File size: 2,705 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 | import OAI.Combinatorics.Crossing.AxisGenericPositions
namespace OAI
noncomputable section
namespace Zarankiewicz
def axisIntervalLo (r : ℕ) (i : Fin r) : ℝ := if i.val < r/2 then 2*(i.val : ℝ)+1 else -(2*(i.val : ℝ)+2)
def axisIntervalHi (r : ℕ) (i : Fin r) : ℝ := if i.val < r/2 then 2*(i.val : ℝ)+2 else -(2*(i.val : ℝ)+1)
theorem axisInterval_nonempty (r : ℕ) (i : Fin r) : axisIntervalLo r i < axisIntervalHi r i := by
unfold axisIntervalLo axisIntervalHi
split_ifs <;> linarith
theorem axisInterval_abs_bounds (r : ℕ) (i : Fin r) (x : ℝ)
(hx : x ∈ Set.Ioo (axisIntervalLo r i) (axisIntervalHi r i)) :
2*(i.val : ℝ)+1 < |x| ∧ |x| < 2*(i.val : ℝ)+2 := by
have hi : 0 ≤ (i.val : ℝ) := Nat.cast_nonneg _
by_cases h : i.val < r/2
· simp only [axisIntervalLo,axisIntervalHi,ite_eq_left h,Set.mem_Ioo] at hx
rw [abs_of_pos (by linarith [hx.1])]
exact hx
· simp only [axisIntervalLo,axisIntervalHi,ite_eq_right h,Set.mem_Ioo] at hx
rw [abs_of_neg (by linarith [hx.2])]
constructor <;> linarith [hx.1,hx.2]
theorem axisInterval_sign (r : ℕ) (i : Fin r) (x : ℝ)
(hx : x ∈ Set.Ioo (axisIntervalLo r i) (axisIntervalHi r i)) :
(0 < x ↔ i.val < r/2) ∧ (x < 0 ↔ r/2 ≤ i.val) := by
have hi : 0 ≤ (i.val : ℝ) := Nat.cast_nonneg _
by_cases h : i.val < r/2
· simp only [axisIntervalLo,axisIntervalHi,ite_eq_left h,Set.mem_Ioo] at hx
have hp : 0 < x := by linarith [hx.1]
simp [hp,h,not_lt_of_ge hp.le,Nat.not_le.mpr h]
· simp only [axisIntervalLo,axisIntervalHi,ite_eq_right h,Set.mem_Ioo] at hx
have hn : x < 0 := by linarith [hx.2]
simp [hn,h,not_lt_of_ge hn.le,Nat.le_of_not_gt h]
theorem axisInterval_ne_zero (r : ℕ) (i : Fin r) (x : ℝ)
(hx : x ∈ Set.Ioo (axisIntervalLo r i) (axisIntervalHi r i)) : x ≠ 0 := by
have hh := (axisInterval_abs_bounds r i x hx).1
intro he
rw [he,abs_zero] at hh
have hi : 0 ≤ (i.val : ℝ) := Nat.cast_nonneg _
linarith
theorem axisInterval_abs_strictMono (r : ℕ) (x : Fin r → ℝ)
(hx : ∀ i, x i ∈ Set.Ioo (axisIntervalLo r i) (axisIntervalHi r i)) :
StrictMono (fun i => |x i|) := by
intro i k hik
have hi := (axisInterval_abs_bounds r i (x i) (hx i)).2
have hk := (axisInterval_abs_bounds r k (x k) (hx k)).1
have hik' : (i.val : ℝ)+1 ≤ (k.val : ℝ) := by exact_mod_cast (Nat.succ_le_of_lt hik)
linarith
theorem axisInterval_injective (r : ℕ) (x : Fin r → ℝ)
(hx : ∀ i, x i ∈ Set.Ioo (axisIntervalLo r i) (axisIntervalHi r i)) : Function.Injective x := by
intro i k hik
exact (axisInterval_abs_strictMono r x hx).injective (congrArg abs hik)
end Zarankiewicz
end
end OAI
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