factorQuotient: the COMPUTED synthetic-division quotient, never extracted by Exists-elimination

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

Recorded description

Complex.factorQuotient c a n k := Nat.rec (fun _ => Complex) zero (fun r' _ => hornerFromTop c a n r') (n - k) -- the synthetic-division quotient of the degree-<n+1 polynomial c by (X - a), COMPUTED via hornerFromTop rather than extracted from an existence proof. The zero-then-hornerFromTop shape (prepending a forced zero base case and reindexing hornerFromTop's own index DOWN by one) is deliberate, not cosmetic: the natural first attempt, reindexing directly as hornerFromTop c a n (n - k), sends every k >= n through truncated Nat.sub to the SAME index 0, and hornerFromTop c a n 0 returns c(n) -- the polynomial's own LEADING coefficient, not zero -- which would make the intended degree bound on factorQuotient flatly false once type-checked against a real k >= n. Reindexing hornerFromTop's own index down by one instead sends every k >= n through n - k = 0 to the forced zero base case, and any k < n unwraps to n - k >= 1 and the correct hornerFromTop c a n (n-k-1).

Formal statement

def Complex.factorQuotient : ((x0 : ((x0 : AxNat) -> Complex)) -> ((x1 : Complex) -> ((x2 : AxNat) -> ((x3 : AxNat) -> Complex))))

Dependencies

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0 direct dependencies 0 direct dependents

Evidence

kernel-Complex.factorQuotient

Kind: kernel-term
Status: checked

Supports: Complex.factorQuotient 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 kernel_declaration_projection -- --require-declaration Complex.factorQuotient --require-kind definition 2>/dev/null | grep -cE '^found[[:space:]]complex[[:space:]]definition[[:space:]]Complex\.factorQuotient[[:space:]]'

Evidence notes

Complex.factorQuotient is a Declaration::Definition. theorem_dependency_inventory / nat_theorem_inventory / prelude_theorem_inventory all filter to Declaration::Theorem and exclude Definitions/Inductives, so this fact uses the DIRECT 'kernel_declaration_projection --require-declaration' checker: it searches every constructed prelude's environment for an exact display-name match and exits non-zero on no match or kind mismatch. Verified on this tree: the command prints exactly one 'found\tcomplex\tdefinition\tComplex.factorQuotient\t...' line. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-Complex.factorQuotient

Kind: exhaustive-enumeration
Status: checked

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

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, 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)",
  "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); crates/ source was not touched to produce this batch."
}