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62291ee | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 | import OAI.NumberTheory.Ostmann.Conclusion.BulkPositionShape
namespace OAI
noncomputable section
namespace Ostmann.Conclusion
open Construction
@[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
@[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
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
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]
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
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]
@[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]
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]
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)
end Ostmann.Conclusion
end
end OAI
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