e <= 3, via a genuine {0, 1, k+2} case split at the mathematical kink

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

Recorded description

le e three, three := add two one -- the classical index-2 split sharpening e <= 4: e = sum 1/n! = 1 + 1 + sum_{n>=2} 1/n!, the first two terms exact, and for n >= 2, 1/n! <= (1/2)^(n-1) so the tail is at most sum_{k>=1} (1/2)^k = 1. Built WITHOUT any new Nat-level factorial fact, via the pure CReal-algebra identity 2 * pow half (succ m) ~ pow half m. UNLIKE e_le_four, this is NOT one uniform forall-n bound but a nested Nat.rec on {0, 1, k+2}: expSeriesPartial 0 = expSeriesPartial 1 = 1, not yet geometric, so the index-2 kink is mathematical, not an artifact of the formalization.

Formal statement

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

Dependencies

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39 direct dependencies 0 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-CReal.e_le_three

Kind: kernel-term
Status: checked

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

Evidence notes

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

footprint-CReal.e_le_three

Kind: exhaustive-enumeration
Status: checked

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

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_e_le_three)",
  "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."
}