hornerFromTop's diagonal equals polyEval: the sum-level bridge

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

Recorded description

For every c, a, n: hornerFromTop c a n n is Equiv to polyEval c (n+1) a -- the bridge between hornerFromTop's nested Nat.rec and polyEval's sumRange fold. The genuinely open part this theorem closes is connecting hornerFromTop's own recursion to polyEval at all; the module doc's original 'row growth' formula, restated correctly, turns out to already BE Complex.hornerFromTop_succ_succ (proved by Eq.refl before this lemma existed). Proved by induction on n: base (n=0), both sides reduce (via hornerFromTop_zero on the left; poly_eval_succ, poly_eval_zero, pow_zero lifted to Equiv and a ring_law_proof collapse on the right) to the same value c(0); step (n=succ j), hornerFromTop_succ_succ unfolds the LHS, the induction hypothesis rewrites the carried summand via add_congr, poly_eval_succ unfolds the RHS, and the two sides then differ only by mul_comm inside the new summand, closed by ring_law_proof.

Formal statement

theorem Complex.hornerFromTop_diag_eq_polyEval : ((x0 : ((x0 : AxNat) -> Complex)) -> ((x1 : Complex) -> ((x2 : AxNat) -> Complex.Equiv (Complex.hornerFromTop x0 x1 x2 x2) (Complex.polyEval x0 (AxNat.succ x2) x1))))

Dependencies

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

Evidence

kernel-Complex.hornerFromTop_diag_eq_polyEval

Kind: kernel-term
Status: checked

Supports: Complex.hornerFromTop_diag_eq_polyEval 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 -- hornerFromTop_diag_eq_polyEval 2>/dev/null | grep -cE '^Complex\.hornerFromTop_diag_eq_polyEval[[:space:]]'

Evidence notes

build_complex_prelude admits Complex.hornerFromTop_diag_eq_polyEval through the trusted Kernel::add_declaration gate. theorem_dependency_inventory exits non-zero for a named filter matching nothing; grep -c asserts the exact tab-anchored line. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-Complex.hornerFromTop_diag_eq_polyEval

Kind: exhaustive-enumeration
Status: checked

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

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_diag_eq_polyEval, 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, declare_horner_from_top_diag_eq_poly_eval)",
  "source": "canonical type read via kernel_declaration_projection's own UNFILTERED emit mode (cargo run -q --release -p axeyum-lean-kernel --example kernel_declaration_projection, no --require-declaration flag), which prints, per constructed prelude, one TSV row per declaration whose last field is kernel.render_lean(declaration.ty()) -- the same Kernel::render_lean canonical form nat_theorem_inventory prints, just not filtered to Declaration::Theorem. That output was piped to a scratchpad file and the exact row for this declaration's own prelude label was extracted and injected here programmatically (a Python script reading the TSV, never hand-transcribed); direct theorem dependencies were cross-read from the same run's direct_theorems column (field 6) and matched against the ledger's own registered kernel_theorem/formal.statement names to populate depends_on. No new probe binary was written for this batch; crates/ source was not touched to produce this batch."
}