File size: 1,397 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 | import OAI.Computability.LoopMatching.BitAdd
namespace OAI
/-! Polynomial-time approximation of weighted matchings with singleton loops. -/
namespace LoopMatching.BitAdd
lemma value_lt_two_pow_length (xs : List Bool) : value xs < 2 ^ xs.length := by
induction xs with
| nil => simp [value]
| cons b xs ih =>
cases b <;>
simp only [value, Bool.toNat_false, Bool.toNat_true, List.length_cons,
pow_succ] <;> omega
lemma length_bits_value_le (xs : List Bool) : (value xs).bits.length ≤ xs.length := by
rw [Nat.size_eq_bits_len]
exact Nat.size_le.mpr (value_lt_two_pow_length xs)
lemma length_bits_value_add (xs ys : List Bool) :
(value xs + value ys).bits.length ≤ max xs.length ys.length + 1 := by
rw [Nat.size_eq_bits_len]
apply Nat.size_le.mpr
have hx : value xs < 2 ^ max xs.length ys.length :=
(value_lt_two_pow_length xs).trans_le
(Nat.pow_le_pow_right (by decide) (Nat.le_max_left _ _))
have hy : value ys < 2 ^ max xs.length ys.length :=
(value_lt_two_pow_length ys).trans_le
(Nat.pow_le_pow_right (by decide) (Nat.le_max_right _ _))
rw [pow_succ]
omega
lemma length_bits_value_add_le (xs ys : List Bool) (a b : ℕ)
(hx : xs.length ≤ a) (hy : ys.length ≤ b) :
(value xs + value ys).bits.length ≤ max a b + 1 := by
exact (length_bits_value_add xs ys).trans (by omega)
end LoopMatching.BitAdd
end OAI
|