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    <p>After accepting Vitus' answer, I discovered a different way to accomplish the goal of proving a function terminates in Agda, namely using "sized types." I am providing my answer here because it seems acceptable, and also for critique of any weak points of this answer.</p> <p>Sized types are described: <a href="http://arxiv.org/pdf/1012.4896.pdf">http://arxiv.org/pdf/1012.4896.pdf</a></p> <p>They are implemented in Agda, not only MiniAgda; see here: <a href="http://www2.tcs.ifi.lmu.de/~abel/talkAIM2008Sendai.pdf">http://www2.tcs.ifi.lmu.de/~abel/talkAIM2008Sendai.pdf</a>.</p> <p>The idea is to augment the data type with a size that allows the typechecker to more easily prove termination. Size is defined in the standard library.</p> <pre><code>open import Size </code></pre> <p>We define sized natural numbers:</p> <pre><code>data Nat : {i : Size} \to Set where zero : {i : Size} \to Nat {\up i} succ : {i : Size} \to Nat {i} \to Nat {\up i} </code></pre> <p>Next, we define predecessor and subtraction (monus):</p> <pre><code>pred : {i : Size} → Nat {i} → Nat {i} pred .{↑ i} (zero {i}) = zero {i} pred .{↑ i} (succ {i} n) = n sub : {i : Size} → Nat {i} → Nat {∞} → Nat {i} sub .{↑ i} (zero {i}) n = zero {i} sub .{↑ i} (succ {i} m) zero = succ {i} m sub .{↑ i} (succ {i} m) (succ n) = sub {i} m n </code></pre> <p>Now, we may define division via Euclid's algorithm:</p> <pre><code>div : {i : Size} → Nat {i} → Nat → Nat {i} div .{↑ i} (zero {i}) n = zero {i} div .{↑ i} (succ {i} m) n = succ {i} (div {i} (sub {i} m n) n) data ⊥ : Set where record ⊤ : Set where notZero : Nat → Set notZero zero = ⊥ notZero _ = ⊤ </code></pre> <p>We give division for nonzero denominators. If the denominator is nonzero, then it is of the form, b+1. We then do divPos a (b+1) = div a b Since div a b returns ceiling (a/(b+1)).</p> <pre><code>divPos : {i : Size} → Nat {i} → (m : Nat) → (notZero m) → Nat {i} divPos a (succ b) p = div a b divPos a zero () </code></pre> <p>As auxiliary:</p> <pre><code>div2 : {i : Size} → Nat {i} → Nat {i} div2 n = divPos n (succ (succ zero)) (record {}) </code></pre> <p>Now we can define a divide and conquer method for computing the n-th Fibonacci number.</p> <pre><code>fibd : {i : Size} → Nat {i} → Nat fibd zero = zero fibd (succ zero) = succ zero fibd (succ (succ zero)) = succ zero fibd (succ n) with even (succ n) fibd .{↑ i} (succ {i} n) | true = let -- When m=n+1, the input, is even, we set k = m/2 -- Note, ceil(m/2) = ceil(n/2) k = div2 {i} n fib[k-1] = fibd {i} (pred {i} k) fib[k] = fibd {i} k fib[k+1] = fib[k-1] + fib[k] in (fib[k+1] * fib[k]) + (fib[k] * fib[k-1]) fibd .{↑ i} (succ {i} n) | false = let -- When m=n+1, the input, is odd, we set k = n/2 = (m-1)/2. k = div2 {i} n fib[k-1] = fibd {i} (pred {i} k) fib[k] = fibd {i} k fib[k+1] = fib[k-1] + fib[k] in (fib[k+1] * fib[k+1]) + (fib[k] * fib[k]) </code></pre>
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    1. POAssisting Agda's termination checker
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    1. USJonathan Gallagher
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    1. COI forgot to mention... for this to work you need to add {-# OPTIONS --sized-types #-} to the top of the file, as sized types are apparently an Agda extension.
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    2. CO`div` doesn't do the thing you think it does. `div m n` is actually `ceiling(m/(n + 1))`, `div2 3 == 1` and consequently `fibd` produces the sequence `0, 1, 1, 2, 3, 5, 8, 5, 8, 13...`. But yes, sized types are a cool concept; however, it's very intrusive to data definitions and it's not used much in standard library making it sort of painful to use.
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    3. COI edited the post to use a correct ceil(n/2).
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