:: Formalization of Ortholattices via Orthoposets
:: by Adam Grabowski and Markus Moschner
::
:: Received December 28, 2004
:: Copyright (c) 2004-2018 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies XBOOLE_0, LATTICES, SUBSET_1, EQREL_1, SHEFFER1, ROBBINS1,
OPOSET_1, QMAX_1, FUNCT_1, WAYBEL_0, XXREAL_0, LATTICE3, STRUCT_0,
ORDERS_2, BINOP_1, RELAT_1, FUNCT_5, ZFMISC_1, RELAT_2, FILTER_1,
ORDERS_1, PBOOLE, TARSKI, YELLOW_0, WAYBEL_1, ROBBINS3, CARD_1;
notations TARSKI, XBOOLE_0, ZFMISC_1, BINOP_1, RELAT_1, RELSET_1, PARTFUN1,
RELAT_2, FUNCT_2, FUNCT_5, ORDINAL1, CARD_1, STRUCT_0, LATTICE3,
LATTICES, ORDERS_1, ORDERS_2, FILTER_1, ROBBINS1, QMAX_1, OPOSET_1,
WAYBEL_0, WAYBEL_1, YELLOW_0, SHEFFER1, PARTIT_2;
constructors BINOP_1, REALSET2, LATTICE3, WAYBEL_1, YELLOW_6, OPOSET_1,
SHEFFER1, FUNCT_5, RELSET_1, PARTIT_2;
registrations RELAT_1, FUNCT_1, PARTFUN1, FUNCT_2, STRUCT_0, LATTICES,
YELLOW_0, YELLOW_1, ROBBINS1, OPOSET_1, SHEFFER1, PARTIT_2, CARD_1;
requirements SUBSET, BOOLE, NUMERALS;
begin :: Another short axiomatization of lattices
:: Originally proved by McCune with the help of OTTER
definition
let L be non empty \/-SemiLattStr;
attr L is join-Associative means
:: ROBBINS3:def 1
for x, y, z being Element of L holds x "\/" (y "\/" z) = y "\/" (x "\/" z);
end;
definition
let L be non empty /\-SemiLattStr;
attr L is meet-Associative means
:: ROBBINS3:def 2
for x, y, z being Element of L holds x "/\" (y "/\" z) = y "/\" (x "/\" z);
end;
definition
let L be non empty LattStr;
attr L is meet-Absorbing means
:: ROBBINS3:def 3
for x, y being Element of L holds x "\/" (x "/\" y) = x;
end;
theorem :: ROBBINS3:1
for L being non empty LattStr holds L is meet-Associative
join-Associative meet-Absorbing join-absorbing implies L is meet-idempotent
join-idempotent;
theorem :: ROBBINS3:2
for L being non empty LattStr holds L is meet-Associative
join-Associative meet-Absorbing join-absorbing implies L is meet-commutative
join-commutative;
theorem :: ROBBINS3:3
for L being non empty LattStr holds L is meet-Associative
join-Associative meet-Absorbing join-absorbing implies L is meet-absorbing;
theorem :: ROBBINS3:4
for L being non empty LattStr holds L is meet-Associative
join-Associative meet-Absorbing join-absorbing implies L is meet-associative
join-associative;
theorem :: ROBBINS3:5
for L being non empty LattStr holds L is Lattice-like iff L is
meet-Associative join-Associative meet-Absorbing join-absorbing;
registration
cluster Lattice-like -> meet-Associative join-Associative meet-Absorbing for
non empty LattStr;
cluster meet-Associative join-Associative meet-Absorbing join-absorbing ->
Lattice-like for non empty LattStr;
end;
begin :: Orthoposets
registration
cluster OrderInvolutive -> Dneg for PartialOrdered non empty OrthoRelStr;
end;
theorem :: ROBBINS3:6
for L being Dneg non empty OrthoRelStr, x being Element of L holds x`` = x;
theorem :: ROBBINS3:7
for O being OrderInvolutive PartialOrdered non empty OrthoRelStr
, x, y being Element of O holds x <= y implies y` <= x`;
registration
cluster with_infima with_suprema strict for PreOrthoPoset;
end;
notation
let L be non empty \/-SemiLattStr, x, y be Element of L;
synonym x |_| y for x "\/" y;
end;
notation
let L be non empty /\-SemiLattStr, x, y be Element of L;
synonym x |^| y for x "/\" y;
end;
notation
let L be non empty RelStr, x, y be Element of L;
synonym x "|^|" y for x "/\" y;
synonym x "|_|" y for x "\/" y;
end;
begin :: Merging relational structures and lattice structures together
definition
struct (\/-SemiLattStr, RelStr) \/-SemiLattRelStr (# carrier -> set, L_join
-> (BinOp of the carrier), InternalRel -> Relation of the carrier #);
end;
definition
struct (/\-SemiLattStr, RelStr) /\-SemiLattRelStr (# carrier -> set, L_meet
-> (BinOp of the carrier), InternalRel -> Relation of the carrier #);
end;
definition
struct (/\-SemiLattRelStr, \/-SemiLattRelStr, LattStr) LattRelStr (# carrier
-> set, L_join, L_meet -> (BinOp of the carrier), InternalRel -> Relation of
the carrier #);
end;
definition
func TrivLattRelStr -> LattRelStr equals
:: ROBBINS3:def 4
LattRelStr (# {0}, op2, op2, id{0} #);
end;
registration
cluster TrivLattRelStr -> 1-element;
end;
registration
cluster non empty for \/-SemiLattRelStr;
cluster non empty for /\-SemiLattRelStr;
cluster non empty for LattRelStr;
end;
theorem :: ROBBINS3:8
for R being non empty RelStr st the InternalRel of R is_reflexive_in
the carrier of R & the InternalRel of R is antisymmetric transitive holds R is
reflexive antisymmetric transitive;
registration
cluster TrivLattRelStr -> reflexive;
end;
registration
cluster antisymmetric reflexive transitive with_suprema with_infima
for LattRelStr;
end;
registration
cluster TrivLattRelStr -> meet-Absorbing;
end;
registration
cluster Lattice-like for non empty LattRelStr;
end;
definition
let L be Lattice;
redefine func LattRel L -> Order of the carrier of L;
end;
begin :: Binary approach to ortholattices
definition
struct (LattRelStr, OrthoLattStr, OrthoRelStr) OrthoLattRelStr (# carrier ->
set, L_join, L_meet -> (BinOp of the carrier), InternalRel -> (Relation of the
carrier), Compl -> UnOp of the carrier #);
end;
definition
func TrivCLRelStr -> OrthoLattRelStr equals
:: ROBBINS3:def 5
OrthoLattRelStr (# {0}, op2, op2, id {0}, op1 #);
end;
:: Axiomatics for ortholattices is the classical one for lattices extended
:: by the three following:
:: x ^ y = c(c(x) v c(y)). % DM de_Morgan from ROBBINS1
:: c(c(x)) = x. % CC involutive from OPOSET_1, too specific
:: x v c(x) = y v c(y). % ONE
definition
let L be non empty ComplStr;
attr L is involutive means
:: ROBBINS3:def 6
for x being Element of L holds x`` = x;
end;
definition
let L be non empty ComplLLattStr;
attr L is with_Top means
:: ROBBINS3:def 7
for x, y being Element of L holds x |_| x` = y |_| y`;
end;
registration
cluster TrivOrtLat -> involutive with_Top;
end;
registration
cluster TrivCLRelStr -> 1-element;
end;
registration
cluster TrivCLRelStr -> reflexive;
end;
registration
cluster TrivCLRelStr -> involutive with_Top;
end;
registration
cluster involutive with_Top de_Morgan Lattice-like for
1-element OrthoLattStr;
end;
definition
mode Ortholattice is involutive with_Top de_Morgan Lattice-like non empty
OrthoLattStr;
end;
begin :: Lemmas
theorem :: ROBBINS3:9
for K, L being non empty LattStr st the LattStr of K = the
LattStr of L & K is join-commutative holds L is join-commutative;
theorem :: ROBBINS3:10
for K, L being non empty LattStr st the LattStr of K = the
LattStr of L & K is meet-commutative holds L is meet-commutative;
theorem :: ROBBINS3:11
for K, L being non empty LattStr st the LattStr of K = the
LattStr of L & K is join-associative holds L is join-associative;
theorem :: ROBBINS3:12
for K, L being non empty LattStr st the LattStr of K = the
LattStr of L & K is meet-associative holds L is meet-associative;
theorem :: ROBBINS3:13
for K, L being non empty LattStr st the LattStr of K = the
LattStr of L & K is join-absorbing holds L is join-absorbing;
theorem :: ROBBINS3:14
for K, L being non empty LattStr st the LattStr of K = the
LattStr of L & K is meet-absorbing holds L is meet-absorbing;
theorem :: ROBBINS3:15
for K, L being non empty LattStr st the LattStr of K = the LattStr of
L & K is Lattice-like holds L is Lattice-like;
theorem :: ROBBINS3:16
for L1,L2 being non empty \/-SemiLattStr st the \/-SemiLattStr of L1 =
the \/-SemiLattStr of L2 for a1,b1 being Element of L1, a2,b2 being Element of
L2 st a1 = a2 & b1 = b2 holds a1 "\/" b1 = a2 "\/" b2;
theorem :: ROBBINS3:17
for L1,L2 being non empty /\-SemiLattStr st the /\-SemiLattStr of L1 =
the /\-SemiLattStr of L2 for a1,b1 being Element of L1, a2,b2 being Element of
L2 st a1 = a2 & b1 = b2 holds a1 "/\" b1 = a2 "/\" b2;
theorem :: ROBBINS3:18
for K, L being non empty ComplStr, x being Element of K, y being
Element of L st the Compl of K = the Compl of L & x = y holds x` = y`;
theorem :: ROBBINS3:19
for K, L being non empty ComplLLattStr st the ComplLLattStr of K =
the ComplLLattStr of L & K is with_Top holds L is with_Top;
theorem :: ROBBINS3:20
for K, L being non empty OrthoLattStr st the OrthoLattStr of K =
the OrthoLattStr of L & K is de_Morgan holds L is de_Morgan;
theorem :: ROBBINS3:21
for K, L being non empty OrthoLattStr st the OrthoLattStr of K =
the OrthoLattStr of L & K is involutive holds L is involutive;
begin :: Structure Extensions
definition
let R be RelStr;
mode RelAugmentation of R -> LattRelStr means
:: ROBBINS3:def 8
the RelStr of it = the RelStr of R;
end;
definition
let R be LattStr;
mode LatAugmentation of R -> LattRelStr means
:: ROBBINS3:def 9
the LattStr of it = the LattStr of R;
end;
registration
let L be non empty LattStr;
cluster -> non empty for LatAugmentation of L;
end;
registration
let L be meet-associative non empty LattStr;
cluster -> meet-associative for LatAugmentation of L;
end;
registration
let L be join-associative non empty LattStr;
cluster -> join-associative for LatAugmentation of L;
end;
registration
let L be meet-commutative non empty LattStr;
cluster -> meet-commutative for LatAugmentation of L;
end;
registration
let L be join-commutative non empty LattStr;
cluster -> join-commutative for LatAugmentation of L;
end;
registration
let L be join-absorbing non empty LattStr;
cluster -> join-absorbing for LatAugmentation of L;
end;
registration
let L be meet-absorbing non empty LattStr;
cluster -> meet-absorbing for LatAugmentation of L;
end;
definition
let L be non empty \/-SemiLattRelStr;
attr L is naturally_sup-generated means
:: ROBBINS3:def 10
for x, y being Element of L holds x <= y iff x |_| y = y;
end;
definition
let L be non empty /\-SemiLattRelStr;
attr L is naturally_inf-generated means
:: ROBBINS3:def 11
for x, y being Element of L holds x <= y iff x |^| y = x;
end;
registration
let L be Lattice;
cluster naturally_sup-generated naturally_inf-generated Lattice-like
for LatAugmentation of L;
end;
registration
cluster 1-element reflexive for LattRelStr;
end;
registration
cluster 1-element reflexive for OrthoLattRelStr;
end;
registration
cluster 1-element reflexive for OrthoRelStr;
end;
registration
cluster -> involutive with_Top de_Morgan well-complemented
for 1-element OrthoLattStr;
end;
registration
cluster -> OrderInvolutive Pure PartialOrdered
for 1-element reflexive OrthoRelStr;
end;
registration
cluster -> naturally_sup-generated naturally_inf-generated
for 1-element reflexive LattRelStr;
end;
registration
cluster with_infima with_suprema naturally_sup-generated
naturally_inf-generated de_Morgan Lattice-like OrderInvolutive Pure
PartialOrdered for non empty OrthoLattRelStr;
end;
registration
cluster with_infima with_suprema naturally_sup-generated
naturally_inf-generated Lattice-like for non empty LattRelStr;
end;
theorem :: ROBBINS3:22
for L being naturally_sup-generated non empty LattRelStr, x, y
being Element of L holds x <= y iff x [= y;
theorem :: ROBBINS3:23
for L being naturally_sup-generated Lattice-like non empty
LattRelStr holds the RelStr of L = LattPOSet L;
registration
cluster naturally_sup-generated Lattice-like -> with_infima with_suprema for
non empty LattRelStr;
end;
begin :: Extending OrthoLattStr
definition
let R be OrthoLattStr;
mode CLatAugmentation of R -> OrthoLattRelStr means
:: ROBBINS3:def 12
the OrthoLattStr of it = the OrthoLattStr of R;
end;
registration
let L be non empty OrthoLattStr;
cluster -> non empty for CLatAugmentation of L;
end;
registration
let L be meet-associative non empty OrthoLattStr;
cluster -> meet-associative for CLatAugmentation of L;
end;
registration
let L be join-associative non empty OrthoLattStr;
cluster -> join-associative for CLatAugmentation of L;
end;
registration
let L be meet-commutative non empty OrthoLattStr;
cluster -> meet-commutative for CLatAugmentation of L;
end;
registration
let L be join-commutative non empty OrthoLattStr;
cluster -> join-commutative for CLatAugmentation of L;
end;
registration
let L be meet-absorbing non empty OrthoLattStr;
cluster -> meet-absorbing for CLatAugmentation of L;
end;
registration
let L be join-absorbing non empty OrthoLattStr;
cluster -> join-absorbing for CLatAugmentation of L;
end;
registration
let L be with_Top non empty OrthoLattStr;
cluster -> with_Top for CLatAugmentation of L;
end;
registration
let L be non empty Ortholattice;
cluster naturally_sup-generated naturally_inf-generated Lattice-like
for CLatAugmentation of L;
end;
registration
cluster involutive with_Top de_Morgan Lattice-like naturally_sup-generated
well-complemented for non empty OrthoLattRelStr;
end;
theorem :: ROBBINS3:24
for L being with_infima with_suprema PartialOrdered non empty
OrthoRelStr for x,y being Element of L holds x <= y implies y = x "|_|" y & x
= x "|^|" y;
definition
let L be meet-commutative non empty /\-SemiLattStr, a, b be Element of L;
redefine func a |^| b;
commutativity;
end;
definition
let L be join-commutative non empty \/-SemiLattStr, a, b be Element of L;
redefine func a |_| b;
commutativity;
end;
registration
cluster meet-absorbing join-absorbing meet-commutative
naturally_sup-generated -> reflexive for non empty LattRelStr;
end;
registration
cluster join-associative naturally_sup-generated -> transitive for non empty
LattRelStr;
end;
registration
cluster join-commutative naturally_sup-generated -> antisymmetric for
non empty
LattRelStr;
end;
theorem :: ROBBINS3:25
for L being with_infima with_suprema naturally_sup-generated
Lattice-like non empty OrthoLattRelStr, x, y being Element of L holds x "|_|"
y = x |_| y;
theorem :: ROBBINS3:26
for L being with_infima with_suprema naturally_sup-generated
Lattice-like non empty OrthoLattRelStr, x, y being Element of L holds x "|^|"
y = x |^| y;
theorem :: ROBBINS3:27
for L being with_infima with_suprema naturally_sup-generated
naturally_inf-generated Lattice-like OrderInvolutive PartialOrdered non empty
OrthoLattRelStr holds L is de_Morgan;
registration
let L be Ortholattice;
cluster -> involutive for CLatAugmentation of L;
end;
registration
let L be Ortholattice;
cluster -> de_Morgan for CLatAugmentation of L;
end;
theorem :: ROBBINS3:28
for L being non empty OrthoLattRelStr st L is involutive
with_Top de_Morgan Lattice-like naturally_sup-generated holds L is
Orthocomplemented PartialOrdered;
theorem :: ROBBINS3:29
for L being Ortholattice, E being naturally_sup-generated
CLatAugmentation of L holds E is Orthocomplemented;
registration
let L be Ortholattice;
cluster -> Orthocomplemented for
naturally_sup-generated CLatAugmentation of L;
end;
theorem :: ROBBINS3:30
for L being non empty OrthoLattStr st L is Boolean
well-complemented Lattice-like holds L is Ortholattice;
registration
cluster Boolean well-complemented Lattice-like -> involutive with_Top
de_Morgan for non empty OrthoLattStr;
end;