Insertion Sort in Java

// Insertion Sort in Java
//
// PURPOSE
// Insertion sort grows a sorted prefix by inserting the next value into its proper position.
// It works well for small or nearly sorted arrays.
//
// METHOD
// Save the next key, shift larger prefix values to the right, then place the key in the gap.
// Using a strict comparison preserves the relative order of equal values.
//
// COMPLEXITY
// O(n squared) worst-case time, O(n) best-case time, O(1) auxiliary space. Stable.
//
// CONTRACT
// Sorts n elements ascending in place; returns 0.
// 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
// Java 17+: primitive int arrays, Arrays.binarySearch for LIS, and Math exact arithmetic.
// Invalid input throws IllegalArgumentException; arithmetic overflow throws
// ArithmeticException.
//
// EXAMPLE
// Expected return: 0.
// Expected a after call: [1, 1, 3, 5].
// Run: javac AlgorithmExample.java && java AlgorithmExample
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

class AlgorithmExample {
    static void alg_require(boolean valid) { if (!valid) throw new IllegalArgumentException("invalid algorithm input"); }
    static int alg_size(int n) { alg_require(n>=0 && n<=4000000); return n; }

    static int insertion_sort(int[] a) {
        java.util.Objects.requireNonNull(a, "a");
        int n = a.length;
        alg_require(((n >= 0) && (n <= 1000000)));
        alg_require((a.length == n));
        int i = 1;
        int j = 0;
        int key = 0;
        while ((i < n)) {
            key = a[i];
            j = Math.subtractExact(i, 1);
            while (((j >= 0) && (a[j] > key))) {
                a[Math.addExact(j, 1)] = a[j];
                j = Math.subtractExact(j, 1);
            }
            a[Math.addExact(j, 1)] = key;
            i = Math.addExact(i, 1);
        }
        return 0;
    }

    public static void main(String[] args) {
        int[] a = new int[] {5, 1, 3, 1};
        int result = insertion_sort(a);
        System.out.println(result);
        System.out.println(java.util.Arrays.toString(a));
        // Expected a: [1, 1, 3, 5]
    }
}