Journal of Formalized Mathematics
Volume 2, 1990
University of Bialystok
Copyright (c) 1990 Association of Mizar Users

### Three-Argument Operations and Four-Argument Operations

by
Michal Muzalewski, and
Wojciech Skaba

Received October 2, 1990

MML identifier: MULTOP_1
[ Mizar article, MML identifier index ]

```environ

vocabulary FUNCT_1, MULTOP_1;
notation XBOOLE_0, ZFMISC_1, SUBSET_1, FUNCT_1, FUNCT_2, MCART_1, DOMAIN_1;
constructors FUNCT_2, DOMAIN_1, MEMBERED, XBOOLE_0;
clusters RELSET_1, SUBSET_1, MEMBERED, ZFMISC_1, XBOOLE_0;
requirements SUBSET, BOOLE;

begin :: THREE ARGUMENT OPERATIONS

definition
let f be Function;
let a,b,c be set;
func f.(a,b,c) -> set equals
:: MULTOP_1:def 1
f.[a,b,c];
end;

reserve A,B,C,D,E for non empty set,
a for Element of A, b for Element of B,
c for Element of C, d for Element of D,
X,Y,Z,S,x,y,z,s,t for set;

definition
let A,B,C,D;
let f be Function of [:A,B,C:],D;
let a,b,c;
redefine func f.(a,b,c) -> Element of D;
end;

canceled;

theorem :: MULTOP_1:2
for f1,f2 being Function of [:X,Y,Z:],D st
for x,y,z st x in X & y in Y & z in Z holds f1.[x,y,z] = f2.[x,y,z]
holds f1 = f2;

theorem :: MULTOP_1:3
for f1,f2 being Function of [:A,B,C:],D
st for a,b,c holds f1.[a,b,c] = f2.[a,b,c]
holds f1 = f2;

theorem :: MULTOP_1:4
for f1,f2 being Function of [:A,B,C:],D st
for a being Element of A
for b being Element of B
for c being Element of C
holds
f1.(a,b,c) = f2.(a,b,c)
holds f1 = f2;

definition let A be set;
mode TriOp of A is Function of [:A,A,A:],A;
end;

scheme FuncEx3D
{ X,Y,Z,T() -> non empty set, P[set,set,set,set] } :
ex f being Function of [:X(),Y(),Z():],T() st
for x being Element of X()
for y being Element of Y()
for z being Element of Z()
holds P[x,y,z,f.[x,y,z]]
provided
for x being Element of X()
for y being Element of Y()
for z being Element of Z()
ex t being Element of T() st P[x,y,z,t];

scheme TriOpEx
{ A()->non empty set,
P[ Element of A(), Element of A(), Element of A(), Element of A()] }:
ex o being TriOp of A() st
for a,b,c being Element of A() holds P[a,b,c,o.(a,b,c)]
provided
for x,y,z being Element of A() ex t being Element of A() st P[x,y,z,t];

scheme Lambda3D
{ X, Y, Z, T()->non empty set,
F( Element of X(), Element of Y(), Element of Z()) -> Element of T()
}:
ex f being Function of [:X(),Y(),Z():],T()
st for x being Element of X()
for y being Element of Y()
for z being Element of Z()
holds f.[x,y,z]=F(x,y,z);

scheme TriOpLambda
{ A,B,C,D()->non empty set,
O( Element of A(), Element of B(), Element of C()) -> Element of D()
}:
ex o being Function of [:A(),B(),C():],D() st
for a being Element of A(),
b being Element of B(),
c being Element of C() holds o.(a,b,c) = O(a,b,c);

:: FOUR ARGUMENT OPERATIONS

definition
let f be Function;
let a,b,c,d be set;
func f.(a,b,c,d) -> set equals
:: MULTOP_1:def 2
f.[a,b,c,d];
end;

definition
let A,B,C,D,E;
let f be Function of [:A,B,C,D:],E;
let a,b,c,d;
redefine func f.(a,b,c,d) -> Element of E;
end;

canceled;

theorem :: MULTOP_1:6
for f1,f2 being Function of [:X,Y,Z,S:],D st
for x,y,z,s st x in X & y in Y & z in Z & s in S
holds f1.[x,y,z,s] = f2.[x,y,z,s]
holds
f1 = f2;

theorem :: MULTOP_1:7
for f1,f2 being Function of [:A,B,C,D:],E
st for a,b,c,d holds f1.[a,b,c,d] = f2.[a,b,c,d]
holds f1 = f2;

theorem :: MULTOP_1:8
for f1,f2 being Function of [:A,B,C,D:],E st
for a,b,c,d holds f1.(a,b,c,d) = f2.(a,b,c,d)
holds f1 = f2;

definition let A;
mode QuaOp of A is Function of [:A,A,A,A:],A;
end;

scheme FuncEx4D
{ X, Y, Z, S, T() -> non empty set, P[set,set,set,set,set] }:
ex f being Function of [:X(),Y(),Z(),S():],T() st
for x being Element of X()
for y being Element of Y()
for z being Element of Z()
for s being Element of S()
holds P[x,y,z,s,f.[x,y,z,s]]
provided
for x being Element of X()
for y being Element of Y()
for z being Element of Z()
for s being Element of S()
ex t being Element of T() st P[x,y,z,s,t];

scheme QuaOpEx
{ A()->non empty set,
P[ Element of A(), Element of A(),
Element of A(), Element of A(), Element of A()] }:
ex o being QuaOp of A() st
for a,b,c,d being Element of A() holds P[a,b,c,d,o.(a,b,c,d)]
provided
for x,y,z,s being Element of A()
ex t being Element of A() st P[x,y,z,s,t];

scheme Lambda4D
{ X, Y, Z, S, T() -> non empty set,
F( Element of X(), Element of Y(),
Element of Z(), Element of S()) -> Element of T() }:
ex f being Function of [:X(),Y(),Z(),S():],T()
st for x being Element of X()
for y being Element of Y()
for z being Element of Z()
for s being Element of S()
holds f.[x,y,z,s]=F(x,y,z,s);

scheme QuaOpLambda
{ A()->non empty set, O( Element of A(), Element of A(),
Element of A(), Element of A()) -> Element of A() }:
ex o being QuaOp of A() st
for a,b,c,d being Element of A() holds o.(a,b,c,d) = O(a,b,c,d);
```