multichoose(1, k) = 1

Identifier: F:nat-multichoose-one
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The number of size-k multisets from a 1-element type is 1, for any k (there is only one multiset, all elements the same).

Formal statement

theorem Nat.multichoose_one : ((x0 : AxNat) -> Eq.{1} AxNat (AxNat.multichoose (AxNat.succ AxNat.zero) x0) (AxNat.succ AxNat.zero))

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

3 direct dependencies 0 direct dependents

Evidence

kernel-Nat.multichoose_one

Kind: kernel-term
Status: checked

Supports: Nat.multichoose 1 k = 1

Checker command

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

Evidence notes

`build_nat_prelude` admits this theorem through the trusted `Kernel::add_declaration` gate. `1.multichoose k` reduces to `choose (pred (add 1 k)) k`; `Nat.add` recurses on its RIGHT argument, so the literal `1` on the LEFT leaves `add 1 k` stuck for symbolic `k` -- bridged via `Nat.succ_add (0, k) : add (succ 0) k = succ (add 0 k)` then `Nat.zero_add k : add 0 k = k`, giving `add 1 k = succ k` propositionally (a `congr`/`trans` chain, not defeq). `pred (succ k)` then reduces to `k` by ι alone once that rewrite is in hand, and `Nat.choose_self : choose k k = 1` closes it. This is the one boundary lemma of the three that genuinely needs propositional rewriting rather than pure reduction, since the literal sits on `add`'s non-recursive side. `nat_theorem_inventory` exits non-zero for a name that does not exist, and `grep -Ec` consumes the pipe and the count is tested, so the exit status depends on what the run found.

compute-Nat.multichoose_one

Kind: kernel-term
Status: checked

Supports: the equation holds at a concrete instance, not merely symbolically

Checker command

cargo test -p axeyum-lean-kernel --lib nat_prelude::nat_prelude_tests::multichoose_evaluates_correctly

Evidence notes

`Kernel::def_eq` reduces `1.multichoose 4` (through `choose 4 4`) to `1`, with a negative reduction control at a different pair (`3.multichoose 2` must NOT be def-eq to `10`, the `pred`-dropped value) so the assertion cannot pass on a vacuous `def_eq`.

footprint-Nat.multichoose_one

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 2026-08-28: `nat: axiom=0 opaque=0 quotient=0 total_trusted=0` with `Nat.ascFactorial` and its three theorems, and `Nat.multichoose` and its three theorems, all declared.

Provenance

{
  "date": "2026-08-28",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane nat-asc-multichoose",
  "source": "declare_multichoose_one (crates/axeyum-lean-kernel/src/nat_prelude/multichoose.rs)"
}