Monitored spherical openings and boundary configurations
DefinitionP2MAssembly_Chapter13V2_Part3cauchy-rigiditydihedral-anglesgeometrylean4polyhedraproofs-from-the-book
This part defines admissible monitored opening families and their supremum parameters, together with target-reaching, support-stopping and base-stopping conditions. It includes reversed and reflected spherical arms, interval-wrap data, tail-fold boundary conditions and strict diagonal-support predicates. The retained proofs organize progress and boundary cases in the arm comparison argument. Conditions described by these intermediate predicates remain conditions; their names do not assert that all such configurations are realizable.
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
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
/-- **The hemisphere-free monitored family `WBS`** (design §7). Three member classes, all opening by
`-θ` (the widening direction):
* `inl (inl c)` — the non-incident support constraint `θ ↦ supportConstraint A K c (-θ)`;
* `inl (inr ())` — the joint slack `θ ↦ jointAngle B k − openedInteriorJointAngle A k (-θ)`;
* `inr ()` — the base cap support `baseCapSupportW A k` (already `-θ`).
Indexed by `(NonIncident n ⊕ Unit) ⊕ Unit`. **No hemisphere member.** -/
def monitoredFamilyWBS {n : ℕ} (A B : Fin (n + 1) → S2) (k : Fin (n - 1)) :
(NonIncident n ⊕ Unit) ⊕ Unit → ℝ → ℝ
| Sum.inl (Sum.inl c) => fun θ => supportConstraint A (openingAxis k) c (-θ)
| Sum.inl (Sum.inr ()) => fun θ => jointAngle B k - openedInteriorJointAngle A k (-θ)
| Sum.inr () => baseCapSupportW A k
/-- Each `WBS`-family member is continuous in `θ`. Supports and slack are the `W`-family members
precomposed with the continuous negation (`continuous_supportConstraint`, `continuous_openedInteriorJointAngle`),
the base member is `continuous_baseCapSupportW`. -/
theorem continuous_monitoredFamilyWBS {n : ℕ} {A B : Fin (n + 1) → S2} {k : Fin (n - 1)}
(hka : ShortArc (A (openingAxis k)) (jointPrev A k))
(hkt : ShortArc (A (openingAxis k)) (jointNext A k)) :
∀ o, Continuous (monitoredFamilyWBS A B k o) := by
rintro ((c | ⟨⟩) | ⟨⟩)
· exact (continuous_supportConstraint A (openingAxis k) c).comp continuous_neg
· exact continuous_const.sub ((continuous_openedInteriorJointAngle hka hkt).comp continuous_neg)
· exact continuous_baseCapSupportW A k
/-- The `WBS` admissible supremum `δ*_WBS := sSup {θ ∈ [0,π] : ∀ o, 0 ≤ monitoredFamilyWBS … o θ}`. -/
def monitoredSupWBS {n : ℕ} (A B : Fin (n + 1) → S2) (k : Fin (n - 1)) : ℝ :=
sSup (admissibleSet (monitoredFamilyWBS A B k) Real.pi)
/-- **Init admissibility at `θ = 0`.** Supports `≥ 0` (`edge_support`), the joint slack `≥ 0` (from the
deficit `jointAngle A k < jointAngle B k`), and the base support `≥ 0` (the banked convex base orientation
`orientedDatum_interior`). At `θ = 0` the support/slack members are the unopened arm's data (`-0 = 0`,
`openedInteriorJointAngle A k 0 = jointAngle A k`), the base member is the unopened base support. -/
theorem monitoredFamilyWBS_zero_nonneg {n : ℕ} {A : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
{B : Fin (n + 1) → S2} {k : Fin (n - 1)}
(hkdef : jointAngle A k < jointAngle B k) :
∀ o, 0 ≤ monitoredFamilyWBS A B k o 0 := by
rintro ((c | ⟨⟩) | ⟨⟩)
· -- support member: `supportConstraint A K c (-0) = supportConstraint A K c 0 ≥ 0`.
show 0 ≤ supportConstraint A (openingAxis k) c (-(0 : ℝ))
rw [neg_zero, supportConstraint_apply, openTail_zero_angle]
exact hA.closed_convex.edge_support c.1.1 c.1.2
· -- slack member: `jointAngle B k − openedInteriorJointAngle A k (-0) = jointAngle B k − jointAngle A k ≥ 0`.
show 0 ≤ jointAngle B k - openedInteriorJointAngle A k (-(0 : ℝ))
rw [neg_zero, openedInteriorJointAngle_zero]
linarith
· -- base member at `0`: `sOrient (A 0)(A K)(rotS2 (A K) 0 (A last)) = sOrient (A 0)(A K)(A last) ≥ 0`.
show 0 ≤ baseCapSupportW A k 0
obtain ⟨hK0, hKn⟩ := openingAxis_interior k
have hrot0 : rotS2 (A (openingAxis k)) (-(0 : ℝ)) (A (Fin.last n)) = A (Fin.last n) := by
apply S2.ext; rw [rotS2_coe, neg_zero, rot_zero]
show 0 ≤ sOrient (A 0) (A (openingAxis k)) (rotS2 (A (openingAxis k)) (-(0 : ℝ)) (A (Fin.last n)))
rw [hrot0]
exact orientedDatum_interior hA hK0 hKn
/-- **The `WBS` admissible supremum is `WBS`-admissible** (closed nonempty bounded family). -/
theorem monitoredSupWBS_mem {n : ℕ} {A B : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
{k : Fin (n - 1)}
(hka : ShortArc (A (openingAxis k)) (jointPrev A k))
(hkt : ShortArc (A (openingAxis k)) (jointNext A k))
(hkdef : jointAngle A k < jointAngle B k) :
monitoredSupWBS A B k ∈ admissibleSet (monitoredFamilyWBS A B k) Real.pi :=
sSup_mem_admissibleSet (continuous_monitoredFamilyWBS hka hkt) Real.pi_nonneg
(monitoredFamilyWBS_zero_nonneg hA hkdef)
/-- `δ*_WBS ∈ [0, π]`. -/
theorem monitoredSupWBS_mem_Icc {n : ℕ} {A B : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
{k : Fin (n - 1)}
(hka : ShortArc (A (openingAxis k)) (jointPrev A k))
(hkt : ShortArc (A (openingAxis k)) (jointNext A k))
(hkdef : jointAngle A k < jointAngle B k) :
monitoredSupWBS A B k ∈ Set.Icc (0 : ℝ) Real.pi :=
(monitoredSupWBS_mem hA hka hkt hkdef).1
/-- **Closure (supports `≥ 0` at `δ*_WBS`)**, support form on the opened triple of `A'_WBS`. -/
theorem supportWBS_sOrient_nonneg {n : ℕ} {A B : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
{k : Fin (n - 1)}
(hka : ShortArc (A (openingAxis k)) (jointPrev A k))
(hkt : ShortArc (A (openingAxis k)) (jointNext A k))
(hkdef : jointAngle A k < jointAngle B k) :
∀ i j : Fin (n + 1), j ≠ i → j ≠ i + 1 →
0 ≤ sOrient (openTail A (openingAxis k) (-(monitoredSupWBS A B k)) i)
(openTail A (openingAxis k) (-(monitoredSupWBS A B k)) (i + 1))
(openTail A (openingAxis k) (-(monitoredSupWBS A B k)) j) := by
intro i j hji hji1
have hmem := monitoredSupWBS_mem hA hka hkt hkdef
have := hmem.2 (Sum.inl (Sum.inl (⟨(i, j), ⟨hji, hji1⟩⟩ : NonIncident n)))
simp only [monitoredFamilyWBS, supportConstraint_apply] at this
exact this
/-- **Closure (joint slack `≥ 0` at `δ*_WBS`)**: the opened interior joint is `≤ jointAngle B k`. -/
theorem openedInteriorJoint_le_at_supWBS {n : ℕ} {A B : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
{k : Fin (n - 1)}
(hka : ShortArc (A (openingAxis k)) (jointPrev A k))
(hkt : ShortArc (A (openingAxis k)) (jointNext A k))
(hkdef : jointAngle A k < jointAngle B k) :
openedInteriorJointAngle A k (-(monitoredSupWBS A B k)) ≤ jointAngle B k := by
have hmem := monitoredSupWBS_mem hA hka hkt hkdef
have := hmem.2 (Sum.inl (Sum.inr ()))
simp only [monitoredFamilyWBS] at this
linarith
/-- **Every `WBS`-admissible `θ` lies in `[0, jointAngle B k − jointAngle A k]`.** Port of
`ZinanFFCT37.admissibleW_le_deficit`, re-instantiated on the `WBS` joint-witness support member and the
`WBS` slack member (both present in the hemisphere-free family). -/
theorem admissibleWBS_le_deficit {n : ℕ} {A B : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
{k : Fin (n - 1)}
(hka : ShortArc (A (openingAxis k)) (jointPrev A k))
(hkt : ShortArc (A (openingAxis k)) (jointNext A k))
{θ : ℝ} (hθadm : θ ∈ admissibleSet (monitoredFamilyWBS A B k) Real.pi) :
θ ≤ jointAngle B k - jointAngle A k := by
obtain ⟨⟨hθ0, hθπ⟩, hmem⟩ := hθadm
-- joint-witness support ≥ 0 (it is a `WBS` support member).
have hsupp : 0 ≤ supportConstraint A (openingAxis k) (jointWitness k) (-θ) := by
have := hmem (Sum.inl (Sum.inl (jointWitness k)))
simpa only [monitoredFamilyWBS] using this
rw [supportConstraint_jointWitness_neg, support_openNeg_eq_sin hA hka hkt] at hsupp
-- norms positive ⟹ `sin (γ + θ) ≥ 0`.
have hunz : tangentTo (A (openingAxis k)) (jointPrev A k) ≠ 0 := (tangentTo_ne_zero_iff _ _).2 hka
have hwnz : tangentTo (A (openingAxis k)) (jointNext A k) ≠ 0 := (tangentTo_ne_zero_iff _ _).2 hkt
have hup : (0 : ℝ) < ‖tangentTo (A (openingAxis k)) (jointPrev A k)‖ := norm_pos_iff.2 hunz
have hwp : (0 : ℝ) < ‖tangentTo (A (openingAxis k)) (jointNext A k)‖ := norm_pos_iff.2 hwnz
have hsin : 0 ≤ Real.sin (jointAngle A k + θ) := by
by_contra hneg
push_neg at hneg
have : ‖tangentTo (A (openingAxis k)) (jointPrev A k)‖
* ‖tangentTo (A (openingAxis k)) (jointNext A k)‖ * Real.sin (jointAngle A k + θ) < 0 :=
mul_neg_of_pos_of_neg (mul_pos hup hwp) hneg
linarith
-- `γ + θ ∈ [0, 2π)` and `sin ≥ 0` ⟹ `γ + θ ≤ π` (the additive branch).
have hγ0 : 0 ≤ jointAngle A k := by
rw [show jointAngle A k = sphAngle (jointPrev A k) (A (openingAxis k)) (jointNext A k) by
simp only [jointAngle, jointPrev, jointNext, openingAxis]]
exact sphAngle_nonneg _ _ _
have hγlt : jointAngle A k < Real.pi := strict_jointAngle_lt_pi hA k
have hbranch : jointAngle A k + θ ≤ Real.pi := by
by_contra hgt
push_neg at hgt
set t : ℝ := jointAngle A k + θ - Real.pi with ht
have ht0 : 0 < t := by rw [ht]; linarith
have htlt : t < Real.pi := by rw [ht]; linarith [hγlt, hθπ]
have hsin_t : Real.sin (jointAngle A k + θ) = - Real.sin t := by
rw [show jointAngle A k + θ = Real.pi + t by rw [ht]; ring, Real.sin_add, Real.sin_pi,
Real.cos_pi]
ring
have hpos : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi ht0 htlt
rw [hsin_t] at hsin; linarith
-- on the branch, the opened joint equals `γ + θ`; slack ≥ 0 pins `γ + θ ≤ jointAngle B k`.
have hopened : openedInteriorJointAngle A k (-θ) = jointAngle A k + θ :=
openedNegJointAngle_eq_add hA hka hkt hθ0 hbranch
have hslack : 0 ≤ jointAngle B k - openedInteriorJointAngle A k (-θ) := by
have := hmem (Sum.inl (Sum.inr ()))
simpa only [monitoredFamilyWBS] using this
rw [hopened] at hslack
linarith
/-- **`δ*_WBS ≤ jointAngle B k − jointAngle A k`.** -/
theorem monitoredSupWBS_le_deficit {n : ℕ} {A B : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
{k : Fin (n - 1)}
(hka : ShortArc (A (openingAxis k)) (jointPrev A k))
(hkt : ShortArc (A (openingAxis k)) (jointNext A k))
(hkdef : jointAngle A k < jointAngle B k) :
monitoredSupWBS A B k ≤ jointAngle B k - jointAngle A k :=
admissibleWBS_le_deficit hA hka hkt (monitoredSupWBS_mem hA hka hkt hkdef)
/-- **`δ*_WBS < π`.** The deficit `jointAngle B k − jointAngle A k < π` (strict `B`, `γ ≥ 0`). -/
theorem monitoredSupWBS_lt_pi {n : ℕ} {A B : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
(hB : StrictConvexSphArm B) {k : Fin (n - 1)}
(hka : ShortArc (A (openingAxis k)) (jointPrev A k))
(hkt : ShortArc (A (openingAxis k)) (jointNext A k))
(hkdef : jointAngle A k < jointAngle B k) :
monitoredSupWBS A B k < Real.pi := by
have hub := monitoredSupWBS_le_deficit hA hka hkt hkdef
have hγ0 : 0 ≤ jointAngle A k := by
rw [show jointAngle A k = sphAngle (jointPrev A k) (A (openingAxis k)) (jointNext A k) by
simp only [jointAngle, jointPrev, jointNext, openingAxis]]
exact sphAngle_nonneg _ _ _
have hBlt : jointAngle B k < Real.pi := strict_jointAngle_lt_pi hB k
linarith
/-- **The base cap at the `WBS` supremum** (`GlueWBaseCap` for `δ*_WBS`): `δ*_WBS + γbase ≤ π`. The `WBS`
supremum is `WBS`-admissible, so the base member is `≥ 0`, hence (norms positive, `γbase < π`, `δ*_WBS ≤ π`)
`admissibleWB_baseCap`'s sine-branch argument closes the cap. Mirror of FFCT41's `GlueWBaseCap_at_supWB`. -/
theorem GlueWBaseCap_at_supWBS {n : ℕ} {A B : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
{k : Fin (n - 1)}
(hka : ShortArc (A (openingAxis k)) (jointPrev A k))
(hkt : ShortArc (A (openingAxis k)) (jointNext A k))
(hkdef : jointAngle A k < jointAngle B k) :
monitoredSupWBS A B k
+ sphAngle (A 0) (A (openingAxis k)) (A (Fin.last n)) ≤ Real.pi := by
obtain ⟨hbase0, hbaseLast⟩ :=
shortArc_interior_base hA (openingAxis_interior k).1 (openingAxis_interior k).2
obtain ⟨hIcc, hmemfun⟩ := monitoredSupWBS_mem hA hka hkt hkdef
-- the sine-branch argument: base member `≥ 0` ⟹ `sin (γbase + δ*) ≥ 0` ⟹ cap.
set γ : ℝ := sphAngle (A 0) (A (openingAxis k)) (A (Fin.last n)) with hγ
set θ : ℝ := monitoredSupWBS A B k with hθ
obtain ⟨hθ0, hθπ⟩ := hIcc
have hbase : 0 ≤ baseCapSupportW A k θ := by simpa only [monitoredFamilyWBS] using hmemfun (Sum.inr ())
rw [baseSupport_openNeg_eq_sin hA k hbase0 hbaseLast, ← hγ] at hbase
have hunz : tangentTo (A (openingAxis k)) (A 0) ≠ 0 := (tangentTo_ne_zero_iff _ _).2 hbase0
have hwnz : tangentTo (A (openingAxis k)) (A (Fin.last n)) ≠ 0 := (tangentTo_ne_zero_iff _ _).2 hbaseLast
have hup : (0 : ℝ) < ‖tangentTo (A (openingAxis k)) (A 0)‖ := norm_pos_iff.2 hunz
have hwp : (0 : ℝ) < ‖tangentTo (A (openingAxis k)) (A (Fin.last n))‖ := norm_pos_iff.2 hwnz
have hsin : 0 ≤ Real.sin (γ + θ) := by
by_contra hneg
push_neg at hneg
have : ‖tangentTo (A (openingAxis k)) (A 0)‖
* ‖tangentTo (A (openingAxis k)) (A (Fin.last n))‖ * Real.sin (γ + θ) < 0 :=
mul_neg_of_pos_of_neg (mul_pos hup hwp) hneg
linarith
have hγ0 : 0 ≤ γ := by rw [hγ]; exact sphAngle_nonneg _ _ _
have hγlt : γ < Real.pi := base_sphAngle_lt_pi hA k
by_contra hgt
push_neg at hgt
set t : ℝ := γ + θ - Real.pi with ht
have ht0 : 0 < t := by rw [ht]; linarith
have htlt : t < Real.pi := by rw [ht]; linarith [hγlt, hθπ]
have hsin_t : Real.sin (γ + θ) = - Real.sin t := by
rw [show γ + θ = Real.pi + t by rw [ht]; ring, Real.sin_add, Real.sin_pi, Real.cos_pi]
ring
have hpos : 0 < Real.sin t := Real.sin_pos_of_pos_of_lt_pi ht0 htlt
rw [hsin_t] at hsin; linarith
/-- The REACH predicate at the `WBS` supremum: the opened-by-`-δ*_WBS` interior joint reaches `B`'s value. -/
def ReachWBS {n : ℕ} (A B : Fin (n + 1) → S2) (k : Fin (n - 1)) : Prop :=
openedInteriorJointAngle A k (-(monitoredSupWBS A B k)) = jointAngle B k
/-- The SUPPORT-STUCK predicate at the `WBS` supremum: a non-incident support of the opened-by-`-δ*_WBS`
arm vanishes. (No hemisphere disjunct — the hemisphere members were dropped.) -/
def SupportStuckWBS {n : ℕ} (A B : Fin (n + 1) → S2) (k : Fin (n - 1)) : Prop :=
∃ c : NonIncident n,
supportConstraint A (openingAxis k) c (-(monitoredSupWBS A B k)) = 0
/-- The BASE-STUCK predicate at the `WBS` supremum: the base monitor vanishes — `sin (γbase + δ*_WBS) = 0`. -/
def BaseStuckWBS {n : ℕ} (A B : Fin (n + 1) → S2) (k : Fin (n - 1)) : Prop :=
baseCapSupportW A k (monitoredSupWBS A B k) = 0
/-- **The `WBS` boundary trichotomy** (here a tetrachotomy *without* a hemi branch). At `δ*_WBS`: either
`δ*_WBS = π`, or `ReachWBS`, or `SupportStuckWBS`, or `BaseStuckWBS`. The generic `reach_or_stuck` engine;
the CAP branch is killed downstream by `monitoredSupWBS_lt_pi`. **No hemisphere-stuck branch exists.** -/
theorem opening_boundary_trichotomyWBS {n : ℕ} {A B : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
{k : Fin (n - 1)}
(hka : ShortArc (A (openingAxis k)) (jointPrev A k))
(hkt : ShortArc (A (openingAxis k)) (jointNext A k))
(hkdef : jointAngle A k < jointAngle B k) :
monitoredSupWBS A B k = Real.pi ∨
ReachWBS A B k ∨ SupportStuckWBS A B k ∨ BaseStuckWBS A B k := by
rcases reach_or_stuck (continuous_monitoredFamilyWBS hka hkt) Real.pi_nonneg
(monitoredFamilyWBS_zero_nonneg hA hkdef) with hcap | ⟨o, ho⟩
· exact Or.inl hcap
· rcases o with (c | ⟨⟩) | ⟨⟩
· -- a support constraint vanishes: SUPPORT-STUCK.
refine Or.inr (Or.inr (Or.inl ⟨c, ?_⟩))
simpa only [monitoredFamilyWBS, monitoredSupWBS] using ho
· -- the joint slack vanishes: REACH.
refine Or.inr (Or.inl ?_)
have hslack : monitoredFamilyWBS A B k (Sum.inl (Sum.inr ())) (monitoredSupWBS A B k) = 0 := ho
simp only [monitoredFamilyWBS] at hslack
unfold ReachWBS
linarith
· -- the base monitor vanishes: BASE-STUCK.
refine Or.inr (Or.inr (Or.inr ?_))
have : monitoredFamilyWBS A B k (Sum.inr ()) (monitoredSupWBS A B k) = 0 := ho
simpa only [monitoredFamilyWBS, monitoredSupWBS] using this
/-- **The `WBS` honest clause (ii): `¬ SupportStuckWBS → ReachWBS ∨ BaseStuckWBS`.** The CAP branch is
impossible (`monitoredSupWBS_lt_pi`), SUPPORT-STUCK is excluded, leaving REACH or BASE-STUCK. (There is no
hemi branch to dispose of — the design's payoff for dropping the hemisphere monitors.) -/
theorem glueWBS_clause_ii {n : ℕ} {A B : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
(hB : StrictConvexSphArm B) {k : Fin (n - 1)}
(hka : ShortArc (A (openingAxis k)) (jointPrev A k))
(hkt : ShortArc (A (openingAxis k)) (jointNext A k))
(hkdef : jointAngle A k < jointAngle B k)
(hnotStuck : ¬ SupportStuckWBS A B k) :
ReachWBS A B k ∨ BaseStuckWBS A B k := by
rcases opening_boundary_trichotomyWBS hA hka hkt hkdef with hcap | hreach | hstuck | hbase
· exact absurd hcap (ne_of_lt (monitoredSupWBS_lt_pi hA hB hka hkt hkdef))
· exact Or.inl hreach
· exact absurd hstuck hnotStuck
· exact Or.inr hbase
/-- **Clause (i) at `δ*_WBS`** — UNCONDITIONAL (the cap is discharged by `GlueWBaseCap_at_supWBS`). Opening
by `-δ*_WBS` does not decrease the endpoint, since `δ*_WBS + γbase ≤ π` holds by admissibility. Mirror of
FFCT41's `glueWB_clause_i`. -/
theorem glueWBS_clause_i {n : ℕ} {A B : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
{k : Fin (n - 1)}
(hka : ShortArc (A (openingAxis k)) (jointPrev A k))
(hkt : ShortArc (A (openingAxis k)) (jointNext A k))
(hkdef : jointAngle A k < jointAngle B k) :
endpt A ≤ endpt (openTail A (openingAxis k) (-(monitoredSupWBS A B k))) := by
have hδ0 : 0 ≤ monitoredSupWBS A B k := (monitoredSupWBS_mem_Icc hA hka hkt hkdef).1
exact endpt_openTail_interior_mono_neg hA k hδ0 (GlueWBaseCap_at_supWBS hA hka hkt hkdef)
/-- **Base-stuck forces a vanishing non-incident support of `A'_WBS`** — re-instantiation of FFCT42's
`baseStuck_forces_vanishingSupport` at the `WBS` sup. The base diagonal zero **is** the wraparound
non-incident support zero at `(last, K)` (the `det3` cyclic identity, family-independent). -/
theorem baseStuckWBS_forces_vanishingSupport {n : ℕ} {A B : Fin (n + 1) → S2} (k : Fin (n - 1))
(hbase : BaseStuckWBS A B k) :
∃ i j : Fin (n + 1), j ≠ i ∧ j ≠ i + 1 ∧
sOrient (openTail A (openingAxis k) (-(monitoredSupWBS A B k)) i)
(openTail A (openingAxis k) (-(monitoredSupWBS A B k)) (i + 1))
(openTail A (openingAxis k) (-(monitoredSupWBS A B k)) j) = 0 := by
-- `BaseStuckWBS` unfolds to `baseCapSupportW A k δ*_WBS = 0`; FFCT42's `baseStuck_eq_openedDiagonal`
-- (free in `δ`) turns this into the opened diagonal `(0,K,last)` zero, and the cyclic identity
-- `baseDiagonal_zero_is_wrapEdgeSupport_zero` (free in `δ`) yields the wrap-edge `(last,K)` payload.
have hdiag : sOrient (openTail A (openingAxis k) (-(monitoredSupWBS A B k)) 0)
(openTail A (openingAxis k) (-(monitoredSupWBS A B k)) (openingAxis k))
(openTail A (openingAxis k) (-(monitoredSupWBS A B k)) (Fin.last n)) = 0 := by
rw [← baseStuck_eq_openedDiagonal]; exact hbase
exact baseDiagonal_zero_is_wrapEdgeSupport_zero A k (monitoredSupWBS A B k) hdiag
/-- **The base-stuck progress residual at `δ*_WBS`** (design §11). Stated in the same shape as FFCT41's
`BaseStuckProgressW`, but at the `WBS` sup and **without** the false-shaped hemi monitoring. Unlike FFCT41,
this is a *theorem*, not a named residual: the FFCT42 cyclic shortcut discharges it by taking the
vanishing-support disjunct (the base diagonal zero **is** the wrap-edge non-incident support zero). -/
def BaseStuckProgressWBS : Prop :=
∀ n : ℕ, ∀ A B : Fin (n + 1) → S2, StrictConvexSphArm A → StrictConvexSphArm B →
∀ k : Fin (n - 1), jointAngle A k < jointAngle B k →
BaseStuckWBS A B k →
ReachWBS A B k ∨
∃ i j : Fin (n + 1), j ≠ i ∧ j ≠ i + 1 ∧
sOrient (openTail A (openingAxis k) (-(monitoredSupWBS A B k)) i)
(openTail A (openingAxis k) (-(monitoredSupWBS A B k)) (i + 1))
(openTail A (openingAxis k) (-(monitoredSupWBS A B k)) j) = 0
/-- **`BaseStuckProgressWBS` holds** — Brick 7, the FFCT42 port DISCHARGED. A base-stuck `WBS` supremum
always makes recursion-ready progress by taking the **vanishing-support** disjunct via the cyclic identity
(`baseStuckWBS_forces_vanishingSupport`). `ReachWBS` is never needed. Unconditional — no
`OpenedClosingEdge*`, no `SupportStuckMargins`, no straightening completion, no sub-arm IH. This is the
honest analogue of FFCT42's `BaseStuckProgressW_holds`, but for the hemisphere-free family. -/
theorem BaseStuckProgressWBS_holds : BaseStuckProgressWBS := by
intro n A B _hA _hB k _hkdef hbase
exact Or.inr (baseStuckWBS_forces_vanishingSupport k hbase)
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
/-- **The endpoint of a strict convex arm is positive.** `endpt A = sDist (A 0)(A last)`; the wraparound
edge `(last, last + 1)` of the closed polygon is short (`StrictConvexSphArm`'s `edge_short`), and
`last + 1 = 0` (`ZinanFFCT42.lastAddOne_eq_zero`), so `A last ≠ A 0`, whence `0 < sDist (A 0)(A last)`. -/
theorem strict_arm_endpt_pos {n : ℕ} {A : Fin (n + 1) → S2} (hA : StrictConvexSphArm A) :
0 < endpt A := by
-- the wraparound edge `(last, last + 1 = 0)` is short ⟹ `A last ≠ A 0`.
have hwrap : A (Fin.last n) ≠ A (Fin.last n + 1) := base_consecutive_ne hA (Fin.last n)
rw [lastAddOne_eq_zero] at hwrap
have hne : A 0 ≠ A (Fin.last n) := fun h => hwrap h.symm
-- `endpt A = sDist (A 0)(A last) > 0`.
exact sDist_pos_of_ne hne
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
/-- **Margins-free open hemisphere from weak supports + open joints + short edges.** For a closed chain
`P : Fin (n+1) → S2` (`2 ≤ n`) with cyclic short-arc edges (`hside`), weak (`≥ 0`) non-incident edge
supports (`hsupp`), and all interior joints in `(0, π)` (`hjopen`), there is a *unit* normal `h'`
strictly positive against every vertex: `∃ h', ‖h'‖ = 1 ∧ ∀ r, 0 < ⟪h', P r⟫`. **No hemisphere-margin
hypothesis is consumed** — the genuine replacement for FFCT44's tilt route, which needed a weak-margin
base normal the WBS family does not supply. -/
theorem openHemisphere_of_weakSupports_jointOpen_full {n : ℕ} {P : Fin (n + 1) → S2}
(hn : 2 ≤ n)
(hside : ∀ i : Fin (n + 1), ShortArc (P i) (P (i + 1)))
(hsupp : ∀ i j : Fin (n + 1), j ≠ i → j ≠ i + 1 →
0 ≤ sOrient (P i) (P (i + 1)) (P j))
(hjopen : ∀ r : Fin (n - 1), 0 < jointAngle P r ∧ jointAngle P r < Real.pi) :
∃ h' : E3, ‖h'‖ = 1 ∧ ∀ r : Fin (n + 1), 0 < (⟪h', (P r : E3)⟫ : ℝ) := by
classical
-- The full vertex image set.
set s : Finset E3 := (Finset.univ : Finset (Fin (n + 1))).image (fun r => ((P r : S2) : E3))
with hs
have hmem_iff : ∀ v, v ∈ s ↔ ∃ m, ((P m : S2) : E3) = v := by
intro v
rw [hs, Finset.mem_image]
constructor
· rintro ⟨m, _hm, rfl⟩; exact ⟨m, rfl⟩
· rintro ⟨m, rfl⟩; exact ⟨m, Finset.mem_univ _, rfl⟩
by_cases h0 : (0 : E3) ∈ convexHull ℝ (s : Set E3)
· -- `0 ∈ hull`: derive a common edge-plane axis and contradict the FFCT44 collapse kernel.
exfalso
rw [Finset.mem_convexHull'] at h0
obtain ⟨w, hw0, hw1, hwsum⟩ := h0
-- For every edge `i` and every member `v`, `w v * det3 (P i)(P i+1) v ≥ 0` (weak supports).
have hterm_nonneg : ∀ (i : Fin (n + 1)) (v : E3), v ∈ s →
0 ≤ w v * det3 (P i : E3) (P (i + 1) : E3) v := by
intro i v hv
obtain ⟨m, rfl⟩ := (hmem_iff v).mp hv
by_cases hmi : m = i
· subst hmi; rw [show det3 (P m : E3) (P (m + 1) : E3) (P m : E3) = 0 from by
simp only [det3]; ring, mul_zero]
· by_cases hmi1 : m = i + 1
· subst hmi1; rw [show det3 (P i : E3) (P (i + 1) : E3) (P (i + 1) : E3) = 0 from by
simp only [det3]; ring, mul_zero]
· exact mul_nonneg (hw0 _ hv)
(le_trans (le_of_eq rfl) (hsupp i m (by simpa [eq_comm] using hmi)
(by simpa [eq_comm] using hmi1)))
-- The edge functional pushed through the convex combination is `0`.
have hedge_zero : ∀ i : Fin (n + 1),
(0 : ℝ) = ∑ v ∈ s, w v * det3 (P i : E3) (P (i + 1) : E3) v := by
intro i
have hkey := det3_edge_centerSum (P i : E3) (P (i + 1) : E3) s w
rw [hwsum] at hkey
rw [show det3 (P i : E3) (P (i + 1) : E3) (0 : E3) = 0 from by simp [det3]] at hkey
exact hkey
have hterm_zero : ∀ i : Fin (n + 1), ∀ v ∈ s,
w v * det3 (P i : E3) (P (i + 1) : E3) v = 0 := by
intro i
exact (Finset.sum_eq_zero_iff_of_nonneg (fun v hv => hterm_nonneg i v hv)).mp
(hedge_zero i).symm
-- Pick a positive-weight member `v₀`.
have hexists_pos : ∃ v ∈ s, 0 < w v := by
by_contra hnone
push_neg at hnone
have : ∑ v ∈ s, w v = 0 :=
Finset.sum_eq_zero (fun v hv => le_antisymm (hnone v hv) (hw0 v hv))
rw [hw1] at this; exact one_ne_zero this
obtain ⟨v0, hv0s, hv0pos⟩ := hexists_pos
obtain ⟨r0, hv0eq⟩ := (hmem_iff v0).mp hv0s
-- `z := P r0` is a common edge-plane axis.
have hallplanes : ∀ i : Fin (n + 1), det3 (P i : E3) (P (i + 1) : E3) (P r0 : E3) = 0 := by
intro i
have := hterm_zero i v0 hv0s
rcases mul_eq_zero.mp this with hw | hd
· exact absurd hw (ne_of_gt hv0pos)
· rw [hv0eq]; exact hd
exact commonLine_collapse_forces_flat_joint hn hside hallplanes hjopen
· -- `0 ∉ hull`: separation gives the strict hemisphere normal directly; normalize to unit length.
obtain ⟨t, ht⟩ := exists_inner_pos_of_zero_notMem_convexHull s.finite_toSet h0
have htpos : ∀ r : Fin (n + 1), 0 < (⟪t, (P r : E3)⟫ : ℝ) := by
intro r
have hrmem : ((P r : S2) : E3) ∈ s := by rw [hmem_iff]; exact ⟨r, rfl⟩
exact ht _ hrmem
-- `t ≠ 0` (it has a strictly positive inner product against `P 0`); normalize.
have htne : t ≠ 0 := by
intro h
have := htpos ⟨0, by omega⟩
rw [h, inner_zero_left] at this
exact lt_irrefl _ this
have htnorm : 0 < ‖t‖ := norm_pos_iff.mpr htne
refine ⟨(‖t‖⁻¹ : ℝ) • t, ?_, fun r => ?_⟩
· rw [norm_smul, norm_inv, norm_norm]; field_simp
· rw [inner_smul_left]
simp only [RCLike.conj_to_real]
exact mul_pos (by positivity) (htpos r)
/-- **Brick 4 — the open hemisphere at the WBS supremum.** At `A'_WBS := openTail A K (-δ*_WBS)`, from
the WBS closure supports (`≥ 0`), the opened ShortArc edges (`hedge`), and joints-in-`(0, π)` (`hjopen`),
there is a *unit* normal `h'` strictly positive against every vertex. **Margins-free** (no fixed-`h₀`
margins are monitored or consumed) — this is the §15.2 keystone replacing all fixed-hemi residuals. -/
theorem openHemisphere_at_WBS_sup {n : ℕ} {A B : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
{k : Fin (n - 1)}
(hka : ShortArc (A (openingAxis k)) (jointPrev A k))
(hkt : ShortArc (A (openingAxis k)) (jointNext A k))
(hkdef : jointAngle A k < jointAngle B k)
(hedge : ∀ i : Fin (n + 1),
ShortArc (openTail A (openingAxis k) (-(monitoredSupWBS A B k)) i)
(openTail A (openingAxis k) (-(monitoredSupWBS A B k)) (i + 1)))
(hjopen : ∀ r : Fin (n - 1),
0 < jointAngle (openTail A (openingAxis k) (-(monitoredSupWBS A B k))) r ∧
jointAngle (openTail A (openingAxis k) (-(monitoredSupWBS A B k))) r < Real.pi) :
∃ h' : E3, ‖h'‖ = 1 ∧ ∀ r : Fin (n + 1),
0 < (⟪h', ((openTail A (openingAxis k) (-(monitoredSupWBS A B k)) r : S2) : E3)⟫ : ℝ) := by
have hn : 2 ≤ n := hA.two_le
exact openHemisphere_of_weakSupports_jointOpen_full hn hedge
(supportWBS_sOrient_nonneg hA hka hkt hkdef) hjopen
/-- A point pair with `0 < sDist < π` is a `ShortArc` (the converse of `ShortArc.sDist_pos`/`sDist_lt_pi`).
Distinctness from `sDist_eq_zero_iff`; non-antipodality because `(p : E3) = -(q : E3)` forces
`sInner p q = -1`, hence `sDist p q = arccos(-1) = π`, contradicting `sDist p q < π`. -/
theorem shortArc_of_sDist_pos_lt_pi {p q : S2}
(hpos : 0 < sDist p q) (hlt : sDist p q < Real.pi) : ShortArc p q := by
refine ⟨fun he => ?_, fun he => ?_⟩
· rw [sDist_eq_zero_iff.2 he] at hpos; exact lt_irrefl 0 hpos
· -- `(p : E3) = -(q : E3)` ⟹ `sInner p q = -1` ⟹ `sDist = π`.
have hcos : sInner p q = -1 := by
simp only [sInner, he, inner_neg_left, S2.inner_self]
have : sDist p q = Real.pi := by rw [sDist, hcos, Real.arccos_neg_one]
rw [this] at hlt; exact lt_irrefl Real.pi hlt
/-- **Opened non-wrap edges stay short** (margins-free). For an edge `(i, i+1)` with `i ≠ Fin.last n`,
the opened edge `(openTail A K δ i, openTail A K δ (i + 1))` has the same spherical length as the base
edge `(A i, A (i + 1))` — both endpoints are on the same side of the axis `K` (or the seam, axis fixed +
isometry), so `sDist` is preserved — and the base edge is a `ShortArc` (`A` strict). The wraparound edge
`i = Fin.last n` (where `i + 1 = 0` straddles the axis) is excluded. -/
theorem openTail_nonwrap_shortArc {n : ℕ} {A : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
(K : Fin (n + 1)) (δ : ℝ) {i : Fin (n + 1)} (hi : i ≠ Fin.last n) :
ShortArc (openTail A K δ i) (openTail A K δ (i + 1)) := by
-- the base edge is short, with a known spherical length in `(0, π)`.
have hbase : ShortArc (A i) (A (i + 1)) := arm_edge_short hA i
-- `i ≠ last` ⟹ `i.val < n` ⟹ `(i + 1).val = i.val + 1` (no wraparound).
have hilt : i.val < n := by
have hival : i.val < n + 1 := i.isLt
have hlast : ((Fin.last n : Fin (n + 1)) : ℕ) = n := Fin.val_last n
rcases Nat.lt_or_ge i.val n with h | h
· exact h
· exact absurd (Fin.ext (by omega)) hi
have hi1 : ((i + 1 : Fin (n + 1)) : ℕ) = i.val + 1 := by
rw [Fin.val_add, Fin.val_one']
rw [Nat.mod_eq_of_lt (by omega : 1 < n + 1)]
exact Nat.mod_eq_of_lt (by omega)
-- the opened edge has the same length as the base edge; ShortArc transfers.
have hsd : sDist (openTail A K δ i) (openTail A K δ (i + 1)) = sDist (A i) (A (i + 1)) := by
rcases lt_trichotomy i.val K.val with hlt | heq | hgt
· -- both endpoints fixed (`i ≤ K` and `i + 1 ≤ K`).
rw [openTail_fixed A K δ (by omega), openTail_fixed A K δ (show (i + 1).val ≤ K.val by omega)]
· -- seam: `i = K` (axis fixed), `i + 1 > K` (tail rotated).
have hiK : i = K := Fin.ext heq
subst hiK
have hgt1 : i.val < (i + 1).val := by omega
exact sDist_openTail_axis_tail A i δ hgt1
· -- both endpoints rotated (`i > K` and `i + 1 > K`).
exact sDist_openTail_tail A K δ hgt (show K.val < (i + 1).val by omega)
-- ShortArc from the preserved length in `(0, π)`.
refine shortArc_of_sDist_pos_lt_pi ?_ ?_
· rw [hsd]; exact hbase.sDist_pos
· rw [hsd]; exact hbase.sDist_lt_pi
/-- **The opened-arm joints lie in `(0, π)` at the WBS supremum** (margins-free). Off-axis joints are
unchanged (`jointAngle_openTail_eq_of_ne`), so positive (`A` strict) and `< π` (`A` strict); the joint
`k` opens to `jointAngle A k + δ*_WBS` (`openedNegJointAngle_eq_add`, branch from the deficit bound), so
`≥ jointAngle A k > 0` and `≤ jointAngle B k < π` (slack closure + `B` strict). -/
theorem openedJoints_in_Ioo_at_supWBS {n : ℕ} {A B : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
(hB : StrictConvexSphArm B) {k : Fin (n - 1)}
(hka : ShortArc (A (openingAxis k)) (jointPrev A k))
(hkt : ShortArc (A (openingAxis k)) (jointNext A k))
(hkdef : jointAngle A k < jointAngle B k) :
∀ r : Fin (n - 1),
0 < jointAngle (openTail A (openingAxis k) (-(monitoredSupWBS A B k))) r ∧
jointAngle (openTail A (openingAxis k) (-(monitoredSupWBS A B k))) r < Real.pi := by
set δ : ℝ := monitoredSupWBS A B k with hδ
-- the deficit bound gives `δ ≤ jointAngle B k − jointAngle A k`, hence the additive branch.
have hδ0 : 0 ≤ δ := (monitoredSupWBS_mem_Icc hA hka hkt hkdef).1
have hub : δ ≤ jointAngle B k - jointAngle A k := monitoredSupWBS_le_deficit hA hka hkt hkdef
have hBlt : jointAngle B k < Real.pi := strict_jointAngle_lt_pi hB k
have hbranch : jointAngle A k + δ ≤ Real.pi := by linarith
intro r
by_cases hrk : r = k
· -- the opened joint `k`.
subst hrk
rw [jointAngle_openTail_eq_openedInterior A r (-δ),
openedNegJointAngle_eq_add hA hka hkt hδ0 hbranch]
refine ⟨?_, by linarith⟩
have hpos : 0 < jointAngle A r := strict_jointAngle_pos hA r
linarith
· -- off-axis joints are unchanged.
rw [jointAngle_openTail_eq_of_ne A k (-δ) hrk]
exact ⟨strict_jointAngle_pos hA r, strict_jointAngle_lt_pi hA r⟩
/-- **The opened-arm wraparound edge is distinct at the WBS supremum** (margins-free, FFCT43 route).
`endpt A'_WBS = sDist (A'_WBS 0)(A'_WBS last)`; `glueWBS_clause_i` gives `endpt A ≤ endpt A'_WBS` and
`strict_arm_endpt_pos` gives `0 < endpt A`, so `A'_WBS 0 ≠ A'_WBS last`. (Non-antipodality of the wrap
edge is the one fact the weak-support branch cannot conclude margins-free; the strict branch closes it.) -/
theorem openedWrap_distinct_at_supWBS {n : ℕ} {A B : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
{k : Fin (n - 1)}
(hka : ShortArc (A (openingAxis k)) (jointPrev A k))
(hkt : ShortArc (A (openingAxis k)) (jointNext A k))
(hkdef : jointAngle A k < jointAngle B k) :
openTail A (openingAxis k) (-(monitoredSupWBS A B k)) 0
≠ openTail A (openingAxis k) (-(monitoredSupWBS A B k)) (Fin.last n) := by
have hpos : 0 < endpt A := strict_arm_endpt_pos hA
have hmono : endpt A ≤ endpt (openTail A (openingAxis k) (-(monitoredSupWBS A B k))) :=
glueWBS_clause_i hA hka hkt hkdef
have hsd : (0 : ℝ) < sDist (openTail A (openingAxis k) (-(monitoredSupWBS A B k)) 0)
(openTail A (openingAxis k) (-(monitoredSupWBS A B k)) (Fin.last n)) :=
lt_of_lt_of_le hpos hmono
intro heq
rw [sDist_eq_zero_iff.2 heq] at hsd
exact lt_irrefl 0 hsd
/-- **Opened ShortArc edges (all `n + 1`) from the wrap residual.** Interior/seam/tail edges are short
margins-free (`openTail_nonwrap_shortArc`); the wraparound edge `i = Fin.last n` (`i + 1 = 0`) is supplied
by the wrap ShortArc (symmetrized to the `(last, last + 1)` orientation). -/
theorem openedEdges_short_at_supWBS_of_wrap {n : ℕ} {A B : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
{k : Fin (n - 1)}
(hwrap : ShortArc (openTail A (openingAxis k) (-(monitoredSupWBS A B k)) (Fin.last n))
(openTail A (openingAxis k) (-(monitoredSupWBS A B k)) 0)) :
∀ i : Fin (n + 1),
ShortArc (openTail A (openingAxis k) (-(monitoredSupWBS A B k)) i)
(openTail A (openingAxis k) (-(monitoredSupWBS A B k)) (i + 1)) := by
intro i
by_cases hi : i = Fin.last n
· subst hi; rw [lastAddOne_eq_zero]; exact hwrap
· exact openTail_nonwrap_shortArc hA (openingAxis k) (-(monitoredSupWBS A B k)) hi
/-- **Brick 5 — support-stuck ⟹ weakly convex.** At a `SupportStuckWBS` supremum (a non-incident support
vanishes, so only weak supports are available), the opened arm `A'_WBS` is `WeakConvexSphArm`. The open
hemisphere (brick 4) is produced margins-free from weak supports + open joints + the opened ShortArc edges;
it gives the edge distinctness `hdist` and feeds `weakConvex_of_supportStuckW_of_hemiPos_anyH`. The wrap
edge's non-antipodality is the residual `OpenedWrapShortArcAtSupWBS`. -/
theorem supportStuckWBS_weakConvex {n : ℕ} {A B : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
(hB : StrictConvexSphArm B) {k : Fin (n - 1)}
(hka : ShortArc (A (openingAxis k)) (jointPrev A k))
(hkt : ShortArc (A (openingAxis k)) (jointNext A k))
(hkdef : jointAngle A k < jointAngle B k)
(hwrap : ShortArc (openTail A (openingAxis k) (-(monitoredSupWBS A B k)) (Fin.last n))
(openTail A (openingAxis k) (-(monitoredSupWBS A B k)) 0)) :
WeakConvexSphArm (openTail A (openingAxis k) (-(monitoredSupWBS A B k))) := by
set δ : ℝ := monitoredSupWBS A B k with hδ
have hedge := openedEdges_short_at_supWBS_of_wrap hA hwrap
have hjopen := openedJoints_in_Ioo_at_supWBS hA hB hka hkt hkdef
have hsupp := supportWBS_sOrient_nonneg hA hka hkt hkdef
-- the open hemisphere (brick 4), margins-free.
obtain ⟨h', hnorm, hhem⟩ := openHemisphere_at_WBS_sup hA hka hkt hkdef hedge hjopen
-- edge distinctness from the ShortArc edges.
have hdist : ∀ i : Fin (n + 1),
openTail A (openingAxis k) (-δ) i ≠ openTail A (openingAxis k) (-δ) (i + 1) :=
fun i => (hedge i).1
exact weakConvex_of_supportStuckW_of_hemiPos_anyH hA hsupp hdist ⟨h', hnorm, hhem⟩
/-- **Brick 6 — reach / no-support-stuck ⟹ strictly convex.** When no non-incident support of `A'_WBS`
vanishes (`¬ SupportStuckWBS`), the weak supports upgrade to **strict** (`lt_of_le_of_ne`). The strict
supports give the opened ShortArc edges margins-free — including the wraparound edge, whose non-antipodality
follows from `antipodal_pair_excluded_of_strict` (so **no wrap residual is needed here**) — and the open
hemisphere (brick 4); `reach_strictConvex_interior` assembles `StrictConvexSphArm A'_WBS`. -/
theorem reachWBS_strictConvex {n : ℕ} {A B : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
(hB : StrictConvexSphArm B) {k : Fin (n - 1)}
(hka : ShortArc (A (openingAxis k)) (jointPrev A k))
(hkt : ShortArc (A (openingAxis k)) (jointNext A k))
(hkdef : jointAngle A k < jointAngle B k)
(hnotStuck : ¬ SupportStuckWBS A B k) :
StrictConvexSphArm (openTail A (openingAxis k) (-(monitoredSupWBS A B k))) := by
set δ : ℝ := monitoredSupWBS A B k with hδ
have hn : 2 ≤ n := hA.two_le
have hsupp := supportWBS_sOrient_nonneg hA hka hkt hkdef
-- strict supports: weak `≥ 0` is `> 0` since none vanishes (else `SupportStuckWBS`).
have hmix : ∀ i j : Fin (n + 1), j ≠ i → j ≠ i + 1 →
0 < sOrient (openTail A (openingAxis k) (-δ) i) (openTail A (openingAxis k) (-δ) (i + 1))
(openTail A (openingAxis k) (-δ) j) := by
intro i j hji hji1
refine lt_of_le_of_ne (hsupp i j hji hji1) (fun heq => hnotStuck ⟨⟨(i, j), ⟨hji, hji1⟩⟩, ?_⟩)
rw [supportConstraint_apply]; exact heq.symm
-- the opened ShortArc wrap edge from strict supports: distinct + non-antipodal.
have hwrapdist := openedWrap_distinct_at_supWBS hA hka hkt hkdef
have hwrap : ShortArc (openTail A (openingAxis k) (-δ) (Fin.last n))
(openTail A (openingAxis k) (-δ) 0) := by
refine ⟨fun he => hwrapdist he.symm, fun he => ?_⟩
-- `A'_WBS last = -(A'_WBS 0)`: antipodal pair `(last, 0)` excluded by strict supports
-- (`last ≠ 0`, `last ≠ 0 + 1 = 1` for `n ≥ 2`).
have hl0 : (Fin.last n : Fin (n + 1)) ≠ (0 : Fin (n + 1)) := by
intro h; have := congrArg Fin.val h; simp only [Fin.val_last, Fin.val_zero] at this; omega
have hl1 : (Fin.last n : Fin (n + 1)) ≠ (0 : Fin (n + 1)) + 1 := by
intro h; have := congrArg Fin.val h
rw [Fin.val_last, zero_add, Fin.val_one'] at this
rw [Nat.mod_eq_of_lt (by omega : 1 < n + 1)] at this; omega
exact antipodal_pair_excluded_of_strict hmix (r := Fin.last n) (s := 0) hl0 hl1 he
have hedge := openedEdges_short_at_supWBS_of_wrap hA hwrap
have hjopen := openedJoints_in_Ioo_at_supWBS hA hB hka hkt hkdef
obtain ⟨h', hnorm, hhem⟩ := openHemisphere_at_WBS_sup hA hka hkt hkdef hedge hjopen
exact reach_strictConvex_interior hA hnorm hmix hhem
/-- A `SupportStuckWBS` witness yields a vanishing non-incident support of `A'_WBS` in `sOrient` form. -/
theorem supportStuckWBS_vanishingSupport {n : ℕ} {A B : Fin (n + 1) → S2} {k : Fin (n - 1)}
(hstuck : SupportStuckWBS A B k) :
∃ i j : Fin (n + 1), j ≠ i ∧ j ≠ i + 1 ∧
sOrient (openTail A (openingAxis k) (-(monitoredSupWBS A B k)) i)
(openTail A (openingAxis k) (-(monitoredSupWBS A B k)) (i + 1))
(openTail A (openingAxis k) (-(monitoredSupWBS A B k)) j) = 0 := by
obtain ⟨c, hc⟩ := hstuck
rw [supportConstraint_apply] at hc
exact ⟨c.1.1, c.1.2, c.2.1, c.2.2, hc⟩
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
/-- The base index of an interior joint, as a `Fin (n+1)`. -/
def jIdx {n : ℕ} (r : Fin (n - 1)) : Fin (n + 1) :=
⟨r.val, by have := r.isLt; omega⟩
theorem jIdx_val {n : ℕ} (r : Fin (n - 1)) : ((jIdx r : Fin (n + 1)) : ℕ) = r.val := rfl
theorem jIdx_succ_val {n : ℕ} (r : Fin (n - 1)) :
((jIdx r + 1 : Fin (n + 1)) : ℕ) = r.val + 1 := by
have hr := r.isLt
rw [Fin.val_add, jIdx_val]
have h1 : ((1 : Fin (n + 1)) : ℕ) = 1 := by
rw [Fin.val_one']; exact Nat.mod_eq_of_lt (by omega)
rw [h1]
exact Nat.mod_eq_of_lt (by omega)
theorem jIdx_succ_succ_val {n : ℕ} (r : Fin (n - 1)) :
(((jIdx r + 1) + 1 : Fin (n + 1)) : ℕ) = r.val + 2 := by
have hr := r.isLt
rw [Fin.val_add, jIdx_succ_val]
have h1 : ((1 : Fin (n + 1)) : ℕ) = 1 := by
rw [Fin.val_one']; exact Nat.mod_eq_of_lt (by omega)
rw [h1]; exact Nat.mod_eq_of_lt (by omega)
theorem P_jIdx_succ {n : ℕ} (P : Fin (n + 1) → S2) (r : Fin (n - 1)) :
P (jIdx r + 1) = P ⟨r.val + 1, by have := r.isLt; omega⟩ := by
congr 1; apply Fin.ext; rw [jIdx_succ_val]
theorem P_jIdx_succ_succ {n : ℕ} (P : Fin (n + 1) → S2) (r : Fin (n - 1)) :
P ((jIdx r + 1) + 1) = P ⟨r.val + 2, by have := r.isLt; omega⟩ := by
congr 1; apply Fin.ext; rw [jIdx_succ_succ_val]
theorem P_jIdx {n : ℕ} (P : Fin (n + 1) → S2) (r : Fin (n - 1)) :
P (jIdx r) = P ⟨r.val, by have := r.isLt; omega⟩ := rfl
theorem jointAngle_eq_consecutive {n : ℕ} (P : Fin (n + 1) → S2) (r : Fin (n - 1)) :
jointAngle P r =
sphAngle (P ⟨r.val, by have := r.isLt; omega⟩) (P ⟨r.val + 1, by have := r.isLt; omega⟩)
(P ⟨r.val + 2, by have := r.isLt; omega⟩) := rfl
/-- An interior-joint base index `jIdx r` is a **real** edge: `jIdx r ≠ Fin.last n` (its value is
`r.val ≤ n - 2 < n`). -/
theorem jIdx_ne_last {n : ℕ} (r : Fin (n - 1)) : (jIdx r : Fin (n + 1)) ≠ Fin.last n := by
intro h
have hr := r.isLt
have := congrArg Fin.val h
rw [jIdx_val, Fin.val_last] at this
omega
/-- The successor index `jIdx r + 1` is also a **real** edge: its value is `r.val + 1 ≤ n - 1 < n`. -/
theorem jIdx_succ_ne_last {n : ℕ} (r : Fin (n - 1)) :
(jIdx r + 1 : Fin (n + 1)) ≠ Fin.last n := by
intro h
have hr := r.isLt
have := congrArg Fin.val h
rw [jIdx_succ_val, Fin.val_last] at this
omega
/-- A flat interior joint is impossible (open-chain copy of FFCT44's `flat_interior_joint_absurd`). -/
theorem flat_interior_joint_absurd {n : ℕ} {P : Fin (n + 1) → S2} (r : Fin (n - 1))
(hsau : ShortArc (P ⟨r.val + 1, by have := r.isLt; omega⟩) (P ⟨r.val, by have := r.isLt; omega⟩))
(hsav : ShortArc (P ⟨r.val + 1, by have := r.isLt; omega⟩)
(P ⟨r.val + 2, by have := r.isLt; omega⟩))
(hcol : det3 (P ⟨r.val, by have := r.isLt; omega⟩ : E3)
(P ⟨r.val + 1, by have := r.isLt; omega⟩ : E3)
(P ⟨r.val + 2, by have := r.isLt; omega⟩ : E3) = 0)
(hjopen : 0 < jointAngle P r ∧ jointAngle P r < Real.pi) :
False := by
have hbridge := sphAngle_eq_zero_or_pi_of_det3_zero
(u := P ⟨r.val, by have := r.isLt; omega⟩) (v := P ⟨r.val + 1, by have := r.isLt; omega⟩)
(w := P ⟨r.val + 2, by have := r.isLt; omega⟩) hsau hsav hcol
rw [jointAngle_eq_consecutive] at hjopen
rcases hbridge with h0 | hπ
· rw [h0] at hjopen; exact lt_irrefl 0 hjopen.1
· rw [hπ] at hjopen; exact lt_irrefl Real.pi hjopen.2
/-- The two real edges adjacent to the apex of interior joint `r` both have a vanishing area form
against the axis `z` (open-chain: `hallplanes` invoked only at the two real indices). -/
theorem edge_planes_at_apex {n : ℕ} {P : Fin (n + 1) → S2} {z : S2}
(hallplanes : ∀ i : Fin (n + 1), i ≠ Fin.last n →
det3 (P i : E3) (P (i + 1) : E3) (z : E3) = 0)
(r : Fin (n - 1)) :
det3 (P ⟨r.val, by have := r.isLt; omega⟩ : E3) (P ⟨r.val + 1, by have := r.isLt; omega⟩ : E3)
(z : E3) = 0 ∧
det3 (P ⟨r.val + 1, by have := r.isLt; omega⟩ : E3)
(P ⟨r.val + 2, by have := r.isLt; omega⟩ : E3) (z : E3) = 0 := by
have h1 := hallplanes (jIdx r) (jIdx_ne_last r)
have h2 := hallplanes (jIdx r + 1) (jIdx_succ_ne_last r)
rw [P_jIdx P r, P_jIdx_succ P r] at h1
rw [P_jIdx_succ P r, P_jIdx_succ_succ P r] at h2
exact ⟨h1, h2⟩
/-- The non-pole apex collapse (open-chain copy of FFCT44's `consecutive_det3_zero_of_nonpole`). -/
theorem consecutive_det3_zero_of_nonpole {n : ℕ} {P : Fin (n + 1) → S2} {z : S2}
(hallplanes : ∀ i : Fin (n + 1), i ≠ Fin.last n →
det3 (P i : E3) (P (i + 1) : E3) (z : E3) = 0)
(r : Fin (n - 1))
(hne : (P ⟨r.val + 1, by have := r.isLt; omega⟩ : E3) ≠ (z : E3))
(hanti : (P ⟨r.val + 1, by have := r.isLt; omega⟩ : E3) ≠ -(z : E3)) :
det3 (P ⟨r.val, by have := r.isLt; omega⟩ : E3) (P ⟨r.val + 1, by have := r.isLt; omega⟩ : E3)
(P ⟨r.val + 2, by have := r.isLt; omega⟩ : E3) = 0 := by
obtain ⟨he1, he2⟩ := edge_planes_at_apex hallplanes r
set apex : E3 := (P ⟨r.val + 1, by have := r.isLt; omega⟩ : E3) with hapex
set x : E3 := (P ⟨r.val, by have := r.isLt; omega⟩ : E3) with hx
set y : E3 := (P ⟨r.val + 2, by have := r.isLt; omega⟩ : E3) with hy
set zz : E3 := (z : E3) with hzz
have hzu : ‖zz‖ = 1 := z.2
have hau : ‖apex‖ = 1 := (P _).2
have hzne : zz ≠ apex := fun h => hne h.symm
have hzanti : zz ≠ -apex := by
intro h
exact hanti (by rw [h]; simp)
have hdetx : det3 zz apex x = 0 := by
have hcyc : det3 zz apex x = -det3 x apex zz := by simp only [det3]; ring
rw [hcyc, he1, neg_zero]
have hdety : det3 zz apex y = 0 := by
have hcyc : det3 zz apex y = det3 apex y zz := by simp only [det3]; ring
rw [hcyc, he2]
obtain ⟨c1, d1, hx'⟩ := lin_indep_span_of_det3_zero hzu hau hzne hzanti hdetx
obtain ⟨c2, d2, hy'⟩ := lin_indep_span_of_det3_zero hzu hau hzne hzanti hdety
have hap' : apex = (0 : ℝ) • zz + (1 : ℝ) • apex := by simp
exact coplanar_triple_det3_zero ⟨c1, d1, hx'.symm⟩ ⟨0, 1, hap'.symm⟩ ⟨c2, d2, hy'.symm⟩
/-- **§1 — the open-chain meridian-pencil collapse kernel (`3 ≤ n`).** A chain `P : Fin (n+1) → S2`
(`3 ≤ n`), every **real** edge of which is a short arc (`hside`, `i ≠ Fin.last n`), every **real** edge
plane of which contains a common unit axis `z` (`hallplanes`, `i ≠ Fin.last n`), with all interior joints
in `(0, π)` (`hjopen`), is impossible. The wrap edge `(P last, P 0)` is **not** used. -/
theorem openChain_collapse_forces_flat_joint_ge3 {n : ℕ} {P : Fin (n + 1) → S2} {z : S2}
(hn : 3 ≤ n)
(hside : ∀ i : Fin (n + 1), i ≠ Fin.last n → ShortArc (P i) (P (i + 1)))
(hallplanes : ∀ i : Fin (n + 1), i ≠ Fin.last n →
det3 (P i : E3) (P (i + 1) : E3) (z : E3) = 0)
(hjopen : ∀ r : Fin (n - 1), 0 < jointAngle P r ∧ jointAngle P r < Real.pi) :
False := by
classical
-- The two short joint arcs at the apex of interior joint `r` (both edges real).
have hshort_apex : ∀ r : Fin (n - 1),
ShortArc (P ⟨r.val + 1, by have := r.isLt; omega⟩) (P ⟨r.val, by have := r.isLt; omega⟩) ∧
ShortArc (P ⟨r.val + 1, by have := r.isLt; omega⟩)
(P ⟨r.val + 2, by have := r.isLt; omega⟩) := by
intro r
have e1 := hside (jIdx r) (jIdx_ne_last r)
have e2 := hside (jIdx r + 1) (jIdx_succ_ne_last r)
rw [P_jIdx P r, P_jIdx_succ P r] at e1
rw [P_jIdx_succ P r, P_jIdx_succ_succ P r] at e2
exact ⟨e1.symm, e2⟩
by_cases hsome : ∃ r : Fin (n - 1),
(P ⟨r.val + 1, by have := r.isLt; omega⟩ : E3) ≠ (z : E3) ∧
(P ⟨r.val + 1, by have := r.isLt; omega⟩ : E3) ≠ -(z : E3)
· -- CASE (a): a non-pole apex. The collapse gives a flat joint.
obtain ⟨r, hne, hanti⟩ := hsome
obtain ⟨hsau, hsav⟩ := hshort_apex r
have hcol := consecutive_det3_zero_of_nonpole hallplanes r hne hanti
exact flat_interior_joint_absurd r hsau hsav hcol (hjopen r)
· -- CASE (b): every interior apex is a pole `P apex = ± z`. With `n ≥ 3`, joints `0`, `1` give
-- adjacent interior poles `P 1`, `P 2` sharing the real edge `(P 1, P 2)` — ShortArc kill.
push_neg at hsome
have hpole : ∀ r : Fin (n - 1),
(P ⟨r.val + 1, by have := r.isLt; omega⟩ : E3) = (z : E3) ∨
(P ⟨r.val + 1, by have := r.isLt; omega⟩ : E3) = -(z : E3) := by
intro r
by_cases h1 : (P ⟨r.val + 1, by have := r.isLt; omega⟩ : E3) = (z : E3)
· exact Or.inl h1
· exact Or.inr (hsome r h1)
have hp0 := hpole ⟨0, by omega⟩
have hp1 := hpole ⟨1, by omega⟩
have hsh := (hshort_apex ⟨0, by omega⟩).2
have hsh1 : (P ⟨0 + 1, by omega⟩ : E3) ≠ (P ⟨0 + 2, by omega⟩ : E3) := by
intro he; exact hsh.1 (S2.ext he)
have hsh2 : (P ⟨0 + 1, by omega⟩ : E3) ≠ -(P ⟨0 + 2, by omega⟩ : E3) := hsh.2
rcases hp0 with hp0z | hp0z <;> rcases hp1 with hp1z | hp1z
· exact hsh1 (by rw [hp0z, hp1z])
· exact hsh2 (by rw [hp0z, hp1z, neg_neg])
· exact hsh2 (by rw [hp0z, hp1z])
· exact hsh1 (by rw [hp0z, hp1z])
/-- **§2 — margins-free, wrap-edge-free open hemisphere.** For a chain `P : Fin (n+1) → S2` (`2 ≤ n`)
with **real-edge** short arcs (`hside`, `i ≠ Fin.last n`), **real-edge** weak supports (`hsupp`,
`i ≠ Fin.last n`), and all interior joints in `(0, π)` (`hjopen`), there is a *unit* normal `h'`
strictly positive against every vertex. **No margins, no wrap edge.** -/
theorem openHemisphere_full_openChain {n : ℕ} {P : Fin (n + 1) → S2}
(hn : 2 ≤ n)
(hside : ∀ i : Fin (n + 1), i ≠ Fin.last n → ShortArc (P i) (P (i + 1)))
(hsupp : ∀ i j : Fin (n + 1), i ≠ Fin.last n → j ≠ i → j ≠ i + 1 →
0 ≤ sOrient (P i) (P (i + 1)) (P j))
(hjopen : ∀ r : Fin (n - 1), 0 < jointAngle P r ∧ jointAngle P r < Real.pi) :
∃ h' : E3, ‖h'‖ = 1 ∧ ∀ r : Fin (n + 1), 0 < (⟪h', (P r : E3)⟫ : ℝ) := by
classical
set s : Finset E3 := (Finset.univ : Finset (Fin (n + 1))).image (fun r => ((P r : S2) : E3))
with hs
have hmem_iff : ∀ v, v ∈ s ↔ ∃ m, ((P m : S2) : E3) = v := by
intro v
rw [hs, Finset.mem_image]
constructor
· rintro ⟨m, _hm, rfl⟩; exact ⟨m, rfl⟩
· rintro ⟨m, rfl⟩; exact ⟨m, Finset.mem_univ _, rfl⟩
by_cases h0 : (0 : E3) ∈ convexHull ℝ (s : Set E3)
· exfalso
rw [Finset.mem_convexHull'] at h0
obtain ⟨w, hw0, hw1, hwsum⟩ := h0
-- For every REAL edge `i` and every member `v`, `w v * det3 (P i)(P i+1) v ≥ 0`.
have hterm_nonneg : ∀ (i : Fin (n + 1)), i ≠ Fin.last n → ∀ (v : E3), v ∈ s →
0 ≤ w v * det3 (P i : E3) (P (i + 1) : E3) v := by
intro i hi v hv
obtain ⟨m, rfl⟩ := (hmem_iff v).mp hv
by_cases hmi : m = i
· subst hmi; rw [show det3 (P m : E3) (P (m + 1) : E3) (P m : E3) = 0 from by
simp only [det3]; ring, mul_zero]
· by_cases hmi1 : m = i + 1
· subst hmi1; rw [show det3 (P i : E3) (P (i + 1) : E3) (P (i + 1) : E3) = 0 from by
simp only [det3]; ring, mul_zero]
· exact mul_nonneg (hw0 _ hv)
(le_trans (le_of_eq rfl) (hsupp i m hi (by simpa [eq_comm] using hmi)
(by simpa [eq_comm] using hmi1)))
-- The REAL-edge functional pushed through the convex combination is `0`.
have hedge_zero : ∀ i : Fin (n + 1), i ≠ Fin.last n →
(0 : ℝ) = ∑ v ∈ s, w v * det3 (P i : E3) (P (i + 1) : E3) v := by
intro i _hi
have hkey := det3_edge_centerSum (P i : E3) (P (i + 1) : E3) s w
rw [hwsum] at hkey
rw [show det3 (P i : E3) (P (i + 1) : E3) (0 : E3) = 0 from by simp [det3]] at hkey
exact hkey
have hterm_zero : ∀ i : Fin (n + 1), i ≠ Fin.last n → ∀ v ∈ s,
w v * det3 (P i : E3) (P (i + 1) : E3) v = 0 := by
intro i hi
exact (Finset.sum_eq_zero_iff_of_nonneg (fun v hv => hterm_nonneg i hi v hv)).mp
(hedge_zero i hi).symm
-- Pick a positive-weight member.
have hexists_pos : ∃ v ∈ s, 0 < w v := by
by_contra hnone
push_neg at hnone
have : ∑ v ∈ s, w v = 0 :=
Finset.sum_eq_zero (fun v hv => le_antisymm (hnone v hv) (hw0 v hv))
rw [hw1] at this; exact one_ne_zero this
-- A positive-weight vertex `v₀ = P r₀` is a common REAL-edge axis.
have hcommon_axis : ∀ (r0 : Fin (n + 1)), 0 < w ((P r0 : S2) : E3) →
∀ i : Fin (n + 1), i ≠ Fin.last n →
det3 (P i : E3) (P (i + 1) : E3) (P r0 : E3) = 0 := by
intro r0 hr0pos i hi
have hv0s : ((P r0 : S2) : E3) ∈ s := by rw [hmem_iff]; exact ⟨r0, rfl⟩
have := hterm_zero i hi ((P r0 : S2) : E3) hv0s
rcases mul_eq_zero.mp this with hw | hd
· exact absurd hw (ne_of_gt hr0pos)
· exact hd
rcases Nat.lt_or_ge 2 n with hn3 | hn2
· -- `3 ≤ n`: a common-axis vertex contradicts the open-chain kernel.
obtain ⟨v0, hv0s, hv0pos⟩ := hexists_pos
obtain ⟨r0, hv0eq⟩ := (hmem_iff v0).mp hv0s
have hr0pos : 0 < w ((P r0 : S2) : E3) := by rw [hv0eq]; exact hv0pos
exact openChain_collapse_forces_flat_joint_ge3 (z := P r0) (by omega) hside
(hcommon_axis r0 hr0pos) hjopen
· -- `n = 2`: a positive-weight vertex `≠ P 1` (index `0` or `2`) gives the flat single joint.
have hneq : n = 2 := by omega
subst hneq
-- There is a positive-weight vertex `v₀` with `v₀ ≠ (P 1 : E3)` (else `0 = (Σw)•(P 1)`).
have hne1 : ∃ v ∈ s, 0 < w v ∧ v ≠ ((P (1 : Fin 3) : S2) : E3) := by
by_contra hcon
push_neg at hcon
-- every member is either zero-weight or equals `P 1`.
have hsum1 : ∑ v ∈ s, w v • v = ((∑ v ∈ s, w v) : ℝ) • ((P (1 : Fin 3) : S2) : E3) := by
rw [Finset.sum_smul]
refine Finset.sum_congr rfl (fun v hv => ?_)
rcases lt_or_eq_of_le (hw0 v hv) with hpos | hzero
· rw [hcon v hv hpos]
· rw [← hzero, zero_smul, zero_smul]
rw [hwsum, hw1, one_smul] at hsum1
-- `0 = P 1` contradicts `‖P 1‖ = 1`.
have hnorm1 : ‖((P (1 : Fin 3) : S2) : E3)‖ = 1 := (P (1 : Fin 3)).2
rw [← hsum1, norm_zero] at hnorm1
exact one_ne_zero hnorm1.symm
obtain ⟨v0, hv0s, hv0pos, hv0ne⟩ := hne1
obtain ⟨r0, hv0eq⟩ := (hmem_iff v0).mp hv0s
have hr0pos : 0 < w ((P r0 : S2) : E3) := by rw [hv0eq]; exact hv0pos
-- `r0 ≠ 1` (else `P r0 = P 1 = v0`, contradicting `hv0ne`).
have hr0ne1 : r0 ≠ (1 : Fin 3) := by
intro h; apply hv0ne; rw [← hv0eq, h]
-- the single interior joint `0`; vertices `P 0, P 1, P 2`.
have hjoint0 := hjopen ⟨0, by omega⟩
-- the two real edges `0` and `1`.
have he0 : (Fin.last 2 : Fin 3) = (2 : Fin 3) := rfl
have h0ne : (0 : Fin 3) ≠ Fin.last 2 := by rw [he0]; decide
have h1ne : (1 : Fin 3) ≠ Fin.last 2 := by rw [he0]; decide
-- `det3 (P 0)(P 1)(P 2) = 0` from the common-axis facts, depending on `r0 ∈ {0, 2}`.
have hcol : det3 (P ⟨0, by omega⟩ : E3) (P ⟨0 + 1, by omega⟩ : E3)
(P ⟨0 + 2, by omega⟩ : E3) = 0 := by
-- normalise the nat-indexed vertices to `P 0, P 1, P 2`.
have e0 : (P ⟨0, by omega⟩ : E3) = (P (0 : Fin 3) : E3) := rfl
have e1 : (P ⟨0 + 1, by omega⟩ : E3) = (P (1 : Fin 3) : E3) := rfl
have e2 : (P ⟨0 + 2, by omega⟩ : E3) = (P (2 : Fin 3) : E3) := rfl
rw [e0, e1, e2]
-- `r0` is `0` or `2`.
have hr0cases : r0 = (0 : Fin 3) ∨ r0 = (2 : Fin 3) := by
fin_cases r0
· exact Or.inl rfl
· exact absurd rfl hr0ne1
· exact Or.inr rfl
rcases hr0cases with hr0 | hr0
· -- `r0 = 0`: edge `1 = (P 1, P 2)` axis gives `det3 (P 1)(P 2)(P 0) = 0`; cyclic.
have hax := hcommon_axis r0 hr0pos (1 : Fin 3) h1ne
rw [hr0] at hax
-- `(1 : Fin 3) + 1 = 2`.
have h11 : ((1 : Fin 3) + 1) = (2 : Fin 3) := by decide
rw [h11] at hax
-- `hax : det3 (P 1)(P 2)(P 0) = 0`; cyclic ⟹ `det3 (P 0)(P 1)(P 2) = 0`.
have hcyc : det3 (P (0 : Fin 3) : E3) (P (1 : Fin 3) : E3) (P (2 : Fin 3) : E3)
= det3 (P (1 : Fin 3) : E3) (P (2 : Fin 3) : E3) (P (0 : Fin 3) : E3) := by
simp only [det3]; ring
rw [hcyc]; exact hax
· -- `r0 = 2`: edge `0 = (P 0, P 1)` axis gives `det3 (P 0)(P 1)(P 2) = 0` directly.
have hax := hcommon_axis r0 hr0pos (0 : Fin 3) h0ne
rw [hr0] at hax
have h01 : ((0 : Fin 3) + 1) = (1 : Fin 3) := by decide
rw [h01] at hax
exact hax
-- the short joint arcs at apex `P 1` (both real edges).
have hside0 := hside (0 : Fin 3) h0ne
have hside1 := hside (1 : Fin 3) h1ne
have h01 : ((0 : Fin 3) + 1) = (1 : Fin 3) := by decide
have h11 : ((1 : Fin 3) + 1) = (2 : Fin 3) := by decide
rw [h01] at hside0
rw [h11] at hside1
-- assemble the apex short arcs in the orientation `flat_interior_joint_absurd` wants.
have hsau : ShortArc (P ⟨0 + 1, by omega⟩ : S2) (P ⟨0, by omega⟩ : S2) := by
have : ShortArc (P (1 : Fin 3)) (P (0 : Fin 3)) := hside0.symm
exact this
have hsav : ShortArc (P ⟨0 + 1, by omega⟩ : S2) (P ⟨0 + 2, by omega⟩ : S2) := by
have : ShortArc (P (1 : Fin 3)) (P (2 : Fin 3)) := hside1
exact this
exact flat_interior_joint_absurd (⟨0, by omega⟩ : Fin (2 - 1)) hsau hsav hcol hjoint0
· -- `0 ∉ hull`: separation gives the strict hemisphere normal directly; normalise to unit length.
obtain ⟨t, ht⟩ := exists_inner_pos_of_zero_notMem_convexHull s.finite_toSet h0
have htpos : ∀ r : Fin (n + 1), 0 < (⟪t, (P r : E3)⟫ : ℝ) := by
intro r
have hrmem : ((P r : S2) : E3) ∈ s := by rw [hmem_iff]; exact ⟨r, rfl⟩
exact ht _ hrmem
have htne : t ≠ 0 := by
intro h
have := htpos ⟨0, by omega⟩
rw [h, inner_zero_left] at this
exact lt_irrefl _ this
have htnorm : 0 < ‖t‖ := norm_pos_iff.mpr htne
refine ⟨(‖t‖⁻¹ : ℝ) • t, ?_, fun r => ?_⟩
· rw [norm_smul, norm_inv, norm_norm]; field_simp
· rw [inner_smul_left]
simp only [RCLike.conj_to_real]
exact mul_pos (by positivity) (htpos r)
/-- **§3 — wrap ShortArc from the open hemisphere.** If `h'` is strictly positive against both wrap
endpoints `P last`, `P 0` (an open-hemisphere certificate) and the two endpoints are distinct
(`hdist`), then the wrap edge `(P last, P 0)` is a `ShortArc`: distinctness is `hdist`, and
non-antipodality is forced because `(P last : E3) = -(P 0 : E3)` would give
`0 < ⟪h', P last⟫ = -⟪h', P 0⟫ < 0`. -/
theorem wrap_shortArc_of_hemisphere {n : ℕ} {P : Fin (n + 1) → S2} {h' : E3}
(hhem : ∀ r : Fin (n + 1), 0 < (⟪h', (P r : E3)⟫ : ℝ))
(hdist : P (Fin.last n) ≠ P 0) :
ShortArc (P (Fin.last n)) (P 0) := by
refine ⟨hdist, fun he => ?_⟩
have hlast : 0 < (⟪h', (P (Fin.last n) : E3)⟫ : ℝ) := hhem (Fin.last n)
have hzero : 0 < (⟪h', (P 0 : E3)⟫ : ℝ) := hhem 0
rw [he, inner_neg_right] at hlast
linarith
/-- **§4 — the opened wraparound edge IS a short arc at the WBS supremum** (the discharge of the
residual content of FFCT46's `OpenedWrapShortArcAtSupWBS`, instantiated). Margins-free,
wrap-edge-free: produced by the open-chain hemisphere on the real-edge data, plus the FFCT43-route
distinctness. -/
theorem openedWrapShortArc_at_supWBS {n : ℕ} {A B : Fin (n + 1) → S2} (hA : StrictConvexSphArm A)
(hB : StrictConvexSphArm B) {k : Fin (n - 1)}
(hka : ShortArc (A (openingAxis k)) (jointPrev A k))
(hkt : ShortArc (A (openingAxis k)) (jointNext A k))
(hkdef : jointAngle A k < jointAngle B k) :
ShortArc (openTail A (openingAxis k) (-(monitoredSupWBS A B k)) (Fin.last n))
(openTail A (openingAxis k) (-(monitoredSupWBS A B k)) 0) := by
set δ : ℝ := monitoredSupWBS A B k with hδ
set K : Fin (n + 1) := openingAxis k with hK
set P : Fin (n + 1) → S2 := openTail A K (-δ) with hP
have hn : 2 ≤ n := hA.two_le
-- real-edge ShortArc edges (the wrap edge `i = Fin.last n` is excluded).
have hside : ∀ i : Fin (n + 1), i ≠ Fin.last n → ShortArc (P i) (P (i + 1)) := by
intro i hi; rw [hP]; exact openTail_nonwrap_shortArc hA K (-δ) hi
-- real-edge weak supports (the FFCT45 supports hold for ALL `(i, j)`; restrict to real `i`).
have hsupp : ∀ i j : Fin (n + 1), i ≠ Fin.last n → j ≠ i → j ≠ i + 1 →
0 ≤ sOrient (P i) (P (i + 1)) (P j) := by
intro i j _hi hji hji1
rw [hP]; exact supportWBS_sOrient_nonneg hA hka hkt hkdef i j hji hji1
-- opened joints in `(0, π)`.
have hjopen : ∀ r : Fin (n - 1), 0 < jointAngle P r ∧ jointAngle P r < Real.pi := by
rw [hP]; exact openedJoints_in_Ioo_at_supWBS hA hB hka hkt hkdef
-- the wrap-edge-free open hemisphere.
obtain ⟨h', _hnorm, hhem⟩ := openHemisphere_full_openChain hn hside hsupp hjopen
-- margins-free distinctness of the wrap endpoints.
have hdist : P (Fin.last n) ≠ P 0 := by
rw [hP]
intro he
exact openedWrap_distinct_at_supWBS hA hka hkt hkdef he.symm
exact wrap_shortArc_of_hemisphere hhem hdist
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
/-- The opened arm `A'_WBS := openTail A (openingAxis k) (-δ*_WBS)` at the WBS supremum (the widening
direction, `= openTailW A (openingAxis k) (monitoredSupWBS A B k)`). All bridge data live on this arm. -/
def openedWBS {n : ℕ} (A B : Fin (n + 1) → S2) (k : Fin (n - 1)) : Fin (n + 1) → S2 :=
openTail A (openingAxis k) (-(monitoredSupWBS A B k))
/-- **Non-antipodality from the open hemisphere.** Two opened vertices `A' a`, `A' b` strictly inside an
open hemisphere (`0 < ⟪h', A' a⟫`, `0 < ⟪h', A' b⟫`) are non-antipodal: `(A' a : E3) = -(A' b : E3)` would
give `0 < ⟪h', A' a⟫ = -⟪h', A' b⟫ < 0`. This is the slot the open hemisphere supplies for `hsa` (the
*non-antipodal* half of `ShortArc`); the *distinctness* half is the named no-repeat residue. -/
theorem hemisphere_nonAntipodal {n : ℕ} {P : Fin (n + 1) → S2} {h' : E3}
(hhem : ∀ r : Fin (n + 1), 0 < (⟪h', (P r : E3)⟫ : ℝ)) (a b : Fin (n + 1)) :
(P a : E3) ≠ -(P b : E3) := by
intro he
have ha : 0 < (⟪h', (P a : E3)⟫ : ℝ) := hhem a
have hb : 0 < (⟪h', (P b : E3)⟫ : ℝ) := hhem b
rw [he, inner_neg_right] at ha
linarith
/-- **The equal first side at a consecutive cut endpoint.** For `i + 1 < j ≤ n` the pair `(i, i+1)` is a
real edge (`i.val < n`), so `SameSides A' B` at index `⟨i⟩ : Fin n` reads as
`sDist (B ⟨i+1⟩)(B ⟨i⟩) = sDist (A' ⟨i+1⟩)(A' ⟨i⟩)` — exactly `StuckAtKData.hside`. -/
theorem hside_of_sameSides {n : ℕ} {A' B : Fin (n + 1) → S2} (hside : SameSides A' B)
{i j : ℕ} (hij1 : i + 1 < j) (hj : j ≤ n) :
sDist (B ⟨i + 1, by omega⟩) (B ⟨i, by omega⟩)
= sDist (A' ⟨i + 1, by omega⟩) (A' ⟨i, by omega⟩) := by
have hilt : i < n := by omega
have hsd : sideLen A' (⟨i, hilt⟩ : Fin n) = sideLen B (⟨i, hilt⟩ : Fin n) := hside ⟨i, hilt⟩
-- `sideLen X ⟨i⟩ = sDist (X ⟨i⟩)(X ⟨i+1⟩)`: castSucc/succ at value `i` are `⟨i⟩`, `⟨i+1⟩`.
have hcs : ∀ X : Fin (n + 1) → S2,
sideLen X (⟨i, hilt⟩ : Fin n) = sDist (X ⟨i, by omega⟩) (X ⟨i + 1, by omega⟩) := by
intro X
have hcast : ((⟨i, hilt⟩ : Fin n).castSucc) = (⟨i, by omega⟩ : Fin (n + 1)) :=
Fin.ext (by simp [Fin.castSucc, Fin.castAdd])
have hsucc : ((⟨i, hilt⟩ : Fin n).succ) = (⟨i + 1, by omega⟩ : Fin (n + 1)) :=
Fin.ext (by simp [Fin.succ])
unfold sideLen
rw [hcast, hsucc]
rw [hcs A', hcs B] at hsd
-- hsd : sDist (A' ⟨i⟩)(A' ⟨i+1⟩) = sDist (B ⟨i⟩)(B ⟨i+1⟩)
-- goal: sDist (B ⟨i+1⟩)(B ⟨i⟩) = sDist (A' ⟨i+1⟩)(A' ⟨i⟩); use sDist_comm + hsd.
rw [sDist_comm (B ⟨i + 1, by omega⟩) (B ⟨i, by omega⟩),
sDist_comm (A' ⟨i + 1, by omega⟩) (A' ⟨i, by omega⟩)]
exact hsd.symm
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
/-- **Distinctness at a normalized cut, derived.** For any arm `P : Fin (n+1) → S2` that is weakly convex
and has no nonadjacent repeat, the normalized cut endpoints `P ⟨i+1⟩`, `P ⟨j⟩` (`i + 1 < j ≤ n`) are
distinct. Adjacent case `j = i + 2`: the edge `(i+1, i+2)` is a `ShortArc` (weak convexity's
`edge_short`), so its two endpoints are distinct. Nonadjacent case `(i+1) + 2 ≤ j`: `NoNonadjacentRepeat`.
This discharges `WBSCutNormalization.hrepeat`. -/
theorem distinctNormalized_of_noRepeat {n : ℕ} {P : Fin (n + 1) → S2}
(hP : WeakConvexSphArm P) (hrep : NoNonadjacentRepeat P) {i j : ℕ}
(hij1 : i + 1 < j) (hj : j ≤ n) :
P ⟨i + 1, by omega⟩ ≠ P ⟨j, by omega⟩ := by
rcases Nat.lt_or_ge (i + 1 + 2) (j + 1) with hadj | hnon
· -- nonadjacent: (i+1) + 2 ≤ j, use NoNonadjacentRepeat.
have hle : (i + 1) + 2 ≤ j := by omega
exact hrep (i + 1) j (by omega) (by omega) hle
· -- adjacent: j = i + 2 (since i+1 < j ≤ i+2). The edge (i+1, i+2) is a real arm edge.
have hjeq : j = i + 2 := by omega
subst hjeq
-- edge_short at the Fin index ⟨i+1⟩: ShortArc (P ⟨i+1⟩) (P (⟨i+1⟩ + 1)).
have hedge : ShortArc (P ⟨i + 1, by omega⟩) (P (⟨i + 1, by omega⟩ + 1)) :=
hP.closed_convex.edge_short ⟨i + 1, by omega⟩
-- (⟨i+1⟩ + 1 : Fin (n+1)) = ⟨i+2⟩ since i + 2 ≤ n < n + 1.
have hsucc : (⟨i + 1, by omega⟩ + 1 : Fin (n + 1)) = ⟨i + 2, by omega⟩ := by
apply Fin.ext
have : ((⟨i + 1, by omega⟩ + 1 : Fin (n + 1)) : ℕ) = (i + 1 + 1) % (n + 1) := by
rw [Fin.add_def]; simp
rw [this]
rw [Nat.mod_eq_of_lt (by omega)]
rw [hsucc] at hedge
exact hedge.1
/-- The Fin reversal `m ↦ ⟨n − m⟩` on `Fin (n+1)`. -/
def revFin {n : ℕ} (m : Fin (n + 1)) : Fin (n + 1) := ⟨n - m.val, by have := m.isLt; omega⟩
@[simp] theorem revFin_val {n : ℕ} (m : Fin (n + 1)) : (revFin m).val = n - m.val := rfl
/-- The reversed arm: `revArm P m = P ⟨n − m⟩`. -/
def revArm {n : ℕ} (P : Fin (n + 1) → S2) : Fin (n + 1) → S2 := fun m => P (revFin m)
/-- `revArm P` at a value index `v ≤ n` reads `P` at `n − v`. -/
theorem revArm_index {n : ℕ} (P : Fin (n + 1) → S2) {v : ℕ} (hv : v < n + 1) :
revArm P ⟨v, hv⟩ = P ⟨n - v, by omega⟩ := rfl
/-- **`sOrient` reversal at a normalized triple.** For a raw binding with `b < a` (so `a + 1 ≤ n`), the
normalized reversed triple `(revArm P ⟨n−a−1⟩, revArm P ⟨n−a⟩, revArm P ⟨n−b⟩)` reads as
`(P ⟨a+1⟩, P ⟨a⟩, P ⟨b⟩)`, whose `sOrient` is `−sOrient (P ⟨a⟩)(P ⟨a+1⟩)(P ⟨b⟩)` (slot-1-2 swap,
`det3_swap12`). Hence a *vanishing* raw support gives a *vanishing* normalized reversed support. -/
theorem sOrient_revArm_normalized {n : ℕ} (P : Fin (n + 1) → S2) {a b : ℕ}
(ha1 : a + 1 < n + 1) (hb : b < a) :
sOrient (revArm P ⟨n - a - 1, by omega⟩) (revArm P ⟨n - a, by omega⟩)
(revArm P ⟨n - b, by omega⟩)
= - sOrient (P ⟨a, by omega⟩) (P ⟨a + 1, by omega⟩) (P ⟨b, by omega⟩) := by
-- evaluate the reversed indices: n−(n−a−1) = a+1, n−(n−a) = a, n−(n−b) = b.
rw [revArm_index P (by omega), revArm_index P (by omega), revArm_index P (by omega)]
have e1 : n - (n - a - 1) = a + 1 := by omega
have e2 : n - (n - a) = a := by omega
have e3 : n - (n - b) = b := by omega
rw [show (⟨n - (n - a - 1), by omega⟩ : Fin (n + 1)) = ⟨a + 1, by omega⟩ from Fin.ext (by omega),
show (⟨n - (n - a), by omega⟩ : Fin (n + 1)) = ⟨a, by omega⟩ from Fin.ext (by omega),
show (⟨n - (n - b), by omega⟩ : Fin (n + 1)) = ⟨b, by omega⟩ from Fin.ext (by omega)]
-- sOrient (P(a+1))(P a)(P b) = -sOrient (P a)(P(a+1))(P b) by slot-1-2 swap.
exact det3_swap12 _ _ _
/-- **`sideLen` reversal.** Side `i` of `revArm P` (`i : Fin n`) is the parent's side `n − 1 − i` with its
two endpoints swapped; `sDist_comm` makes them equal. Concretely `sideLen (revArm P) i =
sDist (P ⟨n−i⟩)(P ⟨n−i−1⟩) = sDist (P ⟨n−i−1⟩)(P ⟨n−i⟩) = sideLen P ⟨n−1−i⟩`. -/
theorem revArm_sideLen {n : ℕ} (P : Fin (n + 1) → S2) (i : Fin n) :
sideLen (revArm P) i = sideLen P ⟨n - 1 - i.val, by have := i.isLt; omega⟩ := by
have hi := i.isLt
unfold sideLen
-- index equalities (as Fin equalities, so rewriting is motive-safe).
have l0 : revArm P i.castSucc = P ⟨n - i.val, by omega⟩ := by
show P (revFin i.castSucc) = _
exact congrArg P (Fin.ext (by simp only [revFin_val, Fin.val_castSucc]))
have l1 : revArm P i.succ = P ⟨n - i.val - 1, by omega⟩ := by
show P (revFin i.succ) = _
exact congrArg P (Fin.ext (by simp only [revFin_val, Fin.val_succ]; omega))
have r0 : P ((⟨n - 1 - i.val, by omega⟩ : Fin n).castSucc) = P ⟨n - i.val - 1, by omega⟩ :=
congrArg P (Fin.ext (by simp only [Fin.val_castSucc]; omega))
have r1 : P ((⟨n - 1 - i.val, by omega⟩ : Fin n).succ) = P ⟨n - i.val, by omega⟩ :=
congrArg P (Fin.ext (by simp only [Fin.val_succ]; omega))
rw [l0, l1, r0, r1]
-- goal: sDist (P ⟨n−i⟩)(P ⟨n−i−1⟩) = sDist (P ⟨n−i−1⟩)(P ⟨n−i⟩).
exact sDist_comm _ _
/-- **`jointAngle` reversal.** Interior joint `i` of `revArm P` (`i : Fin (n−1)`) is the parent's interior
joint `n − 2 − i` read backwards; `sphAngle_comm` (swap the two neighbours) makes the value invariant. -/
theorem revArm_jointAngle {n : ℕ} (P : Fin (n + 1) → S2) (i : Fin (n - 1)) :
jointAngle (revArm P) i = jointAngle P ⟨n - 2 - i.val, by have := i.isLt; omega⟩ := by
have hi := i.isLt
unfold jointAngle
-- LHS = sphAngle (revArm P ⟨i⟩)(revArm P ⟨i+1⟩)(revArm P ⟨i+2⟩); each reads P at n−i, n−i−1, n−i−2.
have v0 : (revArm P ⟨i.val, by omega⟩ : S2) = P ⟨n - 2 - i.val + 2, by omega⟩ :=
congrArg P (Fin.ext (by simp only [revFin_val]; omega))
have v1 : (revArm P ⟨i.val + 1, by omega⟩ : S2) = P ⟨n - 2 - i.val + 1, by omega⟩ :=
congrArg P (Fin.ext (by simp only [revFin_val]; omega))
have v2 : (revArm P ⟨i.val + 2, by omega⟩ : S2) = P ⟨n - 2 - i.val, by omega⟩ :=
congrArg P (Fin.ext (by simp only [revFin_val]; omega))
-- rewrite each LHS vertex; LHS becomes sphAngle (P j+2)(P j+1)(P j) with j = n−2−i.
show sphAngle (revArm P ⟨i.val, _⟩) (revArm P ⟨i.val + 1, _⟩) (revArm P ⟨i.val + 2, _⟩) = _
rw [v0, v1, v2, sphAngle_comm]
/-- The no-nonadjacent-repeat fact transports under reversal: a nonadjacent repeat of `revArm P` at
`r + 2 ≤ s` is a nonadjacent repeat of `P` at the reversed indices `n − s + 2 ≤ n − r`. -/
theorem revArm_noNonadjacentRepeat {n : ℕ} {P : Fin (n + 1) → S2}
(hrep : NoNonadjacentRepeat P) : NoNonadjacentRepeat (revArm P) := by
intro r s hr hs hrs he
-- revArm P ⟨r⟩ = P ⟨n−r⟩, revArm P ⟨s⟩ = P ⟨n−s⟩; he : P ⟨n−r⟩ = P ⟨n−s⟩.
rw [revArm_index P hr, revArm_index P hs] at he
-- nonadjacent on P: (n−s) + 2 ≤ n−r since r + 2 ≤ s.
exact hrep (n - s) (n - r) (by omega) (by omega) (by omega) he.symm
/-- **The orientation-normalized vanishing support, both branches.** From the raw Fin binding
`sOrient (P a)(P (a+1))(P b) = 0` with `b ≠ a` and `b ≠ a+1` (as Fin), the normalized cut exists:
* if `a.val + 1 < b.val`: on `P` itself, `(i, j) = (a, b)`, support unchanged;
* if `b.val < a.val`: on `revArm P`, `(i, j) = (n − a − 1, n − b)`, support flipped (`= 0` is sign-free);
in both branches `i + 1 < j ≤ n` and `sOrient (Q ⟨i⟩)(Q ⟨i+1⟩)(Q ⟨j⟩) = 0` for the chosen arm `Q`. -/
theorem orientationNormalized {n : ℕ} (P : Fin (n + 1) → S2) {a b : Fin (n + 1)}
(hne : b ≠ a) (hne1 : b ≠ a + 1)
(hsupp : sOrient (P a) (P (a + 1)) (P b) = 0)
(hadj : a.val + 1 < n + 1) :
(∃ i j : ℕ, ∃ (hij1 : i + 1 < j) (hj : j ≤ n),
sOrient (P ⟨i, by omega⟩) (P ⟨i + 1, by omega⟩) (P ⟨j, by omega⟩) = 0)
∨ (∃ i j : ℕ, ∃ (hij1 : i + 1 < j) (hj : j ≤ n),
sOrient (revArm P ⟨i, by omega⟩) (revArm P ⟨i + 1, by omega⟩) (revArm P ⟨j, by omega⟩) = 0) := by
have hai := a.isLt
have hbi := b.isLt
-- the Fin (a+1) reads as value a.val + 1 (since a.val + 1 < n + 1, no wrap).
have hav : a.val + 1 < n + 1 := hadj
have hfsucc : (a + 1 : Fin (n + 1)) = ⟨a.val + 1, hav⟩ := by
apply Fin.ext
have : ((a + 1 : Fin (n + 1)) : ℕ) = (a.val + 1) % (n + 1) := by rw [Fin.add_def]; simp
rw [this, Nat.mod_eq_of_lt hav]
-- ℕ-disequalities from the Fin ones.
have hbne : b.val ≠ a.val := fun h => hne (Fin.ext h)
have hbne1 : b.val ≠ a.val + 1 := by
intro h; apply hne1; rw [hfsucc]; exact Fin.ext h
rcases Nat.lt_or_ge (a.val + 1) b.val with hgt | hle
· -- a.val + 1 < b.val: normalize on P directly with (i, j) = (a.val, b.val).
left
refine ⟨a.val, b.val, hgt, by have := b.isLt; omega, ?_⟩
-- rewrite the goal's Fin indices to a, a+1, b.
rw [show (⟨a.val, by omega⟩ : Fin (n + 1)) = a from Fin.ext rfl,
show (⟨a.val + 1, by omega⟩ : Fin (n + 1)) = a + 1 from hfsucc.symm,
show (⟨b.val, by omega⟩ : Fin (n + 1)) = b from Fin.ext rfl]
exact hsupp
· -- b.val < a.val (since b ≠ a, b ≠ a+1 and ¬ a+1 < b): normalize on revArm P.
have hblt : b.val < a.val := by omega
right
refine ⟨n - a.val - 1, n - b.val, by omega, by have := b.isLt; omega, ?_⟩
-- the reversed normalized support = -(raw support) = 0.
have hrev := sOrient_revArm_normalized P (a := a.val) (b := b.val) (by omega) hblt
-- align the goal's middle index `(n-a-1)+1` to `n-a` (the form `hrev` uses).
have hmid : (⟨(n - a.val - 1) + 1, by omega⟩ : Fin (n + 1)) = ⟨n - a.val, by omega⟩ :=
Fin.ext (show (n - a.val - 1) + 1 = n - a.val by omega)
rw [hmid, hrev]
-- raw support in value form = hsupp after aligning a, a+1, b.
rw [show (⟨a.val, by omega⟩ : Fin (n + 1)) = a from Fin.ext rfl,
show (⟨a.val + 1, by omega⟩ : Fin (n + 1)) = a + 1 from hfsucc.symm,
show (⟨b.val, by omega⟩ : Fin (n + 1)) = b from Fin.ext rfl]
rw [hsupp]; ring
/-- The wrap-edge data of the ear `intervalArm A a m` (the diagonal chord `(A ⟨a+m⟩, A ⟨a⟩)` and its
supports): a `ShortArc` on the wrap chord, and the nonnegativity of every support whose base is the wrap
edge `(A ⟨a+m⟩, A ⟨a⟩)`. These are exactly the certificates the parent's weak convexity at an *arbitrary*
interval does NOT supply (the wrap chord is a diagonal, not a parent edge); everything else restricts. -/
structure IntervalWrapData {N : ℕ} (A : Fin (N + 1) → S2) (a m : ℕ) (hb : a + m ≤ N) : Prop where
/-- The wrap (diagonal) edge is a short arc. -/
wrap_short : ShortArc (A ⟨a + m, by omega⟩) (A ⟨a, by omega⟩)
/-- Every vertex is supported on the nonnegative side of the oriented wrap edge. -/
wrap_support : ∀ v : ℕ, (hv : v < m + 1) →
0 ≤ sOrient (A ⟨a + m, by omega⟩) (A ⟨a, by omega⟩) (A ⟨a + v, by have := hv; omega⟩)
/-- **Interior edges of the ear are parent edges, hence short.** For a strictly/weakly convex parent and
an interior ear edge index `t < m`, the ear edge `(ear ⟨t⟩, ear ⟨t+1⟩) = (A ⟨a+t⟩, A ⟨a+t+1⟩)` is the
parent edge `t' = a + t`, which is a `ShortArc` (parent `edge_short`). -/
theorem intervalArm_interiorEdgeShort {N : ℕ} {A : Fin (N + 1) → S2} (hA : WeakConvexSphArm A)
{a m : ℕ} (hb : a + m ≤ N) {t : ℕ} (ht : t < m) :
ShortArc (A ⟨a + t, by omega⟩) (A ⟨a + t + 1, by omega⟩) := by
have hedge := hA.closed_convex.edge_short ⟨a + t, by omega⟩
-- (⟨a+t⟩ + 1 : Fin (N+1)) = ⟨a+t+1⟩ since a+t+1 ≤ N < N+1.
have hsucc : (⟨a + t, by omega⟩ + 1 : Fin (N + 1)) = ⟨a + t + 1, by omega⟩ := by
apply Fin.ext
have : ((⟨a + t, by omega⟩ + 1 : Fin (N + 1)) : ℕ) = (a + t + 1) % (N + 1) := by
rw [Fin.add_def]; simp
rw [this, Nat.mod_eq_of_lt (by omega)]
rwa [hsucc] at hedge
/-- **Interior-base supports of the ear are parent supports, hence ≥ 0.** For a weakly convex parent, an
interior ear edge `(A ⟨a+t⟩, A ⟨a+t+1⟩)` (`t < m`) supports any ear vertex `A ⟨a+v⟩` (`v ≤ m`) on the
nonnegative side, because it is the parent non-incident support `sOrient (A ⟨a+t⟩)(A ⟨a+t+1⟩)(A ⟨a+v⟩) ≥ 0`
(parent `edge_support`). -/
theorem intervalArm_interiorSupport {N : ℕ} {A : Fin (N + 1) → S2} (hA : WeakConvexSphArm A)
{a m : ℕ} (hb : a + m ≤ N) {t v : ℕ} (ht : t < m) (hv : v < m + 1) :
0 ≤ sOrient (A ⟨a + t, by omega⟩) (A ⟨a + t + 1, by omega⟩) (A ⟨a + v, by omega⟩) := by
have hsupp := hA.closed_convex.edge_support ⟨a + t, by omega⟩ ⟨a + v, by omega⟩
have hsucc : (⟨a + t, by omega⟩ + 1 : Fin (N + 1)) = ⟨a + t + 1, by omega⟩ := by
apply Fin.ext
have : ((⟨a + t, by omega⟩ + 1 : Fin (N + 1)) : ℕ) = (a + t + 1) % (N + 1) := by
rw [Fin.add_def]; simp
rw [this, Nat.mod_eq_of_lt (by omega)]
rwa [hsucc] at hsupp
/-- **The open hemisphere restricts to the ear.** The parent's open-hemisphere normal `h` works verbatim
for the ear (the ear vertices are a subset of the parent vertices). -/
theorem intervalArm_openHemisphere {N : ℕ} {A : Fin (N + 1) → S2} (hA : WeakConvexSphArm A)
{a m : ℕ} (hb : a + m ≤ N) :
∃ h : E3, ‖h‖ = 1 ∧ ∀ x : Fin (m + 1), 0 < (⟪h, (intervalArm A a m hb x : E3)⟫ : ℝ) := by
obtain ⟨h, hnorm, hhem⟩ := hA.closed_convex.open_hemisphere
exact ⟨h, hnorm, fun x => hhem _⟩
/-- **Component 3, sharpened.** The ear `intervalArm A a m` (`2 ≤ m`, `a + m ≤ N`) of a weakly convex
parent `A` is itself weakly convex PROVIDED the wrap-edge data `IntervalWrapData` (the diagonal chord's
`ShortArc` and the nonnegativity of the supports based at the diagonal). All other fields restrict from
the parent.
This isolates the genuine Component-3 residue (the wrap diagonal) and discharges the rest. -/
theorem weakConvex_intervalArm_of_wrap {N : ℕ} {A : Fin (N + 1) → S2} (hA : WeakConvexSphArm A)
{a m : ℕ} (hm : 2 ≤ m) (hb : a + m ≤ N)
(hwrap : IntervalWrapData A a m hb) :
WeakConvexSphArm (intervalArm A a m hb) := by
have hNz : NeZero (m + 1) := ⟨by omega⟩
refine ⟨hm, ?_⟩
refine ⟨by omega, ?_, ?_, ?_⟩
· -- edge_short for every cyclic edge `i : Fin (m+1)`: interior (i < m) parent edge, or wrap (i = m).
intro i
by_cases hi : i.val < m
· -- interior edge: (ear i, ear (i+1)) = (A ⟨a+i⟩, A ⟨a+i+1⟩).
have hsucc : (i + 1 : Fin (m + 1)) = ⟨i.val + 1, by have := i.isLt; omega⟩ := by
apply Fin.ext
have : ((i + 1 : Fin (m + 1)) : ℕ) = (i.val + 1) % (m + 1) := by rw [Fin.add_def]; simp
rw [this, Nat.mod_eq_of_lt (by omega)]
rw [intervalArm_apply, intervalArm_apply, hsucc]
have := intervalArm_interiorEdgeShort hA hb (t := i.val) hi
-- align ⟨a + (i+1).val⟩ to ⟨a + i + 1⟩.
simpa only [show a + (⟨i.val + 1, by have := i.isLt; omega⟩ : Fin (m + 1)).val = a + i.val + 1 from rfl]
using this
· -- wrap edge: i.val = m, (i+1) wraps to 0.
have him : i.val = m := by have := i.isLt; omega
have hi0 : (i + 1 : Fin (m + 1)) = 0 := by
apply Fin.ext
have : ((i + 1 : Fin (m + 1)) : ℕ) = (i.val + 1) % (m + 1) := by rw [Fin.add_def]; simp
rw [this, him]; simp [Nat.mod_self]
rw [intervalArm_apply, hi0, intervalArm_apply]
simp only [Fin.val_zero, Nat.add_zero]
have halign : (A ⟨a + i.val, by have := i.isLt; omega⟩ : S2) = A ⟨a + m, by omega⟩ :=
congrArg A (Fin.ext (by simp only [him]))
rw [halign]
exact hwrap.wrap_short
· -- edge_support for every base edge `i` and vertex `j`.
intro i j
by_cases hi : i.val < m
· -- interior base edge: parent support.
have hsucc : (i + 1 : Fin (m + 1)) = ⟨i.val + 1, by have := i.isLt; omega⟩ := by
apply Fin.ext
have : ((i + 1 : Fin (m + 1)) : ℕ) = (i.val + 1) % (m + 1) := by rw [Fin.add_def]; simp
rw [this, Nat.mod_eq_of_lt (by omega)]
rw [intervalArm_apply, intervalArm_apply, intervalArm_apply, hsucc]
have := intervalArm_interiorSupport hA hb (t := i.val) (v := j.val) hi j.isLt
simpa only [show a + (⟨i.val + 1, by have := i.isLt; omega⟩ : Fin (m + 1)).val = a + i.val + 1 from rfl]
using this
· -- wrap base edge: i.val = m, base = (A ⟨a+m⟩, A ⟨a⟩); use wrap_support at v = j.val.
have him : i.val = m := by have := i.isLt; omega
have hi0 : (i + 1 : Fin (m + 1)) = 0 := by
apply Fin.ext
have : ((i + 1 : Fin (m + 1)) : ℕ) = (i.val + 1) % (m + 1) := by rw [Fin.add_def]; simp
rw [this, him]; simp [Nat.mod_self]
rw [intervalArm_apply, hi0, intervalArm_apply, intervalArm_apply]
simp only [Fin.val_zero, Nat.add_zero]
have halign : (A ⟨a + i.val, by have := i.isLt; omega⟩ : S2) = A ⟨a + m, by omega⟩ :=
congrArg A (Fin.ext (by simp only [him]))
rw [halign]
exact hwrap.wrap_support j.val j.isLt
· -- open hemisphere restricts.
obtain ⟨h, hnorm, hhem⟩ := intervalArm_openHemisphere hA hb (a := a) (m := m)
exact ⟨h, hnorm, hhem⟩
/-- **Interior strict supports restrict.** For a *strictly* convex parent, an interior ear edge
`(A ⟨a+t⟩, A ⟨a+t+1⟩)` (`t < m`) supports a non-incident ear vertex `A ⟨a+v⟩` strictly on the positive
side, whenever `a+v` is non-incident to the parent edge `a+t` (i.e. `a+v ≠ a+t` and `a+v ≠ a+t+1`). -/
theorem intervalArm_interiorStrictSupport {N : ℕ} {A : Fin (N + 1) → S2} (hA : StrictConvexSphArm A)
{a m : ℕ} (hb : a + m ≤ N) {t v : ℕ} (ht : t < m) (hv : v < m + 1)
(hne : v ≠ t) (hne1 : v ≠ t + 1) :
0 < sOrient (A ⟨a + t, by omega⟩) (A ⟨a + t + 1, by omega⟩) (A ⟨a + v, by omega⟩) := by
have hj1 : (⟨a + v, by omega⟩ : Fin (N + 1)) ≠ ⟨a + t, by omega⟩ := by
intro h; apply hne
have : a + v = a + t := congrArg Fin.val h
omega
-- the parent edge `a+t` and its `+1` as Fin: align `(⟨a+t⟩ + 1)` to `⟨a+t+1⟩`.
have hsucc : (⟨a + t, by omega⟩ + 1 : Fin (N + 1)) = ⟨a + t + 1, by omega⟩ := by
apply Fin.ext
have : ((⟨a + t, by omega⟩ + 1 : Fin (N + 1)) : ℕ) = (a + t + 1) % (N + 1) := by
rw [Fin.add_def]; simp
rw [this, Nat.mod_eq_of_lt (by omega)]
have hj2 : (⟨a + v, by omega⟩ : Fin (N + 1)) ≠ (⟨a + t, by omega⟩ : Fin (N + 1)) + 1 := by
rw [hsucc]; intro h; apply hne1
have : a + v = a + t + 1 := congrArg Fin.val h
omega
have hstr := hA.closed_convex.strict_nonincident ⟨a + t, by omega⟩ ⟨a + v, by omega⟩ hj1 hj2
rwa [hsucc] at hstr
/-- The strict-ear wrap data: as `IntervalWrapData`, plus the *strict* support of every non-incident
vertex against the wrap (diagonal) edge. This is the only genuinely-new strict field for the ear; the
interior strict non-incidence restricts from the parent. -/
structure IntervalWrapDataStrict {N : ℕ} (A : Fin (N + 1) → S2) (a m : ℕ) (hb : a + m ≤ N) : Prop where
toWeak : IntervalWrapData A a m hb
/-- The wrap (diagonal) edge supports every NON-incident vertex strictly. Non-incidence against the
cyclic wrap edge `(vertex m, vertex 0)` means `v ≠ m` (not the base tail) and `v ≠ 0`... but vertex `0`
IS the wrap edge's head, so the non-incident condition for the wrap edge `(m, 0)` is `v ≠ m ∧ v ≠ 0`.
-/
wrap_strict : ∀ v : ℕ, (hv : v < m + 1) → v ≠ m → v ≠ 0 →
0 < sOrient (A ⟨a + m, by omega⟩) (A ⟨a, by omega⟩) (A ⟨a + v, by have := hv; omega⟩)
/-- **Component 3, strict version.** The ear `intervalArm A a m` (`2 ≤ m`) of a *strictly* convex parent
`A` is strictly convex GIVEN the strict wrap data. The interior strict non-incidences restrict from the
parent; the wrap base's strict non-incidence is the residue (`wrap_strict`). -/
theorem strictConvex_intervalArm_of_wrap {N : ℕ} {A : Fin (N + 1) → S2} (hA : StrictConvexSphArm A)
{a m : ℕ} (hm : 2 ≤ m) (hb : a + m ≤ N)
(hwrap : IntervalWrapDataStrict A a m hb) :
StrictConvexSphArm (intervalArm A a m hb) := by
have hNz : NeZero (m + 1) := ⟨by omega⟩
-- reuse the weak assembly for edge_short / edge_support / open_hemisphere.
have hweak := weakConvex_intervalArm_of_wrap (strictConvexSphArm_toWeak hA) hm hb hwrap.toWeak
refine ⟨hm, ?_⟩
refine ⟨by omega, hweak.closed_convex.edge_short, hweak.closed_convex.edge_support, ?_,
hweak.closed_convex.open_hemisphere⟩
-- strict_nonincident: tested vertex `j` non-incident to base edge `i`.
intro i j hji hji1
by_cases hi : i.val < m
· -- interior base edge: parent strict non-incidence at (a+i, a+j).
have hsucc : (i + 1 : Fin (m + 1)) = ⟨i.val + 1, by have := i.isLt; omega⟩ := by
apply Fin.ext
have : ((i + 1 : Fin (m + 1)) : ℕ) = (i.val + 1) % (m + 1) := by rw [Fin.add_def]; simp
rw [this, Nat.mod_eq_of_lt (by omega)]
-- ear non-incidence j ≠ i, j ≠ i+1 transfers to value non-incidence v ≠ t, v ≠ t+1.
have hvne : j.val ≠ i.val := fun h => hji (Fin.ext h)
have hvne1 : j.val ≠ i.val + 1 := by
intro h; apply hji1; rw [hsucc]; exact Fin.ext h
rw [intervalArm_apply, intervalArm_apply, intervalArm_apply, hsucc]
have := intervalArm_interiorStrictSupport hA hb (t := i.val) (v := j.val) hi j.isLt hvne hvne1
simpa only [show a + (⟨i.val + 1, by have := i.isLt; omega⟩ : Fin (m + 1)).val = a + i.val + 1 from rfl]
using this
· -- wrap base edge: i.val = m, base = (A ⟨a+m⟩, A ⟨a⟩); use wrap_strict at v = j.val (j ≠ m, j ≠ 0).
have him : i.val = m := by have := i.isLt; omega
have hi0 : (i + 1 : Fin (m + 1)) = 0 := by
apply Fin.ext
have : ((i + 1 : Fin (m + 1)) : ℕ) = (i.val + 1) % (m + 1) := by rw [Fin.add_def]; simp
rw [this, him]; simp [Nat.mod_self]
-- j ≠ i means j.val ≠ m; j ≠ i+1 = 0 means j.val ≠ 0.
have hjm : j.val ≠ m := by intro h; apply hji; apply Fin.ext; rw [him]; exact h
have hj0 : j.val ≠ 0 := by intro h; apply hji1; rw [hi0]; apply Fin.ext; simp [h]
rw [intervalArm_apply, hi0, intervalArm_apply, intervalArm_apply]
simp only [Fin.val_zero, Nat.add_zero]
have halign : (A ⟨a + i.val, by have := i.isLt; omega⟩ : S2) = A ⟨a + m, by omega⟩ :=
congrArg A (Fin.ext (by simp only [him]))
rw [halign]
exact hwrap.wrap_strict j.val j.isLt hjm hj0
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
/-- **(Brick 2) Adjacent fold contradicts `PositiveJoints`.** When `j = i + 2` the betweenness
`A i ∈ span≥0 {A (i+1), A (i+2)}` puts the cut vertex `A i` on the minor arc between the apex's two
neighbours, forcing the interior joint at index `i` (apex `A (i+1)`) to angle `0`
(`lastCorner_hcol_forces_joint_zero`), contradicting `PositiveJoints A`. -/
theorem foldedFlat_adjacent_contradiction {n : ℕ} {A : Fin (n + 1) → S2}
(hA : WeakConvexSphArm A) (hpos : PositiveJoints A)
{i : ℕ} (hi2 : i + 2 < n + 1)
(hcol : (A ⟨i, by omega⟩ : E3) ∈
Submodule.span NNReal
({(A ⟨i + 1, by omega⟩ : E3), (A ⟨i + 2, hi2⟩ : E3)} : Set E3)) :
False := by
-- short edge `(A (i+1), A i)` from the arm edge `(A i, A (i+1))` reversed.
have hedge : ShortArc (A ⟨i, by omega⟩) (A ⟨i + 1, by omega⟩) := by
have he := hA.closed_convex.edge_short ⟨i, by omega⟩
have hsucc : ((⟨i, by omega⟩ : Fin (n + 1)) + 1) = (⟨i + 1, by omega⟩ : Fin (n + 1)) := by
apply Fin.ext
have hone : ((1 : Fin (n + 1)) : ℕ) = 1 := by
rw [Fin.val_one']; exact Nat.mod_eq_of_lt (by omega)
rw [Fin.val_add, Fin.val_mk, hone, Nat.mod_eq_of_lt (show i + 1 < n + 1 by omega)]
rwa [hsucc] at he
have hsa : ShortArc (A ⟨i + 1, by omega⟩) (A ⟨i, by omega⟩) := hedge.symm
-- the betweenness forces `jointAngle A ⟨i⟩ = 0`.
have hzero : jointAngle A ⟨i, by omega⟩ = 0 :=
lastCorner_hcol_forces_joint_zero (N := n) (A := A) (k := i) (by omega) hsa hcol
-- contradict `PositiveJoints`.
have hposi := hpos ⟨i, by omega⟩
rw [hzero] at hposi
exact lt_irrefl 0 hposi
/-- **The `(0, n-1)` tail-fold premise** (design §8 residue). At the `(0, n-1)` boundary fold, the
last vertex `A (last)` is folded onto the ray from `A 0` to `A ⟨n-1⟩`:
`sDist (A 0)(A ⟨n-1⟩) = endpt A + sDist (A last)(A ⟨n-1⟩)`. Genuinely exceeds the `(0, n-1)`
betweenness (`A 0 ∈ span≥0 {A 1, A ⟨n-1⟩}`); the design §8 master brick produces it from the two
boundary edge supports + the positive-coefficient fold datum. Named, satisfiable. -/
def TailFoldBoundary {n : ℕ} (A : Fin (n + 1) → S2) (hn1 : n - 1 < n + 1) : Prop :=
sDist (A ⟨0, by omega⟩) (A ⟨n - 1, hn1⟩)
= endpt A + sDist (A (Fin.last n)) (A ⟨n - 1, hn1⟩)
/-- **(Brick 5) The `(0, n-1)` boundary transport.** From the named tail-fold premise, the diagonal
inequality `sDist (A 0)(A ⟨n-1⟩) ≤ sDist (B 0)(B ⟨n-1⟩)`, and the equal last side
(`SameSides` at side `n-1`), `ZinanFFCT18.endpoint_le_of_tail_fold` gives `endpt A ≤ endpt B`. -/
theorem foldedFlat_boundary_j_eq_n_minus_one {n : ℕ} {A B : Fin (n + 1) → S2}
(hn : 2 ≤ n)
(hside : SameSides A B)
(htail : TailFoldBoundary A (by omega))
(hdiag : sDist (A ⟨0, by omega⟩) (A ⟨n - 1, by omega⟩)
≤ sDist (B ⟨0, by omega⟩) (B ⟨n - 1, by omega⟩)) :
endpt A ≤ endpt B := by
-- the tail-fold equation in the `endpoint_le_of_tail_fold` shape.
have hflat : sDist (A ⟨0, by omega⟩) (A ⟨n - 1, by omega⟩)
= sDist (A ⟨0, by omega⟩) (A (Fin.last n)) + sDist (A (Fin.last n)) (A ⟨n - 1, by omega⟩) := by
have h := htail
rw [TailFoldBoundary, endpt] at h
-- `endpt A = sDist (A 0)(A last)`; reconcile the `⟨0,_⟩` and `0` indices.
have h00 : (A ⟨0, by omega⟩ : S2) = A 0 := by congr 1
rw [h00] at h ⊢
linarith [h]
-- the equal last side: `sDist (A last)(A ⟨n-1⟩) = sDist (B last)(B ⟨n-1⟩)`.
have hs := hside ⟨n - 1, by omega⟩
-- `(⟨n-1,_⟩ : Fin n).castSucc = ⟨n-1,_⟩` and `.succ = ⟨n,_⟩` (values, with `hn`).
have hcast : ((⟨n - 1, by omega⟩ : Fin n).castSucc) = (⟨n - 1, by omega⟩ : Fin (n + 1)) :=
Fin.ext (by simp)
have hsucc : ((⟨n - 1, by omega⟩ : Fin n).succ) = (⟨n, by omega⟩ : Fin (n + 1)) :=
Fin.ext (by simp; omega)
have hsA : sideLen A (⟨n - 1, by omega⟩ : Fin n)
= sDist (A ⟨n - 1, by omega⟩) (A ⟨n, by omega⟩) := by
rw [sideLen, hcast, hsucc]
have hsB : sideLen B (⟨n - 1, by omega⟩ : Fin n)
= sDist (B ⟨n - 1, by omega⟩) (B ⟨n, by omega⟩) := by
rw [sideLen, hcast, hsucc]
rw [hsA, hsB] at hs
-- `A (Fin.last n) = A ⟨n, _⟩`, then symmetrise to the `(last, n-1)` orientation.
have hlastA : (A (Fin.last n) : S2) = A ⟨n, by omega⟩ := by congr 1
have hlastB : (B (Fin.last n) : S2) = B ⟨n, by omega⟩ := by congr 1
have hsideLast : sDist (A (Fin.last n)) (A ⟨n - 1, by omega⟩)
= sDist (B (Fin.last n)) (B ⟨n - 1, by omega⟩) := by
rw [hlastA, hlastB, sDist_comm (A ⟨n, by omega⟩) (A ⟨n - 1, by omega⟩),
sDist_comm (B ⟨n, by omega⟩) (B ⟨n - 1, by omega⟩)]
exact hs
-- assemble via the landed FFCT18 arithmetic.
have hkey := endpoint_le_of_tail_fold (A0 := A ⟨0, by omega⟩) (An1 := A ⟨n - 1, by omega⟩)
(An := A (Fin.last n)) (B0 := B ⟨0, by omega⟩) (Bn1 := B ⟨n - 1, by omega⟩)
(Bn := B (Fin.last n)) hflat hdiag hsideLast
-- `hkey : sDist (A 0)(A last) ≤ sDist (B 0)(B last)`, i.e. `endpt A ≤ endpt B`.
have heA : endpt A = sDist (A ⟨0, by omega⟩) (A (Fin.last n)) := by
rw [endpt]; congr 1
have heB : endpt B = sDist (B ⟨0, by omega⟩) (B (Fin.last n)) := by
rw [endpt]; congr 1
rw [heA, heB]; exact hkey
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
/-- **`endpt` is reversal-invariant.** `endpt (revArm A) = sDist (revArm A 0)(revArm A (last)) =
sDist (A n)(A 0) = sDist (A 0)(A n) = endpt A` (`sDist_comm`). -/
theorem endpt_revArm {n : ℕ} (A : Fin (n + 1) → S2) : endpt (revArm A) = endpt A := by
unfold endpt
have h0 : (revArm A 0 : S2) = A (Fin.last n) := by
show A (revFin 0) = _
exact congrArg A (Fin.ext (by simp [revFin_val, Fin.last]))
have hl : (revArm A (Fin.last n) : S2) = A 0 := by
show A (revFin (Fin.last n)) = _
exact congrArg A (Fin.ext (by simp [revFin_val, Fin.last]))
rw [h0, hl, sDist_comm]
/-- **The det3 row-expansion identity.** With `a•A 1 + b•A n = A 0`,
`a·det3 (A n)(A 1)(A m) = det3 (A n)(A 0)(A m)` (the `b•A n` term vanishes, `det3 x x y = 0`). -/
theorem det3_rowExpand_wrap {a b : ℝ} {An A1 An0 Am : E3}
(hcoeff : a • A1 + b • An = An0) :
a * det3 An A1 Am = det3 An An0 Am := by
rw [← hcoeff, det3_add_mid, det3_smul_mid, det3_smul_mid]
have hself : det3 An An Am = 0 := by simp only [det3]; ring
rw [hself, mul_zero, add_zero]
/-- **The A-side wrap supports are derivable from the `(0, n)` betweenness.** Given the nondegenerate
decomposition `a•A 1 + b•A n = A 0` (`0 < a`) and the parent weak convexity, every interior vertex
`A ⟨1 + v⟩` (`v < n`) is on the nonnegative side of the wrap diagonal `(A n, A 1)`:
`0 ≤ sOrient (A n)(A 1)(A ⟨1+v⟩)`. The parent wrap-edge `(n, 0)` support
`0 ≤ sOrient (A n)(A 0)(A ⟨1+v⟩)` row-expands through the betweenness. -/
theorem intervalWrap_support_of_betweenness {n : ℕ} {A : Fin (n + 1) → S2}
(hA : WeakConvexSphArm A) (hn : 2 ≤ n)
{a b : ℝ} (hapos : 0 < a)
(hcoeff : a • (A ⟨1, by omega⟩ : E3) + b • (A ⟨n, by omega⟩ : E3) = (A ⟨0, by omega⟩ : E3))
{v : ℕ} (hv : v < n) :
0 ≤ sOrient (A ⟨n, by omega⟩) (A ⟨1, by omega⟩) (A ⟨1 + v, by omega⟩) := by
-- parent wrap-edge `(n, 0)` support: edge index `⟨n⟩ : Fin (n+1)`, its `+1` wraps to `⟨0⟩`.
have hwrapEdge : (⟨n, by omega⟩ + 1 : Fin (n + 1)) = (⟨0, by omega⟩ : Fin (n + 1)) := by
apply Fin.ext
have : ((⟨n, by omega⟩ + 1 : Fin (n + 1)) : ℕ) = (n + 1) % (n + 1) := by
rw [Fin.add_def]; simp
rw [this, Nat.mod_self]
have hpar : 0 ≤ sOrient (A ⟨n, by omega⟩) (A (⟨n, by omega⟩ + 1)) (A ⟨1 + v, by omega⟩) :=
hA.closed_convex.edge_support ⟨n, by omega⟩ ⟨1 + v, by omega⟩
rw [hwrapEdge] at hpar
-- `hpar : 0 ≤ sOrient (A n)(A 0)(A ⟨1+v⟩)`, i.e. `0 ≤ det3 (A n)(A 0)(A ⟨1+v⟩)`.
-- row expansion: `a · det3 (A n)(A 1)(A ⟨1+v⟩) = det3 (A n)(A 0)(A ⟨1+v⟩) ≥ 0`.
have hrow := det3_rowExpand_wrap (a := a) (b := b)
(An := (A ⟨n, by omega⟩ : E3)) (A1 := (A ⟨1, by omega⟩ : E3))
(An0 := (A ⟨0, by omega⟩ : E3)) (Am := (A ⟨1 + v, by omega⟩ : E3)) hcoeff
-- unfold sOrient = det3 and conclude.
have hpar' : 0 ≤ det3 (A ⟨n, by omega⟩ : E3) (A ⟨0, by omega⟩ : E3) (A ⟨1 + v, by omega⟩ : E3) :=
hpar
have hge : 0 ≤ a * det3 (A ⟨n, by omega⟩ : E3) (A ⟨1, by omega⟩ : E3)
(A ⟨1 + v, by omega⟩ : E3) := by rw [hrow]; exact hpar'
-- `a > 0` and `a * x ≥ 0` ⟹ `x ≥ 0`.
exact (mul_nonneg_iff_of_pos_left hapos).mp hge
/-- **The A-side wrap `ShortArc` is derivable.** `A n ≠ A 1` from `NoNonadjacentRepeat` (the pair
`(1, n)` is nonadjacent for `n ≥ 3`); non-antipodal from the parent open hemisphere. -/
theorem intervalWrap_shortArc_of_parent {n : ℕ} {A : Fin (n + 1) → S2}
(hA : WeakConvexSphArm A) (hnr : NoNonadjacentRepeat A) (hn3 : 3 ≤ n) :
ShortArc (A ⟨n, by omega⟩) (A ⟨1, by omega⟩) := by
-- distinctness from NoNonadjacentRepeat at (1, n): 1 + 2 ≤ n.
have hdist : A ⟨1, by omega⟩ ≠ A ⟨n, by omega⟩ := hnr 1 n (by omega) (by omega) (by omega)
-- non-antipodality from the open hemisphere.
obtain ⟨h, _, hhem⟩ := hA.closed_convex.open_hemisphere
refine ⟨fun he => hdist he.symm, fun hanti => ?_⟩
-- `(A n : E3) = -(A 1 : E3)` would give `0 < ⟪h, A n⟫ = -⟪h, A 1⟫ < 0`.
have hn0 : 0 < (⟪h, (A ⟨n, by omega⟩ : E3)⟫ : ℝ) := hhem ⟨n, by omega⟩
have h10 : 0 < (⟪h, (A ⟨1, by omega⟩ : E3)⟫ : ℝ) := hhem ⟨1, by omega⟩
rw [hanti, inner_neg_right] at hn0
linarith
/-- **The A-side weak interval wrap data, from the `(0, n)` betweenness.** Assembles
`IntervalWrapData A 1 (n-1)` (the wrap diagonal `(A n, A 1)`) from the parent + the betweenness, with
`a > 0` supplied by FFCT23's nondegenerate datum. -/
theorem intervalWrapData_of_betweenness {n : ℕ} {A : Fin (n + 1) → S2}
(hA : WeakConvexSphArm A) (hnr : NoNonadjacentRepeat A) (hn3 : 3 ≤ n)
{a b : ℝ} (hapos : 0 < a)
(hcoeff : a • (A ⟨1, by omega⟩ : E3) + b • (A ⟨n, by omega⟩ : E3) = (A ⟨0, by omega⟩ : E3)) :
IntervalWrapData A 1 (n - 1) (by omega) := by
have hn : 2 ≤ n := by omega
have h1n : 1 + (n - 1) = n := by omega
have hidx : (⟨1 + (n - 1), by omega⟩ : Fin (n + 1)) = (⟨n, by omega⟩ : Fin (n + 1)) := Fin.ext h1n
refine
{ wrap_short := ?_
wrap_support := ?_ }
· -- the wrap diagonal `(A ⟨1+(n-1)⟩, A ⟨1⟩) = (A n, A 1)`.
rw [hidx]
exact intervalWrap_shortArc_of_parent hA hnr hn3
· intro v hv
-- `1 + (n-1) = n`; support at `A ⟨1+v⟩`, `v < (n-1)+1 = n`.
rw [hidx]
exact intervalWrap_support_of_betweenness hA hn hapos hcoeff (v := v) (by have := hv; omega)
/-- **The weak ear `hAe` from the `(0, n)` betweenness, DISCHARGED.** Combining
`intervalWrapData_of_betweenness` with FFCT52's `weakConvex_intervalArm_of_wrap`: the interval `[1..n]`
of `A` is weakly convex, no certificate needed beyond the parent + the betweenness coefficients. -/
theorem weakEar_of_betweenness {n : ℕ} {A : Fin (n + 1) → S2}
(hA : WeakConvexSphArm A) (hnr : NoNonadjacentRepeat A) (hn3 : 3 ≤ n)
{a b : ℝ} (hapos : 0 < a)
(hcoeff : a • (A ⟨1, by omega⟩ : E3) + b • (A ⟨n, by omega⟩ : E3) = (A ⟨0, by omega⟩ : E3)) :
WeakConvexSphArm (intervalArm A 1 (n - 1) (by omega)) :=
weakConvex_intervalArm_of_wrap hA (by omega) (by omega)
(intervalWrapData_of_betweenness hA hnr hn3 hapos hcoeff)
/-- **The A-side wrap from the raw NNReal span membership.** Extracts the FFCT23 nondegenerate datum
(`0 < a`, `0 < b`, `a•A 1 + b•A n = A 0`) from the betweenness `A 0 ∈ span≥0 {A 1, A n}` and feeds
`weakEar_of_betweenness`. This is the end-to-end `hAe` supplier the `(0, n)` branch consumes — fully
discharged. -/
theorem weakEar_of_span {n : ℕ} {A : Fin (n + 1) → S2}
(hA : WeakConvexSphArm A) (hnr : NoNonadjacentRepeat A)
(hn3 : 3 ≤ n)
(hcol : (A ⟨0, by omega⟩ : E3) ∈
Submodule.span NNReal ({(A ⟨1, by omega⟩ : E3), (A ⟨n, by omega⟩ : E3)} : Set E3)) :
WeakConvexSphArm (intervalArm A 1 (n - 1) (by omega)) := by
-- FFCT23's nondegenerate datum at `(i, j) = (0, n)` (needs `0 + 2 < n`, i.e. `n ≥ 3`).
have hcol' : (A ⟨0, by omega⟩ : E3) ∈
Submodule.span NNReal
({(A ⟨0 + 1, by omega⟩ : E3), (A ⟨n, by omega⟩ : E3)} : Set E3) := by
have hidx : (⟨0 + 1, by omega⟩ : Fin (n + 1)) = (⟨1, by omega⟩ : Fin (n + 1)) := Fin.ext rfl
rwa [hidx]
obtain ⟨a, b, hapos, _hbpos, hcoeff⟩ :=
far_fold_nondeg_datum_of_no_repeat hA hnr (i := 0) (j := n) (by omega) (by omega) hcol'
-- align the `0 + 1` index to `1`.
have hcoeff' : (a : ℝ) • (A ⟨1, by omega⟩ : E3) + (b : ℝ) • (A ⟨n, by omega⟩ : E3)
= (A ⟨0, by omega⟩ : E3) := by
have hidx : (⟨0 + 1, by omega⟩ : Fin (n + 1)) = (⟨1, by omega⟩ : Fin (n + 1)) := Fin.ext rfl
rwa [hidx] at hcoeff
exact weakEar_of_betweenness hA hnr hn3 hapos hcoeff'
/-- **The B-side strict diagonal support** — the genuine residue. For a strictly convex `B` and the
interval `[1..n]`, every interior vertex `B ⟨1+v⟩` (`v ≠ 0`, `v ≠ n−1`, i.e. not a diagonal endpoint)
is **strictly** on the positive side of the wrap diagonal `(B n, B 1)`:
`0 < sOrient (B n)(B 1)(B ⟨1+v⟩)`. This is the convex-position fact (diagonal of a strictly convex
polygon); NOT a parent edge support (FFCT52 §4), so genuinely beyond the strict-non-incidence the
substrate banks. -/
def StrictDiagonalSupport {n : ℕ} (B : Fin (n + 1) → S2) (hn3 : 3 ≤ n) : Prop :=
∀ v : ℕ, (hv : v < n) → v ≠ 0 → v ≠ n - 1 →
0 < sOrient (B ⟨n, by omega⟩) (B ⟨1, by omega⟩) (B ⟨1 + v, by have := hv; omega⟩)
/-- **The B-side strict wrap `ShortArc` is derivable** (same hemisphere/no-repeat route as the A side):
`B n ≠ B 1` from strict convexity's distinctness; non-antipodal from the hemisphere. -/
theorem intervalWrap_shortArc_strict {n : ℕ} {B : Fin (n + 1) → S2}
(hB : StrictConvexSphArm B) (hn3 : 3 ≤ n) :
ShortArc (B ⟨n, by omega⟩) (B ⟨1, by omega⟩) := by
-- distinctness: the strict non-incidence at the wrap edge `(n, 0)` with tested vertex `1`.
-- simplest: `(1, n)` strict support is positive ⟹ distinct; but we route through hemisphere + a
-- direct distinctness from strict_nonincident of edge `0` at vertex `n`.
-- distinct: B ⟨1⟩ ≠ B ⟨n⟩ since edge `(0,1)` strictly supports `n` (non-incident), so positive,
-- hence the three points are not collinear ⟹ in particular B 1 ≠ B n.
obtain ⟨h, _, hhem⟩ := hB.closed_convex.open_hemisphere
-- distinctness via strict non-incidence of the parent edge `(1, 2)` is awkward; use the wrap-edge
-- strict support of vertex `1` against edge `(n, 0)`.
have hne_fin : (⟨1, by omega⟩ : Fin (n + 1)) ≠ (⟨n, by omega⟩ : Fin (n + 1)) :=
fun he => by have : (1 : ℕ) = n := congrArg Fin.val he; omega
-- the parent edge `⟨n⟩` with `+1 = ⟨0⟩`; vertex `1` is non-incident (`1 ≠ n`, `1 ≠ 0`).
have hwrapEdge : (⟨n, by omega⟩ + 1 : Fin (n + 1)) = (⟨0, by omega⟩ : Fin (n + 1)) := by
apply Fin.ext
have : ((⟨n, by omega⟩ + 1 : Fin (n + 1)) : ℕ) = (n + 1) % (n + 1) := by
rw [Fin.add_def]; simp
rw [this, Nat.mod_self]
have hjne : (⟨1, by omega⟩ : Fin (n + 1)) ≠ (⟨n, by omega⟩ : Fin (n + 1)) := hne_fin
have hjne1 : (⟨1, by omega⟩ : Fin (n + 1)) ≠ (⟨n, by omega⟩ : Fin (n + 1)) + 1 := by
rw [hwrapEdge]; intro he; have : (1 : ℕ) = 0 := congrArg Fin.val he; omega
have hstr := hB.closed_convex.strict_nonincident ⟨n, by omega⟩ ⟨1, by omega⟩ hjne hjne1
rw [hwrapEdge] at hstr
-- `hstr : 0 < sOrient (B n)(B 0)(B 1)`, so in particular B n, B 1 distinct.
refine ⟨?_, fun hanti => ?_⟩
· -- distinctness: if B n = B 1 then sOrient (B n)(B 0)(B 1) = sOrient (B n)(B 0)(B n) = 0 < 0.
intro he
rw [he] at hstr
have hz : sOrient (B ⟨1, by omega⟩) (B ⟨0, by omega⟩) (B ⟨1, by omega⟩) = 0 := by
simp only [sOrient, det3]; ring
rw [hz] at hstr; exact lt_irrefl 0 hstr
· -- non-antipodality from the hemisphere.
have hn0 : 0 < (⟪h, (B ⟨n, by omega⟩ : E3)⟫ : ℝ) := hhem ⟨n, by omega⟩
have h10 : 0 < (⟪h, (B ⟨1, by omega⟩ : E3)⟫ : ℝ) := hhem ⟨1, by omega⟩
rw [hanti, inner_neg_right] at hn0
linarith
/-- **The B-side strict interval wrap data, from the named residue.** Assembles
`IntervalWrapDataStrict B 1 (n-1)` from the parent (the weak wrap supports of `B` restrict trivially —
`B`'s edge supports are nonneg) and the named `StrictDiagonalSupport`. -/
theorem intervalWrapDataStrict_of_diagonal {n : ℕ} {B : Fin (n + 1) → S2}
(hB : StrictConvexSphArm B) (hn3 : 3 ≤ n)
(hdiag : StrictDiagonalSupport B (by omega)) :
IntervalWrapDataStrict B 1 (n - 1) (by omega) := by
have hn : 2 ≤ n := by omega
have h1n : 1 + (n - 1) = n := by omega
have hidx : (⟨1 + (n - 1), by omega⟩ : Fin (n + 1)) = (⟨n, by omega⟩ : Fin (n + 1)) := Fin.ext h1n
-- weak wrap supports of B: from B's strict (hence weak) edge support at the wrap edge.
have hBw : WeakConvexSphArm B := strictConvexSphArm_toWeak hB
refine
{ toWeak :=
{ wrap_short := ?_
wrap_support := ?_ }
wrap_strict := ?_ }
· rw [hidx]
exact intervalWrap_shortArc_strict hB hn3
· -- weak support: the wrap diagonal `(B n, B 1)` supports every interior vertex nonneg. For the two
-- diagonal endpoints (`v = 0` ⟹ vertex `B 1`; `v = n-1` ⟹ vertex `B n`) the support is `0` (a
-- repeated argument); for the rest it is the strict residue weakened.
intro v hv
rw [hidx]
by_cases hv0 : v = 0
· subst hv0
have hidx0 : (⟨1 + 0, by omega⟩ : Fin (n + 1)) = (⟨1, by omega⟩ : Fin (n + 1)) :=
Fin.ext rfl
rw [hidx0, sOrient]
have hz : det3 (B ⟨n, by omega⟩ : E3) (B ⟨1, by omega⟩ : E3) (B ⟨1, by omega⟩ : E3) = 0 := by
simp only [det3]; ring
rw [hz]
· by_cases hvn : v = n - 1
· subst hvn
rw [sOrient, hidx]
have hz : det3 (B ⟨n, by omega⟩ : E3) (B ⟨1, by omega⟩ : E3) (B ⟨n, by omega⟩ : E3) = 0 := by
simp only [det3]; ring
rw [hz]
· exact le_of_lt (hdiag v (by have := hv; omega) hv0 hvn)
· -- the strict wrap support: exactly the named residue (for `v ≠ 0`, `v ≠ m = n-1`).
intro v hv hvm hv0
rw [hidx]
exact hdiag v (by have := hv; omega) hv0 hvm
/-- **The strict ear `hBe` from the named diagonal residue, DISCHARGED modulo `StrictDiagonalSupport`.**
Combining `intervalWrapDataStrict_of_diagonal` with FFCT52's `strictConvex_intervalArm_of_wrap`. -/
theorem strictEar_of_diagonal {n : ℕ} {B : Fin (n + 1) → S2}
(hB : StrictConvexSphArm B) (hn3 : 3 ≤ n)
(hdiag : StrictDiagonalSupport B (by omega)) :
StrictConvexSphArm (intervalArm B 1 (n - 1) (by omega)) :=
strictConvex_intervalArm_of_wrap hB (by omega) (by omega)
(intervalWrapDataStrict_of_diagonal hB hn3 hdiag)
/-- **The interval certs `(hAe, hBe)` for the `(0, n)` branch.** The weak ear `hAe` is **discharged
unconditionally** from the betweenness (the A-side row expansion); the strict ear `hBe` is supplied by
the named B-side residue `StrictDiagonalSupport`. This is the precise content of FFCT53's `hivl` at the
boundary fold, with the A side eliminated and the B side named. -/
theorem intervalCerts_of_betweenness_and_strictDiagonal {n : ℕ} {A B : Fin (n + 1) → S2}
(hA : WeakConvexSphArm A) (hnr : NoNonadjacentRepeat A)
(hB : StrictConvexSphArm B) (hn3 : 3 ≤ n)
(hcol : (A ⟨0, by omega⟩ : E3) ∈
Submodule.span NNReal ({(A ⟨1, by omega⟩ : E3), (A ⟨n, by omega⟩ : E3)} : Set E3))
(hdiag : StrictDiagonalSupport B (by omega)) :
WeakConvexSphArm (intervalArm A 1 (n - 1) (by omega)) ∧
StrictConvexSphArm (intervalArm B 1 (n - 1) (by omega)) :=
⟨weakEar_of_span hA hnr hn3 hcol, strictEar_of_diagonal hB hn3 hdiag⟩
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
/-- Index arithmetic for `Fin (n + 1)`: the successor of `⟨k, _⟩` is `⟨k+1, _⟩` when `k + 1 < n + 1`. -/
theorem succ_mk {n k : ℕ} (hk : k < n + 1) (hk1 : k + 1 < n + 1) :
((⟨k, hk⟩ : Fin (n + 1)) + 1) = (⟨k + 1, hk1⟩ : Fin (n + 1)) := by
apply Fin.ext
have hone : ((1 : Fin (n + 1)) : ℕ) = 1 := by
rw [Fin.val_one']; exact Nat.mod_eq_of_lt (by omega)
rw [Fin.val_add, Fin.val_mk, hone, Nat.mod_eq_of_lt (show k + 1 < n + 1 by omega)]
/-- **Brick 1 base case (`v = 1`).** The arc-interior vertex `B 2` is strictly on the positive side
of the wrap diagonal `(B n, B 1)`: `0 < sOrient (B n) (B 1) (B 2)`.
Proof: `sOrient (B n)(B 1)(B 2) = det3 (B n)(B 1)(B 2)`. By the cyclic rotation
`det3 (B n)(B 1)(B 2) = det3 (B 1)(B 2)(B n)`, this is the *strict non-incidence* support of the
parent edge `(B 1, B 2)` at the vertex `B n` (which is non-incident: `n ≠ 1` and `n ≠ 2` as `n ≥ 3`). -/
theorem strictDiagonal_base {n : ℕ} {B : Fin (n + 1) → S2}
(hB : StrictConvexSphArm B) (hn3 : 3 ≤ n) :
0 < sOrient (B ⟨n, by omega⟩) (B ⟨1, by omega⟩) (B ⟨2, by omega⟩) := by
-- the parent edge `(1, 2)`: `⟨1⟩ + 1 = ⟨2⟩`.
have hsucc : ((⟨1, by omega⟩ : Fin (n + 1)) + 1) = (⟨2, by omega⟩ : Fin (n + 1)) :=
succ_mk (by omega) (by omega)
-- vertex `n` is non-incident to edge `(1, 2)`.
have hne1 : (⟨n, by omega⟩ : Fin (n + 1)) ≠ (⟨1, by omega⟩ : Fin (n + 1)) := by
intro he; have : n = 1 := congrArg Fin.val he; omega
have hne2 : (⟨n, by omega⟩ : Fin (n + 1)) ≠ (⟨1, by omega⟩ : Fin (n + 1)) + 1 := by
rw [hsucc]; intro he; have : n = 2 := congrArg Fin.val he; omega
have hstr := hB.closed_convex.strict_nonincident ⟨1, by omega⟩ ⟨n, by omega⟩ hne1 hne2
rw [hsucc] at hstr
-- `hstr : 0 < sOrient (B 1) (B 2) (B n)`. Cyclically rotate to `sOrient (B n) (B 1) (B 2)`.
rw [sOrient] at hstr ⊢
rwa [det3_cyclic (B ⟨n, by omega⟩ : E3) (B ⟨1, by omega⟩ : E3) (B ⟨2, by omega⟩ : E3)]
/-- **Brick 1 top case (`v = n − 2`).** The arc-interior vertex `B ⟨n−1⟩` is strictly on the
positive side of the wrap diagonal `(B n, B 1)`: `0 < sOrient (B n) (B 1) (B ⟨n−1⟩)`.
Proof: `det3 (B n)(B 1)(B ⟨n−1⟩) = det3 (B ⟨n−1⟩)(B n)(B 1)` (cyclic), the strict non-incidence
support of the parent edge `(B ⟨n−1⟩, B n)` at the vertex `B 1` (non-incident: `1 ≠ n−1` and `1 ≠ n`
as `n ≥ 3`). -/
theorem strictDiagonal_top {n : ℕ} {B : Fin (n + 1) → S2}
(hB : StrictConvexSphArm B) (hn3 : 3 ≤ n) :
0 < sOrient (B ⟨n, by omega⟩) (B ⟨1, by omega⟩) (B ⟨n - 1, by omega⟩) := by
-- the parent edge `(n-1, n)`: `⟨n-1⟩ + 1 = ⟨n⟩`.
have hsucc : ((⟨n - 1, by omega⟩ : Fin (n + 1)) + 1) = (⟨n, by omega⟩ : Fin (n + 1)) := by
have h := succ_mk (n := n) (k := n - 1) (by omega) (by omega)
have he : (⟨n - 1 + 1, by omega⟩ : Fin (n + 1)) = (⟨n, by omega⟩ : Fin (n + 1)) :=
Fin.ext (show n - 1 + 1 = n by omega)
rw [he] at h; exact h
-- vertex `1` is non-incident to edge `(n-1, n)`.
have hne1 : (⟨1, by omega⟩ : Fin (n + 1)) ≠ (⟨n - 1, by omega⟩ : Fin (n + 1)) := by
intro he; have : (1 : ℕ) = n - 1 := congrArg Fin.val he; omega
have hne2 : (⟨1, by omega⟩ : Fin (n + 1)) ≠ (⟨n - 1, by omega⟩ : Fin (n + 1)) + 1 := by
rw [hsucc]; intro he; have : (1 : ℕ) = n := congrArg Fin.val he; omega
have hstr := hB.closed_convex.strict_nonincident ⟨n - 1, by omega⟩ ⟨1, by omega⟩ hne1 hne2
rw [hsucc] at hstr
-- `hstr : 0 < sOrient (B ⟨n-1⟩) (B n) (B 1)`. Rotate to `sOrient (B n) (B 1) (B ⟨n-1⟩)`.
rw [sOrient] at hstr ⊢
-- det3 (B n)(B 1)(B ⟨n-1⟩) = det3 (B ⟨n-1⟩)(B n)(B 1) by two cyclic rotations.
rw [det3_cyclic (B ⟨n, by omega⟩ : E3) (B ⟨1, by omega⟩ : E3) (B ⟨n - 1, by omega⟩ : E3),
det3_cyclic (B ⟨1, by omega⟩ : E3) (B ⟨n - 1, by omega⟩ : E3) (B ⟨n, by omega⟩ : E3)]
exact hstr
/-- **The genuine interior residue of `StrictDiagonalSupport`.** Only the *strictly interior* arc
vertices (`2 ≤ v`, `v ≤ n − 3`, i.e. excluding both the base vertex `B 2` and the top vertex
`B ⟨n−1⟩`, which are discharged unconditionally) on the positive side of the wrap diagonal. This is
the irreducible spherical convex-position core; the local Grassmann–Plücker route is provably
sign-indeterminate, so a faithful proof needs a 2D-projection / global-hemisphere argument. -/
def StrictDiagonalInteriorSupport {n : ℕ} (B : Fin (n + 1) → S2) (hn3 : 3 ≤ n) : Prop :=
∀ v : ℕ, (hv : v < n) → 2 ≤ v → v ≤ n - 3 →
0 < sOrient (B ⟨n, by omega⟩) (B ⟨1, by omega⟩) (B ⟨1 + v, by have := hv; omega⟩)
/-- **Brick 1 assembled: `StrictDiagonalSupport` from the interior residue + the discharged base/top
cases.** Splitting the arc-interior range `v ∈ {1, …, n−2}`: `v = 1` is `strictDiagonal_base`,
`v = n − 2` is `strictDiagonal_top`, and the strict interior `2 ≤ v ≤ n − 3` is the named residue.
This is the precise content of FFCT54's `StrictDiagonalSupport`, with the two boundary cases of the
arc KILLED unconditionally. -/
theorem strictDiagonalSupport_of_interior {n : ℕ} {B : Fin (n + 1) → S2}
(hB : StrictConvexSphArm B) (hn3 : 3 ≤ n)
(hint : StrictDiagonalInteriorSupport B hn3) :
ProofsInTheBook.ZinanFFCT54.StrictDiagonalSupport B hn3 := by
intro v hv hv0 hvn
-- `v ∈ {1, …, n-2}` (from `v < n`, `v ≠ 0`, `v ≠ n-1`). Split base / interior / top.
rcases Nat.lt_or_ge v 2 with hlt2 | hge2
· -- `v = 1` (since `v ≠ 0`, `v < 2`).
have hv1 : v = 1 := by omega
subst hv1
-- align `B ⟨1+1⟩ = B ⟨2⟩`.
have hidx : (⟨1 + 1, by have := hv; omega⟩ : Fin (n + 1)) = (⟨2, by omega⟩ : Fin (n + 1)) :=
Fin.ext rfl
rw [hidx]
exact strictDiagonal_base hB hn3
· rcases Nat.lt_or_ge (n - 3) v with hgt3 | hle3
· -- `v ≥ n - 2`; with `v ≠ n - 1` and `v < n`, this forces `v = n - 2`.
have hvtop : v = n - 2 := by omega
subst hvtop
-- align `B ⟨1 + (n-2)⟩ = B ⟨n-1⟩`.
have hidx : (⟨1 + (n - 2), by have := hv; omega⟩ : Fin (n + 1))
= (⟨n - 1, by omega⟩ : Fin (n + 1)) := Fin.ext (show 1 + (n - 2) = n - 1 by omega)
rw [hidx]
exact strictDiagonal_top hB hn3
· -- strict interior: `2 ≤ v ≤ n - 3`.
exact hint v hv hge2 hle3
/-- **The genuine core of `TailFoldBoundary`: the tail vertex's ray membership.** At the `(0, n−1)`
boundary fold, the last vertex `A (last)` lies on the *nonnegative cone* (the short geodesic ray)
spanned by `A 0` and `A ⟨n−1⟩`. This is exactly the FFCT22 audited master gap (the out-of-plane cone
re-extraction beyond `det3 = 0`); `far_fold_tail_collinear_step` gives only the line (`det3 = 0`), not
the ray. Named, satisfiable, context-carrying. -/
def TailRayMembership {n : ℕ} (A : Fin (n + 1) → S2) (hn1 : n - 1 < n + 1) : Prop :=
(A (Fin.last n) : E3) ∈ Submodule.span NNReal
({(A ⟨0, by omega⟩ : E3), (A ⟨n - 1, hn1⟩ : E3)} : Set E3)
/-- **Brick 2 reduction (unconditional): `TailFoldBoundary` from `TailRayMembership`.** The metric
collinearity is the betweenness equation `sDist_betweenness_of_collinear` applied to the ray
membership of `A (last)` on `span≥0 {A 0, A ⟨n−1⟩}`, reconciled with `endpt A = sDist (A 0)(A last)`.
Fully discharged modulo the named ray residue. -/
theorem tailFoldBoundary_of_rayMembership {n : ℕ} {A : Fin (n + 1) → S2} (hn1 : n - 1 < n + 1)
(hray : TailRayMembership A hn1) :
ProofsInTheBook.ZinanFFCT53.TailFoldBoundary A hn1 := by
-- the betweenness equation: `sDist p r = sDist p q + sDist q r` with `q = A last` between
-- `p = A 0` and `r = A ⟨n-1⟩`.
have hbtw := sDist_betweenness_of_collinear (p := A ⟨0, by omega⟩) (q := A (Fin.last n))
(r := A ⟨n - 1, hn1⟩) hray
-- unfold `TailFoldBoundary`: `sDist (A 0)(A ⟨n-1⟩) = endpt A + sDist (A last)(A ⟨n-1⟩)`.
rw [ProofsInTheBook.ZinanFFCT53.TailFoldBoundary, endpt]
-- reconcile `A 0` (from `endpt`) with `A ⟨0,_⟩`.
have h00 : (A 0 : S2) = A ⟨0, by omega⟩ := by congr 1
rw [h00]
-- `hbtw : sDist (A 0)(A ⟨n-1⟩) = sDist (A 0)(A last) + sDist (A last)(A ⟨n-1⟩)`.
linarith [hbtw]
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
/-- **The `a`-readout.** Under the weak support of edge `(j-1, j)` at `Aδ i`
(`0 ≤ det3 (Aδ(j-1))(Aδ j)(Aδ i)`) and the span decomposition, `0 ≤ a · E` where
`E := det3 (Aδ(j-1)) (Aδ j) (Aδ(i+1))`.
Expansion: `det3 z'' q p = det3 z'' q (a•mid + b•q) = a·det3 z'' q mid + b·det3 z'' q q
= a·det3 z'' q mid = a·E` (right-argument linearity, `det3 z'' q q = 0`). -/
theorem nearSide_a_readout {n : ℕ} {A : Fin (n + 1) → S2}
{i j : ℕ} (hi1 : i + 1 < n + 1) (hj : j < n + 1) (hjm1 : j - 1 < n + 1)
{a b : ℝ}
(hp : (A ⟨i, by omega⟩ : E3)
= a • (A ⟨i + 1, hi1⟩ : E3) + b • (A ⟨j, hj⟩ : E3))
(hsupp : 0 ≤ det3 (A ⟨j - 1, hjm1⟩ : E3) (A ⟨j, hj⟩ : E3) (A ⟨i, by omega⟩ : E3)) :
0 ≤ a * det3 (A ⟨j - 1, hjm1⟩ : E3) (A ⟨j, hj⟩ : E3) (A ⟨i + 1, hi1⟩ : E3) := by
have hexp : det3 (A ⟨j - 1, hjm1⟩ : E3) (A ⟨j, hj⟩ : E3) (A ⟨i, by omega⟩ : E3)
= a * det3 (A ⟨j - 1, hjm1⟩ : E3) (A ⟨j, hj⟩ : E3) (A ⟨i + 1, hi1⟩ : E3) := by
rw [hp, det3_add_right, det3_smul_right, det3_smul_right]
have h0 : det3 (A ⟨j - 1, hjm1⟩ : E3) (A ⟨j, hj⟩ : E3) (A ⟨j, hj⟩ : E3) = 0 := by
simp only [det3]; ring
rw [h0]; ring
rw [hexp] at hsupp
exact hsupp
/-- **The `j+1`-side `a`-readout.** Under the weak support of edge `(j, j+1)` at `Aδ i`, the same
expansion gives `0 ≤ a · E'` where `E' := det3 (Aδ j) (Aδ(j+1)) (Aδ(i+1))`. -/
theorem nearSide_a_readout_succ {n : ℕ} {A : Fin (n + 1) → S2}
{i j : ℕ} (hi1 : i + 1 < n + 1) (hj : j < n + 1) (hjp1 : j + 1 < n + 1)
{a b : ℝ}
(hp : (A ⟨i, by omega⟩ : E3)
= a • (A ⟨i + 1, hi1⟩ : E3) + b • (A ⟨j, hj⟩ : E3))
(hsupp : 0 ≤ det3 (A ⟨j, hj⟩ : E3) (A ⟨j + 1, hjp1⟩ : E3) (A ⟨i, by omega⟩ : E3)) :
0 ≤ a * det3 (A ⟨j, hj⟩ : E3) (A ⟨j + 1, hjp1⟩ : E3) (A ⟨i + 1, hi1⟩ : E3) := by
have hexp : det3 (A ⟨j, hj⟩ : E3) (A ⟨j + 1, hjp1⟩ : E3) (A ⟨i, by omega⟩ : E3)
= a * det3 (A ⟨j, hj⟩ : E3) (A ⟨j + 1, hjp1⟩ : E3) (A ⟨i + 1, hi1⟩ : E3) := by
rw [hp, det3_add_right, det3_smul_right, det3_smul_right]
have h0 : det3 (A ⟨j, hj⟩ : E3) (A ⟨j + 1, hjp1⟩ : E3) (A ⟨j, hj⟩ : E3) = 0 := by
simp only [det3]; ring
rw [h0]; ring
rw [hexp] at hsupp
exact hsupp
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
/-- From `ShortArc mid q` (`mid ≠ q`, `(mid:E3) ≠ -(q:E3)`), the swapped base pair `(q, mid)` is
independent: `q ≠ mid` (as `S2`) and `(q : E3) ≠ -(mid : E3)`. -/
theorem base_ne_of_shortArc {mid q : S2} (hsa : ShortArc mid q) :
q ≠ mid ∧ (q : E3) ≠ -(mid : E3) := by
refine ⟨fun h => hsa.1 h.symm, fun h => hsa.2 ?_⟩
rw [h, neg_neg]
/-- **`E_{pred} = 0` puts `Aδ(j-1)` in the plane.** `det3 (Aδ(j-1)) (Aδ j) (Aδ(i+1)) = 0` gives
`Aδ(j-1) = c•(Aδ j) + d•(Aδ(i+1))` for reals `c, d`. -/
theorem witnessPred_mem_plane {n : ℕ} {A : Fin (n + 1) → S2}
{i j : ℕ} (hi1 : i + 1 < n + 1) (hj : j < n + 1) (hjm1 : j - 1 < n + 1)
(hsa : ShortArc (A ⟨i + 1, hi1⟩) (A ⟨j, hj⟩))
(hE : det3 (A ⟨j - 1, hjm1⟩ : E3) (A ⟨j, hj⟩ : E3) (A ⟨i + 1, hi1⟩ : E3) = 0) :
∃ c d : ℝ, c • (A ⟨j, hj⟩ : E3) + d • (A ⟨i + 1, hi1⟩ : E3) = (A ⟨j - 1, hjm1⟩ : E3) := by
have hcyc : det3 (A ⟨j, hj⟩ : E3) (A ⟨i + 1, hi1⟩ : E3) (A ⟨j - 1, hjm1⟩ : E3) = 0 := by
rw [show det3 (A ⟨j, hj⟩ : E3) (A ⟨i + 1, hi1⟩ : E3) (A ⟨j - 1, hjm1⟩ : E3)
= det3 (A ⟨j - 1, hjm1⟩ : E3) (A ⟨j, hj⟩ : E3) (A ⟨i + 1, hi1⟩ : E3) by
simp only [det3]; ring]
exact hE
obtain ⟨hne, hanti⟩ := base_ne_of_shortArc hsa
obtain ⟨c, d, hcd⟩ := lin_indep_span_of_det3_zero (A ⟨j, hj⟩).2 (A ⟨i + 1, hi1⟩).2
(fun h => hne (S2.ext h)) hanti hcyc
exact ⟨c, d, hcd.symm⟩
/-- **`E_{succ} = 0` puts `Aδ(j+1)` in the plane.** `det3 (Aδ j) (Aδ(j+1)) (Aδ(i+1)) = 0` gives
`Aδ(j+1) = c•(Aδ j) + d•(Aδ(i+1))`. -/
theorem witnessSucc_mem_plane {n : ℕ} {A : Fin (n + 1) → S2}
{i j : ℕ} (hi1 : i + 1 < n + 1) (hj : j < n + 1) (hjp1 : j + 1 < n + 1)
(hsa : ShortArc (A ⟨i + 1, hi1⟩) (A ⟨j, hj⟩))
(hE' : det3 (A ⟨j, hj⟩ : E3) (A ⟨j + 1, hjp1⟩ : E3) (A ⟨i + 1, hi1⟩ : E3) = 0) :
∃ c d : ℝ, c • (A ⟨j, hj⟩ : E3) + d • (A ⟨i + 1, hi1⟩ : E3) = (A ⟨j + 1, hjp1⟩ : E3) := by
have hswap : det3 (A ⟨j, hj⟩ : E3) (A ⟨i + 1, hi1⟩ : E3) (A ⟨j + 1, hjp1⟩ : E3) = 0 := by
rw [show det3 (A ⟨j, hj⟩ : E3) (A ⟨i + 1, hi1⟩ : E3) (A ⟨j + 1, hjp1⟩ : E3)
= - det3 (A ⟨j, hj⟩ : E3) (A ⟨j + 1, hjp1⟩ : E3) (A ⟨i + 1, hi1⟩ : E3) by
simp only [det3]; ring]
rw [hE']; ring
obtain ⟨hne, hanti⟩ := base_ne_of_shortArc hsa
obtain ⟨c, d, hcd⟩ := lin_indep_span_of_det3_zero (A ⟨j, hj⟩).2 (A ⟨i + 1, hi1⟩).2
(fun h => hne (S2.ext h)) hanti hswap
exact ⟨c, d, hcd.symm⟩
/-- **(The dichotomy core) Both witnesses degenerating is impossible.** At an interior binding
(`i + 2 ≤ j`, `j + 1 < n + 1`) with the independent edge base `ShortArc (Aδ(i+1)) (Aδ j)`, the two
apex arcs at `Aδ j`, `PositiveJoints A`, and the non-flat bound, the witness determinants
`E_{pred} := det3 (Aδ(j-1)) (Aδ j) (Aδ(i+1))` and `E_{succ} := det3 (Aδ j) (Aδ(j+1)) (Aδ(i+1))`
cannot both vanish: both vanishing puts `Aδ(j-1), Aδ j, Aδ(j+1)` coplanar in `Π`, flattening the
joint at `Aδ j`. -/
theorem not_both_witness_zero {n : ℕ} {A B : Fin (n + 1) → S2}
(hposA : PositiveJoints A) (hB : StrictConvexSphArm B) (hangle : JointLe A B)
{i j : ℕ} (hi1 : i + 1 < n + 1) (hj : j < n + 1)
(hjm1 : j - 1 < n + 1) (hjp1 : j + 1 < n + 1)
(hij : i + 2 ≤ j)
(hsa : ShortArc (A ⟨i + 1, hi1⟩) (A ⟨j, hj⟩))
(hsau : ShortArc (A ⟨j, hj⟩) (A ⟨j - 1, hjm1⟩))
(hsav : ShortArc (A ⟨j, hj⟩) (A ⟨j + 1, hjp1⟩))
(hEpred : det3 (A ⟨j - 1, hjm1⟩ : E3) (A ⟨j, hj⟩ : E3) (A ⟨i + 1, hi1⟩ : E3) = 0)
(hEsucc : det3 (A ⟨j, hj⟩ : E3) (A ⟨j + 1, hjp1⟩ : E3) (A ⟨i + 1, hi1⟩ : E3) = 0) :
False := by
have hxpred := witnessPred_mem_plane hi1 hj hjm1 hsa hEpred
have hzsucc := witnessSucc_mem_plane hi1 hj hjp1 hsa hEsucc
have hymid : ∃ c d : ℝ,
c • (A ⟨j, hj⟩ : E3) + d • (A ⟨i + 1, hi1⟩ : E3) = (A ⟨j, hj⟩ : E3) :=
⟨1, 0, by rw [one_smul, zero_smul, add_zero]⟩
have hcol : det3 (A ⟨j - 1, hjm1⟩ : E3) (A ⟨j, hj⟩ : E3) (A ⟨j + 1, hjp1⟩ : E3) = 0 :=
coplanar_triple_det3_zero hxpred hymid hzsucc
exact far_fold_tail_not_interior (B := B) hposA hB hangle
(t := j) (by omega) hjp1 hjm1 hj hjp1 hsau hsav hcol
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
/-- **(Brick C2) `b = 0` is impossible under a short edge `(p, mid)`.** If `p = a • mid + b • q`
with `b = 0`, then `p = a • mid` with `a` a real scalar; taking norms forces `a = ±1`, hence
`p = mid` or `p = -mid`, both excluded by `ShortArc p mid` (whose two conjuncts are `p ≠ mid` and
`(p : E3) ≠ -(mid : E3)`). -/
theorem bcoef_ne_zero_of_short_edge {p mid q : S2} {a b : ℝ}
(hpm : ShortArc p mid)
(hp : (p : E3) = a • (mid : E3) + b • (q : E3))
(hb0 : b = 0) :
False := by
-- `p = a • mid`.
have hpa : (p : E3) = a • (mid : E3) := by rw [hp, hb0, zero_smul, add_zero]
-- norms: `1 = |a| · 1 = |a|`.
have hnp : ‖(p : E3)‖ = 1 := p.2
have hnm : ‖(mid : E3)‖ = 1 := mid.2
have habs : |a| = 1 := by
have := congrArg (fun x : E3 => ‖x‖) hpa
simp only [norm_smul, Real.norm_eq_abs] at this
rw [hnp, hnm, mul_one] at this
linarith [this]
-- `a = 1` (⟹ `p = mid`) or `a = -1` (⟹ `p = -mid`).
rcases abs_eq (by norm_num : (0 : ℝ) ≤ 1) |>.1 habs with ha1 | ham1
· exact hpm.1 (S2.ext (by rw [hpa, ha1, one_smul]))
· exact hpm.2 (by rw [hpa, ham1, neg_one_smul])
/-- **(Brick C1) Rearrange a `b < 0` fold to a positive-coefficient mid-fold.** Given the open
hemisphere (a unit `h` strictly positive against every vertex), `P i = a • P(i+1) + b • P j` and
`b < 0`, the leading coefficient is `a > 0`, and `P(i+1) = (1/a) • P i + (-b/a) • P j` with both
coefficients strictly positive. -/
theorem midFold_coeffs_of_bneg {n : ℕ} {P : Fin (n + 1) → S2} {i j : ℕ}
(hhem : ∃ h : E3, ‖h‖ = 1 ∧ ∀ r : Fin (n + 1), 0 < (⟪h, (P r : E3)⟫ : ℝ))
(hi : i < n + 1) (hi1 : i + 1 < n + 1) (hj : j < n + 1)
{a b : ℝ}
(hp : (P ⟨i, hi⟩ : E3) = a • (P ⟨i + 1, hi1⟩ : E3) + b • (P ⟨j, hj⟩ : E3))
(hb : b < 0) :
∃ c d : ℝ, 0 < c ∧ 0 < d ∧
(P ⟨i + 1, hi1⟩ : E3) = c • (P ⟨i, hi⟩ : E3) + d • (P ⟨j, hj⟩ : E3) := by
obtain ⟨h, _hnorm, hpos⟩ := hhem
-- inner product against `h`: `⟪h, P i⟫ = a ⟪h, P(i+1)⟫ + b ⟪h, P j⟫`.
have hinner : (⟪h, (P ⟨i, hi⟩ : E3)⟫ : ℝ)
= a * (⟪h, (P ⟨i + 1, hi1⟩ : E3)⟫ : ℝ) + b * (⟪h, (P ⟨j, hj⟩ : E3)⟫ : ℝ) := by
rw [hp]
rw [inner_add_right, inner_smul_right, inner_smul_right]
-- all three inner products are strictly positive.
have h0 := hpos ⟨i, hi⟩
have h1 := hpos ⟨i + 1, hi1⟩
have h2 := hpos ⟨j, hj⟩
-- `a > 0`: else `a ⟪h,P(i+1)⟫ ≤ 0` and `b ⟪h,P j⟫ < 0`, so `⟪h,P i⟫ < 0`, contradiction.
have ha : 0 < a := by nlinarith [hinner, h0, h1, h2, hb]
-- rearrange `a • P(i+1) = P i - b • P j`, then scale by `1/a`.
have hane : a ≠ 0 := ne_of_gt ha
refine ⟨1 / a, (-b) / a, by positivity, div_pos (neg_pos.2 hb) ha, ?_⟩
-- `a • P(i+1) = P i - b • P j`.
have hav : a • (P ⟨i + 1, hi1⟩ : E3) = (P ⟨i, hi⟩ : E3) - b • (P ⟨j, hj⟩ : E3) := by
rw [hp]; abel
-- `P(i+1) = (1/a) • (a • P(i+1)) = (1/a) • (P i - b • P j) = (1/a)•P i + ((-b)/a)•P j`.
have hkey : (P ⟨i + 1, hi1⟩ : E3) = (1 / a) • (a • (P ⟨i + 1, hi1⟩ : E3)) := by
rw [smul_smul, one_div, inv_mul_cancel₀ hane, one_smul]
rw [hkey, hav]
match_scalars <;> field_simp
/-- **(Brick C3) The mid-fold local contradiction.** Suppose the between-vertex
`M = P(i+1) = c • P i + d • P j` with `c, d > 0`, the two short edges `(P i, P(i+1))`,
`(P(i+1), P(i+2))`, the weak supports of the successor edge `(P(i+1), P(i+2))` at the two fold
neighbours `P i` and `P j`, `PositiveJoints P` and the non-flat bound `jointAngle P · < π`. Then
`False`.
The audited algebra (`P0 = P i`, `M = P(i+1)`, `R = P(i+2)`, `Q = P j`, `M = c P0 + d Q`):
`det3 M R P0 = d · det3 Q R P0`, `det3 M R Q = -c · det3 Q R P0`; the two supports with `c, d > 0`
force `det3 Q R P0 = 0`, hence `det3 P0 M R = d · det3 P0 Q R = 0` — the adjacent joint triple at
apex `M = P(i+1)` (joint index `i`) vanishes, so the joint is in `{0, π}`, excluded. -/
theorem midFold_interior_contradiction {n : ℕ} {P : Fin (n + 1) → S2}
{i j : ℕ}
(hi : i < n + 1) (hi1 : i + 1 < n + 1) (hi2 : i + 2 < n + 1) (hj : j < n + 1)
{c d : ℝ} (hc : 0 < c) (hd : 0 < d)
(hmid : (P ⟨i + 1, hi1⟩ : E3) = c • (P ⟨i, hi⟩ : E3) + d • (P ⟨j, hj⟩ : E3))
(hsuppP0 : 0 ≤ sOrient (P ⟨i + 1, hi1⟩) (P ⟨i + 2, hi2⟩) (P ⟨i, hi⟩))
(hsuppQ : 0 ≤ sOrient (P ⟨i + 1, hi1⟩) (P ⟨i + 2, hi2⟩) (P ⟨j, hj⟩))
(hposJoint : 0 < jointAngle P ⟨i, by omega⟩)
(hltJoint : jointAngle P ⟨i, by omega⟩ < Real.pi)
(hsmp : ShortArc (P ⟨i + 1, hi1⟩) (P ⟨i, hi⟩))
(hsmr : ShortArc (P ⟨i + 1, hi1⟩) (P ⟨i + 2, hi2⟩)) :
False := by
-- name the four sphere points.
set P0 : E3 := (P ⟨i, hi⟩ : E3) with hP0
set M : E3 := (P ⟨i + 1, hi1⟩ : E3) with hM
set R : E3 := (P ⟨i + 2, hi2⟩ : E3) with hR
set Q : E3 := (P ⟨j, hj⟩ : E3) with hQ
-- the supports are `det3` (unfold `sOrient`).
have hsuppP0' : 0 ≤ det3 M R P0 := hsuppP0
have hsuppQ' : 0 ≤ det3 M R Q := hsuppQ
-- the mid-fold representation: `M = c • P0 + d • Q`.
have hMrep : M = c • P0 + d • Q := hmid
-- `det3 M R P0 = c·det3 P0 R P0 + d·det3 Q R P0 = d·(det3 Q R P0)`.
have hidP0 : det3 M R P0 = d * det3 Q R P0 := by
rw [hMrep]
simp only [det3, PiLp.add_apply, PiLp.smul_apply, smul_eq_mul]; ring
-- `det3 M R Q = c·det3 P0 R Q + d·det3 Q R Q = -c·(det3 Q R P0)`.
have hidQ : det3 M R Q = -c * det3 Q R P0 := by
rw [hMrep]
simp only [det3, PiLp.add_apply, PiLp.smul_apply, smul_eq_mul]; ring
-- from the two supports + positivity: `det3 Q R P0 = 0`.
have hD0 : det3 Q R P0 = 0 := by
have h1 : 0 ≤ d * det3 Q R P0 := hidP0 ▸ hsuppP0'
have h2 : 0 ≤ -c * det3 Q R P0 := hidQ ▸ hsuppQ'
nlinarith [h1, h2, hc, hd, mul_pos hc hd]
-- the adjacent triple `det3 P0 M R = d·det3 P0 Q R = d·(det3 Q R P0) = 0`.
have hadj0 : det3 P0 M R = 0 := by
have hexpand : det3 P0 M R = d * det3 Q R P0 := by
rw [hMrep]
simp only [det3, PiLp.add_apply, PiLp.smul_apply, smul_eq_mul]; ring
rw [hexpand, hD0, mul_zero]
-- bridge: `det3 P0 M R = 0` ⟹ `sphAngle (P i) (P(i+1)) (P(i+2)) ∈ {0, π}`.
-- short arcs at the apex `M = P(i+1)`: `(P(i+1), P i)` and `(P(i+1), P(i+2))`.
have hbridge := sphAngle_eq_zero_or_pi_of_det3_zero (u := P ⟨i, hi⟩) (v := P ⟨i + 1, hi1⟩)
(w := P ⟨i + 2, hi2⟩) hsmp hsmr (by rw [← hP0, ← hM, ← hR]; exact hadj0)
-- the joint angle at index `i` is this spherical angle.
have hjoint_eq : jointAngle P ⟨i, by omega⟩
= sphAngle (P ⟨i, hi⟩) (P ⟨i + 1, hi1⟩) (P ⟨i + 2, hi2⟩) := by
rw [jointAngle]
-- contradiction: the joint is in `(0, π)` but the bridge forces it into `{0, π}`.
rcases hbridge with h0 | hπ
· rw [hjoint_eq, h0] at hposJoint; exact lt_irrefl 0 hposJoint
· rw [hjoint_eq, hπ] at hltJoint; exact lt_irrefl Real.pi hltJoint
/-- **The mid-fold kill from a `b < 0` span datum on a weakly convex `PositiveJoints` arm.** Given a
weakly convex `P` with positive joints, a strictly convex comparison `B` with `JointLe P B`, the
interior apex `i + 2 < n + 1`, the strict open hemisphere, and the `b < 0` span datum
`P i = a • P(i+1) + b • P j`, the configuration is impossible. -/
theorem midFold_bneg_false {n : ℕ} {P B : Fin (n + 1) → S2}
(hP : WeakConvexSphArm P) (hpos : PositiveJoints P)
(hB : StrictConvexSphArm B) (hangle : JointLe P B)
{i j : ℕ}
(hi : i < n + 1) (hi1 : i + 1 < n + 1) (hi2 : i + 2 < n + 1) (hj : j < n + 1)
(hhem : ∃ h : E3, ‖h‖ = 1 ∧ ∀ r : Fin (n + 1), 0 < (⟪h, (P r : E3)⟫ : ℝ))
{a b : ℝ}
(hp : (P ⟨i, hi⟩ : E3) = a • (P ⟨i + 1, hi1⟩ : E3) + b • (P ⟨j, hj⟩ : E3))
(hb : b < 0) :
False := by
-- rearrange to the positive mid-fold.
obtain ⟨c, d, hc, hd, hmid⟩ := midFold_coeffs_of_bneg hhem hi hi1 hj hp hb
have hn2 : 2 ≤ n := hP.two_le
-- the successor edge `(P(i+1), P(i+2))` of the closed polygon.
have hsucc : ((⟨i + 1, hi1⟩ : Fin (n + 1)) + 1) = (⟨i + 2, hi2⟩ : Fin (n + 1)) := by
apply Fin.ext
have hone : ((1 : Fin (n + 1)) : ℕ) = 1 := by
rw [Fin.val_one']; exact Nat.mod_eq_of_lt (by omega)
show ((⟨i + 1, hi1⟩ : Fin (n + 1)) + 1).val = i + 2
rw [Fin.val_add, Fin.val_mk, hone,
Nat.mod_eq_of_lt (show (i + 1) + 1 < n + 1 by omega)]
-- weak supports of the successor edge at the two fold neighbours `P i` and `P j`.
have hsuppP0 : 0 ≤ sOrient (P ⟨i + 1, hi1⟩) (P ⟨i + 2, hi2⟩) (P ⟨i, hi⟩) := by
have h := hP.closed_convex.edge_support ⟨i + 1, hi1⟩ ⟨i, hi⟩
rwa [hsucc] at h
have hsuppQ : 0 ≤ sOrient (P ⟨i + 1, hi1⟩) (P ⟨i + 2, hi2⟩) (P ⟨j, hj⟩) := by
have h := hP.closed_convex.edge_support ⟨i + 1, hi1⟩ ⟨j, hj⟩
rwa [hsucc] at h
-- the short edge `(P i, P(i+1))`.
have hedge1 : ShortArc (P ⟨i, hi⟩) (P ⟨i + 1, hi1⟩) := by
have h := hP.closed_convex.edge_short ⟨i, hi⟩
have hsucc1 : ((⟨i, hi⟩ : Fin (n + 1)) + 1) = (⟨i + 1, hi1⟩ : Fin (n + 1)) := by
apply Fin.ext
have hone : ((1 : Fin (n + 1)) : ℕ) = 1 := by
rw [Fin.val_one']; exact Nat.mod_eq_of_lt (by omega)
rw [Fin.val_add, Fin.val_mk, hone, Nat.mod_eq_of_lt (show i + 1 < n + 1 by omega)]
rwa [hsucc1] at h
have hsmp : ShortArc (P ⟨i + 1, hi1⟩) (P ⟨i, hi⟩) := hedge1.symm
-- the short edge `(P(i+1), P(i+2))`.
have hsmr : ShortArc (P ⟨i + 1, hi1⟩) (P ⟨i + 2, hi2⟩) := by
have h := hP.closed_convex.edge_short ⟨i + 1, hi1⟩
rwa [hsucc] at h
-- positivity and non-flat bound at joint index `i`.
have hposJoint : 0 < jointAngle P ⟨i, by omega⟩ := hpos ⟨i, by omega⟩
have hltJoint : jointAngle P ⟨i, by omega⟩ < Real.pi :=
jointAngle_lt_pi hB hangle ⟨i, by omega⟩
exact midFold_interior_contradiction hi hi1 hi2 hj hc hd hmid hsuppP0 hsuppQ
hposJoint hltJoint hsmp hsmr
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
/-- Reflection in the `z = 0` coordinate plane: `(x,y,z) ↦ (x,y,-z)`. -/
def reflectZ (v : E3) : E3 := !₂[v 0, v 1, -v 2]
@[simp] theorem reflectZ_apply_zero (v : E3) : reflectZ v 0 = v 0 := rfl
@[simp] theorem reflectZ_apply_one (v : E3) : reflectZ v 1 = v 1 := rfl
@[simp] theorem reflectZ_apply_two (v : E3) : reflectZ v 2 = -v 2 := rfl
theorem reflectZ_involutive (v : E3) : reflectZ (reflectZ v) = v := by
apply ext_coord <;> simp [reflectZ]
theorem reflectZ_injective : Function.Injective reflectZ := by
intro x y h
have h' := congrArg reflectZ h
simpa [reflectZ_involutive] using h'
theorem reflectZ_neg (v : E3) : reflectZ (-v) = -reflectZ v := by
apply ext_coord <;> simp [reflectZ]
theorem reflectZ_sub (u v : E3) : reflectZ (u - v) = reflectZ u - reflectZ v := by
apply ext_coord
· simp [reflectZ]
· simp [reflectZ]
· simp [reflectZ]; ring_nf
theorem reflectZ_smul (a : ℝ) (v : E3) : reflectZ (a • v) = a • reflectZ v := by
apply ext_coord <;> simp [reflectZ]
theorem inner_reflectZ_reflectZ (u v : E3) :
(⟪reflectZ u, reflectZ v⟫ : ℝ) = ⟪u, v⟫ := by
rw [inner_eq_coord, inner_eq_coord]
simp [reflectZ]
theorem norm_reflectZ (v : E3) : ‖reflectZ v‖ = ‖v‖ := by
have hsq : ‖reflectZ v‖ ^ 2 = ‖v‖ ^ 2 := by
rw [← real_inner_self_eq_norm_sq, ← real_inner_self_eq_norm_sq,
inner_reflectZ_reflectZ]
rcases sq_eq_sq_iff_eq_or_eq_neg.mp hsq with h | h
· exact h
· have hn1 : (0 : ℝ) ≤ ‖reflectZ v‖ := norm_nonneg _
have hn2 : (0 : ℝ) ≤ ‖v‖ := norm_nonneg _
linarith
theorem det3_reflectZ (x y z : E3) :
det3 (reflectZ x) (reflectZ y) (reflectZ z) = -det3 x y z := by
simp only [det3, reflectZ_apply_zero, reflectZ_apply_one, reflectZ_apply_two]
ring
/-- The induced reflection of `S²`. -/
def mirrorS2 (p : S2) : S2 :=
⟨reflectZ (p : E3), by rw [norm_reflectZ, p.2]⟩
@[simp] theorem mirrorS2_coe (p : S2) : (mirrorS2 p : E3) = reflectZ (p : E3) := rfl
theorem mirrorS2_injective : Function.Injective mirrorS2 := by
intro p q h
apply S2.ext
apply reflectZ_injective
exact congrArg (fun r : S2 => (r : E3)) h
theorem sInner_mirrorS2 (p q : S2) :
sInner (mirrorS2 p) (mirrorS2 q) = sInner p q := by
simp [sInner, inner_reflectZ_reflectZ]
theorem sDist_mirrorS2 (p q : S2) :
sDist (mirrorS2 p) (mirrorS2 q) = sDist p q := by
simp [sDist, sInner_mirrorS2]
theorem shortArc_mirrorS2 {p q : S2} (h : ShortArc p q) :
ShortArc (mirrorS2 p) (mirrorS2 q) := by
refine ⟨?_, ?_⟩
· intro heq
exact h.1 (mirrorS2_injective heq)
· intro hanti
apply h.2
apply reflectZ_injective
rw [reflectZ_neg]
exact hanti
theorem angle_reflectZ (u v : E3) :
InnerProductGeometry.angle (reflectZ u) (reflectZ v) =
InnerProductGeometry.angle u v := by
simp [InnerProductGeometry.angle, inner_reflectZ_reflectZ, norm_reflectZ]
theorem tangentTo_mirrorS2 (p q : S2) :
tangentTo (mirrorS2 p) (mirrorS2 q) = reflectZ (tangentTo p q) := by
rw [tangentTo_eq, tangentTo_eq]
show (mirrorS2 q : E3) - sInner (mirrorS2 q) (mirrorS2 p) • (mirrorS2 p : E3)
= reflectZ ((q : E3) - sInner q p • (p : E3))
rw [mirrorS2_coe, mirrorS2_coe, sInner_mirrorS2, reflectZ_sub, reflectZ_smul]
theorem sphAngle_mirrorS2 (u v w : S2) :
sphAngle (mirrorS2 u) (mirrorS2 v) (mirrorS2 w) = sphAngle u v w := by
rw [sphAngle, sphAngle, tangentTo_mirrorS2, tangentTo_mirrorS2, angle_reflectZ]
theorem sOrient_mirrorS2 (a b c : S2) :
sOrient (mirrorS2 a) (mirrorS2 b) (mirrorS2 c) = -sOrient a b c := by
simp [sOrient, det3_reflectZ]
/-- The corrected tail arm: reverse the index order and then reflect in one coordinate. -/
def mirrorArm {n : ℕ} (P : Fin (n + 1) → S2) : Fin (n + 1) → S2 :=
fun m => mirrorS2 (revArm P m)
@[simp] theorem mirrorArm_apply {n : ℕ} (P : Fin (n + 1) → S2) (m : Fin (n + 1)) :
mirrorArm P m = mirrorS2 (revArm P m) := rfl
theorem sideLen_mirrorArm {n : ℕ} (P : Fin (n + 1) → S2) (i : Fin n) :
sideLen (mirrorArm P) i = sideLen P ⟨n - 1 - i.val, by have := i.isLt; omega⟩ := by
rw [show sideLen (mirrorArm P) i = sideLen (revArm P) i by
unfold sideLen mirrorArm
rw [sDist_mirrorS2]]
exact revArm_sideLen P i
theorem jointAngle_mirrorArm {n : ℕ} (P : Fin (n + 1) → S2) (i : Fin (n - 1)) :
jointAngle (mirrorArm P) i = jointAngle P ⟨n - 2 - i.val, by have := i.isLt; omega⟩ := by
rw [show jointAngle (mirrorArm P) i = jointAngle (revArm P) i by
unfold jointAngle mirrorArm
rw [sphAngle_mirrorS2]]
exact revArm_jointAngle P i
theorem endpt_mirrorArm {n : ℕ} (P : Fin (n + 1) → S2) :
endpt (mirrorArm P) = endpt P := by
rw [show endpt (mirrorArm P) = endpt (revArm P) by
unfold endpt mirrorArm
rw [sDist_mirrorS2]]
exact endpt_revArm P
theorem sameSides_mirrorArm {n : ℕ} {A B : Fin (n + 1) → S2} (h : SameSides A B) :
SameSides (mirrorArm A) (mirrorArm B) := by
intro i
rw [sideLen_mirrorArm, sideLen_mirrorArm]
exact h _
theorem jointLe_mirrorArm {n : ℕ} {A B : Fin (n + 1) → S2} (h : JointLe A B) :
JointLe (mirrorArm A) (mirrorArm B) := by
intro i
rw [jointAngle_mirrorArm, jointAngle_mirrorArm]
exact h _
theorem positiveJoints_mirrorArm {n : ℕ} {A : Fin (n + 1) → S2} (h : PositiveJoints A) :
PositiveJoints (mirrorArm A) := by
intro i
rw [jointAngle_mirrorArm]
exact h _
theorem noNonadjacentRepeat_mirrorArm {n : ℕ} {A : Fin (n + 1) → S2}
(h : NoNonadjacentRepeat A) : NoNonadjacentRepeat (mirrorArm A) := by
intro r s hr hs hrs he
have hrev : revArm A ⟨r, hr⟩ = revArm A ⟨s, hs⟩ := mirrorS2_injective he
exact revArm_noNonadjacentRepeat h r s hr hs hrs hrev
theorem neg_sOrient_eq_swap12 (a b c : S2) :
-sOrient a b c = sOrient b a c := by
rw [sOrient, sOrient, det3_swap12 (b : E3) (a : E3) (c : E3)]
theorem mirrorArm_edge_short {n : ℕ} {P : Fin (n + 1) → S2}
(hshort : ∀ i : Fin (n + 1), ShortArc (P i) (P (i + 1))) :
∀ i : Fin (n + 1), ShortArc (mirrorArm P i) (mirrorArm P (i + 1)) := by
intro i
by_cases hi : i.val < n
· have hsucc_i : (i + 1 : Fin (n + 1)) = ⟨i.val + 1, by omega⟩ := by
apply Fin.ext
simp [Fin.add_def]
omega
have hsucc_e : (⟨n - i.val - 1, by omega⟩ + 1 : Fin (n + 1))
= ⟨n - i.val, by omega⟩ := by
apply Fin.ext
simp [Fin.add_def, Nat.mod_eq_of_lt (show n - i.val - 1 + 1 < n + 1 by omega)]
omega
have hbase := hshort ⟨n - i.val - 1, by omega⟩
rw [hsucc_e] at hbase
have ri : revArm P i = P ⟨n - i.val, by omega⟩ := by
change P (revFin i) = _
exact congrArg P (Fin.ext (by simp [revFin_val]))
have ris : revArm P (i + 1) = P ⟨n - i.val - 1, by omega⟩ := by
rw [hsucc_i]
change P (revFin (⟨i.val + 1, by omega⟩ : Fin (n + 1))) = _
exact congrArg P (Fin.ext (by simp [revFin_val]; omega))
rw [mirrorArm_apply, mirrorArm_apply, ri, ris]
exact shortArc_mirrorS2 hbase.symm
· have hin : i.val = n := by omega
have hsucc_i : (i + 1 : Fin (n + 1)) = 0 := by
apply Fin.ext
simp [Fin.add_def, hin, Nat.mod_self]
have hwrap : (⟨n, by omega⟩ + 1 : Fin (n + 1)) = (0 : Fin (n + 1)) := by
apply Fin.ext
simp [Fin.add_def, Nat.mod_self]
have hbase := hshort ⟨n, by omega⟩
rw [hwrap] at hbase
have ri : revArm P i = P ⟨0, by omega⟩ := by
change P (revFin i) = _
exact congrArg P (Fin.ext (by simp [revFin_val, hin]))
have ris : revArm P (i + 1) = P ⟨n, by omega⟩ := by
rw [hsucc_i]
change P (revFin (0 : Fin (n + 1))) = _
exact congrArg P (Fin.ext (by simp [revFin_val]))
rw [mirrorArm_apply, mirrorArm_apply, ri, ris]
exact shortArc_mirrorS2 hbase.symm
theorem mirrorArm_edge_support {n : ℕ} {P : Fin (n + 1) → S2}
(hsupp : ∀ i j : Fin (n + 1), 0 ≤ sOrient (P i) (P (i + 1)) (P j)) :
∀ i j : Fin (n + 1),
0 ≤ sOrient (mirrorArm P i) (mirrorArm P (i + 1)) (mirrorArm P j) := by
intro i j
by_cases hi : i.val < n
· have hsucc_i : (i + 1 : Fin (n + 1)) = ⟨i.val + 1, by omega⟩ := by
apply Fin.ext
simp [Fin.add_def]
omega
have hsucc_e : (⟨n - i.val - 1, by omega⟩ + 1 : Fin (n + 1))
= ⟨n - i.val, by omega⟩ := by
apply Fin.ext
simp [Fin.add_def, Nat.mod_eq_of_lt (show n - i.val - 1 + 1 < n + 1 by omega)]
omega
have hbase := hsupp ⟨n - i.val - 1, by omega⟩ ⟨n - j.val, by omega⟩
rw [hsucc_e] at hbase
have ri : revArm P i = P ⟨n - i.val, by omega⟩ := by
change P (revFin i) = _
exact congrArg P (Fin.ext (by simp [revFin_val]))
have ris : revArm P (i + 1) = P ⟨n - i.val - 1, by omega⟩ := by
rw [hsucc_i]
change P (revFin (⟨i.val + 1, by omega⟩ : Fin (n + 1))) = _
exact congrArg P (Fin.ext (by simp [revFin_val]; omega))
have rj : revArm P j = P ⟨n - j.val, by omega⟩ := by
change P (revFin j) = _
exact congrArg P (Fin.ext (by simp [revFin_val]))
have heq : sOrient (mirrorArm P i) (mirrorArm P (i + 1)) (mirrorArm P j)
= sOrient (P ⟨n - i.val - 1, by omega⟩) (P ⟨n - i.val, by omega⟩)
(P ⟨n - j.val, by omega⟩) := by
rw [mirrorArm_apply, mirrorArm_apply, mirrorArm_apply, ri, ris, rj,
sOrient_mirrorS2, neg_sOrient_eq_swap12]
rw [heq]
exact hbase
· have hin : i.val = n := by omega
have hsucc_i : (i + 1 : Fin (n + 1)) = 0 := by
apply Fin.ext
simp [Fin.add_def, hin, Nat.mod_self]
have hwrap : (⟨n, by omega⟩ + 1 : Fin (n + 1)) = (0 : Fin (n + 1)) := by
apply Fin.ext
simp [Fin.add_def, Nat.mod_self]
have hbase := hsupp ⟨n, by omega⟩ ⟨n - j.val, by omega⟩
rw [hwrap] at hbase
have ri : revArm P i = P ⟨0, by omega⟩ := by
change P (revFin i) = _
exact congrArg P (Fin.ext (by simp [revFin_val, hin]))
have ris : revArm P (i + 1) = P ⟨n, by omega⟩ := by
rw [hsucc_i]
change P (revFin (0 : Fin (n + 1))) = _
exact congrArg P (Fin.ext (by simp [revFin_val]))
have rj : revArm P j = P ⟨n - j.val, by omega⟩ := by
change P (revFin j) = _
exact congrArg P (Fin.ext (by simp [revFin_val]))
have heq : sOrient (mirrorArm P i) (mirrorArm P (i + 1)) (mirrorArm P j)
= sOrient (P ⟨n, by omega⟩) (P ⟨0, by omega⟩)
(P ⟨n - j.val, by omega⟩) := by
rw [mirrorArm_apply, mirrorArm_apply, mirrorArm_apply, ri, ris, rj,
sOrient_mirrorS2, neg_sOrient_eq_swap12]
rw [heq]
exact hbase
theorem weakConvex_mirrorArm {n : ℕ} {P : Fin (n + 1) → S2}
(hP : WeakConvexSphArm P) : WeakConvexSphArm (mirrorArm P) := by
refine ⟨hP.two_le, ?_⟩
refine
{ three_le := hP.closed_convex.three_le
edge_short := mirrorArm_edge_short hP.closed_convex.edge_short
edge_support := mirrorArm_edge_support hP.closed_convex.edge_support
open_hemisphere := ?_ }
obtain ⟨h, hnorm, hhem⟩ := hP.closed_convex.open_hemisphere
refine ⟨reflectZ h, by rw [norm_reflectZ, hnorm], ?_⟩
intro r
rw [mirrorArm_apply, mirrorS2_coe, inner_reflectZ_reflectZ]
exact hhem (revFin r)
theorem strictConvex_mirrorArm {n : ℕ} {P : Fin (n + 1) → S2}
(hP : StrictConvexSphArm P) : StrictConvexSphArm (mirrorArm P) := by
have hweak : WeakConvexSphArm (mirrorArm P) :=
weakConvex_mirrorArm (strictConvexSphArm_toWeak hP)
refine ⟨hP.two_le, ?_⟩
refine
{ three_le := hweak.closed_convex.three_le
edge_short := hweak.closed_convex.edge_short
edge_support := hweak.closed_convex.edge_support
strict_nonincident := ?_
open_hemisphere := hweak.closed_convex.open_hemisphere }
intro i j hji hji1
by_cases hi : i.val < n
· have hsucc_i : (i + 1 : Fin (n + 1)) = ⟨i.val + 1, by omega⟩ := by
apply Fin.ext
simp [Fin.add_def]
omega
have hsucc_e : (⟨n - i.val - 1, by omega⟩ + 1 : Fin (n + 1))
= ⟨n - i.val, by omega⟩ := by
apply Fin.ext
simp [Fin.add_def, Nat.mod_eq_of_lt (show n - i.val - 1 + 1 < n + 1 by omega)]
omega
have hv_ne_e : (⟨n - j.val, by omega⟩ : Fin (n + 1))
≠ ⟨n - i.val - 1, by omega⟩ := by
intro hv
apply hji1
apply Fin.ext
have hsval : ((i + 1 : Fin (n + 1)) : ℕ) = i.val + 1 := by
rw [hsucc_i]
rw [hsval]
have hvval : n - j.val = n - i.val - 1 := congrArg Fin.val hv
have hjlt := j.isLt
have hilt := i.isLt
omega
have hv_ne_succ : (⟨n - j.val, by omega⟩ : Fin (n + 1))
≠ (⟨n - i.val - 1, by omega⟩ : Fin (n + 1)) + 1 := by
rw [hsucc_e]
intro hv
apply hji
apply Fin.ext
have hvval : n - j.val = n - i.val := congrArg Fin.val hv
omega
have hbase := hP.closed_convex.strict_nonincident
⟨n - i.val - 1, by omega⟩ ⟨n - j.val, by omega⟩ hv_ne_e hv_ne_succ
rw [hsucc_e] at hbase
have ri : revArm P i = P ⟨n - i.val, by omega⟩ := by
change P (revFin i) = _
exact congrArg P (Fin.ext (by simp [revFin_val]))
have ris : revArm P (i + 1) = P ⟨n - i.val - 1, by omega⟩ := by
rw [hsucc_i]
change P (revFin (⟨i.val + 1, by omega⟩ : Fin (n + 1))) = _
exact congrArg P (Fin.ext (by simp [revFin_val]; omega))
have rj : revArm P j = P ⟨n - j.val, by omega⟩ := by
change P (revFin j) = _
exact congrArg P (Fin.ext (by simp [revFin_val]))
have heq : sOrient (mirrorArm P i) (mirrorArm P (i + 1)) (mirrorArm P j)
= sOrient (P ⟨n - i.val - 1, by omega⟩) (P ⟨n - i.val, by omega⟩)
(P ⟨n - j.val, by omega⟩) := by
rw [mirrorArm_apply, mirrorArm_apply, mirrorArm_apply, ri, ris, rj,
sOrient_mirrorS2, neg_sOrient_eq_swap12]
rw [heq]
exact hbase
· have hin : i.val = n := by omega
have hsucc_i : (i + 1 : Fin (n + 1)) = 0 := by
apply Fin.ext
simp [Fin.add_def, hin, Nat.mod_self]
have hwrap : (⟨n, by omega⟩ + 1 : Fin (n + 1)) = (0 : Fin (n + 1)) := by
apply Fin.ext
simp [Fin.add_def, Nat.mod_self]
have hv_ne_e : (⟨n - j.val, by omega⟩ : Fin (n + 1))
≠ ⟨n, by omega⟩ := by
intro hv
apply hji1
apply Fin.ext
have hsval : ((i + 1 : Fin (n + 1)) : ℕ) = 0 := by
rw [hsucc_i]
simp
rw [hsval]
have hvval : n - j.val = n := congrArg Fin.val hv
have hjlt := j.isLt
omega
have hv_ne_succ : (⟨n - j.val, by omega⟩ : Fin (n + 1))
≠ (⟨n, by omega⟩ : Fin (n + 1)) + 1 := by
rw [hwrap]
intro hv
apply hji
apply Fin.ext
have hvval : n - j.val = 0 := congrArg Fin.val hv
omega
have hbase := hP.closed_convex.strict_nonincident
⟨n, by omega⟩ ⟨n - j.val, by omega⟩ hv_ne_e hv_ne_succ
rw [hwrap] at hbase
have ri : revArm P i = P ⟨0, by omega⟩ := by
change P (revFin i) = _
exact congrArg P (Fin.ext (by simp [revFin_val, hin]))
have ris : revArm P (i + 1) = P ⟨n, by omega⟩ := by
rw [hsucc_i]
change P (revFin (0 : Fin (n + 1))) = _
exact congrArg P (Fin.ext (by simp [revFin_val]))
have rj : revArm P j = P ⟨n - j.val, by omega⟩ := by
change P (revFin j) = _
exact congrArg P (Fin.ext (by simp [revFin_val]))
have heq : sOrient (mirrorArm P i) (mirrorArm P (i + 1)) (mirrorArm P j)
= sOrient (P ⟨n, by omega⟩) (P ⟨0, by omega⟩)
(P ⟨n - j.val, by omega⟩) := by
rw [mirrorArm_apply, mirrorArm_apply, mirrorArm_apply, ri, ris, rj,
sOrient_mirrorS2, neg_sOrient_eq_swap12]
rw [heq]
exact hbase
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
/-- If the far coefficient is zero, a span relation across a short edge is impossible. -/
theorem span_bzero_false_of_weak {n : ℕ} {P : Fin (n + 1) → S2}
(hP : WeakConvexSphArm P) {i j : ℕ}
(hi : i < n + 1) (hi1 : i + 1 < n + 1) (hj : j < n + 1)
{a b : ℝ}
(hspan : (P ⟨i, hi⟩ : E3) = a • (P ⟨i + 1, hi1⟩ : E3) + b • (P ⟨j, hj⟩ : E3))
(hb0 : b = 0) :
False := by
have hedge : ShortArc (P ⟨i, hi⟩) (P ⟨i + 1, hi1⟩) := by
have h := hP.closed_convex.edge_short ⟨i, hi⟩
have hsucc : ((⟨i, hi⟩ : Fin (n + 1)) + 1) = (⟨i + 1, hi1⟩ : Fin (n + 1)) := by
apply Fin.ext
have hone : ((1 : Fin (n + 1)) : ℕ) = 1 := by
rw [Fin.val_one']; exact Nat.mod_eq_of_lt (by omega)
rw [Fin.val_add, Fin.val_mk, hone, Nat.mod_eq_of_lt (show i + 1 < n + 1 by omega)]
rwa [hsucc] at h
exact bcoef_ne_zero_of_short_edge hedge hspan hb0
/-- If `b > 0` and `a = 0`, the span relation forces a forbidden nonadjacent repeat. -/
theorem span_azero_bpos_false_of_noRepeat {n : ℕ} {P : Fin (n + 1) → S2}
(hnr : NoNonadjacentRepeat P) {i j : ℕ}
(hi : i < n + 1) (hi1 : i + 1 < n + 1) (hj : j < n + 1)
(hij : i + 2 ≤ j) {a b : ℝ}
(hspan : (P ⟨i, hi⟩ : E3) = a • (P ⟨i + 1, hi1⟩ : E3) + b • (P ⟨j, hj⟩ : E3))
(hbpos : 0 < b) (ha0 : a = 0) :
False := by
have hpj : (P ⟨i, hi⟩ : E3) = b • (P ⟨j, hj⟩ : E3) := by
rw [hspan, ha0, zero_smul, zero_add]
have hbabs : |b| = 1 := by
have hnorm := congrArg (fun x : E3 => ‖x‖) hpj
simp only [norm_smul, Real.norm_eq_abs] at hnorm
rw [(P ⟨i, hi⟩).2, (P ⟨j, hj⟩).2, mul_one] at hnorm
linarith
have hb1 : b = 1 := by
rcases (abs_eq (by norm_num : (0 : ℝ) ≤ 1)).1 hbabs with hb | hb
· exact hb
· linarith
have hpeq : P ⟨i, hi⟩ = P ⟨j, hj⟩ := by
apply S2.ext
rw [hpj, hb1, one_smul]
exact (hnr i j hi hj hij) hpeq
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
/-- The remaining B-side strict-diagonal content after FFCT63: only the strict interior range
`2 ≤ v ≤ n-3`, and only in the nonempty regime `n ≥ 5`. -/
def StrictDiagonalInteriorCore : Prop :=
∀ {n : ℕ} (hn3 : 3 ≤ n) (_hn5 : 5 ≤ n) (B : Fin (n + 1) → S2),
StrictConvexSphArm B → StrictDiagonalInteriorSupport B hn3
/-- For `n = 3, 4`, FFCT63's interior range is empty; for `n ≥ 5`, use the named core. -/
theorem strictDiagonalSupport_of_interiorCore
(hcore : StrictDiagonalInteriorCore) {n : ℕ} {B : Fin (n + 1) → S2}
(hB : StrictConvexSphArm B) (hn3 : 3 ≤ n) :
StrictDiagonalSupport B hn3 := by
apply strictDiagonalSupport_of_interior hB hn3
by_cases hn5 : 5 ≤ n
· exact hcore hn3 hn5 B hB
· intro v hv hv2 hvn
omega
/-- The `(0,n)` interval certificates with FFCT63's smaller B-side core threaded in. The B ear is
free for `n ∈ {3,4}` and uses `StrictDiagonalInteriorCore` only for `n ≥ 5`. -/
theorem intervalCerts_of_betweenness_and_interiorCore
(hcore : StrictDiagonalInteriorCore) {n : ℕ} {A B : Fin (n + 1) → S2}
(hA : WeakConvexSphArm A) (hnr : NoNonadjacentRepeat A)
(hB : StrictConvexSphArm B) (hn3 : 3 ≤ n)
(hcol : (A ⟨0, by omega⟩ : E3) ∈
Submodule.span NNReal ({(A ⟨1, by omega⟩ : E3), (A ⟨n, by omega⟩ : E3)} : Set E3)) :
WeakConvexSphArm (intervalArm A 1 (n - 1) (by omega)) ∧
StrictConvexSphArm (intervalArm B 1 (n - 1) (by omega)) :=
intervalCerts_of_betweenness_and_strictDiagonal hA hnr hB hn3 hcol
(strictDiagonalSupport_of_interiorCore hcore hB hn3)
/-- In FFCT64's normalized surface, the retained `b < 0` tail branch is arithmetically empty:
`i+1<j` and `j<n+1` imply `i+2<n+1`. -/
theorem bneg_tail_closed_by_normalization {n : ℕ} {P B : Fin (n + 1) → S2}
(_hP : WeakConvexSphArm P) (_hpos : PositiveJoints P) (_hB : StrictConvexSphArm B)
(_hside : SameSides P B) (_hangle : JointLe P B) (_hnr : NoNonadjacentRepeat P)
(_hhem : ∃ h : E3, ‖h‖ = 1 ∧ ∀ r : Fin (n + 1), 0 < (⟪h, (P r : E3)⟫ : ℝ))
{i j : ℕ} (hij : i + 1 < j)
(_hi : i < n + 1) (_hi1 : i + 1 < n + 1) (hj : j < n + 1)
{a b : ℝ}
(_hspan : (P ⟨i, by omega⟩ : E3) = a • (P ⟨i + 1, by omega⟩ : E3)
+ b • (P ⟨j, hj⟩ : E3))
(_hbneg : b < 0) (hnot : ¬ i + 2 < n + 1) :
endpt P ≤ endpt B := by
exfalso
apply hnot
omega
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
/-- Cyclic rotation for `sOrient`. -/
theorem sOrient_cyclic (a b c : S2) :
sOrient a b c = sOrient b c a := by
rw [sOrient]
exact ProofsInTheBook.PlanarConvexDiag.det3_cyclic (a : E3) (b : E3) (c : E3)
/-- The wrap diagonal `(B n, B 1)` strictly supports every arc-interior vertex `B (1+v)`. -/
theorem strictDiagonal_arcInterior_of_cyclicTriple
{n : ℕ} {B : Fin (n + 1) → S2}
(hB : StrictConvexSphArm B)
{v : ℕ} (hv1 : 1 ≤ v) (hvn : v ≤ n - 2) :
0 < sOrient
(B ⟨n, by omega⟩)
(B ⟨1, by omega⟩)
(B ⟨1 + v, by omega⟩) := by
have hcyc :
ProofsInTheBook.SphericalCyclicTriple.CyclicTriplePos (n := n + 1) B :=
ProofsInTheBook.PlanarConvexDiag.cyclicTriplePos_unconditional hB.closed_convex
have hpos :
0 < sOrient
(B ⟨1, by omega⟩)
(B ⟨1 + v, by omega⟩)
(B ⟨n, by omega⟩) := by
exact hcyc
⟨1, by omega⟩
⟨1 + v, by omega⟩
⟨n, by omega⟩
(by exact_mod_cast (show (1 : ℕ) < 1 + v by omega))
(by exact_mod_cast (show (1 + v : ℕ) < n by omega))
rw [sOrient_cyclic (B ⟨n, by omega⟩) (B ⟨1, by omega⟩)
(B ⟨1 + v, by omega⟩)]
exact hpos
/-- FFCT63's strict-diagonal interior residue is fully discharged. -/
theorem StrictDiagonalInteriorSupport_holds
{n : ℕ} {B : Fin (n + 1) → S2}
(hB : StrictConvexSphArm B) (hn3 : 3 ≤ n) :
StrictDiagonalInteriorSupport B hn3 := by
intro v hv hv2 hvn
exact strictDiagonal_arcInterior_of_cyclicTriple (B := B) hB (by omega) (by omega)
/-- FFCT65's `StrictDiagonalInteriorCore` field, now supplied unconditionally. -/
theorem StrictDiagonalInteriorCore_holds :
StrictDiagonalInteriorCore := by
intro n hn3 _hn5 B hB
exact StrictDiagonalInteriorSupport_holds hB hn3
/-- Convert the real coefficient signs from the tail-line computation into the desired NNReal cone. -/
theorem tail_rayMembership_of_coeff_signs
{u v p q : E3} {a b α β : ℝ}
(ha : 0 < a)
(hfold : p = a • u + b • v)
(hq : q = α • u + β • v)
(hα : 0 ≤ α)
(hμ : 0 ≤ β * a - α * b) :
q ∈ Submodule.span NNReal ({p, v} : Set E3) := by
have hane : a ≠ 0 := ne_of_gt ha
rw [Submodule.mem_span_pair]
refine ⟨⟨α / a, div_nonneg hα (le_of_lt ha)⟩,
⟨(β * a - α * b) / a, div_nonneg hμ (le_of_lt ha)⟩, ?_⟩
have hcalc :
(α / a) • p + ((β * a - α * b) / a) • v = α • u + β • v := by
rw [hfold, smul_add, smul_smul, smul_smul, add_assoc, ← add_smul]
have hca : (α / a) * a = α := by field_simp [hane]
have hcb : (α / a) * b + (β * a - α * b) / a = β := by
field_simp [hane]
ring
rw [hca, hcb]
change (α / a : ℝ) • p + ((β * a - α * b) / a : ℝ) • v = q
rw [hcalc, ← hq]
/-- The actual `(0,n-1)` tail fold supplies the last vertex on the ray `span≥0 {A0,A(n-1)}` when
the comparison-arm non-flat bound is in scope. -/
theorem TailRayMembership_holds_context
{n : ℕ} {A B : Fin (n + 1) → S2}
(hn3 : 3 ≤ n)
(hA : WeakConvexSphArm A)
(hpos : PositiveJoints A)
(hB : StrictConvexSphArm B)
(hangle : JointLe A B)
(hnr : NoNonadjacentRepeat A)
(hcol : (A ⟨0, by omega⟩ : E3) ∈
Submodule.span NNReal
({(A ⟨1, by omega⟩ : E3), (A ⟨n - 1, by omega⟩ : E3)} : Set E3)) :
TailRayMembership A (by omega) := by
by_cases hn4 : 4 ≤ n
· have h0 : 0 < n + 1 := by omega
have h1 : 1 < n + 1 := by omega
have h2 : 2 < n + 1 := by omega
have hj : n - 1 < n + 1 := by omega
have hnn : n < n + 1 := by omega
have hcol' : (A ⟨0, by omega⟩ : E3) ∈
Submodule.span NNReal
({(A ⟨0 + 1, by omega⟩ : E3), (A ⟨n - 1, hj⟩ : E3)} : Set E3) := by
have hidx1 : (⟨0 + 1, by omega⟩ : Fin (n + 1)) =
(⟨1, by omega⟩ : Fin (n + 1)) := Fin.ext rfl
have hidxj : (⟨n - 1, hj⟩ : Fin (n + 1)) =
(⟨n - 1, by omega⟩ : Fin (n + 1)) := rfl
rwa [hidx1, hidxj]
obtain ⟨a, b, ha, hb, hcoeff⟩ :=
far_fold_nondeg_datum_of_no_repeat hA hnr (i := 0) (j := n - 1)
(by omega) hj hcol'
have hfold : (A ⟨0, h0⟩ : E3) =
(a : ℝ) • (A ⟨1, h1⟩ : E3) + (b : ℝ) • (A ⟨n - 1, hj⟩ : E3) := by
have hidx1 : (⟨0 + 1, by omega⟩ : Fin (n + 1)) =
(⟨1, h1⟩ : Fin (n + 1)) := Fin.ext rfl
have hidx0 : (⟨0, by omega⟩ : Fin (n + 1)) =
(⟨0, h0⟩ : Fin (n + 1)) := rfl
rw [hidx1, hidx0] at hcoeff
exact hcoeff.symm
-- The `A2` witness gives the oriented area `D = det3 A(n-1) A1 A2` strictly positive.
have hDneg :
det3 (A ⟨1, h1⟩ : E3) (A ⟨n - 1, hj⟩ : E3) (A ⟨2, h2⟩ : E3) < 0 :=
fold_A2_witness_negative hA hpos hB hangle hj (by omega) h1 h2 h0 hb hfold
have hDpos :
0 < det3 (A ⟨n - 1, hj⟩ : E3) (A ⟨1, h1⟩ : E3) (A ⟨2, h2⟩ : E3) := by
rw [ProofsInTheBook.ZinanFFCT12.det3_swap12
(A ⟨n - 1, hj⟩ : E3) (A ⟨1, h1⟩ : E3) (A ⟨2, h2⟩ : E3)]
linarith
-- First put `A n` on the real line spanned by `A1` and `A(n-1)`.
have hsucc01 : ((⟨0, h0⟩ : Fin (n + 1)) + 1) = (⟨1, h1⟩ : Fin (n + 1)) :=
succ_mk (by omega) (by omega)
have htn : n - 1 + 1 < n + 1 := by omega
have hidxn : (⟨n - 1 + 1, htn⟩ : Fin (n + 1)) =
(⟨n, hnn⟩ : Fin (n + 1)) := Fin.ext (show n - 1 + 1 = n by omega)
have hsuccj : ((⟨n - 1, hj⟩ : Fin (n + 1)) + 1) = (⟨n, hnn⟩ : Fin (n + 1)) := by
have h := succ_mk (n := n) (k := n - 1) hj htn
exact h.trans hidxn
have hsupp1 : 0 ≤ det3 (A ⟨0, h0⟩ : E3) (A ⟨1, h1⟩ : E3) (A ⟨n, hnn⟩ : E3) := by
have h := hA.closed_convex.edge_support ⟨0, h0⟩ ⟨n, hnn⟩
rwa [hsucc01] at h
have hsupp2 : 0 ≤ det3 (A ⟨n - 1, hj⟩ : E3) (A ⟨n, hnn⟩ : E3)
(A ⟨1, h1⟩ : E3) := by
have h := hA.closed_convex.edge_support ⟨n - 1, hj⟩ ⟨1, h1⟩
rwa [hsuccj] at h
have hseed : (A ⟨n - 1, hj⟩ : E3) =
(0 : ℝ) • (A ⟨1, h1⟩ : E3) + (1 : ℝ) • (A ⟨n - 1, hj⟩ : E3) := by
rw [zero_smul, one_smul, zero_add]
have hline0 : det3 (A ⟨1, h1⟩ : E3) (A ⟨n - 1, hj⟩ : E3)
(A ⟨n - 1 + 1, htn⟩ : E3) = 0 :=
have hsupp1t : 0 ≤ det3 (A ⟨0, h0⟩ : E3) (A ⟨1, h1⟩ : E3)
(A ⟨n - 1 + 1, htn⟩ : E3) := by
rw [hidxn]
exact hsupp1
have hsupp2t : 0 ≤ det3 (A ⟨n - 1, hj⟩ : E3)
(A ⟨n - 1 + 1, htn⟩ : E3) (A ⟨1, h1⟩ : E3) := by
rw [hidxn]
exact hsupp2
tail_step_collinear (j := n - 1) (t := n - 1) hj h1 h0 hj htn hb
(by norm_num : (0 : ℝ) < 1) hfold hseed hsupp1t hsupp2t
have hline : det3 (A ⟨1, h1⟩ : E3) (A ⟨n - 1, hj⟩ : E3)
(A ⟨n, hnn⟩ : E3) = 0 := by
rwa [hidxn] at hline0
obtain ⟨α, β, hq⟩ :=
repr_of_collinear hA hnr h1 hj hnn (by omega) hline
-- Edge `(n-1,n)` at `2` gives `0 ≤ α`.
have hsuppVq : 0 ≤ det3 (A ⟨n - 1, hj⟩ : E3) (A ⟨n, hnn⟩ : E3)
(A ⟨2, h2⟩ : E3) := by
have h := hA.closed_convex.edge_support ⟨n - 1, hj⟩ ⟨2, h2⟩
rwa [hsuccj] at h
have hαexp : det3 (A ⟨n - 1, hj⟩ : E3) (A ⟨n, hnn⟩ : E3)
(A ⟨2, h2⟩ : E3)
= α * det3 (A ⟨n - 1, hj⟩ : E3) (A ⟨1, h1⟩ : E3)
(A ⟨2, h2⟩ : E3) := by
rw [hq]
simp only [det3, PiLp.add_apply, PiLp.smul_apply, smul_eq_mul]
ring
have hα : 0 ≤ α := by
rw [hαexp] at hsuppVq
nlinarith [hsuppVq, hDpos]
-- Wrap edge `(n,0)` at `2` gives `0 ≤ β*a - α*b`.
have hsuccn : ((⟨n, hnn⟩ : Fin (n + 1)) + 1) = (⟨0, h0⟩ : Fin (n + 1)) := by
apply Fin.ext
have : ((⟨n, hnn⟩ + 1 : Fin (n + 1)) : ℕ) = (n + 1) % (n + 1) := by
rw [Fin.add_def]; simp
rw [this, Nat.mod_self]
have hsuppqp : 0 ≤ det3 (A ⟨n, hnn⟩ : E3) (A ⟨0, h0⟩ : E3)
(A ⟨2, h2⟩ : E3) := by
have h := hA.closed_convex.edge_support ⟨n, hnn⟩ ⟨2, h2⟩
rwa [hsuccn] at h
have hμexp : det3 (A ⟨n, hnn⟩ : E3) (A ⟨0, h0⟩ : E3)
(A ⟨2, h2⟩ : E3)
= (β * (a : ℝ) - α * (b : ℝ))
* det3 (A ⟨n - 1, hj⟩ : E3) (A ⟨1, h1⟩ : E3)
(A ⟨2, h2⟩ : E3) := by
rw [hq, hfold]
simp only [det3, PiLp.add_apply, PiLp.smul_apply, smul_eq_mul]
ring
have hμ : 0 ≤ β * (a : ℝ) - α * (b : ℝ) := by
rw [hμexp] at hsuppqp
nlinarith [hsuppqp, hDpos]
-- Convert the signs to the desired ray membership.
unfold TailRayMembership
have hidxLast : (Fin.last n : Fin (n + 1)) = ⟨n, hnn⟩ := rfl
have hidx0 : (⟨0, by omega⟩ : Fin (n + 1)) = ⟨0, h0⟩ := rfl
have hidxj : (⟨n - 1, by omega⟩ : Fin (n + 1)) = ⟨n - 1, hj⟩ := rfl
rw [hidxLast, hidx0, hidxj]
exact tail_rayMembership_of_coeff_signs (a := (a : ℝ)) (b := (b : ℝ))
(α := α) (β := β) ha hfold hq hα hμ
· have hn_eq : n = 3 := by omega
subst hn_eq
exfalso
have hcolAdj : (A ⟨0, by omega⟩ : E3) ∈
Submodule.span NNReal
({(A ⟨0 + 1, by omega⟩ : E3), (A ⟨0 + 2, by omega⟩ : E3)} : Set E3) := by
simpa using hcol
exact foldedFlat_adjacent_contradiction (i := 0) hA hpos (by omega) hcolAdj
/-- Metric tail boundary from the contextual ray-membership proof. -/
theorem TailFoldBoundary_holds_context
{n : ℕ} {A B : Fin (n + 1) → S2}
(hn3 : 3 ≤ n)
(hA : WeakConvexSphArm A)
(hpos : PositiveJoints A)
(hB : StrictConvexSphArm B)
(hangle : JointLe A B)
(hnr : NoNonadjacentRepeat A)
(hcol : (A ⟨0, by omega⟩ : E3) ∈
Submodule.span NNReal
({(A ⟨1, by omega⟩ : E3), (A ⟨n - 1, by omega⟩ : E3)} : Set E3)) :
TailFoldBoundary A (by omega) :=
tailFoldBoundary_of_rayMembership (by omega)
(TailRayMembership_holds_context hn3 hA hpos hB hangle hnr hcol)
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
/-- A strict arm has no repeated vertices at nonadjacent positions. -/
theorem strictConvex_noNonadjacentRepeat {n : ℕ} {A : Fin (n + 1) → S2}
(hA : StrictConvexSphArm A) : NoNonadjacentRepeat A := by
intro r s hr hs hrs heq
have hr1 : r + 1 < n + 1 := by omega
have hsucc : ((⟨r, hr⟩ : Fin (n + 1)) + 1) = (⟨r + 1, hr1⟩ : Fin (n + 1)) := by
apply Fin.ext
have hone : ((1 : Fin (n + 1)) : ℕ) = 1 := by
rw [Fin.val_one']
exact Nat.mod_eq_of_lt (by omega)
rw [Fin.val_add, Fin.val_mk, hone, Nat.mod_eq_of_lt (show r + 1 < n + 1 by omega)]
have hnon :
0 < sOrient (A ⟨r, hr⟩) (A ⟨r + 1, hr1⟩) (A ⟨s, hs⟩) := by
have hbase := hA.closed_convex.strict_nonincident
(⟨r, hr⟩ : Fin (n + 1)) (⟨s, hs⟩ : Fin (n + 1))
(by
intro h
have hv : s = r := by simpa using congrArg Fin.val h
omega)
(by
rw [hsucc]
intro h
have hv : s = r + 1 := by simpa using congrArg Fin.val h
omega)
rwa [hsucc] at hbase
have hzero : sOrient (A ⟨r, hr⟩) (A ⟨r + 1, hr1⟩) (A ⟨s, hs⟩) = 0 := by
rw [← heq, sOrient]
exact ProofsInTheBook.SphericalDiagCut.det3_self_right _ _
linarith
/-- The induced rotation on `S²` is injective. -/
theorem rotS2_injective (k : S2) (θ : ℝ) : Function.Injective (rotS2 k θ) := by
intro p q hpq
apply S2.ext
apply rot_injective k.2 θ
exact congrArg (fun x : S2 => (x : E3)) hpq
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
/-- Reflection converts a reversed-arm zero support into a mirror-arm zero support. -/
theorem mirrorArm_sOrient_zero_of_revArm_zero {n : ℕ} (P : Fin (n + 1) → S2)
{i j : ℕ} (hi : i < n + 1) (hi1 : i + 1 < n + 1) (hj : j < n + 1)
(hzero :
sOrient (revArm P ⟨i, hi⟩) (revArm P ⟨i + 1, hi1⟩)
(revArm P ⟨j, hj⟩) = 0) :
sOrient (mirrorArm P ⟨i, hi⟩) (mirrorArm P ⟨i + 1, hi1⟩)
(mirrorArm P ⟨j, hj⟩) = 0 := by
rw [mirrorArm_apply, mirrorArm_apply, mirrorArm_apply, sOrient_mirrorS2, hzero, neg_zero]
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
/-- In the normalized `b > 0, a < 0` branch, a real successor edge at the far
vertex forces both adjacent witness determinants at `j` to vanish. The
FFCT32 two-witness flat-joint core then contradicts positive, non-flat joints. -/
theorem bpos_aneg_false_of_successor {n : ℕ} {P B : Fin (n + 1) → S2}
(hP : WeakConvexSphArm P) (hpos : PositiveJoints P)
(hB : StrictConvexSphArm B) (hangle : JointLe P B) (hnr : NoNonadjacentRepeat P)
(hhem : ∃ h : E3, ‖h‖ = 1 ∧ ∀ r : Fin (n + 1), 0 < (⟪h, (P r : E3)⟫ : ℝ))
{i j : ℕ} (hij : i + 1 < j)
(hi : i < n + 1) (hi1 : i + 1 < n + 1) (hj : j < n + 1)
(hj1 : j + 1 < n + 1)
{a b : ℝ}
(hspan : (P ⟨i, hi⟩ : E3) =
a • (P ⟨i + 1, hi1⟩ : E3) + b • (P ⟨j, hj⟩ : E3))
(_hb : 0 < b) (ha : a < 0) :
False := by
have hij2 : i + 2 ≤ j := by omega
have hjle : j ≤ n := by omega
have hdist0 : P ⟨i + 1, by omega⟩ ≠ P ⟨j, by omega⟩ :=
distinctNormalized_of_noRepeat hP hnr hij hjle
have hdist : P ⟨i + 1, hi1⟩ ≠ P ⟨j, hj⟩ := by
simpa using hdist0
obtain ⟨h, _hnorm, hhemPos⟩ := hhem
have hanti : (P ⟨i + 1, hi1⟩ : E3) ≠ -(P ⟨j, hj⟩ : E3) :=
hemisphere_nonAntipodal hhemPos ⟨i + 1, hi1⟩ ⟨j, hj⟩
have hbaseShort : ShortArc (P ⟨i + 1, hi1⟩) (P ⟨j, hj⟩) :=
⟨hdist, hanti⟩
have hjm1 : j - 1 < n + 1 := by omega
have hspan' : (P ⟨i, by omega⟩ : E3) =
a • (P ⟨i + 1, hi1⟩ : E3) + b • (P ⟨j, hj⟩ : E3) := by
simpa using hspan
have hpredSucc : ((⟨j - 1, hjm1⟩ : Fin (n + 1)) + 1) =
(⟨j, hj⟩ : Fin (n + 1)) := by
apply Fin.ext
have hone : ((1 : Fin (n + 1)) : ℕ) = 1 := by
rw [Fin.val_one']; exact Nat.mod_eq_of_lt (by omega)
rw [Fin.val_add, Fin.val_mk, hone,
Nat.mod_eq_of_lt (show (j - 1) + 1 < n + 1 by omega)]
change (j - 1) + 1 = j
omega
have hsucc : ((⟨j, hj⟩ : Fin (n + 1)) + 1) =
(⟨j + 1, hj1⟩ : Fin (n + 1)) := by
apply Fin.ext
have hone : ((1 : Fin (n + 1)) : ℕ) = 1 := by
rw [Fin.val_one']; exact Nat.mod_eq_of_lt (by omega)
rw [Fin.val_add, Fin.val_mk, hone,
Nat.mod_eq_of_lt (show j + 1 < n + 1 by omega)]
have hpredShort : ShortArc (P ⟨j, hj⟩) (P ⟨j - 1, hjm1⟩) := by
have h := hP.closed_convex.edge_short ⟨j - 1, hjm1⟩
rw [hpredSucc] at h
exact h.symm
have hsuccShort : ShortArc (P ⟨j, hj⟩) (P ⟨j + 1, hj1⟩) := by
have h := hP.closed_convex.edge_short ⟨j, hj⟩
rwa [hsucc] at h
have hEpred_mid : 0 ≤ det3 (P ⟨j - 1, hjm1⟩ : E3) (P ⟨j, hj⟩ : E3)
(P ⟨i + 1, hi1⟩ : E3) := by
have h := hP.closed_convex.edge_support ⟨j - 1, hjm1⟩ ⟨i + 1, hi1⟩
rw [hpredSucc] at h
exact h
have hEpred_i : 0 ≤ det3 (P ⟨j - 1, hjm1⟩ : E3) (P ⟨j, hj⟩ : E3)
(P ⟨i, by omega⟩ : E3) := by
have h := hP.closed_convex.edge_support ⟨j - 1, hjm1⟩ ⟨i, by omega⟩
rw [hpredSucc] at h
exact h
have hEpred_read :
0 ≤ a * det3 (P ⟨j - 1, hjm1⟩ : E3) (P ⟨j, hj⟩ : E3)
(P ⟨i + 1, hi1⟩ : E3) :=
nearSide_a_readout hi1 hj hjm1 hspan' hEpred_i
have hEpred0 : det3 (P ⟨j - 1, hjm1⟩ : E3) (P ⟨j, hj⟩ : E3)
(P ⟨i + 1, hi1⟩ : E3) = 0 := by
nlinarith [hEpred_read, hEpred_mid, ha]
have hEsucc_mid : 0 ≤ det3 (P ⟨j, hj⟩ : E3) (P ⟨j + 1, hj1⟩ : E3)
(P ⟨i + 1, hi1⟩ : E3) := by
have h := hP.closed_convex.edge_support ⟨j, hj⟩ ⟨i + 1, hi1⟩
rw [hsucc] at h
exact h
have hEsucc_i : 0 ≤ det3 (P ⟨j, hj⟩ : E3) (P ⟨j + 1, hj1⟩ : E3)
(P ⟨i, by omega⟩ : E3) := by
have h := hP.closed_convex.edge_support ⟨j, hj⟩ ⟨i, by omega⟩
rw [hsucc] at h
exact h
have hEsucc_read :
0 ≤ a * det3 (P ⟨j, hj⟩ : E3) (P ⟨j + 1, hj1⟩ : E3)
(P ⟨i + 1, hi1⟩ : E3) :=
nearSide_a_readout_succ hi1 hj hj1 hspan' hEsucc_i
have hEsucc0 : det3 (P ⟨j, hj⟩ : E3) (P ⟨j + 1, hj1⟩ : E3)
(P ⟨i + 1, hi1⟩ : E3) = 0 := by
nlinarith [hEsucc_read, hEsucc_mid, ha]
exact not_both_witness_zero (B := B) hpos hB hangle hi1 hj hjm1 hj1 hij2
hbaseShort hpredShort hsuccShort hEpred0 hEsucc0
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
/-- Row expansion at the parent edge `(Ai, Aip1)`: if
`Ai = a • Aip1 + b • Aj`, then the support of the diagonal `(Aj, Aip1)`
multiplied by `b` is the parent edge support. -/
theorem det3_rowExpand_edge {a b : ℝ} {Ai Aip1 Aj V : E3}
(hspan : Ai = a • Aip1 + b • Aj) :
b * det3 Aj Aip1 V = det3 Ai Aip1 V := by
rw [hspan, det3_add_fst, det3_smul_fst, det3_smul_fst]
have hself : det3 Aip1 Aip1 V = 0 := by
simp only [det3]
ring
rw [hself]
ring
/-- A positive real span relation gives the NNReal betweenness consumed by the
folded-flat forward transport. -/
theorem span_mem_of_positive_coeffs {n : ℕ} {P : Fin (n + 1) → S2}
{i j : ℕ} (hi : i < n + 1) (hi1 : i + 1 < n + 1) (hj : j < n + 1)
{a b : ℝ}
(hspan : (P ⟨i, hi⟩ : E3) =
a • (P ⟨i + 1, hi1⟩ : E3) + b • (P ⟨j, hj⟩ : E3))
(hbpos : 0 < b) (hapos : 0 < a) :
(P ⟨i, hi⟩ : E3) ∈
Submodule.span NNReal
({(P ⟨i + 1, hi1⟩ : E3), (P ⟨j, hj⟩ : E3)} : Set E3) := by
rw [Submodule.mem_span_pair]
refine ⟨⟨a, le_of_lt hapos⟩, ⟨b, le_of_lt hbpos⟩, ?_⟩
rw [NNReal.smul_def, NNReal.smul_def]
exact hspan.symm
/-- The A-side interval wrap data for `P[i+1..j]`, derived from the positive
span relation and the parent weak convexity. -/
theorem intervalWrapData_of_positive_span {n : ℕ} {P : Fin (n + 1) → S2}
(hP : WeakConvexSphArm P) (hnr : NoNonadjacentRepeat P)
{i j : ℕ} (hij : i + 1 < j) (hfar : i + 2 < j)
(hi : i < n + 1) (hi1 : i + 1 < n + 1) (hj : j < n + 1)
{a b : ℝ}
(hspan : (P ⟨i, hi⟩ : E3) =
a • (P ⟨i + 1, hi1⟩ : E3) + b • (P ⟨j, hj⟩ : E3))
(hbpos : 0 < b) :
IntervalWrapData P (i + 1) (j - (i + 1)) (by omega) := by
have hidxj :
(⟨(i + 1) + (j - (i + 1)), by omega⟩ : Fin (n + 1)) = ⟨j, hj⟩ :=
Fin.ext (by simp; omega)
refine
{ wrap_short := ?_
wrap_support := ?_ }
· rw [hidxj]
have hdist0 : P ⟨i + 1, by omega⟩ ≠ P ⟨j, by omega⟩ :=
distinctNormalized_of_noRepeat hP hnr hij (by omega)
obtain ⟨h, _hnorm, hhem⟩ := hP.closed_convex.open_hemisphere
refine ⟨?_, ?_⟩
· intro heq
exact hdist0 (by simpa using heq.symm)
· exact hemisphere_nonAntipodal hhem ⟨j, hj⟩ ⟨i + 1, hi1⟩
· intro v hv
rw [hidxj]
have hsucc : ((⟨i, hi⟩ : Fin (n + 1)) + 1) = (⟨i + 1, hi1⟩ : Fin (n + 1)) := by
apply Fin.ext
have hone : ((1 : Fin (n + 1)) : ℕ) = 1 := by
rw [Fin.val_one']; exact Nat.mod_eq_of_lt (by omega)
rw [Fin.val_add, Fin.val_mk, hone, Nat.mod_eq_of_lt (show i + 1 < n + 1 by omega)]
have hpar :
0 ≤ sOrient (P ⟨i, hi⟩) (P ⟨i + 1, hi1⟩)
(P ⟨i + 1 + v, by have := hv; omega⟩) := by
have hs := hP.closed_convex.edge_support ⟨i, hi⟩
⟨i + 1 + v, by have := hv; omega⟩
rwa [hsucc] at hs
have hrow := det3_rowExpand_edge (a := a) (b := b)
(Ai := (P ⟨i, hi⟩ : E3)) (Aip1 := (P ⟨i + 1, hi1⟩ : E3))
(Aj := (P ⟨j, hj⟩ : E3))
(V := (P ⟨i + 1 + v, by have := hv; omega⟩ : E3)) hspan
have hge :
0 ≤ b * det3 (P ⟨j, hj⟩ : E3) (P ⟨i + 1, hi1⟩ : E3)
(P ⟨i + 1 + v, by have := hv; omega⟩ : E3) := by
rw [hrow]
exact hpar
have hdiag :
0 ≤ det3 (P ⟨j, hj⟩ : E3) (P ⟨i + 1, hi1⟩ : E3)
(P ⟨i + 1 + v, by have := hv; omega⟩ : E3) :=
(mul_nonneg_iff_of_pos_left hbpos).mp hge
simpa [sOrient]
using hdiag
/-- Strict interval wrap data for `B[i+1..j]`, from the unconditional cyclic
triple theorem for strict spherical polygons. -/
theorem intervalWrapDataStrict_of_cyclicTriple {n : ℕ} {B : Fin (n + 1) → S2}
(hB : StrictConvexSphArm B)
{i j : ℕ} (_hij : i + 1 < j) (hfar : i + 2 < j) (hj : j < n + 1) :
IntervalWrapDataStrict B (i + 1) (j - (i + 1)) (by omega) := by
have hcyc :
ProofsInTheBook.SphericalCyclicTriple.CyclicTriplePos (n := n + 1) B :=
ProofsInTheBook.PlanarConvexDiag.cyclicTriplePos_unconditional hB.closed_convex
have hidxj :
(⟨(i + 1) + (j - (i + 1)), by omega⟩ : Fin (n + 1)) = ⟨j, hj⟩ :=
Fin.ext (by simp; omega)
refine
{ toWeak :=
{ wrap_short := ?_
wrap_support := ?_ }
wrap_strict := ?_ }
· rw [hidxj]
have hpos :
0 < sOrient (B ⟨i + 1, by omega⟩) (B ⟨i + 2, by omega⟩) (B ⟨j, hj⟩) :=
hcyc ⟨i + 1, by omega⟩ ⟨i + 2, by omega⟩ ⟨j, hj⟩
(by exact_mod_cast (show i + 1 < i + 2 by omega))
(by exact_mod_cast (show i + 2 < j by omega))
obtain ⟨h, _hnorm, hhem⟩ := hB.closed_convex.open_hemisphere
refine ⟨?_, ?_⟩
· intro heq
rw [heq] at hpos
have hz :
sOrient (B ⟨i + 1, by omega⟩) (B ⟨i + 2, by omega⟩)
(B ⟨i + 1, by omega⟩) = 0 := by
simp only [sOrient, det3]
ring
rw [hz] at hpos
exact lt_irrefl 0 hpos
· exact hemisphere_nonAntipodal hhem ⟨j, hj⟩ ⟨i + 1, by omega⟩
· intro v hv
rw [hidxj]
by_cases hv0 : v = 0
· subst hv0
have hidx0 :
(⟨i + 1 + 0, by omega⟩ : Fin (n + 1)) =
(⟨i + 1, by omega⟩ : Fin (n + 1)) := Fin.ext rfl
rw [hidx0, sOrient]
have hz :
det3 (B ⟨j, hj⟩ : E3) (B ⟨i + 1, by omega⟩ : E3)
(B ⟨i + 1, by omega⟩ : E3) = 0 := by
simp only [det3]
ring
rw [hz]
· by_cases hvm : v = j - (i + 1)
· subst hvm
have hidxv :
(⟨i + 1 + (j - (i + 1)), by omega⟩ : Fin (n + 1)) = ⟨j, hj⟩ :=
Fin.ext (by omega)
rw [hidxv, sOrient]
have hz :
det3 (B ⟨j, hj⟩ : E3) (B ⟨i + 1, by omega⟩ : E3)
(B ⟨j, hj⟩ : E3) = 0 := by
simp only [det3]
ring
rw [hz]
· have hmid_lt : i + 1 < i + 1 + v := by omega
have hmid_j : i + 1 + v < j := by omega
have hpos :
0 < sOrient (B ⟨i + 1, by omega⟩)
(B ⟨i + 1 + v, by have := hv; omega⟩) (B ⟨j, hj⟩) :=
hcyc ⟨i + 1, by omega⟩ ⟨i + 1 + v, by have := hv; omega⟩ ⟨j, hj⟩
(by exact_mod_cast hmid_lt) (by exact_mod_cast hmid_j)
rw [sOrient_cyclic (B ⟨j, hj⟩) (B ⟨i + 1, by omega⟩)
(B ⟨i + 1 + v, by have := hv; omega⟩)]
exact le_of_lt hpos
· intro v hv hvm hv0
rw [hidxj]
have hmid_lt : i + 1 < i + 1 + v := by omega
have hmid_j : i + 1 + v < j := by omega
have hpos :
0 < sOrient (B ⟨i + 1, by omega⟩)
(B ⟨i + 1 + v, by have := hv; omega⟩) (B ⟨j, hj⟩) :=
hcyc ⟨i + 1, by omega⟩ ⟨i + 1 + v, by have := hv; omega⟩ ⟨j, hj⟩
(by exact_mod_cast hmid_lt) (by exact_mod_cast hmid_j)
rw [sOrient_cyclic (B ⟨j, hj⟩) (B ⟨i + 1, by omega⟩)
(B ⟨i + 1 + v, by have := hv; omega⟩)]
exact hpos
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
/-- `MainPlus` with `NoNonadjacentRepeat` threaded on the weak left arm. -/
def MainPlusNR (n : ℕ) : Prop :=
∀ A B : Fin (n + 1) → S2,
WeakConvexSphArm A → PositiveJoints A → NoNonadjacentRepeat A →
StrictConvexSphArm B → SameSides A B → JointLe A B →
endpt A ≤ endpt B
/-- The opening step consumed by the `MainPlusNR` recursion. -/
def SZOpeningStepPlusNR : Prop :=
∀ n : ℕ, 2 ≤ n →
(∀ m : ℕ, m < n → MainPlusNR m) →
(∀ A B : Fin (n + 1) → S2,
WeakConvexSphArm A → PositiveJoints A → NoNonadjacentRepeat A →
StrictConvexSphArm B → SameSides A B → JointLe A B →
(∀ A' B' : Fin (n + 1) → S2,
WeakConvexSphArm A' → PositiveJoints A' → NoNonadjacentRepeat A' →
StrictConvexSphArm B' → SameSides A' B' → JointLe A' B' →
deficitCount A' B' < deficitCount A B → endpt A' ≤ endpt B') →
endpt A ≤ endpt B)
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
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
end ProofsInTheBook.Ch13CyclicSigns
end
/- Original source header (imports hoisted):
import Mathlib
import ProofsInTheBook.PlanarMap
import ProofsInTheBook.Chapter13
import ProofsInTheBook.Ch13CyclicSigns
-/
/- Source module: ProofsInTheBook.Ch13MarkedSphere -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Ch13MarkedSphere
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.Chapter13
open ProofsInTheBook.Ch13CyclicSigns
end ProofsInTheBook.Ch13MarkedSphere
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMap
-/
/- Source module: ProofsInTheBook.PlanarMapEuler -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.PlanarMap
end ProofsInTheBook.PlanarMap.CombMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapEuler
-/
/- Source module: ProofsInTheBook.PlanarMapSimple -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapEuler
-/
/- Source module: ProofsInTheBook.PlanarMapDelete -/
section
set_option autoImplicit true
namespace Equiv.Perm
open Equiv
namespace DeleteSet
end DeleteSet
open DeleteSet
end Equiv.Perm
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
section TwoEdgePathObstruction
end TwoEdgePathObstruction
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapSimple
-/
/- Source module: ProofsInTheBook.PlanarMapBoundary -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace BoundaryPath
end BoundaryPath
namespace BoundaryCycle
namespace Chord
end Chord
end BoundaryCycle
namespace BoundaryArcSplit
end BoundaryArcSplit
namespace BoundaryCycle
end BoundaryCycle
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapBoundary
-/
/- Source module: ProofsInTheBook.PlanarMapNearTriangulation -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace BoundaryCycle
end BoundaryCycle
namespace NearTriangulation
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapNearTriangulation
import ProofsInTheBook.PlanarMapDelete
-/
/- Source module: ProofsInTheBook.PlanarMapFilteredRotation -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace FilteredRotation
namespace ContiguousInterval
end ContiguousInterval
section FreshDart
end FreshDart
end FilteredRotation
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapFilteredRotation
-/
/- Source module: ProofsInTheBook.PlanarMapChordSplitData -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
section ChordDarts
end ChordDarts
namespace ChordSplitData
end ChordSplitData
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapChordSplitData
-/
/- Source module: ProofsInTheBook.PlanarMapChordSplit -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace BoundaryPath
end BoundaryPath
namespace NearTriangulation
namespace ChordSplitData
end ChordSplitData
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapChordSplit
-/
/- Source module: ProofsInTheBook.PlanarMapSeparation -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
namespace ChordSplitData
end ChordSplitData
namespace ChordSplitData
end ChordSplitData
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapNearTriangulation
-/
/- Source module: ProofsInTheBook.PlanarMapBoundaryFan -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
namespace FanTriangle
end FanTriangle
namespace BoundaryVertexFan
end BoundaryVertexFan
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapBoundaryFan
import ProofsInTheBook.PlanarMapDelete
-/
/- Source module: ProofsInTheBook.PlanarMapBoundaryDelete -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
namespace BoundaryDeletionData
end BoundaryDeletionData
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapBoundaryDelete
-/
/- Source module: ProofsInTheBook.PlanarMapFanSurgery -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
namespace NeighborRotationOrder
end NeighborRotationOrder
namespace FanSurgeryReconstruction
end FanSurgeryReconstruction
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
/-
List-coloring primitives (Chapter 35 layer 4).
Design-independent groundwork for the Thomassen five-list-coloring route
(HANDOFF/CH35_DESIGN_ANSWER.md): proper colorings from lists, monotonicity
in the graph and in the lists, and the piecewise gluing lemmas — including
the rooted cut-vertex glue, which is the form that is actually true for
list colorings (naive gluing fails because the two sides may disagree at
the cut vertex).
-/
import Mathlib
-/
/- Source module: ProofsInTheBook.ListColoring -/
section
set_option autoImplicit true
namespace ProofsInTheBook.ListColoring
section Glue
end Glue
end ProofsInTheBook.ListColoring
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapSeparation
import ProofsInTheBook.PlanarMapFanSurgery
import ProofsInTheBook.ListColoring
-/
/- Source module: ProofsInTheBook.ThomassenLists -/
section
set_option autoImplicit true
namespace ProofsInTheBook.ThomassenLists
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.ListColoring
namespace CombMap
open ProofsInTheBook.PlanarMap.CombMap
namespace ThomassenLists
end ThomassenLists
namespace ChordSplitRegions
end ChordSplitRegions
section Deletion
end Deletion
end CombMap
end ProofsInTheBook.ThomassenLists
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapFanSurgery
-/
/- Source module: ProofsInTheBook.PlanarMapFanConnectivity -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
section Reduction
end Reduction
namespace NearTriangulation
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapFanConnectivity
import ProofsInTheBook.PlanarMapFilteredRotation
-/
/- Source module: ProofsInTheBook.PlanarMapFanFaces -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapFanFaces
-/
/- Source module: ProofsInTheBook.PlanarMapFanMergedOrbit -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapBoundary
-/
/- Source module: ProofsInTheBook.PlanarMapBoundaryArcSplit -/
section
set_option autoImplicit true
set_option maxHeartbeats 1600000
set_option linter.unusedVariables false
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace BoundaryCycleData
end BoundaryCycleData
namespace DataDartArc
end DataDartArc
namespace BoundaryCycleData
end BoundaryCycleData
section Casts
end Casts
namespace BoundaryPath
end BoundaryPath
section BPOfDartArc
end BPOfDartArc
namespace BoundaryCycleData
end BoundaryCycleData
namespace BoundaryCycleData
end BoundaryCycleData
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapFanFaces
import ProofsInTheBook.PlanarMapBoundaryArcSplit
-/
/- Source module: ProofsInTheBook.PlanarMapDeletedBoundary -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
namespace DeletedMergedBoundaryCertificate
end DeletedMergedBoundaryCertificate
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapFanMergedOrbit
import ProofsInTheBook.PlanarMapDeletedBoundary
-/
/- Source module: ProofsInTheBook.PlanarMapOuterArc -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
namespace MergedOuterArcData
end MergedOuterArcData
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapOuterArc
-/
/- Source module: ProofsInTheBook.PlanarMapFanExistence -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace NearTriangulation
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ThomassenLists
import ProofsInTheBook.PlanarMapFanExistence
-/
/- Source module: ProofsInTheBook.ThomassenInduction -/
section
set_option autoImplicit true
namespace ProofsInTheBook.ThomassenInduction
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.ListColoring
open ProofsInTheBook.ThomassenLists
open ProofsInTheBook.ThomassenLists.CombMap
universe u
section Base
end Base
section Chord
end Chord
section Chordless
end Chordless
section Induction
end Induction
section Corollaries
end Corollaries
section FiveColor
end FiveColor
end ProofsInTheBook.ThomassenInduction
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ThomassenInduction
import ProofsInTheBook.PlanarMapChordSplit
import ProofsInTheBook.PlanarMapSeparation
-/
/- Source module: ProofsInTheBook.ChordSplitNT -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
namespace ProofsInTheBook.ChordSplitNT
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.ListColoring
open ProofsInTheBook.ThomassenLists
open ProofsInTheBook.ThomassenLists.CombMap
open ProofsInTheBook.ThomassenInduction
universe u
namespace ChordSideReconstruction
end ChordSideReconstruction
namespace ChordRecursionData
end ChordRecursionData
end ProofsInTheBook.ChordSplitNT
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ChordSplitNT
-/
/- Source module: ProofsInTheBook.ChordSplitEuler -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
namespace ProofsInTheBook.ChordSplitEuler
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.PlanarMap.FilteredRotation
universe u
section VertexCount
end VertexCount
section EulerReduction
end EulerReduction
section ChordApplication
end ChordApplication
section NonVacuity
end NonVacuity
end ProofsInTheBook.ChordSplitEuler
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ChordSplitEuler
-/
/- Source module: ProofsInTheBook.ChordSideRecon -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
namespace ProofsInTheBook.ChordSideRecon
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.PlanarMap.FilteredRotation
open ProofsInTheBook.ChordSplitEuler
universe u
section Connectivity
end Connectivity
section SphereAssembly
end SphereAssembly
section ChordApplication
end ChordApplication
section JordanData
end JordanData
section NonVacuity
end NonVacuity
end ProofsInTheBook.ChordSideRecon
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapFilteredRotation
import ProofsInTheBook.PlanarMapSeparation
-/
/- Source module: ProofsInTheBook.PlanarMapCutCap -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace SimplePrimalCycle
end SimplePrimalCycle
namespace SimplePrimalCycle
-- c_i^- ↦ α (dart i)
end SimplePrimalCycle
namespace CutCapSurgery
end CutCapSurgery
namespace NearTriangulation
end NearTriangulation
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapCutCap
-/
/- Source module: ProofsInTheBook.PlanarMapCutCapSigma -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
namespace SimplePrimalCycle
-- c_i^- ↦ p_i
-- c_i^- ↦ ℓ_i^- = σ⁻¹ q_i
end SimplePrimalCycle
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import Mathlib
-/
/- Source module: ProofsInTheBook.PermTranspositionCycleCount -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
set_option linter.unusedSimpArgs false
set_option linter.unnecessarySimpa false
set_option linter.unusedVariables false
open Equiv Equiv.Perm Function
namespace PermTranspositionCycleCount
open scoped Finset
end PermTranspositionCycleCount
end
/- Original source header (imports hoisted):
import Mathlib
-/
/- Source module: ProofsInTheBook.RelationComponentCount -/
section
set_option autoImplicit true
open Classical
universe u
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapEuler
import ProofsInTheBook.PermTranspositionCycleCount
import ProofsInTheBook.RelationComponentCount
-/
/- Source module: ProofsInTheBook.PlanarMapEulerInequality -/
section
set_option autoImplicit true
namespace ProofsInTheBook.PlanarMap
open Equiv
namespace CombMap
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapCutCapSigma
import ProofsInTheBook.PlanarMapEulerInequality
-/
/- Source module: ProofsInTheBook.PlanarMapCutCapCounts -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
set_option maxHeartbeats 1600000
namespace ProofsInTheBook.PlanarMap
open Equiv Equiv.Perm Function
namespace CombMap
namespace CutCapCount
section SumCongr
end SumCongr
end CutCapCount
namespace SimplePrimalCycle
open CutCapCount
end SimplePrimalCycle
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapCutCapCounts
-/
/- Source module: ProofsInTheBook.PlanarMapCutCapV -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
set_option maxHeartbeats 1600000
namespace ProofsInTheBook.PlanarMap
open Equiv Equiv.Perm Function
namespace CombMap
namespace SimplePrimalCycle
open CutCapCount
end SimplePrimalCycle
namespace CutCapCount
end CutCapCount
namespace SimplePrimalCycle
open CutCapCount
end SimplePrimalCycle
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.PlanarMapCutCapV
-/
/- Source module: ProofsInTheBook.PlanarMapCutCapF -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
set_option maxHeartbeats 1600000
namespace ProofsInTheBook.PlanarMap
open Equiv Equiv.Perm Function
namespace CombMap
namespace CutCapCount
end CutCapCount
namespace SimplePrimalCycle
open CutCapCount
end SimplePrimalCycle
end CombMap
end ProofsInTheBook.PlanarMap
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ChordSideRecon
import ProofsInTheBook.PlanarMapCutCapCounts
import ProofsInTheBook.PlanarMapCutCapF
-/
/- Source module: ProofsInTheBook.ChordFaceCount -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
set_option linter.unusedVariables false
namespace ProofsInTheBook.ChordFaceCount
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.PlanarMap.FilteredRotation
open ProofsInTheBook.ChordSplitEuler
open ProofsInTheBook.ChordSideRecon
open ProofsInTheBook.PlanarMap.CombMap.CutCapCount
universe u
section FacePerm
end FacePerm
section FaceBijection
end FaceBijection
section Dichotomy
end Dichotomy
section Genus0
end Genus0
section SphereAssembly
end SphereAssembly
section NonVacuity
end NonVacuity
section ChordApplication
end ChordApplication
section Headline
end Headline
end ProofsInTheBook.ChordFaceCount
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ChordFaceCount
import ProofsInTheBook.PlanarMapEulerInequality
-/
/- Source module: ProofsInTheBook.ChordDisk -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
set_option linter.unusedVariables false
namespace ProofsInTheBook.ChordDisk
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.PlanarMap.FilteredRotation
open ProofsInTheBook.ChordSplitEuler
open ProofsInTheBook.ChordSideRecon
open ProofsInTheBook.ChordFaceCount
universe u
section Facts
end Facts
section LowerHalf
end LowerHalf
section Threading
end Threading
section ChordApplication
end ChordApplication
section NonVacuity
end NonVacuity
section Headline
end Headline
end ProofsInTheBook.ChordDisk
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ChordDisk
-/
/- Source module: ProofsInTheBook.SubmapPlanar -/
section
set_option autoImplicit true
set_option linter.unusedSectionVars false
set_option linter.unusedVariables false
namespace ProofsInTheBook.SubmapPlanar
open Equiv
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
universe u
section OrbitSplit
open scoped Classical
end OrbitSplit
section RawRestrict
open scoped Classical
open scoped Classical
end RawRestrict
section ChordThreading
open ProofsInTheBook.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 Equiv Equiv.Perm
section ListBridge
end ListBridge
section OrbitBridge
end OrbitBridge
section StrictBridge
end StrictBridge
section ActiveComponent
end ActiveComponent
section Obstruction
end Obstruction
end ProofsInTheBook.Ch13MarkedReduction
end
/- Original source header (imports hoisted):
import Mathlib
import ProofsInTheBook.PlanarMap
import ProofsInTheBook.PlanarMapSimple
import ProofsInTheBook.PlanarMapDelete
import ProofsInTheBook.SubmapPlanar
import ProofsInTheBook.Chapter13
import ProofsInTheBook.Ch13CyclicSigns
import ProofsInTheBook.Ch13MarkedSphere
import ProofsInTheBook.Ch13MarkedReduction
-/
/- Source module: ProofsInTheBook.Ch13ActiveComponent -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Ch13ActiveComponent
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.Chapter13
open ProofsInTheBook.Ch13CyclicSigns
open ProofsInTheBook.Ch13MarkedSphere
open ProofsInTheBook.Ch13MarkedReduction
open ProofsInTheBook.SubmapPlanar
-- unreachable on active darts
end ProofsInTheBook.Ch13ActiveComponent
end
/- Original source header (imports hoisted):
import Mathlib
import ProofsInTheBook.PlanarMap
import ProofsInTheBook.PlanarMapSimple
import ProofsInTheBook.PlanarMapDelete
import ProofsInTheBook.SubmapPlanar
import ProofsInTheBook.Chapter13
import ProofsInTheBook.Ch13CyclicSigns
import ProofsInTheBook.Ch13MarkedSphere
import ProofsInTheBook.Ch13MarkedReduction
import ProofsInTheBook.Ch13ActiveComponent
-/
/- Source module: ProofsInTheBook.Ch13FlipTransport -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Ch13FlipTransport
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.Chapter13
open ProofsInTheBook.Ch13CyclicSigns
open ProofsInTheBook.Ch13MarkedSphere
open ProofsInTheBook.Ch13MarkedReduction
open ProofsInTheBook.Ch13ActiveComponent
open ProofsInTheBook.SubmapPlanar
open Equiv Equiv.Perm
open ProofsInTheBook -- for DeleteSet.firstOutside via Equiv.Perm namespace
end ProofsInTheBook.Ch13FlipTransport
end
/- Original source header (imports hoisted):
import Mathlib
import ProofsInTheBook.PlanarMap
import ProofsInTheBook.PlanarMapSimple
import ProofsInTheBook.PlanarMapEuler
import ProofsInTheBook.PlanarMapDelete
import ProofsInTheBook.SubmapPlanar
import ProofsInTheBook.Chapter13
import ProofsInTheBook.Ch13CyclicSigns
import ProofsInTheBook.Ch13MarkedSphere
import ProofsInTheBook.Ch13MarkedReduction
import ProofsInTheBook.Ch13ActiveComponent
import ProofsInTheBook.Ch13FlipTransport
import ProofsInTheBook.PlanarMapNearTriangulation
-/
/- Source module: ProofsInTheBook.Ch13ComponentClose -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Ch13ComponentClose
open ProofsInTheBook.PlanarMap
open ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.Chapter13
open ProofsInTheBook.Ch13CyclicSigns
open ProofsInTheBook.Ch13MarkedSphere
open ProofsInTheBook.Ch13MarkedReduction
open ProofsInTheBook.Ch13ActiveComponent
open ProofsInTheBook.Ch13FlipTransport
open ProofsInTheBook.SubmapPlanar
open Equiv Equiv.Perm
end ProofsInTheBook.Ch13ComponentClose
end
/- Original source header (imports hoisted):
import ProofsInTheBook.Ch13MarkedSphere
import ProofsInTheBook.Ch13ComponentClose
import ProofsInTheBook.Chapter13
-/
/- Source module: ProofsInTheBook.Ch13CauchyAssembly -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Ch13CauchyAssembly
open ProofsInTheBook.PlanarMap ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.Ch13MarkedSphere
open ProofsInTheBook.Chapter13
end ProofsInTheBook.Ch13CauchyAssembly
end
/- Original source header (imports hoisted):
import ProofsInTheBook.ZinanFFCT112
-/
/- Source module: ProofsInTheBook.Ch13LemmaII -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Ch13LemmaII
open ProofsInTheBook.SphericalKernel
open ProofsInTheBook.ZinanFFCT112
end ProofsInTheBook.Ch13LemmaII
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalDiagCut
-/
/- Source module: ProofsInTheBook.Ch13SubArc -/
section
set_option autoImplicit true
noncomputable section
open scoped RealInnerProductSpace NNReal
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.SphericalHingeCut ProofsInTheBook.SphericalDiagCut
open ProofsInTheBook.SphericalSZChain
namespace ProofsInTheBook.Ch13SubArc
end ProofsInTheBook.Ch13SubArc
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.Chapter13
import ProofsInTheBook.Ch13LemmaII
import ProofsInTheBook.Ch13SubArc
-/
/- Source module: ProofsInTheBook.Ch13ArmVertex -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Ch13ArmVertex
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.Chapter13
open ProofsInTheBook.Ch13LemmaII
open ProofsInTheBook.Ch13SubArc
open scoped Classical
end ProofsInTheBook.Ch13ArmVertex
end
/- Original source header (imports hoisted):
import ProofsInTheBook.Ch13ArmVertex
-/
/- Source module: ProofsInTheBook.Ch13ArmVertexFull -/
section
set_option autoImplicit true
namespace ProofsInTheBook.Ch13ArmVertexFull
open ProofsInTheBook.SphericalKernel ProofsInTheBook.SphericalArm
open ProofsInTheBook.Chapter13
open ProofsInTheBook.Ch13LemmaII
open ProofsInTheBook.Ch13ArmVertex
open scoped Classical
end ProofsInTheBook.Ch13ArmVertexFull
end
/- Original source header (imports hoisted):
import ProofsInTheBook.SphericalKernel
-/
/- Source module: ProofsInTheBook.Ch13VertexStar -/
section
set_option autoImplicit true
noncomputable section
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
open scoped RealInnerProductSpace
open ProofsInTheBook.TetPearls ProofsInTheBook.TetDihedral
open ProofsInTheBook.SphericalKernel
namespace ProofsInTheBook.Ch13VertexStar
namespace VertexStar
end VertexStar
end ProofsInTheBook.Ch13VertexStar
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.Ch13VertexStar
-/
/- Source module: ProofsInTheBook.Ch13Dihedral -/
section
set_option autoImplicit true
noncomputable section
set_option synthInstance.maxHeartbeats 400000
set_option maxHeartbeats 1600000
open scoped RealInnerProductSpace
open ProofsInTheBook.TetDihedral
open ProofsInTheBook.SphericalKernel
namespace ProofsInTheBook.Ch13VertexStar
namespace VertexStar
end VertexStar
end ProofsInTheBook.Ch13VertexStar
end
end
/- Original source header (imports hoisted):
import ProofsInTheBook.Ch13CauchyAssembly
import ProofsInTheBook.Ch13ArmVertexFull
import ProofsInTheBook.Ch13VertexStar
import ProofsInTheBook.Ch13Dihedral
import ProofsInTheBook.PlanarMapSimple
-/
/- Source module: ProofsInTheBook.Ch13Realization -/
section
set_option autoImplicit true
noncomputable section
open scoped Classical
open ProofsInTheBook.PlanarMap ProofsInTheBook.PlanarMap.CombMap
open ProofsInTheBook.Chapter13
open ProofsInTheBook.Ch13CyclicSigns
open ProofsInTheBook.Ch13ArmVertex
open ProofsInTheBook.Ch13ArmVertexFull
open ProofsInTheBook.Ch13MarkedSphere
open ProofsInTheBook.Ch13VertexStar
open ProofsInTheBook.SphericalKernel
namespace ProofsInTheBook.Ch13Realization
namespace List
end List
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.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.