Excluded middle is irrefutable

Identifier: F:logic-not-not-em
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

not-(not-(p or not-p)) holds for every proposition p -- excluded middle cannot be refuted, even though it cannot be proved intuitionistically either.

Formal statement

theorem not_not_em : ((x0 : Prop) -> ((x1 : ((x1 : Or x0 (((x1 : x0) -> False))) -> False)) -> False))

Dependencies

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0 direct dependencies 1 direct dependents

Evidence

kernel-logic-not-not-em

Kind: kernel-term
Status: checked

Supports: not-(not-(p or not-p)) holds for every proposition p -- excluded middle cannot be refuted, even though it cannot be proved intuitionistically either.

Checker command

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

Evidence notes

`build_logic_prelude` (built first by `build_nat_prelude`, which `nat_theorem_inventory` constructs) admits `not_not_em` only through the trusted kernel gate, so this line in the tool's output IS the type-check. Anchored with `grep -Ec '^not_not_em[[:space:]]'` on the tool's own substring-filtered output, matching exactly one row; a name that does not exist makes both the tool's argument filter and the anchored grep fail (verified both ways 2026-08-25).

footprint-logic-not-not-em

Kind: instance-pin
Status: checked

Supports: axiom_footprint: [] -- the logic environment admits no trusted declaration

Checker command

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

Evidence notes

Reports `logic: axiom=0 opaque=0 quotient=0 total_trusted=0`, over the full trusted surface (`Axiom` + `Opaque` + `Quotient`), not `Declaration::Axiom` alone. The enumeration is per-environment, so it bounds this theorem's footprint: a proof admitted into an environment cannot depend on a trusted declaration the environment does not contain. `logic` is the isolated, intuitionistic core (`build_logic_prelude` alone) that `nat` nests -- measured 2026-08-25 with `nat_axiom_inventory --require-axiom-free logic`.

Provenance

{
  "date": "2026-08-25",
  "established_by": "nat-axiom-coverage lane, axeyum-lean-kernel logic prelude",
  "source": "read off the kernel by `nat_theorem_inventory`, not transcribed; statement authored here",
  "prior_art": [
    {
      "citation": "Standard intuitionistic/classical propositional logic (natural deduction); no single primary source.",
      "establishes": "Glivenko-style fact: the double negation of excluded middle is intuitionistically provable, though excluded middle itself is not.",
      "attribution": "standard textbook material; this lane did not consult a primary source"
    }
  ]
}