sinTerm is dominated by the shared geometric bound expDominant

Identifier: F:creal-sintermabsledominant
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For every k, |sinTerm(k)| <= expDominant(k) -- the sinTerm analogue of CReal.cosTermAbsLeDominant, using the odd index (2k+1) and the private exp_dominant_odd_le lemma in place of the doubled index and exp_dominant_double_le. expDominant is the SAME geometric bound CReal.e's own convergence machinery uses -- this proof reuses that machinery unchanged rather than deriving a new bound, exactly as CReal.cosOne's own construction does.

Formal statement

theorem CReal.sinTermAbsLeDominant : ((x0 : AxNat) -> CReal.le (CReal.abs (CReal.mul (CReal.pow (CReal.neg CReal.one) x0) (CReal.expTerm (AxNat.add (AxNat.add x0 x0) (AxNat.succ AxNat.zero))))) (CReal.mul (CReal.ofRat (Rat.normalize (Int.ofNat (AxNat.succ (AxNat.succ AxNat.zero))) (AxNat.succ AxNat.zero) (AxNat.le.refl (AxNat.succ AxNat.zero)))) (CReal.pow (CReal.ofRat (Rat.natDivSucc (AxNat.succ AxNat.zero) (AxNat.succ AxNat.zero))) x0)))

Dependencies

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

32 direct dependencies 1 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-CReal.sinTermAbsLeDominant

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.sinTermAbsLeDominant 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-CReal.sinTermAbsLeDominant

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the creal prelude's trusted surface is empty, which bounds CReal.sinTermAbsLeDominant

Checker command

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

Evidence notes

Re-measured on this tree: creal: axiom=0 opaque=0 quotient=0 total_trusted=0, exits 0 printing 'ok: creal trusted surface = 0'. That bounds every declaration in the creal environment, including CReal.sinTermAbsLeDominant, 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_creal_prelude (crates/axeyum-lean-kernel/src/creal/trig.rs, declare_sin_term_abs_le_dominant)",
  "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."
}