feat(Classes): define the polynomial hierarchy - #18
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Define certificate quantifiers over the pair codec (polyExistsLang / polyForallLang and their class operators), the levels SigmaP / PiP by the standard recursion through complement classes, and PH as their union. Prove quantifier and class-level complement duality, monotonicity, the recursion laws, and level inclusions, the latter conditional on the single machine-engineering seam `pairFst ∈ FP` (first-component pair decoding), isolated as a hypothesis following the WitnessNTMConstruction pattern. Ported from leanprover/cslib#192, re-proved against this library's machine model and Set-based languages. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
# Conflicts: # Complexitylib/Classes.lean
# Conflicts: # Complexitylib/Models/RoseTreeMachine/Data.lean
`dev` de-exposed `Encoding/DataEncode.lean` as part of the module-interface minimization, but this branch adds `bitstringEncode` together with `bitstringEncode_def` and `bitstringEncode_injective`, both of which need the definition's body to typecheck. Expose the single definition rather than re-exposing the whole module. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
`dev` gained commit 60118c9 ("revert encoding stuff") via the SamuelSchlesinger#23 merge, which deleted Complexitylib/Encoding.lean and Encoding/Delimit.lean and inlined the delimiting logic back into Encoding/Pairing.lean. That is the exact refactor this branch performs, so the merge conflicts on both files. Resolved in favour of this branch's extraction: keep Encoding.lean and Encoding/Delimit.lean, and restore this branch's Encoding/Pairing.lean (`pair x y = delimit x ++ y`). No declaration is lost — every declaration in dev's Pairing.lean is present in this branch's Pairing.lean plus Delimit.lean. Also restore the `Complexitylib.Encoding` import in Complexitylib.lean, which the merge dropped because dev deleted the line. The two `pair`s produce the same bitstring but associate differently (`(A ++ sep) ++ y` here vs `A ++ (sep ++ y)` on dev), so five consumer proofs needed the shape realigned: - SAT/ThreeSAT/Verifier: unfold `delimit` in the `foldl_append` chain - TuringMachine/Subroutines/PairEmit/Internal: add `delimit` to `simpa` - TuringMachine/UTM/Internal/Init: add `delimit` to `simp` - Classes/NP/Internal/PairBuildTM: drop now-unused `List.append_assoc` - Classes/PPoly/Advice: drop now-unused `List.append_assoc` Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
…ncoding # Conflicts: # Complexitylib/Encoding/Data.lean # Complexitylib/Models.lean
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Revert the incidental docstring rewording and the `@[expose]` widening of `Encoding/Data.lean` introduced by this branch, keeping the diff against dev limited to substantive changes. Docstrings for newly added declarations are retained. Co-Authored-By: Claude Opus 5 (1M context) <noreply@anthropic.com>
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This is an in-progress PR to add the definition of the Polynomial Time hierarchy.
Depends on #21
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