File size: 3,138 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 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | import OAI.Combinatorics.PerfectMatching.SizeBounds
namespace OAI
section
namespace BinaryMatching.Algorithm
open Finset
def tau (G : Input) : ℕ := G.n+B G+1
def Q (G : Input) : ℕ := D₀ G*2^(K G.n)
lemma tau_pos (G : Input) : 0<tau G := by unfold tau; omega
lemma order_le_tau (G : Input) : G.n≤tau G := by unfold tau; omega
lemma B_le_tau (G : Input) : B G≤tau G := by unfold tau; omega
lemma K_le_q (n : ℕ) : K n≤q n :=
Nat.clog_le_of_le_pow (Nat.le_of_lt (Nat.lt_two_pow_self))
lemma q_le_tau_sq (G : Input) : q G.n≤(tau G)^2 :=
q_le_square.trans (Nat.pow_le_pow_left (order_le_tau G) 2)
lemma tau_pow_mono (G : Input) {i j : ℕ} (h : i≤j) : (tau G)^i≤(tau G)^j :=
Nat.pow_le_pow_right (tau_pos G) h
lemma K_L_bound (G : Input) : K G.n+L G≤101*(tau G)^3 := by
have hk : K G.n≤(tau G)^3 :=
(K_le_q G.n).trans ((q_le_tau_sq G).trans (tau_pow_mono G (by omega)))
change K G.n+100*(tau G)^3≤_
omega
lemma A₀_bound (G : Input) : 2+A₀ G.n≤12*(tau G)^2 := by
have hn := Nat.pow_le_pow_left (order_le_tau G) 2
have ht : 1≤(tau G)^2 := one_le_pow₀ (tau_pos G)
unfold A₀
omega
lemma M₀_bound (G : Input) : M₀ G≤1212*(tau G)^7 := by
calc
_ ≤ (tau G)^2*(101*(tau G)^3)*(12*(tau G)^2) :=
Nat.mul_le_mul (Nat.mul_le_mul (q_le_tau_sq G) (K_L_bound G)) (A₀_bound G)
_ = _ := by ring
lemma K_L_M_bound (G : Input) : K G.n+L G+M₀ G≤1313*(tau G)^7 := by
have hk := (K_L_bound G).trans (Nat.mul_le_mul_left 101 (tau_pow_mono G (by omega : 3≤7)))
have hm := M₀_bound G
omega
lemma D₀_bound (G : Input) : D₀ G≤1313^2*(tau G)^16 := by
calc
_ ≤ (tau G)^2*(1313*(tau G)^7)^2 :=
Nat.mul_le_mul (q_le_tau_sq G) (Nat.pow_le_pow_left (K_L_M_bound G) 2)
_ = _ := by ring
lemma iterations_bound (G : Input) : q G.n*D₀ G≤1313^2*(tau G)^18 := by
calc
_ ≤ (tau G)^2*(1313^2*(tau G)^16) := Nat.mul_le_mul (q_le_tau_sq G) (D₀_bound G)
_ = _ := by ring
lemma pow_K_bound (n : ℕ) (hn : 2≤n) : 2^(K n)≤2*q n := by
have hq := q_pos hn
by_cases he : q n=1
· simp [K,he]
· have hq' : 1<q n := by omega
have hk : 0<K n := by
unfold K
rw [Nat.pos_iff_ne_zero]
intro h
have hh := Nat.le_pow_clog (by omega : 1<2) (q n)
rw [h] at hh
simp at hh
omega
have hh := Nat.pow_pred_clog_lt_self (by omega : 1<2) hq'
change 2^(K n-1)<q n at hh
calc
_ = 2^(K n-1)*2 := by rw [←pow_succ]; congr 1; omega
_ ≤ q n*2 := Nat.mul_le_mul_right _ hh.le
_ = _ := by omega
lemma Q_bound (G : Input) (hn : 2≤G.n) : Q G≤2*1313^2*(tau G)^18 := by
calc
_ ≤ (1313^2*(tau G)^16)*(2*(tau G)^2) :=
Nat.mul_le_mul (D₀_bound G) ((pow_K_bound G.n hn).trans (Nat.mul_le_mul_left 2 (q_le_tau_sq G)))
_ = _ := by ring
lemma tau_input_bound (G : Input) (hguard : G.n≤(covered G).card) :
tau G≤3*inputLength G+1 := by
have hn := hguard.trans ((covered_card_le_records G).trans
(Nat.mul_le_mul_left 2 (records_length_le_input G)))
have hb := B_le_input G
unfold tau
omega
end BinaryMatching.Algorithm
end
end OAI
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