| import OAI.NumberTheory.Ostmann.Conclusion.BulkPositionShape |
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| namespace OAI |
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| noncomputable section |
| namespace Ostmann.Conclusion |
| open Construction |
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| @[simp] theorem currentBulkPositionEquiv_fst_val (m k l : ℕ) |
| (x : BulkPosition (Template.current (Template.initial m k) l)) : |
| (currentBulkPositionEquiv m k l x).1.val= |
| x.val.val/(bulkLeafTemplate m k l).length := rfl |
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| @[simp] theorem currentBulkPositionEquiv_snd_val (m k l : ℕ) |
| (x : BulkPosition (Template.current (Template.initial m k) l)) : |
| (currentBulkPositionEquiv m k l x).2.val= |
| x.val.val%(bulkLeafTemplate m k l).length := rfl |
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| theorem currentBulkPositionEquiv_leaf_iff (m k l : ℕ) |
| (x : BulkPosition (Template.current (Template.initial m k) l)) (b : Fin (2^l)) : |
| (currentBulkPositionEquiv m k l x).1=b ↔ |
| b.val*(bulkLeafTemplate m k l).length ≤ x.val.val ∧ |
| x.val.val<(b.val+1)*(bulkLeafTemplate m k l).length := by |
| have hm : 0 < m := Nat.zero_lt_of_lt (currentBulkPositionEquiv m k l x).2.isLt |
| have hn : 0 < (bulkLeafTemplate m k l).length := hm.trans_le (bulkLeafTemplate_length_ge m k l) |
| rw [Fin.ext_iff,currentBulkPositionEquiv_fst_val,Nat.div_eq_iff hn,Nat.add_mul,Nat.one_mul] |
| omega |
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| theorem currentBulkPositionEquiv_slot_lt_iff (m k l : ℕ) (b : Fin (2^l)) (i j : Fin m) : |
| ((currentBulkPositionEquiv m k l).symm (b,i)).val.val < |
| ((currentBulkPositionEquiv m k l).symm (b,j)).val.val ↔ i.val<j.val := by |
| simp only [currentBulkPositionEquiv_symm_val,Nat.add_lt_add_iff_left] |
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| theorem currentBulkPositionEquiv_leaf_lt (m k l : ℕ) (b c : Fin (2^l)) |
| (hbc : b.val<c.val) (i j : Fin m) : |
| ((currentBulkPositionEquiv m k l).symm (b,i)).val.val < |
| ((currentBulkPositionEquiv m k l).symm (c,j)).val.val := by |
| rw [currentBulkPositionEquiv_symm_val,currentBulkPositionEquiv_symm_val] |
| have hi := i.isLt.trans_le (bulkLeafTemplate_length_ge m k l) |
| have hmul := Nat.mul_le_mul_right (bulkLeafTemplate m k l).length hbc |
| nlinarith |
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| theorem current_remainder_eq_bulkLeafBlocks (m k l : ℕ) : |
| Template.remainder (l+1) (Template.current (Template.initial m k) l)= |
| (List.replicate (2^l) (bulkLeafTemplate m k (l+1))).flatten := by |
| rw [current_eq_bulkLeafBlocks] |
| simp only [Template.remainder,List.filter_flatten,List.map_replicate] |
| have he := (bulkLeafTemplate_succ m k l).symm |
| unfold Template.remainder at he |
| rw [he] |
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| @[simp] theorem current_remainder_bulk_block_length (m k l : ℕ) : |
| (Template.remainder (l+1) (Template.current (Template.initial m k) l)).length= |
| 2^l*(bulkLeafTemplate m k (l+1)).length := by |
| rw [current_remainder_eq_bulkLeafBlocks] |
| simp [List.length_flatten,List.map_replicate] |
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| theorem currentBulkPositionEquiv_left_half_iff (m k l : ℕ) |
| (x : BulkPosition (Template.current (Template.initial m k) (l+1))) : |
| (currentBulkPositionEquiv m k (l+1) x).1.val<2^l ↔ |
| x.val.val<(Template.remainder (l+1) (Template.current (Template.initial m k) l)).length := by |
| have hm : 0 < m := Nat.zero_lt_of_lt (currentBulkPositionEquiv m k (l+1) x).2.isLt |
| have hn := hm.trans_le (bulkLeafTemplate_length_ge m k (l+1)) |
| rw [currentBulkPositionEquiv_fst_val,current_remainder_bulk_block_length, |
| Nat.div_lt_iff_lt_mul hn] |
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| theorem currentBulkPositionEquiv_right_half_iff (m k l : ℕ) |
| (x : BulkPosition (Template.current (Template.initial m k) (l+1))) : |
| 2^l ≤ (currentBulkPositionEquiv m k (l+1) x).1.val ↔ |
| (Template.remainder (l+1) (Template.current (Template.initial m k) l)).length ≤ x.val.val := by |
| simpa only [not_lt] using not_congr (currentBulkPositionEquiv_left_half_iff m k l x) |
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| end Ostmann.Conclusion |
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| end |
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| end OAI |
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