Mathlib v4.30 source proposition Nat.add_mod_left

Identifier: F:ml430-nat-add-mod-left-6b337077
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ (x z : ℕ), (x + z) % x = z % x

Dependencies

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9 direct dependencies 0 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.add_mod_left

Kind: kernel-term
Status: checked

Supports: ∀ (x z : ℕ), (x + z) % x = z % x

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- add_mod_left 2>/dev/null | grep -Ec '^Nat\.add_mod_left[[:space:]]')" -ge 1

Evidence notes

`build_nat_prelude` admits `Nat.add_mod_left` through the trusted `Kernel::add_declaration` gate (declared in `nat_prelude/div_mod_lemmas.rs`'s `declare_add_div_mod_shift_family`). Route: no positivity hypothesis is given, so the divisor `x` is case-split via `cases_zero_succ`; at `x=0` the goal collapses via `zero_add`, at `x=succ xpred` it reads `x+z` as `z+x*1` (`add_comm` then `mul_one` backwards) to reach `div_mod_shift`'s reusable `(n+dd*k)%dd = n%dd` fact at `dd:=x`, `n:=z`, `k:=1`. `nat_theorem_inventory`'s rendered type for `Nat.add_mod_left` is `(x0:AxNat)->(x1:AxNat)->Eq (mod (add x0 x1) x0) (mod x1 x0)`, matching this fact's `formal.statement` verbatim (`x0`=x, `x1`=z). `nat_theorem_inventory` exits non-zero for a name that does not exist, and the `grep -c` count (tested `-ge 1`, not piped into `grep -q`) requires the admitted declaration to actually be printed.

footprint-Nat.add_mod_left

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the Nat prelude's trusted surface is empty

Checker command

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

Evidence notes

`nat_axiom_inventory --require-axiom-free nat` enumerates the built Nat environment and exits non-zero unless it admits no Axiom, Opaque or Quotient declaration (measured: axiom=0 opaque=0 quotient=0). A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds every individual theorem's footprint by []. `nat_prelude_tests::every_nat_declaration_is_checked_and_axiom_free` additionally checks this theorem's own `Kernel::axiom_footprint` directly (via `theorem_names`, which now lists it), and `add_div_mod_shift_family_applies_at_concrete_discriminating_instances` re-checks it plus a concrete instantiation (x=4, z=7, so `11%4=3=7%4`) chosen to discriminate a swapped argument via `def_eq`.

Provenance

{
  "date": "2026-08-29",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Nat.add_mod_left` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.add_mod_left`",
      "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"
    }
  ]
}