Mathlib v4.30 source proposition Nat.dist_comm

Identifier: F:ml430-nat-dist-comm-1fa29a04
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The proposition declared as `Nat.dist_comm` in the pinned Mathlib v4.30 source.

Formal statement

∀ (n m : ℕ), n.dist m = m.dist n

Dependencies

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1 direct dependencies 2 direct dependents

Evidence

kernel-Nat.dist_comm

Kind: kernel-term
Status: checked

Supports: ∀ (n m : ℕ), n.dist m = m.dist n

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.dist_comm` through the trusted `Kernel::add_declaration` gate (`nat_prelude/dist.rs`), proved by `add_comm` on the two truncated subtractions -- `dist n m` and `dist m n` are definitionally `add(sub n m, sub m n)` and `add(sub m n, sub n m)`, so commutativity of `dist` is exactly commutativity of `add` applied to those two terms. `nat_theorem_inventory`'s rendered type `((x0:AxNat)->((x1:AxNat)->Eq.{1} AxNat (AxNat.dist x0 x1) (AxNat.dist x1 x0)))` matches this fact's `formal.statement`. Anchored with `^Nat\.dist_comm[[:space:]]` so a sibling name cannot satisfy this row; `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. Run `--release` -- the debug build of this inventory example SIGABRTs on stack depth (unrelated to this proof).

compute-Nat.dist_comm

Kind: kernel-term
Status: checked

Supports: dist(3,5) = dist(5,3) = 2, and BOTH symbolically (free n,m) and at this concrete, non-symmetric pair -- covered by nat_prelude_tests::dist_theorems_apply_at_free_variables_and_concrete_instances and dist_evaluates_correctly.

Checker command

cargo test -p axeyum-lean-kernel --lib nat_prelude::nat_prelude_tests::dist_theorems_apply_at_free_variables_and_concrete_instances nat_prelude::nat_prelude_tests::dist_evaluates_correctly

Evidence notes

`dist_theorems_apply_at_free_variables_and_concrete_instances` applies `dist_comm` at a genuinely free `(n, m)` pushed into an explicit `LocalContext` and confirms the inferred type is `Eq (dist n m) (dist m n)`, not merely that SOME type-checks. `dist_evaluates_correctly` confirms `dist 3 5` and `dist 5 3` both reduce to `2` and NOT to `0` (the value a dropped-reverse-subtraction bug would give).

footprint-Nat.dist_comm

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the Nat prelude's trusted surface is empty

Checker command

cargo run -q --release -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: axiom=0 opaque=0 quotient=0). A theorem cannot depend on a trusted declaration the environment does not contain, so an empty trusted surface bounds `Nat.dist_comm`'s footprint by [].

Provenance

{
  "date": "2026-08-30",
  "established_by": "lane draw9-first-theorems, reconciled against pre-existing nat_prelude/dist.rs",
  "source": "statement-only extraction of `Nat.dist_comm` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.dist_comm`",
      "where": "mathlib4 commit c5ea00351c28e24afc9f0f84379aa41082b1188f (v4.30.0)",
      "year": 2026,
      "attribution": "the proposition was read from the pinned statement-only inventory; the proof term and tactic trace were not consulted"
    }
  ]
}