File size: 2,602 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.BitAdd
namespace OAI
/-! Polynomial-time approximation of weighted matchings with singleton loops. -/
namespace LoopMatching.BitAdd
open BinaryMatching.StackCode
lemma boolList_code_length (l : List Bool) : (encode l).length = 3*l.length := by
induction l with
| nil => rfl
| cons b l ih =>
simp only [code_cons, pairBits_length, encode_bool, List.length_cons,
List.length_nil, ih]
omega
lemma state_code_length (s : State) :
(encode s).length = 12*s.1.1.length + 6*s.1.2.length + 3*s.2.2.length + 6 := by
simp only [encode_prod, pairBits_length, boolList_code_length, encode_bool,
List.length_singleton]
omega
@[fun_prop] theorem xor_poly : Poly (fun z : Bool × Bool => xor z.1 z.2) := by
unfold xor
fun_prop
@[fun_prop] theorem digit_poly : Poly (fun z : (Bool × Bool) × Bool => digit z.1.1 z.1.2 z.2) := by
unfold digit
fun_prop
@[fun_prop] theorem carry_poly : Poly (fun z : (Bool × Bool) × Bool => carry z.1.1 z.1.2 z.2) := by
unfold carry
fun_prop
@[fun_prop] theorem step_poly : Poly step := by
have hi : Poly (fun z : State × Bool =>
((z.1.1.1.headD false, z.1.1.2.headD false), z.1.2.1)) := by fun_prop
have hc : Poly (fun z : State × Bool =>
carry (z.1.1.1.headD false) (z.1.1.2.headD false) z.1.2.1) :=
Poly.comp _ _ carry_poly hi
have hd : Poly (fun z : State × Bool =>
digit (z.1.1.1.headD false) (z.1.1.2.headD false) z.1.2.1) :=
Poly.comp _ _ digit_poly hi
have ho : Poly (fun z : State × Bool =>
z.1.2.2 ++ [digit (z.1.1.1.headD false) (z.1.1.2.headD false) z.1.2.1]) :=
Poly.comp (fun p : List Bool × Bool => p.1 ++ [p.2]) _ (by fun_prop)
(Poly.pair _ _ (by fun_prop) hd)
unfold step
exact Poly.pair _ _ (by fun_prop) (Poly.pair _ _ hc ho)
lemma step_growth (z : State × Bool) :
(encode (step z)).length ≤ (encode z.1).length + 1*(2*(encode z.2).length+1) := by
rw [state_code_length, state_code_length]
simp only [step, encode_bool, List.length_singleton, List.length_append]
have hx : z.1.1.1.tail.length ≤ z.1.1.1.length := by simp
have hy : z.1.1.2.tail.length ≤ z.1.1.2.length := by simp
omega
@[fun_prop] theorem run_poly : Poly (fun z : List Bool × State => run z.1 z.2) := by
exact Poly.listFoldAdd step step_poly 1 step_growth
@[fun_prop] theorem initial_poly : Poly initial := by
unfold initial
fun_prop
@[fun_prop] theorem finish_poly : Poly finish := by
unfold finish
fun_prop
@[fun_prop] theorem add_poly : Poly add := by
unfold add
fun_prop
end LoopMatching.BitAdd
end OAI
|