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== /health ==
{"ok":true,"lean_mode":"fake","llm_mode":"fake","has_key":true,"verify_policy":"strategic","retrieval":"off","prover_backend":false,"persistent":false}
== submitted job job_f6a7c5c5 ==
== final job state ==
status: proved | phase: proved | round: 1
llm_calls: 24 | lean_calls: 15 | events: 53
nodes:
thm_main [verified] attempts=1
lemma_1 [verified] attempts=2
lemma_2 [verified] attempts=3
lemma_2_a [verified] attempts=1
lemma_2_b [verified] attempts=1
--- final_lean (Lean-verified) ---
theorem lemma_1 (n : ℕ) : Even (n * (n + 1)) := by
exact Nat.even_mul_succ_self n
theorem lemma_2_a (n : ℕ) : n ^ 2 = n * n := by
rw [sq]
theorem lemma_2_b (n : ℕ) : n * n + n = n * (n + 1) := by
ring
theorem lemma_2 (n : ℕ) : n ^ 2 + n = n * (n + 1) := by
rw [lemma_2_a]
exact lemma_2_b n
theorem thm_main (n : ℕ) : Even (n ^ 2 + n) := by
rw [lemma_2]
exact lemma_1 n
== agent timeline (paginated via ?since=) ==
[ 1] chef chef start Problem received: Prove that for every natural number n, n^2 + n is even.
[ 2] worker skeptic probe probing 'Eq(Mod({n}**2 + {n}, 2), 0)' at 9 candidates
[ 3] worker skeptic probe_result no counterexample found; proceeding
[ 4] chef chef round === Round 1/2 ===
[ 5] worker brainstormer:algebraist strategy consecutive-product: n²+n = n(n+1): consecutive integers, one is even, so the product is even.
[ 6] worker brainstormer:analyst strategy parity-cases: Split on n even/odd; both cases give an even total by direct computation.
[ 7] worker brainstormer:combinatorialist strategy induction: Base n=0 gives 0; step adds 2n+2, which is even, preserving parity.
[ 8] sous strategist plan strategy 'consecutive-product' with 2 lemma(s). Rewrite the goal with the identity lemma, then a
[ 9] worker formalizer statement_attempt theorem lemma_1 (n : ℕ) : Evenn (n * (n + 1))
[ 10] lean lean_verifier lean_check error: unknown identifier 'Evenn'
[ 11] worker formalizer statement_attempt theorem lemma_1 (n : ℕ) : Even (n * (n + 1))
[ 12] lean lean_verifier lean_check OK with 1 sorry(ies)
[ 13] worker formalizer statement_ok lemma_1: statement elaborates
[ 14] worker formalizer statement_attempt theorem lemma_2 (n : ℕ) : n ^ 2 + n = n * (n + 1)
[ 15] lean lean_verifier lean_check OK with 1 sorry(ies)
[ 16] worker formalizer statement_ok lemma_2: statement elaborates
[ 17] worker formalizer statement_attempt theorem thm_main (n : ℕ) : Even (n ^ 2 + n)
[ 18] lean lean_verifier lean_check OK with 1 sorry(ies)
[ 19] worker formalizer statement_ok thm_main: statement elaborates
[ 20] worker prover prove_attempt lemma_1 attempt 1: | simp
[ 21] lean lean_verifier lean_check error: unsolved goals | ⊢ Even (n * (n + 1))
[ 22] sous critic triage lemma_1: closing tactic too weak for this goal → cite the exact Mathlib lemma or split the state
[ 23] worker prover prove_attempt lemma_1 attempt 2: | exact Nat.even_mul_succ_self n
[ 24] lean lean_verifier lean_check OK (no errors, no sorries)
[ 25] chef chef verified lemma_1 VERIFIED by Lean after 2 attempt(s).
[ 26] worker prover prove_attempt lemma_2 attempt 1: | rfl
[ 27] lean lean_verifier lean_check error: unsolved goals | ⊢ n ^ 2 + n = n * (n + 1)
[ 28] sous critic triage lemma_2: closing tactic too weak for this goal → cite the exact Mathlib lemma or split the state
[ 29] worker prover prove_attempt lemma_2 attempt 2: | decide
[ 30] lean lean_verifier lean_check error: unsolved goals | ⊢ n ^ 2 + n = n * (n + 1)
[ 31] sous critic triage lemma_2: closing tactic too weak for this goal → cite the exact Mathlib lemma or split the state
[ 32] chef chef decompose lemma_2 split into ['lemma_2_a', 'lemma_2_b']. rewrite with the first sub-lemma, close with the
[ 33] worker formalizer statement_attempt theorem lemma_2_a (n : ℕ) : n ^ 2 = n * n
[ 34] lean lean_verifier lean_check OK with 1 sorry(ies)
[ 35] worker formalizer statement_ok lemma_2_a: statement elaborates
[ 36] worker prover prove_attempt lemma_2_a attempt 1: | rw [sq]
[ 37] lean lean_verifier lean_check OK (no errors, no sorries)
[ 38] chef chef verified lemma_2_a VERIFIED by Lean after 1 attempt(s).
[ 39] worker formalizer statement_attempt theorem lemma_2_b (n : ℕ) : n * n + n = n * (n + 1)
[ 40] lean lean_verifier lean_check OK with 1 sorry(ies)
[ 41] worker formalizer statement_ok lemma_2_b: statement elaborates
[ 42] worker prover prove_attempt lemma_2_b attempt 1: | ring
[ 43] lean lean_verifier lean_check OK (no errors, no sorries)
[ 44] chef chef verified lemma_2_b VERIFIED by Lean after 1 attempt(s).
[ 45] worker prover prove_attempt lemma_2 (post-split) attempt 3: | rw [lemma_2_a] | exact lemma_2_b n
[ 46] lean lean_verifier lean_check OK (no errors, no sorries)
[ 47] chef chef verified lemma_2 VERIFIED after decomposition.
[ 48] worker prover prove_attempt thm_main attempt 1: | rw [lemma_2] | exact lemma_1 n
[ 49] lean lean_verifier lean_check OK (no errors, no sorries)
[ 50] chef chef verified thm_main VERIFIED by Lean after 1 attempt(s).
[ 51] lean lean_verifier lean_check OK (no errors, no sorries)
[ 52] lean lean_verifier axiom_audit ok=True axioms=['Classical.choice', 'Quot.sound', 'propext']
[ 53] chef chef proved Final theorem 'thm_main' verified by Lean (round 1, 15 Lean checks, 23 LLM calls).
53 events, strictly increasing seq: True