:: Semantic of MML Query
:: by Grzegorz Bancerek
::
:: Received December 18, 2011
:: Copyright (c) 2011-2018 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 MMLQUERY, TARSKI, XBOOLE_0, SUBSET_1, SETFAM_1, RELAT_1, FUNCT_1,
FUNCOP_1, ZFMISC_1, AOFA_000, CARD_1, FINSEQ_1, NAT_1, NUMBERS, XXREAL_0,
CARD_3, ORDINAL1, FINSET_1, STRUCT_0, QC_LANG1, ARYTM_3;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, SETFAM_1, RELAT_1, FUNCT_1,
RELSET_1, FUNCOP_1, ORDINAL1, FINSET_1, CARD_1, CARD_3, STRUCT_0,
XCMPLX_0, XXREAL_0, NUMBERS, NAT_1, FINSEQ_1, AOFA_000, FINSEQ_4, PROB_3;
constructors TARSKI, SUBSET_1, SETFAM_1, RELSET_1, FUNCT_1, FUNCOP_1,
FINSEQ_1, CARD_1, XXREAL_0, CARD_3, FINSET_1, STRUCT_0, WELLORD2,
FINSEQ_4, XCMPLX_0, NAT_1, PROB_3;
registrations SUBSET_1, RELSET_1, ORDINAL1, FINSET_1, XXREAL_0, FINSEQ_1,
CARD_1, NAT_1, XCMPLX_0;
requirements BOOLE, SUBSET, NUMERALS, ARITHM;
definitions TARSKI, XBOOLE_0, RELAT_1, STRUCT_0;
equalities XBOOLE_0, RELAT_1, ORDINAL1, CARD_1;
expansions TARSKI, XBOOLE_0;
theorems TARSKI, XBOOLE_0, XBOOLE_1, ZFMISC_1, RELAT_1, SYSREL, SETFAM_1,
FUNCOP_1, XXREAL_0, CARD_1, FINSEQ_1, NAT_1, FUNCT_1, CARD_5, FINSEQ_2,
CARD_3, RELSET_2, ORDINAL1, ORDINAL3, XTUPLE_0, CARD_2;
schemes RELAT_1, NAT_1;
begin :: Elementary queries
definition
let X be set;
mode List of X is Subset of X;
mode Operation of X is Relation of X;
end;
definition
let x,y,R be set;
pred x,y in R means [x,y] in R;
end;
notation
let x,y,R be set;
antonym x,y nin R for x,y in R;
end;
reserve X,Y,z,s for set, L,L1,L2,A,B for List of X, x for Element of X,
O,O1,O2,O3 for Operation of X, a,b,y for Element of X, n,m for Nat;
theorem Th1:
for R1,R2 be Relation holds R1 c= R2 iff
for z being object holds Im(R1,z) c= Im(R2,z)
proof
let R1,R2 be Relation;
hereby
assume
A1: R1 c= R2;
let z be object;
thus Im(R1,z) c= Im(R2,z)
proof
let s be object; assume s in Im(R1,z); then
consider v being object such that
A2: [v,s] in R1 & v in {z} by RELAT_1:def 13;
thus thesis by A1,A2,RELAT_1:def 13;
end;
end;
assume
A3: for z being object holds Im(R1,z) c= Im(R2,z);
let a,b be object;
assume [a,b] in R1; then
b in Im(R1,a) & Im(R1,a) c= Im(R2,a) by A3,RELAT_1:169;
hence thesis by RELAT_1:169;
end;
notation
let X,O,x;
synonym x.O for Im(O,x);
end;
definition
let X,O,x;
redefine func x.O -> List of X;
coherence
proof
x.O = O.:{x};
hence x.O is Subset of X;
end;
end;
theorem Th2:
x,y in O iff y in x.O
by RELAT_1:169;
notation
let X,O,L;
synonym L | O for O.:L;
end;
definition
let X,O,L;
redefine func L | O -> List of X equals union {x.O: x in L};
coherence
proof
thus O.:L is Subset of X;
end;
compatibility
proof
union {x.O: x in L} = L|O
proof
thus union {x.O: x in L} c= L|O
proof
let z be object;
assume z in union {x.O: x in L}; then
consider Y being set such that
A1: z in Y & Y in {x.O: x in L} by TARSKI:def 4;
consider x such that
A2: Y = x.O & x in L by A1;
[x,z] in O by A1,A2,RELAT_1:169;
hence thesis by A2,RELAT_1:def 13;
end;
thus L|O c= union {x.O: x in L}
proof
let z be object;
assume z in L|O; then
consider y being object such that
A3: [y,z] in O & y in L by RELAT_1:def 13;
reconsider y as Element of X by A3;
z in y.O & y.O in {x.O: x in L} by A3,RELAT_1:169;
hence thesis by TARSKI:def 4;
end;
end;
hence thesis;
end;
func L \& O -> List of X equals meet {x.O: x in L};
coherence
proof
meet {x.O: x in L} c= X
proof
let z be object;
assume
A4: z in meet {x.O: x in L}; then
{x.O: x in L} <> {} by SETFAM_1:def 1; then
consider Y being object such that
A5: Y in {x.O: x in L} by XBOOLE_0:def 1;
consider x such that
A6: Y = x.O & x in L by A5;
z in x.O by A4,A5,A6,SETFAM_1:def 1;
hence thesis;
end;
hence thesis;
end;
func L WHERE O -> List of X equals {x: ex y st x,y in O & x in L};
coherence
proof
{x: ex y st x,y in O & x in L} c= X
proof
let z be object;
assume z in {x: ex y st x,y in O & x in L}; then
consider x such that
A7: z = x & ex y st x,y in O & x in L;
thus thesis by A7;
end;
hence thesis;
end;
let O2 be Operation of X;
func L WHEREeq(O,O2) -> List of X equals
{x: card(x.O) = card(x.O2) & x in L};
coherence
proof
{x: card(x.O) = card(x.O2) & x in L} c= X
proof
let z be object;
assume z in {x: card(x.O) = card(x.O2) & x in L}; then
ex x st z = x & card(x.O) = card(x.O2) & x in L;
hence thesis;
end;
hence thesis;
end;
func L WHEREle(O,O2) -> List of X equals
{x: card(x.O) c= card(x.O2) & x in L};
coherence
proof
{x: card(x.O) c= card(x.O2) & x in L} c= X
proof
let z be object;
assume z in {x: card(x.O) c= card(x.O2) & x in L}; then
ex x st z = x & card(x.O) c= card(x.O2) & x in L;
hence thesis;
end;
hence thesis;
end;
func L WHEREge(O,O2) -> List of X equals
{x: card(x.O2) c= card(x.O) & x in L};
coherence
proof
{x: card(x.O2) c= card(x.O) & x in L} c= X
proof
let z be object;
assume z in {x: card(x.O2) c= card(x.O) & x in L}; then
ex x st z = x & card(x.O2) c= card(x.O) & x in L;
hence thesis;
end;
hence thesis;
end;
func L WHERElt(O,O2) -> List of X equals
{x: card(x.O) in card(x.O2) & x in L};
coherence
proof
{x: card(x.O) in card(x.O2) & x in L} c= X
proof
let z be object;
assume z in {x: card(x.O) in card(x.O2) & x in L}; then
ex x st z = x & card(x.O) in card(x.O2) & x in L;
hence thesis;
end;
hence thesis;
end;
func L WHEREgt(O,O2) -> List of X equals
{x: card(x.O2) in card(x.O) & x in L};
coherence
proof
{x: card(x.O2) in card(x.O) & x in L} c= X
proof
let z be object;
assume z in {x: card(x.O2) in card(x.O) & x in L}; then
ex x st z = x & card(x.O2) in card(x.O) & x in L;
hence thesis;
end;
hence thesis;
end;
end;
definition
let X,L,O,n;
func L WHEREeq(O,n) -> List of X equals {x: card(x.O) = n & x in L};
coherence
proof
{x: card(x.O) = n & x in L} c= X
proof
let z be object;
assume z in {x: card(x.O) = n & x in L}; then
ex x st z = x & card(x.O) = n & x in L;
hence thesis;
end;
hence thesis;
end;
func L WHEREle(O,n) -> List of X equals {x: card(x.O) c= n & x in L};
coherence
proof
{x: card(x.O) c= n & x in L} c= X
proof
let z be object;
assume z in {x: card(x.O) c= n & x in L}; then
ex x st z = x & card(x.O) c= n & x in L;
hence thesis;
end;
hence thesis;
end;
func L WHEREge(O,n) -> List of X equals {x: n c= card(x.O) & x in L};
coherence
proof
{x: n c= card(x.O) & x in L} c= X
proof
let z be object;
assume z in {x: n c= card(x.O) & x in L}; then
ex x st z = x & n c= card(x.O) & x in L;
hence thesis;
end;
hence thesis;
end;
func L WHERElt(O,n) -> List of X equals {x: card(x.O) in n & x in L};
coherence
proof
{x: card(x.O) in n & x in L} c= X
proof
let z be object;
assume z in {x: card(x.O) in n & x in L}; then
ex x st z = x & card(x.O) in n & x in L;
hence thesis;
end;
hence thesis;
end;
func L WHEREgt(O,n) -> List of X equals {x: n in card(x.O) & x in L};
coherence
proof
{x: n in card(x.O) & x in L} c= X
proof
let z be object;
assume z in {x: n in card(x.O) & x in L}; then
ex x st z = x & n in card(x.O) & x in L;
hence thesis;
end;
hence thesis;
end;
end;
theorem Th3:
x in L WHERE O iff x in L & x.O <> {}
proof
hereby
assume x in L WHERE O; then
consider y such that
A1: x = y & ex a st y,a in O & y in L;
thus x in L & x.O <> {} by A1,Th2;
end;
assume
A2: x in L & x.O <> {};
set y = the Element of x.O;
y in x.O by A2; then
reconsider y as Element of X;
x,y in O by A2,Th2;
hence thesis by A2;
end;
theorem Th4:
L WHERE O c= L
by Th3;
theorem
L c= dom O implies L WHERE O = L
proof
assume
A1: L c= dom O;
thus L WHERE O c= L by Th4;
let z be object; assume
A2: z in L; then
reconsider z as Element of X;
z.O <> {} by A1,A2,RELAT_1:170;
hence thesis by A2,Th3;
end;
theorem Th6:
n <> 0 & L1 c= L2 implies L1 WHEREge(O,n) c= L2 WHERE O
proof
assume
A1: n <> 0 & L1 c= L2;
let z be object; assume z in L1 WHEREge(O,n); then
consider x such that
A2: z = x & n c= card(x.O) & x in L1;
x.O <> {} by A1,A2;
hence thesis by A1,A2,Th3;
end;
theorem
L WHEREge(O,1) = L WHERE O
proof
thus L WHEREge(O,1) c= L WHERE O by Th6;
let z be object; assume
A1: z in L WHERE O; then
reconsider z as Element of X;
A2: z.O <> {} & z in L by A1,Th3; then
succ 0 c= card(z.O) by ORDINAL1:21,ORDINAL3:8;
hence thesis by A2;
end;
theorem Th8:
L1 c= L2 implies L1 WHEREgt(O,n) c= L2 WHERE O
proof assume
A1: L1 c= L2;
let z be object; assume z in L1 WHEREgt(O,n); then
consider x such that
A2: z = x & n in card(x.O) & x in L1;
x.O <> {} by A2;
hence thesis by A1,A2,Th3;
end;
theorem
L WHEREgt(O,0) = L WHERE O
proof
thus L WHEREgt(O,0) c= L WHERE O by Th8;
let z be object; assume
A1: z in L WHERE O; then
reconsider z as Element of X;
A2: z.O <> {} & z in L by A1,Th3; then
0 in card(z.O) by ORDINAL3:8;
hence thesis by A2;
end;
theorem
n <> 0 & L1 c= L2 implies L1 WHEREeq(O,n) c= L2 WHERE O
proof
assume
A1: n <> 0 & L1 c= L2;
let z be object; assume z in L1 WHEREeq(O,n); then
consider x such that
A2: z = x & card(x.O) = n & x in L1;
x.O <> {} by A1,A2;
hence thesis by A1,A2,Th3;
end;
theorem
L WHEREge(O,n+1) = L WHEREgt(O,n)
proof
thus L WHEREge(O,n+1) c= L WHEREgt(O,n)
proof
let z be object; assume z in L WHEREge(O,n+1); then
consider x such that
A1: z = x & n+1 c= card(x.O) & x in L;
Segm(n+1) = succ Segm n by NAT_1:38; then
n in card(x.O) by A1,ORDINAL1:21;
hence thesis by A1;
end;
let z be object; assume z in L WHEREgt(O,n); then
consider x such that
A2: z = x & n in card(x.O) & x in L;
Segm(n+1) = succ Segm n by NAT_1:38; then
n+1 c= card(x.O) by A2,ORDINAL1:21;
hence thesis by A2;
end;
theorem
L WHEREle(O,n) = L WHERElt(O,n+1)
proof
thus L WHEREle(O,n) c= L WHERElt(O,n+1)
proof
let z be object; assume z in L WHEREle(O,n); then
consider x such that
A1: z = x & card(x.O) c= n & x in L;
Segm(n+1) = succ Segm n by NAT_1:38; then
card(x.O) in n+1 by A1,ORDINAL1:22;
hence thesis by A1;
end;
let z be object; assume z in L WHERElt(O,n+1); then
consider x such that
A2: z = x & card(x.O) in n+1 & x in L;
Segm(n+1) = succ Segm n by NAT_1:38; then
card(x.O) c= n by A2,ORDINAL1:22;
hence thesis by A2;
end;
theorem
n <= m & L1 c= L2 & O1 c= O2 implies L1 WHEREge(O1,m) c= L2 WHEREge(O2,n)
proof
assume
A1: n <= m & L1 c= L2 & O1 c= O2;
let z be object; assume z in L1 WHEREge(O1,m); then
consider x such that
A2: z = x & m c= card(x.O1) & x in L1;
Segm n c= Segm m & x.O1 c= x.O2 by A1,Th1,NAT_1:39; then
n c= card (x.O1) & card (x.O1) c= card (x.O2) by A2,CARD_1:11;
then n c= card (x.O2);
hence thesis by A1,A2;
end;
theorem
n <= m & L1 c= L2 & O1 c= O2 implies L1 WHEREgt(O1,m) c= L2 WHEREgt(O2,n)
proof
assume
A1: n <= m & L1 c= L2 & O1 c= O2;
let z be object; assume z in L1 WHEREgt(O1,m); then
consider x such that
A2: z = x & m in card(x.O1) & x in L1;
Segm n c= Segm m & card(x.O1) c= card (x.O2)
by A1,Th1,CARD_1:11,NAT_1:39; then
n in card (x.O2) by A2,ORDINAL1:12;
hence thesis by A1,A2;
end;
theorem
n <= m & L1 c= L2 & O1 c= O2 implies L1 WHEREle(O2,n) c= L2 WHEREle(O1,m)
proof
assume
A1: n <= m & L1 c= L2 & O1 c= O2;
let z be object; assume z in L1 WHEREle(O2,n); then
consider x such that
A2: z = x & card(x.O2) c= n & x in L1;
Segm n c= Segm m & x.O1 c= x.O2 by A1,Th1,NAT_1:39; then
card (x.O1) c= card (x.O2) & card (x.O2) c= m by A2,CARD_1:11;
then card (x.O1) c= m;
hence thesis by A1,A2;
end;
theorem
n <= m & L1 c= L2 & O1 c= O2 implies L1 WHERElt(O2,n) c= L2 WHERElt(O1,m)
proof
assume
A1: n <= m & L1 c= L2 & O1 c= O2;
let z be object; assume z in L1 WHERElt(O2,n); then
consider x such that
A2: z = x & card(x.O2) in n & x in L1;
Segm n c= Segm m & card(x.O1) c= card(x.O2) by A1,Th1,CARD_1:11,NAT_1:39;
then
card(x.O1) in m by A2,ORDINAL1:12;
hence thesis by A1,A2;
end;
theorem
O1 c= O2 & L1 c= L2 & O c= O3 implies L1 WHEREge(O,O2) c= L2 WHEREge(O3,O1)
proof
assume A1: O1 c= O2 & L1 c= L2 & O c= O3;
let z be object; assume z in L1 WHEREge(O,O2); then
consider x such that
A2: z = x & card(x.O2) c= card(x.O) & x in L1;
A3: card(x.O1) c= card(x.O2) & card(x.O) c= card(x.O3)
by A1,Th1,CARD_1:11; then
card(x.O1) c= card(x.O) by A2; then
card(x.O1) c= card(x.O3) by A3;
hence z in L2 WHEREge(O3,O1) by A2,A1;
end;
theorem
O1 c= O2 & L1 c= L2 & O c= O3 implies L1 WHEREgt(O,O2) c= L2 WHEREgt(O3,O1)
proof
assume A1: O1 c= O2 & L1 c= L2 & O c= O3;
let z be object; assume z in L1 WHEREgt(O,O2); then
consider x such that
A2: z = x & card(x.O2) in card(x.O) & x in L1;
x.O1 c= x.O2 & x.O c= x.O3 by A1,Th1; then
card(x.O1) in card(x.O) & card(x.O) c= card(x.O3)
by A2,CARD_1:11,ORDINAL1:12;
hence z in L2 WHEREgt(O3,O1) by A2,A1;
end;
theorem
O1 c= O2 & L1 c= L2 & O c= O3 implies L1 WHEREle(O3,O1) c= L2 WHEREle(O,O2)
proof
assume A1: O1 c= O2 & L1 c= L2 & O c= O3;
let z be object; assume z in L1 WHEREle(O3,O1); then
consider x such that
A2: z = x & card(x.O3) c= card(x.O1) & x in L1;
A3: card(x.O1) c= card(x.O2) & card(x.O) c= card(x.O3)
by A1,Th1,CARD_1:11; then
card(x.O3) c= card(x.O2) by A2; then
card(x.O) c= card(x.O2) by A3;
hence z in L2 WHEREle(O,O2) by A2,A1;
end;
theorem
O1 c= O2 & L1 c= L2 & O c= O3 implies L1 WHERElt(O3,O1) c= L2 WHERElt(O,O2)
proof
assume A1: O1 c= O2 & L1 c= L2 & O c= O3;
let z be object; assume z in L1 WHERElt(O3,O1); then
consider x such that
A2: z = x & card(x.O3) in card(x.O1) & x in L1;
card(x.O1) c= card(x.O2) & card(x.O) c= card(x.O3)
by A1,Th1,CARD_1:11; then
card(x.O) in card(x.O2) by A2,ORDINAL1:12;
hence z in L2 WHERElt(O,O2) by A2,A1;
end;
theorem
L WHEREgt(O,O1) c= L WHERE O
proof
let z be object; assume z in L WHEREgt(O,O1); then
consider x such that
A1: z = x & card(x.O1) in card(x.O) & x in L;
x.O <> {} by A1;
hence thesis by A1,Th3;
end;
theorem
O1 c= O2 & L1 c= L2 implies L1 WHERE O1 c= L2 WHERE O2
proof assume
A1: O1 c= O2 & L1 c= L2;
let z be object; assume
A2: z in L1 WHERE O1; then
reconsider z as Element of X;
A3: z.O1 <> {} & z in L1 by A2,Th3;
z.O1 c= z.O2 by A1,Th1; then
z.O2 <> {} by A3;
hence thesis by A1,A3,Th3;
end;
theorem Th23:
a in L|O iff ex b st a in b.O & b in L
proof
hereby
assume
a in L|O; then
consider b being object such that
A1: [b,a] in O & b in L by RELAT_1:def 13;
reconsider b as Element of X by A1;
take b;
thus a in b.O by A1,RELAT_1:169;
thus b in L by A1;
end;
given b such that
A2: a in b.O & b in L;
[b,a] in O by A2,RELAT_1:169;
hence a in L|O by A2,RELAT_1:def 13;
end;
notation
let X,A,B;
synonym A AND B for A /\ B;
synonym A OR B for A \/ B;
synonym A BUTNOT B for A \ B;
end;
definition
let X,A,B;
redefine func A AND B -> List of X;
coherence
proof
A/\B is Subset of X;
hence thesis;
end;
redefine func A OR B -> List of X;
coherence
proof
A\/B is Subset of X;
hence thesis;
end;
redefine func A BUTNOT B -> List of X;
coherence
proof
A\B is Subset of X;
hence thesis;
end;
end;
theorem Th24:
L1 <> {} & L2 <> {} implies (L1 OR L2)\& O = (L1 \& O)AND(L2 \& O)
proof assume
A1: L1 <> {} & L2 <> {};
thus (L1 OR L2)\& O c= (L1 \& O)AND(L2 \& O)
proof
let z be object; assume
A2: z in (L1 OR L2)\& O;
now
set c = the Element of L1;
c in L1 by A1; then
reconsider c as Element of X;
c.O in {x.O: x in L1} by A1;
hence {x.O: x in L1} <> {};
let Y; assume Y in {x.O: x in L1}; then
consider x such that
A3: Y = x.O & x in L1;
x in L1 OR L2 by A3,XBOOLE_0:def 3; then
Y in {a.O: a in L1 OR L2} by A3;
hence z in Y by A2,SETFAM_1:def 1;
end; then
A4: z in L1\&O by SETFAM_1:def 1;
now
set c = the Element of L2;
c in L2 by A1; then
reconsider c as Element of X;
c.O in {x.O: x in L2} by A1;
hence {x.O: x in L2} <> {};
let Y; assume Y in {x.O: x in L2}; then
consider x such that
A5: Y = x.O & x in L2;
x in L1 OR L2 by A5,XBOOLE_0:def 3; then
Y in {a.O: a in L1 OR L2} by A5;
hence z in Y by A2,SETFAM_1:def 1;
end; then
z in L2\&O by SETFAM_1:def 1;
hence thesis by A4,XBOOLE_0:def 4;
end;
let z be object; assume z in (L1\&O)AND(L2\&O); then
A6: z in L1\&O & z in L2\&O by XBOOLE_0:def 4;
now
set c = the Element of L2;
c in L2 by A1; then
reconsider c as Element of X;
c in L1 OR L2 by A1,XBOOLE_0:def 3; then
c.O in {x.O: x in L1 OR L2};
hence {x.O: x in L1 OR L2} <> {};
let Y; assume Y in {x.O: x in L1 OR L2}; then
consider x such that
A7: Y = x.O & x in L1 OR L2;
x in L1 or x in L2 by A7,XBOOLE_0:def 3; then
Y in {a.O: a in L1} or Y in {b.O: b in L2} by A7;
hence z in Y by A6,SETFAM_1:def 1;
end;
hence thesis by SETFAM_1:def 1;
end;
theorem
L1 c= L2 & O1 c= O2 implies L1|O1 c= L2|O2
proof
assume L1 c= L2 & O1 c= O2; then
L1|O1 c= L2|O1 & L2|O1 c= L2|O2 by RELAT_1:123,124;
hence thesis;
end;
theorem Th26:
O1 c= O2 implies L\&O1 c= L\&O2
proof
assume
A1: O1 c= O2;
let z be object; assume
A2: z in L\&O1; then
{x.O1: x in L} <> {} by SETFAM_1:def 1; then
consider Y being object such that
A3: Y in {x.O1: x in L} by XBOOLE_0:def 1;
consider y such that
A4: Y = y.O1 & y in L by A3;
A5: y.O2 in {x.O2: x in L} by A4;
now
let Y; assume Y in {x.O2: x in L}; then
consider a such that
A6: Y = a.O2 & a in L;
a.O1 in {x.O1: x in L} by A6; then
z in a.O1 & a.O1 c= Y by A1,A2,A6,Th1,SETFAM_1:def 1;
hence z in Y;
end;
hence thesis by A5,SETFAM_1:def 1;
end;
theorem
L\&(O1 AND O2) = (L \& O1)AND(L \& O2)
proof
L\&(O1 AND O2) c= L\&O1 & L\&(O1 AND O2) c= L\&O2 by Th26,XBOOLE_1:17;
hence L\&(O1 AND O2) c= (L\&O1)AND(L\&O2) by XBOOLE_1:19;
let z be object; assume z in (L\&O1)AND(L\&O2); then
A1: z in L\&O1 & z in L\&O2 by XBOOLE_0:def 4;
now
set O = O1 AND O2;
set c = the Element of L;
{x.O1: x in L} <> {} by A1,SETFAM_1:def 1; then
consider c being object such that
A2: c in {x.O1: x in L} by XBOOLE_0:def 1;
consider a such that
A3: c = a.O1 & a in L by A2;
a.O in {x.O: x in L} by A3;
hence {x.O: x in L} <> {};
let Y; assume Y in {x.O: x in L}; then
consider x such that
A4: Y = x.O & x in L;
A5: Y = (x.O1) AND (x.O2) by A4,RELSET_2:11;
x.O1 in {y.O1: y in L} by A4; then
A6: z in x.O1 by A1,SETFAM_1:def 1;
x.O2 in {y.O2: y in L} by A4; then
z in x.O2 by A1,SETFAM_1:def 1;
hence z in Y by A5,A6,XBOOLE_0:def 4;
end;
hence thesis by SETFAM_1:def 1;
end;
theorem
L1 <> {} & L1 c= L2 implies L2 \& O c= L1 \& O
proof
assume A1: L1 <> {};
assume A2: L1 c= L2;
let z be object;
assume
A3: z in L2 \& O;
now
set c = the Element of L1;
c in L1 by A1; then
reconsider c as Element of X;
c.O in {x.O: x in L1} by A1;
hence {x.O: x in L1} <> {};
let Y; assume Y in {x.O: x in L1}; then
consider x such that
A4: Y = x.O & x in L1;
Y in {a.O: a in L2} by A4,A2;
hence z in Y by A3,SETFAM_1:def 1;
end;
hence z in L1 \& O by SETFAM_1:def 1;
end;
begin :: Operations
theorem Th29:
for O1,O2 being Operation of X st for x holds x.O1 = x.O2 holds O1 = O2
proof
let O1,O2 be Operation of X;
assume
A1: for x holds x.O1 = x.O2;
let a,b be object;
thus [a,b] in O1 implies [a,b] in O2
proof
assume
A2: [a,b] in O1;
reconsider a,b as Element of X by A2,ZFMISC_1:87;
b in a.O1 by A2,RELAT_1:169; then
b in a.O2 by A1;
hence thesis by RELAT_1:169;
end;
assume
A3: [a,b] in O2; then
A4: a in X & b in X by ZFMISC_1:87;
reconsider a,b as Element of X by A3,ZFMISC_1:87;
reconsider L = {a} as Subset of X by A4,ZFMISC_1:31;
b in a.O2 by A3,RELAT_1:169; then
b in a.O1 by A1;
hence thesis by RELAT_1:169;
end;
theorem Th30:
for O1,O2 being Operation of X st for L holds L|O1 = L|O2 holds O1 = O2
proof
let O1,O2 be Operation of X;
assume
A1: for L holds L|O1 = L|O2;
now let x;
per cases;
suppose X <> {}; then
reconsider L = {x} as Subset of X by ZFMISC_1:31;
thus x.O1 = L|O1 .= x.O2 by A1;
end;
suppose X = {};
hence x.O1 = x.O2;
end;
end;
hence thesis by Th29;
end;
definition
let X,O;
func NOT O -> Operation of X means: Def15:
for L holds L|it = union {IFEQ(x.O, {}, {x}, {}): x in L};
existence
proof
defpred P[object,object] means Im(O,$1) = {} & $2 = $1;
consider O1 being Relation such that
A1: for a,b being object holds
[a,b] in O1 iff a in X & b in X & P[a,b] from RELAT_1:sch 1;
O1 c= [:X,X:]
proof
let x,y be object;
assume [x,y] in O1; then
x in X & y in X by A1;
hence thesis by ZFMISC_1:87;
end; then
reconsider O1 as Operation of X;
take O1;
let L;
thus L|O1 c= union {IFEQ(x.O, {}, {x}, {}): x in L}
proof let a be object; assume a in L|O1; then
consider b such that
A2: a in b.O1 & b in L by Th23;
[b,a] in O1 by A2,RELAT_1:169; then
A3: a = b & b.O = {} by A1; then
B1: a in {b} by TARSKI:def 1;
reconsider Z = IFEQ(b.O,{},{b},{}) as set;
A4: a in Z by A3,B1,FUNCOP_1:def 8;
IFEQ(b.O,{},{b},{}) in {IFEQ(x.O,{},{x},{}): x in L} by A2;
hence thesis by A4,TARSKI:def 4;
end;
let a be object; assume a in union {IFEQ(x.O, {}, {x}, {}): x in L}; then
consider A be set such that
A5: a in A & A in {IFEQ(x.O, {}, {x}, {}): x in L} by TARSKI:def 4;
consider x such that
A6: A = IFEQ(x.O, {}, {x}, {}) & x in L by A5;
A7: x.O = {} by A5,A6,FUNCOP_1:def 8; then
A = {x} by A6,FUNCOP_1:def 8; then
a = x by A5,TARSKI:def 1; then
[x,a] in O1 by A1,A6,A7;
hence thesis by A6,RELAT_1:def 13;
end;
uniqueness
proof
let O1,O2 be Operation of X such that
A8: for L holds L|O1 = union {IFEQ(x.O, {}, {x}, {}): x in L} and
A9: for L holds L|O2 = union {IFEQ(x.O, {}, {x}, {}): x in L};
now let L;
thus L|O1 = union {IFEQ(x.O, {}, {x}, {}): x in L} by A8 .= L|O2 by A9;
end;
hence thesis by Th30;
end;
end;
notation
let X;
let O1,O2 be Operation of X;
synonym O1 AND O2 for O1 /\ O2;
synonym O1 OR O2 for O1 \/ O2;
synonym O1 BUTNOT O2 for O1 \ O2;
synonym O1 | O2 for O1 * O2;
end;
definition
let X;
let O1,O2 be Operation of X;
redefine func O1 AND O2 -> Operation of X means
for L holds L|it = union {x.O1 AND x.O2: x in L};
coherence
proof
thus O1 /\ O2 is Subset of [:X,X:];
end;
compatibility
proof
A1: for O being Operation of X holds
O = O1 AND O2 implies for L holds L|O = union {x.O1 AND x.O2: x in L}
proof
let O;
assume
A2: O = O1 /\ O2;
defpred P[set,set] means [$1,$2] in O1 & [$1,$2] in O2;
let L;
thus L|O c= union {x.O1 AND x.O2: x in L}
proof
let z be object;
assume z in L|O; then
consider y being object such that
A3: [y,z] in O & y in L by RELAT_1:def 13;
reconsider y,z as Element of X by A3,ZFMISC_1:87;
[y,z] in O1 & [y,z] in O2 by A2,A3,XBOOLE_0:def 4; then
z in y.O1 & z in y.O2 by RELAT_1:169; then
z in y.O1 AND y.O2 & y.O1 AND y.O2 in {x.O1 AND x.O2: x in L}
by A3,XBOOLE_0:def 4;
hence thesis by TARSKI:def 4;
end;
let z be object; assume
z in union {x.O1 AND x.O2: x in L}; then
consider Y being set such that
A4: z in Y & Y in {x.O1 AND x.O2: x in L} by TARSKI:def 4;
consider x such that
A5: Y = x.O1 AND x.O2 & x in L by A4;
A6: z in x.O1 & z in x.O2 by A4,A5,XBOOLE_0:def 4;
reconsider z as Element of X by A4,A5;
[x,z] in O1 & [x,z] in O2 by A6,RELAT_1:169; then
[x,z] in O by A2,XBOOLE_0:def 4;
hence thesis by A5,RELAT_1:def 13;
end;
let O be Operation of X;
thus O = O1 AND O2 implies for L holds L|O = union {x.O1 AND x.O2: x in L}
by A1;
assume
A7: for L holds L|O = union {x.O1 AND x.O2: x in L};
now
let L;
thus L|O = union {x.O1 AND x.O2: x in L} by A7 .= L|(O1 /\ O2) by A1;
end;
hence thesis by Th30;
end;
redefine func O1 OR O2 -> Operation of X means
for L holds L|it = union {x.O1 OR x.O2: x in L};
coherence
proof
thus O1 \/ O2 is Subset of [:X,X:];
end;
compatibility
proof
A8: for O being Operation of X holds
O = O1 OR O2 implies for L holds L|O = union {x.O1 OR x.O2: x in L}
proof
let O;
assume
A9: O = O1 \/ O2;
defpred P[set,set] means [$1,$2] in O1 or [$1,$2] in O2;
let L;
thus L|O c= union {x.O1 OR x.O2: x in L}
proof
let z be object;
assume z in L|O; then
consider y being object such that
A10: [y,z] in O & y in L by RELAT_1:def 13;
reconsider y,z as Element of X by A10,ZFMISC_1:87;
[y,z] in O1 or [y,z] in O2 by A9,A10,XBOOLE_0:def 3; then
z in y.O1 or z in y.O2 by RELAT_1:169; then
z in y.O1 OR y.O2 & y.O1 OR y.O2 in {x.O1 OR x.O2: x in L}
by A10,XBOOLE_0:def 3;
hence thesis by TARSKI:def 4;
end;
let z be object; assume
z in union {x.O1 OR x.O2: x in L}; then
consider Y being set such that
A11: z in Y & Y in {x.O1 OR x.O2: x in L} by TARSKI:def 4;
consider x such that
A12: Y = x.O1 OR x.O2 & x in L by A11;
A13: z in x.O1 or z in x.O2 by A11,A12,XBOOLE_0:def 3;
reconsider z as Element of X by A11,A12;
[x,z] in O1 or [x,z] in O2 by A13,RELAT_1:169; then
[x,z] in O by A9,XBOOLE_0:def 3;
hence thesis by A12,RELAT_1:def 13;
end;
let O be Operation of X;
thus O = O1 OR O2 implies for L holds L|O = union {x.O1 OR x.O2: x in L}
by A8;
assume
A14: for L holds L|O = union {x.O1 OR x.O2: x in L};
now
let L;
thus L|O = union {x.O1 OR x.O2: x in L} by A14 .= L|(O1 \/ O2) by A8;
end;
hence thesis by Th30;
end;
redefine func O1 BUTNOT O2 -> Operation of X means
for L holds L|it = union {x.O1 BUTNOT x.O2: x in L};
coherence
proof
thus O1 \ O2 is Subset of [:X,X:];
end;
compatibility
proof
A15: for O being Operation of X holds
O = O1 BUTNOT O2 implies for L holds L|O = union {x.O1 BUTNOT x.O2: x in L}
proof
let O;
assume
A16: O = O1 \ O2;
defpred P[set,set] means [$1,$2] in O1 & not [$1,$2] in O2;
let L;
thus L|O c= union {x.O1 BUTNOT x.O2: x in L}
proof
let z be object;
assume z in L|O; then
consider y being object such that
A17: [y,z] in O & y in L by RELAT_1:def 13;
reconsider y,z as Element of X by A17,ZFMISC_1:87;
[y,z] in O1 & [y,z] nin O2 by A16,A17,XBOOLE_0:def 5; then
z in y.O1 & z nin y.O2 by RELAT_1:169; then
z in y.O1 BUTNOT y.O2 & y.O1 BUTNOT y.O2 in {x.O1 BUTNOT x.O2: x in L}
by A17,XBOOLE_0:def 5;
hence thesis by TARSKI:def 4;
end;
let z be object; assume
z in union {x.O1 BUTNOT x.O2: x in L}; then
consider Y being set such that
A18: z in Y & Y in {x.O1 BUTNOT x.O2: x in L} by TARSKI:def 4;
consider x such that
A19: Y = x.O1 BUTNOT x.O2 & x in L by A18;
A20: z in x.O1 & not z in x.O2 by A18,A19,XBOOLE_0:def 5;
reconsider z as Element of X by A18,A19;
[x,z] in O1 & [x,z] nin O2 by A20,RELAT_1:169; then
[x,z] in O by A16,XBOOLE_0:def 5;
hence thesis by A19,RELAT_1:def 13;
end;
let O be Operation of X;
thus O = O1 BUTNOT O2 implies
for L holds L|O = union {x.O1 BUTNOT x.O2: x in L}
by A15;
assume
A21: for L holds L|O = union {x.O1 BUTNOT x.O2: x in L};
now
let L;
thus L|O = union {x.O1 BUTNOT x.O2: x in L} by A21 .= L|(O1 \ O2) by A15;
end;
hence thesis by Th30;
end;
redefine func O1 | O2 -> Operation of X means
for L holds L|it = L|O1|O2;
coherence
proof
thus O1 * O2 is Relation of X;
end;
compatibility
proof
let O be Operation of X;
thus O = O1|O2 implies for L holds L|O = L|O1|O2 by RELAT_1:126;
assume
A22: for L holds L|O = L|O1|O2;
now
let L;
thus L|O = L|O1|O2 by A22 .= L|(O1*O2) by RELAT_1:126;
end;
hence thesis by Th30;
end;
func O1 \& O2 -> Operation of X means :Def20:
for L holds L|it = union {x.O1\&O2: x in L};
existence
proof
defpred P[object,object] means ex x st $1 = x & $2 in x.O1\&O2;
consider O being Relation such that
A23: for z,s being object holds [z,s] in O iff z in X & s in X & P[z,s]
from RELAT_1:sch 1;
O c= [:X,X:]
proof
let z,s be object;
assume [z,s] in O; then
z in X & s in X by A23;
hence thesis by ZFMISC_1:87;
end; then
reconsider O as Operation of X;
take O;
let L;
thus L|O c= union {x.O1\&O2: x in L}
proof
let z be object;
assume z in L|O; then
consider y such that
A24: z in y.O & y in L by Th23;
[y,z] in O by A24,RELAT_1:169; then
ex x st y = x & z in x.O1\&O2 by A23; then
z in y.O1\&O2 & y.O1\&O2 in {x.O1\&O2: x in L} by A24;
hence thesis by TARSKI:def 4;
end;
let z be object;
assume z in union {x.O1\&O2: x in L}; then
consider Y such that
A25: z in Y & Y in {x.O1\&O2: x in L} by TARSKI:def 4;
consider x such that
A26: Y = x.O1\&O2 & x in L by A25;
[x,z] in O by A23,A25,A26;
hence thesis by A26,RELAT_1:def 13;
end;
uniqueness
proof
let R1,R2 be Operation of X such that
A27: for L holds L|R1 = union {x.O1\&O2: x in L} and
A28: for L holds L|R2 = union {x.O1\&O2: x in L};
now let L;
thus L|R1 = union {x.O1\&O2: x in L} by A27 .= L|R2 by A28;
end;
hence thesis by Th30;
end;
end;
theorem
x.(O1 AND O2) = (x.O1)AND(x.O2) by RELSET_2:11;
theorem
x.(O1 OR O2) = (x.O1)OR(x.O2) by RELSET_2:10;
theorem
x.(O1 BUTNOT O2) = (x.O1)BUTNOT(x.O2) by RELSET_2:12;
theorem
x.(O1|O2) = (x.O1)|O2 by RELAT_1:126;
theorem Th35:
x.(O1\&O2) = (x.O1)\&O2
proof
per cases;
suppose X = {}; then
x.(O1\&O2) = {} & (x.O1)\&O2 = {};
hence thesis;
end;
suppose
X <> {}; then
reconsider L = {x} as List of X by ZFMISC_1:31;
A1: {a.O1\&O2: a in L} = {x.O1\&O2}
proof
thus {a.O1\&O2: a in L} c= {x.O1\&O2}
proof
let z be object; assume z in {a.O1\&O2: a in L}; then
consider a such that
A2: z = a.O1\&O2 & a in L;
a = x by A2,TARSKI:def 1;
hence thesis by A2,TARSKI:def 1;
end;
let z be object; assume z in {x.O1\&O2}; then
z = x.O1\&O2 & x in L by TARSKI:def 1;
hence thesis;
end;
thus x.(O1\&O2) = union {a.O1\&O2: a in L} by Def20
.= x.O1\&O2 by A1,ZFMISC_1:25;
end;
end;
theorem Th36:
for z,s being object holds [z,s] in NOT O iff z = s & z in X & z nin dom O
proof let z,s be object;
thus [z,s] in NOT O implies z = s & z in X & z nin dom O
proof
assume
A1: [z,s] in NOT O; then
s in Im(NOT O,z) & z in X by RELAT_1:169,ZFMISC_1:87; then
s in (NOT O).:{z} & {z} c= X by ZFMISC_1:31; then
s in union {IFEQ(x.O, {}, {x}, {}): x in {z}} by Def15; then
consider Y such that
A2: s in Y & Y in {IFEQ(x.O, {}, {x}, {}): x in {z}} by TARSKI:def 4;
consider x such that
A3: Y = IFEQ(x.O, {}, {x}, {}) & x in {z} by A2;
A4: x = z by A3,TARSKI:def 1;
A5: x.O = {} by A2,A3,FUNCOP_1:def 8; then
s in {x} by A2,A3,FUNCOP_1:def 8; then
A6: s = x &
for s being object holds [x,s] nin O by A5,RELAT_1:169,TARSKI:def 1;
hence z = s by A4;
thus z in X by A1,ZFMISC_1:87;
x in dom O iff
ex y being object st [x,y] in O by XTUPLE_0:def 12;
hence z nin dom O by A4,A6;
end;
assume
A7: z = s & z in X & z nin dom O;
then reconsider z as Element of X;
z.O c= {}
proof
let y be object; assume y in z.O; then
[z,y] in O by RELAT_1:169;
hence thesis by A7,XTUPLE_0:def 12;
end; then
z.O = {}; then
A8: IFEQ(z.O, {}, {z}, {}) = {z} by FUNCOP_1:def 8;
A9: z in {z} by TARSKI:def 1;
reconsider L = {z} as Subset of X by A7,ZFMISC_1:31;
{z} in {IFEQ(x.O, {}, {x}, {}): x in {z}} by A8,A9; then
z in union {IFEQ(x.O, {}, {x}, {}): x in {z}} by A9,TARSKI:def 4; then
z in L|(NOT O) by Def15; then
z in z.NOT O;
hence thesis by A7,RELAT_1:169;
end;
theorem Th37:
NOT O = id(X\dom O)
proof
let z,s be object;
thus [z,s] in NOT O implies [z,s] in id(X\dom O)
proof
assume [z,s] in NOT O; then
A1: z = s & z in X & z nin dom O by Th36; then
z in X\dom O by XBOOLE_0:def 5;
hence thesis by A1,RELAT_1:def 10;
end;
assume [z,s] in id(X\dom O); then
A2: z = s & z in X\dom O by RELAT_1:def 10; then
z in X & z nin dom O by XBOOLE_0:def 5;
hence thesis by A2,Th36;
end;
theorem Th38:
dom NOT NOT O = dom O
proof
thus dom NOT NOT O = dom id(X\dom NOT O) by Th37
.= X\dom NOT O by RELAT_1:45 .= X\dom id(X\dom O) by Th37
.= X\(X\dom O) by RELAT_1:45 .= X/\dom O by XBOOLE_1:48
.= dom O by XBOOLE_1:28;
end;
theorem
L WHERE NOT NOT O = L WHERE O
proof
thus L WHERE NOT NOT O c= L WHERE O
proof
let z be object; assume
A1: z in L WHERE NOT NOT O; then
reconsider x = z as Element of X;
A2: x in L & x.NOT NOT O <> {} by A1,Th3; then
x in dom NOT NOT O by RELAT_1:170; then
x in dom O by Th38; then
x.O <> {} by RELAT_1:170;
hence thesis by A2,Th3;
end;
let z be object;
assume
A3: z in L WHERE O; then
reconsider x = z as Element of X;
A4: z in L & x.O <> {} by A3,Th3; then
x in dom O by RELAT_1:170; then
x in dom NOT NOT O by Th38; then
x.NOT NOT O <> {} by RELAT_1:170;
hence thesis by A4,Th3;
end;
theorem
L WHEREeq(O,0) = L WHERE NOT O
proof
thus L WHEREeq(O,0) c= L WHERE NOT O
proof
let z be object; assume z in L WHEREeq(O,0); then
consider x such that
A1: z = x & card(x.O) = 0 & x in L;
x.O = {} by A1; then
x nin dom O by RELAT_1:170; then
[x,x] in NOT O by A1,Th36; then
x,x in NOT O;
hence thesis by A1;
end;
let z be object; assume z in L WHERE NOT O; then
consider x such that
A2: z = x & ex y st x,y in NOT O & x in L;
consider y such that
A3: x,y in NOT O & x in L by A2;
[x,y] in NOT O by A3; then
x = y & x in X & x nin dom O by Th36; then
x.O = {} by RELAT_1:170; then
card(x.O) = 0;
hence thesis by A2;
end;
theorem
NOT NOT NOT O = NOT O
proof
thus NOT NOT NOT O = id(X\dom NOT NOT O) by Th37
.= id(X\dom O) by Th38 .= NOT O by Th37;
end;
theorem
(NOT O1) OR NOT O2 c= NOT (O1 AND O2)
proof
let z,s be object;
assume [z,s] in (NOT O1) OR NOT O2; then
[z,s] in NOT O1 or [z,s] in NOT O2 by XBOOLE_0:def 3; then
A1: s = z & z in X & (z nin dom O1 or z nin dom O2) by Th36; then
z nin (dom O1) /\ dom O2 & dom(O1 AND O2) c= (dom O1)/\dom O2
by XBOOLE_0:def 4,XTUPLE_0:24; then
z nin dom(O1 AND O2);
hence thesis by A1,Th36;
end;
theorem
NOT (O1 OR O2) = (NOT O1) AND NOT O2
proof
let z,s be object;
thus [z,s] in NOT (O1 OR O2) implies [z,s] in (NOT O1) AND NOT O2
proof
assume [z,s] in NOT (O1 OR O2); then
A1: z = s & z in X & z nin dom (O1 OR O2) by Th36; then
z nin (dom O1)\/dom O2 by XTUPLE_0:23; then
z nin dom O1 & z nin dom O2 by XBOOLE_0:def 3; then
[z,s] in NOT O1 & [z,s] in NOT O2 by A1,Th36;
hence thesis by XBOOLE_0:def 4;
end;
assume [z,s] in (NOT O1) AND NOT O2; then
[z,s] in NOT O1 & [z,s] in NOT O2 by XBOOLE_0:def 4; then
A2: z = s & z in X & z nin dom O1 & z nin dom O2 by Th36; then
z nin (dom O1)\/dom O2 by XBOOLE_0:def 3; then
z nin dom (O1 OR O2) by XTUPLE_0:23;
hence thesis by A2,Th36;
end;
theorem
dom O1 = X & dom O2 = X implies (O1 OR O2)\& O = (O1 \& O) AND (O2 \& O)
proof assume
A1: dom O1 = X & dom O2 = X;
let z,s be object;
thus [z,s] in (O1 OR O2)\& O implies [z,s] in (O1 \& O) AND (O2 \& O)
proof
assume
A2: [z,s] in (O1 OR O2)\& O; then
reconsider z,s as Element of X by ZFMISC_1:87;
s in z.((O1 OR O2)\& O) by A2,RELAT_1:169; then
s in z.(O1 OR O2)\&O by Th35; then
s in ((z.O1) OR (z.O2))\&O & z.O1 <> {} & z.O2 <> {}
by A1,RELAT_1:170,RELSET_2:10; then
s in (z.O1\&O)AND(z.O2\&O) by Th24; then
s in (z.(O1\&O))AND(z.O2\&O) by Th35; then
s in (z.(O1\&O))AND(z.(O2\&O)) by Th35; then
s in z.((O1\&O)AND(O2\&O)) by RELSET_2:11;
hence thesis by RELAT_1:169;
end;
assume
A3: [z,s] in (O1 \& O) AND (O2 \& O); then
reconsider z,s as Element of X by ZFMISC_1:87;
s in z.((O1 \& O) AND (O2 \& O)) by A3,RELAT_1:169; then
s in (z.(O1\&O))AND(z.(O2\&O)) by RELSET_2:11; then
s in (z.(O1\&O))AND(z.O2\&O) by Th35; then
s in (z.O1\&O)AND(z.O2\&O) & z.O1 <> {} & z.O2 <> {}
by Th35,A1,RELAT_1:170; then
s in ((z.O1) OR (z.O2))\&O by Th24; then
s in (z.(O1 OR O2))\&O by RELSET_2:10; then
s in z.((O1 OR O2)\&O) by Th35;
hence thesis by RELAT_1:169;
end;
definition
let X,O;
attr O is filtering means :Def21: O c= id X;
end;
theorem Th45:
O is filtering iff O = id dom O
proof
thus O is filtering implies O = id dom O
proof
assume
A1: O c= id X;
let z,s be object;
thus [z,s] in O implies [z,s] in id dom O
proof
assume [z,s] in O; then
z in dom O & z = s by A1,RELAT_1:def 10,XTUPLE_0:def 12;
hence thesis by RELAT_1:def 10;
end;
assume [z,s] in id dom O; then
A2: z in dom O & z = s by RELAT_1:def 10; then
consider y being object such that
A3: [z,y] in O by XTUPLE_0:def 12;
thus thesis by A1,A2,A3,RELAT_1:def 10;
end;
assume O = id dom O;
hence O c= id X by SYSREL:15;
end;
registration
let X,O;
cluster NOT O -> filtering;
coherence
proof
NOT O = id(X\dom O) by Th37;
hence NOT O c= id X by SYSREL:15;
end;
end;
registration
let X;
cluster filtering for Operation of X;
existence
proof
set O = the Operation of X;
take NOT O; thus thesis;
end;
end;
reserve F,F1,F2 for filtering Operation of X;
registration
let X,F,O;
cluster F AND O -> filtering for Operation of X;
coherence
proof
let O1; assume O1 = F AND O; then
O1 c= F & F c= id X by Def21,XBOOLE_1:17;
hence O1 c= id X;
end;
cluster O AND F -> filtering for Operation of X;
coherence;
cluster F BUTNOT O -> filtering for Operation of X;
coherence
proof
let O1; assume O1 = F BUTNOT O; then
O1 c= F & F c= id X by Def21,XBOOLE_1:36;
hence O1 c= id X;
end;
end;
registration
let X,F1,F2;
cluster F1 OR F2 -> filtering for Operation of X;
coherence
proof
let O1; assume A1: O1 = F1 OR F2;
F1 c= id X & F2 c= id X by Def21;
hence O1 c= id X by A1,XBOOLE_1:8;
end;
end;
theorem Th46:
z in x.F implies z = x
proof
assume z in x.F; then
[x,z] in F & F c= id X by Def21,RELAT_1:169;
hence thesis by RELAT_1:def 10;
end;
theorem
L|F = L WHERE F
proof
thus L|F c= L WHERE F
proof
let z be object;
assume z in L|F; then
consider y such that
A1: z in y.F & y in L by Th23;
z = y by A1,Th46;
hence thesis by A1,Th3;
end;
let z be object;
assume
A2: z in L WHERE F; then
reconsider x = z as Element of X;
A3: x in L & x.F <> {} by A2,Th3;
set y = the Element of x.F;
A4: [x,y] in F by A3,RELAT_1:169;
F c= id X by Def21; then
x = y by A4,RELAT_1:def 10;
hence thesis by A3,Th23;
end;
theorem
NOT NOT F = F
proof
thus NOT NOT F = id(X\dom NOT F) by Th37
.= id (X\dom id(X\dom F)) by Th37
.= id (X\(X\dom F)) by RELAT_1:45
.= id (X/\dom F) by XBOOLE_1:48
.= id dom F by XBOOLE_1:28
.= F by Th45;
end;
theorem
NOT (F1 AND F2) = (NOT F1) OR (NOT F2)
proof
A1: F1 = id dom F1 & F2 = id dom F2 by Th45;
NOT (F1 AND F2) = id(X\dom(F1 AND F2)) by Th37
.= id (X\dom id ((dom F1) AND dom F2)) by A1,SYSREL:14
.= id (X\((dom F1) AND dom F2)) by RELAT_1:45
.= id ((X\dom F1)\/(X\dom F2)) by XBOOLE_1:54
.= id (X\dom F1) \/ id (X\dom F2) by SYSREL:14
.= (NOT F1) \/ id (X\dom F2) by Th37;
hence thesis by Th37;
end;
theorem Th50:
dom(O OR NOT O) = X
proof
thus dom(O OR NOT O) = (dom O) OR dom NOT O by XTUPLE_0:23
.= (dom O) \/ dom id (X\dom O) by Th37
.= (dom O) OR (X\dom O) by RELAT_1:45
.= (dom O) \/ X by XBOOLE_1:39
.= X by XBOOLE_1:12;
end;
theorem
F OR NOT F = id X
proof
dom(F OR NOT F) = X & F OR NOT F c= id X by Th50,Def21;
hence thesis by SYSREL:17;
end;
theorem Th52:
O AND NOT O = {}
proof
dom (O AND NOT O) = dom(O /\ id(X\dom O)) by Th37; then
A1: dom (O AND NOT O) c= (dom O) /\ dom id(X\dom O) by XTUPLE_0:24;
(dom O) /\ dom id(X\dom O) = (dom O) /\ (X\dom O) by RELAT_1:45
.= ((dom O) /\ X)\dom O by XBOOLE_1:49
.= (dom O)\dom O by XBOOLE_1:28
.= {} by XBOOLE_1:37;
hence thesis by A1,RELAT_1:41,XBOOLE_1:3;
end;
theorem
(O1 OR O2) AND NOT O1 c= O2
proof
(O1 OR O2) AND NOT O1 = (O1 AND NOT O1)OR (O2 AND NOT O1) by XBOOLE_1:23
.= {} \/ (O2 AND NOT O1) by Th52
.= O2 AND NOT O1;
hence thesis by XBOOLE_1:17;
end;
begin :: Rough queries
reserve i for Element of NAT;
definition
let A be FinSequence;
let a be object;
func #occurrences(a,A) -> Nat equals card {i: i in dom A & a in A.i};
coherence
proof
{i: i in dom A & a in A.i} c= dom A
proof
let z be object; assume z in {i: i in dom A & a in A.i}; then
ex i st z = i & i in dom A & a in A.i;
hence thesis;
end;
hence thesis;
end;
end;
theorem Th54:
for A being FinSequence, a being set holds #occurrences(a,A) <= len A
proof
let A be FinSequence;
let a be set;
{i: i in dom A & a in A.i} c= dom A
proof
let z be object; assume z in {i: i in dom A & a in A.i}; then
ex i st z = i & i in dom A & a in A.i;
hence thesis;
end; then
Segm #occurrences(a,A) c= Segm card dom A & dom A = Seg len A &
card Seg len A = len A by CARD_1:11,FINSEQ_1:57,def 3;
hence #occurrences(a,A) <= len A by NAT_1:39;
end;
theorem Th55:
for A being FinSequence, a being set holds
A <> {} & #occurrences(a,A) = len A iff a in meet rng A
proof
let A be FinSequence;
let a be set;
A1: {i: i in dom A & a in A.i} c= dom A
proof
let z be object; assume z in {i: i in dom A & a in A.i}; then
ex i st z = i & i in dom A & a in A.i;
hence thesis;
end;
A2: dom A = Seg len A & card Seg len A = len A by FINSEQ_1:57,def 3;
hereby
assume
A <> {}; then
A3: rng A <> {} by RELAT_1:41;
assume
A4: #occurrences(a,A) = len A;
A5: {i: i in dom A & a in A.i} = dom A by A4,A1,A2,CARD_2:102;
now
let z; assume z in rng A; then
consider s being object such that
A6: s in dom A & z = A.s by FUNCT_1:def 3;
consider i such that
A7: s = i & i in dom A & a in A.i by A5,A6;
thus a in z by A6,A7;
end;
hence a in meet rng A by A3,SETFAM_1:def 1;
end;
assume
A8: a in meet rng A;
thus A <> {} by A8,RELAT_1:38,SETFAM_1:def 1;
dom A c= {i: i in dom A & a in A.i}
proof
let z be object; assume
A9: z in dom A; then
A.z in rng A by FUNCT_1:def 3; then
a in A.z by A8,SETFAM_1:def 1;
hence thesis by A9;
end;
hence #occurrences(a,A) = len A by A2,A1,XBOOLE_0:def 10;
end;
definition
let A be FinSequence;
func max# A -> Nat means :Def23:
(for a being set holds #occurrences(a,A) <= it) &
(for n st for a being set holds #occurrences(a,A) <= n holds it <= n);
existence
proof
defpred P[Nat] means for a being set holds #occurrences(a,A) <= $1;
P[len A] by Th54; then
A1: ex n st P[n];
consider n such that
A2: P[n] & for m being Nat st P[m] holds n <= m from NAT_1:sch 5(A1);
take n; thus thesis by A2;
end;
uniqueness
proof
let n1,n2 be Nat such that
A3: (for a being set holds #occurrences(a,A) <= n1) &
(for n st for a being set holds #occurrences(a,A) <= n holds n1 <= n) and
A4: (for a being set holds #occurrences(a,A) <= n2) &
(for n st for a being set holds #occurrences(a,A) <= n holds n2 <= n);
n1 <= n2 & n2 <= n1 by A3,A4;
hence thesis by XXREAL_0:1;
end;
end;
theorem Th56:
for A being FinSequence holds max# A <= len A
proof
let A be FinSequence;
for a being set holds #occurrences(a,A) <= len A by Th54;
hence max# A <= len A by Def23;
end;
theorem
for A being FinSequence, a being set st #occurrences(a,A) = len A holds
max# A = len A
proof
let A be FinSequence;
let a be set;
assume #occurrences(a,A) = len A; then
len A <= max# A & max# A <= len A by Def23,Th56;
hence max# A = len A by XXREAL_0:1;
end;
definition
let X;
let A be FinSequence of bool X;
let n be Nat;
func ROUGH(A,n) -> List of X equals :Def24:
{x: n <= #occurrences(x,A)} if X <> {};
consistency;
coherence
proof
assume
A1: X <> {};
{x: n <= #occurrences(x,A)} c= X
proof
let z be object; assume z in {x: n <= #occurrences(x,A)}; then
ex x st z = x & n <= #occurrences(x,A);
hence thesis by A1;
end;
hence {x: n <= #occurrences(x,A)} is List of X;
end;
let m be Nat;
func ROUGH(A,n,m) -> List of X equals :Def25:
{x: n <= #occurrences(x,A) & #occurrences(x,A) <= m} if X <> {};
consistency;
coherence
proof
assume
A2: X <> {};
{x: n <= #occurrences(x,A) & #occurrences(x,A) <= m} c= X
proof
let z be object;
assume z in {x: n <= #occurrences(x,A) & #occurrences(x,A) <= m}; then
ex x st z = x & n <= #occurrences(x,A) & #occurrences(x,A) <= m;
hence thesis by A2;
end;
hence thesis;
end;
end;
definition
let X;
let A be FinSequence of bool X;
func ROUGH(A) -> List of X equals ROUGH(A, max# A);
coherence;
end;
theorem
for A being FinSequence of bool X holds
ROUGH(A, n, len A) = ROUGH(A, n)
proof
let A be FinSequence of bool X;
thus ROUGH(A, n, len A) c= ROUGH(A, n)
proof
let z be object; assume
A1: z in ROUGH(A,n,len A); then
z in {x: n <= #occurrences(x,A) & #occurrences(x,A) <= len A} by Def25;
then ex x st z = x & n <= #occurrences(x,A) & #occurrences(x,A) <= len A;
then z in {x: n <= #occurrences(x,A)};
hence thesis by A1,Def24;
end;
let z be object; assume
A2: z in ROUGH(A, n); then
z in {x: n <= #occurrences(x,A)} by Def24; then
consider x such that
A3: z = x & n <= #occurrences(x,A);
#occurrences(x,A) <= len A by Th54; then
z in {x1 where x1 is Element of X: n <= #occurrences(x1,A) &
#occurrences(x1,A) <= len A} by A3;
hence thesis by A2,Def25;
end;
theorem
for A being FinSequence of bool X holds
n <= m implies ROUGH(A,m) c= ROUGH(A,n)
proof
let A be FinSequence of bool X;
assume A1: n <= m;
let z be object; assume
A2: z in ROUGH(A,m); then
z in {x: m <= #occurrences(x,A)} by Def24; then
consider a such that
A3: z = a & m <= #occurrences(a,A);
n <= #occurrences(a,A) by A1,A3,XXREAL_0:2; then
z in {x: n <= #occurrences(x,A)} by A3;
hence z in ROUGH(A,n) by A2,Def24;
end;
theorem
for A being FinSequence of bool X holds
for n1,n2,m1,m2 being Nat st n1 <= m1 & n2 <= m2 holds
ROUGH(A,m1,n2) c= ROUGH(A,n1,m2)
proof
let A be FinSequence of bool X;
let n1,n2,m1,m2 be Nat;
assume A1: n1 <= m1;
assume A2: n2 <= m2;
let z be object; assume
A3: z in ROUGH(A,m1,n2); then
z in {x: m1 <= #occurrences(x,A) & #occurrences(x,A) <= n2} by Def25; then
consider a such that
A4: z = a & m1 <= #occurrences(a,A) & #occurrences(a,A) <= n2;
n1 <= #occurrences(a,A) & #occurrences(a,A) <= m2
by A1,A2,A4,XXREAL_0:2; then
z in {x: n1 <= #occurrences(x,A) & #occurrences(x,A) <= m2} by A4;
hence z in ROUGH(A,n1,m2) by A3,Def25;
end;
theorem
for A being FinSequence of bool X holds
ROUGH(A,n,m) c= ROUGH(A,n)
proof
let A be FinSequence of bool X;
let z be object; assume
A1: z in ROUGH(A,n,m); then
z in {x: n <= #occurrences(x,A) & #occurrences(x,A) <= m} by Def25; then
ex x st z = x & n <= #occurrences(x,A) & #occurrences(x,A) <= m; then
z in {x: n <= #occurrences(x,A)};
hence z in ROUGH(A,n) by A1,Def24;
end;
theorem Th62:
for A being FinSequence of bool X st A <> {} holds
ROUGH(A, len A) = meet rng A
proof
let A be FinSequence of bool X such that
A1: A <> {};
thus ROUGH(A, len A) c= meet rng A
proof
let z be object; assume
z in ROUGH(A, len A); then
z in {x: len A <= #occurrences(x,A)} by Def24; then
consider x such that
A2: z = x & len A <= #occurrences(x,A);
#occurrences(x,A) <= len A by Th54;
hence thesis by A1,A2,Th55,XXREAL_0:1;
end;
let z be object; assume
A3: z in meet rng A; then
#occurrences(z,A) = len A by Th55; then
z in {x: len A <= #occurrences(x,A)} by A3;
hence thesis by A3,Def24;
end;
theorem Th63:
for A being FinSequence of bool X holds
ROUGH(A, 1) = Union A
proof
let A be FinSequence of bool X;
thus ROUGH(A, 1) c= Union A
proof
let z be object; assume
z in ROUGH(A,1); then
z in {x: 1 <= #occurrences(x,A)} by Def24; then
consider x such that
A1: z = x & 1 <= #occurrences(x,A);
1 = 0+1; then
#occurrences(x,A) > 0 by A1,NAT_1:13; then
{i: i in dom A & x in A.i} <> {}; then
consider s being object such that
A2: s in {i: i in dom A & x in A.i} by XBOOLE_0:def 1;
consider i such that
A3: s = i & i in dom A & x in A.i by A2;
thus thesis by A1,A3,CARD_5:2;
end;
let z be object; assume
A4: z in Union A; then
consider s being object such that
A5: s in dom A & z in A.s by CARD_5:2;
s in {i: i in dom A & z in A.i} by A5; then
card {s} c= #occurrences(z,A) by CARD_1:11,ZFMISC_1:31; then
Segm 1 c= Segm #occurrences(z,A) by CARD_1:30; then
1 <= #occurrences(z,A) by NAT_1:39; then
z in {x: 1 <= #occurrences(x,A)} by A4;
hence thesis by A4,Def24;
end;
theorem
for L1,L2 being List of X holds
ROUGH(<*L1,L2*>,2) = L1 AND L2
proof
let L1,L2 be List of X;
len <*L1,L2*> = 2 & rng <*L1,L2*> = {L1,L2} & meet {L1,L2} = L1 AND L2
by FINSEQ_1:44,FINSEQ_2:127,SETFAM_1:11;
hence ROUGH(<*L1,L2*>,2) = L1 AND L2 by Th62;
end;
theorem
for L1,L2 being List of X holds
ROUGH(<*L1,L2*>,1) = L1 OR L2
proof
let L1,L2 be List of X;
len <*L1,L2*> = 2 & rng <*L1,L2*> = {L1,L2} & union {L1,L2} = L1 OR L2 &
Union <*L1,L2*> = union rng <*L1,L2*>
by CARD_3:def 4,FINSEQ_1:44,FINSEQ_2:127,ZFMISC_1:75;
hence ROUGH(<*L1,L2*>,1) = L1 OR L2 by Th63;
end;
begin :: Constructor Database
definition
struct(1-sorted) ConstructorDB(#
carrier -> set,
Constrs -> List of the carrier,
ref-operation -> Relation of the carrier, the Constrs
#);
end;
definition
let X be 1-sorted;
mode List of X is List of the carrier of X;
mode Operation of X is Operation of the carrier of X;
end;
definition
let X;
let S be Subset of X;
let R be Relation of X,S;
func @R -> Relation of X equals R;
coherence
proof
[:X,S:] c= [:X,X:] by ZFMISC_1:96;
hence thesis by XBOOLE_1:1;
end;
end;
definition
let X be ConstructorDB;
let a be Element of X;
func a ref -> List of X equals a.@the ref-operation of X;
coherence;
func a occur -> List of X equals a.((@the ref-operation of X)~);
coherence;
end;
theorem Th66:
for X being ConstructorDB
for x,y being Element of X holds x in y ref iff y in x occur
proof
let X be ConstructorDB;
let x,y be Element of X;
hereby
assume x in y ref; then
[y,x] in @the ref-operation of X by RELAT_1:169; then
[x,y] in (@the ref-operation of X)~ by RELAT_1:def 7;
hence y in x occur by RELAT_1:169;
end;
assume y in x occur; then
[x,y] in (@the ref-operation of X)~ by RELAT_1:169; then
[y,x] in @the ref-operation of X by RELAT_1:def 7;
hence x in y ref by RELAT_1:169;
end;
definition
let X be ConstructorDB;
attr X is ref-finite means :Def30:
for x being Element of X holds x ref is finite;
end;
registration
cluster finite -> ref-finite for ConstructorDB;
coherence
proof
let X be ConstructorDB;
assume
A1: the carrier of X is finite;
let x be Element of X;
thus thesis by A1;
end;
end;
registration
cluster finite non empty for ConstructorDB;
existence
proof
set X = the finite non empty set;
set C = the Subset of X;
set R = the Relation of X,C;
take D = ConstructorDB(#X,C,R#);
thus the carrier of D is finite non empty;
end;
end;
registration
let X be ref-finite ConstructorDB;
let x be Element of X;
cluster x ref -> finite;
coherence by Def30;
end;
definition
let X be ConstructorDB;
let A be FinSequence of the Constrs of X;
func ATLEAST(A) -> List of X equals :Def31:
{x where x is Element of X: rng A c= x ref}
if the carrier of X <> {};
consistency;
coherence
proof
assume
A1: the carrier of X <> {};
{x where x is Element of X: rng A c= x ref} c= the carrier of X
proof
let z be object;
assume z in {x where x is Element of X: rng A c= x ref}; then
ex x being Element of X st z = x & rng A c= x ref;
hence thesis by A1;
end;
hence thesis;
end;
func ATMOST(A) -> List of X equals :Def32:
{x where x is Element of X: x ref c= rng A}
if the carrier of X <> {};
consistency;
coherence
proof
assume
A2: the carrier of X <> {};
{x where x is Element of X: x ref c= rng A} c= the carrier of X
proof
let z be object;
assume z in {x where x is Element of X: x ref c= rng A}; then
ex x being Element of X st z = x & x ref c= rng A;
hence thesis by A2;
end;
hence thesis;
end;
func EXACTLY(A) -> List of X equals :Def33:
{x where x is Element of X: x ref = rng A}
if the carrier of X <> {};
consistency;
coherence
proof
assume
A3: the carrier of X <> {};
{x where x is Element of X: x ref = rng A} c= the carrier of X
proof
let z be object;
assume z in {x where x is Element of X: x ref = rng A}; then
ex x being Element of X st z = x & x ref = rng A;
hence thesis by A3;
end;
hence thesis;
end;
let n be Nat;
func ATLEAST-(A,n) -> List of X equals :Def34:
{x where x is Element of X: card((rng A) \ x ref) <= n}
if the carrier of X <> {};
consistency;
coherence
proof
assume
A4: the carrier of X <> {};
{x where x is Element of X: card((rng A)\x ref) <= n} c= the carrier of X
proof
let z be object; assume
z in {x where x is Element of X: card((rng A)\x ref) <= n}; then
ex x being Element of X st z = x & card((rng A)\x ref) <= n;
hence thesis by A4;
end;
hence thesis;
end;
end;
definition
let X be ref-finite ConstructorDB;
let A be FinSequence of the Constrs of X;
let n be Nat;
func ATMOST+(A,n) -> List of X equals :Def35:
{x where x is Element of X: card((x ref) \ rng A) <= n}
if the carrier of X <> {};
consistency;
coherence
proof
assume
A1: the carrier of X <> {};
{x where x is Element of X: card((x ref)\rng A) <= n} c= the carrier of X
proof
let z be object; assume
z in {x where x is Element of X: card((x ref)\rng A) <= n}; then
ex x being Element of X st z = x & card((x ref)\rng A) <= n;
hence thesis by A1;
end;
hence thesis;
end;
let m be Nat;
func EXACTLY+-(A,n,m) -> List of X equals :Def36:
{x where x is Element of X: card((x ref) \ rng A) <= n &
card((rng A) \ x ref) <= m}
if the carrier of X <> {};
consistency;
coherence
proof
assume
A2: the carrier of X <> {};
{x where x is Element of X: card((x ref) \ rng A) <= n &
card((rng A)\x ref) <= m} c= the carrier of X
proof
let z be object; assume
z in {x where x is Element of X: card((x ref) \ rng A) <= n &
card((rng A)\x ref) <= m}; then
ex x being Element of X st z = x & card((x ref) \ rng A) <= n &
card((rng A)\x ref) <= m;
hence thesis by A2;
end;
hence thesis;
end;
end;
reserve X for ConstructorDB, A for FinSequence of the Constrs of X,
x for Element of X;
reserve Y for ref-finite ConstructorDB,
B for FinSequence of the Constrs of Y,
y for Element of Y;
theorem Th67:
ATLEAST-(A,0) = ATLEAST(A)
proof
per cases;
suppose the carrier of X = {}; then
ATLEAST-(A,0) = {} & ATLEAST(A) = {};
hence thesis;
end;
suppose
A1: the carrier of X <> {}; then
A2: ATLEAST-(A,0) = {x: card((rng A)\x ref) <= 0} by Def34;
A3: ATLEAST(A) = {x: rng A c= x ref} by A1,Def31;
thus ATLEAST-(A,0) c= ATLEAST(A)
proof
let z be object; assume z in ATLEAST-(A,0); then
consider x such that
A4: z = x & card((rng A)\x ref) <= 0 by A2;
(rng A)\x ref = {} by A4,NAT_1:3; then
rng A c= x ref by XBOOLE_1:37;
hence thesis by A4,A3;
end;
let z be object; assume z in ATLEAST(A); then
consider x such that
A5: z = x & rng A c= x ref by A3;
(rng A)\x ref = {} by A5,XBOOLE_1:37; then
card((rng A)\x ref) = 0;
hence z in ATLEAST-(A,0) by A2,A5;
end;
end;
theorem Th68:
ATMOST+(B,0) = ATMOST(B)
proof
per cases;
suppose the carrier of Y = {}; then
ATMOST+(B,0) = {} & ATMOST(B) = {};
hence thesis;
end;
suppose
A1: the carrier of Y <> {}; then
A2: ATMOST+(B,0) = {y: card((y ref)\rng B) <= 0} by Def35;
A3: ATMOST(B) = {y: y ref c= rng B} by A1,Def32;
thus ATMOST+(B,0) c= ATMOST(B)
proof
let z be object; assume z in ATMOST+(B,0); then
consider y such that
A4: z = y & card((y ref)\rng B) <= 0 by A2;
(y ref)\rng B = {} by A4,NAT_1:3; then
y ref c= rng B by XBOOLE_1:37;
hence thesis by A4,A3;
end;
let z be object; assume z in ATMOST(B); then
consider y such that
A5: z = y & y ref c= rng B by A3;
(y ref)\rng B = {} by A5,XBOOLE_1:37; then
card((y ref)\rng B) = 0;
hence z in ATMOST+(B,0) by A2,A5;
end;
end;
theorem Th69:
EXACTLY+-(B,0,0) = EXACTLY(B)
proof
per cases;
suppose the carrier of Y = {}; then
EXACTLY+-(B,0,0) = {} & EXACTLY(B) = {};
hence thesis;
end;
suppose
A1: the carrier of Y <> {}; then
A2: EXACTLY+-(B,0,0) = {y: card((y ref)\rng B) <= 0 & card((rng B)\y ref)<=0}
by Def36;
A3: EXACTLY(B) = {y: y ref = rng B} by A1,Def33;
thus EXACTLY+-(B,0,0) c= EXACTLY(B)
proof
let z be object; assume z in EXACTLY+-(B,0,0); then
consider y such that
A4: z = y & card((y ref)\rng B) <= 0 & card((rng B)\y ref)<=0 by A2;
(y ref)\rng B = {} & (rng B)\y ref = {}
by A4,NAT_1:3; then
y ref c= rng B & rng B c= y ref by XBOOLE_1:37; then
y ref = rng B;
hence thesis by A4,A3;
end;
let z be object; assume z in EXACTLY(B); then
consider y such that
A5: z = y & y ref = rng B by A3;
(y ref)\rng B = {} & (rng B)\y ref = {} by A5,XBOOLE_1:37; then
card((y ref)\rng B) = 0 & card((rng B)\y ref) = 0;
hence z in EXACTLY+-(B,0,0) by A2,A5;
end;
end;
theorem Th70:
n <= m implies ATLEAST-(A,n) c= ATLEAST-(A,m)
proof
assume
A1: n <= m;
let z be object; assume
A2: z in ATLEAST-(A,n); then
z in {x: card((rng A)\x ref) <= n} by Def34; then
consider x such that
A3: z = x & card((rng A)\x ref) <= n;
card((rng A)\x ref) <= m by A1,A3,XXREAL_0:2; then
x in {x1 where x1 is Element of X: card((rng A)\x1 ref) <= m};
hence thesis by A2,A3,Def34;
end;
theorem Th71:
n <= m implies ATMOST+(B,n) c= ATMOST+(B,m)
proof
assume
A1: n <= m;
let z be object; assume
A2: z in ATMOST+(B,n); then
z in {y: card((y ref)\rng B) <= n} by Def35; then
consider y such that
A3: z = y & card((y ref)\rng B) <= n;
card((y ref)\rng B) <= m by A1,A3,XXREAL_0:2; then
y in {x1 where x1 is Element of Y: card((x1 ref)\rng B) <= m};
hence thesis by A2,A3,Def35;
end;
theorem Th72:
for n1,n2,m1,m2 being Nat st n1 <= m1 & n2 <= m2 holds
EXACTLY+-(B,n1,n2) c= EXACTLY+-(B,m1,m2)
proof let n1,n2,m1,m2 be Nat;
assume
A1: n1 <= m1 & n2 <= m2;
let z be object; assume
A2: z in EXACTLY+-(B,n1,n2); then
z in {y: card((y ref)\rng B) <= n1 & card((rng B)\y ref) <= n2}
by Def36; then
consider y such that
A3: z = y & card((y ref)\rng B) <= n1 & card((rng B)\y ref) <= n2;
card((y ref)\rng B) <= m1 & card((rng B)\y ref) <= m2
by A1,A3,XXREAL_0:2; then
y in {x1 where x1 is Element of Y: card((x1 ref)\rng B) <= m1 &
card((rng B)\x1 ref) <= m2};
hence thesis by A2,A3,Def36;
end;
theorem
ATLEAST(A) c= ATLEAST-(A,n)
proof
ATLEAST(A) = ATLEAST-(A,0) & 0 <= n by Th67,NAT_1:2;
hence thesis by Th70;
end;
theorem
ATMOST(B) c= ATMOST+(B,n)
proof
ATMOST(B) = ATMOST+(B,0) & 0 <= n by Th68,NAT_1:2;
hence thesis by Th71;
end;
theorem
EXACTLY(B) c= EXACTLY+-(B,n,m)
proof
EXACTLY(B) = EXACTLY+-(B,0,0) & 0 <= n & 0 <= m by Th69,NAT_1:2;
hence thesis by Th72;
end;
theorem
EXACTLY A = ATLEAST A AND ATMOST A
proof
thus EXACTLY A c= ATLEAST A AND ATMOST A
proof
let z be object; assume
A1: z in EXACTLY A; then
z in {x where x is Element of X: (x ref) = rng A} by Def33; then
consider x being Element of X such that
A2: z = x & (x ref) = rng A;
z in {y where y is Element of X: (rng A) c= y ref} by A2; then
A3: z in ATLEAST A by A1,Def31;
z in {y where y is Element of X: (y ref) c= rng A} by A2; then
z in ATMOST A by A1,Def32;
hence thesis by A3,XBOOLE_0:def 4;
end;
let z be object; assume
A4: z in ATLEAST A AND ATMOST A; then
A5: z in ATLEAST A & z in ATMOST A by XBOOLE_0:def 4; then
z in {y where y is Element of X: (y ref) c= rng A} by Def32; then
consider a being Element of X such that
A6: z = a & (a ref) c= rng A;
z in {y where y is Element of X: (rng A) c= y ref} by A5,Def31; then
consider b being Element of X such that
A7: z = b & (rng A) c= b ref;
z = a & a ref = rng A by A6,A7; then
z in {y where y is Element of X: (y ref) = rng A};
hence thesis by A4,Def33;
end;
theorem
EXACTLY+-(B,n,m) = ATLEAST-(B,m) AND ATMOST+(B,n)
proof
thus EXACTLY+-(B,n,m) c= ATLEAST-(B,m) AND ATMOST+(B,n)
proof
let z be object; assume
A1: z in EXACTLY+-(B,n,m); then
z in {x where x is Element of Y: card((x ref) \ rng B) <= n &
card((rng B) \ x ref) <= m} by Def36; then
consider x being Element of Y such that
A2: z = x & card((x ref) \ rng B) <= n & card((rng B) \ x ref) <= m;
z in {y: card((rng B) \ y ref) <= m} by A2; then
A3: z in ATLEAST-(B,m) by A1,Def34;
z in {y: card((y ref) \ rng B) <= n} by A2; then
z in ATMOST+(B,n) by A1,Def35;
hence z in ATLEAST-(B,m) AND ATMOST+(B,n) by A3,XBOOLE_0:def 4;
end;
let z be object; assume
A4: z in ATLEAST-(B,m) AND ATMOST+(B,n); then
A5: z in ATLEAST-(B,m) & z in ATMOST+(B,n) by XBOOLE_0:def 4; then
z in {y: card((y ref) \ rng B) <= n} by Def35; then
A6: ex y st z = y & card((y ref) \ rng B) <= n;
z in {y: card((rng B) \ y ref) <= m} by A5,Def34; then
ex y st z = y & card((rng B) \ y ref) <= m; then
z in {y: card((y ref) \ rng B) <= n & card((rng B) \ y ref) <= m} by A6;
hence thesis by A4,Def36;
end;
theorem Th78:
A <> {} implies ATLEAST A = meet {x occur: x in rng A}
proof assume
A <> {}; then
rng A <> {} by RELAT_1:41; then
consider s being object such that
A1: s in rng A by XBOOLE_0:def 1;
s in the Constrs of X & the Constrs of X c= the carrier of X
by A1; then
reconsider s as Element of X;
A2: s occur in {x occur: x in rng A} by A1;
thus ATLEAST A c= meet {x occur: x in rng A}
proof
let z be object; assume z in ATLEAST A; then
z in {y where y is Element of X: (rng A) c= y ref} by Def31; then
consider a being Element of X such that
A3: z = a & (rng A) c= a ref;
now let Y be set;
assume Y in {x occur: x in rng A}; then
consider x such that
A4: Y = x occur & x in rng A;
thus z in Y by A3,A4,Th66;
end;
hence thesis by A2,SETFAM_1:def 1;
end;
let z be object; assume
A5: z in meet {x occur: x in rng A};
then
A6: z in s occur by A2,SETFAM_1:def 1;
then reconsider z as Element of X;
rng A c= z ref
proof
let s be object; assume
A7: s in rng A; then
s in the Constrs of X; then
reconsider s as Element of X;
s occur in {x occur: x in rng A} by A7; then
z in s occur by A5,SETFAM_1:def 1;
hence thesis by Th66;
end; then
z in {x: rng A c= x ref};
hence thesis by A6,Def31;
end;
theorem
for c1,c2 being Element of X holds
A = <*c1,c2*> implies ATLEAST A = (c1 occur) AND (c2 occur)
proof
let c1,c2 be Element of X;
assume A1: A = <*c1,c2*>; then
A2: rng A = {c1,c2} by FINSEQ_2:127;
A3: {x occur: x in rng A} = {c1 occur, c2 occur}
proof
thus {x occur: x in rng A} c= {c1 occur, c2 occur}
proof
let z be object; assume z in {x occur: x in rng A}; then
consider x such that
A4: z = x occur & x in rng A;
x = c1 or x = c2 by A2,A4,TARSKI:def 2;
hence thesis by A4,TARSKI:def 2;
end;
let z be object; assume z in {c1 occur,c2 occur}; then
A5: z = c1 occur or z = c2 occur by TARSKI:def 2;
c1 in rng A & c2 in rng A by A2,TARSKI:def 2;
hence thesis by A5;
end;
thus ATLEAST A = meet {x occur: x in rng A} by A1,Th78
.= (c1 occur) AND (c2 occur) by A3,SETFAM_1:11;
end;