# A new proof of the wellfoundednes of the multiset ordering

```Dear colleagues,

Attached you find a new(?) proof of the wellfoundednes of the multiset
ordering due to Wilfried Buchholz. Salient features:
- it is completely inductive and does not involve classical reasoning (in
contrast to the proof by Dershowitz and Manna);
- it is a joy for theorem provers (it takes about 40 lines in Isabelle/HOL).

I would be interested to hear if this proof is known in the literature.

Tobias

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(order)g(due)h(to)g(Wilfried)e(Buc)m(hholz)2279 1980
y Fk(1)2355 2013 y Fl(comm)m(unicated)i(to)g(me)3198
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b(Matthes)976 2093 y Fk(3)1017 2126 y Fl(.)40 b(All)29
b(t)m(yp)s(os)i(are)f(en)m(tirely)g(mine.)382 2413 y
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b Fh(w)m(ell-)382 3317 y(founded)35 b(part)g(induction)p
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b Fl(\()p Fg(8)p Fi(y)d(<)d(x:)31 b(P)13 b Fl(\()p Fi(y)s
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1239 3543 1275 4 v 1603 3626 a Fg(8)p Fi(x)25 b Fg(2)g
Fi(W)m(:)31 b(P)13 b Fl(\()p Fi(x)p Fl(\))382 3812 y(It)30
b(also)h(follo)m(ws)e(that)i Fi(<)f Fl(is)g(w)m(ellfounded)e(i\013)h
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b(The)44 b(pro)t(of)382 4301 y Fl(Let)32 b Fi(<)f Fl(b)s(e)f(a)i(w)m
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4414 y(\014nite)26 b(m)m(ultisets)h(o)m(v)m(er)i Fi(A)p
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b(The)27 b(letters)h Fi(a)f Fl(and)382 4527 y Fi(b)37
b Fl(range)h(o)m(v)m(er)g(A;)g Fi(K)7 b Fl(,)38 b Fi(M)47
b Fl(and)37 b Fi(N)47 b Fl(range)38 b(o)m(v)m(er)g Fg(M)p
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p 382 4611 1196 4 v 486 4665 a Ff(1)520 4697 y Fe(Mathematisc)n(hes)195
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5652 y Fl(1)p eop
%%Page: 2 2
2 1 bop 382 548 a Fl(abbreviations:)709 713 y Fi(M)36
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b Fi(K)27 b Fg(^)20 b Fi(K)32 b(<)25 b(a)878 988 y(W)106
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Fi(M)470 1288 y Fk(0)510 1274 y Fi(:)d(M)f Fg([)19 b(f)p
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b(de\014nition)d(of)i Fi(W)13 b Fl(.)74 b(Let)42 b Fi(N)54
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y Fk(0)2316 1466 y Fg([)27 b(f)p Fi(a)p Fg(g)p Fl(.)75
b(W)-8 b(e)43 b(need)e(to)i(pro)m(v)m(e)382 1579 y Fi(N)35
b Fg(2)25 b Fi(W)13 b Fl(.)40 b(There)30 b(are)h(t)m(w)m(o)g(p)s
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1579 y Fi(M)2454 1593 y Fk(0)2514 1579 y Fg([)20 b(f)p
Fi(a)p Fg(g)31 b Fl(holds:)518 1734 y Fg(\017)46 b Fl(If)32
b Fi(N)37 b Fl(=)28 b Fi(M)j Fg([)21 b(f)p Fi(a)p Fg(g)32
b Fl(for)g(some)g Fi(M)38 b(<)1849 1749 y Fc(mul)q(t)2031
1734 y Fi(M)2119 1748 y Fk(0)2190 1734 y Fl(then)32 b
Fi(N)38 b Fg(2)27 b Fi(W)44 b Fl(follo)m(ws)31 b(from)g(the)609
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b Fi(N)36 b Fl(=)25 b Fi(M)989 2035 y Fk(0)1041 2021
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b(from)g(the)h(\014rst)f(t)m(w)m(o)609 2134 y(assumptions)j(b)m(y)h
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b Fg(8)p Fi(M)35 b Fg(2)25 b Fi(W)m(:)g(M)12 b Fg([)r(f)p
Fi(a)p Fg(g)25 b(2)g Fi(W)13 b Fb(.)382 2653 y Fh(Pro)s(of)66
b Fl(F)-8 b(rom)31 b(Lemma)f(2.1)i(b)m(y)e(w)m(ellfounded)e(part)i
(induction.)668 b Fa(2)382 2826 y Fh(Lemma)33 b(2.3)46
b Fg(8)p Fi(M)35 b Fg(2)24 b Fi(W)m(:)33 b(M)e Fg([)20
b(f)p Fi(a)p Fg(g)26 b(2)f Fi(W)13 b Fb(.)382 3018 y
Fh(Pro)s(of)66 b Fl(F)-8 b(rom)31 b(Lemma)f(2.2)i(b)m(y)e(w)m
(ellfounded)e(induction)g(on)i Fi(a)p Fl(.)663 b Fa(2)382
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b Fb(.)382 3382 y Fh(Pro)s(of)67 b Fl(b)m(y)30 b(induction)f(on)h(the)h
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3495 y(there)g(is)g(no)g Fi(N)35 b(<)1011 3510 y Fc(mul)q(t)1191
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(induction)d(step.)480 b Fa(2)382 3681 y Fl(Th)m(us)29
b(w)m(e)i(kno)m(w)f(that)h Fi(<)1251 3696 y Fc(mul)q(t)1436
3681 y Fl(is)f(w)m(ellfounded)e(on)i(all)f(of)i Fg(M)p
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b(is)h(classically)g(equiv)-5 b(alen)m(t)42 b(with)f(termina-)382
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4390 y(p)s(ersp)s(ectiv)m(e,)34 b(w)m(e)h(can)f(turn)g(things)f(around)
g(and)g(de\014ne)h Fi(W)46 b Fl(as)35 b(the)f(set)h(of)f(all)f(ter-)382
4503 y(minating)39 b(elemen)m(ts)i(of)f Fi(S)5 b Fl(.)71
b(No)m(w)41 b(w)m(ellfounded)d(part)i(induction)e(is)i(simply)e(go)s(o)
s(d)382 4616 y(old)33 b(w)m(ellfounded)e(induction,)h(and)h(the)h
(inductiv)m(e)e(c)m(haracterization)j(of)f Fi(W)46 b
Fl(is)33 b(no)m(w)382 4729 y(a)e(consequence)g(of)f(this)g(direct)f
(de\014nition)f(of)j Fi(W)13 b Fl(.)382 5009 y Fj(References)382
5212 y Fl([1])46 b(P)-8 b(.)28 b(Aczel.)36 b(An)27 b(in)m(tro)s
(duction)e(to)j(inductiv)m(e)e(de\014nitions.)33 b(In)27
b(J.)g(Barwise,)h(editor,)523 5325 y Fb(Handb)-5 b(o)g(ok)35
b(of)e(Mathematic)-5 b(al)34 b(L)-5 b(o)g(gic)p Fl(.)32
b(North-Holland,)d(1977.)1854 5652 y(2)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF
```