File size: 4,006 Bytes
2f887db | 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 | import Mathlib
import OAI.Computability.MinUncut.Graphs.UnboundedGraphBase
namespace OAI
section
noncomputable section
namespace MinUncut.Costed.Arena
open _root_.Turing _root_.OAI.Turing Turing.ToPartrec _root_.Turing.PartrecToTM2 _root_.OAI.Turing.PartrecToTM2 Polynomial
open MinUncutGames.Foundations.Complexity MinUncutGames.BinaryEncoding
def initializedGraphMachine (c : Code) : FinTM2 :=
MachineSequential.machine (codeMachine c) unboundedGraphMachine id .cons
lemma initializedGraphMachine_finite (c : Code) (k : (initializedGraphMachine c).K) :
Finite ((initializedGraphMachine c).Γ k) :=
sequential_finiteAlphabet _ _ _ _ (codeMachine_finiteAlphabet _) unboundedGraphMachine_finiteAlphabet k
lemma magnitude_words (xs : List ℕ) : magnitude xs=(encodeWords xs).length := by
induction xs with
| nil => rfl
| cons a xs ih=> simp only [magnitude_cons,encodeWords,List.length_append,encodeWord_length,ih]
lemma exists_graph_initializer {f : ℕ → List ℕ → List ℕ} (P : ∀K,PolyProgram (f K))
(he : Computable (fun K=>(P K).code)) :
∃c : Code,∀K,∃p : Polynomial ℕ,∀xs n table k junk,
f K xs=n::((table : List Bool).map Bool.toNat++2::k::junk) →
Nonempty (TM2OutputsInTime (initializedGraphMachine c)
(trList (K::(encodeWords xs).map FrameDecode.symbol))
(some (nameBits n++table++nameBits k)) (p.eval (encodeWords xs).length)) := by
obtain ⟨c,hc⟩:=Init.exists_initializer he
refine ⟨c,fun K=>?_⟩
obtain ⟨p,hp⟩:=hc K
let q:Polynomial ℕ:=p.comp (C (K+1)+C 3*X)
let r:Polynomial ℕ:=q+1+C 2*(q+1)+unboundedGraphTime.comp ((P K).bound+1+q)
refine ⟨r,fun xs n table k junk ho=>?_⟩
let v:=(encodeWords xs).map FrameDecode.symbol
let b:=Init.base (P K).code xs K v
let w:=(initialState (P K).code xs b).encode
have hs:magnitude (K::v)≤(C (K+1)+C 3*X : Polynomial ℕ).eval (encodeWords xs).length:=by
have hh:=FrameDecode.magnitude_symbols (encodeWords xs)
simp only [magnitude_cons,eval_add,eval_C,eval_mul,eval_X]
dsimp only [v]
omega
have hpbound:=evalNat_mono p hs
have hir: Runs c (K::v) w (q.eval (encodeWords xs).length):=
(hp xs).1.mono (by simpa only [q,eval_comp] using hpbound)
have his: magnitude w≤q.eval (encodeWords xs).length:=
(hp xs).2.trans (by simpa only [q,eval_comp] using hpbound)
have hg:Runs (P K).code xs (n::(table.map Bool.toNat++2::k::junk))
((P K).bound.eval (encodeWords xs).length):=by
have hh:=(P K).run xs
rw [ho,magnitude_words] at hh
exact hh
have hl:(n::(table.map Bool.toNat++2::k::junk)).length≤(P K).bound.eval (encodeWords xs).length:=by
have hh:=(magnitude_length (f K xs)).trans ((P K).size xs)
simpa only [ho,magnitude_words] using hh
let h2:=unboundedGraphMachine_outputs_base b hg hl
obtain ⟨t,ht,hr⟩:=hir
let h1:=hr.finiteMachine
let h2':TM2OutputsInTime unboundedGraphMachine ((trList w).map id)
(some (nameBits n++table++nameBits k))
(unboundedGraphTime.eval (magnitude ((P K).bound.eval (encodeWords xs).length::w))) := by
have input_eq : @List.map (unboundedGraphMachine.Γ unboundedGraphMachine.k₀)
(unboundedGraphMachine.Γ unboundedGraphMachine.k₀) id (trList w) = trList w :=
List.map_id _
simpa only [input_eq] using h2
have hh:=MachineSequential.execute (codeMachine c) unboundedGraphMachine id Γ'.cons
_ _ _ _ _ h1 h2'
have hwb: (trList w).length≤q.eval (encodeWords xs).length:= (bitsize_le_magnitude w).trans his
have hm:magnitude ((P K).bound.eval (encodeWords xs).length::w)≤
(((P K).bound+1+q).eval (encodeWords xs).length):=by
simp only [magnitude_cons,eval_add,eval_one]
omega
have hb:=evalNat_mono unboundedGraphTime hm
simp only [eval_add,eval_one] at hb
exact ⟨⟨hh.toEvalsTo,hh.steps_le_m.trans (by
simp only [r,eval_add,eval_one,eval_mul,eval_C,eval_comp]
change t+1+2*((trList w).length+1)+_≤_
omega)⟩⟩
end MinUncut.Costed.Arena
end
end
end OAI
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