[Axiom-developer] [MutualRecursion]

[billpage]

??changed:
-Here is an example of defining functions by mutual recursion in Axiom
-
-We start with a "bootstrap" definition of 'parity(n)\$Even'. All that is
-really needed here is a package that exports a function named 'parity'
Here we give two examples of defining functions by mutual recursion in Axiom

Recursion between Separate Domains

  First we show how to define two separate domains **EVEN** and **ODD**
both of which have an attribute represented by the polymorthic function
**parity**.

We must start with a "bootstrap" definition of 'parity(n)\$even'. All that
is really needed here is a package that exports a function named 'parity'

??changed:
-)abbrev package EVEN Even
-Even(): E == I where
)abbrev domain EVEN even
even(): E == I where

??changed:
-Now we can define 'parity(n)\$Odd'. It depends on **EVEN**.
Now we can define 'parity(n)\$odd'. It depends on **EVEN**.

??changed:
-)abbrev package ODD Odd
-Odd(): E == I where
)abbrev domain ODD odd
odd(): E == I where

??changed:
-      (n>0) => parity(n-1)$Even
-      (n<0) => parity(n+1)$Even
      (n>0) => parity(n-1)$even
      (n<0) => parity(n+1)$even

??changed:
-depends (recusively) on **ODD**. So finally we need the full (re-)definition
-of 'parity(n)\$Even'
depends (recusively) on **ODD**. So finally we need the full
(re-)definition of 'parity(n)\$even'

??changed:
-)abbrev package EVEN Even
-Even(): E == I where
)abbrev domain EVEN even
even(): E == I where

??changed:
-      n>0 => parity(n-1)$Odd
-      n<0 => parity(n+1)$Odd
      n>0 => parity(n-1)$odd
      n<0 => parity(n+1)$odd

??changed:
-Now we can test the new functions:
-
Now we can test the new function:

??changed:
-parity(10)$Even
-parity(8)$Odd
-parity(-1111)$Odd
parity(10)$even
parity(8)$odd
parity(-1111)$odd

++added:
Recursion within a Single Domain

  It is possible to write this same recursion as a domain that
exports two functions **Even** and **Odd**. In this case we do not
need to supply any initial *bootstrap* code because the compiler is
able to resolve both functions simultaneously.

\begin{axiom}
)abbrev domain PARITY Parity
Parity(): Exports == Implements where
  Exports == with
    Even: Integer -> Boolean
    Odd: Integer -> Boolean
  Implements == add
    Odd(n: Integer) ==
      n>0 => Even(n-1)
      n<0 => Even(n+1)
      false
    Even(n: Integer) ==
      n>0 => Odd(n-1)
      n<0 => Odd(n+1)
      true
\end{axiom}

Test
\begin{axiom}
Even(10)$Parity
Odd(8)$Parity
Odd(-1111)$Parity
\end{axiom}


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