proofs(lean4): close ET-2 — L1 conversion is decidable#24
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Completes the MECH-1 milestone that Conversion.lean recorded as open.
Purely additive: four new modules under Systemet/L1/, plus append-only
entries in Systemet.lean, MANIFEST and Audit.lean. Syntax/Normal/Hsub/
Conversion.lean are untouched.
Landed (all statements verbatim from the ET-2 ledger entry):
SubstLemmas.lean substitution/weakening exchange family, incl.
nf_substTy — normalization commutes with hereditary
substitution (the beta case's whole content)
Soundness.lean soundness : DefEq t (embNf (nf t))
Completeness.lean completeness : DefEq t u -> nf t = nf u
defEq_iff_nf : DefEq t u <-> nf t = nf u
Decidable.lean decEqNf / decEqSp (hand-rolled mutual DecidableEq --
`deriving` does not work across the Nf/Sp pair)
decDefEq : (t u : Ty G k) -> Decidable (DefEq t u)
Method: exchange coordinates (swp/remSwap) with an Eq.rec cast toolkit
whose push lemmas are all `by subst h; rfl`, so definitional proof
irrelevance makes diagonal casts vanish. The subst/subst exchange needs a
SUM measure (kindSize k + kindSize b, tag, size) because the miss-then-hit
case folds appSp at the second kind.
Evidence (measured, not asserted):
clean rebuild from scratch Build completed successfully (16 jobs)
scripts/check-proofs.sh PASS, 20/20 headline items in trusted base
scripts/scan-dangerous.sh PASS
scripts/check-proof-status PASS after the doc update below
#print axioms on soundness, completeness, defEq_iff_nf, decDefEq,
decEqNf, decEqSp, nf_substTy -> [propext, Quot.sound] only; no
sorryAx, no user axioms, no Classical.choice.
Scope, stated honestly: DefEq here is beta + equivalence + congruence.
eta is NOT included and remains an OPEN row, so ET-3 is unaffected --
its eta and model directions stay open. PROOF-STATUS.adoc moves ET-2 to
proven *for the L1 core calculus* under the file's existing core-vs-full-
theory caveat, and the now-satisfied "ET-2 (finish)" OPEN row is removed.
Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
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July 22, 2026 08:41
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Closes obligation ET-2 (decidability of L1 conversion) for the mechanized core calculus. This was the upstream blocker sitting under ET-14 / Phase 6.
Stacking and provenance
proofs/lean4-l1-l2, notmain. That branch is pushed but has no PR of its own; this stacks on it so the mechanization lands as one reviewable lineage. Retarget tomainifproofs/lean4-l1-l2merges first.lake buildas the judge). Two attempts reached full closure; this is the stronger one (38 theorems, faithful Keller–Altenkirch decomposition). Attempt artefacts were written to a scratch directory and applied here by hand — the campaign's own verify and adversarial-review stages were cut short by a monthly spend limit, so every check below was re-run manually rather than taken on an agent's word.What landed
Purely additive —
Syntax/Normal/Hsub/Conversion.leanare byte-identical; the only edits to existing files are append-only registry entries (18 insertions, 0 deletions).Systemet/L1/SubstLemmas.leannf_substTy— normalization commutes with hereditary substitution (the whole content of the beta case)Systemet/L1/Soundness.leansoundness : DefEq t (embNf (nf t))Systemet/L1/Completeness.leancompleteness : DefEq t u → nf t = nf u;defEq_iff_nf : DefEq t u ↔ nf t = nf uSystemet/L1/Decidable.leandecEqNf/decEqSp(hand-rolled mutualDecidableEq—derivingdoes not work across theNf/Sppair);decDefEq : (t u : Ty Γ k) → Decidable (DefEq t u)File layout matches the slots the ET-2 ledger entry already named.
Evidence (measured)
rm -rf .lake/build: Build completed successfully (16 jobs)scripts/check-proofs.sh lean4— PASS, audit reports 20/20 headline items within the trusted basescripts/scan-dangerous.sh— PASSscripts/check-proof-status.sh— PASS (exit 0) after thePROOF-STATUS.adocupdate#print axiomsonsoundness,completeness,defEq_iff_nf,decDefEq,decEqNf,decEqSp,nf_substTy→[propext, Quot.sound]only. NosorryAx, no user axioms, and not evenClassical.choice.DefEq/nf: none — the new theorems quantify over the originals fromConversion.lean, with generalΓandkand no added hypotheses.Scope — what this does not claim
DefEqhere is β + equivalence + congruence. η is not included, and remains an OPEN row. ET-3 is unaffected — its η and model directions stay open.PROOF-STATUS.adocmarks ET-2 proven for the L1 core calculus, under the caveat that file already carries about core-vs-full-theory claims, and drops the now-satisfied "ET-2 (finish)" OPEN row.ET-14 (TEA erasure) is untouched and remains the headline open problem. The campaign's TEA design track produced nothing — all three panelists and the judge died on the spend limit — so the ET-16 / L0 lowering ADR is still owner-gated and unstarted.
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