:: ROBBINS1 semantic presentation  Show TPTP formulae Show IDV graph for whole article:: Showing IDV graph ... (Click the Palm Trees again to close it)

definition
attr c1 is strict;
struct ComplStr -> 1-sorted ;
aggr ComplStr(# carrier, Compl #) -> ComplStr ;
sel Compl c1 -> UnOp of the carrier of c1;
end;

definition
attr c1 is strict;
struct ComplLattStr -> \/-SemiLattStr , ComplStr ;
aggr ComplLattStr(# carrier, L_join, Compl #) -> ComplLattStr ;
end;

definition
attr c1 is strict;
struct OrthoLattStr -> ComplLattStr , LattStr ;
aggr OrthoLattStr(# carrier, L_join, L_meet, Compl #) -> OrthoLattStr ;
end;

definition
func TrivComplLat -> strict ComplLattStr equals :: ROBBINS1:def 1
ComplLattStr(# {{} },op2 ,op1 #);
coherence
ComplLattStr(# {{} },op2 ,op1 #) is strict ComplLattStr
;
end;

:: deftheorem defines TrivComplLat ROBBINS1:def 1 :
TrivComplLat = ComplLattStr(# {{} },op2 ,op1 #);

definition
func TrivOrtLat -> strict OrthoLattStr equals :: ROBBINS1:def 2
OrthoLattStr(# {{} },op2 ,op2 ,op1 #);
coherence
OrthoLattStr(# {{} },op2 ,op2 ,op1 #) is strict OrthoLattStr
;
end;

:: deftheorem defines TrivOrtLat ROBBINS1:def 2 :
TrivOrtLat = OrthoLattStr(# {{} },op2 ,op2 ,op1 #);

registration
cluster TrivComplLat -> non empty trivial strict ;
coherence
( not TrivComplLat is empty & TrivComplLat is trivial )
by REALSET2:def 5, STRUCT_0:def 1;
cluster TrivOrtLat -> non empty trivial strict ;
coherence
( not TrivOrtLat is empty & TrivOrtLat is trivial )
by REALSET2:def 5, STRUCT_0:def 1;
end;

registration
cluster non empty trivial strict OrthoLattStr ;
existence
ex b1 being OrthoLattStr st
( b1 is strict & not b1 is empty & b1 is trivial )
proof end;
cluster non empty trivial strict ComplLattStr ;
existence
ex b1 being ComplLattStr st
( b1 is strict & not b1 is empty & b1 is trivial )
proof end;
end;

registration
let L be non empty trivial ComplLattStr ;
cluster ComplStr(# the carrier of L,the Compl of L #) -> non empty trivial ;
coherence
( not ComplStr(# the carrier of L,the Compl of L #) is empty & ComplStr(# the carrier of L,the Compl of L #) is trivial )
proof end;
end;

registration
cluster non empty trivial strict ComplStr ;
existence
ex b1 being ComplStr st
( b1 is strict & not b1 is empty & b1 is trivial )
proof end;
end;

definition
let L be non empty ComplStr ;
let x be Element of L;
func x ` -> Element of L equals :: ROBBINS1:def 3
the Compl of L . x;
coherence
the Compl of L . x is Element of L
;
end;

:: deftheorem defines ` ROBBINS1:def 3 :
for L being non empty ComplStr
for x being Element of L holds x ` = the Compl of L . x;

notation
let L be non empty ComplLattStr ;
let x, y be Element of L;
synonym x + y for x "\/" y;
end;

definition
let L be non empty ComplLattStr ;
let x, y be Element of L;
func x *' y -> Element of L equals :: ROBBINS1:def 4
((x ` ) "\/" (y ` )) ` ;
coherence
((x ` ) "\/" (y ` )) ` is Element of L
;
end;

:: deftheorem defines *' ROBBINS1:def 4 :
for L being non empty ComplLattStr
for x, y being Element of L holds x *' y = ((x ` ) "\/" (y ` )) ` ;

definition
let L be non empty ComplLattStr ;
attr L is Robbins means :Def5: :: ROBBINS1:def 5
for x, y being Element of L holds (((x + y) ` ) + ((x + (y ` )) ` )) ` = x;
attr L is Huntington means :Def6: :: ROBBINS1:def 6
for x, y being Element of L holds (((x ` ) + (y ` )) ` ) + (((x ` ) + y) ` ) = x;
end;

:: deftheorem Def5 defines Robbins ROBBINS1:def 5 :
for L being non empty ComplLattStr holds
( L is Robbins iff for x, y being Element of L holds (((x + y) ` ) + ((x + (y ` )) ` )) ` = x );

:: deftheorem Def6 defines Huntington ROBBINS1:def 6 :
for L being non empty ComplLattStr holds
( L is Huntington iff for x, y being Element of L holds (((x ` ) + (y ` )) ` ) + (((x ` ) + y) ` ) = x );

definition
let G be non empty \/-SemiLattStr ;
attr G is join-idempotent means :Def7: :: ROBBINS1:def 7
for x being Element of G holds x "\/" x = x;
end;

:: deftheorem Def7 defines join-idempotent ROBBINS1:def 7 :
for G being non empty \/-SemiLattStr holds
( G is join-idempotent iff for x being Element of G holds x "\/" x = x );

registration
cluster TrivComplLat -> non empty join-commutative join-associative trivial strict Robbins Huntington join-idempotent ;
coherence
( TrivComplLat is join-commutative & TrivComplLat is join-associative & TrivComplLat is Robbins & TrivComplLat is Huntington & TrivComplLat is join-idempotent )
proof end;
cluster TrivOrtLat -> non empty join-commutative join-associative trivial strict Robbins Huntington ;
coherence
( TrivOrtLat is join-commutative & TrivOrtLat is join-associative & TrivOrtLat is Huntington & TrivOrtLat is Robbins )
proof end;
end;

registration
cluster TrivOrtLat -> non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing trivial strict Robbins Huntington ;
coherence
( TrivOrtLat is meet-commutative & TrivOrtLat is meet-associative & TrivOrtLat is meet-absorbing & TrivOrtLat is join-absorbing )
proof end;
end;

registration
cluster non empty join-commutative join-associative strict Robbins Huntington join-idempotent ComplLattStr ;
existence
ex b1 being non empty ComplLattStr st
( b1 is strict & b1 is join-associative & b1 is join-commutative & b1 is Robbins & b1 is join-idempotent & b1 is Huntington )
proof end;
end;

registration
cluster non empty Lattice-like strict Robbins Huntington OrthoLattStr ;
existence
ex b1 being non empty OrthoLattStr st
( b1 is strict & b1 is Lattice-like & b1 is Robbins & b1 is Huntington )
proof end;
end;

definition
let L be non empty join-commutative ComplLattStr ;
let x, y be Element of L;
:: original: +
redefine func x + y -> M2(the carrier of L);
commutativity
for x, y being Element of L holds x + y = y + x
by LATTICES:def 4;
end;

theorem Th1: :: ROBBINS1:1  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a, b being Element of L holds (a *' b) + (a *' (b ` )) = a by Def6;

theorem Th2: :: ROBBINS1:2  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a being Element of L holds a + (a ` ) = (a ` ) + ((a ` ) ` )
proof end;

theorem Th3: :: ROBBINS1:3  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for x being Element of L holds (x ` ) ` = x
proof end;

theorem Th4: :: ROBBINS1:4  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a, b being Element of L holds a + (a ` ) = b + (b ` )
proof end;

theorem Th5: :: ROBBINS1:5  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington join-idempotent ComplLattStr ex c being Element of L st
for a being Element of L holds
( c + a = c & a + (a ` ) = c )
proof end;

theorem Th6: :: ROBBINS1:6  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington join-idempotent ComplLattStr holds L is upper-bounded
proof end;

registration
cluster non empty join-commutative join-associative Huntington join-idempotent -> non empty upper-bounded ComplLattStr ;
coherence
for b1 being non empty ComplLattStr st b1 is join-commutative & b1 is join-associative & b1 is join-idempotent & b1 is Huntington holds
b1 is upper-bounded
by Th6;
end;

definition
let L be non empty join-commutative join-associative Huntington join-idempotent ComplLattStr ;
redefine func Top L means :Def8: :: ROBBINS1:def 8
ex a being Element of L st it = a + (a ` );
compatibility
for b1 being M2(the carrier of L) holds
( b1 = Top L iff ex a being Element of L st b1 = a + (a ` ) )
proof end;
end;

:: deftheorem Def8 defines Top ROBBINS1:def 8 :
for L being non empty join-commutative join-associative Huntington join-idempotent ComplLattStr
for b2 being M2(the carrier of b1) holds
( b2 = Top L iff ex a being Element of L st b2 = a + (a ` ) );

theorem Th7: :: ROBBINS1:7  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington join-idempotent ComplLattStr ex c being Element of L st
for a being Element of L holds
( c *' a = c & (a + (a ` )) ` = c )
proof end;

theorem Th8: :: ROBBINS1:8  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative ComplLattStr
for a, b being Element of L holds a *' b = b *' a
proof end;

definition
let L be non empty join-commutative join-associative ComplLattStr ;
let x, y be Element of L;
:: original: *'
redefine func x *' y -> Element of L;
commutativity
for x, y being Element of L holds x *' y = y *' x
by Th8;
end;

definition
let L be non empty join-commutative join-associative Huntington join-idempotent ComplLattStr ;
func Bot L -> Element of L means :Def9: :: ROBBINS1:def 9
for a being Element of L holds it *' a = it;
existence
ex b1 being Element of L st
for a being Element of L holds b1 *' a = b1
proof end;
uniqueness
for b1, b2 being Element of L st ( for a being Element of L holds b1 *' a = b1 ) & ( for a being Element of L holds b2 *' a = b2 ) holds
b1 = b2
proof end;
end;

:: deftheorem Def9 defines Bot ROBBINS1:def 9 :
for L being non empty join-commutative join-associative Huntington join-idempotent ComplLattStr
for b2 being Element of L holds
( b2 = Bot L iff for a being Element of L holds b2 *' a = b2 );

theorem Th9: :: ROBBINS1:9  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington join-idempotent ComplLattStr
for a being Element of L holds Bot L = (a + (a ` )) `
proof end;

theorem Th10: :: ROBBINS1:10  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington join-idempotent ComplLattStr holds
( (Top L) ` = Bot L & Top L = (Bot L) ` )
proof end;

theorem Th11: :: ROBBINS1:11  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a, b being Element of L st a ` = b ` holds
a = b
proof end;

theorem Th12: :: ROBBINS1:12  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a, b being Element of L holds a + ((b + (b ` )) ` ) = a
proof end;

theorem Th13: :: ROBBINS1:13  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a being Element of L holds a + a = a
proof end;

registration
cluster non empty join-commutative join-associative Huntington -> non empty join-idempotent ComplLattStr ;
coherence
for b1 being non empty ComplLattStr st b1 is join-commutative & b1 is join-associative & b1 is Huntington holds
b1 is join-idempotent
proof end;
end;

theorem Th14: :: ROBBINS1:14  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a being Element of L holds a + (Bot L) = a
proof end;

theorem :: ROBBINS1:15  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a being Element of L holds a *' (Top L) = a
proof end;

theorem Th16: :: ROBBINS1:16  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a being Element of L holds a *' (a ` ) = Bot L
proof end;

theorem Th17: :: ROBBINS1:17  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a, b, c being Element of L holds a *' (b *' c) = (a *' b) *' c
proof end;

theorem Th18: :: ROBBINS1:18  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a, b being Element of L holds a + b = ((a ` ) *' (b ` )) `
proof end;

theorem :: ROBBINS1:19  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a being Element of L holds a *' a = a
proof end;

theorem :: ROBBINS1:20  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a being Element of L holds a + (Top L) = Top L
proof end;

theorem Th21: :: ROBBINS1:21  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a, b being Element of L holds a + (a *' b) = a
proof end;

theorem Th22: :: ROBBINS1:22  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a, b being Element of L holds a *' (a + b) = a
proof end;

theorem Th23: :: ROBBINS1:23  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a, b being Element of L st (a ` ) + b = Top L & (b ` ) + a = Top L holds
a = b
proof end;

theorem Th24: :: ROBBINS1:24  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a, b being Element of L st a + b = Top L & a *' b = Bot L holds
a ` = b
proof end;

theorem Th25: :: ROBBINS1:25  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a, b, c being Element of L holds ((((((((a *' b) *' c) + ((a *' b) *' (c ` ))) + ((a *' (b ` )) *' c)) + ((a *' (b ` )) *' (c ` ))) + (((a ` ) *' b) *' c)) + (((a ` ) *' b) *' (c ` ))) + (((a ` ) *' (b ` )) *' c)) + (((a ` ) *' (b ` )) *' (c ` )) = Top L
proof end;

theorem Th26: :: ROBBINS1:26  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a, b, c being Element of L holds
( (a *' c) *' (b *' (c ` )) = Bot L & ((a *' b) *' c) *' (((a ` ) *' b) *' c) = Bot L & ((a *' (b ` )) *' c) *' (((a ` ) *' b) *' c) = Bot L & ((a *' b) *' c) *' (((a ` ) *' (b ` )) *' c) = Bot L & ((a *' b) *' (c ` )) *' (((a ` ) *' (b ` )) *' (c ` )) = Bot L & ((a *' (b ` )) *' c) *' (((a ` ) *' b) *' c) = Bot L )
proof end;

theorem Th27: :: ROBBINS1:27  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a, b, c being Element of L holds (a *' b) + (a *' c) = (((a *' b) *' c) + ((a *' b) *' (c ` ))) + ((a *' (b ` )) *' c)
proof end;

theorem Th28: :: ROBBINS1:28  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a, b, c being Element of L holds (a *' (b + c)) ` = (((((a *' (b ` )) *' (c ` )) + (((a ` ) *' b) *' c)) + (((a ` ) *' b) *' (c ` ))) + (((a ` ) *' (b ` )) *' c)) + (((a ` ) *' (b ` )) *' (c ` ))
proof end;

theorem Th29: :: ROBBINS1:29  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a, b, c being Element of L holds ((a *' b) + (a *' c)) + ((a *' (b + c)) ` ) = Top L
proof end;

theorem Th30: :: ROBBINS1:30  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a, b, c being Element of L holds ((a *' b) + (a *' c)) *' ((a *' (b + c)) ` ) = Bot L
proof end;

theorem Th31: :: ROBBINS1:31  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a, b, c being Element of L holds a *' (b + c) = (a *' b) + (a *' c)
proof end;

theorem :: ROBBINS1:32  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr
for a, b, c being Element of L holds a + (b *' c) = (a + b) *' (a + c)
proof end;

definition
let L be non empty OrthoLattStr ;
attr L is well-complemented means :Def10: :: ROBBINS1:def 10
for a being Element of L holds a ` is_a_complement_of a;
end;

:: deftheorem Def10 defines well-complemented ROBBINS1:def 10 :
for L being non empty OrthoLattStr holds
( L is well-complemented iff for a being Element of L holds a ` is_a_complement_of a );

registration
cluster TrivOrtLat -> non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing upper-bounded Boolean trivial strict Robbins Huntington join-idempotent well-complemented ;
coherence
( TrivOrtLat is Boolean & TrivOrtLat is well-complemented )
proof end;
end;

definition
mode preOrthoLattice is non empty Lattice-like OrthoLattStr ;
end;

registration
cluster Boolean strict well-complemented OrthoLattStr ;
existence
ex b1 being preOrthoLattice st
( b1 is strict & b1 is Boolean & b1 is well-complemented )
proof end;
end;

theorem Th33: :: ROBBINS1:33  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being distributive well-complemented preOrthoLattice
for x being Element of L holds (x ` ) ` = x
proof end;

theorem Th34: :: ROBBINS1:34  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being distributive bounded well-complemented preOrthoLattice
for x, y being Element of L holds x "/\" y = ((x ` ) "\/" (y ` )) `
proof end;

definition
let L be non empty ComplLattStr ;
func CLatt L -> strict OrthoLattStr means :Def11: :: ROBBINS1:def 11
( the carrier of it = the carrier of L & the L_join of it = the L_join of L & the Compl of it = the Compl of L & ( for a, b being Element of L holds the L_meet of it . a,b = a *' b ) );
existence
ex b1 being strict OrthoLattStr st
( the carrier of b1 = the carrier of L & the L_join of b1 = the L_join of L & the Compl of b1 = the Compl of L & ( for a, b being Element of L holds the L_meet of b1 . a,b = a *' b ) )
proof end;
uniqueness
for b1, b2 being strict OrthoLattStr st the carrier of b1 = the carrier of L & the L_join of b1 = the L_join of L & the Compl of b1 = the Compl of L & ( for a, b being Element of L holds the L_meet of b1 . a,b = a *' b ) & the carrier of b2 = the carrier of L & the L_join of b2 = the L_join of L & the Compl of b2 = the Compl of L & ( for a, b being Element of L holds the L_meet of b2 . a,b = a *' b ) holds
b1 = b2
proof end;
end;

:: deftheorem Def11 defines CLatt ROBBINS1:def 11 :
for L being non empty ComplLattStr
for b2 being strict OrthoLattStr holds
( b2 = CLatt L iff ( the carrier of b2 = the carrier of L & the L_join of b2 = the L_join of L & the Compl of b2 = the Compl of L & ( for a, b being Element of L holds the L_meet of b2 . a,b = a *' b ) ) );

registration
let L be non empty ComplLattStr ;
cluster CLatt L -> non empty strict ;
coherence
not CLatt L is empty
proof end;
end;

registration
let L be non empty join-commutative ComplLattStr ;
cluster CLatt L -> non empty join-commutative strict ;
coherence
CLatt L is join-commutative
proof end;
end;

registration
let L be non empty join-associative ComplLattStr ;
cluster CLatt L -> non empty join-associative strict ;
coherence
CLatt L is join-associative
proof end;
end;

registration
let L be non empty join-commutative join-associative ComplLattStr ;
cluster CLatt L -> non empty join-commutative join-associative meet-commutative strict ;
coherence
CLatt L is meet-commutative
proof end;
end;

theorem Th35: :: ROBBINS1:35  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty ComplLattStr
for a, b being Element of L
for a', b' being Element of (CLatt L) st a = a' & b = b' holds
( a *' b = a' "/\" b' & a + b = a' "\/" b' & a ` = a' ` ) by Def11;

registration
let L be non empty join-commutative join-associative Huntington ComplLattStr ;
cluster CLatt L -> non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing strict ;
coherence
( CLatt L is meet-associative & CLatt L is join-absorbing & CLatt L is meet-absorbing )
proof end;
end;

registration
let L be non empty Huntington ComplLattStr ;
cluster CLatt L -> non empty strict Huntington ;
coherence
CLatt L is Huntington
proof end;
end;

registration
let L be non empty join-commutative join-associative Huntington ComplLattStr ;
cluster CLatt L -> non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing lower-bounded upper-bounded strict Huntington join-idempotent ;
coherence
CLatt L is lower-bounded
proof end;
end;

theorem Th36: :: ROBBINS1:36  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative join-associative Huntington ComplLattStr holds Bot L = Bottom (CLatt L)
proof end;

registration
let L be non empty join-commutative join-associative Huntington ComplLattStr ;
cluster CLatt L -> non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing distributive lower-bounded upper-bounded bounded complemented strict Huntington join-idempotent ;
coherence
( CLatt L is complemented & CLatt L is distributive & CLatt L is bounded )
proof end;
end;

notation
let G be non empty ComplLattStr ;
let x be Element of G;
synonym - x for x ` ;
end;

definition
let G be non empty join-commutative ComplLattStr ;
redefine attr G is Huntington means :Def12: :: ROBBINS1:def 12
for x, y being Element of G holds (- ((- x) + (- y))) + (- (x + (- y))) = y;
compatibility
( G is Huntington iff for x, y being Element of G holds (- ((- x) + (- y))) + (- (x + (- y))) = y )
proof end;
end;

:: deftheorem Def12 defines Huntington ROBBINS1:def 12 :
for G being non empty join-commutative ComplLattStr holds
( G is Huntington iff for x, y being Element of G holds (- ((- x) + (- y))) + (- (x + (- y))) = y );

definition
let G be non empty ComplLattStr ;
attr G is with_idempotent_element means :Def13: :: ROBBINS1:def 13
ex x being Element of G st x + x = x;
correctness
;
end;

:: deftheorem Def13 defines with_idempotent_element ROBBINS1:def 13 :
for G being non empty ComplLattStr holds
( G is with_idempotent_element iff ex x being Element of G st x + x = x );

definition
let G be non empty ComplLattStr ;
let x, y be Element of G;
func \delta x,y -> Element of G equals :: ROBBINS1:def 14
- ((- x) + y);
coherence
- ((- x) + y) is Element of G
;
end;

:: deftheorem defines \delta ROBBINS1:def 14 :
for G being non empty ComplLattStr
for x, y being Element of G holds \delta x,y = - ((- x) + y);

definition
let G be non empty ComplLattStr ;
let x, y be Element of G;
func Expand x,y -> Element of G equals :: ROBBINS1:def 15
\delta (x + y),(\delta x,y);
coherence
\delta (x + y),(\delta x,y) is Element of G
;
end;

:: deftheorem defines Expand ROBBINS1:def 15 :
for G being non empty ComplLattStr
for x, y being Element of G holds Expand x,y = \delta (x + y),(\delta x,y);

definition
let G be non empty ComplLattStr ;
let x be Element of G;
func x _0 -> Element of G equals :: ROBBINS1:def 16
- ((- x) + x);
coherence
- ((- x) + x) is Element of G
;
func Double x -> Element of G equals :: ROBBINS1:def 17
x + x;
coherence
x + x is Element of G
;
end;

:: deftheorem defines _0 ROBBINS1:def 16 :
for G being non empty ComplLattStr
for x being Element of G holds x _0 = - ((- x) + x);

:: deftheorem defines Double ROBBINS1:def 17 :
for G being non empty ComplLattStr
for x being Element of G holds Double x = x + x;

definition
let G be non empty ComplLattStr ;
let x be Element of G;
func x _1 -> Element of G equals :: ROBBINS1:def 18
(x _0 ) + x;
coherence
(x _0 ) + x is Element of G
;
func x _2 -> Element of G equals :: ROBBINS1:def 19
(x _0 ) + (Double x);
coherence
(x _0 ) + (Double x) is Element of G
;
func x _3 -> Element of G equals :: ROBBINS1:def 20
(x _0 ) + ((Double x) + x);
coherence
(x _0 ) + ((Double x) + x) is Element of G
;
func x _4 -> Element of G equals :: ROBBINS1:def 21
(x _0 ) + ((Double x) + (Double x));
coherence
(x _0 ) + ((Double x) + (Double x)) is Element of G
;
end;

:: deftheorem defines _1 ROBBINS1:def 18 :
for G being non empty ComplLattStr
for x being Element of G holds x _1 = (x _0 ) + x;

:: deftheorem defines _2 ROBBINS1:def 19 :
for G being non empty ComplLattStr
for x being Element of G holds x _2 = (x _0 ) + (Double x);

:: deftheorem defines _3 ROBBINS1:def 20 :
for G being non empty ComplLattStr
for x being Element of G holds x _3 = (x _0 ) + ((Double x) + x);

:: deftheorem defines _4 ROBBINS1:def 21 :
for G being non empty ComplLattStr
for x being Element of G holds x _4 = (x _0 ) + ((Double x) + (Double x));

theorem Th37: :: ROBBINS1:37  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x, y being Element of G holds \delta (x + y),(\delta x,y) = y
proof end;

theorem Th38: :: ROBBINS1:38  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x, y being Element of G holds Expand x,y = y by Th37;

theorem :: ROBBINS1:39  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x, y, z being Element of G holds \delta ((- x) + y),z = - ((\delta x,y) + z) ;

theorem :: ROBBINS1:40  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x being Element of G holds \delta x,x = x _0 ;

theorem Th41: :: ROBBINS1:41  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x being Element of G holds \delta (Double x),(x _0 ) = x
proof end;

theorem Th42: :: ROBBINS1:42  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x being Element of G holds \delta (x _2 ),x = x _0
proof end;

theorem Th43: :: ROBBINS1:43  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x being Element of G holds (x _2 ) + x = x _3 by LATTICES:def 5;

theorem Th44: :: ROBBINS1:44  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x being Element of G holds (x _4 ) + (x _0 ) = (x _3 ) + (x _1 )
proof end;

theorem Th45: :: ROBBINS1:45  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x being Element of G holds (x _3 ) + (x _0 ) = (x _2 ) + (x _1 )
proof end;

theorem Th46: :: ROBBINS1:46  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x being Element of G holds (x _3 ) + x = x _4
proof end;

theorem Th47: :: ROBBINS1:47  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x being Element of G holds \delta (x _3 ),(x _0 ) = x
proof end;

theorem :: ROBBINS1:48  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x, y, z being Element of G st - x = - y holds
\delta x,z = \delta y,z ;

theorem Th49: :: ROBBINS1:49  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x, y being Element of G holds \delta x,(- y) = \delta y,(- x)
proof end;

theorem Th50: :: ROBBINS1:50  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x being Element of G holds \delta (x _3 ),x = x _0
proof end;

theorem Th51: :: ROBBINS1:51  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x being Element of G holds \delta ((x _1 ) + (x _3 )),x = x _0
proof end;

theorem Th52: :: ROBBINS1:52  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x being Element of G holds \delta ((x _1 ) + (x _2 )),x = x _0
proof end;

theorem Th53: :: ROBBINS1:53  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x being Element of G holds \delta ((x _1 ) + (x _3 )),(x _0 ) = x
proof end;

definition
let G be non empty join-commutative join-associative Robbins ComplLattStr ;
let x be Element of G;
func \beta x -> Element of G equals :: ROBBINS1:def 22
((- ((x _1 ) + (x _3 ))) + x) + (- (x _3 ));
coherence
((- ((x _1 ) + (x _3 ))) + x) + (- (x _3 )) is Element of G
;
end;

:: deftheorem defines \beta ROBBINS1:def 22 :
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x being Element of G holds \beta x = ((- ((x _1 ) + (x _3 ))) + x) + (- (x _3 ));

theorem Th54: :: ROBBINS1:54  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x being Element of G holds \delta (\beta x),x = - (x _3 )
proof end;

theorem Th55: :: ROBBINS1:55  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr
for x being Element of G holds \delta (\beta x),x = - ((x _1 ) + (x _3 ))
proof end;

theorem :: ROBBINS1:56  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr ex y, z being Element of G st - (y + z) = - z
proof end;

theorem :: ROBBINS1:57  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr st ( for z being Element of G holds - (- z) = z ) holds
G is Huntington
proof end;

theorem Th58: :: ROBBINS1:58  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr st G is with_idempotent_element holds
G is Huntington
proof end;

registration
cluster TrivComplLat -> non empty join-commutative join-associative upper-bounded trivial strict Robbins Huntington join-idempotent with_idempotent_element ;
coherence
TrivComplLat is with_idempotent_element
proof end;
end;

registration
cluster non empty join-commutative join-associative Robbins with_idempotent_element -> non empty join-commutative join-associative upper-bounded Robbins Huntington join-idempotent ComplLattStr ;
coherence
for b1 being non empty join-commutative join-associative Robbins ComplLattStr st b1 is with_idempotent_element holds
b1 is Huntington
by Th58;
end;

theorem Th59: :: ROBBINS1:59  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr st ex c, d being Element of G st c + d = c holds
G is Huntington
proof end;

theorem Th60: :: ROBBINS1:60  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for G being non empty join-commutative join-associative Robbins ComplLattStr ex y, z being Element of G st y + z = z
proof end;

registration
cluster non empty join-commutative join-associative Robbins -> non empty join-commutative join-associative upper-bounded Huntington join-idempotent ComplLattStr ;
coherence
for b1 being non empty join-commutative join-associative ComplLattStr st b1 is Robbins holds
b1 is Huntington
proof end;
end;

definition
let L be non empty OrthoLattStr ;
attr L is de_Morgan means :Def23: :: ROBBINS1:def 23
for x, y being Element of L holds x "/\" y = ((x ` ) "\/" (y ` )) ` ;
end;

:: deftheorem Def23 defines de_Morgan ROBBINS1:def 23 :
for L being non empty OrthoLattStr holds
( L is de_Morgan iff for x, y being Element of L holds x "/\" y = ((x ` ) "\/" (y ` )) ` );

registration
let L be non empty ComplLattStr ;
cluster CLatt L -> non empty strict de_Morgan ;
coherence
CLatt L is de_Morgan
proof end;
end;

theorem Th61: :: ROBBINS1:61  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being non empty join-commutative meet-commutative well-complemented OrthoLattStr
for x being Element of L holds
( x + (x ` ) = Top L & x "/\" (x ` ) = Bottom L )
proof end;

theorem Th62: :: ROBBINS1:62  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being distributive bounded well-complemented preOrthoLattice holds (Top L) ` = Bottom L
proof end;

registration
cluster TrivOrtLat -> non empty join-commutative join-associative meet-commutative meet-associative meet-absorbing join-absorbing upper-bounded Boolean trivial strict Robbins Huntington join-idempotent well-complemented de_Morgan ;
coherence
TrivOrtLat is de_Morgan
proof end;
end;

registration
cluster upper-bounded Boolean strict Robbins Huntington join-idempotent de_Morgan OrthoLattStr ;
existence
ex b1 being preOrthoLattice st
( b1 is strict & b1 is de_Morgan & b1 is Boolean & b1 is Robbins & b1 is Huntington )
proof end;
end;

Lm1: for L being non empty de_Morgan OrthoLattStr
for a, b being Element of L holds a *' b = a "/\" b
by Def23;

registration
cluster non empty join-commutative join-associative de_Morgan -> non empty meet-commutative OrthoLattStr ;
coherence
for b1 being non empty OrthoLattStr st b1 is join-associative & b1 is join-commutative & b1 is de_Morgan holds
b1 is meet-commutative
proof end;
end;

theorem Th63: :: ROBBINS1:63  Show TPTP formulae Show IDV graph:: Showing IDV graph ... (Click the Palm Tree again to close it) Show TPTP problem
for L being Huntington de_Morgan preOrthoLattice holds Bot L = Bottom L
proof end;

registration
cluster Boolean well-complemented -> upper-bounded Huntington join-idempotent well-complemented OrthoLattStr ;
coherence
for b1 being well-complemented preOrthoLattice st b1 is Boolean holds
b1 is Huntington
proof end;
end;

registration
cluster Huntington de_Morgan -> Boolean de_Morgan OrthoLattStr ;
coherence
for b1 being de_Morgan preOrthoLattice st b1 is Huntington holds
b1 is Boolean
proof end;
end;

registration
cluster Robbins de_Morgan -> Boolean OrthoLattStr ;
coherence
for b1 being preOrthoLattice st b1 is Robbins & b1 is de_Morgan holds
b1 is Boolean
proof end;
cluster Boolean well-complemented -> upper-bounded Robbins Huntington join-idempotent well-complemented OrthoLattStr ;
coherence
for b1 being well-complemented preOrthoLattice st b1 is Boolean holds
b1 is Robbins
proof end;
end;