Copyright (c) 1999 Association of Mizar Users
environ vocabulary EQREL_1, PARTIT1, T_1TOPSP, BOOLE, BVFUNC_2, FUNCOP_1, SETFAM_1, RELAT_1, FUNCT_1, CANTOR_1, CAT_1, FUNCT_2, MARGREL1, ZF_LANG, BVFUNC_1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, MARGREL1, VALUAT_1, RELAT_1, SETFAM_1, FUNCT_1, FRAENKEL, EQREL_1, CANTOR_1, CQC_LANG, PARTIT1, BVFUNC_1, BVFUNC_2, FUNCT_4; constructors CANTOR_1, BVFUNC_2, PUA2MSS1, BVFUNC_1, FUNCT_4, MEMBERED; clusters FUNCT_7, PARTIT1, SUBSET_1, MARGREL1, VALUAT_1, FRAENKEL; requirements SUBSET, BOOLE; definitions TARSKI, BVFUNC_1, XBOOLE_0; theorems TARSKI, FUNCT_1, SETFAM_1, EQREL_1, ZFMISC_1, CANTOR_1, T_1TOPSP, MARGREL1, PARTIT1, BVFUNC_1, BVFUNC_2, ENUMSET1, FUNCT_4, CQC_LANG, MSSUBFAM, PARTIT_2, BVFUNC_4, XBOOLE_0, XBOOLE_1, VALUAT_1; begin ::Chap. 1 Preliminaries reserve Y for non empty set; theorem Th1: for z being Element of Y, PA,PB being a_partition of Y st PA '<' PB holds EqClass(z,PA) c= EqClass(z,PB) proof let z be Element of Y; let PA,PB be a_partition of Y; assume A1: PA '<' PB; A2:z in EqClass(z,PA) & EqClass(z,PA) in PA by T_1TOPSP:def 1; ex b being set st b in PB & EqClass(z,PA) c= b by A1,SETFAM_1:def 2; hence thesis by A2,T_1TOPSP:def 1; end; theorem for z being Element of Y, PA,PB being a_partition of Y holds EqClass(z,PA) c= EqClass(z,PA '\/' PB) proof let z be Element of Y; let PA,PB be a_partition of Y; PA '<' PA '\/' PB by PARTIT1:19; hence thesis by Th1; end; theorem for z being Element of Y, PA,PB being a_partition of Y holds EqClass(z,PA '/\' PB) c= EqClass(z,PA) proof let z be Element of Y; let PA,PB be a_partition of Y; PA '>' PA '/\' PB by PARTIT1:17; hence thesis by Th1; end; theorem for z being Element of Y, PA being a_partition of Y holds EqClass(z,PA) c= EqClass(z,%O(Y)) & EqClass(z,%I(Y)) c= EqClass(z,PA) proof let z be Element of Y; let PA be a_partition of Y; %O(Y) '>' PA & PA '>' %I(Y) by PARTIT1:36; hence thesis by Th1; end; theorem for G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G is independent & G={A,B} & A<>B holds for a,b being set st a in A & b in B holds a meets b proof let G be Subset of PARTITIONS(Y); let A,B be a_partition of Y; assume that A1:G is independent and A2: G={A,B} & A<>B; let a,b be set; assume A3: a in A & b in B; set h2 = (A,B) --> (a,b); {a,b} c= bool Y proof let x be set; assume x in {a,b}; then x = a or x = b by TARSKI:def 2; hence thesis by A3; end; then reconsider SS = {a,b} as Subset-Family of Y by SETFAM_1:def 7; A4: dom h2 = {A,B} & rng h2 = SS by A2,FUNCT_4:65,67; for d being set st d in G holds h2.d in d proof let d be set; assume d in G; then d = A or d = B by A2,TARSKI:def 2; hence thesis by A2,A3,FUNCT_4:66; end; then Intersect SS <> {} by A1,A2,A4,BVFUNC_2:def 5; hence a /\ b <> {} by A3,MSSUBFAM:10; end; theorem for G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G is independent & G={A,B} & A <> B holds '/\' G = A '/\' B proof let G be Subset of PARTITIONS(Y); let A,B be a_partition of Y; assume that A1:G is independent and A2: G={A,B} & A <> B; A3:A '/\' B c= '/\' G proof let x be set; assume x in A '/\' B; then x in INTERSECTION(A,B) \ {{}} by PARTIT1:def 4; then x in INTERSECTION(A,B) & not x in {{}} by XBOOLE_0:def 4; then consider X,Z being set such that A4: X in A & Z in B & x = X /\ Z by SETFAM_1:def 5; set h = (A,B) --> (X,Z); {X,Z} c= bool Y proof let m be set; assume m in {X,Z}; then m = X or m = Z by TARSKI:def 2; hence thesis by A4; end; then reconsider SS = {X,Z} as Subset-Family of Y by SETFAM_1:def 7; A5: dom h = {A,B} & rng h = SS by A2,FUNCT_4:65,67; A6: for d being set st d in G holds h.d in d proof let d be set; assume d in G; then d = A or d = B by A2,TARSKI:def 2; hence thesis by A2,A4,FUNCT_4:66; end; then A7: Intersect SS <> {} by A1,A2,A5,BVFUNC_2:def 5; X /\ Z = Intersect SS by A4,MSSUBFAM:10; hence thesis by A2,A4,A5,A6,A7,BVFUNC_2:def 1; end; '/\' G c= A '/\' B proof let x be set; assume x in '/\' G; then consider h being Function, F being Subset-Family of Y such that A8: dom h=G & rng h = F & (for d being set st d in G holds h.d in d) & x=Intersect F & x<>{} by BVFUNC_2:def 1; A9:A in G & B in G by A2,TARSKI:def 2; then A10:h.A in A & h.B in B by A8; A11:h.A in rng h & h.B in rng h by A8,A9,FUNCT_1:def 5; then A12:Intersect F = meet (rng h) by A8,CANTOR_1:def 3; A13:not (x in {{}}) by A8,TARSKI:def 1; A14:h.A /\ h.B c= x proof let m be set; assume m in h.A /\ h.B; then A15:m in h.A & m in h.B by XBOOLE_0:def 3; rng h = {h.A,h.B} proof thus rng h c= {h.A,h.B} proof let m be set; assume m in rng h; then consider w being set such that A16: w in dom h & m = h.w by FUNCT_1:def 5; w = A or w = B by A2,A8,A16,TARSKI:def 2; hence thesis by A16,TARSKI:def 2; end; let m be set; assume m in {h.A,h.B}; then A17: m = h.A or m = h.B by TARSKI:def 2; A in dom h & B in dom h by A2,A8,TARSKI:def 2; hence thesis by A17,FUNCT_1:def 5; end; then for y being set holds y in rng h implies m in y by A15,TARSKI:def 2 ; hence thesis by A8,A11,A12,SETFAM_1:def 1; end; x c= h.A /\ h.B proof let m be set; assume m in x; then m in h.A & m in h.B by A8,A11,A12,SETFAM_1:def 1; hence thesis by XBOOLE_0:def 3; end; then h.A /\ h.B = x by A14,XBOOLE_0:def 10; then x in INTERSECTION(A,B) by A10,SETFAM_1:def 5; then x in INTERSECTION(A,B) \ {{}} by A13,XBOOLE_0:def 4; hence thesis by PARTIT1:def 4; end; hence thesis by A3,XBOOLE_0:def 10; end; theorem for G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G={A,B} & A<>B holds CompF(A,G) = B proof let G be Subset of PARTITIONS(Y); let A,B be a_partition of Y; assume A1: G={A,B} & A<>B; then A2:G \ {A}={A} \/ {B} \ {A} by ENUMSET1:41 .= ({A} \ {A}) \/ ({B} \ {A}) by XBOOLE_1:42 .= ({A} \ {A}) \/ {B} by A1,ZFMISC_1:20; A in {A} by TARSKI:def 1; then {A} \ {A}={} by ZFMISC_1:68; then A3:CompF(A,G)='/\' {B} by A2,BVFUNC_2:def 7; A4:'/\' {B} c= B proof let x be set; assume x in '/\' {B}; then consider h being Function, F being Subset-Family of Y such that A5: dom h={B} & rng h = F & (for d being set st d in {B} holds h.d in d) & x=Intersect F & x<>{} by BVFUNC_2:def 1; rng h = {h.B} by A5,FUNCT_1:14; then A6:x=meet ({h.B}) by A5,CANTOR_1:def 3; B in {B} by TARSKI:def 1; then h.B in B by A5; hence thesis by A6,SETFAM_1:11; end; B c= '/\' {B} proof let x be set; assume A7:x in B; set h0 = B .--> x; A8:dom h0 = {B} & rng h0 = {x} by CQC_LANG:5; A9:for d being set st d in {B} holds h0.d in d proof let d be set; assume d in {B}; then d=B by TARSKI:def 1; hence thesis by A7,CQC_LANG:6; end; rng h0 c= bool Y proof let m be set; assume m in rng h0; then m=x by A8,TARSKI:def 1; hence thesis by A7; end; then reconsider F0 = rng h0 as Subset-Family of Y by SETFAM_1:def 7; meet F0 = Intersect F0 by A8,CANTOR_1:def 3; then A10:x=Intersect F0 by A8,SETFAM_1:11; x<>{} by A7,EQREL_1:def 6; hence thesis by A8,A9,A10,BVFUNC_2:def 1; end; hence thesis by A3,A4,XBOOLE_0:def 10; end; begin Lm1: for a,b being Element of Funcs(Y,BOOLEAN), G being Subset of PARTITIONS(Y), P being a_partition of Y st a '<' b holds All(a,P,G) '<' Ex(b,P,G) proof let a,b be Element of Funcs(Y,BOOLEAN), G be Subset of PARTITIONS(Y), P be a_partition of Y; assume a '<' b; then A1:All(a,P,G) '<' All(b,P,G) by PARTIT_2:13; All(b,P,G) '<' Ex(b,P,G) by BVFUNC_4:18; hence thesis by A1,BVFUNC_1:18; end; canceled 3; theorem for a being Element of Funcs(Y,BOOLEAN), G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G is independent holds All(All(a,A,G),B,G) '<' Ex(All(a,B,G),A,G) proof let a be Element of Funcs(Y,BOOLEAN); let G be Subset of PARTITIONS(Y); let A,B be a_partition of Y; assume G is independent; then All(All(a,A,G),B,G) = All(All(a,B,G),A,G) by PARTIT_2:17; hence thesis by Lm1; end; theorem for a being Element of Funcs(Y,BOOLEAN), G being Subset of PARTITIONS(Y), A,B being a_partition of Y holds All(All(a,A,G),B,G) '<' Ex(Ex(a,B,G),A,G) proof let a be Element of Funcs(Y,BOOLEAN); let G be Subset of PARTITIONS(Y); let A,B be a_partition of Y; A1:All(All(a,A,G),B,G) '<' All(a,A,G) by BVFUNC_2:11; a '<' Ex(a,B,G) by BVFUNC_2:12; then All(a,A,G) '<' Ex(Ex(a,B,G),A,G) by Lm1; hence thesis by A1,BVFUNC_1:18; end; theorem for a being Element of Funcs(Y,BOOLEAN), G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G is independent holds All(All(a,A,G),B,G) '<' All(Ex(a,B,G),A,G) proof let a be Element of Funcs(Y,BOOLEAN); let G be Subset of PARTITIONS(Y); let A,B be a_partition of Y; assume G is independent; then A1:All(All(a,A,G),B,G) = All(All(a,B,G),A,G) by PARTIT_2:17; All(a,B,G) '<' Ex(a,B,G) by Lm1; hence thesis by A1,PARTIT_2:13; end; theorem for a being Element of Funcs(Y,BOOLEAN),G being Subset of PARTITIONS(Y), A,B being a_partition of Y holds All(Ex(a,A,G),B,G) '<' Ex(Ex(a,B,G),A,G) proof let a be Element of Funcs(Y,BOOLEAN); let G be Subset of PARTITIONS(Y); let A,B be a_partition of Y; A1:Ex(a,B,G) = B_SUP(a,CompF(B,G)) by BVFUNC_2:def 10; let z be Element of Y; assume A2:Pj(All(Ex(a,A,G),B,G),z)=TRUE; A3:z in EqClass(z,CompF(B,G)) & EqClass(z,CompF(B,G)) in CompF(B,G) by T_1TOPSP:def 1; A4: now assume not (for x being Element of Y st x in EqClass(z,CompF(B,G)) holds Pj(Ex(a,A,G),x)=TRUE); then Pj(B_INF(Ex(a,A,G),CompF(B,G)),z) = FALSE by BVFUNC_1:def 19; then Pj(All(Ex(a,A,G),B,G),z)=FALSE by BVFUNC_2:def 9; hence contradiction by A2,MARGREL1:43; end; now assume not (ex x being Element of Y st x in EqClass(z,CompF(A,G)) & Pj(a,x)=TRUE); then Pj(B_SUP(a,CompF(A,G)),z) = FALSE by BVFUNC_1:def 20; then Pj(Ex(a,A,G),z)=FALSE by BVFUNC_2:def 10; then Pj(Ex(a,A,G),z)<>TRUE by MARGREL1:43; hence contradiction by A3,A4; end; then consider x1 being Element of Y such that A5: x1 in EqClass(z,CompF(A,G)) & Pj(a,x1)=TRUE; x1 in EqClass(x1,CompF(B,G)) by T_1TOPSP:def 1; then Pj(Ex(a,B,G),x1)=TRUE by A1,A5,BVFUNC_1:def 20; then Pj(B_SUP(Ex(a,B,G),CompF(A,G)),z) = TRUE by A5,BVFUNC_1:def 20; hence thesis by BVFUNC_2:def 10; end; theorem for a being Element of Funcs(Y,BOOLEAN), G being Subset of PARTITIONS(Y), A,B being a_partition of Y holds 'not' Ex(All(a,A,G),B,G) '<' Ex(Ex('not' a,B,G),A,G) proof let a be Element of Funcs(Y,BOOLEAN); let G be Subset of PARTITIONS(Y); let A,B be a_partition of Y; A1:All(a,A,G) = B_INF(a,CompF(A,G)) by BVFUNC_2:def 9; A2:Ex('not' a,B,G) = B_SUP('not' a,CompF(B,G)) by BVFUNC_2:def 10; A3:Ex(All(a,A,G),B,G) = B_SUP(All(a,A,G),CompF(B,G)) by BVFUNC_2:def 10; let z be Element of Y; assume Pj('not' Ex(All(a,A,G),B,G),z)=TRUE; then 'not' Pj(Ex(All(a,A,G),B,G),z)=TRUE by VALUAT_1:def 5; then Pj(Ex(All(a,A,G),B,G),z)=FALSE by MARGREL1:41; then A4: Pj(Ex(All(a,A,G),B,G),z)<>TRUE by MARGREL1:43; z in EqClass(z,CompF(B,G)) & EqClass(z,CompF(B,G)) in CompF(B,G) by T_1TOPSP:def 1; then Pj(All(a,A,G),z)<>TRUE by A3,A4,BVFUNC_1:def 20; then consider x1 being Element of Y such that A5: x1 in EqClass(z,CompF(A,G)) & Pj(a,x1)<>TRUE by A1,BVFUNC_1:def 19; A6:x1 in EqClass(x1,CompF(B,G)) & EqClass(x1,CompF(B,G)) in CompF(B,G) by T_1TOPSP:def 1; Pj(a,x1)=FALSE by A5,MARGREL1:43; then Pj('not' a,x1)='not' FALSE by VALUAT_1:def 5; then Pj('not' a,x1)=TRUE by MARGREL1:41; then Pj(Ex('not' a,B,G),x1)=TRUE by A2,A6,BVFUNC_1:def 20; then Pj(B_SUP(Ex('not' a,B,G),CompF(A,G)),z) = TRUE by A5,BVFUNC_1:def 20; hence thesis by BVFUNC_2:def 10; end; theorem Th16: for a being Element of Funcs(Y,BOOLEAN), G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G is independent holds Ex('not' All(a,A,G),B,G) '<' Ex(Ex('not' a,B,G),A,G) proof let a be Element of Funcs(Y,BOOLEAN); let G be Subset of PARTITIONS(Y); let A,B be a_partition of Y; assume G is independent; then Ex(Ex('not' a,B,G),A,G) = Ex(Ex('not' a,A,G),B,G) by PARTIT_2:18; hence thesis by BVFUNC_2:20; end; theorem Th17: for a being Element of Funcs(Y,BOOLEAN), G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G is independent holds 'not' All(All(a,A,G),B,G) = Ex('not' All(a,B,G),A,G) proof let a be Element of Funcs(Y,BOOLEAN); let G be Subset of PARTITIONS(Y); let A,B be a_partition of Y; assume G is independent; then All(All(a,A,G),B,G) = All(All(a,B,G),A,G) by PARTIT_2:17; hence thesis by BVFUNC_2:20; end; theorem for a being Element of Funcs(Y,BOOLEAN), G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G is independent holds All('not' All(a,A,G),B,G) '<' Ex(Ex('not' a,B,G),A,G) proof let a be Element of Funcs(Y,BOOLEAN); let G be Subset of PARTITIONS(Y); let A,B be a_partition of Y; assume G is independent; then A1:Ex(Ex('not' a,B,G),A,G) = Ex(Ex('not' a,A,G),B,G) by PARTIT_2:18; 'not' All(a,A,G) = Ex('not' a,A,G) by BVFUNC_2:20; hence thesis by A1,Lm1; end; theorem for a being Element of Funcs(Y,BOOLEAN), G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G is independent holds 'not' All(All(a,A,G),B,G) = Ex(Ex('not' a,B,G),A,G) proof let a be Element of Funcs(Y,BOOLEAN); let G be Subset of PARTITIONS(Y); let A,B be a_partition of Y; assume A1:G is independent; Ex('not' a,B,G) = 'not' All(a,B,G) by BVFUNC_2:20; hence thesis by A1,Th17; end; theorem for a being Element of Funcs(Y,BOOLEAN), G being Subset of PARTITIONS(Y), A,B being a_partition of Y st G is independent holds 'not' All(All(a,A,G),B,G) '<' Ex(Ex('not' a,A,G),B,G) proof let a be Element of Funcs(Y,BOOLEAN); let G be Subset of PARTITIONS(Y); let A,B be a_partition of Y; assume A1:G is independent; then Ex('not' All(a,B,G),A,G) '<' Ex(Ex('not' a,A,G),B,G) by Th16; hence thesis by A1,Th17; end;