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namespace OAI
section
namespace MeanPayoff.RegisterBits.Compile
open BinaryData
open scoped Classical
theorem correct_probability {K : ℕ} (P : Program K) (G : Game) (n : ℕ)
(hh : (tree P n (RegisterBits.initial P (input G))).All (fun c => c.control=none)) :
(2:ℚ)^(2*bound P (input G) n)*
(tree P n (RegisterBits.initial P (input G))).probability (fun c => BinaryInput.CorrectList G c.output)≤
((Finset.univ.filter (fun bits : Fin (2*bound P (input G) n)→Bool =>
CorrectOutput G (result (machine P) G (2*bound P (input G) n) bits))).card : ℚ) := by
classical
have hp := machine_probability P (input G) n hh (BinaryInput.CorrectList G)
apply hp.trans
apply Nat.cast_le.mpr
apply Finset.card_le_card
intro bits hb
simp only [Finset.mem_filter,Finset.mem_univ,true_and] at hb ⊢
obtain ⟨out,ho,hhalt,ht⟩ := hb
exact correctList_packet G (machine P) _ out ho hhalt ht
theorem correct_count {K : ℕ} (P : Program K) (G : Game) (n : ℕ)
(hh : (tree P n (RegisterBits.initial P (input G))).All (fun c => c.control=none))
(hp : 7/8≤(tree P n (RegisterBits.initial P (input G))).probability
(fun c => BinaryInput.CorrectList G c.output)) :
7*2^(2*bound P (input G) n)≤8*(Finset.univ.filter
(fun bits : Fin (2*bound P (input G) n)→Bool =>
CorrectOutput G (result (machine P) G (2*bound P (input G) n) bits))).card := by
classical
have hq := correct_probability P G n hh
have hl : (2:ℚ)^(2*bound P (input G) n)*(7/8)≤
((Finset.univ.filter (fun bits : Fin (2*bound P (input G) n)→Bool =>
CorrectOutput G (result (machine P) G (2*bound P (input G) n) bits))).card : ℚ) :=
(mul_le_mul_of_nonneg_left hp (by positivity)).trans hq
have hr : (7:ℚ)*2^(2*bound P (input G) n)≤8*
((Finset.univ.filter (fun bits : Fin (2*bound P (input G) n)→Bool =>
CorrectOutput G (result (machine P) G (2*bound P (input G) n) bits))).card : ℚ) := by
linarith only [hl]
exact_mod_cast hr
theorem halt_all {K : ℕ} (P : Program K) (G : Game) (n : ℕ)
(hh : (tree P n (RegisterBits.initial P (input G))).All (fun c => c.control=none))
(bits : Fin (2*bound P (input G) n)→Bool) :
(result (machine P) G (2*bound P (input G) n) bits).control=none :=
(machine_packet P (input G) n hh bits).1
end MeanPayoff.RegisterBits.Compile
end
section
namespace MeanPayoff
theorem main : MainClaim := by
obtain ⟨K,e,h⟩ := Realizer.post_polynomial
obtain ⟨C,hC,hbound⟩ := Realizer.total_quasi K e
refine ⟨RegisterBits.Compile.machine Realizer.program,C,hC,fun G =>
⟨Realizer.total K e ⟨input G⟩,hbound ⟨input G⟩,?_,?_⟩⟩
· exact RegisterBits.Compile.halt_all Realizer.program G (Realizer.limit K e ⟨input G⟩)
(Realizer.tree_halts K e h ⟨input G⟩)
· exact RegisterBits.Compile.correct_count Realizer.program G (Realizer.limit K e ⟨input G⟩)
(Realizer.tree_halts K e h ⟨input G⟩) (Realizer.tree_success K e h G)
end MeanPayoff
theorem randomized_quasipolynomial_mean_payoff : MeanPayoff.MainClaim := MeanPayoff.main
end
end OAI
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