| import OAI.NumberTheory.Ostmann.Conclusion.BulkPositionShapeOrder |
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| namespace OAI |
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| noncomputable section |
| namespace Ostmann.Conclusion |
| open Construction |
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|
| def leafBulkPermutation (m k l : ℕ) (σ : Equiv.Perm (Fin (2^l) × Fin m)) : |
| Equiv.Perm (Fin (Template.current (Template.initial m k) l).length) := |
| bulkPositionPermutation _ ((currentBulkPositionEquiv m k l).symm.permCongr σ) |
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| @[simp] theorem leafBulkPermutation_apply_bulk (m k l : ℕ) |
| (σ : Equiv.Perm (Fin (2^l) × Fin m)) |
| (x : BulkPosition (Template.current (Template.initial m k) l)) : |
| leafBulkPermutation m k l σ x.val= |
| ((currentBulkPositionEquiv m k l).symm (σ (currentBulkPositionEquiv m k l x))).val := by |
| unfold leafBulkPermutation bulkPositionPermutation |
| rw [Equiv.Perm.ofSubtype_apply_of_mem |
| ((currentBulkPositionEquiv m k l).symm.permCongr σ) x.property] |
| rfl |
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|
| @[simp] theorem leafBulkPermutation_apply_leaf (m k l : ℕ) |
| (σ : Equiv.Perm (Fin (2^l) × Fin m)) (x : Fin (2^l) × Fin m) : |
| leafBulkPermutation m k l σ ((currentBulkPositionEquiv m k l).symm x).val= |
| ((currentBulkPositionEquiv m k l).symm (σ x)).val := by |
| rw [leafBulkPermutation_apply_bulk,Equiv.apply_symm_apply] |
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|
| @[simp] theorem leafBulkPermutation_apply_nonbulk (m k l : ℕ) |
| (σ : Equiv.Perm (Fin (2^l) × Fin m)) |
| (i : Fin (Template.current (Template.initial m k) l).length) |
| (hi : (Template.current (Template.initial m k) l)[i].role≠.bulk) : |
| leafBulkPermutation m k l σ i=i := |
| Equiv.Perm.ofSubtype_apply_of_not_mem _ hi |
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|
| theorem leafBulkPermutation_source (b k l : ℕ) (bulk : PrimeSource) |
| (top : Fin 3 → PrimeSource) (comp : Fin k → Fin 2 → PrimeSource) |
| (σ : Equiv.Perm (Fin (2^l) × Fin (2*b))) |
| (i : Fin (Template.current (Template.initial (2*b) k) l).length) : |
| initialSourceFamily b k bulk top comp |
| ((Template.current (Template.initial (2*b) k) l)[leafBulkPermutation (2*b) k l σ i].origin)= |
| initialSourceFamily b k bulk top comp |
| ((Template.current (Template.initial (2*b) k) l)[i].origin) := |
| bulkPositionPermutation_source _ _ bulk (current_bulk_source b k l bulk top comp) _ i |
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| def leafBulkAssignmentPermutation (b k l : ℕ) (bulk : PrimeSource) |
| (top : Fin 3 → PrimeSource) (comp : Fin k → Fin 2 → PrimeSource) |
| (σ : Equiv.Perm (Fin (2^l) × Fin (2*b))) : |
| SourceAssignment (initialSourceFamily b k bulk top comp) (Template.current (Template.initial (2*b) k) l) ≃ |
| SourceAssignment (initialSourceFamily b k bulk top comp) (Template.current (Template.initial (2*b) k) l) := |
| sourceAssignmentPermutation _ _ (leafBulkPermutation (2*b) k l σ) |
| (leafBulkPermutation_source b k l bulk top comp σ) |
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|
| @[simp] theorem leafBulkAssignmentPermutation_val (b k l : ℕ) (bulk : PrimeSource) |
| (top : Fin 3 → PrimeSource) (comp : Fin k → Fin 2 → PrimeSource) |
| (σ : Equiv.Perm (Fin (2^l) × Fin (2*b))) |
| (x : SourceAssignment (initialSourceFamily b k bulk top comp) (Template.current (Template.initial (2*b) k) l)) |
| (i : Fin (Template.current (Template.initial (2*b) k) l).length) : |
| (leafBulkAssignmentPermutation b k l bulk top comp σ x i).val= |
| (x (leafBulkPermutation (2*b) k l σ i)).val := |
| sourceAssignmentPermutation_val _ _ _ _ _ _ |
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| @[simp] theorem leafBulkAssignmentPermutation_leaf_val (b k l : ℕ) (bulk : PrimeSource) |
| (top : Fin 3 → PrimeSource) (comp : Fin k → Fin 2 → PrimeSource) |
| (σ : Equiv.Perm (Fin (2^l) × Fin (2*b))) |
| (x : SourceAssignment (initialSourceFamily b k bulk top comp) (Template.current (Template.initial (2*b) k) l)) |
| (u : Fin (2^l) × Fin (2*b)) : |
| (leafBulkAssignmentPermutation b k l bulk top comp σ x |
| ((currentBulkPositionEquiv (2*b) k l).symm u).val).val= |
| (x ((currentBulkPositionEquiv (2*b) k l).symm (σ u)).val).val := by |
| rw [leafBulkAssignmentPermutation_val,leafBulkPermutation_apply_leaf] |
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| theorem leafBulkAssignmentPermutation_mass (b k l : ℕ) (bulk : PrimeSource) |
| (top : Fin 3 → PrimeSource) (comp : Fin k → Fin 2 → PrimeSource) |
| (σ : Equiv.Perm (Fin (2^l) × Fin (2*b))) |
| (x : SourceAssignment (initialSourceFamily b k bulk top comp) (Template.current (Template.initial (2*b) k) l)) : |
| (assignmentPrior (initialSourceFamily b k bulk top comp) (Template.current (Template.initial (2*b) k) l)).mass |
| (leafBulkAssignmentPermutation b k l bulk top comp σ x)= |
| (assignmentPrior (initialSourceFamily b k bulk top comp) (Template.current (Template.initial (2*b) k) l)).mass x := |
| sourceAssignmentPermutation_mass _ _ _ _ _ |
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| end Ostmann.Conclusion |
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| end |
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| end OAI |
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