feat: verify String.startsWith and String.skipPrefix?

[TwoFX]

This PR verifies the String.startsWith and String.skipPrefix? functions for our various pattern types.

[mathlib-lean-pr-testing]

Mathlib CI status (docs):

• ❗ Batteries/Mathlib CI will not be attempted unless your PR branches off the nightly-with-mathlib branch. Try git rebase 0f730662de0bbaa6ed07484d8f0e411de2bcd1ea --onto 61a3443a9569d548a235f785b9a33984bb7ff622. You can force Mathlib CI using the force-mathlib-ci label. (2026-03-19 18:45:14)

[leanprover-bot]

Reference manual CI status:

• ❗ Reference manual CI will not be attempted unless your PR branches off the nightly-with-manual branch. Try git rebase 0f730662de0bbaa6ed07484d8f0e411de2bcd1ea --onto cfa8c5a036d6990635c6ec50b02d0e806995cec3. You can force reference manual CI using the force-manual-ci label. (2026-03-19 18:45:15)