| import Mathlib |
| import OAI.Computability.MinUncut.Search.CodeArithmetic |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace MinUncut.Costed.PolyProgram |
| open Turing.ToPartrec Polynomial |
|
|
| def pairPack : PolyProgram (fun v=>[v.headI,v.tail.headI]) := |
| head.cons ((head.comp tail).cons nil) |
|
|
| def binary (c : Code) (f : ℕ → ℕ → ℕ) (p : Polynomial ℕ) |
| (hr : ∀a b,Runs c [a,b] [f a b] (p.eval (magnitude [a,b]))) |
| (hs : ∀a b,magnitude [f a b]≤p.eval (magnitude [a,b])) : |
| PolyProgram (fun v=>[f v.headI v.tail.headI]) where |
| code := .comp c pairPack.code |
| bound := pairPack.bound+p.comp pairPack.bound+1 |
| run v := by |
| have h := run_comp (pairPack.run v) (hr v.headI v.tail.headI) |
| apply h.mono |
| simp only [eval_add,eval_comp,eval_one] |
| have h' := evalNat_mono p (pairPack.size v) |
| omega |
| size v := by |
| have h := (hs v.headI v.tail.headI).trans (evalNat_mono p (pairPack.size v)) |
| simp only [eval_add,eval_comp,eval_one] |
| omega |
|
|
| def add : PolyProgram (fun v=>[v.headI+v.tail.headI]) := |
| binary addCode (fun a b=>a+b) (C 100000*(X+2)^2) |
| (by intro a b; simpa [Nat.add_comm] using run_add a b []) |
| (by intro a b; simp only [magnitude_cons,magnitude_nil,eval_mul,eval_C,eval_pow, |
| eval_add,eval_X,eval_ofNat]; nlinarith) |
|
|
| def sub : PolyProgram (fun v=>[v.headI-v.tail.headI]) := |
| (binary subCode (fun a b=>b-a) (C 100000*(X+2)^2) |
| (by intro a b; simpa using run_sub a b []) |
| (by intro a b; simp only [magnitude_cons,magnitude_nil,eval_mul,eval_C,eval_pow, |
| eval_add,eval_X,eval_ofNat]; |
| have h : b-a≤b := Nat.sub_le .. |
| have hp : b+1≤(a+1+(b+1+0)+2)^2 := |
| (by omega : b+1≤a+1+(b+1+0)+2).trans (Nat.le_self_pow (by omega) _) |
| omega)).comp |
| ((head.comp tail).cons (head.cons nil)) |
|
|
| def mul : PolyProgram (fun v=>[v.headI*v.tail.headI]) := |
| binary mulCode (fun a b=>a*b) (C 10000000*(X+2)^6) |
| (by intro a b; simpa [Nat.mul_comm] using run_mul a b []) |
| (by intro a b |
| simp only [magnitude_cons,magnitude_nil,eval_mul,eval_C,eval_pow, |
| eval_add,eval_X,eval_ofNat] |
| have h₁ : a*b+1≤(a+1+(b+1+0)+2)^2 := by nlinarith |
| have h₂ : (a+1+(b+1+0)+2)^2≤(a+1+(b+1+0)+2)^6 := |
| Nat.pow_le_pow_right (by omega) (by omega) |
| omega) |
|
|
| def addOf {f g : List ℕ → List ℕ} (F : PolyProgram f) (G : PolyProgram g) : |
| PolyProgram (fun v=>[(f v).headI+(g v).headI]) := add.comp (F.cons (G.cons nil)) |
| def subOf {f g : List ℕ → List ℕ} (F : PolyProgram f) (G : PolyProgram g) : |
| PolyProgram (fun v=>[(f v).headI-(g v).headI]) := sub.comp (F.cons (G.cons nil)) |
| def mulOf {f g : List ℕ → List ℕ} (F : PolyProgram f) (G : PolyProgram g) : |
| PolyProgram (fun v=>[(f v).headI*(g v).headI]) := mul.comp (F.cons (G.cons nil)) |
|
|
| def eqOf {f g : List ℕ → List ℕ} (F : PolyProgram f) (G : PolyProgram g) : |
| PolyProgram (fun v=>[if (f v).headI=(g v).headI then 1 else 0]) := |
| (branch (addOf (subOf F G) (subOf G F)) (const 1) (const 0)).ofEq (by |
| intro v |
| simp only [List.headI_cons] |
| have h : (f v).headI-(g v).headI+((g v).headI-(f v).headI)=0 ↔ |
| (f v).headI=(g v).headI := by omega |
| simp only [h] |
| split_ifs <;> rfl) |
|
|
| def leOf {f g : List ℕ → List ℕ} (F : PolyProgram f) (G : PolyProgram g) : |
| PolyProgram (fun v=>[if (f v).headI≤(g v).headI then 1 else 0]) := |
| (branch (subOf F G) (const 1) (const 0)).ofEq (by |
| intro v |
| simp only [List.headI_cons,Nat.sub_eq_zero_iff_le] |
| split_ifs <;> rfl) |
|
|
| end MinUncut.Costed.PolyProgram |
|
|
| end |
| end |
|
|
| end OAI |
|
|
