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$Page 1: lecture-7 - University of Cambridge · Title: C:\Users\andre\CL\Documents\cst\courses\comt\15-16\lectures\lecture-7 Author: Andrew Pitts Created Date: 20190207075831Z$

$Page 2: lecture-7 - University of Cambridge · Title: C:\Users\andre\CL\Documents\cst\courses\comt\15-16\lectures\lecture-7 Author: Andrew Pitts Created Date: 20190207075831Z$

$Page 3: lecture-7 - University of Cambridge · Title: C:\Users\andre\CL\Documents\cst\courses\comt\15-16\lectures\lecture-7 Author: Andrew Pitts Created Date: 20190207075831Z$

$Page 4: lecture-7 - University of Cambridge · Title: C:\Users\andre\CL\Documents\cst\courses\comt\15-16\lectures\lecture-7 Author: Andrew Pitts Created Date: 20190207075831Z$

$Page 5: lecture-7 - University of Cambridge · Title: C:\Users\andre\CL\Documents\cst\courses\comt\15-16\lectures\lecture-7 Author: Andrew Pitts Created Date: 20190207075831Z$

$Page 6: lecture-7 - University of Cambridge · Title: C:\Users\andre\CL\Documents\cst\courses\comt\15-16\lectures\lecture-7 Author: Andrew Pitts Created Date: 20190207075831Z$

$Page 7: lecture-7 - University of Cambridge · Title: C:\Users\andre\CL\Documents\cst\courses\comt\15-16\lectures\lecture-7 Author: Andrew Pitts Created Date: 20190207075831Z$

$Page 8: lecture-7 - University of Cambridge · Title: C:\Users\andre\CL\Documents\cst\courses\comt\15-16\lectures\lecture-7 Author: Andrew Pitts Created Date: 20190207075831Z$

$Page 9: lecture-7 - University of Cambridge · Title: C:\Users\andre\CL\Documents\cst\courses\comt\15-16\lectures\lecture-7 Author: Andrew Pitts Created Date: 20190207075831Z$

$Page 10: lecture-7 - University of Cambridge · Title: C:\Users\andre\CL\Documents\cst\courses\comt\15-16\lectures\lecture-7 Author: Andrew Pitts Created Date: 20190207075831Z$

$Page 11: lecture-7 - University of Cambridge · Title: C:\Users\andre\CL\Documents\cst\courses\comt\15-16\lectures\lecture-7 Author: Andrew Pitts Created Date: 20190207075831Z$

$Page 12: lecture-7 - University of Cambridge · Title: C:\Users\andre\CL\Documents\cst\courses\comt\15-16\lectures\lecture-7 Author: Andrew Pitts Created Date: 20190207075831Z$

$Page 13: lecture-7 - University of Cambridge · Title: C:\Users\andre\CL\Documents\cst\courses\comt\15-16\lectures\lecture-7 Author: Andrew Pitts Created Date: 20190207075831Z$

$Page 14: lecture-7 - University of Cambridge · Title: C:\Users\andre\CL\Documents\cst\courses\comt\15-16\lectures\lecture-7 Author: Andrew Pitts Created Date: 20190207075831Z$

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Primitive recursionTheorem. Given f ∈ Nn

⇀N and g ∈ Nn+2⇀N, there

is a unique h ∈ Nn+1⇀N satisfying

!

h(⃗x, 0) ≡ f (⃗x)

h(⃗x, x + 1) ≡ g(⃗x, x, h(⃗x, x))

for all x⃗ ∈ Nn and x ∈ N.

We write ρn( f , g) for h and call it the partial functiondefined by primitive recursion from f and g.

L8 87

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