Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Computability.VertexCover.Analysis.BatchGradientForm

namespace OAI

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namespace VertexCover.LabelCover

def affineSign (b : Bool) : ℝ := if b then 1 else -1

@[simp] theorem affineSign_true : affineSign true = 1 := rfl
@[simp] theorem affineSign_false : affineSign false = -1 := rfl

theorem norm_select_affine (Φ : LabelCover) {d : ℕ} (x : Φ.Coordinate d → ℝ) :
    ∃ I, Φ.Compatible I ∧ ∃ e : Bool,
      Φ.compatibilityNorm x = affineSign e * Φ.selectionForm I x := by
  obtain ⟨I, hI, heq⟩ := (Φ.compatibleSelections d).exists_mem_eq_sup'
    (Φ.compatibleSelections_nonempty d) (fun I => |∑ p ∈ I, x p|)
  change Φ.compatibilityNorm x = |∑ p ∈ I, x p| at heq
  rw [← Φ.selectionForm_apply] at heq
  refine ⟨I, (Φ.mem_compatibleSelections I).mp hI, ?_⟩
  by_cases h : 0 ≤ Φ.selectionForm I x
  · refine ⟨true, ?_⟩
    simpa only [affineSign_true, one_mul, ← Φ.selectionForm_apply, abs_of_nonneg h] using heq
  · refine ⟨false, ?_⟩
    simpa only [affineSign_false, neg_one_mul, ← Φ.selectionForm_apply,
      abs_of_neg (lt_of_not_ge h)] using heq

abbrev AffineIndex (Φ : LabelCover) {d : ℕ} (A : Finset (Φ.Coordinate d → ℝ)) :=
  A × (Φ.compatibleSelections d) × Bool × Bool

noncomputable def affineSlope (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (J : Finset (Fin d)) {A : Finset (Φ.Coordinate d → ℝ)}
    (i : Φ.AffineIndex A) : Φ.BatchWeights J →L[ℝ] ℝ :=
  affineSign i.2.2.2 • (Φ.selectionForm i.2.1.val).comp (Φ.batchSumCLM seed J)

noncomputable def affineOffset (Φ : LabelCover) {d : ℕ}
    (c0 : Φ.Coordinate d → ℝ) {A : Finset (Φ.Coordinate d → ℝ)}
    (i : Φ.AffineIndex A) : ℝ :=
  affineSign i.2.2.2 * Φ.selectionForm i.2.1.val (c0 - affineSign i.2.2.1 • i.1.val)

theorem affine_value (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (J : Finset (Fin d)) (c0 : Φ.Coordinate d → ℝ)
    {A : Finset (Φ.Coordinate d → ℝ)} (i : Φ.AffineIndex A) (s : Φ.BatchWeights J) :
    Φ.affineSlope seed J i s + Φ.affineOffset c0 i =
      affineSign i.2.2.2 * Φ.selectionForm i.2.1.val
        (c0 + Φ.batchSum seed J s - affineSign i.2.2.1 • i.1.val) := by
  simp only [affineSlope, affineOffset, smul_apply,
    ContinuousLinearMap.comp_apply, Φ.batchSumCLM_apply, smul_eq_mul, map_sub, map_add]
  ring

theorem distance_plus_select (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (J : Finset (Fin d)) (c0 : Φ.Coordinate d → ℝ)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (s : Φ.BatchWeights J) :
    ∃ i : Φ.AffineIndex A, Φ.distanceTo A hA (c0 + Φ.batchSum seed J s) =
      Φ.affineSlope seed J i s + Φ.affineOffset c0 i := by
  classical
  obtain ⟨z, hz, heq⟩ := A.exists_mem_eq_inf' hA
    (fun z => Φ.compatibilityNorm (c0 + Φ.batchSum seed J s - z))
  obtain ⟨I, hI, e, hnorm⟩ := Φ.norm_select_affine (c0 + Φ.batchSum seed J s - z)
  let i : Φ.AffineIndex A := (⟨z,hz⟩, ⟨I,(Φ.mem_compatibleSelections I).mpr hI⟩, true,e)
  change Φ.distanceTo A hA (c0 + Φ.batchSum seed J s) = _ at heq
  refine ⟨i, ?_⟩
  rw [Φ.affine_value]
  simpa only [i, affineSign_true, one_smul] using heq.trans hnorm

theorem distance_minus_select (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (J : Finset (Fin d)) (c0 : Φ.Coordinate d → ℝ)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (s : Φ.BatchWeights J) :
    ∃ i : Φ.AffineIndex A, Φ.distanceTo A hA (-(c0 + Φ.batchSum seed J s)) =
      Φ.affineSlope seed J i s + Φ.affineOffset c0 i := by
  classical
  obtain ⟨z, hz, heq⟩ := A.exists_mem_eq_inf' hA
    (fun z => Φ.compatibilityNorm (-(c0 + Φ.batchSum seed J s)-z))
  have heq' : Φ.distanceTo A hA (-(c0 + Φ.batchSum seed J s)) =
      Φ.compatibilityNorm (c0 + Φ.batchSum seed J s + z) := by
    change _ = _ at heq
    rw [show -(c0 + Φ.batchSum seed J s)-z =
      -(c0 + Φ.batchSum seed J s + z) by abel, Φ.norm_neg] at heq
    exact heq
  obtain ⟨I, hI, e, hnorm⟩ := Φ.norm_select_affine (c0 + Φ.batchSum seed J s + z)
  let i : Φ.AffineIndex A := (⟨z,hz⟩, ⟨I,(Φ.mem_compatibleSelections I).mpr hI⟩, false,e)
  refine ⟨i, ?_⟩
  rw [Φ.affine_value]
  simpa only [i, affineSign_false, neg_one_smul, sub_neg_eq_add] using heq'.trans hnorm

end VertexCover.LabelCover


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end OAI

	import Mathlib
	import OAI.Computability.VertexCover.Analysis.BatchGradientForm

	namespace OAI

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	namespace VertexCover.LabelCover

	def affineSign (b : Bool) : ℝ := if b then 1 else -1

	@[simp] theorem affineSign_true : affineSign true = 1 := rfl
	@[simp] theorem affineSign_false : affineSign false = -1 := rfl

	theorem norm_select_affine (Φ : LabelCover) {d : ℕ} (x : Φ.Coordinate d → ℝ) :
	∃ I, Φ.Compatible I ∧ ∃ e : Bool,
	Φ.compatibilityNorm x = affineSign e * Φ.selectionForm I x := by
	obtain ⟨I, hI, heq⟩ := (Φ.compatibleSelections d).exists_mem_eq_sup'
	(Φ.compatibleSelections_nonempty d) (fun I => \|∑ p ∈ I, x p\|)
	change Φ.compatibilityNorm x = \|∑ p ∈ I, x p\| at heq
	rw [← Φ.selectionForm_apply] at heq
	refine ⟨I, (Φ.mem_compatibleSelections I).mp hI, ?_⟩
	by_cases h : 0 ≤ Φ.selectionForm I x
	· refine ⟨true, ?_⟩
	simpa only [affineSign_true, one_mul, ← Φ.selectionForm_apply, abs_of_nonneg h] using heq
	· refine ⟨false, ?_⟩
	simpa only [affineSign_false, neg_one_mul, ← Φ.selectionForm_apply,
	abs_of_neg (lt_of_not_ge h)] using heq

	abbrev AffineIndex (Φ : LabelCover) {d : ℕ} (A : Finset (Φ.Coordinate d → ℝ)) :=
	A × (Φ.compatibleSelections d) × Bool × Bool

	noncomputable def affineSlope (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
	(J : Finset (Fin d)) {A : Finset (Φ.Coordinate d → ℝ)}
	(i : Φ.AffineIndex A) : Φ.BatchWeights J →L[ℝ] ℝ :=
	affineSign i.2.2.2 • (Φ.selectionForm i.2.1.val).comp (Φ.batchSumCLM seed J)

	noncomputable def affineOffset (Φ : LabelCover) {d : ℕ}
	(c0 : Φ.Coordinate d → ℝ) {A : Finset (Φ.Coordinate d → ℝ)}
	(i : Φ.AffineIndex A) : ℝ :=
	affineSign i.2.2.2 * Φ.selectionForm i.2.1.val (c0 - affineSign i.2.2.1 • i.1.val)

	theorem affine_value (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
	(J : Finset (Fin d)) (c0 : Φ.Coordinate d → ℝ)
	{A : Finset (Φ.Coordinate d → ℝ)} (i : Φ.AffineIndex A) (s : Φ.BatchWeights J) :
	Φ.affineSlope seed J i s + Φ.affineOffset c0 i =
	affineSign i.2.2.2 * Φ.selectionForm i.2.1.val
	(c0 + Φ.batchSum seed J s - affineSign i.2.2.1 • i.1.val) := by
	simp only [affineSlope, affineOffset, smul_apply,
	ContinuousLinearMap.comp_apply, Φ.batchSumCLM_apply, smul_eq_mul, map_sub, map_add]
	ring

	theorem distance_plus_select (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
	(J : Finset (Fin d)) (c0 : Φ.Coordinate d → ℝ)
	(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (s : Φ.BatchWeights J) :
	∃ i : Φ.AffineIndex A, Φ.distanceTo A hA (c0 + Φ.batchSum seed J s) =
	Φ.affineSlope seed J i s + Φ.affineOffset c0 i := by
	classical
	obtain ⟨z, hz, heq⟩ := A.exists_mem_eq_inf' hA
	(fun z => Φ.compatibilityNorm (c0 + Φ.batchSum seed J s - z))
	obtain ⟨I, hI, e, hnorm⟩ := Φ.norm_select_affine (c0 + Φ.batchSum seed J s - z)
	let i : Φ.AffineIndex A := (⟨z,hz⟩, ⟨I,(Φ.mem_compatibleSelections I).mpr hI⟩, true,e)
	change Φ.distanceTo A hA (c0 + Φ.batchSum seed J s) = _ at heq
	refine ⟨i, ?_⟩
	rw [Φ.affine_value]
	simpa only [i, affineSign_true, one_smul] using heq.trans hnorm

	theorem distance_minus_select (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
	(J : Finset (Fin d)) (c0 : Φ.Coordinate d → ℝ)
	(A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) (s : Φ.BatchWeights J) :
	∃ i : Φ.AffineIndex A, Φ.distanceTo A hA (-(c0 + Φ.batchSum seed J s)) =
	Φ.affineSlope seed J i s + Φ.affineOffset c0 i := by
	classical
	obtain ⟨z, hz, heq⟩ := A.exists_mem_eq_inf' hA
	(fun z => Φ.compatibilityNorm (-(c0 + Φ.batchSum seed J s)-z))
	have heq' : Φ.distanceTo A hA (-(c0 + Φ.batchSum seed J s)) =
	Φ.compatibilityNorm (c0 + Φ.batchSum seed J s + z) := by
	change _ = _ at heq
	rw [show -(c0 + Φ.batchSum seed J s)-z =
	-(c0 + Φ.batchSum seed J s + z) by abel, Φ.norm_neg] at heq
	exact heq
	obtain ⟨I, hI, e, hnorm⟩ := Φ.norm_select_affine (c0 + Φ.batchSum seed J s + z)
	let i : Φ.AffineIndex A := (⟨z,hz⟩, ⟨I,(Φ.mem_compatibleSelections I).mpr hI⟩, false,e)
	refine ⟨i, ?_⟩
	rw [Φ.affine_value]
	simpa only [i, affineSign_false, neg_one_smul, sub_neg_eq_add] using heq'.trans hnorm

	end VertexCover.LabelCover


	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
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	end
	end
	end

	end OAI