land is associative

Identifier: F:nat-land-assoc
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For every natural a, b, c: land(land(a, b), c) = land(a, land(b, c)).

Formal statement

theorem Nat.land_assoc : ((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat) -> Eq.{1} AxNat (AxNat.land (AxNat.land x0 x1) x2) (AxNat.land x0 (AxNat.land x1 x2)))))

Dependencies

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6 direct dependencies 0 direct dependents

Evidence

kernel-Nat.land_assoc

Kind: kernel-term
Status: checked

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

Checker command

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

Evidence notes

`build_nat_prelude` admits this theorem only through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type -- producing this row at all is a machine-checked proof. `nat_theorem_inventory` exits non-zero for a name that does not exist, and `grep -Ec` consumes the pipe so the tested count decides the exit status. Anchored with `^Nat\.land_assoc[[:space:]]` so a sibling name cannot satisfy this row. Run `--release` -- the debug build of this inventory example SIGABRTs on stack depth (unrelated to this proof).

compute-Nat.land_assoc

Kind: kernel-term
Status: checked

Supports: land(land(3, 7), 5) = land(3, land(7, 5)) = 1, with BOTH intermediate values (land(3,7)=3, land(7,5)=5) genuinely NONZERO -- the fully-generic sub-case of the hard leaf (both nested landAux applications stuck, neither settling for the easy zero-corner), not a degenerate instance a wrong proof of the easy leaves alone could pass.

Checker command

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

Evidence notes

Applies `Nat.land_assoc` symbolically (against free `a`, `b`, `c`) and at the concrete instance `(a, b, c) = (3, 7, 5)`, confirming the admitted type matches `Eq (land (land 3 7) 5) (land 3 (land 7 5))`, that `land 3 7` and `land 7 5` are each nonzero (non-vacuity of the hard-leaf exercise), and that both sides compute to `1`.

footprint-Nat.land_assoc

Kind: instance-pin
Status: checked

Supports: axiom_footprint: [] -- the Nat environment admits no trusted declaration

Checker command

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

Evidence notes

Reports `nat: axiom=0 opaque=0 quotient=0 total_trusted=0`, over the FULL trusted surface rather than `Declaration::Axiom` alone. The enumeration is per-environment, not per-theorem; it bounds this theorem's footprint because a proof cannot depend on a trusted declaration the environment does not contain.

Provenance

{
  "date": "2026-08-29",
  "established_by": "axeyum-lean-kernel build_nat_prelude, lane nat-land-assoc-finish",
  "source": "theorem name and canonical type read directly via nat_theorem_inventory, which prints render_lean of the admitted type."
}