File size: 3,572 Bytes
97c3405 | 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 79 80 81 82 83 84 85 | import OAI.Combinatorics.ContingencyTables.Dense.DenseWordProgram
namespace OAI
/-! # Polynomial partition of a supplied Boolean word into fixed-width rows -/
namespace ContingencyTables.DensePrograms
open MatchingFPRAS.TreeTyped Polynomial
def wordChunk (z : (ℕ×List Bool)×ℕ) : List Bool :=
(z.1.2.drop (z.2*z.1.1)).take z.1.1
def wordChunkRealizer : Realizer wordChunk :=
composition (pair
(composition (pair (composition first second)
(composition (pair second (composition first first)) multiply)) dropFast)
(composition first first)) takeFast
theorem polynomial_wordChunk : PolynomialTime wordChunkRealizer :=
polynomial_composition (polynomial_pair
(polynomial_composition (polynomial_pair
(polynomial_composition polynomial_first polynomial_second)
(polynomial_composition (polynomial_pair polynomial_second
(polynomial_composition polynomial_first polynomial_first)) polynomial_multiply))
polynomial_dropFast)
(polynomial_composition polynomial_first polynomial_first)) polynomial_takeFast
def completeWordRows (z : ℕ×List Bool) : List ℕ := List.range (z.2.length/z.1)
def completeWordRowsRealizer : Realizer completeWordRows :=
composition (composition (pair (composition second listLength) first) quotient) range
theorem polynomial_completeWordRows : PolynomialTime completeWordRowsRealizer := by
apply polynomial_boundedRange
(polynomial_composition (polynomial_pair
(polynomial_composition polynomial_second polynomial_listLength) polynomial_first)
polynomial_quotient) X
intro z
simp only [eval_X,Function.comp_apply]
exact (Nat.div_le_self _ _).trans ((list_length_le_weight _).trans (weight_snd z))
def wordChunks (z : ℕ×List Bool) : List (List Bool) :=
(completeWordRows z).map (fun i => wordChunk (z,i))
def wordChunksRealizer : Realizer wordChunks :=
composition (pair identity completeWordRowsRealizer) (mapWith wordChunkRealizer)
theorem polynomial_wordChunks : PolynomialTime wordChunksRealizer :=
polynomial_composition (polynomial_pair polynomial_identity polynomial_completeWordRows)
(polynomial_mapWith polynomial_wordChunk)
lemma wordChunk_ofFn {n k : ℕ} (bits : Fin (n*k) → Bool) (i : Fin n) :
wordChunk ((k,List.ofFn bits),i.val)=
List.ofFn (fun j : Fin k => bits (finProdFinEquiv (i,j))) := by
have hbound : k ≤ n*k-i.val*k := by
apply Nat.le_sub_of_add_le
simpa only [Nat.succ_mul,Nat.add_comm] using
Nat.mul_le_mul_right k (Nat.succ_le_of_lt i.isLt)
apply List.ext_getElem
· simp only [wordChunk,List.length_take,List.length_drop,List.length_ofFn,min_eq_left hbound]
· intro j _ _
simp only [wordChunk,List.getElem_take,List.getElem_drop,List.getElem_ofFn]
apply congrArg bits
apply Fin.ext
change i.val*k+j=j+k*i.val
ac_rfl
theorem wordChunks_ofFn {n k : ℕ} (hk : 0<k) (bits : Fin (n*k) → Bool) :
wordChunks (k,List.ofFn bits)=
List.ofFn (fun i : Fin n => List.ofFn (fun j : Fin k => bits (finProdFinEquiv (i,j)))) := by
have hrows : completeWordRows (k,List.ofFn bits)=List.range n := by
simp only [completeWordRows,List.length_ofFn,Nat.mul_div_cancel _ hk]
unfold wordChunks
rw [hrows]
apply List.ext_getElem
· simp only [List.length_map,List.length_range,List.length_ofFn]
· intro i hi _
have hin : i<n := by simpa only [List.length_map,List.length_range] using hi
simp only [List.getElem_map,List.getElem_range,List.getElem_ofFn]
exact wordChunk_ofFn bits ⟨i,hin⟩
end ContingencyTables.DensePrograms
end OAI
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