hornerFromTop's recursive step: hornerFromTop c a (m+1) (j+1) = hornerFromTop c a m j + a^(j+1)*c(m+1)

Identifier: F:complex-hornerfromtop-succsucc
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

Complex.hornerFromTop c a (m+1) (j+1) = hornerFromTop c a m j + pow a (j+1) * c(m+1) -- the third and final defining equation of hornerFromTop, its genuine recursive step.

Formal statement

theorem Complex.hornerFromTop_succ_succ : ((x0 : ((x0 : AxNat) -> Complex)) -> ((x1 : Complex) -> ((x2 : AxNat) -> ((x3 : AxNat) -> Eq.{1} Complex (Complex.hornerFromTop x0 x1 (AxNat.succ x2) (AxNat.succ x3)) (Complex.add (Complex.hornerFromTop x0 x1 x2 x3) (Complex.mul (Complex.pow x1 (AxNat.succ x3)) (x0 (AxNat.succ x2))))))))

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

0 direct dependencies 2 direct dependents

Evidence

kernel-Complex.hornerFromTop_succ_succ

Kind: kernel-term
Status: checked

Supports: Complex.hornerFromTop_succ_succ is admitted by the trusted kernel gate with the type recorded in formal.statement.

Checker command

cargo run -q --release -p axeyum-lean-kernel --example theorem_dependency_inventory -- Complex.hornerFromTop_succ_succ 2>/dev/null | grep -cE '^Complex\.hornerFromTop_succ_succ[[:space:]]'

Evidence notes

The producing prelude admits Complex.hornerFromTop_succ_succ through the trusted Kernel::add_declaration gate, which re-checks the proof term against the stated type. theorem_dependency_inventory exits non-zero for a named filter matching nothing (a deleted/renamed theorem cannot read as a re-derived one), and grep -c (never -q) both avoids a SIGPIPE-under-pipefail false negative and independently asserts the exact tab-anchored line is present. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-Complex.hornerFromTop_succ_succ

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the complex prelude's trusted surface is empty, which bounds Complex.hornerFromTop_succ_succ

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_axiom_inventory -- --include-constructed --require-axiom-free complex

Evidence notes

Re-measured on this tree: complex: axiom=0 opaque=0 quotient=0 total_trusted=0, exits 0 printing 'ok: complex trusted surface = 0'. That bounds every declaration in the complex environment, including Complex.hornerFromTop_succ_succ, since a declaration cannot depend on a trusted declaration the environment does not contain. --require-axiom-free <name> errors for a prelude never built by this run rather than silently passing on zero rows. --release is MANDATORY here.

Provenance

{
  "date": "2026-08-27",
  "established_by": "axeyum-lean-kernel build_complex_prelude (crates/axeyum-lean-kernel/src/complex/poly.rs)",
  "source": "theorem name, canonical type and direct dependency edges read via theorem_dependency_inventory's own per-name output (cargo run -q --release -p axeyum-lean-kernel --example theorem_dependency_inventory -- <Name>), which prints '<Name>\\t<comma-separated direct deps>'; the canonical type itself was cross-extracted from kernel_declaration_projection's unfiltered TSV dump the same way as the Definition facts in this batch, and confirmed byte-for-byte identical between the two tools' output for every name in this batch. crates/ source was not touched to produce this batch."
}