The Cauchy-Schwarz inequality

Identifier: F:cpoint-cauchy-schwarz
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

For points a, b of the constructed plane, dot(a,b) * dot(a,b) is CReal.le dot(a,a) * dot(b,b) -- the squared form of the Cauchy-Schwarz inequality, proved here directly from CReal order and ring laws (CReal.sq_nonneg on the discriminant) rather than by citing F:cpoint-dot-self-add/sub.

Formal statement

theorem CPoint.cauchy_schwarz : ((x0 : CPoint) -> ((x1 : CPoint) -> CReal.le (CReal.mul (CPoint.dot x0 x1) (CPoint.dot x0 x1)) (CReal.mul (CPoint.dot x0 x0) (CPoint.dot x1 x1))))

Dependencies

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21 direct dependencies 2 direct dependents Graph shows the first 8 on each side.

Evidence

kernel-CPoint.cauchy_schwarz

Kind: kernel-term
Status: checked

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

Evidence notes

`build_cpoint_prelude` admits CPoint.cauchy_schwarz 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 (extended 2026-08-25); 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) -- the same resource-limit gotcha documented for `prelude_theorem_inventory --include-constructed`.

footprint-CPoint.cauchy_schwarz

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- CPoint.cauchy_schwarz's narrow AND widened trusted-declaration closure is empty

Checker command

cargo run -q --release -p axeyum-lean-kernel --example footprint_closure_audit -- --include-constructed 2>/dev/null | grep -cE '^cpoint[[:space:]]CPoint\.cauchy_schwarz[[:space:]][^[:space:]]*[[:space:]]0[[:space:]][[:space:]][[:space:]]0[[:space:]]$'

Evidence notes

`nat_axiom_inventory` does NOT cover `cpoint` at all (verified: `--include-constructed --require-axiom-free cpoint` errors '"cpoint" is not enumerated by this run (known: ... creal, complex)' -- cpoint was never added to that tool's group list, an empty/absent-coverage trap in its own right). `footprint_closure_audit --include-constructed` DOES build a `cpoint` group (verified in its `build_groups`) and, per admitted theorem, emits a row `cpoint<TAB>NAME<TAB>narrow_kinds<TAB>narrow_dangling<TAB>narrow_trusted<TAB>widened_trusted<TAB>widened_dangling<TAB>inductive_edge_diffs`. An axiom-free theorem has narrow_dangling=0, narrow_trusted empty, widened_trusted empty (the WIDENED closure additionally walks inductive constructor/recursor types one step further than `render_deps`, so this is stronger than the narrow-only check the `nat_axiom_inventory`-style facts use), widened_dangling=0, and no inductive edge diff -- the anchored regex below requires all of that in one line. `--include-constructed` is REQUIRED: without it the cpoint group is never built and BOTH the row-match and the `cpoint: theorems=` coverage line are absent (verified: 0 matches for either, vs 1 for both with the flag), which is the same coverage trap CLAUDE.md already documents for `nat_axiom_inventory`. `--release` is MANDATORY for the same deep-recursion reason as `theorem_dependency_inventory` (this tool's own doc example uses it); measured full-audit wall time on this tree ~20s release.

Provenance

{
  "date": "2026-08-25",
  "established_by": "axeyum-lean-kernel build_cpoint_prelude",
  "source": "theorem name and dependency edges from theorem_dependency_inventory (builds creal/complex/cpoint); canonical type read via a standalone probe binary depending on axeyum-lean-kernel by path, calling only its public Kernel API (environment(), display_name(), render_lean(), axiom_footprint()) -- no in-tree example prints Complex/CPoint theorem types; crates/ source was not touched to produce this batch."
}