multichoose(n, 1) = n

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

Recorded description

The number of size-1 multisets from an n-element type is n.

Formal statement

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

Dependencies

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

1 direct dependencies 0 direct dependents

Evidence

kernel-Nat.multichoose_one_right

Kind: kernel-term
Status: checked

Supports: n.multichoose 1 = n

Checker command

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

Evidence notes

`build_nat_prelude` admits this theorem through the trusted `Kernel::add_declaration` gate. `n.multichoose 1` reduces to `choose (pred (add n 1)) 1`, and `add n 1 ≡ add n (succ zero) ≡ succ (add n zero) ≡ succ n` reduces FULLY by ι alone -- the literal `1` sits on `Nat.add`'s right/recursive side, and `add n zero ≡ n` is its base case holding for any `n` -- then `pred (succ n) ≡ n` reduces the same way, so the whole index collapses to `n` with no lemma at all, and this theorem's proof term is literally `Nat.choose_one_right`'s own proof, applied at `n`. Unlike `Nat.multichoose_one`, no `congr`/`trans` chain is needed here, because the literal is on `add`'s recursive side rather than its 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_right

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 `4.multichoose 1` (through `choose 4 1`) to `4`, 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_right

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_right (crates/axeyum-lean-kernel/src/nat_prelude/multichoose.rs)"
}