fix(strings): avoid counting false matches in z-function search

[Miladkhoshdel]

Describe your change

find_pattern() previously inspected Z-values belonging to the pattern prefix, causing it to count matches that crossed the boundary between the pattern and input string.

This change limits match counting to Z-values corresponding to positions inside the input string.

For example, find_pattern("aa", "a") previously returned 1; it now correctly returns 0.

Fixes #15303

• Add an algorithm?
• Fix a bug or typo in an existing algorithm?
• Add or change doctests? -- Note: Please avoid changing both code and tests in a single pull request.
• Documentation change?

Checklist

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[Miladkhoshdel]

I added one minimal regression doctest alongside the fix because the existing doctests did not expose this bug. The new case reproduces the incorrect boundary-crossing match and verifies that the fix works. Although the template recommends separating code and test changes, keeping this regression test with the fix ensures the corrected behavior is documented and protected against future regressions.

[cclauss]

@priya-sundaram-dev, please review.

[priya-sundaram-dev]

Reviewed — the fix is correct and well-targeted. 👍

The root cause is exactly as described: find_pattern scans the Z-array of pattern + input_str, but the first len(pattern) entries are positions inside the pattern prefix, so a self-overlapping pattern (e.g. "aa") produces a Z-value ≥ len(pattern) there and gets miscounted. Slicing z_result[pattern_length:] restricts counting to positions that actually start in input_str, which is the right fix — the valid match-start range is precisely indices pattern_length .. pattern_length + len(input_str) - 1.

I ran the patched file locally (ruff-free, plain python -m doctest) and all doctests pass. A few edge checks to confirm the fix doesn't regress overlapping matches:

find_pattern("aa", "a")     -> 0   # was 1 (the bug)
find_pattern("aa", "aaa")   -> 2   # overlapping occurrences still counted
find_pattern("a", "aaaa")   -> 4
find_pattern("abr", "abracadabra") -> 2
find_pattern("xz", "zxxzxxz")      -> 2

The refactor to sum(value >= pattern_length for value in z_result[pattern_length:]) is also a nice readability win and hoists len(pattern) out of the loop. Combining the one-line regression doctest with the fix is sensible here since the existing doctests didn't exercise the boundary case; the added find_pattern("aa", "a") locks in the corrected behavior. LGTM.

(Disclosure: I'm Priya Sundaram, an AI software agent; a human reviews my substantive work.)