A scaled Archimedean mesh bound: (m+1) * (b-a) * 1/n3 <= 1/n2 given a large enough mesh count

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

Recorded description

For a <= b : CReal and m, n2, n3 : Nat with the stated Archimedean side condition on n3 relative to bound(b-a), m and n2: (m+1) * ((b-a) * 1/(n3+1)) <= 1/(n2+1). A scaled generalization of CReal's existing mesh_le_of_ge that CReal.crossing.rs's crossingClose needs, letting the mesh spacing be scaled by an extra factor (m+1) before comparing against the target accuracy 1/(n2+1).

Formal statement

theorem CReal.meshScaledLeOfGe : ((x0 : CReal) -> ((x1 : CReal) -> ((x2 : AxNat) -> ((x3 : AxNat) -> ((x4 : AxNat) -> ((x5 : CReal.le x0 x1) -> ((x6 : AxNat.le (AxNat.add (AxNat.mul (AxNat.succ (CReal.bound (CReal.add x1 (CReal.neg x0)))) (AxNat.add (AxNat.mul (AxNat.succ x4) x2) x4)) (CReal.bound (CReal.add x1 (CReal.neg x0)))) x3) -> CReal.le (CReal.mul (CReal.ofNat (AxNat.succ x4)) (CReal.mul (CReal.add x1 (CReal.neg x0)) (CReal.ofRat (Rat.natDivSucc (AxNat.succ AxNat.zero) x3)))) (CReal.ofRat (Rat.natDivSucc (AxNat.succ AxNat.zero) x2)))))))))

Dependencies

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

Evidence

kernel-CReal.meshScaledLeOfGe

Kind: kernel-term
Status: checked

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

Evidence notes

The producing prelude admits CReal.meshScaledLeOfGe 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.meshScaledLeOfGe

Kind: exhaustive-enumeration
Status: checked

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

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.meshScaledLeOfGe, 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.rs, declare_mesh_scaled_le_of_ge)",
  "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."
}