File size: 2,730 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 | import OAI.Combinatorics.Crossing.AxisSegments
import Mathlib.Analysis.Convex.PathConnected
namespace OAI
noncomputable section
namespace Zarankiewicz
variable {m n : ℕ}
def axisVertex (a : Fin m → ℝ) (b : Fin n → ℝ) : BipartiteVertex m n → Plane
| .inl i => (0,a i)
| .inr p => (b p,0)
theorem axisVertex_injective (a : Fin m → ℝ) (b : Fin n → ℝ)
(hai : Function.Injective a) (hbi : Function.Injective b) (ha : ∀ i, a i ≠ 0) :
Function.Injective (axisVertex a b) := by
rintro (i|p) (k|q) he
· exact congrArg Sum.inl (hai (congrArg Prod.snd he))
· exact False.elim (ha i (congrArg Prod.snd he))
· exact False.elim (ha k (congrArg Prod.snd he).symm)
· exact congrArg Sum.inr (hbi (congrArg Prod.fst he))
def axisContinuousDrawing (a : Fin m → ℝ) (b : Fin n → ℝ)
(hai : Function.Injective a) (hbi : Function.Injective b)
(ha : ∀ i, a i ≠ 0) (hb : ∀ p, b p ≠ 0) : ContinuousDrawing m n where
vertex := axisVertex a b
vertex_injective := axisVertex_injective a b hai hbi ha
edge e := Path.segment (0,a e.1) (b e.2,0)
edge_injective := by
intro e t u he
apply Subtype.ext
apply axisSegmentPoint_injective (a e.1) (b e.2) (hb e.2)
change AffineMap.lineMap (0,a e.1) (b e.2,0) t.val = AffineMap.lineMap (0,a e.1) (b e.2,0) u.val at he
simpa only [AffineMap.lineMap_apply_module,axisSegmentPoint,segmentPoint] using he
interior_avoids_vertices := by
intro e t ht j he
have hh := axisSegmentPoint_interior_off_axes (a e.1) (b e.2) t.val (ha e.1) (hb e.2) ⟨ht.1,ht.2⟩
have hv : axisSegmentPoint (a e.1) (b e.2) t.val=axisVertex a b j := by
change AffineMap.lineMap (0,a e.1) (b e.2,0) t.val = axisVertex a b j at he
simpa only [AffineMap.lineMap_apply_module,axisSegmentPoint,segmentPoint] using he
rcases j with i | p
· exact hh.1 (congrArg Prod.fst hv)
· exact hh.2 (congrArg Prod.snd hv)
theorem axisContinuousDrawing_interior (a : Fin m → ℝ) (b : Fin n → ℝ)
(hai : Function.Injective a) (hbi : Function.Injective b)
(ha : ∀ i, a i ≠ 0) (hb : ∀ p, b p ≠ 0) (e : BipartiteEdge m n) :
edgeInterior (axisContinuousDrawing a b hai hbi ha hb) e=openSegmentTrace (0,a e.1) (b e.2,0) := by
ext x
constructor
· rintro ⟨t,ht,he⟩
refine ⟨t.val,⟨ht.1,ht.2⟩,?_⟩
change AffineMap.lineMap (0,a e.1) (b e.2,0) t.val = x at he
simpa only [AffineMap.lineMap_apply_module,segmentPoint] using he
· rintro ⟨t,ht,he⟩
refine ⟨⟨t,⟨ht.1.le,ht.2.le⟩⟩,⟨ht.1,ht.2⟩,?_⟩
change AffineMap.lineMap (0,a e.1) (b e.2,0) t = x
simpa only [AffineMap.lineMap_apply_module,segmentPoint] using he
end Zarankiewicz
end
end OAI
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