Mathlib v4.30 source proposition Nat.dist_eq_zero

Identifier: F:ml430-nat-dist-eq-zero-5ae5b706
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n m : ℕ}, n = m → n.dist m = 0

Dependencies

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1 direct dependencies 0 direct dependents

Evidence

kernel-Nat.dist_eq_zero

Kind: kernel-term
Status: checked

Supports: ∀ {n m : ℕ}, n = m → n.dist m = 0

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.dist_eq_zero` through the trusted `Kernel::add_declaration` gate (`nat_prelude/dist.rs`, `declare_dist_more_all`, lane draw9-first-theorems). `nat_theorem_inventory`'s rendered type matches this fact's `formal.statement`. Anchored with `^Nat\.dist_eq_zero[[:space:]]` so a sibling name cannot satisfy this row; `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_eq_zero

Kind: kernel-term
Status: checked

Supports: dist_eq_zero applied at a free n (via Eq.refl) and at the concrete instance n=m=5 (dist 5 5 = 0).

Checker command

cargo test -p axeyum-lean-kernel --lib nat_prelude::nat_prelude_tests::dist_draw9_additions_apply_at_concrete_discriminating_instances

footprint-Nat.dist_eq_zero

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",
  "source": "statement-only extraction of `Nat.dist_eq_zero` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.dist_eq_zero`",
      "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"
    }
  ]
}