mathlib docs: data.real.pi

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theorem real.pi_gt_sqrt_two_add_series (n : ℕ) :

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

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theorem real.pi_lt_sqrt_two_add_series (n : ℕ) :

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

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theorem real.pi_lower_bound_start (n : ℕ) {a : ℝ} :

(↑0 / ↑1).sqrt_two_add_series n ≤ 2 - (a / 2 ^ (n + 1)) ^ 2 → a < real.pi

From an upper bound on sqrt_two_add_series 0 n = 2 cos (pi / 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 < pi thanks to basic trigonometric inequalities as expressed in pi_gt_sqrt_two_add_series.

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theorem real.sqrt_two_add_series_step_up (c d : ℕ) {a b n : ℕ} {z : ℝ} :

(↑c / ↑d).sqrt_two_add_series n ≤ z → 0 < b → 0 < d → (2 * b + a) * d ^ 2 ≤ c ^ 2 * b → (↑a / ↑b).sqrt_two_add_series (n + 1) ≤ z

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meta def real.pi_lower_bound :

list ℚ → tactic unit

Create a proof of a < pi 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).

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theorem real.pi_upper_bound_start (n : ℕ) {a : ℝ} :

2 - ((a - 1 / 4 ^ n) / 2 ^ (n + 1)) ^ 2 ≤ (↑0 / ↑1).sqrt_two_add_series n → 1 / 4 ^ n ≤ a → real.pi < a

From a lower bound on sqrt_two_add_series 0 n = 2 cos (pi / 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 pi < a thanks to basic trigonometric formulas as expressed in pi_lt_sqrt_two_add_series.

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theorem real.sqrt_two_add_series_step_down (a b : ℕ) {c d n : ℕ} {z : ℝ} :

z ≤ (↑a / ↑b).sqrt_two_add_series n → 0 < b → 0 < d → a ^ 2 * d ≤ (2 * d + c) * b ^ 2 → z ≤ (↑c / ↑d).sqrt_two_add_series (n + 1)

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meta def real.pi_upper_bound :

list ℚ → tactic unit

Create a proof of pi < 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).

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theorem real.pi_gt_three :

3 < real.pi

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theorem real.pi_gt_314 :

157 / 50 < real.pi

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theorem real.pi_lt_315 :

real.pi < 63 / 20

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theorem real.pi_gt_31415 :

6283 / 2000 < real.pi