:: Formalization of Generalized Almost Distributive Lattices
:: by Adam Grabowski
::
:: Received September 26, 2014
:: Copyright (c) 2014-2021 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 STRUCT_0, XBOOLE_0, ZFMISC_1, SUBSET_1, EQREL_1, PBOOLE,
LATTICES, ROBBINS3, LATTAD_1, QUANTAL1, RELAT_1, RELAT_2, PARTFUN1,
TARSKI, ORDERS_1, ORDERS_2, WAYBEL_0, XXREAL_0, BINOP_1, FUNCT_1,
REALSET1, LATTICE4, FILTER_0;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, STRUCT_0, BINOP_1, RELAT_1,
RELAT_2, RELSET_1, PARTFUN1, ORDERS_1, ORDERS_2, LATTICES, ROBBINS3,
WAYBEL_0, REALSET1, FUNCT_1, ENUMSET1;
constructors BINOP_1, STRUCT_0, LATTICES, ROBBINS3, RELAT_1, RELAT_2,
RELSET_1, PARTFUN1, ORDERS_1, ORDERS_2, WAYBEL_0, REALSET1, ENUMSET1;
registrations SUBSET_1, STRUCT_0, LATTICES, RELSET_1, RELAT_1, PARTFUN1,
ORDERS_2;
requirements SUBSET, BOOLE;
begin :: Preliminaries
theorem :: LATTAD_1:1
for L being non empty 1-sorted,
R being total Relation of the carrier of L holds
R is reflexive iff for x being Element of L holds [x,x] in R;
registration
cluster trivial -> distributive for non empty LattStr;
end;
begin :: Almost Distributive Lattices
:: Almost Distributive Lattices satisfy:
:: a "/\" (b "\/" c) = (a "/\" b) "\/" (a "/\" c);
:: (x \/ y) /\ z = (x /\ z) \/ (y /\ z)
:: (x \/ y) /\ y = y
:: (x \/ y) /\ x = x
:: x \/ (x /\ y) = x (meet-Absorbing)
definition let L be non empty LattStr;
attr L is Distributive means
:: LATTAD_1:def 1
for x,y,z being Element of L holds
(x "\/" y) "/\" z = (x "/\" z) "\/" (y "/\" z);
attr L is Meet-absorbing means
:: LATTAD_1:def 2
for x,y being Element of L holds
(x "\/" y) "/\" y = y;
attr L is Meet-Absorbing means
:: LATTAD_1:def 3
for x,y being Element of L holds
(x "\/" y) "/\" x = x;
end;
registration
cluster trivial -> Distributive Meet-absorbing
Meet-Absorbing meet-Absorbing for non empty LattStr;
cluster trivial -> Lattice-like for non empty LattStr;
end;
registration
cluster trivial for Lattice;
end;
registration
cluster Distributive distributive Meet-absorbing
Meet-Absorbing meet-Absorbing for non empty LattStr;
end;
definition
mode AD_Lattice is Distributive distributive Meet-absorbing
Meet-Absorbing meet-Absorbing non empty LattStr;
end;
begin :: Properties of Almost Distributive Lattices
reserve L for AD_Lattice;
reserve x,y,z for Element of L;
theorem :: LATTAD_1:2 :: Lemma 2.3. (1)
x "\/" y = x iff x "/\" y = y;
theorem :: LATTAD_1:3 :: Lemma 2.3. (11)
x "\/" x = x;
theorem :: LATTAD_1:4 :: Lemma 2.3. (11)
x "/\" x = x;
theorem :: LATTAD_1:5 :: Lemma 2.3. (9)
(x "/\" y) "\/" y = y;
theorem :: LATTAD_1:6 :: Lemma 2.3 (2)
x "\/" y = y iff x "/\" y = x;
theorem :: LATTAD_1:7 :: Lemma 2.3. (9)
x "/\" (x "\/" y) = x;
theorem :: LATTAD_1:8 :: Lemma 2.3. (9)
x "\/" (y "/\" x) = x;
theorem :: LATTAD_1:9 :: Lemma 2.3. (10)
x [= x "\/" y & x "/\" y [= y;
theorem :: LATTAD_1:10
x [= y iff x "/\" y = x;
theorem :: LATTAD_1:11
x "/\" (y "/\" x) = y "/\" x;
theorem :: LATTAD_1:12 :: Theorem 1.7. (1) <=> (2)
(x "/\" y) "\/" x = x iff x "/\" (y "\/" x) = x;
theorem :: LATTAD_1:13 :: Theorem 1.7. (3) <=> (4)
(y "/\" x) "\/" y = y iff y "/\" (x "\/" y) = y;
theorem :: LATTAD_1:14 :: Theorem 1.7. (1) => (5)
(x "/\" y) "\/" x = x implies x "/\" y = y "/\" x;
theorem :: LATTAD_1:15 :: Theorem 1.7. (2) => (6)
x "/\" (y "\/" x) = x implies x "\/" y = y "\/" x;
theorem :: LATTAD_1:16 :: Lemma 2.3. (13)
(ex z being Element of L st x [= z & y [= z) implies
x "\/" y = y "\/" x;
theorem :: LATTAD_1:17 :: Lemma 2.3. (3)
x [= y implies x "\/" y = y "\/" x;
begin :: Generalization of Almost Distributive Lattices
:: A Generalized Almost Distributive Lattice satisfies:
::
:: (x /\ y) /\ z = x /\ (y /\ z)
:: x /\ (y \/ z) = (x /\ y) \/ (x /\ z) (distributive)
:: x \/ (y /\ z) = (x \/ y) /\ (x \/ z)
:: x /\ (x \/ y) = x (join-absorbing)
:: (x \/ y) /\ x = x (Meet-Absorbing)
:: (x /\ y) \/ y = y (meet-absorbing)
definition let L be non empty LattStr;
attr L is left-Distributive means
:: LATTAD_1:def 4
for x,y,z being Element of L holds
x "\/" (y "/\" z) = (x "\/" y) "/\" (x "\/" z);
attr L is ADL-absorbing means
:: LATTAD_1:def 5
for x,y being Element of L holds
x "/\" (y "\/" x) = x;
end;
registration
cluster trivial -> meet-associative distributive left-Distributive
Meet-Absorbing for non empty LattStr;
end;
registration
cluster meet-associative distributive left-Distributive
join-absorbing Meet-Absorbing meet-absorbing
for non empty LattStr;
end;
definition
mode GAD_Lattice is meet-associative distributive left-Distributive
join-absorbing Meet-Absorbing meet-absorbing non empty LattStr;
end;
reserve L for GAD_Lattice;
reserve x,y,z for Element of L;
theorem :: LATTAD_1:18 :: Lemma 3.4. (I \/)
x "\/" x = x;
theorem :: LATTAD_1:19 :: Lemma 3.4. (I /\)
x "/\" x = x;
theorem :: LATTAD_1:20 :: Lemma 3.4. (A4)
x "\/" (x "/\" y) = x;
theorem :: LATTAD_1:21 :: Lemma 3.4. (A5)
x "\/" (y "/\" x) = x;
theorem :: LATTAD_1:22 :: Lemma 3.5. (1)
x "/\" y = y implies x "\/" y = x;
theorem :: LATTAD_1:23 :: Lemma 3.5. (2)
x "\/" y = y iff x "/\" y = x;
begin :: Order Properties of the Generated Relation on GADLs
theorem :: LATTAD_1:24
x [= x;
theorem :: LATTAD_1:25
x [= y & y [= z implies x [= z;
definition let L be non empty LattStr;
::: LattRel could be used, if generalized - TODO
func LatOrder L -> Relation equals
:: LATTAD_1:def 6
{ [a,b] where a,b is Element of L : a [= b };
end;
theorem :: LATTAD_1:26
dom LatOrder L = the carrier of L &
rng LatOrder L = the carrier of L &
field LatOrder L = the carrier of L;
definition let L;
redefine func LatOrder L -> Relation of the carrier of L;
end;
registration let L;
cluster LatOrder L -> total for Relation of the carrier of L;
end;
theorem :: LATTAD_1:27
[x,y] in LatOrder L iff x [= y;
definition let L be non empty LattStr;
func ThetaOrder L -> Relation equals
:: LATTAD_1:def 7
{ [a,b] where a,b is Element of L : a "/\" b = b };
end;
theorem :: LATTAD_1:28
dom ThetaOrder L = the carrier of L &
rng ThetaOrder L = the carrier of L &
field ThetaOrder L = the carrier of L;
definition let L;
redefine func ThetaOrder L -> Relation of the carrier of L;
end;
registration let L;
cluster ThetaOrder L -> total for Relation of the carrier of L;
end;
theorem :: LATTAD_1:29
[x,y] in ThetaOrder L iff x "/\" y = y;
registration let L;
cluster LatOrder L -> reflexive;
cluster LatOrder L -> transitive;
end;
registration let L;
cluster ThetaOrder L -> reflexive;
cluster ThetaOrder L -> transitive;
end;
begin :: Formalization of Rao et al.'s paper
theorem :: LATTAD_1:30 :: Lemma 3.6. (1)
x "\/" (x "\/" y) = x "\/" y;
theorem :: LATTAD_1:31 :: Lemma 3.6. (3)
x "/\" (y "/\" x) = y "/\" x;
theorem :: LATTAD_1:32 :: Lemma 3.6. (2)
y "/\" (x "/\" y) = x "/\" y;
definition let L; let a,b be Element of L;
pred ex_lub_of a,b means
:: LATTAD_1:def 8
ex c being Element of L st
a [= c & b [= c &
for x being Element of L st a [= x & b [= x holds c [= x;
pred ex_glb_of a,b means
:: LATTAD_1:def 9
ex c being Element of L st
c [= a & c [= b &
for x being Element of L st x [= a & x [= b holds x [= c;
end;
definition let L; let a,b be Element of L;
assume
ex_lub_of a,b;
func lub (a,b) -> Element of L means
:: LATTAD_1:def 10
a [= it & b [= it &
for x being Element of L st a [= x & b [= x holds it [= x;
end;
definition let L; let a,b be Element of L;
assume
ex_glb_of a,b;
func glb (a,b) -> Element of L means
:: LATTAD_1:def 11
it [= a & it [= b &
for x being Element of L st x [= a & x [= b holds x [= it;
end;
:: Theorem 3.7.
theorem :: LATTAD_1:33 :: Theorem 3.7. (1) <=> (2)
(x "/\" y) "\/" x = x iff x "/\" (y "\/" x) = x;
theorem :: LATTAD_1:34 :: Theorem 3.7. (1) <=> (3)
(x "/\" y) "\/" x = x iff (y "/\" x) "\/" y = y;
theorem :: LATTAD_1:35 :: Theorem 3.7. (1) <=> (4)
(x "/\" y) "\/" x = x iff y "/\" (x "\/" y) = y;
theorem :: LATTAD_1:36 :: Theorem 3.7. (1) <=> (5)
(x "/\" y) "\/" x = x iff x "/\" y = y "/\" x;
theorem :: LATTAD_1:37 :: Theorem 3.7. (1) <=> (6)
(x "/\" y) "\/" x = x iff x "\/" y = y "\/" x;
theorem :: LATTAD_1:38
x [= y iff x "/\" y = x;
:: Theorem 3.8.
theorem :: LATTAD_1:39 :: Theorem 3.8. 1) <=> 2)
x "\/" y = y "\/" x iff y [= x "\/" y;
theorem :: LATTAD_1:40 :: Theorem 3.8. 1) <=> 3)
x "\/" y = y "\/" x iff ex z st x [= z & y [= z;
theorem :: LATTAD_1:41 :: Theorem 3.8. 1) <=> 4)
x "\/" y = y "\/" x iff ex_lub_of x,y & x "\/" y = lub (x,y);
theorem :: LATTAD_1:42 :: Theorem 3.8. 1) <=> 5)
x "\/" y = y "\/" x iff x [= y "\/" x;
theorem :: LATTAD_1:43 :: Theorem 3.8. 1) <=> 6)
x "\/" y = y "\/" x iff ex_lub_of x,y & y "\/" x = lub (x,y);
:: Theorem 3.9.
theorem :: LATTAD_1:44 :: Theorem 3.9. (2) => (?)
x "/\" y [= x implies ex z st z [= x & z [= y;
theorem :: LATTAD_1:45 :: Theorem 3.9. (1) <=> (4)
x "/\" y = y "/\" x iff y "/\" x [= y;
theorem :: LATTAD_1:46 :: Theorem 3.9. (1) <=> (5)
x "/\" y = y "/\" x iff ex_glb_of x,y & y "/\" x = glb (x,y);
theorem :: LATTAD_1:47 :: Theorem 3.9. (1) <=> (2)
x "/\" y = y "/\" x iff x "/\" y [= x;
theorem :: LATTAD_1:48 :: Theorem 3.9. (1) <=> (3)
x "/\" y = y "/\" x iff ex_glb_of x,y & x "/\" y = glb (x,y);
theorem :: LATTAD_1:49 :: Lemma 3.10.
x "/\" y "/\" z = y "/\" x "/\" z;
definition
let L be GAD_Lattice;
func LatRelStr L -> strict RelStr equals
:: LATTAD_1:def 12
RelStr (# the carrier of L, LatOrder L #);
end;
registration
let L be GAD_Lattice;
cluster LatRelStr L -> reflexive transitive;
end;
theorem :: LATTAD_1:50
for a,b being Element of L,
x,y being Element of LatRelStr L st
a = x & b = y holds
x <= y iff a [= b;
theorem :: LATTAD_1:51 :: Theorem 3.11. (4) <=> (1)
L is join-commutative iff L is Lattice-like distributive;
theorem :: LATTAD_1:52 :: Theorem 3.11. (4) <=> (2)
L is join-commutative iff LatRelStr L is directed;
theorem :: LATTAD_1:53 :: Theorem 3.11. (4) <=> (3)
L is join-commutative iff L is ADL-absorbing;
theorem :: LATTAD_1:54 :: Theorem 3.11. (4) <=> (5)
L is join-commutative iff L is meet-commutative;
theorem :: LATTAD_1:55 :: Theorem 3.11. (4) <=> (6)
L is join-commutative iff ThetaOrder L is antisymmetric;
theorem :: LATTAD_1:56 :: Theorem 3.11. (4) <=> (7)
L is join-commutative iff ThetaOrder L is being_partial-order;
registration let L be join-commutative GAD_Lattice;
cluster ThetaOrder L -> antisymmetric;
end;
registration
cluster join-commutative -> ADL-absorbing for GAD_Lattice;
cluster ADL-absorbing -> join-commutative for GAD_Lattice;
end;
registration
cluster join-commutative -> meet-commutative for GAD_Lattice;
cluster meet-commutative -> join-commutative for GAD_Lattice;
end;
:: Theorem 3.13.
theorem :: LATTAD_1:57 :: Theorem 3.13. (3) => (1)
(for a,b,c being Element of L holds
(a "\/" b) "/\" c = (b "\/" a) "/\" c) implies
for a,b,c being Element of L holds
(a "\/" b) "/\" c = (a "/\" c) "\/" (b "/\" c);
theorem :: LATTAD_1:58 :: Theorem 3.13. (1) => (2)
(for a,b,c being Element of L holds
(a "\/" b) "/\" c = (a "/\" c) "\/" (b "/\" c)) implies
for a,b being Element of L holds (a "\/" b) "/\" b = b;
theorem :: LATTAD_1:59 :: Theorem 3.13. (2) => (3)
(for a,b being Element of L holds (a "\/" b) "/\" b = b) implies
for a,b,c being Element of L holds
(a "\/" b) "/\" c = (b "\/" a) "/\" c;
begin :: Generalized Almost Distributive Lattices with Zero
definition let L;
attr L is with_zero means
:: LATTAD_1:def 13
ex x being Element of L st
for a being Element of L holds x "/\" a = x;
end;
registration
cluster trivial -> with_zero for non empty GAD_Lattice;
end;
registration
cluster with_zero for non empty GAD_Lattice;
end;
definition let L; :: Definition 3.14.
assume
L is with_zero;
func bottom L -> Element of L means
:: LATTAD_1:def 14
for a being Element of L holds it "/\" a = it;
end;
reserve L for with_zero GAD_Lattice,
x,y for Element of L;
theorem :: LATTAD_1:60 :: Lemma 3.15. (2)
x "\/" bottom L = x;
theorem :: LATTAD_1:61 :: Lemma 3.15. (3)
bottom L "\/" x = x;
theorem :: LATTAD_1:62 :: Lemma 3.15. (4)
x "/\" bottom L = bottom L;
theorem :: LATTAD_1:63 :: Lemma 3.16.
x "/\" y = bottom L iff y "/\" x = bottom L;
theorem :: LATTAD_1:64 :: Lemma 3.17.
x "/\" y = bottom L implies x "\/" y = y "\/" x;
begin
definition
let x,y be Element of {1,2,3};
func x example32"/\" y -> Element of {1,2,3} equals
:: LATTAD_1:def 15
1 if y = 1 or (y = 2 & (x = 1 or x = 3)),
2 if x = 2 & y = 2,
3 if y = 3;
func x example32"\/" y -> Element of {1,2,3} equals
:: LATTAD_1:def 16
1 if x = 1 & (y = 1 or y = 3),
2 if x = 2 or (x = 1 & y = 2),
3 if x = 3;
end;
definition
func example32\/ -> BinOp of {1,2,3} means
:: LATTAD_1:def 17
for x,y being Element of {1,2,3} holds it.(x,y) = x example32"\/" y;
func example32/\ -> BinOp of {1,2,3} means
:: LATTAD_1:def 18
for x,y being Element of {1,2,3} holds it.(x,y) = x example32"/\" y;
end;
theorem :: LATTAD_1:65 :: Example 3.2
ex L being non empty LattStr st
(for x being Element of L holds x = 1 or x = 2 or x = 3) &
(for x,y being Element of L holds
(x "/\" y = 1 iff y = 1 or (y = 2 & (x = 1 or x = 3))) &
(x "/\" y = 2 iff x = 2 & y = 2) &
(x "/\" y = 3 iff y = 3)) &
(for x,y being Element of L holds
(x "\/" y = 1 iff x = 1 & (y = 1 or y = 3)) &
(x "\/" y = 2 iff x = 2 or (x = 1 & y = 2)) &
(x "\/" y = 3 iff x = 3)) &
L is GAD_Lattice & L is not AD_Lattice;
definition
let x,y be Element of {1,2,3};
func x example33"/\" y -> Element of {1,2,3} equals
:: LATTAD_1:def 19
1 if x = 1 & y = 1,
2 if y = 2 or (y = 1 & (x = 2 or x = 3)),
3 if y = 3;
func x example33"\/" y -> Element of {1,2,3} equals
:: LATTAD_1:def 20
1 if x = 1 or (x = 2 & y = 1),
2 if x = 2 & (y = 2 or y = 3),
3 if x = 3;
end;
definition
func example33\/ -> BinOp of {1,2,3} means
:: LATTAD_1:def 21
for x,y being Element of {1,2,3} holds it.(x,y) = x example33"\/" y;
func example33/\ -> BinOp of {1,2,3} means
:: LATTAD_1:def 22
for x,y being Element of {1,2,3} holds it.(x,y) = x example33"/\" y;
end;
theorem :: LATTAD_1:66 :: Example 3.3
ex L being non empty LattStr st
(for x being Element of L holds x = 1 or x = 2 or x = 3) &
(for x,y being Element of L holds
(x "/\" y = 1 iff x = 1 & y = 1) &
(x "/\" y = 2 iff y = 2 or (y = 1 & (x = 2 or x = 3))) &
(x "/\" y = 3 iff y = 3)) &
(for x,y being Element of L holds
(x "\/" y = 1 iff x = 1 or (x = 2 & y = 1)) &
(x "\/" y = 2 iff x = 2 & (y = 2 or y = 3)) &
(x "\/" y = 3 iff x = 3)) &
L is GAD_Lattice;
:: like NAT_LAT ::
definition
let L be non empty LattStr;
mode SubLattStr of L -> LattStr means
:: LATTAD_1:def 23
the carrier of it c= the carrier of L &
the L_join of it = (the L_join of L)||the carrier of it &
the L_meet of it = (the L_meet of L)||the carrier of it;
end;
registration
let L be non empty LattStr;
cluster strict for SubLattStr of L;
end;
definition
::$CD 2
end;
registration
let L be non empty LattStr;
cluster meet-closed join-closed non empty for Subset of L;
end;
definition
let L be non empty LattStr;
mode ClosedSubset of L is meet-closed join-closed Subset of L;
end;
definition
let L be non empty LattStr;
let P be ClosedSubset of L;
func latt(L,P) -> strict SubLattStr of L means
:: LATTAD_1:def 26
the carrier of it = P;
end;
registration
let L be non empty LattStr;
let S be non empty ClosedSubset of L;
cluster latt(L,S) -> non empty;
end;
registration
let L be non empty LattStr;
cluster non empty for SubLattStr of L;
end;
theorem :: LATTAD_1:67
for L be non empty LattStr, S be non empty SubLattStr of L,
x1,x2 be Element of L, y1,y2 be Element of S
st x1 = y1 & x2 = y2 holds x1 "\/" x2 = y1 "\/" y2;
theorem :: LATTAD_1:68
for L be non empty LattStr, S be non empty SubLattStr of L,
x1,x2 be Element of L, y1,y2 be Element of S
st x1 = y1 & x2 = y2 holds x1 "/\" x2 = y1 "/\" y2;
theorem :: LATTAD_1:69
for L being non empty LattStr, S being non empty ClosedSubset of L holds
(L is meet-associative implies latt(L,S) is meet-associative) &
(L is meet-absorbing implies latt(L,S) is meet-absorbing) &
(L is meet-commutative implies latt(L,S) is meet-commutative) &
(L is join-associative implies latt(L,S) is join-associative) &
(L is join-absorbing implies latt(L,S) is join-absorbing) &
(L is join-commutative implies latt(L,S) is join-commutative) &
(L is Meet-Absorbing implies latt(L,S) is Meet-Absorbing) &
(L is distributive implies latt(L,S) is distributive) &
(L is left-Distributive implies latt(L,S) is left-Distributive);
theorem :: LATTAD_1:70 :: Corollary 3.12
for a being Element of L, X being set st
X = the set of all x "/\" a where x is Element of L
holds
X = {x where x is Element of L: x [= a} & X is ClosedSubset of L;
theorem :: LATTAD_1:71 :: Corollary 3.12
for a being Element of L, S being non empty ClosedSubset of L,
b being Element of latt(L,S) st b = a &
S = the set of all x "/\" a where x is Element of L
holds latt(L,S) is Lattice-like distributive &
(for c being Element of latt(L,S) holds b "\/" c = b & c "\/" b = b & c [= b)
;