feat(Classes): define the polynomial hierarchy - #18
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SamuelSchlesinger merged 14 commits intoAug 30, 2026
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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
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…c lemmas Close the `pairFst ∈ FP` seam: the projection is the `fstBlockTM` scanner of Cobham's algebra, so `pairFst_mem_FP` is `Cobham.fstBlock_mem_FP`, and the base and level inclusions of the hierarchy no longer take a hypothesis. The polynomial-time facts about the pairing codec are broadly useful, so they get their own surface file `Classes/P/Pairing.lean` (`pairFst_mem_FP`, `pairSnd_mem_FP`, `mem_FP_pair`, `mem_FP_pair_right`, `mem_P_preimage_pairFst/Snd`). `complClass_mono` moves next to `complClass` in `Classes/Time.lean` and `complClass_P` next to `P_compl` in `Classes/Containments.lean`. The unused `SigmaP_one` / `PiP_one` are dropped. Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
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Defines the polynomial hierarchy via certificate quantifiers
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