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97c3405 | 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 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 | import OAI.Combinatorics.CompleteCrossing.CycleIntersections
import OAI.Combinatorics.CompleteCrossing.Refinement
import OAI.Combinatorics.CompleteCrossing.Tensor
namespace OAI
/-! The geometric cycle identity deduced from endpoint and intersection coordinates. -/
section
/-! Actual vertex-coordinate orders give the frozen signed endpoint contribution. -/
noncomputable section
open scoped BigOperators Classical
namespace Paper170
variable {Vertex Edge : Type} {graph : Graph Vertex Edge}
theorem endpointSign_eq_sub_indicators (v : Vertex) (e : Edge) :
endpointSign graph v e =
(if graph.src e=v then 1 else 0) - (if graph.dst e=v then 1 else 0) := by
by_cases hs : graph.src e=v
· have ht : graph.dst e≠v := fun h => graph.loopless e (hs.trans h.symm)
simp [endpointSign,hs,ht]
· by_cases ht : graph.dst e=v <;> simp [endpointSign,hs,ht]
theorem sum_endpointSign [Fintype Vertex] (e : Edge) (f : Vertex → ℝ) :
(∑v, endpointSign graph v e * f v) = f (graph.src e)-f (graph.dst e) := by
simp [endpointSign_eq_sub_indicators,sub_mul,ite_mul,Finset.sum_sub_distrib]
end Paper170
namespace Paper170Geometry.TruncatedSpokeCoordinates
open Zarankiewicz
variable {Vertex Edge : Type} [Fintype Vertex] {graph : Paper170.Graph Vertex Edge}
{N : Paper170.NormalizedDrawing graph} {V : VertexTruncation N.toContinuousDrawing}
(C : TruncatedSpokeCoordinates V) (O : Paper170.SpokeOrders N)
omit [Fintype Vertex] in
theorem flagPairing_eq_spokeComparison
(hnode : ∀v (e f : IncidentEdges graph v),
orderSign (C.node v e) (C.node v f) = Paper170.spokeComparison O v e.val f.val)
(a b : IncidentFlag graph) : C.flagPairing a b =
if b.1=a.1 then Paper170.spokeComparison O a.1 a.2.val b.2.val / 2 else 0 := by
unfold flagPairing
split_ifs with h
· rw [hnode]
rfl
· rfl
theorem edgeEndpointPairing_eq_signed_sum
(hnode : ∀v (e f : IncidentEdges graph v),
orderSign (C.node v e) (C.node v f) = Paper170.spokeComparison O v e.val f.val)
(e f : Edge) : C.edgeEndpointPairing e f =
(1/2 : ℝ) * ∑v, Paper170.endpointSign graph v e * Paper170.endpointSign graph v f *
Paper170.spokeComparison O v e f := by
have hsum := Paper170.sum_endpointSign (graph := graph) e
(fun v => Paper170.endpointSign graph v f * Paper170.spokeComparison O v e f)
simp only [← mul_assoc] at hsum
rw [hsum]
unfold edgeEndpointPairing
rw [C.flagPairing_eq_spokeComparison O hnode,C.flagPairing_eq_spokeComparison O hnode,
C.flagPairing_eq_spokeComparison O hnode,C.flagPairing_eq_spokeComparison O hnode]
simp only [dartSourceFlag,dartTargetFlag,dartSourceIncident,dartTargetIncident,
forwardDart,OrientedDart.fst,OrientedDart.snd,Paper170.endpointSign_eq_sub_indicators]
split_ifs <;> ring
end Paper170Geometry.TruncatedSpokeCoordinates
end
end
section
/-! The geometric closed-polygon calculation annihilates the real graph cycle space. -/
noncomputable section
namespace Paper170Geometry.TruncatedSpokeCoordinates
variable {Vertex Edge : Type} [Fintype Vertex] [Fintype Edge]
{graph : Paper170.Graph Vertex Edge} {D : Paper170.AdmissibleDrawing graph}
{V : VertexTruncation D.toContinuousDrawing} (C : TruncatedSpokeCoordinates V)
theorem cycleMatrix_pairing_zero (hnorm : NoAdjacentCrossings D)
(hstraight : LocallyStraightCrossings D) (htrans : LocallyTransverseCrossings D)
(z z' : Edge → ℝ) (hz : ∀v, Paper170.boundary graph z v = 0)
(hz' : ∀v, Paper170.boundary graph z' v = 0) :
Paper170.edgePairing C.cycleIntersectionMatrix z z' = 0 := by
apply Paper170.edgePairing_eq_zero_of_simple_cycles graph C.cycleIntersectionMatrix ?_ z z' hz hz'
intro P Q
rw [Paper170.edgePairing_simple_cycles_range]
exact C.closed_cycle_matrix_sum hnorm hstraight htrans P.cyclicDart Q.cyclicDart
P.cyclicDart_seam Q.cyclicDart_seam P.length Q.length P.cyclicDart_period Q.cyclicDart_period
(fun i _ => P.cyclicDart_next_edge_ne i) (fun j _ => Q.cyclicDart_next_edge_ne j)
end Paper170Geometry.TruncatedSpokeCoordinates
end
end
section
/-! Reconstruction of the frozen cycle pairing from faithful local coordinates. -/
noncomputable section
namespace Paper170
open Paper170Geometry
variable {Vertex Edge : Type} [Fintype Vertex] [Fintype Edge]
{graph : Graph Vertex Edge}
theorem cycle_pairing_of_coordinates (N : NormalizedDrawing graph) (O : SpokeOrders N)
(V : VertexTruncation N.toContinuousDrawing) (C : TruncatedSpokeCoordinates V)
(hnode : ∀v (e f : IncidentEdges graph v),
Zarankiewicz.orderSign (C.node v e) (C.node v f) = spokeComparison O v e.val f.val)
(z z' : Edge → ℝ) (hz : ∀v, boundary graph z v = 0) (hz' : ∀v, boundary graph z' v = 0) :
edgePairing (cycleIntersection O) z z' = 0 := by
have hmatrix : C.cycleIntersectionMatrix = cycleIntersection O := by
funext e f
unfold TruncatedSpokeCoordinates.cycleIntersectionMatrix cycleIntersection
rw [VertexTruncation.signedIntersection_eq_interiorIntersection N V,
C.edgeEndpointPairing_eq_signed_sum O hnode]
rw [← hmatrix]
exact C.cycleMatrix_pairing_zero (normalized_noAdjacent N) (normalized_locallyStraight N)
(normalized_locallyTransverse N) z z' hz hz'
end Paper170
end
end
section
/-! The source cycle identity for every normalized ordinary drawing. -/
noncomputable section
namespace Paper170
open Paper170Geometry
theorem cycle_pairing {Vertex Edge : Type} [Fintype Vertex] [Fintype Edge]
(G : Graph Vertex Edge) (N : NormalizedDrawing G) (O : SpokeOrders N)
(z z' : Edge → ℝ) (hz : ∀v, boundary G z v = 0) (hz' : ∀v, boundary G z' v = 0) :
edgePairing (cycleIntersection O) z z' = 0 := by
obtain ⟨V⟩ := normalized_vertexTruncation N
obtain ⟨C,hC⟩ := exists_truncatedSpokeCoordinates_compatible N V O
exact cycle_pairing_of_coordinates N O V C hC z z' hz hz'
theorem raw_cycle_pairing {n : ℕ} (N : NormalizedDrawing (completeGraph n)) (O : SpokeOrders N)
(z z' : Fin n → Fin n → ℝ) (hz : ∀v, chainBoundary z v = 0) (hz' : ∀v, chainBoundary z' v = 0) :
chainPairing (rawCycleIntersection O) z z' = 0 := by
rw [chainPairing_rawCycleIntersection]
apply cycle_pairing (completeGraph n) N O (liftedChain z) (liftedChain z')
· simpa only [boundary_liftedChain] using hz
· simpa only [boundary_liftedChain] using hz'
end Paper170
end
end
end OAI
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