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"]]
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