1/n! is dominated by the geometric term 2/2^n

Identifier: F:creal-expterm-le-geom
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

forall n, le (expTerm n) (ofRat (2 / 2^n)) -- the pointwise domination that lets the exponential series' Cauchy property and e's numeric bounds all be derived from the geometric series' own Cauchy property, without ever constructing a Nat-level factorial growth rate directly. Part of the domination bridge (with expDominantCauchy and cauchyOfPointwiseEquiv) this batch's brief names explicitly.

Formal statement

theorem CReal.expTerm_le_geom : ((x0 : AxNat) -> CReal.le (CReal.expTerm x0) (CReal.ofRat (Rat.normalize (Int.ofNat (AxNat.succ (AxNat.succ AxNat.zero))) (AxNat.pow (AxNat.succ (AxNat.succ AxNat.zero)) x0) (AxNat.pow_pos (AxNat.succ (AxNat.succ AxNat.zero)) x0 (AxNat.le_succ_succ AxNat.zero (AxNat.succ AxNat.zero) (AxNat.zero_le (AxNat.succ AxNat.zero)))))))

Dependencies

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

Evidence

kernel-CReal.expTerm_le_geom

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.expTerm_le_geom through the trusted Kernel::add_declaration gate. theorem_dependency_inventory exits non-zero for a named filter matching nothing; grep -c (never -q) both consumes the pipe and asserts the exact tab-anchored line is present. Verified on this tree: the command currently prints exactly one matching line for CReal.expTerm_le_geom. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.expTerm_le_geom

Kind: exhaustive-enumeration
Status: checked

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

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. --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/exponential.rs, declare_exp_term_le_geom)",
  "source": "theorem name and canonical type read via a standalone probe binary depending on axeyum-lean-kernel by path, calling only its public Kernel API (environment(), display_name(), render_lean(), axiom_footprint(), theorem_dependencies()) -- no in-tree example prints this theorem's own type directly. Direct dependency edges cross-checked against theorem_dependency_inventory's own output on the same tree. The probe was built and run in the session scratchpad and deleted after use; crates/ source was not touched to produce this batch."
}