Mathlib v4.30 source proposition Nat.ModEq.cancel_left_of_coprime

Identifier: F:ml430-nat-modeq-cancel-left-of-coprime-f89af373
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m a b c : ℕ}, m.gcd c = 1 → c * a ≡ c * b [MOD m] → a ≡ b [MOD m]

Dependencies

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1 direct dependencies 0 direct dependents

Evidence

kernel-Nat.mod_eq_cancel_left

Kind: kernel-term
Status: checked

Supports: `Nat.mod_eq_cancel_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) -> ((x4 : Eq.{1} AxNat (AxNat.gcd x0 x3) (AxNat.succ AxNat.zero)) -> ((x5 : AxNat.modEq x0 (AxNat.mul x3 x1) (AxNat.mul x3 x2)) -> AxNat.modEq x0 x1 x2))))))`.

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example nat_theorem_inventory 2>/dev/null | awk -F'	' '$1 == "Nat.mod_eq_cancel_left" && $3 == "((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat) -> ((x3 : AxNat) -> ((x4 : Eq.{1} AxNat (AxNat.gcd x0 x3) (AxNat.succ AxNat.zero)) -> ((x5 : AxNat.modEq x0 (AxNat.mul x3 x1) (AxNat.mul x3 x2)) -> AxNat.modEq x0 x1 x2))))))"' | wc -l)" -ge 1

footprint-Nat.mod_eq_cancel_left

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] for `Nat.mod_eq_cancel_left`. The `nat` prelude's trusted surface is empty (no Axiom, no Opaque, no Quotient), which bounds every declaration in it, including `Nat.mod_eq_cancel_left`.

Checker command

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

Provenance

{
  "date": "2026-08-29",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.ModEq.cancel_left_of_coprime` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.ModEq.cancel_left_of_coprime`",
      "where": "mathlib4 commit c5ea00351c28e24afc9f0f84379aa41082b1188f (v4.30.0)",
      "year": 2026,
      "attribution": "the proposition was read from the pinned statement-only inventory; the proof term and tactic trace were not consulted"
    }
  ]
}