The rank of a rational matrix is at most its column count, given one property of the echelon form

Identifier: F:rat-rank-le-cols-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.Le (Rat.rank M rows cols) cols. 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-1555 stated this bound as OPEN and explained why: Rat.rank is a count over [0, rows), so bounding it by cols is a claim that the echelon form has at most one pivot per column. The column-form counterpart Rat.rankCols_le_cols is free -- one Nat.countRange_le, because a count over [0, cols) cannot exceed cols whatever the predicate does -- and this is that free bound transported across the bridge. Nothing is re-proved; the asymmetry ADR-1558 recorded between the row and column forms is exactly the asymmetry the bridge removes.

Formal statement

theorem Rat.rank_le_cols_of_pivotSection : ((x0 : ((x0 : AxNat) -> ((x1 : AxNat) -> Rat))) -> ((x1 : AxNat) -> ((x2 : AxNat) -> ((x3 : <the section hypothesis, as in Rat.rank_eq_rankCols_of_pivotSection>) -> AxNat.le (Rat.rank x0 x1 x2) x2))))

Dependencies

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2 direct dependencies 1 direct dependents

Evidence

rat-rank-le-cols-of-pivot-section-1

Kind: kernel-term
Status: checked

Supports: `Rat.rank_le_cols_of_pivotSection` is a checked `Declaration::Theorem` with an EMPTY axiom footprint in all four preludes, and the pattern pins the CONCLUSION to `AxNat.le (Rat.rank x0 x1 x2) x2` anchored to end of line. `x2` is the COLUMN binder and `x1` the row binder, so the already-free `rank <= rows` (`Rat.rank_le_rows`, bounding by `x1`) does not match this pattern -- which is the distinction the fact is about, and the one a name alone cannot make.

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_le_cols_of_pivotSection[[:space:]]+0[[:space:]].*-> AxNat\.le \(Rat\.rank x0 x1 x2\) 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"
}