Congruence modulo m is an equivalence relation

Identifier: F:nat-modeq-equivalence-on
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For every natural modulus m and every bound n, congruence modulo m (a ≡ b mod m) is reflexive, symmetric and transitive on natural numbers below n.

Formal statement

theorem Nat.modEq_equivalence_on : ((x0 : AxNat) -> ((x1 : AxNat) -> AxNat.equivalenceOn (AxNat.modEq x0) x1))

Dependencies

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3 direct dependencies 0 direct dependents

Evidence

kernel-Nat.modEq_equivalence_on

Kind: kernel-term
Status: checked

Supports: For every natural modulus m and every bound n, congruence modulo m (a ≡ b mod m) is reflexive, symmetric and transitive on natural numbers below n.

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- modEq_equivalence_on 2>/dev/null | grep -Ec '^Nat\.modEq_equivalence_on[[:space:]]')" -ge 1

Evidence notes

`build_nat_prelude` admits this theorem through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type, so producing this row at all is a machine-checked proof. `nat_theorem_inventory` exits non-zero for a name that does not exist, and the `grep -q` requires the admitted declaration to be printed -- both halves of the check depend on the theorem actually being present.

footprint-Nat.modEq_equivalence_on

Kind: exhaustive-enumeration
Status: checked

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

Checker command

cargo run -q -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free nat

Evidence notes

`nat_axiom_inventory --require-axiom-free nat` enumerates the built Nat environment and exits non-zero unless it admits no Axiom, Opaque or Quotient declaration (measured: axiom=0 opaque=0 quotient=0). A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds every individual theorem's footprint by [].

Provenance

{
  "date": "2026-08-25",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane rado-claim-ledger",
  "source": "one of the theorems build_nat_prelude admits; registered to close the flywheel's fact-ledger arrow for recently landed Nat theorems"
}