| import Mathlib |
|
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| namespace OAI |
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| namespace PiExponent.NumericalAmpleness |
| noncomputable section |
| open Polynomial Finset |
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| def mixedSlopePolynomial (d : ℕ) (c : ℕ → ℝ) : Polynomial ℝ := |
| ∑ k ∈ range (d+1), C ((d.choose k : ℝ) * c k) * X^k |
|
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| lemma mixedSlopePolynomial_eval (d : ℕ) (c : ℕ → ℝ) (t : ℝ) : |
| (mixedSlopePolynomial d c).eval t = |
| ∑ k ∈ range (d+1), (d.choose k : ℝ) * c k * t^k := by |
| simp only [mixedSlopePolynomial, Polynomial.eval_finsetSum, Polynomial.eval_mul, |
| Polynomial.eval_C, Polynomial.eval_pow, Polynomial.eval_X] |
|
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| lemma mixedSlopePolynomial_eval_zero (d : ℕ) (c : ℕ → ℝ) : |
| (mixedSlopePolynomial d c).eval 0 = c 0 := by |
| rw [mixedSlopePolynomial_eval, Finset.sum_range_succ'] |
| simp |
|
|
| theorem mixedSlopePolynomial_derivative (d : ℕ) (c : ℕ → ℝ) : |
| (mixedSlopePolynomial (d+1) c).derivative = |
| C ((d+1 : ℕ) : ℝ) * mixedSlopePolynomial d (fun k => c (k+1)) := by |
| unfold mixedSlopePolynomial |
| rw [Polynomial.derivative_sum, Finset.sum_range_succ'] |
| simp only [Polynomial.derivative_C_mul_X_pow, Nat.cast_zero, mul_zero, |
| C_0, zero_mul, add_zero, Nat.add_sub_cancel] |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro k hk |
| have h : (((d+1).choose (k+1) : ℕ) : ℝ) * ((k+1 : ℕ) : ℝ) = |
| ((d+1 : ℕ) : ℝ) * (d.choose k : ℝ) := by |
| exact_mod_cast (Nat.add_one_mul_choose_eq d k).symm |
| rw [← mul_assoc, ← C_mul] |
| congr 2 |
| calc |
| _ = (((d+1).choose (k+1) : ℝ) * ((k+1 : ℕ) : ℝ)) * c (k+1) := by ring |
| _ = _ := by rw [h]; ring |
|
|
| lemma mixedSlopePolynomial_derivative_eval (d : ℕ) (c : ℕ → ℝ) (t : ℝ) : |
| (mixedSlopePolynomial (d+1) c).derivative.eval t = |
| (d+1 : ℕ) * (mixedSlopePolynomial d (fun k => c (k+1))).eval t := by |
| rw [mixedSlopePolynomial_derivative] |
| simp |
|
|
| theorem mixedSlopePolynomial_eval_div (d : ℕ) (c : ℕ → ℝ) |
| (a b : ℝ) (hb : b ≠ 0) : |
| b^d * (mixedSlopePolynomial d c).eval (a/b) = |
| ∑ k ∈ range (d+1), (d.choose k : ℝ) * c k * b^(d-k) * a^k := by |
| rw [mixedSlopePolynomial_eval, Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro k hk |
| have hk' : k ≤ d := Nat.lt_succ_iff.mp (Finset.mem_range.mp hk) |
| have hp : b^d = b^(d-k) * b^k := by |
| rw [← pow_add, Nat.sub_add_cancel hk'] |
| rw [div_pow, hp] |
| field_simp |
|
|
| lemma mixedSlopePolynomial_natDegree_le (d : ℕ) (c : ℕ → ℝ) : |
| (mixedSlopePolynomial d c).natDegree ≤ d := by |
| apply Polynomial.natDegree_sum_le_of_forall_le |
| intro k hk |
| exact (Polynomial.natDegree_C_mul_X_pow_le _ _).trans |
| (Nat.lt_succ_iff.mp (Finset.mem_range.mp hk)) |
|
|
| end |
| end PiExponent.NumericalAmpleness |
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| end OAI |
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