# Documentation

Mathlib.Data.Real.Pi.Bounds

# Pi #

This file contains lemmas which establish bounds on real.pi. Notably, these include pi_gt_sqrtTwoAddSeries and pi_lt_sqrtTwoAddSeries, 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 Mathlib/Data/Real/Pi/Leibniz.lean and Mathlib/Data/Real/Pi/Wallis.lean for infinite formulas for π.

theorem Real.pi_lt_sqrtTwoAddSeries (n : ) :
Real.pi < 2 ^ (n + 1) * Real.sqrt () + 1 / 4 ^ n
theorem Real.pi_lower_bound_start (n : ) {a : } (h : Real.sqrtTwoAddSeries (0 / 1) n 2 - (a / 2 ^ (n + 1)) ^ 2) :

From an upper bound on sqrtTwoAddSeries 0 n = 2 cos (π / 2 ^ (n+1)) of the form sqrtTwoAddSeries 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_sqrtTwoAddSeries.

theorem Real.sqrtTwoAddSeries_step_up (c : ) (d : ) {a : } {b : } {n : } {z : } (hz : Real.sqrtTwoAddSeries (c / d) n z) (hb : 0 < b) (hd : 0 < d) (h : (2 * b + a) * d ^ 2 c ^ 2 * b) :
Real.sqrtTwoAddSeries (a / b) (n + 1) z

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).

Instances For
theorem Real.pi_upper_bound_start (n : ) {a : } (h : 2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 Real.sqrtTwoAddSeries (0 / 1) n) (h₂ : 1 / 4 ^ n a) :

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

theorem Real.sqrtTwoAddSeries_step_down (a : ) (b : ) {c : } {d : } {n : } {z : } (hz : z Real.sqrtTwoAddSeries (a / b) n) (hb : 0 < b) (hd : 0 < d) (h : a ^ 2 * d (2 * d + c) * b ^ 2) :
z Real.sqrtTwoAddSeries (c / d) (n + 1)

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).

Instances For