Mathlib v4.30 source proposition Int.neg_modEq_neg

Identifier: F:ml430-int-neg-modeq-neg-30d98479
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n a b : ℤ}, -a ≡ -b [ZMOD n] ↔ a ≡ b [ZMOD n]

Dependencies

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6 direct dependencies 0 direct dependents

Evidence

kernel-Int.neg_modEq_neg

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

`declare_neg_modeq_neg` (`crates/axeyum-lean-kernel/src/int_prelude/modeq.rs`), a genuine `Iff` (this kernel's `Iff.intro`/`iff_mp`/`iff_mpr`, the same machinery `modEq_iff_dvd` already uses), unconditional in `n`. `mpr` is exactly `Int.modEq_neg`; `mp` applies `Int.modEq_neg` again to a `ModEq n (-a) (-b)` hypothesis and simplifies the resulting `-(-a)`/`-(-b)` back down via a double-negation helper (`ineg_neg`, derived from `Int.add_neg` and `Int.cancel_neg_add`-style cancellation).

footprint-Int.neg_modEq_neg

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 -- neg_modEq_neg 2>/dev/null | grep -Ec '^theorem[[:space:]]+Int\.neg_modEq_neg[[: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.neg_modEq_neg` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.neg_modEq_neg`",
      "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"
    }
  ]
}