Mathlib v4.30 source proposition Nat.lt_of_mul_lt_mul_left

Identifier: F:ml430-nat-lt-of-mul-lt-mul-left-234e8530
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {a b c : ℕ}, a * b < a * c → b < c

Dependencies

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4 direct dependencies 2 direct dependents

Evidence

kernel-Nat.lt_of_mul_lt_mul_left

Kind: kernel-term
Status: checked

Supports: `Nat.lt_of_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.mul x0 x1) (AxNat.mul x0 x2)) -> AxNat.lt x1 x2))))`. Built in the same new file with NO positivity hypothesis, by contradiction: `Nat.lt_or_ge b c` splits into `Lt b c` (the goal directly) and `Le c b`, the latter refuted by chaining `Nat.mul_le_mul_left` against the strict hypothesis via `Nat.le_trans` into `Lt (mul a b) (mul a b)`, discharged by `Nat.lt_irrefl`. No `Nat.rec` case split.

Checker command

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

footprint-Nat.lt_of_mul_lt_mul_left

Kind: kernel-term
Status: checked

Supports: axiom_footprint: [] for `Nat.lt_of_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.lt_of_mul_lt_mul_left

Kind: kernel-term
Status: checked

Supports: Coverage for `Nat.lt_of_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.lt_of_mul_lt_mul_left`",
      "where": "mathlib4 commit c5ea00351c28e24afc9f0f84379aa41082b1188f (v4.30.0)",
      "who": "the Mathlib contributors",
      "year": 2026
    }
  ],
  "source": "statement-only extraction of `Nat.lt_of_mul_lt_mul_left` from Mathlib v4.30.0; no proof value was exposed"
}