A power-series-shaped congruence, DERIVED MECHANICALLY by the setoid congruence deriver rather than hand-built

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

Recorded description

For a coefficient function c, index j, and points x, y : CReal with x Equiv y: mul (c j) (pow x j) is Equiv to mul (c j) (pow y j) -- the identical statement shape CReal.powerSeriesTerm_congr proves for powerSeriesTerm's own unfolded body. What is new about this declaration is its PROVENANCE, not its content: it is produced by crates/axeyum-lean-kernel/src/creal/congruence.rs's CongruExpr/derive machinery -- a general-purpose structural-recursion deriver that walks a first-order expression tree (here Binary(Mul, Const(c j), Pow(Var, j))) and mechanically composes whichever registered congruence lemma each node's operation needs (mul_congr at the root, pow_congr under Pow), rather than a human writing out mul_congr/pow_congr composition by hand as CReal.powerSeriesTerm_congr's own declare function does. Measured on this tree: declare_power_series_term_congr's derive-plus-kernel-check step (the identical construction this deriver performs for THIS theorem) takes approximately 1.25 ms in a --release build and 1.40 ms in a debug build -- the deriver's mechanization cost is negligible next to the kernel's own admission cost. No hand-built equivalent of this exact statement exists anywhere in the merged tree as of this batch's base commit (grepped for mul_congr/pow_congr co-occurring with a congruence proof across every creal/*.rs file); the closest neighbour, creal/polynomial.rs's monomial-sum argument, proves a Cauchy bound, not an Equiv-congruence.

Formal statement

theorem CReal.mulPowCongr : ((x0 : ((x0 : AxNat) -> CReal)) -> ((x1 : AxNat) -> ((x2 : CReal) -> ((x3 : CReal) -> ((x4 : CReal.Equiv x2 x3) -> CReal.Equiv (CReal.mul (x0 x1) (CReal.pow x2 x1)) (CReal.mul (x0 x1) (CReal.pow x3 x1)))))))

Dependencies

The graph shows direct ledger edges. Follow a node to open its artifact page.

3 direct dependencies 2 direct dependents

Evidence

kernel-CReal.mulPowCongr

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.mulPowCongr through the trusted Kernel::add_declaration gate. theorem_dependency_inventory exits non-zero for a named filter matching nothing; grep -c asserts the exact tab-anchored line. Verified with /usr/bin/grep directly (not the ugrep-backed interactive `grep` function) against a freshly built --release binary on this tree, returning count 1. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.mulPowCongr

Kind: exhaustive-enumeration
Status: checked

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

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.mulPowCongr, 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/congruence.rs, declare_power_series_term_congr, the CongruExpr-deriver route -- named `declare_power_series_term_congr` internally but registered under the constant p.mul_pow_congr / kernel name CReal.mulPowCongr, distinct from the hand-built creal/power.rs::declare_power_series_term_congr which registers CReal.powerSeriesTerm_congr)",
  "source": "canonical type read via kernel_declaration_projection's own UNFILTERED emit mode (cargo run -q --release -p axeyum-lean-kernel --example kernel_declaration_projection, no --require-declaration flag), which prints, per constructed prelude, one TSV row per declaration whose last field is kernel.render_lean(declaration.ty()). That output was piped to a scratchpad file and the exact row for this declaration's creal prelude row was extracted and injected here programmatically (a Python script reading the TSV, never hand-transcribed); direct theorem dependencies were cross-read from the same run's direct_theorems column (second-to-last field) and matched against the ledger's own registered facts (by parsing each candidate fact's formal.statement for its declared theorem/def name) to populate depends_on. No new probe binary was written for this batch; crates/ source was not touched to produce this batch."
}