Mathlib v4.30 source proposition Int.ModEq.cancel_right_div_gcd

Identifier: F:ml430-int-modeq-cancel-right-div-gcd-00cd73fa
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {m a b c : ℤ}, 0 < m → a * c ≡ b * c [ZMOD m] → a ≡ b [ZMOD m / ↑(m.gcd c)]

Dependencies

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2 direct dependencies 0 direct dependents

Evidence

kernel-Int.mod_eq_cancel_right_div_gcd

Kind: kernel-term
Status: checked

Supports: ∀ {m a b c : ℤ}, 0 < m → a * c ≡ b * c [ZMOD m] → a ≡ b [ZMOD m / ↑(m.gcd c)]

Checker command

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

Evidence notes

build_int_prelude admits Int.mod_eq_cancel_right_div_gcd through the trusted Kernel::add_declaration gate. New proof, lane modeq-div-gcd, in int_prelude/modeq_cancel_div_gcd.rs. Not re-derived from scratch: commutes both sides (a*c to c*a, b*c to c*b via Int.mul_comm) and applies Int.mod_eq_cancel_left_div_gcd (F:ml430-int-modeq-cancel-left-div-gcd-b2d407e8, same lane) directly, exactly Mathlib's own proof and exactly how the Nat mirror closes the same way. int_theorem_inventory's rendered type matches this fact's formal.statement exactly. int_theorem_inventory exits non-zero (0 rows) for a bogus name (measured).

footprint-Int.mod_eq_cancel_right_div_gcd

Kind: exhaustive-enumeration
Status: checked

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

Checker command

cargo run -q -p axeyum-lean-kernel --example prelude_axiom_inventory -- --require-axiom-free integer

Evidence notes

prelude_axiom_inventory --require-axiom-free integer enumerates the built Int environment and exits non-zero unless it admits no Axiom, Opaque or Quotient declaration (measured: integer axiom=0). int_prelude_tests::every_int_declaration_is_checked_and_axiom_free and derived_laws_have_no_axiom_footprint additionally confirm this theorem's own Kernel::axiom_footprint directly, and mod_eq_cancel_div_gcd_family_applies_at_a_discriminating_concrete_instance_and_symbolically checks it applies at a concrete DISCRIMINATING instance (m,c)=(6,4), gcd=2 -- this development has no Int-level "cancel a coprime factor" lemma at all, so a coprime instance could not even be a control -- with a wrong-modulus negative control, and symbolically at a genuinely free (m,a,b,c) via a fresh restating theorem.

Provenance

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