Mathlib v4.30 source proposition Nat.eq_of_beq_eq_true

Identifier: F:ml430-nat-eq-of-beq-eq-true-6ec35ee4
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

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

Formal statement

∀ {n m : ℕ}, n.beq m = true → n = m

Dependencies

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0 direct dependencies 26 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-Nat.eq_of_beq_eq_true

Kind: kernel-term
Status: checked

Supports: `Nat.eq_of_beq_eq_true` is admitted as a Theorem with EXACTLY the stated shape, pinned verbatim via the kernel's own renderer: `((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : Eq.{1} Bool (AxNat.beq x0 x1) Bool.true) -> Eq.{1} AxNat x0 x1)))`.

Checker command

test "$(cargo run -q --release -p axeyum-lean-kernel --example nat_theorem_inventory 2>/dev/null | awk -F'\t' '$1 == "Nat.eq_of_beq_eq_true" && $3 == "((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : Eq.{1} Bool (AxNat.beq x0 x1) Bool.true) -> Eq.{1} AxNat x0 x1)))"' | wc -l)" -ge 1

footprint-Nat.eq_of_beq_eq_true

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] for `Nat.eq_of_beq_eq_true`. The `nat` prelude's trusted surface is empty (no Axiom, no Opaque, no Quotient), which bounds every declaration in it, including `Nat.eq_of_beq_eq_true`.

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free nat

Provenance

{
  "date": "2026-08-29",
  "established_by": "kernel prelude build (crates/axeyum-lean-kernel/src/), predating this mirror's preregistration; found and verified by lane already-proved-sweep (2026-08-29), not proved by this lane",
  "source": "statement-only extraction of `Nat.eq_of_beq_eq_true` from Mathlib v4.30.0; no proof value was exposed",
  "prior_art": [
    {
      "who": "the Mathlib contributors",
      "what": "the theorem declaration `Nat.eq_of_beq_eq_true`",
      "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"
    }
  ]
}