| import OAI.Computability.MatchingCount.OptionCases |
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| namespace OAI |
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| section |
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| namespace BinaryMatching.Raw |
| open StackCode Polynomial |
| abbrev SearchContext := ℕ×(Matrix ℕ×(Matrix ℤ×ℕ)) |
| abbrev SearchState := SearchContext×Paths |
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| def searchStep (s : SearchState) : SearchState := |
| (s.1,pathStep s.1.1 s.1.2.1 s.1.2.2.1 s.2) |
| def searchInitial (z : SearchContext×ℕ) : SearchState := |
| (z.1,initial z.1.1 z.1.2.2.2) |
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| lemma searchStep_iterate (z : SearchContext×ℕ) (j : ℕ) : |
| searchStep^[j] (searchInitial z)=(z.1,search z.1.1 z.1.2.1 z.1.2.2.1 z.1.2.2.2 j) := by |
| induction j with |
| | zero => rfl |
| | succ j ih => simp only [Function.iterate_succ_apply',ih,searchStep,search] |
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| namespace Compiler |
| open StackCode.Poly |
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| @[fun_prop] theorem residual_poly : |
| Poly (fun z : (Matrix ℕ×Matrix ℤ)×(ℕ×ℕ) => residual z.1.1 z.1.2 z.2.1 z.2.2) := by |
| unfold residual |
| fun_prop |
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| @[fun_prop] theorem initial_poly : Poly (fun z : ℕ×ℕ => initial z.1 z.2) := by |
| unfold initial |
| apply Poly.rawTab Prod.fst (fun z : (ℕ×ℕ)×ℕ => if z.2=z.1.2 then Option.some [z.1.2] else none) |
| · fun_prop |
| · have h : Poly (fun z : (ℕ×ℕ)×ℕ => if decide (z.2=z.1.2) then Option.some [z.1.2] else none) := by fun_prop |
| exact h.congr (fun z => by simp) |
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| @[fun_prop] theorem predecessor_poly : |
| Poly (fun z : ((ℕ×(Matrix ℕ×Matrix ℤ))×Paths)×ℕ => |
| predecessor z.1.1.1 z.1.1.2.1 z.1.1.2.2 z.1.2 z.2) := by |
| unfold predecessor |
| apply Poly.findWith (fun z => List.range z.1.1.1) |
| (fun z : (((ℕ×(Matrix ℕ×Matrix ℤ))×Paths)×ℕ)×ℕ => |
| residual z.1.1.1.2.1 z.1.1.1.2.2 z.2 z.1.2 && (nth z.1.1.2 z.2).isSome) |
| · fun_prop |
| · have hr := Poly.comp _ (fun z : (((ℕ×(Matrix ℕ×Matrix ℤ))×Paths)×ℕ)×ℕ => |
| ((z.1.1.1.2.1,z.1.1.1.2.2),(z.2,z.1.2))) residual_poly (by fun_prop) |
| have hs : Poly (fun z : (((ℕ×(Matrix ℕ×Matrix ℤ))×Paths)×ℕ)×ℕ => |
| (nth z.1.1.2 z.2).isSome) := by fun_prop |
| exact Poly.comp _ _ Poly.boolAnd (Poly.pair _ _ hr hs) |
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| def pathEntry (z : ((ℕ×(Matrix ℕ×Matrix ℤ))×Paths)×ℕ) : Option (List ℕ) := |
| match nth z.1.2 z.2 with |
| | Option.some l => Option.some l |
| | none => match predecessor z.1.1.1 z.1.1.2.1 z.1.1.2.2 z.1.2 z.2 with |
| | none => none |
| | Option.some u => (nth z.1.2 u).map (fun l => z.2::l) |
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| @[fun_prop] theorem pathEntry_poly : Poly pathEntry := by |
| have h := Poly.optionCasesWith |
| (fun z : ((ℕ×(Matrix ℕ×Matrix ℤ))×Paths)×ℕ => nth z.1.2 z.2) |
| (fun z => match predecessor z.1.1.1 z.1.1.2.1 z.1.1.2.2 z.1.2 z.2 with |
| | none => none |
| | Option.some u => (nth z.1.2 u).map (fun l => z.2::l)) |
| (fun z => Option.some z.2) |
| (by fun_prop) |
| (by |
| have hi := Poly.optionCasesWith |
| (fun z : ((ℕ×(Matrix ℕ×Matrix ℤ))×Paths)×ℕ => predecessor z.1.1.1 z.1.1.2.1 z.1.1.2.2 z.1.2 z.2) |
| (fun _ => (none:Option (List ℕ))) |
| (fun z => (nth z.1.1.2 z.2).map (fun l => z.1.2::l)) |
| predecessor_poly (by fun_prop) |
| (Poly.optionMapWith (fun z : (((ℕ×(Matrix ℕ×Matrix ℤ))×Paths)×ℕ)×ℕ => nth z.1.1.2 z.2) |
| (fun z => z.1.1.2::z.2) (by fun_prop) (by fun_prop)) |
| exact hi.congr (fun z => by cases predecessor z.1.1.1 z.1.1.2.1 z.1.1.2.2 z.1.2 z.2 <;> rfl)) |
| (by fun_prop) |
| exact h.congr (fun z => by unfold pathEntry; cases nth z.1.2 z.2 <;> rfl) |
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| @[fun_prop] theorem pathStep_poly : |
| Poly (fun z : (ℕ×(Matrix ℕ×Matrix ℤ))×Paths => pathStep z.1.1 z.1.2.1 z.1.2.2 z.2) := by |
| exact Poly.rawTab (fun z => z.1.1) pathEntry (by fun_prop) pathEntry_poly |
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| @[fun_prop] theorem searchStep_poly : Poly searchStep := by |
| unfold searchStep |
| apply Poly.pair |
| · fun_prop |
| · exact Poly.comp _ (fun s : SearchState => ((s.1.1,(s.1.2.1,s.1.2.2.1)),s.2)) |
| pathStep_poly (by fun_prop) |
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| @[fun_prop] theorem searchInitial_poly : Poly searchInitial := by |
| unfold searchInitial |
| apply Poly.pair |
| · fun_prop |
| · exact Poly.comp _ (fun z : SearchContext×ℕ => (z.1.1,z.1.2.2.2)) initial_poly (by fun_prop) |
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| @[fun_prop] theorem search_poly : Poly (fun z : SearchContext×ℕ => |
| search z.1.1 z.1.2.1 z.1.2.2.1 z.1.2.2.2 z.2) := by |
| have hi := Poly.iterateFrom searchStep searchStep_poly Prod.snd searchInitial (by fun_prop) |
| searchInitial_poly (100*(X+1)^3) (by |
| intro z j hj |
| rw [searchStep_iterate] |
| let N := (encode (z.2,searchInitial z)).length |
| have hctx : (encode z.1).length≤N := by |
| simp only [N,searchInitial,code_pair_length,code_nat_length]; omega |
| have hn : z.1.1≤N := by |
| have := (pair_left_le z.1).trans hctx |
| simpa only [code_nat_length] using this |
| have ha : z.1.2.2.2≤N := by |
| have := (pair_right_le z.1.2.2).trans ((pair_right_le z.1.2).trans ((pair_right_le z.1).trans hctx)) |
| simpa only [code_nat_length] using this |
| have hk : j≤N := by |
| simp only [N,code_pair_length,code_nat_length]; omega |
| have hb := search_code_bound z.1.1 z.1.2.1 z.1.2.2.1 z.1.2.2.2 j |
| have hm : z.1.1*(2*(4*z.1.2.2.2+3+j*(4*z.1.1+2))+1)≤ |
| N*(2*(4*N+3+N*(4*N+2))+1) := by gcongr |
| simp only [eval_mul,eval_ofNat,eval_pow,eval_add,eval_X,eval_one] |
| change (encode (z.1,search z.1.1 z.1.2.1 z.1.2.2.1 z.1.2.2.2 j)).length≤100*(N+1)^3 |
| rw [code_pair_length] |
| have hbb := hb.trans hm |
| nlinarith) |
| have h := Poly.second _ hi |
| exact h.congr (fun z => by rw [searchStep_iterate]) |
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| end Compiler |
| end BinaryMatching.Raw |
| end |
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| end OAI |
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