Mathlib v4.30 source proposition Int.modEq_one

Identifier: F:ml430-int-modeq-one-01d9de39
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Int.modEq_one` in the pinned Mathlib v4.30 source.

Formal statement

∀ {a b : ℤ}, a ≡ b [ZMOD 1]

Dependencies

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

3 direct dependencies 0 direct dependents

Evidence

statement-Int.modEq_one

Kind: kernel-term
Status: checked

Supports: `Int.modEq_one` is admitted with the kernel-rendered type `((x0 : Int) -> ((x1 : Int) -> Int.ModEq Int.one x0 x1))`, which is this ledger row's statement with the binders made explicit -- Mathlib's `{a b}` are implicit, a surface-syntax difference with no mathematical content. `Int.ModEq n a b` is defined here as `Int.emod a n = Int.emod b n`, which is Mathlib's `a % n = b % n`. CRUCIALLY, the rendered type carries NO `Int.lt Int.zero` premise: the ledger row quantifies over all arguments with no positivity hypothesis, and so does the theorem.

Checker command

test "$(scripts/cargo-serialized.sh test -p axeyum-lean-kernel --lib the_modeq_ledger_rows_are_stated_without_a_positivity_hypothesis 2>&1 | grep -Ec 'test result: ok\. 1 passed; 0 failed')" -ge 1

Evidence notes

The statement pin exists because the footprint check cannot carry this claim: a theorem stating something WEAKER -- carrying the `0 < n` hypothesis that every other congruence lemma in `modeq.rs` carries, or with `a`/`b` swapped -- has exactly the same empty footprint and passes the footprint sweep unchanged. Mutation-verified: changing `Int.one` to `Int.zero` in the pinned string kills exactly one test, this one, and nothing else (23 passed / 1 failed). Anchored on the exact `1 passed; 0 failed` count.

footprint-modEq_one

Kind: kernel-term
Status: checked

Supports: axiom_footprint: [] for `Int.modEq_one` and for `Int.emod_neg` beneath them. The `integer` prelude's trusted surface is 0: no Axiom, no Opaque, no Quotient. Presence is checked in the same sweep -- axiom_footprint of a name that is interned but never DECLARED is vacuously empty, so a footprint check alone cannot distinguish an axiom-free theorem from an absent one.

Checker command

test "$(scripts/cargo-serialized.sh test -p axeyum-lean-kernel --lib int_prelude::int_prelude_tests::derived_laws_have_no_axiom_footprint -- --exact 2>&1 | grep -Ec 'test result: ok\. 1 passed; 0 failed')" -ge 1

Evidence notes

Counting Declaration::Axiom alone would not suffice in this kernel (Opaque has no proof body and Quotient admits Quot.sound); axiom_footprint covers all three. Anchored on the exact `1 passed; 0 failed` count -- verified that an unmatched filter prints `0 passed` and makes the identical command exit 1 rather than reporting a vacuous pass.

Provenance

{
  "date": "2026-08-23",
  "established_by": "lane agent-capability-assurance (2026-08-23) via a dispatched int_prelude lane: `Int.modEq_one` in crates/axeyum-lean-kernel/src/int_prelude/modeq_family.rs, proved by instantiating the existing `Int.modEq_iff_dvd` bridge at the literal modulus 1 -- `0 < 1` is `Int.zero_lt_one`, already proved -- and supplying the divisibility witness `b - a = 1 * (b - a)` from `Int.one_mul`. Kernel theorem, empty axiom_footprint.",
  "source": "statement-only extraction of `Int.modEq_one` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.modEq_one`",
      "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"
    }
  ]
}