%------------------------------------------------------------------- set(auto). assign(max_seconds,32700). %-----------------------AXIOMS ON ORDERED GROUPS-------------------- %------------------------------------------------------------------- formula_list(usable). % abelian group axioms (think that minuz(X,Y)=pluz(X,rvz(Y)) by def.) all x y z ( pluz(pluz(x,y),z) = pluz(x,pluz(y,z))). % associativity all x ( pluz(x,e) = x). % right unit all x ( pluz(x,rvz(x)) = e). % right inverse all x y ( pluz(x,y) = pluz(y,x) ). % commutativity % ordering axioms (axioms concerning non-negativeness) all x y ( (nneg(x) & nneg(y)) -> nneg(pluz(x,y)) ). all x ( nneg(x) | nneg(rvz(x)) ). all x ( (nneg(x) & nneg(rvz(x))) -> x=e ). %------------------------------------------------------------------- end_of_list. %-----------------------DEFINITIONAL EXTENSIONS--------------------- %------------------------------------------------------------------- formula_list(usable). all x ( nneg(x) -> abs(x)=x ). % definition of the absolute value... all x ( -(nneg(x)) -> abs(x)=rvz(x) ). % ...def.n of the absolute value all x y ( leq(x,y) <-> nneg(pluz(y,rvz(x))) ). % def.n of comparison %------------------------------------------------------------------- end_of_list. %------------------------------------------------------------------- %------------------------------------------------------------------- %-----------------------LEMMAS-------------------------------------- %------------------------------------------------------------------- formula_list(usable). % laws concerning the relation "leq" %% statements A1,Ba,Bb,B below proved without definitional extensions ( all x y z ( pluz(x,y)=pluz(x,z) -> y=z )). % A1: cancellation law %%( all x y ( pluz(pluz(x,rvz(y)),pluz(y,rvz(x)))=e )). % Ba %%( all x y ( pluz(pluz(x,rvz(y)),rvz(pluz(x,rvz(y))))=e )). % Bb ( all x y ( rvz(pluz(x,rvz(y)))=pluz(y,rvz(x)) )). % B (from A1,Ba,Bb alone) ( all x y ( leq(x,y) | leq(y,x) )). % C: totality ( all x ( leq(x,x) ) ). % D: reflexivity ( all x y z ( (leq(x,y) & leq(y,z)) -> leq(x,z) )). % E: transitivity ( all x y z ( (leq(x,y) & x != y & leq(y,z)) -> x != z )). % E1: transitivity ( all x y z ( (leq(x,y) & leq(y,z) & y != z) -> x != z )). % E2: transitivity ( all x y z ( leq(x,y) -> leq(pluz(x,z),pluz(y,z)) )). % F: additivity ( all x y z ( pluz(x,z)=pluz(y,z) -> x=y )). % A2: cancellation law ( all x y z ( leq(x,y) & x != y -> % F1, strict additivity pluz(x,z) != pluz(y,z) )). end_of_list. formula_list(usable). % laws concerning the operations "abs" and "nneg" ( all x ( abs(pluz(x,rvz(x)))=e )). %1 ( all x ( leq(x,abs(x)) )). %2 ( all x ( abs(abs(x))=abs(x) )). %3 ( all x ( abs(x)=e <-> x=e )). %4 ( all x ( abs(rvz(x))=abs(x) )). %5 ( all x y ( leq(pluz(x,y),pluz(abs(x),abs(y))) )). %6 ( all x y ( nneg(pluz(x,y)) -> leq(abs(pluz(x,y)),pluz(abs(x),abs(y))) )). %7a ( all x y ( -(nneg(pluz(x,y))) -> %7b nneg(pluz(rvz(x),rvz(y))) )). ( all x y ( -(nneg(pluz(x,y))) -> %7c (proved without earlier laws on "leq") leq(abs(pluz(rvz(x),rvz(y))),pluz(abs(rvz(x)),abs(rvz(y)))) )). ( all x y ( -(nneg(pluz(x,y))) -> %7d leq(abs(pluz(x,y)),pluz(abs(x),abs(y))) )). ( all x y ( leq(abs(pluz(x,y)),pluz(abs(x),abs(y))) )). %7 ( all x y z ( pluz(x,z)=pluz(pluz(x,rvz(y)),pluz(y,z)) )). %8a ( all x y z ( leq(abs(pluz(x,rvz(z))), %8 (proved without the axioms) pluz(abs(pluz(x,rvz(y))),abs(pluz(y,rvz(z))))) )). ( all x y ( -(nneg(x)) -> leq(x,abs(y)) & x != abs(y) )). %9 ( all x y ( nneg(y) -> leq(pluz(x,rvz(y)),pluz(x,y)) )). %10 ( all x y ( ( nneg(x) & -(nneg(y)) ) -> %11a leq(abs(pluz(abs(x),rvz(abs(y)))),abs(pluz(x,rvz(y)))) )). ( all x y ( ( nneg(x) & nneg(y) ) -> %11b abs(pluz(abs(x),rvz(abs(y))))=abs(pluz(x,rvz(y))) )). ( all x y ( ( -(nneg(x)) & -(nneg(y)) ) -> %11c abs(pluz(abs(rvz(x)),rvz(abs(rvz(y)))))=abs(pluz(rvz(x),y)) )). % to prove the next lemma, it turned out useful to temporarily inhibit: % * all axioms save commutativity; % * all definitional extensions; % * all of the laws concerning "leq" save D (reflexivity) and B; % * all of the laws concerning "abs" and "nneg" save 5 and 11a--11c ( all x y ( %11 leq(abs(pluz(abs(x),rvz(abs(y)))),abs(pluz(x,rvz(y)))) )). (all x y ( %12 leq(pluz(abs(x),rvz(abs(pluz(abs(y),rvz(abs(x)))))),abs(y)) )). end_of_list. %-------------------------------------------------------------------