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namespace OAI
/-!
# Uniform finite-cut embeddings at recursive capacities
-/
namespace EditUpperParameters
open scoped BigOperators
lemma log_two_bounds : (1/2 : ℝ) ≤ Real.log 2 ∧ Real.log 2 ≤ 1 := by
constructor
· have hh := Real.one_sub_inv_le_log_of_pos (by norm_num : (0 : ℝ) < 2)
norm_num at hh ⊢
exact hh
· have hh := Real.log_le_sub_one_of_pos (by norm_num : (0 : ℝ) < 2)
norm_num at hh ⊢
exact hh
lemma scale_count_bound {l : ℕ} (hl : 0 < l) :
((Nat.log 2 l+1 : ℕ) : ℝ) ≤ 2*(1+Real.log (l : ℝ)) := by
have hpow := Nat.pow_log_le_self 2 (Nat.ne_of_gt hl)
have hpr : (2 : ℝ)^(Nat.log 2 l) ≤ l := by exact_mod_cast hpow
have hh := Real.log_le_log (pow_pos (by norm_num : (0 : ℝ)<2) _) hpr
rw [Real.log_pow] at hh
have hlog : 0 ≤ Real.log (l : ℝ) := Real.log_nonneg (by exact_mod_cast hl)
have hm : 0 ≤ (Nat.log 2 l : ℝ) := Nat.cast_nonneg _
have h2 := log_two_bounds
push_cast
nlinarith
lemma log_square_bound {l : ℕ} (hl : 0 < l) :
Real.log (2*(l : ℝ)^2) ≤ 2*(1+Real.log (l : ℝ)) := by
have hp : (0 : ℝ)<l := by exact_mod_cast hl
rw [Real.log_mul (by norm_num) (pow_ne_zero 2 hp.ne'),Real.log_pow]
have h2 := log_two_bounds
norm_num only [Nat.cast_ofNat]
linarith
lemma step_constant_bound {b l : ℕ} (hl : 0 < l) {D : ℝ} (hD : 0 ≤ D) :
16*((Nat.log 2 l+1 : ℕ) : ℝ)*(2*(b : ℝ)+4*Real.log (2*(l : ℝ)^2)*D) ≤
512*(1+Real.log (l : ℝ))^2*((b : ℝ)+D) := by
have hlog : 0 ≤ Real.log (l : ℝ) := Real.log_nonneg (by exact_mod_cast hl)
have hcount := scale_count_bound hl
have hsq := log_square_bound hl
have hsnn : 0 ≤ Real.log (2*(l : ℝ)^2) := Real.log_nonneg (by
have hh : (1 : ℝ) ≤ l := by exact_mod_cast hl
nlinarith)
have hb := Nat.cast_nonneg (α := ℝ) b
have hm := Nat.cast_nonneg (α := ℝ) (Nat.log 2 l+1)
have hleft : 2*(b : ℝ)+4*Real.log (2*(l : ℝ)^2)*D ≤ 2*b+8*(1+Real.log (l : ℝ))*D := by nlinarith
calc
_ ≤ 16*(2*(1+Real.log (l : ℝ)))*(2*b+8*(1+Real.log (l : ℝ))*D) := by
apply mul_le_mul
· nlinarith
· exact hleft
· positivity
· positivity
_ ≤ _ := by nlinarith [mul_nonneg hlog hb,mul_nonneg (sq_nonneg (1+Real.log (l : ℝ))) hD,
mul_nonneg (sq_nonneg (Real.log (l : ℝ))) hb]
end EditUpperParameters
namespace EditUpper
universe u
attribute [local instance] Classical.propDecidable
open scoped BigOperators
open Finset EditDistortion EditWindow EditCutModel
noncomputable def growth (b N : ℕ) : ℝ := 1024*(1+(N : ℝ)*Real.log (b : ℝ))^2
lemma growth_ge {b N : ℕ} (hb : 1 ≤ b) : 1 ≤ growth b N := by
have hlog : 0 ≤ Real.log (b : ℝ) := Real.log_nonneg (by exact_mod_cast hb)
have hn := Nat.cast_nonneg (α := ℝ) N
dsimp [growth]
nlinarith [mul_nonneg hn hlog]
theorem exists_level (b N : ℕ) (hb : 1 ≤ b) (k : ℕ) (hk : k ≤ N)
(A : Type u) [Fintype A] :
∃ M : Model (List A),
(∀ x y : List A, x.length ≤ b^k → y.length ≤ b^k → (indel x y : ℝ) ≤ M.dist x y) ∧
(∀ x y : List A, M.dist x y ≤ ((b : ℝ)+2)*(growth b N)^k*(indel x y : ℝ)) := by
induction k generalizing A with
| zero =>
refine ⟨base,?_,?_⟩
· intro x y hx hy
exact base_lower x y (by simpa using hx) (by simpa using hy)
· intro x y
have hh := base_upper x y
have hi := Nat.cast_nonneg (α := ℝ) (indel x y)
simp only [pow_zero,mul_one]
nlinarith [Nat.cast_nonneg (α := ℝ) b]
| succ k ih =>
obtain ⟨M,hlo,hup⟩ := ih (by omega) (Option A)
let D := ((b : ℝ)+2)*(growth b N)^k
have hP := growth_ge (N := N) hb
have hD : 0 ≤ D := by dsimp [D]; positivity
have hpow : 1 ≤ (growth b N)^k := one_le_pow₀ hP
have hbD : (b : ℝ) ≤ D := by
dsimp [D]
have hb0 := Nat.cast_nonneg (α := ℝ) b
nlinarith
have hl : 0 < b^k := pow_pos (by omega) _
have hkN : (k : ℝ) ≤ N := by exact_mod_cast (show k ≤ N by omega)
have hlog : 0 ≤ Real.log (b : ℝ) := Real.log_nonneg (by exact_mod_cast hb)
have hT : 1+Real.log ((b^k : ℕ) : ℝ) ≤ 1+(N : ℝ)*Real.log (b : ℝ) := by
rw [Nat.cast_pow,Real.log_pow]
nlinarith
have hTnn : 0 ≤ 1+Real.log ((b^k : ℕ) : ℝ) := by
have hh : 0 ≤ Real.log ((b^k : ℕ) : ℝ) := Real.log_nonneg (by exact_mod_cast (Nat.one_le_pow k b (by omega)))
linarith
have hconst : 512*(1+Real.log ((b^k : ℕ) : ℝ))^2*((b : ℝ)+D) ≤
((b : ℝ)+2)*(growth b N)^(k+1) := by
calc
_ ≤ 512*(1+(N : ℝ)*Real.log (b : ℝ))^2*(2*D) := by
apply mul_le_mul
· gcongr
· linarith
· positivity
· positivity
_ = _ := by dsimp [D,growth]; rw [pow_succ]; ring
refine ⟨lift M b (b^k),?_,?_⟩
· intro x y hx hy
apply lift_lower M b (b^k) hl hlo x y
· simpa only [pow_succ,Nat.mul_comm] using hx
· simpa only [pow_succ,Nat.mul_comm] using hy
· apply model_indel_lipschitz
intro p q a
exact (lift_insert_raw M hD hup p q a b (b^k) hl).trans
((EditUpperParameters.step_constant_bound hl hD).trans hconst)
end EditUpper
end OAI
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