File size: 2,315 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 | import OAI.Computability.MatchingCount.RawSearch
namespace OAI
section
namespace BinaryMatching.Raw
open StackCode
def PathSize (p : Paths) (M : ℕ) : Prop := ∀ q∈p,(encode q).length≤M
lemma initial_size (n a : ℕ) : PathSize (initial n a) (4*a+3) := by
intro q hq
obtain ⟨v,hv,rfl⟩ := List.mem_map.mp hq
split_ifs
· simp only [code_option_some,code_cons,code_nat_length,code_nil,List.length_nil,pairBits_length]
omega
· simp only [code_option_none,List.length_nil]; omega
lemma pathStep_size (n : ℕ) (c : Matrix ℕ) (b : Matrix ℤ) (p : Paths) (M : ℕ)
(hp : PathSize p M) : PathSize (pathStep n c b p) (M+4*n+2) := by
intro q hq
obtain ⟨v,hv,rfl⟩ := List.mem_map.mp hq
have hvn := List.mem_range.mp hv
have hsize (u : ℕ) : (encode (nth p u)).length≤M :=
nth_code_bound p u M (Nat.zero_le M) hp
change (encode (match nth p v with
| some l => some l
| none => match predecessor n c b p v with
| none => none
| some u => (nth p u).map (fun l => v::l))).length≤_
cases h : nth p v with
| some l => dsimp only; have hh := hsize v; rw [h] at hh; omega
| none =>
cases hu : predecessor n c b p v with
| none => simp only [code_option_none,List.length_nil]; omega
| some u =>
dsimp only
have hh := option_prepend_bound v (nth p u)
have he := hsize u
omega
lemma search_size (n : ℕ) (c : Matrix ℕ) (b : Matrix ℤ) (a k : ℕ) :
PathSize (search n c b a k) (4*a+3+k*(4*n+2)) := by
induction k with
| zero => simpa only [search,Function.iterate_zero,id_eq,Nat.zero_mul,Nat.add_zero] using initial_size n a
| succ k ih =>
have hh := pathStep_size n c b _ _ ih
simpa only [search,Function.iterate_succ_apply',Nat.add_mul,Nat.one_mul,Nat.add_assoc] using hh
lemma search_length (n : ℕ) (c : Matrix ℕ) (b : Matrix ℤ) (a k : ℕ) :
(search n c b a k).length=n := by
cases k with
| zero => simp [search,initial,tab]
| succ k => simp [search,Function.iterate_succ_apply',pathStep,tab]
lemma search_code_bound (n : ℕ) (c : Matrix ℕ) (b : Matrix ℤ) (a k : ℕ) :
(encode (search n c b a k)).length≤n*(2*(4*a+3+k*(4*n+2))+1) := by
have h := list_code_bound _ _ (search_size n c b a k)
rwa [search_length] at h
end BinaryMatching.Raw
end
end OAI
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