import Mathlib
import OAI.Computability.VertexCover.Analysis.RestrictionSum

namespace OAI

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namespace VertexCover.Restriction

theorem choose_step (d h : ℕ) (hd : 0 < d) (hh : 0 < h) :
    (d:ℝ)*((d-1).choose (h-1):ℝ) = (h:ℝ)*(d.choose h:ℝ) := by
  have he := Nat.add_one_mul_choose_eq (d-1) (h-1)
  rw [Nat.sub_add_cancel hd, Nat.sub_add_cancel hh] at he
  exact_mod_cast he.trans (Nat.mul_comm _ _)

theorem batch_pair_scale (d h : ℕ) (hd : 4 ≤ d) (hh : 3 ≤ h) :
    (d:ℝ)*((d-2).choose (h-2):ℝ) ≤ 2*(h:ℝ)*((d-1).choose (h-1):ℝ) := by
  have he := choose_step (d-1) (h-1) (by omega) (by omega)
  have hed : d-1-1 = d-2 := by omega
  have heh : h-1-1 = h-2 := by omega
  rw [hed, heh, Nat.cast_sub (by omega : 1 ≤ d), Nat.cast_sub (by omega : 1 ≤ h)] at he
  norm_num only [Nat.cast_one] at he
  have hdR : (4:ℝ) ≤ d := by exact_mod_cast hd
  have hC1 : (0:ℝ) ≤ (d-1).choose (h-1) := Nat.cast_nonneg _
  have hC2 : (0:ℝ) ≤ (d-2).choose (h-2) := Nat.cast_nonneg _
  nlinarith [mul_nonneg (by linarith : (0:ℝ) ≤ d-2) hC2]

theorem batch_triple_scale (d h : ℕ) (hd : 4 ≤ d) (hh : 3 ≤ h) :
    (d:ℝ)^2*((d-3).choose (h-3):ℝ) ≤ 4*(h:ℝ)^2*((d-1).choose (h-1):ℝ) := by
  have he := choose_step (d-2) (h-2) (by omega) (by omega)
  have hed : d-2-1 = d-3 := by omega
  have heh : h-2-1 = h-3 := by omega
  rw [hed, heh, Nat.cast_sub (by omega : 2 ≤ d), Nat.cast_sub (by omega : 2 ≤ h)] at he
  norm_num only [Nat.cast_ofNat] at he
  have hdR : (4:ℝ) ≤ d := by exact_mod_cast hd
  have hhR : (0:ℝ) ≤ h := Nat.cast_nonneg _
  have hC2 : (0:ℝ) ≤ (d-2).choose (h-2) := Nat.cast_nonneg _
  have hC3 : (0:ℝ) ≤ (d-3).choose (h-3) := Nat.cast_nonneg _
  have hl : (d:ℝ)*((d-3).choose (h-3):ℝ) ≤ 2*(h:ℝ)*((d-2).choose (h-2):ℝ) := by
    nlinarith [mul_nonneg (by linarith : (0:ℝ) ≤ d-4) hC3]
  have hl' := mul_le_mul_of_nonneg_left hl (Nat.cast_nonneg d : (0:ℝ) ≤ d)
  have hp := mul_le_mul_of_nonneg_left (batch_pair_scale d h hd hh) (by positivity : (0:ℝ) ≤ 2*h)
  nlinarith

end VertexCover.Restriction

namespace VertexCover.LabelCover
open VertexCover.Restriction
open EnergyForm.Projection
open scoped BigOperators

theorem privateProjection_batch_lower (Φ : LabelCover) {d h : ℕ}
    (hd : 4 ≤ d) (hh : 3 ≤ h) (ν : ℝ)
    (hscale : 16*(h:ℝ)+32*(h:ℝ)^2 ≤ ν*d/2)
    (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty)
    (henergy : ν*d ≤ (Φ.restrictionForm d).energy (Φ.restrictionFunction A hA)) :
    (d.choose h:ℝ)*ν*h/2 ≤
      ∑ J ∈ batches d h, ∑ j ∈ J,
        (Φ.restrictionForm d).energy (Φ.privateProjection J j (Φ.restrictionFunction A hA)) := by
  have hb := Φ.privateProjection_batch_bound hh A hA
  have hC1 : (0:ℝ) ≤ (d-1).choose (h-1) := Nat.cast_nonneg _
  have hl := mul_le_mul_of_nonneg_left henergy hC1
  have hs := mul_le_mul_of_nonneg_left hscale hC1
  have hp := batch_pair_scale d h hd hh
  have ht := batch_triple_scale d h hd hh
  have he := choose_step d h (by omega) (by omega)
  have he' : ((d-1).choose (h-1):ℝ)*ν*d = (d.choose h:ℝ)*ν*h := by
    calc
      _ = ν*((d:ℝ)*((d-1).choose (h-1):ℝ)) := by ring
      _ = _ := by rw [he]; ring
  nlinarith

end VertexCover.LabelCover


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end OAI