cos(1) <= 1/0! = 1, the exact real upper bound on cos(1)

Identifier: F:creal-cosone-le-exp-term-zero
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

CReal.cosOne_alternating_upper (this batch) at m := 0: sumRange cosTerm (succ(0+0)) is ι-defeq to add(zero, cosTerm 0) (both Nat.add's and CReal.sumRange's own recursion steps), and cosTerm(0) is itself ι-defeq to mul(one, expTerm(0)) (CReal.pow_zero closes pow(-1,0) to one by Eq.refl alone). Folding add(zero, mul(one, expTerm(0))) down to expTerm(0) needs two genuine Equiv steps (add_comm+add_zero, one_mul), composed and transported across the inequality via CReal.le_congr; reducing expTerm(0) further to the literal constant CReal.one (which F:creal-expterm-zero-eq-one, this batch, separately does) is not attempted inside this proof. THIS SUPERSEDES the loose upper bound in F:creal-cosone-le-four without deleting it: this is the exact real interval [0, expTerm(0)] (paired with F:creal-cosone-nonneg), not a coarse [-4,4]-style envelope.

Formal statement

theorem CReal.cosOne_le_exp_term_zero : CReal.le CReal.cosOne (CReal.expTerm AxNat.zero)

Dependencies

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8 direct dependencies 0 direct dependents

Evidence

kernel-CReal.cosOne_le_exp_term_zero

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.cosOne_le_exp_term_zero 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.cosOne_le_exp_term_zero

Kind: exhaustive-enumeration
Status: checked

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

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.cosOne_le_exp_term_zero, 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_cos_one_le_exp_term_zero)",
  "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."
}