multichoose(n, 0) = 1

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

Recorded description

The number of size-0 multisets from an n-element type is 1 (the empty multiset), for any n.

Formal statement

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

Dependencies

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1 direct dependencies 0 direct dependents

Evidence

kernel-Nat.multichoose_zero_right

Kind: kernel-term
Status: checked

Supports: n.multichoose 0 = 1

Checker command

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

Evidence notes

`build_nat_prelude` admits this theorem through the trusted `Kernel::add_declaration` gate. `Nat.multichoose` is declared (`nat_prelude/multichoose.rs`) as the plain, non-recursive abbreviation `fun n k => choose (pred (add n k)) k`, so `n.multichoose 0` reduces by β/δ alone to `choose (pred (add n 0)) 0`, and `Nat.choose_zero_right : ∀ m, choose m 0 = 1` proves `choose _ 0 = 1` for ANY first argument -- instantiating it at `pred (add n 0)` closes the goal directly, with no need to reduce that expression further and no dependency on `Nat.multichoose_one`/`_one_right`. `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_zero_right

Kind: kernel-term
Status: checked

Supports: the abbreviation computes the right value at a concrete instance, not merely an opaque application

Checker command

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

Evidence notes

`Kernel::def_eq` reduces `0.multichoose 0` directly (definition unfolding through `choose 0 0`) and confirms it is `1` (the empty multiset), with a negative reduction control at a different pair (`3.multichoose 2` must NOT be def-eq to `10`, the value a `pred`-dropping copy-paste would compute) so the assertion cannot pass on a vacuous `def_eq`.

footprint-Nat.multichoose_zero_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. `Nat.multichoose` is not a recursion at all -- a plain abbreviation over already-declared `Nat.add`/`Nat.pred`/`Nat.choose` -- so nothing here was ever at risk of needing a fuel device or `WellFounded`. 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_zero_right (crates/axeyum-lean-kernel/src/nat_prelude/multichoose.rs)"
}