feat: verification of String.dropWhile and String.takeWhile

[TwoFX]

This PR verifies the String.dropWhile and String.takeWhile functions.

It also includes a refactor of the PatternModel class so that the not_matches_empty condition is moved into a separate typeclass StrictPatternModel. This allows string patterns to implement LawfulForwardPatternModel unconditionally, which means that more of the general theory about patterns directly applies to string patterns without having to do a case distinction for empty strings.

This PR also includes a study of the PatternModel machinery given to slices s and t such that s.copy = t.copy. From these results, we deduce statements like s.copy.startsWith pat = s.startsWith pat (which is far from obvious!).

TODOs:

• the material in TakeDrop/Basic.lean still needs to be dualized to dropEndWhile/takeEndWhile and friends
• specialized lemmas for our pattern types are still missing
• there are some TODO: move comments on displaced lemmas

[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 a8bbc95d9fe665255a5c98fb402546233e760636 --onto 4786e082dca22873d14d2a5b9b7c8843380c6e78. You can force Mathlib CI using the force-mathlib-ci label. (2026-03-27 13:25:49)

[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 a8bbc95d9fe665255a5c98fb402546233e760636 --onto cfa8c5a036d6990635c6ec50b02d0e806995cec3. You can force reference manual CI using the force-manual-ci label. (2026-03-27 13:25:51)