Adding a nonnegative rational to a constructed real does not decrease it

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

Recorded description

For a constructed real x0 and a rational x1 with 0 <= x1: x0 <= x0 + ofRat(x1).

Formal statement

theorem CReal.le_add_of_nonneg : ((x0 : CReal) -> ((x1 : Rat) -> ((x2 : Rat.le Rat.zero x1) -> CReal.le x0 (CReal.add x0 (CReal.ofRat x1)))))

Dependencies

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12 direct dependencies 6 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-CReal.le_add_of_nonneg

Kind: kernel-term
Status: checked

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

Evidence notes

`build_creal_prelude` admits CReal.le_add_of_nonneg through the trusted `Kernel::add_declaration` gate, which re-checks the proof term against the stated type, so a line in this tool's output for the exact name is a machine-checked proof having been admitted. `theorem_dependency_inventory` builds creal/complex/cpoint in one run; it exits non-zero for a named filter that matches nothing, and `grep -c` (never `-q`) independently asserts the exact tab-anchored line is present. `--release` is MANDATORY: building the constructed carriers recurses deep enough in a debug build to overflow the default thread stack (measured on this tree: release exits 0, debug SIGABRTs at 134).

footprint-CReal.le_add_of_nonneg

Kind: exhaustive-enumeration
Status: checked

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

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_axiom_inventory -- --include-constructed --require-axiom-free creal

Evidence notes

`nat_axiom_inventory --include-constructed --require-axiom-free creal` builds the creal environment as its own group and reports `creal: axiom=0 opaque=0 quotient=0 total_trusted=0` (re-measured on this tree), covering Axiom, Opaque AND Quotient, not just Declaration::Axiom -- this bounds CReal.le_add_of_nonneg's individual footprint by []. `--include-constructed` is REQUIRED: without it the command exits 1 naming `creal` as known but not built this run rather than a silent pass. `--release` is MANDATORY for the same deep-recursion reason as `theorem_dependency_inventory` (measured on this tree: debug SIGABRTs at 134).

Provenance

{
  "date": "2026-08-25",
  "established_by": "axeyum-lean-kernel build_creal_prelude",
  "source": "theorem name and dependency edges from theorem_dependency_inventory (builds nat/int/rat/creal/complex/cpoint/string/characterization); canonical type and top-level binder count read via a standalone probe binary depending on axeyum-lean-kernel by path, calling only its public Kernel API (environment(), display_name(), render_lean()) -- no in-tree example prints Rat/CReal theorem types beyond a substring filter; crates/ source was not touched to produce this batch; the probe was built and run in the session scratchpad and deleted after use."
}