| 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 |
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|
