Copyright (c) 1999 Association of Mizar Users
environ vocabulary FUNCT_2, MARGREL1, PARTIT1, EQREL_1, ZF_LANG, BVFUNC_2, BVFUNC_1, T_1TOPSP; notation XBOOLE_0, SUBSET_1, MARGREL1, VALUAT_1, FRAENKEL, EQREL_1, BVFUNC_1, BVFUNC_2; constructors BVFUNC_2, BVFUNC_1; clusters MARGREL1, VALUAT_1, FRAENKEL; definitions BVFUNC_1; theorems T_1TOPSP, MARGREL1, BVFUNC_1, BVFUNC_2, BVFUNC11, BVFUNC12, VALUAT_1, PARTIT_2, BVFUNC_4; begin :: Chap. 1 Predicate Calculus reserve Y for non empty set, a,b for Element of Funcs(Y,BOOLEAN), G for Subset of PARTITIONS(Y), A, B for a_partition of Y; Lm1: a '<' b implies All(a,A,G) '<' Ex(b,A,G) proof assume a '<' b; then A1:All(a,A,G) '<' All(b,A,G) by PARTIT_2:13; All(b,A,G) '<' Ex(b,A,G) by BVFUNC_4:18; hence thesis by A1,BVFUNC_1:18; end; theorem Th1: G is independent implies All('not' All(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G) proof assume G is independent; then A1:All(All(a,B,G),A,G) = All(All(a,A,G),B,G) by PARTIT_2:17; A2:All('not' All(a,A,G),B,G) = 'not' Ex(All(a,A,G),B,G) by BVFUNC_2:21; All(All(a,B,G),A,G) '<' Ex(All(a,A,G),B,G) by A1,Lm1; hence thesis by A2,PARTIT_2:11; end; theorem Th2: All(All('not' a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G) proof A1:All('not' a,A,G) = B_INF('not' a,CompF(A,G)) by BVFUNC_2:def 9; let z be Element of Y; assume A2:Pj(All(All('not' a,A,G),B,G),z)=TRUE; A3:z in EqClass(z,CompF(A,G)) & EqClass(z,CompF(A,G)) in CompF(A,G) by T_1TOPSP:def 1; A4:z in EqClass(z,CompF(B,G)) & EqClass(z,CompF(B,G)) in CompF(B,G) by T_1TOPSP:def 1; now assume not (for x being Element of Y st x in EqClass(z,CompF(B,G)) holds Pj(All('not' a,A,G),x)=TRUE); then Pj(B_INF(All('not' a,A,G),CompF(B,G)),z) = FALSE by BVFUNC_1:def 19 ; then Pj(All(All('not' a,A,G),B,G),z)=FALSE by BVFUNC_2:def 9; hence contradiction by A2,MARGREL1:43; end; then A5:Pj(All('not' a,A,G),z)=TRUE by A4; now assume not (for x being Element of Y st x in EqClass(z,CompF(A,G)) holds Pj('not' a,x)=TRUE); then Pj(B_INF('not' a,CompF(A,G)),z) = FALSE by BVFUNC_1:def 19; hence contradiction by A1,A5,MARGREL1:43; end; then Pj('not' a,z)=TRUE by A3; then 'not' Pj(a,z)=TRUE by VALUAT_1:def 5; then Pj(a,z)=FALSE by MARGREL1:41; then Pj(a,z)<>TRUE by MARGREL1:43; then Pj(B_INF(a,CompF(B,G)),z) = FALSE by A4,BVFUNC_1:def 19; then Pj(All(a,B,G),z)=FALSE by BVFUNC_2:def 9; then Pj(All(a,B,G),z)<>TRUE by MARGREL1:43; then Pj(B_INF(All(a,B,G),CompF(A,G)),z) = FALSE by A3,BVFUNC_1:def 19; then Pj(All(All(a,B,G),A,G),z)=FALSE by BVFUNC_2:def 9; then 'not' Pj(All(All(a,B,G),A,G),z)=TRUE by MARGREL1:41; hence thesis by VALUAT_1:def 5; end; theorem Th3: All('not' Ex(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G) proof All('not' Ex(a,A,G),B,G) = All(All('not' a,A,G),B,G) by BVFUNC_2:21; hence thesis by Th2; end; theorem Th4: G is independent implies All(Ex('not' a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G) proof assume A1:G is independent; Ex('not' a,A,G) = 'not' All(a,A,G) by BVFUNC_2:20; then A2:All(Ex('not' a,A,G),B,G) = 'not' Ex(All(a,A,G),B,G) by BVFUNC_2:21; All(All(a,B,G),A,G) '<' Ex(All(a,A,G),B,G) by A1,BVFUNC11:11; hence thesis by A2,PARTIT_2:11; end; canceled; theorem G is independent implies Ex(All('not' a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G) proof assume A1:G is independent; then A2:Ex(All('not' a,A,G),B,G) '<' All(Ex('not' a,B,G),A,G) by PARTIT_2:19; All(Ex('not' a,B,G),A,G) '<' 'not' All(All(a,A,G),B,G) by A1,Th4; then Ex(All('not' a,A,G),B,G) '<' 'not' All(All(a,A,G),B,G) by A2,BVFUNC_1:18 ; hence thesis by A1,PARTIT_2:17; end; theorem Th7: G is independent implies Ex('not' Ex(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G) proof assume A1:G is independent; A2:Ex('not' Ex(a,A,G),B,G) = 'not' All(Ex(a,A,G),B,G) by BVFUNC_2:20; A3:All(All(a,B,G),A,G) = All(All(a,A,G),B,G) by A1,PARTIT_2:17; All(a,A,G) '<' Ex(a,A,G) by Lm1; then All(All(a,A,G),B,G) '<' All(Ex(a,A,G),B,G) by PARTIT_2:13; hence thesis by A2,A3,PARTIT_2:11; end; canceled; theorem Th9: G is independent implies 'not' All(Ex(a,A,G),B,G) '<' 'not' Ex(All(a,B,G),A,G) proof assume G is independent; then Ex(All(a,B,G),A,G) '<' All(Ex(a,A,G),B,G) by PARTIT_2:19; hence thesis by PARTIT_2:11; end; theorem Th10: G is independent implies 'not' Ex(Ex(a,A,G),B,G) '<' 'not' Ex(All(a,B,G),A,G) proof assume G is independent; then A1:Ex(Ex(a,A,G),B,G) = Ex(Ex(a,B,G),A,G) by PARTIT_2:18; All(a,B,G) '<' Ex(a,B,G) by Lm1; then Ex(All(a,B,G),A,G) '<' Ex(Ex(a,A,G),B,G) by A1,PARTIT_2:15; hence thesis by PARTIT_2:11; end; theorem Th11: 'not' Ex(Ex(a,A,G),B,G) '<' 'not' All(Ex(a,B,G),A,G) proof A1:Ex(a,A,G) = B_SUP(a,CompF(A,G)) by BVFUNC_2:def 10; let z be Element of Y; assume Pj('not' Ex(Ex(a,A,G),B,G),z)=TRUE; then 'not' Pj(Ex(Ex(a,A,G),B,G),z)=TRUE by VALUAT_1:def 5; then A2:Pj(Ex(Ex(a,A,G),B,G),z)=FALSE by MARGREL1:41; A3:now assume ex x being Element of Y st x in EqClass(z,CompF(B,G)) & Pj(Ex(a,A,G),x)=TRUE; then Pj(B_SUP(Ex(a,A,G),CompF(B,G)),z) = TRUE by BVFUNC_1:def 20; then Pj(Ex(Ex(a,A,G),B,G),z)=TRUE by BVFUNC_2:def 10; hence contradiction by A2,MARGREL1:43; end; A4:for x being Element of Y st x in EqClass(z,CompF(B,G)) holds (for y being Element of Y st y in EqClass(x,CompF(A,G)) holds Pj(a,y)<>TRUE) proof let x be Element of Y; assume x in EqClass(z,CompF(B,G)); then Pj(Ex(a,A,G),x)<>TRUE by A3; hence thesis by A1,BVFUNC_1:def 20; end; A5:z in EqClass(z,CompF(A,G)) by T_1TOPSP:def 1; for x being Element of Y st x in EqClass(z,CompF(B,G)) holds Pj(a,x)<> TRUE proof let x be Element of Y; assume A6:x in EqClass(z,CompF(B,G)); x in EqClass(x,CompF(A,G)) by T_1TOPSP:def 1; hence thesis by A4,A6; end; then Pj(B_SUP(a,CompF(B,G)),z) = FALSE by BVFUNC_1:def 20; then Pj(Ex(a,B,G),z)=FALSE by BVFUNC_2:def 10; then Pj(Ex(a,B,G),z)<>TRUE by MARGREL1:43; then Pj(B_INF(Ex(a,B,G),CompF(A,G)),z) = FALSE by A5,BVFUNC_1:def 19; then Pj(All(Ex(a,B,G),A,G),z)=FALSE by BVFUNC_2:def 9; then 'not' Pj(All(Ex(a,B,G),A,G),z)=TRUE by MARGREL1:41; hence thesis by VALUAT_1:def 5; end; canceled 2; theorem G is independent implies 'not' Ex(All(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G) proof assume G is independent; then All('not' All(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G) by Th1; hence thesis by BVFUNC_2:21; end; theorem G is independent implies 'not' All(Ex(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G) proof assume G is independent; then A1:Ex('not' Ex(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G) by Th7; Ex('not' Ex(a,A,G),B,G) = Ex(All('not' a,A,G),B,G) by BVFUNC_2:21; hence thesis by A1,BVFUNC12:5; end; theorem 'not' Ex(Ex(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G) proof All('not' Ex(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G) by Th3; hence thesis by BVFUNC_2:21; end; theorem Th17: G is independent implies 'not' Ex(All(a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G) proof assume G is independent; then A1:All('not' All(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G) by Th1; 'not' Ex(All(a,A,G),B,G) = All('not' All(a,A,G),B,G) by BVFUNC_2:21; hence thesis by A1,BVFUNC_2:20; end; theorem Th18: G is independent implies 'not' All(Ex(a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G) proof assume G is independent; then A1:Ex('not' Ex(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G) by Th7; A2:'not' All(Ex(a,A,G),B,G) = Ex(All('not' a,A,G),B,G) by BVFUNC12:5; Ex('not' Ex(a,A,G),B,G) = Ex(All('not' a,A,G),B,G) by BVFUNC_2:21; hence thesis by A1,A2,BVFUNC_2:20; end; theorem Th19: 'not' Ex(Ex(a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G) proof A1:All('not' Ex(a,A,G),B,G) '<' 'not' All(All(a,B,G),A,G) by Th3; 'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,A,G),B,G) by BVFUNC_2:21; hence thesis by A1,BVFUNC_2:20; end; theorem Th20: G is independent implies 'not' All(Ex(a,A,G),B,G) '<' All('not' All(a,B,G),A,G) proof assume G is independent; then 'not' All(Ex(a,A,G),B,G) '<' 'not' Ex(All(a,B,G),A,G) by Th9; hence thesis by BVFUNC_2:21; end; theorem Th21: G is independent implies 'not' Ex(Ex(a,A,G),B,G) '<' All('not' All(a,B,G),A,G) proof assume G is independent; then 'not' Ex(Ex(a,A,G),B,G) '<' 'not' Ex(All(a,B,G),A,G) by Th10; hence thesis by BVFUNC_2:21; end; theorem Th22: 'not' Ex(Ex(a,A,G),B,G) '<' Ex('not' Ex(a,B,G),A,G) proof A1:'not' Ex(Ex(a,A,G),B,G) '<' 'not' All(Ex(a,B,G),A,G) by Th11; Ex('not' Ex(a,B,G),A,G) = Ex(All('not' a,B,G),A,G) by BVFUNC_2:21; hence thesis by A1,BVFUNC12:5; end; theorem Th23: G is independent implies 'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,B,G),A,G) proof assume G is independent; then 'not' Ex(Ex(a,A,G),B,G) = 'not' Ex(Ex(a,B,G),A,G) by PARTIT_2:18; hence thesis by BVFUNC_2:21; end; theorem Th24: G is independent implies 'not' All(Ex(a,A,G),B,G) '<' Ex(Ex('not' a,B,G),A,G) proof assume G is independent; then 'not' All(Ex(a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G) by Th18; hence thesis by BVFUNC_2:20; end; theorem Th25: 'not' Ex(Ex(a,A,G),B,G) '<' Ex(Ex('not' a,B,G),A,G) proof 'not' Ex(Ex(a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G) by Th19; hence thesis by BVFUNC_2:20; end; theorem Th26: G is independent implies 'not' All(Ex(a,A,G),B,G) '<' All(Ex('not' a,B,G),A,G) proof assume G is independent; then 'not' All(Ex(a,A,G),B,G) '<' All('not' All(a,B,G),A,G) by Th20; hence thesis by BVFUNC_2:20; end; theorem Th27: G is independent implies 'not' Ex(Ex(a,A,G),B,G) '<' All(Ex('not' a,B,G),A,G) proof assume G is independent; then 'not' Ex(Ex(a,A,G),B,G) '<' All('not' All(a,B,G),A,G) by Th21; hence thesis by BVFUNC_2:20; end; theorem Th28: 'not' Ex(Ex(a,A,G),B,G) '<' Ex(All('not' a,B,G),A,G) proof 'not' Ex(Ex(a,A,G),B,G) '<' Ex('not' Ex(a,B,G),A,G) by Th22; hence thesis by BVFUNC_2:21; end; theorem Th29: G is independent implies 'not' Ex(Ex(a,A,G),B,G) = All(All('not' a,B,G),A,G) proof assume G is independent; then 'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,B,G),A,G) by Th23; hence thesis by BVFUNC_2:21; end; theorem G is independent implies Ex('not' Ex(a,A,G),B,G) '<' 'not' Ex(All(a,B,G),A,G) proof assume G is independent; then A1:'not' All(Ex(a,A,G),B,G) '<' 'not' Ex(All(a,B,G),A,G) by Th9; 'not' All(Ex(a,A,G),B,G) = Ex(All('not' a,A,G),B,G) by BVFUNC12:5; hence thesis by A1,BVFUNC_2:21; end; theorem All('not' Ex(a,A,G),B,G) '<' 'not' Ex(All(a,B,G),A,G) proof A1:Ex(a,A,G) = B_SUP(a,CompF(A,G)) by BVFUNC_2:def 10; A2:All('not' Ex(a,A,G),B,G) = B_INF('not' Ex(a,A,G),CompF(B,G)) by BVFUNC_2:def 9; A3:All(a,B,G) = B_INF(a,CompF(B,G)) by BVFUNC_2:def 9; A4:Ex(All(a,B,G),A,G) = B_SUP(All(a,B,G),CompF(A,G)) by BVFUNC_2:def 10; let z be Element of Y; assume A5:Pj(All('not' Ex(a,A,G),B,G),z)=TRUE; A6:z in EqClass(z,CompF(B,G)) & EqClass(z,CompF(B,G)) in CompF(B,G) by T_1TOPSP:def 1; now assume not (for x being Element of Y st x in EqClass(z,CompF(B,G)) holds Pj('not' Ex(a,A,G),x)=TRUE); then Pj(All('not' Ex(a,A,G),B,G),z)=FALSE by A2,BVFUNC_1:def 19; hence contradiction by A5,MARGREL1:43; end; then Pj('not' Ex(a,A,G),z)=TRUE by A6; then 'not' Pj(Ex(a,A,G),z)=TRUE by VALUAT_1:def 5; then A7:Pj(Ex(a,A,G),z)=FALSE by MARGREL1:41; A8: now assume 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) = TRUE by BVFUNC_1:def 20; hence contradiction by A1,A7,MARGREL1:43; end; for x being Element of Y st x in EqClass(z,CompF(A,G)) holds Pj(All(a,B,G),x)<>TRUE proof let x be Element of Y; assume x in EqClass(z,CompF(A,G)); then A9:Pj(a,x)<>TRUE by A8; x in EqClass(x,CompF(B,G)) & EqClass(x,CompF(B,G)) in CompF(B,G) by T_1TOPSP:def 1; then Pj(All(a,B,G),x)=FALSE by A3,A9,BVFUNC_1:def 19; hence thesis by MARGREL1:43; end; then Pj(B_SUP(All(a,B,G),CompF(A,G)),z) = FALSE by BVFUNC_1:def 20; then 'not' Pj(Ex(All(a,B,G),A,G),z)=TRUE by A4,MARGREL1:41; hence thesis by VALUAT_1:def 5; end; theorem All('not' Ex(a,A,G),B,G) '<' 'not' All(Ex(a,B,G),A,G) proof 'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,A,G),B,G) by BVFUNC_2:21; hence thesis by Th11; end; theorem G is independent implies All('not' Ex(a,A,G),B,G) = 'not' Ex(Ex(a,B,G),A,G) proof assume A1:G is independent; 'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,A,G),B,G) by BVFUNC_2:21; hence thesis by A1,PARTIT_2:18; end; theorem G is independent implies Ex('not' All(a,A,G),B,G) = Ex('not' All(a,B,G),A,G) proof assume A1:G is independent; 'not' All(All(a,A,G),B,G) = Ex('not' All(a,A,G),B,G) by BVFUNC_2:20; hence thesis by A1,BVFUNC11:17; end; theorem G is independent implies All('not' All(a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G) proof assume A1:G is independent; 'not' Ex(All(a,A,G),B,G) = All('not' All(a,A,G),B,G) by BVFUNC_2:21; hence thesis by A1,Th17; end; theorem G is independent implies Ex('not' Ex(a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G) proof assume A1:G is independent; A2:'not' All(Ex(a,A,G),B,G) = Ex(All('not' a,A,G),B,G) by BVFUNC12:5; Ex('not' Ex(a,A,G),B,G) = Ex(All('not' a,A,G),B,G) by BVFUNC_2:21; hence thesis by A1,A2,Th18; end; theorem All('not' Ex(a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G) proof 'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,A,G),B,G) by BVFUNC_2:21; hence thesis by Th19; end; theorem G is independent implies Ex('not' Ex(a,A,G),B,G) '<' All('not' All(a,B,G),A,G) proof assume A1:G is independent; A2:'not' All(Ex(a,A,G),B,G) = Ex(All('not' a,A,G),B,G) by BVFUNC12:5; Ex('not' Ex(a,A,G),B,G) = Ex(All('not' a,A,G),B,G) by BVFUNC_2:21; hence thesis by A1,A2,Th20; end; theorem G is independent implies All('not' Ex(a,A,G),B,G) '<' All('not' All(a,B,G),A,G) proof assume G is independent; then 'not' Ex(Ex(a,A,G),B,G) '<' All('not' All(a,B,G),A,G) by Th21; hence thesis by BVFUNC_2:21; end; theorem All('not' Ex(a,A,G),B,G) '<' Ex('not' Ex(a,B,G),A,G) proof 'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,A,G),B,G) by BVFUNC_2:21; hence thesis by Th22; end; theorem G is independent implies All('not' Ex(a,A,G),B,G) = All('not' Ex(a,B,G),A,G) proof assume A1:G is independent; 'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,A,G),B,G) by BVFUNC_2:21; hence thesis by A1,Th23; end; theorem G is independent implies Ex('not' Ex(a,A,G),B,G) '<' Ex(Ex('not' a,B,G),A,G) proof assume A1:G is independent; A2:'not' All(Ex(a,A,G),B,G) = Ex(All('not' a,A,G),B,G) by BVFUNC12:5; Ex('not' Ex(a,A,G),B,G) = Ex(All('not' a,A,G),B,G) by BVFUNC_2:21; hence thesis by A1,A2,Th24; end; theorem All('not' Ex(a,A,G),B,G) '<' Ex(Ex('not' a,B,G),A,G) proof 'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,A,G),B,G) by BVFUNC_2:21; hence thesis by Th25; end; theorem G is independent implies Ex('not' Ex(a,A,G),B,G) '<' All(Ex('not' a,B,G),A,G) proof assume A1:G is independent; A2:'not' All(Ex(a,A,G),B,G) = Ex(All('not' a,A,G),B,G) by BVFUNC12:5; Ex('not' Ex(a,A,G),B,G) = Ex(All('not' a,A,G),B,G) by BVFUNC_2:21; hence thesis by A1,A2,Th26; end; theorem G is independent implies All('not' Ex(a,A,G),B,G) '<' All(Ex('not' a,B,G),A,G) proof assume A1:G is independent; 'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,A,G),B,G) by BVFUNC_2:21; hence thesis by A1,Th27; end; theorem All('not' Ex(a,A,G),B,G) '<' Ex(All('not' a,B,G),A,G) proof 'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,A,G),B,G) by BVFUNC_2:21; hence thesis by Th28; end; theorem G is independent implies All('not' Ex(a,A,G),B,G) = All(All('not' a,B,G),A,G) proof assume A1:G is independent; 'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,A,G),B,G) by BVFUNC_2:21; hence thesis by A1,Th29; end; theorem G is independent implies Ex(All('not' a,A,G),B,G) '<' 'not' Ex(All(a,B,G),A,G) proof assume A1:G is independent; 'not' All(Ex(a,A,G),B,G) = Ex(All('not' a,A,G),B,G) by BVFUNC12:5; hence thesis by A1,Th9; end; theorem G is independent implies All(All('not' a,A,G),B,G) '<' 'not' Ex(All(a,B,G),A,G) proof assume A1:G is independent; A2:'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,A,G),B,G) by BVFUNC_2:21; All('not' Ex(a,A,G),B,G) = All(All('not' a,A,G),B,G) by BVFUNC_2:21; hence thesis by A1,A2,Th10; end; theorem All(All('not' a,A,G),B,G) '<' 'not' All(Ex(a,B,G),A,G) proof A1:'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,A,G),B,G) by BVFUNC_2:21; All('not' Ex(a,A,G),B,G) = All(All('not' a,A,G),B,G) by BVFUNC_2:21; hence thesis by A1,Th11; end; theorem G is independent implies All(All('not' a,A,G),B,G) '<' 'not' Ex(Ex(a,B,G),A,G) proof assume A1:G is independent; A2:'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,A,G),B,G) by BVFUNC_2:21; All('not' Ex(a,A,G),B,G) = All(All('not' a,A,G),B,G) by BVFUNC_2:21; hence thesis by A1,A2,PARTIT_2:18; end; theorem G is independent implies Ex(Ex('not' a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G) proof assume A1:G is independent; 'not' All(All(a,A,G),B,G) = Ex(Ex('not' a,A,G),B,G) by BVFUNC12:7; hence thesis by A1,BVFUNC11:17; end; theorem G is independent implies All(Ex('not' a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G) proof assume A1:G is independent; 'not' Ex(All(a,A,G),B,G) = All(Ex('not' a,A,G),B,G) by BVFUNC12:6; hence thesis by A1,Th17; end; theorem G is independent implies Ex(All('not' a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G) proof assume A1:G is independent; 'not' All(Ex(a,A,G),B,G) = Ex(All('not' a,A,G),B,G) by BVFUNC12:5; hence thesis by A1,Th18; end; theorem All(All('not' a,A,G),B,G) '<' Ex('not' All(a,B,G),A,G) proof A1:'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,A,G),B,G) by BVFUNC_2:21; All('not' Ex(a,A,G),B,G) = All(All('not' a,A,G),B,G) by BVFUNC_2:21; hence thesis by A1,Th19; end; theorem G is independent implies Ex(All('not' a,A,G),B,G) '<' All('not' All(a,B,G),A,G) proof assume A1:G is independent; 'not' All(Ex(a,A,G),B,G) = Ex(All('not' a,A,G),B,G) by BVFUNC12:5; hence thesis by A1,Th20; end; theorem G is independent implies All(All('not' a,A,G),B,G) '<' All('not' All(a,B,G),A,G) proof assume A1:G is independent; A2:'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,A,G),B,G) by BVFUNC_2:21; All('not' Ex(a,A,G),B,G) = All(All('not' a,A,G),B,G) by BVFUNC_2:21; hence thesis by A1,A2,Th21; end; theorem All(All('not' a,A,G),B,G) '<' Ex('not' Ex(a,B,G),A,G) proof A1:'not' Ex(Ex(a,A,G),B,G) = All('not' Ex(a,A,G),B,G) by BVFUNC_2:21; All('not' Ex(a,A,G),B,G) = All(All('not' a,A,G),B,G) by BVFUNC_2:21; hence thesis by A1,Th22; end; theorem G is independent implies All(All('not' a,A,G),B,G) = All('not' Ex(a,B,G),A,G) proof assume G is independent; then All(All('not' a,A,G),B,G) = All(All('not' a,B,G),A,G) by PARTIT_2:17; hence thesis by BVFUNC_2:21; end; canceled; theorem All(Ex('not' a,A,G),B,G) '<' Ex(Ex('not' a,B,G),A,G) proof 'not' Ex(All(a,A,G),B,G) = All(Ex('not' a,A,G),B,G) by BVFUNC12:6; hence thesis by BVFUNC11:15; end; theorem G is independent implies Ex(All('not' a,A,G),B,G) '<' Ex(Ex('not' a,B,G),A,G) proof assume A1:G is independent; 'not' All(Ex(a,A,G),B,G) = Ex(All('not' a,A,G),B,G) by BVFUNC12:5; hence thesis by A1,Th24; end;