| import OAI.Computability.BinPacking.Computation.MachineCanonicalOutput | |
| namespace OAI | |
| namespace BinPackingGap.ExtensionMachineSpace | |
| open Turing BinPackingGames.Foundations.Complexity | |
| def lengthSum {K : Type} {Γ : K → Type} (chosen : List K) | |
| (tapes : ∀ k, List (Γ k)) : Nat := | |
| (chosen.map fun k => (tapes k).length).sum | |
| theorem execution_stackLength (tm : FinTM2) (k : tm.K) | |
| {start finish : tm.Cfg} {budget : Nat} | |
| (run : StateTransition.EvalsToInTime tm.step start (some finish) budget) : | |
| (finish.stk k).length ≤ (start.stk k).length + budget * Runtime.programPushBound tm := | |
| Runtime.executionSizeBound tm.step (fun cfg => (cfg.stk k).length) | |
| (Runtime.programPushBound tm) (Runtime.stepStackLength tm k) run | |
| theorem execution_lengthSum (tm : FinTM2) (chosen : List tm.K) | |
| {start finish : tm.Cfg} {budget : Nat} | |
| (run : StateTransition.EvalsToInTime tm.step start (some finish) budget) : | |
| lengthSum chosen finish.stk ≤ | |
| lengthSum chosen start.stk + chosen.length * (budget * Runtime.programPushBound tm) := by | |
| induction chosen with | |
| | nil => simp [lengthSum] | |
| | cons k chosen ih => | |
| have stepBound := execution_stackLength tm k run | |
| simp only [lengthSum, List.map_cons, List.sum_cons, List.length_cons, | |
| Nat.add_mul, Nat.one_mul] at ih ⊢ | |
| omega | |
| theorem execution_uniformStackLength (tm : FinTM2) | |
| {start finish : tm.Cfg} {budget initialBound : Nat} | |
| (initial : ∀ k, (start.stk k).length ≤ initialBound) | |
| (run : StateTransition.EvalsToInTime tm.step start (some finish) budget) : | |
| ∀ k, (finish.stk k).length ≤ initialBound + budget * Runtime.programPushBound tm := by | |
| intro k | |
| exact (execution_stackLength tm k run).trans (Nat.add_le_add_right (initial k) _) | |
| theorem lengthSum_le_uniform {K : Type} {Γ : K → Type} (chosen : List K) | |
| (tapes : ∀ k, List (Γ k)) (bound : Nat) (bounded : ∀ k, (tapes k).length ≤ bound) : | |
| lengthSum chosen tapes ≤ chosen.length * bound := by | |
| induction chosen with | |
| | nil => simp [lengthSum] | |
| | cons k chosen ih => | |
| have head := bounded k | |
| simp only [lengthSum, List.map_cons, List.sum_cons, List.length_cons, | |
| Nat.add_mul, Nat.one_mul] at ih ⊢ | |
| omega | |
| theorem lengthSum_eq_cleanup {K : Type} (chosen : List K) (tapes : K → List Bool) : | |
| lengthSum chosen tapes = MachineDrainMany.lengthSum chosen tapes := rfl | |
| end BinPackingGap.ExtensionMachineSpace | |
| end OAI | |