:: Formalizing Two Generalized Approximation Operators
:: by Adam Grabowski and Micha{\l} Sielwiesiuk
::
:: Received June 29, 2018
:: Copyright (c) 2018-2019 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, ORDERS_2, RELAT_1, TARSKI, ZFMISC_1, RELAT_2, XBOOLE_0,
PARTFUN1, SUBSET_1, FUNCT_1, EQREL_1, ROUGHS_1, ROUGHS_2, ROUGHS_5,
ALTCAT_2, FIB_NUM, FUNCT_7, COHSP_1, ROUGHS_4, SETWISEO;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, RELAT_2, FUNCT_1,
RELSET_1, PARTFUN1, FUNCT_2, DOMAIN_1, STRUCT_0, ORDERS_2, SETWISEO,
EQREL_1, PRE_TOPC, ROUGHS_1, YELLOW_3, ROUGHS_2, ALTCAT_2, ROUGHS_4;
constructors EQREL_1, REALSET2, RELSET_1, ROUGHS_1, YELLOW_3, TOPS_1,
ROUGHS_2, ALTCAT_2, COHSP_1, ROUGHS_3, ROUGHS_4, RELAT_2, SETWISEO,
PARTIT1, BVFUNC_1;
registrations XBOOLE_0, SUBSET_1, RELAT_1, FINSET_1, NAT_1, STRUCT_0,
ORDERS_2, RELSET_1, FUNCT_1, FUNCT_2, YELLOW_3, TOPS_1, PRE_TOPC,
ROUGHS_2, PARTFUN1, ROUGHS_1;
requirements BOOLE, SUBSET, NUMERALS;
definitions TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, RELAT_1, FUNCT_2, ROUGHS_1,
ROUGHS_2;
equalities TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, RELAT_1, FUNCT_2, ROUGHS_1;
expansions TARSKI, XBOOLE_0, SUBSET_1, STRUCT_0, RELAT_1, FUNCT_2, ROUGHS_1,
ROUGHS_3, ROUGHS_2, ROUGHS_4;
theorems XBOOLE_0, XBOOLE_1, SUBSET_1, TARSKI, ZFMISC_1, RELAT_1, FUNCT_2,
FUNCT_1, ROUGHS_2, ALTCAT_2, PREFER_1, LATTAD_1;
schemes FUNCT_2;
begin :: Definitions of the mappings defined in the paper by Gomolinska
definition :: property (1) p. 105
let R be non empty set;
let I be Function of R, bool R;
attr I is map-reflexive means :MapRefl:
for u being Element of R holds u in I.u;
end;
definition :: stolen from SETWISEO
let R be non empty set;
func singleton R -> Function of R, bool R means :SingleFunc:
for x being Element of R holds it.x = {x};
existence
proof
deffunc U(object) = {$1};
A1: now
let x be Element of R;
{x} c= R;
hence U(x) in bool R;
end;
thus ex f being Function of R, bool R st
for x being Element of R holds f.x = U(x) from FUNCT_2:sch 8(A1);
end;
uniqueness
proof
let IT,IT9 be Function of R, bool R such that
A3: for x being Element of R holds IT.x = {x} and
A4: for x being Element of R holds IT9.x = {x};
now
let x be Element of R;
IT.x = {x} by A3;
hence IT.x = IT9.x by A4;
end;
hence thesis;
end;
correctness;
end;
registration let R be non empty set;
cluster singleton R -> map-reflexive;
coherence
proof
for r being Element of R holds r in (singleton R).r
proof
let r be Element of R;
r in {r} by TARSKI:def 1;
hence thesis by SingleFunc;
end;
hence thesis;
end;
end;
theorem :: property (2)
for R being non empty RelStr,
I being Function of the carrier of R, bool the carrier of R
st I is map-reflexive holds
the carrier of R = union (I.:[#]R)
proof
let R be non empty RelStr,
I be Function of the carrier of R, bool the carrier of R;
assume AA: I is map-reflexive;
thus the carrier of R c= union (I.:[#]R)
proof
let x be object;
assume A0: x in the carrier of R; then
reconsider y = x as Element of R;
A2: y in I.x by AA;
x in dom I by A0,FUNCT_2:def 1; then
I.x in I.:[#]R by FUNCT_1:def 6;
hence thesis by A2,TARSKI:def 4;
end;
let x be object;
assume x in union (I.:[#]R); then
consider y being set such that
T1: x in y & y in I.:[#]R by TARSKI:def 4;
thus thesis by T1;
end;
reserve f,g for Function;
reserve R for non empty reflexive RelStr;
theorem LApId: :: Theorem 4.1 b)
LAp R cc= id bool the carrier of R
proof
set f = LAp R;
set g = id bool the carrier of R;
A1: dom f c= dom g;
for i being set st i in dom f holds f.i c= g.i
proof
let i be set;
assume i in dom f; then
reconsider ii = i as Subset of R;
f.ii = LAp ii by ROUGHS_2:def 10;
hence thesis by ROUGHS_2:35;
end;
hence thesis by A1,ALTCAT_2:def 1;
end;
theorem :: 4.1 b)
id bool the carrier of R cc= UAp R
proof
set f = id bool the carrier of R;
set g = UAp R;
A1: dom f c= dom g by FUNCT_2:def 1;
for i being set st i in dom f holds f.i c= g.i
proof
let i be set;
assume i in dom f; then
reconsider ii = i as Subset of R;
g.ii = UAp ii by ROUGHS_2:def 11;
hence f.i c= g.i by ROUGHS_2:36;
end;
hence thesis by A1,ALTCAT_2:def 1;
end;
reserve R for non empty RelStr;
theorem :: Proposition 1.1 a)
for f being map of R,
x,y being Subset of R holds
Flip (Flip f) = f by ROUGHS_2:23;
::: $f^d$
theorem FlipCompose:
for f,g being map of R holds
Flip (f * g) = Flip f * Flip g
proof
let f,g be map of R;
set fg = Flip (f * g);
set ff = Flip f;
set gg = Flip g;
for x being Subset of R holds fg.x = (ff * gg).x
proof
let x be Subset of R;
x` in bool the carrier of R; then
A1: x` in dom g by FUNCT_2:def 1;
a2: dom Flip g = bool the carrier of R by FUNCT_2:def 1;
fg.x = ((f * g).x`)` by ROUGHS_2:def 14
.= (f.((g.x`)`)`)` by FUNCT_1:13,A1
.= ff.((g.x`)`) by ROUGHS_2:def 14
.= ff.(gg.x) by ROUGHS_2:def 14
.= (ff * gg).x by a2,FUNCT_1:13;
hence thesis;
end;
hence thesis;
end;
theorem :: c)
for f being map of R holds
f.{} = {} iff (Flip f).(the carrier of R) = the carrier of R
proof
let f be map of R;
thus f.{} = {} implies (Flip f).(the carrier of R) = the carrier of R
by ROUGHS_2:18;
set g = Flip f;
A1: Flip Flip f = f by ROUGHS_2:23;
thus (Flip f).(the carrier of R) = the carrier of R implies f.{} = {}
by A1,ROUGHS_2:19;
end;
begin :: Formalization of I and tau mappings
definition
let R be non empty RelStr;
func UncertaintyMap R -> Function of the carrier of R,
bool the carrier of R means :DefUnc:
for x being Element of R holds
it.x = Coim(the InternalRel of R,x);
existence
proof
deffunc F(Element of R) = In(Coim(the InternalRel of R,$1),
bool the carrier of R);
consider f being Function of the carrier of R, bool the carrier of R
such that
A1: for x being Element of R holds f.x = F(x) from FUNCT_2:sch 4;
take f;
let x be Element of R;
A2: f.x = In(Coim(the InternalRel of R,x),bool the carrier of R) by A1;
(the InternalRel of R)"{x} c= the carrier of R;
hence thesis by A2;
end;
uniqueness
proof
let f1, f2 be Function of the carrier of R, bool the carrier of R
such that
A1: for x being Element of R holds
f1.x = Coim(the InternalRel of R,x) and
A2: for x being Element of R holds
f2.x = Coim(the InternalRel of R,x);
for x being Element of R holds f1.x = f2.x
proof
let x be Element of R;
f1.x = Coim(the InternalRel of R,x) by A1
.= f2.x by A2;
hence thesis;
end;
hence thesis;
end;
correctness;
end;
theorem For3: :: property (3)
for w,u being Element of R holds
[w,u] in the InternalRel of R iff w in (UncertaintyMap R).u
proof
let w,u be Element of R;
thus [w,u] in the InternalRel of R implies w in (UncertaintyMap R).u
proof
assume S1: [w,u] in the InternalRel of R;
u in {u} by TARSKI:def 1; then
w in Coim(the InternalRel of R,u) by S1,RELAT_1:def 14;
hence thesis by DefUnc;
end;
assume w in (UncertaintyMap R).u; then
w in Coim(the InternalRel of R,u) by DefUnc; then
consider x being object such that
S1: [w,x] in the InternalRel of R & x in {u} by RELAT_1:def 14;
thus thesis by S1,TARSKI:def 1;
end;
definition let R be non empty RelStr;
func tau R -> Function of the carrier of R, bool the carrier of R means
:DefTau:
for u being Element of R holds
it.u = Im(the InternalRel of R,u);
existence
proof
deffunc F(Element of R) = In(Im(the InternalRel of R,$1),
bool the carrier of R);
consider f being Function of the carrier of R, bool the carrier of R
such that
A1: for x being Element of R holds f.x = F(x) from FUNCT_2:sch 4;
take f;
let x be Element of R;
A2: f.x = In(Im(the InternalRel of R,x),bool the carrier of R) by A1;
(the InternalRel of R).:{x} c= the carrier of R;
hence thesis by A2;
end;
uniqueness
proof
let f1, f2 be Function of the carrier of R, bool the carrier of R
such that
A1: for x being Element of R holds
f1.x = Im(the InternalRel of R,x) and
A2: for x being Element of R holds
f2.x = Im(the InternalRel of R,x);
for x being Element of R holds f1.x = f2.x
proof
let x be Element of R;
f1.x = Im(the InternalRel of R,x) by A1
.= f2.x by A2;
hence thesis;
end;
hence thesis;
end;
end;
theorem ImCoim:
for u,w being Element of R holds
u in Im (the InternalRel of R,w) iff
w in Coim (the InternalRel of R,u)
proof
let u,w be Element of R;
thus u in Im (the InternalRel of R,w) implies
w in Coim (the InternalRel of R,u)
proof
assume u in Im (the InternalRel of R,w); then
Z1: [w,u] in the InternalRel of R by RELAT_1:169;
u in {u} by TARSKI:def 1;
hence thesis by Z1,RELAT_1:def 14;
end;
assume w in Coim (the InternalRel of R,u); then
consider t being object such that
W1: [w,t] in the InternalRel of R & t in {u} by RELAT_1:def 14;
t = u by W1,TARSKI:def 1;
hence thesis by RELAT_1:169,W1;
end;
theorem For3A: :: formula (3)
for w,u being Element of R holds
[w,u] in the InternalRel of R iff u in (tau R).w
proof
let w,u be Element of R;
thus [w,u] in the InternalRel of R implies u in (tau R).w
proof
assume [w,u] in the InternalRel of R; then
u in Im(the InternalRel of R,w) by RELAT_1:169;
hence thesis by DefTau;
end;
assume u in (tau R).w; then
u in Im(the InternalRel of R,w) by DefTau; then
w in Coim(the InternalRel of R,u) by ImCoim; then
consider x being object such that
S1: [w,x] in the InternalRel of R & x in {u} by RELAT_1:def 14;
thus thesis by S1,TARSKI:def 1;
end;
:: General version of the induced mapping
definition let R be non empty RelStr;
let f be Function of the carrier of R, bool the carrier of R;
func ff_0 f -> map of R means :Defff:
for x being Subset of R holds
it.x = { u where u is Element of R : f.u meets x };
existence
proof
deffunc F(Subset of R) =
In({ u where u is Element of R : f.u meets $1 },
bool the carrier of R);
consider g being map of R such that
A1: for x being Element of bool the carrier of R holds
g.x = F(x) from FUNCT_2:sch 4;
take g;
let x be Subset of R;
A2: g.x = In({ u where u is Element of R : f.u meets x },
bool the carrier of R) by A1;
{ u where u is Element of R : f.u meets x } c= the carrier of R
proof
let y be object;
assume y in { u where u is Element of R : f.u meets x }; then
consider u being Element of R such that
A3: y = u & f.u meets x;
thus thesis by A3;
end;
hence thesis by A2,SUBSET_1:def 8;
end;
uniqueness
proof
let f1, f2 be map of R;
assume that
A1: for x being Subset of R holds
f1.x = { u where u is Element of R : f.u meets x } and
A2: for x being Subset of R holds
f2.x = { u where u is Element of R : f.u meets x };
for y being Element of bool the carrier of R holds f1.y = f2.y
proof
let y be Element of bool the carrier of R;
f1.y = { u where u is Element of R : f.u meets y } by A1
.= f2.y by A2;
hence thesis;
end;
hence thesis;
end;
end;
definition
let R be non empty RelStr;
func f_0 R -> map of R equals :F0Unc:
ff_0 tau R;
coherence;
func f_1 R -> map of R equals :F1Unc:
ff_0 UncertaintyMap R;
coherence;
end;
theorem UncEqTau:
the InternalRel of R is symmetric implies
UncertaintyMap R = tau R
proof
assume
AA: the InternalRel of R is symmetric;
set f = UncertaintyMap R,
g = tau R;
for x being Element of R holds f.x = g.x
proof
let x be Element of R;
Z2: f.x = Coim(the InternalRel of R,x) by DefUnc;
ZZ: Im(the InternalRel of R,x) c= Coim(the InternalRel of R,x)
proof
let y be object;
assume y in Im(the InternalRel of R,x); then
[x,y] in the InternalRel of R by RELAT_1:169; then
B2: [y,x] in the InternalRel of R by AA,PREFER_1:20;
x in {x} by TARSKI:def 1;
hence thesis by B2,RELAT_1:def 14;
end;
Coim(the InternalRel of R,x) c= Im(the InternalRel of R,x)
proof
let y be object;
assume y in Coim(the InternalRel of R,x); then
consider yy being object such that
B2: [y,yy] in the InternalRel of R & yy in {x} by RELAT_1:def 14;
yy = x by TARSKI:def 1,B2; then
[x,y] in the InternalRel of R by B2,PREFER_1:20,AA;
hence thesis by RELAT_1:169;
end;
hence thesis by ZZ,DefTau,Z2;
end;
hence thesis;
end;
theorem :: Proposition 4.2
the InternalRel of R is symmetric implies
f_0 R = f_1 R
proof
assume the InternalRel of R is symmetric;
hence f_0 R = f_1 R by UncEqTau;
end;
Lemacik:
for a,b being object,
RR being Relation of the carrier of R st [a,b] in RR holds
a is Element of R & b is Element of R
proof
let a,b be object;
let RR be Relation of the carrier of R;
assume [a,b] in RR;
hence thesis by ZFMISC_1:87;
end;
theorem :: (6)
the InternalRel of R is symmetric iff
for u,v being Element of R holds
u in (tau R).v implies v in (tau R).u
proof
hereby assume
A1: the InternalRel of R is symmetric;
let u,v be Element of R;
assume u in (tau R).v; then
u in (UncertaintyMap R).v by A1,UncEqTau; then
[u,v] in the InternalRel of R by For3;
hence v in (tau R).u by For3A;
end;
assume
Z0: for u,v being Element of R holds
u in (tau R).v implies v in (tau R).u;
for a,b being object st [a,b] in the InternalRel of R holds
[b,a] in the InternalRel of R
proof
let a,b be object;
assume
Z1: [a,b] in the InternalRel of R; then
reconsider aa = a, bb = b as Element of R by Lemacik;
bb in (tau R).aa by Z1,For3A; then
aa in (tau R).bb by Z0;
hence thesis by For3A;
end;
hence thesis by PREFER_1:20;
end;
theorem UApF0:
f_0 R = UAp R
proof
set ff = f_0 R;
set gg = UAp R;
for x being Subset of R holds ff.x = gg.x
proof
let x be Subset of R;
WW: { u where u is Element of R : (tau R).u meets x } =
{ w where w is Element of R :
Class (the InternalRel of R, w) meets x }
proof
thus { u where u is Element of R : (tau R).u meets x } c=
{ w where w is Element of R :
Class (the InternalRel of R, w) meets x }
proof
let t be object;
assume t in { u where u is Element of R :
(tau R).u meets x }; then
consider u being Element of R such that
W1: u = t & (tau R).u meets x;
consider tt being object such that
W2: tt in (tau R).u & tt in x by XBOOLE_0:3,W1;
W4: (tau R).u = Im (the InternalRel of R, u) by DefTau;
reconsider ttt = tt as Element of R by W2;
thus thesis by W1,W4;
end;
let t be object;
assume t in { w where w is Element of R :
Class (the InternalRel of R, w) meets x }; then
consider tt being Element of R such that
H1: tt = t & Class (the InternalRel of R, tt) meets x;
consider wx being object such that
H2: wx in Class (the InternalRel of R, tt) & wx in x by H1,XBOOLE_0:3;
reconsider wxx = wx as Element of R by H2;
(tau R).tt = Im (the InternalRel of R, tt) by DefTau;
hence thesis by H1;
end;
ff.x = UAp x by WW,Defff
.= gg.x by ROUGHS_2:def 11;
hence thesis;
end;
hence thesis;
end;
theorem FlipLAp:
Flip f_0 R = LAp R
proof
f_0 R = UAp R by UApF0;
hence thesis by ROUGHS_2:27;
end;
theorem :: Theorem 4.1 c)
for R being Approximation_Space
for x being Subset of R holds
(f_0 R).x is exact
proof
let R be Approximation_Space;
let x be Subset of R;
(f_0 R).x = (UAp R).x by UApF0
.= UAp x by ROUGHS_2:def 11;
hence thesis;
end;
theorem :: 4.1 a)
the InternalRel of R is total reflexive implies
id bool the carrier of R cc= f_0 R
proof
assume zz: the InternalRel of R is total reflexive;
set f = id bool the carrier of R;
set g = f_0 R;
A1: dom f c= dom g by FUNCT_2:def 1;
for i being set st i in dom f holds f.i c= g.i
proof
let i be set;
assume k2: i in dom f; then
reconsider ii = i as Subset of R;
i c= { u where u is Element of R : (tau R).u meets ii }
proof
let y be object;
assume D1: y in i; then
reconsider wy = y as Element of R by k2;
[wy,wy] in the InternalRel of R by zz,LATTAD_1:1; then
wy in (tau R).wy by For3A; then
(tau R).wy meets ii by XBOOLE_0:3,D1;
hence thesis;
end;
hence f.i c= g.i by Defff;
end;
hence thesis by A1,ALTCAT_2:def 1;
end;
theorem :: 4.1 a)
R is reflexive implies
Flip (f_0 R) cc= id bool the carrier of R
proof
assume A0: R is reflexive;
Flip (f_0 R) = LAp R by FlipLAp;
hence thesis by A0,LApId;
end;
theorem :: 4.1 b)
the InternalRel of R is total reflexive implies
id bool the carrier of R cc= f_1 R
proof
assume zz: the InternalRel of R is total reflexive;
set f = id bool the carrier of R;
set g = f_1 R;
A1: dom f c= dom g by FUNCT_2:def 1;
for i being set st i in dom f holds f.i c= g.i
proof
let i be set;
assume k2: i in dom f; then
reconsider ii = i as Subset of R;
i c= { u where u is Element of R : (UncertaintyMap R).u meets ii }
proof
let y be object;
assume D1: y in i; then
reconsider wy = y as Element of R by k2;
[wy,wy] in the InternalRel of R by zz,LATTAD_1:1; then
wy in (UncertaintyMap R).wy by For3; then
(UncertaintyMap R).wy meets ii by XBOOLE_0:3,D1;
hence thesis;
end;
hence f.i c= g.i by Defff;
end;
hence thesis by A1,ALTCAT_2:def 1;
end;
reserve f for Function of the carrier of R, bool the carrier of R;
theorem Proph: :: 4.1 h)
(ff_0 f).{} = {}
proof
{ u where u is Element of R : f.u meets {}R } c= {}
proof
let y be object;
assume y in { u where u is Element of R : f.u meets {}R };
then consider u being Element of R such that
A1: u = y & f.u meets {}R;
thus thesis by A1;
end;
hence thesis by Defff;
end;
registration let R;
let f;
cluster ff_0 f -> empty-preserving;
coherence by Proph;
end;
theorem :: 4.1 h)
(f_0 R).{} = {}
proof
{ u where u is Element of R : (tau R).u meets {}R } c= {}
proof
let y be object;
assume y in { u where u is Element of R : (tau R).u meets {}R };
then consider u being Element of R such that
A1: u = y & (tau R).u meets {}R;
thus thesis by A1;
end;
hence thesis by Defff;
end;
theorem :: 4.1 h)
(f_1 R).{} = {}
proof
{ u where u is Element of R : (UncertaintyMap R).u meets {}R } c= {}
proof
let y be object;
assume y in { u where u is Element of R :
(UncertaintyMap R).u meets {}R };
then consider u being Element of R such that
A1: u = y & (UncertaintyMap R).u meets {}R;
thus thesis by A1;
end;
hence thesis by Defff;
end;
registration let R be non empty reflexive RelStr;
cluster the InternalRel of R -> total reflexive;
coherence;
end;
theorem :: 4.1 h)
f is map-reflexive implies
(ff_0 f).the carrier of R = the carrier of R
proof
assume RR: f is map-reflexive;
A0: (ff_0 f).([#]R) =
{ u where u is Element of R : f.u meets [#]R } by Defff;
the carrier of R c= { u where u is Element of R : f.u meets [#]R }
proof
let y be object;
assume y in the carrier of R; then
reconsider u = y as Element of R;
u in f.u by RR;
then consider u being Element of R such that
A1: u = y & f.u meets [#]R by XBOOLE_0:3;
thus thesis by A1;
end;
hence thesis by A0;
end;
theorem :: 4.1 h)
the InternalRel of R is reflexive total implies
(f_0 R).the carrier of R = the carrier of R
proof
assume RR: the InternalRel of R is reflexive total;
A0: (f_0 R).([#]R) =
{ u where u is Element of R : (tau R).u meets [#]R } by Defff;
the carrier of R c= { u where u is Element of R : (tau R).u meets [#]R }
proof
let y be object;
assume y in the carrier of R; then
reconsider u = y as Element of R;
ZZ: (tau R).u = Im(the InternalRel of R,u) by DefTau;
[u,u] in the InternalRel of R by LATTAD_1:1,RR; then
u in (tau R).u by RELAT_1:169,ZZ;
then consider u being Element of R such that
A1: u = y & (tau R).u meets [#]R by XBOOLE_0:3;
thus thesis by A1;
end;
hence thesis by A0;
end;
theorem :: 4.1 h)
the InternalRel of R is reflexive total implies
(f_1 R).the carrier of R = the carrier of R
proof
assume RR: the InternalRel of R is reflexive total;
A0: (f_1 R).([#]R) =
{ u where u is Element of R : (UncertaintyMap R).u meets [#]R } by Defff;
the carrier of R c= { u where u is Element of R :
(UncertaintyMap R).u meets [#]R }
proof
let y be object;
assume y in the carrier of R; then
reconsider u = y as Element of R;
[u,u] in the InternalRel of R by LATTAD_1:1,RR; then
u in (UncertaintyMap R).u by For3; then
consider u being Element of R such that
A1: u = y & (UncertaintyMap R).u meets [#]R by XBOOLE_0:3;
thus thesis by A1;
end;
hence thesis by A0;
end;
theorem :: 4.1 f) g) general version
for u,w being Element of R,
x being Subset of R st
f.u = f.w holds u in (ff_0 f).x iff w in (ff_0 f).x
proof
let u,w be Element of R,
x be Subset of R;
assume AA: f.u = f.w;
A3: (ff_0 f).x = { w where w is Element of R : f.w meets x }
by Defff;
thus u in (ff_0 f).x implies w in (ff_0 f).x
proof
assume
A1: u in (ff_0 f).x;
consider v being Element of R such that
A2: u = v & f.v meets x by A1,A3;
thus w in (ff_0 f).x by A3,AA,A2;
end;
assume w in (ff_0 f).x; then
consider v being Element of R such that
A2: w = v & f.v meets x by A3;
thus thesis by A3,AA,A2;
end;
theorem :: 4.1 g)
for u,w being Element of R,
x being Subset of R st
(UncertaintyMap R).u = (UncertaintyMap R).w holds
u in (f_1 R).x iff w in (f_1 R).x
proof
let u,w be Element of R,
x be Subset of R;
assume AA: (UncertaintyMap R).u = (UncertaintyMap R).w;
A3: (f_1 R).x = { w where w is Element of R : (UncertaintyMap R).w meets x }
by Defff;
thus u in (f_1 R).x implies w in (f_1 R).x
proof
assume
A1: u in (f_1 R).x;
consider v being Element of R such that
A2: u = v & (UncertaintyMap R).v meets x by A1,A3;
thus w in (f_1 R).x by A3,AA,A2;
end;
assume w in (f_1 R).x; then
consider v being Element of R such that
A2: w = v & (UncertaintyMap R).v meets x by A3;
thus thesis by A3,AA,A2;
end;
theorem :: 4.1 f)
for u,w being Element of R,
x being Subset of R st
(tau R).u = (tau R).w holds
u in (f_0 R).x iff w in (f_0 R).x
proof
let u,w be Element of R,
x be Subset of R;
assume
AA: (tau R).u = (tau R).w;
A3: (f_0 R).x = { w where w is Element of R : (tau R).w meets x } by Defff;
thus u in (f_0 R).x implies w in (f_0 R).x
proof
assume
A1: u in (f_0 R).x; then
consider v being Element of R such that
A2: u = v & (tau R).v meets x by A3;
thus w in (f_0 R).x by A3,AA,A2;
end;
assume w in (f_0 R).x; then
A2: ex v being Element of R st
w = v & (tau R).v meets x by A3;
thus thesis by A3,AA,A2;
end;
theorem FlipFF:
for f being Function of the carrier of R, bool the carrier of R
for x being Subset of R holds
(Flip ff_0 f).x = { w where w is Element of R : f.w c= x }
proof
let f be Function of the carrier of R, bool the carrier of R;
let x be Subset of R;
ZZ: (ff_0 f).(x`) = { w where w is Element of R : f.w meets x` }
by Defff;
thus (Flip ff_0 f).x c= { w where w is Element of R : f.w c= x }
proof
let y be object;
assume
S1: y in (Flip ff_0 f).x; then
y in ((ff_0 f).(x`))` by ROUGHS_2:def 14; then
Z1: not y in (ff_0 f).(x`) by XBOOLE_0:def 5;
reconsider yy = y as Element of R by S1;
f.yy misses x` by Z1,ZZ; then
f.yy c= x by SUBSET_1:24;
hence thesis;
end;
let y be object;
assume y in { w where w is Element of R : f.w c= x }; then
consider w being Element of R such that
L1: y = w & f.w c= x;
reconsider yy = y as Element of R by L1;
not yy in ((ff_0 f).x`)
proof
assume yy in (ff_0 f).x`; then
consider v being Element of R such that
L2: yy = v & f.v meets x` by ZZ;
thus thesis by L1,L2,SUBSET_1:24;
end; then
yy in ((ff_0 f).(x`))` by XBOOLE_0:def 5;
hence thesis by ROUGHS_2:def 14;
end;
theorem FlipF0:
for x being Subset of R holds
(Flip f_0 R).x = { w where w is Element of R : (tau R).w c= x }
proof
let x be Subset of R;
ZZ: (f_0 R).(x`) = { w where w is Element of R : (tau R).w meets x` }
by Defff;
thus (Flip f_0 R).x c= { w where w is Element of R : (tau R).w c= x }
proof
let y be object;
assume
S1: y in (Flip f_0 R).x; then
y in ((f_0 R).(x`))` by ROUGHS_2:def 14; then
Z1: not y in (f_0 R).(x`) by XBOOLE_0:def 5;
reconsider yy = y as Element of R by S1;
(tau R).yy misses x` by Z1,ZZ; then
(tau R).yy c= x by SUBSET_1:24;
hence thesis;
end;
let y be object;
assume y in { w where w is Element of R : (tau R).w c= x }; then
consider w being Element of R such that
L1: y = w & (tau R).w c= x;
reconsider yy = y as Element of R by L1;
not yy in ((f_0 R).x`)
proof
assume yy in (f_0 R).x`; then
ex v being Element of R st
yy = v & (tau R).v meets x` by ZZ;
hence thesis by L1,SUBSET_1:24;
end; then
yy in ((f_0 R).(x`))` by XBOOLE_0:def 5;
hence thesis by ROUGHS_2:def 14;
end;
theorem FlipF1:
for x being Subset of R holds
(Flip f_1 R).x = { w where w is Element of R :
(UncertaintyMap R).w c= x }
proof
let x be Subset of R;
ZZ: (f_1 R).(x`) = { w where w is Element of R :
(UncertaintyMap R).w meets x` } by Defff;
thus (Flip f_1 R).x c= { w where w is Element of R :
(UncertaintyMap R).w c= x }
proof
let y be object;
assume
S1: y in (Flip f_1 R).x; then
y in ((f_1 R).(x`))` by ROUGHS_2:def 14; then
Z1: not y in (f_1 R).(x`) by XBOOLE_0:def 5;
reconsider yy = y as Element of R by S1;
(UncertaintyMap R).yy misses x` by Z1,ZZ; then
(UncertaintyMap R).yy c= x by SUBSET_1:24;
hence thesis;
end;
let y be object;
assume y in { w where w is Element of R :
(UncertaintyMap R).w c= x }; then
consider w being Element of R such that
L1: y = w & (UncertaintyMap R).w c= x;
reconsider yy = y as Element of R by L1;
not yy in ((f_1 R).x`)
proof
assume yy in (f_1 R).x`; then
ex v being Element of R st
yy = v & (UncertaintyMap R).v meets x` by ZZ;
hence thesis by L1,SUBSET_1:24;
end; then
yy in ((f_1 R).(x`))` by XBOOLE_0:def 5;
hence thesis by ROUGHS_2:def 14;
end;
theorem :: 4.1 f) Flip
for u,w being Element of R,
x being Subset of R st
f.u = f.w holds
u in (Flip ff_0 f).x iff w in (Flip ff_0 f).x
proof
let u,w be Element of R,
x be Subset of R;
assume
AA: f.u = f.w;
A3: (Flip ff_0 f).x = { w where w is Element of R : f.w c= x } by FlipFF;
thus u in (Flip ff_0 f).x implies w in (Flip ff_0 f).x
proof
assume u in (Flip ff_0 f).x; then
consider v being Element of R such that
A2: u = v & f.v c= x by A3;
thus w in (Flip ff_0 f).x by A3,AA,A2;
end;
assume w in (Flip ff_0 f).x; then
ex v being Element of R st w = v & f.v c= x by A3;
hence thesis by A3,AA;
end;
theorem :: 4.1 f) Flip
for u,w being Element of R,
x being Subset of R st
(tau R).u = (tau R).w holds
u in (Flip f_0 R).x iff w in (Flip f_0 R).x
proof
let u,w be Element of R,
x be Subset of R;
assume AA: (tau R).u = (tau R).w;
A3: (Flip f_0 R).x = { w where w is Element of R : (tau R).w c= x }
by FlipF0;
thus u in (Flip f_0 R).x implies w in (Flip f_0 R).x
proof
assume u in (Flip f_0 R).x; then
consider v being Element of R such that
A2: u = v & (tau R).v c= x by A3;
thus w in (Flip f_0 R).x by A3,AA,A2;
end;
assume w in (Flip f_0 R).x; then
ex v being Element of R st w = v & (tau R).v c= x by A3;
hence thesis by A3,AA;
end;
theorem :: 4.1 g) Flip
for u,w being Element of R,
x being Subset of R st
(UncertaintyMap R).u = (UncertaintyMap R).w holds
u in (Flip f_1 R).x iff w in (Flip f_1 R).x
proof
let u,w be Element of R,
x be Subset of R;
assume AA: (UncertaintyMap R).u = (UncertaintyMap R).w;
A3: (Flip f_1 R).x = { w where w is Element of R :
(UncertaintyMap R).w c= x } by FlipF1;
thus u in (Flip f_1 R).x implies w in (Flip f_1 R).x
proof
assume u in (Flip f_1 R).x; then
consider v being Element of R such that
A2: u = v & (UncertaintyMap R).v c= x by A3;
thus w in (Flip f_1 R).x by A3,AA,A2;
end;
assume w in (Flip f_1 R).x; then
ex v being Element of R st w = v & (UncertaintyMap R).v c= x by A3;
hence thesis by A3,AA;
end;
theorem ReflUnc:
R is reflexive implies
for w being Element of R holds w in (UncertaintyMap R).w
proof
assume aa: R is reflexive;
let w be Element of R;
[w,w] in the InternalRel of R by aa,LATTAD_1:1;
hence thesis by For3;
end;
theorem ReflTau:
R is reflexive implies
for w being Element of R holds w in (tau R).w
proof
assume aa: R is reflexive;
let w be Element of R;
[w,w] in the InternalRel of R by aa,LATTAD_1:1;
hence thesis by For3A;
end;
registration let R be reflexive non empty RelStr;
cluster UncertaintyMap R -> map-reflexive;
coherence by ReflUnc;
cluster tau R -> map-reflexive;
coherence by ReflTau;
end;
theorem :: 4.1 b)
R is reflexive implies
Flip f_1 R cc= id bool the carrier of R
proof
assume TT: R is reflexive;
set f = Flip f_1 R;
set g = id bool the carrier of R;
A1: dom f c= dom g;
for i being set st i in dom f holds f.i c= g.i
proof
let i be set;
assume i in dom f; then
reconsider ii = i as Subset of R;
{ w where w is Element of R : (UncertaintyMap R).w c= ii } c= ii
proof
let r be object;
assume r in { w where w is Element of R : (UncertaintyMap R).w c= ii };
then consider w being Element of R such that
C4: r = w & (UncertaintyMap R).w c= ii;
thus thesis by C4,TT,ReflUnc;
end;
hence thesis by FlipF1;
end;
hence thesis by A1,ALTCAT_2:def 1;
end;
theorem :: Theorem 4.2 (a) <=> (b)
(f_0 R) * (f_0 R) = f_0 R
iff
Flip (f_0 R) * Flip (f_0 R) = Flip (f_0 R)
proof
thus (f_0 R) * (f_0 R) = f_0 R implies
Flip (f_0 R) * Flip (f_0 R) = Flip (f_0 R) by FlipCompose;
assume Flip (f_0 R) * Flip (f_0 R) = Flip (f_0 R); then
Flip (Flip (f_0 R) * Flip (f_0 R)) = f_0 R by ROUGHS_2:23; then
f_0 R = (Flip Flip (f_0 R)) * (Flip Flip (f_0 R)) by FlipCompose
.= (f_0 R) * (Flip Flip (f_0 R)) by ROUGHS_2:23
.= (f_0 R) * (f_0 R) by ROUGHS_2:23;
hence thesis;
end;
theorem
R is reflexive implies
union ((UncertaintyMap R).:[#]R) = the carrier of R
proof
assume AA: R is reflexive;
B1: the carrier of R c= union ((UncertaintyMap R).:[#]R)
proof
let t be object;
assume t in the carrier of R; then
reconsider tt = t as Element of R;
dom UncertaintyMap R = the carrier of R by FUNCT_2:def 1; then
A2: tt in dom UncertaintyMap R & tt in [#]R;
A3: tt in (UncertaintyMap R).t by AA,ReflUnc;
(UncertaintyMap R).t in (UncertaintyMap R).:[#]R by A2,FUNCT_1:def 6;
hence thesis by TARSKI:def 4,A3;
end;
union ((UncertaintyMap R).:[#]R) c= the carrier of R
proof
let f be object;
assume f in union ((UncertaintyMap R).:[#]R); then
ex tt being set st
f in tt & tt in ((UncertaintyMap R).:[#]R) by TARSKI:def 4;
hence thesis;
end;
hence thesis by B1;
end;
F0Mono:
f_0 R is c=-monotone
proof
set f = f_0 R;
for a,b being Subset of R st a c= b holds f.a c= f.b
proof
let a,b be Subset of R;
assume
A0: a c= b;
A1: f.a = (UAp R).a by UApF0 .= UAp a by ROUGHS_2:def 11;
f.b = (UAp R).b by UApF0 .= UAp b by ROUGHS_2:def 11;
hence thesis by A1,A0,ROUGHS_2:15;
end;
hence thesis;
end;
F1Mono:
f_1 R is c=-monotone
proof
set f = f_1 R;
for a,b being Subset of R st a c= b holds f.a c= f.b
proof
let a,b be Subset of R;
assume
A0: a c= b;
A2: { u where u is Element of R : (UncertaintyMap R).u meets a } c=
{ u where u is Element of R : (UncertaintyMap R).u meets b }
proof
let t be object;
assume t in { u where u is Element of R :
(UncertaintyMap R).u meets a }; then
consider u being Element of R such that
B1: u = t & (UncertaintyMap R).u meets a;
(UncertaintyMap R).u meets b by A0,B1,XBOOLE_1:63;
hence thesis by B1;
end;
f.a = { u where u is Element of R : (UncertaintyMap R).u meets a }
by Defff;
hence thesis by A2,Defff;
end;
hence thesis;
end;
registration let R be non empty RelStr; :: i)
cluster f_0 R -> c=-monotone;
coherence by F0Mono;
cluster f_1 R -> c=-monotone;
coherence by F1Mono;
end;
theorem FlipMono:
for f being map of R st f is c=-monotone holds
Flip f is c=-monotone
proof
let f be map of R;
set g = Flip f;
assume
A1: f is c=-monotone;
for A,B being Subset of R st A c= B holds g.A c= g.B
proof
let A,B be Subset of R;
assume A c= B; then
B` c= A` by SUBSET_1:12; then
f.B` c= f.A` by A1; then
(f.A`)` c= (f.B`)` by SUBSET_1:12; then
g.A c= (f.B`)` by ROUGHS_2:def 14;
hence thesis by ROUGHS_2:def 14;
end;
hence thesis;
end;
theorem :: i)
Flip f_0 R is c=-monotone by FlipMono;
theorem :: i)
Flip f_1 R is c=-monotone by FlipMono;
theorem Propj: :: j)
for f being Function of the carrier of R, bool the carrier of R
for x,y being Subset of R holds
(ff_0 f).(x \/ y) = (ff_0 f).x \/ (ff_0 f).y
proof
let f be Function of the carrier of R, bool the carrier of R;
let x,y be Subset of R;
AA: (ff_0 f).(x \/ y) =
{ u where u is Element of R : f.u meets (x \/ y) } by Defff;
AB: (ff_0 f).x =
{ u where u is Element of R : f.u meets x } by Defff;
AC: (ff_0 f).y =
{ u where u is Element of R : f.u meets y } by Defff;
thus (ff_0 f).(x \/ y) c= (ff_0 f).x \/ (ff_0 f).y
proof
let t be object;
assume t in (ff_0 f).(x \/ y); then
consider u being Element of R such that
A1: t = u & f.u meets (x \/ y) by AA;
f.u meets x or f.u meets y by A1,XBOOLE_1:70; then
t in (ff_0 f).x or t in (ff_0 f).y by A1,AB,AC;
hence thesis by XBOOLE_0:def 3;
end;
let t be object;
assume t in (ff_0 f).x \/ (ff_0 f).y; then
per cases by XBOOLE_0:def 3;
suppose t in (ff_0 f).x; then
consider u being Element of R such that
A1: t = u & f.u meets x by AB;
f.u meets (x \/ y) by XBOOLE_1:70,A1;
hence thesis by A1,AA;
end;
suppose t in (ff_0 f).y; then
consider u being Element of R such that
A1: t = u & f.u meets y by AC;
f.u meets (x \/ y) by XBOOLE_1:70,A1;
hence thesis by A1,AA;
end;
end;
theorem :: j) version
for x,y being Subset of R holds
(f_0 R).(x \/ y) = (f_0 R).x \/ (f_0 R).y
proof
let x,y be Subset of R;
set f = tau R;
thus thesis by Propj;
end;
theorem :: j) version
for x,y being Subset of R holds
(f_1 R).(x \/ y) = (f_1 R).x \/ (f_1 R).y
proof
let x,y be Subset of R;
set f = UncertaintyMap R;
thus thesis by Propj;
end;
theorem Propk: :: k)
for f being Function of the carrier of R, bool the carrier of R
for x,y being Subset of R holds
(Flip ff_0 f).x \/ (Flip ff_0 f).y c= (Flip ff_0 f).(x \/ y)
proof
let f be Function of the carrier of R, bool the carrier of R;
let x,y be Subset of R;
AA: (Flip ff_0 f).(x \/ y) =
{ u where u is Element of R : f.u c= (x \/ y) } by FlipFF;
AB: (Flip ff_0 f).x =
{ u where u is Element of R : f.u c= x } by FlipFF;
AC: (Flip ff_0 f).y =
{ u where u is Element of R : f.u c= y } by FlipFF;
XX: x c= x \/ y & y c= x \/ y by XBOOLE_1:7;
let t be object;
assume t in (Flip ff_0 f).x \/ (Flip ff_0 f).y; then
per cases by XBOOLE_0:def 3;
suppose t in (Flip ff_0 f).x; then
consider u being Element of R such that
A1: t = u & f.u c= x by AB;
f.u c= (x \/ y) by A1,XX;
hence thesis by A1,AA;
end;
suppose t in (Flip ff_0 f).y; then
consider u being Element of R such that
A1: t = u & f.u c= y by AC;
f.u c= (x \/ y) by A1,XX;
hence thesis by A1,AA;
end;
end;
theorem :: k)
for x,y being Subset of R holds
((Flip f_0 R).x \/ (Flip f_0 R).y) c= (Flip f_0 R).(x \/ y)
proof
let x,y be Subset of R;
set f = tau R;
thus thesis by Propk;
end;
theorem :: k)
for x,y being Subset of R holds
((Flip f_1 R).x \/ (Flip f_1 R).y) c= (Flip f_1 R).(x \/ y)
proof
let x,y be Subset of R;
set f = UncertaintyMap R;
thus thesis by Propk;
end;
theorem Propl: :: l)
for f being Function of the carrier of R, bool the carrier of R
for x,y being Subset of R holds
(ff_0 f).(x /\ y) c= (ff_0 f).x /\ (ff_0 f).y
proof
let f be Function of the carrier of R, bool the carrier of R;
let x,y be Subset of R;
AB: (ff_0 f).x =
{ u where u is Element of R : f.u meets x } by Defff;
AC: (ff_0 f).y =
{ u where u is Element of R : f.u meets y } by Defff;
let t be object;
assume t in (ff_0 f).(x /\ y); then
t in { u where u is Element of R : f.u meets (x /\ y) } by Defff; then
consider u being Element of R such that
A1: t = u & f.u meets (x /\ y);
f.u meets x & f.u meets y by A1,XBOOLE_1:74; then
t in (ff_0 f).x & t in (ff_0 f).y by A1,AB,AC;
hence thesis by XBOOLE_0:def 4;
end;
theorem :: l)
for x,y being Subset of R holds
(f_0 R).(x /\ y) c= (f_0 R).x /\ (f_0 R).y
proof
let x,y be Subset of R;
set f = tau R;
thus thesis by Propl;
end;
theorem :: l)
for x,y being Subset of R holds
(f_1 R).(x /\ y) c= (f_1 R).x /\ (f_1 R).y
proof
let x,y be Subset of R;
set f = UncertaintyMap R;
thus thesis by Propl;
end;
theorem Propm: :: m)
for f being Function of the carrier of R, bool the carrier of R
for x,y being Subset of R holds
(Flip ff_0 f).x /\ (Flip ff_0 f).y = (Flip ff_0 f).(x /\ y)
proof
let f be Function of the carrier of R, bool the carrier of R;
let x,y be Subset of R;
AA: (ff_0 f).(x /\ y)` =
{ u where u is Element of R : f.u meets (x /\ y)` } by Defff;
AB: (ff_0 f).x` =
{ u where u is Element of R : f.u meets x` } by Defff;
AC: (ff_0 f).y` =
{ u where u is Element of R : f.u meets y` } by Defff;
thus (Flip ff_0 f).x /\ (Flip ff_0 f).y c= (Flip ff_0 f).(x /\ y)
proof
let t be object;
assume zz: t in (Flip ff_0 f).x /\ (Flip ff_0 f).y; then
ZZ: t in (Flip ff_0 f).x & t in (Flip ff_0 f).y by XBOOLE_0:def 4; then
t in ((ff_0 f).x`)` by ROUGHS_2:def 14; then
Z1: not t in ((ff_0 f).x`) by XBOOLE_0:def 5;
t in ((ff_0 f).y`)` by ZZ,ROUGHS_2:def 14; then
V1: not t in ((ff_0 f).y`) by XBOOLE_0:def 5;
not t in (ff_0 f).(x /\ y)`
proof
assume t in (ff_0 f).(x /\ y)`; then
consider w being Element of R such that
A1: t = w & f.w meets (x /\ y)` by AA;
f.w meets (x` \/ y`) by XBOOLE_1:54,A1; then
per cases by XBOOLE_1:70;
suppose f.w meets x`;
hence thesis by Z1,A1,AB;
end;
suppose f.w meets y`;
hence thesis by V1,AC,A1;
end;
end; then
t in ((ff_0 f).(x /\ y)`)` by zz,XBOOLE_0:def 5;
hence thesis by ROUGHS_2:def 14;
end;
let t be object;
assume v0: t in (Flip ff_0 f).(x /\ y); then
t in ((ff_0 f).(x /\ y)`)` by ROUGHS_2:def 14; then
ww: not t in { u where u is Element of R : f.u meets (x /\ y)` }
by AA,XBOOLE_0:def 5;
vc: (x /\ y)` = x` \/ y` by XBOOLE_1:54;
w1: not t in (ff_0 f).x`
proof
assume t in (ff_0 f).x`; then
consider w being Element of R such that
A1: t = w & f.w meets x` by AB;
f.w meets (x /\ y)` by vc,A1,XBOOLE_1:63,XBOOLE_1:7;
hence thesis by ww,A1;
end;
z1: not t in (ff_0 f).y`
proof
assume t in (ff_0 f).y`; then
consider w being Element of R such that
A1: t = w & f.w meets y` by AC;
f.w meets (x /\ y)` by vc,A1,XBOOLE_1:63,7;
hence thesis by ww,A1;
end;
t in ((ff_0 f).x`)` by w1,v0,XBOOLE_0:def 5; then
W1: t in (Flip ff_0 f).x by ROUGHS_2:def 14;
t in ((ff_0 f).y`)` by z1,v0,XBOOLE_0:def 5; then
t in (Flip ff_0 f).y by ROUGHS_2:def 14;
hence thesis by W1,XBOOLE_0:def 4;
end;
theorem :: m)
for x,y being Subset of R holds
(Flip f_0 R).x /\ (Flip f_0 R).y = (Flip f_0 R).(x /\ y)
proof
let x,y be Subset of R;
set f = tau R;
thus thesis by Propm;
end;
theorem :: m)
for x,y being Subset of R holds
(Flip f_1 R).x /\ (Flip f_1 R).y = (Flip f_1 R).(x /\ y)
proof
let x,y be Subset of R;
set f = UncertaintyMap R;
thus thesis by Propm;
end;