Grid Path Counting in PHP

<?php
declare(strict_types=1);
// Grid Path Counting in PHP
//
// PURPOSE
// Grid path counting counts ways to reach a cell when each step moves only right or down.
// Every path arrives from the cell above or the cell to the left.
//
// METHOD
// Initialize one path along the first row. Update a single row left to right, adding the
// path count from the left to the retained count from above.
//
// COMPLEXITY
// O(rows * cols) time and O(cols) auxiliary space.
//
// CONTRACT
// rows and cols are at least one. Grid has no obstacles. Counts must fit the integer type.
// Public APIs infer array lengths except in C, which accepts pointer/length pairs. Vertex
// counts and matrix dimensions remain explicit. Arrays must contain signed 32-bit integers.
// Sorted order, correct tree topology, and caller-owned data-structure invariants are
// preconditions; checking them on every query would change query complexity.
// Scalar and shape checks are included. Value-range checks are included where needed to
// prevent invalid indexing or nontermination. Allocation sizes are capped at 4,000,000
// workspace entries. Arithmetic outside Python raises an error on 32-bit overflow; Python
// uses arbitrary precision. Native allocation failures propagate. In-place outputs may be
// partly updated after arithmetic failure. These are single-threaded, in-memory
// implementations.
//
// LANGUAGE DETAILS
// PHP 8.1+ on a 64-bit runtime: packed arrays, strict scalar types, read-only inputs by
// value using copy-on-write, references only for outputs, and checked 32-bit arithmetic.
// Invalid input and overflow throw exceptions. PHP arrays have higher overhead than native
// packed buffers in C/Java.
//
// EXAMPLE
// Expected return: 6.
// Run: php example.php
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

function alg_require(bool $valid): void { if (!$valid) throw new InvalidArgumentException("invalid algorithm input"); }
function alg_checked(int|float $v): int { if ($v < -2147483648 || $v > 2147483647 || floor($v) != $v) throw new OverflowException("32-bit result overflow"); return (int)$v; }
function alg_add(int $a,int $b): int { return alg_checked($a+$b); }
function alg_sub(int $a,int $b): int { return alg_checked($a-$b); }
function alg_mul(int $a,int $b): int { return alg_checked($a*$b); }
function alg_size(int $n): int { alg_require($n>=0 && $n<=4000000); return $n; }

function grid_paths(int $rows, int $cols): int {
    alg_require((($rows >= 1) && ($cols >= 1) && ($rows <= 1000000) && ($cols <= 1000000)));
    $dp = array_fill(0, alg_size($cols), 0);
    $c = 0;
    $r = 1;
    while (($c < $cols)) {
        $dp[$c] = 1;
        $c = alg_add($c, 1);
    }
    while (($r < $rows)) {
        $c = 1;
        while (($c < $cols)) {
            $dp[$c] = alg_add($dp[$c], $dp[alg_sub($c, 1)]);
            $c = alg_add($c, 1);
        }
        $r = alg_add($r, 1);
    }
    return $dp[alg_sub($cols, 1)];
}

$rows = 3;
$cols = 3;
$result = grid_paths($rows, $cols);
echo $result, PHP_EOL;