
all(x,p(x) iff exists(y,p(y))) iff 
    (all(z,p(z)) iff exists(v,p(v))).

exists(x,all(y,p(x) iff p(y))) iff 
    (all(z,p(z)) iff exists(v,p(v))).

all(x,p(x) iff all(y,p(y))) iff 
    (exists(z,p(z)) iff all(v,p(v))).

all(x,all(y,~s(y) or ~(x = f(g(y)))) or
	  exists(z,exists(w,s(w) and (z = g(w))) and (x = f(z)))).

(exists(x,all(y,p(x) iff p(y))) iff
   (exists(u,q(u)) iff all(v,p(v)))) iff
   (exists(w,all(z,q(z) iff q(w))) iff
   (exists(s,p(s)) iff all(t,q(t)))).

all(vv1,all(vv2,exists(ww1,~supplyold(vv1,vv2,handguns) or supplyold(vv1,ww1,bullets)))) and all(zz1,~supplyold(a,zz1,bullets) or (zz1=b)) and exists(uu3,(~(uu3=b) or ~(handguns=c)) and supplyold(a,uu3,handguns)) and (bullets=c).

all(vv1,all(vv2,all(vv3,exists(ww1,exists(ww2,(~supplyold(vv1,vv2,vv3) or ~class(vv3,type1) or supplyold(vv1,ww2,ww1)) and (~supplyold(vv1,vv2,vv3) or ~class(vv3,type1) or class(ww1,type2))))))) and class(c,type1) and all(zz1,all(zz2,~supplyold(a,zz1,zz2) or ~class(zz2,type2))) and ~class(c,type2).

all(vv1,all(vv2,all(vv3,exists(ww1,exists(ww2,(~supply(vv1,vv2,vv3) or ~classold(vv3,type1) or supply(vv1,ww2,ww1)) and (~supply(vv1,vv2,vv3) or ~classold(vv3,type1) or classold(ww1,type2))))))) and ~(type2=b) and exists(uu7,exists(uu8,all(zz1,all(zz2,(~supply(uu7,zz1,zz2) or ~classold(zz2,type2)) and supply(uu7,uu8,a))))) and (type1=b).

all(zz1,all(zz2,all(zz3,~supplyold(zz1,zz2,item1) or ~supplyold(zz1,zz3,item1) or (zz2=zz3)))) and exists(ww1,exists(ww2,(supplyold(a,ww1,item1) or (ww1=b)) and ((supplyold(a,ww2,item1) or (ww2=b)) and ~(ww1=ww2)))) and (item1=c).

all(zz1,all(zz2,all(zz3,~subold(zz1,zz2) or ~subold(zz2,zz3) or subold(zz1,zz3)))) and exists(ww1,exists(ww2,exists(ww3,(subold(ww1,ww2) or (ww1=a)) and (subold(ww1,ww2) or (ww2=b)) and (subold(ww2,ww3) or (ww2=a)) and (subold(ww2,ww3) or (ww3=b)) and (~(ww1=a) or ~(ww3=b)) and ~subold(ww1,ww3)))).

all(vv1,exists(ww1,exists(ww2,exists(ww3,all(zz1,(~class(vv1,type3) or supplyold(ww1,ww2,ww3) or (ww1=a)) and (~class(vv1,type3) or supplyold(ww1,ww2,ww3) or (ww2=b)) and (~class(vv1,type3) or supplyold(ww1,ww2,ww3) or (ww3=c)) and (~class(vv1,type3) or ~supplyold(ww1,zz1,vv1)) and (~class(vv1,type3) or ~(ww1=a) or ~(zz1=b) or ~(vv1=c))))))) and exists(uu41,all(uu42,all(vv2,all(vv3,exists(uu43,(~supplyold(uu42,vv2,vv3) or supplyold(uu42,uu43,uu41)) and class(uu41,type3)))))).

all(vv1,exists(ww1,exists(ww2,exists(ww3,all(zz1,(~classold(vv1,type3) or (vv1=a) or supply2(ww1,ww2,ww3,1)) and (~classold(vv1,type3) or (vv1=a) or ~supply2(ww1,zz1,vv1,2))))))) and all(uu67,all(vv2,all(vv3,exists(uu68,~supply2(uu67,vv2,vv3,1) or supply2(uu67,uu68,a,2))))) and classold(a,type3) and (type3=b).

all(vv1,exists(ww1,exists(ww2,exists(ww3,all(zz1,(~class(vv1,type3) or supplyold(ww1,ww2,ww3)) and (~class(vv1,type3) or ~(ww1=a) or ~(ww2=b) or ~(ww3=c)) and (~class(vv1,type3) or ~supplyold(ww1,zz1,vv1) or (ww1=a)) and (~class(vv1,type3) or ~supplyold(ww1,zz1,vv1) or (zz1=b)) and (~class(vv1,type3) or ~supplyold(ww1,zz1,vv1) or (vv1=c))))))) and all(uu1,all(vv2,all(vv3,exists(uu2,~supplyold(uu1,vv2,vv3) or supplyold(uu1,uu2,c))))) and class(c,type3).
