∀i, j ≥ 0, R i+j = R i ◦ R jw

(1)

24365798:.;=<?>@BA9ACEDF<GHI@<JCKML

◦

^N

OQPSRSTSUVWYXZP [T4\^]Y_[`_PaVWb_\c[

degfihjikljfSm 'on0

R

^p ) qrn0tsu

p p

t)vsq

E

^w ^xn.n0)

R ⁰ = id, R ⁱ⁺¹ = R ◦ R ⁱ .

!$n0i y

p

∀i, j ≥ 0, R ^i+j = R ⁱ ◦ R ^j

^w

degfihjikljf{z 'on0

M

q}|~)JvsqFn

p

q*vn0

p p

Squs

A

ⁿ⁰⁾ ^w ^{!$n0i y} ^p ^,n ^p

n

p

Bvn0

u

v

⁾⁾

M

ⁿ⁰

u · v = v · u ⇐⇒ ∃w ∈ M, ∃p, q ∈ N | u = w ^p

v = w ^q .

degfihjikljf ()cn0)n0 *

p

f

^p ?I v*n

p

qrn0 v|

p

)*y

p

n0 *

p

qr,n0 v

f (x) = g(x, f(h(x)))

h

g

n0i*n0)n0)cqq*cy p

h(x) < x

^w v) qrn0)n0

f

^))cq*0 ^p ^.i

w

f : N → N

f (0) = 1

h(x) = x − 1

g(x, y) = 2y

+ w

f : N → N

f (0) = 0

h(x) = x − 1

g(x, y) = y + 2x − 1

w

f : N → N

f (0) = 0

h(x) = [x/2]

g(x, y) =

4y :

x

^u

4y + 9(2x − 1) :

⁾ⁿ⁰

degfihjikljf Zqq

F n

q

n

}v()n0vs (xsn0u0*x

F 0 = F 1 = 1

F n+2 = F n+1 + F n

^w

w

p

)n0)n04¡i¢.£¥¤¢I¦¨§)©bªI«¬y

p

=q

p

q

F n

^w

+ w

'on0

r n

q )n0v,s |®q§u©YªI«n

p

Bqq

p

q=

F n

^w

!$n0i y

p

r n = 2F n+1 − 1

^w

w

n0) * ¯

p

vii

p

°

p

}y

p

±q±v)|~?*

p

n0²±qran0)n0²§)©Yª«$*

?In0)iq

w

/

w

p

)³n0)n0´¡µ¢0£9¤I¢¦¶©b£I¢.¤¶¬·§¶¸¹y

p

¨q

p

q

f ⁿ (x)

^º

f ⁰ (x) = x, f ¹ (x) = f (x), . . . , f ⁱ⁺¹ (x) = f f ⁱ (x)

^»

w

1 w t v± y

p

,y

p

(F n+2 , F n+1 ) = s(F n+1 , F n )

ⁿ

s

^* ^p )n0)n0ty

p

Q¼

p

) u

(x, y)

0Jnµcqru

(x +y, x)

^w ^{!n0µ y} ^p

(F n , F n − 1 ) = s ⁿ ⁻ ¹ (F 1 , F 0 )

ⁿ

p

n ≥ 1

^w ^* ^p

p

gq

p

q

F n

p

av)q)

w

¾ w

'on0(qrFn0)n0

f ib(n) = (F n , F n − 1 )

^w ^{I v}

f ib(n)

ⁿ ^p BqrFn0 v |

p

)*y

p

n0M *

p

¿

w

degfihjikljf$À 'on0cÁ)«ÂoÃbÁqrFn0)n0t?Ä)*n0vv

p

w×p

q

p

q*n0)n0Ø

+ w

o v Á)«.ÂIÃbÁan

p

Bqr,n0 v|

p

)*y

p

n0M *

p w

w

!$n0i y

p

* n0)n0M v)

w

∀i, j ≥ 0, R i+j = R i ◦ R jw

◦

R

E

R 0 = id, R i+1 = R ◦ R i .

∀i, j ≥ 0, R i+j = R i ◦ R j

M

A

u

v

M

u · v = v · u ⇐⇒ ∃w ∈ M, ∃p, q ∈ N | u = w p

v = w q .

f

f (x) = g(x, f(h(x)))

h

g

h(x) < x

f

f : N → N

f (0) = 1

h(x) = x − 1

g(x, y) = 2y

f : N → N

f (0) = 0

h(x) = x − 1

g(x, y) = y + 2x − 1

f : N → N

f (0) = 0

h(x) = [x/2]

g(x, y) =

4y :

x

4y + 9(2x − 1) :

F n

n

F 0 = F 1 = 1

F n+2 = F n+1 + F n

F n

r n

F n

r n = 2F n+1 − 1

f n (x)

f 0 (x) = x, f 1 (x) = f (x), . . . , f i+1 (x) = f f i (x)

(F n+2 , F n+1 ) = s(F n+1 , F n )

s

(x, y)

(x +y, x)

(F n , F n − 1 ) = s n − 1 (F 1 , F 0 )

n ≥ 1

F n

f ib(n) = (F n , F n − 1 )

f ib(n)

R ⁰ = id, R ⁱ⁺¹ = R ◦ R ⁱ .

∀i, j ≥ 0, R ^i+j = R ⁱ ◦ R ^j

u · v = v · u ⇐⇒ ∃w ∈ M, ∃p, q ∈ N | u = w ^p

v = w ^q .

f ⁿ (x)

f ⁰ (x) = x, f ¹ (x) = f (x), . . . , f ⁱ⁺¹ (x) = f f ⁱ (x)

(F n , F n − 1 ) = s ⁿ ⁻ ¹ (F 1 , F 0 )