Mathlib v4.30 source proposition Int.ModEq.add_left

Identifier: F:ml430-int-modeq-add-left-6e17c69a
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

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

Dependencies

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2 direct dependencies 0 direct dependents

Evidence

kernel-Int.modEq_add_left

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

This is a GENERALIZATION, not a new proof route: `declare_modeq_add_right`/`declare_modeq_add_left` (`crates/axeyum-lean-kernel/src/int_prelude/modeq.rs`) previously carried a `0 < n` hypothesis Mathlib's own statement does not have. Re-reading `modEq_iff_dvd`'s `mp` half showed the positivity hypothesis is used only by `mpr` (which needs `ediv_emod_unique`'s `0<=r<n` bound); `mp` itself is unconditional in `n`, extracted as `modeq_to_dvd`. The converse (`dvd n (b-a) -> ModEq n a b`) is also unconditional via a different route -- `Int.modEq_add_mul_left : ModEq n (n*q+a) a` (already unconditional, `modeq_family.rs`) applied directly to the witness, extracted as `dvd_to_modeq` -- so `mpr`'s positivity was never load-bearing either, it was simply the wrong bridge. The two old declarations were REPLACED (not duplicated) with the general form; no other file in the tree called the old 5-argument signature.

footprint-Int.modEq_add_left

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_add_left 2>/dev/null | grep -Ec '^theorem[[:space:]]+Int\.modEq_add_left[[:space:]]+\(\(')" -ge 1

Evidence notes

The regex requires the tab-separated footprint column to be EMPTY (whitespace runs straight into the type's opening `((`), which discriminates a real empty footprint from a nonempty one -- `Int.euclidean_decomposition`, the one still-asserted `Int` law, prints its own name in that column and fails the same regex. `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.add_left` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Int.ModEq.add_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"
    }
  ]
}