import Mathlib
import OAI.Combinatorics.IndependentSets.Expansion.DegreeReplacement
import OAI.Combinatorics.IndependentSets.Expansion.CloudRounding

namespace OAI

namespace IndependentSetsGames.Foundations.PCP.CloudRounding
open scoped BigOperators
section GraphRounding

open DegreeReplacement PoweringWalks

variable {V E A D : Type*} [Fintype V] [Fintype E] [Fintype A] [Fintype D]
variable [DecidableEq V] [DecidableEq E] [DecidableEq A]

def mismatchSet (G : ConstraintGraph V E A) (ell : E → A) (rounded : V → A) :
    Finset E :=
  Finset.univ.filter (fun e => ell e ≠ rounded (G.tail e))

def copiedRejectionCount (G : ConstraintGraph V E A) (ell : E → A) : Nat :=
  (Finset.univ.filter (fun e => G.accepts e (ell e) (ell (G.reverse e)) = false)).card

def innerDisagreementSet (G : ConstraintGraph V E A)
    (H : ∀ v, PortGraph (Cloud G v) D) (ell : E → A) : Finset (E × D) :=
  Finset.univ.filter (fun ed => ell ed.1 ≠ ell (innerRotation G H ed).1)

noncomputable def roundLabels [Nonempty A]
    (G : ConstraintGraph V E A) (ell : E → A) : V → A :=
  fun v => majority (fun e : Cloud G v => ell e.val)

def mismatchFiberEquiv (G : ConstraintGraph V E A) (ell : E → A)
    (rounded : V → A) :
    (Σ v, ↥(minoritySet (fun e : Cloud G v => ell e.val) (rounded v))) ≃
      ↥(mismatchSet G ell rounded) where
  toFun := by
    rintro ⟨v, ⟨⟨e, he⟩, hbad⟩⟩
    cases he
    exact ⟨e, by simpa [mismatchSet, minoritySet] using hbad⟩
  invFun e :=
    ⟨G.tail e.val, ⟨⟨e.val, rfl⟩, by
      simpa [mismatchSet, minoritySet] using e.property⟩⟩
  left_inv := by
    rintro ⟨v, ⟨⟨e, he⟩, hbad⟩⟩
    cases he
    rfl
  right_inv := by intro e; apply Subtype.ext; rfl

omit [Fintype A] [DecidableEq E] in
theorem mismatch_card_eq_sum (G : ConstraintGraph V E A) (ell : E → A)
    (rounded : V → A) :
    (mismatchSet G ell rounded).card =
      ∑ v, (minoritySet (fun e : Cloud G v => ell e.val) (rounded v)).card := by
  have h := Fintype.card_congr (mismatchFiberEquiv G ell rounded)
  simpa only [Fintype.card_sigma, Fintype.card_coe] using h.symm

def disagreementFiberEquiv (G : ConstraintGraph V E A)
    (H : ∀ v, PortGraph (Cloud G v) D) (ell : E → A) :
    (Σ v, ↥(disagreementSet (fun e : Cloud G v => ell e.val)
      (fun ed => ((H v).rot ed).1))) ≃ ↥(innerDisagreementSet G H ell) where
  toFun := by
    rintro ⟨v, ⟨⟨⟨e, he⟩, d⟩, hbad⟩⟩
    cases he
    refine ⟨(e, d), ?_⟩
    simpa [disagreementSet, innerDisagreementSet, innerRotation,
      cloudIndexEquiv, cloudRotation] using hbad
  invFun := by
    rintro ⟨⟨e, d⟩, hbad⟩
    refine ⟨G.tail e, ⟨(⟨e, rfl⟩, d), ?_⟩⟩
    simpa [disagreementSet, innerDisagreementSet, innerRotation,
      cloudIndexEquiv, cloudRotation] using hbad
  left_inv := by
    rintro ⟨v, ⟨⟨⟨e, he⟩, d⟩, hbad⟩⟩
    cases he
    rfl
  right_inv := by intro ed; apply Subtype.ext; rfl

omit [Fintype A] [DecidableEq E] in
theorem disagreement_card_eq_sum (G : ConstraintGraph V E A)
    (H : ∀ v, PortGraph (Cloud G v) D) (ell : E → A) :
    (innerDisagreementSet G H ell).card =
      ∑ v, (disagreementSet (fun e : Cloud G v => ell e.val)
        (fun ed => ((H v).rot ed).1)).card := by
  have h := Fintype.card_congr (disagreementFiberEquiv G H ell)
  simpa only [Fintype.card_sigma, Fintype.card_coe] using h.symm

theorem rounded_mismatch_expansion [Nonempty A]
    (G : ConstraintGraph V E A) (H : ∀ v, PortGraph (Cloud G v) D)
    (ell : E → A) (h : ℚ)
    (expansion : ∀ v (S : Finset (Cloud G v)),
      S.card ≤ Fintype.card (Cloud G v) / 2 →
      h * (S.card : ℚ) ≤ ((directedCut (fun ed => ((H v).rot ed).1) S).card : ℚ)) :
    h * ((mismatchSet G ell (roundLabels G ell)).card : ℚ) ≤
      ((innerDisagreementSet G H ell).card : ℚ) := by
  rw [mismatch_card_eq_sum, disagreement_card_eq_sum, Nat.cast_sum, Nat.cast_sum,
    Finset.mul_sum]
  apply Finset.sum_le_sum
  intro v _
  exact minority_expansion_bound (fun e : Cloud G v => ell e.val)
    (fun ed => ((H v).rot ed).1) (roundLabels G ell v) h
    (majority_maximal (fun e : Cloud G v => ell e.val)) (expansion v)

omit [Fintype V] [Fintype A] [DecidableEq V] [DecidableEq E] in
theorem rounded_endpoint_bound (G : ConstraintGraph V E A)
    (ell : E → A) (rounded : V → A) :
    G.rejectionCount rounded ≤ copiedRejectionCount G ell +
      2 * (mismatchSet G ell rounded).card :=
  dart_rejection_count_le G.tail G.reverse G.accepts ell rounded

omit [Fintype V] [Fintype E] [Fintype A] [Fintype D] [DecidableEq V] [DecidableEq E] in
@[simp] theorem copied_satisfied_inl (G : ConstraintGraph V E A)
    (H : ∀ v, PortGraph (Cloud G v) D) (ell : E → A) (e : E) (d : D) :
    (replacementGraph G H).edgeSatisfied ell (e, Sum.inl d) =
      decide (ell e = ell (innerRotation G H (e, d)).1) := rfl

omit [Fintype V] [Fintype E] [Fintype A] [Fintype D] [DecidableEq V] [DecidableEq E] in
@[simp] theorem copied_satisfied_inr (G : ConstraintGraph V E A)
    (H : ∀ v, PortGraph (Cloud G v) D) (ell : E → A) (e : E) (u : Unit) :
    (replacementGraph G H).edgeSatisfied ell (e, Sum.inr u) =
      G.accepts e (ell e) (ell (G.reverse e)) := rfl

omit [Fintype V] [Fintype A] [DecidableEq V] [DecidableEq E] in
theorem replacement_rejection_split (G : ConstraintGraph V E A)
    (H : ∀ v, PortGraph (Cloud G v) D) (ell : E → A) :
    (replacementGraph G H).rejectionCount ell =
      copiedRejectionCount G ell + (innerDisagreementSet G H ell).card := by
  classical
  simp only [rejectionCount_eq_sum, Fintype.sum_prod_type, Fintype.sum_sum_type,
    copied_satisfied_inl, copied_satisfied_inr, decide_eq_false_iff_not,
    Fintype.sum_unique, Finset.sum_add_distrib]
  simp only [copiedRejectionCount, innerDisagreementSet, Finset.card_eq_sum_ones,
    Finset.sum_filter, Fintype.sum_prod_type]
  exact Nat.add_comm _ _

theorem replacement_soundness [Nonempty A]
    (G : ConstraintGraph V E A) (H : ∀ v, PortGraph (Cloud G v) D)
    (expansion : ∀ v (S : Finset (Cloud G v)),
      S.card ≤ Fintype.card (Cloud G v) / 2 →
      2 * S.card ≤ (directedCut (fun ed => ((H v).rot ed).1) S).card)
    (ell : E → A) :
    G.rejectionCount (roundLabels G ell) ≤ (replacementGraph G H).rejectionCount ell := by
  have hc := rounded_mismatch_expansion G H ell 2 (by
    intro v S hS
    exact_mod_cast expansion v S hS)
  have hcNat : 2 * (mismatchSet G ell (roundLabels G ell)).card ≤
      (innerDisagreementSet G H ell).card := by exact_mod_cast hc
  rw [replacement_rejection_split]
  exact (rounded_endpoint_bound G ell (roundLabels G ell)).trans
    (Nat.add_le_add_left hcNat _)

theorem padded_replacement_soundness [Nonempty A]
    (G : ConstraintGraph V E A) (dummy : V → Type*) [∀ v, Fintype (dummy v)]
    [∀ v, DecidableEq (dummy v)]
    (H : ∀ v, PortGraph (Cloud (paddedGraph G dummy) v) D)
    (expansion : ∀ v (S : Finset (Cloud (paddedGraph G dummy) v)),
      S.card ≤ Fintype.card (Cloud (paddedGraph G dummy) v) / 2 →
      2 * S.card ≤ (directedCut (fun ed => ((H v).rot ed).1) S).card)
    (ell : PaddedDart G dummy → A) :
    G.rejectionCount (roundLabels (paddedGraph G dummy) ell) ≤
      (paddedReplacementGraph G dummy H).rejectionCount ell := by
  simpa only [paddedReplacementGraph, padded_rejectionCount] using
    replacement_soundness (paddedGraph G dummy) H expansion ell

end GraphRounding

theorem combine_counts (R b M c : Nat) (h : ℚ) (hh : 0 < h)
    (endpoint : R ≤ b + 2 * M) (cloud : h * (M : ℚ) ≤ (c : ℚ)) :
    (R : ℚ) ≤ max 1 (2 / h) * ((b : ℚ) + (c : ℚ)) := by
  let K : ℚ := max 1 (2 / h)
  have hK : 1 ≤ K := le_max_left _ _
  have hK0 : 0 ≤ K := le_trans zero_le_one hK
  have hcoef : 2 ≤ K * h := (div_le_iff₀ hh).mp (le_max_right _ _)
  have hm : (2 : ℚ) * M ≤ K * c := by
    calc
      (2 : ℚ) * M ≤ (K * h) * M :=
        mul_le_mul_of_nonneg_right hcoef (Nat.cast_nonneg M)
      _ = K * (h * M) := mul_assoc _ _ _
      _ ≤ K * c := mul_le_mul_of_nonneg_left cloud hK0
  have hb : (b : ℚ) ≤ K * b := by
    simpa only [one_mul] using mul_le_mul_of_nonneg_right hK (Nat.cast_nonneg b)
  have hr : (R : ℚ) ≤ (b : ℚ) + 2 * (M : ℚ) := by exact_mod_cast endpoint
  exact hr.trans ((add_le_add hb hm).trans_eq (mul_add K (b : ℚ) (c : ℚ)).symm)

end IndependentSetsGames.Foundations.PCP.CloudRounding

end OAI