Volume 2, 1990

University of Bialystok

Copyright (c) 1990 Association of Mizar Users

### The abstract of the Mizar article:

### 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);

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