Rank-nullity in the row form over the rationals, given one property of the echelon form

Identifier: F:rat-rank-nullity-rows-of-pivot-section
Proof route: kernel-lean
External status: unknown
Axiom footprint: Empty

Recorded description

For every matrix M : Nat -> Nat -> Rat and all dimensions rows, cols, IF the section hypothesis holds for the row-echelon form of M, THEN Nat.add (Rat.rank M rows cols) (Rat.nullity M rows cols) = cols -- rank-nullity in the ROW form, where rank counts the independent ROWS of the echelon form rather than its pivot columns. The SECTION HYPOTHESIS is: for every row r below rows whose leading index in E := Rat.rowEchelon M rows cols is below cols (that is, Rat.nonzeroRowB E cols r = true), Rat.pivotRowOfCol E rows cols (Rat.pivotColOfRow E cols r) = r -- in words, the FIRST row whose leading index equals row r's leading index is r itself. It is the weakest form of ADR-1554 obligation 4 (rowEchelon_isEchelon) that the bridge consumes: it says nothing about zero rows sitting last, only that no earlier nonzero row shares a nonzero row's leading index. ADR-1558 landed rank-nullity in the COLUMN form (rankCols + nullity = cols, one application of Nat.countRange_compl, using no property of Rat.rowEchelon whatever) and said the obligation was relocated, entirely, into one bridge. This is the receipt for that claim: the row form is the column form with rankCols rewritten to rank across Rat.rank_eq_rankCols_of_pivotSection, and nothing else.

Formal statement

theorem Rat.rank_nullity_rows_of_pivotSection : ((x0 : ((x0 : AxNat) -> ((x1 : AxNat) -> Rat))) -> ((x1 : AxNat) -> ((x2 : AxNat) -> ((x3 : <the section hypothesis, as in Rat.rank_eq_rankCols_of_pivotSection>) -> Eq.{1} AxNat (AxNat.add (Rat.rank x0 x1 x2) (Rat.nullity x0 x1 x2)) x2))))

Dependencies

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2 direct dependencies 1 direct dependents

Evidence

rat-rank-nullity-rows-of-pivot-section-1

Kind: kernel-term
Status: checked

Supports: `Rat.rank_nullity_rows_of_pivotSection` is a checked `Declaration::Theorem` with an EMPTY axiom footprint in all four preludes, and the pattern pins the summands: `AxNat.add (Rat.rank x0 x1 x2) (Rat.nullity x0 x1 x2)` equal to `x2`, anchored to end of line. `Rat.rank_nullity` -- the column form, which was already proved and is UNCONDITIONAL -- adds `Rat.rankCols` instead and does not match, so this checker cannot be satisfied by the pre-existing theorem under a new name.

Checker command

out=$(target/release/examples/kernel_declaration_projection 2>&1) && test "$(printf "%s\n" "$out" | grep -cE '^(rat|creal|complex|cpoint)[[:space:]]+theorem[[:space:]]+Rat\.rank_nullity_rows_of_pivotSection[[:space:]]+0[[:space:]].*-> Eq\.\{1\} AxNat \(AxNat\.add \(Rat\.rank x0 x1 x2\) \(Rat\.nullity x0 x1 x2\)\) x2\)\)\)\)$')" = 4

Evidence notes

Run 2026-09-02: the pattern matches the real output at exactly the required count. `scripts/new-fact.py` verified it FAILS on mutated output (digits perturbed, true -> false, 0 -> 1) before this file was written, so a pattern anchored on the surrounding prose rather than on the finding would have been rejected there.

Provenance

{
  "date": "2026-09-02",
  "established_by": "axeyum-lean-kernel build_rat_prelude (crates/axeyum-lean-kernel/src/rat_prelude/rank_bridge.rs)",
  "source": "docs/research/09-decisions/adr-1562-the-bridge-orientation-is-the-finding-and-obligation-4-collapses-to-one-section-equation.md"
}