ABI Layer 3: div/mod completeness of the block partition — closes the Layer-2 residual#42
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Adds Chapeliser.ABI.Invariants, a new machine-checked theorem deeper than
and distinct from the Layer-2 Partition tiling proof. Partition.idr proved
the block partition is a gapless, non-overlapping tiling for all n,k but
explicitly left open the arithmetic residual `sumNat (perItemCounts n k) = n`
("the only div/mod obligation"). This module discharges exactly that.
blockCountsComplete : (n,k') -> sumNat (blockCounts n k') = n proves every
item is covered exactly once, for ALL n and all k>0, via the Euclidean
division theorem (contrib Data.Nat.Division) plus a self-contained count of
remainder slots. Counts are expressed with the public-export divNatNZ/modNatNZ
(the reducing form of Prelude div/mod on a positive divisor) so the proof and
its concrete controls reduce at the type level.
Includes: a sound+complete Dec (decCoversExactly), positive controls
(covers10over3, covers12over4 by Refl + via the general theorem), and
non-vacuity/negative controls (notCovers10as9, dec10over3as9No,
remainderCountMatters). Genuine proof: no believe_me/postulate/assert_total/
sorry/admitted. Builds clean (0 warnings); a deliberately false variant is
rejected by the type checker.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01A6PSzJWpRxtzGDjUCEh7Mx
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Summary
Layer 3 (second, deeper invariant): proves
sumNat (blockCounts n k) = nfor all n and all k>0 — every item is covered exactly once. This discharges the exactdiv/modarithmetic residual the Layer-2Partitiontiling proof explicitly left open, via the Euclidean division theorem plus a self-contained count of remainder slots.New module
Chapeliser.ABI.Invariants(imports the Layer-2Partitionmodel; usesdivNatNZ/modNatNZbecause Preludediv/moddon't reduce at the type level). Sound+completeDec, corollaries tying back toPartitionComplete, positive + non-vacuity controls.Testing
Idris2 0.7.0
--build→ exit 0, zero warnings (6 modules). Adversarial: false variants (=9,=11) rejected.build/removed. Nobelieve_me/postulate/sorry.🤖 Generated with Claude Code
https://claude.ai/code/session_01A6PSzJWpRxtzGDjUCEh7Mx
Generated by Claude Code