File size: 3,538 Bytes
760f90e | 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 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | import OAI.Computability.Scheduling.MachineBridge
namespace OAI
section
namespace ThreeMachine.StackCompiler
theorem enc_request {n : ℕ} (x : Request n) :
enc x = enc (ThreeMachine.encodeInput x.graph x.deadline) := rfl
theorem requestAnswer_time_charge {n : ℕ} (x : Request n) :
Uniform.requestAnswer.time n x =
Uniform.requestAnswer.charge (enc (ThreeMachine.encodeInput x.graph x.deadline)) := by
unfold Uniform.time
exact congrArg Uniform.requestAnswer.charge (enc_request x)
theorem Uniform.exists_requestMachine
(R : Uniform (fun n (x : Request n) => Algorithm.output x.graph x.deadline)) :
∃ k q g, ∃ M : Machine k q g, ∀ n (x : Request n),
M.Produces (ThreeMachine.encodeInput x.graph x.deadline)
(Algorithm.output x.graph x.deadline)
(2000*(R.time n x+(ThreeMachine.encodeInput x.graph x.deadline).length+
(Algorithm.output x.graph x.deadline).length+1)) := by
obtain ⟨k,q,g,M,hM⟩ := R.routine.exists_rawMachine
refine ⟨k,q,g,M,?_⟩
intro n x
have hc := R.correct n x
rw [enc_request] at hc
have hr := hM (ThreeMachine.encodeInput x.graph x.deadline)
(Algorithm.output x.graph x.deadline) hc
simpa only [Uniform.time,enc_request] using hr
def requestRawBound (x : RequestData.Index) : ℕ :=
2000*(Uniform.requestAnswer.time x.1 x.2+(RequestData.bits x).length+
(Algorithm.output x.2.graph x.2.deadline).length+1)
theorem requestRawBound_poly : Poly RequestData.size requestRawBound 150011 := by
have ht := Costs.requestAnswer
have hb := RequestData.bitsPoly
have ho := Costs.requestOutputLength
unfold requestRawBound
poly_auto
abbrev SmallDeadline := {x : RequestData.Index // x.2.deadline.getD 0 ≤ x.1}
theorem requestRawBound_length : Poly (fun x : SmallDeadline => (RequestData.bits x.1).length)
(fun x => requestRawBound x.1) 150020 := by
let s (x : SmallDeadline) := (RequestData.bits x.1).length
have hsize : Poly s (fun x => RequestData.size x.1) 1 := by
apply Poly.of_le (fun x : SmallDeadline => requestSize_le x.1.2 x.2)
exact (Poly.const s 4).mul (Poly.size s)
have h := Poly.reparam (r := 1) requestRawBound_poly (Subtype.val : SmallDeadline → RequestData.Index) hsize
exact h.lift (by omega)
end ThreeMachine.StackCompiler
namespace ThreeMachine
open StackCompiler
theorem main_theorem : ReleaseTheorem := by
obtain ⟨k,q,g,M,hM⟩ := Uniform.exists_requestMachine Uniform.requestAnswer
obtain ⟨C,hC⟩ := requestRawBound_length
refine ⟨k,q,g,M,C+1,by omega,?_⟩
intro n G hn hG deadline hd
let x : Request n := ⟨G,deadline⟩
have hsmall : deadline.getD 0 ≤ n := by
cases deadline with
| none => simp only [Option.getD_none]; omega
| some T => exact (hd T (by simp)).2
let y : SmallDeadline := ⟨⟨n,x⟩,hsmall⟩
refine ⟨Algorithm.output G deadline,Algorithm.output_correct G hn hG deadline,?_⟩
have hr := hM n x
have hb : 2000*(Uniform.requestAnswer.time n x+
(encodeInput G deadline).length+(Algorithm.output G deadline).length+1) ≤
(C+1)*((encodeInput G deadline).length+2)^150020 := by
calc
_ = requestRawBound y.1 := rfl
_ ≤ C*((encodeInput G deadline).length+2)^150020 := by
have hy := hC y
dsimp only [RequestData.bits,y,x] at hy
exact hy
_ ≤ (C+1)*((encodeInput G deadline).length+2)^150020 :=
Nat.mul_le_mul_right _ (Nat.le_succ C)
obtain ⟨t,ht,halt,out⟩ := hr
exact ⟨t,ht.trans hb,halt,out⟩
end ThreeMachine
end
end OAI
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