Convex Euclidean realizations and derived spherical links
DefinitionP2MAssembly_Chapter13V2_Part6cauchy-rigiditydihedral-anglesgeometrylean4polyhedraproofs-from-the-book
This part defines triangulated Euclidean realizations with vertex coordinates, triangular face representatives, nondegenerate edges and faces, supporting planes and strict separation of vertices not belonging to each face. Face-local outward orientation specifies a positive multiple of the ordered cross product. ConvexEuclideanPolyhedron additionally carries degree at least three, a connected Euler-characteristic-two map, triangular faces and graph simplicity. Derived links list neighbors in reverse vertex-rotation order. CongruentFaces means equality of corresponding dart-edge lengths. The part also defines the signed dihedral differences and the correspondence between darts and vertex-star positions. These precise geometric input conditions are retained; arbitrary polygon-faced polyhedra are not introduced.
Definition code
import Init
import Mathlib
import Mathlib.Analysis.LocallyConvex.Separation
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.LinearAlgebra.LinearIndependent.Lemmas
import Mathlib.Data.Fin.Tuple.Reflection
import Mathlib.Data.Fin.Rev
import Mathlib.Geometry.Euclidean.Triangle
import Definitions.Def_P2MAssembly_Chapter13V2_Part1
import Definitions.Def_P2MAssembly_Chapter13V2_Part2
import Definitions.Def_P2MAssembly_Chapter13V2_Part3
import Definitions.Def_P2MAssembly_Chapter13V2_Part4
import Definitions.Def_P2MAssembly_Chapter13V2_Part5
set_option autoImplicit true
/- Original source header (imports hoisted):
import Mathlib
-/
/- Source module: ProofsInTheBook.PlanarMap -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import Mathlib
-/
/- Source module: ProofsInTheBook.TetPearls -/
section
set_option autoImplicit true
noncomputable section
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1000000
open scoped Classical
open Set
namespace ProofsInTheBook.TetPearls
namespace Tet
end Tet
namespace TetSolid
end TetSolid
namespace Segment3
end Segment3
namespace Tet
end Tet
namespace Pearl
end Pearl
end ProofsInTheBook.TetPearls
end
end
/- Original source header (imports hoisted):
import Mathlib
-/
/- Source module: ProofsInTheBook.Chapter09 -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Chapter09
open scoped BigOperators TensorProduct
open Polynomial Chebyshev
-- (`angleClassQ_arccos_one_third_ne_zero` defined below, after
-- `arccos_one_third_irrational_over_pi`.)
end ProofsInTheBook.Chapter09
end
/- Original source header (imports hoisted):
import ProofsInTheBook.TetPearls
import ProofsInTheBook.Chapter09
-/
/- Source module: ProofsInTheBook.TetDihedral -/
section
set_option autoImplicit true
noncomputable section
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
open scoped RealInnerProductSpace
open ProofsInTheBook.TetPearls
namespace ProofsInTheBook.TetDihedral
end ProofsInTheBook.TetDihedral
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.TetDihedral
-/
/- Source module: ProofsInTheBook.SphericalKernel -/
section
set_option autoImplicit true
noncomputable section
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
open scoped RealInnerProductSpace
open ProofsInTheBook.TetPearls ProofsInTheBook.TetDihedral
namespace ProofsInTheBook.SphericalKernel
end ProofsInTheBook.SphericalKernel
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalKernel
-/
/- Source module: ProofsInTheBook.SphericalArm -/
section
set_option autoImplicit true
noncomputable section
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
open scoped RealInnerProductSpace
open ProofsInTheBook.TetPearls ProofsInTheBook.TetDihedral
open ProofsInTheBook.SphericalKernel
namespace ProofsInTheBook.SphericalArm
end ProofsInTheBook.SphericalArm
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalArm
-/
/- Source module: ProofsInTheBook.SphericalRotation -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace
open ProofsInTheBook.TetPearls ProofsInTheBook.TetDihedral
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
namespace ProofsInTheBook.SphericalRotation
end ProofsInTheBook.SphericalRotation
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalRotation
-/
/- Source module: ProofsInTheBook.SphericalSZ -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.TetPearls ProofsInTheBook.TetDihedral
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm ProofsInTheBook.SphericalRotation
namespace ProofsInTheBook.SphericalSZ
end ProofsInTheBook.SphericalSZ
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalSZ
-/
/- Source module: ProofsInTheBook.SphericalCore -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.TetPearls ProofsInTheBook.TetDihedral
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
namespace ProofsInTheBook.SphericalCore
end ProofsInTheBook.SphericalCore
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalCore
-/
/- Source module: ProofsInTheBook.SphericalFinish -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.TetPearls ProofsInTheBook.TetDihedral
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore
namespace ProofsInTheBook.SphericalFinish
end ProofsInTheBook.SphericalFinish
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalFinish
-/
/- Source module: ProofsInTheBook.SphericalOpening -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.TetPearls ProofsInTheBook.TetDihedral
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
namespace ProofsInTheBook.SphericalOpening
end ProofsInTheBook.SphericalOpening
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalOpening
-/
/- Source module: ProofsInTheBook.SphericalHinge -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.TetPearls ProofsInTheBook.TetDihedral
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening
namespace ProofsInTheBook.SphericalHinge
end ProofsInTheBook.SphericalHinge
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalHinge
-/
/- Source module: ProofsInTheBook.SphericalSZChain -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.TetPearls ProofsInTheBook.TetDihedral
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
namespace ProofsInTheBook.SphericalSZChain
end ProofsInTheBook.SphericalSZChain
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalSZChain
-/
/- Source module: ProofsInTheBook.SphericalCyclicTriple -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.TetPearls ProofsInTheBook.TetDihedral
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain
namespace ProofsInTheBook.SphericalCyclicTriple
end ProofsInTheBook.SphericalCyclicTriple
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalCyclicTriple
-/
/- Source module: ProofsInTheBook.SphericalGnomonic -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.TetPearls ProofsInTheBook.TetDihedral
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
namespace ProofsInTheBook.SphericalGnomonic
end ProofsInTheBook.SphericalGnomonic
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalGnomonic
-/
/- Source module: ProofsInTheBook.PlanarConvexDiag -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalArm ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCyclicTriple ProofsInTheBook.SphericalSZChain
open ProofsInTheBook.SphericalGnomonic
namespace ProofsInTheBook.PlanarConvexDiag
end ProofsInTheBook.PlanarConvexDiag
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarConvexDiag
-/
/- Source module: ProofsInTheBook.SphericalSZStep -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
namespace ProofsInTheBook.SphericalSZStep
end ProofsInTheBook.SphericalSZStep
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalSZStep
-/
/- Source module: ProofsInTheBook.SphericalHingeCut -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep
namespace ProofsInTheBook.SphericalHingeCut
end ProofsInTheBook.SphericalHingeCut
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalHingeCut
-/
/- Source module: ProofsInTheBook.SphericalDiagCut -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
namespace ProofsInTheBook.SphericalDiagCut
end ProofsInTheBook.SphericalDiagCut
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalDiagCut
-/
/- Source module: ProofsInTheBook.SphericalOpeningProcess -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut
namespace ProofsInTheBook.SphericalOpeningProcess
end ProofsInTheBook.SphericalOpeningProcess
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalOpeningProcess
-/
/- Source module: ProofsInTheBook.SphericalReachStuck -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut ProofsInTheBook.SphericalOpeningProcess
namespace ProofsInTheBook.SphericalReachStuck
end ProofsInTheBook.SphericalReachStuck
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalReachStuck
-/
/- Source module: ProofsInTheBook.SphericalAdmissibleSup -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalReachStuck
namespace ProofsInTheBook.SphericalAdmissibleSup
end ProofsInTheBook.SphericalAdmissibleSup
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalAdmissibleSup
-/
/- Source module: ProofsInTheBook.SphericalArmClose -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalReachStuck ProofsInTheBook.SphericalAdmissibleSup
namespace ProofsInTheBook.SphericalArmClose
end ProofsInTheBook.SphericalArmClose
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalArmClose
-/
/- Source module: ProofsInTheBook.SphericalArmFinal -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalReachStuck ProofsInTheBook.SphericalAdmissibleSup
open ProofsInTheBook.SphericalArmClose
namespace ProofsInTheBook.SphericalArmFinal
end ProofsInTheBook.SphericalArmFinal
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalArmFinal
-/
/- Source module: ProofsInTheBook.SphericalSZComplete -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalReachStuck ProofsInTheBook.SphericalAdmissibleSup
open ProofsInTheBook.SphericalArmClose
namespace ProofsInTheBook.SphericalSZComplete
end ProofsInTheBook.SphericalSZComplete
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalSZComplete
-/
/- Source module: ProofsInTheBook.SphericalStuckWitness -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalReachStuck ProofsInTheBook.SphericalAdmissibleSup
open ProofsInTheBook.SphericalArmClose ProofsInTheBook.SphericalSZComplete
namespace ProofsInTheBook.SphericalStuckWitness
end ProofsInTheBook.SphericalStuckWitness
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalStuckWitness
-/
/- Source module: ProofsInTheBook.SphericalTerminalVis -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalReachStuck ProofsInTheBook.SphericalAdmissibleSup
open ProofsInTheBook.SphericalArmClose ProofsInTheBook.SphericalSZComplete
open ProofsInTheBook.SphericalStuckWitness
namespace ProofsInTheBook.SphericalTerminalVis
end ProofsInTheBook.SphericalTerminalVis
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalTerminalVis
-/
/- Source module: ProofsInTheBook.SphericalArmUncond -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalReachStuck ProofsInTheBook.SphericalAdmissibleSup
open ProofsInTheBook.SphericalArmClose ProofsInTheBook.SphericalSZComplete
open ProofsInTheBook.SphericalTerminalVis
namespace ProofsInTheBook.SphericalArmUncond
end ProofsInTheBook.SphericalArmUncond
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalArmUncond
-/
/- Source module: ProofsInTheBook.SphericalMatchedCut -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalReachStuck ProofsInTheBook.SphericalAdmissibleSup
open ProofsInTheBook.SphericalArmClose ProofsInTheBook.SphericalSZComplete
open ProofsInTheBook.SphericalTerminalVis ProofsInTheBook.SphericalArmUncond
namespace ProofsInTheBook.SphericalMatchedCut
end ProofsInTheBook.SphericalMatchedCut
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalMatchedCut
-/
/- Source module: ProofsInTheBook.SphericalCornerStep -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalReachStuck ProofsInTheBook.SphericalAdmissibleSup
open ProofsInTheBook.SphericalArmClose ProofsInTheBook.SphericalSZComplete
open ProofsInTheBook.SphericalTerminalVis ProofsInTheBook.SphericalArmUncond
open ProofsInTheBook.SphericalMatchedCut
namespace ProofsInTheBook.SphericalCornerStep
end ProofsInTheBook.SphericalCornerStep
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalCornerStep
import ProofsInTheBook.PlanarConvexDiag
-/
/- Source module: ProofsInTheBook.SphericalConeMembership -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalReachStuck ProofsInTheBook.SphericalAdmissibleSup
open ProofsInTheBook.SphericalArmClose ProofsInTheBook.SphericalSZComplete
open ProofsInTheBook.SphericalTerminalVis ProofsInTheBook.SphericalArmUncond
open ProofsInTheBook.SphericalMatchedCut ProofsInTheBook.SphericalCornerStep
namespace ProofsInTheBook.SphericalConeMembership
end ProofsInTheBook.SphericalConeMembership
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalConeMembership
-/
/- Source module: ProofsInTheBook.SphericalArmDone -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalReachStuck ProofsInTheBook.SphericalAdmissibleSup
open ProofsInTheBook.SphericalArmClose ProofsInTheBook.SphericalSZComplete
open ProofsInTheBook.SphericalTerminalVis ProofsInTheBook.SphericalArmUncond
open ProofsInTheBook.SphericalMatchedCut ProofsInTheBook.SphericalCornerStep
open ProofsInTheBook.SphericalConeMembership
namespace ProofsInTheBook.SphericalArmDone
end ProofsInTheBook.SphericalArmDone
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalArmDone
-/
/- Source module: ProofsInTheBook.SphericalArmFinish -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalReachStuck ProofsInTheBook.SphericalAdmissibleSup
open ProofsInTheBook.SphericalArmClose ProofsInTheBook.SphericalSZComplete
open ProofsInTheBook.SphericalTerminalVis ProofsInTheBook.SphericalArmUncond
open ProofsInTheBook.SphericalMatchedCut ProofsInTheBook.SphericalCornerStep
open ProofsInTheBook.SphericalConeMembership ProofsInTheBook.SphericalArmDone
namespace ProofsInTheBook.SphericalArmFinish
end ProofsInTheBook.SphericalArmFinish
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalArmFinish
-/
/- Source module: ProofsInTheBook.SphericalArmClose2 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalReachStuck ProofsInTheBook.SphericalAdmissibleSup
open ProofsInTheBook.SphericalArmClose ProofsInTheBook.SphericalSZComplete
open ProofsInTheBook.SphericalTerminalVis ProofsInTheBook.SphericalArmUncond
open ProofsInTheBook.SphericalMatchedCut ProofsInTheBook.SphericalCornerStep
open ProofsInTheBook.SphericalConeMembership ProofsInTheBook.SphericalArmDone
open ProofsInTheBook.SphericalArmFinish
namespace ProofsInTheBook.SphericalArmClose2
end ProofsInTheBook.SphericalArmClose2
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalArmClose2
-/
/- Source module: ProofsInTheBook.SphericalStuckCollinear -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalReachStuck ProofsInTheBook.SphericalAdmissibleSup
open ProofsInTheBook.SphericalArmClose ProofsInTheBook.SphericalSZComplete
open ProofsInTheBook.SphericalTerminalVis ProofsInTheBook.SphericalArmUncond
open ProofsInTheBook.SphericalMatchedCut ProofsInTheBook.SphericalCornerStep
open ProofsInTheBook.SphericalConeMembership ProofsInTheBook.SphericalArmDone
open ProofsInTheBook.SphericalArmFinish ProofsInTheBook.SphericalArmClose2
namespace ProofsInTheBook.SphericalStuckCollinear
end ProofsInTheBook.SphericalStuckCollinear
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalStuckCollinear
-/
/- Source module: ProofsInTheBook.SphericalOpenedArmCore -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalReachStuck ProofsInTheBook.SphericalAdmissibleSup
open ProofsInTheBook.SphericalArmClose ProofsInTheBook.SphericalSZComplete
open ProofsInTheBook.SphericalTerminalVis ProofsInTheBook.SphericalArmUncond
open ProofsInTheBook.SphericalMatchedCut ProofsInTheBook.SphericalCornerStep
open ProofsInTheBook.SphericalConeMembership ProofsInTheBook.SphericalArmDone
open ProofsInTheBook.SphericalArmFinish ProofsInTheBook.SphericalArmClose2
open ProofsInTheBook.SphericalStuckCollinear
namespace ProofsInTheBook.SphericalOpenedArmCore
end ProofsInTheBook.SphericalOpenedArmCore
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalStuckCollinear
-/
/- Source module: ProofsInTheBook.SphericalSZInduction -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalDiagCut
open ProofsInTheBook.SphericalSZChain
open ProofsInTheBook.SphericalTerminalVis
open ProofsInTheBook.SphericalArmUncond
open ProofsInTheBook.SphericalStuckCollinear
namespace ProofsInTheBook.SphericalSZInduction
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.SphericalSZInduction
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalSZInduction
-/
/- Source module: ProofsInTheBook.SphericalSZStepClose -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCyclicTriple ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZChain
open ProofsInTheBook.SphericalStuckCollinear
open ProofsInTheBook.SphericalSZInduction
namespace ProofsInTheBook.SphericalSZStepClose
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.SphericalSZStepClose
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalSZStepClose
-/
/- Source module: ProofsInTheBook.SphericalSZFinal -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCyclicTriple ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalSZChain
open ProofsInTheBook.SphericalDiagCut
open ProofsInTheBook.SphericalSZStep
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
namespace ProofsInTheBook.SphericalSZFinal
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.SphericalSZFinal
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalSZFinal
-/
/- Source module: ProofsInTheBook.SphericalSZClose -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCyclicTriple ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalSZChain
open ProofsInTheBook.SphericalDiagCut
open ProofsInTheBook.SphericalSZStep
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalSZFinal
namespace ProofsInTheBook.SphericalSZClose
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.SphericalSZClose
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalSZClose
-/
/- Source module: ProofsInTheBook.SphericalCutTransport -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalSZClose
namespace ProofsInTheBook.SphericalCutTransport
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.SphericalCutTransport
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalCutTransport
-/
/- Source module: ProofsInTheBook.ZinanFFCT -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalCutTransport
namespace ProofsInTheBook.ZinanFFCT
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT
-/
/- Source module: ProofsInTheBook.ZinanFFCT2 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.SphericalConeMembership
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.ZinanFFCT
namespace ProofsInTheBook.ZinanFFCT2
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT2
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT2
-/
/- Source module: ProofsInTheBook.ZinanFFCT3 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.SphericalConeMembership
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.ZinanFFCT
open ProofsInTheBook.ZinanFFCT2
namespace ProofsInTheBook.ZinanFFCT3
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT3
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT3
-/
/- Source module: ProofsInTheBook.ZinanFFCT4 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.SphericalConeMembership
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalGnomonic
open ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.ZinanFFCT
open ProofsInTheBook.ZinanFFCT2
open ProofsInTheBook.ZinanFFCT3
namespace ProofsInTheBook.ZinanFFCT4
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT4
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT4
-/
/- Source module: ProofsInTheBook.ZinanFFCT5 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.SphericalConeMembership
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalGnomonic
open ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.ZinanFFCT
open ProofsInTheBook.ZinanFFCT3
open ProofsInTheBook.ZinanFFCT4
namespace ProofsInTheBook.ZinanFFCT5
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT5
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT5
-/
/- Source module: ProofsInTheBook.ZinanFFCT6 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.SphericalConeMembership
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalGnomonic
open ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.ZinanFFCT
open ProofsInTheBook.ZinanFFCT2
open ProofsInTheBook.ZinanFFCT3
open ProofsInTheBook.ZinanFFCT5
namespace ProofsInTheBook.ZinanFFCT6
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT6
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT6
-/
/- Source module: ProofsInTheBook.ZinanFFCT7 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.SphericalConeMembership
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalGnomonic
open ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.ZinanFFCT
open ProofsInTheBook.ZinanFFCT2
open ProofsInTheBook.ZinanFFCT3
open ProofsInTheBook.ZinanFFCT5
open ProofsInTheBook.ZinanFFCT6
namespace ProofsInTheBook.ZinanFFCT7
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT7
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT7
import ProofsInTheBook.PlanarConvexDiag
-/
/- Source module: ProofsInTheBook.ZinanFFCT8 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZ ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalCutTransport ProofsInTheBook.SphericalGnomonic
open ProofsInTheBook.SphericalConeMembership
open ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.ZinanFFCT ProofsInTheBook.ZinanFFCT2 ProofsInTheBook.ZinanFFCT3
open ProofsInTheBook.ZinanFFCT4 ProofsInTheBook.ZinanFFCT5 ProofsInTheBook.ZinanFFCT6
open ProofsInTheBook.ZinanFFCT7
namespace ProofsInTheBook.ZinanFFCT8
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT8
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT8
import ProofsInTheBook.SphericalRotation
-/
/- Source module: ProofsInTheBook.ZinanFFCT9 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalConeMembership
open ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.ZinanFFCT8
namespace ProofsInTheBook.ZinanFFCT9
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT9
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT9
-/
/- Source module: ProofsInTheBook.ZinanFFCT10 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace
open ProofsInTheBook.SphericalKernel
open ProofsInTheBook.ZinanFFCT8 ProofsInTheBook.ZinanFFCT9
namespace ProofsInTheBook.ZinanFFCT10
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT10
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT10
-/
/- Source module: ProofsInTheBook.ZinanFFCT17 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.ZinanFFCT10
namespace ProofsInTheBook.ZinanFFCT17
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT17
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT17
-/
/- Source module: ProofsInTheBook.ZinanFFCT18 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.ZinanFFCT3 ProofsInTheBook.ZinanFFCT17
namespace ProofsInTheBook.ZinanFFCT18
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT18
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalStuckWitness
import ProofsInTheBook.SphericalCutTransport
-/
/- Source module: ProofsInTheBook.SphericalStuckGeneral -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalReachStuck ProofsInTheBook.SphericalAdmissibleSup
open ProofsInTheBook.SphericalArmClose ProofsInTheBook.SphericalSZComplete
open ProofsInTheBook.SphericalStuckWitness ProofsInTheBook.SphericalTerminalVis
open ProofsInTheBook.SphericalSZInduction ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalCutTransport
namespace ProofsInTheBook.SphericalStuckGeneral
end ProofsInTheBook.SphericalStuckGeneral
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalStuckGeneral
-/
/- Source module: ProofsInTheBook.SphericalLastCornerStuck -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpening ProofsInTheBook.SphericalHinge
open ProofsInTheBook.SphericalSZChain ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalGnomonic ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZStep ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalReachStuck ProofsInTheBook.SphericalAdmissibleSup
open ProofsInTheBook.SphericalArmClose ProofsInTheBook.SphericalSZComplete
open ProofsInTheBook.SphericalCutTransport ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose ProofsInTheBook.SphericalStuckGeneral
namespace ProofsInTheBook.SphericalLastCornerStuck
end ProofsInTheBook.SphericalLastCornerStuck
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT18
import ProofsInTheBook.SphericalLastCornerStuck
-/
/- Source module: ProofsInTheBook.ZinanFFCT19 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalCutTransport ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.TetDihedral
open ProofsInTheBook.ZinanFFCT18
namespace ProofsInTheBook.ZinanFFCT19
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT19
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalSZClose
-/
/- Source module: ProofsInTheBook.SphericalMonitoredSup -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalCyclicTriple ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalSZChain
open ProofsInTheBook.SphericalDiagCut
open ProofsInTheBook.SphericalSZStep
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
namespace ProofsInTheBook.SphericalMonitoredSup
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.SphericalMonitoredSup
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalSZClose
-/
/- Source module: ProofsInTheBook.SphericalSpliceTransport -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
namespace ProofsInTheBook.SphericalSpliceTransport
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.SphericalSpliceTransport
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalRotation
import ProofsInTheBook.SphericalCyclicTriple
-/
/- Source module: ProofsInTheBook.SphericalCongruence -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace
open ProofsInTheBook.TetPearls ProofsInTheBook.TetDihedral
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalCyclicTriple
namespace ProofsInTheBook.SphericalCongruence
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.SphericalCongruence
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalMonitoredSup
import ProofsInTheBook.SphericalSpliceTransport
import ProofsInTheBook.SphericalCongruence
-/
/- Source module: ProofsInTheBook.SphericalArmAssembly -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalCyclicTriple ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalSpliceTransport
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalCongruence
namespace ProofsInTheBook.SphericalArmAssembly
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.SphericalArmAssembly
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalArmAssembly
-/
/- Source module: ProofsInTheBook.SphericalOpeningOutcome -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalArmAssembly
namespace ProofsInTheBook.SphericalOpeningOutcome
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.SphericalOpeningOutcome
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT19
import ProofsInTheBook.SphericalSZClose
import ProofsInTheBook.SphericalOpeningOutcome
import ProofsInTheBook.ZinanFFCT18
-/
/- Source module: ProofsInTheBook.ZinanFFCT20 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.ZinanFFCT18
namespace ProofsInTheBook.ZinanFFCT20
end ProofsInTheBook.ZinanFFCT20
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT10
-/
/- Source module: ProofsInTheBook.ZinanFFCT12 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace
open ProofsInTheBook.SphericalKernel
open ProofsInTheBook.ZinanFFCT8 ProofsInTheBook.ZinanFFCT9 ProofsInTheBook.ZinanFFCT10
namespace ProofsInTheBook.ZinanFFCT12
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT12
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT20
import ProofsInTheBook.ZinanFFCT12
-/
/- Source module: ProofsInTheBook.ZinanFFCT21 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.ZinanFFCT3 ProofsInTheBook.ZinanFFCT9 ProofsInTheBook.ZinanFFCT10
open ProofsInTheBook.ZinanFFCT12 ProofsInTheBook.ZinanFFCT18
namespace ProofsInTheBook.ZinanFFCT21
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT21
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT21
-/
/- Source module: ProofsInTheBook.ZinanFFCT22 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.ZinanFFCT3 ProofsInTheBook.ZinanFFCT9 ProofsInTheBook.ZinanFFCT10
open ProofsInTheBook.ZinanFFCT12 ProofsInTheBook.ZinanFFCT18 ProofsInTheBook.ZinanFFCT21
namespace ProofsInTheBook.ZinanFFCT22
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT22
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT22
-/
/- Source module: ProofsInTheBook.ZinanFFCT23 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.ZinanFFCT3 ProofsInTheBook.ZinanFFCT9 ProofsInTheBook.ZinanFFCT10
open ProofsInTheBook.ZinanFFCT12 ProofsInTheBook.ZinanFFCT18 ProofsInTheBook.ZinanFFCT21
namespace ProofsInTheBook.ZinanFFCT23
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT23
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT23
-/
/- Source module: ProofsInTheBook.ZinanFFCT24 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.ZinanFFCT3 ProofsInTheBook.ZinanFFCT9 ProofsInTheBook.ZinanFFCT10
open ProofsInTheBook.ZinanFFCT12 ProofsInTheBook.ZinanFFCT18 ProofsInTheBook.ZinanFFCT21
open ProofsInTheBook.ZinanFFCT22 ProofsInTheBook.ZinanFFCT23
namespace ProofsInTheBook.ZinanFFCT24
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT24
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT24
-/
/- Source module: ProofsInTheBook.ZinanFFCT25 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction ProofsInTheBook.SphericalRotation
open ProofsInTheBook.ZinanFFCT3 ProofsInTheBook.ZinanFFCT9 ProofsInTheBook.ZinanFFCT10
open ProofsInTheBook.ZinanFFCT12 ProofsInTheBook.ZinanFFCT18 ProofsInTheBook.ZinanFFCT21
open ProofsInTheBook.ZinanFFCT22 ProofsInTheBook.ZinanFFCT23 ProofsInTheBook.ZinanFFCT24
namespace ProofsInTheBook.ZinanFFCT25
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT25
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT25
import ProofsInTheBook.SphericalCore
-/
/- Source module: ProofsInTheBook.ZinanFFCT26 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalCore
open ProofsInTheBook.ZinanFFCT10
namespace ProofsInTheBook.ZinanFFCT26
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT26
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT26
import ProofsInTheBook.SphericalStuckGeneral
-/
/- Source module: ProofsInTheBook.ZinanFFCT27 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalCore
open ProofsInTheBook.ZinanFFCT9 ProofsInTheBook.ZinanFFCT10 ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT26 ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalSZ
namespace ProofsInTheBook.ZinanFFCT27
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT27
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT27
import ProofsInTheBook.ZinanFFCT25
import ProofsInTheBook.SphericalMonitoredSup
import ProofsInTheBook.SphericalOpeningOutcome
-/
/- Source module: ProofsInTheBook.ZinanFFCT28 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.ZinanFFCT18 ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT26 ProofsInTheBook.ZinanFFCT27
open ProofsInTheBook.SphericalStuckGeneral ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.SphericalMonitoredSup ProofsInTheBook.SphericalSZFinal
namespace ProofsInTheBook.ZinanFFCT28
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT28
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalOpeningOutcome
-/
/- Source module: ProofsInTheBook.SphericalOpeningGlue -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalDiagCut
open ProofsInTheBook.SphericalSZChain
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalOpeningOutcome
namespace ProofsInTheBook.SphericalOpeningGlue
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.SphericalOpeningGlue
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT28
import ProofsInTheBook.SphericalOpeningGlue
-/
/- Source module: ProofsInTheBook.ZinanFFCT30 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalMonitoredSup ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalOpeningGlue
namespace ProofsInTheBook.ZinanFFCT30
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT30
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT30
import ProofsInTheBook.ZinanFFCT22
-/
/- Source module: ProofsInTheBook.ZinanFFCT33 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.ZinanFFCT18 ProofsInTheBook.ZinanFFCT30
namespace ProofsInTheBook.ZinanFFCT33
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT33
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT33
-/
/- Source module: ProofsInTheBook.ZinanFFCT34 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.ZinanFFCT18 ProofsInTheBook.ZinanFFCT30 ProofsInTheBook.ZinanFFCT33
namespace ProofsInTheBook.ZinanFFCT34
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT34
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT34
import Mathlib.Analysis.LocallyConvex.Separation
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Convex.Combination
-/
/- Source module: ProofsInTheBook.ZinanFFCT36 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.ZinanFFCT18 ProofsInTheBook.ZinanFFCT30
open ProofsInTheBook.ZinanFFCT33 ProofsInTheBook.ZinanFFCT34
namespace ProofsInTheBook.ZinanFFCT36
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT36
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT36
-/
/- Source module: ProofsInTheBook.ZinanFFCT44 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalCore
open ProofsInTheBook.ZinanFFCT21 ProofsInTheBook.ZinanFFCT22
open ProofsInTheBook.ZinanFFCT25 ProofsInTheBook.ZinanFFCT30
open ProofsInTheBook.ZinanFFCT36
namespace ProofsInTheBook.ZinanFFCT44
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT44
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT20
import ProofsInTheBook.ZinanFFCT3
import ProofsInTheBook.SphericalOpeningGlue
-/
/- Source module: ProofsInTheBook.ZinanFFCT37 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalOpeningGlue
open ProofsInTheBook.ZinanFFCT20
open ProofsInTheBook.ZinanFFCT3
namespace ProofsInTheBook.ZinanFFCT37
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT37
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT37
import ProofsInTheBook.ZinanFFCT36
-/
/- Source module: ProofsInTheBook.ZinanFFCT38 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalSpliceTransport
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalOpeningGlue
open ProofsInTheBook.ZinanFFCT30
open ProofsInTheBook.ZinanFFCT36
open ProofsInTheBook.ZinanFFCT37
namespace ProofsInTheBook.ZinanFFCT38
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT38
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT38
-/
/- Source module: ProofsInTheBook.ZinanFFCT39 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalOpeningGlue
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.ZinanFFCT37
open ProofsInTheBook.ZinanFFCT38
namespace ProofsInTheBook.ZinanFFCT39
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT39
-- Brick 1 (positive content + assembly + audit)
-- Brick 2 (audit + positive content)
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT39
-/
/- Source module: ProofsInTheBook.ZinanFFCT40 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalSpliceTransport
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalOpeningGlue
open ProofsInTheBook.SphericalDiagCut
open ProofsInTheBook.ZinanFFCT30
open ProofsInTheBook.ZinanFFCT36
open ProofsInTheBook.ZinanFFCT37
open ProofsInTheBook.ZinanFFCT38
open ProofsInTheBook.ZinanFFCT39
namespace ProofsInTheBook.ZinanFFCT40
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT40
-- §1 the any-h assembler
-- §3 the pure-hemi strict certificate + repaired stuck outcome + repaired clause (iii)
-- §3 the corrected outcome + repaired headline
-- refutation-resistance witnesses
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT40
-/
/- Source module: ProofsInTheBook.ZinanFFCT41 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalSZChain
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalSpliceTransport
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalOpeningGlue
open ProofsInTheBook.ZinanFFCT3
open ProofsInTheBook.ZinanFFCT20
open ProofsInTheBook.ZinanFFCT30
open ProofsInTheBook.ZinanFFCT36
open ProofsInTheBook.ZinanFFCT37
open ProofsInTheBook.ZinanFFCT38
open ProofsInTheBook.ZinanFFCT39
open ProofsInTheBook.ZinanFFCT40
namespace ProofsInTheBook.ZinanFFCT41
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT41
-- §1 the WB family + W-admissibility bridge
-- §2 the base sinusoid
-- §3 the cap by admissibility (the central new content)
-- §5 the WB trichotomy
-- §6/§7 the clauses at the WB sup
-- §8/§9 the base-capped outcome + headline (GlueWBaseCap discharged)
-- refutation-resistance witness
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT41
-/
/- Source module: ProofsInTheBook.ZinanFFCT42 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalSpliceTransport
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.ZinanFFCT41
namespace ProofsInTheBook.ZinanFFCT42
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT42
-- §1 the algebra/index micro-lemmas
-- §2 base-stuck = opened diagonal
-- §3 Brick 1 (the cyclic-identity bridge) + the vanishing-support payload
-- §4 the residual DISCHARGED + the base-stuck-free headline
-- non-vacuity guards
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT42
-/
/- Source module: ProofsInTheBook.ZinanFFCT45 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalOpeningGlue
open ProofsInTheBook.ZinanFFCT3
open ProofsInTheBook.ZinanFFCT20
open ProofsInTheBook.ZinanFFCT37
open ProofsInTheBook.ZinanFFCT41
open ProofsInTheBook.ZinanFFCT42
namespace ProofsInTheBook.ZinanFFCT45
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT45
-- §1 the WBS family + closure facts
-- §2 init admissibility
-- §3 deficit bound + base cap
-- §4 the trichotomy + clauses
-- §5 Brick 7: the FFCT42 base-stuck port DISCHARGED
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT42
-/
/- Source module: ProofsInTheBook.ZinanFFCT43 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalSpliceTransport
open ProofsInTheBook.ZinanFFCT39
open ProofsInTheBook.ZinanFFCT41
open ProofsInTheBook.ZinanFFCT42
namespace ProofsInTheBook.ZinanFFCT43
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT43
-- §1 endpoint positivity
-- §2 closing edge distinct at the WB supremum
-- §3 the residual DISCHARGED + the closing-edge-free headline
-- non-vacuity guards
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT44
import ProofsInTheBook.ZinanFFCT45
import ProofsInTheBook.ZinanFFCT43
-/
/- Source module: ProofsInTheBook.ZinanFFCT46 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalSpliceTransport
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalOpeningGlue
open ProofsInTheBook.ZinanFFCT3
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT34
open ProofsInTheBook.ZinanFFCT36
open ProofsInTheBook.ZinanFFCT37
open ProofsInTheBook.ZinanFFCT40
open ProofsInTheBook.ZinanFFCT42
open ProofsInTheBook.ZinanFFCT43
open ProofsInTheBook.ZinanFFCT44
open ProofsInTheBook.ZinanFFCT45
namespace ProofsInTheBook.ZinanFFCT46
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT46
-- §1 the margins-free open-hemisphere production (THE keystone mechanism)
-- §2 brick 4
-- §2′ the opened side / joint geometry
-- §3 bricks 5–6
-- §4 brick 8
-- §5 brick 9 + non-vacuity
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT46
-/
/- Source module: ProofsInTheBook.ZinanFFCT47 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalSpliceTransport
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.ZinanFFCT21 ProofsInTheBook.ZinanFFCT22
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT36
open ProofsInTheBook.ZinanFFCT42
open ProofsInTheBook.ZinanFFCT43
open ProofsInTheBook.ZinanFFCT44
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
namespace ProofsInTheBook.ZinanFFCT47
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT47
-- §1 the open-chain collapse kernel (3 ≤ n)
-- §2 the wrap-edge-free open-hemisphere production
-- §3 wrap ShortArc from the hemisphere
-- §4 the residual discharged
-- §5 the wrap-free headline
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT47
import ProofsInTheBook.ZinanFFCT28
import ProofsInTheBook.SphericalStuckGeneral
-/
/- Source module: ProofsInTheBook.ZinanFFCT49 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.ZinanFFCT28
open ProofsInTheBook.ZinanFFCT37
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT47
namespace ProofsInTheBook.ZinanFFCT49
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT49
-- §0 the opened arm
-- §2 discharged pieces
-- §4 the bridge
-- §5 non-vacuity guards
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT49
import ProofsInTheBook.ZinanFFCT23
-/
/- Source module: ProofsInTheBook.ZinanFFCT52 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.ZinanFFCT12
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT47
open ProofsInTheBook.ZinanFFCT49
namespace ProofsInTheBook.ZinanFFCT52
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT52
-- §1 component 2
-- §2 reversal infra
-- §3 orientation normalization
-- §4 interval convexity
-- §5 assembly
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT19
import ProofsInTheBook.ZinanFFCT46
import ProofsInTheBook.ZinanFFCT47
-/
/- Source module: ProofsInTheBook.ZinanFFCT48 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT19
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT47
namespace ProofsInTheBook.ZinanFFCT48
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT48
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT25
import ProofsInTheBook.ZinanFFCT48
-/
/- Source module: ProofsInTheBook.ZinanFFCT53 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.ZinanFFCT18 ProofsInTheBook.ZinanFFCT19
open ProofsInTheBook.ZinanFFCT23 ProofsInTheBook.ZinanFFCT25
namespace ProofsInTheBook.ZinanFFCT53
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT53
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT52
import ProofsInTheBook.ZinanFFCT53
-/
/- Source module: ProofsInTheBook.ZinanFFCT54 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.ZinanFFCT18 ProofsInTheBook.ZinanFFCT19
open ProofsInTheBook.ZinanFFCT21 ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT52 ProofsInTheBook.ZinanFFCT53
namespace ProofsInTheBook.ZinanFFCT54
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT54
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT54
-/
/- Source module: ProofsInTheBook.ZinanFFCT63 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.ZinanFFCT12
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT53
open ProofsInTheBook.ZinanFFCT54
namespace ProofsInTheBook.ZinanFFCT63
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT63
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT28
-/
/- Source module: ProofsInTheBook.ZinanFFCT29 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalSZ
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.ZinanFFCT10 ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT26 ProofsInTheBook.ZinanFFCT27
open ProofsInTheBook.ZinanFFCT28
namespace ProofsInTheBook.ZinanFFCT29
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT29
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT29
-/
/- Source module: ProofsInTheBook.ZinanFFCT31 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.ZinanFFCT3 ProofsInTheBook.ZinanFFCT9 ProofsInTheBook.ZinanFFCT10
open ProofsInTheBook.ZinanFFCT12 ProofsInTheBook.ZinanFFCT18 ProofsInTheBook.ZinanFFCT21
open ProofsInTheBook.ZinanFFCT22 ProofsInTheBook.ZinanFFCT23 ProofsInTheBook.ZinanFFCT24
open ProofsInTheBook.ZinanFFCT25 ProofsInTheBook.ZinanFFCT27 ProofsInTheBook.ZinanFFCT29
namespace ProofsInTheBook.ZinanFFCT31
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT31
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT31
-/
/- Source module: ProofsInTheBook.ZinanFFCT32 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.ZinanFFCT3 ProofsInTheBook.ZinanFFCT9 ProofsInTheBook.ZinanFFCT10
open ProofsInTheBook.ZinanFFCT12 ProofsInTheBook.ZinanFFCT18 ProofsInTheBook.ZinanFFCT21
open ProofsInTheBook.ZinanFFCT22 ProofsInTheBook.ZinanFFCT23 ProofsInTheBook.ZinanFFCT24
open ProofsInTheBook.ZinanFFCT25 ProofsInTheBook.ZinanFFCT27 ProofsInTheBook.ZinanFFCT29
open ProofsInTheBook.ZinanFFCT31
namespace ProofsInTheBook.ZinanFFCT32
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT32
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT49
import ProofsInTheBook.ZinanFFCT32
-/
/- Source module: ProofsInTheBook.ZinanFFCT51 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.ZinanFFCT3 ProofsInTheBook.ZinanFFCT18 ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT27 ProofsInTheBook.ZinanFFCT29 ProofsInTheBook.ZinanFFCT31
open ProofsInTheBook.ZinanFFCT32
open ProofsInTheBook.ZinanFFCT45 ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT49
namespace ProofsInTheBook.ZinanFFCT51
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT51
-- §1 the sharp residue
-- §2 the corner sign verification
-- §3 the main near-side line
-- §4 non-vacuity guards
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT51
-/
/- Source module: ProofsInTheBook.ZinanFFCT55 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.ZinanFFCT26 ProofsInTheBook.ZinanFFCT27
open ProofsInTheBook.ZinanFFCT29
open ProofsInTheBook.ZinanFFCT45 ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT51
namespace ProofsInTheBook.ZinanFFCT55
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT55
-- §R1/R2 the constant-binding contradiction at the WBS family
-- §δ*=0 edge
-- §R3 slot normalization
-- §R4 the derivative + the sign finding
-- §R4′ the forced collapse
-- §5 non-vacuity guards
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT21
import ProofsInTheBook.ZinanFFCT55
-/
/- Source module: ProofsInTheBook.ZinanFFCT56 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.ZinanFFCT3
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT21
open ProofsInTheBook.ZinanFFCT45 ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT55
namespace ProofsInTheBook.ZinanFFCT56
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT56
-- §A the coefficient bricks
-- §B the master mid-fold kill
-- §C the WBS axis-edge elimination
-- §D the honest dispatch + residue
-- §E the consequence wiring
-- §F non-vacuity guards
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT48
import ProofsInTheBook.ZinanFFCT56
-/
/- Source module: ProofsInTheBook.ZinanFFCT57 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalSpliceTransport
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT19
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT47
open ProofsInTheBook.ZinanFFCT48
open ProofsInTheBook.ZinanFFCT56
namespace ProofsInTheBook.ZinanFFCT57
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT57
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT10
import ProofsInTheBook.SphericalSpliceTransport
import ProofsInTheBook.ZinanFFCT48
import ProofsInTheBook.ZinanFFCT57
-/
/- Source module: ProofsInTheBook.ZinanFFCT58 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalSpliceTransport
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.ZinanFFCT10
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT19
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT47
open ProofsInTheBook.ZinanFFCT48
open ProofsInTheBook.ZinanFFCT57
namespace ProofsInTheBook.ZinanFFCT58
set_option maxHeartbeats 1600000
set_option linter.unnecessarySeqFocus false
end ProofsInTheBook.ZinanFFCT58
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT58
-/
/- Source module: ProofsInTheBook.ZinanFFCT59 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT19
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT47
open ProofsInTheBook.ZinanFFCT48
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT56
open ProofsInTheBook.ZinanFFCT57
open ProofsInTheBook.ZinanFFCT58
namespace ProofsInTheBook.ZinanFFCT59
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT59
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT54
import ProofsInTheBook.ZinanFFCT59
-/
/- Source module: ProofsInTheBook.ZinanFFCT60 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT21
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT53
open ProofsInTheBook.ZinanFFCT54
open ProofsInTheBook.ZinanFFCT56
open ProofsInTheBook.ZinanFFCT59
namespace ProofsInTheBook.ZinanFFCT60
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT60
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT60
import ProofsInTheBook.SphericalRotation
-/
/- Source module: ProofsInTheBook.ZinanFFCT61 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.ZinanFFCT12
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT21
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT53
open ProofsInTheBook.ZinanFFCT54
open ProofsInTheBook.ZinanFFCT56
open ProofsInTheBook.ZinanFFCT59
open ProofsInTheBook.ZinanFFCT60
namespace ProofsInTheBook.ZinanFFCT61
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT61
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT61
-/
/- Source module: ProofsInTheBook.ZinanFFCT62 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT19
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT48
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT53
open ProofsInTheBook.ZinanFFCT54
open ProofsInTheBook.ZinanFFCT56
open ProofsInTheBook.ZinanFFCT57
open ProofsInTheBook.ZinanFFCT58
open ProofsInTheBook.ZinanFFCT59
open ProofsInTheBook.ZinanFFCT61
namespace ProofsInTheBook.ZinanFFCT62
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT62
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT62
-/
/- Source module: ProofsInTheBook.ZinanFFCT64 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT47
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT53
open ProofsInTheBook.ZinanFFCT56
open ProofsInTheBook.ZinanFFCT57
open ProofsInTheBook.ZinanFFCT61
open ProofsInTheBook.ZinanFFCT62
namespace ProofsInTheBook.ZinanFFCT64
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT64
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT63
import ProofsInTheBook.ZinanFFCT64
-/
/- Source module: ProofsInTheBook.ZinanFFCT65 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalCore ProofsInTheBook.SphericalFinish
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT19
open ProofsInTheBook.ZinanFFCT12
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT47
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT53
open ProofsInTheBook.ZinanFFCT54
open ProofsInTheBook.ZinanFFCT56
open ProofsInTheBook.ZinanFFCT59
open ProofsInTheBook.ZinanFFCT61
open ProofsInTheBook.ZinanFFCT62
open ProofsInTheBook.ZinanFFCT63
open ProofsInTheBook.ZinanFFCT64
namespace ProofsInTheBook.ZinanFFCT65
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT65
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT65
import ProofsInTheBook.PlanarConvexDiag
-/
/- Source module: ProofsInTheBook.ZinanFFCT66 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.ZinanFFCT3
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT21
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT24
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT53
open ProofsInTheBook.ZinanFFCT54
open ProofsInTheBook.ZinanFFCT63
open ProofsInTheBook.ZinanFFCT64
open ProofsInTheBook.ZinanFFCT65
namespace ProofsInTheBook.ZinanFFCT66
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT66
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT66
-/
/- Source module: ProofsInTheBook.ZinanFFCT67 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.ZinanFFCT64
open ProofsInTheBook.ZinanFFCT65
open ProofsInTheBook.ZinanFFCT66
namespace ProofsInTheBook.ZinanFFCT67
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT67
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT67
import ProofsInTheBook.ZinanFFCT26
-/
/- Source module: ProofsInTheBook.ZinanFFCT68 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT19
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT26
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT53
open ProofsInTheBook.ZinanFFCT54
open ProofsInTheBook.ZinanFFCT64
open ProofsInTheBook.ZinanFFCT65
open ProofsInTheBook.ZinanFFCT66
open ProofsInTheBook.ZinanFFCT67
namespace ProofsInTheBook.ZinanFFCT68
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT68
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT68
-/
/- Source module: ProofsInTheBook.ZinanFFCT69 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.ZinanFFCT12
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT47
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT61
open ProofsInTheBook.ZinanFFCT62
open ProofsInTheBook.ZinanFFCT64
open ProofsInTheBook.ZinanFFCT65
open ProofsInTheBook.ZinanFFCT67
open ProofsInTheBook.ZinanFFCT68
namespace ProofsInTheBook.ZinanFFCT69
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT69
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT69
import ProofsInTheBook.ZinanFFCT32
-/
/- Source module: ProofsInTheBook.ZinanFFCT70 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.ZinanFFCT3
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT31
open ProofsInTheBook.ZinanFFCT32
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT64
open ProofsInTheBook.ZinanFFCT65
open ProofsInTheBook.ZinanFFCT68
open ProofsInTheBook.ZinanFFCT69
namespace ProofsInTheBook.ZinanFFCT70
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT70
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT70
-/
/- Source module: ProofsInTheBook.ZinanFFCT71 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT61
open ProofsInTheBook.ZinanFFCT64
open ProofsInTheBook.ZinanFFCT66
open ProofsInTheBook.ZinanFFCT68
open ProofsInTheBook.ZinanFFCT69
open ProofsInTheBook.ZinanFFCT70
namespace ProofsInTheBook.ZinanFFCT71
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT71
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT71
-/
/- Source module: ProofsInTheBook.ZinanFFCT72 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT19
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT24
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT47
open ProofsInTheBook.ZinanFFCT48
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT56
open ProofsInTheBook.ZinanFFCT57
open ProofsInTheBook.ZinanFFCT58
open ProofsInTheBook.ZinanFFCT61
open ProofsInTheBook.ZinanFFCT64
open ProofsInTheBook.ZinanFFCT65
open ProofsInTheBook.ZinanFFCT66
open ProofsInTheBook.ZinanFFCT68
open ProofsInTheBook.ZinanFFCT69
open ProofsInTheBook.ZinanFFCT70
open ProofsInTheBook.ZinanFFCT71
namespace ProofsInTheBook.ZinanFFCT72
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT72
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT72
-/
/- Source module: ProofsInTheBook.ZinanFFCT73 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT19
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT24
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT47
open ProofsInTheBook.ZinanFFCT48
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT53
open ProofsInTheBook.ZinanFFCT54
open ProofsInTheBook.ZinanFFCT56
open ProofsInTheBook.ZinanFFCT57
open ProofsInTheBook.ZinanFFCT58
open ProofsInTheBook.ZinanFFCT61
open ProofsInTheBook.ZinanFFCT64
open ProofsInTheBook.ZinanFFCT65
open ProofsInTheBook.ZinanFFCT66
open ProofsInTheBook.ZinanFFCT68
open ProofsInTheBook.ZinanFFCT69
open ProofsInTheBook.ZinanFFCT70
open ProofsInTheBook.ZinanFFCT71
open ProofsInTheBook.ZinanFFCT72
namespace ProofsInTheBook.ZinanFFCT73
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT73
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT73
-/
/- Source module: ProofsInTheBook.ZinanFFCT74 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT19
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT47
open ProofsInTheBook.ZinanFFCT48
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT53
open ProofsInTheBook.ZinanFFCT54
open ProofsInTheBook.ZinanFFCT56
open ProofsInTheBook.ZinanFFCT57
open ProofsInTheBook.ZinanFFCT58
open ProofsInTheBook.ZinanFFCT61
open ProofsInTheBook.ZinanFFCT64
open ProofsInTheBook.ZinanFFCT65
open ProofsInTheBook.ZinanFFCT66
open ProofsInTheBook.ZinanFFCT68
open ProofsInTheBook.ZinanFFCT69
open ProofsInTheBook.ZinanFFCT70
open ProofsInTheBook.ZinanFFCT71
open ProofsInTheBook.ZinanFFCT73
namespace ProofsInTheBook.ZinanFFCT74
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT74
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT74
-/
/- Source module: ProofsInTheBook.ZinanFFCT75 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT70
open ProofsInTheBook.ZinanFFCT71
open ProofsInTheBook.ZinanFFCT74
namespace ProofsInTheBook.ZinanFFCT75
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT75
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT75
import ProofsInTheBook.ZinanFFCT44
-/
/- Source module: ProofsInTheBook.ZinanFFCT76 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.ZinanFFCT3
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT21
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT24
open ProofsInTheBook.ZinanFFCT44
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT56
open ProofsInTheBook.ZinanFFCT64
open ProofsInTheBook.ZinanFFCT70
open ProofsInTheBook.ZinanFFCT71
open ProofsInTheBook.ZinanFFCT74
open ProofsInTheBook.ZinanFFCT75
namespace ProofsInTheBook.ZinanFFCT76
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT76
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT76
-/
/- Source module: ProofsInTheBook.ZinanFFCT77 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT19
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT47
open ProofsInTheBook.ZinanFFCT48
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT56
open ProofsInTheBook.ZinanFFCT57
open ProofsInTheBook.ZinanFFCT58
open ProofsInTheBook.ZinanFFCT61
open ProofsInTheBook.ZinanFFCT64
open ProofsInTheBook.ZinanFFCT65
open ProofsInTheBook.ZinanFFCT66
open ProofsInTheBook.ZinanFFCT68
open ProofsInTheBook.ZinanFFCT69
open ProofsInTheBook.ZinanFFCT70
open ProofsInTheBook.ZinanFFCT71
open ProofsInTheBook.ZinanFFCT74
open ProofsInTheBook.ZinanFFCT76
namespace ProofsInTheBook.ZinanFFCT77
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT77
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT77
-/
/- Source module: ProofsInTheBook.ZinanFFCT78 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT19
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT47
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT56
open ProofsInTheBook.ZinanFFCT61
open ProofsInTheBook.ZinanFFCT64
open ProofsInTheBook.ZinanFFCT68
open ProofsInTheBook.ZinanFFCT70
open ProofsInTheBook.ZinanFFCT71
open ProofsInTheBook.ZinanFFCT74
open ProofsInTheBook.ZinanFFCT75
open ProofsInTheBook.ZinanFFCT76
open ProofsInTheBook.ZinanFFCT77
namespace ProofsInTheBook.ZinanFFCT78
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT78
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT78
-/
/- Source module: ProofsInTheBook.ZinanFFCT79 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT68
open ProofsInTheBook.ZinanFFCT76
open ProofsInTheBook.ZinanFFCT77
open ProofsInTheBook.ZinanFFCT78
namespace ProofsInTheBook.ZinanFFCT79
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT79
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT79
-/
/- Source module: ProofsInTheBook.ZinanFFCT80 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.ZinanFFCT12
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT47
open ProofsInTheBook.ZinanFFCT48
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT68
open ProofsInTheBook.ZinanFFCT75
open ProofsInTheBook.ZinanFFCT76
open ProofsInTheBook.ZinanFFCT77
open ProofsInTheBook.ZinanFFCT78
open ProofsInTheBook.ZinanFFCT79
namespace ProofsInTheBook.ZinanFFCT80
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT80
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT80
-/
/- Source module: ProofsInTheBook.ZinanFFCT81 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.ZinanFFCT12
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT56
open ProofsInTheBook.ZinanFFCT64
open ProofsInTheBook.ZinanFFCT65
open ProofsInTheBook.ZinanFFCT68
open ProofsInTheBook.ZinanFFCT70
open ProofsInTheBook.ZinanFFCT74
open ProofsInTheBook.ZinanFFCT76
open ProofsInTheBook.ZinanFFCT77
open ProofsInTheBook.ZinanFFCT78
open ProofsInTheBook.ZinanFFCT79
open ProofsInTheBook.ZinanFFCT80
namespace ProofsInTheBook.ZinanFFCT81
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT81
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT81
-/
/- Source module: ProofsInTheBook.ZinanFFCT82 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.ZinanFFCT12
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT56
open ProofsInTheBook.ZinanFFCT64
open ProofsInTheBook.ZinanFFCT65
open ProofsInTheBook.ZinanFFCT68
open ProofsInTheBook.ZinanFFCT70
open ProofsInTheBook.ZinanFFCT74
open ProofsInTheBook.ZinanFFCT76
open ProofsInTheBook.ZinanFFCT77
open ProofsInTheBook.ZinanFFCT78
open ProofsInTheBook.ZinanFFCT79
open ProofsInTheBook.ZinanFFCT80
open ProofsInTheBook.ZinanFFCT81
namespace ProofsInTheBook.ZinanFFCT82
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT82
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT82
-/
/- Source module: ProofsInTheBook.ZinanFFCT83 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.ZinanFFCT12
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT22
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT24
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT56
open ProofsInTheBook.ZinanFFCT64
open ProofsInTheBook.ZinanFFCT65
open ProofsInTheBook.ZinanFFCT68
open ProofsInTheBook.ZinanFFCT70
open ProofsInTheBook.ZinanFFCT74
open ProofsInTheBook.ZinanFFCT76
open ProofsInTheBook.ZinanFFCT77
open ProofsInTheBook.ZinanFFCT78
open ProofsInTheBook.ZinanFFCT79
open ProofsInTheBook.ZinanFFCT80
open ProofsInTheBook.ZinanFFCT81
open ProofsInTheBook.ZinanFFCT82
namespace ProofsInTheBook.ZinanFFCT83
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT83
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT83
-/
/- Source module: ProofsInTheBook.ZinanFFCT84 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.ZinanFFCT12
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT22
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT24
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT56
open ProofsInTheBook.ZinanFFCT64
open ProofsInTheBook.ZinanFFCT65
open ProofsInTheBook.ZinanFFCT68
open ProofsInTheBook.ZinanFFCT70
open ProofsInTheBook.ZinanFFCT74
open ProofsInTheBook.ZinanFFCT76
open ProofsInTheBook.ZinanFFCT77
open ProofsInTheBook.ZinanFFCT78
open ProofsInTheBook.ZinanFFCT79
open ProofsInTheBook.ZinanFFCT80
open ProofsInTheBook.ZinanFFCT81
open ProofsInTheBook.ZinanFFCT82
open ProofsInTheBook.ZinanFFCT83
namespace ProofsInTheBook.ZinanFFCT84
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT84
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT84
-/
/- Source module: ProofsInTheBook.ZinanFFCT85 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.ZinanFFCT12
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT22
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT24
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT47
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT56
open ProofsInTheBook.ZinanFFCT61
open ProofsInTheBook.ZinanFFCT64
open ProofsInTheBook.ZinanFFCT65
open ProofsInTheBook.ZinanFFCT68
open ProofsInTheBook.ZinanFFCT69
open ProofsInTheBook.ZinanFFCT70
open ProofsInTheBook.ZinanFFCT71
open ProofsInTheBook.ZinanFFCT74
open ProofsInTheBook.ZinanFFCT75
open ProofsInTheBook.ZinanFFCT76
open ProofsInTheBook.ZinanFFCT77
open ProofsInTheBook.ZinanFFCT78
open ProofsInTheBook.ZinanFFCT79
open ProofsInTheBook.ZinanFFCT80
open ProofsInTheBook.ZinanFFCT81
open ProofsInTheBook.ZinanFFCT82
open ProofsInTheBook.ZinanFFCT83
open ProofsInTheBook.ZinanFFCT84
namespace ProofsInTheBook.ZinanFFCT85
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1800000
end ProofsInTheBook.ZinanFFCT85
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT85
-/
/- Source module: ProofsInTheBook.ZinanFFCT86 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.ZinanFFCT12
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT22
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT24
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT47
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT56
open ProofsInTheBook.ZinanFFCT61
open ProofsInTheBook.ZinanFFCT64
open ProofsInTheBook.ZinanFFCT65
open ProofsInTheBook.ZinanFFCT68
open ProofsInTheBook.ZinanFFCT69
open ProofsInTheBook.ZinanFFCT70
open ProofsInTheBook.ZinanFFCT71
open ProofsInTheBook.ZinanFFCT74
open ProofsInTheBook.ZinanFFCT75
open ProofsInTheBook.ZinanFFCT76
open ProofsInTheBook.ZinanFFCT77
open ProofsInTheBook.ZinanFFCT78
open ProofsInTheBook.ZinanFFCT79
open ProofsInTheBook.ZinanFFCT80
open ProofsInTheBook.ZinanFFCT81
open ProofsInTheBook.ZinanFFCT82
open ProofsInTheBook.ZinanFFCT83
open ProofsInTheBook.ZinanFFCT84
open ProofsInTheBook.ZinanFFCT85
namespace ProofsInTheBook.ZinanFFCT86
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1800000
end ProofsInTheBook.ZinanFFCT86
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT86
-/
/- Source module: ProofsInTheBook.ZinanFFCT100 -/
section
set_option autoImplicit true
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT78
open ProofsInTheBook.ZinanFFCT86
namespace ProofsInTheBook.ZinanFFCT100
end ProofsInTheBook.ZinanFFCT100
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT100
-/
/- Source module: ProofsInTheBook.ZinanFFCT111 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation
open ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZStepClose
open ProofsInTheBook.SphericalSZFinal
open ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalArmAssembly
open ProofsInTheBook.SphericalMonitoredSup
open ProofsInTheBook.SphericalOpeningOutcome
open ProofsInTheBook.SphericalReachStuck
open ProofsInTheBook.SphericalStuckGeneral
open ProofsInTheBook.SphericalCutTransport
open ProofsInTheBook.PlanarConvexDiag
open ProofsInTheBook.SphericalCyclicTriple
open ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalDiagCut
open ProofsInTheBook.SphericalGnomonic
open ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.ZinanFFCT3
open ProofsInTheBook.ZinanFFCT20
open ProofsInTheBook.ZinanFFCT37
open ProofsInTheBook.ZinanFFCT12
open ProofsInTheBook.ZinanFFCT18
open ProofsInTheBook.ZinanFFCT22
open ProofsInTheBook.ZinanFFCT23
open ProofsInTheBook.ZinanFFCT24
open ProofsInTheBook.ZinanFFCT25
open ProofsInTheBook.ZinanFFCT45
open ProofsInTheBook.ZinanFFCT46
open ProofsInTheBook.ZinanFFCT47
open ProofsInTheBook.ZinanFFCT49
open ProofsInTheBook.ZinanFFCT52
open ProofsInTheBook.ZinanFFCT56
open ProofsInTheBook.ZinanFFCT61
open ProofsInTheBook.ZinanFFCT64
open ProofsInTheBook.ZinanFFCT65
open ProofsInTheBook.ZinanFFCT66
open ProofsInTheBook.ZinanFFCT68
open ProofsInTheBook.ZinanFFCT69
open ProofsInTheBook.ZinanFFCT70
open ProofsInTheBook.ZinanFFCT71
open ProofsInTheBook.ZinanFFCT74
open ProofsInTheBook.ZinanFFCT75
open ProofsInTheBook.ZinanFFCT76
open ProofsInTheBook.ZinanFFCT77
open ProofsInTheBook.ZinanFFCT78
open ProofsInTheBook.ZinanFFCT79
open ProofsInTheBook.ZinanFFCT80
open ProofsInTheBook.ZinanFFCT81
open ProofsInTheBook.ZinanFFCT82
open ProofsInTheBook.ZinanFFCT83
open ProofsInTheBook.ZinanFFCT84
open ProofsInTheBook.ZinanFFCT85
open ProofsInTheBook.ZinanFFCT86
open ProofsInTheBook.ZinanFFCT100
namespace ProofsInTheBook.ZinanFFCT111
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1800000
end ProofsInTheBook.ZinanFFCT111
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalReachStuck
import ProofsInTheBook.SphericalSZFinal
import ProofsInTheBook.SphericalSZClose
import ProofsInTheBook.ZinanFFCT111
-/
/- Source module: ProofsInTheBook.ZinanFFCT113 -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalRotation ProofsInTheBook.SphericalCore
open ProofsInTheBook.SphericalHinge ProofsInTheBook.SphericalHingeCut
open ProofsInTheBook.SphericalFinish ProofsInTheBook.SphericalSZStep
open ProofsInTheBook.SphericalReachStuck ProofsInTheBook.SphericalSZInduction
open ProofsInTheBook.SphericalSZFinal ProofsInTheBook.SphericalSZClose
open ProofsInTheBook.SphericalMonitoredSup ProofsInTheBook.SphericalOpeningProcess
open ProofsInTheBook.ZinanFFCT78 ProofsInTheBook.ZinanFFCT111
namespace ProofsInTheBook.ZinanFFCT113
set_option maxHeartbeats 1600000
end ProofsInTheBook.ZinanFFCT113
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalOpenedArmCore
import ProofsInTheBook.ZinanFFCT111
import ProofsInTheBook.ZinanFFCT113
-/
/- Source module: ProofsInTheBook.ZinanFFCT112 -/
section
set_option autoImplicit true
namespace ProofsInTheBook.ZinanFFCT112
open ProofsInTheBook.SphericalKernel
open ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalOpenedArmCore
open ProofsInTheBook.SphericalOpeningProcess (StuckWitnessExists)
end ProofsInTheBook.ZinanFFCT112
end
/- Original source header (imports hoisted):
import Mathlib
import ProofsInTheBook.ZinanFFCT112
-/
/- Source module: ProofsInTheBook.Chapter13 -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Chapter13
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open EdgeSign
namespace StrictTriangleSigns
end StrictTriangleSigns
namespace CauchyArmOpeningObstruction
end CauchyArmOpeningObstruction
namespace CauchyArmClosingObstruction
end CauchyArmClosingObstruction
namespace CauchyArmFixedChordObstruction
end CauchyArmFixedChordObstruction
namespace CauchyArmVertex
end CauchyArmVertex
namespace CauchyRigidityCertificate
end CauchyRigidityCertificate
end ProofsInTheBook.Chapter13
end
/- Original source header (imports hoisted):
import Mathlib
import ProofsInTheBook.Chapter13
-/
/- Source module: ProofsInTheBook.Ch13CyclicSigns -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Ch13CyclicSigns
open ProofsInTheBook.Chapter13
open EdgeSign
end ProofsInTheBook.Ch13CyclicSigns
end
/- Original source header (imports hoisted):
import Mathlib
import ProofsInTheBook.PlanarMap
import ProofsInTheBook.Chapter13
import ProofsInTheBook.Ch13CyclicSigns
-/
/- Source module: ProofsInTheBook.Ch13MarkedSphere -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Ch13MarkedSphere
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.Chapter13
open ProofsInTheBook.Ch13CyclicSigns
end ProofsInTheBook.Ch13MarkedSphere
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMap
-/
/- Source module: ProofsInTheBook.PlanarMapEuler -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.PlanarMap
end ProofsInTheBook.PlanarMap.CombMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapEuler
-/
/- Source module: ProofsInTheBook.PlanarMapSimple -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapEuler
-/
/- Source module: ProofsInTheBook.PlanarMapDelete -/
section
set_option autoImplicit true
namespace Equiv.Perm
open Equiv
namespace DeleteSet
end DeleteSet
open DeleteSet
end Equiv.Perm
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
section TwoEdgePathObstruction
end TwoEdgePathObstruction
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapSimple
-/
/- Source module: ProofsInTheBook.PlanarMapBoundary -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace BoundaryPath
end BoundaryPath
namespace BoundaryCycle
namespace Chord
end Chord
end BoundaryCycle
namespace BoundaryArcSplit
end BoundaryArcSplit
namespace BoundaryCycle
end BoundaryCycle
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapBoundary
-/
/- Source module: ProofsInTheBook.PlanarMapNearTriangulation -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace BoundaryCycle
end BoundaryCycle
namespace NearTriangulation
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapNearTriangulation
import ProofsInTheBook.PlanarMapDelete
-/
/- Source module: ProofsInTheBook.PlanarMapFilteredRotation -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace FilteredRotation
namespace ContiguousInterval
end ContiguousInterval
section FreshDart
end FreshDart
end FilteredRotation
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapFilteredRotation
-/
/- Source module: ProofsInTheBook.PlanarMapChordSplitData -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
section ChordDarts
end ChordDarts
namespace ChordSplitData
end ChordSplitData
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapChordSplitData
-/
/- Source module: ProofsInTheBook.PlanarMapChordSplit -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace BoundaryPath
end BoundaryPath
namespace NearTriangulation
namespace ChordSplitData
end ChordSplitData
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapChordSplit
-/
/- Source module: ProofsInTheBook.PlanarMapSeparation -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
namespace ChordSplitData
end ChordSplitData
namespace ChordSplitData
end ChordSplitData
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapNearTriangulation
-/
/- Source module: ProofsInTheBook.PlanarMapBoundaryFan -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
namespace FanTriangle
end FanTriangle
namespace BoundaryVertexFan
end BoundaryVertexFan
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapBoundaryFan
import ProofsInTheBook.PlanarMapDelete
-/
/- Source module: ProofsInTheBook.PlanarMapBoundaryDelete -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
namespace BoundaryDeletionData
end BoundaryDeletionData
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapBoundaryDelete
-/
/- Source module: ProofsInTheBook.PlanarMapFanSurgery -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
namespace NeighborRotationOrder
end NeighborRotationOrder
namespace FanSurgeryReconstruction
end FanSurgeryReconstruction
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
/-
List-coloring primitives (Chapter 35 layer 4).
Design-independent groundwork for the Thomassen five-list-coloring route
(HANDOFF/CH35_DESIGN_ANSWER.md): proper colorings from lists, monotonicity
in the graph and in the lists, and the piecewise gluing lemmas — including
the rooted cut-vertex glue, which is the form that is actually true for
list colorings (naive gluing fails because the two sides may disagree at
the cut vertex).
-/
import Mathlib
-/
/- Source module: ProofsInTheBook.ListColoring -/
section
set_option autoImplicit true
namespace ProofsInTheBook.ListColoring
section Glue
end Glue
end ProofsInTheBook.ListColoring
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapSeparation
import ProofsInTheBook.PlanarMapFanSurgery
import ProofsInTheBook.ListColoring
-/
/- Source module: ProofsInTheBook.ThomassenLists -/
section
set_option autoImplicit true
namespace ProofsInTheBook.ThomassenLists
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.ListColoring
namespace CombMap
open ProofsInTheBook.PlanarMap.CombMap
namespace ThomassenLists
end ThomassenLists
namespace ChordSplitRegions
end ChordSplitRegions
section Deletion
end Deletion
end CombMap
end ProofsInTheBook.ThomassenLists
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapFanSurgery
-/
/- Source module: ProofsInTheBook.PlanarMapFanConnectivity -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
section Reduction
end Reduction
namespace NearTriangulation
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapFanConnectivity
import ProofsInTheBook.PlanarMapFilteredRotation
-/
/- Source module: ProofsInTheBook.PlanarMapFanFaces -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapFanFaces
-/
/- Source module: ProofsInTheBook.PlanarMapFanMergedOrbit -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapBoundary
-/
/- Source module: ProofsInTheBook.PlanarMapBoundaryArcSplit -/
section
set_option autoImplicit true
set_option maxHeartbeats 1600000
set_option linter.unusedVariables false
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace BoundaryCycleData
end BoundaryCycleData
namespace DataDartArc
end DataDartArc
namespace BoundaryCycleData
end BoundaryCycleData
section Casts
end Casts
namespace BoundaryPath
end BoundaryPath
section BPOfDartArc
end BPOfDartArc
namespace BoundaryCycleData
end BoundaryCycleData
namespace BoundaryCycleData
end BoundaryCycleData
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapFanFaces
import ProofsInTheBook.PlanarMapBoundaryArcSplit
-/
/- Source module: ProofsInTheBook.PlanarMapDeletedBoundary -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
namespace DeletedMergedBoundaryCertificate
end DeletedMergedBoundaryCertificate
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapFanMergedOrbit
import ProofsInTheBook.PlanarMapDeletedBoundary
-/
/- Source module: ProofsInTheBook.PlanarMapOuterArc -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
namespace MergedOuterArcData
end MergedOuterArcData
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapOuterArc
-/
/- Source module: ProofsInTheBook.PlanarMapFanExistence -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ThomassenLists
import ProofsInTheBook.PlanarMapFanExistence
-/
/- Source module: ProofsInTheBook.ThomassenInduction -/
section
set_option autoImplicit true
namespace ProofsInTheBook.ThomassenInduction
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.ListColoring
open ProofsInTheBook.ThomassenLists
open ProofsInTheBook.ThomassenLists.CombMap
universe u
section Base
end Base
section Chord
end Chord
section Chordless
end Chordless
section Induction
end Induction
section Corollaries
end Corollaries
section FiveColor
end FiveColor
end ProofsInTheBook.ThomassenInduction
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ThomassenInduction
import ProofsInTheBook.PlanarMapChordSplit
import ProofsInTheBook.PlanarMapSeparation
-/
/- Source module: ProofsInTheBook.ChordSplitNT -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
namespace ProofsInTheBook.ChordSplitNT
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.ListColoring
open ProofsInTheBook.ThomassenLists
open ProofsInTheBook.ThomassenLists.CombMap
open ProofsInTheBook.ThomassenInduction
universe u
attribute [instance] ChordSideReconstruction.fintypeDₛ ChordSideReconstruction.decEqDₛ
namespace ChordSideReconstruction
end ChordSideReconstruction
namespace ChordRecursionData
end ChordRecursionData
end ProofsInTheBook.ChordSplitNT
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ChordSplitNT
-/
/- Source module: ProofsInTheBook.ChordSplitEuler -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
namespace ProofsInTheBook.ChordSplitEuler
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.PlanarMap.FilteredRotation
universe u
section VertexCount
end VertexCount
section EulerReduction
end EulerReduction
section ChordApplication
end ChordApplication
section NonVacuity
end NonVacuity
end ProofsInTheBook.ChordSplitEuler
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ChordSplitEuler
-/
/- Source module: ProofsInTheBook.ChordSideRecon -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
namespace ProofsInTheBook.ChordSideRecon
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.PlanarMap.FilteredRotation
open ProofsInTheBook.ChordSplitEuler
universe u
section Connectivity
end Connectivity
section SphereAssembly
end SphereAssembly
section ChordApplication
end ChordApplication
section JordanData
end JordanData
section NonVacuity
end NonVacuity
end ProofsInTheBook.ChordSideRecon
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapFilteredRotation
import ProofsInTheBook.PlanarMapSeparation
-/
/- Source module: ProofsInTheBook.PlanarMapCutCap -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace SimplePrimalCycle
end SimplePrimalCycle
namespace SimplePrimalCycle
-- c_i^- ↦ α (dart i)
end SimplePrimalCycle
namespace CutCapSurgery
end CutCapSurgery
namespace NearTriangulation
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapCutCap
-/
/- Source module: ProofsInTheBook.PlanarMapCutCapSigma -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace SimplePrimalCycle
-- c_i^- ↦ p_i
-- c_i^- ↦ ℓ_i^- = σ⁻¹ q_i
end SimplePrimalCycle
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import Mathlib
-/
/- Source module: ProofsInTheBook.PermTranspositionCycleCount -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
set_option linter.unusedSimpArgs false
set_option linter.unnecessarySimpa false
set_option linter.unusedVariables false
open Equiv Equiv.Perm Function
namespace PermTranspositionCycleCount
open scoped Finset
end PermTranspositionCycleCount
end
/- Original source header (imports hoisted):
import Mathlib
-/
/- Source module: ProofsInTheBook.RelationComponentCount -/
section
set_option autoImplicit true
open Classical
universe u
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapEuler
import ProofsInTheBook.PermTranspositionCycleCount
import ProofsInTheBook.RelationComponentCount
-/
/- Source module: ProofsInTheBook.PlanarMapEulerInequality -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapCutCapSigma
import ProofsInTheBook.PlanarMapEulerInequality
-/
/- Source module: ProofsInTheBook.PlanarMapCutCapCounts -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
set_option maxHeartbeats 1600000
namespace ProofsInTheBook.PlanarMap
open Equiv Equiv.Perm Function
namespace CombMap
namespace CutCapCount
section SumCongr
end SumCongr
end CutCapCount
namespace SimplePrimalCycle
open CutCapCount
end SimplePrimalCycle
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapCutCapCounts
-/
/- Source module: ProofsInTheBook.PlanarMapCutCapV -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
set_option maxHeartbeats 1600000
namespace ProofsInTheBook.PlanarMap
open Equiv Equiv.Perm Function
namespace CombMap
namespace SimplePrimalCycle
open CutCapCount
end SimplePrimalCycle
namespace CutCapCount
end CutCapCount
namespace SimplePrimalCycle
open CutCapCount
end SimplePrimalCycle
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapCutCapV
-/
/- Source module: ProofsInTheBook.PlanarMapCutCapF -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
set_option maxHeartbeats 1600000
namespace ProofsInTheBook.PlanarMap
open Equiv Equiv.Perm Function
namespace CombMap
namespace CutCapCount
end CutCapCount
namespace SimplePrimalCycle
open CutCapCount
end SimplePrimalCycle
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ChordSideRecon
import ProofsInTheBook.PlanarMapCutCapCounts
import ProofsInTheBook.PlanarMapCutCapF
-/
/- Source module: ProofsInTheBook.ChordFaceCount -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
set_option linter.unusedVariables false
namespace ProofsInTheBook.ChordFaceCount
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.PlanarMap.FilteredRotation
open ProofsInTheBook.ChordSplitEuler
open ProofsInTheBook.ChordSideRecon
open ProofsInTheBook.PlanarMap.CombMap.CutCapCount
universe u
section FacePerm
end FacePerm
section FaceBijection
end FaceBijection
section Dichotomy
end Dichotomy
section Genus0
end Genus0
section SphereAssembly
end SphereAssembly
section NonVacuity
end NonVacuity
section ChordApplication
end ChordApplication
section Headline
end Headline
end ProofsInTheBook.ChordFaceCount
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ChordFaceCount
import ProofsInTheBook.PlanarMapEulerInequality
-/
/- Source module: ProofsInTheBook.ChordDisk -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
set_option linter.unusedVariables false
namespace ProofsInTheBook.ChordDisk
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.PlanarMap.FilteredRotation
open ProofsInTheBook.ChordSplitEuler
open ProofsInTheBook.ChordSideRecon
open ProofsInTheBook.ChordFaceCount
universe u
section Facts
end Facts
section LowerHalf
end LowerHalf
section Threading
end Threading
section ChordApplication
end ChordApplication
section NonVacuity
end NonVacuity
section Headline
end Headline
end ProofsInTheBook.ChordDisk
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ChordDisk
-/
/- Source module: ProofsInTheBook.SubmapPlanar -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
set_option linter.unusedVariables false
namespace ProofsInTheBook.SubmapPlanar
open Equiv
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
universe u
section OrbitSplit
open scoped Classical
end OrbitSplit
section RawRestrict
open scoped Classical
open scoped Classical
end RawRestrict
section ChordThreading
open ProofsInTheBook.PlanarMap.CombMap.NearTriangulation
open ProofsInTheBook.ChordSideRecon
end ChordThreading
end ProofsInTheBook.SubmapPlanar
end
/- Original source header (imports hoisted):
import Mathlib
import ProofsInTheBook.PlanarMap
import ProofsInTheBook.Chapter13
import ProofsInTheBook.Ch13CyclicSigns
import ProofsInTheBook.Ch13MarkedSphere
-/
/- Source module: ProofsInTheBook.Ch13MarkedReduction -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Ch13MarkedReduction
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.Chapter13
open ProofsInTheBook.Ch13CyclicSigns
open ProofsInTheBook.Ch13MarkedSphere
open EdgeSign
open Equiv Equiv.Perm
section ListBridge
end ListBridge
section OrbitBridge
end OrbitBridge
section StrictBridge
end StrictBridge
section ActiveComponent
end ActiveComponent
section Obstruction
end Obstruction
end ProofsInTheBook.Ch13MarkedReduction
end
/- Original source header (imports hoisted):
import Mathlib
import ProofsInTheBook.PlanarMap
import ProofsInTheBook.PlanarMapSimple
import ProofsInTheBook.PlanarMapDelete
import ProofsInTheBook.SubmapPlanar
import ProofsInTheBook.Chapter13
import ProofsInTheBook.Ch13CyclicSigns
import ProofsInTheBook.Ch13MarkedSphere
import ProofsInTheBook.Ch13MarkedReduction
-/
/- Source module: ProofsInTheBook.Ch13ActiveComponent -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Ch13ActiveComponent
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.Chapter13
open ProofsInTheBook.Ch13CyclicSigns
open ProofsInTheBook.Ch13MarkedSphere
open ProofsInTheBook.Ch13MarkedReduction
open EdgeSign
open ProofsInTheBook.SubmapPlanar
-- unreachable on active darts
end ProofsInTheBook.Ch13ActiveComponent
end
/- Original source header (imports hoisted):
import Mathlib
import ProofsInTheBook.PlanarMap
import ProofsInTheBook.PlanarMapSimple
import ProofsInTheBook.PlanarMapDelete
import ProofsInTheBook.SubmapPlanar
import ProofsInTheBook.Chapter13
import ProofsInTheBook.Ch13CyclicSigns
import ProofsInTheBook.Ch13MarkedSphere
import ProofsInTheBook.Ch13MarkedReduction
import ProofsInTheBook.Ch13ActiveComponent
-/
/- Source module: ProofsInTheBook.Ch13FlipTransport -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Ch13FlipTransport
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.Chapter13
open ProofsInTheBook.Ch13CyclicSigns
open ProofsInTheBook.Ch13MarkedSphere
open ProofsInTheBook.Ch13MarkedReduction
open ProofsInTheBook.Ch13ActiveComponent
open ProofsInTheBook.SubmapPlanar
open EdgeSign
open Equiv Equiv.Perm
open ProofsInTheBook -- for DeleteSet.firstOutside via Equiv.Perm namespace
end ProofsInTheBook.Ch13FlipTransport
end
/- Original source header (imports hoisted):
import Mathlib
import ProofsInTheBook.PlanarMap
import ProofsInTheBook.PlanarMapSimple
import ProofsInTheBook.PlanarMapEuler
import ProofsInTheBook.PlanarMapDelete
import ProofsInTheBook.SubmapPlanar
import ProofsInTheBook.Chapter13
import ProofsInTheBook.Ch13CyclicSigns
import ProofsInTheBook.Ch13MarkedSphere
import ProofsInTheBook.Ch13MarkedReduction
import ProofsInTheBook.Ch13ActiveComponent
import ProofsInTheBook.Ch13FlipTransport
import ProofsInTheBook.PlanarMapNearTriangulation
-/
/- Source module: ProofsInTheBook.Ch13ComponentClose -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Ch13ComponentClose
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.Chapter13
open ProofsInTheBook.Ch13CyclicSigns
open ProofsInTheBook.Ch13MarkedSphere
open ProofsInTheBook.Ch13MarkedReduction
open ProofsInTheBook.Ch13ActiveComponent
open ProofsInTheBook.Ch13FlipTransport
open ProofsInTheBook.SubmapPlanar
open EdgeSign
open Equiv Equiv.Perm
end ProofsInTheBook.Ch13ComponentClose
end
/- Original source header (imports hoisted):
import ProofsInTheBook.Ch13MarkedSphere
import ProofsInTheBook.Ch13ComponentClose
import ProofsInTheBook.Chapter13
-/
/- Source module: ProofsInTheBook.Ch13CauchyAssembly -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Ch13CauchyAssembly
open ProofsInTheBook.PlanarMap ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.Ch13MarkedSphere
open ProofsInTheBook.Chapter13
end ProofsInTheBook.Ch13CauchyAssembly
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT112
-/
/- Source module: ProofsInTheBook.Ch13LemmaII -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Ch13LemmaII
open ProofsInTheBook.SphericalKernel
open ProofsInTheBook.ZinanFFCT112
end ProofsInTheBook.Ch13LemmaII
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalDiagCut
-/
/- Source module: ProofsInTheBook.Ch13SubArc -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalHingeCut ProofsInTheBook.SphericalDiagCut
open ProofsInTheBook.SphericalSZChain
namespace ProofsInTheBook.Ch13SubArc
end ProofsInTheBook.Ch13SubArc
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.Chapter13
import ProofsInTheBook.Ch13LemmaII
import ProofsInTheBook.Ch13SubArc
-/
/- Source module: ProofsInTheBook.Ch13ArmVertex -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Ch13ArmVertex
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.Chapter13
open ProofsInTheBook.Ch13LemmaII
open ProofsInTheBook.Ch13SubArc
open scoped Classical
end ProofsInTheBook.Ch13ArmVertex
end
/- Original source header (imports hoisted):
import ProofsInTheBook.Ch13ArmVertex
-/
/- Source module: ProofsInTheBook.Ch13ArmVertexFull -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Ch13ArmVertexFull
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.Chapter13
open ProofsInTheBook.Ch13LemmaII
open ProofsInTheBook.Ch13ArmVertex
open scoped Classical
end ProofsInTheBook.Ch13ArmVertexFull
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalKernel
-/
/- Source module: ProofsInTheBook.Ch13VertexStar -/
section
set_option autoImplicit true
noncomputable section
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
open scoped RealInnerProductSpace
open ProofsInTheBook.TetPearls ProofsInTheBook.TetDihedral
open ProofsInTheBook.SphericalKernel
namespace ProofsInTheBook.Ch13VertexStar
namespace VertexStar
end VertexStar
end ProofsInTheBook.Ch13VertexStar
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.Ch13VertexStar
-/
/- Source module: ProofsInTheBook.Ch13Dihedral -/
section
set_option autoImplicit true
noncomputable section
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
open scoped RealInnerProductSpace
open ProofsInTheBook.TetDihedral
open ProofsInTheBook.SphericalKernel
namespace ProofsInTheBook.Ch13VertexStar
namespace VertexStar
end VertexStar
end ProofsInTheBook.Ch13VertexStar
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.Ch13CauchyAssembly
import ProofsInTheBook.Ch13ArmVertexFull
import ProofsInTheBook.Ch13VertexStar
import ProofsInTheBook.Ch13Dihedral
import ProofsInTheBook.PlanarMapSimple
-/
/- Source module: ProofsInTheBook.Ch13Realization -/
section
set_option autoImplicit true
noncomputable section
open scoped Classical
open ProofsInTheBook.PlanarMap ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.Chapter13 EdgeSign
open ProofsInTheBook.Ch13CyclicSigns
open ProofsInTheBook.Ch13ArmVertex
open ProofsInTheBook.Ch13ArmVertexFull
open ProofsInTheBook.Ch13MarkedSphere
open ProofsInTheBook.Ch13VertexStar
open ProofsInTheBook.SphericalKernel
namespace ProofsInTheBook.Ch13Realization
variable {D : Type*} [Fintype D] [DecidableEq D]
namespace List
end List
/-- Reindexing a strict convex arm by a propositionally-trivial `Fin.cast` preserves it. -/
theorem strictArm_reindex {n m : ℕ} (h : n = m) (A : Fin (m + 1) → S2)
(hA : StrictConvexSphArm A) :
StrictConvexSphArm (fun i : Fin (n + 1) => A (Fin.cast (by rw [h]) i)) := by
subst h
simpa using hA
/-- The `Q`-link reindex is a strict convex arm (from the genuine `(starQ Q).vertexLink_strictArm`). -/
theorem linkQcast_strictArm (M : CombMap D) (starP starQ : M.Vertex → VertexStar)
(hnn : ∀ Q, (starQ Q).n = (starP Q).n) (Q : M.Vertex) :
StrictConvexSphArm (linkQcast M starP starQ hnn Q) :=
strictArm_reindex (hnn Q).symm (starQ Q).vertexLink (starQ Q).vertexLink_strictArm
namespace ConvexPolytopeRealization
variable {M : CombMap D} (R : ConvexPolytopeRealization M)
/-- Shorthand: the `Q`-link (reindexed) at `Q`. -/
@[reducible] def linkQ (Q : M.Vertex) : Fin ((R.starP Q).n + 1) → S2 :=
linkQcast M R.starP R.starQ R.hnn Q
/-- The `Q`-link reindex is a strict convex arm. -/
theorem linkQ_strictArm (Q : M.Vertex) : StrictConvexSphArm (R.linkQ Q) :=
linkQcast_strictArm M R.starP R.starQ R.hnn Q
/-- A dart `d` is in the `σ`-orbit of its vertex's representative `dartRep (tail d)`. -/
theorem sameCycle_dartRep (d : D) : M.σ.SameCycle (R.dartRep (M.tail d)) d := by
have h : Quotient.mk (cycleSetoid M.σ) (R.dartRep (M.tail d)) = Quotient.mk (cycleSetoid M.σ) d := by
rw [show Quotient.mk (cycleSetoid M.σ) (R.dartRep (M.tail d)) = M.tail (R.dartRep (M.tail d)) from rfl,
R.dartRep_tail]
rfl
exact Quotient.exact h
/-- `ActiveVertex` is constant on a `σ`-orbit. -/
theorem activeVertex_congr {d d' : D} (h : M.σ.SameCycle d d') :
ActiveVertex M R.edgeSign d ↔ ActiveVertex M R.edgeSign d' := by
constructor
· rintro ⟨x, hx, hxne⟩; exact ⟨x, h.symm.trans hx, hxne⟩
· rintro ⟨x, hx, hxne⟩; exact ⟨x, h.trans hx, hxne⟩
/-- **The crux bridge (DERIVED, not posited).** At the representative dart of vertex `Q`, the closed-link
cyclic flip count equals the `σ`-cyclic skip-zeros count of the edge signs. Chain:
`signChangesFull = cyclicFlips (nzSigns linkDiff)` (def) `= cyclicFlipCountSkipZeros (real signs)`
(reconciliation) `= cyclicFlipCountSkipZeros (σ-edge-sign list)` (`linkOrder`) `= vertexFlipCountSkipZeros`
(def). -/
theorem signChangesFull_eq_vertexFlip_rep (Q : M.Vertex) :
signChangesFull (R.starP Q).vertexLink (R.linkQ Q)
= vertexFlipCountSkipZeros M R.edgeSign (R.dartRep Q) := by
unfold signChangesFull
rw [cyclicFlips_nzSigns_eq_cyclicFlipCountSkipZeros]
exact (cyclicFlipCountSkipZeros_of_dihedralRotated (R.linkOrder Q)).symm
/-- **The crux bridge at an arbitrary active dart.** By `σ`-orbit invariance of
`vertexFlipCountSkipZeros`, the bridge at the representative transfers to every dart of the vertex. -/
theorem signChangesFull_eq_vertexFlip (d : D) :
signChangesFull (R.starP (M.tail d)).vertexLink (R.linkQ (M.tail d))
= vertexFlipCountSkipZeros M R.edgeSign d := by
rw [R.signChangesFull_eq_vertexFlip_rep (M.tail d)]
exact vertexFlipCountSkipZeros_sameCycle M R.edgeSign (R.sameCycle_dartRep d)
/-- **The genuine vertex-arm datum at an active dart** (Bridge: `vertexArm`). Built from the two real
spherical links via `cauchyArmVertexFull_of_links`: equal sides (`sides_eq`), equal closing chord
(`close_eq`), the interior strict witness (`interiorActive`), and the two-arc residual (`twoArc`). Its
`signChanges` is the genuine FULL closed-link cyclic count `signChangesFull`. -/
noncomputable def vertexArm (d : D) (hd : ActiveVertex M R.edgeSign d) :
Chapter13.CauchyArmVertex :=
cauchyArmVertexFull_of_links (R.starP (M.tail d)).n (R.starP (M.tail d)).hn
(R.starP (M.tail d)).vertexLink (R.linkQ (M.tail d))
(R.starP (M.tail d)).vertexLink_strictArm (R.linkQ_strictArm (M.tail d))
(R.sides_eq (M.tail d)) (R.close_eq (M.tail d))
(R.interiorActive (M.tail d)
((R.activeVertex_congr (R.sameCycle_dartRep d)).mpr hd))
(R.twoArc (M.tail d))
/-- `vertexArm`'s `signChanges` is `signChangesFull` (by construction). -/
theorem vertexArm_signChanges (d : D) (hd : ActiveVertex M R.edgeSign d) :
(R.vertexArm d hd).signChanges
= signChangesFull (R.starP (M.tail d)).vertexLink (R.linkQ (M.tail d)) := rfl
/-- **The crux bridge as the assembly field** (DERIVED): the arm-datum's sign-change count equals the
`σ`-cycle skip-zeros flip count at the vertex. -/
theorem vertexArm_signChanges_eq (d : D) (hd : ActiveVertex M R.edgeSign d) :
(R.vertexArm d hd).signChanges = vertexFlipCountSkipZeros M R.edgeSign d := by
rw [R.vertexArm_signChanges d hd, R.signChangesFull_eq_vertexFlip d]
/-- **`realization_marked`** — the faithful `CauchyMarkedTriangulatedSphere` of the realization `R`.
All four bridges are derived theorems: `edgeSign`/`edgeSign_inv` (interface), `vertexArm` (real links),
and the crux `vertexArm_signChanges_eq` (derived via `linkOrder` + reconciliation + orbit invariance). -/
def realization_marked :
Ch13CauchyAssembly.CauchyMarkedTriangulatedSphere M where
isSphere := R.isSphere
triangleFaces := R.triangle
isSimple := R.isSimple
edgeSign := R.edgeSign
edgeSign_inv := R.edgeSign_inv
vertexArm := R.vertexArm
vertexArm_signChanges_eq := R.vertexArm_signChanges_eq
end ConvexPolytopeRealization
end ProofsInTheBook.Ch13Realization
namespace ProofsInTheBook.Ch13Realization
end ProofsInTheBook.Ch13Realization
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.Ch13Realization
import ProofsInTheBook.Ch13ComponentClose
import ProofsInTheBook.SphericalRotation
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.LinearAlgebra.LinearIndependent.Lemmas
import Mathlib.Data.Fin.Tuple.Reflection
-/
/- Source module: ProofsInTheBook.ZinanCh13Euclidean -/
section
set_option autoImplicit true
noncomputable section
open scoped Classical
open ProofsInTheBook.PlanarMap ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.Ch13MarkedSphere
open ProofsInTheBook.SphericalRotation
namespace ProofsInTheBook.Ch13Euclidean
variable {D : Type*} [Fintype D] [DecidableEq D]
/-- The ambient Euclidean space for the chapter-13 geometric witness. -/
abbrev E3 : Type := EuclideanSpace ℝ (Fin 3)
/-- Three vertices assigned to a face, in cyclic combinatorial order. -/
abbrev FaceVertices (M : CombMap D) := M.Face → Fin 3 → M.Vertex
/--
A triangulated Euclidean polyhedron carried by a combinatorial map.
The field `faceDart` chooses a dart on every face; `faceVertex` is required to
be the three tails of that face in the `φ`-order from the chosen dart. The
supporting halfspace certificate orients every face normal outward, so all
vertices lie in the non-positive halfspace of the face plane.
The vertex-link cyclic order is deliberately not part of this b1 interface;
that is the later b3 residue.
-/
structure TriangulatedEuclideanPolyhedron (M : CombMap D) where
/-- Vertex coordinates in `ℝ³`. -/
pos : M.Vertex → E3
/-- A representative dart on each face. -/
faceDart : M.Face → D
/-- The representative really lies on the face it represents. -/
faceDart_face : ∀ f, M.dartFace (faceDart f) = f
/-- The three combinatorial vertices of a face, in cyclic order. -/
faceVertex : FaceVertices M
/-- Face vertices are exactly the tails along one `φ`-cycle from `faceDart`. -/
face_vertices_match : ∀ f,
faceVertex f =
![M.tail (faceDart f),
M.tail (M.φ (faceDart f)),
M.tail (M.φ (M.φ (faceDart f)))]
/-- Every combinatorial face is triangular. -/
every_face_triangle : M.FaceRegular 3
/-- Edges are realized by distinct points. -/
edge_nondegenerate : ∀ d, pos (M.tail d) ≠ pos (M.head d)
/-- The three points of each face are affinely independent. -/
face_nondegenerate : ∀ f,
AffineIndependent ℝ
(![pos (M.tail (faceDart f)),
pos (M.tail (M.φ (faceDart f))),
pos (M.tail (M.φ (M.φ (faceDart f))))] : Fin 3 → E3)
/-- A selected point on each supporting face plane. -/
face_point : M.Face → E3
/-- An outward normal for each face plane. -/
outward_normal : M.Face → E3
/-- The three face vertices lie on the chosen plane. -/
face_plane : ∀ f i,
inner ℝ (outward_normal f) (pos (faceVertex f i) - face_point f) = 0
/-- Convexity as a supporting halfspace certificate for every face. -/
face_supporting_halfspace : ∀ f v,
inner ℝ (outward_normal f) (pos v - face_point f) ≤ 0
/-- Strict support: only the three vertices of the face lie on its supporting plane. -/
face_support_strict : ∀ (f : M.Face) (v : M.Vertex),
(∀ i, v ≠ faceVertex f i) →
inner ℝ (outward_normal f) (pos v - face_point f) < 0
/-- The Euclidean edge vector carried by an oriented dart. -/
def edgeVec {M : CombMap D} (P : TriangulatedEuclideanPolyhedron M) (d : D) : E3 :=
P.pos (M.head d) - P.pos (M.tail d)
/-- The supporting face used for the reverse-`σ` vertex-link side ending at `d`. -/
def reverseFaceBetween (M : CombMap D) (d : D) : M.Face :=
M.dartFace d
/--
The stored map rotation is faithful to the outward orientation used by the
existing Chapter 13 vertex-link convention.
The current link builder reads neighbours in reverse `σ` order. Thus the face
of `d` is oriented by the two outgoing edge vectors
`edgeVec (σ⁻¹ d), edgeVec d`, and the outward normal is a positive multiple of
that reversed cross product.
-/
structure RotationFaithful {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) : Prop where
outward_normal_eq_pos_smul_reverse_cross :
∀ d : D,
∃ lam : ℝ, 0 < lam ∧
P.outward_normal (reverseFaceBetween M d) =
lam • cross (edgeVec P (M.σ.symm d)) (edgeVec P d)
/--
Face-local outward orientation, stated directly in the cyclic order of the
triangular face containing `d`.
-/
def FaceOrientationFaithful {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) : Prop :=
∀ d : D,
∃ lam : ℝ, 0 < lam ∧
P.outward_normal (M.dartFace d) =
lam • cross
(P.pos (M.tail (M.φ (M.φ d))) - P.pos (M.tail d))
(P.pos (M.tail (M.φ d)) - P.pos (M.tail d))
theorem faceDart_phi_ne_self {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) (f : M.Face) :
M.φ (P.faceDart f) ≠ P.faceDart f := by
intro h
let p : Fin 3 → E3 :=
![P.pos (M.tail (P.faceDart f)),
P.pos (M.tail (M.φ (P.faceDart f))),
P.pos (M.tail (M.φ (M.φ (P.faceDart f))))]
have hinj : Function.Injective p := (P.face_nondegenerate f).injective
have hpts : p 1 = p 0 := by
simp [p, h]
have h10 : (1 : Fin 3) = 0 := hinj hpts
norm_num at h10
theorem faceDart_phi_cube_eq_self {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) (f : M.Face) :
(M.φ ^ 3) (P.faceDart f) = P.faceDart f := by
let fd := P.faceDart f
have hφne : M.φ fd ≠ fd := by
simpa [fd] using faceDart_phi_ne_self P f
have hlen : M.faceLen f = 3 := by
simpa [CombMap.faceLen] using P.every_face_triangle f
have hcard : (M.φ.cycleOf fd).support.card = 3 := by
rw [← faceLen_dartFace_eq_card_support_cycleOf M hφne]
simpa [fd, P.faceDart_face f] using hlen
have hpow := Equiv.Perm.pow_mod_card_support_cycleOf_self_apply M.φ 3 fd
rw [hcard, Nat.mod_self] at hpow
simpa [fd] using hpow.symm
/-- A dart on a triangular Euclidean face is one of the three `φ`-successive
darts from the stored representative of that face. -/
theorem dart_eq_faceDart_or_phi_or_phi2_of_dartFace_eq {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) {f : M.Face} {d : D}
(hd : M.dartFace d = f) :
d = P.faceDart f ∨
d = M.φ (P.faceDart f) ∨
d = M.φ (M.φ (P.faceDart f)) := by
let fd := P.faceDart f
have hφne : M.φ fd ≠ fd := by
simpa [fd] using faceDart_phi_ne_self P f
have hlen : M.faceLen f = 3 := by
simpa [CombMap.faceLen] using P.every_face_triangle f
have hcard : (M.φ.cycleOf fd).support.card = 3 := by
rw [← faceLen_dartFace_eq_card_support_cycleOf M hφne]
simpa [fd, P.faceDart_face f] using hlen
have hsame : M.φ.SameCycle fd d := by
have hq : M.dartFace d = M.dartFace fd := by
rw [hd, P.faceDart_face f]
exact (Quotient.exact hq).symm
have hsupp : fd ∈ M.φ.support := Equiv.Perm.mem_support.mpr hφne
obtain ⟨i, hi, hpow⟩ := hsame.exists_pow_eq_of_mem_support hsupp
rw [hcard] at hi
interval_cases i
· left
simpa [fd] using hpow.symm
· right
left
simpa [fd] using hpow.symm
· right
right
simpa [fd, pow_succ] using hpow.symm
/-- Every dart tail is one of the three stored vertices of its dart face. -/
theorem tail_mem_faceVertex {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) (d : D) :
∃ k : Fin 3, M.tail d = P.faceVertex (M.dartFace d) k := by
rcases dart_eq_faceDart_or_phi_or_phi2_of_dartFace_eq
P (f := M.dartFace d) (d := d) rfl with h | h | h
· refine ⟨0, ?_⟩
rw [h]
have hv := congrFun (P.face_vertices_match (M.dartFace d)) 0
simpa [P.faceDart_face (M.dartFace d)] using hv.symm
· refine ⟨1, ?_⟩
rw [h]
have hv := congrFun (P.face_vertices_match (M.dartFace d)) 1
simpa [P.faceDart_face (M.dartFace d)] using hv.symm
· refine ⟨2, ?_⟩
rw [h]
have hv := congrFun (P.face_vertices_match (M.dartFace d)) 2
simpa [P.faceDart_face (M.dartFace d)] using hv.symm
/-- On a triangular Euclidean face, two `φ` steps reach the head of the
previous dart in the reverse `σ` order. -/
theorem tail_phi_phi_eq_head_sigma_symm_of_triangular_euclidean {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) (d : D) :
M.tail (M.φ (M.φ d)) = M.head (M.σ.symm d) := by
have hcube : (M.φ ^ 3) d = d := by
rcases dart_eq_faceDart_or_phi_or_phi2_of_dartFace_eq
P (f := M.dartFace d) (d := d) rfl with h | h | h
· rw [h]
exact faceDart_phi_cube_eq_self P (M.dartFace d)
· rw [h]
exact congrArg M.φ (faceDart_phi_cube_eq_self P (M.dartFace d))
· rw [h]
exact congrArg (fun x => M.φ (M.φ x))
(faceDart_phi_cube_eq_self P (M.dartFace d))
have hpred : M.φ (M.φ d) = M.φ.symm d := by
apply M.φ.injective
rw [Equiv.apply_symm_apply]
simpa [pow_succ, Equiv.Perm.coe_mul, Function.comp_apply] using hcube
have hsymm : M.φ.symm d = M.α (M.σ.symm d) := by
apply M.φ.injective
rw [Equiv.apply_symm_apply]
symm
change (M.σ * M.α) (M.α (M.σ.symm d)) = d
rw [Equiv.Perm.mul_apply, M.alpha_alpha, Equiv.apply_symm_apply]
rw [hpred, hsymm, M.tail_alpha]
theorem tail_sigma_symm {M : CombMap D} (d : D) :
M.tail (M.σ.symm d) = M.tail d := by
have h := M.tail_sigma (M.σ.symm d)
simpa using h.symm
/-- Face-local orientation implies the reverse-`σ` rotation-faithful convention
used by the vertex-link construction. -/
theorem rotationFaithful_of_faceOrientationFaithful {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M)
(hface : FaceOrientationFaithful P) :
RotationFaithful P where
outward_normal_eq_pos_smul_reverse_cross := by
intro d
obtain ⟨lam, hlam, hnormal⟩ := hface d
refine ⟨lam, hlam, ?_⟩
have hprev :
edgeVec P (M.σ.symm d) =
P.pos (M.tail (M.φ (M.φ d))) - P.pos (M.tail d) := by
simp [edgeVec, tail_sigma_symm,
← tail_phi_phi_eq_head_sigma_symm_of_triangular_euclidean P d]
have hnext :
edgeVec P d =
P.pos (M.tail (M.φ d)) - P.pos (M.tail d) := by
simp [edgeVec, M.tail_phi]
simpa [reverseFaceBetween, hprev, hnext] using hnormal
/-- The selected face plane contains the tail of every dart on that face. -/
theorem face_plane_dart {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) (d : D) :
inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.tail d) - P.face_point (M.dartFace d)) = 0 := by
obtain ⟨k, hk⟩ := tail_mem_faceVertex P d
rw [hk]
exact P.face_plane (M.dartFace d) k
-- The regular tetrahedron satisfies the reverse-`σ` rotation-faithfulness convention.
-- The regular tetrahedron satisfies the face-local outward-orientation convention.
/-- The outward normal on the face to the left of a dart. -/
def dartNormal {M : CombMap D} (P : TriangulatedEuclideanPolyhedron M) (d : D) : E3 :=
P.outward_normal (M.dartFace d)
/--
The interior dihedral angle along a dart-represented edge.
With outward normals `n_f,n_g`, this is `π - angle n_f n_g`.
-/
def dihedralAngleAtDart {M : CombMap D} (P : TriangulatedEuclideanPolyhedron M) (d : D) : ℝ :=
Real.pi - InnerProductGeometry.angle (dartNormal P d) (dartNormal P (M.α d))
theorem dihedralAngleAtDart_alpha {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) (d : D) :
dihedralAngleAtDart P (M.α d) = dihedralAngleAtDart P d := by
unfold dihedralAngleAtDart dartNormal
rw [M.alpha_alpha, InnerProductGeometry.angle_comm]
/-- The dihedral-difference sign carried by a dart. -/
def dihedralSignAtDart {M : CombMap D}
(P Q : TriangulatedEuclideanPolyhedron M) (d : D) : ProofsInTheBook.Chapter13.EdgeSign :=
ProofsInTheBook.Ch13Realization.realSignToEdgeSign
(dihedralAngleAtDart Q d - dihedralAngleAtDart P d)
theorem dihedralSignAtDart_alpha {M : CombMap D}
(P Q : TriangulatedEuclideanPolyhedron M) (d : D) :
dihedralSignAtDart P Q (M.α d) = dihedralSignAtDart P Q d := by
unfold dihedralSignAtDart
rw [dihedralAngleAtDart_alpha P d, dihedralAngleAtDart_alpha Q d]
end ProofsInTheBook.Ch13Euclidean
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanCh13Euclidean
import ProofsInTheBook.Ch13VertexStar
import ProofsInTheBook.Ch13Realization
import ProofsInTheBook.SphericalRotation
import Mathlib.Data.Fin.Rev
-/
/- Source module: ProofsInTheBook.ZinanCh13EuclLink -/
section
set_option autoImplicit true
noncomputable section
set_option maxHeartbeats 3000000
open scoped Classical RealInnerProductSpace
open ProofsInTheBook.PlanarMap ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.Ch13Euclidean
open ProofsInTheBook.Ch13VertexStar
open ProofsInTheBook.Ch13MarkedSphere
open ProofsInTheBook.SphericalRotation
namespace ProofsInTheBook.Ch13EuclLink
variable {D : Type*} [Fintype D] [DecidableEq D]
variable {M : CombMap D}
lemma prev_mod_succ_mod {N k : ℕ} (hN : 0 < N) (hk : k < N) :
(((k + N - 1) % N + 1) % N) = k := by
by_cases hk0 : k = 0
· subst hk0
have hNm1 : (N - 1) % N = N - 1 := Nat.mod_eq_of_lt (Nat.sub_lt hN Nat.zero_lt_one)
rw [zero_add, hNm1]
have hN' : N - 1 + 1 = N := Nat.sub_add_cancel (Nat.succ_le_of_lt hN)
rw [hN', Nat.mod_self]
· have hkpos : 0 < k := Nat.pos_of_ne_zero hk0
have hsplit : k + N - 1 = (k - 1) + N := by omega
rw [hsplit, Nat.add_mod_right]
have hkpred : k - 1 < N := by omega
rw [Nat.mod_eq_of_lt hkpred]
have hks : k - 1 + 1 = k := Nat.sub_add_cancel hkpos
rw [hks, Nat.mod_eq_of_lt hk]
def finOneOfThree {N : ℕ} (hN : 3 ≤ N) : Fin N :=
⟨1, by omega⟩
lemma rev_add_one_rev_val {N : ℕ} (hN : 3 ≤ N) (i : Fin N) :
((Fin.rev (Fin.rev i + finOneOfThree hN) : Fin N) : ℕ) =
(i.val + N - 1) % N := by
rw [Fin.val_rev, Fin.val_add, Fin.val_rev]
simp only [finOneOfThree, Fin.val_mk]
by_cases hi0 : i.val = 0
· rw [hi0]
have hinner : (N - (0 + 1) + 1) % N = 0 := by
have hNpos : 0 < N := by omega
have heq : N - (0 + 1) + 1 = N := by omega
rw [heq, Nat.mod_self]
rw [hinner]
have hrhs : (0 + N - 1) % N = N - 1 := by
rw [zero_add, Nat.mod_eq_of_lt (by omega)]
rw [hrhs]
· have hipos : 0 < i.val := Nat.pos_of_ne_zero hi0
have hinner : (N - (i.val + 1) + 1) % N = N - i.val := by
have heq : N - (i.val + 1) + 1 = N - i.val := by omega
rw [heq, Nat.mod_eq_of_lt (by omega)]
have hrhs : (i.val + N - 1) % N = i.val - 1 := by
have heq : i.val + N - 1 = (i.val - 1) + N := by omega
rw [heq, Nat.add_mod_right, Nat.mod_eq_of_lt (by omega)]
rw [hinner, hrhs]
omega
lemma rev_sub_one_rev_val {N : ℕ} (hN : 3 ≤ N) (i : Fin N) :
((Fin.rev (Fin.rev i - finOneOfThree hN) : Fin N) : ℕ) =
(i.val + 1) % N := by
rw [Fin.val_rev, Fin.sub_def, Fin.val_rev]
simp only [finOneOfThree, Fin.val_mk]
have hinner : (N - 1 + (N - (i.val + 1))) % N =
(N - (i.val + 1) + (N - 1)) % N := by
rw [Nat.add_comm]
rw [hinner]
by_cases hilast : i.val + 1 = N
· have hi : i.val = N - 1 := by omega
rw [hi]
have hmod : (N - (N - 1 + 1) + (N - 1)) % N = N - 1 := by
have heq : N - (N - 1 + 1) + (N - 1) = N - 1 := by omega
rw [heq, Nat.mod_eq_of_lt (by omega)]
rw [hmod]
rw [show (N - 1 + 1) % N = 0 by rw [Nat.sub_add_cancel (by omega), Nat.mod_self]]
omega
· have hi1lt : i.val + 1 < N := by omega
have hmod1 : (N - (i.val + 1) + (N - 1)) % N = N - (i.val + 2) := by
have hsum : N - (i.val + 1) + (N - 1) = (N - (i.val + 2)) + N := by omega
rw [hsum, Nat.add_mod_right, Nat.mod_eq_of_lt (by omega)]
rw [hmod1]
have htarget : (i.val + 1) % N = i.val + 1 := Nat.mod_eq_of_lt hi1lt
rw [htarget]
omega
/-- The incident darts at a vertex, rooted at `Quotient.out v` and ordered by `σ`. -/
def incidentDarts (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex) : List D :=
M.σ.toList (Quotient.out v)
/-- The combinatorial degree read from the `σ`-cycle list. -/
def vDeg (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex) : ℕ :=
(incidentDarts P v).length
/-- The `i`-th incident dart in the `σ`-cycle. -/
def incidentDart (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(i : Fin (vDeg P v)) : D :=
(incidentDarts P v).get i
/-- Every dart read from the incident list has tail `v`. -/
theorem incidentDart_tail (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(i : Fin (vDeg P v)) :
M.tail (incidentDart P v i) = v := by
unfold incidentDart
have hmem : (incidentDarts P v).get i ∈ incidentDarts P v :=
List.get_mem _ _
unfold incidentDarts at hmem ⊢
have hsame : M.σ.SameCycle (Quotient.out v) ((M.σ.toList (Quotient.out v)).get i) :=
(Equiv.Perm.mem_toList_iff.mp hmem).1
calc
M.tail ((M.σ.toList (Quotient.out v)).get i) = M.tail (Quotient.out v) :=
Quotient.sound hsame.symm
_ = v := Quotient.out_eq v
/-- The `VertexStar.n` associated to a vertex of degree `vDeg`: there are `n + 1` neighbours. -/
def starN (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex) : ℕ :=
vDeg P v - 1
/-- A `VertexStar` index converted to the corresponding degree-list index. -/
def starIndexToDeg (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) (i : Fin (starN P v + 1)) : Fin (vDeg P v) :=
⟨i.1, by
have hi := i.2
unfold starN at hi
omega⟩
/-- The `i`-th incident dart, indexed in the eventual `VertexStar` convention. -/
def incidentDartOfStarIndex (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) (i : Fin (starN P v + 1)) : D :=
incidentDart P v (starIndexToDeg P v hdeg i)
theorem incidentDartOfStarIndex_tail (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) (i : Fin (starN P v + 1)) :
M.tail (incidentDartOfStarIndex P v hdeg i) = v := by
unfold incidentDartOfStarIndex
exact incidentDart_tail P v (starIndexToDeg P v hdeg i)
theorem starN_add_one_eq_vDeg (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) :
starN P v + 1 = vDeg P v := by
unfold starN
omega
def incidentIndexOfDart (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(d : D) (hd : d ∈ incidentDarts P v) : Fin (vDeg P v) :=
⟨(incidentDarts P v).idxOf d, by
unfold vDeg
exact List.idxOf_lt_length_iff.mpr hd⟩
theorem incidentDart_incidentIndexOfDart
(P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(d : D) (hd : d ∈ incidentDarts P v) :
incidentDart P v (incidentIndexOfDart P v d hd) = d := by
unfold incidentDart incidentIndexOfDart
exact List.idxOf_get (List.idxOf_lt_length_iff.mpr hd)
def reverseStarIndexOfDart (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) (d : D) (hd : d ∈ incidentDarts P v) :
Fin (starN P v + 1) :=
Fin.rev (Fin.cast (starN_add_one_eq_vDeg P v hdeg).symm
(incidentIndexOfDart P v d hd))
theorem incidentDartOfStarIndex_reverseStarIndexOfDart
(P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) (d : D) (hd : d ∈ incidentDarts P v) :
incidentDartOfStarIndex P v hdeg
(Fin.rev (reverseStarIndexOfDart P v hdeg d hd)) = d := by
unfold reverseStarIndexOfDart incidentDartOfStarIndex starIndexToDeg
simpa [Fin.rev_rev, Fin.cast_trans, Fin.cast_eq_self] using
incidentDart_incidentIndexOfDart P v d hd
/-- The cyclic step `1` in the eventual vertex-star index type. -/
def starOne (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) : Fin (starN P v + 1) :=
⟨1, by
unfold starN
omega⟩
theorem incidentDartOfStarIndex_reverseStarIndexOfDart_add_one
(P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) (d : D) (hd : d ∈ incidentDarts P v) :
incidentDartOfStarIndex P v hdeg
(Fin.rev (reverseStarIndexOfDart P v hdeg d hd + starOne P v hdeg)) = M.σ.symm d := by
set root : D := Quotient.out v
set L : List D := incidentDarts P v
set N : ℕ := vDeg P v
set k : ℕ := L.idxOf d
have hL : L = M.σ.toList root := by
simp [L, incidentDarts, root]
have hN : N = L.length := by
simp [N, vDeg, L]
have hNpos : 0 < N := by
have := hdeg
omega
have hklt : k < N := by
rw [hN]
exact List.idxOf_lt_length_iff.mpr (by simpa [L] using hd)
have hroot_support : root ∈ M.σ.support := by
have hmem : d ∈ M.σ.toList root := by simpa [hL] using (by simpa [L] using hd)
exact (Equiv.Perm.mem_toList_iff.mp hmem).2
have hcard : (M.σ.cycleOf root).support.card = N := by
rw [← Equiv.Perm.length_toList M.σ root, ← hL, hN]
have hd_pow : d = (M.σ ^ k) root := by
have hkltL : k < L.length := by rwa [← hN]
have hget_idx : L.get ⟨k, hkltL⟩ = d := by
simpa [k] using List.idxOf_get hkltL
have hget_pow :
L.get ⟨k, hkltL⟩ = (M.σ ^ k) root := by
simpa [hL] using Equiv.Perm.getElem_toList M.σ root k (by simpa [hL] using hkltL)
exact hget_idx.symm.trans hget_pow
apply M.σ.injective
rw [Equiv.apply_symm_apply]
unfold incidentDartOfStarIndex incidentDart starIndexToDeg reverseStarIndexOfDart
set j : Fin N := incidentIndexOfDart P v d hd
have hjval : j.val = k := by
simp [j, incidentIndexOfDart, k, L]
have hN3 : 3 ≤ N := hdeg
have hval :
((starIndexToDeg P v hdeg
(Fin.rev (Fin.rev (Fin.cast (starN_add_one_eq_vDeg P v hdeg).symm j)
+ starOne P v hdeg))) : ℕ) = (k + N - 1) % N := by
unfold starIndexToDeg starOne
have hrev :
(((Fin.rev (Fin.rev (Fin.cast (starN_add_one_eq_vDeg P v hdeg).symm j)
+ ⟨1, by unfold starN; omega⟩)) :
Fin (starN P v + 1)) : ℕ)
=
(((Fin.cast (starN_add_one_eq_vDeg P v hdeg).symm j).val + (starN P v + 1) - 1)
% (starN P v + 1)) :=
rev_add_one_rev_val (by unfold starN; omega)
(Fin.cast (starN_add_one_eq_vDeg P v hdeg).symm j)
rw [hrev]
simp [hjval, starN_add_one_eq_vDeg P v hdeg, N]
have hget :
(incidentDarts P v).get
(starIndexToDeg P v hdeg
(Fin.rev (Fin.rev (Fin.cast (starN_add_one_eq_vDeg P v hdeg).symm j)
+ starOne P v hdeg))) =
(M.σ ^ ((k + N - 1) % N)) root := by
have hget0 := Equiv.Perm.getElem_toList M.σ root
((starIndexToDeg P v hdeg
(Fin.rev (Fin.rev (Fin.cast (starN_add_one_eq_vDeg P v hdeg).symm j)
+ starOne P v hdeg))) : ℕ)
(by
have hlt := (starIndexToDeg P v hdeg
(Fin.rev (Fin.rev (Fin.cast (starN_add_one_eq_vDeg P v hdeg).symm j)
+ starOne P v hdeg))).2
have hlenTo : (M.σ.toList root).length = vDeg P v := by
rw [← hL]
simp [vDeg, L]
simpa [hlenTo] using hlt)
simpa [hL, hval] using hget0
change M.σ ((incidentDarts P v).get
(starIndexToDeg P v hdeg
(Fin.rev (Fin.rev (Fin.cast (starN_add_one_eq_vDeg P v hdeg).symm j)
+ starOne P v hdeg)))) = d
rw [hget]
change ((M.σ * (M.σ ^ ((k + N - 1) % N))) root) = d
rw [← pow_succ']
have hmod : (((k + N - 1) % N + 1) % (M.σ.cycleOf root).support.card) = k := by
rw [hcard]
exact prev_mod_succ_mod hNpos hklt
calc
(M.σ ^ (((k + N - 1) % N) + 1)) root
= (M.σ ^ ((((k + N - 1) % N) + 1) % (M.σ.cycleOf root).support.card)) root := by
rw [Equiv.Perm.pow_mod_card_support_cycleOf_self_apply]
_ = (M.σ ^ k) root := by rw [hmod]
_ = d := hd_pow.symm
theorem incidentDartOfStarIndex_reverseStarIndexOfDart_sub_one
(P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) (d : D) (hd : d ∈ incidentDarts P v) :
incidentDartOfStarIndex P v hdeg
(Fin.rev (reverseStarIndexOfDart P v hdeg d hd - starOne P v hdeg)) = M.σ d := by
set root : D := Quotient.out v
set L : List D := incidentDarts P v
set N : ℕ := vDeg P v
set k : ℕ := L.idxOf d
have hL : L = M.σ.toList root := by
simp [L, incidentDarts, root]
have hN : N = L.length := by
simp [N, vDeg, L]
have hklt : k < N := by
rw [hN]
exact List.idxOf_lt_length_iff.mpr (by simpa [L] using hd)
have hcard : (M.σ.cycleOf root).support.card = N := by
rw [← Equiv.Perm.length_toList M.σ root, ← hL, hN]
have hd_pow : d = (M.σ ^ k) root := by
have hkltL : k < L.length := by rwa [← hN]
have hget_idx : L.get ⟨k, hkltL⟩ = d := by
simpa [k] using List.idxOf_get hkltL
have hget_pow :
L.get ⟨k, hkltL⟩ = (M.σ ^ k) root := by
simpa [hL] using Equiv.Perm.getElem_toList M.σ root k (by simpa [hL] using hkltL)
exact hget_idx.symm.trans hget_pow
unfold incidentDartOfStarIndex incidentDart starIndexToDeg reverseStarIndexOfDart
set j : Fin N := incidentIndexOfDart P v d hd
have hjval : j.val = k := by
simp [j, incidentIndexOfDart, k, L]
have hval :
((starIndexToDeg P v hdeg
(Fin.rev (Fin.rev (Fin.cast (starN_add_one_eq_vDeg P v hdeg).symm j)
- starOne P v hdeg))) : ℕ) = (k + 1) % N := by
unfold starIndexToDeg starOne
have hrev :
(((Fin.rev (Fin.rev (Fin.cast (starN_add_one_eq_vDeg P v hdeg).symm j)
- ⟨1, by unfold starN; omega⟩)) :
Fin (starN P v + 1)) : ℕ)
=
(((Fin.cast (starN_add_one_eq_vDeg P v hdeg).symm j).val + 1)
% (starN P v + 1)) :=
rev_sub_one_rev_val (by unfold starN; omega)
(Fin.cast (starN_add_one_eq_vDeg P v hdeg).symm j)
rw [hrev]
simp [hjval, starN_add_one_eq_vDeg P v hdeg, N]
have hget :
(incidentDarts P v).get
(starIndexToDeg P v hdeg
(Fin.rev (Fin.rev (Fin.cast (starN_add_one_eq_vDeg P v hdeg).symm j)
- starOne P v hdeg))) =
(M.σ ^ ((k + 1) % N)) root := by
have hget0 := Equiv.Perm.getElem_toList M.σ root
((starIndexToDeg P v hdeg
(Fin.rev (Fin.rev (Fin.cast (starN_add_one_eq_vDeg P v hdeg).symm j)
- starOne P v hdeg))) : ℕ)
(by
have hlt := (starIndexToDeg P v hdeg
(Fin.rev (Fin.rev (Fin.cast (starN_add_one_eq_vDeg P v hdeg).symm j)
- starOne P v hdeg))).2
have hlenTo : (M.σ.toList root).length = vDeg P v := by
rw [← hL]
simp [vDeg, L]
simpa [hlenTo] using hlt)
simpa [hL, hval] using hget0
change (incidentDarts P v).get
(starIndexToDeg P v hdeg
(Fin.rev (Fin.rev (Fin.cast (starN_add_one_eq_vDeg P v hdeg).symm j)
- starOne P v hdeg))) = M.σ d
rw [hget]
calc
(M.σ ^ ((k + 1) % N)) root
= (M.σ ^ ((k + 1) % (M.σ.cycleOf root).support.card)) root := by rw [hcard]
_ = (M.σ ^ (k + 1)) root := by
rw [Equiv.Perm.pow_mod_card_support_cycleOf_self_apply]
_ = M.σ d := by
rw [hd_pow]
rw [pow_succ', Equiv.Perm.coe_mul, Function.comp_apply]
theorem starN_ge_two (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) : 2 ≤ starN P v := by
unfold starN
omega
theorem incidentDartOfStarIndex_injective
(P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) :
Function.Injective (incidentDartOfStarIndex P v hdeg) := by
intro i j hij
have hnodup : (incidentDarts P v).Nodup := by
unfold incidentDarts
exact Equiv.Perm.nodup_toList M.σ (Quotient.out v)
unfold incidentDartOfStarIndex incidentDart at hij
have hidx :=
(List.Nodup.getElem_inj_iff hnodup).mp hij
exact Fin.ext hidx
theorem dart_eq_of_same_tail_head_of_isSimpleGraph
(hsimple : M.IsSimpleGraph) {d e : D}
(htail : M.tail d = M.tail e) (hhead : M.head d = M.head e) :
d = e := by
have hsc : M.α.SameCycle d e :=
M.alpha_sameCycle_of_same_endpoints hsimple htail hhead
rcases (M.alpha_sameCycle_iff d e).mp hsc with heq | halpha
· exact heq.symm
· exfalso
apply hsimple.no_loop d
have hloop : M.tail d = M.head d := by
calc
M.tail d = M.tail e := htail
_ = M.tail (M.α d) := by rw [halpha]
_ = M.head d := M.tail_alpha d
exact hloop
/-- The dart read by the Euclidean bridge's reverse-`σ` link order. -/
def reverseLinkDart (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) (i : Fin (starN P v + 1)) : D :=
incidentDartOfStarIndex P v hdeg (Fin.rev i)
/-- The neighbour read by the Euclidean bridge's reverse-`σ` link order. -/
def reverseLinkNbr (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) (i : Fin (starN P v + 1)) : M.Vertex :=
M.head (reverseLinkDart P v hdeg i)
theorem reverseLinkDart_tail (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) (i : Fin (starN P v + 1)) :
M.tail (reverseLinkDart P v hdeg i) = v := by
exact incidentDartOfStarIndex_tail P v hdeg (Fin.rev i)
theorem reverseLinkDart_mem_incident (P : TriangulatedEuclideanPolyhedron M)
(v : M.Vertex) (hdeg : 3 ≤ vDeg P v) (i : Fin (starN P v + 1)) :
reverseLinkDart P v hdeg i ∈ incidentDarts P v := by
unfold reverseLinkDart incidentDartOfStarIndex incidentDart
exact List.get_mem _ _
theorem reverseLinkNbr_apex_ne (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) (i : Fin (starN P v + 1)) :
P.pos (reverseLinkNbr P v hdeg i) ≠ P.pos v := by
intro h
exact P.edge_nondegenerate (reverseLinkDart P v hdeg i) (by
rw [reverseLinkDart_tail P v hdeg i]
exact h.symm)
theorem reverseLinkDart_injective (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) :
Function.Injective (reverseLinkDart P v hdeg) := by
intro i j h
have hidx := incidentDartOfStarIndex_injective P v hdeg h
exact Fin.rev_injective hidx
theorem reverseLinkDart_add_one (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) (i : Fin (starN P v + 1)) :
reverseLinkDart P v hdeg (i + 1) = M.σ.symm (reverseLinkDart P v hdeg i) := by
let d := reverseLinkDart P v hdeg i
have hdmem : d ∈ incidentDarts P v := by
simpa [d] using reverseLinkDart_mem_incident P v hdeg i
have hidx : reverseStarIndexOfDart P v hdeg d hdmem = i := by
have hbase :=
incidentDartOfStarIndex_reverseStarIndexOfDart P v hdeg d hdmem
have hsame :
incidentDartOfStarIndex P v hdeg
(Fin.rev (reverseStarIndexOfDart P v hdeg d hdmem)) =
incidentDartOfStarIndex P v hdeg (Fin.rev i) := by
simpa [d, reverseLinkDart] using hbase
have hrev := incidentDartOfStarIndex_injective P v hdeg hsame
exact Fin.rev_injective hrev
have hstep :=
incidentDartOfStarIndex_reverseStarIndexOfDart_add_one P v hdeg d hdmem
have hidx_add :
reverseStarIndexOfDart P v hdeg d hdmem + starOne P v hdeg = i + 1 := by
rw [hidx]
apply Fin.ext
simp [Fin.add_def, starOne]
rw [hidx_add] at hstep
simpa [reverseLinkDart, d] using hstep
theorem reverseLinkNbr_add_one (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) (i : Fin (starN P v + 1)) :
reverseLinkNbr P v hdeg (i + 1) =
M.head (M.σ.symm (reverseLinkDart P v hdeg i)) := by
unfold reverseLinkNbr
rw [reverseLinkDart_add_one P v hdeg i]
theorem reverseLinkNbr_eq_apex_false_of_simple
(P : TriangulatedEuclideanPolyhedron M) (hsimple : M.IsSimpleGraph)
(v : M.Vertex) (hdeg : 3 ≤ vDeg P v) (i : Fin (starN P v + 1)) :
reverseLinkNbr P v hdeg i ≠ v := by
intro h
exact hsimple.no_loop (reverseLinkDart P v hdeg i)
((reverseLinkDart_tail P v hdeg i).trans h.symm)
theorem reverseLinkNbr_injective_of_simple
(P : TriangulatedEuclideanPolyhedron M) (hsimple : M.IsSimpleGraph)
(v : M.Vertex) (hdeg : 3 ≤ vDeg P v) :
Function.Injective (reverseLinkNbr P v hdeg) := by
intro i j hhead
have hdart : reverseLinkDart P v hdeg i = reverseLinkDart P v hdeg j :=
dart_eq_of_same_tail_head_of_isSimpleGraph hsimple
((reverseLinkDart_tail P v hdeg i).trans (reverseLinkDart_tail P v hdeg j).symm)
hhead
exact reverseLinkDart_injective P v hdeg hdart
theorem reverseLink_nonincident_of_simple
(P : TriangulatedEuclideanPolyhedron M) (hsimple : M.IsSimpleGraph)
(v : M.Vertex) (hdeg : 3 ≤ vDeg P v) :
∀ i j : Fin (starN P v + 1), j ≠ i → j ≠ i + 1 →
¬(reverseLinkNbr P v hdeg j = v ∨
reverseLinkNbr P v hdeg j = reverseLinkNbr P v hdeg i ∨
reverseLinkNbr P v hdeg j = reverseLinkNbr P v hdeg (i + 1)) := by
intro i j hji hjnext hbad
rcases hbad with hapex | heq | hnext
· exact reverseLinkNbr_eq_apex_false_of_simple P hsimple v hdeg j hapex
· exact hji (reverseLinkNbr_injective_of_simple P hsimple v hdeg heq)
· exact hjnext (reverseLinkNbr_injective_of_simple P hsimple v hdeg hnext)
theorem exists_fin_not_incident_edge {n : ℕ} (hn : 2 ≤ n) (i : Fin (n + 1)) :
∃ j : Fin (n + 1), j ≠ i ∧ j ≠ i + 1 := by
by_contra hcon
push_neg at hcon
have hsub : (Finset.univ : Finset (Fin (n + 1))) ⊆ {i, i + 1} := by
intro j _
by_cases hji : j = i
· simp [hji]
· have hjnext := hcon j hji
simp [hjnext]
have hle := Finset.card_le_card hsub
have hcard : ({i, i + 1} : Finset (Fin (n + 1))).card ≤ 2 :=
le_trans (Finset.card_insert_le _ _) (by simp)
simp only [Finset.card_univ, Fintype.card_fin] at hle
have hle2 : n + 1 ≤ 2 := le_trans hle hcard
omega
/-- Local scalar triple product, in the same coordinate convention as `SphericalKernel.det3`. -/
def det3 (u v w : E3) : ℝ :=
u 0 * (v 1 * w 2 - v 2 * w 1)
- u 1 * (v 0 * w 2 - v 2 * w 0)
+ u 2 * (v 0 * w 1 - v 1 * w 0)
theorem det3_eq_spherical (u v w : E3) :
det3 u v w = ProofsInTheBook.SphericalKernel.det3 u v w := rfl
theorem det3_eq_inner_cross (u v z : E3) :
det3 u v z = (⟪cross u v, z⟫ : ℝ) := by
rw [det3_eq_spherical]
calc
ProofsInTheBook.SphericalKernel.det3 u v z = (⟪u, cross v z⟫ : ℝ) := by
rw [inner_cross_eq_det3]
_ = (⟪z, cross u v⟫ : ℝ) := inner_cross_cyclic u v z
_ = (⟪cross u v, z⟫ : ℝ) := (real_inner_comm z (cross u v)).symm
theorem faceDart_phi_ne_self_link
(P : TriangulatedEuclideanPolyhedron M) (f : M.Face) :
M.φ (P.faceDart f) ≠ P.faceDart f := by
intro h
let p : Fin 3 → E3 :=
![P.pos (M.tail (P.faceDart f)),
P.pos (M.tail (M.φ (P.faceDart f))),
P.pos (M.tail (M.φ (M.φ (P.faceDart f))))]
have hinj : Function.Injective p := (P.face_nondegenerate f).injective
have hpts : p 1 = p 0 := by
simp [p, h]
have h10 : (1 : Fin 3) = 0 := hinj hpts
norm_num at h10
theorem faceDart_phi_cube_eq_self
(P : TriangulatedEuclideanPolyhedron M) (f : M.Face) :
(M.φ ^ 3) (P.faceDart f) = P.faceDart f := by
let fd := P.faceDart f
have hφne : M.φ fd ≠ fd := by
simpa [fd] using faceDart_phi_ne_self_link P f
have hlen : M.faceLen f = 3 := by
simpa [CombMap.faceLen] using P.every_face_triangle f
have hcard : (M.φ.cycleOf fd).support.card = 3 := by
rw [← faceLen_dartFace_eq_card_support_cycleOf M hφne]
simpa [fd, P.faceDart_face f] using hlen
have hpow := Equiv.Perm.pow_mod_card_support_cycleOf_self_apply M.φ 3 fd
rw [hcard, Nat.mod_self] at hpow
simpa [fd] using hpow.symm
theorem phi_cube_eq_self_of_triangular_euclidean
(P : TriangulatedEuclideanPolyhedron M) (d : D) :
(M.φ ^ 3) d = d := by
rcases dart_eq_faceDart_or_phi_or_phi2_of_dartFace_eq
P (f := M.dartFace d) (d := d) rfl with h | h | h
· rw [h]
exact faceDart_phi_cube_eq_self P (M.dartFace d)
· rw [h]
exact congrArg M.φ (faceDart_phi_cube_eq_self P (M.dartFace d))
· rw [h]
exact congrArg (fun x => M.φ (M.φ x))
(faceDart_phi_cube_eq_self P (M.dartFace d))
theorem tail_phi_phi_eq_head_sigma_symm_of_triangular_euclidean
(P : TriangulatedEuclideanPolyhedron M) (d : D) :
M.tail (M.φ (M.φ d)) = M.head (M.σ.symm d) := by
have hcube := phi_cube_eq_self_of_triangular_euclidean P d
have hpred : M.φ (M.φ d) = M.φ.symm d := by
apply M.φ.injective
rw [Equiv.apply_symm_apply]
simpa [pow_succ, Equiv.Perm.coe_mul, Function.comp_apply] using hcube
have hsymm : M.φ.symm d = M.α (M.σ.symm d) := by
apply M.φ.injective
rw [Equiv.apply_symm_apply]
symm
change (M.σ * M.α) (M.α (M.σ.symm d)) = d
rw [Equiv.Perm.mul_apply, M.alpha_alpha, Equiv.apply_symm_apply]
rw [hpred, hsymm, M.tail_alpha]
/-- The stored face vertices are the three tails of any dart on the same
triangular face, up to cyclic rotation. -/
theorem faceVertex_eq_tail_or_head_or_tail_phi2_of_dartFace_eq
(P : TriangulatedEuclideanPolyhedron M) {f : M.Face} {e : D}
(he : M.dartFace e = f) (k : Fin 3) :
P.faceVertex f k = M.tail e ∨
P.faceVertex f k = M.head e ∨
P.faceVertex f k = M.tail (M.φ (M.φ e)) := by
rcases dart_eq_faceDart_or_phi_or_phi2_of_dartFace_eq P (f := f) (d := e) he with
h | h | h
· subst h
fin_cases k
· left
have hv := congrFun (P.face_vertices_match f) 0
simpa using hv
· right; left
have hv := congrFun (P.face_vertices_match f) 1
simpa [M.tail_phi] using hv
· right; right
have hv := congrFun (P.face_vertices_match f) 2
simpa using hv
· subst h
fin_cases k
· right; right
have hv := congrFun (P.face_vertices_match f) 0
have hcube := faceDart_phi_cube_eq_self P f
have hcube' : M.φ (M.φ (M.φ (P.faceDart f))) = P.faceDart f := by
simpa [pow_succ, Equiv.Perm.coe_mul, Function.comp_apply] using hcube
have htail : M.tail (P.faceDart f) =
M.tail (M.φ (M.φ (M.φ (P.faceDart f)))) := by
rw [hcube']
exact hv.trans htail
· left
have hv := congrFun (P.face_vertices_match f) 1
simpa [M.tail_phi] using hv
· right; left
have hv := congrFun (P.face_vertices_match f) 2
simpa [M.tail_phi] using hv
· subst h
fin_cases k
· right; left
have hv := congrFun (P.face_vertices_match f) 0
have hcube := faceDart_phi_cube_eq_self P f
have hcube' : M.φ (M.φ (M.φ (P.faceDart f))) = P.faceDart f := by
simpa [pow_succ, Equiv.Perm.coe_mul, Function.comp_apply] using hcube
have htail : M.tail (P.faceDart f) =
M.head (M.φ (M.φ (P.faceDart f))) := by
rw [← M.tail_phi (M.φ (M.φ (P.faceDart f))), hcube']
exact hv.trans htail
· right; right
have hv := congrFun (P.face_vertices_match f) 1
have hcube := faceDart_phi_cube_eq_self P f
have hcube' : M.φ (M.φ (M.φ (P.faceDart f))) = P.faceDart f := by
simpa [pow_succ, Equiv.Perm.coe_mul, Function.comp_apply] using hcube
have htail : M.tail (M.φ (P.faceDart f)) =
M.tail (M.φ (M.φ (M.φ (M.φ (P.faceDart f))))) := by
rw [hcube']
exact hv.trans htail
· left
have hv := congrFun (P.face_vertices_match f) 2
simpa using hv
theorem reverseFaceBetween_support_edgeVec_lt
(P : TriangulatedEuclideanPolyhedron M) (d e : D)
(he_tail : M.tail e = M.tail d)
(hoff : ∀ i, M.head e ≠ P.faceVertex (reverseFaceBetween M d) i) :
inner ℝ (P.outward_normal (reverseFaceBetween M d)) (edgeVec P e) < 0 := by
have htail_plane :
inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.tail d) - P.face_point (M.dartFace d)) = 0 :=
face_plane_dart P d
have hstrict := P.face_support_strict (M.dartFace d) (M.head e) (by
simpa [reverseFaceBetween] using hoff)
have hrewrite :
inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.head e) - P.pos (M.tail e))
=
inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.head e) - P.face_point (M.dartFace d))
-
inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.tail d) - P.face_point (M.dartFace d)) := by
rw [he_tail]
calc
inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.head e) - P.pos (M.tail d))
=
inner ℝ (P.outward_normal (M.dartFace d))
((P.pos (M.head e) - P.face_point (M.dartFace d))
- (P.pos (M.tail d) - P.face_point (M.dartFace d))) := by
congr 1
module
_ =
inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.head e) - P.face_point (M.dartFace d))
-
inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.tail d) - P.face_point (M.dartFace d)) := by
rw [inner_sub_right]
simpa [reverseFaceBetween, edgeVec, hrewrite, htail_plane] using hstrict
/--
Strict face support plus reverse-`σ` rotation faithfulness gives strict
determinant positivity once the tested vertex is off the supporting triangle.
-/
theorem link_side_strict_of_rotationFaithful
(P : TriangulatedEuclideanPolyhedron M) (hfaith : RotationFaithful P)
(d e : D) (he_tail : M.tail e = M.tail d)
(hoff : ∀ i, M.head e ≠ P.faceVertex (reverseFaceBetween M d) i) :
0 < det3 (edgeVec P d) (edgeVec P (M.σ.symm d)) (edgeVec P e) := by
obtain ⟨lam, hlam, hnormal⟩ :=
hfaith.outward_normal_eq_pos_smul_reverse_cross d
have hs := reverseFaceBetween_support_edgeVec_lt P d e he_tail hoff
rw [hnormal, real_inner_smul_left] at hs
have hcross :
inner ℝ (cross (edgeVec P (M.σ.symm d)) (edgeVec P d)) (edgeVec P e) < 0 := by
nlinarith
rw [det3_eq_inner_cross, cross_antisymm, inner_neg_left]
nlinarith
theorem face_support_from_dart_tail
(P : TriangulatedEuclideanPolyhedron M) (d : D) (w : M.Vertex) :
inner ℝ (P.outward_normal (M.dartFace d)) (P.pos w - P.pos (M.tail d)) ≤ 0 := by
have htail_plane := face_plane_dart P d
have hsupport := P.face_supporting_halfspace (M.dartFace d) w
have hrewrite :
inner ℝ (P.outward_normal (M.dartFace d)) (P.pos w - P.pos (M.tail d)) =
inner ℝ (P.outward_normal (M.dartFace d)) (P.pos w - P.face_point (M.dartFace d)) -
inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.tail d) - P.face_point (M.dartFace d)) := by
calc
inner ℝ (P.outward_normal (M.dartFace d)) (P.pos w - P.pos (M.tail d))
= inner ℝ (P.outward_normal (M.dartFace d))
((P.pos w - P.face_point (M.dartFace d)) -
(P.pos (M.tail d) - P.face_point (M.dartFace d))) := by
congr 1
module
_ = inner ℝ (P.outward_normal (M.dartFace d)) (P.pos w - P.face_point (M.dartFace d)) -
inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.tail d) - P.face_point (M.dartFace d)) := by
rw [inner_sub_right]
rw [hrewrite, htail_plane, sub_zero]
exact hsupport
theorem face_plane_head_sub_tail
(P : TriangulatedEuclideanPolyhedron M) (d : D) :
inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.head d) - P.pos (M.tail d)) = 0 := by
have htail := face_plane_dart P d
have hhead0 := face_plane_dart P (M.φ d)
have hhead :
inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.head d) - P.face_point (M.dartFace d)) = 0 := by
simpa [M.tail_phi] using hhead0
calc
inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.head d) - P.pos (M.tail d))
= inner ℝ (P.outward_normal (M.dartFace d))
((P.pos (M.head d) - P.face_point (M.dartFace d)) -
(P.pos (M.tail d) - P.face_point (M.dartFace d))) := by
congr 1
module
_ = inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.head d) - P.face_point (M.dartFace d)) -
inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.tail d) - P.face_point (M.dartFace d)) := by
rw [inner_sub_right]
_ = 0 := by rw [hhead, htail, sub_self]
theorem face_plane_head_sigma_symm_sub_tail
(P : TriangulatedEuclideanPolyhedron M) (d : D) :
inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.head (M.σ.symm d)) - P.pos (M.tail d)) = 0 := by
have htail := face_plane_dart P d
have hhead0 := face_plane_dart P (M.φ (M.φ d))
have htail_phi2 := tail_phi_phi_eq_head_sigma_symm_of_triangular_euclidean P d
have hhead :
inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.head (M.σ.symm d)) - P.face_point (M.dartFace d)) = 0 := by
simpa [htail_phi2] using hhead0
calc
inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.head (M.σ.symm d)) - P.pos (M.tail d))
= inner ℝ (P.outward_normal (M.dartFace d))
((P.pos (M.head (M.σ.symm d)) - P.face_point (M.dartFace d)) -
(P.pos (M.tail d) - P.face_point (M.dartFace d))) := by
congr 1
module
_ = inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.head (M.σ.symm d)) - P.face_point (M.dartFace d)) -
inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.tail d) - P.face_point (M.dartFace d)) := by
rw [inner_sub_right]
_ = 0 := by rw [hhead, htail, sub_self]
/--
An oriented supporting triangle through `v,a,b`.
This is the non-circular local geometry: the determinant functional of the
oriented triangle is identified with a supporting face normal, with a positive
scale and exact equality set.
-/
structure OrientedTriangleSupport (P : TriangulatedEuclideanPolyhedron M)
(v a b : M.Vertex) where
normal : E3
normal_unit : ‖normal‖ = 1
c : ℝ
c_pos : 0 < c
det_eq :
∀ z : E3,
det3 (P.pos a - P.pos v) (P.pos b - P.pos v) z = -c * inner ℝ normal z
support : ∀ w : M.Vertex, inner ℝ normal (P.pos w - P.pos v) ≤ 0
eq_iff :
∀ w : M.Vertex,
inner ℝ normal (P.pos w - P.pos v) = 0 ↔ (w = v ∨ w = a ∨ w = b)
noncomputable def orientedTriangleSupport_of_rotationFaithful
(P : TriangulatedEuclideanPolyhedron M) (hfaith : RotationFaithful P)
(d e : D) (he_tail : M.tail e = M.tail d)
(hoff : ∀ i, M.head e ≠ P.faceVertex (M.dartFace d) i) :
OrientedTriangleSupport P (M.tail d) (M.head d) (M.head (M.σ.symm d)) := by
classical
let N : E3 := P.outward_normal (M.dartFace d)
let rot := hfaith.outward_normal_eq_pos_smul_reverse_cross d
let lam : ℝ := Classical.choose rot
have hlam : 0 < lam := (Classical.choose_spec rot).1
have hnormal :
P.outward_normal (reverseFaceBetween M d) =
lam • cross (edgeVec P (M.σ.symm d)) (edgeVec P d) :=
(Classical.choose_spec rot).2
have hlam_ne : lam ≠ 0 := ne_of_gt hlam
have hdetpos :
0 < det3 (edgeVec P d) (edgeVec P (M.σ.symm d)) (edgeVec P e) :=
link_side_strict_of_rotationFaithful P hfaith d e he_tail (by
simpa [reverseFaceBetween] using hoff)
have hNne : N ≠ 0 := by
intro hNzero
have hprev0 : cross (edgeVec P (M.σ.symm d)) (edgeVec P d) = 0 := by
have hsmul : lam • cross (edgeVec P (M.σ.symm d)) (edgeVec P d) = 0 := by
rw [← hnormal]
exact hNzero
rcases smul_eq_zero.mp hsmul with hlam0 | hcross0
· exact False.elim (hlam_ne hlam0)
· exact hcross0
have hcross0 : cross (edgeVec P d) (edgeVec P (M.σ.symm d)) = 0 := by
rw [cross_antisymm, hprev0, neg_zero]
have hdet0 : det3 (edgeVec P d) (edgeVec P (M.σ.symm d)) (edgeVec P e) = 0 := by
rw [det3_eq_inner_cross, hcross0, inner_zero_left]
nlinarith
have hnorm_pos : 0 < ‖N‖ := norm_pos_iff.mpr hNne
have hnorm_ne : ‖N‖ ≠ 0 := ne_of_gt hnorm_pos
refine
{ normal := (‖N‖)⁻¹ • N
normal_unit := ?_
c := ‖N‖ * lam⁻¹
c_pos := ?_
det_eq := ?_
support := ?_
eq_iff := ?_ }
· rw [norm_smul, Real.norm_eq_abs,
abs_of_nonneg (inv_nonneg.mpr (norm_nonneg N))]
exact inv_mul_cancel₀ hnorm_ne
· exact mul_pos hnorm_pos (inv_pos.mpr hlam)
· intro z
rw [det3_eq_inner_cross]
have hprev :
cross (edgeVec P (M.σ.symm d)) (edgeVec P d) = lam⁻¹ • N := by
have hN :
N = lam • cross (edgeVec P (M.σ.symm d)) (edgeVec P d) := by
simpa [N, reverseFaceBetween] using hnormal
rw [hN]
rw [smul_smul]
rw [inv_mul_cancel₀ hlam_ne, one_smul]
have hcross :
cross (edgeVec P d) (edgeVec P (M.σ.symm d)) = -(lam⁻¹) • N := by
rw [cross_antisymm, hprev, neg_smul]
have htail_symm : M.tail (M.σ.symm d) = M.tail d := by
rw [← M.tail_sigma (M.σ.symm d), Equiv.apply_symm_apply]
have hcross' :
cross (P.pos (M.head d) - P.pos (M.tail d))
(P.pos (M.head (M.σ.symm d)) - P.pos (M.tail d)) =
-(lam⁻¹) • N := by
simpa [edgeVec, htail_symm] using hcross
rw [hcross', real_inner_smul_left, real_inner_smul_left]
field_simp [hlam_ne, hnorm_ne]
· intro w
have hs := face_support_from_dart_tail P d w
rw [real_inner_smul_left]
have hnonneg : 0 ≤ (‖N‖)⁻¹ := inv_nonneg.mpr (norm_nonneg N)
nlinarith
· intro w
constructor
· intro hz
have hNzero :
inner ℝ N (P.pos w - P.pos (M.tail d)) = 0 := by
rw [real_inner_smul_left] at hz
exact (mul_eq_zero.mp hz).resolve_left (ne_of_gt (inv_pos.mpr hnorm_pos))
by_cases hgoal :
w = M.tail d ∨ w = M.head d ∨ w = M.head (M.σ.symm d)
· exact hgoal
· exfalso
push_neg at hgoal
have hnotFace : ∀ k, w ≠ P.faceVertex (M.dartFace d) k := by
intro k hw
have hcases :=
faceVertex_eq_tail_or_head_or_tail_phi2_of_dartFace_eq P (e := d) rfl k
rw [← hw] at hcases
rcases hcases with htail | hhead | hphi2
· exact hgoal.1 htail
· exact hgoal.2.1 hhead
· have htail_phi2 := tail_phi_phi_eq_head_sigma_symm_of_triangular_euclidean P d
exact hgoal.2.2 (hphi2.trans htail_phi2)
have hstrict0 := P.face_support_strict (M.dartFace d) w hnotFace
have htail_plane := face_plane_dart P d
have hstrict :
inner ℝ N (P.pos w - P.pos (M.tail d)) < 0 := by
have hrewrite :
inner ℝ N (P.pos w - P.pos (M.tail d)) =
inner ℝ N (P.pos w - P.face_point (M.dartFace d)) -
inner ℝ N (P.pos (M.tail d) - P.face_point (M.dartFace d)) := by
calc
inner ℝ N (P.pos w - P.pos (M.tail d))
= inner ℝ N
((P.pos w - P.face_point (M.dartFace d)) -
(P.pos (M.tail d) - P.face_point (M.dartFace d))) := by
congr 1
module
_ = inner ℝ N (P.pos w - P.face_point (M.dartFace d)) -
inner ℝ N (P.pos (M.tail d) - P.face_point (M.dartFace d)) := by
rw [inner_sub_right]
rw [hrewrite, htail_plane, sub_zero]
exact hstrict0
nlinarith
· rintro (rfl | rfl | rfl)
· simp
· rw [real_inner_smul_left, face_plane_head_sub_tail P d, mul_zero]
· rw [real_inner_smul_left, face_plane_head_sigma_symm_sub_tail P d, mul_zero]
/--
Local vertex-link geometry in the outward-normal orientation, i.e. the reverse
of the map's `σ` order at the vertex. The determinant and hemisphere fields of
`VertexStar` are derived from the oriented triangle supports below.
-/
structure VertexLinkGeometry (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex) where
n : ℕ
hn : 2 ≤ n
nbr : Fin (n + 1) → M.Vertex
nbr_is_sigma :
∃ hdeg : 3 ≤ vDeg P v, ∃ e : n = starN P v,
∀ i : Fin (n + 1),
nbr i =
M.head (incidentDartOfStarIndex P v hdeg
(Fin.rev (Fin.cast (by rw [← e]) i)))
oriented : ∀ i : Fin (n + 1), OrientedTriangleSupport P v (nbr i) (nbr (i + 1))
nbr_apex_ne : ∀ i : Fin (n + 1), P.pos (nbr i) ≠ P.pos v
nonincident :
∀ i j : Fin (n + 1), j ≠ i → j ≠ i + 1 →
¬(nbr j = v ∨ nbr j = nbr i ∨ nbr j = nbr (i + 1))
noncomputable def vertexLinkGeometryOfEuclidean
(P : TriangulatedEuclideanPolyhedron M)
(hfaith : RotationFaithful P) (hsimple : M.IsSimpleGraph)
(v : M.Vertex) (hdeg : 3 ≤ vDeg P v) :
VertexLinkGeometry P v := by
classical
refine
{ n := starN P v
hn := starN_ge_two P v hdeg
nbr := reverseLinkNbr P v hdeg
nbr_is_sigma := ?_
oriented := ?_
nbr_apex_ne := reverseLinkNbr_apex_ne P v hdeg
nonincident := reverseLink_nonincident_of_simple P hsimple v hdeg }
· refine ⟨hdeg, rfl, ?_⟩
intro i
rfl
· intro i
let hex := exists_fin_not_incident_edge (starN_ge_two P v hdeg) i
let j : Fin (starN P v + 1) := Classical.choose hex
have hji : j ≠ i := (Classical.choose_spec hex).1
have hjnext : j ≠ i + 1 := (Classical.choose_spec hex).2
let d := reverseLinkDart P v hdeg i
let e := reverseLinkDart P v hdeg j
have he_tail : M.tail e = M.tail d := by
simp [d, e, reverseLinkDart_tail P v hdeg j, reverseLinkDart_tail P v hdeg i]
have hnonincident :=
reverseLink_nonincident_of_simple P hsimple v hdeg i j hji hjnext
have hoff : ∀ k, M.head e ≠ P.faceVertex (M.dartFace d) k := by
intro k hk
have hcases :=
faceVertex_eq_tail_or_head_or_tail_phi2_of_dartFace_eq P (e := d) rfl k
apply hnonincident
rcases hcases with htail | hhead | hphi2
· left
change M.head e = v
calc
M.head e = P.faceVertex (M.dartFace d) k := hk
_ = M.tail d := htail
_ = v := by simpa [d] using reverseLinkDart_tail P v hdeg i
· right; left
change M.head e = M.head d
rw [hk, hhead]
· right; right
change M.head e = reverseLinkNbr P v hdeg (i + 1)
calc
M.head e = P.faceVertex (M.dartFace d) k := hk
_ = M.tail (M.φ (M.φ d)) := hphi2
_ = M.head (M.σ.symm d) :=
tail_phi_phi_eq_head_sigma_symm_of_triangular_euclidean P d
_ = reverseLinkNbr P v hdeg (i + 1) := by
simpa [d] using (reverseLinkNbr_add_one P v hdeg i).symm
have hsupp := orientedTriangleSupport_of_rotationFaithful P hfaith d e he_tail hoff
simpa [d, reverseLinkNbr, reverseLinkDart_tail P v hdeg i,
reverseLinkDart_add_one P v hdeg i] using hsupp
namespace VertexLinkGeometry
variable {P : TriangulatedEuclideanPolyhedron M} {v : M.Vertex}
variable (LG : VertexLinkGeometry P v)
/-- Sum of oriented supporting normals around the vertex. -/
def normalSum : E3 :=
∑ i : Fin (LG.n + 1), (LG.oriented i).normal
theorem exists_nonincident (j : Fin (LG.n + 1)) :
∃ i : Fin (LG.n + 1), j ≠ i ∧ j ≠ i + 1 := by
by_contra hcon
push_neg at hcon
have hsub : (Finset.univ : Finset (Fin (LG.n + 1))) ⊆ {j, j - 1} := by
intro i _
rcases eq_or_ne j i with hji | hji
· simp [hji]
· have hnext := hcon i hji
have him1 : i = j - 1 := by
rw [eq_sub_iff_add_eq]
exact hnext.symm
simp [him1]
have hle := Finset.card_le_card hsub
have hcard : ({j, j - 1} : Finset (Fin (LG.n + 1))).card ≤ 2 :=
le_trans (Finset.card_insert_le _ _) (by simp)
simp only [Finset.card_univ, Fintype.card_fin] at hle
have hle2 : LG.n + 1 ≤ 2 := le_trans hle hcard
have hn : 2 ≤ LG.n := LG.hn
omega
theorem normal_inner_nbr_lt (j : Fin (LG.n + 1)) :
inner ℝ LG.normalSum (P.pos (LG.nbr j) - P.pos v) < 0 := by
rw [normalSum, sum_inner]
have hle :
∀ i ∈ (Finset.univ : Finset (Fin (LG.n + 1))),
inner ℝ (LG.oriented i).normal (P.pos (LG.nbr j) - P.pos v) ≤ 0 := by
intro i _
exact (LG.oriented i).support (LG.nbr j)
have hlt :
∃ i ∈ (Finset.univ : Finset (Fin (LG.n + 1))),
inner ℝ (LG.oriented i).normal (P.pos (LG.nbr j) - P.pos v) < 0 := by
obtain ⟨i, hji, hjnext⟩ := LG.exists_nonincident j
refine ⟨i, by simp, ?_⟩
have hle_i := (LG.oriented i).support (LG.nbr j)
have hne :
inner ℝ (LG.oriented i).normal (P.pos (LG.nbr j) - P.pos v) ≠ 0 := by
intro hz
exact LG.nonincident i j hji hjnext (((LG.oriented i).eq_iff (LG.nbr j)).1 hz)
exact lt_of_le_of_ne hle_i hne
have hsum := Finset.sum_lt_sum hle hlt
simpa using hsum
theorem normalSum_ne_zero : LG.normalSum ≠ 0 := by
intro hzero
have hlt := LG.normal_inner_nbr_lt (0 : Fin (LG.n + 1))
rw [hzero] at hlt
simp at hlt
theorem open_hemi :
∃ h : E3, ‖h‖ = 1 ∧
∀ i : Fin (LG.n + 1), 0 < inner ℝ h (P.pos (LG.nbr i) - P.pos v) := by
let N := LG.normalSum
have hN : N ≠ 0 := LG.normalSum_ne_zero
refine ⟨-(‖N‖)⁻¹ • N, ?_, ?_⟩
· rw [norm_smul, norm_neg, Real.norm_eq_abs,
abs_of_nonneg (inv_nonneg.mpr (norm_nonneg N))]
exact inv_mul_cancel₀ (norm_ne_zero_iff.mpr hN)
· intro i
have hlt : inner ℝ N (P.pos (LG.nbr i) - P.pos v) < 0 := LG.normal_inner_nbr_lt i
rw [real_inner_smul_left]
have hpos : 0 < (‖N‖)⁻¹ := inv_pos.mpr (norm_pos_iff.mpr hN)
nlinarith [mul_pos hpos (neg_pos.mpr hlt)]
theorem turn_support (i j : Fin (LG.n + 1)) :
0 ≤ ProofsInTheBook.SphericalKernel.det3
(P.pos (LG.nbr i) - P.pos v)
(P.pos (LG.nbr (i + 1)) - P.pos v)
(P.pos (LG.nbr j) - P.pos v) := by
have hdet := (LG.oriented i).det_eq (P.pos (LG.nbr j) - P.pos v)
rw [det3_eq_spherical] at hdet
rw [hdet]
have hs := (LG.oriented i).support (LG.nbr j)
nlinarith [(LG.oriented i).c_pos, hs]
theorem turn_strict (i j : Fin (LG.n + 1)) (hji : j ≠ i) (hjnext : j ≠ i + 1) :
0 < ProofsInTheBook.SphericalKernel.det3
(P.pos (LG.nbr i) - P.pos v)
(P.pos (LG.nbr (i + 1)) - P.pos v)
(P.pos (LG.nbr j) - P.pos v) := by
have hdet := (LG.oriented i).det_eq (P.pos (LG.nbr j) - P.pos v)
rw [det3_eq_spherical] at hdet
rw [hdet]
have hs_le := (LG.oriented i).support (LG.nbr j)
have hs_ne :
inner ℝ (LG.oriented i).normal (P.pos (LG.nbr j) - P.pos v) ≠ 0 := by
intro hz
exact LG.nonincident i j hji hjnext (((LG.oriented i).eq_iff (LG.nbr j)).1 hz)
have hs_lt : inner ℝ (LG.oriented i).normal (P.pos (LG.nbr j) - P.pos v) < 0 :=
lt_of_le_of_ne hs_le hs_ne
nlinarith [(LG.oriented i).c_pos, hs_lt]
/-- Assemble the `VertexStar` from honest local vertex-link geometry. -/
def toVertexStar : VertexStar where
n := LG.n
hn := LG.hn
o := P.pos v
p := fun i => P.pos (LG.nbr i)
apex_ne := LG.nbr_apex_ne
open_hemi := LG.open_hemi
turn_support := LG.turn_support
turn_strict := LG.turn_strict
end VertexLinkGeometry
end ProofsInTheBook.Ch13EuclLink
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanCh13EuclLink
import ProofsInTheBook.SphericalCongruence
import ProofsInTheBook.Ch13ArmVertexFull
-/
/- Source module: ProofsInTheBook.ZinanCh13SphAngle -/
section
set_option autoImplicit true
noncomputable section
open scoped Classical RealInnerProductSpace
open ProofsInTheBook.PlanarMap ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.Ch13Euclidean
open ProofsInTheBook.Ch13EuclLink
open ProofsInTheBook.Ch13VertexStar
open ProofsInTheBook.Ch13ArmVertexFull (linkAngle)
open ProofsInTheBook.SphericalKernel
(S2 ShortArc tangentTo tangentTo_eq tangentTo_eq_zero_iff jointAngle sphAngle)
open ProofsInTheBook.SphericalRotation
namespace ProofsInTheBook.Ch13SphAngle
/-- Compatibility abbreviation for assembly lemmas that already carry a `VertexLinkGeometry`. -/
abbrev vertexStarOfEuclidean
{D : Type*} [Fintype D] [DecidableEq D] {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(LG : VertexLinkGeometry P v) : VertexStar :=
LG.toVertexStar
/-- Projection of `u` to the tangent plane at a unit vector `v`. -/
def tangentToVec (v u : E3) : E3 :=
u - (inner ℝ u v) • v
lemma eq_smul_axis_of_cross_zero_unit {k v : E3} (hk : ‖k‖ = 1) (h : cross k v = 0) :
v = (⟪v, k⟫ : ℝ) • k := by
have hkk : (⟪k, k⟫ : ℝ) = 1 := by
rw [real_inner_self_eq_norm_sq, hk]
norm_num
set t : E3 := v - (⟪v, k⟫ : ℝ) • k with ht
have htk : (⟪t, k⟫ : ℝ) = 0 := by
rw [ht, inner_sub_left, real_inner_smul_left, hkk, mul_one, sub_self]
have hct : cross k t = 0 := by
rw [ht, show v - (⟪v, k⟫ : ℝ) • k = v + (-(⟪v, k⟫ : ℝ)) • k by module,
cross_add_right, cross_smul_right, cross_self, smul_zero, add_zero, h]
have hkt : (⟪k, t⟫ : ℝ) = 0 := by
rw [real_inner_comm]
exact htk
have hnorm : ‖t‖ ^ 2 = 0 := by
have hl := norm_sq_cross k t
rw [hct, norm_zero] at hl
rw [hk, hkt] at hl
nlinarith [hl]
have : t = 0 := by
have := pow_eq_zero_iff (n := 2) (by norm_num) |>.mp hnorm
exact norm_eq_zero.mp this
rw [ht] at this
linear_combination (norm := module) this
lemma cross_ne_zero_of_linearIndependent_pair {u v : E3}
(hli : LinearIndependent ℝ ![u, v]) :
cross u v ≠ 0 := by
intro hcross
have hu : u ≠ 0 := by
intro hu0
exact hli.ne_zero 0 (by simpa using hu0)
have hnormu : ‖u‖ ≠ 0 := by
simpa [norm_eq_zero] using hu
set k : E3 := ‖u‖⁻¹ • u with hkdef
have hk : ‖k‖ = 1 := by
rw [hkdef, norm_smul, norm_inv, norm_norm, inv_mul_cancel₀ hnormu]
have hkv : cross k v = 0 := by
rw [hkdef, cross_smul_left, hcross, smul_zero]
have hvk : v = (⟪v, k⟫ : ℝ) • k :=
eq_smul_axis_of_cross_zero_unit hk hkv
have hvu : v = ((⟪v, k⟫ : ℝ) * ‖u‖⁻¹) • u := by
calc
v = (⟪v, k⟫ : ℝ) • k := hvk
_ = ((⟪v, k⟫ : ℝ) * ‖u‖⁻¹) • u := by
rw [hkdef, smul_smul]
have hcontra := ((LinearIndependent.pair_iff' (K := ℝ) (x := u) (y := v) hu).mp hli)
(((⟪v, k⟫ : ℝ) * ‖u‖⁻¹))
exact hcontra hvu.symm
lemma perp_two_imp_parallel_cross {u v n : E3}
(hnu : (⟪n, u⟫ : ℝ) = 0) (hnv : (⟪n, v⟫ : ℝ) = 0)
(hli : LinearIndependent ℝ ![u, v]) :
∃ s : ℝ, n = s • cross u v := by
set X : E3 := cross u v with hXdef
have hX : X ≠ 0 := by
rw [hXdef]
exact cross_ne_zero_of_linearIndependent_pair hli
let s : ℝ := (⟪n, X⟫ : ℝ) / ‖X‖ ^ 2
refine ⟨s, ?_⟩
apply ProofsInTheBook.SphericalCongruence.eq_of_inner_frame_eq hX
· have hux : (⟪u, cross u v⟫ : ℝ) = 0 := by
rw [real_inner_comm]
exact inner_cross_left u v
rw [real_inner_smul_right, hXdef, hux, mul_zero]
rw [real_inner_comm]
exact hnu
· have hvx : (⟪v, cross u v⟫ : ℝ) = 0 := by
rw [real_inner_comm]
exact inner_cross_right u v
rw [real_inner_smul_right, hXdef, hvx, mul_zero]
rw [real_inner_comm]
exact hnv
· have hcrossne : cross u v ≠ 0 := by
rwa [← hXdef]
rw [real_inner_smul_right, hXdef]
change (⟪cross u v, n⟫ : ℝ) =
s * (⟪cross u v, cross u v⟫ : ℝ)
rw [show (⟪cross u v, n⟫ : ℝ) = ⟪n, cross u v⟫ by rw [real_inner_comm],
real_inner_self_eq_norm_sq]
unfold s
have hnorm : ‖cross u v‖ ^ 2 ≠ 0 := by
positivity
field_simp [hnorm]
ring
lemma inner_sub_of_plane_eq {normal point x y : E3}
(hx : inner ℝ normal (x - point) = 0)
(hy : inner ℝ normal (y - point) = 0) :
inner ℝ normal (x - y) = 0 := by
calc
inner ℝ normal (x - y)
= inner ℝ normal ((x - point) - (y - point)) := by
congr 1
module
_ = inner ℝ normal (x - point) - inner ℝ normal (y - point) := by
rw [inner_sub_right]
_ = 0 := by rw [hx, hy, sub_self]
theorem faceDart_phi_ne_self
{D : Type*} [Fintype D] [DecidableEq D] {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) (f : M.Face) :
M.φ (P.faceDart f) ≠ P.faceDart f := by
intro h
let p : Fin 3 → E3 :=
![P.pos (M.tail (P.faceDart f)),
P.pos (M.tail (M.φ (P.faceDart f))),
P.pos (M.tail (M.φ (M.φ (P.faceDart f))))]
have hinj : Function.Injective p := (P.face_nondegenerate f).injective
have hpts : p 1 = p 0 := by
simp [p, h]
have h10 : (1 : Fin 3) = 0 := hinj hpts
norm_num at h10
/-- A dart on a triangular face is one of the three `φ`-successive darts from the
stored representative of that face. -/
theorem dart_eq_faceDart_or_phi_or_phi2_of_dartFace_eq
{D : Type*} [Fintype D] [DecidableEq D] {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) {f : M.Face} {d : D}
(hd : M.dartFace d = f) :
d = P.faceDart f ∨
d = M.φ (P.faceDart f) ∨
d = M.φ (M.φ (P.faceDart f)) := by
let fd := P.faceDart f
have hφne : M.φ fd ≠ fd := by
simpa [fd] using faceDart_phi_ne_self P f
have hlen : M.faceLen f = 3 := by
simpa [CombMap.faceLen] using P.every_face_triangle f
have hcard : (M.φ.cycleOf fd).support.card = 3 := by
rw [← faceLen_dartFace_eq_card_support_cycleOf M hφne]
simpa [fd, P.faceDart_face f] using hlen
have hsame : M.φ.SameCycle fd d := by
have hq : M.dartFace d = M.dartFace fd := by
rw [hd, P.faceDart_face f]
exact (Quotient.exact hq).symm
have hsupp : fd ∈ M.φ.support := Equiv.Perm.mem_support.mpr hφne
obtain ⟨i, hi, hpow⟩ := hsame.exists_pow_eq_of_mem_support hsupp
rw [hcard] at hi
interval_cases i
· left
simpa [fd] using hpow.symm
· right
left
simpa [fd] using hpow.symm
· right
right
simpa [fd, pow_succ] using hpow.symm
/-- Every dart tail is one of the three stored vertices of its dart face. -/
theorem tail_mem_faceVertex
{D : Type*} [Fintype D] [DecidableEq D] {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) (d : D) :
∃ k : Fin 3, M.tail d = P.faceVertex (M.dartFace d) k := by
rcases dart_eq_faceDart_or_phi_or_phi2_of_dartFace_eq P (f := M.dartFace d) (d := d) rfl with
h | h | h
· refine ⟨0, ?_⟩
rw [h]
have hv := congrFun (P.face_vertices_match (M.dartFace d)) 0
simpa [P.faceDart_face (M.dartFace d)] using hv.symm
· refine ⟨1, ?_⟩
rw [h]
have hv := congrFun (P.face_vertices_match (M.dartFace d)) 1
simpa [P.faceDart_face (M.dartFace d)] using hv.symm
· refine ⟨2, ?_⟩
rw [h]
have hv := congrFun (P.face_vertices_match (M.dartFace d)) 2
simpa [P.faceDart_face (M.dartFace d)] using hv.symm
/-- The selected face plane contains the tail of every dart on that face, not only
the stored representative's three tails. -/
theorem face_plane_dart
{D : Type*} [Fintype D] [DecidableEq D] {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) (d : D) :
inner ℝ (P.outward_normal (M.dartFace d))
(P.pos (M.tail d) - P.face_point (M.dartFace d)) = 0 := by
obtain ⟨k, hk⟩ := tail_mem_faceVertex P d
rw [hk]
exact P.face_plane (M.dartFace d) k
theorem faceDart_phi_cube_eq_self
{D : Type*} [Fintype D] [DecidableEq D] {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) (f : M.Face) :
(M.φ ^ 3) (P.faceDart f) = P.faceDart f := by
let fd := P.faceDart f
have hφne : M.φ fd ≠ fd := by
simpa [fd] using faceDart_phi_ne_self P f
have hlen : M.faceLen f = 3 := by
simpa [CombMap.faceLen] using P.every_face_triangle f
have hcard : (M.φ.cycleOf fd).support.card = 3 := by
rw [← faceLen_dartFace_eq_card_support_cycleOf M hφne]
simpa [fd, P.faceDart_face f] using hlen
have hpow := Equiv.Perm.pow_mod_card_support_cycleOf_self_apply M.φ 3 fd
rw [hcard, Nat.mod_self] at hpow
simpa [fd] using hpow.symm
theorem phi_cube_eq_self_of_triangular_euclidean
{D : Type*} [Fintype D] [DecidableEq D] {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) (d : D) :
(M.φ ^ 3) d = d := by
rcases dart_eq_faceDart_or_phi_or_phi2_of_dartFace_eq P (f := M.dartFace d) (d := d) rfl with
h | h | h
· rw [h]
exact faceDart_phi_cube_eq_self P (M.dartFace d)
· rw [h]
exact congrArg M.φ (faceDart_phi_cube_eq_self P (M.dartFace d))
· rw [h]
exact congrArg (fun x => M.φ (M.φ x))
(faceDart_phi_cube_eq_self P (M.dartFace d))
theorem tail_phi_phi_eq_head_sigma_symm_of_triangular_euclidean
{D : Type*} [Fintype D] [DecidableEq D] {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) (d : D) :
M.tail (M.φ (M.φ d)) = M.head (M.σ.symm d) := by
have hcube := phi_cube_eq_self_of_triangular_euclidean P d
have hpred : M.φ (M.φ d) = M.φ.symm d := by
apply M.φ.injective
rw [Equiv.apply_symm_apply]
simpa [pow_succ, Equiv.Perm.coe_mul, Function.comp_apply] using hcube
have hsymm : M.φ.symm d = M.α (M.σ.symm d) := by
apply M.φ.injective
rw [Equiv.apply_symm_apply]
symm
change (M.σ * M.α) (M.α (M.σ.symm d)) = d
rw [Equiv.Perm.mul_apply, M.alpha_alpha, Equiv.apply_symm_apply]
rw [hpred, hsymm, M.tail_alpha]
/-- The stored face vertices are the three tails of any dart on the same
triangular face, up to cyclic rotation. -/
theorem faceVertex_eq_tail_or_head_or_tail_phi2_of_dartFace_eq
{D : Type*} [Fintype D] [DecidableEq D] {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) {f : M.Face} {e : D}
(he : M.dartFace e = f) (k : Fin 3) :
P.faceVertex f k = M.tail e ∨
P.faceVertex f k = M.head e ∨
P.faceVertex f k = M.tail (M.φ (M.φ e)) := by
rcases dart_eq_faceDart_or_phi_or_phi2_of_dartFace_eq P (f := f) (d := e) he with
h | h | h
· subst h
fin_cases k
· left
have hv := congrFun (P.face_vertices_match f) 0
simpa using hv
· right; left
have hv := congrFun (P.face_vertices_match f) 1
simpa [M.tail_phi] using hv
· right; right
have hv := congrFun (P.face_vertices_match f) 2
simpa using hv
· subst h
fin_cases k
· right; right
have hv := congrFun (P.face_vertices_match f) 0
have hcube := faceDart_phi_cube_eq_self P f
have hcube' : M.φ (M.φ (M.φ (P.faceDart f))) = P.faceDart f := by
simpa [pow_succ, Equiv.Perm.coe_mul, Function.comp_apply] using hcube
have htail : M.tail (P.faceDart f) =
M.tail (M.φ (M.φ (M.φ (P.faceDart f)))) := by
rw [hcube']
exact hv.trans htail
· left
have hv := congrFun (P.face_vertices_match f) 1
simpa [M.tail_phi] using hv
· right; left
have hv := congrFun (P.face_vertices_match f) 2
simpa [M.tail_phi] using hv
· subst h
fin_cases k
· right; left
have hv := congrFun (P.face_vertices_match f) 0
have hcube := faceDart_phi_cube_eq_self P f
have hcube' : M.φ (M.φ (M.φ (P.faceDart f))) = P.faceDart f := by
simpa [pow_succ, Equiv.Perm.coe_mul, Function.comp_apply] using hcube
have htail : M.tail (P.faceDart f) =
M.head (M.φ (M.φ (P.faceDart f))) := by
rw [← M.tail_phi (M.φ (M.φ (P.faceDart f))), hcube']
exact hv.trans htail
· right; right
have hv := congrFun (P.face_vertices_match f) 1
have hcube := faceDart_phi_cube_eq_self P f
have hcube' : M.φ (M.φ (M.φ (P.faceDart f))) = P.faceDart f := by
simpa [pow_succ, Equiv.Perm.coe_mul, Function.comp_apply] using hcube
have htail : M.tail (M.φ (P.faceDart f)) =
M.tail (M.φ (M.φ (M.φ (M.φ (P.faceDart f))))) := by
rw [hcube']
exact hv.trans htail
· left
have hv := congrFun (P.face_vertices_match f) 2
simpa using hv
lemma linearIndependent_pair_of_cross_ne_zero {u v : E3}
(hcross : cross u v ≠ 0) :
LinearIndependent ℝ ![u, v] := by
by_cases hu : u = 0
· exfalso
apply hcross
rw [hu]
have hzero := cross_smul_left (0 : ℝ) v v
simpa using hzero
rw [LinearIndependent.pair_iff' hu]
intro a hv
apply hcross
rw [← hv, cross_smul_right, cross_self, smul_zero]
theorem normal_eq_pos_smul_neg_cross_of_support {n u v w : E3}
(hnu : (⟪n, u⟫ : ℝ) = 0) (hnv : (⟪n, v⟫ : ℝ) = 0)
(hli : LinearIndependent ℝ ![u, v])
(hopp : (⟪n, w⟫ : ℝ) < 0)
(hdet : 0 < (⟪cross u v, w⟫ : ℝ)) :
∃ lam : ℝ, 0 < lam ∧ n = lam • (-(cross u v)) := by
obtain ⟨s, hs⟩ := perp_two_imp_parallel_cross hnu hnv hli
have hsneg : s < 0 := by
have hdot : (⟪n, w⟫ : ℝ) = s * ⟪cross u v, w⟫ := by
rw [hs, real_inner_smul_left]
nlinarith
refine ⟨-s, by linarith, ?_⟩
rw [hs]
module
theorem face_normal_eq_pos_smul_neg_cross_of_coplanar_edges
{D : Type*} [Fintype D] [DecidableEq D] {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) (f : M.Face)
(v a b w : M.Vertex)
(hv : inner ℝ (P.outward_normal f) (P.pos v - P.face_point f) = 0)
(ha : inner ℝ (P.outward_normal f) (P.pos a - P.face_point f) = 0)
(hb : inner ℝ (P.outward_normal f) (P.pos b - P.face_point f) = 0)
(hli : LinearIndependent ℝ ![P.pos a - P.pos v, P.pos b - P.pos v])
(hw : ∀ i, w ≠ P.faceVertex f i)
(hdet : 0 < inner ℝ (cross (P.pos a - P.pos v) (P.pos b - P.pos v))
(P.pos w - P.pos v)) :
∃ lam : ℝ, 0 < lam ∧
P.outward_normal f = lam • (-(cross (P.pos a - P.pos v) (P.pos b - P.pos v))) := by
have hstrict0 := P.face_support_strict f w hw
have hstrict : inner ℝ (P.outward_normal f) (P.pos w - P.pos v) < 0 := by
have hrewrite :
inner ℝ (P.outward_normal f) (P.pos w - P.pos v) =
inner ℝ (P.outward_normal f) (P.pos w - P.face_point f) -
inner ℝ (P.outward_normal f) (P.pos v - P.face_point f) := by
calc
inner ℝ (P.outward_normal f) (P.pos w - P.pos v)
= inner ℝ (P.outward_normal f)
((P.pos w - P.face_point f) - (P.pos v - P.face_point f)) := by
congr 1
module
_ = inner ℝ (P.outward_normal f) (P.pos w - P.face_point f) -
inner ℝ (P.outward_normal f) (P.pos v - P.face_point f) := by
rw [inner_sub_right]
rw [hrewrite, hv, sub_zero]
exact hstrict0
have hperp1 : inner ℝ (P.outward_normal f) (P.pos a - P.pos v) = 0 :=
inner_sub_of_plane_eq ha hv
have hperp2 : inner ℝ (P.outward_normal f) (P.pos b - P.pos v) = 0 :=
inner_sub_of_plane_eq hb hv
exact normal_eq_pos_smul_neg_cross_of_support hperp1 hperp2 hli hstrict hdet
/-- The outward normal of `dartFace e` is the negative cross product of the two
face edges emanating from `tail e`, up to a positive scalar, once a strict
off-face vertex supplies the sign. -/
theorem face_normal_eq_pos_smul_neg_cross_of_dart_edges
{D : Type*} [Fintype D] [DecidableEq D] {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) (e : D) (w : M.Vertex)
(hw : ∀ i, w ≠ P.faceVertex (M.dartFace e) i)
(hdet : 0 < inner ℝ
(cross
(P.pos (M.head e) - P.pos (M.tail e))
(P.pos (M.tail (M.φ (M.φ e))) - P.pos (M.tail e)))
(P.pos w - P.pos (M.tail e))) :
∃ lam : ℝ, 0 < lam ∧
P.outward_normal (M.dartFace e) =
lam • (-
cross
(P.pos (M.head e) - P.pos (M.tail e))
(P.pos (M.tail (M.φ (M.φ e))) - P.pos (M.tail e))) := by
have hv : inner ℝ (P.outward_normal (M.dartFace e))
(P.pos (M.tail e) - P.face_point (M.dartFace e)) = 0 :=
face_plane_dart P e
have ha : inner ℝ (P.outward_normal (M.dartFace e))
(P.pos (M.head e) - P.face_point (M.dartFace e)) = 0 := by
have h := face_plane_dart P (M.φ e)
simpa [M.tail_phi] using h
have hb : inner ℝ (P.outward_normal (M.dartFace e))
(P.pos (M.tail (M.φ (M.φ e))) - P.face_point (M.dartFace e)) = 0 := by
have h := face_plane_dart P (M.φ (M.φ e))
simpa using h
have hcrossne :
cross
(P.pos (M.head e) - P.pos (M.tail e))
(P.pos (M.tail (M.φ (M.φ e))) - P.pos (M.tail e)) ≠ 0 := by
intro hzero
rw [hzero] at hdet
simp at hdet
have hli : LinearIndependent ℝ
![P.pos (M.head e) - P.pos (M.tail e),
P.pos (M.tail (M.φ (M.φ e))) - P.pos (M.tail e)] :=
linearIndependent_pair_of_cross_ne_zero hcrossne
exact face_normal_eq_pos_smul_neg_cross_of_coplanar_edges
(P := P) (f := M.dartFace e)
(v := M.tail e) (a := M.head e)
(b := M.tail (M.φ (M.φ e))) (w := w)
hv ha hb hli hw hdet
lemma fin_sub_one_add_one {n : ℕ} [NeZero n] (i : Fin n) :
i - 1 + 1 = i := by
rw [sub_add_cancel]
lemma vertexLinkGeometry_exists_noninc_face
{D : Type*} [Fintype D] [DecidableEq D] {M : CombMap D}
{P : TriangulatedEuclideanPolyhedron M} {v : M.Vertex}
(LG : VertexLinkGeometry P v) (i : Fin (LG.n + 1)) :
∃ j : Fin (LG.n + 1), j ≠ i ∧ j ≠ i + 1 := by
by_contra hcon
push_neg at hcon
have hsub : (Finset.univ : Finset (Fin (LG.n + 1))) ⊆ {i, i + 1} := by
intro j _
rcases eq_or_ne j i with hji | hji
· simp [hji]
· have := hcon j hji
simp [this]
have hle := Finset.card_le_card hsub
simp only [Finset.card_univ, Fintype.card_fin] at hle
have hle2 : ({i, i + 1} : Finset (Fin (LG.n + 1))).card ≤ 2 :=
le_trans (Finset.card_insert_le _ _) (by simp)
have := LG.hn
omega
lemma normal_neg_raw_cross_to_edgeDir_cross
(S : VertexStar) (i j : Fin (S.n + 1)) {normal : E3}
{lam : ℝ} (hlam : 0 < lam)
(hraw : normal = lam • (-(cross (S.rawDir i) (S.rawDir j)))) :
∃ lam' : ℝ, 0 < lam' ∧
normal = lam' • (-(cross (S.edgeDir i : E3) (S.edgeDir j : E3))) := by
let c : ℝ := ‖S.rawDir i‖⁻¹ * ‖S.rawDir j‖⁻¹
have hcpos : 0 < c := mul_pos (S.inv_norm_pos i) (S.inv_norm_pos j)
have hcross :
cross (S.edgeDir i : E3) (S.edgeDir j : E3) =
c • cross (S.rawDir i) (S.rawDir j) := by
rw [S.edgeDir_coe i, S.edgeDir_coe j, cross_smul_left, cross_smul_right]
simp [c, smul_smul, mul_comm, mul_left_comm, mul_assoc]
refine ⟨lam / c, div_pos hlam hcpos, ?_⟩
rw [hraw, hcross]
have hcne : c ≠ 0 := ne_of_gt hcpos
have hcoef : lam / c * c = lam := by
field_simp [hcne]
have hcoef_neg : lam / c * -c = -lam := by
nlinarith
conv_lhs => rw [smul_neg, ← neg_smul]
conv_rhs => rw [← neg_smul, smul_smul]
rw [hcoef_neg]
lemma inner_tangent_tangent_unit {a b c : E3} (hb : ‖b‖ = 1) :
inner ℝ (tangentToVec b a) (tangentToVec b c)
= inner ℝ a c - inner ℝ a b * inner ℝ b c := by
unfold tangentToVec
have hb_inner : inner ℝ b b = (1 : ℝ) := by
rw [real_inner_self_eq_norm_sq, hb]
norm_num
simp [inner_sub_left, inner_sub_right, real_inner_smul_left, real_inner_smul_right,
hb_inner, real_inner_comm]
rw [hb]
ring_nf
lemma inner_cross_cross_unit {a b c : E3} (hb : ‖b‖ = 1) :
inner ℝ (cross a b) (cross b c)
= inner ℝ a b * inner ℝ b c - inner ℝ a c := by
have hb_inner : inner ℝ b b = (1 : ℝ) := by
rw [real_inner_self_eq_norm_sq, hb]
norm_num
rw [inner_cross_cross, hb_inner]
ring
lemma inner_tangent_eq_neg_inner_cross {a b c : E3}
(ha : ‖a‖ = 1) (hb : ‖b‖ = 1) (hc : ‖c‖ = 1) :
inner ℝ (tangentToVec b a) (tangentToVec b c)
= - inner ℝ (cross a b) (cross b c) := by
rw [inner_tangent_tangent_unit hb, inner_cross_cross_unit hb]
ring
lemma norm_tangent_sq_unit {a b : E3} (ha : ‖a‖ = 1) (hb : ‖b‖ = 1) :
‖tangentToVec b a‖ ^ 2 = 1 - (inner ℝ a b) ^ 2 := by
rw [← real_inner_self_eq_norm_sq, inner_tangent_tangent_unit (a := a) (b := b) (c := a) hb]
have ha_inner : inner ℝ a a = (1 : ℝ) := by
rw [real_inner_self_eq_norm_sq, ha]
norm_num
rw [ha_inner]
rw [real_inner_comm b a]
ring
lemma norm_cross_sq_unit {a b : E3} (ha : ‖a‖ = 1) (hb : ‖b‖ = 1) :
‖cross a b‖ ^ 2 = 1 - (inner ℝ a b) ^ 2 := by
rw [norm_sq_cross, ha, hb]
ring
lemma norm_tangent_eq_norm_cross_left {a b : E3} (ha : ‖a‖ = 1) (hb : ‖b‖ = 1) :
‖tangentToVec b a‖ = ‖cross a b‖ := by
have hsq : ‖tangentToVec b a‖ ^ 2 = ‖cross a b‖ ^ 2 := by
rw [norm_tangent_sq_unit ha hb, norm_cross_sq_unit ha hb]
rcases sq_eq_sq_iff_eq_or_eq_neg.mp hsq with h | h
· exact h
· have h1 : 0 ≤ ‖tangentToVec b a‖ := norm_nonneg _
have h2 : 0 ≤ ‖cross a b‖ := norm_nonneg _
linarith
lemma norm_tangent_eq_norm_cross_right {b c : E3} (hb : ‖b‖ = 1) (hc : ‖c‖ = 1) :
‖tangentToVec b c‖ = ‖cross b c‖ := by
have hsq : ‖tangentToVec b c‖ ^ 2 = ‖cross b c‖ ^ 2 := by
rw [norm_tangent_sq_unit hc hb, norm_cross_sq_unit hb hc]
rw [real_inner_comm c b]
rcases sq_eq_sq_iff_eq_or_eq_neg.mp hsq with h | h
· exact h
· have h1 : 0 ≤ ‖tangentToVec b c‖ := norm_nonneg _
have h2 : 0 ≤ ‖cross b c‖ := norm_nonneg _
linarith
lemma cos_tangent_eq_neg_cos_cross {a b c : E3}
(ha : ‖a‖ = 1) (hb : ‖b‖ = 1) (hc : ‖c‖ = 1)
(hta : tangentToVec b a ≠ 0) (htc : tangentToVec b c ≠ 0)
(hX : cross a b ≠ 0) (hY : cross b c ≠ 0) :
Real.cos (InnerProductGeometry.angle (tangentToVec b a) (tangentToVec b c))
=
- Real.cos (InnerProductGeometry.angle (cross a b) (cross b c)) := by
rw [InnerProductGeometry.cos_angle, InnerProductGeometry.cos_angle,
inner_tangent_eq_neg_inner_cross ha hb hc,
norm_tangent_eq_norm_cross_left ha hb,
norm_tangent_eq_norm_cross_right hb hc]
field_simp [norm_pos_iff.mpr hX, norm_pos_iff.mpr hY]
lemma tangent_angle_eq_pi_sub_cross_angle {a b c : E3}
(ha : ‖a‖ = 1) (hb : ‖b‖ = 1) (hc : ‖c‖ = 1)
(hta : tangentToVec b a ≠ 0) (htc : tangentToVec b c ≠ 0)
(hX : cross a b ≠ 0) (hY : cross b c ≠ 0) :
InnerProductGeometry.angle (tangentToVec b a) (tangentToVec b c)
=
Real.pi - InnerProductGeometry.angle (cross a b) (cross b c) := by
apply Real.injOn_cos.eq_iff
(s := Set.Icc (0 : ℝ) Real.pi)
⟨InnerProductGeometry.angle_nonneg _ _, InnerProductGeometry.angle_le_pi _ _⟩
⟨by linarith [InnerProductGeometry.angle_le_pi (cross a b) (cross b c)],
by linarith [InnerProductGeometry.angle_nonneg (cross a b) (cross b c)]⟩ |>.1
rw [cos_tangent_eq_neg_cos_cross ha hb hc hta htc hX hY]
rw [Real.cos_pi_sub]
lemma angle_normals_eq_cross {a b c n_f n_g : E3}
(horient :
∃ l m : ℝ,
0 < l ∧ 0 < m ∧
((n_f = l • cross a b ∧ n_g = m • cross b c) ∨
(n_f = l • (-(cross a b)) ∧ n_g = m • (-(cross b c))))) :
InnerProductGeometry.angle n_f n_g
=
InnerProductGeometry.angle (cross a b) (cross b c) := by
rcases horient with ⟨l, m, hl, hm, hcase⟩
rcases hcase with ⟨hf, hg⟩ | ⟨hf, hg⟩
· rw [hf, hg]
rw [InnerProductGeometry.angle_smul_left_of_pos _ _ hl]
rw [InnerProductGeometry.angle_smul_right_of_pos _ _ hm]
· rw [hf, hg]
rw [InnerProductGeometry.angle_smul_left_of_pos _ _ hl]
rw [InnerProductGeometry.angle_smul_right_of_pos _ _ hm]
rw [InnerProductGeometry.angle_neg_neg]
theorem sphAngle_eq_pi_sub_normal_angle {a b c n_f n_g : E3}
(ha : ‖a‖ = 1) (hb : ‖b‖ = 1) (hc : ‖c‖ = 1)
(hta : tangentToVec b a ≠ 0) (htc : tangentToVec b c ≠ 0)
(hX : cross a b ≠ 0) (hY : cross b c ≠ 0)
(horient :
∃ l m : ℝ,
0 < l ∧ 0 < m ∧
((n_f = l • cross a b ∧ n_g = m • cross b c) ∨
(n_f = l • (-(cross a b)) ∧ n_g = m • (-(cross b c))))) :
InnerProductGeometry.angle (tangentToVec b a) (tangentToVec b c)
=
Real.pi - InnerProductGeometry.angle n_f n_g := by
calc
InnerProductGeometry.angle (tangentToVec b a) (tangentToVec b c)
= Real.pi - InnerProductGeometry.angle (cross a b) (cross b c) :=
tangent_angle_eq_pi_sub_cross_angle ha hb hc hta htc hX hY
_ = Real.pi - InnerProductGeometry.angle n_f n_g := by
rw [angle_normals_eq_cross horient]
lemma tangentTo_eq_tangentToVec (p q : S2) :
tangentTo p q = tangentToVec (p : E3) (q : E3) := by
rw [tangentTo_eq]
rfl
theorem linkAngle_vertexLink_eq_pi_sub_normal_angle
(S : VertexStar) (i : Fin (S.n + 1)) {n_f n_g : E3}
(hta :
tangentToVec (S.edgeDir i : E3) (S.edgeDir (i - 1) : E3) ≠ 0)
(htc :
tangentToVec (S.edgeDir i : E3) (S.edgeDir (i + 1) : E3) ≠ 0)
(hX : cross (S.edgeDir (i - 1) : E3) (S.edgeDir i : E3) ≠ 0)
(hY : cross (S.edgeDir i : E3) (S.edgeDir (i + 1) : E3) ≠ 0)
(horient :
∃ l m : ℝ,
0 < l ∧ 0 < m ∧
((n_f = l • cross (S.edgeDir (i - 1) : E3) (S.edgeDir i : E3) ∧
n_g = m • cross (S.edgeDir i : E3) (S.edgeDir (i + 1) : E3)) ∨
(n_f = l • (-(cross (S.edgeDir (i - 1) : E3) (S.edgeDir i : E3))) ∧
n_g = m • (-(cross (S.edgeDir i : E3) (S.edgeDir (i + 1) : E3)))))) :
linkAngle S.vertexLink i = Real.pi - InnerProductGeometry.angle n_f n_g := by
rw [linkAngle]
simp only [VertexStar.vertexLink_apply]
rw [sphAngle]
show InnerProductGeometry.angle
(tangentTo (S.edgeDir i) (S.edgeDir (i - 1)))
(tangentTo (S.edgeDir i) (S.edgeDir (i + 1)))
= Real.pi - InnerProductGeometry.angle n_f n_g
rw [tangentTo_eq_tangentToVec, tangentTo_eq_tangentToVec]
exact sphAngle_eq_pi_sub_normal_angle
(a := (S.edgeDir (i - 1) : E3))
(b := (S.edgeDir i : E3))
(c := (S.edgeDir (i + 1) : E3))
(n_f := n_f) (n_g := n_g)
(S.edgeDir (i - 1)).2 (S.edgeDir i).2 (S.edgeDir (i + 1)).2
hta htc hX hY horient
theorem dihedralAngleAtDart_eq_linkAngle_of_neighbors
{D : Type*} [Fintype D] [DecidableEq D] {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) (d : D)
(LG : VertexLinkGeometry P (M.tail d)) (J : Fin (LG.n + 1))
(hprev : LG.nbr (J - 1) = M.head (M.σ d))
(hcenter : LG.nbr J = M.head d)
(hnext : LG.nbr (J + 1) = M.head (M.σ.symm d)) :
dihedralAngleAtDart P d =
linkAngle (vertexStarOfEuclidean P (M.tail d) LG).vertexLink J := by
let S : VertexStar := vertexStarOfEuclidean P (M.tail d) LG
let cJ : Fin (S.n + 1) := (show Fin (S.n + 1) from J)
change dihedralAngleAtDart P d = linkAngle S.vertexLink cJ
have hJprev_next : (J - 1) + 1 = J := fin_sub_one_add_one J
have hcJprev_next : (cJ - 1) + 1 = cJ := fin_sub_one_add_one cJ
obtain ⟨wF, hwF_ne0, hwF_ne1⟩ := vertexLinkGeometry_exists_noninc_face LG (J - 1)
obtain ⟨wG, hwG_ne0, hwG_ne1⟩ := vertexLinkGeometry_exists_noninc_face LG J
have hface_sigma : M.dartFace (M.σ d) = M.dartFace (M.α d) := by
have hσφ : M.σ d = M.φ (M.α d) := by
change M.σ d = (M.σ * M.α) (M.α d)
rw [Equiv.Perm.mul_apply, M.alpha_alpha]
rw [hσφ, M.dartFace_phi]
have htail_sigma : M.tail (M.σ d) = M.tail d := M.tail_sigma d
have htail_phi2_sigma :
M.tail (M.φ (M.φ (M.σ d))) = M.head d := by
rw [tail_phi_phi_eq_head_sigma_symm_of_triangular_euclidean P (M.σ d),
Equiv.symm_apply_apply]
have hhead_phi_sigma : M.head (M.φ (M.σ d)) = M.head d := by
simpa [M.tail_phi] using htail_phi2_sigma
have htail_phi2_d :
M.tail (M.φ (M.φ d)) = M.head (M.σ.symm d) :=
tail_phi_phi_eq_head_sigma_symm_of_triangular_euclidean P d
have hhead_phi_d : M.head (M.φ d) = M.head (M.σ.symm d) := by
simpa [M.tail_phi] using htail_phi2_d
have hdetF : 0 < inner ℝ
(cross
(P.pos (M.head (M.σ d)) - P.pos (M.tail d))
(P.pos (M.head d) - P.pos (M.tail d)))
(P.pos (LG.nbr wF) - P.pos (M.tail d)) := by
have h := LG.turn_strict (J - 1) wF hwF_ne0 hwF_ne1
rw [← det3_eq_spherical, det3_eq_inner_cross] at h
simpa [hprev, hcenter, hJprev_next] using h
have hwF_face : ∀ k, LG.nbr wF ≠ P.faceVertex (M.dartFace (M.σ d)) k := by
intro k heq
have hcases := faceVertex_eq_tail_or_head_or_tail_phi2_of_dartFace_eq
(P := P) (e := M.σ d) (f := M.dartFace (M.σ d)) rfl k
rw [← heq, htail_sigma, htail_phi2_sigma] at hcases
have hnon := LG.nonincident (J - 1) wF hwF_ne0 hwF_ne1
have hnon' :
¬LG.nbr wF = M.tail d ∧
¬LG.nbr wF = M.head (M.σ d) ∧ ¬LG.nbr wF = M.head d := by
simpa [hprev, hJprev_next, hcenter] using hnon
rcases hcases with htail | hsig | hhead
· exact hnon'.1 htail
· exact hnon'.2.1 hsig
· exact hnon'.2.2 hhead
obtain ⟨lamF, hlamF, hnormalFraw⟩ :=
face_normal_eq_pos_smul_neg_cross_of_dart_edges P (M.σ d) (LG.nbr wF)
hwF_face (by simpa [htail_sigma, htail_phi2_sigma] using hdetF)
have hrawF_left :
S.rawDir (cJ - 1) =
P.pos (M.head (M.σ d)) - P.pos (M.tail d) := by
unfold S cJ vertexStarOfEuclidean VertexLinkGeometry.toVertexStar VertexStar.rawDir
change P.pos (LG.nbr (J - 1)) - P.pos (M.tail d) =
P.pos (M.head (M.σ d)) - P.pos (M.tail d)
rw [hprev]
have hrawF_right :
S.rawDir cJ =
P.pos (M.head d) - P.pos (M.tail d) := by
unfold S cJ vertexStarOfEuclidean VertexLinkGeometry.toVertexStar VertexStar.rawDir
change P.pos (LG.nbr J) - P.pos (M.tail d) =
P.pos (M.head d) - P.pos (M.tail d)
rw [hcenter]
have hnormalFraw' :
dartNormal P (M.α d) =
lamF • (-(cross
(S.rawDir (cJ - 1))
(S.rawDir cJ))) := by
rw [hrawF_left, hrawF_right]
unfold dartNormal
rw [← hface_sigma]
simpa [htail_sigma, hhead_phi_sigma, smul_neg] using hnormalFraw
obtain ⟨lamF', hlamF', hnormalF⟩ :=
normal_neg_raw_cross_to_edgeDir_cross S
(cJ - 1) cJ hlamF hnormalFraw'
have hdetG : 0 < inner ℝ
(cross
(P.pos (M.head d) - P.pos (M.tail d))
(P.pos (M.head (M.σ.symm d)) - P.pos (M.tail d)))
(P.pos (LG.nbr wG) - P.pos (M.tail d)) := by
have h := LG.turn_strict J wG hwG_ne0 hwG_ne1
rw [← det3_eq_spherical, det3_eq_inner_cross] at h
simpa [hcenter, hnext] using h
have hwG_face : ∀ k, LG.nbr wG ≠ P.faceVertex (M.dartFace d) k := by
intro k heq
have hcases := faceVertex_eq_tail_or_head_or_tail_phi2_of_dartFace_eq
(P := P) (e := d) (f := M.dartFace d) rfl k
rw [← heq, htail_phi2_d] at hcases
have hnon := LG.nonincident J wG hwG_ne0 hwG_ne1
have hnon' :
¬LG.nbr wG = M.tail d ∧
¬LG.nbr wG = M.head d ∧ ¬LG.nbr wG = M.head (M.σ.symm d) := by
simpa [hcenter, hnext] using hnon
rcases hcases with htail | hhead | hnext'
· exact hnon'.1 htail
· exact hnon'.2.1 hhead
· exact hnon'.2.2 hnext'
obtain ⟨lamG, hlamG, hnormalGraw⟩ :=
face_normal_eq_pos_smul_neg_cross_of_dart_edges P d (LG.nbr wG)
hwG_face (by simpa [htail_phi2_d] using hdetG)
have hrawG_right :
S.rawDir (cJ + 1) =
P.pos (M.head (M.σ.symm d)) - P.pos (M.tail d) := by
unfold S cJ vertexStarOfEuclidean VertexLinkGeometry.toVertexStar VertexStar.rawDir
change P.pos (LG.nbr (J + 1)) - P.pos (M.tail d) =
P.pos (M.head (M.σ.symm d)) - P.pos (M.tail d)
rw [hnext]
have hnormalGraw' :
dartNormal P d =
lamG • (-(cross
(S.rawDir cJ)
(S.rawDir (cJ + 1)))) := by
rw [hrawF_right, hrawG_right]
unfold dartNormal
simpa [hhead_phi_d, smul_neg] using hnormalGraw
obtain ⟨lamG', hlamG', hnormalG⟩ :=
normal_neg_raw_cross_to_edgeDir_cross S
cJ (cJ + 1) hlamG hnormalGraw'
have hta :
tangentToVec (S.edgeDir cJ : E3)
(S.edgeDir (cJ - 1) : E3) ≠ 0 := by
intro hzero
have ht : tangentTo (S.edgeDir cJ)
(S.edgeDir (cJ - 1)) = 0 := by
simpa [tangentTo_eq_tangentToVec] using hzero
have hnot := (tangentTo_eq_zero_iff
(S.edgeDir cJ)
(S.edgeDir (cJ - 1))).mp ht
have hsa : ShortArc (S.edgeDir cJ)
(S.edgeDir (cJ - 1)) := by
have h0 := S.edgeDir_shortArc (cJ - 1)
simpa [hcJprev_next] using h0.symm
exact hnot hsa
have htc :
tangentToVec (S.edgeDir cJ : E3)
(S.edgeDir (cJ + 1) : E3) ≠ 0 := by
intro hzero
have ht : tangentTo (S.edgeDir cJ)
(S.edgeDir (cJ + 1)) = 0 := by
simpa [tangentTo_eq_tangentToVec] using hzero
have hnot := (tangentTo_eq_zero_iff
(S.edgeDir cJ)
(S.edgeDir (cJ + 1))).mp ht
exact hnot (S.edgeDir_shortArc cJ)
have hX :
cross (S.edgeDir (cJ - 1) : E3)
(S.edgeDir cJ : E3) ≠ 0 := by
have hsa := S.edgeDir_shortArc (cJ - 1)
exact ProofsInTheBook.SphericalCongruence.cross_ne_zero_of_shortArc
(by simpa [hcJprev_next] using hsa)
have hY :
cross (S.edgeDir cJ : E3)
(S.edgeDir (cJ + 1) : E3) ≠ 0 :=
ProofsInTheBook.SphericalCongruence.cross_ne_zero_of_shortArc
(S.edgeDir_shortArc cJ)
have hlink := linkAngle_vertexLink_eq_pi_sub_normal_angle
(S := S) (i := cJ)
(n_f := dartNormal P (M.α d)) (n_g := dartNormal P d)
hta htc hX hY
⟨lamF', lamG', hlamF', hlamG', Or.inr ⟨hnormalF, hnormalG⟩⟩
unfold dihedralAngleAtDart
rw [hlink, InnerProductGeometry.angle_comm]
theorem dihedralAngleAtDart_eq_linkAngle
{D : Type*} [Fintype D] [DecidableEq D] {M : CombMap D}
(P : TriangulatedEuclideanPolyhedron M) (d : D)
(LG : VertexLinkGeometry P (M.tail d)) (J : Fin (LG.n + 1))
(hprev : LG.nbr (J - 1) = M.head (M.σ d))
(hcenter : LG.nbr J = M.head d)
(hnext : LG.nbr (J + 1) = M.head (M.σ.symm d)) :
dihedralAngleAtDart P d =
linkAngle (vertexStarOfEuclidean P (M.tail d) LG).vertexLink J :=
dihedralAngleAtDart_eq_linkAngle_of_neighbors P d LG J hprev hcenter hnext
end ProofsInTheBook.Ch13SphAngle
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.Ch13VertexStar
-/
/- Source module: ProofsInTheBook.Ch13LinkSides -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Ch13VertexStar
open scoped RealInnerProductSpace
open ProofsInTheBook.SphericalKernel
/-- **Equal face angles ⟹ equal link side lengths.** Two vertex stars `S`, `T` with the same edge
count and matching corresponding face angles at the apex have, by Bridge A, equal link side lengths.
This is the `equal_sides` input of the Cauchy arm obstruction, derived (not posited). -/
theorem sideLen_vertexLink_eq_of_faceAngle_eq
(S T : VertexStar) (hnT : T.n = S.n)
(hface : ∀ i : Fin S.n,
EuclideanGeometry.angle (S.p i.castSucc) S.o (S.p i.succ)
= EuclideanGeometry.angle (T.p (i.cast hnT.symm).castSucc) T.o (T.p (i.cast hnT.symm).succ)) :
∀ i : Fin S.n,
sideLen S.vertexLink i = sideLen T.vertexLink (i.cast hnT.symm) := by
intro i
rw [S.sideLen_vertexLink, T.sideLen_vertexLink, hface i]
end ProofsInTheBook.Ch13VertexStar
end
/- Original source header (imports hoisted):
import ProofsInTheBook.Ch13ArmVertex
-/
/- Source module: ProofsInTheBook.Ch13SubArcWrap -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.Ch13SubArc
open ProofsInTheBook.Ch13ArmVertex
namespace ProofsInTheBook.Ch13SubArcWrap
/-- **Cyclic rotation preserves interior joints.** The `i`-th joint of `rotPoly A k` is the
spherical angle at the parent triple starting at `⟨i.val⟩ + k`. -/
theorem rotPoly_jointAngle {n : ℕ} (A : Fin (n + 1) → S2) (k : Fin (n + 1)) (i : Fin (n - 1)) :
jointAngle (rotPoly A k) i
= sphAngle (A (⟨i.val, by have := i.isLt; omega⟩ + k))
(A (⟨i.val + 1, by have := i.isLt; omega⟩ + k))
(A (⟨i.val + 2, by have := i.isLt; omega⟩ + k)) := by
unfold jointAngle rotPoly
rfl
/-- The wrap parameter `M = (n+1) - s + t` for cut indices `t < s ≤ n`. The wrapped arc has `M + 1`
vertices `A s, …, A n, A 0, …, A t`. -/
def wrapLen (n s t : ℕ) : ℕ := (n + 1) - s + t
/-- For `t < s ≤ n`, the wrap parameter satisfies `0 < M ≤ n`, so `subArc (rotPoly A s) 0 M` is a
valid contiguous sub-arc of the rotated polygon. -/
theorem wrapLen_lt_le {n s t : ℕ} (hts : t < s) (hsn : s ≤ n) :
0 < wrapLen n s t ∧ wrapLen n s t ≤ n := by
unfold wrapLen; omega
/-- The wrapped contiguous sub-arc `A s, A (s+1), …, A n, A 0, …, A t` (`t < s ≤ n`), realized as the
non-wrapping range `[0 .. M]` of the rotated polygon `rotPoly A s` (`M = (n+1) - s + t`). -/
def subArcWrap {n : ℕ} (A : Fin (n + 1) → S2) (t s : ℕ) (hts : t < s) (hsn : s ≤ n) :
Fin (wrapLen n s t + 1) → S2 :=
subArc (rotPoly A ⟨s, by omega⟩) 0 (wrapLen n s t)
(wrapLen_lt_le hts hsn).1 (wrapLen_lt_le hts hsn).2
/-- `subArcWrap A t s i = A (⟨i.val⟩ + s)` (cyclic `Fin (n+1)` addition). -/
theorem subArcWrap_apply {n : ℕ} (A : Fin (n + 1) → S2) (t s : ℕ) (hts : t < s) (hsn : s ≤ n)
(i : Fin (wrapLen n s t + 1)) :
subArcWrap A t s hts hsn i
= A (⟨i.val, by have := i.isLt; unfold wrapLen at this; omega⟩ + ⟨s, by omega⟩) := by
unfold subArcWrap
rw [subArc_apply]
show rotPoly A ⟨s, by omega⟩ ⟨0 + i.val, by have := i.isLt; unfold wrapLen at this; omega⟩ = _
unfold rotPoly
congr 2
apply Fin.ext
exact Nat.zero_add i.val
/-- The starting vertex of the wrapped sub-arc is `A s`. -/
@[simp] theorem subArcWrap_zero {n : ℕ} (A : Fin (n + 1) → S2) (t s : ℕ) (hts : t < s)
(hsn : s ≤ n) :
subArcWrap A t s hts hsn 0 = A ⟨s, by omega⟩ := by
rw [subArcWrap_apply]
congr 1
apply Fin.ext
show ((0 : Fin (wrapLen n s t + 1)).val + s) % (n + 1) = s
simp only [Fin.val_zero, Nat.zero_add, Nat.mod_eq_of_lt (show s < n + 1 by omega)]
/-- The final vertex of the wrapped sub-arc is `A t`. -/
@[simp] theorem subArcWrap_last {n : ℕ} (A : Fin (n + 1) → S2) (t s : ℕ) (hts : t < s)
(hsn : s ≤ n) :
subArcWrap A t s hts hsn (Fin.last (wrapLen n s t)) = A ⟨t, by omega⟩ := by
rw [subArcWrap_apply]
congr 1
apply Fin.ext
show ((Fin.last (wrapLen n s t)).val + s) % (n + 1) = t
rw [Fin.val_last]
unfold wrapLen
-- ((n+1) - s + t + s) % (n+1) = (n + 1 + t) % (n+1) = t
rw [show (n + 1) - s + t + s = (n + 1) + t by omega, Nat.add_comm (n + 1) t,
Nat.add_mod_right, Nat.mod_eq_of_lt (show t < n + 1 by omega)]
/-- **The wrapped sub-arc is a strictly convex arm.** For `t < s ≤ n` with `2 ≤ M = (n+1) - s + t`,
the wrapping range `A s, …, A n, A 0, …, A t` of a strictly convex spherical arm `A` is again a
`StrictConvexSphArm`, closing diagonal `A s → A t`. -/
theorem subArcWrap_strictConvexArm {n : ℕ} (A : Fin (n + 1) → S2)
(hA : StrictConvexSphArm A) (t s : ℕ) (hts : t < s) (hsn : s ≤ n)
(hm : 2 ≤ wrapLen n s t) :
StrictConvexSphArm (subArcWrap A t s hts hsn) := by
unfold subArcWrap
exact subArc_strictConvexArm (rotPoly A ⟨s, by omega⟩)
(rotPoly_strictConvexArm hA ⟨s, by omega⟩) 0 (wrapLen n s t)
(wrapLen_lt_le hts hsn).1 (wrapLen_lt_le hts hsn).2 (by simpa using hm)
/-- **Interior wrapped sides = rotated parent sides.** The `i`-th side of `subArcWrap A t s` is the
side `sideLen (rotPoly A s) ⟨i.val⟩` of the rotated polygon, i.e. the cyclic edge
`A (⟨i.val⟩ + s) → A (⟨i.val⟩ + s + 1)`. -/
theorem subArcWrap_sideLen {n : ℕ} (A : Fin (n + 1) → S2) (t s : ℕ) (hts : t < s) (hsn : s ≤ n)
(i : Fin (wrapLen n s t)) :
sideLen (subArcWrap A t s hts hsn) i
= sideLen (rotPoly A ⟨s, by omega⟩)
⟨i.val, by have := i.isLt; unfold wrapLen at this; omega⟩ := by
unfold subArcWrap
rw [subArc_sideLen]
congr 1
apply Fin.ext
show 0 + i.val = i.val
exact Nat.zero_add i.val
/-- **Interior wrapped joints = rotated parent joints.** The `i`-th joint of `subArcWrap A t s` is
the joint `jointAngle (rotPoly A s) ⟨i.val⟩` of the rotated polygon. -/
theorem subArcWrap_jointAngle {n : ℕ} (A : Fin (n + 1) → S2) (t s : ℕ) (hts : t < s) (hsn : s ≤ n)
(i : Fin (wrapLen n s t - 1)) :
jointAngle (subArcWrap A t s hts hsn) i
= jointAngle (rotPoly A ⟨s, by omega⟩)
⟨i.val, by have := i.isLt; unfold wrapLen at this; omega⟩ := by
unfold subArcWrap
rw [subArc_jointAngle]
congr 1
apply Fin.ext
show 0 + i.val = i.val
exact Nat.zero_add i.val
/-- **The wrapped arm endpoint chord is the diagonal `A t → A s`.** `sDist (subArcWrap … 0)
(subArcWrap … (Fin.last M)) = sDist (A t) (A s)`. -/
theorem subArcWrap_endpt {n : ℕ} (A : Fin (n + 1) → S2) (t s : ℕ) (hts : t < s) (hsn : s ≤ n) :
sDist (subArcWrap A t s hts hsn 0)
(subArcWrap A t s hts hsn (Fin.last (wrapLen n s t)))
= sDist (A ⟨t, by omega⟩) (A ⟨s, by omega⟩) := by
rw [subArcWrap_zero, subArcWrap_last, sDist_comm]
open ProofsInTheBook.Ch13ArmVertex in
/-- **`TwoArcSplitData` from a cyclic cut at `t < s`.** Given the link pair `(A, B)` with equal
sides and equal closing chord, two cut indices `t < s ≤ n` with both arcs non-degenerate
(`2 ≤ s - t`, `2 ≤ wrapLen n s t`), and the per-arc joint monotonicity of the `signChanges = 2`
pattern, assemble the genuine `TwoArcSplitData A B`. Feeding it to `TwoArcSplitData.contradiction`
yields `False`. -/
noncomputable def twoArcSplitData_of_indices {n : ℕ} (hn : 1 ≤ n) (A B : Fin (n + 1) → S2)
(hA : StrictConvexSphArm A) (hB : StrictConvexSphArm B)
(hsides : ∀ i : Fin n, sideLen A i = sideLen B i)
(hclose : sDist (A 0) (A (Fin.last n)) = sDist (B 0) (B (Fin.last n)))
(t s : ℕ) (hts : t < s) (hsn : s ≤ n)
(hm1 : 2 ≤ s - t) (hm2 : 2 ≤ wrapLen n s t)
-- the non-wrap arc opens (`A ≤ B` joints), strictly somewhere; the wrap arc closes (`B ≤ A`).
(hmono1 : ∀ i : Fin (s - t - 1),
jointAngle (subArc A t s hts hsn) i ≤ jointAngle (subArc B t s hts hsn) i)
(hstrict1 : ∃ i : Fin (s - t - 1),
jointAngle (subArc A t s hts hsn) i < jointAngle (subArc B t s hts hsn) i)
(hmono2 : ∀ i : Fin (wrapLen n s t - 1),
jointAngle (subArcWrap B t s hts hsn) i ≤ jointAngle (subArcWrap A t s hts hsn) i) :
ProofsInTheBook.Ch13ArmVertex.TwoArcSplitData A B where
m₁ := s - t
m₂ := wrapLen n s t
hm₁ := hm1
hm₂ := hm2
Arc1 := subArc A t s hts hsn
Brc1 := subArc B t s hts hsn
Arc2 := subArcWrap A t s hts hsn
Brc2 := subArcWrap B t s hts hsn
harc1A := subArc_strictConvexArm A hA t s hts hsn hm1
harc1B := subArc_strictConvexArm B hB t s hts hsn hm1
harc2A := subArcWrap_strictConvexArm A hA t s hts hsn hm2
harc2B := subArcWrap_strictConvexArm B hB t s hts hsn hm2
hsides1 := by
intro i
rw [subArc_sideLen, subArc_sideLen]
exact hsides ⟨t + i.val, by have := i.isLt; omega⟩
hsides2 := by
intro i
rw [subArcWrap_sideLen, subArcWrap_sideLen]
exact rotPoly_sideLen_eq hn A B hsides hclose ⟨s, by omega⟩ ⟨i.val, by
have := i.isLt; unfold wrapLen at this; omega⟩
hshareA := by
rw [subArc_endpt, subArcWrap_endpt, sDist_comm (A ⟨t, by omega⟩) (A ⟨s, by omega⟩)]
hshareB := by
rw [subArc_endpt, subArcWrap_endpt, sDist_comm (B ⟨t, by omega⟩) (B ⟨s, by omega⟩)]
hmono1 := hmono1
hstrict1 := hstrict1
hmono2 := hmono2
end ProofsInTheBook.Ch13SubArcWrap
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanCh13SphAngle
import ProofsInTheBook.ZinanCh13EuclLink
import ProofsInTheBook.Ch13Realization
import ProofsInTheBook.Ch13LinkSides
import ProofsInTheBook.Ch13SubArcWrap
import Mathlib.Geometry.Euclidean.Triangle
-/
/- Source module: ProofsInTheBook.ZinanCh13Cauchy3D -/
section
set_option autoImplicit true
noncomputable section
open scoped Classical RealInnerProductSpace
open ProofsInTheBook.PlanarMap ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.Chapter13
open ProofsInTheBook.Ch13Euclidean
open ProofsInTheBook.Ch13EuclLink
open ProofsInTheBook.Ch13Realization
open ProofsInTheBook.Ch13VertexStar
open ProofsInTheBook.Ch13ArmVertexFull
open ProofsInTheBook.Ch13ArmVertex
open ProofsInTheBook.Ch13SubArc
open ProofsInTheBook.Ch13SubArcWrap
open ProofsInTheBook.Ch13MarkedSphere
open ProofsInTheBook.SphericalKernel
namespace ProofsInTheBook.Ch13VertexStar
namespace VertexStar
/-- Rotate the cyclic neighbor order of a vertex star. -/
noncomputable abbrev rotate (S : VertexStar) (k : Fin (S.n + 1)) : VertexStar where
n := S.n
hn := S.hn
o := S.o
p := fun i => S.p (i + k)
apex_ne := fun i => S.apex_ne (i + k)
open_hemi := by
rcases S.open_hemi with ⟨h, hnorm, hpos⟩
exact ⟨h, hnorm, fun i => hpos (i + k)⟩
turn_support := by
intro i j
have hnext : (i + 1 : Fin (S.n + 1)) + k = (i + k) + 1 := by
rw [add_right_comm]
simpa [hnext] using S.turn_support (i + k) (j + k)
turn_strict := by
intro i j hji hji1
have hnext : (i + 1 : Fin (S.n + 1)) + k = (i + k) + 1 := by
rw [add_right_comm]
have hne0 : j + k ≠ i + k := by
intro h
exact hji (add_right_cancel h)
have hne1 : j + k ≠ (i + k) + 1 := by
intro h
apply hji1
apply add_right_cancel (b := k)
rw [hnext]
exact h
simpa [hnext] using S.turn_strict (i + k) (j + k) hne0 hne1
theorem vertexLink_rotate (S : VertexStar) (k : Fin (S.n + 1)) :
(S.rotate k).vertexLink = rotPoly S.vertexLink k := by
funext i
rfl
end VertexStar
end ProofsInTheBook.Ch13VertexStar
namespace ProofsInTheBook.Ch13Cauchy3D
variable {D : Type*} [Fintype D] [DecidableEq D]
variable {M : CombMap D}
/-- Compatibility abbreviation for this assembly layer: it already carries the local link geometry. -/
abbrev vertexStarOfEuclidean
(P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(LG : VertexLinkGeometry P v) : VertexStar :=
LG.toVertexStar
/--
A convex Euclidean polyhedron is a triangulated Euclidean realization together
with the face-local outward orientation and simple-graph hypotheses from which
the local vertex-link geometry is derived.
The `isSimple` field is included because the downstream Cauchy realization
interface requires a simple triangulated sphere.
-/
structure ConvexEuclideanPolyhedron (M : CombMap D)
extends TriangulatedEuclideanPolyhedron M where
degree_ge_three :
∀ (v : M.Vertex), 3 ≤ vDeg toTriangulatedEuclideanPolyhedron v
face_orientation_faithful :
FaceOrientationFaithful toTriangulatedEuclideanPolyhedron
sphere : M.IsSphereMap
triangle : M.FaceRegular 3
isSimple : M.IsSimpleGraph
namespace ConvexEuclideanPolyhedron
/-- The underlying triangulated Euclidean realization. -/
abbrev toTri (P : ConvexEuclideanPolyhedron M) : TriangulatedEuclideanPolyhedron M :=
P.toTriangulatedEuclideanPolyhedron
/-- The derived reverse-`σ` rotation faithfulness used by the link builder. -/
theorem faithful (P : ConvexEuclideanPolyhedron M) :
RotationFaithful P.toTri :=
rotationFaithful_of_faceOrientationFaithful P.toTri P.face_orientation_faithful
/-- The derived local vertex-link geometry at a vertex. -/
def linkGeom (P : ConvexEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P.toTri v) : VertexLinkGeometry P.toTri v :=
vertexLinkGeometryOfEuclidean P.toTri P.faithful P.isSimple v hdeg
/-- The derived vertex-link geometry with the stored degree lower bound supplied. -/
def linkGeomAt (P : ConvexEuclideanPolyhedron M) (v : M.Vertex) :
VertexLinkGeometry P.toTri v :=
P.linkGeom v (P.degree_ge_three v)
end ConvexEuclideanPolyhedron
/-- Edge-length congruence for two Euclidean realizations on the same combinatorial map. -/
def CongruentFaces (P Q : TriangulatedEuclideanPolyhedron M) : Prop :=
∀ d : D,
‖P.pos (M.head d) - P.pos (M.tail d)‖ =
‖Q.pos (M.head d) - Q.pos (M.tail d)‖
theorem euclidean_angle_eq_of_three_dist_eq
{a b c a' b' c' : Ch13Euclidean.E3}
(hab : dist b a = dist b' a')
(hac : dist c a = dist c' a')
(hbc : dist b c = dist b' c')
(hba : b ≠ a) (hca : c ≠ a) :
EuclideanGeometry.angle b a c = EuclideanGeometry.angle b' a' c' := by
have hcos₁ := EuclideanGeometry.law_cos b a c
have hcos₂ := EuclideanGeometry.law_cos b' a' c'
rw [← hab, ← hac, ← hbc] at hcos₂
have hprod : 2 * dist b a * dist c a ≠ 0 := by
exact mul_ne_zero (mul_ne_zero (by norm_num) (dist_ne_zero.mpr hba))
(dist_ne_zero.mpr hca)
have hcos :
Real.cos (EuclideanGeometry.angle b a c) =
Real.cos (EuclideanGeometry.angle b' a' c') := by
have hmul :
(2 * dist b a * dist c a) * Real.cos (EuclideanGeometry.angle b a c) =
(2 * dist b a * dist c a) * Real.cos (EuclideanGeometry.angle b' a' c') := by
nlinarith
exact mul_left_cancel₀ hprod hmul
exact Real.injOn_cos
⟨EuclideanGeometry.angle_nonneg b a c, EuclideanGeometry.angle_le_pi b a c⟩
⟨EuclideanGeometry.angle_nonneg b' a' c', EuclideanGeometry.angle_le_pi b' a' c'⟩
hcos
theorem congruentFaces_face_angle_at_dart
(P Q : TriangulatedEuclideanPolyhedron M) (hcong : CongruentFaces P Q) (d : D) :
EuclideanGeometry.angle
(P.pos (M.head d)) (P.pos (M.tail d)) (P.pos (M.head (M.σ.symm d)))
=
EuclideanGeometry.angle
(Q.pos (M.head d)) (Q.pos (M.tail d)) (Q.pos (M.head (M.σ.symm d))) := by
have htail_symm : M.tail (M.σ.symm d) = M.tail d := by
have h := M.tail_sigma (M.σ.symm d)
simpa using h.symm
have htail_phi2 :
M.tail (M.φ (M.φ d)) = M.head (M.σ.symm d) :=
Ch13SphAngle.tail_phi_phi_eq_head_sigma_symm_of_triangular_euclidean P d
have hhead_phi :
M.head (M.φ d) = M.head (M.σ.symm d) := by
simpa [M.tail_phi] using htail_phi2
have hab :
dist (P.pos (M.head d)) (P.pos (M.tail d)) =
dist (Q.pos (M.head d)) (Q.pos (M.tail d)) := by
simpa [dist_eq_norm] using hcong d
have hac :
dist (P.pos (M.head (M.σ.symm d))) (P.pos (M.tail d)) =
dist (Q.pos (M.head (M.σ.symm d))) (Q.pos (M.tail d)) := by
have h := hcong (M.σ.symm d)
simpa [dist_eq_norm, htail_symm] using h
have hbc :
dist (P.pos (M.head d)) (P.pos (M.head (M.σ.symm d))) =
dist (Q.pos (M.head d)) (Q.pos (M.head (M.σ.symm d))) := by
have h := hcong (M.φ d)
simpa [dist_eq_norm, M.tail_phi, hhead_phi, norm_sub_rev] using h
have hba : P.pos (M.head d) ≠ P.pos (M.tail d) := by
exact (P.edge_nondegenerate d).symm
have hca : P.pos (M.head (M.σ.symm d)) ≠ P.pos (M.tail d) := by
have hnd := P.edge_nondegenerate (M.σ.symm d)
simpa [htail_symm] using hnd.symm
exact euclidean_angle_eq_of_three_dist_eq hab hac hbc hba hca
/-- The Euclidean dart sign used by the Cauchy marked sphere. -/
def euclideanEdgeSign (P Q : TriangulatedEuclideanPolyhedron M) : D → EdgeSign :=
dihedralSignAtDart P Q
theorem euclideanEdgeSign_alpha
(P Q : TriangulatedEuclideanPolyhedron M) (d : D) :
euclideanEdgeSign P Q (M.α d) = euclideanEdgeSign P Q d := by
unfold euclideanEdgeSign
exact dihedralSignAtDart_alpha P Q d
/-- A canonical representative dart for a vertex. -/
def vertexDartRep (v : M.Vertex) : D :=
Quotient.out v
theorem vertexDartRep_tail (v : M.Vertex) :
M.tail (vertexDartRep (M := M) v) = v :=
Quotient.out_eq v
/-- Reading a finite list through `Fin.rev` gives its reverse. -/
theorem ofFn_get_rev {α : Type*} (L : List α) :
List.ofFn (fun i : Fin L.length => L.get (Fin.rev i)) = L.reverse := by
apply (List.ext_get_iff).2
constructor
· simp
· intro n hn₁ hn₂
simp only [List.length_ofFn, List.length_reverse] at hn₁ hn₂
rw [List.get_ofFn]
rw [List.get_reverse' L ⟨n, by simpa using hn₂⟩ (by omega)]
simp [Fin.rev]
have hidx : L.length - (n + 1) = L.length - 1 - n := by omega
simp [hidx]
theorem ofFn_cast {α : Type*} {n m : ℕ} (e : n = m) (f : Fin m → α) :
List.ofFn (fun i : Fin n => f (Fin.cast e i)) = List.ofFn f := by
subst e
rfl
/-- The dart in the actual Euclidean vertex-star order, i.e. the reverse of the
combinatorial `σ` order used by `incidentDartOfStarIndex`. -/
def starDart (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) (i : Fin (starN P v + 1)) : D :=
incidentDartOfStarIndex P v hdeg (Fin.rev i)
theorem starDart_tail (P : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdeg : 3 ≤ vDeg P v) (i : Fin (starN P v + 1)) :
M.tail (starDart P v hdeg i) = v := by
unfold starDart
exact incidentDartOfStarIndex_tail P v hdeg (Fin.rev i)
theorem starDart_eq_of_index
(P Q : TriangulatedEuclideanPolyhedron M) (v : M.Vertex)
(hdegP : 3 ≤ vDeg P v) (hdegQ : 3 ≤ vDeg Q v)
{i : Fin (starN P v + 1)} {j : Fin (starN Q v + 1)}
(hij : HEq i j) :
starDart P v hdegP i = starDart Q v hdegQ j := by
cases hij
simp [starDart, incidentDartOfStarIndex, incidentDart, starIndexToDeg,
incidentDarts, vDeg, starN]
namespace ListCyclicOrder
end ListCyclicOrder
namespace RotTwoBlockCert
end RotTwoBlockCert
end ProofsInTheBook.Ch13Cauchy3D
end
end
Source
Exact reviewed local source: proof_in_the_book commit 873d52e0c88cd351f594221e70c3c5b3559777a9, ProofsInTheBook/ZinanCh13Cauchy3D.lean:4470 (headline), :96 (ConvexEuclideanPolyhedron), :143 (edge-length congruence), :1157 (adaptive offset), :3467 (rotated stars); ProofsInTheBook/ZinanCh13Euclidean.lean:46 (realization) and :115 (face orientation). These staged files match git show at that local commit. PUBLIC SOURCE GAP: the raw GitHub URL for this commit returned HTTP 404; the older public commit 88d88d141768cded75e782c525ef1bf04b8fe220 differs in these two files and is not an exact source citation for this artifact. Unchanged supporting definitions are publicly byte-verified at https://github.com/xiangyazi24/proof_in_the_book/blob/88d88d141768cded75e782c525ef1bf04b8fe220/ProofsInTheBook/PlanarMap.lean#L24 and https://github.com/xiangyazi24/proof_in_the_book/blob/88d88d141768cded75e782c525ef1bf04b8fe220/ProofsInTheBook/PlanarMapSimple.lean#L97. Repository topic: Cauchy rigidity; no edition-specific chapter mapping asserted.