<?php
 
class Fish
{
    protected function B($m, $n, $f)
    {
        if ($m == 0) {
            // B(0, n) = f(n)
            return $f($n);
        } else if ($n == 0) {
            // B(m+1, 0) = B(m, 1)
            return $this->B($m - 1, 1, $f);
        } else {
            // B(m+1, n+1) = B(m, B(m+1, n))
            return $this->B($m - 1, $this->B($m, $n - 1, $f), $f);
        }
    }
 
    protected function g($f)
    {
        // g(x) = B(x, x)
        return function ($x) use ($f) {
            return $this->B($x, $x, $f);
        };
    }
 
    /**
     * S 変換
     * S(m, f(x)) = (g(m), g(x))
     * 自然数と関数のペアから、
     * 自然数と関数のペアへの写像 S を定義。
     */
    public function S()
    {
        return function ($m, $f) {
            $g = $this->g($f);
            return [$g($m), $g];
        };
    }
 
    /**
     * S2 (SS変換の一部)
     * S^f(m) を定義。
     */
    protected function S2($S)
    {
        return function ($m, $f) use ($S) {
            foreach (range(1, $f($m)) as $i) {
                list($m, $f) = $S($m, $f);
            }
        };
    }
 
    /**
     * SS 変換
     * SS(m, f, S) = (S^f(m)[m, f], S^f(m))
     * 自然数、関数、変換の組から、
     * 同様の組を生み出す写像 SS を定義。
     *
     * @param $m    int     自然数
     * @param $f    Closure 関数
     * @param $S    Closure 変換
     * @return array (
     *      $n      int     自然数
     *      $g      Closure 関数
     *      $S2     Closure 変換
     * )
     */
    public function SS($m, $f, $S)
    {
        $S2 = $this->S2($S);
        list($n, $g) = $S2($m, $f);
        return [$n, $g, $S2];
    }
}
 
/**
 * ふぃっしゅ数 バージョン 1
 *
 * $m = 0
 * $f = $x + 1
 * $S = S 変換
 *
 * 初期値を上記の通り設定した上で、
 * SS 変換を 63 回行った後の、
 * 第 1 成分をふぃっしゅ数バージョン 1、
 * 第 2 成分をふぃっしゅ関数バージョン 1
 * と定義する。
 */
$fish = new Fish();
 
$m = 3;
$f = function ($x) {
    return $x + 1;
};
$S = $fish->S();
 
foreach (range(1, 63) as $i) {
    list($m, $f, $S) = $fish->SS($m, $f, $S);
}
 
echo 'ふぃっしゅ数 = ' . $m;
 

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