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5697766 | 1 2 | {"text": "[GOAL]\nm n k : ℕ\n⊢ choose (m + n) k = ∑ ij in antidiagonal k, choose m ij.fst * choose n ij.snd\n[PROOFSTEP]\ncalc\n (m + n).choose k = ((X + 1) ^ (m + n)).coeff k := by rw [coeff_X_add_one_pow, Nat.cast_id]\n _ = ((X + 1) ^ m * (X + 1) ^ n).coeff k := by rw [pow_add]\n _ = ∑ ij : ℕ × ℕ in antidiagonal k, m.choose ij.1 * n.choose ij.2 :=\n by\n rw [coeff_mul, Finset.sum_congr rfl]\n simp only [coeff_X_add_one_pow, Nat.cast_id, eq_self_iff_true, imp_true_iff]\n[GOAL]\nm n k : ℕ\n⊢ choose (m + n) k = coeff ((X + 1) ^ (m + n)) k\n[PROOFSTEP]\nrw [coeff_X_add_one_pow, Nat.cast_id]\n[GOAL]\nm n k : ℕ\n⊢ coeff ((X + 1) ^ (m + n)) k = coeff ((X + 1) ^ m * (X + 1) ^ n) k\n[PROOFSTEP]\nrw [pow_add]\n[GOAL]\nm n k : ℕ\n⊢ coeff ((X + 1) ^ m * (X + 1) ^ n) k = ∑ ij in antidiagonal k, choose m ij.fst * choose n ij.snd\n[PROOFSTEP]\nrw [coeff_mul, Finset.sum_congr rfl]\n[GOAL]\nm n k : ℕ\n⊢ ∀ (x : ℕ × ℕ),\n x ∈ antidiagonal k → coeff ((X + 1) ^ m) x.fst * coeff ((X + 1) ^ n) x.snd = choose m x.fst * choose n x.snd\n[PROOFSTEP]\nsimp only [coeff_X_add_one_pow, Nat.cast_id, eq_self_iff_true, imp_true_iff]\n", "meta": {"mathlib_filename": "Mathlib.Data.Nat.Choose.Vandermonde", "llama_tokens": 573, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.9353465170505205, "lm_q2_score": 0.8558511506439708, "lm_q1q2_score": 0.8005173928685184}}
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