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### Rewrites Tutorial Exercise 4

Here is a suitable set of rules to solve the first part of the problem:

```[ seq(n, c) := seq(n/2,  c+1) :: n%2 = 0,
seq(n, c) := seq(3n+1, c+1) :: n%2 = 1 :: n > 1 ]
```

Given the initial formula `seq(6, 0)', application of these rules produces the following sequence of formulas:

```seq( 3, 1)
seq(10, 2)
seq( 5, 3)
seq(16, 4)
seq( 8, 5)
seq( 4, 6)
seq( 2, 7)
seq( 1, 8)
```

whereupon neither of the rules match, and rewriting stops.

We can pretty this up a bit with a couple more rules:

```[ seq(n) := seq(n, 0),
seq(1, c) := c,
... ]
```

Now, given `seq(6)' as the starting configuration, we get 8 as the result.

The change to return a vector is quite simple:

```[ seq(n) := seq(n, []) :: integer(n) :: n > 0,
seq(1, v) := v | 1,
seq(n, v) := seq(n/2,  v | n) :: n%2 = 0,
seq(n, v) := seq(3n+1, v | n) :: n%2 = 1 ]
```

Given `seq(6)', the result is `[6, 3, 10, 5, 16, 8, 4, 2, 1]'.

Notice that the n > 1 guard is no longer necessary on the last rule since the n = 1 case is now detected by another rule. But a guard has been added to the initial rule to make sure the initial value is suitable before the computation begins.

While still a good idea, this guard is not as vitally important as it was for the `fib` function, since calling, say, `seq(x, [])' will not get into an infinite loop. Calc will not be able to prove the symbol `x' is either even or odd, so none of the rules will apply and the rewrites will stop right away.

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