A bounded Boolean universal reflects back to a pointwise fact

Identifier: F:nat-finset-all-below-true-at
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

If the bounded loop allBelow f n evaluates to true then f i = true at every index i below n. The reflection direction of a Bool-valued bounded universal quantifier.

Formal statement

theorem Nat.Finset.allBelow_true_at : ((x0 : ((x0 : AxNat) -> Bool)) -> ((x1 : AxNat) -> ((x2 : Eq.{1} Bool (AxNat.Finset.allBelow x0 x1) Bool.true) -> ((x3 : AxNat) -> ((x4 : AxNat.lt x3 x1) -> Eq.{1} Bool (x0 x3) Bool.true)))))

Dependencies

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4 direct dependencies 7 direct dependents

Evidence

kernel-nat-finset-all-below-true-at

Kind: kernel-term
Status: checked

Supports: Nat.Finset.allBelow_true_at is admitted by the trusted kernel gate with the type recorded in formal.statement.

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example nat_theorem_inventory -- Nat.Finset.allBelow_true_at 2>/dev/null | grep -Fc '((x2 : Eq.{1} Bool (AxNat.Finset.allBelow x0 x1) Bool.true) -> ((x3 : AxNat) -> ((x4 : AxNat.lt x3 x1) -> Eq.{1} Bool (x0 x3) Bool.true)))')" -ge 1

Evidence notes

The pattern is a DISTINGUISHING substring of the admitted type, not the theorem's name: an existence check would still pass if the statement drifted. Verified to fail on a perturbed pattern before this row was written. `build_nat_prelude` admits this theorem only through the trusted kernel gate, so a successful build IS the type-check.

footprint-nat-finset-all-below-true-at

Kind: instance-pin
Status: checked

Supports: axiom_footprint: [] -- read from Kernel::axiom_footprint for this declaration, not from a maintained list

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --example theorem_axiom_footprint -- Finset 2>/dev/null | awk -F'\t' '$1 == "nat" && $2 == "Nat.Finset.allBelow_true_at" && $3 == "0"' | wc -l)" -ge 1

Evidence notes

`theorem_axiom_footprint` prints, per declaration, the size and contents of `Kernel::axiom_footprint`. The pattern pins BOTH the declaration name and the size 0 with an empty axiom column, so a proof that reached for a trusted declaration would fail the check rather than be reported as axiom-free. The `nat` prelude's whole trusted surface is empty, which bounds this independently.

evaluation-nat-finset-all-below-true-at

Kind: instance-pin
Status: checked

Supports: The definitions this statement is about COMPUTE the intended values, with negative controls -- the kernel cannot tell a Definition is wrong

Checker command

cargo test -q --release -p axeyum-lean-kernel --lib -- nat_prelude::finset_tests --test-threads=4

Evidence notes

`Nat.Finset`'s operations are admitted on their TYPE; a `card` that computed the wrong number would have the right type, an empty axiom footprint, and would pass every other check in this ledger. `finset_tests.rs` reduces each operation to a numeral or a Bool at tiny discriminating arguments and pairs every positive with the specific wrong formula its negative control rules out. It caught one wrong hand-computed expectation while being written.

Provenance

{
  "date": "2026-09-03",
  "established_by": "axeyum-lean-kernel build_nat_prelude",
  "source": "theorem name and canonical type read directly via nat_theorem_inventory, which prints render_lean of the admitted type; declared by `nat_prelude/finset.rs` (lane finset-role, ADR-1577)."
}