maxfuncreal A) . (r,q)),p) & (maxfuncreal A) . (p,((maxfuncreal A) . (q,r))) = (maxfuncreal A) . (((maxfuncreal A) . (p,r)),q) )

proof end;

theorem :: REAL_LAT:43

for A being non empty set
for p, q, r being Element of (RealFunc_Lattice A) holds
( (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (q,r)),p) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (p,q)),r) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (q,p)),r) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (r,p)),q) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (r,q)),p) & (minfuncreal A) . (p,((minfuncreal A) . (q,r))) = (minfuncreal A) . (((minfuncreal A) . (p,r)),q) )

proof end;

theorem :: REAL_LAT:44

for A being non empty set
for p, q being Element of (RealFunc_Lattice A) holds
( (maxfuncreal A) . (((minfuncreal A) . (p,q)),q) = q & (maxfuncreal A) . (q,((minfuncreal A) . (p,q))) = q & (maxfuncreal A) . (q,((minfuncreal A) . (q,p))) = q & (maxfuncreal A) . (((minfuncreal A) . (q,p)),q) = q )

proof end;

theorem :: REAL_LAT:45

for A being non empty set
for q, p being Element of (RealFunc_Lattice A) holds
( (minfuncreal A) . (q,((maxfuncreal A) . (q,p))) = q & (minfuncreal A) . (((maxfuncreal A) . (p,q)),q) = q & (minfuncreal A) . (q,((maxfuncreal A) . (p,q))) = q & (minfuncreal A) . (((maxfuncreal A) . (q,p)),q) = q )

proof end;

theorem :: REAL_LAT:46

for A being non empty set
for q, p, r being Element of (RealFunc_Lattice A) holds (minfuncreal A) . (q,((maxfuncreal A) . (p,r))) = (maxfuncreal A) . (((minfuncreal A) . (q,p)),((minfuncreal A) . (q,r))) by Th32;

theorem :: REAL_LAT:47

for A being non empty set holds RealFunc_Lattice A is D_Lattice

proof end;