Upper Bound in C#

using System;
// Upper Bound in C#
//
// 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
// C# / .NET 10: ReadOnlySpan<int> inputs, Span<int> outputs, primitive workspace arrays,
// Array.BinarySearch for LIS, and checked integer arithmetic. Invalid arguments and overflow
// throw exceptions.
//
// EXAMPLE
// Expected return: 3.
// Run: Use Program.cs in a net10.0 console project; dotnet run.
// Tests for this exact source artifact run before publication. The accompanying source
// archive includes unit cases, runtime results, and source hashes.

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

    static int upper_bound(ReadOnlySpan<int> a, int target) {
        checked {
            int n = a.Length;
            alg_require(((n >= 0) && (n <= 1000000)));
            alg_require((a.Length == n));
            int lo = 0;
            int hi = n;
            int mid = 0;
            while ((lo < hi)) {
                mid = (lo + ((hi - lo) / 2));
                if ((a[mid] <= target)) {
                    lo = (mid + 1);
                } else {
                    hi = mid;
                }
            }
            return lo;
        }
    }

    static void Main() {
        int[] a = new int[] {1, 2, 2, 4};
        int target = 2;
        int result = upper_bound(a, target);
        Console.WriteLine(result);
    }
}