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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 | import OAI.Combinatorics.Crossing.AxisSegments
import OAI.Combinatorics.Crossing.GenericPolynomialBox
namespace OAI
noncomputable section
namespace Zarankiewicz
open MvPolynomial
variable {m n : ℕ}
def axisConcurrency (a b c d e f : ℝ) : ℝ :=
a*d*e*f+b*c*d*e+a*b*c*f-a*b*d*e-b*c*e*f-a*c*d*f
theorem axisSegmentPoint_line_equation (a b t : ℝ) :
a*(axisSegmentPoint a b t).1+b*(axisSegmentPoint a b t).2=a*b := by
rw [axisSegmentPoint_coordinates]
ring
theorem axisConcurrency_zero_of_meeting (a b c d e f : ℝ) (z : Plane)
(h₀ : a*z.1+b*z.2=a*b) (h₁ : c*z.1+d*z.2=c*d) (h₂ : e*z.1+f*z.2=e*f) :
axisConcurrency a b c d e f=0 := by
unfold axisConcurrency
linear_combination -(c*f-d*e)*h₀-(b*e-a*f)*h₁-(a*d-b*c)*h₂
def axisConcurrencyPolynomial (e f g : BipartiteEdge m n) : MvPolynomial (BipartiteVertex m n) ℝ :=
let a := X (Sum.inl e.1); let b := X (Sum.inr e.2)
let c := X (Sum.inl f.1); let d := X (Sum.inr f.2)
let u := X (Sum.inl g.1); let v := X (Sum.inr g.2)
a*d*u*v+b*c*d*u+a*b*c*v-a*b*d*u-b*c*u*v-a*c*d*v
theorem axisConcurrencyPolynomial_eval (e f g : BipartiteEdge m n) (x : BipartiteVertex m n → ℝ) :
eval x (axisConcurrencyPolynomial e f g)=
axisConcurrency (x (Sum.inl e.1)) (x (Sum.inr e.2)) (x (Sum.inl f.1)) (x (Sum.inr f.2))
(x (Sum.inl g.1)) (x (Sum.inr g.2)) := by
simp [axisConcurrencyPolynomial,axisConcurrency]
theorem axisConcurrencyPolynomial_ne_zero (e f g : BipartiteEdge m n)
(hef : e.1 ≠ f.1) (hgf : g.1 ≠ f.1) (hfe : f.2 ≠ e.2) (hge : g.2 ≠ e.2) :
axisConcurrencyPolynomial e f g ≠ 0 := by
let x : BipartiteVertex m n → ℝ
| .inl i => if i=f.1 then 0 else 1
| .inr p => if p=e.2 then 0 else 1
intro h
have hh := congrArg (eval x) h
rw [axisConcurrencyPolynomial_eval] at hh
norm_num [x,axisConcurrency,hef,hgf,hfe,hge] at hh
end Zarankiewicz
end
end OAI
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