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  • Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev

Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev/Extremal.lean

Lines changed: 2 additions & 2 deletions
Original file line numberDiff line numberDiff line change
@@ -200,7 +200,7 @@ theorem eval_iterate_derivative_eq_sum_chebyshevNode {n k : ℕ} (hk : k ≤ n)
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∑ i ≤ n, P.eval (chebyshevNode n i) *
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(k.factorial *
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(∏ j ∈ (Finset.range (n + 1)).erase i, ((chebyshevNode n i) - (chebyshevNode n j)))⁻¹ *
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∑ t ∈ ((Finset.range (n + 1)).erase i).powerset with t.card = n - k,
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∑ t ∈ ((Finset.range (n + 1)).erase i).powersetCard (n - k),
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∏ a ∈ t, (x - chebyshevNode n a)) := by
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rw [Lagrange.eval_iterate_derivative_eq_sum (strictAntiOn_chebyshevNode n).injOn (by simp [hP])
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(le_of_le_of_eq (Nat.cast_le.mpr hk) hP.symm) x, Finset.mul_sum, Finset.card_range,
@@ -213,7 +213,7 @@ theorem eval_iterate_derivative_eq_sum_chebyshevNode_coeff_pos
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0 < (-1) ^ i *
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(k.factorial *
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(∏ j ∈ (Finset.range (n + 1)).erase i, ((chebyshevNode n i) - (chebyshevNode n j)))⁻¹ *
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∑ t ∈ ((Finset.range (n + 1)).erase i).powerset with t.card = n - k,
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∑ t ∈ ((Finset.range (n + 1)).erase i).powersetCard (n - k),
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∏ a ∈ t, (x - chebyshevNode n a)) := by
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rw [← mul_assoc]
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refine mul_pos ?_ (Finset.sum_pos' ?_ ?_)

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