Mathlib v4.30 source proposition Nat.dist_pos_of_ne

Identifier: F:ml430-nat-dist-pos-of-ne-00f5e22f
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {i j : ℕ}, i ≠ j → 0 < i.dist j

Dependencies

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7 direct dependencies 0 direct dependents

Evidence

kernel-Nat.dist_pos_of_ne

Kind: kernel-term
Status: checked

Supports: ∀ {i j : ℕ}, i ≠ j → 0 < i.dist j

Checker command

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

Evidence notes

`build_nat_prelude` admits `Nat.dist_pos_of_ne` (`nat_prelude/dist_more2.rs`) through the trusted `Kernel::add_declaration` gate. `nat_theorem_inventory`'s rendered type is `(x0 x1 : AxNat) -> (x2 : Not (Eq x0 x1)) -> AxNat.lt AxNat.zero (AxNat.dist x0 x1)`, matching this fact's `formal.statement` up to variable naming. Anchored with `^Nat\.dist_pos_of_ne[[: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_pos_of_ne

Kind: kernel-term
Status: checked

Supports: dist(3, 7) = 4 > 0 and dist(7, 3) = 4 > 0 -- two instances exercising BOTH branches of the internal `Lt i j` / `Lt j i` case split, not just one order.

Checker command

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

footprint-Nat.dist_pos_of_ne

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_pos_of_ne`'s footprint by [].

Provenance

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