The bounded universal search is congruent in its predicate

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

Recorded description

For all Boolean predicates f and g on the naturals and every bound n, if f and g agree at every index then the bounded searches `allBelow f n` and `allBelow g n` return the same Boolean.

Formal statement

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

Dependencies

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1 direct dependencies 1 direct dependents

Evidence

kernel-Nat.Finset.allBelow_congr

Kind: kernel-term
Status: checked

Supports: theorem Nat.Finset.allBelow_congr : ((x0 : ((x0 : AxNat) -> Bool)) -> ((x1 : ((x1 : AxNat) -> Bool)) -> ((x2 : AxNat) -> ((x3 : ((x3 : AxNat) -> Eq.{1} Bool (x0 x3) (x1 x3))) -> Eq.{1} Bool (AxNat.Finset.allBelow x0 x2) (AxNat.Finset.allBelow x1 x2)))))

Checker command

test "$(cargo run -q -p axeyum-lean-kernel --release --example nat_theorem_inventory -- Nat.Finset.allBelow_congr 2>/dev/null | grep -xFc 'Nat.Finset.allBelow_congr	7	((x0 : ((x0 : AxNat) -> Bool)) -> ((x1 : ((x1 : AxNat) -> Bool)) -> ((x2 : AxNat) -> ((x3 : ((x3 : AxNat) -> Eq.{1} Bool (x0 x3) (x1 x3))) -> Eq.{1} Bool (AxNat.Finset.allBelow x0 x2) (AxNat.Finset.allBelow x1 x2)))))')" -ge 1

Evidence notes

`build_nat_prelude` admits this through the trusted `Kernel::add_declaration` gate, which re-derives the type from the proof term; the checker greps for the inventory's exact tab-separated row, so a single wrong token in the rendered type makes it exit 1. Mutation-verified: reversing the hypothesis direction to `g i = f i` while leaving the proof unchanged makes the kernel reject the declaration with `TypeMismatch`, and all 11 tests in `nat_prelude::hall_theorem_tests` die (measured 2026-09-05, exit 101).

footprint-Nat.Finset.allBelow_congr

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the Nat prelude's trusted surface is empty

Checker command

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

Evidence notes

`nat_axiom_inventory` reports `nat: axiom=0 opaque=0 quotient=0 total_trusted=0`, and this theorem is declared inside `build_nat_prelude`, so it cannot rest on a trusted declaration the Nat environment does not contain. Demonstrated to discriminate: `--require-axiom-free axreal` against the SAME kernel build exits 1 (`axreal trusted surface = 30, expected 0`), so the flag is not satisfied unconditionally.

Provenance

{
  "date": "2026-09-05",
  "established_by": "lane hall-theorem (ADR-1644): proved in `crates/axeyum-lean-kernel/src/nat_prelude/hall_theorem.rs` and admitted through `Kernel::add_declaration`",
  "source": "proved in-tree; no import, no replay"
}