INTRODUCTION To FORMAL PROOF
> foyunok “proof, is one in vohich Abp y Abp produce
th used Jo Adlve She prdtom i
> Wee Jae oath sae stant fs Amo yc
Seypanee Stops ord. Steadons
> Whig ne _proptom sda impartant, wud Ff swe Pfr”
Js wey tonplat shal solving re, Yerwion o¥ etalon
dhen our code betome inicwvteck
> To make the itnration oo Heuwrsion to be Comrart dee
wie tad. Jo Aebvp on Jndudinve hypo ssuris
“> 1h folpS ty Jahon rah dhe hypo unas Js contin
ith Lbrabor of Hewrsion i
PF This provers i Set any a consett Proj
is rio 8 pg ne inte
dn duchiow
Auborola {i used bo coven tne rathadalogies ¢f
pets. |
Ta past voy eh,
> tanbists of o Mgpance of Suits Sop
Sa gti on
Sadan oleh
> th tha Revive proof aahich Consals of
morte 4 Poramelinintd Stxtamurts
S Thab Use Aor Abateniank itself wide awe
Volucs of is Pasamebsh,
@ scanned with OKEN Scanner @ scanned with OKEN Scanner i 2 Modus Ponens oc. a
Si: me ad4ts +d os fastest
$2 Ot, bet cot dd! Choy anther )
6a OF 21,01 Pd1,44d1 ,
BA KEP HHP HI? Oy ;
Heh (01) HH Ctheofem(t-3)
8s out
Reduckiovt To
> Tp owe one nf Suse howdibo Mano push, dren Corwerd
ol Jota nw hy pasnests Jn hath, definitions
"op 4
Napposhenis dy Definite ;
i
®O Dae a S>Sek | ee
5 Alam, at we r ;
ee ) "gor 20 a
fel fe aaspacnnadiae pe of oor
@ scanned with OKEN Scanner Pret
sen
rn es’
\rilem
SUT =U
SORT =U
S4T cu
ye nam —> False
tprdodichien,
powed
Werte [ Beachive pe
cate hoster 13
THEOREM 1b
The Stobemant of ‘Thustarn 13 woul appeat
Jn whe fouk formes
1
af - Tran | ao
ww i h eae |
a \2 a ae
1 heres o \3.9 eee a
2 mony Ae A wher Dh,
3 ch oS a? filow
4 qghaniyen H holds C fl |
THEOREM 14
Jot 1 be a Ho umber Then J} = fr]
An only of ¥' i an Joe
@ scanned with OKEN Scanner Piaf
2 Notation
J = loon 4 Heat no x
fx] = Ceiting ee no %
By dhe Pepnibion
Lule ep Bn :
Pelee beade2 Ga \- Tx
Ta41=3 a Aue! fon]
Sen
len
PuTs%
aries ayaek bape
fol >=%
Jus BY dhe,
saci po me sweat baron fr
\ observations
7 tnducbive prop dash with Hewnavely ee
abfeuts Sike Snook and, enprenions ¢ Vani oud Sots
Aah a Pogpaate Sxpre nis
> Tnduchen en Jinligess
fp lew Sy chuctisal unduclicrs
—> mudual dnductic
@ scanned with OKEN Scanner ‘The prtanf bY Mathumadica? vinducbien, Can be Codsuss cot
sing Ateps
\. Bobts
In Jhis owe asdume she lowest pomitle value
This Mon initio! Sp dn the past of, Mathenetic
Jn duckie,
7. For. znomple
We Can purse that Xe Hesull is dae
fr zo or n=l l
2 Trduckion hyptheis >
>In this Shp we usin Nalin qn ty Aome
vole K- That Moan que awil chetk rwhether she
Hosult is Sure fon nek lov) nat.
3 Inductee stp, ao
—>4n shi sep, 3 nek Js due ben we
Check whether the Selle is eae toh nzk-) of nob
ah onbe get ke, Anime yeaawune ob ne K+!
‘un nye stoke Dra given, prcfis Arua prnople
4 ARCADE: Sinduobiam one pvdoubyr
’ Jones bur donald dd oajdo
© apduckiary: ow Inbege on joy)
> Suppose 44 owe one Given a Akodemant St)
about art Jntegor n’ which is Jo be -prwved ord We
Con Prove two Duras
V BobiCA > Whe pouye SOr) then Stns)
\ 2 Indughion : 2f Stn) dun Stn)
@ scanned with OKEN Scanner Basis
Assume ndL avhale Lis the batts sinless
Thducken,
*
Here ewe have 0 number ef bast Cases Such a
“sci) Sci) SCL42). = $n"
Example
Prove Shot Psa +P 4. Ze aM mena) (anh
wing Nathumatient nde e
Jat per: Pasha Bae Pe ncneidCona)
b
Shp
0 prove PO) ts dure
Pe stand (20241)
vey
So, pu) As dure
Sbp 2
Assume Jthod, p(x) is due
PCK) is drwe
Parra ry ke KCK DCLEED
sts 6
To prove pCKH co Drow
ane
[Prada de.- J+ Chae Pxownox0)
a tend
2 lk CK )CKH) | + (ka?
My
= KCKat) (2k 0+6 Ck+))
cg
$
@ scanned with OKEN Scanner = (ky) Ik Coxer)eb Len)
6
(ks) Lavtax+brtt |
Ba aan
; 6
, a ee] yy
b yi
3 4
Qa WKASK +b
DK KD43 LED)
(0x43) Cea)
(kt) Ck+n) (2k+3)
bide thse os
6
oy bie
Pl) Js Sue atheraver pew ts bust :
by athe pwinsiple & Modhomabita Jnducbien POT? 48
gnut for oll ane positive nti 1.
Prove by induction mre hak 2 b= nlm
120 2
Job pie jadages oth echt)
a2
Soe
TO prime pl) is Sue
1, pa Wertt)
a 24
Ve)
So pen is dye
Step a i A
Assume hat PCK) 4 true
\ PC) Js deve
\
@ scanned with OKEN Scanner Jerta+--tkK = kCKH)
Step 3
To PROVE PCK a) Lh due
Jao aar--tkK ACK HD =KUKH) 4 CK)
z 1
= CCD
1
=kKCK+t) 40¢KH)
a
a
= AK AaKE2
a a
=k oK+2 Wa
Sa all 4
2 2
= (kai CK2d
2
2
P CKD Jk doe oahenever pCk) Jb Aene
@ scanned with OKEN Scanner Concepis OF NuToMaAta
© Adphabels
>A gtd sete
> nen empty, fU
Fs =h0,y} > birang Ogee
2 Say eee
Sok AS J asey cranncton’
Fir, Lish of Sols am alphas
i Bq ool S Eh 012
“bankas S- baby
) ay “4
i)
4 i of an nl
> Pee oceuiniances
eee
oh Ec ee as
> dinotd by FoeeTtal
&q > lou =3
fr fo as
= Peay dondid enporenbil
totahion.
@ scanned with OKEN Scanner [Nile dak ss fey Ngan
baa “Pi isp
te doy. Madea ) qpebsrminibbic Finite Awtomada Comsss of
ibe 50d ¢ states onda fire Seb dnpet
Srymbots
>
>in ti ad hea
alediog lls a
—> DEA ees not acceph Ane null move es
Yi (she Charge Atala, withoub any 4
Unodactes.
ACG ordain multiple Pino stabs
7 Te Lb whed in Jerndicod arate Jn Cg
Fohmak ae (
_ A abebehminisbic 4 Avdomate (FA) is
Oapinad bY five Juptes
M*(8, 2,5, %0,F.)
Whehe
M4 - Finite Auborroda,
Tt, oh eri eka ae
aa Sek A
Ant he ae
F> Final Stale §-stranditian fun (b22%4)
@ scanned with OKEN Scanner Lrrophito. Ripredontalion. ef DPA
A DFA ton he Supreyenteol, bv dinraphs Called state
i nn, in achch Yao om
> The Stole Ls yiopprerentes bey yeouices Cocncle)_
> The oe Codrtow) oF ealgea Jobeled with oun srvpub
Chotactes. how the drunvyitions
| > The inbAP State is marked with on ovo
| > The Final Make i oendted by a double corde
| ride Thanntion Funttion
“Pittising dw Acgonte portion
= Oe :
Pass: $C, 8) =o
Thduocbin SNS Ws xo
A
SC ao,w)= £400)
2 $C €(4p0),0) whore
= $ Cra) f stownt]
€ = ly
W-10
S°(%0,£)=%
8° C40,19= 8 08 C40,83,19
= §l%)d
=
£ (qo,0092 SCS°C40.15,0)
> 8 (%,0)
a
amt Geom
The Jon fA M= i
by LEM) and tae i | a
Lt) = SCw) dba) da in pS
@ scanned with OKEN Scanner M= (8,2; 8,.%0/F)
Be 0A, 8; 6,04
Ebony a Mead
fat. whey
Pehod Ale |&
§:hxz! 8 | DIA
= AXD cl A\D
o[ Ble
Pe ee
Gorwbuk a DPA that acces Aah fal Abo
over’ $0,1% q lange 2 oe
) 1
OO FS BPO.
he GOSS G5 2O2O+O Kes x
@ scanned with OKEN Scanner yer a,by
(orhtrmt o DFA Shak oxceptt any Noung 6
hal doos nit tombs jhe + Nooing ahh inti
T
Flip the Atote
| Fig SMFS
| NE Sic F Sui) olin Q +“
of Bayard
AOA OrY
Neon pebrtmninittic Finke Awtormaba.
> NEA Sods fe Non sedan siristic rite Awkomala
> ab fh ony by corbbuuck an NEA Bar DFA DAS
yo pt meget
> The finite antoroln oho tolled NFA when dene
at a
etl pbs hon A parte, anpuk, fem Lhe, pent
> Evo NFA I raf DFA bul eath NEA Con be borulabel,
indo PFA
7 NA Ab, dened i the Jame uny of FA, but with she
| fularoitg tioo Exaptions at Confais mulbiple nant
| otal and ik Contoins ¢ tAavsibions
| An NIA hod Bente Ack ef; Mates) on Rone Sete unpul
eps in ba Hl Lr ge
Sf tod ok ahi ee
@ scanned with OKEN Scanner > Tn wea, § iso Dansilinm fan hign Ahad tokos 0 Mabe
ard. Job ng Symi 0 gut at hes
Seb of ox ona ov rate Stabs,
> NPA adows owe ov moke Jaan Per 0 Hake ore
dhe Same Jou ayet
Fema pep initin of NFA
NFA dog vy Bove haps M:(.8,2, $,40,8)
M-> Fini, oulamade
BS Ghoteh > Spat
o> aniting Stoke > Bl Sa,
poe urchin (5: xe *)
Pista fonan wpa)
show ton be Mulkipe fal shit
Extinding Thonbiten funttis
> frothing Arps Fore
pis? S400) 40
Trducbow 7 n> | were
Say = 8 derted
<8 Ha0.x)) a)
odtere] (Cay x) =
> # GP. paies.- aha) Soups Pet
8( Pia) u8CPaja). -8 (Ka)
“hat yaa 7k
4
@ scanned with OKEN Scanner Example ww =10
S'C40,¢.)=%
04,1926 C0 C49¢3,1)
= & (%,1)
" Sadi}
$C 40,10) = & £040,190)
$4, 09 us(1,09
= Hays yuh
24 w4y
Lonsuage of, NFA
7C Az, 6,407)
rs longs copa nA CE
Ju dundoh'by Lt) ond SE Abel os
L905 Cif nad eit oe Jn 4
(or) .
Joy] Saag OF = #4
ters .
eS 5)
M2 (62,8, %0,F)
a= FABy
Z-Soury
do = 5n%
F-4SBy
§: ORE PP
AXoza \ gxo-g
A yn=8 Bue p
AxrsA
@ scanned with OKEN Scanner Y sot qa bugs Arak er nat ah
= ho, ©0, 10,000, 018 a
SO
“Jone te
ABy AB DA a> shi pa
\ Modus end avith,
—b-\, re ; pate
he fs As SE
he 00, nee cd
10
ee
(onkbustk 0 NFA spob, apts Heed, all Amey!
Over hoy 4, st
a hony
de le, ol, 10, uy
@ scanned with OKEN Scanner eee
5 hel af on Abin’ thal Ord voit 1
L- 401, ae We -4
heG seh q ol Absngs ha terbabs'0”f
L- 4 00,0), 10, 000,--» §
>t ie oo"
be oa au aeaps t
Le 10,101 OP. a
0,1
Nosh with 10 }
\
a oe 7
he hie of a, sosings A Cob ort
hho), \ol, oor. - 3
! ou
\
oho
se oon burps ond its 1
de Au, 1M, ooll,- =
Ree
@ scanned with OKEN Scanner >O-4OO)
§ heb of, ott Abvings yak end attr 1
L- 401, N90), m-~ 4
=O ..
bef sbg ob ng ane et}
L-= 4 00,01, 19, 000,--- q
fit,”
bebo a me
Le 10,10" WP “4
0;
Atos with 10}
he tes ee
Lr 40), 101, 001- dy
pyar
foe
fer oo. oor! ond with 1b
be hu, 1M ott, - ©
ps
@ scanned with OKEN Scanner NEA Jn DEA
pada ab write tha NEA given be
\ Ca), $A, (Ep | how J A gin
DFA
desler on NFA iv a pabeaagon
ow fo ain ohh he Jatond, lott Aymbel obonyé
Sve hon ter J to il explant DA
Le hor, 110, 1010 ~ 4
¢ 0 |
au LAO! _jo4
- al A AB
Bic
| o |.
AT Aone
ne} Ac ABC
fel AL
ABC Ac ARE
© scanned with OKEN Scanner (tiven below 1h the NFA fon & cue Leh Sebo} oll
Coodbud cayaivelont
‘rags oWWtC0,0) tha ondd quith’gy/
OFA Soy, t01,---4
oe fdoning NEA bb APA
a i G
qhantibion Goble fot. NFA
(for}.0?
= 8 (4,0) S449)
2 haovtus
ors] oun fous
SoH.
Bf hat 0) 6 Lyo,rrv9 (2
= §l4e,0) vdl¥ar0) 2 ou42
a 2 hnova}
@ scanned with OKEN Scanner NFA Jy DEA wg Swrseb Cont ckbr Mathodl,
NEA i b 6
& =f 4 de}
W 2 Ps9
Ag Ga aegiu a
pet fioait YB rh
cfg |b, 0 Proarfog
Disk [ga A $4043
a ‘Yova}
Fina lait, doe
OF 54 a gn ao _| on
H fant lyon: [Yoang A] AA
E F E> By |"E, 8.
clea >
oR AOA
VEL ILB a Fi y
FAL €, 8
OA
i ce
wea |
tonibuitl NPA “dhak ocaph alt Ab fe Jal prt wth
vrive JH rintlbine Anble ard “enbardbl losin
porin ph due nga Absing evict - < Mio bendbruk oAFA
opr oleve, ne wang Subset tonbbuachizn matreeL
@ scanned with OKEN Scanner 6-048
\
9% fea E
u
af ; ’
Fok w= oto!
§ (4,0): 340,915
§ (4,00) $( 8(40,0),0)
*6($404,0)
£ §(4o0)u stro)
> Sw tig gd > S%49
$(4o,001) = 8(§ (40,009,-1)
28 (fang 1)
70 (41) v 8C41,19
= vgs
Mo 4s 4
8(4p,o0r0 ) = 8( 8 (4, v01),0)
c 8 (fho,a24, 03)
*finadu§ > S409
§C%, o0101 = $( 804, 0010),1)
=$(} 403,19
* (bu osn
> flo} ofa
= M40, Dh > $e, ea) Ae Fed
Moke Js included tn the Ars
Abtpie ouaptio
@ scanned with OKEN Scanner Sayan OMI
p
2 Us? 00101
4 Vv
oy Na
maf
1
Yo % .
Lu ye he sig, ep OY
i | NFA
1] 4 qo
| Epailew NPA te DAA
y ashore”
PFA ao
| Az ¢- closure Co)
@:- Fo.1, aay
| § Caja) =| €- closure ($(0)47,2,4),2)
: > &- closwe( 2,3) = $0367, aay 26°
3 CAB) = &- closure (8 (0,112.4) 6) :
2 $- Ooswu (5)= 55,672,452 ¢
\-~
@ scanned with OKEN Scanner £(Ba)= 2 dosure (8 ( 8,3,6,1,1,2,4),a)
= €-chagwre (83) = $8.3,67, 12,43 Me
SCBib)> 2-closude ($ (9,3,b,7,'12,449,6)
= &-clasue (5) = 45,61, 124% =C
$(¢.a) = f-clesuse (S.C 5)6/7.124),4)
| =f. dssuxe (5,8) => BY
| S(O br= Q-clasue CPC Shit! 2,6)
= £-chgwre (5) => C
S IS [inne an]
ws
2. &-NFA to DPA
& 2
A> E- closure Cao)
Ae 8 %0,41 %04
(Aa) = &- closure (8 (40/4142),2)
> €- dlatuxe Cp) = Suaidey = aA
8 (A.B = e-doae (C009 4598)" hl
= E. chose (44) = Suay > B-
§ (Arc) = ¢- choose (40,414'9),c)
7 £- plosute (429 4a? => ¢
© scanned with OKEN Scanner $ (Bb) = &-closude ($0 %1,%2),b)
= E> elntwle (49 = $4404 >B
§ (Bra) = g. clesute ($ (4143), 4)
=. Chosite (GJ =D JP (0, abc) =9
§(Bc)= 8- cosure (6(41,4),6)
£- Ansure (9220
§Ceyay= ¢- cswre ($42, 0)) =p 2D
§ eb E- clomude (E(42b))= 6 SP
$(t,0) =e tosmee C2044,09)
=8-closute (42) 2 C
a
Gphilen NEA to NFA
ae
© scanned with OKEN Scanner 8- Cosuhe
qo~hrorrra f
Gis han de 7
427 had
§(4e,a 2 € -caswre [8 (¢-dawe(p),)
: g-eloswee ( §(41929,4)
2 € -tswre (4o)2 $ 10,9424
§ t4oyb) = E-losure (8 Ce-cosue (40) 2)
= Ge ebosune (8 (40)41,429, 6)
og closunetti)> 4414.5,
$tdocd> 2 closwowe ( § (8-claswce (40), 0
=-ehoome (§ (4091,423,¢ 9
= - closure Ca) fat
8 (41a) = ¢-choture ($C ¢-chosuee (C11) a.)
29- closure (§C41,429,0) = 2
= AUR
g (arp)s €-ehasune 6Ce- dana (6)
© = geclawte (8(4),429,b) dase)
paijen s &-losuee (8 Ce-elocwt (10,6)
= ee elpgure (8C41929,8) °
rg close (42) shut
§ Corre) = &-choture (8 (e-clonue (42.9,0)
2b csuee (30497,0) = Sod
$ (42,6) 2 9- chosure (8 (¢- closure (424)
sop cha (2 (42,092 B
Stqarcds ¢. dncusie (4 (e-claswte (42)
= b- closure cane) Bey
@ scanned with OKEN Scanner €
oOo
Yo [samen handed hey
“S404
Stement io
iy mh a
[DER] which alho
Proof
Jeb Mz (H2 ye, emertings pnp
We congue DER M=(81, 2,0, 6, 6)
a= 22] pu 8
Cony Atake, Jn 8 Ht ascaiegl a
whore, :
(Vad 2H 6
ii) 98 LM]:
ii) pik sak Bal bas
whe dafnng & ok ws ose ab dhe trsbrichiong
ek “e “4 : fo
8,40, F
@ scanned with OKEN Scanner Mis dnitially ok Ye
on ee symoct Say a, Can 30h ony
a cohen of the np inpu Symbal a.
qo destsibe M, Je a ak Ze
Wwe Hogue aul te side Soles ak opie
an ene
oa me iat
tntainang oul Pa Aloles ~
Now Ure dibure
a yp acne $CUA)
8 (fsa ail) ): at
esyivolaty
SEEM -4i),0) > [Pha al]
(Fayge -4i5,9)° Ermey
proof iy anduchin Renal
npuk Sing 2, ssh
£(%, ® > [m, Y- i]
4 and onby z
$4002 9 > -4if
NFA
@ scanned with OKEN Scanner AN
Boia
The Stout is Baivin Sy Abring bongth 0
ie [aloo
Sint, ©
we fo thing
ne must be emp mt
tnduchion | !
Suppose dne hyppotasis As dre Me spt ie
Jeb to eo Aine ae m4) aislthy 0 in.
‘Than
ECR xad 28°Caeas wad
anduction hyp theses
rte. Pre Py]
: Hand only sp |
80) = 5 PhP - my
By Ufinshin, 8 gl
8 CL Pie, p75 ade [vt nae
BNA sed
8c hry, D> =f fa) = 47m
us
8 (td, na)= Jryra.- kJ oo
only
8 (4,402 fran. ax} nae
‘whith establishes Je Irduchie hy poanentt
Thus,
Teas Ltn) |
@ scanned with OKEN Scanner Theokem
TE @ ts occopted by NEA] avithe ‘e_pansitions Shor
Lis cee td by NFA osithouk © buansihions
prot
Jok Mz (A,2, §,40,2) be on NFA with e- Lansilions
Consbuck mM! aohith ib NFA without 2. dranitions
M=(0,z,8, 40,1) nohwe
Fl [Eds B- daswne (40) tonbins a Mabe oF
© OtneHWOl se
By induction,
qe $ axe Aame
ond $f au aioe
Lot Re be any i
“8'(40.) 2 8 Caoc) a
This Moderont Lt mat Aue o- @ becouse 840824409
$ (20,2) = €- dodure (409
Basis Atop
wae
oe Lh a Symbol whore value Js a
§ (40a)- § (40,0)
Enduchon Shep .
“dob y= Wa mphote a as wn 4
$04, un) = 8( 8CAo.0}a)
8 Cf Cey00),00)
s CP.)
u
a
@ scanned with OKEN Scanner To Mhow » 9/(Pad= $'( 4,000)
Sra v8 (Wad- Ve, Pty)
| - SCC wa)
| = 8 (gp wa)
| = Sdn)
os § (ann) = $ (40,73)
® scanned with OKEN Scanner
