Commit d8d9a52
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>1 parent afa0e03 commit d8d9a52
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