lean/OAI/Probability/RadialTransition/BoundedCore.lean

import OAI.Probability.RadialTransition.ShellGeometry

namespace OAI

universe uV

noncomputable section

open scoped BigOperators NNReal ContDiff ENNReal
open Filter MeasureTheory Set

namespace RadialTransition

def clamp01 (x : ℝ) : ℝ := max 0 (min 1 x)

theorem clamp01_nonneg (x : ℝ) : 0 ≤ clamp01 x := le_max_left _ _
theorem clamp01_le_one (x : ℝ) : clamp01 x ≤ 1 :=
  max_le (by norm_num) (min_le_left _ _)
theorem clamp01_eq_zero {x : ℝ} (hx : x ≤ 0) : clamp01 x = 0 := by
  exact max_eq_left (le_trans (min_le_right _ _) hx)
theorem clamp01_eq_one {x : ℝ} (hx : 1 ≤ x) : clamp01 x = 1 := by
  simp [clamp01, min_eq_left hx]
theorem clamp01_mono : Monotone clamp01 := by
  intro x y hxy
  exact max_le_max le_rfl (min_le_min le_rfl hxy)
theorem continuous_clamp01 : Continuous clamp01 := by
  unfold clamp01
  fun_prop

def microLength (t : ℝ) : ℝ := Real.exp (-t)
def shellStep (t : ℝ) : ℝ := microLength t / 3000

def coreReal (t r : ℝ) : ℝ :=
  t ^ 2 * min
    (1 - clamp01 ((r - microLength t) / shellStep t) +
      clamp01 ((r - microLength t - 2 * shellStep t) / shellStep t))
    (clamp01 (1 + t * (1 - r)))

def corePotential (t : ℝ) : Potential := fun r => coreReal t r

theorem microLength_pos (t : ℝ) : 0 < microLength t := Real.exp_pos _
theorem shellStep_pos (t : ℝ) : 0 < shellStep t := div_pos (microLength_pos t) (by norm_num)

theorem coreReal_nonneg (t r : ℝ) : 0 ≤ coreReal t r := by
  apply mul_nonneg (sq_nonneg t)
  apply le_min
  · linarith [clamp01_le_one ((r - microLength t) / shellStep t),
      clamp01_nonneg ((r - microLength t - 2 * shellStep t) / shellStep t)]
  · exact clamp01_nonneg _

theorem coreReal_le (t r : ℝ) : coreReal t r ≤ t ^ 2 := by
  have hh : min
      (1 - clamp01 ((r - microLength t) / shellStep t) +
        clamp01 ((r - microLength t - 2 * shellStep t) / shellStep t))
      (clamp01 (1 + t * (1 - r))) ≤ 1 :=
    (min_le_right _ _).trans (clamp01_le_one _)
  simpa only [coreReal, mul_one] using mul_le_mul_of_nonneg_left hh (sq_nonneg t)

theorem continuous_coreReal (t : ℝ) : Continuous (coreReal t) := by
  unfold coreReal
  exact continuous_const.mul
    (((continuous_const.sub (continuous_clamp01.comp
      ((continuous_id.sub continuous_const).div_const _))).add
      (continuous_clamp01.comp
        (((continuous_id.sub continuous_const).sub continuous_const).div_const _))).min
      (continuous_clamp01.comp
        (continuous_const.add (continuous_const.mul (continuous_const.sub continuous_id)))))

theorem continuous_corePotential (t : ℝ) : Continuous (corePotential t) :=
  (continuous_coreReal t).comp NNReal.continuous_coe

theorem bounded_corePotential (t : ℝ) : IsBoundedPotential (corePotential t) := by
  refine ⟨t ^ 2, sq_nonneg t, fun r => ?_⟩
  change |coreReal t r| ≤ t ^ 2
  rw [abs_of_nonneg (coreReal_nonneg t r)]
  exact coreReal_le t r

theorem coreReal_zero_plateau (t r : ℝ)
    (hlo : microLength t + shellStep t ≤ r)
    (hhi : r ≤ microLength t + 2 * shellStep t) : coreReal t r = 0 := by
  have hd := shellStep_pos t
  have hdown : clamp01 ((r - microLength t) / shellStep t) = 1 :=
    clamp01_eq_one ((le_div_iff₀ hd).mpr (by linarith))
  have hup : clamp01 ((r - microLength t - 2 * shellStep t) / shellStep t) = 0 :=
    clamp01_eq_zero (div_nonpos_of_nonpos_of_nonneg (by linarith) hd.le)
  simp only [coreReal, hdown, hup, sub_self, add_zero,
    min_eq_left (clamp01_nonneg _), mul_zero]

theorem coreReal_zero_outer {t r : ℝ} (ht : 0 < t) (hr : 1 + 1 / t ≤ r) :
    coreReal t r = 0 := by
  have hc : clamp01 (1 + t * (1 - r)) = 0 := by
    apply clamp01_eq_zero
    have hm := mul_le_mul_of_nonneg_left hr ht.le
    have hd := div_mul_cancel₀ (1 : ℝ) (ne_of_gt ht)
    nlinarith [hm, hd]
  have hs : 0 ≤ 1 - clamp01 ((r - microLength t) / shellStep t) +
      clamp01 ((r - microLength t - 2 * shellStep t) / shellStep t) := by
    linarith [clamp01_le_one ((r - microLength t) / shellStep t),
      clamp01_nonneg ((r - microLength t - 2 * shellStep t) / shellStep t)]
  simp only [coreReal, hc, min_eq_right hs, mul_zero]

theorem coreReal_full_left {t r : ℝ} (ht : 0 ≤ t)
    (hl : r ≤ microLength t) (hr : r ≤ 1) : coreReal t r = t ^ 2 := by
  have hd := shellStep_pos t
  have hdown : clamp01 ((r - microLength t) / shellStep t) = 0 :=
    clamp01_eq_zero (div_nonpos_of_nonpos_of_nonneg (by linarith) hd.le)
  have hup : clamp01 ((r - microLength t - 2 * shellStep t) / shellStep t) = 0 :=
    clamp01_eq_zero (div_nonpos_of_nonpos_of_nonneg (by linarith) hd.le)
  have hc : clamp01 (1 + t * (1 - r)) = 1 :=
    clamp01_eq_one (by nlinarith [mul_nonneg ht (sub_nonneg.mpr hr)])
  simp [coreReal, hdown, hup, hc]

theorem coreReal_full_right {t r : ℝ} (ht : 0 ≤ t)
    (hl : microLength t + 3 * shellStep t ≤ r) (hr : r ≤ 1) :
    coreReal t r = t ^ 2 := by
  have hd := shellStep_pos t
  have hdown : clamp01 ((r - microLength t) / shellStep t) = 1 :=
    clamp01_eq_one ((le_div_iff₀ hd).mpr (by linarith))
  have hup : clamp01 ((r - microLength t - 2 * shellStep t) / shellStep t) = 1 :=
    clamp01_eq_one ((le_div_iff₀ hd).mpr (by linarith))
  have hc : clamp01 (1 + t * (1 - r)) = 1 :=
    clamp01_eq_one (by nlinarith [mul_nonneg ht (sub_nonneg.mpr hr)])
  simp [coreReal, hdown, hup, hc]

theorem coreReal_full_of_not_shell {t r : ℝ} (ht : 0 ≤ t) (hr : r ≤ 1)
    (hs : ¬(microLength t ≤ r ∧ r ≤ (1 + (1 : ℝ) / 1000) * microLength t)) :
    coreReal t r = t ^ 2 := by
  by_cases hlo : microLength t ≤ r
  · apply coreReal_full_right ht ?_ hr
    have hhi : (1 + (1 : ℝ) / 1000) * microLength t < r :=
      lt_of_not_ge (fun hh => hs ⟨hlo, hh⟩)
    dsimp [shellStep]
    linarith [hhi]
  · exact coreReal_full_left ht (le_of_not_ge hlo) hr

theorem twice_pair_sum {N : ℕ} (F : Fin N → Fin N → ℝ)
    (hF : ∀ i j, F i j = F j i) :
    2 * (∑ i, ∑ j with i < j, F i j) =
      ∑ i, ∑ j, if i ≠ j then F i j else 0 := by
  classical
  simp only [Finset.sum_filter]
  have hswap : (∑ i, ∑ j, if j < i then F i j else 0) =
      ∑ i, ∑ j, if i < j then F i j else 0 := by
    rw [Finset.sum_comm]
    apply Finset.sum_congr rfl
    intro i _
    apply Finset.sum_congr rfl
    intro j _
    rw [hF j i]
  calc
    2 * (∑ i, ∑ j, if i < j then F i j else 0) =
      (∑ i, ∑ j, if i < j then F i j else 0) +
      (∑ i, ∑ j, if j < i then F i j else 0) := by rw [hswap]; ring
    _ = _ := by
      rw [← Finset.sum_add_distrib]
      apply Finset.sum_congr rfl
      intro i _
      rw [← Finset.sum_add_distrib]
      apply Finset.sum_congr rfl
      intro j _
      rcases lt_trichotomy i j with h | h | h
      · simp [h, ne_of_lt h, not_lt_of_ge h.le]
      · simp [h]
      · simp [h, ne_of_gt h, not_lt_of_ge h.le]

theorem sum_adj_indicator {V : Type uV} [Fintype V] (G : SimpleGraph V)
    [DecidableRel G.Adj] :
    (∑ i, ∑ j, if G.Adj i j then (1 : ℝ) else 0) = 2 * G.edgeFinset.card := by
  classical
  have hi (i : V) : (∑ j, if G.Adj i j then (1 : ℝ) else 0) = G.degree i := by
    rw [← G.card_neighborFinset_eq_degree, G.neighborFinset_eq_filter]
    simp
  simp_rw [hi]
  exact_mod_cast G.sum_degrees_eq_twice_card_edges

theorem core_energy_lower_small_diameter {N : ℕ} {t : ℝ} (ht : 0 ≤ t)
    (x : Fin N → Position) (hx : ∀ i j, dist (x i) (x j) ≤ 1) :
    t ^ 2 / 8 * (N : ℝ) ^ 2 - t ^ 2 / 2 * N ≤ energy (corePotential t) x := by
  classical
  let G := shellGraph x (microLength t)
  have hG : G.CliqueFree 5 := shellGraph_cliqueFree x (microLength_pos t)
  have hbound : (8 : ℝ) * G.edgeFinset.card ≤ 3 * (N : ℝ) ^ 2 := by
    have hh : 8 * G.edgeFinset.card ≤ 3 * N ^ 2 := by
      simpa using shell_edge_bound G hG
    exact_mod_cast hh
  have hpair := twice_pair_sum
    (fun i j => coreReal t (dist (x i) (x j))) (fun i j => by rw [dist_comm])
  have heq : (∑ i, ∑ j with i < j, coreReal t (dist (x i) (x j))) =
      energy (corePotential t) x := by rfl
  rw [heq] at hpair
  have hterm (i j : Fin N) :
      t ^ 2 * ((if i ≠ j then (1 : ℝ) else 0) - (if G.Adj i j then 1 else 0)) ≤
      (if i ≠ j then coreReal t (dist (x i) (x j)) else 0) := by
    by_cases hij : i = j
    · subst j
      simp
    · simp only [ite_eq_left hij]
      by_cases hA : G.Adj i j
      · simp only [hA, ite_true, sub_self, mul_zero]
        exact coreReal_nonneg _ _
      · simp only [hA, ite_false, sub_zero, mul_one]
        rw [coreReal_full_of_not_shell ht (hx i j)]
        intro h
        exact hA ⟨hij, h⟩
  have hsum := Finset.sum_le_sum (s := Finset.univ) fun i _ =>
    Finset.sum_le_sum (s := Finset.univ) fun j _ => hterm i j
  have hcount : (∑ i : Fin N, ∑ j : Fin N, if i ≠ j then (1 : ℝ) else 0) =
      (N : ℝ) ^ 2 - N := by
    have hite (i j : Fin N) : (if i ≠ j then (1 : ℝ) else 0) =
        1 - (if i = j then 1 else 0) := by by_cases h : i = j <;> simp [h]
    simp_rw [hite]
    simp only [Finset.sum_sub_distrib, Finset.sum_const, Finset.card_univ,
      Fintype.card_fin, nsmul_eq_mul, mul_one, Finset.sum_ite_eq,
      Finset.mem_univ, ite_true]
    ring
  simp only [mul_sub, Finset.sum_sub_distrib, ← Finset.mul_sum] at hsum
  rw [hcount, sum_adj_indicator G, ← hpair] at hsum
  have hb := mul_le_mul_of_nonneg_left hbound (sq_nonneg t)
  nlinarith [hsum, hb]

end RadialTransition

end

end OAI