import OAI.Combinatorics.ContingencyTables.Rooted.RootedCut
import OAI.Combinatorics.ContingencyTables.Rooted.ForestAffineCoefficients

namespace OAI

/-! # Cut coefficients for an explicitly selected forest root

The same signed cut calculation applies to the executed root selector.
In particular, coefficient size and the maximum-forest capacity bound
do not depend on a classical choice of cut representative.
-/

namespace ContingencyTables.RootedAffine

open scoped BigOperators Classical
open CompanionFactorialApplication RootedCompletion RootedCut GraphCompletionCounts
open ForestCapacity ForestCut MaximumForest

variable {I J : Type*} [Fintype I] [Fintype J]

noncomputable def coefficient (A : Instance I J) (S : Selection A.T)
    (t : TreeCell A.T) (a : I×J) : ℝ :=
  if A.small a.1 a.2=false ∧ ¬A.T.Adj (.inl a.1) (.inr a.2)
  then -sign A.T S t a/sign A.T S t t.val else 0

lemma coefficient_abs_le_one (A : Instance I J) (S : Selection A.T)
    (t : TreeCell A.T) (a : I×J) : |coefficient A S t a|≤1 := by
  unfold coefficient
  split_ifs
  · rw [abs_div,abs_neg,sign_abs_self A.T S A.hT,div_one]
    exact sign_abs_le_one A.T S t a
  · norm_num

lemma capacity_le_of_coefficient_ne_zero (A : Instance I J) (S : Selection A.T)
    (t : TreeCell A.T) (a : I×J) (ha : coefficient A S t a≠0) :
    A.b a.1 a.2≤A.b t.val.1 t.val.2 := by
  have hf : A.small a.1 a.2=false ∧ ¬A.T.Adj (.inl a.1) (.inr a.2) := by
    by_contra hn
    exact ha (by simp only [coefficient,hn,ite_false])
  have hcross : treeCut A.T S t (.inl a.1)≠treeCut A.T S t (.inr a.2) := by
    intro he
    apply ha
    simp [coefficient,hf,sign,he]
  have h := crossing_free_edge_le (cellGraph A.small) A.T (capacity A.b)
    A.hTG A.hT A.hmax (s(Sum.inl t.val.1,Sum.inr t.val.2))
    (by simpa using t.property)
    (away A.T (.inl t.val.1) (.inr t.val.2) (S.root (.inl t.val.1)))
    (show (cellGraph A.small).Adj (.inl a.1) (.inr a.2) from hf.1) hf.2 hcross
  simpa only [capacity_cell] using h

theorem normalized_coefficient_bound (A : Instance I J) (S : Selection A.T)
    (t : TreeCell A.T) (a : I×J) (B : ℝ) (hB : 0<B) :
    |coefficient A S t a*((A.b a.1 a.2:ℝ)+1)/B/(A.b t.val.1 t.val.2:ℝ)|≤2/B := by
  have hb : (1:ℝ)≤A.b t.val.1 t.val.2 := by
    exact_mod_cast (A.hlarge _ _ (A.hTG t.property)).2
  by_cases ha : coefficient A S t a=0
  · simp only [ha,zero_mul,zero_div,abs_zero]
    positivity
  have hcap : (A.b a.1 a.2:ℝ)≤A.b t.val.1 t.val.2 := by
    exact_mod_cast capacity_le_of_coefficient_ne_zero A S t a ha
  rw [abs_div,abs_div,abs_mul,abs_of_pos hB,
    abs_of_pos (by linarith : (0:ℝ)<A.b t.val.1 t.val.2),
    abs_of_nonneg (by positivity : (0:ℝ)≤(A.b a.1 a.2:ℝ)+1)]
  apply (div_le_iff₀ (by linarith : (0:ℝ)<A.b t.val.1 t.val.2)).mpr
  rw [div_le_iff₀ hB]
  have hh := mul_le_mul_of_nonneg_right (coefficient_abs_le_one A S t a)
    (by positivity : (0:ℝ)≤(A.b a.1 a.2:ℝ)+1)
  field_simp [hB.ne']
  nlinarith

noncomputable def smallContribution (A : Instance I J) (S : Selection A.T)
    (t : TreeCell A.T) (x q : I → J → ℝ) : ℝ :=
  ∑ a : I×J,if A.small a.1 a.2 then
    indicator (treeCut A.T S t (.inl a.1))*(x a.1 a.2-A.X a.1 a.2)-
      indicator (treeCut A.T S t (.inr a.2))*(q a.1 a.2-A.X a.1 a.2) else 0

noncomputable def offset (A : Instance I J) (S : Selection A.T)
    (t : TreeCell A.T) (x q : I → J → ℝ) : ℝ :=
  A.X t.val.1 t.val.2-smallContribution A S t x q/sign A.T S t t.val-
    ∑ a : I×J,coefficient A S t a*(A.X a.1 a.2:ℝ)

noncomputable def affineTree (A : Instance I J) (S : Selection A.T)
    (t : TreeCell A.T) (x q : I → J → ℝ) (z : I×J → ℝ) : ℝ :=
  AffinePenalty.affineValue (offset A S t x q) (coefficient A S t) z

theorem affine_of_nonroot_equations (A : Instance I J) (S : Selection A.T)
    (t : TreeCell A.T) (x q z : I → J → ℝ)
    (hr : ∀ i,Sum.inl i≠S.root (.inl i) →
      rowDifference A.small x z (fun i j => (A.X i j:ℝ)) i=0)
    (hc : ∀ j,Sum.inr j≠S.root (.inr j) →
      columnDifference A.small q z (fun i j => (A.X i j:ℝ)) j=0) :
    z t.val.1 t.val.2=affineTree A S t x q (fun a => z a.1 a.2) := by
  let f : I×J → ℝ := fun a => cutTerm A.small
    (fun i => treeCut A.T S t (.inl i)) (fun j => treeCut A.T S t (.inr j))
    x q z (fun i j => (A.X i j:ℝ)) a.1 a.2
  have hsum : (∑ a,f a)=0 := by
    rw [Fintype.sum_prod_type]
    exact cut_sum_zero A.small _ _ x q z _
      (fun i hi => hr i (treeCut_nonroot A.T S A.hT t (.inl i) hi))
      (fun j hj => hc j (treeCut_nonroot A.T S A.hT t (.inr j) hj))
  have hterm (a : I×J) : f a=
      (if a=t.val then sign A.T S t t.val*(z a.1 a.2-A.X a.1 a.2) else 0)+
      (if A.small a.1 a.2 then
        indicator (treeCut A.T S t (.inl a.1))*(x a.1 a.2-A.X a.1 a.2)-
          indicator (treeCut A.T S t (.inr a.2))*(q a.1 a.2-A.X a.1 a.2) else 0)-
      sign A.T S t t.val*coefficient A S t a*(z a.1 a.2-A.X a.1 a.2) := by
    by_cases he : a=t.val
    · subst a
      have hs : A.small t.val.1 t.val.2=false := A.hTG t.property
      simp only [f,cutTerm,hs,Bool.false_eq_true,ite_false,ite_true,coefficient,t.property,
        not_true_eq_false,and_false,sign,mul_zero,zero_mul,add_zero,sub_zero]
    · by_cases hs : A.small a.1 a.2=true
      · simp [f,cutTerm,hs,he,coefficient]
      · have hs' : A.small a.1 a.2=false := Bool.eq_false_iff.mpr hs
        by_cases ht : A.T.Adj (.inl a.1) (.inr a.2)
        · have hz := other_tree_sign_zero A.T S t a ht he
          simp only [f,cutTerm,hs',Bool.false_eq_true,ite_false,he,coefficient,ht,
            not_true_eq_false,and_false,zero_add,mul_zero,zero_mul,sub_zero]
          change sign A.T S t a*(z a.1 a.2-(A.X a.1 a.2:ℝ))=0
          rw [hz,zero_mul]
        · simp only [f,cutTerm,hs',Bool.false_eq_true,ite_false,he,coefficient,ht,
            not_false_eq_true,and_self,ite_true,zero_add,zero_sub]
          change sign A.T S t a*(z a.1 a.2-(A.X a.1 a.2:ℝ))=
            -(sign A.T S t t.val*(-sign A.T S t a/sign A.T S t t.val)*
              (z a.1 a.2-A.X a.1 a.2))
          field_simp [sign_self_ne_zero A.T S A.hT t]
  simp_rw [hterm] at hsum
  rw [Finset.sum_sub_distrib,Finset.sum_add_distrib] at hsum
  have hsingle : (∑ a : I×J,
      if a=t.val then sign A.T S t t.val*(z a.1 a.2-A.X a.1 a.2) else 0)=
      sign A.T S t t.val*(z t.val.1 t.val.2-A.X t.val.1 t.val.2) := by simp
  rw [hsingle] at hsum
  change sign A.T S t t.val*(z t.val.1 t.val.2-A.X t.val.1 t.val.2)+
    smallContribution A S t x q-
    (∑ a : I×J,sign A.T S t t.val*coefficient A S t a*(z a.1 a.2-A.X a.1 a.2))=0 at hsum
  have hprod : (∑ a : I×J,sign A.T S t t.val*coefficient A S t a*(z a.1 a.2-A.X a.1 a.2))=
      sign A.T S t t.val*((∑ a : I×J,coefficient A S t a*z a.1 a.2)-
        ∑ a : I×J,coefficient A S t a*(A.X a.1 a.2:ℝ)) := by
    rw [mul_sub,Finset.mul_sum,Finset.mul_sum,←Finset.sum_sub_distrib]
    apply Finset.sum_congr rfl
    intro a _
    ring
  rw [hprod] at hsum
  unfold affineTree AffinePenalty.affineValue offset
  apply mul_left_cancel₀ (sign_self_ne_zero A.T S A.hT t)
  rw [mul_add,mul_sub,mul_sub,←mul_div_assoc,
    mul_div_cancel_left₀ _ (sign_self_ne_zero A.T S A.hT t)]
  nlinarith [hsum]

end ContingencyTables.RootedAffine

end OAI