Upper Bound in PHP

<?php
declare(strict_types=1);
// Upper Bound in PHP
//
// PURPOSE
// A boundary search finds an insertion position in a sorted array, including arrays with
// repeated values. This is useful for maintaining sorted collections and counting
// occurrences.
//
// METHOD
// Use a half-open interval [lo, hi). Move lo beyond elements that belong before the
// requested boundary; otherwise move hi to mid. The interval shrinks until both bounds
// coincide.
//
// COMPLEXITY
// O(log n) time, O(1) auxiliary space.
//
// CONTRACT
// Input is sorted ascending. Returns the first index whose value is greater than target, or
// n if none exists. Does not change the input.
// 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: 3.
// 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 upper_bound(array $a, int $target): int {
    alg_require(array_is_list($a));
    $n = count($a);
    alg_require((($n >= 0) && ($n <= 1000000)));
    alg_require((count($a) == $n));
    $lo = 0;
    $hi = $n;
    $mid = 0;
    while (($lo < $hi)) {
        $mid = alg_add($lo, intdiv(alg_sub($hi, $lo), 2));
        if (($a[$mid] <= $target)) {
            $lo = alg_add($mid, 1);
        } else {
            $hi = $mid;
        }
    }
    return $lo;
}

$a = [1, 2, 2, 4];
$target = 2;
$result = upper_bound($a, $target);
echo $result, PHP_EOL;