Rupert Counterexample

8. Computational Step🔗

Theorem8.1
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There exists a valid solution table whose zeroth row covers

\begin{align*} \theta_1,\theta_2&\in[0,2\pi/15] \subset [0,0.42], \\ \varphi_1&\in [0,\pi] \subset [0,3.15],\\ \varphi_2&\in [0,\pi/2] \subset [0,1.58],\\ \alpha &\in [-\pi/2,\pi/2] \subset [-1.58,1.58]. \end{align*}

Lean code for Theorem8.1●1 theorem, incomplete
  • theoremdefined in Noperthedron/ComputationalStep.lean
    contains sorry
    theorem Noperthedron.exists_solution_table : ∃ x, True
    theorem Noperthedron.exists_solution_table :
  ∃ x, True
Proof for Theorem 8.1
uses 0

By exhibiting the table and running the validity checking algorithm.

Theorem8.2
Group: Soundness of table rows and propagated non-Rupert certificates. (3)
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Theorem 8.3
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If a global node in the solution tree is valid, then there is no Rupert solution for its interval.

Lean code for Theorem8.2●1 theorem
  • theoremdefined in Noperthedron/SolutionTable/Global.lean
    complete
    theorem Noperthedron.Solution.valid_global_imp_no_rupert (row : Solution.Row)
  (hrow : row.ValidGlobal) :
  ¬∃ q ∈ row.interval.toReal, RupertPose q exactPolyhedron.hull
    theorem Noperthedron.Solution.valid_global_imp_no_rupert
  (row : Solution.Row)
  (hrow : row.ValidGlobal) :
  ¬∃ q ∈ row.interval.toReal,
      RupertPose q exactPolyhedron.hull
Proof for Theorem 8.2
uses 1
Theorem8.3
Group: Soundness of table rows and propagated non-Rupert certificates. (3)
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Theorem 8.2
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If a local node in the solution tree is valid, then there is no Rupert solution for its interval.

Lean code for Theorem8.3●1 theorem
  • theoremdefined in Noperthedron/SolutionTable/Local.lean
    complete
    theorem Noperthedron.Solution.valid_local_imp_no_rupert (row : Solution.Row)
  (hrow : row.ValidLocal) :
  ¬∃ q ∈ row.interval.toReal, RupertPose q exactPolyhedron.hull
    theorem Noperthedron.Solution.valid_local_imp_no_rupert
  (row : Solution.Row)
  (hrow : row.ValidLocal) :
  ¬∃ q ∈ row.interval.toReal,
      RupertPose q exactPolyhedron.hull
Proof for Theorem 8.3
Proof uses 3
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Lemma 2.1.2
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Theorem8.4
Group: Soundness of table rows and propagated non-Rupert certificates. (3)
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Theorem 8.2
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If we have a valid solution table, and in particular its ith row is valid, then there is no Rupert solution of the interval of its ith row.

Lean code for Theorem8.4●5 theorems
  • theoremdefined in Noperthedron/SolutionTable.lean
    complete
    theorem Noperthedron.Solution.Row.valid_imp_not_rupert_ix (tab : Solution.Table)
  (i : ℕ) (tab_valid : tab.RowsValid) (row : Solution.Row)
  (row_valid : Solution.Row.ValidIx tab i row) :
  ¬∃ q ∈ row.interval.toReal, RupertPose q exactPolyhedron.hull
    theorem Noperthedron.Solution.Row.valid_imp_not_rupert_ix
  (tab : Solution.Table) (i : ℕ)
  (tab_valid : tab.RowsValid)
  (row : Solution.Row)
  (row_valid :
    Solution.Row.ValidIx tab i row) :
  ¬∃ q ∈ row.interval.toReal,
      RupertPose q exactPolyhedron.hull
  • theoremdefined in Noperthedron/SolutionTable.lean
    complete
    theorem Noperthedron.Solution.valid_split_imp_no_rupert (tab : Solution.Table)
  (row : Solution.Row) (htab : tab.RowsValid)
  (hr : Solution.Row.ValidSplit tab row)
  (hlt : row.ID < Array.size tab) :
  ¬∃ q ∈ row.interval.toReal, RupertPose q exactPolyhedron.hull
    theorem Noperthedron.Solution.valid_split_imp_no_rupert
  (tab : Solution.Table)
  (row : Solution.Row)
  (htab : tab.RowsValid)
  (hr : Solution.Row.ValidSplit tab row)
  (hlt : row.ID < Array.size tab) :
  ¬∃ q ∈ row.interval.toReal,
      RupertPose q exactPolyhedron.hull
  • theoremdefined in Noperthedron/SolutionTable.lean
    complete
    theorem Noperthedron.Solution.valid_single_param_split_imp_no_rupert
  (tab : Solution.Table) (row : Solution.Row) (htab : tab.RowsValid)
  (hr : Solution.Row.ValidSingleParamSplit tab row) :
  ¬∃ q ∈ row.interval.toReal, RupertPose q exactPolyhedron.hull
    theorem Noperthedron.Solution.valid_single_param_split_imp_no_rupert
  (tab : Solution.Table)
  (row : Solution.Row)
  (htab : tab.RowsValid)
  (hr :
    Solution.Row.ValidSingleParamSplit tab
      row) :
  ¬∃ q ∈ row.interval.toReal,
      RupertPose q exactPolyhedron.hull
  • theoremdefined in Noperthedron/SolutionTable.lean
    complete
    theorem Noperthedron.Solution.valid_full_split_imp_no_rupert
  (tab : Solution.Table) (row : Solution.Row) (htab : tab.RowsValid)
  (_hgt : row.ID < row.IDfirstChild) (_hlt : row.ID < Array.size tab)
  (hi :
    tab.HasIntervals row.IDfirstChild
      (Solution.cubeFold
        [Solution.Interval.lower_half, Solution.Interval.upper_half]
        row.interval
        [Solution.Param.θ₁, Solution.Param.φ₁, Solution.Param.θ₂,
          Solution.Param.φ₂, Solution.Param.α])) :
  ¬∃ q ∈ row.interval.toReal, RupertPose q exactPolyhedron.hull
    theorem Noperthedron.Solution.valid_full_split_imp_no_rupert
  (tab : Solution.Table)
  (row : Solution.Row)
  (htab : tab.RowsValid)
  (_hgt : row.ID < row.IDfirstChild)
  (_hlt : row.ID < Array.size tab)
  (hi :
    tab.HasIntervals row.IDfirstChild
      (Solution.cubeFold
        [Solution.Interval.lower_half,
          Solution.Interval.upper_half]
        row.interval
        [Solution.Param.θ₁,
          Solution.Param.φ₁,
          Solution.Param.θ₂,
          Solution.Param.φ₂,
          Solution.Param.α])) :
  ¬∃ q ∈ row.interval.toReal,
      RupertPose q exactPolyhedron.hull
  • theoremdefined in Noperthedron/SolutionTable.lean
    complete
    theorem Noperthedron.Solution.valid_param_split_imp_no_rupert
  (tab : Solution.Table) (row : Solution.Row) (htab : tab.RowsValid)
  (p : Solution.Param) (h : Solution.Row.ValidSplitParam tab row p) :
  ¬∃ q ∈ row.interval.toReal, RupertPose q exactPolyhedron.hull
    theorem Noperthedron.Solution.valid_param_split_imp_no_rupert
  (tab : Solution.Table)
  (row : Solution.Row)
  (htab : tab.RowsValid)
  (p : Solution.Param)
  (h :
    Solution.Row.ValidSplitParam tab row
      p) :
  ¬∃ q ∈ row.interval.toReal,
      RupertPose q exactPolyhedron.hull
Proof for Theorem 8.4
Proof uses 2
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Theorem 8.2
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By a mutual induction on the number of rows left in the table following the ith. This is because validity constrains each row to only refer to later entries. Appeal inductively to this same theorem if the row splits into other nodes in the tree, or appeal to Theorem or Theorem at the leaves.

Corollary8.5
Group: Soundness of table rows and propagated non-Rupert certificates. (3)
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Theorem 8.2
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If we have a valid solution table, then there is no Rupert solution of the interval of its zeroth row.

Lean code for Corollary8.5●1 theorem
  • theoremdefined in Noperthedron/SolutionTable.lean
    complete
    theorem Noperthedron.Solution.Row.valid_imp_not_rupert (tab : Solution.Table)
  (tab_valid : tab.RowsValid) (hz : 0 < Array.size tab) :
  ¬∃ q ∈ tab[0].interval.toReal, RupertPose q exactPolyhedron.hull
    theorem Noperthedron.Solution.Row.valid_imp_not_rupert
  (tab : Solution.Table)
  (tab_valid : tab.RowsValid)
  (hz : 0 < Array.size tab) :
  ¬∃ q ∈ tab[0].interval.toReal,
      RupertPose q exactPolyhedron.hull
Proof for Corollary 8.5
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Immediate special case of Theorem 8.4.