feat(proofs): program equalities over all inputs: integer order, linear_arithmetic over the runtime, fold laws, and the law form (PF-20..PF-23, #117)

[afflom]

Closes #117 (plan item L3.4 of #106).

The probe (first step of the issue)

Re-run on main at 32afe5da with Lean 4.32.1 (recorded in tests/runtime-library/LAWS.md):

• Integer kernel (clampStep monotonicity, written with less_than) under linear_arithmetic{definitions: [clampStep]} fails with omega … No usable constraints found. The script unfolds none of less_than, quotient or negate, and nothing splits the clamp's ifs. This confirms the three gaps from the earlier probe on the issue.
• Two folds (doubledTotal xs = total xs + total xs): omega sees two unrelated listFold atoms. The missing lemma is a normal form for additive folds: a fold whose step adds k x equals initial + Σ k, and Σ is linear in k. With both sides in that form, the equality is linear over one atom.

What changes

1. Integer order (PF-20). Under language 1.2, <= and < take two operands of one integer type (Nat, Int, or one fixed-width type) and lower to Lean's order on that type. Language 1.1 keeps Nat only and is byte-identical.

2. linear_arithmetic reads the runtime (PF-21). The script is still one fixed script (SPEC.md §17.12). Its simplifier pass now covers:

   • the runtime bridges for less_than on every integer key type, equal, negate, quotient and remainder (a zero divisor becomes an if; truncating division by a positive or negative literal becomes Euclidean division with a sign case);
   • the registered rewrite set of the laws module;
   • fixed-width order and equality on mathematical values, with literals evaluated first.

   A second pass reads every remaining fixed-width value as the bounded natural number of its bits, so omega knows its range. Then all_goals (repeat' split) splits every if/match left in the goal, and all_goals omega decides each case.

3. The fold laws as LexLean source (PF-22). LexLeanRuntime.Laws (namespace LexLeanLaws) is the fifth native runtime module, importing Portable and Collections. It contains:

   • sumNat/sumInt;
   • the bridges;
   • the additive-fold normal form for list, set and map folds (a map's fold is the fold of its entries);
   • linearity of sums, and constant contributions as length;
   • invariance of sums under permutation (sumNat_perm, sumInt_perm), append and reverse, plus the cons- and snoc-folds;
   • fusion, the fold of a pair, and the fold of an append.

   Every row pins its exact axioms, and none uses Classical.choice. The registry language/runtime-1.2/laws.toml names the rewrite set and the applicable laws; the runtime build refuses a registry that names anything the module lacks.

4. The law proof form (PF-23). law{name, type_arguments, arguments, premises} applies a registered law (list_fold_fusion, list_fold_pair, list_fold_append). Linking checks the counts, the types and that each premise names a prior theorem; Lean checks the statement. It lowers to exact @L ….

5. Corpus. examples/fold-laws covers integer kernels (clamped requantization monotone and in range for every Int, truncating division by ±7, UInt8/Int8 order and equality) and fold equalities over lists, sets and maps (doubling, counting, weighting, append both ways, reverse, debits against credits, a map's keys and values). It includes a one-pass statistics fold by list_fold_pair and a fused parity fold by list_fold_fusion; decide appears only on closed instances.

Provenance of the laws module

Commit b58f1952 is an oracle checkpoint: the authored Laws.lean plus the runtime migration's exporter narrowed to Laws. The next commit removes it, as #144's migration did, and RT-01 now also requires laws-oracle to be absent. Running run.sh from the checkpoint gives:

• the export printed in the notation is the committed Laws.lex.tex, byte for byte;
• the regenerated environment matches in 118 of 119 rows, and the remaining row (a sparse casesOn) is equal up to binder names.

Identity

semantic_ir 11, lean_backend 18, proof_lowering 3. Every language-1.2 lock, golden, fixture expectation, calculus package, production export and verified record was regenerated by the repository's writers. The existing linear_arithmetic proofs of models, recursion, reasoning, production(-coverage) and compiler all verify under the extended script. Languages 1.0 and 1.1, Atlas included, are byte-identical to main. New reserved names (BitVec, LexLeanLaws, beq_iff_eq, decide_eq_true_eq, decide_eq_false_iff_not, reduceIte) keep a module or binder from capturing what the script writes. The register has 391 IDs after merging main; there is no new diagnostic code.

Evidence

• R3 order in the history: register rows, scenarios and failing tests (cb02f8d3), then the checkpoint, then the implementation.
• Local runs on Linux x86-64 with pinned Lean:
  • fmt, validate-model, validate-spec-links, clippy, test, features, check-test-inventory, check-golden and check-reproducibility passed;
  • verify-examples --write verified every example except Atlas;
  • cargo test -p repo-conformance: 384 cases passed and 6 failed on the first run; all 6 now pass, and the runtime cases pass against the merge of main.
• Planted defects for each case are in VERIFICATION.md ("the fold laws can fail").

Full just vv runs in CI.

🤖 Generated with Claude Code

https://claude.ai/code/session_018risLc2kcmeqW2hL5pcQNs

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[afflom]

atlas (cert4-replay) (job 117145703530) failed before any verification ran. Its actions/download-artifact step reported Unable to download artifact(s): Failed to ListArtifacts: Unable to make request: ECONNRESET. The next step, ci-parts/atlas-xtask/xtask ci-summary, then exited 126 (Permission denied) because the download had not completed.

This is the runner's connection to the artifact service, not this PR: the job never reached a test body, and this PR does not change the Atlas or the workflow. The other Atlas parts downloaded the same artifacts successfully.

Re-running the failed jobs from this session is refused (403 Resource not accessible by integration), so it needs one re-run from someone with Actions write access. I'm keeping the PR watched until it is green.

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[afflom]

Run 38941418389 on aa58c1d7 has finished: 21 of 22 checks are green, including the 14 Atlas jobs but atlas (join). The one red check is atlas (cert4-replay), which died at "Take the gate's executable and the shards before this one from the run" with Failed to ListArtifacts: ECONNRESET, before any shard ran. atlas (join) was skipped because it needs it, so nothing about this head has been judged by a test.

The failed job needs a re-run, and neither this session nor the one that wrote the PR can do it: rerun-failed-jobs returns 403 (Resource not accessible by integration) for both of us. @humuhumu33 or @afflom, could one of you with Actions access press "Re-run failed jobs" on the run? That is the one allowed re-run (a failure before any test body). Nothing else on this head needs to change, and it still contains main 80331cb4.

If nobody has re-run it by 21:15Z, I will instead push a small, real change to this branch (RT-01 and RT-04 do not yet list LexLeanLaws, the fifth runtime namespace this PR adds, so a module or a source literal under it would escape both gates) and take the fresh run that push starts. That costs about 500 runner-minutes against about 60 for the re-run, so I would rather not.

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[humuhumu33]

The atlas (cert4-replay) failure was infrastructure, not the change: its first step could not download the run's artifacts (Failed to ListArtifacts: Unable to make request: ECONNRESET, exit 126), so no verification ran. I re-ran the run's failed jobs.

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[afflom]

Maintainer review: approved at aa58c1d7.

Checked against #117's acceptance and the repository rules, on the head that ran:

• CI. Run 38941418389, attempt 2 (hh re-ran the one job that died at artifact download): all 22 checks green, including the 14 Atlas jobs and atlas (join). The head contains main 80331cb4.
• [0.5.0] Lemmas for universal equalities over list folds #117, first step. The probe is recorded before the change, in tests/runtime-library/LAWS.md: the integer kernel fails under linear_arithmetic with No usable constraints found, and the two folds need a normal form for additive folds. The missing lemma is named, not assumed.
• [0.5.0] Lemmas for universal equalities over list folds #117, accepted when. examples/fold-laws states equalities over lists, sets and maps and integer kernels for all inputs. decide appears only on the two closed instances (doubled_total_closed, requantize_closed).
• R3. The register rows, scenarios and failing tests come first (cb02f8d3), then the oracle checkpoint, then the implementation.
• R8 / provenance. Laws.lean is an oracle checkpoint removed by the next commit, as feat(#90): the runtime of generated Lean is native modules (RT-01..RT-04) #144's migration was, and RT-01 now requires its absence. The exported notation is byte-identical to the committed Laws.lex.tex.
• R9 / R10. Every row of the laws module pins its exact axioms and none uses Classical.choice. The language-1.2 identity moved once (semantic_ir 11, lean_backend 18, proof_lowering 3) and every record was regenerated by the repository's writers. Languages 1.0 and 1.1 are byte-identical.

One gap found in review, not a blocker: RT-01 and RT-04 do not list LexLeanLaws, the fifth runtime namespace, so a copy of the laws module in a generated module or a source literal would pass them. I have the fix and its plants (the list is now asserted equal to the namespaces the shipped runtime declares; both halves fail without it). It is not pushed here, because a push would void this run; it rides with my next PR.

Merging with a merge commit, pinned to this head.

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