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Add partial _<ᵇ_ order on Buchholz terms (WF-0, partial) (#11)
First step of the Buchholz well-foundedness workstream. Introduces the binary strict order `_<ᵇ_ : BT → BT → Set` together with the auxiliary strict order `_<Ω_` on Ω-indices. Irreflexivity and transitivity are both proved for the constructors that ship. What lands: OmegaMarkers.agda * `_<Ω_` with constructors `fin<fin` and `fin<ω` (no `ω <Ω _` case since ω is the top marker). * `<Ω-irrefl`, `<Ω-trans`, `<Ω→≤Ω`. * Named strict witnesses: `Omega0<Omega1`, `Omega0<Omegaω`, `Omega1<Omegaω`. Ordinal/Buchholz/Order.agda (new) * `_<ᵇ_` as a 7-constructor inductive relation covering: - bzero vs every non-zero head (three cases) - bOmega μ <ᵇ bOmega ν via μ <Ω ν - bOmega μ <ᵇ bpsi ν α via μ <Ω ν - bpsi μ α <ᵇ bpsi ν β via μ <Ω ν - bplus x₁ x₂ <ᵇ bplus y₁ y₂ via x₁ <ᵇ y₁ * `<ᵇ-irrefl` and `<ᵇ-trans` proved by case analysis. * Named strict-below-ψ witnesses: `bzero<Ω0`, `Ω0<Ω1`, `Ω1<Ωω`, `Ω0<ψ1-zero`. * Wired into All.agda; all above names pinned in Smoke.agda. What is *explicitly deferred* (documented in-file): * bOmega vs bplus (either direction) — needs CNF-normal-form comparison between atomic heads and additive forms. * bpsi vs bplus (either direction) — same reason. * bpsi ν α <ᵇ bOmega μ with ν ≤Ω μ — the Buchholz-1986 admissibility ordering; deferred. * Two same-binder sub-cases whose natural shape shares a binder on both sides of `_<ᵇ_` and runs into Agda 2.6.3's `--without-K` restriction on reflexive-equation elimination at irreflexivity-matching time: - bpsi ν α <ᵇ bpsi ν β with α <ᵇ β (same Ω-index). - bplus x y₂ <ᵇ bplus x z₂ with y₂ <ᵇ z₂ (same left summand). A rank-function embedding or a non-propositional Ω-equality is the next follow-up on this branch. Totality of `_<ᵇ_` and well-foundedness (`Acc _<ᵇ_`) are the subject of the follow-up WF-1 milestone once the deferred cases land. Verified: `agda proofs/agda/All.agda` and `agda proofs/agda/Smoke.agda` both exit 0 under --safe --without-K. No postulates introduced. https://claude.ai/code/session_01JRLz84fAaWvRBKyXuc4tyK Co-authored-by: Claude <noreply@anthropic.com>
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