| import OAI.Combinatorics.Crossing.StepEvents |
| import Mathlib.Tactic.FieldSimp |
|
|
| namespace OAI |
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|
| noncomputable section |
| namespace Zarankiewicz |
| open scoped BigOperators |
|
|
| def positiveIndicator (x : ℝ) : ℝ := if 0 < x then 1 else 0 |
|
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| def slopeSign (b : ℝ) : ℝ := if 0 < b then 1 else -1 |
|
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| def sideBefore (b : ℝ) : ℝ := if 0 < b then 0 else 1 |
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|
| def sideAfter (b : ℝ) : ℝ := if 0 < b then 1 else 0 |
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| theorem sideAfter_sub_before (b : ℝ) : sideAfter b-sideBefore b = slopeSign b := by |
| unfold sideAfter sideBefore slopeSign |
| split_ifs <;> norm_num |
|
|
| theorem positiveIndicator_factor (b r t : ℝ) (hb : b ≠ 0) (ht : t ≠ r) : |
| positiveIndicator (b*(t-r)) = stepAt r (sideBefore b) (sideAfter b) t := by |
| rcases lt_or_gt_of_ne hb with hb | hb |
| · by_cases htr : t<r |
| · have hprod := mul_pos_of_neg_of_neg hb (sub_neg.mpr htr) |
| simp [positiveIndicator,stepAt,sideBefore,htr,hprod,not_lt.mpr hb.le] |
| · have hrt : r<t := lt_of_le_of_ne (le_of_not_gt htr) ht.symm |
| have hprod := mul_neg_of_neg_of_pos hb (sub_pos.mpr hrt) |
| simp [positiveIndicator,stepAt,sideAfter,htr,not_lt.mpr hprod.le,not_lt.mpr hb.le] |
| · by_cases htr : t<r |
| · have hprod := mul_neg_of_pos_of_neg hb (sub_neg.mpr htr) |
| simp [positiveIndicator,stepAt,sideBefore,htr,not_lt.mpr hprod.le,hb] |
| · have hrt : r<t := lt_of_le_of_ne (le_of_not_gt htr) ht.symm |
| have hprod := mul_pos hb (sub_pos.mpr hrt) |
| simp [positiveIndicator,stepAt,sideAfter,htr,hprod,hb] |
|
|
| theorem sum_affine_side_events (n : ℕ) (r b : Fin n → ℝ) |
| (hr : StrictMono r) (hb : ∀ i, b i ≠ 0) (h0 : ∀ i, r i ≠ 0) (h1 : ∀ i, r i ≠ 1) : |
| (∑ i, if 0<r i ∧ r i<1 then slopeSign (b i) * |
| (∏ j, if j=i then 1 else positiveIndicator (b j*(r i-r j))) else 0) = |
| (∏ i, positiveIndicator (b i*(1-r i))) - ∏ i, positiveIndicator (b i*(0-r i)) := by |
| have hh := sum_stepEvent n r (fun i => sideBefore (b i)) (fun i => sideAfter (b i)) hr h0 h1 |
| have he : ∀ i : Fin n, stepEvent r (fun i => sideBefore (b i)) (fun i => sideAfter (b i)) i = |
| if 0<r i ∧ r i<1 then slopeSign (b i) * |
| (∏ j, if j=i then 1 else positiveIndicator (b j*(r i-r j))) else 0 := by |
| intro i |
| unfold stepEvent |
| rw [sideAfter_sub_before] |
| congr 1 |
| congr 1 |
| apply Finset.prod_congr rfl |
| intro j hj |
| by_cases hji : j=i |
| · simp [hji] |
| · rw [ite_eq_right hji,ite_eq_right hji] |
| exact (positiveIndicator_factor (b j) (r j) (r i) (hb j) (hr.injective.ne (Ne.symm hji))).symm |
| simp_rw [he] at hh |
| rw [hh] |
| congr 1 |
| · apply Finset.prod_congr rfl |
| intro i hi |
| exact (positiveIndicator_factor (b i) (r i) 1 (hb i) (h1 i).symm).symm |
| · apply Finset.prod_congr rfl |
| intro i hi |
| exact (positiveIndicator_factor (b i) (r i) 0 (hb i) (h0 i).symm).symm |
|
|
| end Zarankiewicz |
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|
| end |
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| end OAI |
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|
