File size: 3,133 Bytes
ea76aac | 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 | import OAI.Computability.MatchingCount.RawRoutine
namespace OAI
section
namespace BinaryMatching.StackCode
open Polynomial
namespace Block
variable {r : ℕ}
def lengthInto (src dst : Fin r) : Block r := loop src ((pop src).seq (push dst true))
lemma lengthInto_runs (src dst : Fin r) (hne : src≠dst) (s : Store r) :
(lengthInto src dst).Runs s
(Function.update (Function.update s src []) dst (unary (s src).length++s dst))
(3*(s src).length+1) := by
generalize hl : s src=l
induction l generalizing s with
| nil =>
have he : Function.update (Function.update s src []) dst (unary 0++s dst)=s := by
simp only [unary_zero,List.nil_append]
rw [←hl,Function.update_eq_self,Function.update_eq_self]
change (lengthInto src dst).Runs s
(Function.update (Function.update s src []) dst (unary 0++s dst)) 1
rw [he]
exact loop_runs_empty src _ s hl
| cons b l ih =>
let s' := Function.update (Function.update s src l) dst (true::s dst)
have hp : (pop src).Runs s (Function.update s src l) 1 := by
simpa [hl] using pop_runs src s
have ht : (push dst true).Runs (Function.update s src l) s' 1 := by
simpa [s',hne.symm] using push_runs dst true (Function.update s src l)
have hi := ih s' (by simp [s',hne])
have he : Function.update (Function.update s' src []) dst (unary l.length++s' dst)=
Function.update (Function.update s src []) dst (unary (b::l).length++s dst) := by
funext k
by_cases hk : k=dst
· subst k
simp only [Function.update_self,s',List.length_cons,unary]
simp only [List.replicate_succ',List.append_assoc,List.singleton_append]
· by_cases hks : k=src
· subst k; simp [s',hne]
· simp [s',Function.update_of_ne,hk,hks]
rw [he] at hi
have h := loop_runs_step src ((pop src).seq (push dst true)) (by simp [hl])
(seq_runs _ _ hp ht) hi
have heq : 2+(3*l.length+1)+1=3*(b::l).length+1 := by simp; omega
rw [heq] at h
exact h
end Block
namespace Routine
variable {α : Type}
noncomputable def codeLength (ea : α → List Bool) : Routine ea unary (fun a => (ea a).length) where
scratch := 1
block := (Block.lengthInto 0 1).seq (Block.reverseInto 1 0)
time := 7*X+2
correct a := by
let s₀ : Store 2 := Function.update (fun _ => []) 0 (ea a)
let s₁ := Function.update (Function.update s₀ 0 []) 1 (unary (ea a).length)
have h₁ : (Block.lengthInto 0 1).Runs s₀ s₁ (3*(ea a).length+1) := by
simpa [s₀,s₁] using Block.lengthInto_runs (0:Fin 2) 1 (by decide) s₀
have h₂ := Block.reverseInto_runs (1:Fin 2) 0 (by decide) s₁
have he : Function.update (Function.update s₁ 1 []) 0 ((s₁ 1).reverse++s₁ 0)=
Function.update (fun _ => []) 0 (unary (ea a).length) := by
funext k
fin_cases k <;> simp [s₁,s₀]
rw [he] at h₂
have hl : (s₁ 1).length=(ea a).length := by simp [s₁]
rw [hl] at h₂
refine ⟨_,?_,Block.seq_runs _ _ h₁ h₂⟩
simp only [eval_add,eval_mul,eval_ofNat,eval_X]
omega
end Routine
end BinaryMatching.StackCode
end
end OAI
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