proof-of-false

2025 April Fools' Day joke

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/F8 9.96 Tf 0 0 Td[(the)-402(function)-401(to)-402(the)-402(left)-401(con)28(tains)-402(a)-402(b)-28(ound)]TJ/F11 9.96 Tf 186.54 0 Td[(y)]TJ/F8 9.96 Tf 5.25 0 Td[(,)-419(whereas)-401(the)]TJ/F11 9.96 Tf 63.19 0 Td[(y)]TJ/F8 9.96 Tf 9.25 0 Td[(at)-403(the)-402(righ)29(t)-403(is)-402(free.)-649(An)]TJ -264.23 -11.95 Td[(incorrect)-332(substitution)-331(w)28(ould)-333(mix)-333(the)-333(t)28(w)28(o)-334(iden)29(ti\014ers)-332(in)-333(the)-332(erroneous)-332(result)]TJ 168.23 -21.11 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(y)-37(:y)-36(y)]TJ/F8 9.96 Tf 24.31 0 Td[(\051)]TJ -196.41 -21.11 Td[(Simply)-332(b)28(y)-333(renaming)-332(the)-333(b)-27(ound)]TJ/F11 9.96 Tf 138.95 0 Td[(y)]TJ/F8 9.96 Tf 8.57 0 Td[(to)]TJ/F11 9.96 Tf 12.17 0 Td[(t)]TJ/F8 9.96 Tf 6.92 0 Td[(w)27(e)-333(obtain)]TJ -31.62 -21.11 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(x:)]TJ/F8 9.96 Tf 14.28 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(t:xt)]TJ/F8 9.96 Tf 21.47 0 Td[(\051\051)]TJ/F11 9.96 Tf 7.75 0 Td[(y)]TJ/F8 9.96 Tf 8.01 0 Td[(=)-278(\050)]TJ/F11 9.96 Tf 14.39 0 Td[(t:y)-37(t)]TJ/F8 9.96 Tf 21.01 0 Td[(\051)]TJ -229.64 -21.11 Td[(whic)28(h)-333(is)-333(a)-333(completely)-332(dieren)30(t)-334(result)-332(but)-333(nev)29(ertheless)-330(the)-333(correct)-332(one.)]TJ 14.95 -11.95 Td[(Therefore,)-315(if)-312(the)-313(function)]TJ/F11 9.96 Tf 112.47 0 Td[(x:)-278(<)]TJ/F8 9.96 Tf 24.79 0 Td[(exp)]TJ/F11 9.96 Tf 15.22 0 Td[(>)]TJ/F8 9.96 Tf 10.87 0 Td[(is)-312(applied)-312(to)]TJ/F11 9.96 Tf 56.46 0 Td[(E)]TJ/F8 9.96 Tf 7.92 0 Td[(,)-317(w)27(e)-313(substitute)-310(all)]TJ/F34 9.96 Tf 80.43 0 Td[(fr)51(e)51(e)]TJ/F8 9.96 Tf 18.52 0 Td[(o)-28(ccur-)]TJ -341.63 -11.96 Td[(rences)-338(of)]TJ/F11 9.96 Tf 41.46 0 Td[(x)]TJ/F8 9.96 Tf 9.08 0 Td[(in)]TJ/F11 9.96 Tf 11.69 0 Td[(<)]TJ/F8 9.96 Tf 7.75 0 Td[(exp)]TJ/F11 9.96 Tf 15.22 0 Td[(>)]TJ/F8 9.96 Tf 11.14 0 Td[(with)]TJ/F11 9.96 Tf 22.76 0 Td[(E)]TJ/F8 9.96 Tf 7.93 0 Td[(.)-465(If)-340(the)-340(substitution)-338(w)28(ould)-340(bring)-340(a)-340(free)-339(v)56(ariable)-339(of)]TJ/F11 9.96 Tf 221.85 0 Td[(E)]TJ/F8 9.96 Tf 11.32 0 Td[(in)]TJ -360.2 -11.95 Td[(an)-294(expression)-291(where)-292(this)-293(v)56(ariable)-293(o)-28(ccurs)-291(b)-28(ound,)-301(w)28(e)-293(rename)-292(the)-293(b)-28(ound)-292(v)55(ariable)-292(b)-28(efore)]TJ 0 -11.96 Td[(p)-28(erforming)-331(the)-333(substitution.)-442(F)83(or)-333(example,)-332(in)-333(the)-332(expression)]TJ 136.64 -21.11 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(x:)]TJ/F8 9.96 Tf 14.27 0 Td[(\050)]TJ/F11 9.96 Tf 3.88 0 Td[(y)-37(:)]TJ/F8 9.96 Tf 13.82 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(x)]TJ/F8 9.96 Tf 5.7 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(x:xy)]TJ/F8 9.96 Tf 25.21 0 Td[(\051\051\051\051)]TJ/F11 9.96 Tf 15.5 0 Td[(y)]TJ/F8 9.96 Tf -226.63 -21.11 Td[(w)27(e)-333(asso)-27(ciate)-333(the)-332(argumen)29(t)]TJ/F11 9.96 Tf 118.31 0 Td[(x)]TJ/F8 9.96 Tf 9.01 0 Td[(with)]TJ/F11 9.96 Tf 22.7 0 Td[(y)]TJ/F8 9.96 Tf 5.24 0 Td[(.)-444(In)-334(the)-332(b)-28(o)-28(dy)]TJ -4.99 -21.1 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(y)-37(:)]TJ/F8 9.96 Tf 13.82 0 Td[(\050)]TJ/F11 9.96 Tf 3.88 0 Td[(x)]TJ/F8 9.96 Tf 5.69 0 Td[(\050)]TJ/F11 9.96 Tf 3.88 0 Td[(x:xy)]TJ/F8 9.96 Tf 25.21 0 Td[(\051\051\051)]TJ -206.62 -21.11 Td[(only)-371(the)-370(\014rst)]TJ/F11 9.96 Tf 60.71 0 Td[(x)]TJ/F8 9.96 Tf 9.39 0 Td[(is)-370(free)-370(and)-370(can)-371(b)-27(e)-371(substituted.)-555(Before)-369(substituting)-369(though,)-379(w)27(e)-371(ha)28(v)28(e)]TJ -70.1 -11.96 Td[(to)-334(rename)-331(the)-333(v)56(ariable)]TJ/F11 9.96 Tf 101.62 0 Td[(y)]TJ/F8 9.96 Tf 8.56 0 Td[(to)-334(a)28(v)28(oid)-334(mixing)-332(its)-333(b)-27(ound)-333(with)-333(is)-333(free)-332(o)-28(ccurrence:)]TJ -7.75 -21.11 Td[([)]TJ/F11 9.96 Tf 2.77 0 Td[(y)-36(=x)]TJ/F8 9.96 Tf 15.92 0 Td[(]\050)]TJ/F11 9.96 Tf 6.64 0 Td[(t:)]TJ/F8 9.96 Tf 12.17 0 Td[(\050)]TJ/F11 9.96 Tf 3.88 0 Td[(x)]TJ/F8 9.96 Tf 5.69 0 Td[(\050)]TJ/F11 9.96 Tf 3.88 0 Td[(x:xt)]TJ/F8 9.96 Tf 23.56 0 Td[(\051\051\051)-278(=)-278(\050)]TJ/F11 9.96 Tf 28.78 0 Td[(t:)]TJ/F8 9.96 Tf 12.18 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(y)]TJ/F8 9.96 Tf 5.24 0 Td[(\050)]TJ/F11 9.96 Tf 3.88 0 Td[(x:xt)]TJ/F8 9.96 Tf 23.56 0 Td[(\051\051\051)]TJ -239.5 -21.11 Td[(In)-383(normal)-382(order)-382(reduction)-381(w)27(e)-383(try)-383(to)-383(reduce)-381(alw)27(a)28(ys)-383(the)-382(left)-382(most)-383(expression)-380(of)-383(a)]TJ -14.95 -11.95 Td[(series)-331(of)-334(applications.)-442(W)83(e)-333(con)28(tin)29(ue)-333(un)28(til)-333(no)-333(further)-332(reductions)-330(are)-333(p)-28(ossible.)]TJ/F61 14.35 Tf 0 -32.81 Td[(2)-1125(Arithmetic)]TJ/F8 9.96 Tf 0 -21.82 Td[(W)83(e)-406(exp)-27(ect)-405(from)-405(a)-406(programming)-405(language)-405(that)-406(it)-406(should)-404(b)-28(e)-406(capable)-404(of)-406(doing)-406(arith-)]TJ 0 -11.95 Td[(metical)-383(calculations.)-596(Num)28(b)-27(ers)-384(can)-384(b)-27(e)-384(represen)30(ted)-383(in)-385(lam)29(b)-28(da)-384(calculus)-383(starting)-383(from)]TJ 0 -11.96 Td[(zero)-381(and)-381(writing)-381(\134suc\050zero\051")-379(to)-382(represen)31(t)-381(1,)-394(\134suc\050suc\050zero\051\051")-378(to)-382(represen)31(t)-381(2,)-394(and)-381(so)]TJ 0 -11.95 Td[(on.)-422(In)-265(the)-264(lam)28(b)-27(da)-265(calculus)-264(w)28(e)-265(can)-264(only)-265(de\014ne)-264(new)-264(functions.)-420(Num)28(b)-27(ers)-264(will)-265(b)-28(e)-265(de\014ned)]TJ 0 -11.96 Td[(as)-333(functions)-332(using)-332(the)-333(follo)29(wing)-334(approac)29(h:)-444(zero)-333(can)-333(b)-27(e)-333(de\014ned)-331(as)]TJ/F11 9.96 Tf 164.39 -21.11 Td[(s:)]TJ/F8 9.96 Tf 13.25 0 Td[(\050)]TJ/F11 9.96 Tf 3.88 0 Td[(z)-44(:z)]TJ/F8 9.96 Tf 18.72 0 Td[(\051)]TJ -200.24 -21.11 Td[(This)-421(is)-421(a)-422(function)-419(of)-422(t)28(w)28(o)-422(argumen)29(ts)]TJ/F11 9.96 Tf 164.9 0 Td[(s)]TJ/F8 9.96 Tf 8.87 0 Td[(and)]TJ/F11 9.96 Tf 20.25 0 Td[(z)]TJ/F8 9.96 Tf 5.07 0 Td[(.)-709(W)84(e)-421(will)-422(abbreviate)-420(suc)29(h)-421(expressions)]TJ -199.09 -11.95 Td[(with)-334(more)-332(than)-333(one)-333(argumen)29(t)-333(as)]TJ/F11 9.96 Tf 172.56 -11.96 Td[(sz)-44(:z)]TJ/F8 9.96 Tf -172.56 -17.44 Td[(It)-332(is)-331(understo)-26(o)-28(d)-331(here)-330(that)]TJ/F11 9.96 Tf 116.03 0 Td[(s)]TJ/F8 9.96 Tf 7.97 0 Td[(is)-331(the)-331(\014rst)-331(argumen)29(t)-331(to)-332(b)-27(e)-331(substituted)-329(during)-331(the)-331(ev)56(alu-)]TJ -124 -11.96 Td[(ation)-306(and)]TJ/F11 9.96 Tf 44.28 0 Td[(z)]TJ/F8 9.96 Tf 8.12 0 Td[(the)-305(second.)-434(Using)-305(this)-305(notation,)-311(the)-305(\014rst)-305(natural)-305(n)28(um)29(b)-28(ers)-304(can)-306(b)-27(e)-306(de\014ned)]TJ -52.4 -11.95 Td[(as)]TJ/F38 9.96 Tf 137.6 -9.37 Td[(1)]TJ/F14 9.96 Tf 14.95 0 Td[()]TJ/F11 9.96 Tf 17.71 0 Td[(sz)-44(:s)]TJ/F8 9.96 Tf 22.99 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(z)]TJ/F8 9.96 Tf 5.07 0 Td[(\051)]TJ/F38 9.96 Tf -64.59 -11.95 Td[(2)]TJ/F14 9.96 Tf 14.95 0 Td[()]TJ/F11 9.96 Tf 17.71 0 Td[(sz)-44(:s)]TJ/F8 9.96 Tf 22.99 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(s)]TJ/F8 9.96 Tf 4.67 0 Td[(\050)]TJ/F11 9.96 Tf 3.88 0 Td[(z)]TJ/F8 9.96 Tf 5.07 0 Td[(\051\051)]TJ/F38 9.96 Tf -73.14 -11.96 Td[(3)]TJ/F14 9.96 Tf 14.95 0 Td[()]TJ/F11 9.96 Tf 17.71 0 Td[(sz)-44(:s)]TJ/F8 9.96 Tf 22.99 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(s)]TJ/F8 9.96 Tf 4.67 0 Td[(\050)]TJ/F11 9.96 Tf 3.88 0 Td[(s)]TJ/F8 9.96 Tf 4.67 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(z)]TJ/F8 9.96 Tf 5.07 0 Td[(\051\051\051)]TJ -219.28 -16.99 Td[(and)-333(so)-333(on.)]TJ 181.76 -11.34 Td[(4)]TJ
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1 0 0 1 128.69 721.07 cm
BT
/F8 9.96 Tf 0 0 Td[(Our)-333(\014rst)-333(in)29(teresting)-332(function)-332(is)-332(the)-333(successor)-330(function.)-443(This)-333(can)-333(b)-27(e)-333(de\014ned)-332(as)]TJ/F38 9.96 Tf 145.72 -20.37 Td[(S)]TJ/F14 9.96 Tf 8.31 0 Td[()]TJ/F11 9.96 Tf 10.51 0 Td[(w)-28(y)-36(x:y)]TJ/F8 9.96 Tf 32.16 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(w)-27(y)-37(x)]TJ/F8 9.96 Tf 18.34 0 Td[(\051)]TJ -218.91 -20.36 Td[(The)-333(successor)-331(function)-331(applied)-332(to)-334(our)-333(represen)31(tation)-333(for)-333(zero)-333(yields)]TJ/F38 9.96 Tf 123.79 -20.37 Td[(S0)]TJ/F14 9.96 Tf 13.28 0 Td[()]TJ/F8 9.96 Tf 10.52 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(w)-28(y)-36(x:y)]TJ/F8 9.96 Tf 32.16 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(w)-27(y)-37(x)]TJ/F8 9.96 Tf 18.34 0 Td[(\051\051\050)]TJ/F11 9.96 Tf 11.62 0 Td[(sz)-45(:z)]TJ/F8 9.96 Tf 23.39 0 Td[(\051)]TJ -240.84 -20.36 Td[(In)-285(the)-284(b)-27(o)-28(dy)-285(of)-284(the)-284(\014rst)-284(expression)-282(w)27(e)-284(substitute)-282(all)-285(o)-28(ccurrences)-281(of)]TJ/F11 9.96 Tf 286.03 0 Td[(w)]TJ/F8 9.96 Tf 10.24 0 Td[(with)-285(\050)]TJ/F11 9.96 Tf 26.08 0 Td[(sz)-45(:z)]TJ/F8 9.96 Tf 23.39 0 Td[(\051)-285(and)]TJ -345.74 -11.96 Td[(this)-333(yields)]TJ/F11 9.96 Tf 73.11 -11.95 Td[(y)-37(x:y)]TJ/F8 9.96 Tf 24.76 0 Td[(\050\050)]TJ/F11 9.96 Tf 7.75 0 Td[(sz)-44(:z)]TJ/F8 9.96 Tf 23.39 0 Td[(\051)]TJ/F11 9.96 Tf 3.87 0 Td[(y)-37(x)]TJ/F8 9.96 Tf 10.94 0 Td[(\051)-278(=)]TJ/F11 9.96 Tf 17.16 0 Td[(y)-37(x:y)]TJ/F8 9.96 Tf 24.75 0 Td[(\050\050)]TJ/F11 9.96 Tf 7.75 0 Td[(z)-45(:z)]TJ/F8 9.96 Tf 18.72 0 Td[(\051)]TJ/F11 9.96 Tf 3.88 0 Td[(x)]TJ/F8 9.96 Tf 5.69 0 Td[(\051)-278(=)]TJ/F11 9.96 Tf 17.16 0 Td[(y)-37(x:y)]TJ/F8 9.96 Tf 24.76 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(x)]TJ/F8 9.96 Tf 5.7 0 Td[(\051)]TJ/F14 9.96 Tf 6.64 0 Td[()]TJ/F38 9.96 Tf 10.51 0 Td[(1)]TJ/F8 9.96 Tf -290.41 -17.01 Td[(That)-280(is,)-290(w)27(e)-280(obtain)-279(the)-279(represen)30(tation)-279(of)-280(the)-280(n)29(um)28(b)-27(er)]TJ/F38 9.96 Tf 225.76 0 Td[(1)]TJ/F8 9.96 Tf 7.77 0 Td[(\050remem)30(b)-28(er)-279(that)-280(v)56(ariable)-279(names)]TJ -233.53 -11.95 Td[(are)-333(\134dummies"\051.)]TJ 14.95 -11.96 Td[(Successor)-330(applied)-332(to)-334(1)-333(yields:)]TJ/F38 9.96 Tf 11.12 -20.36 Td[(S1)]TJ/F14 9.96 Tf 13.28 0 Td[()]TJ/F8 9.96 Tf 10.52 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(w)-28(y)-36(x:y)]TJ/F8 9.96 Tf 32.16 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(w)-27(y)-37(x)]TJ/F8 9.96 Tf 18.34 0 Td[(\051\051\050)]TJ/F11 9.96 Tf 11.62 0 Td[(sz)-45(:s)]TJ/F8 9.96 Tf 22.99 0 Td[(\050)]TJ/F11 9.96 Tf 3.88 0 Td[(z)]TJ/F8 9.96 Tf 5.07 0 Td[(\051\051)-278(=)]TJ/F11 9.96 Tf 21.03 0 Td[(y)-37(x:y)]TJ/F8 9.96 Tf 24.76 0 Td[(\050\050)]TJ/F11 9.96 Tf 7.75 0 Td[(sz)-44(:s)]TJ/F8 9.96 Tf 22.99 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(z)]TJ/F8 9.96 Tf 5.07 0 Td[(\051\051)]TJ/F11 9.96 Tf 7.75 0 Td[(y)-36(x)]TJ/F8 9.96 Tf 10.94 0 Td[(\051)-277(=)]TJ/F11 9.96 Tf 17.15 0 Td[(y)-37(x:y)]TJ/F8 9.96 Tf 24.76 0 Td[(\050)]TJ/F11 9.96 Tf 3.88 0 Td[(y)]TJ/F8 9.96 Tf 5.24 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(x)]TJ/F8 9.96 Tf 5.7 0 Td[(\051\051)]TJ/F14 9.96 Tf 10.51 0 Td[()]TJ/F38 9.96 Tf 10.52 0 Td[(2)]TJ/F8 9.96 Tf -337.46 -20.37 Td[(Notice)-362(that)-362(the)-362(only)-362(purp)-26(ose)-362(of)-362(applying)-361(the)-362(n)28(um)29(b)-28(er)-362(\050)]TJ/F11 9.96 Tf 241.26 0 Td[(sz)-44(:s)]TJ/F8 9.96 Tf 22.99 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(z)]TJ/F8 9.96 Tf 5.07 0 Td[(\051\051)-363(to)-362(the)-362(argumen)29(ts)]TJ/F11 9.96 Tf 90.07 0 Td[(y)]TJ/F8 9.96 Tf -363.26 -11.95 Td[(and)]TJ/F11 9.96 Tf 19.37 0 Td[(x)]TJ/F8 9.96 Tf 9.02 0 Td[(is)-333(to)-333(\134rename")-332(the)-333(v)56(ariables)-332(used)-332(in)-333(the)-333(de\014nition)-331(of)-334(our)-333(n)28(um)29(b)-27(er.)]TJ/F54 11.96 Tf -28.39 -27.63 Td[(2.1)-1126(Addition)]TJ/F8 9.96 Tf 0 -18.38 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Td[(create)-332(a)-334(pair)-333(\050)]TJ/F11 9.96 Tf 62.05 0 Td[(n;)-167(n)]TJ/F14 9.96 Tf 18.6 0 Td[(�)]TJ/F8 9.96 Tf 9.96 0 Td[(1\051)-334(and)-333(then)-332(pic)29(k)-334(the)-332(second)-332(elemen)30(t)-333(of)-334(the)-332(pair)-333(as)-333(the)-333(result.)]TJ -75.66 -11.96 Td[(A)-333(pair)-333(\050)]TJ/F11 9.96 Tf 35.17 0 Td[(a;)-166(b)]TJ/F8 9.96 Tf 13.97 0 Td[(\051)-333(can)-333(b)-28(e)-333(represen)31(ted)-333(in)]TJ/F11 9.96 Tf 103.61 0 Td[()]TJ/F8 9.96 Tf 5.81 0 Td[(-calculus)-332(using)-332(the)-333(function)]TJ -7.26 -20.75 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(z)-45(:z)-44(ab)]TJ/F8 9.96 Tf 28.27 0 Td[(\051)]TJ -198.39 -20.75 Td[(W)83(e)-268(can)-268(extract)-267(the)-268(\014rst)-268(elemen)30(t)-268(of)-268(the)-268(pair)-268(from)-268(the)-267(expression)-266(applying)-268(this)-267(function)]TJ 0 -11.96 Td[(to)]TJ/F38 9.96 Tf 12.18 0 Td[(T)]TJ/F8 9.96 Tf 126.6 -11.95 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(z)-45(:z)-44(ab)]TJ/F8 9.96 Tf 28.27 0 Td[(\051)]TJ/F38 9.96 Tf 3.87 0 Td[(T)]TJ/F8 9.96 Tf 9.55 0 Td[(=)]TJ/F38 9.96 Tf 10.52 0 Td[(T)]TJ/F11 9.96 Tf 6.78 0 Td[(ab)]TJ/F8 9.96 Tf 12.3 0 Td[(=)]TJ/F11 9.96 Tf 10.52 0 Td[(a)]TJ/F8 9.96 Tf -224.46 -17.23 Td[(and)-333(the)-333(second)-331(applying)-333(the)-333(function)-331(to)]TJ/F38 9.96 Tf 177.72 0 Td[(F)]TJ/F8 9.96 Tf -37.34 -20.76 Td[(\050)]TJ/F11 9.96 Tf 3.88 0 Td[(z)-44(:z)-44(ab)]TJ/F8 9.96 Tf 28.26 0 Td[(\051)]TJ/F38 9.96 Tf 3.87 0 Td[(F)]TJ/F8 9.96 Tf 8.44 0 Td[(=)]TJ/F38 9.96 Tf 10.52 0 Td[(F)]TJ/F11 9.96 Tf 5.67 0 Td[(ab)]TJ/F8 9.96 Tf 12.31 0 Td[(=)]TJ/F11 9.96 Tf 10.52 0 Td[(b)]TJ/F8 9.96 Tf -223.85 -20.75 Td[(The)-313(follo)28(wing)-313(function)-311(generates)-311(from)-312(the)-312(pair)-312(\050)]TJ/F11 9.96 Tf 209.03 0 Td[(n;)-167(n)]TJ/F14 9.96 Tf 18.2 0 Td[(�)]TJ/F8 9.96 Tf 9.55 0 Td[(1\051)-313(\050whic)28(h)-312(is)-313(the)-312(argumen)29(t)]TJ/F11 9.96 Tf 115.29 0 Td[(p)]TJ/F8 9.96 Tf 8.13 0 Td[(in)]TJ -360.2 -11.95 Td[(the)-333(function\051)-332(the)-332(pair)-333(\050)]TJ/F11 9.96 Tf 101.59 0 Td[(n)]TJ/F8 9.96 Tf 8.2 0 Td[(+)-222(1)]TJ/F11 9.96 Tf 14.94 0 Td[(;)-167(n)]TJ/F14 9.96 Tf 12.62 0 Td[(�)]TJ/F8 9.96 Tf 9.97 0 Td[(1\051:)]TJ -15.23 -20.76 Td[(\010)]TJ/F14 9.96 Tf 9.96 0 Td[()]TJ/F8 9.96 Tf 10.52 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(pz)-45(:z)]TJ/F8 9.96 Tf 23.74 0 Td[(\050)]TJ/F38 9.96 Tf 3.87 0 Td[(S)]TJ/F8 9.96 Tf 5.54 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(p)]TJ/F38 9.96 Tf 5.01 0 Td[(T)]TJ/F8 9.96 Tf 6.78 0 Td[(\051\051\050)]TJ/F11 9.96 Tf 11.63 0 Td[(p)]TJ/F38 9.96 Tf 5.01 0 Td[(T)]TJ/F8 9.96 Tf 6.78 0 Td[(\051\051)]TJ -228.67 -20.75 Td[(The)-275(sub)-27(expression)]TJ/F11 9.96 Tf 82.61 0 Td[(p)]TJ/F38 9.96 Tf 5.01 0 Td[(T)]TJ/F8 9.96 Tf 9.52 0 Td[(extracts)-274(the)-275(\014rst)-274(elemen)30(t)-275(from)-274(the)-275(pair)]TJ/F11 9.96 Tf 169.68 0 Td[(p)]TJ/F8 9.96 Tf 5.01 0 Td[(.)-425(A)-276(new)-275(pair)-274(is)-275(formed)]TJ -271.83 -11.95 Td[(using)-282(this)-282(elemen)30(t,)-292(whic)28(h)-283(is)-282(incremen)31(ted)-282(for)-282(the)-282(\014rst)-282(p)-27(osition)-282(of)-283(the)-282(new)-282(pair)-282(and)-283(just)]TJ 0 -11.96 Td[(copied)-332(for)-333(the)-333(second)-331(p)-28(osition)-333(of)-333(the)-332(new)-333(pair.)]TJ 14.95 -11.95 Td[(The)-320(predecessor)-318(of)-320(a)-321(n)28(um)28(b)-27(er)]TJ/F11 9.96 Tf 128.53 0 Td[(n)]TJ/F8 9.96 Tf 9.18 0 Td[(is)-320(obtained)-320(b)28(y)-320(applying)]TJ/F11 9.96 Tf 104.99 0 Td[(n)]TJ/F8 9.96 Tf 9.18 0 Td[(times)-319(the)-321(function)-319(\010)-321(to)]TJ -266.83 -11.96 Td[(the)-333(pair)-333(\050)]TJ/F11 9.96 Tf 41.54 0 Td[(:z)]TJ/F38 9.96 Tf 13.65 0 Td[(00)]TJ/F8 9.96 Tf 9.96 0 Td[(\051)-334(and)-333(then)-332(selecting)-331(the)-333(second)-332(mem)30(b)-28(er)-333(of)-333(the)-333(new)-333(pair:)]TJ/F38 9.96 Tf 72.42 -20.75 Td[(P)]TJ/F14 9.96 Tf 9.13 0 Td[()]TJ/F8 9.96 Tf 10.52 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(n:n)]TJ/F8 9.96 Tf 20.54 0 Td[(\010\050)]TJ/F11 9.96 Tf 11.07 0 Td[(z)-45(:z)]TJ/F38 9.96 Tf 18.72 0 Td[(00)]TJ/F8 9.96 Tf 9.97 0 Td[(\051)]TJ/F38 9.96 Tf 3.87 0 Td[(F)]TJ/F8 9.96 Tf -225.26 -20.75 Td[(Notice)-267(that)-267(using)-266(this)-267(approac)29(h)-267(the)-267(predecessor)-263(of)-268(zero)-266(is)-267(zero.)-421(This)-267(prop)-27(ert)29(y)-268(is)-266(useful)]TJ 0 -11.96 Td[(for)-333(the)-333(de\014nition)-331(of)-334(other)-332(functions.)]TJ/F54 11.96 Tf 0 -27.69 Td[(3.4)-1126(Equalit)32(y)-375(and)-374(inequalities)]TJ/F8 9.96 Tf 0 -18.38 Td[(With)-410(the)-410(predecessor)-407(function)-409(as)-410(the)-410(building)-409(blo)-28(c)29(k,)-430(w)27(e)-410(can)-410(no)28(w)-411(de\014ne)-408(a)-411(function)]TJ 0 -11.96 Td[(whic)28(h)-333(tests)-332(if)-333(a)-334(n)28(um)29(b)-27(er)]TJ/F11 9.96 Tf 105.58 0 Td[(x)]TJ/F8 9.96 Tf 9.01 0 Td[(is)-333(greater)-332(than)-333(or)-334(equal)-332(to)-334(a)-333(n)28(um)28(b)-27(er)]TJ/F11 9.96 Tf 162.34 0 Td[(y)]TJ/F8 9.96 Tf 5.24 0 Td[(:)]TJ/F38 9.96 Tf -137.08 -20.75 Td[(G)]TJ/F14 9.96 Tf 9.41 0 Td[()]TJ/F8 9.96 Tf 10.52 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(xy)-36(:)]TJ/F38 9.96 Tf 19.51 0 Td[(Z)]TJ/F8 9.96 Tf 6.09 0 Td[(\050)]TJ/F11 9.96 Tf 3.88 0 Td[(x)]TJ/F38 9.96 Tf 5.69 0 Td[(P)]TJ/F11 9.96 Tf 6.37 0 Td[(y)]TJ/F8 9.96 Tf 5.24 0 Td[(\051\051)]TJ -33.91 -11.34 Td[(7)]TJ
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/F8 9.96 Tf 0 0 Td[(If)-297(the)-296(predecessor)-293(function)-295(applied)]TJ/F11 9.96 Tf 151.93 0 Td[(x)]TJ/F8 9.96 Tf 8.65 0 Td[(times)-296(to)]TJ/F11 9.96 Tf 38.07 0 Td[(y)]TJ/F8 9.96 Tf 8.2 0 Td[(yields)-295(zero,)-303(then)-296(it)-296(is)-297(true)-295(that)]TJ/F11 9.96 Tf 134.67 0 Td[(x)]TJ/F14 9.96 Tf 8.46 0 Td[()]TJ/F11 9.96 Tf 10.52 0 Td[(y)]TJ/F8 9.96 Tf 5.24 0 Td[(.)]TJ -350.79 -11.95 Td[(If)]TJ/F11 9.96 Tf 10.9 0 Td[(x)]TJ/F14 9.96 Tf 10.02 0 Td[()]TJ/F11 9.96 Tf 12.08 0 Td[(y)]TJ/F8 9.96 Tf 9.5 0 Td[(and)]TJ/F11 9.96 Tf 20.32 0 Td[(y)]TJ/F14 9.96 Tf 9.57 0 Td[()]TJ/F11 9.96 Tf 12.08 0 Td[(x)]TJ/F8 9.96 Tf 5.69 0 Td[(,)-451(then)]TJ/F11 9.96 Tf 30.9 0 Td[(x)]TJ/F8 9.96 Tf 10.02 0 Td[(=)]TJ/F11 9.96 Tf 12.08 0 Td[(y)]TJ/F8 9.96 Tf 5.24 0 Td[(.)-727(This)-428(leads)-426(to)-428(the)-427(follo)28(wing)-427(de\014nition)-426(of)-428(the)]TJ -163.35 -11.96 Td[(function)]TJ/F38 9.96 Tf 39.02 0 Td[(E)]TJ/F8 9.96 Tf 9.27 0 Td[(whic)28(h)-333(tests)-332(if)-333(t)28(w)27(o)-334(n)29(um)28(b)-27(ers)-333(are)-333(equal:)]TJ/F38 9.96 Tf 68.29 -18.6 Td[(E)]TJ/F14 9.96 Tf 8.72 0 Td[()]TJ/F8 9.96 Tf 10.52 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(xy)-37(:)]TJ/F14 9.96 Tf 21.73 0 Td[(^)]TJ/F8 9.96 Tf 8.86 0 Td[(\050)]TJ/F38 9.96 Tf 3.87 0 Td[(Z)]TJ/F8 9.96 Tf 6.09 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(x)]TJ/F38 9.96 Tf 5.7 0 Td[(P)]TJ/F11 9.96 Tf 6.36 0 Td[(y)]TJ/F8 9.96 Tf 5.24 0 Td[(\051\051\050)]TJ/F38 9.96 Tf 11.63 0 Td[(Z)]TJ/F8 9.96 Tf 6.09 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(y)]TJ/F38 9.96 Tf 5.24 0 Td[(P)]TJ/F11 9.96 Tf 6.37 0 Td[(x)]TJ/F8 9.96 Tf 5.69 0 Td[(\051\051\051)]TJ -240.3 -18.59 Td[(In)-334(a)-333(similar)-332(manner)-332(w)27(e)-333(can)-332(de\014ne)-332(functions)-331(to)-334(test)-332(whether)]TJ/F11 9.96 Tf 263.98 0 Td[(x)-278(>)-278(y)]TJ/F8 9.96 Tf 24.22 0 Td[(,)]TJ/F11 9.96 Tf 6.09 0 Td[(x)-278(<)-278(y)]TJ/F8 9.96 Tf 27.54 0 Td[(or)]TJ/F11 9.96 Tf 12.21 0 Td[(x)]TJ/F14 9.96 Tf 8.46 0 Td[(6)]TJ/F8 9.96 Tf 0 0 Td[(=)]TJ/F11 9.96 Tf 10.51 0 Td[(y)]TJ/F8 9.96 Tf 5.24 0 Td[(.)]TJ/F61 14.35 Tf -358.25 -32.38 Td[(4)-1125(Recursion)]TJ/F8 9.96 Tf 0 -21.82 Td[(Recursiv)29(e)-402(functions)-401(can)-402(b)-28(e)-402(de\014ned)-401(in)-402(the)]TJ/F11 9.96 Tf 187.77 0 Td[()]TJ/F8 9.96 Tf 9.82 0 Td[(calculus)-401(using)-402(a)-403(function)-401(whic)28(h)-402(calls)-402(a)]TJ -197.59 -11.95 Td[(function)]TJ/F11 9.96 Tf 39 0 Td[(y)]TJ/F8 9.96 Tf 8.54 0 Td[(and)-331(then)-331(regenerates)-328(itself.)-442(This)-331(can)-331(b)-27(e)-331(b)-28(etter)-330(understo)-26(o)-28(d)-331(b)28(y)-331(considering)]TJ -47.54 -11.96 Td[(the)-333(follo)28(wing)-333(function)]TJ/F38 9.96 Tf 98.24 0 Td[(Y)]TJ/F8 9.96 Tf 6.9 0 Td[(:)]TJ/F38 9.96 Tf 11.84 -18.59 Td[(Y)]TJ/F14 9.96 Tf 9.66 0 Td[()]TJ/F8 9.96 Tf 10.52 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(y)-37(:)]TJ/F8 9.96 Tf 13.82 0 Td[(\050)]TJ/F11 9.96 Tf 3.88 0 Td[(x:y)]TJ/F8 9.96 Tf 19.51 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(xx)]TJ/F8 9.96 Tf 11.39 0 Td[(\051\051\050)]TJ/F11 9.96 Tf 11.63 0 Td[(x:y)]TJ/F8 9.96 Tf 19.51 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(xx)]TJ/F8 9.96 Tf 11.39 0 Td[(\051\051\051)]TJ -239.9 -18.6 Td[(This)-333(function)-332(applied)-332(to)-333(a)-334(function)]TJ/F38 9.96 Tf 156.14 0 Td[(R)]TJ/F8 9.96 Tf 9.75 0 Td[(yields:)]TJ/F38 9.96 Tf -42.41 -18.6 Td[(YR)]TJ/F8 9.96 Tf 15.84 0 Td[(=)-278(\050)]TJ/F11 9.96 Tf 14.4 0 Td[(x:)]TJ/F38 9.96 Tf 14.27 0 Td[(R)]TJ/F8 9.96 Tf 6.43 0 Td[(\050)]TJ/F11 9.96 Tf 3.88 0 Td[(xx)]TJ/F8 9.96 Tf 11.38 0 Td[(\051\051\050)]TJ/F11 9.96 Tf 11.63 0 Td[(x:)]TJ/F38 9.96 Tf 14.27 0 Td[(R)]TJ/F8 9.96 Tf 6.43 0 Td[(\050)]TJ/F11 9.96 Tf 3.88 0 Td[(xx)]TJ/F8 9.96 Tf 11.39 0 Td[(\051\051)]TJ -237.28 -18.59 Td[(whic)28(h)-333(further)-332(reduced)-331(yields:)]TJ/F11 9.96 Tf 127.03 -18.6 Td[(R)]TJ/F8 9.96 Tf 7.64 0 Td[(\050\050)]TJ/F11 9.96 Tf 7.75 0 Td[(x:)]TJ/F38 9.96 Tf 14.27 0 Td[(R)]TJ/F8 9.96 Tf 6.44 0 Td[(\050)]TJ/F11 9.96 Tf 3.87 0 Td[(xx)]TJ/F8 9.96 Tf 11.39 0 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%!PS-AdobeFont-1.1: CMSY10 1.0
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11 dict begin
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% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
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%!PS-AdobeFont-1.1: CMR10 1.00B
%%CreationDate: 1992 Feb 19 19:54:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /OARYRS+CMR10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
dup 8 /Phi put
dup 11 /ff put
dup 12 /fi put
dup 33 /exclam put
dup 34 /quotedblright put
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 58 /colon put
dup 61 /equal put
dup 63 /question put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 87 /W put
dup 89 /Y put
dup 91 /bracketleft put
dup 92 /quotedblleft put
dup 93 /bracketright put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
dup 123 /endash put
readonly def
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currentdict end
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%!PS-AdobeFont-1.1: CMBX12 1.0
%%CreationDate: 1991 Aug 20 16:34:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /PCGITJ+CMBX12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
dup 12 /fi put
dup 46 /period put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 65 /A put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 76 /L put
dup 77 /M put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 121 /y put
readonly def
/FontBBox{-53 -251 1139 750}readonly def
/UniqueID 5000769 def
currentdict end
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%!PS-AdobeFont-1.1: CMR12 1.0
%%CreationDate: 1991 Aug 20 16:38:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
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/FullName (CMR12) readonly def
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