The integral of a constant function is base times height

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

Recorded description

For a constant c, endpoints a <= b, and a uniform-continuity witness for the constant function: integral (fun _ => c) a b hab u is Equiv to c * (b - a). Spivak Ch.14's opening evaluation of the integral, exact for every valid choice of witness, with no error term.

Formal statement

theorem CReal.integral_const : ((x0 : CReal) -> ((x1 : CReal) -> ((x2 : CReal) -> ((x3 : CReal.le x1 x2) -> ((x4 : CReal.UniformlyContinuousOn (fun (x4 : CReal) => x0) x1 x2) -> CReal.Equiv (CReal.integral (fun (x5 : CReal) => x0) x1 x2 x3 x4) (CReal.mul x0 (CReal.add x2 (CReal.neg x1))))))))

Dependencies

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5 direct dependencies 4 direct dependents

Evidence

kernel-CReal.integral_const

Kind: kernel-term
Status: checked

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

Evidence notes

build_creal_prelude admits CReal.integral_const 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 and CReal.sharedIndexToCanonical's display names corrupted in an isolated /data0 snapshot, this exact grep still returned 1 (exit 0) against the rebuilt binary -- confirming the pattern is not vacuously green, i.e. it discriminates on THIS theorem's own name rather than on the build succeeding in general. --release is MANDATORY: this tool also builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-CReal.integral_const

Kind: exhaustive-enumeration
Status: checked

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

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_const)",
  "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."
}