Marked-map reductions and three-dimensional vertex stars
DefinitionP2MAssembly_Chapter13V2_Part5cauchy-rigiditydihedral-anglesgeometrylean4polyhedraproofs-from-the-book
This part contains near-triangulation, chord-side and component-counting interfaces, marked-sphere reductions, and spherical subarc comparisons. A vertex star has an apex and at least three neighbors with nonzero raw directions in a common open hemisphere; determinant support is nonnegative and strict at nonincident directions. Its internal dihedral angle is the angle of outer raw vectors projected perpendicular to the middle raw vector. The intermediate ConvexPolytopeRealization interface records two star families, a common signed map, equal link sides and closing chords, and compatibility conditions used to combine the geometry with sign counting. These fields are an intermediate interface, not extra arguments of the final headline.
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
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
variable {D : Type*} [Fintype D] [DecidableEq D]
/-- A near-triangulation is a simple sphere map with one distinguished simple
outer boundary cycle of length at least three; every other face has length
three. -/
structure NearTriangulation (M : CombMap D) where
sphere : M.IsSphereMap
simpleGraph : M.IsSimpleGraph
outerFace : M.Face
outerCycle : BoundaryCycle M outerFace
outer_simple : outerCycle.VertexNodup
outer_len : 3 ≤ outerCycle.length
inner_tri : ∀ f : M.Face, f ≠ outerFace → M.faceLen f = 3
@[simp]
lemma dartFace_phi (M : CombMap D) (d : D) :
M.dartFace (M.φ d) = M.dartFace d := by
unfold dartFace
exact Quotient.sound ⟨-1, by simp⟩
lemma faceLen_dartFace_eq_card_support_cycleOf (M : CombMap D) {d : D}
(hφ : M.φ d ≠ d) :
M.faceLen (M.dartFace d) = (M.φ.cycleOf d).support.card := by
have hset :
(Finset.univ.filter
(fun x => Quotient.mk (cycleSetoid M.φ) x = M.dartFace d))
= (M.φ.cycleOf d).support := by
ext x
simp only [Finset.mem_filter, Finset.mem_univ, true_and, dartFace,
Equiv.Perm.mem_support_cycleOf_iff' hφ, Quotient.eq]
exact Equiv.Perm.sameCycle_comm
simpa [faceLen] using congrArg Finset.card hset
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
variable {D : Type*} [Fintype D] [DecidableEq D]
variable {α : Type*} [DecidableEq α]
namespace CombMap
open ProofsInTheBook.PlanarMap.CombMap
/--
The Thomassen list hypotheses for a near-triangulation `M` with a distinguished
*precolored boundary edge* `s(p, q)`.
The two endpoints `p, q` are adjacent on the outer cycle, precolored by singleton
lists `{cp}`, `{cq}` with distinct colors; every other boundary vertex has a list
of size at least `3`; every interior (non-boundary) vertex has a list of size at
least `5`.
-/
structure ThomassenLists {M : CombMap D} (hNT : NearTriangulation M)
(p q : M.Vertex) (L : M.Vertex → Finset α) (cp cq : α) : Prop where
/-- `p` is on the outer boundary. -/
p_boundary : hNT.outerCycle.IsBoundaryVertex p
/-- `q` is on the outer boundary. -/
q_boundary : hNT.outerCycle.IsBoundaryVertex q
/-- `p` and `q` are joined by an outer-boundary edge: a precolored edge. -/
pq_boundary_edge : hNT.outerCycle.IsBoundaryEdge s(p, q)
/-- The two precolored endpoints carry distinct colors. -/
colors_ne : cp ≠ cq
/-- `p` is precolored to the singleton `{cp}`. -/
list_p : L p = {cp}
/-- `q` is precolored to the singleton `{cq}`. -/
list_q : L q = {cq}
/-- Every other boundary vertex has list size at least three. -/
boundary_ge_three : ∀ v : M.Vertex,
hNT.outerCycle.IsBoundaryVertex v → v ≠ p → v ≠ q → 3 ≤ (L v).card
/-- Every interior (non-boundary) vertex has list size at least five. -/
interior_ge_five : ∀ v : M.Vertex,
¬ hNT.outerCycle.IsBoundaryVertex v → 5 ≤ (L v).card
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
variable {D : Type u} [Fintype D] [DecidableEq D] {α : Type u} [DecidableEq α]
variable {M : CombMap D} {hNT : NearTriangulation M}
/-- The reconstruction datum for ONE side of a chord split, relative to a side
region `s ⊆ M.Vertex` and side lists. `Dₛ` is the side dart type (in the surgery,
`keptSideᵢ ⊕ Fin 2`).
The structure isolates the Jordan/Euler outputs that the combinatorial-map layer
cannot synthesize, exactly as `FanSurgeryReconstruction` does for the chordless
branch:
* `N` — the side as a near-triangulation (carries `IsSphereMap`/genus-0);
* `ι` — the side-vertex-to-`M` correspondence, injective and adjacency-reflecting
*onto* the region `s` (`ι_surj`), so it is a graph isomorphism `N ≃ M⟦s⟧`;
* `ι_lists` — the side near-triangulation's lists are the pullback of `L` along
`ι`, so a list coloring of `N` transports to a region coloring of `M`;
* `smaller` — strict vertex decrease, the recursion fuel.
`Lₛ` and the precolored data (`pₛ qₛ cpₛ cqₛ`) are the side's Thomassen inputs. -/
structure ChordSideReconstruction (hNT : NearTriangulation M)
(s : Set M.Vertex) (L : M.Vertex → Finset α) where
/-- The side dart type. -/
Dₛ : Type u
/-- Side dart type is a fintype. -/
[fintypeDₛ : Fintype Dₛ]
/-- Side dart type has decidable equality. -/
[decEqDₛ : DecidableEq Dₛ]
/-- The side combinatorial map. -/
N : CombMap Dₛ
/-- The side is a near-triangulation (carries `IsSphereMap`: genus-0/Euler-2). -/
hN : NearTriangulation N
/-- The side-vertex-to-`M`-vertex correspondence. -/
ι : N.Vertex → M.Vertex
/-- The correspondence is injective. -/
ι_inj : Function.Injective ι
/-- The correspondence lands in the region. -/
ι_mem : ∀ x : N.Vertex, ι x ∈ s
/-- The correspondence is *surjective onto the region* (the side-vertex
classification: every region vertex is a side vertex). -/
ι_surj : ∀ ⦃w : M.Vertex⦄, w ∈ s → ∃ x : N.Vertex, ι x = w
/-- The correspondence carries side adjacency to `M`-adjacency (graph hom). -/
ι_adj : ∀ ⦃x y : N.Vertex⦄, N.toSimpleGraph.Adj x y →
M.toSimpleGraph.Adj (ι x) (ι y)
/-- The correspondence *reflects* `M`-adjacency on the region (graph iso onto the
induced subgraph): two side vertices adjacent in `M` are adjacent in `N`. -/
ι_adj_reflect : ∀ ⦃x y : N.Vertex⦄, M.toSimpleGraph.Adj (ι x) (ι y) →
N.toSimpleGraph.Adj x y
/-- The side lists are the pullback of `L` along `ι`. -/
Lₛ : N.Vertex → Finset α
/-- The pullback identity for the side lists. -/
Lₛ_eq : ∀ x : N.Vertex, Lₛ x = L (ι x)
/-- The side's precolored boundary edge. -/
pₛ : N.Vertex
/-- The side's precolored boundary edge. -/
qₛ : N.Vertex
/-- The side's precolors. -/
cpₛ : α
/-- The side's precolors. -/
cqₛ : α
/-- The side near-triangulation carries the Thomassen list hypotheses, so the
Thomassen induction can recurse on it. (This is part of the list transport the
classification produces alongside the correspondence.) -/
hLₛ : ThomassenLists hN pₛ qₛ Lₛ cpₛ cqₛ
/-- Strict vertex decrease (the recursion fuel). -/
smaller : N.V < M.V
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
variable {D : Type*} [Fintype D] [DecidableEq D]
noncomputable def numCycles (p : Equiv.Perm D) : ℕ := by
classical
exact Fintype.card (Quotient (SameCycle.setoid p))
namespace PermTranspositionCycleCount
open scoped Finset
def mergeRel (p : Equiv.Perm D) (a b x y : D) : Prop :=
p.SameCycle x y ∨
(p.SameCycle x a ∧ p.SameCycle y b) ∨
(p.SameCycle x b ∧ p.SameCycle y a)
noncomputable def orbitEquivFixedOrCycles (p : Equiv.Perm D) :
Quotient (SameCycle.setoid p) ≃ Function.fixedPoints p ⊕ p.cycleFactorsFinset := by
classical
refine
{ toFun := ?toFun
invFun := ?invFun
left_inv := ?left
right_inv := ?right }
· refine Quotient.lift ?_ ?_
· intro x
by_cases hx : p x = x
· exact Sum.inl ⟨x, by simpa [Function.mem_fixedPoints_iff] using hx⟩
· exact Sum.inr ⟨p.cycleOf x, cycleOf_mem_cycleFactorsFinset_iff.mpr (mem_support.mpr hx)⟩
· intro x y hxy
change p.SameCycle x y at hxy
by_cases hx : p x = x
· have hy : p y = y := (hxy.apply_eq_self_iff).mp hx
have hxy' : x = y := hxy.eq_of_left hx
subst hxy'
simp [hx, hy]
· have hy : p y ≠ y := fun hy => hx ((hxy.apply_eq_self_iff).mpr hy)
simp [hx, hy]
exact hxy.cycleOf_eq
· intro s
rcases s with fp | c
· exact Quotient.mk (SameCycle.setoid p) fp.1
· let y : D := Classical.choose
(IsCycle.nonempty_support (mem_cycleFactorsFinset_iff.mp c.2).1)
exact Quotient.mk (SameCycle.setoid p) y
· intro q
refine Quotient.inductionOn q ?_
intro x
by_cases hx : p x = x
· simp [hx, Function.mem_fixedPoints_iff]
· dsimp
simp [hx]
let c : p.cycleFactorsFinset :=
⟨p.cycleOf x, cycleOf_mem_cycleFactorsFinset_iff.mpr (mem_support.mpr hx)⟩
let y : D := Classical.choose
(IsCycle.nonempty_support (mem_cycleFactorsFinset_iff.mp c.2).1)
have hy : y ∈ (p.cycleOf x).support :=
Classical.choose_spec
(IsCycle.nonempty_support (mem_cycleFactorsFinset_iff.mp c.2).1)
have hy' : p.SameCycle x y := by
have := (mem_support_cycleOf_iff (f := p) (x := x) (y := y)).mp hy
exact this.1
exact Quotient.sound hy'.symm
· intro s
rcases s with fp | c
· have hfp : p fp.1 = fp.1 := Function.mem_fixedPoints_iff.mp fp.2
simp [hfp]
· let y : D := Classical.choose
(IsCycle.nonempty_support (mem_cycleFactorsFinset_iff.mp c.2).1)
have hyc : y ∈ (c : Equiv.Perm D).support :=
Classical.choose_spec
(IsCycle.nonempty_support (mem_cycleFactorsFinset_iff.mp c.2).1)
have hyp : y ∈ p.support := mem_cycleFactorsFinset_support_le c.2 hyc
have hy : p y ≠ y := mem_support.mp hyp
have hcy : c.1 = p.cycleOf y := cycle_is_cycleOf hyc c.2
dsimp
rw [dif_neg hy]
exact congrArg (fun e : p.cycleFactorsFinset => Sum.inr e) (Subtype.ext hcy.symm)
end PermTranspositionCycleCount
end
/- Original source header (imports hoisted):
import Mathlib
-/
/- Source module: ProofsInTheBook.RelationComponentCount -/
section
set_option autoImplicit true
open Classical
universe u
variable {V : Type u} [Fintype V]
def compSetoid (r : V → V → Prop) : Setoid V :=
⟨Relation.EqvGen r, Relation.EqvGen.is_equivalence r⟩
noncomputable def numComp (r : V → V → Prop) : ℕ :=
Nat.card (Quotient (compSetoid r))
def addEdge (r : V → V → Prop) (a b : V) : V → V → Prop :=
fun x y => r x y ∨ (x = a ∧ y = b) ∨ (x = b ∧ y = a)
def pairRel {α : Type u} (a b : α) (x y : α) : Prop :=
x = y ∨ (x = a ∧ y = b) ∨ (x = b ∧ y = a)
theorem pairRel_refl {α : Type u} (a b : α) (x : α) :
pairRel a b x x := by
exact Or.inl rfl
theorem pairRel_symm {α : Type u} (a b : α) {x y : α} :
pairRel a b x y → pairRel a b y x := by
intro h
rcases h with rfl | ⟨rfl, rfl⟩ | ⟨rfl, rfl⟩
· exact Or.inl rfl
· exact Or.inr (Or.inr ⟨rfl, rfl⟩)
· exact Or.inr (Or.inl ⟨rfl, rfl⟩)
theorem pairRel_trans {α : Type u} (a b : α) {x y z : α} :
pairRel a b x y → pairRel a b y z → pairRel a b x z := by
intro hxy hyz
rcases hxy with hEq | hAB | hBA
· subst y
exact hyz
· rcases hAB with ⟨rfl, rfl⟩
rcases hyz with hEq | hAB' | hBA'
· subst z
exact Or.inr (Or.inl ⟨rfl, rfl⟩)
· rcases hAB' with ⟨_, hz⟩
subst z
exact Or.inr (Or.inl ⟨rfl, rfl⟩)
· rcases hBA' with ⟨_, hz⟩
subst z
exact Or.inl rfl
· rcases hBA with ⟨rfl, rfl⟩
rcases hyz with hEq | hAB' | hBA'
· subst z
exact Or.inr (Or.inr ⟨rfl, rfl⟩)
· rcases hAB' with ⟨_, hz⟩
subst z
exact Or.inl rfl
· rcases hBA' with ⟨_, hz⟩
subst z
exact Or.inr (Or.inr ⟨rfl, rfl⟩)
def pairSetoid {α : Type u} (a b : α) : Setoid α where
r := pairRel a b
iseqv :=
⟨pairRel_refl a b, fun h => pairRel_symm a b h,
fun h₁ h₂ => pairRel_trans a b h₁ h₂⟩
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
variable {D : Type*} [Fintype D] [DecidableEq D]
/-- The dart-incidence relation of a raw pair `(σ, α)`: two darts are adjacent if
they share a `σ`-orbit (same vertex) or are joined by the `α`-edge. -/
def dartStepRel (σ α : Equiv.Perm D) (a b : D) : Prop :=
σ.SameCycle a b ∨ b = α a
/-- Number of `α`-transpositions. -/
noncomputable def Ehalf (α : Equiv.Perm D) : ℕ := (Equiv.Perm.support α).card / 2
/-- Component count of the raw pair `(σ, α)`. -/
noncomputable def numComponents (σ α : Equiv.Perm D) : ℕ :=
_root_.numComp (dartStepRel σ α)
/-- Genus slack `2c - V + Ehalf - F` of a raw involution pair. It is `≥ 0`; for a
connected fixed-point-free map this yields `χ ≤ 2`. -/
noncomputable def genusSlack (σ α : Equiv.Perm D) : ℤ :=
2 * (numComponents σ α : ℤ) - (numCycles σ : ℤ) + (Ehalf α : ℤ)
- (numCycles (σ * α) : ℤ)
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
variable {D : Type u} [Fintype D] [DecidableEq D]
/-- `α'` is an **edge-deletion sub-involution** of `α`: an involution that agrees with `α` on
its own support (so its edge set is a subset of `α`'s edge set). -/
def SubInvolution (α α' : Equiv.Perm D) : Prop :=
α' * α' = 1 ∧ ∀ x, α' x ≠ x → α' x = α x
section OrbitSplit
variable (p : Equiv.Perm D) (S : Finset D)
open scoped Classical
/-- A `p`-orbit is **deleted** if all its darts lie in `S`. -/
def DeletedOrbit (q : Quotient (cycleSetoid p)) : Prop :=
∀ x : D, Quotient.mk (cycleSetoid p) x = q → x ∈ S
/-- The kept subtype's filtered orbit quotient, as `p`-orbits via the `SameCycle` coincidence. -/
noncomputable def keptToFull :
Quotient (cycleSetoid (Equiv.Perm.deleteSet p S)) → Quotient (cycleSetoid p) :=
Quotient.lift (fun x => Quotient.mk (cycleSetoid p) (x.1 : D)) (by
intro x y hxy
apply Quotient.sound
exact (Equiv.Perm.sameCycle_deleteSet_iff p S x y).1 hxy)
/-- The number of **deleted** `p`-orbits (orbits entirely inside `S`). -/
noncomputable def numDeletedOrbits : ℕ :=
Fintype.card {q : Quotient (cycleSetoid p) // DeletedOrbit p S q}
end OrbitSplit
section RawRestrict
variable (M : CombMap D) (Del : Finset D)
open scoped Classical
/-- The raw restricted function: identity on deleted darts, `M.α` elsewhere. -/
noncomputable def rawAlphaFun : D → D := fun d => if d ∈ Del then d else M.α d
/-- `rawAlphaFun` is an involution (uses `α`-closedness of `Del`). -/
lemma rawAlphaFun_involutive (hclosed : ∀ d : D, d ∈ Del → M.α d ∈ Del) :
Function.Involutive (rawAlphaFun M Del) := by
classical
intro d
unfold rawAlphaFun
by_cases hd : d ∈ Del
· simp [hd]
· have hαd : M.α d ∉ Del := by
intro h
apply hd
have := hclosed _ h
rwa [M.alpha_alpha] at this
simp [hd, hαd, M.alpha_alpha]
/-- The **raw restricted involution**: fixes every deleted dart, equals `M.α` on kept darts. -/
noncomputable def rawAlpha (hclosed : ∀ d : D, d ∈ Del → M.α d ∈ Del) : Equiv.Perm D :=
Function.Involutive.toPerm (rawAlphaFun M Del) (rawAlphaFun_involutive M Del hclosed)
@[simp] lemma rawAlpha_apply (hclosed : ∀ d : D, d ∈ Del → M.α d ∈ Del) (d : D) :
rawAlpha M Del hclosed d = if d ∈ Del then d else M.α d := rfl
/-- `rawAlpha` equals `M.α` on kept darts. -/
lemma rawAlpha_eq_alpha_of_notMem (hclosed : ∀ d : D, d ∈ Del → M.α d ∈ Del) {d : D}
(hd : d ∉ Del) : rawAlpha M Del hclosed d = M.α d := by simp [rawAlpha_apply, hd]
/-- Abbreviation: the kept combinatorial map's face permutation is
`(deleteSet M.σ Del) * (M.α.subtypePerm)`. -/
noncomputable def keptFacePerm (hclosed : ∀ d : D, d ∈ Del → M.α d ∈ Del)
(hsub : ∀ d, d ∈ Del ↔ M.α d ∈ Del) : Equiv.Perm {d : D // d ∉ Del} :=
(Equiv.Perm.deleteSet M.σ Del) * (M.α.subtypePerm (fun d => by
rw [← hsub d]))
variable (hclosed : ∀ d : D, d ∈ Del → M.α d ∈ Del)
(hsub : ∀ d, d ∈ Del ↔ M.α d ∈ Del)
open scoped Classical
/-- The kept edge involution: `M.α` restricted to the kept subtype. -/
noncomputable def keptAlpha : Equiv.Perm {d : D // d ∉ Del} :=
M.α.subtypePerm (p := fun d => d ∉ Del) (fun d => by
constructor
· intro hd hc; exact hd ((hsub d).1 hc)
· intro hd hc; exact hd ((hsub d).2 hc))
@[simp] lemma keptAlpha_apply_coe (d : {d : D // d ∉ Del}) :
(keptAlpha M Del hsub d : D) = M.α d.1 := rfl
lemma keptAlpha_invol : keptAlpha M Del hsub * keptAlpha M Del hsub = 1 := by
ext z
simp only [Equiv.Perm.coe_mul, Equiv.Perm.coe_one, Function.comp_apply, id_eq,
keptAlpha_apply_coe]
exact M.alpha_alpha z.1
/-- The kept combinatorial map's dart-step relation, on the kept subtype. -/
noncomputable def keptStepRel : {d : D // d ∉ Del} → {d : D // d ∉ Del} → Prop :=
dartStepRel (Equiv.Perm.deleteSet M.σ Del) (keptAlpha M Del hsub)
/-- **A kept dart-step lifts to a raw dart-step on the underlying darts.** -/
lemma keptStepRel_imp_raw {x y : {d : D // d ∉ Del}}
(h : keptStepRel M Del hsub x y) :
dartStepRel M.σ (rawAlpha M Del hclosed) x.1 y.1 := by
classical
rcases h with hσ | hα
· -- same `deleteSet M.σ`-cycle ⇒ same `M.σ`-cycle on coercions.
exact Or.inl ((Equiv.Perm.sameCycle_deleteSet_iff M.σ Del x y).1 hσ)
· -- α-edge: `y = (M.α.subtypePerm) x`, so `y.1 = M.α x.1 = rawAlpha x.1` (x kept).
refine Or.inr ?_
have hxval : (rawAlpha M Del hclosed) x.1 = M.α x.1 :=
rawAlpha_eq_alpha_of_notMem M Del hclosed x.2
rw [hxval]
have := congrArg Subtype.val hα
simpa using this
/-- `EqvGen` of the kept dart-step relation lifts to `EqvGen` of the raw relation. -/
lemma eqvGen_keptStepRel_imp_raw {x y : {d : D // d ∉ Del}}
(h : Relation.EqvGen (keptStepRel M Del hsub) x y) :
Relation.EqvGen (dartStepRel M.σ (rawAlpha M Del hclosed)) x.1 y.1 := by
induction h with
| rel x y hxy => exact Relation.EqvGen.rel _ _ (keptStepRel_imp_raw M Del hclosed hsub hxy)
| refl x => exact Relation.EqvGen.refl _
| symm x y _ ih => exact Relation.EqvGen.symm _ _ ih
| trans x y z _ _ ih1 ih2 => exact Relation.EqvGen.trans _ _ _ ih1 ih2
/-- The lifted map `⟦x⟧_kept ↦ ⟦x.1⟧_raw` of component quotients. -/
noncomputable def keptCompToRaw :
Quotient (_root_.compSetoid (keptStepRel M Del hsub))
→ Quotient (_root_.compSetoid (dartStepRel M.σ (rawAlpha M Del hclosed))) :=
Quotient.lift (fun x => Quotient.mk _ (x.1 : D)) (by
intro x y hxy
apply Quotient.sound
show Relation.EqvGen (dartStepRel M.σ (rawAlpha M Del hclosed)) x.1 y.1
exact eqvGen_keptStepRel_imp_raw M Del hclosed hsub hxy)
/-- A raw component is **deleted** if all its darts lie in `Del`. -/
def DeletedComp (q : Quotient (_root_.compSetoid (dartStepRel M.σ (rawAlpha M Del hclosed)))) :
Prop :=
∀ x : D, Quotient.mk _ x = q → x ∈ Del
/-- The number of deleted raw components. -/
noncomputable def numDeletedComp : ℕ :=
Fintype.card {q : Quotient (_root_.compSetoid (dartStepRel M.σ (rawAlpha M Del hclosed)))
// DeletedComp M Del hclosed q}
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
variable {D : Type*} [Fintype D] [DecidableEq D]
/-- Embed a strict sign into `EdgeSign`. -/
def strictToEdge : StrictEdgeSign → EdgeSign
| StrictEdgeSign.plus => EdgeSign.plus
| StrictEdgeSign.minus => EdgeSign.minus
end StrictBridge
section ActiveComponent
variable {D : Type*} [Fintype D] [DecidableEq D]
/-- The degree of a face (`φ`-orbit): the number of darts on its boundary walk. -/
def faceDeg (M : CombMap D) (Q : Quotient (cycleSetoid M.φ)) : ℕ :=
(Finset.univ.filter (fun x => Quotient.mk (cycleSetoid M.φ) x = Q)).card
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
variable {D : Type*} [Fintype D] [DecidableEq D]
open ProofsInTheBook.SubmapPlanar
/-- The kept (active sub-)combinatorial map on the surviving darts `{d ∉ Del}`. -/
noncomputable def keptMap (M : CombMap D) (Del : Finset D)
(hsub : ∀ d, d ∈ Del ↔ M.α d ∈ Del) : CombMap {d : D // d ∉ Del} where
α := keptAlpha M Del hsub
σ := Equiv.Perm.deleteSet M.σ Del
α_invol := keptAlpha_invol M Del hsub
α_no_fixed := by
intro d hd
apply M.α_no_fixed d.1
have := congrArg Subtype.val hd
rwa [keptAlpha_apply_coe] at this
/-- A nonzero edge sign as a strict sign. -/
def edgeToStrict : EdgeSign → StrictEdgeSign
| EdgeSign.plus => StrictEdgeSign.plus
| EdgeSign.minus => StrictEdgeSign.minus
| EdgeSign.zero => StrictEdgeSign.plus -- unreachable on active darts
/-- The strict signing on kept darts: `es` read as a `±`-sign (zeros, absent on active darts,
are sent to `plus` and never arise). -/
def keptSign (M : CombMap D) (es : D → EdgeSign) (Del : Finset D) :
{d : D // d ∉ Del} → StrictEdgeSign :=
fun d => edgeToStrict (es d.1)
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
variable {D : Type*} [Fintype D] [DecidableEq D]
open ProofsInTheBook -- for DeleteSet.firstOutside via Equiv.Perm namespace
/-- The underlying dart of the `n`-th iterate of `deleteSet p S` from `x` is a `p`-power of `x.1`,
recorded with its explicit exponent `gOffset`. `gOffset p S x n` is the cumulative sum of the
`firstOutside` steps along the first `n` `deleteSet`-iterates. -/
noncomputable def gOffset (p : Equiv.Perm D) (S : Finset D) (x : {d : D // d ∉ S}) : ℕ → ℕ
| 0 => 0
| n + 1 => gOffset p S x n + Equiv.Perm.DeleteSet.firstOutside p S (((deleteSet p S) ^ n) x)
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
variable {D : Type*} [Fintype D] [DecidableEq D]
/-- The active step relation: a vertex- or edge-step between two **active** darts. -/
def activeStep (M : CombMap D) (es : D → EdgeSign) (d e : D) : Prop :=
(M.σ.SameCycle d e ∨ e = M.α d) ∧ es d ≠ EdgeSign.zero ∧ es e ≠ EdgeSign.zero
open scoped Classical in
/-- The active-component deletion: everything **not** `EqvGen activeStep`-related to the seed `d₀`. -/
noncomputable def compDel (M : CombMap D) (es : D → EdgeSign) (d₀ : D) : Finset D :=
Finset.univ.filter (fun d => ¬ Relation.EqvGen (activeStep M es) d₀ d)
@[simp] theorem mem_compDel (M : CombMap D) (es : D → EdgeSign) (d₀ d : D) :
d ∈ compDel M es d₀ ↔ ¬ Relation.EqvGen (activeStep M es) d₀ d := by
classical
rw [compDel, Finset.mem_filter]
simp
/-- Any `EqvGen activeStep` pair is either equal or both active. -/
theorem eqvGen_activeStep_active (M : CombMap D) (es : D → EdgeSign) {a b : D}
(h : Relation.EqvGen (activeStep M es) a b) :
(es a ≠ EdgeSign.zero ∧ es b ≠ EdgeSign.zero) ∨ a = b := by
induction h with
| rel x y hxy => exact Or.inl ⟨hxy.2.1, hxy.2.2⟩
| refl x => exact Or.inr rfl
| symm x y _ ih =>
rcases ih with ⟨hx, hy⟩ | hxy
· exact Or.inl ⟨hy, hx⟩
· exact Or.inr hxy.symm
| trans x y z _ _ ih1 ih2 =>
rcases ih1 with ⟨hx, hy⟩ | hxy
· rcases ih2 with ⟨_, hz⟩ | hyz
· exact Or.inl ⟨hx, hz⟩
· subst hyz; exact Or.inl ⟨hx, hy⟩
· subst hxy; exact ih2
/-- Every dart `EqvGen`-reached from the active seed `d₀` is itself active. -/
theorem reached_active (M : CombMap D) (es : D → EdgeSign) {d₀ : D} (hd₀ : es d₀ ≠ EdgeSign.zero)
{d : D} (h : Relation.EqvGen (activeStep M es) d₀ d) : es d ≠ EdgeSign.zero := by
rcases eqvGen_activeStep_active M es h with ⟨_, hd⟩ | hd
· exact hd
· exact hd ▸ hd₀
/-- The seed `d₀` is reached from itself. -/
theorem reached_refl (M : CombMap D) (es : D → EdgeSign) (d₀ : D) :
Relation.EqvGen (activeStep M es) d₀ d₀ := Relation.EqvGen.refl _
variable (M : CombMap D) (es : D → EdgeSign)
/-- `compDel` is `α`-closed (the `hsub` input) when `es` is edge-invariant: a dart and its edge
partner are reachable together (both active, or both inactive hence both deleted). -/
theorem compDel_hsub (hes : ∀ d, es (M.α d) = es d) {d₀ : D} (hd₀ : es d₀ ≠ EdgeSign.zero) :
∀ d, d ∈ compDel M es d₀ ↔ M.α d ∈ compDel M es d₀ := by
intro d
rw [mem_compDel, mem_compDel, not_iff_not]
constructor
· intro h
have hda : es d ≠ EdgeSign.zero := reached_active M es hd₀ h
have hαa : es (M.α d) ≠ EdgeSign.zero := by rw [hes d]; exact hda
refine Relation.EqvGen.trans _ _ _ h (Relation.EqvGen.rel _ _ ?_)
exact ⟨Or.inr rfl, hda, hαa⟩
· intro h
have hαa : es (M.α d) ≠ EdgeSign.zero := reached_active M es hd₀ h
have hda : es d ≠ EdgeSign.zero := by rw [← hes d]; exact hαa
refine Relation.EqvGen.trans _ _ _ h (Relation.EqvGen.rel _ _ ?_)
refine ⟨Or.inr ?_, hαa, hda⟩
rw [M.alpha_alpha]
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
variable {D : Type*} [Fintype D] [DecidableEq D]
/-- **The faithful geometric input** for Cauchy rigidity (ChatGPT's Q5 replacement for the unfaithful
`CauchyRigidityCertificate`). A triangulated-sphere combinatorial map with a ±/0 edge signing, carrying
the GENUINE per-active-vertex arm data: at each active vertex the spherical-link `CauchyArmVertex` (whose
`four_le_signChanges` ≥4 is real), bridged to the skip-zeros σ-cycle count. It does NOT carry any
Euler/`3F=2E`/face-count field — those live inside the combinatorial lemma. -/
structure CauchyMarkedTriangulatedSphere (M : CombMap D) where
/-- The surface is a triangulated sphere. -/
isSphere : M.IsSphereMap
triangleFaces : M.FaceRegular 3
/-- The edge graph is simple (no loops / no parallel edges). This is a GENUINE property of every
convex 3-polytope's boundary graph (Steinitz: the graph of a convex polytope is simple and
3-connected), supplied by the ℝ³ realization — not a combinatorial convenience. It is what rules
out the digon degeneracy in the combinatorial lemma. -/
isSimple : M.IsSimpleGraph
/-- The dihedral-difference signing (±/0). -/
edgeSign : D → EdgeSign
/-- The signing is edge-invariant (`α`-stable): both darts of an edge carry the same sign. -/
edgeSign_inv : ∀ d, edgeSign (M.α d) = edgeSign d
/-- The GENUINE vertex-link arm datum at each active vertex (from real spherical links). -/
vertexArm : ∀ d, ActiveVertex M edgeSign d → CauchyArmVertex
/-- The bridge: the arm-datum's sign-change count is exactly the skip-zeros σ-cycle count at the vertex
(the geometric σ-order = the link rotational order). -/
vertexArm_signChanges_eq :
∀ d (hd : ActiveVertex M edgeSign d),
(vertexArm d hd).signChanges = vertexFlipCountSkipZeros M edgeSign d
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
/-- **Cauchy's Lemma II, the `signChanges = 2` case — the two-arc contradiction.**
`Arc1 : Fin (m₁+1) → S2` and `Arc2 : Fin (m₂+1) → S2` are the two sub-arcs of `A` cut at the two
sign-change vertices; `Brc1`, `Brc2` the corresponding sub-arcs of `B`. Both arcs are strictly convex
spherical arms (`harc1A …`). The arcs share the chord endpoints: arc 1's endpoint chord and arc 2's
endpoint chord are the SAME spherical distance in `A` (`hshareA`) and in `B` (`hshareB`) — both equal
the diagonal `sDist (A s) (A t)` resp. `sDist (B s) (B t)`. On each arc the arm sides match between
the `A`-side and `B`-side (`hsides1`, `hsides2`). On arc 1 the joints are nondecreasing `A → B`
(`hmono1`) with one strictly opened (`hstrict1`); on arc 2 they are nonincreasing `A → B` (i.e.
nondecreasing `B → A`, `hmono2`). This is impossible.
The proof: the `≤` arm lemma on arc 2 (applied `Brc2 → Arc2`) gives `chord(Brc2) ≤ chord(Arc2)`,
i.e. via the shared-chord identities `chord_B ≤ chord_A`; the **strict** arm lemma on arc 1 gives
`chord_A < chord_B`; chaining contradicts `<`. -/
theorem cauchy_two_signchange_split
{m₁ m₂ : ℕ} (hm₁ : 2 ≤ m₁) (hm₂ : 2 ≤ m₂)
(Arc1 Brc1 : Fin (m₁ + 1) → S2) (Arc2 Brc2 : Fin (m₂ + 1) → S2)
(harc1A : StrictConvexSphArm Arc1) (harc1B : StrictConvexSphArm Brc1)
(harc2A : StrictConvexSphArm Arc2) (harc2B : StrictConvexSphArm Brc2)
-- arm sides match across A/B on each arc (both arcs inherit `hsides` of the closed polygons)
(hsides1 : ∀ i : Fin m₁, sideLen Arc1 i = sideLen Brc1 i)
(hsides2 : ∀ i : Fin m₂, sideLen Arc2 i = sideLen Brc2 i)
-- the two arcs share the SAME chord (the diagonal `A s → A t`, resp. `B s → B t`)
(hshareA : sDist (Arc1 0) (Arc1 (Fin.last m₁)) = sDist (Arc2 0) (Arc2 (Fin.last m₂)))
(hshareB : sDist (Brc1 0) (Brc1 (Fin.last m₁)) = sDist (Brc2 0) (Brc2 (Fin.last m₂)))
-- arc 1: joints open A → B with one strictly open; arc 2: joints close A → B
(hmono1 : ∀ i : Fin (m₁ - 1), jointAngle Arc1 i ≤ jointAngle Brc1 i)
(hstrict1 : ∃ i : Fin (m₁ - 1), jointAngle Arc1 i < jointAngle Brc1 i)
(hmono2 : ∀ i : Fin (m₂ - 1), jointAngle Brc2 i ≤ jointAngle Arc2 i) :
False := by
-- strict arm lemma on arc 1: chord(Arc1) < chord(Brc1)
have h1 : sDist (Arc1 0) (Arc1 (Fin.last m₁)) < sDist (Brc1 0) (Brc1 (Fin.last m₁)) :=
spherical_arm_mono_strict_uncond hm₁ Arc1 Brc1 harc1A harc1B hsides1 hmono1 hstrict1
-- ≤ arm lemma on arc 2, applied Brc2 → Arc2: chord(Brc2) ≤ chord(Arc2)
have h2 : sDist (Brc2 0) (Brc2 (Fin.last m₂)) ≤ sDist (Arc2 0) (Arc2 (Fin.last m₂)) :=
ProofsInTheBook.ZinanFFCT111.spherical_arm_mono_final_ch13 hm₂ Brc2 Arc2 harc2B harc2A
(fun i => (hsides2 i).symm) hmono2
-- chain through the shared-chord identities:
-- chord_A := chord(Arc1) = chord(Arc2); chord_B := chord(Brc1) = chord(Brc2)
-- h1 : chord(Arc1) < chord(Brc1)
-- h2 : chord(Brc2) ≤ chord(Arc2)
-- hshareA : chord(Arc1) = chord(Arc2); hshareB : chord(Brc1) = chord(Brc2)
rw [hshareA] at h1 -- h1 : chord(Arc2) < chord(Brc1)
rw [hshareB] at h1 -- h1 : chord(Arc2) < chord(Brc2)
exact absurd (lt_of_lt_of_le h1 h2) (lt_irrefl _)
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
/-- The contiguous sub-arc: keep `A s, A (s+1), …, A t`, re-indexed by `i ↦ A (s + i)`. -/
def subArc {n : ℕ} (A : Fin (n + 1) → S2) (s t : ℕ) (hst : s < t) (htn : t ≤ n) :
Fin ((t - s) + 1) → S2 :=
fun i => A ⟨s + i.val, by have := i.isLt; omega⟩
@[simp] theorem subArc_apply {n : ℕ} (A : Fin (n + 1) → S2) (s t : ℕ) (hst : s < t) (htn : t ≤ n)
(i : Fin ((t - s) + 1)) :
subArc A s t hst htn i = A ⟨s + i.val, by have := i.isLt; omega⟩ := rfl
@[simp] theorem subArc_zero {n : ℕ} (A : Fin (n + 1) → S2) (s t : ℕ) (hst : s < t) (htn : t ≤ n) :
subArc A s t hst htn 0 = A ⟨s, by omega⟩ := by
simp only [subArc, Fin.val_zero, Nat.add_zero]
/-- The last vertex of the sub-arc is `A t` (`= A ⟨s + (t - s)⟩`). -/
@[simp] theorem subArc_last {n : ℕ} (A : Fin (n + 1) → S2) (s t : ℕ) (hst : s < t) (htn : t ≤ n) :
subArc A s t hst htn (Fin.last (t - s)) = A ⟨t, by omega⟩ := by
simp only [subArc, Fin.val_last]
congr 1
apply Fin.ext
show s + (t - s) = t
omega
/-- The value of `1 : Fin ((t-s)+1)` is `1` (since `1 ≤ t - s`, so `(t-s)+1 ≥ 2 > 1`). -/
theorem one_val_subArc {t s : ℕ} (hm : 1 ≤ t - s) : ((1 : Fin ((t - s) + 1)) : ℕ) = 1 := by
rw [Fin.val_one']; exact Nat.mod_eq_of_lt (by omega)
/-- `i ≠ Fin.last (t - s)` forces `i.val < t - s`. -/
theorem subArc_lt_of_ne_last {m : ℕ} {i : Fin (m + 1)} (hi : i ≠ Fin.last m) :
i.val < m := by
rcases Nat.lt_or_ge i.val m with h | h
· exact h
· exact absurd (Fin.ext (by simp only [Fin.val_last]; omega)) hi
/-- At a NON-last index `i ≠ Fin.last m` the cyclic successor of the sub-arc is `A ⟨s + i.val + 1⟩`
(no wraparound). -/
theorem subArc_succ_of_ne_last {n : ℕ} (A : Fin (n + 1) → S2) {s t : ℕ}
(hst : s < t) (htn : t ≤ n) {i : Fin ((t - s) + 1)} (hi : i ≠ Fin.last (t - s)) :
subArc A s t hst htn (i + 1) = A ⟨s + i.val + 1, by
have := subArc_lt_of_ne_last hi; omega⟩ := by
have hiv : i.val < t - s := subArc_lt_of_ne_last hi
show A ⟨s + (i + 1).val, _⟩ = A ⟨s + i.val + 1, _⟩
congr 1
apply Fin.ext
show s + (i + 1).val = s + i.val + 1
have hsucc : (i + 1 : Fin ((t - s) + 1)).val = i.val + 1 := by
rw [Fin.val_add, one_val_subArc (by omega)]
exact Nat.mod_eq_of_lt (by omega)
rw [hsucc]; omega
/-- At the last index the cyclic successor wraps to `A s`: `subArc A (Fin.last m + 1) = A s`. -/
theorem subArc_succ_last {n : ℕ} (A : Fin (n + 1) → S2) {s t : ℕ}
(hst : s < t) (htn : t ≤ n) :
subArc A s t hst htn (Fin.last (t - s) + 1) = A ⟨s, by omega⟩ := by
have hzero : (Fin.last (t - s) + 1 : Fin ((t - s) + 1)) = 0 := by
apply Fin.ext
simp only [Fin.val_add, Fin.val_last, Fin.val_zero, one_val_subArc (show 1 ≤ t - s by omega)]
rw [Nat.mod_self]
rw [hzero, subArc_zero]
/-- The diagonal `A s → A t` strictly supports the predecessor `A s`'s next interior vertex via an
interior witness. Direct instance of `cut_diagonal_supports`: for `s < k < t` (in `Fin (n+1)`),
`0 < sOrient (A s) (A k) (A t)`. -/
theorem subArc_diag_support {n : ℕ} {A : Fin (n + 1) → S2}
(hP : StrictConvexSphPolygon A) {s t : ℕ} (hst : s < t) (htn : t ≤ n)
{k : ℕ} (hsk : s < k) (hkt : k < t) :
0 < sOrient (A ⟨s, by omega⟩) (A ⟨k, by omega⟩) (A ⟨t, by omega⟩) := by
have hab : (⟨s, by omega⟩ : Fin (n + 1)) < ⟨k, by omega⟩ := by
rw [Fin.lt_def]; exact hsk
have hbk : (⟨k, by omega⟩ : Fin (n + 1)) < ⟨t, by omega⟩ := by
rw [Fin.lt_def]; exact hkt
exact cut_diagonal_supports hP hab hbk
/-- The closing diagonal edge `A t → A s` is a short arc, when the range has an interior vertex
(`2 ≤ t - s`): pick the interior witness `A (s+1)` strictly supported by the diagonal `A s → A t`,
which forces `A s ≠ A t` and `A s` non-antipodal to `A t`. -/
theorem subArc_shortArc_closing {n : ℕ} {A : Fin (n + 1) → S2}
(hP : StrictConvexSphPolygon A) {s t : ℕ} (hst : s < t) (htn : t ≤ n) (hm : 2 ≤ t - s) :
ShortArc (A ⟨t, by omega⟩) (A ⟨s, by omega⟩) := by
-- interior witness `k = s + 1` (exists since `2 ≤ t - s` makes `s + 1 < t`).
have hsk : s < s + 1 := by omega
have hkt : s + 1 < t := by omega
have hpos := subArc_diag_support hP hst htn hsk hkt
-- `0 < sOrient (A s)(A (s+1))(A t)` ⟹ `A s ≠ A t`, non-antipodal; symmetrise to the edge order.
have hne : A ⟨s, by omega⟩ ≠ A ⟨t, by omega⟩ := ne_of_sOrient_pos_ac hpos
have hnanti : (A ⟨s, by omega⟩ : E3) ≠ -(A ⟨t, by omega⟩ : E3) :=
not_antipodal_of_sOrient_pos_ac hpos
refine ⟨?_, ?_⟩
· exact fun he => hne he.symm
· intro he
apply hnanti
rw [he, neg_neg]
/-- The parent-index of a sub-arc vertex: `gidx i = ⟨s + i.val⟩`. -/
def gidx {n : ℕ} (s t : ℕ) (hst : s < t) (htn : t ≤ n) (i : Fin ((t - s) + 1)) : Fin (n + 1) :=
⟨s + i.val, by have := i.isLt; omega⟩
theorem subArc_eq_gidx {n : ℕ} (A : Fin (n + 1) → S2) (s t : ℕ) (hst : s < t) (htn : t ≤ n)
(i : Fin ((t - s) + 1)) :
subArc A s t hst htn i = A (gidx s t hst htn i) := rfl
theorem gidx_val {n : ℕ} (s t : ℕ) (hst : s < t) (htn : t ≤ n) (i : Fin ((t - s) + 1)) :
(gidx s t hst htn i).val = s + i.val := rfl
/-- `gidx` is injective. -/
theorem gidx_injective {n : ℕ} (s t : ℕ) (hst : s < t) (htn : t ≤ n) :
Function.Injective (gidx (n := n) s t hst htn) := by
intro a b hab
have hv : (gidx s t hst htn a).val = (gidx s t hst htn b).val := by rw [hab]
rw [gidx_val, gidx_val] at hv
exact Fin.ext (by omega)
/-- **The sub-arc polygon is strictly convex.** The contiguous range `A s, …, A t` of a strictly
convex spherical arm `A`, with `2 ≤ t - s`, is a strictly convex polygon. -/
theorem subArc_strictConvexPolygon {n : ℕ} (A : Fin (n + 1) → S2)
(hA : StrictConvexSphArm A) (s t : ℕ) (hst : s < t) (htn : t ≤ n) (hm : 2 ≤ t - s) :
StrictConvexSphPolygon (subArc A s t hst htn) := by
have hP := hA.closed_convex
have hinj : Function.Injective (gidx (n := n) s t hst htn) := gidx_injective s t hst htn
refine
{ three_le := by omega
edge_short := ?_
edge_support := ?_
strict_nonincident := ?_
open_hemisphere := ?_ }
· -- edge_short
intro i
by_cases hi : i = Fin.last (t - s)
· -- closing edge: A t → A s
subst hi
rw [subArc_last, subArc_succ_last]
exact subArc_shortArc_closing hP hst htn hm
· -- interior edge: A ⟨s+i⟩ → A ⟨s+i+1⟩, a parent edge
rw [subArc_apply, subArc_succ_of_ne_last A hst htn hi]
have hiv : i.val < t - s := subArc_lt_of_ne_last hi
have := hP.edge_short ⟨s + i.val, by omega⟩
rwa [show (⟨s + i.val, by omega⟩ + 1 : Fin (n + 1)) = ⟨s + i.val + 1, by omega⟩ by
apply Fin.ext
simp only [Fin.val_add, Fin.val_one']
rw [Nat.mod_eq_of_lt (show 1 < n + 1 by omega),
Nat.mod_eq_of_lt (show s + i.val + 1 < n + 1 by omega)]] at this
· -- edge_support
intro i j
rw [subArc_eq_gidx A s t hst htn j]
by_cases hi : i = Fin.last (t - s)
· -- closing edge: A t → A s
subst hi
rw [subArc_last, subArc_succ_last]
-- closing-edge support: `0 ≤ sOrient (A t)(A s)(A (gidx j))`.
have hjv : (gidx s t hst htn j).val = s + j.val := gidx_val s t hst htn j
have hjlt : j.val ≤ t - s := by have := j.isLt; omega
rcases Nat.lt_trichotomy (gidx s t hst htn j).val s with hlt | heq | hgt
· -- impossible: gidx j ≥ s
omega
· -- gidx j = s: repeated column (third = second)
have hjs : gidx s t hst htn j = ⟨s, by omega⟩ := Fin.ext (by rw [heq])
rw [hjs]; simp only [sOrient]; rw [det3_self_mid]
· -- gidx j > s: either gidx j = t (repeated first/third) or s < gidx j < t (strict, via cyclic)
by_cases hjt : (gidx s t hst htn j).val = t
· have hjeqt : gidx s t hst htn j = ⟨t, by omega⟩ := Fin.ext (by rw [hjt])
rw [hjeqt]; simp only [sOrient]; rw [det3_self_right]
· -- interior j: strict via cyclic re-indexing of the diagonal support
have hsk : s < (gidx s t hst htn j).val := hgt
have hkt : (gidx s t hst htn j).val < t := by
rw [hjv] at hjt ⊢; have := j.isLt; omega
have hpos := subArc_diag_support hP hst htn hsk hkt
-- `sOrient (A t)(A s)(A k) = sOrient (A s)(A k)(A t)`
have hcyc : sOrient (A ⟨t, by omega⟩) (A ⟨s, by omega⟩) (A (gidx s t hst htn j))
= sOrient (A ⟨s, by omega⟩) (A (gidx s t hst htn j)) (A ⟨t, by omega⟩) :=
(sOrient_cyclic _ _ _).1
rw [hcyc]
rw [show (A (gidx s t hst htn j)) = A ⟨(gidx s t hst htn j).val, by have := j.isLt; omega⟩ from rfl]
exact le_of_lt hpos
· -- interior edge: A's edge at index ⟨s+i⟩
rw [subArc_apply, subArc_succ_of_ne_last A hst htn hi]
have hiv : i.val < t - s := subArc_lt_of_ne_last hi
have hsucc : (⟨s + i.val, by omega⟩ + 1 : Fin (n + 1)) = ⟨s + i.val + 1, by omega⟩ := by
apply Fin.ext
simp only [Fin.val_add, Fin.val_one']
rw [Nat.mod_eq_of_lt (show 1 < n + 1 by omega),
Nat.mod_eq_of_lt (show s + i.val + 1 < n + 1 by omega)]
have := hP.edge_support ⟨s + i.val, by omega⟩ (gidx s t hst htn j)
rwa [hsucc] at this
· -- strict_nonincident
intro i j hji hji1
rw [subArc_eq_gidx A s t hst htn j]
by_cases hi : i = Fin.last (t - s)
· -- closing edge: A t → A s; non-incident means gidx j ∉ {t, s}, so s < gidx j < t.
subst hi
rw [subArc_last, subArc_succ_last]
have hjv : (gidx s t hst htn j).val = s + j.val := gidx_val s t hst htn j
-- j ≠ last ⟹ gidx j ≠ t; the wraparound successor of last is 0 ⟹ j ≠ 0 ⟹ gidx j ≠ s.
have hjne_last : j ≠ Fin.last (t - s) := hji
have hjval_lt : j.val < t - s := subArc_lt_of_ne_last hjne_last
have hjne_zero : j ≠ 0 := by
intro h; apply hji1; rw [h]
symm
apply Fin.ext
simp only [Fin.val_add, Fin.val_last, Fin.val_zero,
one_val_subArc (show 1 ≤ t - s by omega)]
rw [Nat.mod_self]
have hjpos : 0 < j.val := Nat.pos_of_ne_zero (fun hc => hjne_zero (Fin.ext (by simp [hc])))
have hsk : s < (gidx s t hst htn j).val := by rw [hjv]; omega
have hkt : (gidx s t hst htn j).val < t := by rw [hjv]; omega
have hpos := subArc_diag_support hP hst htn hsk hkt
have hcyc : sOrient (A ⟨t, by omega⟩) (A ⟨s, by omega⟩) (A (gidx s t hst htn j))
= sOrient (A ⟨s, by omega⟩) (A (gidx s t hst htn j)) (A ⟨t, by omega⟩) :=
(sOrient_cyclic _ _ _).1
rw [hcyc]
rw [show (A (gidx s t hst htn j)) = A ⟨(gidx s t hst htn j).val, by have := j.isLt; omega⟩ from rfl]
exact hpos
· -- interior edge: A's edge at index ⟨s+i⟩; transport non-incidence via gidx injectivity.
rw [subArc_apply, subArc_succ_of_ne_last A hst htn hi]
have hiv : i.val < t - s := subArc_lt_of_ne_last hi
have hsucc : (⟨s + i.val, by omega⟩ + 1 : Fin (n + 1)) = ⟨s + i.val + 1, by omega⟩ := by
apply Fin.ext
simp only [Fin.val_add, Fin.val_one']
rw [Nat.mod_eq_of_lt (show 1 < n + 1 by omega),
Nat.mod_eq_of_lt (show s + i.val + 1 < n + 1 by omega)]
-- non-incidence: gidx j ≠ gidx i = ⟨s+i⟩ and gidx j ≠ ⟨s+i+1⟩
have hne1 : gidx s t hst htn j ≠ (⟨s + i.val, by omega⟩ : Fin (n + 1)) := by
intro h
apply hji
have hgi : gidx s t hst htn i = (⟨s + i.val, by omega⟩ : Fin (n + 1)) := rfl
have : gidx s t hst htn j = gidx s t hst htn i := by rw [h, hgi]
exact hinj this
have hne2 : gidx s t hst htn j ≠ (⟨s + i.val + 1, by omega⟩ : Fin (n + 1)) := by
intro h
apply hji1
have hi1ne : (i + 1 : Fin ((t - s) + 1)) ≠ 0 := by
intro hc
have : ((i + 1 : Fin ((t - s) + 1)) : ℕ) = 0 := by rw [hc]; rfl
rw [Fin.val_add, one_val_subArc (show 1 ≤ t - s by omega),
Nat.mod_eq_of_lt (show i.val + 1 < t - s + 1 by omega)] at this
omega
have hgi1 : gidx s t hst htn (i + 1) = (⟨s + i.val + 1, by omega⟩ : Fin (n + 1)) := by
apply Fin.ext
rw [gidx_val]
simp only [Fin.val_add, one_val_subArc (show 1 ≤ t - s by omega),
Nat.mod_eq_of_lt (show i.val + 1 < t - s + 1 by omega)]
omega
have : gidx s t hst htn j = gidx s t hst htn (i + 1) := by rw [h, hgi1]
exact hinj this
have hsupp := hP.strict_nonincident ⟨s + i.val, by omega⟩ (gidx s t hst htn j) hne1
(by rwa [hsucc])
rwa [hsucc] at hsupp
· -- open_hemisphere: subArc A = A ∘ gidx, so reindex
obtain ⟨hh, hhn, hhpos⟩ := open_hemisphere_reindex hP (gidx (n := n) s t hst htn)
refine ⟨hh, hhn, ?_⟩
intro j
rw [show ((subArc A s t hst htn j : S2) : E3) = ((A (gidx s t hst htn j) : S2) : E3) by
rw [subArc_eq_gidx]]
exact hhpos j
/-- **The sub-arc is a strictly convex arm.** The contiguous range `A s, …, A t` of a strictly
convex spherical arm `A`, with `2 ≤ t - s`, is again a `StrictConvexSphArm` (parameter `m = t - s`,
closing diagonal `A s → A t`). -/
theorem subArc_strictConvexArm {n : ℕ} (A : Fin (n + 1) → S2)
(hA : StrictConvexSphArm A) (s t : ℕ) (hst : s < t) (htn : t ≤ n) (hm : 2 ≤ t - s) :
StrictConvexSphArm (subArc A s t hst htn) :=
{ two_le := hm
closed_convex := subArc_strictConvexPolygon A hA s t hst htn hm }
/-- **Interior arm sides = parent sides.** The `i`-th side of `subArc A s t` is the parent side
`sideLen A ⟨s + i.val⟩` (the edge `A(s+i) → A(s+i+1)`). -/
theorem subArc_sideLen {n : ℕ} (A : Fin (n + 1) → S2) (s t : ℕ) (hst : s < t) (htn : t ≤ n)
(i : Fin (t - s)) :
sideLen (subArc A s t hst htn) i = sideLen A ⟨s + i.val, by have := i.isLt; omega⟩ := by
unfold sideLen
have hc : (subArc A s t hst htn) i.castSucc = A ⟨s + i.val, by have := i.isLt; omega⟩ := rfl
have hs : (subArc A s t hst htn) i.succ = A ⟨s + i.val + 1, by have := i.isLt; omega⟩ := rfl
have hrc : (⟨s + i.val, by have := i.isLt; omega⟩ : Fin n).castSucc
= (⟨s + i.val, by have := i.isLt; omega⟩ : Fin (n + 1)) := by
apply Fin.ext; simp
have hrs : (⟨s + i.val, by have := i.isLt; omega⟩ : Fin n).succ
= (⟨s + i.val + 1, by have := i.isLt; omega⟩ : Fin (n + 1)) := by
apply Fin.ext; simp
rw [hc, hs, hrc, hrs]
/-- **Interior arm joints = parent joints.** The `i`-th joint of `subArc A s t` is the parent joint
`jointAngle A ⟨s + i.val⟩` (the spherical angle at `A(s+i+1)`). The new closing diagonal contributes
no interior joint. -/
theorem subArc_jointAngle {n : ℕ} (A : Fin (n + 1) → S2) (s t : ℕ) (hst : s < t) (htn : t ≤ n)
(i : Fin (t - s - 1)) :
jointAngle (subArc A s t hst htn) i = jointAngle A ⟨s + i.val, by have := i.isLt; omega⟩ := by
unfold jointAngle
have hb : i.val < t - s - 1 := i.isLt
have e0 : (subArc A s t hst htn) ⟨i.val, by omega⟩ = A ⟨s + i.val, by omega⟩ := rfl
have e1 : (subArc A s t hst htn) ⟨i.val + 1, by omega⟩ = A ⟨s + i.val + 1, by omega⟩ := rfl
have e2 : (subArc A s t hst htn) ⟨i.val + 2, by omega⟩ = A ⟨s + i.val + 2, by omega⟩ := rfl
rw [e0, e1, e2]
/-- **The arm endpoint chord is the diagonal `A s → A t`.** `sDist (subArc A 0) (subArc A (last m)) =
sDist (A s) (A t)`. -/
theorem subArc_endpt {n : ℕ} (A : Fin (n + 1) → S2) (s t : ℕ) (hst : s < t) (htn : t ≤ n) :
sDist (subArc A s t hst htn 0) (subArc A s t hst htn (Fin.last (t - s)))
= sDist (A ⟨s, by omega⟩) (A ⟨t, by omega⟩) := by
rw [subArc_zero, subArc_last]
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
/-- Number of adjacent-differing pairs in a boolean list (a "linear" sign-change count). -/
def flips : List Bool → ℕ
| [] => 0
| [_] => 0
| a :: b :: rest => (if a ≠ b then 1 else 0) + flips (b :: rest)
/-- The cyclic flip count: close the cycle by appending the head, then count linear flips. -/
def cyclicFlips (l : List Bool) : ℕ :=
match l with
| [] => 0
| h :: t => flips ((h :: t) ++ [h])
/-- **The parity engine.** `flips (a :: rest)` is even iff the first and last entries agree. -/
theorem flips_even_iff_first_eq_last :
∀ (a : Bool) (rest : List Bool), Even (flips (a :: rest)) ↔ a = (a :: rest).getLast (by simp)
| a, [] => by simp [flips]
| a, b :: rest => by
rw [show flips (a :: b :: rest) = (if a ≠ b then 1 else 0) + flips (b :: rest) from rfl]
rw [List.getLast_cons (by simp)]
have ih := flips_even_iff_first_eq_last b rest
by_cases hab : a = b
· subst hab; simp only [ne_eq, not_true_eq_false, if_false, Nat.zero_add]; rw [ih]
· simp only [ne_eq, hab, not_false_eq_true, if_true]
rw [Nat.add_comm, Nat.even_add_one, ih]
revert hab; cases a <;> cases (b :: rest).getLast (by simp) <;> cases b <;> simp
/-- **Cyclic parity.** The number of cyclic sign flips around any boolean cycle is even. -/
theorem cyclicFlips_even (l : List Bool) : Even (cyclicFlips l) := by
cases l with
| nil => simp [cyclicFlips]
| cons h t =>
show Even (flips ((h :: t) ++ [h]))
rw [show (h :: t) ++ [h] = h :: (t ++ [h]) by simp, flips_even_iff_first_eq_last]
rw [List.getLast_cons (by simp)]
simp
/-- `flips l = 0` exactly when all entries of `l` are equal. -/
theorem flips_eq_zero_iff_all_eq :
∀ (l : List Bool), flips l = 0 ↔ ∀ x ∈ l, ∀ y ∈ l, x = y
| [] => by simp [flips]
| [a] => by simp [flips]
| a :: b :: rest => by
rw [show flips (a :: b :: rest) = (if a ≠ b then 1 else 0) + flips (b :: rest) from rfl]
have ih := flips_eq_zero_iff_all_eq (b :: rest)
constructor
· intro h
have hab : a = b := by by_contra hne; simp [hne] at h
have hrest : flips (b :: rest) = 0 := by omega
have hall := ih.mp hrest
intro x hx y hy
simp only [List.mem_cons] at hx hy
have hxb : x = b := by
rcases hx with h' | h'
· rw [h', hab]
· exact hall x (by simp [List.mem_cons, h']) b (by simp)
have hyb : y = b := by
rcases hy with h' | h'
· rw [h', hab]
· exact hall y (by simp [List.mem_cons, h']) b (by simp)
rw [hxb, hyb]
· intro h
have hab : a = b := h a (by simp) b (by simp)
have hrest : flips (b :: rest) = 0 := by
rw [ih]; intro x hx y hy; exact h x (by simp [hx]) y (by simp [hy])
rw [hrest]; simp [hab]
/-- `cyclicFlips l = 0` forces all entries of `l` equal (the closed cycle has no flip). -/
theorem cyclicFlips_eq_zero_iff_all_eq (l : List Bool) :
cyclicFlips l = 0 ↔ ∀ x ∈ l, ∀ y ∈ l, x = y := by
cases l with
| nil => simp [cyclicFlips]
| cons h t =>
show flips ((h :: t) ++ [h]) = 0 ↔ _
rw [flips_eq_zero_iff_all_eq]
constructor
· intro hall x hx y hy
exact hall x (List.mem_append_left _ hx) y (List.mem_append_left _ hy)
· intro hall x hx y hy
-- every element of (h::t)++[h] lies in h::t (the trailing [h] is just h again)
have hxl : x ∈ h :: t := by
rcases List.mem_append.mp hx with hx' | hx'
· exact hx'
· rw [List.mem_singleton.mp hx']; exact List.mem_cons_self
have hyl : y ∈ h :: t := by
rcases List.mem_append.mp hy with hy' | hy'
· exact hy'
· rw [List.mem_singleton.mp hy']; exact List.mem_cons_self
exact hall x hxl y hyl
open scoped Classical
/-- The signed dihedral change at joint `i`. -/
noncomputable def jointDiff {n : ℕ} (A B : Fin (n + 1) → S2) (i : Fin (n - 1)) : ℝ :=
jointAngle B i - jointAngle A i
/-- The nonzero-sign boolean list of a real-valued sequence, in index order
(`true` = positive, `false` = negative; zeros skipped). -/
noncomputable def nzSigns {m : ℕ} (d : Fin m → ℝ) : List Bool :=
((List.finRange m).filter (fun i => decide (d i ≠ 0))).map (fun i => decide (0 < d i))
theorem mem_nzSigns {m : ℕ} (d : Fin m → ℝ) (b : Bool) :
b ∈ nzSigns d ↔ ∃ i, d i ≠ 0 ∧ b = decide (0 < d i) := by
simp only [nzSigns, List.mem_map, List.mem_filter, List.mem_finRange, true_and,
decide_eq_true_eq]
constructor
· rintro ⟨i, hi, rfl⟩; exact ⟨i, hi, rfl⟩
· rintro ⟨i, hi, rfl⟩; exact ⟨i, hi, rfl⟩
theorem nzSigns_ne_nil {m : ℕ} (d : Fin m → ℝ) (h : ∃ i, d i ≠ 0) : nzSigns d ≠ [] := by
obtain ⟨i, hi⟩ := h
intro hnil
have : decide (0 < d i) ∈ nzSigns d := (mem_nzSigns d _).mpr ⟨i, hi, rfl⟩
rw [hnil] at this; simp at this
/-- **All nonzero signs equal ⟹ monotone in one direction.** If `nzSigns d` is nonempty and all its
entries agree, then either every `d i ≥ 0` or every `d i ≤ 0`. -/
theorem all_same_sign {m : ℕ} (d : Fin m → ℝ)
(hne : nzSigns d ≠ [])
(hall : ∀ x ∈ nzSigns d, ∀ y ∈ nzSigns d, x = y) :
(∀ i, 0 ≤ d i) ∨ (∀ i, d i ≤ 0) := by
obtain ⟨head, ht, hhead⟩ := List.exists_cons_of_ne_nil hne
have hb : head ∈ nzSigns d := by rw [hhead]; exact List.mem_cons_self
by_cases hcase : head = true
· left
intro i
by_cases hzi : d i = 0
· rw [hzi]
· have hmem : decide (0 < d i) ∈ nzSigns d := (mem_nzSigns d _).mpr ⟨i, hzi, rfl⟩
have heq := hall _ hmem _ hb
rw [hcase] at heq
have : (0 : ℝ) < d i := by simpa using heq
linarith
· right
intro i
have hcase' : head = false := by cases head with | false => rfl | true => exact absurd rfl hcase
by_cases hzi : d i = 0
· rw [hzi]
· have hmem : decide (0 < d i) ∈ nzSigns d := (mem_nzSigns d _).mpr ⟨i, hzi, rfl⟩
have heq := hall _ hmem _ hb
rw [hcase'] at heq
have hnpos : ¬ (0 < d i) := by simpa using heq
linarith [lt_of_le_of_ne (not_lt.mp hnpos) hzi]
/-- Cyclic rotation of a closed arm: `rotPoly A k i = A (i + k)`. -/
def rotPoly {n : ℕ} (A : Fin (n + 1) → S2) (k : Fin (n + 1)) : Fin (n + 1) → S2 :=
fun i => A (i + k)
/-- **Cyclic rotation preserves strict convexity (`rotateArm`).** The four geometric fields of
`StrictConvexSphPolygon` are cyclic, so relabelling the vertices by a fixed shift preserves them; the
arm bound `two_le` is index-free. -/
theorem rotPoly_strictConvexArm {n : ℕ} {A : Fin (n + 1) → S2}
(hA : StrictConvexSphArm A) (k : Fin (n + 1)) :
StrictConvexSphArm (rotPoly A k) := by
refine { two_le := hA.two_le, closed_convex := ?_ }
have hP := hA.closed_convex
have key : ∀ i : Fin (n + 1), (i + 1) + k = (i + k) + 1 := fun i => by rw [add_right_comm]
refine { three_le := hP.three_le, edge_short := ?_, edge_support := ?_,
strict_nonincident := ?_, open_hemisphere := ?_ }
· intro i
show ShortArc (A (i + k)) (A ((i + 1) + k))
rw [key i]; exact hP.edge_short (i + k)
· intro i j
show 0 ≤ sOrient (A (i + k)) (A ((i + 1) + k)) (A (j + k))
rw [key i]; exact hP.edge_support (i + k) (j + k)
· intro i j hji hji1
show 0 < sOrient (A (i + k)) (A ((i + 1) + k)) (A (j + k))
rw [key i]
apply hP.strict_nonincident (i + k) (j + k)
· intro h; exact hji (add_right_cancel h)
· intro h; apply hji1
have h2 : j + k = (i + 1) + k := by rw [key i]; exact h
exact add_right_cancel h2
· obtain ⟨h, hn, hpos⟩ := hP.open_hemisphere
exact ⟨h, hn, fun i => hpos (i + k)⟩
/-- **Every cyclic edge of `A` equals the corresponding edge of `B`.** The `n` interior edges are
equal by `hsides`; the closing edge `A n → A 0` is equal by `hclose`. -/
theorem all_cyclic_edges_eq {n : ℕ} (hn : 1 ≤ n) (A B : Fin (n + 1) → S2)
(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))) :
∀ k : Fin (n + 1), sDist (A k) (A (k + 1)) = sDist (B k) (B (k + 1)) := by
intro k
rcases Nat.lt_or_ge k.val n with hk | hk
· have hkk := hsides ⟨k.val, hk⟩
unfold sideLen at hkk
have e1 : (⟨k.val, hk⟩ : Fin n).castSucc = k := by apply Fin.ext; simp
have hk1v : (k + 1 : Fin (n + 1)).val = k.val + 1 := by
rw [Fin.val_add, Fin.val_one', Nat.mod_eq_of_lt (show 1 < n + 1 by omega),
Nat.mod_eq_of_lt (show k.val + 1 < n + 1 by omega)]
have e2 : (⟨k.val, hk⟩ : Fin n).succ = k + 1 := by
apply Fin.ext; rw [Fin.val_succ, hk1v]
rw [e1, e2] at hkk; exact hkk
· have hkn : k.val = n := by have := k.isLt; omega
have hklast : k = Fin.last n := Fin.ext (by simp [hkn])
have hk1 : k + 1 = 0 := by
apply Fin.ext
show (k.val + (1 : Fin (n + 1)).val) % (n + 1) = 0
rw [hkn, Fin.val_one', Nat.mod_eq_of_lt (show 1 < n + 1 by omega), Nat.mod_self]
rw [hk1, hklast]
rw [sDist_comm (A (Fin.last n)) (A 0), sDist_comm (B (Fin.last n)) (B 0)]
exact hclose
/-- **The rotated arms have equal corresponding sides.** `rotPoly A k` and `rotPoly B k` agree on
every arm side, because every cyclic edge of `A` equals the corresponding edge of `B`. -/
theorem rotPoly_sideLen_eq {n : ℕ} (hn : 1 ≤ n) (A B : Fin (n + 1) → S2)
(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)))
(k : Fin (n + 1)) :
∀ i : Fin n, sideLen (rotPoly A k) i = sideLen (rotPoly B k) i := by
intro i
unfold sideLen rotPoly
-- side i of rotPoly A k = sDist (A (i.castSucc + k)) (A (i.succ + k)); and i.succ = i.castSucc + 1.
have key := all_cyclic_edges_eq hn A B hsides hclose (i.castSucc + k)
have hcast : (i.succ : Fin (n + 1)) = i.castSucc + 1 := by
apply Fin.ext
rw [Fin.val_succ, Fin.val_add, Fin.val_castSucc, Fin.val_one',
Nat.mod_eq_of_lt (show 1 < n + 1 by omega),
Nat.mod_eq_of_lt (show i.val + 1 < n + 1 by have := i.isLt; omega)]
have heq : (i.succ : Fin (n + 1)) + k = (i.castSucc + k) + 1 := by
rw [hcast, add_right_comm]
rw [heq]; exact key
structure TwoArcSplitData {n : ℕ} (A B : Fin (n + 1) → S2) where
/-- arc-1 parameter. -/
m₁ : ℕ
/-- arc-2 parameter. -/
m₂ : ℕ
hm₁ : 2 ≤ m₁
hm₂ : 2 ≤ m₂
Arc1 : Fin (m₁ + 1) → S2
Brc1 : Fin (m₁ + 1) → S2
Arc2 : Fin (m₂ + 1) → S2
Brc2 : Fin (m₂ + 1) → S2
harc1A : StrictConvexSphArm Arc1
harc1B : StrictConvexSphArm Brc1
harc2A : StrictConvexSphArm Arc2
harc2B : StrictConvexSphArm Brc2
hsides1 : ∀ i : Fin m₁, sideLen Arc1 i = sideLen Brc1 i
hsides2 : ∀ i : Fin m₂, sideLen Arc2 i = sideLen Brc2 i
hshareA : sDist (Arc1 0) (Arc1 (Fin.last m₁)) = sDist (Arc2 0) (Arc2 (Fin.last m₂))
hshareB : sDist (Brc1 0) (Brc1 (Fin.last m₁)) = sDist (Brc2 0) (Brc2 (Fin.last m₂))
hmono1 : ∀ i : Fin (m₁ - 1), jointAngle Arc1 i ≤ jointAngle Brc1 i
hstrict1 : ∃ i : Fin (m₁ - 1), jointAngle Arc1 i < jointAngle Brc1 i
hmono2 : ∀ i : Fin (m₂ - 1), jointAngle Brc2 i ≤ jointAngle Arc2 i
/-- **The two-arc cut yields `False`.** Genuine two-arc split data is refuted by the proven Cauchy
Lemma II two-arc contradiction. -/
theorem TwoArcSplitData.contradiction {n : ℕ} {A B : Fin (n + 1) → S2}
(d : TwoArcSplitData A B) : False :=
cauchy_two_signchange_split d.hm₁ d.hm₂ d.Arc1 d.Brc1 d.Arc2 d.Brc2
d.harc1A d.harc1B d.harc2A d.harc2B d.hsides1 d.hsides2 d.hshareA d.hshareB
d.hmono1 d.hstrict1 d.hmono2
/-- **The `signChanges = 2` fixed-chord obstruction.** From genuine two-arc split data the two-arc
argument gives `False`; any `CauchyArmFixedChordObstruction` then follows (its only use is its
`.contradiction`, which is exactly this `False`). -/
noncomputable def twoSignChanges_obstruction {n : ℕ} {A B : Fin (n + 1) → S2}
(d : TwoArcSplitData A B) : CauchyArmFixedChordObstruction :=
(d.contradiction).elim
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
/-- The spherical angle of the closed link polygon at vertex `i : Fin (n+1)`, taken between its two
**cyclic** neighbours `A (i-1)` and `A (i+1)`. -/
noncomputable def linkAngle {n : ℕ} (A : Fin (n + 1) → S2) (i : Fin (n + 1)) : ℝ :=
sphAngle (A (i - 1)) (A i) (A (i + 1))
/-- **Interior link angles ARE arm joints.** For an interior joint index `i : Fin (n-1)`, the link
angle at vertex `i+1` equals the arm joint angle `jointAngle A i`. (No wraparound: `1 ≤ i+1 ≤ n-1`.) -/
theorem linkAngle_interior {n : ℕ} (A : Fin (n + 1) → S2) (i : Fin (n - 1)) :
linkAngle A ⟨i.val + 1, by have := i.isLt; omega⟩ = jointAngle A i := by
have hi := i.isLt
unfold linkAngle jointAngle
-- the three cyclic indices i+1-1, i+1, i+1+1 are ⟨i⟩, ⟨i+1⟩, ⟨i+2⟩ (no wrap)
have e0 : (⟨i.val + 1, by omega⟩ : Fin (n + 1)) - 1 = ⟨i.val, by omega⟩ := by
apply Fin.ext
rw [Fin.sub_def, Fin.val_one', Nat.mod_eq_of_lt (show 1 < n + 1 by omega)]
show (n + 1 - 1 + (i.val + 1)) % (n + 1) = i.val
rw [show n + 1 - 1 + (i.val + 1) = i.val + (n + 1) by omega, Nat.add_mod_right,
Nat.mod_eq_of_lt (by omega)]
have e2 : (⟨i.val + 1, by omega⟩ : Fin (n + 1)) + 1 = ⟨i.val + 2, by omega⟩ := by
apply Fin.ext
rw [Fin.val_add, Fin.val_one', Nat.mod_eq_of_lt (show 1 < n + 1 by omega),
Nat.mod_eq_of_lt (show i.val + 1 + 1 < n + 1 by omega)]
rw [e0, e2]
/-- The closing angle at link vertex `0`: between the closing edge `n → 0` and edge `0 → 1`. -/
theorem linkAngle_zero {n : ℕ} (A : Fin (n + 1) → S2) :
linkAngle A 0 = sphAngle (A (Fin.last n)) (A 0) (A 1) := by
unfold linkAngle
have em1 : (0 : Fin (n + 1)) - 1 = Fin.last n :=
sub_eq_of_eq_add (Fin.last_add_one n).symm
have ep1 : (0 : Fin (n + 1)) + 1 = 1 := by simp
rw [em1, ep1]
/-- The closing angle at link vertex `n` (= `Fin.last n`): between edge `n-1 → n` and the closing
edge `n → 0`. -/
theorem linkAngle_last {n : ℕ} (A : Fin (n + 1) → S2) :
linkAngle A (Fin.last n) = sphAngle (A (Fin.last n - 1)) (A (Fin.last n)) (A 0) := by
unfold linkAngle
rw [Fin.last_add_one]
/-- The signed change of the link angle at vertex `i : Fin (n+1)`. -/
noncomputable def linkDiff {n : ℕ} (A B : Fin (n + 1) → S2) (i : Fin (n + 1)) : ℝ :=
linkAngle B i - linkAngle A i
/-- **The FULL closed-link cyclic sign-change count.** The genuine cyclic flip count of the nonzero
dihedral-change signs around ALL `n+1` link vertices (the closing angles included). -/
noncomputable def signChangesFull {n : ℕ} (A B : Fin (n + 1) → S2) : ℕ :=
cyclicFlips (nzSigns (linkDiff A B))
/-- `signChangesFull` is even (cyclic parity), a genuine theorem. -/
theorem signChangesFull_even {n : ℕ} (A B : Fin (n + 1) → S2) : Even (signChangesFull A B) :=
cyclicFlips_even _
/-- **The full link-diff restricts to the arm joint-diff.** At the interior link vertex `i+1`, the
full `linkDiff` equals the arm `jointDiff` (`Ch13ArmVertex.jointDiff`). -/
theorem linkDiff_interior {n : ℕ} (A B : Fin (n + 1) → S2) (i : Fin (n - 1)) :
linkDiff A B ⟨i.val + 1, by have := i.isLt; omega⟩ = jointDiff A B i := by
unfold linkDiff jointDiff
rw [linkAngle_interior, linkAngle_interior]
/-- **The `signChangesFull = 0` fixed-chord obstruction.** Same conclusion as
`Ch13ArmVertex.zeroSignChanges_obstruction`, but the hypothesis is now `0` changes over the FULL
`n+1`-cycle (closing angles included), which is *stronger* and so still yields the obstruction. -/
noncomputable def zeroSignChangesFull_obstruction {n : ℕ} (hn : 2 ≤ 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)))
(hactive : ∃ i : Fin (n - 1), jointAngle A i ≠ jointAngle B i)
(hzero : signChangesFull A B = 0) :
CauchyArmFixedChordObstruction := by
-- Some interior arm joint differs ⟹ the corresponding FULL link-diff is nonzero.
have hactive' : ∃ k, linkDiff A B k ≠ 0 := by
obtain ⟨i, hi⟩ := hactive
refine ⟨⟨i.val + 1, by have := i.isLt; omega⟩, ?_⟩
rw [linkDiff_interior]
exact sub_ne_zero.mpr (Ne.symm hi)
have hne : nzSigns (linkDiff A B) ≠ [] := nzSigns_ne_nil _ hactive'
have hall : ∀ x ∈ nzSigns (linkDiff A B), ∀ y ∈ nzSigns (linkDiff A B), x = y :=
(cyclicFlips_eq_zero_iff_all_eq _).mp hzero
have hdir := all_same_sign (linkDiff A B) hne hall
by_cases hpos : ∀ k, 0 ≤ linkDiff A B k
· -- opening: every interior joint A ≤ B (read off from the full diff at vertex i+1)
refine CauchyArmFixedChordObstruction.opening
{ n := n, hn := hn, A := A, B := B, hA := hA, hB := hB,
equal_sides := hsides
opened := fun i => ?_
some_angle_strictly_opened := ?_
fixed_chord := hclose }
· have hk := hpos ⟨i.val + 1, by have := i.isLt; omega⟩
rw [linkDiff_interior] at hk
simp only [jointDiff] at hk; linarith
· obtain ⟨i, hi⟩ := hactive
refine ⟨i, ?_⟩
have hk := hpos ⟨i.val + 1, by have := i.isLt; omega⟩
rw [linkDiff_interior] at hk
simp only [jointDiff] at hk
have hne' : jointAngle B i - jointAngle A i ≠ 0 := sub_ne_zero.mpr (Ne.symm hi)
linarith [lt_of_le_of_ne hk (Ne.symm hne')]
· -- closing: every interior joint B ≤ A
have hneg : ∀ k, linkDiff A B k ≤ 0 := hdir.resolve_left hpos
refine CauchyArmFixedChordObstruction.closing
{ n := n, hn := hn, A := A, B := B, hA := hA, hB := hB,
equal_sides := hsides
closed := fun i => ?_
some_angle_strictly_closed := ?_
fixed_chord := hclose }
· have hk := hneg ⟨i.val + 1, by have := i.isLt; omega⟩
rw [linkDiff_interior] at hk
simp only [jointDiff] at hk; linarith
· obtain ⟨i, hi⟩ := hactive
refine ⟨i, ?_⟩
have hk := hneg ⟨i.val + 1, by have := i.isLt; omega⟩
rw [linkDiff_interior] at hk
simp only [jointDiff] at hk
have hne' : jointAngle B i - jointAngle A i ≠ 0 := sub_ne_zero.mpr (Ne.symm hi)
linarith [lt_of_le_of_ne hk hne']
noncomputable def cauchyArmVertexFull_of_links (n : ℕ) (hn : 2 ≤ 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)))
(hactive : ∃ i : Fin (n - 1), jointAngle A i ≠ jointAngle B i)
(htwoArc : signChangesFull A B = 2 → TwoArcSplitData A B) :
Chapter13.CauchyArmVertex where
signChanges := signChangesFull A B
signChanges_even := signChangesFull_even A B
zero_sign_changes_obstruction :=
fun hzero => zeroSignChangesFull_obstruction hn A B hA hB hsides hclose hactive hzero
two_sign_changes_obstruction :=
fun htwo => twoSignChanges_obstruction (htwoArc htwo)
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
/-- The **local ℝ³ data of one convex-polytope vertex**.
`n + 1` neighbours `p 0, …, p n` are arranged cyclically around the apex `o`. The convexity data is
phrased entirely on the **raw** edge vectors `p i - o` (no posited spherical structure):
* `apex_ne` — every neighbour differs from the apex (so the edge direction is well defined);
* `open_hemi` — the raw edge vectors all lie strictly on the positive side of a common unit
functional `h` (one open hemisphere);
* `turn_support` — each oriented raw edge `(p i - o, p (i+1) - o)` keeps **every** other raw
direction on the nonnegative side of the plane it spans through the apex;
* `turn_strict` — the same, strictly, for the non-incident directions.
These are exactly the predicates `vertexLink_strictArm` consumes, and they are satisfiable
(see `cubeCornerStar`). -/
structure VertexStar where
/-- Arm parameter: there are `n + 1` neighbours / edges, with `n ≥ 2`. -/
n : ℕ
/-- At least three edges (`n + 1 ≥ 3`): a convex-polytope vertex has `≥ 3` incident edges. -/
hn : 2 ≤ n
/-- The apex of the star. -/
o : E3
/-- The cyclically ordered neighbours. -/
p : Fin (n + 1) → E3
/-- Each neighbour differs from the apex. -/
apex_ne : ∀ i : Fin (n + 1), p i ≠ o
/-- The raw edge vectors lie in one open hemisphere. -/
open_hemi : ∃ h : E3, ‖h‖ = 1 ∧ ∀ i : Fin (n + 1), 0 < ⟪h, p i - o⟫
/-- Each oriented raw edge supports every raw direction on the nonnegative side. -/
turn_support : ∀ i j : Fin (n + 1), 0 ≤ det3 (p i - o) (p (i + 1) - o) (p j - o)
/-- Non-incident raw directions are strictly on the positive side. -/
turn_strict : ∀ i j : Fin (n + 1), j ≠ i → j ≠ i + 1 →
0 < det3 (p i - o) (p (i + 1) - o) (p j - o)
namespace VertexStar
variable (S : VertexStar)
/-- The raw edge vector `p i - o`. -/
def rawDir (i : Fin (S.n + 1)) : E3 := S.p i - S.o
theorem rawDir_ne_zero (i : Fin (S.n + 1)) : S.rawDir i ≠ 0 := by
simp only [rawDir, sub_ne_zero]
exact S.apex_ne i
theorem norm_rawDir_pos (i : Fin (S.n + 1)) : 0 < ‖S.rawDir i‖ :=
norm_pos_iff.mpr (S.rawDir_ne_zero i)
/-- The unit edge direction at the apex toward `p i`, a point of `S²`. -/
def edgeDir (i : Fin (S.n + 1)) : S2 :=
⟨‖S.rawDir i‖⁻¹ • S.rawDir i, by
rw [norm_smul, norm_inv, norm_norm]
exact inv_mul_cancel₀ (ne_of_gt (S.norm_rawDir_pos i))⟩
/-- `edgeDir i` is the positive multiple `‖rawDir i‖⁻¹` of the raw direction. -/
theorem edgeDir_coe (i : Fin (S.n + 1)) :
(S.edgeDir i : E3) = ‖S.rawDir i‖⁻¹ • S.rawDir i := rfl
theorem inv_norm_pos (i : Fin (S.n + 1)) : 0 < ‖S.rawDir i‖⁻¹ :=
inv_pos.mpr (S.norm_rawDir_pos i)
/-- The **vertex link**: the cyclic edge-direction family, presented as a `Fin (n+1) → S²` tuple in
the `StrictConvexSphArm` convention. -/
def vertexLink : Fin (S.n + 1) → S2 := S.edgeDir
@[simp] theorem vertexLink_apply (i : Fin (S.n + 1)) : S.vertexLink i = S.edgeDir i := rfl
/-- `det3` is multilinear: scaling each argument multiplies the determinant by the product. -/
theorem det3_smul (ca cb cc : ℝ) (a b c : E3) :
det3 (ca • a) (cb • b) (cc • c) = (ca * cb * cc) * det3 a b c := by
simp only [det3, PiLp.smul_apply, smul_eq_mul]; ring
/-- The signed volume of three link directions is a positive multiple of the raw `det3`. -/
theorem sOrient_edgeDir (a b c : Fin (S.n + 1)) :
sOrient (S.edgeDir a) (S.edgeDir b) (S.edgeDir c)
= (‖S.rawDir a‖⁻¹ * ‖S.rawDir b‖⁻¹ * ‖S.rawDir c‖⁻¹)
* det3 (S.rawDir a) (S.rawDir b) (S.rawDir c) := by
rw [sOrient, edgeDir_coe, edgeDir_coe, edgeDir_coe, det3_smul]
/-- The product of the three inverse norms is positive. -/
theorem inv_norm_prod_pos (a b c : Fin (S.n + 1)) :
0 < ‖S.rawDir a‖⁻¹ * ‖S.rawDir b‖⁻¹ * ‖S.rawDir c‖⁻¹ :=
mul_pos (mul_pos (S.inv_norm_pos a) (S.inv_norm_pos b)) (S.inv_norm_pos c)
/-- Sign transfer (nonnegative): the raw support hypothesis gives nonnegative link orientation. -/
theorem sOrient_edgeDir_nonneg (a b c : Fin (S.n + 1))
(h : 0 ≤ det3 (S.rawDir a) (S.rawDir b) (S.rawDir c)) :
0 ≤ sOrient (S.edgeDir a) (S.edgeDir b) (S.edgeDir c) := by
rw [sOrient_edgeDir]
exact mul_nonneg (le_of_lt (S.inv_norm_prod_pos a b c)) h
/-- Sign transfer (strict). -/
theorem sOrient_edgeDir_pos (a b c : Fin (S.n + 1))
(h : 0 < det3 (S.rawDir a) (S.rawDir b) (S.rawDir c)) :
0 < sOrient (S.edgeDir a) (S.edgeDir b) (S.edgeDir c) := by
rw [sOrient_edgeDir]
exact mul_pos (S.inv_norm_prod_pos a b c) h
/-- Since `n ≥ 2`, every edge index `i` admits a non-incident index `j` (`j ≠ i`, `j ≠ i+1`). -/
theorem exists_noninc (i : Fin (S.n + 1)) : ∃ j : Fin (S.n + 1), j ≠ i ∧ j ≠ i + 1 := by
-- The set `{i, i+1}` has at most 2 elements; with `≥ 3` total there is a third.
by_contra hcon
push_neg at hcon
-- every j is i or i+1
have hsub : (Finset.univ : Finset (Fin (S.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 (S.n + 1))).card ≤ 2 :=
le_trans (Finset.card_insert_le _ _) (by simp)
have := S.hn
omega
/-- If `det3 a b c > 0` for some `c`, then `b` is not a scalar multiple of `a`. -/
theorem not_smul_of_det3_pos {a b : E3} (c : E3) (h : 0 < det3 a b c)
(t : ℝ) : b ≠ t • a := by
intro hb
rw [hb] at h
have : det3 a (t • a) c = 0 := by
simp only [det3, PiLp.smul_apply, smul_eq_mul]; ring
rw [this] at h; exact lt_irrefl _ h
/-- Consecutive link directions are not equal. -/
theorem edgeDir_ne (i : Fin (S.n + 1)) : S.edgeDir i ≠ S.edgeDir (i + 1) := by
obtain ⟨j, hji, hjip⟩ := S.exists_noninc i
intro heq
-- edgeDir i = edgeDir (i+1) ⟹ rawDir (i+1) = (‖raw(i+1)‖/‖raw i‖) • rawDir i
have hcoe : (S.edgeDir i : E3) = (S.edgeDir (i + 1) : E3) := by rw [heq]
rw [edgeDir_coe, edgeDir_coe] at hcoe
-- rawDir (i+1) = (‖raw(i+1)‖ * ‖raw i‖⁻¹) • rawDir i
have hraw : S.rawDir (i + 1) = (‖S.rawDir (i + 1)‖ * ‖S.rawDir i‖⁻¹) • S.rawDir i := by
have h1 : ‖S.rawDir (i + 1)‖ • (‖S.rawDir i‖⁻¹ • S.rawDir i)
= ‖S.rawDir (i + 1)‖ • (‖S.rawDir (i + 1)‖⁻¹ • S.rawDir (i + 1)) := by
rw [hcoe]
rw [smul_smul, smul_smul, mul_inv_cancel₀ (ne_of_gt (S.norm_rawDir_pos (i + 1))),
one_smul] at h1
exact h1.symm
have hpos := S.turn_strict i j hji hjip
exact (not_smul_of_det3_pos (S.rawDir j) hpos (‖S.rawDir (i + 1)‖ * ‖S.rawDir i‖⁻¹)) hraw
/-- Consecutive link directions are not antipodal. -/
theorem edgeDir_not_antipodal (i : Fin (S.n + 1)) :
(S.edgeDir i : E3) ≠ -(S.edgeDir (i + 1) : E3) := by
obtain ⟨j, hji, hjip⟩ := S.exists_noninc i
intro heq
rw [edgeDir_coe, edgeDir_coe] at heq
-- ‖raw i‖⁻¹ • raw i = -(‖raw(i+1)‖⁻¹ • raw(i+1)) ⟹ raw(i+1) = (-(‖raw(i+1)‖ * ‖raw i‖⁻¹)) • raw i
have hraw : S.rawDir (i + 1) = (-(‖S.rawDir (i + 1)‖ * ‖S.rawDir i‖⁻¹)) • S.rawDir i := by
have h1 := congrArg (fun z : E3 => ‖S.rawDir (i + 1)‖ • z) heq
simp only [smul_neg, smul_smul] at h1
rw [mul_inv_cancel₀ (ne_of_gt (S.norm_rawDir_pos (i + 1))), one_smul] at h1
-- h1 : (‖r(i+1)‖ * ‖r i‖⁻¹) • r i = -r(i+1)
rw [neg_smul, h1, neg_neg]
have hpos := S.turn_strict i j hji hjip
exact (not_smul_of_det3_pos (S.rawDir j) hpos (-(‖S.rawDir (i + 1)‖ * ‖S.rawDir i‖⁻¹))) hraw
/-- The consecutive link edge is a short arc. -/
theorem edgeDir_shortArc (i : Fin (S.n + 1)) : ShortArc (S.edgeDir i) (S.edgeDir (i + 1)) :=
⟨S.edgeDir_ne i, S.edgeDir_not_antipodal i⟩
theorem inner_h_edgeDir_pos {h : E3} (i : Fin (S.n + 1))
(hraw : 0 < ⟪h, S.rawDir i⟫) : 0 < ⟪h, (S.edgeDir i : E3)⟫ := by
rw [edgeDir_coe, real_inner_smul_right]
exact mul_pos (S.inv_norm_pos i) hraw
/-- **The vertex link is a strictly convex spherical arm.**
Each of the five `StrictConvexSphPolygon` fields is derived from the ℝ³ convexity fields of `S`:
`edge_short` from `turn_strict` (via `edgeDir_shortArc`); `edge_support`/`strict_nonincident` from
`turn_support`/`turn_strict` (via the `det3` sign-transfer lemmas); `open_hemisphere` from
`open_hemi`; `three_le` from `hn`. -/
theorem vertexLink_strictArm : StrictConvexSphArm S.vertexLink where
two_le := S.hn
closed_convex :=
{ three_le := by have := S.hn; omega
edge_short := fun i => by
simpa only [vertexLink_apply] using S.edgeDir_shortArc i
edge_support := fun i j => by
simp only [vertexLink_apply]
exact S.sOrient_edgeDir_nonneg i (i + 1) j (S.turn_support i j)
strict_nonincident := fun i j hji hjip => by
simp only [vertexLink_apply]
exact S.sOrient_edgeDir_pos i (i + 1) j (S.turn_strict i j hji hjip)
open_hemisphere := by
obtain ⟨h, hh, hpos⟩ := S.open_hemi
exact ⟨h, hh, fun i => by
simpa only [vertexLink_apply] using S.inner_h_edgeDir_pos i (hpos i)⟩ }
/-- The spherical side length of the link equals `arccos` of the inner product of the two link unit
directions. -/
theorem sideLen_vertexLink_eq_iAngle (i : Fin S.n) :
sideLen S.vertexLink i
= InnerProductGeometry.angle (S.edgeDir i.castSucc : E3) (S.edgeDir i.succ : E3) := by
-- sideLen = sDist (edgeDir castSucc) (edgeDir succ) = arccos (sInner …)
unfold sideLen
rw [vertexLink_apply, vertexLink_apply]
rw [sDist, InnerProductGeometry.angle]
congr 1
-- sInner = ⟪·,·⟫ ; and ‖edgeDir‖ = 1 so the denominator is 1
rw [sInner]
rw [S2.norm_coe, S2.norm_coe, mul_one, div_one]
/-- **Bridge A.** The link side length is the Euclidean angle at the apex between the two incident
neighbours. Here `i.castSucc` and `i.succ = i.castSucc + 1` are consecutive neighbour indices. -/
theorem sideLen_vertexLink (i : Fin S.n) :
sideLen S.vertexLink i
= EuclideanGeometry.angle (S.p i.castSucc) S.o (S.p i.succ) := by
rw [sideLen_vertexLink_eq_iAngle]
-- edgeDir = (positive scalar) • rawDir, and rawDir k = p k - o = p k -ᵥ o
rw [edgeDir_coe, edgeDir_coe]
rw [InnerProductGeometry.angle_smul_left_of_pos _ _ (S.inv_norm_pos _),
InnerProductGeometry.angle_smul_right_of_pos _ _ (S.inv_norm_pos _)]
-- rawDir k = p k - o = p k -ᵥ o
show InnerProductGeometry.angle (S.rawDir i.castSucc) (S.rawDir i.succ) = _
rw [EuclideanGeometry.angle]
rfl
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
variable (S : VertexStar)
/-- The lower index `⟨i.val, _⟩ : Fin (S.n + 1)` of the consecutive triple at internal joint `i`. -/
def jIdx0 (i : Fin (S.n - 1)) : Fin (S.n + 1) := ⟨i.val, by have := i.isLt; omega⟩
/-- The middle index `⟨i.val + 1, _⟩ : Fin (S.n + 1)` (the dihedral edge). -/
def jIdx1 (i : Fin (S.n - 1)) : Fin (S.n + 1) := ⟨i.val + 1, by have := i.isLt; omega⟩
/-- The upper index `⟨i.val + 2, _⟩ : Fin (S.n + 1)`. -/
def jIdx2 (i : Fin (S.n - 1)) : Fin (S.n + 1) := ⟨i.val + 2, by have := i.isLt; omega⟩
/-- **The extrinsic dihedral angle** of the star `S` along the middle edge `o → p ⟨i+1⟩` of the
consecutive triple `⟨i⟩, ⟨i+1⟩, ⟨i+2⟩`. Defined directly from the ℝ³ data, mirroring
`TetDihedral.dihedralAngle`: project the two outer raw edge vectors `p ⟨i⟩ - o` and `p ⟨i+2⟩ - o`
onto the plane perpendicular to the middle raw edge direction `p ⟨i+1⟩ - o`, and take the angle.
This is *independent* of the spherical link `vertexLink`; Bridge B (`jointAngle_vertexLink_eq_dihedral`)
identifies it with the link's `jointAngle`. -/
def dihedral (i : Fin (S.n - 1)) : ℝ :=
InnerProductGeometry.angle
(projOut (S.rawDir (S.jIdx1 i)) (S.rawDir (S.jIdx0 i)))
(projOut (S.rawDir (S.jIdx1 i)) (S.rawDir (S.jIdx2 i)))
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
/-- Cyclic list agreement up to orientation reversal. -/
def DihedralRotated {α : Type*} (l m : List α) : Prop :=
l ~r m ∨ l.reverse ~r m
end List
/-- The two-valued bijection `true ↦ plus`, `false ↦ minus`. -/
def boolToStrict : Bool → StrictEdgeSign
| true => StrictEdgeSign.plus
| false => StrictEdgeSign.minus
theorem boolToStrict_inj : Function.Injective boolToStrict := by
intro a b h; cases a <;> cases b <;> simp_all [boolToStrict]
/-- The real-sign map into `EdgeSign`, matching `nzSigns`' `0 < d ↦ plus`, `d < 0 ↦ minus`,
`d = 0 ↦ zero` convention. -/
def realSignToEdgeSign (x : ℝ) : EdgeSign :=
if 0 < x then EdgeSign.plus else if x < 0 then EdgeSign.minus else EdgeSign.zero
theorem realSignToEdgeSign_eq_zero_iff (x : ℝ) : realSignToEdgeSign x = EdgeSign.zero ↔ x = 0 := by
unfold realSignToEdgeSign
rcases lt_trichotomy x 0 with h | h | h
· simp only [not_lt.mpr h.le, if_false, h, if_true]
constructor <;> intro h' <;> first | exact absurd h' (by decide) | (rw [h'] at h; exact absurd h (lt_irrefl 0))
· simp [h]
· simp only [h, if_true]
constructor <;> intro h' <;> first | exact absurd h' (by decide) | (rw [h'] at h; exact absurd h (lt_irrefl 0))
theorem toStrict_realSign_of_ne (x : ℝ) (hx : x ≠ 0) :
(realSignToEdgeSign x).toStrict = some (boolToStrict (decide (0 < x))) := by
unfold realSignToEdgeSign
rcases lt_trichotomy x 0 with h | h | h
· have hnp : ¬ (0 < x) := not_lt.mpr h.le
simp only [hnp, if_false, h, if_true]
simp [EdgeSign.toStrict, boolToStrict]
· exact absurd h hx
· simp only [h, if_true]
simp [EdgeSign.toStrict, boolToStrict]
theorem toStrict_realSign_of_zero {x : ℝ} (hx : x = 0) :
(realSignToEdgeSign x).toStrict = none := by
subst hx; simp [realSignToEdgeSign, EdgeSign.toStrict]
/-- `flips` of a Bool list with its cyclic closer `[first]` appended equals `flipAux` of the mapped
list (single structural induction). -/
theorem flips_append_eq_flipAux (first : Bool) :
∀ (prev : Bool) (t : List Bool),
flips ((prev :: t) ++ [first])
= flipAux (boolToStrict first) (boolToStrict prev) (t.map boolToStrict)
| prev, [] => by
simp only [List.nil_append, List.cons_append, List.map_nil, flips, flipAux]
by_cases h : prev = first
· simp [h]
· simp only [ne_eq, h, not_false_eq_true, if_pos]
rw [if_pos]
intro hc; exact h (boolToStrict_inj hc)
| prev, x :: t => by
have ih := flips_append_eq_flipAux first x t
simp only [List.cons_append, List.map_cons, flips, flipAux] at *
rw [ih]
by_cases h : prev = x
· simp [h]
· rw [if_pos h, if_pos]
intro hc; exact h (boolToStrict_inj hc)
/-- The Bool-based cyclic flip count equals the `StrictEdgeSign`-based one under `boolToStrict`. -/
theorem cyclicFlips_eq_cyclicFlipCount_map (bs : List Bool) :
cyclicFlips bs = cyclicFlipCount (bs.map boolToStrict) := by
cases bs with
| nil => simp [cyclicFlips, cyclicFlipCount]
| cons h t =>
show flips ((h :: t) ++ [h]) = cyclicFlipCount ((h :: t).map boolToStrict)
rw [flips_append_eq_flipAux h h t]
simp only [List.map_cons, cyclicFlipCount]
theorem filter_map_eq_filterMap {β γ : Type*} (p : β → Bool) (g : β → γ) :
∀ (L : List β),
(L.filter p).map g = L.filterMap (fun a => if p a then some (g a) else none)
| [] => by simp
| a :: L => by
simp only [List.filter_cons, List.filterMap_cons]
by_cases h : p a
· simp [h, filter_map_eq_filterMap p g L]
· simp [h, filter_map_eq_filterMap p g L]
/-- Both machineries skip zeros identically: `(nzSigns d).map boolToStrict` is the strict
sub-sequence of `(List.ofFn d).map realSignToEdgeSign`. -/
theorem nzSigns_map_boolToStrict {m : ℕ} (d : Fin m → ℝ) :
(nzSigns d).map boolToStrict
= ((List.ofFn d).map realSignToEdgeSign).filterMap EdgeSign.toStrict := by
rw [List.ofFn_eq_map, List.map_map, List.filterMap_map]
unfold nzSigns
rw [List.map_map, filter_map_eq_filterMap]
apply List.filterMap_congr
intro i _
simp only [Function.comp_apply]
by_cases hi : d i = 0
· rw [if_neg (by simp [hi]), toStrict_realSign_of_zero hi]
· rw [if_pos (by simp [hi]), toStrict_realSign_of_ne (d i) hi]
/-- **The count-reconciliation identity (local copy).** -/
theorem cyclicFlips_nzSigns_eq_cyclicFlipCountSkipZeros {m : ℕ} (d : Fin m → ℝ) :
cyclicFlips (nzSigns d)
= cyclicFlipCountSkipZeros ((List.ofFn d).map realSignToEdgeSign) := by
rw [cyclicFlips_eq_cyclicFlipCount_map, cyclicFlipCountSkipZeros_eq_strict,
nzSigns_map_boolToStrict]
/-- The cyclic adjacency indicator sum: `1` for each cyclically adjacent unequal pair. -/
def cyclicSum {α : Type*} [DecidableEq α] (l : List α) : ℕ :=
(List.zipWith (fun a b => if a ≠ b then 1 else 0) l (l.rotate 1)).sum
/-- `cyclicFlipCount` equals the cyclic adjacency sum (`flipAux` telescopes into the zipWith sum). -/
theorem cyclicFlipCount_eq_cyclicSum {α : Type*} [DecidableEq α] (l : List α) :
cyclicFlipCount l = cyclicSum l := by
cases l with
| nil => simp [cyclicFlipCount, cyclicSum]
| cons h t =>
simp only [cyclicSum, List.rotate_cons_succ, List.rotate_zero, cyclicFlipCount]
-- General: flipAux h p xs = sum (zipWith f (p::xs) (xs ++ [h]))
have key : ∀ (p : α) (xs : List α),
flipAux h p xs
= (List.zipWith (fun a b => if a ≠ b then 1 else 0) (p :: xs) (xs ++ [h])).sum := by
intro p xs
induction xs generalizing p with
| nil => simp [flipAux]
| cons a xs ih =>
simp only [flipAux, List.cons_append, List.zipWith_cons_cons, List.sum_cons]
rw [ih a]
rw [key h t]
/-- `List.sum` is invariant under rotation (rotation is a permutation). -/
theorem sum_rotate {α : Type*} [AddCommMonoid α] (l : List α) (n : ℕ) :
(l.rotate n).sum = l.sum :=
(List.rotate_perm l n).sum_eq
/-- The cyclic adjacency sum is invariant under one rotation. -/
theorem cyclicSum_rotate_one {α : Type*} [DecidableEq α] (l : List α) :
cyclicSum (l.rotate 1) = cyclicSum l := by
unfold cyclicSum
rw [← List.zipWith_rotate_distrib (fun a b => if a ≠ b then 1 else 0) l (l.rotate 1) 1
(List.length_rotate l 1).symm, sum_rotate]
/-- `cyclicFlipCount` is invariant under one rotation. -/
theorem cyclicFlipCount_rotate_one {α : Type*} [DecidableEq α] (l : List α) :
cyclicFlipCount (l.rotate 1) = cyclicFlipCount l := by
rw [cyclicFlipCount_eq_cyclicSum, cyclicFlipCount_eq_cyclicSum, cyclicSum_rotate_one]
/-- `cyclicFlipCount` is invariant under any rotation. -/
theorem cyclicFlipCount_rotate {α : Type*} [DecidableEq α] (l : List α) (k : ℕ) :
cyclicFlipCount (l.rotate k) = cyclicFlipCount l := by
induction k with
| zero => simp
| succ k ih =>
rw [show l.rotate (k + 1) = (l.rotate k).rotate 1 by rw [List.rotate_rotate]]
rw [cyclicFlipCount_rotate_one, ih]
/-- `cyclicFlipCount` is invariant under `IsRotated`. -/
theorem cyclicFlipCount_of_isRotated {α : Type*} [DecidableEq α] {l l' : List α}
(h : l ~r l') : cyclicFlipCount l = cyclicFlipCount l' := by
obtain ⟨k, rfl⟩ := h
exact (cyclicFlipCount_rotate l k).symm
/-- `filterMap` of a singly-rotated list is a rotation of `filterMap` of the list. -/
theorem filterMap_rotate_one_isRotated {α β : Type*} (f : α → Option β) (l : List α) :
(l.rotate 1).filterMap f ~r l.filterMap f := by
cases l with
| nil => simp
| cons h t =>
rw [List.rotate_cons_succ, List.rotate_zero, List.filterMap_append, List.filterMap_cons]
-- (t.filterMap f) ++ (f h).toList? vs (f h).toList? ++ t.filterMap f — a rotation
rw [List.filterMap_cons]
-- goal: (t.filterMap f ++ Option.toList' (f h)) ~r (match f h with ... )
cases hf : f h with
| none => simp [List.IsRotated.refl]
| some b =>
simp only [List.filterMap_nil]
have := List.isRotated_append (l := t.filterMap f) (l' := [b])
simpa using this
/-- `filterMap` is invariant-up-to-rotation under rotation of its source. -/
theorem filterMap_rotate_isRotated {α β : Type*} (f : α → Option β) (l : List α) (k : ℕ) :
(l.rotate k).filterMap f ~r l.filterMap f := by
induction k with
| zero => simp [List.IsRotated.refl]
| succ k ih =>
rw [show l.rotate (k + 1) = (l.rotate k).rotate 1 by rw [List.rotate_rotate]]
exact (filterMap_rotate_one_isRotated f (l.rotate k)).trans ih
/-- `filterMap` carries `IsRotated` to `IsRotated`. -/
theorem filterMap_isRotated {α β : Type*} {l l' : List α} (f : α → Option β) (h : l ~r l') :
l.filterMap f ~r l'.filterMap f := by
obtain ⟨k, rfl⟩ := h
exact (filterMap_rotate_isRotated f l k).symm
/-- **`cyclicFlipCountSkipZeros` is invariant under `IsRotated`.** Rotating the cyclic sign list
leaves the skip-zeros cyclic flip count unchanged. -/
theorem cyclicFlipCountSkipZeros_of_isRotated {l l' : List EdgeSign} (h : l ~r l') :
cyclicFlipCountSkipZeros l = cyclicFlipCountSkipZeros l' := by
unfold cyclicFlipCountSkipZeros
exact cyclicFlipCount_of_isRotated (filterMap_isRotated EdgeSign.toStrict h)
theorem filterMap_reverse {α β : Type*} (f : α → Option β) :
∀ l : List α, l.reverse.filterMap f = (l.filterMap f).reverse
| [] => by simp
| a :: t => by
simp only [List.reverse_cons, List.filterMap_append, filterMap_reverse f t,
List.filterMap_cons, List.filterMap_nil]
cases f a <;> simp
theorem zipWith_append_eq {α β γ : Type*} (f : α → β → γ) :
∀ {l₁ : List α} {l₂ : List β} (r₁ : List α) (r₂ : List β),
l₁.length = l₂.length →
List.zipWith f (l₁ ++ r₁) (l₂ ++ r₂) =
List.zipWith f l₁ l₂ ++ List.zipWith f r₁ r₂
| [], [], r₁, r₂, _ => by rfl
| [], _ :: _, _, _, h => by simp at h
| _ :: _, [], _, _, h => by simp at h
| a :: as, b :: bs, r₁, r₂, h => by
have ht : as.length = bs.length := Nat.succ.inj h
simp only [List.cons_append, List.zipWith_cons_cons, List.cons.injEq, true_and]
exact zipWith_append_eq f r₁ r₂ ht
theorem zipWith_reverse_eq {α β γ : Type*} (f : α → β → γ) :
∀ {l₁ : List α} {l₂ : List β}, l₁.length = l₂.length →
(List.zipWith f l₁ l₂).reverse = List.zipWith f l₁.reverse l₂.reverse
| [], [], _ => by simp
| [], _ :: _, h => by simp at h
| _ :: _, [], h => by simp at h
| a :: as, b :: bs, h => by
have ht : as.length = bs.length := Nat.succ.inj h
simp only [List.zipWith_cons_cons, List.reverse_cons]
rw [zipWith_reverse_eq f ht]
rw [zipWith_append_eq f [a] [b] (by simpa [List.length_reverse] using ht)]
simp
theorem zipWith_comm_of_comm_eq {α γ : Type*} (f : α → α → γ)
(hf : ∀ a b, f a b = f b a) :
∀ {l₁ l₂ : List α}, l₁.length = l₂.length →
List.zipWith f l₁ l₂ = List.zipWith f l₂ l₁
| [], [], _ => by simp
| [], _ :: _, h => by simp at h
| _ :: _, [], h => by simp at h
| a :: as, b :: bs, h => by
have ht : as.length = bs.length := Nat.succ.inj h
simp [hf a b, zipWith_comm_of_comm_eq f hf ht]
theorem cyclicFlipCount_reverse {α : Type*} [DecidableEq α] (l : List α) :
cyclicFlipCount l.reverse = cyclicFlipCount l := by
rw [cyclicFlipCount_eq_cyclicSum, cyclicFlipCount_eq_cyclicSum]
unfold cyclicSum
let f : α → α → ℕ := fun a b => if a ≠ b then 1 else 0
have hfcomm : ∀ a b, f a b = f b a := by
intro a b
by_cases h : a = b
· simp [f, h]
· have hba : b ≠ a := fun hb => h hb.symm
simp [f, h, hba]
let k := l.length - 1 % l.length
rw [List.rotate_reverse]
change (List.zipWith f l.reverse ((l.rotate k).reverse)).sum =
(List.zipWith f l (l.rotate 1)).sum
rw [← zipWith_reverse_eq f (by rw [List.length_rotate]), List.sum_reverse,
zipWith_comm_of_comm_eq f hfcomm (by rw [List.length_rotate])]
have hlen : (l.rotate k).length = l.length := List.length_rotate l k
have hzip := List.zipWith_rotate_distrib f l (l.rotate 1) k
(by rw [List.length_rotate])
have hrot : (l.rotate 1).rotate k = l := by
by_cases hnil : l = []
· subst hnil
simp [k]
· have hlenpos : 0 < l.length := Nat.pos_of_ne_zero (by
intro hlen0
exact hnil (List.eq_nil_of_length_eq_zero hlen0))
rw [List.rotate_rotate]
unfold k
by_cases hlen1 : l.length = 1
· have hmod : 1 % l.length = 0 := by simp [hlen1]
rw [hmod]
have hsum : 1 + (l.length - 0) = l.length * 2 := by omega
rw [hsum, List.rotate_length_mul]
· have hlt : 1 < l.length := by omega
have hmod : 1 % l.length = 1 := Nat.mod_eq_of_lt hlt
rw [hmod]
have hsum : 1 + (l.length - 1) = l.length := by omega
rw [hsum, List.rotate_length]
calc
(List.zipWith f (l.rotate k) l).sum
= (List.zipWith f (l.rotate k) ((l.rotate 1).rotate k)).sum := by rw [hrot]
_ = ((List.zipWith f l (l.rotate 1)).rotate k).sum := by rw [hzip]
_ = (List.zipWith f l (l.rotate 1)).sum := sum_rotate _ _
theorem cyclicFlipCountSkipZeros_reverse (l : List EdgeSign) :
cyclicFlipCountSkipZeros l.reverse = cyclicFlipCountSkipZeros l := by
unfold cyclicFlipCountSkipZeros
rw [filterMap_reverse]
exact cyclicFlipCount_reverse _
theorem cyclicFlipCountSkipZeros_of_dihedralRotated {l m : List EdgeSign}
(h : List.DihedralRotated l m) :
cyclicFlipCountSkipZeros l = cyclicFlipCountSkipZeros m := by
rcases h with hrot | hrev
· exact cyclicFlipCountSkipZeros_of_isRotated hrot
· calc
cyclicFlipCountSkipZeros l
= cyclicFlipCountSkipZeros l.reverse := (cyclicFlipCountSkipZeros_reverse l).symm
_ = cyclicFlipCountSkipZeros m := cyclicFlipCountSkipZeros_of_isRotated hrev
/-- **σ-orbit invariance of the per-vertex flip count.** Two darts in the same `σ`-orbit have the
same `vertexFlipCountSkipZeros`, because their `σ`-`toList`s are rotations of each other. -/
theorem vertexFlipCountSkipZeros_sameCycle (M : CombMap D) (es : D → EdgeSign) {d d' : D}
(h : M.σ.SameCycle d d') :
vertexFlipCountSkipZeros M es d = vertexFlipCountSkipZeros M es d' := by
unfold vertexFlipCountSkipZeros vertexSignList
exact cyclicFlipCountSkipZeros_of_isRotated
((h.toList_isRotated).map es)
/-- The `Q`-realization link at vertex `Q`, reindexed onto `Fin ((starP Q).n + 1)` via the
degree-match `deg_eq`. (Both links have the same number of edges, `vertexDeg`.) -/
@[reducible] def linkQcast (M : CombMap D) (starP starQ : M.Vertex → VertexStar)
(hnn : ∀ Q, (starQ Q).n = (starP Q).n) (Q : M.Vertex) :
Fin ((starP Q).n + 1) → S2 :=
fun i => (starQ Q).vertexLink (Fin.cast (by rw [hnn Q]) i)
/-- **The faithful convex-polytope realization interface.**
Two congruent-faced convex-vertex realizations `P, Q` of the same triangulated-sphere combinatorial map
`M`, presented as a per-vertex family of vertex stars whose links agree on side lengths (congruent
faces) and closing chord, together with the **order bridge** `linkOrder` (the σ-dart order carries the
geometric link order) and the per-vertex two-arc datum `twoArc` (the single isolated geometric residual,
exactly as in `Ch13ArmVertexFull`).
There is **no `active` field**: rigidity is unconditional. -/
structure ConvexPolytopeRealization (M : CombMap D) where
/-- `M` is a triangulated sphere. -/
isSphere : M.IsSphereMap
triangle : M.FaceRegular 3
/-- The edge graph is simple (no loops / no parallel edges) — a genuine property of every convex
3-polytope's boundary graph (Steinitz), supplied by the ℝ³ realization. It is what rules out the
digon degeneracy in the combinatorial low-active-vertex lemma. -/
isSimple : M.IsSimpleGraph
/-- The `P`-realization vertex star at each vertex. -/
starP : M.Vertex → VertexStar
/-- The `Q`-realization vertex star at each vertex. -/
starQ : M.Vertex → VertexStar
/-- Both realizations have the same incident-edge count at each vertex (degree match). -/
hnn : ∀ (Q : M.Vertex), (starQ Q).n = (starP Q).n
/-- The per-edge dihedral-difference signing: `edgeSign d = sign(dihedral_Q − dihedral_P)` at the
edge of `d`. Edge-invariant (`α`-stable): both darts of an edge carry the same sign. -/
edgeSign : D → EdgeSign
edgeSign_inv : ∀ d, edgeSign (M.α d) = edgeSign d
/-- Congruent faces: corresponding link side lengths agree. -/
sides_eq : ∀ (Q : M.Vertex) (i : Fin (starP Q).n),
sideLen (starP Q).vertexLink i = sideLen (linkQcast M starP starQ hnn Q) i
/-- Shared closing chord at each vertex. -/
close_eq : ∀ (Q : M.Vertex),
sDist ((starP Q).vertexLink 0) ((starP Q).vertexLink (Fin.last (starP Q).n))
= sDist ((linkQcast M starP starQ hnn Q) 0)
((linkQcast M starP starQ hnn Q) (Fin.last (starP Q).n))
/-- A representative dart at each vertex (`tail = Q`). -/
dartRep : M.Vertex → D
dartRep_tail : ∀ (Q : M.Vertex), M.tail (dartRep Q) = Q
/-- **Interior activeness bridge.** When some incident edge at `Q` carries a nonzero dihedral-change
sign, some *interior* joint of the link genuinely differs. This is the geometric input feeding the
arm lemma's strict witness (closing angles are determined by the equal sides/chord; only the interior
joints are the free Cauchy variables). It is an interface field exactly like `twoArc`; it does **not**
make rigidity conditional (the conclusion of `realization_rigid` is unconditional). -/
interiorActive : ∀ (Q : M.Vertex),
ActiveVertex M edgeSign (dartRep Q) →
∃ i : Fin ((starP Q).n - 1),
jointAngle (starP Q).vertexLink i ≠ jointAngle (linkQcast M starP starQ hnn Q) i
/-- The per-vertex two-arc split datum for the `signChangesFull = 2` case (the single isolated
geometric residual, exactly as `Ch13ArmVertexFull.cauchyArmVertexFull_of_links` takes). -/
twoArc : ∀ (Q : M.Vertex),
signChangesFull (starP Q).vertexLink (linkQcast M starP starQ hnn Q) = 2 →
TwoArcSplitData (starP Q).vertexLink (linkQcast M starP starQ hnn Q)
/-- **The order bridge (`linkOrder`).** The `σ`-ordered list of edge signs around vertex `Q` (read
from `dartRep Q`) agrees with the link-ordered real-sign list of the dihedral differences up to
cyclic rotation and reversal. This is the honest unoriented cyclic-order bridge; positing
`vertexArm_signChanges_eq` directly instead is the §3.3 trap. -/
linkOrder : ∀ (Q : M.Vertex),
List.DihedralRotated
((M.σ.toList (dartRep Q)).map edgeSign)
((List.ofFn
(linkDiff (starP Q).vertexLink (linkQcast M starP starQ hnn Q))).map realSignToEdgeSign)
namespace ConvexPolytopeRealization
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
-- The regular tetrahedron satisfies the reverse-`σ` rotation-faithfulness convention.
-- The regular tetrahedron satisfies the face-local outward-orientation convention.
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
namespace VertexLinkGeometry
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
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
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
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
end VertexStar
end ProofsInTheBook.Ch13VertexStar
namespace ProofsInTheBook.Ch13Cauchy3D
namespace ConvexEuclideanPolyhedron
end ConvexEuclideanPolyhedron
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.