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import OAI.Computability.VertexCover.Reduction.HighVectorsSeparated
namespace OAI
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namespace VertexCover.LabelCover
theorem norm_sum_le (Φ : LabelCover) {d : ℕ} {ι : Type*}
(T : Finset ι) (f : ι → Φ.Coordinate d → ℝ) :
Φ.compatibilityNorm (∑ i ∈ T, f i) ≤ ∑ i ∈ T, Φ.compatibilityNorm (f i) := by
classical
induction T using Finset.induction_on with
| empty =>
simp only [Finset.sum_empty]
exact (Φ.norm_zero d).le
| @insert i T hi ih =>
simp only [Finset.sum_insert hi]
exact (Φ.norm_add_le _ _).trans (add_le_add_right ih _)
theorem increment_sub (Φ : LabelCover) {d : ℕ} (i : Φ.Query d)
(s t : Fin (Φ.WeightDimension d) → ℝ) :
Φ.increment i (s-t) = Φ.increment i s - Φ.increment i t := by
have hn : Φ.increment i (-t) = -Φ.increment i t := by
ext p
change Φ.increment i (fun k => -t k) p = -Φ.increment i t p
have h := congrFun (Φ.increment_mul i (-1) t) p
simpa only [neg_one_mul] using h
rw [sub_eq_add_neg, Φ.increment_add, hn, sub_eq_add_neg]
noncomputable def continuousSum (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
(s : Fin d → Fin (Φ.WeightDimension d) → ℝ) : Φ.Coordinate d → ℝ :=
∑ j, Φ.increment (Φ.query seed j) (s j)
theorem sumVector_eq_continuousSum (Φ : LabelCover) {d : ℕ} (v : Φ.Vertex d) :
Φ.sumVector v = Φ.continuousSum v.1 (fun j k => (gridRational d (v.2 j k) : ℝ)) := rfl
theorem continuousSum_neg (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
(s : Fin d → Fin (Φ.WeightDimension d) → ℝ) :
Φ.continuousSum seed (-s) = -Φ.continuousSum seed s := by
unfold continuousSum
rw [← Finset.sum_neg_distrib]
apply Finset.sum_congr rfl
intro j _
ext p
change Φ.increment (Φ.query seed j) (fun k => -s j k) p =
-Φ.increment (Φ.query seed j) (s j) p
have h := congrFun (Φ.increment_mul (Φ.query seed j) (-1) (s j)) p
simpa only [neg_one_mul] using h
theorem continuousSum_sub_same_seed (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
(s t : Fin d → Fin (Φ.WeightDimension d) → ℝ) :
Φ.continuousSum seed s - Φ.continuousSum seed t = Φ.continuousSum seed (s-t) := by
simp only [continuousSum, ← Finset.sum_sub_distrib, Pi.sub_apply, increment_sub]
theorem continuousSum_close (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
(s t : Fin d → Fin (Φ.WeightDimension d) → ℝ) {b : ℝ} (hb : 0 ≤ b)
(hclose : ∀ j k, |s j k - t j k| ≤ b) :
Φ.compatibilityNorm (Φ.continuousSum seed s - Φ.continuousSum seed t) ≤ d*b := by
rw [Φ.continuousSum_sub_same_seed]
apply (Φ.norm_sum_le _ _).trans
calc
(∑ j, Φ.compatibilityNorm (Φ.increment (Φ.query seed j) ((s-t) j))) ≤
∑ _j : Fin d, b := by
apply Finset.sum_le_sum
intro j _
exact Φ.increment_norm_le _ _ hb (hclose j)
_ = d*b := by simp
def InWeightCube (Φ : LabelCover) {d : ℕ}
(s : Fin d → Fin (Φ.WeightDimension d) → ℝ) : Prop :=
∀ j k, |s j k| ≤ 1
theorem continuousSum_change_blocks (Φ : LabelCover) {d : ℕ}
(seed seed' : Φ.Seeds d) (s t : Fin d → Fin (Φ.WeightDimension d) → ℝ)
(hs : Φ.InWeightCube s) (ht : Φ.InWeightCube t) (B : Finset (Fin d))
(heq : ∀ j ∉ B, Φ.query seed j = Φ.query seed' j ∧ s j = t j) :
Φ.compatibilityNorm (Φ.continuousSum seed s - Φ.continuousSum seed' t) ≤ 2*B.card := by
classical
unfold continuousSum
rw [← Finset.sum_sub_distrib]
apply (Φ.norm_sum_le _ _).trans
calc
(∑ j, Φ.compatibilityNorm
(Φ.increment (Φ.query seed j) (s j) - Φ.increment (Φ.query seed' j) (t j))) ≤
∑ j : Fin d, if j ∈ B then (2 : ℝ) else 0 := by
apply Finset.sum_le_sum
intro j _
by_cases hj : j ∈ B
· rw [ite_eq_left hj]
have hb := Φ.norm_sub_triangle (Φ.increment (Φ.query seed j) (s j)) 0
(Φ.increment (Φ.query seed' j) (t j))
have hh1 := Φ.increment_norm_le (Φ.query seed j) (s j) (by norm_num : (0:ℝ) ≤ 1) (hs j)
have hh2 := Φ.increment_norm_le (Φ.query seed' j) (t j) (by norm_num : (0:ℝ) ≤ 1) (ht j)
simp only [sub_zero, zero_sub, Φ.norm_neg] at hb
linarith
· obtain ⟨hq, hw⟩ := heq j hj
rw [hq, hw, sub_self, ite_eq_right hj]
exact (Φ.norm_zero d).le
_ = 2*B.card := by simp [Finset.sum_ite_mem, mul_comm]
theorem separator_continuousSum_odd (Φ : LabelCover) {d : ℕ}
(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (seed : Φ.Seeds d)
(s : Fin d → Fin (Φ.WeightDimension d) → ℝ) :
Φ.separator A hA (Φ.continuousSum seed (-s)) =
-Φ.separator A hA (Φ.continuousSum seed s) := by
rw [Φ.continuousSum_neg, Φ.separator_odd]
end VertexCover.LabelCover
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI
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