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Summary: Control Operators and Natural Numbers are
Degenerate
Robin Adams
September 20, 2010
We work in a type theory that has both µ-operators, and a type of natural
numbers. Suppose we have a term iszero such that
iszero0 1
iszero(Sn) 0
In addition to the above, our type theory has the following reduction rules:
S(µ.n) µ.[S[]/]n (1)
iszero(µ.n) µ.[iszero[]/]n (2)
[]µ.n [/]n (3)
where, for C[] a context, the term
[C[]/]n
is defined to be the result of replacing every subterm of the form []M with
[]C[M].
(There may be other reduction rules, too; we do not care.)
Consider the term
M µ.[]S(µ.[]0)
Theorem 1 The term iszeroM is not confluent.
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