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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 | import OAI.Combinatorics.Crossing.IntervalSegments
import Mathlib.Tactic.Ring
namespace OAI
noncomputable section
namespace Zarankiewicz
def intervalLeft (t : unitInterval) : unitInterval := ⟨t.val/2,by constructor <;> linarith [t.property.1,t.property.2]⟩
def intervalRight (t : unitInterval) : unitInterval := ⟨(1+t.val)/2,by constructor <;> linarith [t.property.1,t.property.2]⟩
theorem intervalLeft_strictMono : StrictMono intervalLeft := by
intro s t hst
change s.val/2 < t.val/2
exact div_lt_div_of_pos_right hst (by norm_num)
theorem intervalRight_strictMono : StrictMono intervalRight := by
intro s t hst
change (1+s.val)/2 < (1+t.val)/2
have hh : s.val < t.val := hst
linarith
theorem intervalSegment_left (a b t : unitInterval) :
intervalSegment (intervalLeft a) (intervalLeft b) t=intervalLeft (intervalSegment a b t) := by
apply Subtype.ext
change (1-t.val)*(a.val/2)+t.val*(b.val/2)=((1-t.val)*a.val+t.val*b.val)/2
ring
theorem intervalSegment_right (a b t : unitInterval) :
intervalSegment (intervalRight a) (intervalRight b) t=intervalRight (intervalSegment a b t) := by
apply Subtype.ext
change (1-t.val)*((1+a.val)/2)+t.val*((1+b.val)/2)=(1+((1-t.val)*a.val+t.val*b.val))/2
ring
theorem path_trans_intervalLeft {a b c : Plane} (γ : Path a b) (η : Path b c) (t : unitInterval) :
(γ.trans η) (intervalLeft t)=γ t := by
rw [← (γ.trans η).extend_extends' (intervalLeft t)]
rw [Path.extend_trans_of_le_half γ η (show (intervalLeft t).val ≤ 1/2 by
change t.val/2 ≤ 1/2
linarith [t.property.2])]
have he : 2*(intervalLeft t).val=t.val := by change 2*(t.val/2)=t.val; ring
rw [he,γ.extend_extends']
theorem path_trans_intervalRight {a b c : Plane} (γ : Path a b) (η : Path b c) (t : unitInterval) :
(γ.trans η) (intervalRight t)=η t := by
rw [← (γ.trans η).extend_extends' (intervalRight t)]
rw [Path.extend_trans_of_half_le γ η (show 1/2 ≤ (intervalRight t).val by
change 1/2 ≤ (1+t.val)/2
linarith [t.property.1])]
have he : 2*(intervalRight t).val-1=t.val := by change 2*((1+t.val)/2)-1=t.val; ring
rw [he,η.extend_extends']
theorem intervalSegment_zero_one (t : unitInterval) : intervalSegment 0 1 t=t := by
apply Subtype.ext
change (1-t.val)*0+t.val*1=t.val
ring
end Zarankiewicz
end
end OAI
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