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https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_neg_ofNat_lt | [20, 1] | [26, 13] | exact b0 | case hb
a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (0 < b) _auto✝
h0 : autoParam (b⁻¹ < 2.7182818283 ^ a) _auto✝
⊢ 0 < b | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_ofNat | [28, 1] | [32, 78] | rw [←Real.exp_one_pow a] | a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
h0 : autoParam (b < 2.7182818283 ^ a) _auto✝
⊢ b < (↑a).exp | a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
h0 : autoParam (b < 2.7182818283 ^ a) _auto✝
⊢ b < exp 1 ^ a |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_ofNat | [28, 1] | [32, 78] | exact _root_.trans h0 (pow_lt_pow_left Real.exp_one_gt_d9 (by norm_num) a0) | a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
h0 : autoParam (b < 2.7182818283 ^ a) _auto✝
⊢ b < exp 1 ^ a | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_ofNat | [28, 1] | [32, 78] | norm_num | a : ℕ
b : ℝ
a0 : autoParam (a ≠ 0) _auto✝
h0 : autoParam (b < 2.7182818283 ^ a) _auto✝
⊢ 0 ≤ 2.7182818283 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_div | [34, 1] | [44, 60] | have e : exp (a/b : ℝ) = exp 1 ^ (a / b : ℝ) := by rw [Real.exp_one_rpow] | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
⊢ c < (↑a / ↑b).exp | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ c < (↑a / ↑b).exp |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_div | [34, 1] | [44, 60] | rw [e, div_eq_mul_inv, Real.rpow_mul (Real.exp_nonneg _), Real.lt_rpow_inv_iff_of_pos,
Real.rpow_natCast, Real.rpow_natCast] | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ c < (↑a / ↑b).exp | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ c ^ b < exp 1 ^ a
case hx
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ c
case hy
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ exp 1 ^ ↑a
case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 < ↑b |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_div | [34, 1] | [44, 60] | rw [Real.exp_one_rpow] | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
⊢ (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_div | [34, 1] | [44, 60] | exact _root_.trans h0 (pow_lt_pow_left Real.exp_one_gt_d9 (by norm_num) a0) | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ c ^ b < exp 1 ^ a | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_div | [34, 1] | [44, 60] | norm_num | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ 2.7182818283 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_div | [34, 1] | [44, 60] | exact c0.le | case hx
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ c | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_div | [34, 1] | [44, 60] | apply Real.rpow_nonneg (Real.exp_nonneg _) | case hy
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ exp 1 ^ ↑a | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_div | [34, 1] | [44, 60] | simp only [Nat.cast_pos, Nat.pos_iff_ne_zero] | case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 < ↑b | case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ b ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_div | [34, 1] | [44, 60] | exact b0 | case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c ^ b < 2.7182818283 ^ a) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ b ≠ 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_div_lt | [46, 1] | [56, 60] | have e : exp (a/b : ℝ) = exp 1 ^ (a / b : ℝ) := by rw [Real.exp_one_rpow] | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
⊢ (↑a / ↑b).exp < c | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ (↑a / ↑b).exp < c |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_div_lt | [46, 1] | [56, 60] | rw [e, div_eq_mul_inv, Real.rpow_mul (Real.exp_nonneg _), Real.rpow_inv_lt_iff_of_pos,
Real.rpow_natCast, Real.rpow_natCast] | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ (↑a / ↑b).exp < c | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ exp 1 ^ a < c ^ b
case hx
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ exp 1 ^ ↑a
case hy
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ c
case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 < ↑b |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_div_lt | [46, 1] | [56, 60] | rw [Real.exp_one_rpow] | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
⊢ (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_div_lt | [46, 1] | [56, 60] | exact _root_.trans (pow_lt_pow_left Real.exp_one_lt_d9 (Real.exp_nonneg _) a0) h0 | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ exp 1 ^ a < c ^ b | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_div_lt | [46, 1] | [56, 60] | apply Real.rpow_nonneg (Real.exp_nonneg _) | case hx
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ exp 1 ^ ↑a | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_div_lt | [46, 1] | [56, 60] | exact c0.le | case hy
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 ≤ c | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_div_lt | [46, 1] | [56, 60] | simp only [Nat.cast_pos, Nat.pos_iff_ne_zero] | case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ 0 < ↑b | case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ b ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_div_lt | [46, 1] | [56, 60] | exact b0 | case hz
a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c ^ b) _auto✝
e : (↑a / ↑b).exp = exp 1 ^ (↑a / ↑b)
⊢ b ≠ 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_neg_div_lt | [58, 1] | [63, 45] | rw [Real.exp_neg, inv_lt (Real.exp_pos _) c0] | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c⁻¹ ^ b < 2.7182818283 ^ a) _auto✝
⊢ (-(↑a / ↑b)).exp < c | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c⁻¹ ^ b < 2.7182818283 ^ a) _auto✝
⊢ c⁻¹ < (↑a / ↑b).exp |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | exp_neg_div_lt | [58, 1] | [63, 45] | exact lt_exp_div a0 b0 (inv_pos.mpr c0) h0 | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (c⁻¹ ^ b < 2.7182818283 ^ a) _auto✝
⊢ c⁻¹ < (↑a / ↑b).exp | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_neg_div | [65, 1] | [70, 45] | rw [Real.exp_neg, lt_inv c0 (Real.exp_pos _)] | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c⁻¹ ^ b) _auto✝
⊢ c < (-(↑a / ↑b)).exp | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c⁻¹ ^ b) _auto✝
⊢ (↑a / ↑b).exp < c⁻¹ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_exp_neg_div | [65, 1] | [70, 45] | exact exp_div_lt a0 b0 (inv_pos.mpr c0) h0 | a b : ℕ
c : ℝ
a0 : autoParam (a ≠ 0) _auto✝
b0 : autoParam (b ≠ 0) _auto✝
c0 : autoParam (0 < c) _auto✝
h0 : autoParam (2.7182818286 ^ a < c⁻¹ ^ b) _auto✝
⊢ (↑a / ↑b).exp < c⁻¹ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_2 | [77, 1] | [79, 29] | rw [Real.lt_log_iff_exp_lt (by norm_num)] | ⊢ 0.693 < log 2 | ⊢ exp 0.693 < 2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_2 | [77, 1] | [79, 29] | norm_num | ⊢ exp 0.693 < 2 | ⊢ (693 / 1000).exp < 2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_2 | [77, 1] | [79, 29] | exact exp_div_lt | ⊢ (693 / 1000).exp < 2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_2 | [77, 1] | [79, 29] | norm_num | ⊢ 0 < 2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_3 | [80, 1] | [82, 29] | rw [Real.lt_log_iff_exp_lt (by norm_num)] | ⊢ 1.098 < log 3 | ⊢ exp 1.098 < 3 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_3 | [80, 1] | [82, 29] | norm_num | ⊢ exp 1.098 < 3 | ⊢ (549 / 500).exp < 3 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_3 | [80, 1] | [82, 29] | exact exp_div_lt | ⊢ (549 / 500).exp < 3 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_3 | [80, 1] | [82, 29] | norm_num | ⊢ 0 < 3 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | log_3_lt | [83, 1] | [85, 29] | rw [Real.log_lt_iff_lt_exp (by norm_num)] | ⊢ log 3 < 1.099 | ⊢ 3 < exp 1.099 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | log_3_lt | [83, 1] | [85, 29] | norm_num | ⊢ 3 < exp 1.099 | ⊢ 3 < (1099 / 1000).exp |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | log_3_lt | [83, 1] | [85, 29] | exact lt_exp_div | ⊢ 3 < (1099 / 1000).exp | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | log_3_lt | [83, 1] | [85, 29] | norm_num | ⊢ 0 < 3 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_4 | [86, 1] | [88, 29] | rw [Real.lt_log_iff_exp_lt (by norm_num)] | ⊢ 1.386 < log 4 | ⊢ exp 1.386 < 4 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_4 | [86, 1] | [88, 29] | norm_num | ⊢ exp 1.386 < 4 | ⊢ (693 / 500).exp < 4 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_4 | [86, 1] | [88, 29] | exact exp_div_lt | ⊢ (693 / 500).exp < 4 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | lt_log_4 | [86, 1] | [88, 29] | norm_num | ⊢ 0 < 4 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | log_4_lt | [89, 1] | [91, 29] | rw [Real.log_lt_iff_lt_exp (by norm_num)] | ⊢ log 4 < 1.387 | ⊢ 4 < exp 1.387 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | log_4_lt | [89, 1] | [91, 29] | norm_num | ⊢ 4 < exp 1.387 | ⊢ 4 < (1387 / 1000).exp |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | log_4_lt | [89, 1] | [91, 29] | exact lt_exp_div | ⊢ 4 < (1387 / 1000).exp | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | log_4_lt | [89, 1] | [91, 29] | norm_num | ⊢ 0 < 4 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | rw [neg_lt, Real.lt_log_iff_exp_lt (lt_of_lt_of_le (by norm_num) le)] | x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ -(1 - 1 / x).log < 0.41 | x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ (-0.41).exp < 1 - 1 / x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | refine lt_of_lt_of_le ?_ le | x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ (-0.41).exp < 1 - 1 / x | x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ (-0.41).exp < 2 / 3 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | norm_num | x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ (-0.41).exp < 2 / 3 | x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ (-(41 / 100)).exp < 2 / 3 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | exact exp_neg_div_lt | x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ (-(41 / 100)).exp < 2 / 3 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | rw [le_tsub_iff_le_tsub (by norm_num), one_div, inv_le (by positivity)] | x : ℝ
x3 : 3 ≤ x
⊢ 2 / 3 ≤ 1 - 1 / x | x : ℝ
x3 : 3 ≤ x
⊢ (1 - 2 / 3)⁻¹ ≤ x
x : ℝ
x3 : 3 ≤ x
⊢ 0 < 1 - 2 / 3
x : ℝ
x3 : 3 ≤ x
⊢ 1 / x ≤ 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | norm_num | x : ℝ
x3 : 3 ≤ x
⊢ 2 / 3 ≤ 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | positivity | x : ℝ
x3 : 3 ≤ x
⊢ 0 < x | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | exact le_trans (by norm_num) x3 | x : ℝ
x3 : 3 ≤ x
⊢ (1 - 2 / 3)⁻¹ ≤ x | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | norm_num | x : ℝ
x3 : 3 ≤ x
⊢ (1 - 2 / 3)⁻¹ ≤ 3 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | norm_num | x : ℝ
x3 : 3 ≤ x
⊢ 0 < 1 - 2 / 3 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | rw [one_div, inv_le_one_iff] | x : ℝ
x3 : 3 ≤ x
⊢ 1 / x ≤ 1 | x : ℝ
x3 : 3 ≤ x
⊢ x ≤ 0 ∨ 1 ≤ x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | right | x : ℝ
x3 : 3 ≤ x
⊢ x ≤ 0 ∨ 1 ≤ x | case h
x : ℝ
x3 : 3 ≤ x
⊢ 1 ≤ x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | exact le_trans (by norm_num) x3 | case h
x : ℝ
x3 : 3 ≤ x
⊢ 1 ≤ x | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | norm_num | x : ℝ
x3 : 3 ≤ x
⊢ 1 ≤ 3 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Specific.lean | neg_log_one_sub_lt | [93, 1] | [102, 33] | norm_num | x : ℝ
x3 : 3 ≤ x
le : 2 / 3 ≤ 1 - 1 / x
⊢ 0 < 2 / 3 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt | [100, 1] | [107, 9] | rw [iter_sqrt] | a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
⊢ a ^ 2 ^ (-↑n) ∈ approx (x.iter_sqrt n) | a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
⊢ a ^ 2 ^ (-↑n) ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt | [100, 1] | [107, 9] | by_cases a0 : a ≤ 0 | a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
⊢ a ^ 2 ^ (-↑n) ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp | case pos
a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
a0 : a ≤ 0
⊢ a ^ 2 ^ (-↑n) ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp
case neg
a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
a0 : ¬a ≤ 0
⊢ a ^ 2 ^ (-↑n) ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt | [100, 1] | [107, 9] | simp only [log_nonpos a0 ax, nan_scaleB', exp_nan, approx_nan, mem_univ] | case pos
a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
a0 : a ≤ 0
⊢ a ^ 2 ^ (-↑n) ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt | [100, 1] | [107, 9] | rw [Real.rpow_def_of_pos (not_le.mp a0)] | case neg
a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
a0 : ¬a ≤ 0
⊢ a ^ 2 ^ (-↑n) ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp | case neg
a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
a0 : ¬a ≤ 0
⊢ (a.log * 2 ^ (-↑n)).exp ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt | [100, 1] | [107, 9] | mono | case neg
a : ℝ
x : Interval
n : ℕ
ax : a ∈ approx x
a0 : ¬a ≤ 0
⊢ (a.log * 2 ^ (-↑n)).exp ∈ approx (x.log.scaleB' (-Fixed.ofNat0 n)).exp | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt' | [109, 1] | [118, 32] | generalize hb : a^(2^n) = b at ax | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
ax : a ^ 2 ^ n ∈ approx x
⊢ a ∈ approx (x.iter_sqrt n) | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
⊢ a ∈ approx (x.iter_sqrt n) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt' | [109, 1] | [118, 32] | have ab : a = b ^ (2 ^ (-n : ℝ) : ℝ) := by
have e : (2:ℝ)^n = (2^n : ℕ) := by norm_num
rw [←hb, Real.rpow_neg (by norm_num), Real.rpow_natCast, e,
Real.pow_rpow_inv_natCast a0 (pow_ne_zero _ (by norm_num))] | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
⊢ a ∈ approx (x.iter_sqrt n) | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
ab : a = b ^ 2 ^ (-↑n)
⊢ a ∈ approx (x.iter_sqrt n) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt' | [109, 1] | [118, 32] | rw [ab] | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
ab : a = b ^ 2 ^ (-↑n)
⊢ a ∈ approx (x.iter_sqrt n) | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
ab : a = b ^ 2 ^ (-↑n)
⊢ b ^ 2 ^ (-↑n) ∈ approx (x.iter_sqrt n) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt' | [109, 1] | [118, 32] | exact mem_approx_iter_sqrt ax | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
ab : a = b ^ 2 ^ (-↑n)
⊢ b ^ 2 ^ (-↑n) ∈ approx (x.iter_sqrt n) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt' | [109, 1] | [118, 32] | have e : (2:ℝ)^n = (2^n : ℕ) := by norm_num | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
⊢ a = b ^ 2 ^ (-↑n) | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
e : 2 ^ n = ↑(2 ^ n)
⊢ a = b ^ 2 ^ (-↑n) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt' | [109, 1] | [118, 32] | rw [←hb, Real.rpow_neg (by norm_num), Real.rpow_natCast, e,
Real.pow_rpow_inv_natCast a0 (pow_ne_zero _ (by norm_num))] | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
e : 2 ^ n = ↑(2 ^ n)
⊢ a = b ^ 2 ^ (-↑n) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt' | [109, 1] | [118, 32] | norm_num | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
⊢ 2 ^ n = ↑(2 ^ n) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt' | [109, 1] | [118, 32] | norm_num | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
e : 2 ^ n = ↑(2 ^ n)
⊢ 0 ≤ 2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Interval.mem_approx_iter_sqrt' | [109, 1] | [118, 32] | norm_num | a : ℝ
x : Interval
n : ℕ
a0 : 0 ≤ a
b : ℝ
hb : a ^ 2 ^ n = b
ax : b ∈ approx x
e : 2 ^ n = ↑(2 ^ n)
⊢ 2 ≠ 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_small | [120, 1] | [130, 33] | refine ss ⟨?_, ?_⟩ | c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
ss : Icc 0.216 1 ⊆ approx potential_small
⊢ ⋯.potential c' ↑z' ∈ approx potential_small | case refine_1
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
ss : Icc 0.216 1 ⊆ approx potential_small
⊢ 0.216 ≤ ⋯.potential c' ↑z'
case refine_2
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
ss : Icc 0.216 1 ⊆ approx potential_small
⊢ ⋯.potential c' ↑z' ≤ 1 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_small | [120, 1] | [130, 33] | rw [potential_small] | c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
⊢ Icc 0.216 1 ⊆ approx potential_small | c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
⊢ Icc 0.216 1 ⊆ approx (0.216 ∪ 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_small | [120, 1] | [130, 33] | apply Icc_subset_approx | c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
⊢ Icc 0.216 1 ⊆ approx (0.216 ∪ 1) | case ax
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
⊢ 0.216 ∈ approx (0.216 ∪ 1)
case bx
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
⊢ 1 ∈ approx (0.216 ∪ 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_small | [120, 1] | [130, 33] | exact Interval.approx_union_left (Interval.approx_ofScientific _ _ _) | case ax
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
⊢ 0.216 ∈ approx (0.216 ∪ 1) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_small | [120, 1] | [130, 33] | exact Interval.approx_union_right Interval.mem_approx_one | case bx
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
⊢ 1 ∈ approx (0.216 ∪ 1) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_small | [120, 1] | [130, 33] | exact le_potential c4 z4 | case refine_1
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
ss : Icc 0.216 1 ⊆ approx potential_small
⊢ 0.216 ≤ ⋯.potential c' ↑z' | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_small | [120, 1] | [130, 33] | apply Super.potential_le_one | case refine_2
c' z' : ℂ
c4 : Complex.abs c' ≤ 4
z4 : Complex.abs z' ≤ 4
ss : Icc 0.216 1 ⊆ approx potential_small
⊢ ⋯.potential c' ↑z' ≤ 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [potential_large] | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ ⋯.potential c' ↑z' ∈ approx z.potential_large | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ ⋯.potential c' ↑z' ∈
approx
(let s := z.normSq;
let log_s := s.log;
(-log_s.div2).exp.grow (0.8095 * (log_s * -0.9635).exp).hi) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | apply Interval.approx_grow (potential_approx 2 (le_trans (by norm_num) z6) cz) | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ ⋯.potential c' ↑z' ∈
approx
(let s := z.normSq;
let log_s := s.log;
(-log_s.div2).exp.grow (0.8095 * (log_s * -0.9635).exp).hi) | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ (0.8095 * (z.normSq.log * -0.9635).exp).hi ≠ nan →
potential_error 2 c' z' ≤ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val
case zx
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ 1 / Complex.abs z' ∈ approx (-z.normSq.log.div2).exp |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | norm_num | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ 4 ≤ 6 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | intro n | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ (0.8095 * (z.normSq.log * -0.9635).exp).hi ≠ nan →
potential_error 2 c' z' ≤ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : (0.8095 * (z.normSq.log * -0.9635).exp).hi ≠ nan
⊢ potential_error 2 c' z' ≤ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [Ne, Interval.hi_eq_nan] at n | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : (0.8095 * (z.normSq.log * -0.9635).exp).hi ≠ nan
⊢ potential_error 2 c' z' ≤ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ potential_error 2 c' z' ≤ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | refine le_trans (potential_error_le_of_z6 _ z6 cz) ?_ | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ potential_error 2 c' z' ≤ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ 0.8095 / Complex.abs z' ^ 1.927 ≤ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | apply Interval.le_hi n | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ 0.8095 / Complex.abs z' ^ 1.927 ≤ (0.8095 * (z.normSq.log * -0.9635).exp).hi.val | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ 0.8095 / Complex.abs z' ^ 1.927 ∈ approx (0.8095 * (z.normSq.log * -0.9635).exp) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [div_eq_mul_inv, ←Real.rpow_neg (Complex.abs.nonneg _), Real.rpow_def_of_pos (by linarith)] | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ 0.8095 / Complex.abs z' ^ 1.927 ∈ approx (0.8095 * (z.normSq.log * -0.9635).exp) | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ 0.8095 * ((Complex.abs z').log * -1.927).exp ∈ approx (0.8095 * (z.normSq.log * -0.9635).exp) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | have e : Real.log (Complex.abs z') * -1.927 = Real.log (Complex.abs z' ^ 2) * -0.9635 := by
rw [Real.log_pow, Nat.cast_two, mul_comm (2:ℝ), mul_assoc]; norm_num | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ 0.8095 * ((Complex.abs z').log * -1.927).exp ∈ approx (0.8095 * (z.normSq.log * -0.9635).exp) | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
e : (Complex.abs z').log * -1.927 = (Complex.abs z' ^ 2).log * -0.9635
⊢ 0.8095 * ((Complex.abs z').log * -1.927).exp ∈ approx (0.8095 * (z.normSq.log * -0.9635).exp) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [e] | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
e : (Complex.abs z').log * -1.927 = (Complex.abs z' ^ 2).log * -0.9635
⊢ 0.8095 * ((Complex.abs z').log * -1.927).exp ∈ approx (0.8095 * (z.normSq.log * -0.9635).exp) | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
e : (Complex.abs z').log * -1.927 = (Complex.abs z' ^ 2).log * -0.9635
⊢ 0.8095 * ((Complex.abs z' ^ 2).log * -0.9635).exp ∈ approx (0.8095 * (z.normSq.log * -0.9635).exp) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | mono | case be
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
e : (Complex.abs z').log * -1.927 = (Complex.abs z' ^ 2).log * -0.9635
⊢ 0.8095 * ((Complex.abs z' ^ 2).log * -0.9635).exp ∈ approx (0.8095 * (z.normSq.log * -0.9635).exp) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | linarith | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ 0 < Complex.abs z' | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [Real.log_pow, Nat.cast_two, mul_comm (2:ℝ), mul_assoc] | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ (Complex.abs z').log * -1.927 = (Complex.abs z' ^ 2).log * -0.9635 | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ (Complex.abs z').log * -1.927 = (Complex.abs z').log * (2 * -0.9635) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | norm_num | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
n : ¬0.8095 * (z.normSq.log * -0.9635).exp = nan
⊢ (Complex.abs z').log * -1.927 = (Complex.abs z').log * (2 * -0.9635) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | have e : 1 / Complex.abs z' = Real.exp (-(Real.log (Complex.abs z' ^ 2) / 2)) := by
simp only [one_div, Real.log_pow, Nat.cast_ofNat, neg_mul, Real.rpow_neg zero_le_two,
Real.rpow_one, ←mul_assoc, mul_comm _ (2:ℝ)⁻¹]
rw [mul_div_cancel_left₀ _ two_ne_zero, Real.exp_neg, Real.exp_log (by linarith)] | case zx
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ 1 / Complex.abs z' ∈ approx (-z.normSq.log.div2).exp | case zx
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
e : 1 / Complex.abs z' = (-((Complex.abs z' ^ 2).log / 2)).exp
⊢ 1 / Complex.abs z' ∈ approx (-z.normSq.log.div2).exp |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [e] | case zx
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
e : 1 / Complex.abs z' = (-((Complex.abs z' ^ 2).log / 2)).exp
⊢ 1 / Complex.abs z' ∈ approx (-z.normSq.log.div2).exp | case zx
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
e : 1 / Complex.abs z' = (-((Complex.abs z' ^ 2).log / 2)).exp
⊢ (-((Complex.abs z' ^ 2).log / 2)).exp ∈ approx (-z.normSq.log.div2).exp |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | mono | case zx
c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
e : 1 / Complex.abs z' = (-((Complex.abs z' ^ 2).log / 2)).exp
⊢ (-((Complex.abs z' ^ 2).log / 2)).exp ∈ approx (-z.normSq.log.div2).exp | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | simp only [one_div, Real.log_pow, Nat.cast_ofNat, neg_mul, Real.rpow_neg zero_le_two,
Real.rpow_one, ←mul_assoc, mul_comm _ (2:ℝ)⁻¹] | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ 1 / Complex.abs z' = (-((Complex.abs z' ^ 2).log / 2)).exp | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ (Complex.abs z')⁻¹ = (-(2 * (Complex.abs z').log / 2)).exp |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Render/Potential.lean | Box.approx_potential_large | [132, 1] | [151, 9] | rw [mul_div_cancel_left₀ _ two_ne_zero, Real.exp_neg, Real.exp_log (by linarith)] | c' z' : ℂ
z : Box
cz : Complex.abs c' ≤ Complex.abs z'
z6 : 6 ≤ Complex.abs z'
zm : z' ∈ approx z
⊢ (Complex.abs z')⁻¹ = (-(2 * (Complex.abs z').log / 2)).exp | no goals |
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