(-1)^(2k) = 1 for every k, by plain induction (no parity case split)

Identifier: F:creal-negonepowdouble
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For every k : Nat, CReal.Equiv (CReal.pow (CReal.neg CReal.one) (k+k)) CReal.one. The parity fact underneath the Ch.22-23 alternating-series bracket theorems. Proved by plain induction on k using Nat.succ_add to relate pow(-1)^(succ k + succ k) to pow(-1)^(k+k) times (neg one)*(neg one), never by a decidable-parity case split on an arbitrary Nat (which this kernel does not have machinery for).

Formal statement

theorem CReal.negOnePowDouble : ((x0 : AxNat) -> CReal.Equiv (CReal.pow (CReal.neg CReal.one) (AxNat.add x0 x0)) CReal.one)

Dependencies

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7 direct dependencies 3 direct dependents

Evidence

kernel-CReal.negOnePowDouble

Kind: kernel-term
Status: checked

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

Evidence notes

The producing prelude admits CReal.negOnePowDouble through the trusted Kernel::add_declaration gate, which re-checks the proof term against the stated type. theorem_dependency_inventory exits non-zero for a named filter matching nothing (a deleted/renamed theorem cannot read as a re-derived one), and grep -c (never -q) both avoids a SIGPIPE-under-pipefail false negative and independently asserts the exact tab-anchored line is present. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.negOnePowDouble

Kind: exhaustive-enumeration
Status: checked

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

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.negOnePowDouble, 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/alternating.rs, declare_neg_one_pow_double)",
  "source": "theorem name, canonical type and direct dependency edges read via theorem_dependency_inventory's own per-name output (cargo run -q --release -p axeyum-lean-kernel --example theorem_dependency_inventory -- <Name>), which prints '<Name>\\t<comma-separated direct deps>'; the canonical type itself was cross-extracted from kernel_declaration_projection's unfiltered TSV dump the same way as the Definition facts in this batch, and confirmed byte-for-byte identical between the two tools' output for every name in this batch. crates/ source was not touched to produce this batch."
}