The integral of a sum is the sum of the integrals

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

Recorded description

For functions F, G, endpoints a <= b, and uniform-continuity witnesses for F, G and F+G on [a,b]: integral (fun t => F t + G t) a b hab uFG is Equiv to integral F a b hab uF + integral G a b hab uG.

Formal statement

theorem CReal.integral_add : ((x0 : ((x0 : CReal) -> CReal)) -> ((x1 : ((x1 : CReal) -> CReal)) -> ((x2 : CReal) -> ((x3 : CReal) -> ((x4 : CReal.le x2 x3) -> ((x5 : CReal.UniformlyContinuousOn (fun (x5 : CReal) => CReal.add (x0 x5) (x1 x5)) x2 x3) -> ((x6 : CReal.UniformlyContinuousOn x0 x2 x3) -> ((x7 : CReal.UniformlyContinuousOn x1 x2 x3) -> CReal.Equiv (CReal.integral (fun (x8 : CReal) => CReal.add (x0 x8) (x1 x8)) x2 x3 x4 x5) (CReal.add (CReal.integral x0 x2 x3 x4 x6) (CReal.integral x1 x2 x3 x4 x7))))))))))

Dependencies

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

5 direct dependencies 2 direct dependents

Evidence

kernel-CReal.integral_add

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.integral_add 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 as an unmutated CONTROL during this batch's mutation testing: with CReal.integral_converges's display name corrupted in an isolated /data0 snapshot, this grep still returned 1 (exit 0) for CReal.integral_add itself in the rebuilt binary (its dependency line changed to name CReal.integral_converges_MUTATED, but its own name-anchored line was untouched), confirming the checker discriminates on THIS theorem's presence, not on its dependencies' names. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.integral_add

Kind: exhaustive-enumeration
Status: checked

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

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'. --require-axiom-free <name> errors for a prelude never built by this run. --release is MANDATORY here.

Provenance

{
  "date": "2026-08-27",
  "established_by": "axeyum-lean-kernel build_creal_prelude (crates/axeyum-lean-kernel/src/creal/integral.rs, declare_integral_add)",
  "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. 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."
}