Mathlib v4.30 source proposition Nat.mul_lt_mul_left

Identifier: F:ml430-nat-mul-lt-mul-left-af33301e
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Nat.mul_lt_mul_left` in the pinned Mathlib v4.30 source.

Formal statement

∀ {a b c : ℕ}, 0 < a → (a * b < a * c ↔ b < c)

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

4 direct dependencies 4 direct dependents

Evidence

kernel-Nat.mul_lt_mul_left

Kind: kernel-term
Status: checked

Supports: `Nat.mul_lt_mul_left` is admitted as a Theorem with EXACTLY the stated shape, pinned verbatim via the kernel's own renderer: `((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat) -> ((x3 : AxNat.lt AxNat.zero x0) -> Iff (AxNat.lt (AxNat.mul x0 x1) (AxNat.mul x0 x2)) (AxNat.lt x1 x2)))))`. Built in a new file, nat_prelude/mul_order_lemmas.rs, as `mp` = `Nat.lt_of_mul_lt_mul_left` (the positivity-free cancellation lemma) plus `mpr` = a positive-monotone core built from `Nat.mul_le_mul_left` at `succ b` bridged into `Lt (mul a b) (mul a c)` via `Nat.add_le_add_left`/`Nat.le_trans`, using `mul_succ`/`add_succ`/`add_zero`'s refl-provable defining equations to bridge the two ends with no explicit rewrite.

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example nat_theorem_inventory 2>/dev/null | awk -F'\t' '$1 == "Nat.mul_lt_mul_left" && $3 == "((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat) -> ((x3 : AxNat.lt AxNat.zero x0) -> Iff (AxNat.lt (AxNat.mul x0 x1) (AxNat.mul x0 x2)) (AxNat.lt x1 x2)))))"' | wc -l)" -ge 1

footprint-Nat.mul_lt_mul_left

Kind: kernel-term
Status: checked

Supports: axiom_footprint: [] for `Nat.mul_lt_mul_left`. The `nat` prelude's trusted surface stays 0: no Axiom, no Opaque, no Quotient.

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_axiom_inventory -- --include-constructed --require-axiom-free nat

coverage-Nat.mul_lt_mul_left

Kind: kernel-term
Status: checked

Supports: Coverage for `Nat.mul_lt_mul_left` derived from `kernel.environment()` directly, not from an inventory list -- a Nat-namespace declaration missing from `theorem_names` fails this test by name.

Checker command

test "$(cargo test -p axeyum-lean-kernel --lib nat_prelude::nat_prelude_tests::every_nat_declaration_is_checked_and_axiom_free -- --exact 2>&1 | grep -Ec 'test result: ok\. 1 passed; 0 failed')" -ge 1

Provenance

{
  "date": "2026-08-29",
  "established_by": "lane nat-mul-order (2026-08-29): built directly as a kernel declaration in the new file nat_prelude/mul_order_lemmas.rs. Verified against the pinned Mathlib v4.30.0 source (commit c5ea00351c28e24afc9f0f84379aa41082b1188f) that this is a standard Nat order/multiplication/division lemma about the SAME Nat.mul/Nat.lt/Nat.div this kernel already has (no new definition introduced), so mirroring it is honest under the mirror-flip criterion.",
  "prior_art": [
    {
      "attribution": "the proposition was read from the pinned statement-only inventory; the proof term and tactic trace were not consulted",
      "what": "the theorem declaration `Nat.mul_lt_mul_left`",
      "where": "mathlib4 commit c5ea00351c28e24afc9f0f84379aa41082b1188f (v4.30.0)",
      "who": "the Mathlib contributors",
      "year": 2026
    }
  ],
  "source": "statement-only extraction of `Nat.mul_lt_mul_left` from Mathlib v4.30.0; no proof value was exposed"
}