:: Introduction to Formal Preference Spaces
:: by Eliza Niewiadomska and Adam Grabowski
::
:: Received October 7, 2013
:: Copyright (c) 2013-2018 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 PREFER_1, PCS_0, ORDERS_1, WELLORD1, WAYBEL_4, CARD_1, SUBSET_1,
RELAT_1, XBOOLE_0, PARTFUN1, RELAT_2, STRUCT_0, ZFMISC_1, TARSKI,
ORDERS_2, EQREL_1;
notations TARSKI, XBOOLE_0, XTUPLE_0, SUBSET_1, ZFMISC_1, MCART_1, DOMAIN_1,
CARD_1, NAT_1, RELAT_1, FUNCT_1, PARTFUN1, RELSET_1, RELAT_2, EQREL_1,
CARD_3, PARTIT_2, FUNCT_4, ORDINAL1, NUMBERS, FUNCOP_1, WELLORD1,
STRUCT_0, ORDERS_1, ORDERS_2, YELLOW_3, ENUMSET1, PCS_0, XCMPLX_0,
FUNCT_2, VALUED_0;
constructors PRALG_1, PARTIT_2, TSEP_1, YELLOW16, YELLOW_3, DOMAIN_1,
RELSET_1, FUNCT_4, AFINSQ_1, XTUPLE_0, NUMBERS, PARTFUN2, NAT_1,
ORDERS_1, PCS_0, WELLORD1, WELLORD2, FUNCOP_1, XXREAL_0, XREAL_0, CARD_1,
XCMPLX_0, INT_1, VALUED_0;
registrations EQREL_1, PARTFUN1, SUBSET_1, XBOOLE_0, WAYBEL_3, RELAT_1,
ORDERS_1, WELLORD1, ORDERS_2, STRUCT_0, YELLOW16, YELLOW_3, RELSET_1,
FUNCOP_1, RELAT_2, XTUPLE_0, ORDINAL1, PARTIT_2, TOLER_1, ZFMISC_1,
NAT_1, FOMODEL3, REWRITE2, PRE_POLY, FUNCT_1, FINSET_1, XXREAL_0,
XREAL_0, CARD_1, NUMBERS, VALUED_0;
requirements BOOLE, SUBSET, NUMERALS, REAL, ARITHM;
begin :: Preliminaries
definition let X, Y, Z be set;
pred X, Y, Z are_mutually_disjoint means
:: PREFER_1:def 1
X misses Y & Y misses Z & X misses Z;
end;
theorem :: PREFER_1:1
for A being set holds
{}, A, {} are_mutually_disjoint;
registration
cluster 2-element -> non empty for set;
end;
theorem :: PREFER_1:2
for a, b being set st a <> b holds
{[a, a], [b, b]} <> {[a, a], [a, b], [b, a], [b, b]};
theorem :: PREFER_1:3
for A being 2-element set, a, b being Element of A st
a <> b holds A = {a, b};
theorem :: PREFER_1:4
for A being 2-element set holds
ex a, b being Element of A st a <> b & A = {a, b};
theorem :: PREFER_1:5
for A being non trivial set holds
ex a, b being Element of A st a <> b;
theorem :: PREFER_1:6
for x1, x2, x3, x4 being set holds
{x1} \/ {x2} \/ {x3, x4} = {x3, x1, x2, x4};
theorem :: PREFER_1:7
for a, b being set st a <> b holds
{[a, a], [b, b]} misses {[a, b], [b, a]};
theorem :: PREFER_1:8
for A being 2-element set, a, b being Element of A st a <> b holds
id A = {[a, a], [b, b]};
theorem :: PREFER_1:9
for a, b being object,
R being Relation st R = {[a, b]} holds
R~ = {[b, a]};
theorem :: PREFER_1:10
for a, b being set holds
a <> b iff {[a, b]} misses {[a, a], [b, b]};
theorem :: PREFER_1:11
for X being non empty set,
R being (Relation of X),
x,y being Element of X holds
not [x,y] in R` implies [x,y] in R;
theorem :: PREFER_1:12
for X being non empty set,
R being Relation of X holds
R /\ (R~)`, R /\ R~, R` /\ (R~)` are_mutually_disjoint;
theorem :: PREFER_1:13
for P,R being Relation st P misses R holds
P~ misses R~;
theorem :: PREFER_1:14
for X being non empty set,
R being Relation of X holds
R = R~`~`;
theorem :: PREFER_1:15
for X being non empty set,
R being Relation of X holds
R~ = R`~`;
theorem :: PREFER_1:16
for X being non empty set,
R being Relation of X holds
R~`~ = R`;
begin :: Properties of Binary Relations
registration let X be set;
cluster connected being_linear-order for Order of X;
end;
theorem :: PREFER_1:17
for X being non empty set,
R being total reflexive Relation of X holds
R~ is total;
theorem :: PREFER_1:18
for X being non empty set,
R being total Relation of X holds
field R = X;
theorem :: PREFER_1:19
for R being Relation holds
R is irreflexive iff
for x being object st x in field R holds not [x, x] in R;
theorem :: PREFER_1:20
for R being Relation holds
R is symmetric iff
for x, y being object st [x,y] in R holds [y,x] in R;
theorem :: PREFER_1:21
for X being set, R being Relation of X holds
R /\ R~ is symmetric;
theorem :: PREFER_1:22
for R being Relation holds
R is asymmetric iff
for x, y being object st [x,y] in R holds not [y,x] in R;
theorem :: PREFER_1:23
for a, b being object st a <> b holds
{[a, b]} is asymmetric;
theorem :: PREFER_1:24
for X being non empty set,
R being Relation of X holds
R /\ (R~)` is asymmetric;
theorem :: PREFER_1:25
for X being non empty set,
R being total reflexive Relation of X holds
R /\ R~ is reflexive;
theorem :: PREFER_1:26
for X being non empty set,
R being total reflexive Relation of X holds
R /\ R~ is total;
theorem :: PREFER_1:27
for a, b being object st a <> b holds
{[a, b], [b, a]} is irreflexive symmetric;
theorem :: PREFER_1:28
for X being non empty set,
R being total Relation of X holds
for S being Relation of X holds
R \/ S is total;
theorem :: PREFER_1:29
for X being non empty set,
R being total reflexive Relation of X holds
R` /\ R~` is irreflexive symmetric;
theorem :: PREFER_1:30
for X being set, R being Relation of X st R is symmetric holds
R` is symmetric;
theorem :: PREFER_1:31
for X being object, R being Relation holds
R is antisymmetric iff
for x, y being object st [x,y] in R & [y,x] in R holds x = y;
theorem :: PREFER_1:32
for A being set,
R being asymmetric Relation of A holds
R \/ id A is antisymmetric;
theorem :: PREFER_1:33
for X being object, R being Relation holds
R is connected iff
for x, y being object st x <> y & x in field R & y in field R
holds [x,y] in R or [y,x] in R;
theorem :: PREFER_1:34
for R being Relation holds
R is connected iff [:field R,field R:] = R \/ R~ \/ id (field R);
theorem :: PREFER_1:35
for A being set,
R being asymmetric Relation of A holds
R misses R~;
theorem :: PREFER_1:36
for R,P being Relation st R misses P & P is symmetric holds
R~ misses P;
theorem :: PREFER_1:37
for X being set, R being asymmetric Relation of X holds
R misses id X;
theorem :: PREFER_1:38
for X being set, R being asymmetric Relation of X holds
R * R misses id X;
definition let X be set, R be Relation of X;
func SymCl R -> Relation of X equals
:: PREFER_1:def 2
R \/ R~;
end;
registration let X be set, R be total Relation of X;
cluster SymCl R -> total;
end;
registration let X be set,
R be Relation of X;
cluster SymCl R -> symmetric;
end;
begin :: Preference Structures
definition
struct (1-sorted) PrefStr
(# carrier -> set,
PrefRel -> Relation of the carrier
#);
end;
definition
struct (PrefStr, TolStr) PIStr
(# carrier -> set,
PrefRel,
ToleranceRel -> Relation of the carrier
#);
end;
definition
struct (PIStr, RelStr, PrefStr) PreferenceStr
(# carrier -> set,
PrefRel,
ToleranceRel,
InternalRel -> (Relation of the carrier)
#);
end;
registration
cluster non empty strict for PIStr;
cluster empty strict for PIStr;
end;
registration
cluster non empty strict for PrefStr;
cluster empty strict for PrefStr;
cluster non empty strict for PIStr;
cluster non empty strict for PreferenceStr;
end;
definition let X be PreferenceStr;
attr X is preference-like means
:: PREFER_1:def 3
the PrefRel of X is asymmetric &
the ToleranceRel of X is Tolerance of the carrier of X &
the InternalRel of X is irreflexive symmetric &
the PrefRel of X, the ToleranceRel of X,
the InternalRel of X are_mutually_disjoint &
(the PrefRel of X) \/ (the PrefRel of X)~ \/
(the ToleranceRel of X) \/
(the InternalRel of X) = nabla the carrier of X;
end;
definition let X be set;
func PrefSpace X -> strict PreferenceStr equals
:: PREFER_1:def 4
PreferenceStr (# X, {}(X,X), nabla X, {}(X,X) #);
end;
registration
let A be non empty set;
cluster PrefSpace A -> non empty preference-like;
end;
registration
cluster non empty strict preference-like for PreferenceStr;
end;
definition
mode PreferenceSpace is preference-like PreferenceStr;
end;
registration
cluster empty -> preference-like for PreferenceStr;
end;
registration
cluster PrefSpace {} -> empty preference-like;
end;
registration
cluster empty for PreferenceSpace;
end;
registration
let A be trivial non empty set;
cluster PrefSpace A -> trivial;
end;
registration
let A be trivial non empty set;
cluster PrefSpace A -> non empty preference-like;
end;
begin :: Constructing Examples
definition
let A be set;
func IdPrefSpace A -> strict PreferenceStr means
:: PREFER_1:def 5
the carrier of it = A &
the PrefRel of it = {} &
the ToleranceRel of it = id A &
the InternalRel of it = {};
end;
registration
let A be non trivial set;
cluster IdPrefSpace A -> non preference-like;
end;
definition
let A be 2-element set, a, b be Element of A;
func PrefSpace (A,a,b) -> strict PreferenceStr means
:: PREFER_1:def 6
the carrier of it = A &
the PrefRel of it = {[a, b]} &
the ToleranceRel of it = {[a, a], [b, b]} &
the InternalRel of it = {};
end;
theorem :: PREFER_1:39
for A be 2-element set, a, b be Element of A st a <> b holds
PrefSpace (A,a,b) is preference-like;
definition
let A be non empty set, a, b be Element of A;
func IntPrefSpace (A,a,b) -> strict PreferenceStr means
:: PREFER_1:def 7
the carrier of it = A &
the PrefRel of it = {} &
the ToleranceRel of it = {[a, a], [b, b]} &
the InternalRel of it = {[a, b], [b, a]};
end;
theorem :: PREFER_1:40
for A be 2-element set, a, b be Element of A st a <> b holds
IntPrefSpace (A,a,b) is non empty preference-like;
begin :: Characteristic Relation of a Preference Space
definition let P be PIStr;
func CharRel P -> Relation of the carrier of P equals
:: PREFER_1:def 8
(the PrefRel of P) \/ (the ToleranceRel of P);
end;
definition let P be PIStr;
attr P is PI-preference-like means
:: PREFER_1:def 9
the PrefRel of P is asymmetric &
the ToleranceRel of P is Tolerance of the carrier of P &
(the PrefRel of P) /\ (the ToleranceRel of P) = {} &
(the PrefRel of P) \/ (the PrefRel of P)~ \/
(the ToleranceRel of P) = nabla the carrier of P;
end;
registration
cluster PI-preference-like for non empty strict PIStr;
cluster PI-preference-like for empty strict PIStr;
end;
theorem :: PREFER_1:41
for P being non empty PIStr st P is PI-preference-like holds
the PrefRel of P = CharRel P /\ ((CharRel P)~)`;
theorem :: PREFER_1:42
for P being non empty PIStr st P is PI-preference-like holds
the ToleranceRel of P = CharRel P /\ (CharRel P)~;
theorem :: PREFER_1:43
for P being non empty PreferenceStr st P is preference-like holds
the PrefRel of P = CharRel P /\ ((CharRel P)~)`;
theorem :: PREFER_1:44
for P being non empty PreferenceStr st P is preference-like holds
the ToleranceRel of P = CharRel P /\ (CharRel P)~;
theorem :: PREFER_1:45
for P being non empty PreferenceStr st P is preference-like holds
the InternalRel of P = (CharRel P)` /\ ((CharRel P)~)`;
begin :: Generating Preference Space from Arbitrary (Characteristic) Relation
definition let X be set,
R be Relation of X;
func Aux R -> Relation of X equals
:: PREFER_1:def 10
SymCl (((R /\ R~`) \/ (R /\ R~`)~ \/ (R /\ R~))`);
end;
theorem :: PREFER_1:46
for X being non empty set,
R being Relation of X holds
(R /\ (R~)`) \/ (R /\ (R~)`)~ \/ (R /\ R~) \/ Aux R = nabla X;
theorem :: PREFER_1:47
for X being non empty set,
R being total reflexive Relation of X holds
Aux R = (R~` /\ R`) \/ (R`~ /\ (R` \/ R~));
theorem :: PREFER_1:48
for X being non empty set,
R being total reflexive Relation of X holds
(R /\ R~`) misses Aux R;
theorem :: PREFER_1:49
for X being non empty set,
R being total reflexive Relation of X holds
Aux R is irreflexive symmetric;
registration
let X be non empty set,
R be total reflexive Relation of X;
cluster Aux R -> irreflexive symmetric;
end;
theorem :: PREFER_1:50
for X being non empty set,
R being total reflexive Relation of X holds
(R /\ R~) misses Aux R;
theorem :: PREFER_1:51
for X being non empty set,
R being total reflexive Relation of X holds
R /\ (R~)`, R /\ R~, Aux R are_mutually_disjoint;
definition
let X be set;
let P be Relation of X;
func CharPrefSpace P -> strict PreferenceStr means
:: PREFER_1:def 11
the carrier of it = X &
the PrefRel of it = P /\ (P~)` &
the ToleranceRel of it = P /\ P~ &
the InternalRel of it = Aux P;
end;
theorem :: PREFER_1:52
for A being non empty set,
R being total reflexive Relation of A holds
CharPrefSpace R is preference-like;
registration
let X be non empty set;
let P be Relation of X;
cluster CharPrefSpace P -> non empty;
end;
registration
let X be non empty set;
let P be total reflexive Relation of X;
cluster CharPrefSpace P -> preference-like;
end;
begin :: Flat Preference Spaces
definition let P be PreferenceStr;
attr P is flat means
:: PREFER_1:def 12
the ToleranceRel of P = id the carrier of P &
ex a being Element of P st
the PrefRel of P = [:{a}, (the carrier of P) \ {a}:] &
the InternalRel of P =
[:(the carrier of P) \ {a}, (the carrier of P) \ {a}:];
end;
theorem :: PREFER_1:53
for A being trivial set holds
IdPrefSpace A = PrefSpace A;
registration
let A be trivial non empty set;
cluster IdPrefSpace A -> non empty preference-like;
end;
registration
let A be trivial non empty set;
cluster IdPrefSpace A -> flat;
end;
begin :: Tournament Preference Spaces
definition let P be PreferenceStr;
attr P is tournament-like means
:: PREFER_1:def 13
the ToleranceRel of P = id the carrier of P &
the InternalRel of P = {};
end;
registration
cluster empty -> tournament-like for PreferenceStr;
end;
registration
cluster tournament-like -> void for PreferenceStr;
end;
registration
cluster tournament-like for empty PreferenceSpace;
cluster tournament-like for non empty PreferenceSpace;
end;
theorem :: PREFER_1:54 :: The Connection Between Tournament Spaces and Orders
for P being non empty PreferenceSpace holds
P is tournament-like iff
CharRel P is connected antisymmetric total;
begin :: Total Preference Spaces
definition let P be PreferenceStr;
attr P is total means
:: PREFER_1:def 14
the PrefRel of P is transitive &
the ToleranceRel of P = id the carrier of P &
the InternalRel of P = {};
end;
registration
cluster total -> void for PreferenceStr;
end;
registration
cluster total -> tournament-like for PreferenceStr;
end;
registration
cluster PrefSpace {} -> total;
end;
registration
let A be set;
cluster IdPrefSpace A -> total;
end;
registration
let A be trivial non empty set;
cluster PrefSpace A -> total;
end;
registration
cluster total for empty PreferenceSpace;
cluster total for non empty PreferenceSpace;
end;
theorem :: PREFER_1:55
for P being non empty PreferenceSpace holds
P is total iff
CharRel P is connected Order of the carrier of P;