File size: 2,830 Bytes
d5b7741 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 | import OAI.Combinatorics.Crossing.StepEvents
import Mathlib.Tactic.FieldSimp
namespace OAI
noncomputable section
namespace Zarankiewicz
open scoped BigOperators
def positiveIndicator (x : ℝ) : ℝ := if 0 < x then 1 else 0
def slopeSign (b : ℝ) : ℝ := if 0 < b then 1 else -1
def sideBefore (b : ℝ) : ℝ := if 0 < b then 0 else 1
def sideAfter (b : ℝ) : ℝ := if 0 < b then 1 else 0
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
end
end OAI
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