Volume 10, 1998

University of Bialystok

Copyright (c) 1998 Association of Mizar Users

### The abstract of the Mizar article:

### A Theory of Boolean Valued Functions and Quantifiers with Respect to Partitions

**by****Shunichi Kobayashi, and****Yatsuka Nakamura**- Received December 21, 1998
- MML identifier: BVFUNC_2

- [ Mizar article, MML identifier index ]

environ vocabulary PARTIT1, T_1TOPSP, EQREL_1, FUNCT_1, SETFAM_1, RELAT_1, CANTOR_1, BOOLE, PARTFUN1, FINSEQ_4, TARSKI, SUBSET_1, GROUP_4, FUNCT_2, MARGREL1, BVFUNC_1, ZF_LANG, BINARITH, BVFUNC_2; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, MARGREL1, VALUAT_1, RELAT_1, FUNCT_1, SETFAM_1, FRAENKEL, RELSET_1, PARTFUN1, FINSEQ_4, EQREL_1, CANTOR_1, T_1TOPSP, BINARITH, PARTIT1, BVFUNC_1; constructors FINSEQ_4, BINARITH, CANTOR_1, PARTIT1, BVFUNC_1, PUA2MSS1; clusters RELSET_1, PARTIT1, T_1TOPSP, SUBSET_1, MARGREL1, VALUAT_1, BINARITH, XBOOLE_0; requirements SUBSET, BOOLE; begin :: Chap. 1 Preliminaries reserve Y for non empty set, G for Subset of PARTITIONS(Y); definition let X be set; redefine func PARTITIONS X -> Part-Family of X; end; definition let X be set; let F be non empty Part-Family of X; redefine mode Element of F -> a_partition of X; end; theorem :: BVFUNC_2:1 for y being Element of Y ex X being Subset of Y st y in X & ex h being Function, F being Subset-Family of Y st dom h=G & rng h = F & (for d being set st d in G holds h.d in d) & X=Intersect F & X<>{}; definition let Y;let G be Subset of PARTITIONS(Y); func '/\' G -> a_partition of Y means :: BVFUNC_2:def 1 for x being set holds x in it iff ex h being Function, F being Subset-Family of Y st dom h=G & rng h = F & (for d being set st d in G holds h.d in d) & x=Intersect F & x<>{}; end; definition let Y;let G be Subset of PARTITIONS(Y),b be set; pred b is_upper_min_depend_of G means :: BVFUNC_2:def 2 (for d being a_partition of Y st d in G holds b is_a_dependent_set_of d) & (for e being set st (e c= b & (for d being a_partition of Y st d in G holds e is_a_dependent_set_of d)) holds e=b); end; theorem :: BVFUNC_2:2 for y being Element of Y st G<>{} holds (ex X being Subset of Y st y in X & X is_upper_min_depend_of G); definition let Y;let G be Subset of PARTITIONS(Y); func '\/' G -> a_partition of Y means :: BVFUNC_2:def 3 (for x being set holds x in it iff x is_upper_min_depend_of G) if G<>{} otherwise it = %I(Y); end; theorem :: BVFUNC_2:3 for G being Subset of PARTITIONS(Y),PA being a_partition of Y st PA in G holds PA '>' ('/\' G); theorem :: BVFUNC_2:4 for G being Subset of PARTITIONS(Y),PA being a_partition of Y st PA in G holds PA '<' ('\/' G); begin :: Chap. 2 Coordinate and Quantifiers definition let Y;let G be Subset of PARTITIONS(Y); attr G is generating means :: BVFUNC_2:def 4 ('/\' G) = %I(Y); end; definition let Y;let G be Subset of PARTITIONS(Y); attr G is independent means :: BVFUNC_2:def 5 for h being Function, F being Subset-Family of Y st dom h=G & rng h=F & (for d being set st d in G holds h.d in d) holds (Intersect F)<>{}; end; definition let Y;let G be Subset of PARTITIONS(Y); pred G is_a_coordinate means :: BVFUNC_2:def 6 G is independent generating; end; definition let Y;let PA be a_partition of Y; redefine func {PA} -> Subset of PARTITIONS(Y); end; definition let Y;let PA be a_partition of Y;let G be Subset of PARTITIONS(Y); func CompF(PA,G) -> a_partition of Y equals :: BVFUNC_2:def 7 '/\' (G \ {PA}); end; definition let Y;let a be Element of Funcs(Y,BOOLEAN); let G be Subset of PARTITIONS(Y),PA be a_partition of Y; pred a is_independent_of PA,G means :: BVFUNC_2:def 8 a is_dependent_of CompF(PA,G); end; definition let Y;let a be Element of Funcs(Y,BOOLEAN), G be Subset of PARTITIONS(Y), PA be a_partition of Y; func All(a,PA,G) -> Element of Funcs(Y,BOOLEAN) equals :: BVFUNC_2:def 9 B_INF(a,CompF(PA,G)); end; definition let Y;let a be Element of Funcs(Y,BOOLEAN), G be Subset of PARTITIONS(Y), PA be a_partition of Y; func Ex(a,PA,G) -> Element of Funcs(Y,BOOLEAN) equals :: BVFUNC_2:def 10 B_SUP(a,CompF(PA,G)); end; theorem :: BVFUNC_2:5 for a being Element of Funcs(Y,BOOLEAN), PA being a_partition of Y holds All(a,PA,G) is_dependent_of CompF(PA,G); theorem :: BVFUNC_2:6 for a being Element of Funcs(Y,BOOLEAN), PA being a_partition of Y holds Ex(a,PA,G) is_dependent_of CompF(PA,G); theorem :: BVFUNC_2:7 for a being Element of Funcs(Y,BOOLEAN), PA being a_partition of Y holds All(I_el(Y),PA,G) = I_el(Y); theorem :: BVFUNC_2:8 for a being Element of Funcs(Y,BOOLEAN), PA being a_partition of Y holds Ex(I_el(Y),PA,G) = I_el(Y); theorem :: BVFUNC_2:9 for a being Element of Funcs(Y,BOOLEAN), PA being a_partition of Y holds All(O_el(Y),PA,G) = O_el(Y); theorem :: BVFUNC_2:10 for a being Element of Funcs(Y,BOOLEAN), PA being a_partition of Y holds Ex(O_el(Y),PA,G) = O_el(Y); theorem :: BVFUNC_2:11 for a being Element of Funcs(Y,BOOLEAN), PA being a_partition of Y holds All(a,PA,G) '<' a; theorem :: BVFUNC_2:12 for a being Element of Funcs(Y,BOOLEAN), PA being a_partition of Y holds a '<' Ex(a,PA,G); theorem :: BVFUNC_2:13 for a,b being Element of Funcs(Y,BOOLEAN), PA being a_partition of Y holds All(a '&' b,PA,G) = All(a,PA,G) '&' All(b,PA,G); theorem :: BVFUNC_2:14 for a,b being Element of Funcs(Y,BOOLEAN), PA being a_partition of Y holds All(a,PA,G) 'or' All(b,PA,G) '<' All(a 'or' b,PA,G); theorem :: BVFUNC_2:15 for a,b being Element of Funcs(Y,BOOLEAN), PA being a_partition of Y holds All(a 'imp' b,PA,G) '<' All(a,PA,G) 'imp' All(b,PA,G); theorem :: BVFUNC_2:16 for a,b being Element of Funcs(Y,BOOLEAN), PA being a_partition of Y holds Ex(a 'or' b,PA,G) = Ex(a,PA,G) 'or' Ex(b,PA,G); theorem :: BVFUNC_2:17 for a,b being Element of Funcs(Y,BOOLEAN),G being Subset of PARTITIONS(Y), PA being a_partition of Y holds Ex(a '&' b,PA,G) '<' Ex(a,PA,G) '&' Ex(b,PA,G); theorem :: BVFUNC_2:18 for a,b being Element of Funcs(Y,BOOLEAN), PA being a_partition of Y holds Ex(a,PA,G) 'xor' Ex(b,PA,G) '<' Ex(a 'xor' b,PA,G); theorem :: BVFUNC_2:19 for a,b being Element of Funcs(Y,BOOLEAN), PA being a_partition of Y holds Ex(a,PA,G) 'imp' Ex(b,PA,G) '<' Ex(a 'imp' b,PA,G); reserve a, u for Element of Funcs(Y,BOOLEAN); theorem :: BVFUNC_2:20 :: De'Morgan's Law for PA being a_partition of Y holds 'not' All(a,PA,G) = Ex('not' a,PA,G); theorem :: BVFUNC_2:21 :: De'Morgan's Law for PA being a_partition of Y holds 'not' Ex(a,PA,G) = All('not' a,PA,G); theorem :: BVFUNC_2:22 for PA being a_partition of Y st u is_independent_of PA,G holds All(u 'imp' a,PA,G) = u 'imp' All(a,PA,G); theorem :: BVFUNC_2:23 for PA being a_partition of Y st u is_independent_of PA,G holds All(a 'imp' u,PA,G) = Ex(a,PA,G) 'imp' u; theorem :: BVFUNC_2:24 for PA being a_partition of Y st u is_independent_of PA,G holds All(u 'or' a,PA,G) = u 'or' All(a,PA,G); theorem :: BVFUNC_2:25 for PA being a_partition of Y st u is_independent_of PA,G holds All(a 'or' u,PA,G) = All(a,PA,G) 'or' u; theorem :: BVFUNC_2:26 for PA being a_partition of Y st u is_independent_of PA,G holds All(a 'or' u,PA,G) '<' Ex(a,PA,G) 'or' u; theorem :: BVFUNC_2:27 for PA being a_partition of Y st u is_independent_of PA,G holds All(u '&' a,PA,G) = u '&' All(a,PA,G); theorem :: BVFUNC_2:28 for PA being a_partition of Y st u is_independent_of PA,G holds All(a '&' u,PA,G) = All(a,PA,G) '&' u; theorem :: BVFUNC_2:29 for PA being a_partition of Y holds All(a '&' u,PA,G) '<' Ex(a,PA,G) '&' u; theorem :: BVFUNC_2:30 for PA being a_partition of Y st u is_independent_of PA,G holds All(u 'xor' a,PA,G) '<' u 'xor' All(a,PA,G); theorem :: BVFUNC_2:31 for PA being a_partition of Y st u is_independent_of PA,G holds All(a 'xor' u,PA,G) '<' All(a,PA,G) 'xor' u; theorem :: BVFUNC_2:32 for PA being a_partition of Y st u is_independent_of PA,G holds All(u 'eqv' a,PA,G) '<' u 'eqv' All(a,PA,G); theorem :: BVFUNC_2:33 for PA being a_partition of Y st u is_independent_of PA,G holds All(a 'eqv' u,PA,G) '<' All(a,PA,G) 'eqv' u; theorem :: BVFUNC_2:34 for PA being a_partition of Y st u is_independent_of PA,G holds Ex(u 'or' a,PA,G) = u 'or' Ex(a,PA,G); theorem :: BVFUNC_2:35 for PA being a_partition of Y st u is_independent_of PA,G holds Ex(a 'or' u,PA,G) = Ex(a,PA,G) 'or' u; theorem :: BVFUNC_2:36 for PA being a_partition of Y st u is_independent_of PA,G holds Ex(u '&' a,PA,G) = u '&' Ex(a,PA,G); theorem :: BVFUNC_2:37 for PA being a_partition of Y st u is_independent_of PA,G holds Ex(a '&' u,PA,G) = Ex(a,PA,G) '&' u; theorem :: BVFUNC_2:38 for PA being a_partition of Y holds u 'imp' Ex(a,PA,G) '<' Ex(u 'imp' a,PA,G); theorem :: BVFUNC_2:39 for PA being a_partition of Y holds Ex(a,PA,G) 'imp' u '<' Ex(a 'imp' u,PA,G); theorem :: BVFUNC_2:40 for PA being a_partition of Y st u is_independent_of PA,G holds u 'xor' Ex(a,PA,G) '<' Ex(u 'xor' a,PA,G); theorem :: BVFUNC_2:41 for PA being a_partition of Y st u is_independent_of PA,G holds Ex(a,PA,G) 'xor' u '<' Ex(a 'xor' u,PA,G);

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