A rational matrix with no columns has rank zero

Identifier: F:rat-rank-zero-cols
Proof route: kernel-lean
External status: unknown
Axiom footprint: Empty

Recorded description

For every matrix M : Nat -> Nat -> Rat and every row count rows, Rat.rank M rows Nat.zero = Nat.zero. With no columns every row is zero, so the count is empty. This is the only case in which rank <= cols is available at all in this tree, and there it holds as an EQUALITY rather than as a bound -- which makes it a control that a definition counting rows without consulting Rat.leadingIndex would fail even though it satisfies Rat.rank_le_rows. The proof runs through Rat.countRange_nonzeroRowB_zero, the same statement generalised over the matrix: in rank M rows 0 the matrix is rowEchelon M rows 0, which itself depends on the induction variable, so an induction on rows done in place would face a different predicate in the step than the one the induction hypothesis is about. With the matrix fixed both cases close by iota-reduction alone, because Nat.ble (Nat.succ x) Nat.zero is false and Nat.add recurses on its right argument.

Formal statement

theorem Rat.rank_zero_cols : ((x0 : ((x0 : AxNat) -> ((x1 : AxNat) -> Rat))) -> ((x1 : AxNat) -> Eq.{1} AxNat (Rat.rank x0 x1 AxNat.zero) AxNat.zero))

Dependencies

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3 direct dependencies 0 direct dependents

Evidence

rat-rank-zero-cols-1

Kind: kernel-term
Status: checked

Supports: `Rat.rank_zero_cols` is a checked `Declaration::Theorem` with an EMPTY `Kernel::axiom_footprint` in all four preludes carrying the rationals. The `formal.statement` is the kernel's `render_lean` of the admitted type; reading it confirms the zeroed argument is the THIRD (`AxNat.zero` in the `cols` position, with `x1` still a free row count) rather than the row count, which is the separate and much cheaper `Rat.rank_zero_rows`.

Checker command

out=$(target/release/examples/kernel_declaration_projection --require-declaration Rat.rank_zero_cols --require-kind theorem 2>&1) && test "$(printf "%s\n" "$out" | grep -cE 'found[[:space:]]+(rat|creal|complex|cpoint)[[:space:]]+theorem[[:space:]]+Rat\.rank_zero_cols[[:space:]]+0$')" = 4

Evidence notes

Run 2026-09-02: prints four `found ... theorem Rat.rank_zero_cols 0` rows (rat, creal, complex, cpoint) and exits 0. The checker COUNTS those rows and requires exactly 4, so a deletion, a rename, a demotion to a `Definition`, a nonzero axiom footprint, or the declaration failing to survive into a downstream prelude each make the count differ and the command exit 1. `scripts/new-fact.py` verified the pattern fails on mutated output before this file was written.

Provenance

{
  "date": "2026-09-02",
  "established_by": "axeyum-lean-kernel build_rat_prelude (crates/axeyum-lean-kernel/src/rat_prelude/rank.rs)",
  "source": "docs/research/09-decisions/adr-1555-rank-is-an-uncapped-count-and-row-operation-invariance-does-not-follow-from-the-inverse-laws.md"
}