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8df23f4 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 | 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
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