lean-gym

This repository lets you interact with Lean through a REPL. See Formal Mathematics Statement Curriculum Learning for a presentation of lean-gym.

Setup

# Download pre-built binaries and build the project (targeting mathlib).
bash ./scripts/setup.sh

Usage

lean --run src/repl.lean

Starts a fresh REPL. Once started, the REPL accepts the following commands:

• init_search: takes a declaration name as well as a list of open namespaces to initialize a search at the given declaration opening the provided namespaces, and returning the initial tactic state (along with a fresh search_id and tactic_state_id).
• run_tac: takes a search_id, a tactic_state_id and a tactic to apply at the tactic state denoted by the provided ids.
• clear_search: takes a search_id to clear all state related to a search.

The commands can be interleaved freely enabling the parallelization of multiple proof searches against the same REPL.

$ lean --run src/repl.lean

["init_search", ["int.prime.dvd_mul", ""]]
{"error":null,"proof_steps":[],"search_id":"0","tactic_state":"⊢ ∀ {m n : ℤ} {p : ℕ}, nat.prime p → ↑p ∣ m * n → p ∣ m.nat_abs ∨ p ∣ n.nat_abs","tactic_state_id":"0"}

["run_tac",["0","0","intros"]]
{"error":null,"proof_steps":[],"search_id":"0","tactic_state":"m n : ℤ,\np : ℕ,\nhp : nat.prime p,\nh : ↑p ∣ m * n\n⊢ p ∣ m.nat_abs ∨ p ∣ n.nat_abs","tactic_state_id":"1"}

["run_tac",["0","1","apply (nat.prime.dvd_mul hp).mp"]]
{"error":null,"proof_steps":[],"search_id":"0","tactic_state":"m n : ℤ,\np : ℕ,\nhp : nat.prime p,\nh : ↑p ∣ m * n\n⊢ p ∣ m.nat_abs * n.nat_abs","tactic_state_id":"2"}

["run_tac",["0","2","rw ← int.nat_abs_mul"]]
{"error":null,"proof_steps":[],"search_id":"0","tactic_state":"m n : ℤ,\np : ℕ,\nhp : nat.prime p,\nh : ↑p ∣ m * n\n⊢ p ∣ (m * n).nat_abs","tactic_state_id":"3"}

["run_tac",["0","3","exact int.coe_nat_dvd_left.mp h"]]
{"error":null,"proof_steps":[],"search_id":"0","tactic_state":"no goals","tactic_state_id":"4"}

Declaration names

Declaration names and open namespaces are available in the data/ directory to be used with the init_search command.

Notes

The REPL is subject to crashes in rare cases. Empirically such crash happens no more than ~0.01% of the time.

When a tactic state is reached with no left goals, some custom logic is run to check that the resulting proof's type matches the top level goal type and does not rely on sorry. We also check for the presence of undefined in the proof term. As an example, the following MiniF2F proofs will safely fail with error proof_validation_failed.

["init_search", ["mathd_algebra_35", ""]]
["run_tac", ["0", "0", "intros"]]
["run_tac", ["0", "1", "sorry"]]

["init_search", ["induction_divisibility_3divnto3m2n", ""]]
["run_tac", ["0", "0", "intros"]]
["run_tac", ["0", "1", "rw [add_comm]"]]
["run_tac", ["0", "2", "have h3 : 1 * (n + 1) ≤ (n + 1)"]]
["run_tac", ["0", "3", "rw one_mul"]]
["run_tac", ["0", "4", "apply dvd_trans"]]
["run_tac", ["0", "5", "swap"]]
["run_tac", ["0", "6", "simp []"]]

["init_search", ["mathd_numbertheory_13", ""]]
["run_tac", ["0", "0", "intros u v hu hv hsum"]]
["run_tac", ["0", "1", "intro h"]]
["run_tac", ["0", "2", "contrapose h"]]
["run_tac", ["0", "3", "intro hn"]]
["run_tac", ["0", "4", "exact not_lt_of_lt hn undefined"]]