factorQuotient's recursive relationship across a degree increment

Identifier: F:complex-factorquotient-succ-eq
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For c, a, n, k with k < n: factorQuotient c a (n+1) k is Equiv to factorQuotient c a n k plus the correction term pow(a, n-k) * c(n+1) -- the correction Complex.hornerFromTop_diag_eq_polyEval's own doc names as the reason a naive induction reusing the smaller factorQuotient wholesale does not work, made precise and proved. This turns out to need NO fresh induction at all: once k < n gives n-k a successor shape (Nat.succ j, via succ_pred_of_pos + le_of_lt_succ from the PUBLIC nat prelude -- deliberately not the module-private nat_prelude::choose::sub_succ_of_lt, which proves the identical fact but is not reachable across the module boundary, so this re-derives it from public building blocks), a public Nat.succ_sub_of_le-style reindexing puts (n+1)-k at the same successor-of-successor shape, and BOTH sides reduce (one Nat.rec iota-step each) to hornerFromTop c a (n+1) (j+1) and hornerFromTop c a n j respectively -- at which point Complex.hornerFromTop_succ_succ (an Eq.refl fact, proved before this theorem existed) IS the correction term, verbatim.

Formal statement

theorem Complex.factorQuotient_succ_eq : ((x0 : ((x0 : AxNat) -> Complex)) -> ((x1 : Complex) -> ((x2 : AxNat) -> ((x3 : AxNat) -> ((x4 : AxNat.lt x3 x2) -> Complex.Equiv (Complex.factorQuotient x0 x1 (AxNat.succ x2) x3) (Complex.add (Complex.factorQuotient x0 x1 x2 x3) (Complex.mul (Complex.pow x1 (AxNat.sub x2 x3)) (x0 (AxNat.succ x2)))))))))

Dependencies

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

Evidence

kernel-Complex.factorQuotient_succ_eq

Kind: kernel-term
Status: checked

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

Evidence notes

build_complex_prelude admits Complex.factorQuotient_succ_eq 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.factorQuotient_succ_eq

Kind: exhaustive-enumeration
Status: checked

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

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.factorQuotient_succ_eq, 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_factor_quotient_succ_eq)",
  "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."
}