Mathlib v4.30 source proposition Nat.ModEq.add

Identifier: F:ml430-nat-modeq-add-1561afa8
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n a b c d : ℕ}, a ≡ b [MOD n] → c ≡ d [MOD n] → a + c ≡ b + d [MOD n]

Dependencies

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3 direct dependencies 2 direct dependents

Evidence

kernel-Nat.mod_eq_add

Kind: kernel-term
Status: checked

Supports: `Nat.mod_eq_add` 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 : AxNat) -> ((x5 : AxNat.modEq x0 x1 x2) -> ((x6 : AxNat.modEq x0 x3 x4) -> AxNat.modEq x0 (AxNat.add x1 x3) (AxNat.add x2 x4))))))))`.

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example nat_theorem_inventory 2>/dev/null | awk -F'	' '$1 == "Nat.mod_eq_add" && $3 == "((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat) -> ((x3 : AxNat) -> ((x4 : AxNat) -> ((x5 : AxNat.modEq x0 x1 x2) -> ((x6 : AxNat.modEq x0 x3 x4) -> AxNat.modEq x0 (AxNat.add x1 x3) (AxNat.add x2 x4))))))))"' | wc -l)" -ge 1

footprint-Nat.mod_eq_add

Kind: exhaustive-enumeration
Status: checked

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

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.add` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.ModEq.add`",
      "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"
    }
  ]
}