Subtracting a truncated difference recovers the subtrahend, over the constructed naturals

Identifier: F:nat-sub-sub-self
Proof route: kernel-lean
External status: Not recorded
Axiom footprint: Empty

Recorded description

For constructed naturals n and k with k <= n, n - (n - k) = k, where `-` is this kernel's truncating `Nat.sub`. `Kernel::axiom_footprint` for this declaration is EMPTY. WHAT THIS DOES NOT CLAIM. The bound k <= n is essential and the unbounded form is FALSE: `Nat.sub` truncates, so at n = 3, k = 5 the left side is 3 and the right side is 5. Equivalently: the reflection k -> n - k is an involution ON [0,n] and not globally, which is why the injectivity argument it feeds must be the bounded one.

Formal statement

theorem Nat.sub_sub_self : ((x0 : AxNat) -> ((x1 : AxNat) -> ((x2 : AxNat.le x1 x0) -> Eq.{1} AxNat (AxNat.sub x0 (AxNat.sub x0 x1)) x1)))

Dependencies

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2 direct dependencies 1 direct dependents

Evidence

kernel-Nat.sub_sub_self

Kind: kernel-term
Status: checked

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

Evidence notes

Two independent failure modes, so the exit status depends on the finding rather than on the run completing: theorem_dependency_inventory exits non-zero when a NAMED filter matches nothing, and grep -c exits 1 printing 0 when the anchored line is absent. Anchored with [[:space:]], never \t -- in a scripted (GNU) grep \t is a literal t. grep -c rather than grep -q, which would SIGPIPE the producer under pipefail. --release is MANDATORY: this tool builds creal/complex/cpoint, which overflow the default debug thread stack.

footprint-Nat.sub_sub_self

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the nat prelude's trusted surface is empty, which bounds Nat.sub_sub_self.

Checker command

cargo run -q --release -p axeyum-lean-kernel --example nat_axiom_inventory -- --require-axiom-free nat

Evidence notes

--require-axiom-free exits non-zero when the named prelude's trusted surface (Axiom + Opaque + Quotient) is not empty, and errors rather than silently passing for a prelude the run never built. A declaration cannot depend on a trusted declaration the environment does not contain, so an empty nat surface bounds every declaration in it, including Nat.sub_sub_self. This is a whole-prelude bound, not a per-declaration measurement; the per-declaration figure is the footprint column of kernel_declaration_projection, measured 0 for this row.

Provenance

{
  "date": "2026-08-31",
  "curation": "generated-unreviewed",
  "generated_by": "scripts/gen-kernel-facts.py",
  "established_by": "axeyum-lean-kernel build_nat_prelude (crates/axeyum-lean-kernel/src/nat_prelude/)",
  "source": "Derived mechanically from the unfiltered emit of `cargo run -q --release -p axeyum-lean-kernel --example kernel_declaration_projection`, which prints one TSV row per declaration whose fields are (prelude, kind, display name, axiom-footprint size, direct type declarations, direct declarations, direct theorems, Kernel::render_lean(declaration.ty())). formal.statement is that last field verbatim; depends_on is the direct-theorem column intersected with this ledger's registered facts; axiom_footprint is the footprint-size column, cross-checked by the whole-prelude nat_axiom_inventory run recorded in the second evidence row. No field was hand-transcribed and no prose was authored. `title`, `statement` and `notes` were then REPLACED by hand (lane first-supplementary-residue, ADR-1235); every other field is the generator's output unmodified."
}