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 freshsearch_idandtactic_state_id).run_tac: takes asearch_id, atactic_state_idand a tactic to apply at the tactic state denoted by the provided ids.clear_search: takes asearch_idto 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", ["intermediate_field.adjoin.range_algebra_map_subset", "intermediate_field finite_dimensional polynomial"]]
{"error":null,"search_id":"0","tactic_state":"⊢ ∀ (F : Type u_1) [_inst_1 : field F] {E : Type u_2} [_inst_2 : field E] [_inst_3 : algebra F E] (S : set E),\tset.range ⇑(algebra_map F E) ⊆ ↑(intermediate_field.adjoin F S)","tactic_state_id":"0"}
["init_search", ["int.prime.dvd_mul", ""]]
{"error":null,"search_id":"1","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,"search_id":"0","tactic_state":"F : Type u_1,\t_inst_1 : field F,\tE : Type u_2,\t_inst_2 : field E,\t_inst_3 : algebra F E,\tS : set E\t⊢ set.range ⇑(algebra_map F E) ⊆ ↑(intermediate_field.adjoin F S)","tactic_state_id":"1"}
["run_tac",["1","0","intros"]]
{"error":null,"search_id":"1","tactic_state":"m n : ℤ,\tp : ℕ,\thp : nat.prime p,\th : ↑p ∣ m * n\t⊢ p ∣ m.nat_abs ∨ p ∣ n.nat_abs","tactic_state_id":"1"}
["run_tac",["1","1","apply (nat.prime.dvd_mul hp).mp"]]
{"error":null,"search_id":"1","tactic_state":"m n : ℤ,\tp : ℕ,\thp : nat.prime p,\th : ↑p ∣ m * n\t⊢ p ∣ m.nat_abs * n.nat_abs","tactic_state_id":"2"}
["run_tac",["1","2","rw ← int.nat_abs_mul"]]
{"error":null,"search_id":"1","tactic_state":"m n : ℤ,\tp : ℕ,\thp : nat.prime p,\th : ↑p ∣ m * n\t⊢ p ∣ (m * n).nat_abs","tactic_state_id":"3"}
["run_tac",["1","3","simp"]]
{"error":"run_tac_failed: pos=(some ⟨1, 2⟩) msg=simplify tactic failed to simplify","search_id":null,"tactic_state":null,"tactic_state_id":null}
["run_tac",["1","5","exact int.coe_nat_dvd_left.mp h"]]
{"error":"unknown_id: search_id=1 tactic_state_id=5","search_id":null,"tactic_state":null,"tactic_state_id":null}
["run_tac",["1","3","exact int.coe_nat_dvd_left.mp h"]]
{"error":null,"search_id":"1","tactic_state":"no goals","tactic_state_id":"4"}
["clear_search",["1"]]
{"error":null,"search_id":"1","tactic_state":null,"tactic_state_id":null}
["run_tac",["0","1","intros x hx,"]]
{"error":null,"search_id":"0","tactic_state":"F : Type u_1,\t_inst_1 : field F,\tE : Type u_2,\t_inst_2 : field E,\t_inst_3 : algebra F E,\tS : set E,\tx : E,\thx : x ∈ set.range ⇑(algebra_map F E)\t⊢ x ∈ ↑(intermediate_field.adjoin F S)","tactic_state_id":"2"}
["run_tac",["0","2","cases hx with f hf"]]
{"error":null,"search_id":"0","tactic_state":"F : Type u_1,\t_inst_1 : field F,\tE : Type u_2,\t_inst_2 : field E,\t_inst_3 : algebra F E,\tS : set E,\tx : E,\tf : F,\thf : ⇑(algebra_map F E) f = x\t⊢ x ∈ ↑(intermediate_field.adjoin F S)","tactic_state_id":"3"}
["run_tac",["0","3","rw ← hf"]]
{"error":null,"search_id":"0","tactic_state":"F : Type u_1,\t_inst_1 : field F,\tE : Type u_2,\t_inst_2 : field E,\t_inst_3 : algebra F E,\tS : set E,\tx : E,\tf : F,\thf : ⇑(algebra_map F E) f = x\t⊢ ⇑(algebra_map F E) f ∈ ↑(intermediate_field.adjoin F S)","tactic_state_id":"4"}
["run_tac",["0","4","exact adjoin.algebra_map_mem F S f"]]
{"error":null,"search_id":"0","tactic_state":"no goals","tactic_state_id":"5"}
["clear_search",["0"]]
{"error":null,"search_id":"0","tactic_state":null,"tactic_state_id":null}
Declaration names
Declaration names and open namespaces as recorded by lean_proof_recording 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"]]
373 Dec 20, 2022
125 Dec 23, 2022
906 Jan 03, 2023
156 Dec 27, 2022
21 Jul 30, 2022
189 Dec 07, 2022
130 Dec 11, 2022
19 Dec 01, 2022
24 Jun 27, 2022
217 Jan 04, 2023
182 Dec 19, 2022
583 Jan 03, 2023
148 Dec 30, 2022
210 Dec 30, 2022
86 Dec 31, 2022
59 Dec 12, 2022
239 Dec 28, 2022
190 Dec 08, 2022
303 Dec 09, 2022
108 Jan 01, 2023