Scaling a row of a rational matrix by a nonzero factor is inverted by scaling by its reciprocal

Identifier: F:rat-row-scale-inverse
Proof route: kernel-lean
External status: unknown
Axiom footprint: Empty

Recorded description

For every row index i, every scalar k : Rat with Not (Eq Rat k Rat.zero), and every matrix M : Nat -> Nat -> Rat, rowScale i (inv k) (rowScale i k M) r c = M r c for all r, c. The side condition is stated as k != 0 rather than 0 < k so that it covers the NEGATIVE pivots row reduction actually produces; Rat.mul_inv_cancel_of_ne_zero is the form of the field law that carries it. The rational prelude has no one_mul, so the final step runs 1 * a = a * 1 = a through Rat.mul_comm and Rat.mul_one.

Formal statement

theorem Rat.rowScale_inverse : ((x0 : AxNat) -> ((x1 : Rat) -> ((x2 : ((x2 : AxNat) -> ((x3 : AxNat) -> Rat))) -> ((x3 : ((x3 : Eq.{1} Rat x1 Rat.zero) -> False)) -> ((x4 : AxNat) -> ((x5 : AxNat) -> Eq.{1} Rat (Rat.rowScale x0 (Rat.inv x1) (Rat.rowScale x0 x1 x2) x4 x5) (x2 x4 x5)))))))

Dependencies

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5 direct dependencies 0 direct dependents

Evidence

rat-row-scale-inverse-1

Kind: kernel-term
Status: checked

Supports: `Rat.rowScale_inverse` 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 side condition is the negation `((x3 : Eq Rat x1 Rat.zero) -> False)` rather than a positivity hypothesis, and that the outer scaling factor is literally `Rat.inv x1` rather than an unrelated second scalar.

Checker command

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

Evidence notes

Run 2026-09-02: prints four `found ... theorem Rat.rowScale_inverse 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/echelon.rs)",
  "source": "docs/research/09-decisions/adr-1554-the-pivot-is-computed-not-extracted-and-the-fuel-is-exact.md"
}