Datasets:

roseking
/

openAI-math-dataset

import Mathlib

namespace OAI

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namespace VertexCover.AntiConcentration
noncomputable section

def centralProbability (n : ℕ) : ℝ := (n.centralBinom : ℝ) / 4^n

theorem centralProbability_nonneg (n : ℕ) : 0 ≤ centralProbability n := by
  unfold centralProbability
  positivity

theorem centralProbability_step (n : ℕ) :
    centralProbability (n+1) = centralProbability n * ((2*(n:ℝ)+1)/(2*(n:ℝ)+2)) := by
  have hn : (n : ℝ) + 1 ≠ 0 := by positivity
  have hn2 : 2*(n : ℝ)+2 ≠ 0 := by positivity
  have h4 : (4 : ℝ)^n ≠ 0 := by positivity
  have hrec := congrArg (fun k : ℕ => (k : ℝ)) (Nat.succ_mul_centralBinom_succ n)
  push_cast at hrec
  have hc : ((n+1).centralBinom : ℝ) = 2*(2*(n:ℝ)+1)*(n.centralBinom : ℝ)/((n:ℝ)+1) := by
    apply (eq_div_iff hn).mpr
    nlinarith [hrec]
  unfold centralProbability
  rw [hc, pow_succ]
  field_simp
  ; ring

theorem centralProbability_sq (n : ℕ) :
    centralProbability n ^ 2 * ((n : ℝ)+1) ≤ 1 := by
  induction n with
  | zero => norm_num [centralProbability]
  | succ n ih =>
      have hn : 0 ≤ (n : ℝ) := Nat.cast_nonneg _
      have hden : 0 < (2*(n : ℝ)+2)^2 := by positivity
      have hstep : ((2*(n : ℝ)+1)/(2*(n : ℝ)+2))^2 * ((n : ℝ)+2) ≤ (n : ℝ)+1 := by
        rw [div_pow, div_mul_eq_mul_div, div_le_iff₀ hden]
        nlinarith
      rw [centralProbability_step]
      push_cast
      calc
        _ = centralProbability n ^2 *
            ((((2*(n:ℝ)+1)/(2*(n:ℝ)+2))^2) * ((n:ℝ)+2)) := by ring
        _ ≤ centralProbability n ^2 * ((n:ℝ)+1) :=
          mul_le_mul_of_nonneg_left hstep (sq_nonneg _)
        _ ≤ 1 := ih

theorem centralProbability_le (n : ℕ) :
    centralProbability n ≤ 1 / Real.sqrt ((n : ℝ)+1) := by
  have hs : 0 < Real.sqrt ((n : ℝ)+1) := Real.sqrt_pos.mpr (by positivity)
  apply (le_div_iff₀ hs).mpr
  have hsq : (centralProbability n * Real.sqrt ((n : ℝ)+1))^2 ≤ (1:ℝ)^2 := by
    rw [mul_pow, Real.sq_sqrt (by positivity)]
    simpa only [one_pow] using centralProbability_sq n
  exact (sq_le_sq₀ (mul_nonneg (centralProbability_nonneg n) hs.le) zero_le_one).mp hsq

theorem centralProbability_lt (n : ℕ) (hn : 0 < n) :
    centralProbability n < Real.sqrt 2 / Real.sqrt (2*(n : ℝ)) := by
  have hnR : 0 < (n : ℝ) := Nat.cast_pos.mpr hn
  have hnS : 0 < Real.sqrt (n : ℝ) := Real.sqrt_pos.mpr hnR
  have hs2 : 0 < Real.sqrt (2:ℝ) := Real.sqrt_pos.mpr (by norm_num)
  have hstrict : 1 / Real.sqrt ((n : ℝ)+1) < 1 / Real.sqrt (n : ℝ) := by
    exact one_div_lt_one_div_of_lt hnS (Real.sqrt_lt_sqrt hnR.le (by linarith))
  have heq : 1 / Real.sqrt (n : ℝ) = Real.sqrt 2 / Real.sqrt (2*(n : ℝ)) := by
    rw [Real.sqrt_mul (by norm_num : (0:ℝ) ≤ 2)]
    field_simp
  exact lt_of_le_of_lt (centralProbability_le n) (hstrict.trans_eq heq)

theorem sign_atom_le (n v : ℕ) :
    ((2*n).choose v : ℝ) / 2^(2*n) ≤ centralProbability n := by
  have hb : (2 : ℝ)^(2*n) = (4 : ℝ)^n := by rw [pow_mul]; norm_num
  rw [hb]
  exact div_le_div_of_nonneg_right (Nat.cast_le.mpr (Nat.choose_le_centralBinom v n)) (by positivity)

end
end VertexCover.AntiConcentration


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end OAI

	import Mathlib

	namespace OAI

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	namespace VertexCover.AntiConcentration
	noncomputable section

	def centralProbability (n : ℕ) : ℝ := (n.centralBinom : ℝ) / 4^n

	theorem centralProbability_nonneg (n : ℕ) : 0 ≤ centralProbability n := by
	unfold centralProbability
	positivity

	theorem centralProbability_step (n : ℕ) :
	centralProbability (n+1) = centralProbability n * ((2(n:ℝ)+1)/(2(n:ℝ)+2)) := by
	have hn : (n : ℝ) + 1 ≠ 0 := by positivity
	have hn2 : 2*(n : ℝ)+2 ≠ 0 := by positivity
	have h4 : (4 : ℝ)^n ≠ 0 := by positivity
	have hrec := congrArg (fun k : ℕ => (k : ℝ)) (Nat.succ_mul_centralBinom_succ n)
	push_cast at hrec
	have hc : ((n+1).centralBinom : ℝ) = 2(2(n:ℝ)+1)*(n.centralBinom : ℝ)/((n:ℝ)+1) := by
	apply (eq_div_iff hn).mpr
	nlinarith [hrec]
	unfold centralProbability
	rw [hc, pow_succ]
	field_simp
	; ring

	theorem centralProbability_sq (n : ℕ) :
	centralProbability n ^ 2 * ((n : ℝ)+1) ≤ 1 := by
	induction n with
	\| zero => norm_num [centralProbability]
	\| succ n ih =>
	have hn : 0 ≤ (n : ℝ) := Nat.cast_nonneg _
	have hden : 0 < (2*(n : ℝ)+2)^2 := by positivity
	have hstep : ((2(n : ℝ)+1)/(2(n : ℝ)+2))^2 * ((n : ℝ)+2) ≤ (n : ℝ)+1 := by
	rw [div_pow, div_mul_eq_mul_div, div_le_iff₀ hden]
	nlinarith
	rw [centralProbability_step]
	push_cast
	calc
	_ = centralProbability n ^2 *
	((((2(n:ℝ)+1)/(2(n:ℝ)+2))^2) * ((n:ℝ)+2)) := by ring
	_ ≤ centralProbability n ^2 * ((n:ℝ)+1) :=
	mul_le_mul_of_nonneg_left hstep (sq_nonneg _)
	_ ≤ 1 := ih

	theorem centralProbability_le (n : ℕ) :
	centralProbability n ≤ 1 / Real.sqrt ((n : ℝ)+1) := by
	have hs : 0 < Real.sqrt ((n : ℝ)+1) := Real.sqrt_pos.mpr (by positivity)
	apply (le_div_iff₀ hs).mpr
	have hsq : (centralProbability n * Real.sqrt ((n : ℝ)+1))^2 ≤ (1:ℝ)^2 := by
	rw [mul_pow, Real.sq_sqrt (by positivity)]
	simpa only [one_pow] using centralProbability_sq n
	exact (sq_le_sq₀ (mul_nonneg (centralProbability_nonneg n) hs.le) zero_le_one).mp hsq

	theorem centralProbability_lt (n : ℕ) (hn : 0 < n) :
	centralProbability n < Real.sqrt 2 / Real.sqrt (2*(n : ℝ)) := by
	have hnR : 0 < (n : ℝ) := Nat.cast_pos.mpr hn
	have hnS : 0 < Real.sqrt (n : ℝ) := Real.sqrt_pos.mpr hnR
	have hs2 : 0 < Real.sqrt (2:ℝ) := Real.sqrt_pos.mpr (by norm_num)
	have hstrict : 1 / Real.sqrt ((n : ℝ)+1) < 1 / Real.sqrt (n : ℝ) := by
	exact one_div_lt_one_div_of_lt hnS (Real.sqrt_lt_sqrt hnR.le (by linarith))
	have heq : 1 / Real.sqrt (n : ℝ) = Real.sqrt 2 / Real.sqrt (2*(n : ℝ)) := by
	rw [Real.sqrt_mul (by norm_num : (0:ℝ) ≤ 2)]
	field_simp
	exact lt_of_le_of_lt (centralProbability_le n) (hstrict.trans_eq heq)

	theorem sign_atom_le (n v : ℕ) :
	((2n).choose v : ℝ) / 2^(2n) ≤ centralProbability n := by
	have hb : (2 : ℝ)^(2*n) = (4 : ℝ)^n := by rw [pow_mul]; norm_num
	rw [hb]
	exact div_le_div_of_nonneg_right (Nat.cast_le.mpr (Nat.choose_le_centralBinom v n)) (by positivity)

	end
	end VertexCover.AntiConcentration


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	end OAI