a^m is 1 mod n exactly when the order divides m (Int.pow_modeq_one_iff_order_dvd)

Identifier: F:int-pow-modeq-one-iff-order-dvd
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For n a : Int with 0 < n and k : Nat an order of a mod n (`Int.IsOrder n a k`), and for every m : Nat, `a^m = 1 (mod n)` if and only if k divides m. This is the characterization that makes the order usable: it converts a congruence about an arbitrary exponent into a divisibility about naturals. The forward direction is the one that needs minimality -- it runs the division algorithm (`Nat.div_mod_exists`, m = k*q + r with r < k), splits a^m as a^(k*q) * a^r via `Int.pow_add`, kills the first factor, and refutes a positive remainder against the minimality conjunct. The backward direction never uses minimality and is therefore stated separately and more generally as `Int.pow_modeq_one_of_dvd`, whose only hypothesis about k is that a^k = 1. What this does NOT claim: the hypothesis 0 < n is load-bearing (the cancellation and power congruences it composes both require it), and no claim is made that an order exists -- that is `Int.order_exists`.

Formal statement

theorem Int.pow_modeq_one_iff_order_dvd : ((x0 : Int) -> ((x1 : Int) -> ((x2 : AxNat) -> ((x3 : AxNat) -> ((x4 : Int.lt Int.zero x0) -> ((x5 : Int.IsOrder x0 x1 x2) -> Iff (Int.ModEq x0 (Int.pow x1 x3) Int.one) (AxNat.dvd x2 x3)))))))

Dependencies

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0 direct dependencies 1 direct dependents

Evidence

kernel-Int.pow_modeq_one_iff_order_dvd

Kind: kernel-term
Status: checked

Supports: Int.pow_modeq_one_iff_order_dvd is admitted by the trusted kernel gate with the type recorded in formal.statement.

Checker command

cargo run -q --release -p axeyum-lean-kernel --example theorem_dependency_inventory -- Int.pow_modeq_one_iff_order_dvd 2>/dev/null | grep -cE '^Int\.pow_modeq_one_iff_order_dvd[[:space:]]'

Evidence notes

Two independent failure modes, so the exit status depends on the finding rather than on the run completing: theorem_dependency_inventory exits non-zero when a NAMED filter matches nothing, and grep -c exits 1 printing 0 when the anchored line is absent. Anchored with [[:space:]], never \t -- in a scripted (GNU) grep \t is a literal t. grep -c rather than grep -q, which would SIGPIPE the producer under pipefail. --release is MANDATORY: this tool builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-Int.pow_modeq_one_iff_order_dvd

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the integer prelude admits no Axiom, Opaque or Quotient declaration, which bounds Int.pow_modeq_one_iff_order_dvd.

Checker command

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

Evidence notes

--require-axiom-free exits non-zero when the named prelude's trusted surface (Axiom + Opaque + Quotient) is not empty, and errors rather than silently passing for a prelude the run never built. A declaration cannot depend on a trusted declaration the environment does not contain, so an empty integer surface bounds every declaration in it. The per-declaration figure is the footprint column of kernel_declaration_projection, measured 0 for this row.

concrete-Int.pow_modeq_one_iff_order_dvd

Kind: instance-pin
Status: checked

Supports: The definitions this statement is about compute the intended values at concrete small arguments, and the discriminating negative cases FAIL: ord_8(3) = 2 (not phi(8) = 4), ord_7(3) = 6, and 3 is refutably NOT a primitive root mod 8.

Checker command

cargo test --release -p axeyum-lean-kernel --lib int_prelude::mult_order_tests -- --test-threads=4

Evidence notes

The trusted gate cannot tell you a Definition is wrong -- it type-checks either way -- so Int.IsOrder and Int.IsPrimitiveRoot are pinned by reduction at concrete small arguments. Each battery asserts BOTH what must hold and what must fail, so no run can be vacuous: three_mod_eight_is_killed_by_exactly_the_even_exponents requires hits at m in {0,2,4} and misses at {1,3,5} (the misses are the negative control for Int.pow_modeq_one_of_dvd); is_order_unfolds_to_the_intended_conjunction requires the definition to be def_eq to the intended conjunction and NOT def_eq to the variant with the positivity comparison transposed; three_has_multiplicative_order_two_mod_eight builds a witness for IsOrder 8 3 2 and a REFUTATION of IsOrder 8 3 4, which only the minimality conjunct separates. Confirm a NONZERO test count (6).

Provenance

{
  "date": "2026-09-04",
  "curation": "curated",
  "established_by": "axeyum-lean-kernel build_int_prelude (crates/axeyum-lean-kernel/src/int_prelude/mult_order.rs)",
  "source": "formal.statement is Kernel::render_lean of the declared type, taken verbatim from the unfiltered emit of `cargo run -q --release -p axeyum-lean-kernel --example kernel_declaration_projection`; axiom_footprint is that same row's footprint column, measured 0. The prose statement and the \"what this does NOT claim\" clause are hand-written by the `primitive-roots` lane and are NOT generator output. Design record: ADR-1598.",
  "prior_art": "Classical; Gauss, Disquisitiones Arithmeticae (1801), art. 48. Stated here over this kernel's own `Int.ModEq` and `Nat.dvd`."
}