File size: 3,280 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 74 75 76 77 78 79 80 81 82 83 | import OAI.Combinatorics.Crossing.FloorArithmetic
import Mathlib.Data.Finset.Prod
import Mathlib.Order.Interval.Finset.Fin
namespace OAI
namespace Zarankiewicz
open scoped Finset
def positiveRayIndices (r : ℕ) : Finset (Fin r) := Finset.univ.filter (fun i => i.val < r/2)
def negativeRayIndices (r : ℕ) : Finset (Fin r) := Finset.univ.filter (fun i => ¬ i.val < r/2)
def balancedRayPairs (r : ℕ) : Finset (Fin r × Fin r) :=
((positiveRayIndices r ×ˢ positiveRayIndices r).filter (fun p => p.1 < p.2)) ∪
((negativeRayIndices r ×ˢ negativeRayIndices r).filter (fun p => p.1 < p.2))
theorem mem_balancedRayPairs (r : ℕ) (i k : Fin r) :
(i,k) ∈ balancedRayPairs r ↔ i < k ∧ (i.val < r/2 ↔ k.val < r/2) := by
simp only [balancedRayPairs,positiveRayIndices,negativeRayIndices,Finset.mem_union,
Finset.mem_filter,Finset.mem_product,Finset.mem_univ,true_and]
tauto
theorem positiveRayIndices_card (r : ℕ) : (positiveRayIndices r).card=r/2 := by
cases r with
| zero => simp [positiveRayIndices]
| succ r =>
let c : Fin (r+1) := ⟨(r+1)/2,by omega⟩
have he : positiveRayIndices (r+1)=Finset.Iio c := by
ext i
simp only [positiveRayIndices,Finset.mem_filter,Finset.mem_univ,true_and,Finset.mem_Iio]
rfl
rw [he,Fin.card_Iio]
theorem negativeRayIndices_card (r : ℕ) : (negativeRayIndices r).card=r-r/2 := by
have hh := Finset.card_filter_add_card_filter_not (s := Finset.univ) (fun i : Fin r => i.val < r/2)
change (positiveRayIndices r).card+(negativeRayIndices r).card=Finset.univ.card at hh
rw [positiveRayIndices_card,Finset.card_univ,Fintype.card_fin] at hh
omega
theorem twice_choose_two (s : ℕ) : 2*s.choose 2=s*(s-1) := by
induction s with
| zero => simp
| succ s ih =>
rw [Nat.choose_succ_succ,Nat.choose_one_right,Nat.add_sub_cancel]
have hh : s*(s-1)+2*s=(s+1)*s := by
cases s with
| zero => simp
| succ s => simp only [Nat.add_sub_cancel]; ring
nlinarith
theorem balanced_choose_two (r : ℕ) : (r/2).choose 2+(r-r/2).choose 2=axisPairs r := by
have hr : r=2*(r/2) ∨ r=2*(r/2)+1 := by omega
rcases hr with hr | hr
· generalize r/2=s at hr
subst r
rw [show 2*s-s=s by omega,axisPairs_even]
have hh := twice_choose_two s
omega
· generalize r/2=s at hr
subst r
rw [show 2*s+1-s=s+1 by omega,axisPairs_odd,
Nat.choose_succ_succ,Nat.choose_one_right]
have hh := twice_choose_two s
cases s with
| zero => simp
| succ s => simp only [Nat.add_sub_cancel] at hh; nlinarith
theorem balancedRayPairs_card (r : ℕ) : (balancedRayPairs r).card=axisPairs r := by
have hd : Disjoint
((positiveRayIndices r ×ˢ positiveRayIndices r).filter (fun p => p.1 < p.2))
((negativeRayIndices r ×ˢ negativeRayIndices r).filter (fun p => p.1 < p.2)) := by
apply Finset.disjoint_left.mpr
rintro ⟨i,k⟩ hp hn
simp only [Finset.mem_filter,Finset.mem_product,positiveRayIndices,negativeRayIndices,
Finset.mem_univ,true_and] at hp hn
exact hn.1.1 hp.1.1
rw [balancedRayPairs,Finset.card_union_of_disjoint hd,Finset.card_product_filter_lt,
Finset.card_product_filter_lt,positiveRayIndices_card,negativeRayIndices_card,balanced_choose_two]
end Zarankiewicz
end OAI
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