Mathlib v4.30 source proposition Int.ModEq.dvd_iff

Identifier: F:ml430-int-modeq-dvd-iff-b7ffeff8
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n a b : ℤ}, a ≡ b [ZMOD n] → (n ∣ a ↔ n ∣ b)

Dependencies

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

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

Evidence

kernel-Int.modEq_dvd_iff

Kind: kernel-term
Status: checked

Supports: The proposition declared as `Int.ModEq.dvd_iff` in the pinned Mathlib v4.30 source.

Checker command

cargo test -p axeyum-lean-kernel --lib int_prelude::

Evidence notes

`declare_modeq_dvd_iff` (`crates/axeyum-lean-kernel/src/int_prelude/modeq.rs`), a genuine `Iff`, unconditional in `n`. `modeq_to_dvd` applied to `h` and to `Int.ModEq.symm h` gives `dvd n (b-a)` and `dvd n (a-b)` directly -- no separate 'negate the dividend' lemma needed -- and `Int.dvd_add` plus a small algebra helper (`eq_add_sub`: `x+(y-x)=y`, via `cancel_neg_add`/`Int.add_comm`) close each direction.

footprint-Int.modEq_dvd_iff

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- this theorem's trusted closure is empty

Checker command

test "$(cargo run --release -q -p axeyum-lean-kernel --example int_theorem_inventory -- modEq_dvd_iff 2>/dev/null | grep -Ec '^theorem[[:space:]]+Int\.modEq_dvd_iff[[:space:]]+\(\(')" -ge 1

Evidence notes

The regex requires the tab-separated footprint column to be EMPTY. `int_theorem_inventory` must be run `--release`; in debug it stack-overflows building the full constructed environment (unrelated to this theorem).

Provenance

{
  "date": "2026-08-18",
  "established_by": "not established in this ledger",
  "source": "statement-only extraction of `Int.ModEq.dvd_iff` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.ModEq.dvd_iff`",
      "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"
    }
  ]
}