mathlib3 documentation

data.real.pi.bounds

Pi #

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This file contains lemmas which establish bounds on real.pi. Notably, these include pi_gt_sqrt_two_add_series and pi_lt_sqrt_two_add_series, which bound π using series; numerical bounds on π such as pi_gt_314and pi_lt_315 (more precise versions are given, too).

See also data.real.pi.leibniz and data.real.pi.wallis for infinite formulas for π.

theorem real.pi_gt_sqrt_two_add_series (n : ℕ) :

2 ^ (n + 1) * √ (2 - real.sqrt_two_add_series 0 n) < real.pi

theorem real.pi_lt_sqrt_two_add_series (n : ℕ) :

real.pi < 2 ^ (n + 1) * √ (2 - real.sqrt_two_add_series 0 n) + 1 / 4 ^ n

theorem real.pi_lower_bound_start (n : ℕ) {a : ℝ} (h : real.sqrt_two_add_series (↑0 / ↑1) n ≤ 2 - (a / 2 ^ (n + 1)) ^ 2) :

From an upper bound on sqrt_two_add_series 0 n = 2 cos (π / 2 ^ (n+1)) of the form sqrt_two_add_series 0 n ≤ 2 - (a / 2 ^ (n + 1)) ^ 2), one can deduce the lower bound a < π thanks to basic trigonometric inequalities as expressed in pi_gt_sqrt_two_add_series.

theorem real.sqrt_two_add_series_step_up (c d : ℕ) {a b n : ℕ} {z : ℝ} (hz : real.sqrt_two_add_series (↑c / ↑d) n ≤ z) (hb : 0 < b) (hd : 0 < d) (h : (2 * b + a) * d ^ 2 ≤ c ^ 2 * b) :

real.sqrt_two_add_series (↑a / ↑b) (n + 1) ≤ z

meta def real.pi_lower_bound (l : list ℚ) :

Create a proof of a < π for a fixed rational number a, given a witness, which is a sequence of rational numbers sqrt 2 < r 1 < r 2 < ... < r n < 2 satisfying the property that sqrt (2 + r i) ≤ r(i+1), where r 0 = 0 and sqrt (2 - r n) ≥ a/2^(n+1).

theorem real.pi_upper_bound_start (n : ℕ) {a : ℝ} (h : 2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ real.sqrt_two_add_series (↑0 / ↑1) n) (h₂ : 1 / 4 ^ n ≤ a) :

From a lower bound on sqrt_two_add_series 0 n = 2 cos (π / 2 ^ (n+1)) of the form 2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ sqrt_two_add_series 0 n, one can deduce the upper bound π < a thanks to basic trigonometric formulas as expressed in pi_lt_sqrt_two_add_series.

theorem real.sqrt_two_add_series_step_down (a b : ℕ) {c d n : ℕ} {z : ℝ} (hz : z ≤ real.sqrt_two_add_series (↑a / ↑b) n) (hb : 0 < b) (hd : 0 < d) (h : a ^ 2 * d ≤ (2 * d + c) * b ^ 2) :

z ≤ real.sqrt_two_add_series (↑c / ↑d) (n + 1)

meta def real.pi_upper_bound (l : list ℚ) :

Create a proof of π < a for a fixed rational number a, given a witness, which is a sequence of rational numbers sqrt 2 < r 1 < r 2 < ... < r n < 2 satisfying the property that sqrt (2 + r i) ≥ r(i+1), where r 0 = 0 and sqrt (2 - r n) ≥ (a - 1/4^n) / 2^(n+1).

theorem real.pi_gt_three :

theorem real.pi_gt_314 :

157 / 50 < real.pi

theorem real.pi_lt_315 :

real.pi < 63 / 20

theorem real.pi_gt_31415 :

6283 / 2000 < real.pi

theorem real.pi_lt_31416 :

real.pi < 3927 / 1250

theorem real.pi_gt_3141592 :

392699 / 125000 < real.pi

theorem real.pi_lt_3141593 :

real.pi < 3141593 / 1000000