Mathlib v4.30 source proposition Nat.dist_eq_intro

Identifier: F:ml430-nat-dist-eq-intro-294b44ad
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n m k l : ℕ}, n + m = k + l → n.dist k = l.dist m

Dependencies

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8 direct dependencies 0 direct dependents

Evidence

kernel-Nat.dist_eq_intro

Kind: kernel-term
Status: checked

Supports: ∀ {n m k l : ℕ}, n + m = k + l → n.dist k = l.dist m

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.dist_eq_intro` (`nat_prelude/dist_more2.rs`) through the trusted `Kernel::add_declaration` gate. `nat_theorem_inventory`'s rendered type is `(x0 x1 x2 x3 : AxNat) -> (x4 : Eq (add x0 x1) (add x2 x3)) -> Eq (dist x0 x2) (dist x3 x1)`, matching this fact's `formal.statement` up to variable naming. Anchored with `^Nat\.dist_eq_intro[[:space:]]`; `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_intro

Kind: kernel-term
Status: checked

Supports: n=5, m=2, k=3, l=4 (5+2=3+4=7, a DISCRIMINATING instance with n<>k and m<>l): dist(5,3)=2=dist(4,2).

Checker command

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

footprint-Nat.dist_eq_intro

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_eq_intro`'s footprint by [].

Provenance

{
  "date": "2026-08-30",
  "established_by": "lane draw9-second-theorems",
  "source": "statement-only extraction of `Nat.dist_eq_intro` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.dist_eq_intro`",
      "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"
    }
  ]
}