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b(In)m(tro)s(duction)2166 b(4)515 2004 y(2)76 b(Getting)31
b(Started)2031 b(6)639 2144 y Fq(2.1)84 b(Y)-7 b(our)27
b(First)h(Circuit)79 b(.)41 b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h
(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)131
b(6)639 2283 y(2.2)84 b(The)28 b(La)n(v)-5 b(a)26 b(In)n(terpreter)39
b(.)j(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h
(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)131 b(7)639 2423 y(2.3)84
b(Y)-7 b(our)27 b(Second)h(Circuit)64 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f
(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)
131 b(8)639 2562 y(2.4)84 b(Generating)27 b(VHDL)70 b(.)41
b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h
(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)131 b(9)639 2702 y(2.5)84
b(Exercises)e(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)
h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90
b(10)515 2924 y Fr(3)76 b(Bigger)31 b(Circuits)2008 b(12)639
3064 y Fq(3.1)84 b(Recursion)27 b(o)n(v)n(er)f(Lists)67
b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)
h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(12)639 3203 y(3.2)84
b(Connection)27 b(P)n(atterns)55 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h
(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90
b(13)639 3342 y(3.3)84 b(Arithmetic)22 b(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h
(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)
f(.)h(.)f(.)h(.)f(.)h(.)90 b(14)639 3482 y(3.4)84 b(Compilation)32
b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)
h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90
b(16)639 3621 y(3.5)84 b(Exercises)e(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f
(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)
h(.)f(.)h(.)f(.)h(.)90 b(16)515 3844 y Fr(4)76 b(V)-8
b(eri\014cation)2165 b(18)639 3983 y Fq(4.1)84 b(Simple)28
b(Prop)r(erties)f(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)
f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90
b(18)639 4123 y(4.2)84 b(Quan)n(ti\014cation)25 b(.)41
b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f
(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(19)639
4262 y(4.3)84 b(General)27 b(Prop)r(erties)54 b(.)41
b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h
(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(20)639 4402 y(4.4)84
b(Helping)28 b(V)-7 b(eri\014cation)80 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f
(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)
90 b(20)639 4541 y(4.5)84 b(Exercises)e(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h
(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)
f(.)h(.)f(.)h(.)f(.)h(.)90 b(22)515 4764 y Fr(5)76 b(Sequen)m(tial)32
b(Circuits)1850 b(23)639 4903 y Fq(5.1)84 b(The)28 b(Dela)n(y)f(Comp)r
(onen)n(t)43 b(.)e(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h
(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(23)1926 5255
y(1)p eop
%%Page: 2 3
2 2 bop 639 523 a Fq(5.2)84 b(Multiple)29 b(Dela)n(ys)34
b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)
f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(24)639
663 y(5.3)84 b(Coun)n(ters)24 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f
(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)
h(.)f(.)h(.)f(.)h(.)90 b(25)639 802 y(5.4)84 b(Sequen)n(tialization)55
b(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)
h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(25)639 941
y(5.5)84 b(V)-7 b(ariations)27 b(on)g Fp(rowSeq)f Fq(.)42
b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g
(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(26)639 1081 y(5.6)84
b(Exercises)e(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)
h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90
b(27)515 1303 y Fr(6)76 b(Sequen)m(tial)32 b(V)-8 b(eri\014cation)1702
b(29)639 1443 y Fq(6.1)84 b(Sequen)n(tial)27 b(Safet)n(y)h(Prop)r
(erties)33 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g
(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(29)639 1582 y(6.2)84
b(Sequen)n(tial)27 b(Logic)77 b(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f
(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)
90 b(30)639 1722 y(6.3)84 b(V)-7 b(eri\014cation)64 b(.)42
b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g
(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(30)639
1861 y(6.4)84 b(Induction)68 b(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h
(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)
f(.)h(.)f(.)h(.)90 b(31)639 2001 y(6.5)84 b(Induction)28
b(With)h(Depth)53 b(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g
(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(32)639
2140 y(6.6)84 b(Induction)28 b(With)h(Restricted)e(States)44
b(.)e(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h
(.)90 b(35)639 2280 y(6.7)84 b(Exercises)e(.)41 b(.)h(.)f(.)h(.)g(.)f
(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)
g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(36)515 2502 y Fr(7)76
b(Time)30 b(T)-8 b(ransformations)1717 b(38)639 2642
y Fq(7.1)84 b(Timing)28 b(Issues)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f
(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)
h(.)f(.)h(.)90 b(38)639 2781 y(7.2)84 b(Slo)n(wing)27
b(Do)n(wn)i(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h
(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90
b(39)639 2921 y(7.3)84 b(Sp)r(eeding)28 b(Up)84 b(.)41
b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f
(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(42)639
3060 y(7.4)84 b(Exercises)e(.)41 b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f
(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)
h(.)f(.)h(.)90 b(43)515 3283 y Fr(8)76 b(More)32 b(connection)f
(patterns)1566 b(45)639 3422 y Fq(8.1)84 b(Connection)27
b(patterns)h(revisited)51 b(.)42 b(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h
(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(45)639 3562
y(8.2)84 b(T)-7 b(ree)27 b(shap)r(ed)h(circuits)72 b(.)42
b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g
(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(48)639 3701 y(8.3)84
b(Describing)27 b(Butter\015y)h(Circuits)72 b(.)42 b(.)f(.)h(.)f(.)h(.)
f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90
b(50)639 3841 y(8.4)84 b(Batc)n(her's)26 b(Bitonic)i(Merger)83
b(.)41 b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)
h(.)f(.)h(.)f(.)h(.)90 b(54)639 3980 y(8.5)84 b(Exercises)e(.)41
b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h
(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90
b(56)515 4203 y Fr(9)76 b(Syn)m(thesizing)32 b(La)m(v)-5
b(a)33 b(Circuits)1535 b(59)639 4342 y Fq(9.1)84 b(State)28
b(Mac)n(hines)54 b(.)42 b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f
(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90
b(59)639 4482 y(9.2)84 b(Beha)n(vioral)25 b(Descriptions)70
b(.)42 b(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)
f(.)h(.)f(.)h(.)f(.)h(.)90 b(63)639 4621 y(9.3)84 b(Exercises)e(.)41
b(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h
(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90
b(67)1926 5255 y(2)p eop
%%Page: 3 4
3 3 bop 515 523 a Fr(10)28 b(T)m(yp)s(es)2394 b(68)639
663 y Fq(10.1)42 b(Signals)27 b(and)g(Circuits)73 b(.)42
b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g
(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(68)639 802 y(10.2)42
b(Connection)27 b(P)n(atterns)55 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h
(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90
b(69)639 941 y(10.3)42 b(Ov)n(erloading)d(.)j(.)f(.)h(.)g(.)f(.)h(.)f
(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)
h(.)f(.)h(.)f(.)h(.)90 b(69)515 1164 y Fr(A)52 b(Quic)m(k)33
b(Reference)e(Guide)1677 b(72)639 1303 y Fq(A.1)64 b(The)28
b Fp(lava)e Fq(command)79 b(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h
(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90
b(72)639 1443 y(A.2)64 b(Logical)26 b(Gates)49 b(.)41
b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f
(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(72)639
1582 y(A.3)64 b(Arithmetical)28 b(Gates)43 b(.)e(.)h(.)f(.)h(.)f(.)h(.)
g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f
(.)h(.)90 b(73)639 1722 y(A.4)64 b(Generic)27 b(Gates)k(.)41
b(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f
(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(74)639
1861 y(A.5)64 b(Mo)r(dule:)37 b Fp(Patterns)55 b Fq(.)41
b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h
(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(74)639 2001 y(A.6)64
b(Mo)r(dule:)37 b Fp(Arithmetic)31 b Fq(.)42 b(.)f(.)h(.)f(.)h(.)g(.)f
(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)
90 b(75)639 2140 y(A.7)64 b(Mo)r(dule:)37 b Fp(SequentialCircuit)o(s)68
b Fq(.)41 b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h
(.)f(.)h(.)90 b(76)639 2280 y(A.8)64 b(In)n(terpretations)j(.)42
b(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h
(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)90 b(76)639 2419
y(A.9)64 b(Errors)56 b(.)41 b(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h
(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)f(.)h(.)g(.)f(.)h(.)
f(.)h(.)f(.)h(.)90 b(77)515 2642 y Fr(B)56 b(Answ)m(ers)2295
b(79)639 2864 y(Bibliograph)m(y)2109 b(88)639 3087 y(Index)2409
b(90)1926 5255 y Fq(3)p eop
%%Page: 4 5
4 4 bop 515 1186 a Fo(Chapter)64 b(1)515 1641 y Fs(In)-6
b(tro)6 b(duction)515 2112 y Fq(La)n(v)-5 b(a)27 b(is)h(an)g(exp)r
(erimen)n(tal)f(to)r(ol)h(for)g(hardw)n(are)e(design)h(and)h(v)n
(eri\014cation.)37 b(Using)28 b(La)n(v)-5 b(a,)515 2212
y(one)36 b(can)f(describ)r(e)h(circuits)g(using)g(a)g(simple)g
(functional)h(hardw)n(are)d(description)i(lan-)515 2311
y(guage.)51 b(The)33 b(descriptions)f(are)g(short)h(and)f(sw)n(eet,)i
(and)f(do)g(not)g(su\013er)f(from)h(the)g(v)n(er-)515
2411 y(b)r(osit)n(y)22 b(of)g(more)f(standard)g(hardw)n(are)f
(description)i(languages)e(\(HDLs\))j(lik)n(e)f(VHDL)h(and)515
2511 y(V)-7 b(erilog.)34 b(On)22 b(the)g(other)f(hand,)j(w)n(e)d
(cannot)h(express)f(the)h(same)g(things)g(as)f(in)h(these)g(large,)515
2610 y(expressiv)n(e)32 b(\(and)i(complicated\))g(languages.)54
b(F)-7 b(or)34 b(example,)h(w)n(e)e(cannot)h(express)f(lo)n(w)515
2710 y(lev)n(el)22 b(details)g(ab)r(out)h(timing.)36
b(What)23 b(w)n(e)f(can)g(express)g(v)n(ery)g(nicely)-7
b(,)23 b(though,)h(is)e(the)i(w)n(a)n(ys)515 2809 y(in)29
b(whic)n(h)g(circuits)g(are)f(built)h(from)g(sub-circuits.)41
b(La)n(v)-5 b(a)27 b(facilitates)i(the)h(description)e(of)515
2909 y Fn(c)l(onne)l(ction)g(p)l(atterns)e Fq(so)f(that)i(they)f(are)f
(easily)h(reusable.)35 b(F)-7 b(or)26 b(some)f(kinds)h(of)g(circuits,)
515 3009 y(for)h(example)h(in)g(signal)f(pro)r(cessing,)f(this)i(is)g
(exactly)g(what)f(w)n(e)h(w)n(an)n(t)f(to)h(do.)37 b(La)n(v)-5
b(a)27 b(also)515 3108 y(pro)n(vides)d(man)n(y)g(di\013eren)n(t)h(w)n
(a)n(ys)f(of)h(analysing)f(our)h(circuit)g(descriptions.)35
b(W)-7 b(e)26 b(can)e(sim-)515 3208 y(ulate)29 b(circuits,)g(just)g(as)
g(with)g(more)f(standard)g(HDLs,)i(but)g(w)n(e)e(can)h(also)f(use)h
(sym)n(b)r(olic)515 3308 y(metho)r(ds)e(to)g(generate)f(input)h(to)g
(analysis)f(to)r(ols)g(suc)n(h)h(as)g(automatic)f(theorem)g(pro)n(v)n
(ers)515 3407 y(and)h(mo)r(del)h(c)n(hec)n(k)n(ers.)36
b(Indeed,)28 b(the)h(same)e(metho)r(ds)h(are)f(used)h(to)f(generate)g
(structural)515 3507 y(VHDL)39 b(from)g(La)n(v)-5 b(a)37
b(circuit)i(descriptions.)69 b(Our)38 b(aim)h(in)f(this)h(tutorial)g
(is)f(to)h(gen)n(tly)515 3606 y(in)n(tro)r(duce)27 b(this)h(new)g(st)n
(yle)f(of)g(circuit)h(design)f(and)g(analysis,)g(b)n(y)g(means)g(of)h
(examples.)515 3746 y(La)n(v)-5 b(a)20 b(is)h(used)g(at)g(Chalmers)f
(as)h(a)g(platform)g(for)f(exp)r(erimen)n(ts)h(in)g(the)h(formal)e(v)n
(eri\014cation)515 3846 y(of)30 b(hardw)n(are)e([3,)i(2].)45
b(\(Note,)31 b(ho)n(w)n(ev)n(er,)e(that)i(b)r(oth)f(of)g(these)h
(references)e(are)g(ab)r(out)h(an)515 3945 y(older)24
b(v)n(ersion)g(of)h(La)n(v)-5 b(a,)25 b(in)h(whic)n(h)f(circuit)g
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1304 y(sequen)n(tial)d(circuits,)g(they)h(are)e(essen)n(tial)h(to)h
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2560 y(has)e(t)n(w)n(o)f(parameters,)h Fj(n)p Fq(,)h(the)g(n)n(um)n(b)r
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1386 y(at)36 b(di\013eren)n(t)g(rates)g(equiv)-5 b(alen)n(t.)62
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722 1486 y(c)n(ho)r(ose.)575 1652 y(7.5)k(Can)28 b(y)n(ou)e(\014nd)i(a)
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45 45 bop 515 1186 a Fo(Chapter)64 b(8)515 1641 y Fs(More)78
b(connection)g(patterns)515 2112 y Fq(In)34 b(this)h(c)n(hapter,)h(w)n
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Fp(f)43 b(->-)f(g)p Fq(,)27 b(instead)g(of)f Fp(serial)41
b(f)j(g)p Fq(,)26 b(see)h(\014gure)f(8.1.)35 b(Note)27
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3332 y Fp(f)43 b(->-)f(\(g)h(->-)g(h\))129 b(===)h(\(f)43
b(->-)f(g\))h(->-)f(h)515 3518 y Fq(Sometimes)27 b(w)n(e)h(w)n(an)n(t)e
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b([])478 b(inp)43 b(=)g(inp)602 3804 y(compose)e(\(circ:circs\))d(inp)
43 b(=)g(out)689 3903 y(where)776 4003 y(x)131 b(=)43
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4289 y Fq(Note)34 b(that)g(w)n(e)f(could)h(ha)n(v)n(e)f(written)h(this)
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4388 y(serial)26 b(connection)h(pattern.)602 4574 y Fp(compose1)40
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j(=)h(out)689 4774 y(where)776 4873 y(out)g(=)g(\(circ)e(->-)i
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46 46 bop 1615 1272 56 4 v 1615 1161 V 1615 1051 V 1615
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468 284 4 v 1667 1298 4 831 v 1784 904 a Fj(f)p 1947
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1947 829 V 1947 718 V 1999 661 229 4 v 1999 1104 4 443
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i(\()p Fp(inp)p Fq(\))e(from)g(eac)n(h)g(side)g(of)515
1875 y(the)26 b(de\014nitions.)36 b(The)26 b(iden)n(tit)n(y)f(circuit)h
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(connection)515 2075 y(patterns.)602 2261 y Fp(compose2)40
b([])479 b(=)43 b(id)602 2360 y(compose2)d(\(circ:circs\))f(=)k(circ)f
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(just)i(a)g(matter)f(of)515 2646 y(taste.)h(In)25 b(fact)f(it)h(is)f
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(the)g(other.)515 2885 y(Out)22 b(of)f Fp(compose)p Fq(,)g(w)n(e)g(can)
h(easily)f(mak)n(e)g(a)g(connection)h(pattern,)h(called)e
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(copies)h(of)h(the)g(same)f(circuit)g(in)h(sequence.)602
3171 y Fp(composeN)40 b(n)j(circ)f(=)h(compose)e(\(replicate)f(n)j
(circ\))602 3373 y(Main>)e Fl(simulateSeq)e(\(composeN)h(5)j(inc\))f
([0,2,4,6])602 3473 y Fp([5,7,9,11])515 3675 y Fq(Here)27
b Fp(inc)f Fq(is)i(the)g(circuit)f(that)h(adds)f(one)h(to)f(its)h(in)n
(teger)f(input.)515 3815 y(W)-7 b(e)22 b(also)f(sa)n(w)g(the)h
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b Fq(tak)n(es)g(a)g(pair)h(of)g(inputs,)h(passing)515
3914 y(the)31 b(\014rst)f(to)h Fp(f)f Fq(and)h(the)g(second)f(to)h
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46 b(The)515 4014 y(in\014x)28 b(v)n(ersion)e(of)h Fp(par)42
b(f)i(g)27 b Fq(is)g(written)h Fp(f)43 b(-|-)g(g)p Fq(.)515
4154 y(A)25 b(v)n(ersion)f(of)h Fp(par)f Fq(that)i(\\do)r(es")e
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Fp(g)g Fq(to)g(the)g(second)g(half)515 4253 y(also)g(turns)h(out)g(to)g
(b)r(e)h(useful.)36 b(W)-7 b(e)27 b(call)f(this)g(pattern)g
Fp(parl)p Fq(.)35 b(First,)26 b(w)n(e)g(de\014ne)g(a)g(help)r(er)515
4353 y(function,)i Fp(halveList)p Fq(,)c(whic)n(h)k(divides)f(a)g(list)
h(in)g(t)n(w)n(o.)602 4539 y Fp(halveList)40 b(inps)i(=)h
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b(take)f(half)g(inps)776 4838 y(right)g(=)h(drop)f(half)g(inps)776
4937 y(half)86 b(=)43 b(length)e(inps)h(`div`)f(2)1905
5255 y Fq(46)p eop
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47 47 bop 1808 1272 56 4 v 1808 1161 V 1808 1051 V 1808
940 V 1808 829 V 1808 718 V 1808 608 V 1808 497 V 1861
1132 173 4 v 1861 1298 4 167 v 1926 1225 a Fj(g)p 2030
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634 V 1861 637 173 4 v 2030 1272 56 4 v 2030 1161 V 2030
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b(g)602 1776 y(Main>)f Fl(simulate)g(halveList)f([high,low,high,)o(low)
o(])602 1875 y Fp(\([high,low],[hig)o(h,)o(low)o(]\))515
2078 y Fq(Then,)i(w)n(e)d(de\014ne)h(the)g(circuit)f
Fp(append)p Fq(,)h(whic)n(h)f(tak)n(es)g(a)g(pair)g(of)g(lists)g(of)h
(length)f Fj(m)515 2178 y Fq(and)29 b Fj(n)p Fq(,)g(and)g(joins)g(them)
g(together)f(\(or)h Fn(c)l(onc)l(atenates)g Fq(them\),)h(to)f(giv)n(e)f
(a)h(list)g(of)g(length)515 2277 y Fj(m)11 b Fq(+)g Fj(n)p
Fq(.)36 b(This)24 b(circuit)f(is)h(de\014ned)h(in)f(terms)g(of)g(Hask)n
(ell's)f(built-in)i(in\014x)f(list)g(concatenate)515
2377 y(op)r(erator)i(\()p Fp(++)p Fq(\).)602 2563 y Fp(append)41
b(\(a,b\))h(=)h(a)g(++)g(b)515 2749 y Fq(Lastly)-7 b(,)27
b(w)n(e)g(de\014ne)h Fp(parl)p Fq(:)602 2935 y Fp(parl)42
b(circ1)f(circ2)h(=)689 3034 y(halveList)e(->-)i(\(circ1)f(-|-)i
(circ2\))e(->-)h(append)602 3237 y(Main>)f Fl(simulate)g(\(parl)g
(reverse)g(id\))h([1..16])602 3337 y Fp([8,7,6,5,4,3,2,1)o(,9)o(,10)o
(,1)o(1,1)o(2,)o(13)o(,14)o(,1)o(5,1)o(6])515 3539 y
Fq(Sometimes,)26 b(w)n(e)f(w)n(an)n(t)g(to)g(p)r(erform)g(an)h(op)r
(eration)e(of)i(eac)n(h)e(elemen)n(t)i(of)g(a)f(list)h(of)f(signals)515
3639 y(or)31 b(bus.)50 b(F)-7 b(or)31 b(this)h(w)n(e)g(use)g(the)g
(connection)g(pattern)f Fp(map)p Fq(,)h(whic)n(h)g(y)n(ou)f(will)i(ha)n
(v)n(e)d(seen)515 3738 y(if)36 b(y)n(ou)f(ha)n(v)n(e)f(used)i(a)f
(functional)h(programming)d(language.)60 b(F)-7 b(or)35
b(example,)i Fp(map)42 b(inv)515 3838 y Fq(in)n(v)n(erts)26
b(eac)n(h)h(of)h(a)f(list)h(of)f(bits.)602 4024 y Fp(Main>)41
b Fl(simulate)g(\(map)h(inv\))g([high,)f(low,)h(high,)f(low])602
4124 y Fp([low,high,low,hi)o(gh)o(])515 4310 y Fq(Buses)d(need)g(not)h
(con)n(tain)e(only)h(lists)h(of)f(bits.)70 b(They)38
b(can)g(b)r(e)h(more)f(structured,)j(so)515 4409 y(that)29
b(our)f(circuit)h(descriptions)f(can)h(matc)n(h)g(the)g
Fn(lo)l(gic)l(al)i Fq(structure)d(of)h(the)h(circuit.)41
b(F)-7 b(or)515 4509 y(example,)27 b(the)h(circuit)f
Fp(map)43 b(fullAdd)25 b Fq(mak)n(es)h(p)r(erfect)i(sense.)602
4695 y Fp(Main>)41 b Fl(simulate)g(\(map)h(fullAdd\))951
4794 y([\(low,\(high,low)o(\)\),)o(\(h)o(ig)o(h,\()o(hi)o(gh,)o(hi)o
(gh\))o(\),)o(\(l)o(ow,)o(\(h)o(igh)o(,h)o(ig)o(h\)\))o(])602
4894 y Fp([\(high,low\),\(hig)o(h,)o(hig)o(h\))o(,\(l)o(ow)o(,h)o(igh)o
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48 48 bop 1382 1383 1130 4 v 1759 1100 753 4 v 2135 818
377 4 v 1382 1100 95 4 v 1665 1100 V 1473 1005 195 4
v 1473 1193 4 189 v 1546 1119 a Fj(f)p 1665 1193 V 1473
1196 195 4 v 1382 818 95 4 v 1665 818 V 1473 722 195
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628 4 189 v 1546 555 a(f)p 1665 628 V 1473 631 195 4
v 1759 818 95 4 v 2041 818 V 1849 722 195 4 v 1849 910
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536 95 4 v 2041 536 V 1849 440 195 4 v 1849 628 4 189
v 1922 555 a(f)p 2041 628 V 1849 631 195 4 v 2135 536
95 4 v 2417 536 V 2226 440 195 4 v 2226 628 4 189 v 335
w(f)p 2417 628 V 2226 631 195 4 v 1624 1658 a Fq(Figure)27
b(8.3:)36 b Fp(tri)42 b(f)515 1921 y Fq(Figure)22 b(8.2)g(sho)n(ws)g(a)
h Fp(map)e Fq(in)j(the)f(case)f(where)g(the)i(input)g(is)e(a)h(4-list)f
(\(of)i(pairs)d(or)i(2-lists\).)515 2060 y(Strangely)34
b(enough,)j(the)f(connection)g(pattern)f(that)h(places)f(zero)f(copies)
h(of)h(a)f(circuit)515 2160 y(on)29 b(the)i(\014rst)e(signal)g(in)i(a)e
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(next,)h(and)f(so)f(on,)515 2260 y(is)i(one)f(that)i(arises)e(often)h
(in)g(hardw)n(are)e(design.)48 b(It)31 b(is)g(a)g(sort)f(of)h(mixture)g
(of)g Fp(map)f Fq(and)515 2359 y Fp(composeN)p Fq(.)g(W)-7
b(e)33 b(call)g(it)g Fp(tri)f Fq(for)h Fn(triangle)p
Fq(.)54 b(Y)-7 b(ou)33 b(should)g(understand)g(wh)n(y)g(when)g(y)n(ou)
515 2459 y(lo)r(ok)25 b(at)g(the)h(diagram)e(in)i(\014gure)f(8.3.)35
b(W)-7 b(e)26 b(lea)n(v)n(e)f(the)h(de\014nition)g(of)f
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(use)f(of)h(triangle)e(is)602 2783 y Fp(Main>)41 b Fl(simulate)g(\(tri)
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(,9])515 3066 y Fq(The)24 b(connection)g(patterns)g(that)g(w)n(e)g(ha)n
(v)n(e)f(seen)h(in)h(this)f(section)g(are)f(all)h(useful)h(in)g(man)n
(y)515 3166 y(di\013eren)n(t)38 b(kinds)g(of)g(circuits.)68
b(No)n(w)37 b(let)i(us)f(consider)e(ho)n(w)i(to)g(describ)r(e)f(tree)h
(shap)r(ed)515 3266 y(circuits.)515 3580 y Fm(8.2)135
b(T)-11 b(ree)45 b(shap)t(ed)f(circuits)515 3801 y Fq(Circuits)27
b(in)h(the)g(shap)r(e)f(of)g(trees,)g(lik)n(e)g(that)h(sho)n(wn)f(in)h
(\014gure)e(8.4,)h(can)g(b)r(e)h(used)g(to)f(sys-)515
3901 y(tematically)g(apply)g(a)g(function)g(that)h(com)n(bines)f(data)f
(v)-5 b(alues)27 b(together)g(to)g(a)g(collection)515
4000 y(of)e(data.)36 b(A)26 b(binary)f(tree)g(circuit)h(\014rst)f(com)n
(bines)g(eac)n(h)g(half)h(of)f(the)h(input)h(v)-5 b(alues,)25
b(using)515 4100 y(t)n(w)n(o)31 b(smaller)g(trees)g(and)h(then)h(com)n
(bines)e(the)h(t)n(w)n(o)f(remaining)g(results.)50 b(One)31
b(example)515 4200 y(of)c(suc)n(h)h(a)f(circuit)g(is)h(an)f(adder)g
(tree)g(that)h(adds)f(up)h(a)f(list)h(of)g(n)n(umhers.)515
4339 y(The)f(outline)h(of)g(the)g(recursiv)n(e)e(de\014nition)i(of)f(a)
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b(circ)g([inp])f(=)i(...)g(inp)f(...)602 4623 y(tree)g(circ)g(inps)85
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(inps)515 4807 y Fq(W)-7 b(e)26 b(call)f(the)g(parameter)f
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4907 y(de\014nes)c(what)g(should)g(b)r(e)h(done)f(when)g(w)n(e)g(get)g
(do)n(wn)f(to)h(the)h(base)f(case)f(of)h(the)h(recursion.)515
5006 y(The)29 b(second)g(line)h(should)g(use)f(t)n(w)n(o)g(copies)g(of)
g Fp(tree)42 b(circ)28 b Fq(and)i(com)n(bine)f(their)h(results)1905
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584 V 1947 523 a(6)1350 1774 y Fq(Figure)27 b(8.4:)36
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(should)h(lo)r(ok)f(dep)r(ends)h(partly)f(on)g(what)515
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(in)h(particular)f(on)g(its)h(t)n(yp)r(e.)515 2275 y(F)-7
b(or)27 b(example,)h(if)h Fp(circ)d Fq(is)i(a)g(binary)f(function)i
(taking)e(a)h(pair)f(of)h(inputs)h(and)f(returning)515
2374 y(a)c(single)g(output,)h(then)g(it)g(mak)n(es)f(sense)f(to)i(mak)n
(e)e(the)i(follo)n(wing)f(de\014nition)g(of)h(a)f(binary)515
2474 y(tree)j(connection)g(pattern,)h Fp(binTree)p Fq(.)602
2657 y Fp(binTree)41 b(circ)h([inp])f(=)i(inp)602 2756
y(binTree)e(circ)h(inps)85 b(=)689 2856 y(\(halveList)40
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(inps)515 3039 y Fq(This)23 b(giv)n(es)g(the)h(b)r(eha)n(viour)f(that)h
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Fp(circ)e Fq(comp)r(onen)n(ts)i(gets)515 3138 y(built.)515
3278 y(An)i(example)g(use)g(of)h(a)e(tree)h(connection)g(pattern)g(is)g
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3377 y(that)f(adds)g(up)h(a)e(lot)i(of)f(n)n(um)n(b)r(ers.)35
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(so-called)e Fn(adder)515 3477 y(tr)l(e)l(e)p Fq(.)42
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4706 y Fq(T)-7 b(o)27 b(test)h(it,)g(w)n(e)f(wrap)g(the)h(circuit)f(in)
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602 4888 y Fp(wrapAddTree)39 b(n)k(=)689 4988 y(map)f(\(int2bin)f(n\))h
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(length)g(is)g(a)f(p)r(o)n(w)n(er)g(of)515 925 y(t)n(w)n(o.)j(Exercise)
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(same)g(recursiv)n(e)e(structure.)53 b(It)34 b(turns)f(out)g(that)g
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f(can)h(b)r(e)g(recursiv)n(ely)e(describ)r(ed)515 1959
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2059 y(more)f(w)n(a)n(ys)f(to)h(describ)r(e)g(the)h(same)f(net)n(w)n
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2198 y(Butter\015y)c(circuits)g(are)g(used)g(for)g(example)g(to)g
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2298 y(and)38 b(in)h(building)f(e\016cien)n(t)h(sorting)e(circuits.)69
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2397 y(lik)n(e)29 b(circuit)h(is)g(the)h(standard)e(Co)r(oley-T)-7
b(uk)n(ey)28 b(algorithm)h([4)o(])h(for)g(computing)g(the)g(F)-7
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3135 y(that)41 b(butter\015y)g(circuits)g(can)f(b)r(e)i(made)f(with)g
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3234 y(comp)r(osition.)515 3374 y(The)21 b(\014rst)f(of)h(these)g
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3958 y Fp(two)42 b(circ)g(=)h(parl)f(circ)g(circ)602
4161 y(Main>)f Fl(simulate)g(\(two)h(reverse\))e([1..16])602
4260 y Fp([8,7,6,5,4,3,2,1)o(,1)o(6,1)o(5,)o(14,)o(13)o(,1)o(2,1)o(1,)o
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b Fp(two)515 4862 y(f)21 b Fq(applies)g Fp(f)f Fq(to)i(the)f(top)g(and)
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4961 y(ev)n(en)h(elemen)n(ts.)35 b(W)-7 b(e)23 b(de\014ne)h(it)f(in)g
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3223 y(elemen)n(ts.)36 b(W)-7 b(e)28 b(use)g(this)g(fact)g(to)f
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2699 y(a)d(larger)f(comp)r(onen)n(t:)602 2885 y Fp(bfly2)41
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3469 y Fq(The)d(surprising)f(thing)h(is)g(that)g(all)g(of)g(these)g
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3054 y(the)34 b(circuit)g(that)g(w)n(e)f(call)h Fp(s2)p
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eop
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b(error:)g(INTERNAL)g(ERROR)g(...)722 4589 y Fq(Oops!)34
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(that)h(w)n(e)f(can)g(\014x)h(it.)515 4890 y(If)f(y)n(ou)e(ha)n(v)n(e)h
(some)f(t)n(ypical)h(error)f(that)i(y)n(ou)e(w)n(ould)h(ha)n(v)n(e)f
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y(e-mail)h(us)g(so)g(that)h(w)n(e)f(can)h(mak)n(e)e(this)i(list)g(more)
f(complete.)1905 5255 y(78)p eop
%%Page: 79 80
79 79 bop 515 1186 a Fo(App)5 b(endix)65 b(B)515 1641
y Fs(Answ)-6 b(ers)575 2105 y Fq(2.1)40 b(Here)28 b(is)f(ho)n(w)g(w)n
(e)g(de\014ne)h Fp(swap)e Fq(and)i Fp(copy)p Fq(:)809
2305 y Fp(swap)42 b(\(a,)h(b\))f(=)i(\(b,)e(a\))809 2404
y(copy)g(a)261 b(=)44 b(\(a,)e(a\))575 2604 y Fq(2.2)e(W)-7
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Fq(in)28 b(the)g(follo)n(wing)f(w)n(a)n(y:)809 2803 y
Fp(twoBitSort)40 b(\(a,)i(b\))h(=)g(\(min,)f(max\))897
2902 y(where)984 3002 y(min)g(=)h(and2)f(\(a,)h(b\))984
3102 y(max)f(=)h(or2)86 b(\(a,)43 b(b\))575 3301 y Fq(2.3)d(Here)28
b(is)f(the)h(constan)n(t)f Fp(alwaysHigh)c Fq(circuit:)809
3500 y Fp(alwaysHigh)40 b(\(\))j(=)g(high)575 3699 y
Fq(2.4)d(One)28 b(could)f(de\014ne)h(a)f Fp(multiplexer)c
Fq(as)k(follo)n(ws:)809 3899 y Fp(multiplexer)39 b(\(c,\(x,y\)\))h(=)k
(out)897 3998 y(where)984 4098 y(out)129 b(=)43 b(or2)86
b(\(left,)42 b(right\))984 4198 y(left)85 b(=)43 b(and2)f(\(inv)g(c,)h
(x\))984 4297 y(right)e(=)i(and2)f(\(c,)h(y\))722 4496
y Fq(There)34 b(is)h(a)f(built-in)h(m)n(ultiplexer)g(in)g(La)n(v)-5
b(a,)35 b(called)f Fp(mux)p Fq(.)57 b(Using)34 b(that)h(one,)h(w)n(e)
722 4596 y(could)28 b(de\014ne:)809 4795 y Fp(multiplexer')39
b(\(c,\(x,y\)\))h(=)j(mux)f(\(c,\(x,)g(y\)\))575 4995
y Fq(2.5)e(A)28 b Fp(threeBitAdder)23 b Fq(can)k(b)r(e)h(de\014ned)g
(as)f(follo)n(ws:)1905 5255 y(79)p eop
%%Page: 80 81
80 80 bop 809 523 a Fp(threeBitAdder)39 b(\(carryIn,)h(\(\(a1,b1,c1\),)
e(\(a2,b2,c2\)\)\))h(=)897 623 y(\(\(a3,)i(b3,)h(c3\),)g(carryOut\))853
722 y(where)897 822 y(\(a3,)g(carryA\))127 b(=)44 b(fullAdd)c
(\(carryIn,)g(\(a1,)i(a2\)\))897 922 y(\(b3,)g(carryB\))127
b(=)44 b(fullAdd)c(\(carryA,)84 b(\(b1,)42 b(b2\)\))897
1021 y(\(c3,)g(carryOut\))d(=)44 b(fullAdd)c(\(carryB,)84
b(\(c1,)42 b(c2\)\))575 1220 y Fq(3.2)e(W)-7 b(e)28 b(can)f(mak)n(e)g
(use)h(of)f(the)h(adder)f(w)n(e)g(already)f(ha)n(v)n(e:)809
1420 y Fp(adder2)42 b(\(as,)f(bs\))i(=)g(cs)897 1519
y(where)984 1619 y(\(cs,)f(carryOut\))e(=)j(adder)e(\(low,)h(\(as,)g
(bs\)\))575 1851 y Fq(3.3)e(The)c Fp(adder)e Fq(circuit)h(tak)n(es)g
(as)g(an)g(input)i(a)e(pair)g(of)h(lists)g(of)f(bits,)j(whereas)d(the)
722 1951 y Fp(adder')25 b Fq(circuit)j(gets)f(a)g(list)h(of)g(pairs)e
(of)i(bits.)575 2117 y(3.4)40 b(Here)28 b(is)f(a)g(binary)g(n)n(um)n(b)
r(er)g(to)h(in)n(teger)f(con)n(v)n(erter)e Fp(bin2int)p
Fq(:)809 2316 y Fp(bin2int)41 b([])217 b(=)43 b(0)809
2416 y(bin2int)e(\(b:bs\))g(=)i(num)897 2516 y(where)984
2615 y(num')f(=)h(bin2int)e(bs)984 2715 y(num)86 b(=)43
b(bit2int)e(b)i(+)g(2)g(*)g(num')575 2914 y Fq(3.5)d(Here)28
b(is)f(ho)n(w)g(w)n(e)g(can)g(de\014ne)h Fp(zipp)p Fq(:)809
3113 y Fp(zipp)42 b(\([],)129 b([]\))h(=)43 b([])809
3213 y(zipp)f(\(a:as,)f(b:bs\))h(=)h(\(a,b\))f(:)h(rest)897
3313 y(where)984 3412 y(rest)f(=)h(zipp)f(\(as,)g(bs\))722
3611 y Fq(And)28 b(here)g(is)f(ho)n(w)g(w)n(e)g(de\014ne)h
Fp(unzipp)p Fq(:)809 3811 y Fp(unzipp)42 b([])435 b(=)43
b(\([],)129 b([]\))809 3910 y(unzipp)42 b(\(\(a,b\):abs\))d(=)k
(\(a:as,)e(b:bs\))897 4010 y(where)984 4110 y(\(as,)h(bs\))g(=)h
(unzipp)e(abs)575 4309 y Fq(3.6)f(Here)28 b(is)f(ho)n(w)g(w)n(e)g(can)g
(de\014ne)h Fp(pair)p Fq(:)809 4508 y Fp(pair)42 b(\(x:y:xs\))f(=)i
(\(x,y\))e(:)j(pair)d(xs)809 4608 y(pair)h(xs)305 b(=)43
b([])722 4807 y Fq(W)-7 b(e)27 b(c)n(ho)r(ose)f(to)h(ignore)e(the)i
(last)f(input)i(if)f(the)g(n)n(um)n(b)r(er)g(of)g(elemen)n(ts)f(is)h(o)
r(dd.)36 b(And)722 4907 y(here)27 b(is)h(ho)n(w)f(w)n(e)g(de\014ne)h
Fp(unpair)p Fq(:)1905 5255 y(80)p eop
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81 81 bop 809 523 a Fp(unpair)42 b(\(\(x,y\):xys\))d(=)k(x)g(:)g(y)g(:)
g(unpair)f(xys)809 623 y(unpair)g([])435 b(=)43 b([])575
821 y Fq(3.7)d(This)28 b(is)f(ho)n(w)g(w)n(e)h(can)f(de\014ne)h
(parallel)e(comp)r(osition)h(of)h(circuits)f Fp(par)p
Fq(:)809 1019 y Fp(par)43 b(circ1)e(circ2)h(\(a,)g(b\))h(=)g(\(c,)f
(d\))897 1118 y(where)984 1218 y(c)h(=)g(circ1)f(a)984
1317 y(d)h(=)g(circ2)f(b)575 1515 y Fq(3.9)e(Here)28
b(is)f(ho)n(w)g(w)n(e)g(can)g(de\014ne)h Fp(column)p
Fq(:)809 1713 y Fp(column)42 b(circ)f(\([],)h(carryIn\))128
b(=)43 b(\(carryIn,)d([]\))809 1913 y(column)i(circ)f(\(a:as,)h
(carryIn\))e(=)j(\(carryOut,)c(b:bs\))897 2012 y(where)984
2112 y(\(carry,)h(b\))218 b(=)43 b(circ)f(\(a,)g(carryIn\))984
2211 y(\(carryOut,)d(bs\))k(=)g(column)e(circ)h(\(as,)g(carry\))722
2409 y Fq(Here)37 b(is)g(ho)n(w)g(w)n(e)g(can)g(de\014ne)h
Fp(column)d Fq(in)i(terms)g(of)g Fp(row)p Fq(.)65 b(First,)40
b(w)n(e)d(de\014ne)g(a)722 2509 y(connection)27 b(pattern)f(called)h
Fp(mirror)p Fq(,)e(whic)n(h)h(sw)n(aps)g(the)h(left)h(and)f(righ)n(t)f
(parts)g(of)722 2609 y(input)j(and)e(output:)809 2807
y Fp(mirror)42 b(circ)f(\(a,)i(b\))g(=)g(\(c,)f(d\))897
2906 y(where)984 3006 y(\(d,)g(c\))h(=)g(circ)f(\(b,)g(a\))722
3204 y Fq(And)28 b(then,)h(w)n(e)e(use)g Fp(row)g Fq(and)g(mirror)f
(the)i(input)h(to)e Fp(row)p Fq(:)809 3402 y Fp(column)42
b(circ)f(\(as,)h(carryIn\))f(=)i(\(carryOut,)c(bs\))897
3501 y(where)984 3601 y(\(bs,)j(carryOut\))e(=)j(row)f(\(mirror)f
(circ\))g(\(carryIn,)f(as\))722 3799 y Fq(W)-7 b(e)28
b(could)g(ev)n(en)f(sa)n(y:)809 3997 y Fp(column)42 b(circ)f(=)j
(mirror)d(\(row)h(\(mirror)e(circ\)\))533 4195 y Fq(3.10)g(W)-7
b(e)28 b(could)g(de\014ne)g Fp(grid)e Fq(as:)809 4393
y Fp(grid)42 b(circ)g(\(as,)g(bs\))h(=)g(\(cs,)f(ds\))897
4492 y(where)984 4592 y(\(cs,)g(ds\))g(=)h(row)g(\(column)d(circ\))i
(\(as,)g(bs\))722 4790 y Fq(Or,)27 b(ev)n(en)g(shorter:)809
4988 y Fp(grid)42 b(circ)g(=)h(row)g(\(column)e(circ\))1905
5255 y Fq(81)p eop
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82 82 bop 533 523 a Fq(3.13)40 b(Here)28 b(is)f(ho)n(w)g(w)n(e)g
(de\014ne)h(a)f Fp(swapper)p Fq(:)809 722 y Fp(swapper)41
b(\(swap,)g(\(a,)i(b\)\))f(=)h(\(x,)f(y\))897 822 y(where)984
922 y(\(x,)g(y\))h(=)g(mux)f(\(swap,)g(\(\(a,)f(b\),)i(\(b,)f(a\)\)\))
575 1121 y Fq(4.1)e(The)28 b(\014rst)f(prop)r(ert)n(y)g(can)g(b)r(e)h
(de\014ned)g(as:)809 1320 y Fp(prop_SorterHasSor)o(ted)o(Ou)o(tpu)o(t)
37 b(\(a,)43 b(b\))f(=)i(ok)897 1420 y(where)984 1519
y(\(x,)e(y\))h(=)g(twoBitSort)d(\(a,)i(b\))984 1619 y(ok)217
b(=)43 b(or2)f(\(inv)g(x,)h(y\))86 b(--)43 b Fl(x)g(<=)g(y)722
1818 y Fq(The)28 b(second)f(prop)r(ert)n(y)f(can)i(b)r(e)g(stated)f
(as:)809 2017 y Fp(prop_SorterHasSam)o(eBi)o(ts)37 b(\(a,)42
b(b\))h(=)g(ok)897 2117 y(where)984 2217 y(\(x,)f(y\))86
b(=)44 b(twoBitSort)39 b(\(a,)j(b\))984 2316 y(same)172
b(=)44 b(\(a,)e(b\))h(<==>)f(\(x,)g(y\))984 2416 y(swapped)e(=)k(\(a,)e
(b\))h(<==>)f(\(y,)g(x\))984 2516 y(ok)260 b(=)44 b(or2)e(\(same,)f
(swapped\))575 2715 y Fq(4.4)f(T)-7 b(o)28 b(c)n(hec)n(k)e(that)i(the)g
(subtractor)e(really)h(subtracts,)g(w)n(e)g(can)g(de\014ne:)809
2914 y Fp(prop_SubtractorSu)o(btr)o(ac)o(ts)37 b(\(as,)42
b(bs\))g(=)i(ok)897 3014 y(where)984 3113 y(cs)86 b(=)43
b(subtractor)d(\(as,)i(bs\))984 3213 y(as')g(=)h(adder2)e(\(cs,)h(bs\))
984 3313 y(ok)86 b(=)43 b(as)g(<==>)f(as')575 3512 y
Fq(4.5)e(Here)28 b(is)f(the)h(general)e(prop)r(ert)n(y)h(of)g(asso)r
(ciativit)n(y:)809 3711 y Fp(prop_Associative)37 b(circ)42
b(\(as,)g(bs,)h(cs\))f(=)h(ok)897 3811 y(where)984 3910
y(out1)f(=)h(circ)f(\(as,)g(circ)g(\(bs,)g(cs\)\))984
4010 y(out2)g(=)h(circ)f(\(circ)f(\(as,)h(bs\),)g(cs\))984
4110 y(ok)130 b(=)43 b(out1)f(<==>)g(out2)575 4309 y
Fq(4.8)e(One)28 b(can)f(de\014ne)h(a)f(general)f(v)n(erify)h(function,)
h(as)f(follo)n(ws:)809 4508 y Fp(verifyFor)40 b(prop)i(ns)h(=)g
(sequence)d([)k(prop)e(n)h(|)g(n)g(<-)g(ns)g(])722 4707
y Fq(and)28 b(v)n(erify)f(a)g(prop)r(ert)n(y)f(b)n(y)i(sa)n(ying)e(for)
h(example:)809 4907 y Fp(Main>)42 b Fl(verifyFor)e(prop_AdderCommut)o
(at)o(ive)o(_F)o(or)o(Siz)o(e)d([1..32])809 5006 y Fp(...)1905
5255 y Fq(82)p eop
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83 83 bop 575 523 a Fq(5.1)40 b(W)-7 b(e)28 b(can)f(de\014ne)h
Fp(evenSoFar)c Fq(as)j(follo)n(ws:)809 694 y Fp(evenSoFar)40
b(inp)j(=)g(out)897 794 y(where)984 893 y(out)86 b(=)43
b(delay)e(high)h(even)984 993 y(even)g(=)h(xor2)f(\(inp,)f(even\))722
1164 y Fq(This)28 b(is)f Fn(almost)i Fq(the)f(same)f(as)f(the)i
Fp(toggle)e Fq(circuit.)575 1319 y(5.2)40 b(W)-7 b(e)28
b(can)f(de\014ne)h Fp(flipFlop)d Fq(as)i(follo)n(ws:)809
1490 y Fp(flipFlop)41 b(\(set,)g(reset\))g(=)j(state)897
1589 y(where)984 1689 y(state')d(=)i(delay)f(low)g(state)984
1789 y(state)85 b(=)43 b(and2)f(\(up,)g(inv)g(reset\))984
1888 y(up)217 b(=)43 b(or2)f(\(state',)f(set\))575 2059
y Fq(5.3)f(W)-7 b(e)28 b(can)f(de\014ne)h Fp(delayClk)d
Fq(as)i(follo)n(ws:)809 2230 y Fp(delayClk)41 b(init)h(\(clk,)f(inp\))h
(=)h(out)897 2330 y(where)984 2429 y(out)f(=)h(delay)f(init)g(val)984
2529 y(val)g(=)h(mux)g(\(clk,)e(\(out,)h(inp\)\))575
2700 y Fq(5.4)e(The)28 b(circuit)f Fp(always)f Fq(can)h(b)r(e)h
(de\014ned)g(as)f(follo)n(ws:)809 2871 y Fp(always)42
b(inp)g(=)h(ok)897 2971 y(where)984 3070 y(sofar)e(=)i(delay)f(high)g
(ok)984 3170 y(ok)173 b(=)43 b(and2)f(\(inp,)g(sofar\))575
3341 y Fq(5.5)e(The)28 b(circuits)f(are:)809 3512 y Fp(pulsSix6)41
b(\(\))h(=)i(out)897 3611 y(where)984 3711 y(out)e(=)h(puls)f(6)h(\(\))
741 b(--)42 b Fl(000001...)809 3910 y Fp(pulsSix5)f(\(\))h(=)i(out)897
4010 y(where)984 4110 y(a)130 b(=)43 b(puls)f(2)h(\(\))741
b(--)42 b Fl(010101...)984 4209 y Fp(b)130 b(=)43 b(puls)f(3)h(\(\))741
b(--)42 b Fl(001001...)984 4309 y Fp(out)g(=)h(and2)f(\(a,)h(b\))809
4508 y(pulsSix3)e(\(\))h(=)i(out)897 4608 y(where)984
4707 y(a)130 b(=)43 b(delay)f(low)g(\(inv)g(a\))392 b(--)42
b Fl(010101...)984 4807 y Fp(b)130 b(=)43 b(delay)f(low)g(\(xor2)g
(\(b,)g(c\)\))130 b(--)42 b Fl(001001...)984 4907 y Fp(c)130
b(=)43 b(delay)f(low)g(\(nand2)f(\(b,)i(c\)\))86 b(--)42
b Fl(011011...)984 5006 y Fp(out)g(=)h(and2)f(\(a,)h(b\))1905
5255 y Fq(83)p eop
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84 84 bop 575 523 a Fq(5.6)40 b(Using)28 b(a)f(coun)n(ter,)g(w)n(e)g
(can)g(de\014ne)h Fp(puls2)e Fq(as)g(follo)n(ws:)809
722 y Fp(puls2)42 b(k)h(\(\))g(=)g(out)897 822 y(where)984
922 y(number)e(=)i(counter)e(k)i(\(\))984 1021 y(out)173
b(=)43 b(norl)f(number)575 1220 y Fq(5.7)e(W)-7 b(e)28
b(can)f(de\014ne)h(the)g(circuit)g Fp(counterUpDown)22
b Fq(as)27 b(follo)n(ws:)809 1420 y Fp(counterUpDown)39
b(n)k(\(up,)f(down\))f(=)i(number)897 1519 y(where)940
1619 y(number')e(=)i(delay)f(\(zeroList)d(n\))k(number)940
1719 y(number)85 b(=)43 b(adder2)e(\(diff,)g(number'\))940
1818 y(diff)173 b(=)43 b(one)f(:)i(replicate)39 b(\(n-1\))j(rest)940
1918 y(one)217 b(=)43 b(or2)f(\(up,)g(down\))303 b(--)43
b Fl(should)e(I)i(change?)940 2017 y Fp(rest)173 b(=)43
b(and2)f(\(inv)g(up,)g(down\))85 b(--)43 b Fl(+1)g(or)g(-1?)575
2217 y Fq(5.9)d(Here)28 b(is)f(ho)n(w)g(w)n(e)g(could)h(de\014ne)f
Fp(synchronize)p Fq(:)809 2416 y Fp(synchronize)39 b(\(go1,)j(go2\))g
(=)h(go)897 2516 y(where)984 2615 y(both)f(=)h(and2)f(\(go1,)f(go2\))
984 2715 y(one)86 b(=)43 b(xor2)f(\(go1,)f(go2\))984
2814 y(wait)h(=)h(delay)e(low)i(\(xor2)e(\(one,)h(wait\)\))984
2914 y(go)130 b(=)43 b(or2)f(\(both,)f(and2)h(\(wait,)f(one\)\))533
3113 y Fq(5.10)f(First,)25 b(w)n(e)g(de\014ne)f(the)h(follo)n(wing)f
(help)r(er)g(circuit)h Fp(outputDone)p Fq(.)32 b(It)25
b(do)r(es)f(the)h(same)722 3213 y(as)g(output,)i(but)f(tak)n(es)e(an)h
(extra)g(parameter)f Fp(done)p Fq(,)g(the)i(signal)f(to)g(output)h(at)g
(the)722 3313 y(time)i(when)g(the)g(list)g(is)f(empt)n(y)-7
b(.)809 3512 y Fp(outputDone)40 b([])j(done)f(\(\))g(=)i(done)809
3711 y(outputDone)c(\(sig:sigs\))f(done)j(\(\))h(=)g(out)897
3811 y(where)984 3910 y(out)86 b(=)43 b(delay)e(sig)i(rest)984
4010 y(rest)f(=)h(outputDone)c(sigs)j(done)g(\(\))722
4209 y Fq(No)n(w,)27 b(w)n(e)h(can)f(de\014ne)h(the)g(circuit)f
Fp(outputList)d Fq(as)j(follo)n(ws:)809 4408 y Fp(outputList)40
b(sigs)i(\(\))h(=)g(out)897 4508 y(where)984 4608 y(out)f(=)h
(outputDone)d(sigs)i(out)g(\(\))575 4807 y Fq(6.2)e(Here)28
b(is)f(a)g(prop)r(ert)n(y)g(that)h(c)n(hec)n(ks)e(that:)1905
5255 y(84)p eop
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85 85 bop 809 523 a Fp(prop_Edge_vs_Even)37 b(inp)42
b(=)i(ok)897 623 y(where)984 722 y(out1)e(=)h(edge)f(inp)984
822 y(out2)g(=)h(evenSoFar)d(inp)984 922 y(ok)130 b(=)43
b(inv)f(\(out1)g(<==>)g(out2\))575 1089 y Fq(6.3)e(Here)28
b(is)f(a)g(prop)r(ert)n(y)g(that)h(c)n(hec)n(ks)e(if)i(they)g(are)f
(equiv)-5 b(alen)n(t:)809 1257 y Fp(prop_PulsSixEquiv)o(ale)o(nt)37
b(\(\))43 b(=)g(ok)897 1356 y(where)984 1456 y(out3)f(=)h(pulsSix3)d
(\(\))984 1556 y(out5)i(=)h(pulsSix5)d(\(\))984 1655
y(out6)i(=)h(pulsSix6)d(\(\))984 1855 y(ok35)i(=)h(out3)f(<==>)g(out5)
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Fq(,)d(75)2092 1414 y Fp(bin2int)p Fq(,)f(76,)i(80)2092
1513 y Fp(binAdder)p Fq(,)e(49,)h(76)2092 1613 y Fp(binTree)p
Fq(,)f(49)2092 1713 y Fp(bit2int)p Fq(,)g(73)2092 1812
y Fp(bitAdder)p Fq(,)g(12,)h(76)2092 1912 y Fp(bitMulti)p
Fq(,)f(76)2092 2011 y(bitonic,)j(55)2092 2111 y Fp(bitSubber)p
Fq(,)c(16)2092 2211 y(butter\015y)-7 b(,)28 b(45)2092
2393 y Fp(column)p Fq(,)d(75,)i(81)2092 2493 y(command-line)g(options,)
g(72)2092 2593 y(comm)n(utativit)n(y)-7 b(,)27 b(19)2092
2692 y(compilation,)g(16)2092 2792 y Fp(compose)p Fq(,)e(45,)i(75)2092
2891 y Fp(composeN)p Fq(,)e(46,)h(75)2092 2991 y(comp)r(osition)2258
3091 y(parallel,)h(46,)f(81)2258 3190 y(serial,)h(13,)f(45)2092
3290 y(connection)h(pattern,)h(13)2258 3390 y(prop)r(erties,)f(20)2258
3489 y(sequen)n(tial,)g(26)2092 3589 y Fp(copy)p Fq(,)f(74,)h(79)2092
3688 y(coun)n(ter,)g(25)2092 3788 y Fp(counterUpDown)p
Fq(,)c(84)2092 3971 y Fp(delay)p Fq(,)j(74)2092 4070
y Fp(delayClk)p Fq(,)f(76,)h(83)2092 4170 y Fp(delayN)p
Fq(,)f(24,)i(76)2092 4270 y Fp(Depth)p Fq(,)f(34)2092
4452 y Fp(edge)p Fq(,)g(23,)h(76)2092 4552 y Fp(equal)p
Fq(,)f(19,)h(74)2092 4652 y(equalit)n(y)-7 b(,)27 b(19)2092
4751 y Fp(equiv)p Fq(,)f(73)2092 4851 y(error)g(messages,)g(77)2092
4950 y Fp(evenSoFar)p Fq(,)e(83)1905 5255 y(90)p eop
%%Page: 91 92
91 91 bop 515 523 a Fp(fixit)p Fq(,)25 b(77)515 623 y
Fp(flipFlop)p Fq(,)f(83)515 722 y Fp(fullAdd)p Fq(,)h(8,)i(75)515
902 y(garbage)e(collection,)i(77)515 1001 y Fp(grid)p
Fq(,)f(75,)h(81)515 1101 y Fp(gte)p Fq(,)f(74)515 1280
y Fp(halfAdd)p Fq(,)f(6,)i(75)515 1380 y Fp(halveList)p
Fq(,)d(46,)j(75)515 1479 y Fp(high)p Fq(,)f(72)515 1579
y(Hugs,)h(7)515 1758 y Fp(id)p Fq(,)g(46,)f(72,)h(73)515
1858 y Fp(idiv)p Fq(,)f(73)515 1958 y Fp(ilv)p Fq(,)g(75)515
2057 y Fp(ilvN)p Fq(,)g(75)515 2157 y Fp(ilvTree)p Fq(,)f(53)515
2257 y Fp(imax)p Fq(,)h(73)515 2356 y Fp(imin)p Fq(,)g(73)515
2456 y Fp(imod)p Fq(,)g(73)515 2555 y Fp(impl)p Fq(,)g(30,)h(73)515
2655 y(implication,)g(30)515 2755 y Fp(inc)p Fq(,)f(46)515
2854 y Fp(Increasing)p Fq(,)d(34)515 2954 y(induction,)28
b(30,)f(31)681 3054 y(restricted)g(states,)g(35)681 3153
y(with)h(depth,)g(32)515 3253 y Fp(int2bin)p Fq(,)d(15,)h(76)515
3352 y Fp(int2bit)p Fq(,)f(15,)h(73)515 3452 y Fp(inv)p
Fq(,)g(72)515 3552 y Fp(iter)p Fq(,)g(75)515 3731 y(La)n(v)-5
b(a)26 b(in)n(terpreter,)h(7)515 3831 y Fp(lava.vhd)p
Fq(,)d(9)515 3930 y Fp(length)p Fq(,)h(43)515 4030 y(list,)j(12)515
4130 y Fp(listTree)p Fq(,)c(53)515 4229 y Fp(low)p Fq(,)i(72)515
4408 y Fp(map)p Fq(,)g(47)515 4508 y Fp(mirror)p Fq(,)f(75,)i(81)515
4608 y Fp(multi)p Fq(,)e(76)515 4707 y(m)n(ultiplexer,)i(10)515
4807 y Fp(multiplexer)p Fq(,)c(79)515 4907 y Fp(mux)p
Fq(,)j(27,)h(74,)g(79)515 5006 y Fp(mux)p Fq(,)f Fn(se)l(e)i
Fq(m)n(ultiplexer)2092 523 y Fp(nand2)p Fq(,)e(73)2092
623 y Fp(nandl)p Fq(,)g(73)2092 722 y Fp(neg)p Fq(,)h(73)2092
822 y Fp(nor2)p Fq(,)f(73)2092 922 y Fp(norl)p Fq(,)g(73)2092
1021 y Fp(numBreak)p Fq(,)f(15,)h(76)2092 1200 y Fp(or2)p
Fq(,)h(73)2092 1300 y Fp(orl)p Fq(,)g(73)2092 1400 y
Fp(outputList)p Fq(,)d(76,)j(84)2092 1579 y(P)n(ace,)f(63)2092
1679 y Fp(pair)p Fq(,)g(75,)h(80)2092 1778 y Fp(par)p
Fq(,)g(46,)f(75,)h(81)2092 1878 y Fp(parl)p Fq(,)f(47,)h(75)2092
1978 y Fp(plus)p Fq(,)f(73)2092 2077 y Fp(plusl)p Fq(,)g(74)2092
2177 y Fp(pmap)p Fq(,)g(75)2092 2276 y(program)g(error,)f(77)2092
2376 y(prop)r(erties,)i(18)2092 2476 y Fp(puls)p Fq(,)f(24,)h(76)2092
2575 y Fp(puls2)p Fq(,)f(84)2092 2755 y(quan)n(ti\014cation,)h(19)2092
2934 y(recursion,)f(12)2092 3034 y Fp(RestrictStates)p
Fq(,)c(36)2092 3133 y Fp(riffle)p Fq(,)j(74)2092 3233
y Fp(row)p Fq(,)i(14,)f(75)2092 3333 y Fp(rowSeq)p Fq(,)f(26,)i(76)2092
3432 y Fp(rowSeqPeriod)p Fq(,)c(76)2092 3532 y Fp(rowSeqReset)p
Fq(,)g(27,)k(76)2092 3711 y(safet)n(y)g(prop)r(erties,)g(18)2092
3811 y Fp(serial)p Fq(,)e(13,)i(45,)g(75)2092 3910 y(sim)n(ulate)2258
4010 y(com)n(binational,)g(7)2258 4110 y(sequen)n(tial,)g(9,)g(23)2092
4209 y Fp(simulate)p Fq(,)e(77)2092 4309 y Fp(simulateCon)p
Fq(,)e(77)2092 4408 y Fp(simulateSeq)p Fq(,)g(77)2092
4508 y(step,)28 b(induction,)g(31)2092 4608 y Fp(sub)p
Fq(,)f(73)2092 4707 y Fp(swap)p Fq(,)f(74,)h(79)2092
4807 y Fp(swapl)p Fq(,)f(74)2092 4907 y Fp(swapper)p
Fq(,)f(82)2092 5006 y Fp(synchronize)p Fq(,)e(84)1905
5255 y(91)p eop
%%Page: 92 93
92 92 bop 515 523 a Fp(test)p Fq(,)26 b(77)515 623 y
Fp(threeBitAdder)p Fq(,)c(79)515 722 y Fp(times)p Fq(,)j(73)515
822 y Fp(timesl)p Fq(,)g(74)515 922 y Fp(timeTransform)p
Fq(,)d(43)515 1021 y Fp(toggle)p Fq(,)j(23,)i(76)515
1121 y(transition)g(relation,)f(31)515 1220 y(tree)h(shap)r(ed)h
(circuits,)f(45)515 1320 y Fp(tri)p Fq(,)f(48,)h(75,)g(86)515
1420 y Fp(two)p Fq(,)f(75)515 1519 y Fp(twoBitSort)p
Fq(,)d(79)515 1619 y Fp(twoBitSortl)p Fq(,)g(56)515 1719
y Fp(twoN)p Fq(,)j(75)515 1901 y Fp(unpair)p Fq(,)f(75,)i(80)515
2001 y Fp(unriffle)p Fq(,)d(74)515 2100 y Fp(unzipp)p
Fq(,)h(74,)i(80)515 2283 y Fp(varList)p Fq(,)e(74)515
2383 y(v)n(eri\014cation,)h(30)515 2482 y(v)n(erify)681
2582 y(com)n(binational,)g(18)681 2682 y(one-b)n(y-one,)g(21)681
2781 y(options,)h(77)681 2881 y(sequen)n(tial,)g(29)515
2980 y Fp(verify)p Fq(,)e(18,)i(77)515 3080 y Fp(verifyWith)p
Fq(,)c(34,)k(77)515 3180 y(VHDL)681 3279 y(generation,)f(9)515
3462 y Fp(writeVhdl)p Fq(,)e(9,)j(77)515 3562 y Fp(writeVhdlInput)p
Fq(,)22 b(9,)27 b(77)515 3661 y Fp(writeVhdlInputOu)o(tp)o(ut)o
Fq(,)22 b(9,)27 b(77)515 3844 y Fp(xnor2)p Fq(,)e(73)515
3944 y Fp(xor2)p Fq(,)h(73)515 4043 y Fp(xorl)p Fq(,)g(73)515
4226 y Fp(zero)p Fq(,)g(26,)h(69)515 4325 y Fp(zeroList)p
Fq(,)d(25)515 4425 y Fp(zipp)p Fq(,)i(74,)h(80)1905 5255
y(92)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF