Mathlib v4.30 source proposition Nat.dist_self

Identifier: F:ml430-nat-dist-self-0cfa5426
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ (n : ℕ), n.dist n = 0

Dependencies

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2 direct dependencies 1 direct dependents

Evidence

kernel-Nat.dist_self

Kind: kernel-term
Status: checked

Supports: ∀ (n : ℕ), n.dist n = 0

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.dist_self` through the trusted `Kernel::add_declaration` gate (`nat_prelude/dist.rs`), proved via `sub_self : sub n n = 0` congr'd into `add(sub n n, sub n n)` then `zero_add`. `nat_theorem_inventory`'s rendered type `((x0:AxNat)->Eq.{1} AxNat (AxNat.dist x0 x0) AxNat.zero)` matches this fact's `formal.statement`. Anchored with `^Nat\.dist_self[[:space:]]`; `grep -Ec` consumes the pipe so the tested count decides the exit status. Run `--release` -- the debug build SIGABRTs on stack depth (unrelated to this proof).

compute-Nat.dist_self

Kind: kernel-term
Status: checked

Supports: dist(4,4) = 0, both symbolically (free n) and at this concrete instance -- 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_self` at a genuinely free `n` and confirms the inferred type is `Eq (dist n n) 0`. `dist_evaluates_correctly` confirms `dist 4 4` reduces to `0`.

footprint-Nat.dist_self

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).

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_self` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.dist_self`",
      "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"
    }
  ]
}