File size: 4,179 Bytes
e5a97c6 | 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 | import Mathlib
import OAI.Combinatorics.MatchingCount.Tapes.StackWords
import OAI.Combinatorics.MatchingCount.Tapes.FixedEncoding
import OAI.Combinatorics.MatchingCount.Arithmetic.BinaryCompare
namespace OAI
namespace MatchingFPRAS.PostComparison
open Turing StackCompare
local instance initialOrder : Inhabited Ordering := ⟨.eq⟩
abbrev Alphabet := TM2to1.Γ' (Fin 3) (fun _=>Bool)
lemma support : TM2.Supports program Finset.univ := by
simp [TM2.Supports,program,TM2.SupportsStmt]
noncomputable def machine : RandomMachine :=
StackWords.compiled (FixedEncoding.enc Alphabet) (FixedEncoding.dec Alphabet)
program Finset.univ support
noncomputable def overhead : ℕ :=
StackWords.cost (FixedEncoding.enc Alphabet) (FixedEncoding.dec Alphabet)
program Finset.univ
noncomputable def input (A B : List Bool) : List Symbol :=
StackWords.encoded (FixedEncoding.enc Alphabet) (StackAddition.stackData A B []) (A.length+B.length)
noncomputable def output (c : Ordering) : List Symbol :=
StackWords.encoded (FixedEncoding.enc Alphabet)
(StackAddition.stackData [] [] (flags c)) 2
noncomputable def steps (A B : List Bool) : ℕ :=
overhead*((A.length+B.length+1)*(1+8*(A.length+B.length+4*(A.length+B.length+1)+1)))+2
theorem computes (A B : List Bool) (t : ℕ) (ht : steps A B≤t) (bits : Fin t → Bool) :
(run machine (input A B) bits).state=none ∧
(run machine (input A B) bits).tape.right₀=
ListBlank.mk (output (compare (StackAddition.value A) (StackAddition.value B))) := by
obtain ⟨n,hn,hr⟩:=run_comparison A B [] .eq default
have h₀ : StackCost.StackBound (StackAddition.stackData A B []) (A.length+B.length) := by
intro i
fin_cases i <;> simp [StackAddition.stackData]
have he : MachineCost.run (TM2.step program) n
⟨some (default : Ordering),(default : StackAddition.Store),StackAddition.stackData A B []⟩=
cfg none [] [] (flags (compareDigits .eq A B)) default := by
change MachineCost.run (TM2.step program) n (cfg (some .eq) A B [] default)=_
simpa only [List.append_nil] using hr
have hh : (MachineCost.run (TM2.step program) n
⟨some (default : Ordering),(default : StackAddition.Store),StackAddition.stackData A B []⟩).l=none := by rw [he]; rfl
have ho : StackCost.StackBound (MachineCost.run (TM2.step program) n
⟨some (default : Ordering),(default : StackAddition.Store),StackAddition.stackData A B []⟩).stk 2 := by
rw [he]
change StackCost.StackBound (StackAddition.stackData [] [] (flags (compareDigits .eq A B))) 2
intro i
fin_cases i <;> simp [StackAddition.stackData,flags]
have htime : StackWords.cost (FixedEncoding.enc Alphabet) (FixedEncoding.dec Alphabet)
program Finset.univ*(n*(1+2*4*(A.length+B.length+n*4+1)))+2≤t := by
apply le_trans _ ht
unfold steps overhead
apply Nat.add_le_add_right
apply Nat.mul_le_mul_left
apply Nat.mul_le_mul hn
omega
have hc:=StackWords.compiled_execution (FixedEncoding.enc Alphabet)
(FixedEncoding.dec Alphabet) (FixedEncoding.blank Alphabet) (FixedEncoding.left_inverse Alphabet)
program Finset.univ support 4 accesses_bound
(StackAddition.stackData A B []) (A.length+B.length) h₀ n 2 hh ho t htime bits
change (run machine (input A B) bits).state=none ∧
(run machine (input A B) bits).tape.right₀=ListBlank.mk
(StackWords.encoded (FixedEncoding.enc Alphabet)
(MachineCost.run (TM2.step program) n
⟨some (default : Ordering),(default : StackAddition.Store),StackAddition.stackData A B []⟩).stk
2) at hc
rw [he,compares_values] at hc
exact hc
lemma time_quadratic (A B : List Bool) :
steps A B=overhead*(40*(A.length+B.length)^2+81*(A.length+B.length)+41)+2 := by
unfold steps
congr 2
ring
lemma computes_numerals (m n : ℕ) (t : ℕ) (ht : steps m.bits n.bits≤t) (bits : Fin t → Bool) :
(run machine (input m.bits n.bits) bits).state=none ∧
(run machine (input m.bits n.bits) bits).tape.right₀=ListBlank.mk (output (compare m n)) := by
simpa only [StackAddition.value_bits] using computes m.bits n.bits t ht bits
end MatchingFPRAS.PostComparison
end OAI
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