| import OAI.NumberTheory.Ostmann.Conclusion.BulkPositionShapeBasic |
| |
| namespace OAI |
|
|
| noncomputable section |
| namespace Ostmann.Conclusion |
| open Construction |
|
|
| def currentPositionBlockEquiv (m k l : ℕ) : |
| Fin (Template.current (Template.initial m k) l).length ≃ |
| Fin (2^l) × Fin (bulkLeafTemplate m k l).length := |
| (finCongr (current_bulk_block_length m k l)).trans finProdFinEquiv.symm |
|
|
| @[simp] theorem currentPositionBlockEquiv_symm_val (m k l : ℕ) |
| (b : Fin (2^l)) (i : Fin (bulkLeafTemplate m k l).length) : |
| ((currentPositionBlockEquiv m k l).symm (b,i)).val= |
| b.val*(bulkLeafTemplate m k l).length+i.val := by |
| simp [currentPositionBlockEquiv,finProdFinEquiv,Nat.mul_comm,Nat.add_comm] |
| |
| theorem currentPositionBlockEquiv_get (m k l : ℕ) |
| (b : Fin (2^l)) (i : Fin (bulkLeafTemplate m k l).length) : |
| (Template.current (Template.initial m k) l)[((currentPositionBlockEquiv m k l).symm (b,i)).val]= |
| (bulkLeafTemplate m k l)[i.val] := by |
| have he : Template.current (Template.initial m k) l= |
| List.ofFn (Fin.repeat (2^l) (fun j : Fin (bulkLeafTemplate m k l).length => |
| (bulkLeafTemplate m k l)[j.val])) := by |
| rw [List.ofFn_fin_repeat,List.ofFn_getElem] |
| exact current_eq_bulkLeafBlocks m k l |
| simp only [he,List.getElem_ofFn,Fin.repeat,Fin.modNat] |
| simp [currentPositionBlockEquiv_symm_val,Nat.add_mod,Nat.mod_eq_of_lt i.isLt] |
|
|
| theorem bulkLeafTemplate_length_ge (m k l : ℕ) : m ≤ (bulkLeafTemplate m k l).length := by |
| simp [bulkLeafTemplate] |
| |
| def currentBulkPositionEquiv (m k l : ℕ) : |
| BulkPosition (Template.current (Template.initial m k) l) ≃ Fin (2^l) × Fin m where |
| toFun x := ((currentPositionBlockEquiv m k l x.val).1, |
| ⟨(currentPositionBlockEquiv m k l x.val).2.val,by |
| apply (bulkLeafTemplate_bulk_iff m k l _).mp |
| have hh := currentPositionBlockEquiv_get m k l |
| (currentPositionBlockEquiv m k l x.val).1 (currentPositionBlockEquiv m k l x.val).2 |
| simp only [Prod.mk.eta,Equiv.symm_apply_apply] at hh |
| have hp := x.property |
| simp only [Fin.getElem_fin] at hp |
| simpa only [hh] using hp⟩) |
| invFun x := ⟨(currentPositionBlockEquiv m k l).symm |
| (x.1,⟨x.2.val,x.2.isLt.trans_le (bulkLeafTemplate_length_ge m k l)⟩),by |
| simp only [Fin.getElem_fin] |
| rw [currentPositionBlockEquiv_get] |
| exact (bulkLeafTemplate_bulk_iff m k l _).mpr x.2.isLt⟩ |
| left_inv x := by |
| apply Subtype.ext |
| apply (currentPositionBlockEquiv m k l).injective |
| simp only [Equiv.apply_symm_apply] |
| right_inv x := by |
| apply Prod.ext |
| · simp |
| · apply Fin.ext |
| simp |
|
|
| @[simp] theorem currentBulkPositionEquiv_symm_val (m k l : ℕ) |
| (b : Fin (2^l)) (i : Fin m) : |
| ((currentBulkPositionEquiv m k l).symm (b,i)).val.val= |
| b.val*(bulkLeafTemplate m k l).length+i.val := by |
| change ((currentPositionBlockEquiv m k l).symm (b, |
| ⟨i.val,i.isLt.trans_le (bulkLeafTemplate_length_ge m k l)⟩)).val=_ |
| exact currentPositionBlockEquiv_symm_val m k l b _ |
|
|
| @[simp] theorem currentBulkPositionEquiv_origin (m k l : ℕ) |
| (b : Fin (2^l)) (i : Fin m) : |
| (Template.current (Template.initial m k) l)[((currentBulkPositionEquiv m k l).symm (b,i)).val.val].origin=i.val := by |
| change (Template.current (Template.initial m k) l)[((currentPositionBlockEquiv m k l).symm (b, |
| ⟨i.val,i.isLt.trans_le (bulkLeafTemplate_length_ge m k l)⟩)).val].origin=i.val |
| rw [currentPositionBlockEquiv_get] |
| exact bulkLeafTemplate_bulk_origin m k l _ i.isLt |
|
|
| theorem currentBulkPosition_card (m k l : ℕ) : |
| Fintype.card (BulkPosition (Template.current (Template.initial m k) l))=2^l*m := by |
| rw [Fintype.card_congr (currentBulkPositionEquiv m k l)] |
| simp |
|
|
| end Ostmann.Conclusion |
|
|
| end |
|
|
| end OAI |
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|
