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Z,B, . . .

f : Z Z f(x) =

1 x Z x 0

0

g : (Z Z) Z g(h) =

0

h(x)= x Z

t s t s

f S={, 0}

f(S) ={0, 1} f(S) =f(S)

fi(x) = 0 x i

S = {f1, f2, . . .} S = f f(x) = 0 x Z g(S) = = 0 = g(S)

g

(x.42) bot bot bot

42

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> 1

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D f : D D {fi()| i N}

f

D = 2N

D

S D

D

(D, )

f : D D f

f mult nf mult

() n N

n

f

mult

f mult

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S(S (S))S (S Z)

Z

S(S)S Z

S

fi() fi+1()

i N

i= 0 f0() = f1()

i >0

fi1() fi() f

f

f(fi1()) f(fi()) fi() fi+1()

{fi() | i N} D {fi() | i N}

f

f({fi()| i N}) =f({fi()| i N}) (f )

={fi+1

()| i N}=({fi+1()| i N} { })

={fi()| i N}

d

f {fi()| i N} d

fi() d f0() = d

fi() d f f(fi()) f(d) d f fi+1()d

S= {M1, M2, . . .} Mi Mi+1

S=

Mi Mi

Mi

B Mi B e Mi\ B k

e Mk eB Mk B B S

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Mi D

D

Ni := {k N | k i} Ni Ni+1 {N1, N2, . . .}

f(M) =

M

{42}

g

(nf

mult())(x, y) =

0

x= 0 n > 0

x y 0< x < n y=

f mult

g(x, y) =

0

x= 0

x y 0< x y=

h(x, y) =

0

x= 0

x y x= y=

x= (x= 0 y = ) (

)

f f(c1, c2) = f mult(f)(c1, c2)

f(c1, c2) = 0 c1 = 0

c1 = 0 f(c1, c2) =c2+f(c11, c2) c1 =

c1 1 c2 = c1 = 0

c2+f(c1 1, c2) h

c1, c2 Z f(c1, c2) = c2+ f(c1 1, c2)

B H

A (A ) A (E)A (B)

A B

B(E)A H

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Lam

t= fact.(x.(If(LessThanOrEx 1) 1 (Timesx (fact (Minusx 1)))))

= { If True x y.x,

If Falsex y.y,

fix f.f(fixf)}

{ Minus x y z | x, y Z z= x y}

{ Times x y z | x, y Z z= x y}

{ LessThanOrE x y b | x, y Z ((x y b= True) (x > y b= False))}

fixt 1 t

n

f x.fn x

fx.x

fx.fx

Succ n n+ 1 Succ (f x.t) f x.f t t

fn x

Plus

n

m

Succ

n+m m n

(fix(quot x y.If(LessThanx y) 0 (Plus1 (quot (Minusx y) y )))) v w

fixt 1

(f.(f (fixf))) t 1

t (fixt) 1

(x.(If(LessThanOrEx 1) 1 (Timesx (fixt (Minusx 1))))) 1

If(LessThanOrE1 1) 1 (Times1 (fixt (Minus1 1))) ()

If True1 (Times1 (fixt (Minus1 1)))

(x.(y.x)) 1 (Times1 (fixt (Minus1 1)))

(y.1) (Times1 (fixt (Minus1 1)))

1

If if

()

Succ= n f x.f (n f x)

Plus= n m f x.n f (m f x)

Plus= n m.n Succ m

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A0 := {x:: a.a Int} y.y x W

W(A0, y.y x)W(A0+ {y:: b1}, y x)

W(A0+ {y:: b1}, y) = (id,b1)W(A0+ {y:: b1}, x) = (id,b2 Int)

mgu(b1, (b2 Int)b3) = [b1/(b2 Int) b3])= ([b1/(b2 Int) b3], b3)

= ([b1/(b2 Int) b3], ((b2 Int) b3) b3)