:: Duality in Relation Structures
:: by Grzegorz Bancerek
::
:: Received November 12, 1996
:: Copyright (c) 1996-2012 Association of Mizar Users


begin

notation
let L be RelStr ;
synonym L opp for L ~ ;
end;

theorem Th1: :: YELLOW_7:1
for L being RelStr
for x, y being Element of (L opp) holds
( x <= y iff ~ x >= ~ y )
proofend;

theorem Th2: :: YELLOW_7:2
for L being RelStr
for x being Element of L
for y being Element of (L opp) holds
( ( x <= ~ y implies x ~ >= y ) &amp; ( x ~ >= y implies x <= ~ y ) &amp; ( x >= ~ y implies x ~ <= y ) &amp; ( x ~ <= y implies x >= ~ y ) )
proofend;

theorem :: YELLOW_7:3
for L being RelStr holds
( L is empty iff L opp is empty ) ;

theorem Th4: :: YELLOW_7:4
for L being RelStr holds
( L is reflexive iff L opp is reflexive )
proofend;

theorem Th5: :: YELLOW_7:5
for L being RelStr holds
( L is antisymmetric iff L opp is antisymmetric )
proofend;

theorem Th6: :: YELLOW_7:6
for L being RelStr holds
( L is transitive iff L opp is transitive )
proofend;

theorem Th7: :: YELLOW_7:7
for L being non empty RelStr holds
( L is connected iff L opp is connected )
proofend;

registration
let L be reflexive RelStr ;
clusterL opp  ->  reflexive ;
coherence
 L opp is reflexive by Th4;
end;

registration
let L be transitive RelStr ;
clusterL opp  ->  transitive ;
coherence
 L opp is transitive by Th6;
end;

registration
let L be antisymmetric RelStr ;
clusterL opp  ->  antisymmetric ;
coherence
 L opp is antisymmetric by Th5;
end;

registration
let L be non empty connected RelStr ;
clusterL opp  ->  connected ;
coherence
 L opp is connected by Th7;
end;

theorem Th8: :: YELLOW_7:8
for L being RelStr
for x being Element of L
for X being set holds
( ( x is_<=_than X implies x ~ is_>=_than X ) &amp; ( x ~ is_>=_than X implies x is_<=_than X ) &amp; ( x is_>=_than X implies x ~ is_<=_than X ) &amp; ( x ~ is_<=_than X implies x is_>=_than X ) )
proofend;

theorem Th9: :: YELLOW_7:9
for L being RelStr
for x being Element of (L opp)
for X being set holds
( ( x is_<=_than X implies ~ x is_>=_than X ) &amp; ( ~ x is_>=_than X implies x is_<=_than X ) &amp; ( x is_>=_than X implies ~ x is_<=_than X ) &amp; ( ~ x is_<=_than X implies x is_>=_than X ) )
proofend;

theorem Th10: :: YELLOW_7:10
for L being RelStr
for X being set holds
( ex_sup_of X,L iff ex_inf_of X,L opp )
proofend;

theorem Th11: :: YELLOW_7:11
for L being RelStr
for X being set holds
( ex_sup_of X,L opp iff ex_inf_of X,L )
proofend;

theorem Th12: :: YELLOW_7:12
for L being non empty RelStr
for X being set st ( ex_sup_of X,L or ex_inf_of X,L opp ) holds
"\/" (X,L) = "/\" (X,(L opp))
proofend;

theorem Th13: :: YELLOW_7:13
for L being non empty RelStr
for X being set st ( ex_inf_of X,L or ex_sup_of X,L opp ) holds
"/\" (X,L) = "\/" (X,(L opp))
proofend;

theorem Th14: :: YELLOW_7:14
for L1, L2 being RelStr st RelStr(# the carrier of L1, the InternalRel of L1 #) = RelStr(# the carrier of L2, the InternalRel of L2 #) &amp; L1 is with_infima holds
L2 is with_infima
proofend;

theorem :: YELLOW_7:15
for L1, L2 being RelStr st RelStr(# the carrier of L1, the InternalRel of L1 #) = RelStr(# the carrier of L2, the InternalRel of L2 #) &amp; L1 is with_suprema holds
L2 is with_suprema
proofend;

theorem Th16: :: YELLOW_7:16
for L being RelStr holds
( L is with_infima iff L opp is with_suprema )
proofend;

:: LATTICE3:10
:: for L being RelStr holds L opp is with_infima iff L is with_suprema;
theorem Th17: :: YELLOW_7:17
for L being non empty RelStr holds
( L is complete iff L opp is complete )
proofend;

registration
let L be with_infima RelStr ;
clusterL opp  ->  with_suprema ;
coherence
 L opp is with_suprema by Th16;
end;

registration
let L be with_suprema RelStr ;
clusterL opp  ->  with_infima ;
coherence
 L opp is with_infima by LATTICE3:10;
end;

registration
let L be non empty complete RelStr ;
clusterL opp  ->  complete ;
coherence
 L opp is complete by Th17;
end;

theorem :: YELLOW_7:18
for L being non empty RelStr
for X being Subset of L
for Y being Subset of (L opp) st X = Y holds
( fininfs X = finsups Y &amp; finsups X = fininfs Y )
proofend;

theorem Th19: :: YELLOW_7:19
for L being RelStr
for X being Subset of L
for Y being Subset of (L opp) st X = Y holds
( downarrow X = uparrow Y &amp; uparrow X = downarrow Y )
proofend;

theorem :: YELLOW_7:20
for L being non empty RelStr
for x being Element of L
for y being Element of (L opp) st x = y holds
( downarrow x = uparrow y &amp; uparrow x = downarrow y ) by Th19;

theorem Th21: :: YELLOW_7:21
for L being with_infima Poset
for x, y being Element of L holds x "/\" y = (x ~) "\/" (y ~)
proofend;

theorem Th22: :: YELLOW_7:22
for L being with_infima Poset
for x, y being Element of (L opp) holds (~ x) "/\" (~ y) = x "\/" y
proofend;

theorem Th23: :: YELLOW_7:23
for L being with_suprema Poset
for x, y being Element of L holds x "\/" y = (x ~) "/\" (y ~)
proofend;

theorem Th24: :: YELLOW_7:24
for L being with_suprema Poset
for x, y being Element of (L opp) holds (~ x) "\/" (~ y) = x "/\" y
proofend;

theorem Th25: :: YELLOW_7:25
for L being LATTICE holds
( L is distributive iff L opp is distributive )
proofend;

registration
let L be distributive LATTICE;
clusterL opp  ->  distributive ;
coherence
 L opp is distributive by Th25;
end;

theorem Th26: :: YELLOW_7:26
for L being RelStr
for x being set holds
( x is directed Subset of L iff x is filtered Subset of (L opp) )
proofend;

theorem :: YELLOW_7:27
for L being RelStr
for x being set holds
( x is directed Subset of (L opp) iff x is filtered Subset of L )
proofend;

theorem Th28: :: YELLOW_7:28
for L being RelStr
for x being set holds
( x is lower Subset of L iff x is upper Subset of (L opp) )
proofend;

theorem :: YELLOW_7:29
for L being RelStr
for x being set holds
( x is lower Subset of (L opp) iff x is upper Subset of L )
proofend;

theorem Th30: :: YELLOW_7:30
for L being RelStr holds
( L is lower-bounded iff L opp is upper-bounded )
proofend;

theorem Th31: :: YELLOW_7:31
for L being RelStr holds
( L opp is lower-bounded iff L is upper-bounded )
proofend;

theorem :: YELLOW_7:32
for L being RelStr holds
( L is bounded iff L opp is bounded )
proofend;

theorem :: YELLOW_7:33
for L being non empty antisymmetric lower-bounded RelStr holds
( (Bottom L) ~ = Top (L opp) &amp; ~ (Top (L opp)) = Bottom L ) by Th12, YELLOW_0:42;

theorem :: YELLOW_7:34
for L being non empty antisymmetric upper-bounded RelStr holds
( (Top L) ~ = Bottom (L opp) &amp; ~ (Bottom (L opp)) = Top L ) by Th13, YELLOW_0:43;

theorem Th35: :: YELLOW_7:35
for L being bounded LATTICE
for x, y being Element of L holds
( y is_a_complement_of x iff y ~ is_a_complement_of x ~ )
proofend;

theorem Th36: :: YELLOW_7:36
for L being bounded LATTICE holds
( L is complemented iff L opp is complemented )
proofend;

registration
let L be lower-bounded RelStr ;
clusterL opp  ->  upper-bounded ;
coherence
 L opp is upper-bounded by Th30;
end;

registration
let L be upper-bounded RelStr ;
clusterL opp  ->  lower-bounded ;
coherence
 L opp is lower-bounded by Th31;
end;

registration
let L be bounded complemented LATTICE;
clusterL opp  ->  complemented ;
coherence
 L opp is complemented by Th36;
end;

:: Collorary: L is Boolean -> L opp is Boolean
theorem :: YELLOW_7:37
for L being Boolean LATTICE
for x being Element of L holds 'not' (x ~) = 'not' x
proofend;

definition
let L be non empty RelStr ;
func ComplMap L ->  Function of L,(L opp) means :Def1: :: YELLOW_7:def 1
 for x being Element of L holds it . x = 'not' x;
existence
 ex b1 being Function of L,(L opp) st
for x being Element of L holds b1 . x = 'not' x
proofend;
correctness
uniqueness
for b1, b2 being Function of L,(L opp) st ( for x being Element of L holds b1 . x = 'not' x ) &amp; ( for x being Element of L holds b2 . x = 'not' x ) holds
b1 = b2;
proofend;
end;

:: deftheorem Def1 defines ComplMap YELLOW_7:def_1_:_
 for L being non empty RelStr
for b2 being Function of L,(L opp) holds
( b2 = ComplMap L iff for x being Element of L holds b2 . x = 'not' x );

registration
let L be Boolean LATTICE;
cluster ComplMap L -> V13() ;
coherence
 ComplMap L is one-to-one
proofend;
end;

registration
let L be Boolean LATTICE;
cluster ComplMap L ->  isomorphic ;
coherence
 ComplMap L is isomorphic
proofend;
end;

theorem :: YELLOW_7:38
for L being Boolean LATTICE holds L,L opp are_isomorphic
proofend;

theorem :: YELLOW_7:39
for S, T being non empty RelStr
for f being set holds
( ( f is Function of S,T implies f is Function of (S opp),T ) &amp; ( f is Function of (S opp),T implies f is Function of S,T ) &amp; ( f is Function of S,T implies f is Function of S,(T opp) ) &amp; ( f is Function of S,(T opp) implies f is Function of S,T ) &amp; ( f is Function of S,T implies f is Function of (S opp),(T opp) ) &amp; ( f is Function of (S opp),(T opp) implies f is Function of S,T ) ) ;

theorem :: YELLOW_7:40
for S, T being non empty RelStr
for f being Function of S,T
for g being Function of S,(T opp) st f = g holds
( ( f is monotone implies g is antitone ) &amp; ( g is antitone implies f is monotone ) &amp; ( f is antitone implies g is monotone ) &amp; ( g is monotone implies f is antitone ) )
proofend;

theorem :: YELLOW_7:41
for S, T being non empty RelStr
for f being Function of S,(T opp)
for g being Function of (S opp),T st f = g holds
( ( f is monotone implies g is monotone ) &amp; ( g is monotone implies f is monotone ) &amp; ( f is antitone implies g is antitone ) &amp; ( g is antitone implies f is antitone ) )
proofend;

theorem Th42: :: YELLOW_7:42
for S, T being non empty RelStr
for f being Function of S,T
for g being Function of (S opp),(T opp) st f = g holds
( ( f is monotone implies g is monotone ) &amp; ( g is monotone implies f is monotone ) &amp; ( f is antitone implies g is antitone ) &amp; ( g is antitone implies f is antitone ) )
proofend;

theorem :: YELLOW_7:43
for S, T being non empty RelStr
for f being set holds
( ( f is Connection of S,T implies f is Connection of S ~ ,T ) &amp; ( f is Connection of S ~ ,T implies f is Connection of S,T ) &amp; ( f is Connection of S,T implies f is Connection of S,T ~ ) &amp; ( f is Connection of S,T ~ implies f is Connection of S,T ) &amp; ( f is Connection of S,T implies f is Connection of S ~ ,T ~ ) &amp; ( f is Connection of S ~ ,T ~ implies f is Connection of S,T ) )
proofend;

theorem :: YELLOW_7:44
for S, T being non empty Poset
for f1 being Function of S,T
for g1 being Function of T,S
for f2 being Function of (S ~),(T ~)
for g2 being Function of (T ~),(S ~) st f1 = f2 &amp; g1 = g2 holds
( [f1,g1] is Galois iff [g2,f2] is Galois )
proofend;

theorem Th45: :: YELLOW_7:45
for J being set
for D being non empty set
for K being ManySortedSet of J
for F being DoubleIndexedSet of K,D holds doms F = K
proofend;

definition
let J, D be non empty set ;
let K be V8() ManySortedSet of J;
let F be DoubleIndexedSet of K,D;
let j be Element of J;
let k be Element of K . j;
:: original:..
redefine funcF .. (j,k) ->  Element of D;
coherence
 F .. (j,k) is Element of D
proofend;
end;

theorem :: YELLOW_7:46
for L being non empty RelStr
for J being set
for K being ManySortedSet of J
for x being set holds
( x is DoubleIndexedSet of K,L iff x is DoubleIndexedSet of K,(L opp) ) ;

theorem Th47: :: YELLOW_7:47
for L being complete LATTICE
for J being non empty set
for K being V8() ManySortedSet of J
for F being DoubleIndexedSet of K,L holds Sup <= Inf
proofend;

theorem Th48: :: YELLOW_7:48
for L being complete LATTICE holds
( L is completely-distributive iff for J being non empty set
for K being V8() ManySortedSet of J
for F being DoubleIndexedSet of K,L holds Sup = Inf )
proofend;

theorem Th49: :: YELLOW_7:49
for L being non empty antisymmetric complete RelStr
for F being Function holds
( \\/ (F,L) = //\ (F,(L opp)) &amp; //\ (F,L) = \\/ (F,(L opp)) )
proofend;

theorem Th50: :: YELLOW_7:50
for L being non empty antisymmetric complete RelStr
for F being Function-yielding Function holds
( \// (F,L) = /\\ (F,(L opp)) &amp; /\\ (F,L) = \// (F,(L opp)) )
proofend;

registration
cluster non empty completely-distributive  ->  non empty complete for RelStr ;
coherence
 for b1 being non empty RelStr st b1 is completely-distributive holds
b1 is complete by WAYBEL_5:def_3;
end;

registration
cluster non empty V45() finite 1 -element strict reflexive transitive antisymmetric with_suprema with_infima complete lower-bounded upper-bounded bounded connected up-complete /\-complete distributive V222() completely-distributive for RelStr ;
existence
 ex b1 being 1 -element Poset st
( b1 is completely-distributive &amp; b1 is strict )
proofend;
end;

theorem :: YELLOW_7:51
for L being non empty Poset holds
( L is completely-distributive iff L opp is completely-distributive )
proofend;