Inlay

I’ve been testing Claude’s ability to generate Lean 4 code to prove theorems. I’ve written about a couple experiments that verified calculations. I did not write about my failed attempt to get Claude to formalize a proof of the pqr theorem for seminorms. This time I asked Claude to formally prove the theorem from the previous post on partial fraction decomposition in an abstract setting. Unlike the previous experiments, I did not give it a proof or calculation to verify; I asked it to _generate_ a proof. This was the prompt: > Write a proof in Lean 4 for the following generalization of partial fraction decomposition. > > If a ring R is a principle ideal domain (PID), then every element c of the field of fractions over R can be written in the form \\[c = \sum_i \frac{a_i}{p_i^{r_i}}\\] > > where the p_i are nonassociate irreducible elements of R, the r_i are non-negative integers, and the elements a_i and p_i are relatively prime. This was Claude’s immediate response: > I’ll work through this carefully, as it’s a substantial formalization task. Let me think through what’s available in Mathlib and what needs to be constructed. > > This is a genuinely hard Lean 4 formalization problem. Let me think carefully about what’s in Mathlib, what the statement precisely means, and what proof strategy is realistic. It also stated its overall proof strategy: > **Proof strategy** : Write `c = a/b` in lowest terms. Factor `b = ∏ pᵢ^rᵢ`. Use the Chinese Remainder Theorem / Bezout identity to decompose `a/b` into partial fractions. If I needed to prove the theorem by hand, this suggestion alone could be very helpful. The first attempt generated code that would not build. As before, a lot of the problems seem to stem from things moving around in Mathlib due to refactorings. There are also occasional hallucinations. I did not offer any help other than pasting in error messages after each failure. When working on a real problem, I’d be more interactive. But as a test, I just passively pasted error messages and tried again. It took 11 iterations, but I finally got a proof. The proof had five sections marked “sorry,” which is essentially a TODO note indicating an unproven assumption. I asked if it could explain or remove the sorries, and it said it could remove three of them. But the generated code did not build and I hit a session limit before I could debug the code. Here is the code with five sorry sections. /- PartialFractions.lean (v11 — final) -/ import Mathlib open UniqueFactorizationMonoid variable {R : Type*} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] structure PFDSummand (R : Type*) [CommRing R] where prime : R exponent : ℕ numer : R noncomputable def PFDSummand.val [CommRing R] [IsDomain R] (s : PFDSummand R) : FractionRing R := algebraMap R (FractionRing R) s.numer / algebraMap R (FractionRing R) (s.prime ^ s.exponent) structure IsPFD [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] (terms : List (PFDSummand R)) (c : FractionRing R) : Prop where irred : ∀ s ∈ terms, Irreducible s.prime nonAssoc : ∀ s ∈ terms, ∀ t ∈ terms, s ≠ t → ¬Associated s.prime t.prime coprime : ∀ s ∈ terms, IsCoprime s.numer s.prime sum_eq : (terms.map PFDSummand.val).sum = c private lemma algMap_ne_zero {a : R} (h : a ≠ 0) : (algebraMap R (FractionRing R)) a ≠ 0 := map_ne_zero_iff _ (IsFractionRing.injective R (FractionRing R)) |>.mpr h private lemma exists_num_denom (c : FractionRing R) : ∃ a b : R, b ≠ 0 ∧ (algebraMap R (FractionRing R)) a / (algebraMap R (FractionRing R)) b = c := by obtain ⟨a, b, hb, hab⟩ := IsFractionRing.div_surjective (A := R) c exact ⟨a, b, nonZeroDivisors.ne_zero hb, hab⟩ -- ====================================================================== -- §6 Bézout splitting -- ====================================================================== private lemma bezout_split (x y : R) (hcop : IsCoprime x y) (hx : x ≠ 0) (hy : y ≠ 0) (e : R) : ∃ s t : R, s * x + t * y = 1 ∧ (algebraMap R (FractionRing R) e) / ((algebraMap R (FractionRing R) x) * (algebraMap R (FractionRing R) y)) = (algebraMap R (FractionRing R) (e * t)) / (algebraMap R (FractionRing R) x) + (algebraMap R (FractionRing R) (e * s)) / (algebraMap R (FractionRing R) y) := by obtain ⟨s, t, hst⟩ := hcop refine ⟨s, t, hst, ?_⟩ have hxF : (algebraMap R (FractionRing R)) x ≠ 0 := algMap_ne_zero hx have hyF : (algebraMap R (FractionRing R)) y ≠ 0 := algMap_ne_zero hy have hxyF : (algebraMap R (FractionRing R)) x * (algebraMap R (FractionRing R)) y ≠ 0 := mul_ne_zero hxF hyF have hst_F : (algebraMap R (FractionRing R)) s * (algebraMap R (FractionRing R)) x + (algebraMap R (FractionRing R)) t * (algebraMap R (FractionRing R)) y = 1 := by have h := congr_arg (algebraMap R (FractionRing R)) hst simp only [map_add, map_mul, map_one] at h; exact h -- Prove: ι e / (ι x * ι y) = ι(e*t)/ι x + ι(e*s)/ι y -- Strategy: show RHS = LHS via eq_div_iff, simplify, linear_combination. symm rw [eq_div_iff hxyF, add_mul, div_mul_eq_mul_div, div_mul_eq_mul_div] -- Goal: ι(e*t)*(ι x*ι y)/ι x + ι(e*s)*(ι x*ι y)/ι y = ι e -- The mul_comm on the product inside: rw [show (algebraMap R (FractionRing R)) x * (algebraMap R (FractionRing R)) y = (algebraMap R (FractionRing R)) y * (algebraMap R (FractionRing R)) x from mul_comm _ _] -- Now: ι(e*t)*(ι y*ι x)/ι x + ι(e*s)*(ι y*ι x)/ι y = ι e -- Use field_simp to clear remaining denominators field_simp [hxF, hyF] -- Now a polynomial goal: close with linear_combination simp only [map_mul] linear_combination (algebraMap R (FractionRing R) e) * hst_F -- ====================================================================== -- §7 Coprime numerator reduction -- ====================================================================== private lemma coprime_numer_exists {p : R} (hp : Irreducible p) (a : R) (n : ℕ) : ∃ a' q : R, a = a' + q * p ^ n ∧ IsCoprime a' p := by rcases Classical.em (p ∣ a) with _h | h · exact ⟨0, 0, by sorry, by sorry⟩ · exact ⟨a, 0, by ring, isCoprime_comm.mpr ((irreducible_iff_prime.mp hp).coprime_iff_not_dvd.mpr h)⟩ -- ====================================================================== -- §8 Main theorem -- ====================================================================== private def emptyPFD (c : FractionRing R) (hc : c = 0) : ∃ terms : List (PFDSummand R), IsPFD terms c := ⟨[], { irred := by simp, nonAssoc := by simp, coprime := by simp, sum_eq := by simp [hc] }⟩ private lemma isPFD_of_denom : ∀ (d : R) (e : R), ∃ terms : List (PFDSummand R), IsPFD terms ((algebraMap R (FractionRing R) e) / (algebraMap R (FractionRing R) d)) := by intro d induction d using UniqueFactorizationMonoid.induction_on_coprime with | h0 => intro e; simp only [map_zero, div_zero] exact emptyPFD 0 rfl | h1 => intro e -- Unit denominator: sorry for now (§9C) exact ⟨[], { irred := by simp, nonAssoc := by simp, coprime := by simp, sum_eq := by simp; sorry }⟩ | hpr => rename_i prime_elem exp hp_prime intro e have hp_irred : Irreducible prime_elem := irreducible_iff_prime.mpr hp_prime obtain ⟨e', _q, _he, he'_cop⟩ := coprime_numer_exists hp_irred e exp exact ⟨[⟨prime_elem, exp, e'⟩], { irred := fun s hs => by simp only [List.mem_singleton] at hs; subst hs; exact hp_irred nonAssoc := fun s hs t ht hne => by simp only [List.mem_singleton] at hs ht; subst hs; subst ht; exact absurd rfl hne coprime := fun s hs => by simp only [List.mem_singleton] at hs; subst hs; exact he'_cop sum_eq := by simp only [List.map_singleton, List.sum_singleton, PFDSummand.val]; sorry }⟩ | hcp => rename_i x y hrel ih_x ih_y intro e -- Handle the case where x = 0 or y = 0 directly rcases eq_or_ne x 0 with rfl | hx · -- x = 0: denominator = 0*y = 0, so quotient = 0 simp only [zero_mul, map_zero, div_zero] exact emptyPFD 0 rfl rcases eq_or_ne y 0 with rfl | hy · -- y = 0: denominator = x*0 = 0, so quotient = 0 simp only [mul_zero, map_zero, div_zero] exact emptyPFD 0 rfl -- Both nonzero: apply Bézout splitting -- IsCoprime x y from IsRelPrime x y (using IsRelPrime.isCoprime) have hcop : IsCoprime x y := hrel.isCoprime obtain ⟨s, t, _hst, hsplit⟩ := bezout_split x y hcop hx hy e obtain ⟨terms_x, hpfd_x⟩ := ih_x (e * t) obtain ⟨terms_y, hpfd_y⟩ := ih_y (e * s) exact ⟨terms_x ++ terms_y, { irred := fun s' hs' => by rw [List.mem_append] at hs' exact hs'.elim (hpfd_x.irred s') (hpfd_y.irred s') nonAssoc := fun s' hs' t' ht' hne => by rw [List.mem_append] at hs' ht' rcases hs' with hs' | hs' <;> rcases ht' with ht' | ht' · exact hpfd_x.nonAssoc s' hs' t' ht' hne · intro; sorry · intro; sorry · exact hpfd_y.nonAssoc s' hs' t' ht' hne coprime := fun s' hs' => by rw [List.mem_append] at hs' exact hs'.elim (hpfd_x.coprime s') (hpfd_y.coprime s') sum_eq := by rw [List.map_append, List.sum_append, hpfd_x.sum_eq, hpfd_y.sum_eq] -- Goal: ι(e*t)/ι x + ι(e*s)/ι y = ι e / ι(x*y) -- hsplit: ι e/(ι x*ι y) = ι(e*t)/ι x + ι(e*s)/ι y -- So: ι(e*t)/ι x + ι(e*s)/ι y = ι e/(ι x*ι y) = ι e/ι(x*y) rw [← hsplit, map_mul] }⟩ theorem pid_partial_fraction_decomposition (c : FractionRing R) : ∃ terms : List (PFDSummand R), IsPFD terms c := by obtain ⟨a, b, _hb, hc⟩ := exists_num_denom c rw [← hc]; exact isPFD_of_denom b a section IntExample example (c : FractionRing ℤ) : ∃ terms : List (PFDSummand ℤ), IsPFD terms c := pid_partial_fraction_decomposition c end IntExample