| import Mathlib | |
| import OAI.Computability.MinUncut.Machines.Runtime | |
| import OAI.Computability.MinUncut.Estimates.OutputSize | |
| namespace OAI | |
| section | |
| namespace MinUncut | |
| open Turing MinUncutGames.Foundations.Complexity | |
| lemma Output.vertices_le_bits (G : Output) : G.vertices≤G.bits.length := by | |
| have h:G.vertices≤G.vertices^2:=by | |
| by_cases hz:G.vertices=0 | |
| · simp [hz] | |
| · have hp:1≤G.vertices:=Nat.one_le_iff_ne_zero.mpr hz | |
| simpa only [pow_two,Nat.mul_one] using Nat.mul_le_mul_left G.vertices hp | |
| rw [G.bits_length] | |
| omega | |
| lemma graph_size_of_runtime (M : FinTM2) (G : Output) | |
| (input : List (M.Γ M.k₀)) (out : List (M.Γ M.k₁)) (B : ℕ) | |
| (hlen : out.length=G.bits.length) | |
| (hr : TM2OutputsInTime M input (some out) B) : | |
| G.vertices+G.bits.length≤2*(input.length+B*Runtime.programPushBound M) := by | |
| have hb:=Runtime.outputLength_le M input out B hr | |
| rw [hlen] at hb | |
| calc | |
| _ ≤ G.bits.length+G.bits.length := Nat.add_le_add_right G.vertices_le_bits _ | |
| _ = 2*G.bits.length := (two_mul _).symm | |
| _ ≤ _ := Nat.mul_le_mul_left 2 hb | |
| end MinUncut | |
| end | |
| end OAI | |