| import Mathlib |
| import OAI.Computability.VertexCover.Reduction.SeparatorStarOscillation |
| import OAI.Computability.VertexCover.Analysis.SeparatorRoundingErrorAe |
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| namespace OAI |
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| namespace VertexCover.LabelCover |
| open MeasureTheory ProbabilityTheory |
|
|
| theorem query_splice_outside (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d)) |
| (frozen : Φ.Seeds d) (hidden : Φ.HiddenSeeds J) (j : Fin d) (hj : j ∉ J) : |
| Φ.query (Φ.spliceSeeds J frozen hidden) j = Φ.query frozen j := by |
| classical |
| apply Φ.query_eq_of_questions |
| · intro k h |
| have hn : (⟨(j,k),h⟩ : PositionPair d) ∉ internalPairs J := by |
| simp [internalPairs, hj] |
| simp [spliceSeeds, hn] |
| · intro k h |
| have hn : (⟨(k,j),h⟩ : PositionPair d) ∉ internalPairs J := by |
| simp [internalPairs, hj] |
| simp [spliceSeeds, hn] |
|
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| noncomputable def spliceWeights (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d)) |
| (frozen : Fin d → Fin (Φ.WeightDimension d) → ℝ) (s : Φ.BatchWeights J) : |
| Fin d → Fin (Φ.WeightDimension d) → ℝ := by |
| classical |
| exact fun j => if h : j ∈ J then s ⟨j,h⟩ else frozen j |
| |
| noncomputable def outsideSum (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d)) |
| (frozen : Φ.Seeds d) (weights : Fin d → Fin (Φ.WeightDimension d) → ℝ) : |
| Φ.Coordinate d → ℝ := ∑ j ∈ Jᶜ, Φ.increment (Φ.query frozen j) (weights j) |
| |
| theorem continuousSum_splice (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d)) |
| (frozen : Φ.Seeds d) (weights : Fin d → Fin (Φ.WeightDimension d) → ℝ) |
| (hidden : Φ.HiddenSeeds J) (s : Φ.BatchWeights J) : |
| Φ.continuousSum (Φ.spliceSeeds J frozen hidden) (Φ.spliceWeights J weights s) = |
| Φ.outsideSum J frozen weights + Φ.batchSum (Φ.spliceSeeds J frozen hidden) J s := by |
| classical |
| unfold continuousSum outsideSum batchSum |
| rw [← Finset.sum_add_sum_compl J] |
| rw [add_comm] |
| congr 1 |
| · apply Finset.sum_congr rfl |
| intro j hj |
| have hnj : j ∉ J := Finset.mem_compl.mp hj |
| simp only [spliceWeights, dite_eq_right hnj, Φ.query_splice_outside J frozen hidden j hnj] |
| · rw [← Finset.sum_coe_sort] |
| apply Finset.sum_congr rfl |
| intro j _ |
| simp only [spliceWeights, dite_eq_left j.2] |
| |
| theorem batchFunction_is_restriction (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d)) |
| (frozen : Φ.Seeds d) (weights : Fin d → Fin (Φ.WeightDimension d) → ℝ) |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) |
| (hidden : Φ.HiddenSeeds J) (s : Φ.BatchWeights J) : |
| Φ.batchFunction (Φ.spliceSeeds J frozen hidden) J (Φ.outsideSum J frozen weights) A hA s = |
| Φ.separator A hA (Φ.continuousSum (Φ.spliceSeeds J frozen hidden) |
| (Φ.spliceWeights J weights s)) := by |
| rw [Φ.continuousSum_splice] |
| rfl |
|
|
| theorem separator_bound_on_cube (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) (hs : Φ.InWeightCube s) : |
| |Φ.separator A hA (Φ.continuousSum seed s)| ≤ d := by |
| have hzero : Φ.continuousSum seed 0 = 0 := by |
| change Φ.continuousSumCLM seed 0 = 0 |
| exact map_zero _ |
| have hl := Φ.separator_lipschitz A hA (Φ.continuousSum seed s) 0 |
| have hn := Φ.continuousSum_close seed s 0 (by norm_num : (0:ℝ)≤1) |
| (by simpa only [InWeightCube, Pi.zero_apply, sub_zero] using hs) |
| rw [hzero] at hn |
| simpa only [Φ.separator_zero, sub_zero, mul_one] using hl.trans hn |
|
|
| end VertexCover.LabelCover |
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| end |
| end |
| end |
| end |
| end |
| end |
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| end |
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| end |
| end |
| end |
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| end OAI |
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