2 <= pi, the cheapest honest lower bound

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

Recorded description

le two pi. piHalfSeriesPartial 1 is add zero (piHalfTerm 0), which delta-iota-reduces to CReal.one's own Rat.mk (Rat.add Rat.zero (natDivSucc 1 0) normalises to 1/1), so CReal.le_refl at one type-checks against le one (sumRange piHalfTerm 1) with no rewrite -- the exact analogue of how CReal.two_le_e reads expSeriesPartial 2 as two. Every later partial sum is at least as large (CReal.sumRange_mono_outer at the nonnegative piHalfTerm), so CReal.converges_lower_bound_shift at shift 1 gives 1 <= piHalf, and scaling by two gives 2 <= pi. Registered alongside the sharper F:creal-three-le-pi because it is FREE -- it forms no rational larger than 1, whereas the sharper bound is the entire measured cost of this family on the prelude build.

Formal statement

theorem CReal.twoLePi : 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.pi

Dependencies

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

Evidence

kernel-CReal.twoLePi

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits this theorem 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. ONE NAME PER INVOCATION: this tool silently discards every name argument after the first. --release is MANDATORY.

footprint-CReal.twoLePi

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the creal prelude's trusted surface is empty, which bounds this declaration

Checker command

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

Evidence notes

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 this one. --require-axiom-free <name> errors for a prelude never built by this run rather than silently passing on zero rows. --release is MANDATORY: this tool builds creal/complex/cpoint, which overflow the default debug thread stack. Independently re-measured this session by kernel_declaration_projection, whose footprint-size column reads 0 for all fourteen CReal.pi* declarations.

Provenance

{
  "date": "2026-08-31",
  "established_by": "axeyum-lean-kernel build_creal_prelude (crates/axeyum-lean-kernel/src/creal/pi.rs, declare_pi_family)",
  "source": "canonical type and axiom-footprint size read from kernel_declaration_projection --release on this tree (the creal-prelude row), not from source text or a doc comment."
}