:: ..Kleene Algebra of Partial Predicates
:: by Artur Korni{\l}owicz , Ievgen Ivanov and Mykola Nikitchenko
::
:: Received March 27, 2018
:: Copyright (c) 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 PARTPR_1, SUBSET_1, XBOOLE_0, RELAT_1, FUNCT_1, PARTFUN1,
MARGREL1, XBOOLEAN, TARSKI, MONOID_0, FUNCOP_1, ZFMISC_1, BINOP_1,
LATTICES, EQREL_1, XXREAL_2, ROBBINS1, PBOOLE, STRUCT_0;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, MARGREL1, RELAT_1, RELSET_1,
FUNCT_1, PARTFUN1, FUNCT_2, FUNCOP_1, BINOP_1, STRUCT_0, MONOID_0,
LATTICES, ROBBINS1;
constructors RFUNCT_3, RELSET_1, MARGREL1, LATTICES, MONOID_0, ROBBINS1;
registrations XBOOLE_0, RELAT_1, FUNCT_1, PARTFUN1, FUNCOP_1, RELSET_1, NAT_1,
MONOID_0, LATTICES, ROBBINS1;
requirements NUMERALS, BOOLE, SUBSET;
definitions TARSKI, XBOOLE_0, FUNCT_1, FUNCT_2, BINOP_1, STRUCT_0, LATTICES,
ROBBINS1;
equalities MARGREL1, XBOOLEAN, LATTICES, ROBBINS1;
expansions TARSKI, MONOID_0, LATTICES, ROBBINS1;
theorems TARSKI, RELAT_1, RELSET_1, FUNCOP_1, BINOP_1, PARTFUN1, FUNCT_1,
FUNCT_2, ZFMISC_1, LATTICES;
schemes FUNCT_1, FUNCT_2, BINOP_1;
begin :: Partial predicates
reserve x for object;
reserve D for set;
definition
let D;
func Pr(D) -> set equals
PFuncs(D,BOOLEAN);
coherence;
end;
registration
let D;
cluster Pr(D) -> non empty functional;
coherence;
end;
definition
let D;
mode PartialPredicate of D is PartFunc of D,BOOLEAN;
end;
reserve p for PartialPredicate of D;
theorem
x in Pr(D) implies x is PartialPredicate of D by PARTFUN1:46;
theorem
p in Pr(D) by PARTFUN1:45;
theorem Th3:
x in dom p implies p.x = TRUE or p.x = FALSE
proof
assume that
A1: x in dom p;
A2: rng p c= BOOLEAN by RELAT_1:def 19;
p.x in rng p by A1,FUNCT_1:3;
hence thesis by A2,TARSKI:def 2;
end;
definition
let D;
defpred Dn[object,Function,object] means $1 in dom $2 & $2.$1 = $3;
func PPnegation(D) -> Function of Pr(D),Pr(D) means :Def2:
for p being PartialPredicate of D holds
dom(it.p) = dom p &
for d being object holds
(d in dom p & p.d = TRUE implies it.p.d = FALSE) &
(d in dom p & p.d = FALSE implies it.p.d = TRUE);
existence
proof
defpred P[object,object] means
for p being PartialPredicate of D st p = $1 holds
for f being Function st f = $2 holds
dom p = dom f &
for d being object holds
(Dn[d,p,TRUE] implies f.d = FALSE) & (Dn[d,p,FALSE] implies f.d = TRUE);
A1: for x being object st x in Pr(D)
ex y being object st y in Pr(D) & P[x,y]
proof
let x;
assume x in Pr(D);
then reconsider X = x as PartFunc of D,BOOLEAN by PARTFUN1:46;
defpred Q[object,object] means
for d being object st d = $1 holds
(Dn[d,X,TRUE] implies $2 = FALSE) & (Dn[d,X,FALSE] implies $2 = TRUE);
A2: for a being object st a in dom X
ex b being object st b in BOOLEAN & Q[a,b]
proof
let a be object;
assume a in dom X;
then X.a in BOOLEAN by PARTFUN1:4;
then per cases by TARSKI:def 2;
suppose
A3: X.a = TRUE;
take FALSE;
thus thesis by A3;
end;
suppose
A4: X.a = FALSE;
take TRUE;
thus thesis by A4;
end;
end;
consider g being Function such that
A5: dom g = dom X and
A6: rng g c= BOOLEAN and
A7: for x being object st x in dom X holds Q[x,g.x] from FUNCT_1:sch 6(A2);
take g;
g is PartFunc of D,BOOLEAN by A5,A6,RELSET_1:4;
hence g in Pr(D) by PARTFUN1:45;
thus thesis by A5,A7;
end;
consider F being Function of Pr(D),Pr(D) such that
A8: for x being object st x in Pr(D) holds P[x,F.x] from FUNCT_2:sch 1(A1);
take F;
let p be PartialPredicate of D;
p in Pr(D) by PARTFUN1:45;
hence thesis by A8;
end;
uniqueness
proof
let f,g be Function of Pr(D),Pr(D) such that
A9: for p being PartialPredicate of D holds
dom p = dom(f.p) &
for d being object holds
(Dn[d,p,TRUE] implies f.p.d = FALSE) & (Dn[d,p,FALSE] implies f.p.d = TRUE)
and
A10: for p being PartialPredicate of D holds
dom p = dom(g.p) &
for d being object holds
(Dn[d,p,TRUE] implies g.p.d = FALSE) &
(Dn[d,p,FALSE] implies g.p.d = TRUE);
let x be Element of Pr(D);
reconsider p = x as PartialPredicate of D by PARTFUN1:46;
thus dom(f.x) = dom p by A9
.= dom(g.x) by A10;
let a be object;
assume a in dom(f.x);
then
A11: a in dom p by A9;
then p.a in BOOLEAN by PARTFUN1:4;
then per cases by TARSKI:def 2;
suppose
A12: p.a = TRUE;
hence f.x.a = FALSE by A9,A11
.= g.x.a by A10,A11,A12;
end;
suppose
A13: p.a = FALSE;
hence f.x.a = TRUE by A9,A11
.= g.x.a by A10,A11,A13;
end;
end;
end;
definition
let D,p;
func PP_not(p) -> PartialPredicate of D equals
PPnegation(D).p;
coherence
proof
p in Pr(D) by PARTFUN1:45;
hence thesis by PARTFUN1:46,FUNCT_2:5;
end;
involutiveness
proof
let P,Q be PartialPredicate of D;
assume
A1: P = PPnegation(D).Q;
set Z = PPnegation(D).P;
A2: dom P = dom Q by A1,Def2;
hence dom Q = dom Z by Def2;
let x;
assume
A3: x in dom Q;
then per cases by A2,Th3;
suppose
A4: P.x = FALSE;
Q.x = TRUE
proof
assume Q.x <> TRUE;
then Q.x = FALSE by A3,Th3;
hence contradiction by A1,A3,A4,Def2;
end;
hence thesis by A2,A3,A4,Def2;
end;
suppose
A5: P.x = TRUE;
Q.x = FALSE
proof
assume Q.x <> FALSE;
then Q.x = TRUE by A3,Th3;
hence contradiction by A1,A3,A5,Def2;
end;
hence thesis by A2,A3,A5,Def2;
end;
end;
end;
theorem Th4:
x in dom p & PP_not(p).x = FALSE implies p.x = TRUE
proof
assume that
A1: x in dom p and
A2: PP_not(p).x = FALSE;
assume p.x <> TRUE;
then p.x = FALSE by A1,Th3;
hence thesis by A1,A2,Def2;
end;
theorem Th5:
x in dom p & PP_not(p).x = TRUE implies p.x = FALSE
proof
assume that
A1: x in dom p and
A2: PP_not(p).x = TRUE;
assume p.x <> FALSE;
then p.x = TRUE by A1,Th3;
hence thesis by A1,A2,Def2;
end;
theorem
x in dom PP_not(p) & PP_not(p).x = FALSE implies x in dom p & p.x = TRUE
proof
assume that
A1: x in dom PP_not(p) and
A2: PP_not(p).x = FALSE;
thus
A3: x in dom p by A1,Def2;
assume p.x <> TRUE;
then p.x = FALSE by A3,Th3;
hence thesis by A2,A3,Def2;
end;
theorem
x in dom PP_not(p) & PP_not(p).x = TRUE implies x in dom p & p.x = FALSE
proof
assume that
A1: x in dom PP_not(p) and
A2: PP_not(p).x = TRUE;
thus
A3: x in dom p by A1,Def2;
assume p.x <> FALSE;
then p.x = TRUE by A3,Th3;
hence thesis by A2,A3,Def2;
end;
reserve D for non empty set;
reserve p,q,r for PartialPredicate of D;
definition
let D;
defpred D1[object,Function] means $1 in dom $2 & $2.$1 = TRUE;
defpred D2[object,Function,Function] means
$1 in dom $2 & $2.$1 = FALSE & $1 in dom $3 & $3.$1 = FALSE;
deffunc Dany(Function,Function) =
{d where d is Element of D:
D1[d,$1] or D1[d,$2] or D2[d,$1,$2]};
func PPdisjunction(D) -> Function of [:Pr(D),Pr(D):],Pr(D) means
:Def4:
for p,q being PartialPredicate of D holds
dom(it.(p,q)) = {d where d is Element of D:
d in dom p & p.d = TRUE or d in dom q & q.d = TRUE
or d in dom p & p.d = FALSE & d in dom q & q.d = FALSE} &
for d being object holds
(d in dom p & p.d = TRUE or d in dom q & q.d = TRUE implies
it.(p,q).d = TRUE) &
(d in dom p & p.d = FALSE & d in dom q & q.d = FALSE implies
it.(p,q).d = FALSE);
existence
proof
defpred P[object,object,object] means
for p,q being PartialPredicate of D st $1 = p & $2 = q holds
for f being Function st f = $3 holds
dom(f) = Dany(p,q) &
for d being object holds
(D1[d,p] or D1[d,q] implies f.d = TRUE) &
(D2[d,p,q] implies f.d = FALSE);
A1: for x1,x2 being object st x1 in Pr(D) & x2 in Pr(D)
ex y being object st y in Pr(D) & P[x1,x2,y]
proof
let x1,x2 be object;
assume x1 in Pr(D) & x2 in Pr(D);
then reconsider X1 = x1, X2 = x2 as PartFunc of D,BOOLEAN
by PARTFUN1:46;
defpred Q[object,object] means
for d being Element of D st d = $1 holds
(D1[d,X1] or D1[d,X2] implies $2 = TRUE) &
(D2[d,X1,X2] implies $2 = FALSE);
A2: for a being object st a in Dany(X1,X2)
ex b being object st b in BOOLEAN & Q[a,b]
proof
let a be object;
assume a in Dany(X1,X2);
then consider d being Element of D such that
A3: d = a and
A4: D1[d,X1] or D1[d,X2] or D2[d,X1,X2];
per cases by A4;
suppose
A5: D1[d,X1] or D1[d,X2];
take TRUE;
thus thesis by A3,A5;
end;
suppose
A6: D2[d,X1,X2];
take FALSE;
thus thesis by A3,A6;
end;
end;
consider g being Function such that
A7: dom g = Dany(X1,X2) and
A8: rng g c= BOOLEAN and
A9: for x being object st x in Dany(X1,X2) holds Q[x,g.x]
from FUNCT_1:sch 6(A2);
take g;
Dany(X1,X2) c= D
proof
let x;
assume x in Dany(X1,X2);
then ex d being Element of D st d = x &
(D1[d,X1] or D1[d,X2] or D2[d,X1,X2]);
hence thesis;
end;
then g is PartFunc of D,BOOLEAN by A7,A8,RELSET_1:4;
hence g in Pr(D) by PARTFUN1:45;
let p,q be PartialPredicate of D such that
A10: x1 = p & x2 = q;
let f be Function such that
A11: f = g;
thus dom(f) = Dany(p,q) by A7,A10,A11;
let d be object;
hereby
assume
A12: D1[d,p] or D1[d,q];
then d in dom p or d in dom q;
then
A13: d is Element of D;
then d in Dany(X1,X2) by A10,A12;
hence f.d = TRUE by A9,A10,A11,A12,A13;
end;
assume
A14: D2[d,p,q];
then d in dom p;
then
A15: d is Element of D;
then d in Dany(X1,X2) by A10,A14;
hence f.d = FALSE by A9,A10,A11,A14,A15;
end;
consider F being Function of [:Pr(D),Pr(D):],Pr(D) such that
A16: for x,y being object st x in Pr(D) & y in Pr(D) holds P[x,y,F.(x,y)]
from BINOP_1:sch 1(A1);
take F;
let p,q be PartialPredicate of D;
p in Pr(D) & q in Pr(D) by PARTFUN1:45;
hence thesis by A16;
end;
uniqueness
proof
let f,g be Function of [:Pr(D),Pr(D):],Pr(D) such that
A17: for p,q being PartialPredicate of D holds
dom(f.(p,q)) = Dany(p,q) &
for d being object holds
(D1[d,p] or D1[d,q] implies f.(p,q).d = TRUE) &
(D2[d,p,q] implies f.(p,q).d = FALSE) and
A18: for p,q being PartialPredicate of D holds
dom(g.(p,q)) = Dany(p,q) &
for d being object holds
(D1[d,p] or D1[d,q] implies g.(p,q).d = TRUE) &
(D2[d,p,q] implies g.(p,q).d = FALSE);
let x1,x2 be set such that
A19: x1 in Pr(D) and
A20: x2 in Pr(D);
reconsider p1 = x1, p2 = x2 as PartialPredicate of D
by A19,A20,PARTFUN1:46;
A21: dom(f.(x1,x2)) = Dany(p1,p2) by A17;
hence dom(f.(x1,x2)) = dom(g.(x1,x2)) by A18;
let a be object;
assume a in dom(f.(x1,x2));
then consider d being Element of D such that
A22: a = d and
A23: D1[d,p1] or D1[d,p2] or D2[d,p1,p2] by A21;
per cases by A23;
suppose
A24: D1[d,p1] or D1[d,p2];
thus f.(x1,x2).a = f.(p1,p2).a
.= TRUE by A17,A22,A24
.= g.(p1,p2).a by A18,A22,A24
.= g.(x1,x2).a;
end;
suppose
A25: D2[d,p1,p2];
thus f.(x1,x2).a = FALSE by A17,A22,A25
.= g.(x1,x2).a by A18,A22,A25;
end;
end;
end;
definition
let D,p,q;
func PP_or(p,q) -> PartialPredicate of D equals
PPdisjunction(D).(p,q);
coherence
proof
p in Pr(D) & q in Pr(D) by PARTFUN1:45;
hence thesis by PARTFUN1:46,BINOP_1:17;
end;
commutativity
proof
let P,p,q be PartialPredicate of D;
assume
A1: P = PPdisjunction(D).(p,q);
set Q = PPdisjunction(D).(q,p);
A2: dom P = {d where d is Element of D:
d in dom p & p.d = TRUE or d in dom q & q.d = TRUE
or d in dom p & p.d = FALSE & d in dom q & q.d = FALSE} by A1,Def4;
A3: dom Q = {d where d is Element of D:
d in dom q & q.d = TRUE or d in dom p & p.d = TRUE
or d in dom q & q.d = FALSE & d in dom p & p.d = FALSE} by Def4;
thus dom P = dom Q
proof
thus dom P c= dom Q
proof
let x;
assume x in dom P;
then ex d being Element of D st x = d &
(d in dom p & p.d = TRUE or d in dom q & q.d = TRUE
or d in dom p & p.d = FALSE & d in dom q & q.d = FALSE) by A2;
hence thesis by A3;
end;
let x;
assume x in dom Q;
then ex d being Element of D st x = d &
(d in dom q & q.d = TRUE or d in dom p & p.d = TRUE
or d in dom q & q.d = FALSE & d in dom p & p.d = FALSE) by A3;
hence thesis by A2;
end;
let x;
assume x in dom P;
then consider d being Element of D such that
A4: x = d and
A5: d in dom p & p.d = TRUE or d in dom q & q.d = TRUE
or d in dom p & p.d = FALSE & d in dom q & q.d = FALSE by A2;
per cases by A5;
suppose
A6: d in dom p & p.d = TRUE;
hence P.x = TRUE by A1,A4,Def4
.= Q.x by A4,A6,Def4;
end;
suppose
A7: d in dom q & q.d = TRUE;
hence P.x = TRUE by A1,A4,Def4
.= Q.x by A4,A7,Def4;
end;
suppose
A8: d in dom p & d in dom q & p.d = FALSE & q.d = FALSE;
hence P.x = FALSE by A1,A4,Def4
.= Q.x by A4,A8,Def4;
end;
end;
idempotence
proof
let P be PartialPredicate of D;
set Q = PPdisjunction(D).(P,P);
A9: dom Q = {d where d is Element of D:
d in dom P & P.d = TRUE or d in dom P & P.d = TRUE
or d in dom P & P.d = FALSE & d in dom P & P.d = FALSE} by Def4;
thus dom Q = dom P
proof
thus dom Q c= dom P
proof
let x;
assume x in dom Q;
then ex d being Element of D st
x = d & (d in dom P & P.d = TRUE or d in dom P & P.d = TRUE
or d in dom P & P.d = FALSE & d in dom P & P.d = FALSE) by A9;
hence thesis;
end;
let x;
assume
A10: x in dom P;
then P.x = TRUE or P.x = FALSE by Th3;
hence thesis by A9,A10;
end;
let x;
assume
A11: x in dom P;
then P.x = TRUE or P.x = FALSE by Th3;
hence thesis by A11,Def4;
end;
end;
theorem Th8:
x in dom(PP_or(p,q)) implies
x in dom p & p.x = TRUE or x in dom q & q.x = TRUE
or x in dom p & p.x = FALSE & x in dom q & q.x = FALSE
proof
assume
A1: x in dom(PP_or(p,q));
dom(PP_or(p,q)) = {d where d is Element of D:
d in dom p & p.d = TRUE or d in dom q & q.d = TRUE
or d in dom p & p.d = FALSE & d in dom q & q.d = FALSE} by Def4;
then ex d1 being Element of D st
d1 = x & (d1 in dom p & p.d1 = TRUE
or d1 in dom q & q.d1 = TRUE
or d1 in dom p & p.d1 = FALSE & d1 in dom q & q.d1 = FALSE) by A1;
hence thesis;
end;
theorem Th9:
x in dom p & x in dom q & PP_or(p,q).x = TRUE implies
p.x = TRUE or q.x = TRUE
proof
assume that
A1: x in dom p and
A2: x in dom q and
A3: PP_or(p,q).x = TRUE;
p.x <> FALSE or q.x <> FALSE by A1,A2,A3,Def4;
hence thesis by A1,A2,Th3;
end;
theorem Th10:
x in dom PP_or(p,q) & PP_or(p,q).x = TRUE implies
x in dom p & p.x = TRUE or x in dom q & q.x = TRUE
proof
assume x in dom PP_or(p,q);
then x in dom p & p.x = TRUE or x in dom q & q.x = TRUE
or x in dom p & p.x = FALSE & x in dom q & q.x = FALSE by Th8;
hence thesis by Th9;
end;
theorem Th11:
x in dom p & PP_or(p,q).x = FALSE implies p.x = FALSE
proof
assume that
A1: x in dom p and
A2: PP_or(p,q).x = FALSE;
p.x <> TRUE by A1,A2,Def4;
hence thesis by A1,Th3;
end;
theorem Th12:
x in dom q & PP_or(p,q).x = FALSE implies q.x = FALSE
proof
assume that
A1: x in dom q and
A2: PP_or(p,q).x = FALSE;
q.x <> TRUE by A1,A2,Def4;
hence thesis by A1,Th3;
end;
theorem Th13:
x in dom PP_or(p,q) & PP_or(p,q).x = FALSE implies
x in dom p & p.x = FALSE & x in dom q & q.x = FALSE
proof
assume x in dom PP_or(p,q);
then x in dom p & p.x = TRUE or x in dom q & q.x = TRUE
or x in dom p & p.x = FALSE & x in dom q & q.x = FALSE by Th8;
hence thesis by Th12;
end;
::$N Associativity law
theorem Th14:
PP_or(p,PP_or(q,r)) = PP_or(PP_or(p,q),r)
proof
set qr = PP_or(q,r);
set a = PP_or(p,qr);
set pq = PP_or(p,q);
set b = PP_or(pq,r);
A1: dom qr = {d where d is Element of D:
d in dom q & q.d = TRUE or d in dom r & r.d = TRUE
or d in dom q & q.d = FALSE & d in dom r & r.d = FALSE} by Def4;
A2: dom a = {d where d is Element of D:
d in dom p & p.d = TRUE or d in dom qr & qr.d = TRUE
or d in dom p & p.d = FALSE & d in dom qr & qr.d = FALSE} by Def4;
A3: dom pq = {d where d is Element of D:
d in dom p & p.d = TRUE or d in dom q & q.d = TRUE
or d in dom p & p.d = FALSE & d in dom q & q.d = FALSE} by Def4;
A4: dom b = {d where d is Element of D:
d in dom pq & pq.d = TRUE or d in dom r & r.d = TRUE
or d in dom pq & pq.d = FALSE & d in dom r & r.d = FALSE} by Def4;
thus dom a = dom b
proof
thus dom a c= dom b
proof
let d be object;
assume d in dom a;
then per cases by Th8;
suppose d in dom p & p.d = TRUE;
then d in dom pq & pq.d = TRUE by A3,Def4;
hence thesis by A4;
end;
suppose that
A5: d in dom qr and
A6: qr.d = TRUE;
d in dom q & q.d = TRUE or d in dom r & r.d = TRUE
or d in dom q & q.d = FALSE & d in dom r & r.d = FALSE
by A5,Th8;
then per cases by A6,Def4;
suppose d in dom q & q.d = TRUE;
then d in dom pq & pq.d = TRUE by A3,Def4;
hence thesis by A4;
end;
suppose d in dom r & r.d = TRUE;
hence thesis by A4;
end;
end;
suppose that
A7: d in dom p and
A8: p.d = FALSE and
A9: d in dom qr and
A10: qr.d = FALSE;
A11: d in dom q & q.d = TRUE or d in dom r & r.d = TRUE
or d in dom q & q.d = FALSE & d in dom r & r.d = FALSE
by A9,Th8;
then d in dom pq & pq.d = FALSE by A3,A7,A8,A10,Def4;
hence thesis by A4,A11,Th11;
end;
end;
let d be object;
assume d in dom b;
then per cases by Th8;
suppose that
A12: d in dom pq and
A13: pq.d = TRUE;
d in dom p & p.d = TRUE or d in dom q & q.d = TRUE
or d in dom p & p.d = FALSE & d in dom q & q.d = FALSE
by A12,Th8;
then d in dom p & p.d = TRUE or d in dom qr & qr.d = TRUE
by A1,A13,Def4;
hence thesis by A2;
end;
suppose d in dom r & r.d = TRUE;
then d in dom qr & qr.d = TRUE by A1,Def4;
hence thesis by A2;
end;
suppose that
A14: d in dom pq and
A15: pq.d = FALSE and
A16: d in dom r and
A17: r.d = FALSE;
A18: d in dom p & p.d = TRUE or d in dom q & q.d = TRUE
or d in dom p & p.d = FALSE & d in dom q & q.d = FALSE
by A14,Th8;
then d in dom qr & qr.d = FALSE by A1,A15,A16,A17,Def4;
hence thesis by A2,A18,Def4;
end;
end;
let d be object;
assume d in dom a;
then per cases by Th8;
suppose
A19: d in dom p & p.d = TRUE;
then
A20: d in dom pq by A3;
pq.d = TRUE by A19,Def4;
hence b.d = TRUE by A20,Def4
.= a.d by A19,Def4;
end;
suppose
A21: d in dom qr & qr.d = TRUE;
then d in dom q & q.d = TRUE or d in dom r & r.d = TRUE
or d in dom q & q.d = FALSE & d in dom r & r.d = FALSE
by Th8;
then d in dom pq & pq.d = TRUE or d in dom r & r.d = TRUE
by A3,A21,Def4;
hence b.d = TRUE by Def4
.= a.d by A21,Def4;
end;
suppose that
A22: d in dom p & p.d = FALSE and
A23: d in dom qr & qr.d = FALSE;
A24: d in dom q & q.d = TRUE or d in dom r & r.d = TRUE
or d in dom q & q.d = FALSE & d in dom r & r.d = FALSE
by A23,Th8;
then d in dom pq & pq.d = FALSE by A3,A22,A23,Def4;
hence b.d = FALSE by A23,A24,Def4
.= a.d by A22,A23,Def4;
end;
end;
theorem Th15:
PP_or(PP_or(p,q),PP_or(p,r)) = PP_or(PP_or(p,q),r)
proof
PP_or(p,PP_or(p,q)) = PP_or(PP_or(p,p),q) by Th14;
hence thesis by Th14;
end;
definition
let D,p,q;
func PP_and(p,q) -> PartialPredicate of D equals
PP_not PP_or(PP_not(p),PP_not(q));
coherence;
commutativity;
idempotence;
func PP_imp(p,q) -> PartialPredicate of D equals
PP_or(PP_not(p),q);
coherence;
end;
theorem Th16:
dom(PP_and(p,q)) =
{d where d is Element of D:
d in dom p & p.d = FALSE or d in dom q & q.d = FALSE
or d in dom p & p.d = TRUE & d in dom q & q.d = TRUE}
proof
set F = PP_and(p,q);
set P = PP_not(p);
set Q = PP_not(q);
set Dand = {d where d is Element of D:
d in dom p & p.d = FALSE or d in dom q & q.d = FALSE
or d in dom p & p.d = TRUE & d in dom q & q.d = TRUE};
A1: dom F = dom PP_or(P,Q) by Def2;
A2: dom PP_or(P,Q) =
{d where d is Element of D:
d in dom P & P.d = TRUE or d in dom Q & Q.d = TRUE
or d in dom P & P.d = FALSE & d in dom Q & Q.d = FALSE} by Def4;
A3: dom P = dom p by Def2;
A4: dom Q = dom q by Def2;
thus dom F c= Dand
proof
let x;
assume x in dom F;
then consider d being Element of D such that
A5: x = d and
A6: d in dom P & P.d = TRUE or d in dom Q & Q.d = TRUE
or d in dom P & P.d = FALSE & d in dom Q & Q.d = FALSE by A1,A2;
per cases by A6;
suppose that
A7: d in dom P and
A8: P.d = TRUE;
p.d = FALSE by A3,A7,A8,Th5;
hence thesis by A3,A5,A7;
end;
suppose that
A9: d in dom Q and
A10: Q.d = TRUE;
q.d = FALSE by A4,A9,A10,Th5;
hence thesis by A4,A5,A9;
end;
suppose that
A11: d in dom P & d in dom Q and
A12: P.d = FALSE & Q.d = FALSE;
p.d = TRUE & q.d = TRUE by A3,A4,A11,A12,Th4;
hence thesis by A3,A4,A5,A11;
end;
end;
let x;
assume x in Dand;
then consider d being Element of D such that
A13: x = d and
A14: d in dom p & p.d = FALSE or d in dom q & q.d = FALSE
or d in dom p & p.d = TRUE & d in dom q & q.d = TRUE;
per cases by A14;
suppose that
A15: d in dom p and
A16: p.d = FALSE;
P.d = TRUE by A15,A16,Def2;
hence thesis by A1,A2,A3,A13,A15;
end;
suppose that
A17: d in dom q and
A18: q.d = FALSE;
Q.d = TRUE by A17,A18,Def2;
hence thesis by A1,A2,A4,A13,A17;
end;
suppose that
A19: d in dom p & d in dom q and
A20: p.d = TRUE & q.d = TRUE;
P.d = FALSE & Q.d = FALSE by A19,A20,Def2;
hence thesis by A1,A2,A3,A4,A13,A19;
end;
end;
theorem Th17:
x in dom(PP_and(p,q)) implies
x in dom p & p.x = FALSE or x in dom q & q.x = FALSE
or x in dom p & p.x = TRUE & x in dom q & q.x = TRUE
proof
assume
A1: x in dom(PP_and(p,q));
dom(PP_and(p,q)) =
{d where d is Element of D:
d in dom p & p.d = FALSE or d in dom q & q.d = FALSE
or d in dom p & p.d = TRUE & d in dom q & q.d = TRUE} by Th16;
then ex d1 being Element of D st x = d1 &
(d1 in dom p & p.d1 = FALSE or d1 in dom q & q.d1 = FALSE
or d1 in dom p & p.d1 = TRUE & d1 in dom q & q.d1 = TRUE) by A1;
hence thesis;
end;
theorem Th18:
x in dom p & p.x = TRUE & x in dom q & q.x = TRUE implies
PP_and(p,q).x = TRUE
proof
assume that
A1: x in dom p and
A2: p.x = TRUE and
A3: x in dom q and
A4: q.x = TRUE;
A5: PP_not(p).x = FALSE & PP_not(q).x = FALSE by A1,A3,A2,A4,Def2;
A6: dom PP_not(p) = dom p & dom PP_not(q) = dom q by Def2;
dom PP_or(PP_not(p),PP_not(q)) = {d where d is Element of D:
d in dom PP_not(p) & PP_not(p).d = TRUE or
d in dom PP_not(q) & PP_not(q).d = TRUE
or d in dom PP_not(p) & PP_not(p).d = FALSE &
d in dom PP_not(q) & PP_not(q).d = FALSE} by Def4;
then
A7: x in dom PP_or(PP_not(p),PP_not(q)) by A1,A3,A5,A6;
(PP_or(PP_not(p),PP_not(q))).x = FALSE by A1,A3,A5,A6,Def4;
hence thesis by A7,Def2;
end;
theorem Th19:
x in dom p & p.x = FALSE implies PP_and(p,q).x = FALSE
proof
assume that
A1: x in dom p and
A2: p.x = FALSE;
A3: PP_not(p).x = TRUE by A1,A2,Def2;
A4: dom PP_not(p) = dom p by Def2;
A5: dom PP_or(PP_not(p),PP_not(q)) = {d where d is Element of D:
d in dom PP_not(p) & PP_not(p).d = TRUE or
d in dom PP_not(q) & PP_not(q).d = TRUE
or d in dom PP_not(p) & PP_not(p).d = FALSE &
d in dom PP_not(q) & PP_not(q).d = FALSE} by Def4;
A6: x in dom PP_or(PP_not(p),PP_not(q)) by A1,A3,A4,A5;
(PP_or(PP_not(p),PP_not(q))).x = TRUE by A1,A3,A4,Def4;
hence thesis by A6,Def2;
end;
theorem
x in dom q & q.x = FALSE implies PP_and(p,q).x = FALSE by Th19;
theorem
x in dom p & PP_and(p,q).x = TRUE implies p.x = TRUE by Th3,Th19;
theorem
x in dom q & PP_and(p,q).x = TRUE implies q.x = TRUE by Th3,Th19;
theorem Th23:
x in dom PP_and(p,q) & PP_and(p,q).x = TRUE implies
x in dom p & p.x = TRUE & x in dom q & q.x = TRUE
proof
assume x in dom PP_and(p,q);
then x in dom p & p.x = FALSE or x in dom q & q.x = FALSE
or x in dom p & p.x = TRUE & x in dom q & q.x = TRUE by Th17;
hence thesis by Th19;
end;
theorem Th24:
x in dom p & x in dom q & PP_and(p,q).x = FALSE implies
p.x = FALSE or q.x = FALSE
proof
assume that
A1: x in dom p and
A2: x in dom q and
A3: PP_and(p,q).x = FALSE;
p.x <> TRUE or q.x <> TRUE by A1,A2,A3,Th18;
hence p.x = FALSE or q.x = FALSE by A1,A2,Th3;
end;
theorem Th25:
x in dom PP_and(p,q) & PP_and(p,q).x = FALSE implies
x in dom p & p.x = FALSE or x in dom q & q.x = FALSE
proof
assume x in dom PP_and(p,q);
then x in dom p & p.x = FALSE or x in dom q & q.x = FALSE
or x in dom p & p.x = TRUE & x in dom q & q.x = TRUE by Th17;
hence thesis by Th24;
end;
::$N Associativity law
theorem
PP_and(p,PP_and(q,r)) = PP_and(PP_and(p,q),r) by Th14;
theorem
PP_and(PP_and(p,q),PP_and(p,r)) = PP_and(PP_and(p,q),r) by Th15;
::$N Meet-absorbing law
theorem Th28:
PP_or(PP_and(p,q),q) = q
proof
set a = PP_and(p,q);
set o = PP_or(a,q);
A1: dom a = {d where d is Element of D:
d in dom p & p.d = FALSE or d in dom q & q.d = FALSE
or d in dom p & p.d = TRUE & d in dom q & q.d = TRUE} by Th16;
A2: dom o = {d where d is Element of D:
d in dom a & a.d = TRUE or d in dom q & q.d = TRUE
or d in dom a & a.d = FALSE & d in dom q & q.d = FALSE} by Def4;
thus dom o = dom q
proof
thus dom o c= dom q
proof
let d be object;
assume d in dom o;
then per cases by Th8;
suppose that
A3: d in dom a and
A4: a.d = TRUE;
per cases by A3,Th17;
suppose d in dom p & p.d = FALSE;
hence thesis by A4,Th19;
end;
suppose d in dom q & q.d = FALSE
or d in dom p & p.d = TRUE & d in dom q & q.d = TRUE;
hence thesis;
end;
end;
suppose d in dom q & q.d = TRUE
or d in dom a & a.d = FALSE & d in dom q & q.d = FALSE;
hence thesis;
end;
end;
let d be object;
assume
A5: d in dom q;
then per cases by Th3;
suppose
A6: q.d = FALSE;
then
A7: a.d = FALSE by A5,Th19;
d in dom a by A1,A5,A6;
hence thesis by A2,A5,A6,A7;
end;
suppose q.d = TRUE;
hence thesis by A2,A5;
end;
end;
let d be object;
assume d in dom o;
then per cases by Th8;
suppose that
A8: d in dom a and
A9: a.d = TRUE;
per cases by A8,Th17;
suppose d in dom p & p.d = FALSE;
hence thesis by A9,Th19;
end;
suppose d in dom q & q.d = FALSE;
hence thesis by A9,Th19;
end;
suppose d in dom p & p.d = TRUE & d in dom q & q.d = TRUE;
hence thesis by Def4;
end;
end;
suppose d in dom q & q.d = TRUE;
hence thesis by Def4;
end;
suppose d in dom a & a.d = FALSE & d in dom q & q.d = FALSE;
hence thesis by Def4;
end;
end;
::$N Join-absorbing law
theorem Th29:
PP_and(p,PP_or(p,q)) = p
proof
set o = PP_or(p,q);
set a = PP_and(p,o);
A1: dom a = {d where d is Element of D:
d in dom p & p.d = FALSE or d in dom o & o.d = FALSE
or d in dom p & p.d = TRUE & d in dom o & o.d = TRUE} by Th16;
A2: dom o = {d where d is Element of D:
d in dom p & p.d = TRUE or d in dom q & q.d = TRUE
or d in dom p & p.d = FALSE & d in dom q & q.d = FALSE} by Def4;
thus dom a = dom p
proof
thus dom a c= dom p
proof
let d be object;
assume d in dom a;
then per cases by Th17;
suppose that
A3: d in dom o and
A4: o.d = FALSE;
per cases by A3,Th8;
suppose d in dom p & p.d = TRUE;
hence thesis;
end;
suppose d in dom q & q.d = TRUE
or d in dom p & p.d = FALSE & d in dom q & q.d = FALSE;
hence thesis by A4,Def4;
end;
end;
suppose d in dom p & p.d = FALSE
or d in dom p & p.d = TRUE & d in dom o & o.d = TRUE;
hence thesis;
end;
end;
let d be object;
assume
A5: d in dom p;
then per cases by Th3;
suppose
A6: p.d = TRUE;
then
A7: o.d = TRUE by A5,Def4;
d in dom o by A2,A5,A6;
hence thesis by A1,A5,A6,A7;
end;
suppose p.d = FALSE;
hence thesis by A1,A5;
end;
end;
let d be object;
assume d in dom a;
then per cases by Th17;
suppose that
A8: d in dom o and
A9: o.d = FALSE;
per cases by A8,Th8;
suppose d in dom p & p.d = TRUE;
hence thesis by A9,Def4;
end;
suppose d in dom q & q.d = TRUE;
hence thesis by A9,Def4;
end;
suppose d in dom p & p.d = FALSE & d in dom q & q.d = FALSE;
hence thesis by Th19;
end;
end;
suppose d in dom p & p.d = FALSE;
hence thesis by Th19;
end;
suppose d in dom p & p.d = TRUE & d in dom o & o.d = TRUE;
hence thesis by Th18;
end;
end;
::$N Distributivity law
theorem Th30:
PP_and(p,PP_or(q,r)) = PP_or(PP_and(p,q),PP_and(p,r))
proof
set qr = PP_or(q,r);
set a = PP_and(p,qr);
set pq = PP_and(p,q);
set pr = PP_and(p,r);
set b = PP_or(pq,pr);
A1: dom qr = {d where d is Element of D:
d in dom q & q.d = TRUE or d in dom r & r.d = TRUE
or d in dom q & q.d = FALSE & d in dom r & r.d = FALSE} by Def4;
A2: dom a = {d where d is Element of D:
d in dom p & p.d = FALSE or d in dom qr & qr.d = FALSE
or d in dom p & p.d = TRUE & d in dom qr & qr.d = TRUE} by Th16;
A3: dom pq = {d where d is Element of D:
d in dom p & p.d = FALSE or d in dom q & q.d = FALSE
or d in dom p & p.d = TRUE & d in dom q & q.d = TRUE} by Th16;
A4: dom pr = {d where d is Element of D:
d in dom p & p.d = FALSE or d in dom r & r.d = FALSE
or d in dom p & p.d = TRUE & d in dom r & r.d = TRUE} by Th16;
A5: dom b = {d where d is Element of D:
d in dom pq & pq.d = TRUE or d in dom pr & pr.d = TRUE
or d in dom pq & pq.d = FALSE & d in dom pr & pr.d = FALSE} by Def4;
thus dom a = dom b
proof
thus dom a c= dom b
proof
let d be object;
assume d in dom a;
then per cases by Th17;
suppose d in dom p & p.d = FALSE;
then d in dom pq & pq.d = FALSE & d in dom pr & pr.d = FALSE
by A3,A4,Th19;
hence thesis by A5;
end;
suppose d in dom qr & qr.d = FALSE;
then d in dom q & q.d = FALSE & d in dom r & r.d = FALSE
by Th13;
then d in dom pq & pq.d = FALSE & d in dom pr & pr.d = FALSE
by A3,A4,Th19;
hence thesis by A5;
end;
suppose that
A6: d in dom p & p.d = TRUE and
A7: d in dom qr & qr.d = TRUE;
d in dom q & q.d = TRUE or d in dom r & r.d = TRUE
by A7,Th10;
then d in dom pq & pq.d = TRUE or d in dom pr & pr.d = TRUE
by A3,A4,A6,Th18;
hence thesis by A5;
end;
end;
let d be object;
assume d in dom b;
then per cases by Th8;
suppose
A8: d in dom pq & pq.d = TRUE;
then
A9: d in dom p & p.d = TRUE by Th23;
d in dom q & q.d = TRUE by A8,Th23;
then d in dom qr & qr.d = TRUE by A1,Def4;
hence thesis by A2,A9;
end;
suppose
A10: d in dom pr & pr.d = TRUE;
then
A11: d in dom p & p.d = TRUE by Th23;
d in dom r & r.d = TRUE by A10,Th23;
then d in dom qr & qr.d = TRUE by A1,Def4;
hence thesis by A2,A11;
end;
suppose d in dom pq & pq.d = FALSE & d in dom pr & pr.d = FALSE;
then (d in dom p & p.d = FALSE or d in dom q & q.d = FALSE)
& (d in dom p & p.d = FALSE or d in dom r & r.d = FALSE)
by Th25;
then d in dom p & p.d = FALSE or d in dom qr & qr.d = FALSE
by A1,Def4;
hence thesis by A2;
end;
end;
let d be object;
assume d in dom a;
then per cases by Th17;
suppose
A12: d in dom p & p.d = FALSE;
then d in dom pq & pq.d = FALSE & d in dom pr & pr.d = FALSE
by A3,A4,Th19;
hence b.d = FALSE by Def4
.= a.d by A12,Th19;
end;
suppose
A13: d in dom qr & qr.d = FALSE;
then d in dom q & q.d = FALSE & d in dom r & r.d = FALSE
by Th13;
then d in dom pq & pq.d = FALSE & d in dom pr & pr.d = FALSE
by A3,A4,Th19;
hence b.d = FALSE by Def4
.= a.d by A13,Th19;
end;
suppose that
A14: d in dom p & p.d = TRUE and
A15: d in dom qr & qr.d = TRUE;
d in dom q & q.d = TRUE or d in dom r & r.d = TRUE
by A15,Th10;
then d in dom pq & pq.d = TRUE or d in dom pr & pr.d = TRUE
by A3,A4,A14,Th18;
hence b.d = TRUE by Def4
.= a.d by A14,A15,Th18;
end;
end;
theorem Th31:
dom(PP_imp(p,q)) =
{d where d is Element of D:
d in dom p & p.d = FALSE or d in dom q & q.d = TRUE
or d in dom p & p.d = TRUE & d in dom q & q.d = FALSE}
proof
set F = PP_imp(p,q);
set P = PP_not(p);
set Dimp = {d1 where d1 is Element of D:
d1 in dom p & p.d1 = FALSE or d1 in dom q & q.d1 = TRUE
or d1 in dom p & p.d1 = TRUE & d1 in dom q & q.d1 = FALSE};
A1: dom(F) = {d where d is Element of D:
d in dom P & P.d = TRUE or d in dom q & q.d = TRUE
or d in dom P & P.d = FALSE & d in dom q & q.d = FALSE} by Def4;
A2: dom P = dom p by Def2;
thus dom F c= Dimp
proof
let x;
assume x in dom F;
then consider d being Element of D such that
A3: x = d and
A4: d in dom P & P.d = TRUE or d in dom q & q.d = TRUE
or d in dom P & P.d = FALSE & d in dom q & q.d = FALSE by A1;
per cases by A4;
suppose that
A5: d in dom P and
A6: P.d = TRUE;
p.d = FALSE by A2,A5,A6,Th5;
hence thesis by A2,A3,A5;
end;
suppose d in dom q & q.d = TRUE;
hence thesis by A3;
end;
suppose that
A7: d in dom P & d in dom q and
A8: P.d = FALSE and
A9: q.d = FALSE;
p.d = TRUE by A2,A7,A8,Th4;
hence thesis by A2,A3,A7,A9;
end;
end;
let x;
assume x in Dimp;
then consider d being Element of D such that
A10: x = d and
A11: d in dom p & p.d = FALSE or d in dom q & q.d = TRUE
or d in dom p & p.d = TRUE & d in dom q & q.d = FALSE;
per cases by A11;
suppose that
A12: d in dom p and
A13: p.d = FALSE;
P.d = TRUE by A12,A13,Def2;
hence thesis by A1,A2,A10,A12;
end;
suppose d in dom q & q.d = TRUE;
hence thesis by A1,A10;
end;
suppose that
A14: d in dom p and
A15: d in dom q and
A16: p.d = TRUE and
A17: q.d = FALSE;
P.d = FALSE by A14,A16,Def2;
hence thesis by A1,A2,A10,A14,A15,A17;
end;
end;
theorem Th32:
x in dom(PP_imp(p,q)) implies
x in dom p & p.x = FALSE or x in dom q & q.x = TRUE
or x in dom p & p.x = TRUE & x in dom q & q.x = FALSE
proof
assume
A1: x in dom(PP_imp(p,q));
dom(PP_imp(p,q)) =
{d where d is Element of D:
d in dom p & p.d = FALSE or d in dom q & q.d = TRUE
or d in dom p & p.d = TRUE & d in dom q & q.d = FALSE} by Th31;
then ex d1 being Element of D st d1 = x &
(d1 in dom p & p.d1 = FALSE or d1 in dom q & q.d1 = TRUE
or d1 in dom p & p.d1 = TRUE & d1 in dom q & q.d1 = FALSE) by A1;
hence thesis;
end;
theorem Th33:
x in dom p & p.x = FALSE implies PP_imp(p,q).x = TRUE
proof
assume that
A1: x in dom p and
A2: p.x = FALSE;
A3: dom PP_not(p) = dom p by Def2;
PP_not(p).x = TRUE by A1,A2,Def2;
hence thesis by A1,A3,Def4;
end;
theorem Th34:
x in dom q & q.x = TRUE implies PP_imp(p,q).x = TRUE by Def4;
theorem Th35:
x in dom p & p.x = TRUE & x in dom q & q.x = FALSE implies
PP_imp(p,q).x = FALSE
proof
assume that
A1: x in dom p and
A2: p.x = TRUE and
A3: x in dom q and
A4: q.x = FALSE;
A5: dom PP_not(p) = dom p by Def2;
PP_not(p).x = FALSE by A1,A2,Def2;
hence thesis by A1,A3,A4,A5,Def4;
end;
theorem
x in dom p & x in dom q & PP_imp(p,q).x = TRUE implies
p.x = FALSE or q.x = TRUE
proof
assume that
A1: x in dom p and
A2: x in dom q and
A3: PP_imp(p,q).x = TRUE;
p.x <> TRUE or q.x <> FALSE by A1,A2,A3,Th35;
hence thesis by A1,A2,Th3;
end;
theorem
x in dom p & PP_imp(p,q).x = FALSE implies p.x = TRUE by Th3,Th33;
theorem
x in dom q & PP_imp(p,q).x = FALSE implies q.x = FALSE by Th3,Th34;
theorem
x in dom PP_imp(p,q) & PP_imp(p,q).x = FALSE implies
x in dom p & p.x = TRUE & x in dom q & q.x = FALSE
proof
assume x in dom PP_imp(p,q);
then x in dom p & p.x = FALSE or x in dom q & q.x = TRUE
or x in dom p & p.x = TRUE & x in dom q & q.x = FALSE by Th32;
hence thesis by Th33,Def4;
end;
theorem
x in dom PP_imp(p,q) & PP_imp(p,q).x = TRUE implies
x in dom p & p.x = FALSE or x in dom q & q.x = TRUE
proof
assume x in dom PP_imp(p,q);
then x in dom p & p.x = FALSE or x in dom q & q.x = TRUE
or x in dom p & p.x = TRUE & x in dom q & q.x = FALSE by Th32;
hence thesis by Th35;
end;
theorem
PP_and(PP_imp(p,r),PP_imp(q,r)) = PP_imp(PP_or(p,q),r)
proof
thus PP_and(PP_imp(p,r),PP_imp(q,r))
= PP_not PP_or(PP_and(p,PP_not(r)),PP_and(q,PP_not(r)))
.= PP_not PP_and(PP_not(r),PP_or(p,q)) by Th30
.= PP_imp(PP_or(p,q),r);
end;
theorem
PP_or(PP_imp(p,r),PP_imp(q,r)) = PP_imp(PP_and(p,q),r)
proof
PP_or(PP_or(r,PP_not(p)),PP_or(r,PP_not(q)))
= PP_or(PP_or(r,PP_not(p)),PP_not(q)) by Th15;
hence thesis by Th14;
end;
::$N True constant predicate
definition
let D be set;
func PP_True(D) -> PartialPredicate of D equals
D --> TRUE;
coherence
proof
per cases;
suppose
A1: D is non empty;
set f = D --> TRUE;
A2: dom f = D by FUNCOP_1:13;
rng f = {TRUE} by A1,FUNCOP_1:8;
hence thesis by A2,RELSET_1:4,ZFMISC_1:31;
end;
suppose D is empty;
then D --> TRUE = {};
hence thesis by RELSET_1:12;
end;
end;
end;
::$N False constant predicate
definition
let D be set;
func PP_False(D) -> PartialPredicate of D equals
D --> FALSE;
coherence
proof
per cases;
suppose
A1: D is non empty;
set f = D --> FALSE;
A2: dom f = D by FUNCOP_1:13;
rng f = {FALSE} by A1,FUNCOP_1:8;
hence thesis by A2,RELSET_1:4,ZFMISC_1:31;
end;
suppose D is empty;
then D --> FALSE = {};
hence thesis by RELSET_1:12;
end;
end;
end;
theorem Th43:
for D being set holds PP_not(PP_False(D)) = PP_True(D)
proof
let D be set;
set T = PP_True(D);
set B = PP_False(D);
set n = PP_not(B);
A1: dom B = D by FUNCOP_1:13;
dom T = D by FUNCOP_1:13;
hence dom n = dom T by A1,Def2;
let x;
assume
A2: x in dom n;
then B.x = FALSE by FUNCOP_1:7;
hence n.x = TRUE by A1,A2,Def2
.= T.x by A2,FUNCOP_1:7;
end;
theorem
for D being set holds PP_not(PP_True(D)) = PP_False(D)
proof
let D be set;
PP_not(PP_False(D)) = PP_True(D) by Th43;
hence thesis;
end;
theorem Th45:
PP_or(p,PP_True(D)) = PP_True(D)
proof
set q = PP_True(D);
set f = PP_or(p,q);
A1: dom f = {d where d is Element of D:
d in dom p & p.d = TRUE or d in dom q & q.d = TRUE
or d in dom p & p.d = FALSE & d in dom q & q.d = FALSE} by Def4;
A2: dom q = D by FUNCOP_1:13;
thus
A3: dom f = dom q
proof
thus dom f c= dom q by A2;
let x;
assume
A4: x in dom q;
then reconsider d = x as Element of D;
q.d = TRUE by FUNCOP_1:7;
hence thesis by A1,A4;
end;
let x;
assume
A5: x in dom f;
then q.x = TRUE by FUNCOP_1:7;
hence f.x = q.x by A3,A5,Def4;
end;
theorem
PP_or(PP_True(D),p) = PP_True(D) by Th45;
theorem Th47:
PP_and(p,PP_False(D)) = PP_False(D)
proof
set q = PP_False(D);
set f = PP_and(p,q);
A1: dom f = {d where d is Element of D:
d in dom p & p.d = FALSE or d in dom q & q.d = FALSE
or d in dom p & p.d = TRUE & d in dom q & q.d = TRUE} by Th16;
A2: dom q = D by FUNCOP_1:13;
thus
A3: dom f = dom q
proof
thus dom f c= dom q by A2;
let x;
assume
A4: x in dom q;
then reconsider d = x as Element of D;
q.d = FALSE by FUNCOP_1:7;
hence thesis by A1,A4;
end;
let x;
assume
A5: x in dom f;
then q.x = FALSE by FUNCOP_1:7;
hence f.x = q.x by A3,A5,Th19;
end;
theorem
PP_and(PP_False(D),p) = PP_False(D) by Th47;
theorem Th49:
PP_or(p,PP_not(p)) = PP_True(D) | dom p
proof
set n = PP_not(p);
set a = PP_or(p,n);
set T = PP_True(D);
set t = T | dom p;
A1: dom n = dom p by Def2;
A2: dom a = {d where d is Element of D:
d in dom p & p.d = TRUE or d in dom n & n.d = TRUE
or d in dom p & p.d = FALSE & d in dom n & n.d = FALSE} by Def4;
dom T = D by FUNCOP_1:13;
then
A3: dom t = dom p by RELAT_1:62;
thus
A4: dom a = dom t
proof
thus dom a c= dom t by A1,A3,Th8;
let x;
assume
A5: x in dom t;
then per cases by A3,Th3;
suppose p.x = TRUE;
hence thesis by A2,A3,A5;
end;
suppose p.x = FALSE;
then n.x = TRUE by A5,A3,Def2;
hence thesis by A1,A2,A3,A5;
end;
end;
let x;
assume
A6: x in dom a;
then
A7: TRUE = T.x by FUNCOP_1:7
.= t.x by A3,A4,A6,FUNCT_1:49;
per cases by A3,A4,A6,Th3;
suppose p.x = TRUE;
hence thesis by A3,A4,A6,A7,Def4;
end;
suppose p.x = FALSE;
then n.x = TRUE by A3,A4,A6,Def2;
hence thesis by A1,A3,A4,A6,A7,Def4;
end;
end;
theorem
PP_or(PP_not(p),p) = PP_True(D) | dom p by Th49;
theorem Th51:
PP_and(p,PP_not(p)) = PP_False(D) | dom p
proof
set n = PP_not(p);
set a = PP_and(p,n);
set B = PP_False(D);
set t = B | dom p;
A1: dom n = dom p by Def2;
A2: dom a = {d where d is Element of D:
d in dom p & p.d = FALSE or d in dom n & n.d = FALSE
or d in dom p & p.d = TRUE & d in dom n & n.d = TRUE} by Th16;
dom B = D by FUNCOP_1:13;
then
A3: dom t = dom p by RELAT_1:62;
thus
A4: dom a = dom t
proof
thus dom a c= dom t by A1,A3,Th17;
let x;
assume
A5: x in dom t;
then per cases by A3,Th3;
suppose p.x = FALSE;
hence thesis by A2,A3,A5;
end;
suppose p.x = TRUE;
then n.x = FALSE by A3,A5,Def2;
hence thesis by A1,A2,A3,A5;
end;
end;
let x;
assume
A6: x in dom a;
then
A7: FALSE = B.x by FUNCOP_1:7
.= t.x by A3,A4,A6,FUNCT_1:49;
per cases by A3,A4,A6,Th3;
suppose p.x = FALSE;
hence thesis by A3,A4,A6,A7,Th19;
end;
suppose p.x = TRUE;
then n.x = FALSE by A3,A4,A6,Def2;
hence thesis by A1,A3,A4,A6,A7,Th19;
end;
end;
theorem
PP_and(PP_not(p),p) = PP_False(D) | dom p by Th51;
theorem
PP_imp(PP_False(D),p) = PP_True(D)
proof
PP_not(PP_False(D)) = PP_True(D) by Th43;
hence thesis by Th45;
end;
theorem
PP_imp(p,PP_True(D)) = PP_True(D) by Th45;
theorem Th55:
PP_or(PP_False(D)|dom p,PP_True(D)|dom q) = PP_True(D) | dom q
proof
set F = PP_False(D);
set T = PP_True(D);
set f = F|dom p;
set g = T|dom q;
set o = PP_or(f,g);
A1: dom o = {d where d is Element of D:
d in dom f & f.d = TRUE or d in dom g & g.d = TRUE
or d in dom f & f.d = FALSE & d in dom g & g.d = FALSE} by Def4;
dom T = D by FUNCOP_1:13;
then
A2: dom g = dom q by RELAT_1:62;
thus dom o = dom g
proof
thus dom o c= dom g
proof
let x;
assume
A3: x in dom o;
then per cases by Th8;
suppose that
A4: x in dom f and
A5: f.x = TRUE;
f.x = F.x by A4,FUNCT_1:47;
hence thesis by A3,A5,FUNCOP_1:7;
end;
suppose x in dom g & g.x = TRUE or
x in dom f & f.x = FALSE & x in dom g & g.x = FALSE;
hence thesis;
end;
end;
let x;
assume
A6: x in dom g;
then g.x = T.x by A2,FUNCT_1:49
.= TRUE by A6,FUNCOP_1:7;
hence thesis by A1,A6;
end;
let x;
assume
A7: x in dom o;
then per cases by Th8;
suppose that
A8: x in dom f and
A9: f.x = TRUE;
f.x = F.x by A8,FUNCT_1:47;
hence thesis by A7,A9,FUNCOP_1:7;
end;
suppose x in dom g & g.x = TRUE or
x in dom f & f.x = FALSE & x in dom g & g.x = FALSE;
hence thesis by Def4;
end;
end;
::$N Empty predicate
definition
let D be set;
func PP_BottomPred(D) -> PartialPredicate of D equals
{};
coherence by RELSET_1:12;
end;
theorem Th56:
for D being set holds PP_not(PP_BottomPred(D)) = PP_BottomPred(D)
proof
let D be set;
thus dom PP_not(PP_BottomPred(D)) = dom PP_BottomPred(D) by Def2;
hence thesis;
end;
theorem Th57:
PP_or(PP_BottomPred(D),PP_True(D)) = PP_True(D)
proof
set B = PP_BottomPred(D);
set T = PP_True(D);
set o = PP_or(B,T);
A1: dom(o) = {d where d is Element of D:
d in dom B & B.d = TRUE or d in dom T & T.d = TRUE
or d in dom B & B.d = FALSE & d in dom T & T.d = FALSE} by Def4;
thus dom o = dom T
proof
dom T = D by FUNCOP_1:13;
hence dom o c= dom T;
thus dom T c= dom o
proof
let x;
assume
A2: x in dom T;
then T.x = TRUE by FUNCOP_1:7;
hence thesis by A1,A2;
end;
end;
let x;
assume x in dom o;
then per cases by Th8;
suppose x in dom B & B.x = TRUE or
x in dom B & B.x = FALSE & x in dom T & T.x = FALSE;
hence thesis;
end;
suppose x in dom T & T.x = TRUE;
hence thesis by Def4;
end;
end;
theorem
PP_and(PP_BottomPred(D),PP_False(D)) = PP_False(D)
proof
set B = PP_BottomPred(D);
set F = PP_False(D);
set o = PP_and(B,F);
A1: dom(o) = {d where d is Element of D:
d in dom B & B.d = FALSE or d in dom F & F.d = FALSE
or d in dom B & B.d = TRUE & d in dom F & F.d = TRUE} by Th16;
thus dom o = dom F
proof
dom F = D by FUNCOP_1:13;
hence dom o c= dom F;
thus dom F c= dom o
proof
let x;
assume
A2: x in dom F;
then F.x = FALSE by FUNCOP_1:7;
hence thesis by A1,A2;
end;
end;
let x;
assume x in dom o;
then per cases by Th17;
suppose x in dom B & B.x = FALSE or
x in dom B & B.x = TRUE & x in dom F & F.x = TRUE;
hence thesis;
end;
suppose x in dom F & F.x = FALSE;
hence thesis by Th19;
end;
end;
theorem
PP_imp(PP_BottomPred(D),PP_True(D)) = PP_True(D)
proof
thus PP_imp(PP_BottomPred(D),PP_True(D))
= PP_or(PP_BottomPred(D),PP_True(D)) by Th56
.= PP_True(D) by Th57;
end;
begin :: Algebra of partial connectives with (strong) Kleene logical connectives
definition
let D;
func PartialPredConnectivesMeet(D) -> BinOp of Pr(D) means :Def11:
for p,q being PartialPredicate of D holds it.(p,q) = PP_and(p,q);
existence
proof
defpred P[object,object,object] means
for p,q being PartialPredicate of D st p = $1 & q = $2
holds $3 = PP_and(p,q);
A1: for x,y being object st x in Pr(D) & y in Pr(D)
ex z being object st z in Pr(D) & P[x,y,z]
proof
let x,y be object;
assume x in Pr(D);
then reconsider x as PartialPredicate of D by PARTFUN1:46;
assume y in Pr(D);
then reconsider y as PartialPredicate of D by PARTFUN1:46;
take PP_and(x,y);
thus thesis by PARTFUN1:45;
end;
consider f being Function of [:Pr(D),Pr(D):],Pr(D) such that
A2: for x,y being object st x in Pr(D) & y in Pr(D) holds P[x,y,f.(x,y)]
from BINOP_1:sch 1(A1);
take f;
let p,q be PartialPredicate of D;
p in Pr(D) & q in Pr(D) by PARTFUN1:45;
hence thesis by A2;
end;
uniqueness
proof
let f1,f2 be BinOp of Pr(D) such that
A3: for p,q being PartialPredicate of D holds
f1.(p,q) = PP_and(p,q) and
A4: for p,q being PartialPredicate of D holds
f2.(p,q) = PP_and(p,q);
let x,y be set;
assume x in Pr(D);
then reconsider x1 = x as PartialPredicate of D
by PARTFUN1:46;
assume y in Pr(D);
then reconsider y1 = y as PartialPredicate of D
by PARTFUN1:46;
thus f1.(x,y) = PP_and(x1,y1) by A3
.= f2.(x,y) by A4;
end;
func PartialPredConnectivesJoin(D) -> BinOp of Pr(D) means :Def12:
for p,q being PartialPredicate of D holds it.(p,q) = PP_or(p,q);
existence
proof
defpred P[object,object,object] means
for p,q being PartialPredicate of D st p = $1 & q = $2
holds $3 = PP_or(p,q);
A5: for x,y being object st x in Pr(D) & y in Pr(D)
ex z being object st z in Pr(D) & P[x,y,z]
proof
let x,y be object;
assume x in Pr(D);
then reconsider x as PartialPredicate of D by PARTFUN1:46;
assume y in Pr(D);
then reconsider y as PartialPredicate of D by PARTFUN1:46;
take PP_or(x,y);
thus thesis by PARTFUN1:45;
end;
consider f being Function of [:Pr(D),Pr(D):],Pr(D) such that
A6: for x,y being object st x in Pr(D) & y in Pr(D) holds P[x,y,f.(x,y)]
from BINOP_1:sch 1(A5);
take f;
let p,q be PartialPredicate of D;
p in Pr(D) & q in Pr(D) by PARTFUN1:45;
hence thesis by A6;
end;
uniqueness
proof
let f1,f2 be BinOp of Pr(D) such that
A7: for p,q being PartialPredicate of D holds
f1.(p,q) = PP_or(p,q) and
A8: for p,q being PartialPredicate of D holds
f2.(p,q) = PP_or(p,q);
let x,y be set;
assume x in Pr(D);
then reconsider x1 = x as PartialPredicate of D
by PARTFUN1:46;
assume y in Pr(D);
then reconsider y1 = y as PartialPredicate of D
by PARTFUN1:46;
thus f1.(x,y) = PP_or(x1,y1) by A7
.= f2.(x,y) by A8;
end;
func PartialPredConnectivesCompl(D) -> UnOp of Pr(D) means :Def13:
for p being PartialPredicate of D holds it.p = PP_not(p);
existence
proof
defpred P[object,object] means
for p being PartialPredicate of D st p = $1
holds $2 = PP_not(p);
A9: for x being object st x in Pr(D)
ex z being object st z in Pr(D) & P[x,z]
proof
let x be object;
assume x in Pr(D);
then reconsider x as PartialPredicate of D by PARTFUN1:46;
take PP_not(x);
thus thesis by PARTFUN1:45;
end;
consider f being Function of Pr(D),Pr(D) such that
A10: for x being object st x in Pr(D) holds P[x,f.x]
from FUNCT_2:sch 1(A9);
take f;
thus thesis by A10,PARTFUN1:45;
end;
uniqueness
proof
let f1,f2 be UnOp of Pr(D) such that
A11: for p being PartialPredicate of D holds
f1.p = PP_not(p) and
A12: for p being PartialPredicate of D holds
f2.p = PP_not(p);
let x be Element of Pr(D);
reconsider x1 = x as PartialPredicate of D by PARTFUN1:46;
thus f1.x = PP_not(x1) by A11
.= f2.x by A12;
end;
end;
definition
let D;
func PartialPredConnectivesLatt(D) -> strict OrthoLattStr equals
OrthoLattStr(# Pr(D) , PartialPredConnectivesJoin(D) ,
PartialPredConnectivesMeet(D) ,
PartialPredConnectivesCompl(D) #);
coherence;
end;
registration
let D be non empty set;
let f,g be BinOp of D;
let h be UnOp of D;
cluster OrthoLattStr(#D,f,g,h#) -> non empty;
coherence
proof
thus the carrier of OrthoLattStr(#D,f,g,h#) is non empty;
end;
end;
registration
let D;
cluster PartialPredConnectivesLatt(D) ->
non empty constituted-Functions;
coherence;
end;
registration
cluster non empty constituted-Functions for LattStr;
existence
proof
take PartialPredConnectivesLatt(the non empty set);
thus thesis;
end;
end;
registration
cluster strict non empty constituted-Functions for OrthoLattStr;
existence
proof
take PartialPredConnectivesLatt(the non empty set);
thus thesis;
end;
end;
registration
let D;
cluster PartialPredConnectivesLatt(D) -> Lattice-like;
coherence
proof
set L = PartialPredConnectivesLatt(D);
thus L is join-commutative
proof
let a,b be Element of L;
reconsider a1 = a, b1 = b as PartialPredicate of D
by PARTFUN1:46;
thus a"\/"b = PP_or(b1,a1) by Def12
.= b"\/"a by Def12;
end;
thus L is join-associative
proof
let a,b,c be Element of L;
reconsider a1 = a, b1 = b, c1 = c as PartialPredicate of D
by PARTFUN1:46;
A1: a"\/"b = PP_or(a1,b1) by Def12;
b"\/"c = PP_or(b1,c1) by Def12;
hence a"\/"(b"\/"c) = PP_or(a1,PP_or(b1,c1)) by Def12
.= PP_or(PP_or(a1,b1),c1) by Th14
.= (a"\/"b)"\/"c by A1,Def12;
end;
thus L is meet-absorbing
proof
let a,b be Element of L;
reconsider a1 = a, b1 = b as PartialPredicate of D
by PARTFUN1:46;
a"/\"b = PP_and(a1,b1) by Def11;
hence (a"/\"b)"\/"b = PP_or(PP_and(a1,b1),b1) by Def12
.= b by Th28;
end;
thus L is meet-commutative
proof
let a,b be Element of L;
reconsider a1 = a, b1 = b as PartialPredicate of D
by PARTFUN1:46;
thus a"/\"b = PP_and(b1,a1) by Def11
.= b"/\"a by Def11;
end;
thus L is meet-associative
proof
let a,b,c be Element of L;
reconsider a1 = a, b1 = b, c1 = c as PartialPredicate of D
by PARTFUN1:46;
A2: a"/\"b = PP_and(a1,b1) by Def11;
b"/\"c = PP_and(b1,c1) by Def11;
hence a"/\"(b"/\"c) = PP_and(a1,PP_and(b1,c1)) by Def11
.= PP_and(PP_and(a1,b1),c1) by Th14
.= (a"/\"b)"/\"c by A2,Def11;
end;
thus L is join-absorbing
proof
let a,b be Element of L;
reconsider a1 = a, b1 = b as PartialPredicate of D
by PARTFUN1:46;
a"\/"b = PP_or(a1,b1) by Def12;
hence a"/\"(a"\/"b) = PP_and(a1,PP_or(a1,b1)) by Def11
.= a by Th29;
end;
end;
cluster PartialPredConnectivesLatt(D) -> bounded;
coherence
proof
set L = PartialPredConnectivesLatt(D);
thus L is lower-bounded
proof
set c1 = PP_False(D);
reconsider c = c1 as Element of L by PARTFUN1:45;
take c;
let a be Element of L;
reconsider a1 = a as PartialPredicate of D by PARTFUN1:46;
thus c"/\"a = PP_and(c1,a1) by Def11
.= c by Th47;
thus a"/\"c = PP_and(a1,c1) by Def11
.= c by Th47;
end;
thus L is upper-bounded
proof
set c1 = PP_True(D);
reconsider c = c1 as Element of L by PARTFUN1:45;
take c;
let a be Element of L;
reconsider a1 = a as PartialPredicate of D by PARTFUN1:46;
thus c"\/"a = PP_or(c1,a1) by Def12
.= c by Th45;
thus a"\/"c = PP_or(a1,c1) by Def12
.= c by Th45;
end;
end;
cluster PartialPredConnectivesLatt(D) ->
de_Morgan join-idempotent with_idempotent_element;
coherence
proof
set L = PartialPredConnectivesLatt(D);
thus L is de_Morgan
proof
let x,y be Element of L;
reconsider p = x, q = y as PartialPredicate of D
by PARTFUN1:46;
x` = PP_not(p) & y` = PP_not(q) by Def13;
then x`"\/"y` = PP_or(PP_not(p),PP_not(q)) by Def12;
hence (x` "\/" y`)` = PP_and(p,q) by Def13
.= x "/\" y by Def11;
end;
thus L is join-idempotent;
take x = the Element of L;
thus x"\/"x = x;
end;
end;
theorem Th60:
Top PartialPredConnectivesLatt(D) = PP_True(D)
proof
set L = PartialPredConnectivesLatt(D);
reconsider f = PP_True(D) as Element of L by PARTFUN1:45;
for a being Element of L holds f "\/" a = f & a "\/" f = f
proof
let a be Element of L;
reconsider a1 = a as PartialPredicate of D by PARTFUN1:46;
thus f"\/"a = PP_or(PP_True(D),a1) by Def12
.= f by Th45;
hence thesis;
end;
hence thesis by LATTICES:def 17;
end;
theorem Th61:
Bottom PartialPredConnectivesLatt(D) = PP_False(D)
proof
set L = PartialPredConnectivesLatt(D);
reconsider f = PP_False(D) as Element of L by PARTFUN1:45;
for a being Element of L holds f "/\" a = f & a "/\" f = f
proof
let a be Element of L;
reconsider a1 = a as PartialPredicate of D by PARTFUN1:46;
thus f"/\"a = PP_and(PP_False(D),a1) by Def11
.= f by Th47;
hence thesis;
end;
hence thesis by LATTICES:def 16;
end;
definition
let L be non empty constituted-Functions LattStr, a, b be Element of L;
pred a is_a_partial_complement_of b means
a"\/"b = Top L | dom b & b"\/"a = Top L | dom b &
a"/\"b = Bottom L | dom b & b"/\"a = Bottom L | dom b;
end;
definition
let L be non empty constituted-Functions LattStr;
attr L is partially_complemented means
for b being Element of L
ex a being Element of L st a is_a_partial_complement_of b;
end;
definition
let IT be constituted-Functions non empty LattStr;
attr IT is partially_Boolean means
IT is bounded partially_complemented distributive;
end;
registration
cluster partially_Boolean -> bounded partially_complemented distributive
for constituted-Functions non empty LattStr;
coherence;
cluster bounded partially_complemented distributive -> partially_Boolean
for constituted-Functions non empty LattStr;
coherence;
end;
theorem Th62:
for a,b being Element of PartialPredConnectivesLatt(D)
st a = p & b = PP_not(p) holds b is_a_partial_complement_of a
proof
set L = PartialPredConnectivesLatt(D);
let a,b be Element of L such that
A1: a = p & b = PP_not(p);
Top L = PP_True(D) by Th60;
hence Top L | dom a = PP_or(PP_not(p),p) by A1,Th49
.= b"\/"a by A1,Def12;
hence a"\/"b = Top L | dom a;
Bottom L = PP_False(D) by Th61;
hence Bottom L | dom a = PP_and(PP_not(p),p) by A1,Th51
.= b"/\"a by A1,Def11;
hence a"/\"b = Bottom L | dom a;
end;
registration
let D;
cluster PartialPredConnectivesLatt(D) -> partially_Boolean;
coherence
proof
set L = PartialPredConnectivesLatt(D);
thus L is bounded;
thus L is partially_complemented
proof
let b be Element of L;
reconsider b1 = b as PartialPredicate of D by PARTFUN1:46;
reconsider a = PP_not(b1) as Element of L by PARTFUN1:45;
take a;
thus thesis by Th62;
end;
thus L is distributive
proof
let a,b,c be Element of L;
reconsider a1 = a, b1 = b, c1 = c as PartialPredicate of D
by PARTFUN1:46;
A1: a"/\"b = PP_and(a1,b1) & a"/\"c = PP_and(a1,c1) by Def11;
b"\/"c = PP_or(b1,c1) by Def12;
hence a"/\"(b"\/"c) = PP_and(a1,PP_or(b1,c1)) by Def11
.= PP_or(PP_and(a1,b1),PP_and(a1,c1)) by Th30
.= (a"/\"b)"\/"(a"/\"c) by A1,Def12;
end;
end;
end;
::$N Distributivity law
theorem
PP_or(p,PP_and(q,r)) = PP_and(PP_or(p,q),PP_or(p,r))
proof
reconsider a = p, b = q, c = r
as Element of PartialPredConnectivesLatt(D) by PARTFUN1:45;
A1: PP_or(p,q) = a"\/"b & PP_or(p,r) = a"\/"c by Def12;
PP_and(q,r) = b"/\"c by Def11;
then
A2: PP_or(p,PP_and(q,r)) = a"\/"(b"/\"c) by Def12;
PP_and(PP_or(p,q),PP_or(p,r)) = (a"\/"b)"/\"(a"\/"c) by A1,Def11;
hence thesis by A2,LATTICES:11;
end;
definition
let L be non empty OrthoLattStr;
attr L is Kleene means
for x,y being Element of L holds x"/\"x` [= y"\/"y`;
end;
registration
cluster Boolean well-complemented -> Kleene for
meet-absorbing join-absorbing meet-commutative non empty OrthoLattStr;
coherence
proof
let L be meet-absorbing join-absorbing meet-commutative
non empty OrthoLattStr such that
A1: L is Boolean and
A2: L is well-complemented;
let x,y be Element of L;
x` is_a_complement_of x & y` is_a_complement_of y by A2;
hence (x"/\"x`) "\/" (y"\/"y`) = (y"\/"y`) by A1,LATTICES:def 17;
end;
end;
registration
let D;
cluster PartialPredConnectivesLatt(D) -> Kleene;
coherence
proof
set L = PartialPredConnectivesLatt(D);
let x,y be Element of L;
reconsider p = x, q = y as PartialPredicate of D
by PARTFUN1:46;
x` = PP_not(p) by Def13;
then
A1: x"/\"x` = PP_and(p,PP_not(p)) by Def11;
y` = PP_not(q) by Def13;
then
A2: y"\/"y` = PP_or(q,PP_not(q)) by Def12;
A3: PP_or(q,PP_not(q)) = PP_True(D) | dom q by Th49;
PP_and(p,PP_not(p)) = PP_False(D) | dom p by Th51;
then PP_or(PP_and(p,PP_not(p)),PP_or(q,PP_not(q))) = PP_or(q,PP_not(q))
by A3,Th55;
hence (x"/\"x`) "\/" (y"\/"y`) = y"\/"y` by A1,A2,Def12;
end;
end;
registration
cluster partially_Boolean join-idempotent Lattice-like
for non empty constituted-Functions LattStr;
existence
proof
take PartialPredConnectivesLatt(the non empty set);
thus thesis;
end;
end;
registration
cluster Kleene de_Morgan join-idempotent with_idempotent_element
Lattice-like strict
for non empty OrthoLattStr;
existence
proof
take PartialPredConnectivesLatt(the non empty set);
thus thesis;
end;
end;
registration
cluster partially_Boolean Kleene de_Morgan
join-idempotent with_idempotent_element Lattice-like strict
for non empty constituted-Functions OrthoLattStr;
existence
proof
take PartialPredConnectivesLatt(the non empty set);
thus thesis;
end;
end;