import Mathlib
import OAI.Computability.VertexCover.Reduction.HighVectorsSeparated
import OAI.Computability.VertexCover.Analysis.ListMassHitMaximum

namespace OAI

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namespace VertexCover.LabelCover

noncomputable def selectionForm (Φ : LabelCover) {d : ℕ}
    (I : Finset (Φ.Coordinate d)) : (Φ.Coordinate d → ℝ) →L[ℝ] ℝ :=
  ∑ p ∈ I, ContinuousLinearMap.proj p

@[simp] theorem selectionForm_apply (Φ : LabelCover) {d : ℕ}
    (I : Finset (Φ.Coordinate d)) (x : Φ.Coordinate d → ℝ) :
    Φ.selectionForm I x = ∑ p ∈ I, x p := by
  simp [selectionForm]

theorem compatibilityNorm_continuous (Φ : LabelCover) (d : ℕ) :
    Continuous (Φ.compatibilityNorm (d := d)) := by
  unfold compatibilityNorm
  apply Continuous.finset_sup'_apply
  intro I hI
  simpa only [Φ.selectionForm_apply] using (Φ.selectionForm I).continuous.abs

theorem distanceTo_continuous (Φ : LabelCover) {d : ℕ}
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) :
    Continuous (Φ.distanceTo A hA) := by
  unfold distanceTo
  apply Continuous.finset_inf'_apply
  intro a ha
  exact (Φ.compatibilityNorm_continuous d).comp (continuous_id.sub continuous_const)

abbrev BatchWeights (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d)) :=
  J → Fin (Φ.WeightDimension d) → ℝ

noncomputable def batchSum (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (J : Finset (Fin d)) (s : Φ.BatchWeights J) : Φ.Coordinate d → ℝ :=
  ∑ j : J, Φ.increment (Φ.query seed j) (s j)

noncomputable def batchSumCLM (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (J : Finset (Fin d)) : Φ.BatchWeights J →L[ℝ] (Φ.Coordinate d → ℝ) :=
  LinearMap.toContinuousLinearMap {
    toFun := Φ.batchSum seed J
    map_add' := by
      intro s t
      simp only [batchSum, Pi.add_apply, Φ.increment_add, Finset.sum_add_distrib]
    map_smul' := by
      intro a s
      change (∑ j : J, Φ.increment (Φ.query seed j) (fun k => a*s j k)) = _
      simp only [Φ.increment_mul]
      ext p
      simp [batchSum, Finset.mul_sum]
  }

@[simp] theorem batchSumCLM_apply (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (J : Finset (Fin d)) (s : Φ.BatchWeights J) :
    Φ.batchSumCLM seed J s = Φ.batchSum seed J s := rfl

noncomputable def batchGradient (Φ : LabelCover) {d : ℕ} (J : Finset (Fin d))
    (D : Φ.BatchWeights J →L[ℝ] ℝ) : Φ.BatchWeights J := by
  classical
  exact fun j k => D (Pi.single j (Pi.single k 1))

theorem selectionForm_increment (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    {I : Finset (Φ.Coordinate d)} (hI : Φ.Compatible I) (j : Fin d)
    (s : Fin (Φ.WeightDimension d) → ℝ) :
    Φ.selectionForm I (Φ.increment (Φ.query seed j) s) =
      ∑ k, Φ.incidence seed I j k * s k := by
  classical
  rw [Φ.selectionForm_apply]
  by_cases h : ∃ a : (Φ.query seed j).LocalLabel,
      (⟨Φ.query seed j, a⟩ : Φ.Coordinate d) ∈ I
  · obtain ⟨a, ha⟩ := h
    rw [Finset.sum_eq_single (⟨Φ.query seed j, a⟩ : Φ.Coordinate d)]
    · rw [Φ.increment_same]
      simp_rw [Φ.incidence_eq_single seed hI j a ha]
      simp [truthValue, eq_comm]
    · intro p hp hpa
      apply Φ.increment_other
      intro heq
      exact hpa (hI.1 p hp ⟨_, a⟩ ha heq)
    · intro hn; exact (hn ha).elim
  · have hz : ∀ p ∈ I, Φ.increment (Φ.query seed j) s p = 0 := by
      intro p hp
      apply Φ.increment_other
      intro heq
      rcases p with ⟨i,a⟩
      change i = Φ.query seed j at heq
      subst i
      exact h ⟨a,hp⟩
    rw [Finset.sum_eq_zero hz]
    apply Eq.symm
    apply Finset.sum_eq_zero
    intro k hk
    have hi : Φ.incidence seed I j k = 0 := by
      simp only [incidence, truthValue]
      rw [ite_eq_right (fun ⟨label, hlabel, _⟩ => h ⟨label, hlabel⟩)]
    rw [hi, zero_mul]

theorem batchGradient_form (Φ : LabelCover) {d : ℕ} (seed : Φ.Seeds d)
    (J : Finset (Fin d)) {I : Finset (Φ.Coordinate d)} (hI : Φ.Compatible I)
    (j : J) (k : Fin (Φ.WeightDimension d)) :
    Φ.batchGradient J ((Φ.selectionForm I).comp (Φ.batchSumCLM seed J)) j k =
      Φ.incidence seed I j k := by
  classical
  change Φ.selectionForm I (Φ.batchSum seed J (Pi.single j (Pi.single k 1))) = _
  unfold batchSum
  rw [map_sum]
  simp_rw [Φ.selectionForm_increment seed hI]
  rw [Finset.sum_eq_single j]
  · simp [Pi.single_apply, mul_ite]
  · intro j' hj' hne
    simp [Pi.single_eq_of_ne hne]
  · simp

end VertexCover.LabelCover


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end OAI