The constructed Int is a discretely ordered ring generated by 1, with unique maps out

Identifier: F:int-characterization
Proof route: kernel-lean
External status: proved
Axiom footprint: Empty

Recorded description

The constructed `Int` has no elements beyond its two constructors (every element is +/- a natural number), is generated by 0 under +1 and -1 (an induction principle), is discretely ordered at EVERY point (nothing strictly between a and a+1), is totally ordered, is non-trivial (0 != 1), and any two maps out of it that agree at 0 and satisfy the same +1 and -1 recurrences are equal. These are the properties that separate the integers from the rationals (which fail discreteness) and from lexicographic Z[x] (which is discretely ordered but is not generated by 1).

Formal statement

theorem Int.Characterization.induction : ((x0 : ((x0 : Int) -> Prop)) -> ((x1 : x0 Int.zero) -> ((x2 : ((x2 : Int) -> ((x3 : x0 x2) -> x0 (Int.add x2 Int.one)))) -> ((x3 : ((x3 : Int) -> ((x4 : x0 x3) -> x0 (Int.add x3 (Int.neg Int.one))))) -> ((x4 : Int) -> x0 x4)))))

Dependencies

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2 direct dependencies 1 direct dependents

Evidence

kernel-Int.Characterization.induction

Kind: kernel-term
Status: checked

Supports: Int is generated by 0 under +1 and -1: the property lexicographic Z[x] fails.

Checker command

out=$(cargo run -q -p axeyum-lean-kernel --example characterization_status 2>/dev/null) && test "$(printf '%s\n' "$out" | grep -Ec '^int-generation[[:space:]]+Int[.]Characterization[.]induction[[:space:]]+\[\][[:space:]]')" -ge 1

Evidence notes

Both steps are definitional in the normalized ofNat/negSucc representation -- `ofNat n + 1` iota-reduces to `ofNat (succ n)`, `0 + (-1)` to `negSucc 0`, and `negSucc n + (-1)` to `negSucc (succ n)` -- so each branch is `Nat.rec` applied to the hypotheses with no rewriting. Two `Weakening` variants replace the +1 and the -1 step with `True`; the kernel refuses this declaration in both cases.

kernel-Int.Characterization.of_nat_or_neg

Kind: kernel-term
Status: checked

Supports: No junk: every integer is + or - a natural number.

Checker command

out=$(cargo run -q -p axeyum-lean-kernel --example characterization_status 2>/dev/null) && test "$(printf '%s\n' "$out" | grep -Ec '^int-no-junk[[:space:]]+Int[.]Characterization[.]of_nat_or_neg[[:space:]]+\[\][[:space:]]')" -ge 1

Evidence notes

Together with `Int.Characterization.cases` (exhaustive constructor case analysis, admitted in the same package) this says the carrier has exactly the elements the construction puts in it.

kernel-Int.Characterization.discrete-everywhere

Kind: kernel-term
Status: checked

Supports: Discreteness at every point: nothing strictly between a and a+1. This is the property the rationals fail.

Checker command

out=$(cargo run -q -p axeyum-lean-kernel --example characterization_status 2>/dev/null) && test "$(printf '%s\n' "$out" | grep -Ec '^int-order[[:space:]]+Int[.]Characterization[.]discrete_everywhere[[:space:]]+\[\][[:space:]]')" -ge 1

Evidence notes

Derived, not assumed: the interval (a, a+1) is translated down to (0, 1) through -a with `add_lt_add_of_le_of_lt` and `le_refl`, and the endpoints normalise to 0 and 1 by the ring laws, landing in the prelude's `no_int_between`. The 0/1 instance alone would leave the general statement to a reader's translation-invariance argument; this is the machine-checked one.

kernel-Int.Characterization.rec-unique

Kind: kernel-term
Status: checked

Supports: The UNIQUENESS half of the universal property: two maps out of Int agreeing at 0 with the same +/-1 recurrences are equal. Paired with the existence half (F:int-categoricity) this makes Int the initial Z-structure.

Checker command

out=$(cargo run -q -p axeyum-lean-kernel --example characterization_status 2>/dev/null) && test "$(printf '%s\n' "$out" | grep -Ec '^int-universal-property[[:space:]]+Int[.]Characterization[.]rec_unique[[:space:]]+\[\][[:space:]]')" -ge 1

Evidence notes

Proved from `Int.Characterization.induction`. `characterization_tests::the_int_hypothesis_shapes_are_satisfiable` instantiates the whole premise list and admits the result, so the theorem is not vacuous. Since 2026-08-18 it is also what proves the Int-side half of `Int.Characterization.iso`: `psi . iter` and the identity map agree at 0 with the same recurrences, so they are equal.

integer-trusted-surface-empty

Kind: exhaustive-enumeration
Status: checked

Supports: axiom_footprint: [] -- the integer prelude's whole trusted surface is empty.

Checker command

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

Evidence notes

Measured 2026-08-17: integer=0 over Axiom/Opaque/Quotient.

Provenance

{
  "date": "2026-08-17",
  "established_by": "axeyum-lean-kernel characterization module (agent-characterization lane)",
  "source": "hand-built foundational library; the property list is the standard characterization of the integers among ordered rings, authored here"
}