File size: 3,411 Bytes
ea76aac | 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.Computability.LoopMatching.Model
import OAI.Computability.MatchingCount.BinaryParser
namespace OAI
/-! Polynomial-time approximation of weighted matchings with singleton loops. -/
namespace LoopMatching.InputMachine
open BinaryMatching BinaryMatching.StackCode Polynomial
def countedRecords (k : ℕ) (l : List Bool) : List Bool × List Parse.RecordBits :=
Parse.recordStep^[k] (l,[])
lemma recordStep_records_suffix (rs : List BinaryMatching.Record)
(suffix : List Bool) (acc : List Parse.RecordBits) :
Parse.recordStep^[rs.length] (rs.flatMap encodeRecord ++ suffix,acc) =
(suffix,acc ++ rs.map Parse.recordRepr) := by
induction rs generalizing acc with
| nil => simp
| cons r rs ih =>
rw [List.length_cons,Function.iterate_succ_apply,List.flatMap_cons,
List.append_assoc,Parse.recordStep_encode,ih]
simp [List.append_assoc]
lemma countedRecords_encoded (rs : List BinaryMatching.Record) (suffix : List Bool) :
countedRecords rs.length (rs.flatMap encodeRecord ++ suffix) =
(suffix,rs.map Parse.recordRepr) := by
simpa only [countedRecords,List.nil_append] using
recordStep_records_suffix rs suffix []
@[fun_prop] theorem countedRecords_poly :
Poly (fun z : ℕ × List Bool => countedRecords z.1 z.2) := by
have hs : Poly Parse.recordStep := by unfold Parse.recordStep; fun_prop
have hi := Poly.boundedIterate Parse.recordStep hs (100*(X+1)^2) (by
intro z j hj
obtain ⟨h₁,h₂⟩ := Parse.recordStep_iterate_size j z.2
have he : (encode (Parse.recordStep^[j] z.2)).length ≤
6*z.2.1.length+(encode z.2.2).length+j*(40*z.2.1.length+21)+1 := by
simp only [code_pair_length,cut_code_length]
omega
have hl : z.2.1.length≤(encode z).length := by
simp only [code_pair_length,code_nat_length,cut_code_length]; omega
have hk : j≤(encode z).length := by
simp only [code_pair_length,code_nat_length]; omega
have ha : (encode z.2.2).length≤(encode z).length := by
simp only [code_pair_length]; omega
have hm := Nat.mul_le_mul hk hl
simp only [eval_mul,eval_ofNat,eval_pow,eval_add,eval_X,eval_one]
nlinarith)
exact Poly.comp _ (fun z : ℕ × List Bool => (z.1,(z.2,([]:List Parse.RecordBits))))
hi (by fun_prop)
def ordinaryPrefix (l : List Bool) : List Bool × (List Bool × List Parse.RecordBits) :=
let n := Parse.numeral l
let r := Parse.numeral n.2
(n.1,countedRecords (Parse.boundedValue l.length r.1) r.2)
@[fun_prop] theorem ordinaryPrefix_poly : Poly ordinaryPrefix := by
unfold ordinaryPrefix
fun_prop
lemma ordinaryPrefix_encoded (G : LoopMatching.Input) :
ordinaryPrefix (LoopMatching.encodeInput G) =
(G.n.bits,(encodeNat G.loops.length ++ G.loops.flatMap encodeLoopRecord,
G.ordinary.records.map Parse.recordRepr)) := by
have hr := Parse.records_length_le_bits G.ordinary.records
have hbound : G.ordinary.records.length ≤ (LoopMatching.encodeInput G).length := by
simp only [LoopMatching.encodeInput,BinaryMatching.encodeInput,List.length_append]
omega
unfold ordinaryPrefix LoopMatching.encodeInput BinaryMatching.encodeInput
simp only [List.append_assoc,Parse.numeral_encode,Parse.boundedValue_bits]
simp only [LoopMatching.encodeInput,BinaryMatching.encodeInput,List.append_assoc] at hbound
rw [min_eq_right hbound,countedRecords_encoded]
end LoopMatching.InputMachine
end OAI
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