2 <= e, by an EVENTUAL argument

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

Recorded description

le two e -- the first numeric bound on Euler's number. Needs an EVENTUAL argument (converges_lower_bound_shift), NOT converges_lower_bound directly: expSeriesPartial 0 = 0 < 2, so the bound does not hold at index 0 and only holds from index 2 on, where monotonicity of the nonnegative-summand partial sums (sumRange_mono_outer at expTerm) takes over and expSeriesPartial 2 reduces, by Kernel::def_eq, to the same Rat.mk normal form as the literal two.

Formal statement

theorem CReal.two_le_e : CReal.le (CReal.ofRat (Rat.normalize (Int.ofNat (AxNat.succ (AxNat.succ AxNat.zero))) (AxNat.succ AxNat.zero) (AxNat.le.refl (AxNat.succ AxNat.zero)))) CReal.e

Dependencies

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8 direct dependencies 0 direct dependents

Evidence

kernel-CReal.two_le_e

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.two_le_e 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.two_le_e. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.two_le_e

Kind: exhaustive-enumeration
Status: checked

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

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_two_le_e)",
  "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."
}