We constructed a neural theorem prover for Lean that realized to unravel quite a lot of difficult high-school olympiad issues, together with issues from the AMC12 and AIME competitions, in addition to two issues tailored from the IMO. The prover makes use of a language mannequin to seek out proofs of formal statements. Every time we discover a new proof, we use it as new coaching knowledge, which improves the neural community and permits it to iteratively discover options to tougher and tougher statements.

We achieved a brand new state-of-the-art (41.2% vs 29.3%) on the miniF2F benchmark, a difficult assortment of high-school olympiad issues. Our method, which we name *assertion curriculum studying*, consists of manually gathering a set of statements of various problem ranges (with out proof) the place the toughest statements are just like the benchmark we goal. Initially our neural prover is weak and may solely show a number of of them. We iteratively seek for new proofs and re-train our neural community on the newly found proofs, and after 8 iterations, our prover finally ends up being vastly superior when examined on miniF2F.

Formal arithmetic is an thrilling area to check due to (i) its richness, letting you show arbitrary theorems which require reasoning, creativity and perception and (ii) its similarity to video games—the place AI has been spectacularly profitable—in that it has an automatic method of figuring out whether or not a proof is profitable (i.e., verified by the formal system). As demonstrated within the trivial instance beneath, proving a proper assertion requires producing a sequence of proof steps, every proof step consisting in a name to a tactic. These ways take mathematical phrases as arguments and every tactic name will rework the present assertion to show, into statements which can be simpler to show, till nothing is left to show.

Drawback 1

Tailored from AMC12 2000 Drawback 5

Show that if $|x – 2| = p$, the place $x

```
theorem amc12_2000_p5
(x p : ℝ)
(h₀ : x -- to show
(h₁ : abs (x - 2) = p) :
x - p = 2 - 2 * p :=
start
have h₂ : abs (x - 2) = -(x - 2), {
apply abs_of_neg,
linarith,
},
rw h₁ at h₂,
linarith,
finish
```

We observe that the potential to generate unique mathematical phrases required as arguments of ways, which can’t be executed with no neural language mannequin, emerges from our coaching process. The proof beneath is an instance of it: the proof step `use n + 1`

(fully generated by our fashions) proposes to make use of `n + 1`

as an answer, the remainder of the formal proof counting on the `ring_exp`

tactic to confirm that it’s certainly legitimate.

Drawback 2

Tailored from AMC12B 2020 Drawback 6

For all integers $n ≥ 9$, show that $((n + 2)! −(n + 1)!) / n!$ is an ideal sq..

```
theorem amc12b_2020_p6
(n : ℕ)
(h0 : 9 ≤ n) :
∃ x : ℕ, (x:ℝ)^2 =
(nat.factorial (n + 2) - nat.factorial (n + 1))
/ nat.factorial n :=
start
use n + 1,
field_simp [nat.factorial_ne_zero, pow_succ'],
ring_exp
finish
```

We additionally observe that our fashions and search process are able to producing proofs that chain a number of non-trivial reasoning steps. Within the proof beneath, the mannequin begins by utilizing contraposition resulting in the existential assertion (`∃ (x : ℝ), f x ≠ a * x + b`

). It then generates a witness for it with `use (0 : ℝ)`

and finishes the proof by leveraging the `norm_num`

tactic.

Drawback 3

Tailored from the MATH dataset

Let $f(x) = Ax + B$ and $g(x) = Bx + A$, the place $A ne B$. If $f(g(x)) – g(f(x)) = B – A$, show that $A + B = 0$.

```
theorem mathd_train_algebra_217
(a b : ℝ)
(f g : ℝ → ℝ)
(h₀ : ∀ x, f x = a * x + b)
(h₁ : ∀ x, f x = b * x + a)
(h₂ : a ≠ b)
(h₃ : ∀ x, f (g x) - g (f x) = b - a) :
a + b = 0 :=
start
revert h₀ h₁ h₂ h₃,
contrapose!,
rintro ⟨h₀, ⟨h₁, h₂⟩⟩,
use (0 : ℝ),
simp solely [sub_eq_iff_eq_add, h₀, mul_zero, zero_add],
norm_num at h₀,
finish
```

Our fashions, educated with *assertion curriculum studying*, have been in a position to shut quite a lot of issues from coaching textbooks in addition to AMC12 and AIME competitions, and a pair of issues tailored from the IMO. We current beneath three examples of such generated proofs.

Drawback 4

Tailored from IMO 1964 Drawback 2

Suppose $a$, $b$, $c$ are the edges of a triangle.

Show that $a^2(b + c − a) + b^2(c + a − b) + c^2(a + b − c) leq 3abc$.

```
theorem imo_1964_p2
(a b c : ℝ)
(h₀ : 0 start
nlinarith [sq_nonneg (b - a),
sq_nonneg (c - b),
sq_nonneg (c - a)]
finish
```

Drawback 5

Tailored from AIME 1984 Drawback 1

Show that $a2 + a4 + a6 + a8 + …+ a98 = 93$ if $a1$, $a2$, $a3…$ is an arithmetic development with widespread distinction $1$, and $a1 + a2 + a3 + … + a98 = 137$.

```
theorem aime_1984_p1
(u : ℕ → ℚ)
(h₀ : ∀ n, u (n + 1) = u n + 1)
(h₁ : ∑ okay in finset.vary 98, u okay.succ = 137) :
∑ okay in finset.vary 49, u (2 * okay.succ) = 93 :=
start
rw finset.sum_eq_multiset_sum,
dsimp [finset.range] at h₁,
simp [h₀],
ring,
norm_num at h₁,
norm_num,
apply eq_of_sub_eq_zero,
{ simp solely [*, abs_of_pos, add_zero] at *, linarith },
finish
```

Drawback 6

Tailored from IMO Longlist 1990 Drawback 77

For $a, b, c$ reals, show that $(a^2 + ab + b^2)(b^2 + bc + c^2)(c^2 + ca + a^2) geq (ab + bc + ca)^3$.

```
theorem imo_longlist_1990_p77
(a b c : ℝ) :
(a * b + b * c + c * a)^3 ≤
(a^2 + a * b + b^2) * (b^2 + b * c + c^2) *
(c^2 + c * a + a^2) :=
start
let u : euclidean_space ℝ (fin 2) := ![a, b],
let v : euclidean_space ℝ (fin 2) := ![b, c],
have h₀ := real_inner_mul_inner_self_le u v,
simp [u, v, fin.sum_univ_succ,
←pow_two, ←pow_two, le_of_lt, mul_assoc] at h₀,
have h₃ : 0 ≤ (c + a) * (c + a),
{ nlinarith, },
have h₄ := sq_nonneg (a * b + b * c + c * a),
simp [sq, h₀, h₃, mul_add, add_mul] at h₄ ⊢,
nlinarith [sq_nonneg (b - a),
sq_nonneg (c - b),
sq_nonneg (a - c)]
finish
```

Formal arithmetic entails two fundamental challenges that make a naive software of reinforcement studying unlikely to succeed.

- (i)
**Infinite motion house**: not solely does formal arithmetic have an especially massive search house (like Go for instance), it additionally has an infinite motion house. At every step of a proof search, the mannequin should select not from a well-behaved finite set of actions, however a posh and infinite set of ways, involving exogenous mathematical phrases that need to be generated (e.g., producing a mathematical assertion for use as a witness, an object utilized in steps equivalent to “there exists an $x$ s.t. …”, or a reduce, the introduction and the chaining of a lemma in the midst of a proof). - (ii)
**Lack of self-play**: conversely to 2-player video games, a prover shouldn’t be enjoying in opposition to an opponent however in opposition to a set of statements to show. When confronted with a press release that’s simply too arduous, there isn’t a apparent reframing that can let the prover generate middleman simpler statements to sort out first. This asymmetry prevents naive software of the self-play algorithms that have been profitable with 2-player video games.

In our work, we handle the infinite motion house downside by sampling actions from a language mannequin as we seek for a proof. Language fashions have the potential to generate the tactic calls in addition to the unique mathematical phrases usually required as arguments. Our foundation for addressing the shortage of self-play is the remark that the important thing position of self-play in 2-player video games is to supply an unsupervised curriculum. Our methodology proposes to exchange this unsupervised curriculum with an auxiliary set of downside statements (with out requiring proofs) of various problem. We empirically present that, when the problem of those auxiliary issues is assorted sufficient, our coaching process is ready to remedy a curriculum of more and more troublesome issues, finally generalizing to the set of issues we care about.

Whereas these outcomes are extraordinarily thrilling, as they show that deep studying fashions are able to non-trivial mathematical reasoning when interacting with a proper system, we’re nonetheless very removed from best-student efficiency on these competitions, solely often, fairly than persistently, closing difficult olympiad issues. We hope nonetheless that our work will inspire analysis on this area, particularly in direction of the IMO Grand Problem and that the *assertion curriculum studying* methodology we suggest will assist speed up progress in automated reasoning generally.