state stringlengths 0 159k | srcUpToTactic stringlengths 387 167k | nextTactic stringlengths 3 9k | declUpToTactic stringlengths 22 11.5k | declId stringlengths 38 95 | decl stringlengths 16 1.89k | file_tag stringlengths 17 73 |
|---|---|---|---|---|---|---|
x : ℝ
hx : abs' x ≤ 1
n : ℕ
hn : 0 < n
hxc : Complex.abs ↑x ≤ 1
⊢ abs' (rexp x - ∑ m in range n, x ^ m / ↑(Nat.factorial m)) ≤ abs' x ^ n * (↑(Nat.succ n) / (↑(Nat.factorial n) * ↑n)) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | convert exp_bound hxc hn using 2 | nonrec theorem exp_bound {x : ℝ} (hx : |x| ≤ 1) {n : ℕ} (hn : 0 < n) :
|exp x - ∑ m in range n, x ^ m / m.factorial| ≤ |x| ^ n * (n.succ / (n.factorial * n)) := by
have hxc : Complex.abs x ≤ 1 := mod_cast hx
| Mathlib.Data.Complex.Exponential.1730_0.1bixbwhfBeJKySp | nonrec theorem exp_bound {x : ℝ} (hx : |x| ≤ 1) {n : ℕ} (hn : 0 < n) :
|exp x - ∑ m in range n, x ^ m / m.factorial| ≤ |x| ^ n * (n.succ / (n.factorial * n)) | Mathlib_Data_Complex_Exponential |
case h.e'_3
x : ℝ
hx : abs' x ≤ 1
n : ℕ
hn : 0 < n
hxc : Complex.abs ↑x ≤ 1
⊢ abs' (rexp x - ∑ m in range n, x ^ m / ↑(Nat.factorial m)) =
Complex.abs (cexp ↑x - ∑ m in range n, ↑x ^ m / ↑(Nat.factorial m)) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp only [← abs_ofReal, ← ofReal_sub, ← ofReal_exp, ← ofReal_sum, ← ofReal_pow,
← ofReal_div, ← ofReal_nat_cast] | nonrec theorem exp_bound {x : ℝ} (hx : |x| ≤ 1) {n : ℕ} (hn : 0 < n) :
|exp x - ∑ m in range n, x ^ m / m.factorial| ≤ |x| ^ n * (n.succ / (n.factorial * n)) := by
have hxc : Complex.abs x ≤ 1 := mod_cast hx
convert exp_bound hxc hn using 2 <;>
--Porting note: was `norm_cast`
| Mathlib.Data.Complex.Exponential.1730_0.1bixbwhfBeJKySp | nonrec theorem exp_bound {x : ℝ} (hx : |x| ≤ 1) {n : ℕ} (hn : 0 < n) :
|exp x - ∑ m in range n, x ^ m / m.factorial| ≤ |x| ^ n * (n.succ / (n.factorial * n)) | Mathlib_Data_Complex_Exponential |
case h.e'_4.h.e'_5
x : ℝ
hx : abs' x ≤ 1
n : ℕ
hn : 0 < n
hxc : Complex.abs ↑x ≤ 1
⊢ abs' x ^ n = Complex.abs ↑x ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp only [← abs_ofReal, ← ofReal_sub, ← ofReal_exp, ← ofReal_sum, ← ofReal_pow,
← ofReal_div, ← ofReal_nat_cast] | nonrec theorem exp_bound {x : ℝ} (hx : |x| ≤ 1) {n : ℕ} (hn : 0 < n) :
|exp x - ∑ m in range n, x ^ m / m.factorial| ≤ |x| ^ n * (n.succ / (n.factorial * n)) := by
have hxc : Complex.abs x ≤ 1 := mod_cast hx
convert exp_bound hxc hn using 2 <;>
--Porting note: was `norm_cast`
| Mathlib.Data.Complex.Exponential.1730_0.1bixbwhfBeJKySp | nonrec theorem exp_bound {x : ℝ} (hx : |x| ≤ 1) {n : ℕ} (hn : 0 < n) :
|exp x - ∑ m in range n, x ^ m / m.factorial| ≤ |x| ^ n * (n.succ / (n.factorial * n)) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 ≤ x
h2 : x ≤ 1
n : ℕ
hn : 0 < n
⊢ rexp x ≤ ∑ m in range n, x ^ m / ↑(Nat.factorial m) + x ^ n * (↑n + 1) / (↑(Nat.factorial n) * ↑n) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have h3 : |x| = x := by simpa | theorem exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :
Real.exp x ≤ (∑ m in Finset.range n, x ^ m / m.factorial) +
x ^ n * (n + 1) / (n.factorial * n) := by
| Mathlib.Data.Complex.Exponential.1739_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :
Real.exp x ≤ (∑ m in Finset.range n, x ^ m / m.factorial) +
x ^ n * (n + 1) / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 ≤ x
h2 : x ≤ 1
n : ℕ
hn : 0 < n
⊢ abs' x = x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simpa | theorem exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :
Real.exp x ≤ (∑ m in Finset.range n, x ^ m / m.factorial) +
x ^ n * (n + 1) / (n.factorial * n) := by
have h3 : |x| = x := by | Mathlib.Data.Complex.Exponential.1739_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :
Real.exp x ≤ (∑ m in Finset.range n, x ^ m / m.factorial) +
x ^ n * (n + 1) / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 ≤ x
h2 : x ≤ 1
n : ℕ
hn : 0 < n
h3 : abs' x = x
⊢ rexp x ≤ ∑ m in range n, x ^ m / ↑(Nat.factorial m) + x ^ n * (↑n + 1) / (↑(Nat.factorial n) * ↑n) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have h4 : |x| ≤ 1 := by rwa [h3] | theorem exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :
Real.exp x ≤ (∑ m in Finset.range n, x ^ m / m.factorial) +
x ^ n * (n + 1) / (n.factorial * n) := by
have h3 : |x| = x := by simpa
| Mathlib.Data.Complex.Exponential.1739_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :
Real.exp x ≤ (∑ m in Finset.range n, x ^ m / m.factorial) +
x ^ n * (n + 1) / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 ≤ x
h2 : x ≤ 1
n : ℕ
hn : 0 < n
h3 : abs' x = x
⊢ abs' x ≤ 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rwa [h3] | theorem exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :
Real.exp x ≤ (∑ m in Finset.range n, x ^ m / m.factorial) +
x ^ n * (n + 1) / (n.factorial * n) := by
have h3 : |x| = x := by simpa
have h4 : |x| ≤ 1 := by | Mathlib.Data.Complex.Exponential.1739_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :
Real.exp x ≤ (∑ m in Finset.range n, x ^ m / m.factorial) +
x ^ n * (n + 1) / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 ≤ x
h2 : x ≤ 1
n : ℕ
hn : 0 < n
h3 : abs' x = x
h4 : abs' x ≤ 1
⊢ rexp x ≤ ∑ m in range n, x ^ m / ↑(Nat.factorial m) + x ^ n * (↑n + 1) / (↑(Nat.factorial n) * ↑n) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have h' := Real.exp_bound h4 hn | theorem exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :
Real.exp x ≤ (∑ m in Finset.range n, x ^ m / m.factorial) +
x ^ n * (n + 1) / (n.factorial * n) := by
have h3 : |x| = x := by simpa
have h4 : |x| ≤ 1 := by rwa [h3]
| Mathlib.Data.Complex.Exponential.1739_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :
Real.exp x ≤ (∑ m in Finset.range n, x ^ m / m.factorial) +
x ^ n * (n + 1) / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 ≤ x
h2 : x ≤ 1
n : ℕ
hn : 0 < n
h3 : abs' x = x
h4 : abs' x ≤ 1
h' :
abs' (rexp x - ∑ m in range n, x ^ m / ↑(Nat.factorial m)) ≤ abs' x ^ n * (↑(Nat.succ n) / (↑(Nat.factorial n) * ↑n))
⊢ rexp x ≤ ∑ m in range n, x ^ m / ↑(Nat.factorial m) + x ^ n * (↑n + 1) / (↑(Nat.factorial n) * ↑n) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [h3] at h' | theorem exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :
Real.exp x ≤ (∑ m in Finset.range n, x ^ m / m.factorial) +
x ^ n * (n + 1) / (n.factorial * n) := by
have h3 : |x| = x := by simpa
have h4 : |x| ≤ 1 := by rwa [h3]
have h' := Real.exp_bound h4 hn
| Mathlib.Data.Complex.Exponential.1739_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :
Real.exp x ≤ (∑ m in Finset.range n, x ^ m / m.factorial) +
x ^ n * (n + 1) / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 ≤ x
h2 : x ≤ 1
n : ℕ
hn : 0 < n
h3 : abs' x = x
h4 : abs' x ≤ 1
h' : abs' (rexp x - ∑ m in range n, x ^ m / ↑(Nat.factorial m)) ≤ x ^ n * (↑(Nat.succ n) / (↑(Nat.factorial n) * ↑n))
⊢ rexp x ≤ ∑ m in range n, x ^ m / ↑(Nat.factorial m) + x ^ n * (↑n + 1) / (↑(Nat.factorial n) * ↑n) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have h'' := (abs_sub_le_iff.1 h').1 | theorem exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :
Real.exp x ≤ (∑ m in Finset.range n, x ^ m / m.factorial) +
x ^ n * (n + 1) / (n.factorial * n) := by
have h3 : |x| = x := by simpa
have h4 : |x| ≤ 1 := by rwa [h3]
have h' := Real.exp_bound h4 hn
rw [h3] at h'
| Mathlib.Data.Complex.Exponential.1739_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :
Real.exp x ≤ (∑ m in Finset.range n, x ^ m / m.factorial) +
x ^ n * (n + 1) / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 ≤ x
h2 : x ≤ 1
n : ℕ
hn : 0 < n
h3 : abs' x = x
h4 : abs' x ≤ 1
h' : abs' (rexp x - ∑ m in range n, x ^ m / ↑(Nat.factorial m)) ≤ x ^ n * (↑(Nat.succ n) / (↑(Nat.factorial n) * ↑n))
h'' : rexp x - ∑ m in range n, x ^ m / ↑(Nat.factorial m) ≤ x ^ n * (↑(Nat.succ n) / (↑(Nat.factorial n) * ↑n))
⊢ rexp x ≤ ∑ ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have t := sub_le_iff_le_add'.1 h'' | theorem exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :
Real.exp x ≤ (∑ m in Finset.range n, x ^ m / m.factorial) +
x ^ n * (n + 1) / (n.factorial * n) := by
have h3 : |x| = x := by simpa
have h4 : |x| ≤ 1 := by rwa [h3]
have h' := Real.exp_bound h4 hn
rw [h3] at h'
have h'' := (... | Mathlib.Data.Complex.Exponential.1739_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :
Real.exp x ≤ (∑ m in Finset.range n, x ^ m / m.factorial) +
x ^ n * (n + 1) / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
x : ℝ
h1 : 0 ≤ x
h2 : x ≤ 1
n : ℕ
hn : 0 < n
h3 : abs' x = x
h4 : abs' x ≤ 1
h' : abs' (rexp x - ∑ m in range n, x ^ m / ↑(Nat.factorial m)) ≤ x ^ n * (↑(Nat.succ n) / (↑(Nat.factorial n) * ↑n))
h'' : rexp x - ∑ m in range n, x ^ m / ↑(Nat.factorial m) ≤ x ^ n * (↑(Nat.succ n) / (↑(Nat.factorial n) * ↑n))
t : rexp x ≤ ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simpa [mul_div_assoc] using t | theorem exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :
Real.exp x ≤ (∑ m in Finset.range n, x ^ m / m.factorial) +
x ^ n * (n + 1) / (n.factorial * n) := by
have h3 : |x| = x := by simpa
have h4 : |x| ≤ 1 := by rwa [h3]
have h' := Real.exp_bound h4 hn
rw [h3] at h'
have h'' := (... | Mathlib.Data.Complex.Exponential.1739_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℝ} (h1 : 0 ≤ x) (h2 : x ≤ 1) {n : ℕ} (hn : 0 < n) :
Real.exp x ≤ (∑ m in Finset.range n, x ^ m / m.factorial) +
x ^ n * (n + 1) / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ abs' (rexp x - 1) ≤ 2 * abs' x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have : abs' x ≤ 1 := mod_cast hx | theorem abs_exp_sub_one_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1| ≤ 2 * |x| := by
| Mathlib.Data.Complex.Exponential.1751_0.1bixbwhfBeJKySp | theorem abs_exp_sub_one_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1| ≤ 2 * |x| | Mathlib_Data_Complex_Exponential |
x : ℝ
hx this : abs' x ≤ 1
⊢ abs' (rexp x - 1) ≤ 2 * abs' x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have := Complex.abs_exp_sub_one_le (x := x) (by simpa using this) | theorem abs_exp_sub_one_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1| ≤ 2 * |x| := by
have : abs' x ≤ 1 := mod_cast hx
--Porting note: was
--exact_mod_cast Complex.abs_exp_sub_one_le (x := x) this
| Mathlib.Data.Complex.Exponential.1751_0.1bixbwhfBeJKySp | theorem abs_exp_sub_one_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1| ≤ 2 * |x| | Mathlib_Data_Complex_Exponential |
x : ℝ
hx this : abs' x ≤ 1
⊢ Complex.abs ↑x ≤ 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simpa using this | theorem abs_exp_sub_one_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1| ≤ 2 * |x| := by
have : abs' x ≤ 1 := mod_cast hx
--Porting note: was
--exact_mod_cast Complex.abs_exp_sub_one_le (x := x) this
have := Complex.abs_exp_sub_one_le (x := x) (by | Mathlib.Data.Complex.Exponential.1751_0.1bixbwhfBeJKySp | theorem abs_exp_sub_one_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1| ≤ 2 * |x| | Mathlib_Data_Complex_Exponential |
x : ℝ
hx this✝ : abs' x ≤ 1
this : Complex.abs (cexp ↑x - 1) ≤ 2 * Complex.abs ↑x
⊢ abs' (rexp x - 1) ≤ 2 * abs' x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_exp, ← ofReal_one, ← ofReal_sub, abs_ofReal, abs_ofReal] at this | theorem abs_exp_sub_one_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1| ≤ 2 * |x| := by
have : abs' x ≤ 1 := mod_cast hx
--Porting note: was
--exact_mod_cast Complex.abs_exp_sub_one_le (x := x) this
have := Complex.abs_exp_sub_one_le (x := x) (by simpa using this)
| Mathlib.Data.Complex.Exponential.1751_0.1bixbwhfBeJKySp | theorem abs_exp_sub_one_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1| ≤ 2 * |x| | Mathlib_Data_Complex_Exponential |
x : ℝ
hx this✝ : abs' x ≤ 1
this : abs' (rexp x - 1) ≤ 2 * abs' x
⊢ abs' (rexp x - 1) ≤ 2 * abs' x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | exact this | theorem abs_exp_sub_one_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1| ≤ 2 * |x| := by
have : abs' x ≤ 1 := mod_cast hx
--Porting note: was
--exact_mod_cast Complex.abs_exp_sub_one_le (x := x) this
have := Complex.abs_exp_sub_one_le (x := x) (by simpa using this)
rw [← ofReal_exp, ← ofReal_one, ← ofReal_sub, abs_ofR... | Mathlib.Data.Complex.Exponential.1751_0.1bixbwhfBeJKySp | theorem abs_exp_sub_one_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1| ≤ 2 * |x| | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ abs' (rexp x - 1 - x) ≤ x ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← _root_.sq_abs] | theorem abs_exp_sub_one_sub_id_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1 - x| ≤ x ^ 2 := by
| Mathlib.Data.Complex.Exponential.1760_0.1bixbwhfBeJKySp | theorem abs_exp_sub_one_sub_id_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1 - x| ≤ x ^ 2 | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ abs' (rexp x - 1 - x) ≤ abs' x ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have : Complex.abs x ≤ 1 := mod_cast hx | theorem abs_exp_sub_one_sub_id_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1 - x| ≤ x ^ 2 := by
rw [← _root_.sq_abs]
--Porting note: was
-- exact_mod_cast Complex.abs_exp_sub_one_sub_id_le this
| Mathlib.Data.Complex.Exponential.1760_0.1bixbwhfBeJKySp | theorem abs_exp_sub_one_sub_id_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1 - x| ≤ x ^ 2 | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
this : Complex.abs ↑x ≤ 1
⊢ abs' (rexp x - 1 - x) ≤ abs' x ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have := Complex.abs_exp_sub_one_sub_id_le this | theorem abs_exp_sub_one_sub_id_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1 - x| ≤ x ^ 2 := by
rw [← _root_.sq_abs]
--Porting note: was
-- exact_mod_cast Complex.abs_exp_sub_one_sub_id_le this
have : Complex.abs x ≤ 1 := mod_cast hx
| Mathlib.Data.Complex.Exponential.1760_0.1bixbwhfBeJKySp | theorem abs_exp_sub_one_sub_id_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1 - x| ≤ x ^ 2 | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
this✝ : Complex.abs ↑x ≤ 1
this : Complex.abs (cexp ↑x - 1 - ↑x) ≤ Complex.abs ↑x ^ 2
⊢ abs' (rexp x - 1 - x) ≤ abs' x ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← ofReal_one, ← ofReal_exp, ← ofReal_sub, ← ofReal_sub, abs_ofReal, abs_ofReal] at this | theorem abs_exp_sub_one_sub_id_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1 - x| ≤ x ^ 2 := by
rw [← _root_.sq_abs]
--Porting note: was
-- exact_mod_cast Complex.abs_exp_sub_one_sub_id_le this
have : Complex.abs x ≤ 1 := mod_cast hx
have := Complex.abs_exp_sub_one_sub_id_le this
| Mathlib.Data.Complex.Exponential.1760_0.1bixbwhfBeJKySp | theorem abs_exp_sub_one_sub_id_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1 - x| ≤ x ^ 2 | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
this✝ : Complex.abs ↑x ≤ 1
this : abs' (rexp x - 1 - x) ≤ abs' x ^ 2
⊢ abs' (rexp x - 1 - x) ≤ abs' x ^ 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | exact this | theorem abs_exp_sub_one_sub_id_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1 - x| ≤ x ^ 2 := by
rw [← _root_.sq_abs]
--Porting note: was
-- exact_mod_cast Complex.abs_exp_sub_one_sub_id_le this
have : Complex.abs x ≤ 1 := mod_cast hx
have := Complex.abs_exp_sub_one_sub_id_le this
rw [← ofReal_one, ← ofReal_exp, ← ... | Mathlib.Data.Complex.Exponential.1760_0.1bixbwhfBeJKySp | theorem abs_exp_sub_one_sub_id_le {x : ℝ} (hx : |x| ≤ 1) : |exp x - 1 - x| ≤ x ^ 2 | Mathlib_Data_Complex_Exponential |
x r : ℝ
⊢ expNear 0 x r = r | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [expNear] | @[simp]
theorem expNear_zero (x r) : expNear 0 x r = r := by | Mathlib.Data.Complex.Exponential.1778_0.1bixbwhfBeJKySp | @[simp]
theorem expNear_zero (x r) : expNear 0 x r = r | Mathlib_Data_Complex_Exponential |
n : ℕ
x r : ℝ
⊢ expNear (n + 1) x r = expNear n x (1 + x / (↑n + 1) * r) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [expNear, range_succ, mul_add, add_left_comm, add_assoc, pow_succ, div_eq_mul_inv,
mul_inv, Nat.factorial] | @[simp]
theorem expNear_succ (n x r) : expNear (n + 1) x r = expNear n x (1 + x / (n + 1) * r) := by
| Mathlib.Data.Complex.Exponential.1782_0.1bixbwhfBeJKySp | @[simp]
theorem expNear_succ (n x r) : expNear (n + 1) x r = expNear n x (1 + x / (n + 1) * r) | Mathlib_Data_Complex_Exponential |
n : ℕ
x r : ℝ
⊢ x * x ^ n * ((↑(Nat.factorial n))⁻¹ * (↑n + 1)⁻¹) * r = x ^ n * (↑(Nat.factorial n))⁻¹ * (x * (↑n + 1)⁻¹ * r) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ac_rfl | @[simp]
theorem expNear_succ (n x r) : expNear (n + 1) x r = expNear n x (1 + x / (n + 1) * r) := by
simp [expNear, range_succ, mul_add, add_left_comm, add_assoc, pow_succ, div_eq_mul_inv,
mul_inv, Nat.factorial]
| Mathlib.Data.Complex.Exponential.1782_0.1bixbwhfBeJKySp | @[simp]
theorem expNear_succ (n x r) : expNear (n + 1) x r = expNear n x (1 + x / (n + 1) * r) | Mathlib_Data_Complex_Exponential |
n : ℕ
x r₁ r₂ : ℝ
⊢ expNear n x r₁ - expNear n x r₂ = x ^ n / ↑(Nat.factorial n) * (r₁ - r₂) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [expNear, mul_sub] | theorem expNear_sub (n x r₁ r₂) : expNear n x r₁ -
expNear n x r₂ = x ^ n / n.factorial * (r₁ - r₂) := by
| Mathlib.Data.Complex.Exponential.1789_0.1bixbwhfBeJKySp | theorem expNear_sub (n x r₁ r₂) : expNear n x r₁ -
expNear n x r₂ = x ^ n / n.factorial * (r₁ - r₂) | Mathlib_Data_Complex_Exponential |
n m : ℕ
x : ℝ
e₁ : n + 1 = m
h : abs' x ≤ 1
⊢ abs' (rexp x - expNear m x 0) ≤ abs' x ^ m / ↑(Nat.factorial m) * ((↑m + 1) / ↑m) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp only [expNear, mul_zero, add_zero] | theorem exp_approx_end (n m : ℕ) (x : ℝ) (e₁ : n + 1 = m) (h : |x| ≤ 1) :
|exp x - expNear m x 0| ≤ |x| ^ m / m.factorial * ((m + 1) / m) := by
| Mathlib.Data.Complex.Exponential.1794_0.1bixbwhfBeJKySp | theorem exp_approx_end (n m : ℕ) (x : ℝ) (e₁ : n + 1 = m) (h : |x| ≤ 1) :
|exp x - expNear m x 0| ≤ |x| ^ m / m.factorial * ((m + 1) / m) | Mathlib_Data_Complex_Exponential |
n m : ℕ
x : ℝ
e₁ : n + 1 = m
h : abs' x ≤ 1
⊢ abs' (rexp x - ∑ m in range m, x ^ m / ↑(Nat.factorial m)) ≤ abs' x ^ m / ↑(Nat.factorial m) * ((↑m + 1) / ↑m) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | convert exp_bound (n := m) h ?_ using 1 | theorem exp_approx_end (n m : ℕ) (x : ℝ) (e₁ : n + 1 = m) (h : |x| ≤ 1) :
|exp x - expNear m x 0| ≤ |x| ^ m / m.factorial * ((m + 1) / m) := by
simp only [expNear, mul_zero, add_zero]
| Mathlib.Data.Complex.Exponential.1794_0.1bixbwhfBeJKySp | theorem exp_approx_end (n m : ℕ) (x : ℝ) (e₁ : n + 1 = m) (h : |x| ≤ 1) :
|exp x - expNear m x 0| ≤ |x| ^ m / m.factorial * ((m + 1) / m) | Mathlib_Data_Complex_Exponential |
case h.e'_4
n m : ℕ
x : ℝ
e₁ : n + 1 = m
h : abs' x ≤ 1
⊢ abs' x ^ m / ↑(Nat.factorial m) * ((↑m + 1) / ↑m) = abs' x ^ m * (↑(Nat.succ m) / (↑(Nat.factorial m) * ↑m))
n m : ℕ x : ℝ e₁ : n + 1 = m h : abs' x ≤ 1 ⊢ 0 < m | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | field_simp [mul_comm] | theorem exp_approx_end (n m : ℕ) (x : ℝ) (e₁ : n + 1 = m) (h : |x| ≤ 1) :
|exp x - expNear m x 0| ≤ |x| ^ m / m.factorial * ((m + 1) / m) := by
simp only [expNear, mul_zero, add_zero]
convert exp_bound (n := m) h ?_ using 1
| Mathlib.Data.Complex.Exponential.1794_0.1bixbwhfBeJKySp | theorem exp_approx_end (n m : ℕ) (x : ℝ) (e₁ : n + 1 = m) (h : |x| ≤ 1) :
|exp x - expNear m x 0| ≤ |x| ^ m / m.factorial * ((m + 1) / m) | Mathlib_Data_Complex_Exponential |
n m : ℕ
x : ℝ
e₁ : n + 1 = m
h : abs' x ≤ 1
⊢ 0 < m | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | linarith | theorem exp_approx_end (n m : ℕ) (x : ℝ) (e₁ : n + 1 = m) (h : |x| ≤ 1) :
|exp x - expNear m x 0| ≤ |x| ^ m / m.factorial * ((m + 1) / m) := by
simp only [expNear, mul_zero, add_zero]
convert exp_bound (n := m) h ?_ using 1
field_simp [mul_comm]
| Mathlib.Data.Complex.Exponential.1794_0.1bixbwhfBeJKySp | theorem exp_approx_end (n m : ℕ) (x : ℝ) (e₁ : n + 1 = m) (h : |x| ≤ 1) :
|exp x - expNear m x 0| ≤ |x| ^ m / m.factorial * ((m + 1) / m) | Mathlib_Data_Complex_Exponential |
n : ℕ
x a₁ b₁ : ℝ
m : ℕ
e₁ : n + 1 = m
a₂ b₂ : ℝ
e : abs' (1 + x / ↑m * a₂ - a₁) ≤ b₁ - abs' x / ↑m * b₂
h : abs' (rexp x - expNear m x a₂) ≤ abs' x ^ m / ↑(Nat.factorial m) * b₂
⊢ abs' (rexp x - expNear n x a₁) ≤ abs' x ^ n / ↑(Nat.factorial n) * b₁ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | refine' (abs_sub_le _ _ _).trans ((add_le_add_right h _).trans _) | theorem exp_approx_succ {n} {x a₁ b₁ : ℝ} (m : ℕ) (e₁ : n + 1 = m) (a₂ b₂ : ℝ)
(e : |1 + x / m * a₂ - a₁| ≤ b₁ - |x| / m * b₂)
(h : |exp x - expNear m x a₂| ≤ |x| ^ m / m.factorial * b₂) :
|exp x - expNear n x a₁| ≤ |x| ^ n / n.factorial * b₁ := by
| Mathlib.Data.Complex.Exponential.1802_0.1bixbwhfBeJKySp | theorem exp_approx_succ {n} {x a₁ b₁ : ℝ} (m : ℕ) (e₁ : n + 1 = m) (a₂ b₂ : ℝ)
(e : |1 + x / m * a₂ - a₁| ≤ b₁ - |x| / m * b₂)
(h : |exp x - expNear m x a₂| ≤ |x| ^ m / m.factorial * b₂) :
|exp x - expNear n x a₁| ≤ |x| ^ n / n.factorial * b₁ | Mathlib_Data_Complex_Exponential |
n : ℕ
x a₁ b₁ : ℝ
m : ℕ
e₁ : n + 1 = m
a₂ b₂ : ℝ
e : abs' (1 + x / ↑m * a₂ - a₁) ≤ b₁ - abs' x / ↑m * b₂
h : abs' (rexp x - expNear m x a₂) ≤ abs' x ^ m / ↑(Nat.factorial m) * b₂
⊢ abs' x ^ m / ↑(Nat.factorial m) * b₂ + abs' (expNear m x a₂ - expNear n x a₁) ≤ abs' x ^ n / ↑(Nat.factorial n) * b₁ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | subst e₁ | theorem exp_approx_succ {n} {x a₁ b₁ : ℝ} (m : ℕ) (e₁ : n + 1 = m) (a₂ b₂ : ℝ)
(e : |1 + x / m * a₂ - a₁| ≤ b₁ - |x| / m * b₂)
(h : |exp x - expNear m x a₂| ≤ |x| ^ m / m.factorial * b₂) :
|exp x - expNear n x a₁| ≤ |x| ^ n / n.factorial * b₁ := by
refine' (abs_sub_le _ _ _).trans ((add_le_add_right h _).... | Mathlib.Data.Complex.Exponential.1802_0.1bixbwhfBeJKySp | theorem exp_approx_succ {n} {x a₁ b₁ : ℝ} (m : ℕ) (e₁ : n + 1 = m) (a₂ b₂ : ℝ)
(e : |1 + x / m * a₂ - a₁| ≤ b₁ - |x| / m * b₂)
(h : |exp x - expNear m x a₂| ≤ |x| ^ m / m.factorial * b₂) :
|exp x - expNear n x a₁| ≤ |x| ^ n / n.factorial * b₁ | Mathlib_Data_Complex_Exponential |
n : ℕ
x a₁ b₁ a₂ b₂ : ℝ
e : abs' (1 + x / ↑(n + 1) * a₂ - a₁) ≤ b₁ - abs' x / ↑(n + 1) * b₂
h : abs' (rexp x - expNear (n + 1) x a₂) ≤ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂
⊢ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂ + abs' (expNear (n + 1) x a₂ - expNear n x a₁) ≤
abs' x ^ n / ↑(Nat.factorial n) ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [expNear_succ, expNear_sub, abs_mul] | theorem exp_approx_succ {n} {x a₁ b₁ : ℝ} (m : ℕ) (e₁ : n + 1 = m) (a₂ b₂ : ℝ)
(e : |1 + x / m * a₂ - a₁| ≤ b₁ - |x| / m * b₂)
(h : |exp x - expNear m x a₂| ≤ |x| ^ m / m.factorial * b₂) :
|exp x - expNear n x a₁| ≤ |x| ^ n / n.factorial * b₁ := by
refine' (abs_sub_le _ _ _).trans ((add_le_add_right h _).... | Mathlib.Data.Complex.Exponential.1802_0.1bixbwhfBeJKySp | theorem exp_approx_succ {n} {x a₁ b₁ : ℝ} (m : ℕ) (e₁ : n + 1 = m) (a₂ b₂ : ℝ)
(e : |1 + x / m * a₂ - a₁| ≤ b₁ - |x| / m * b₂)
(h : |exp x - expNear m x a₂| ≤ |x| ^ m / m.factorial * b₂) :
|exp x - expNear n x a₁| ≤ |x| ^ n / n.factorial * b₁ | Mathlib_Data_Complex_Exponential |
n : ℕ
x a₁ b₁ a₂ b₂ : ℝ
e : abs' (1 + x / ↑(n + 1) * a₂ - a₁) ≤ b₁ - abs' x / ↑(n + 1) * b₂
h : abs' (rexp x - expNear (n + 1) x a₂) ≤ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂
⊢ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂ +
abs' (x ^ n / ↑(Nat.factorial n)) * abs' (1 + x / (↑n + 1) * a₂ - a₁) ≤
a... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | convert mul_le_mul_of_nonneg_left (a := abs' x ^ n / ↑(Nat.factorial n))
(le_sub_iff_add_le'.1 e) ?_ using 1 | theorem exp_approx_succ {n} {x a₁ b₁ : ℝ} (m : ℕ) (e₁ : n + 1 = m) (a₂ b₂ : ℝ)
(e : |1 + x / m * a₂ - a₁| ≤ b₁ - |x| / m * b₂)
(h : |exp x - expNear m x a₂| ≤ |x| ^ m / m.factorial * b₂) :
|exp x - expNear n x a₁| ≤ |x| ^ n / n.factorial * b₁ := by
refine' (abs_sub_le _ _ _).trans ((add_le_add_right h _).... | Mathlib.Data.Complex.Exponential.1802_0.1bixbwhfBeJKySp | theorem exp_approx_succ {n} {x a₁ b₁ : ℝ} (m : ℕ) (e₁ : n + 1 = m) (a₂ b₂ : ℝ)
(e : |1 + x / m * a₂ - a₁| ≤ b₁ - |x| / m * b₂)
(h : |exp x - expNear m x a₂| ≤ |x| ^ m / m.factorial * b₂) :
|exp x - expNear n x a₁| ≤ |x| ^ n / n.factorial * b₁ | Mathlib_Data_Complex_Exponential |
case h.e'_3
n : ℕ
x a₁ b₁ a₂ b₂ : ℝ
e : abs' (1 + x / ↑(n + 1) * a₂ - a₁) ≤ b₁ - abs' x / ↑(n + 1) * b₂
h : abs' (rexp x - expNear (n + 1) x a₂) ≤ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂
⊢ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂ +
abs' (x ^ n / ↑(Nat.factorial n)) * abs' (1 + x / (↑n + 1) * a₂ -... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [mul_add, pow_succ', div_eq_mul_inv, abs_mul, abs_inv, ← pow_abs, mul_inv, Nat.factorial] | theorem exp_approx_succ {n} {x a₁ b₁ : ℝ} (m : ℕ) (e₁ : n + 1 = m) (a₂ b₂ : ℝ)
(e : |1 + x / m * a₂ - a₁| ≤ b₁ - |x| / m * b₂)
(h : |exp x - expNear m x a₂| ≤ |x| ^ m / m.factorial * b₂) :
|exp x - expNear n x a₁| ≤ |x| ^ n / n.factorial * b₁ := by
refine' (abs_sub_le _ _ _).trans ((add_le_add_right h _).... | Mathlib.Data.Complex.Exponential.1802_0.1bixbwhfBeJKySp | theorem exp_approx_succ {n} {x a₁ b₁ : ℝ} (m : ℕ) (e₁ : n + 1 = m) (a₂ b₂ : ℝ)
(e : |1 + x / m * a₂ - a₁| ≤ b₁ - |x| / m * b₂)
(h : |exp x - expNear m x a₂| ≤ |x| ^ m / m.factorial * b₂) :
|exp x - expNear n x a₁| ≤ |x| ^ n / n.factorial * b₁ | Mathlib_Data_Complex_Exponential |
case h.e'_3
n : ℕ
x a₁ b₁ a₂ b₂ : ℝ
e : abs' (1 + x / ↑(n + 1) * a₂ - a₁) ≤ b₁ - abs' x / ↑(n + 1) * b₂
h : abs' (rexp x - expNear (n + 1) x a₂) ≤ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂
⊢ abs' x ^ n * abs' x * ((↑(Nat.factorial n))⁻¹ * (↑n + 1)⁻¹) * b₂ =
abs' x ^ n * (↑(Nat.factorial n))⁻¹ * (abs' x * (↑n... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ac_rfl | theorem exp_approx_succ {n} {x a₁ b₁ : ℝ} (m : ℕ) (e₁ : n + 1 = m) (a₂ b₂ : ℝ)
(e : |1 + x / m * a₂ - a₁| ≤ b₁ - |x| / m * b₂)
(h : |exp x - expNear m x a₂| ≤ |x| ^ m / m.factorial * b₂) :
|exp x - expNear n x a₁| ≤ |x| ^ n / n.factorial * b₁ := by
refine' (abs_sub_le _ _ _).trans ((add_le_add_right h _).... | Mathlib.Data.Complex.Exponential.1802_0.1bixbwhfBeJKySp | theorem exp_approx_succ {n} {x a₁ b₁ : ℝ} (m : ℕ) (e₁ : n + 1 = m) (a₂ b₂ : ℝ)
(e : |1 + x / m * a₂ - a₁| ≤ b₁ - |x| / m * b₂)
(h : |exp x - expNear m x a₂| ≤ |x| ^ m / m.factorial * b₂) :
|exp x - expNear n x a₁| ≤ |x| ^ n / n.factorial * b₁ | Mathlib_Data_Complex_Exponential |
n : ℕ
x a₁ b₁ a₂ b₂ : ℝ
e : abs' (1 + x / ↑(n + 1) * a₂ - a₁) ≤ b₁ - abs' x / ↑(n + 1) * b₂
h : abs' (rexp x - expNear (n + 1) x a₂) ≤ abs' x ^ (n + 1) / ↑(Nat.factorial (n + 1)) * b₂
⊢ 0 ≤ abs' x ^ n / ↑(Nat.factorial n) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [div_nonneg, abs_nonneg] | theorem exp_approx_succ {n} {x a₁ b₁ : ℝ} (m : ℕ) (e₁ : n + 1 = m) (a₂ b₂ : ℝ)
(e : |1 + x / m * a₂ - a₁| ≤ b₁ - |x| / m * b₂)
(h : |exp x - expNear m x a₂| ≤ |x| ^ m / m.factorial * b₂) :
|exp x - expNear n x a₁| ≤ |x| ^ n / n.factorial * b₁ := by
refine' (abs_sub_le _ _ _).trans ((add_le_add_right h _).... | Mathlib.Data.Complex.Exponential.1802_0.1bixbwhfBeJKySp | theorem exp_approx_succ {n} {x a₁ b₁ : ℝ} (m : ℕ) (e₁ : n + 1 = m) (a₂ b₂ : ℝ)
(e : |1 + x / m * a₂ - a₁| ≤ b₁ - |x| / m * b₂)
(h : |exp x - expNear m x a₂| ≤ |x| ^ m / m.factorial * b₂) :
|exp x - expNear n x a₁| ≤ |x| ^ n / n.factorial * b₁ | Mathlib_Data_Complex_Exponential |
n : ℕ
x a b : ℝ
m : ℕ
e₁ : n + 1 = m
rm : ℝ
er : ↑m = rm
h : abs' x ≤ 1
e : abs' (1 - a) ≤ b - abs' x / rm * ((rm + 1) / rm)
⊢ abs' (rexp x - expNear n x a) ≤ abs' x ^ n / ↑(Nat.factorial n) * b | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | subst er | theorem exp_approx_end' {n} {x a b : ℝ} (m : ℕ) (e₁ : n + 1 = m) (rm : ℝ) (er : ↑m = rm)
(h : |x| ≤ 1) (e : |1 - a| ≤ b - |x| / rm * ((rm + 1) / rm)) :
|exp x - expNear n x a| ≤ |x| ^ n / n.factorial * b := by
| Mathlib.Data.Complex.Exponential.1815_0.1bixbwhfBeJKySp | theorem exp_approx_end' {n} {x a b : ℝ} (m : ℕ) (e₁ : n + 1 = m) (rm : ℝ) (er : ↑m = rm)
(h : |x| ≤ 1) (e : |1 - a| ≤ b - |x| / rm * ((rm + 1) / rm)) :
|exp x - expNear n x a| ≤ |x| ^ n / n.factorial * b | Mathlib_Data_Complex_Exponential |
n : ℕ
x a b : ℝ
m : ℕ
e₁ : n + 1 = m
h : abs' x ≤ 1
e : abs' (1 - a) ≤ b - abs' x / ↑m * ((↑m + 1) / ↑m)
⊢ abs' (rexp x - expNear n x a) ≤ abs' x ^ n / ↑(Nat.factorial n) * b | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | exact exp_approx_succ _ e₁ _ _ (by simpa using e) (exp_approx_end _ _ _ e₁ h) | theorem exp_approx_end' {n} {x a b : ℝ} (m : ℕ) (e₁ : n + 1 = m) (rm : ℝ) (er : ↑m = rm)
(h : |x| ≤ 1) (e : |1 - a| ≤ b - |x| / rm * ((rm + 1) / rm)) :
|exp x - expNear n x a| ≤ |x| ^ n / n.factorial * b := by
subst er
| Mathlib.Data.Complex.Exponential.1815_0.1bixbwhfBeJKySp | theorem exp_approx_end' {n} {x a b : ℝ} (m : ℕ) (e₁ : n + 1 = m) (rm : ℝ) (er : ↑m = rm)
(h : |x| ≤ 1) (e : |1 - a| ≤ b - |x| / rm * ((rm + 1) / rm)) :
|exp x - expNear n x a| ≤ |x| ^ n / n.factorial * b | Mathlib_Data_Complex_Exponential |
n : ℕ
x a b : ℝ
m : ℕ
e₁ : n + 1 = m
h : abs' x ≤ 1
e : abs' (1 - a) ≤ b - abs' x / ↑m * ((↑m + 1) / ↑m)
⊢ abs' (1 + x / ↑m * 0 - a) ≤ b - abs' x / ↑m * ((↑m + 1) / ↑m) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simpa using e | theorem exp_approx_end' {n} {x a b : ℝ} (m : ℕ) (e₁ : n + 1 = m) (rm : ℝ) (er : ↑m = rm)
(h : |x| ≤ 1) (e : |1 - a| ≤ b - |x| / rm * ((rm + 1) / rm)) :
|exp x - expNear n x a| ≤ |x| ^ n / n.factorial * b := by
subst er
exact exp_approx_succ _ e₁ _ _ (by | Mathlib.Data.Complex.Exponential.1815_0.1bixbwhfBeJKySp | theorem exp_approx_end' {n} {x a b : ℝ} (m : ℕ) (e₁ : n + 1 = m) (rm : ℝ) (er : ↑m = rm)
(h : |x| ≤ 1) (e : |1 - a| ≤ b - |x| / rm * ((rm + 1) / rm)) :
|exp x - expNear n x a| ≤ |x| ^ n / n.factorial * b | Mathlib_Data_Complex_Exponential |
n : ℕ
a₁ b₁ : ℝ
m : ℕ
en : n + 1 = m
rm : ℝ
er : ↑m = rm
h : abs' (rexp 1 - expNear m 1 ((a₁ - 1) * rm)) ≤ abs' 1 ^ m / ↑(Nat.factorial m) * (b₁ * rm)
⊢ abs' (rexp 1 - expNear n 1 a₁) ≤ abs' 1 ^ n / ↑(Nat.factorial n) * b₁ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | subst er | theorem exp_1_approx_succ_eq {n} {a₁ b₁ : ℝ} {m : ℕ} (en : n + 1 = m) {rm : ℝ} (er : ↑m = rm)
(h : |exp 1 - expNear m 1 ((a₁ - 1) * rm)| ≤ |1| ^ m / m.factorial * (b₁ * rm)) :
|exp 1 - expNear n 1 a₁| ≤ |1| ^ n / n.factorial * b₁ := by
| Mathlib.Data.Complex.Exponential.1822_0.1bixbwhfBeJKySp | theorem exp_1_approx_succ_eq {n} {a₁ b₁ : ℝ} {m : ℕ} (en : n + 1 = m) {rm : ℝ} (er : ↑m = rm)
(h : |exp 1 - expNear m 1 ((a₁ - 1) * rm)| ≤ |1| ^ m / m.factorial * (b₁ * rm)) :
|exp 1 - expNear n 1 a₁| ≤ |1| ^ n / n.factorial * b₁ | Mathlib_Data_Complex_Exponential |
n : ℕ
a₁ b₁ : ℝ
m : ℕ
en : n + 1 = m
h : abs' (rexp 1 - expNear m 1 ((a₁ - 1) * ↑m)) ≤ abs' 1 ^ m / ↑(Nat.factorial m) * (b₁ * ↑m)
⊢ abs' (rexp 1 - expNear n 1 a₁) ≤ abs' 1 ^ n / ↑(Nat.factorial n) * b₁ | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | refine' exp_approx_succ _ en _ _ _ h | theorem exp_1_approx_succ_eq {n} {a₁ b₁ : ℝ} {m : ℕ} (en : n + 1 = m) {rm : ℝ} (er : ↑m = rm)
(h : |exp 1 - expNear m 1 ((a₁ - 1) * rm)| ≤ |1| ^ m / m.factorial * (b₁ * rm)) :
|exp 1 - expNear n 1 a₁| ≤ |1| ^ n / n.factorial * b₁ := by
subst er
| Mathlib.Data.Complex.Exponential.1822_0.1bixbwhfBeJKySp | theorem exp_1_approx_succ_eq {n} {a₁ b₁ : ℝ} {m : ℕ} (en : n + 1 = m) {rm : ℝ} (er : ↑m = rm)
(h : |exp 1 - expNear m 1 ((a₁ - 1) * rm)| ≤ |1| ^ m / m.factorial * (b₁ * rm)) :
|exp 1 - expNear n 1 a₁| ≤ |1| ^ n / n.factorial * b₁ | Mathlib_Data_Complex_Exponential |
n : ℕ
a₁ b₁ : ℝ
m : ℕ
en : n + 1 = m
h : abs' (rexp 1 - expNear m 1 ((a₁ - 1) * ↑m)) ≤ abs' 1 ^ m / ↑(Nat.factorial m) * (b₁ * ↑m)
⊢ abs' (1 + 1 / ↑m * ((a₁ - 1) * ↑m) - a₁) ≤ b₁ - abs' 1 / ↑m * (b₁ * ↑m) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | field_simp [show (m : ℝ) ≠ 0 by norm_cast; linarith] | theorem exp_1_approx_succ_eq {n} {a₁ b₁ : ℝ} {m : ℕ} (en : n + 1 = m) {rm : ℝ} (er : ↑m = rm)
(h : |exp 1 - expNear m 1 ((a₁ - 1) * rm)| ≤ |1| ^ m / m.factorial * (b₁ * rm)) :
|exp 1 - expNear n 1 a₁| ≤ |1| ^ n / n.factorial * b₁ := by
subst er
refine' exp_approx_succ _ en _ _ _ h
| Mathlib.Data.Complex.Exponential.1822_0.1bixbwhfBeJKySp | theorem exp_1_approx_succ_eq {n} {a₁ b₁ : ℝ} {m : ℕ} (en : n + 1 = m) {rm : ℝ} (er : ↑m = rm)
(h : |exp 1 - expNear m 1 ((a₁ - 1) * rm)| ≤ |1| ^ m / m.factorial * (b₁ * rm)) :
|exp 1 - expNear n 1 a₁| ≤ |1| ^ n / n.factorial * b₁ | Mathlib_Data_Complex_Exponential |
n : ℕ
a₁ b₁ : ℝ
m : ℕ
en : n + 1 = m
h : abs' (rexp 1 - expNear m 1 ((a₁ - 1) * ↑m)) ≤ abs' 1 ^ m / ↑(Nat.factorial m) * (b₁ * ↑m)
⊢ ↑m ≠ 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | norm_cast | theorem exp_1_approx_succ_eq {n} {a₁ b₁ : ℝ} {m : ℕ} (en : n + 1 = m) {rm : ℝ} (er : ↑m = rm)
(h : |exp 1 - expNear m 1 ((a₁ - 1) * rm)| ≤ |1| ^ m / m.factorial * (b₁ * rm)) :
|exp 1 - expNear n 1 a₁| ≤ |1| ^ n / n.factorial * b₁ := by
subst er
refine' exp_approx_succ _ en _ _ _ h
field_simp [show (m : ℝ)... | Mathlib.Data.Complex.Exponential.1822_0.1bixbwhfBeJKySp | theorem exp_1_approx_succ_eq {n} {a₁ b₁ : ℝ} {m : ℕ} (en : n + 1 = m) {rm : ℝ} (er : ↑m = rm)
(h : |exp 1 - expNear m 1 ((a₁ - 1) * rm)| ≤ |1| ^ m / m.factorial * (b₁ * rm)) :
|exp 1 - expNear n 1 a₁| ≤ |1| ^ n / n.factorial * b₁ | Mathlib_Data_Complex_Exponential |
n : ℕ
a₁ b₁ : ℝ
m : ℕ
en : n + 1 = m
h : abs' (rexp 1 - expNear m 1 ((a₁ - 1) * ↑m)) ≤ abs' 1 ^ m / ↑(Nat.factorial m) * (b₁ * ↑m)
⊢ ¬m = 0 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | linarith | theorem exp_1_approx_succ_eq {n} {a₁ b₁ : ℝ} {m : ℕ} (en : n + 1 = m) {rm : ℝ} (er : ↑m = rm)
(h : |exp 1 - expNear m 1 ((a₁ - 1) * rm)| ≤ |1| ^ m / m.factorial * (b₁ * rm)) :
|exp 1 - expNear n 1 a₁| ≤ |1| ^ n / n.factorial * b₁ := by
subst er
refine' exp_approx_succ _ en _ _ _ h
field_simp [show (m : ℝ)... | Mathlib.Data.Complex.Exponential.1822_0.1bixbwhfBeJKySp | theorem exp_1_approx_succ_eq {n} {a₁ b₁ : ℝ} {m : ℕ} (en : n + 1 = m) {rm : ℝ} (er : ↑m = rm)
(h : |exp 1 - expNear m 1 ((a₁ - 1) * rm)| ≤ |1| ^ m / m.factorial * (b₁ * rm)) :
|exp 1 - expNear n 1 a₁| ≤ |1| ^ n / n.factorial * b₁ | Mathlib_Data_Complex_Exponential |
x a b : ℝ
h : abs' (rexp x - expNear 0 x a) ≤ abs' x ^ 0 / ↑(Nat.factorial 0) * b
⊢ abs' (rexp x - a) ≤ b | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simpa using h | theorem exp_approx_start (x a b : ℝ) (h : |exp x - expNear 0 x a| ≤ |x| ^ 0 / Nat.factorial 0 * b) :
|exp x - a| ≤ b := by | Mathlib.Data.Complex.Exponential.1830_0.1bixbwhfBeJKySp | theorem exp_approx_start (x a b : ℝ) (h : |exp x - expNear 0 x a| ≤ |x| ^ 0 / Nat.factorial 0 * b) :
|exp x - a| ≤ b | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ abs' (cos x - (1 - x ^ 2 / 2)) = Complex.abs (Complex.cos ↑x - (1 - ↑x ^ 2 / 2)) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← abs_ofReal] | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
| Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs ↑(cos x - (1 - x ^ 2 / 2)) = Complex.abs (Complex.cos ↑x - (1 - ↑x ^ 2 / 2)) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
rw [← abs_ofReal]; | Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs (Complex.cos ↑x - (1 - ↑x ^ 2 / 2)) = Complex.abs ((cexp (↑x * I) + cexp (-↑x * I) - (2 - ↑x ^ 2)) / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [Complex.cos, sub_div, add_div, neg_div, div_self (two_ne_zero' ℂ)] | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
... | Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ cexp (↑x * I) + cexp (-↑x * I) - (2 - ↑x ^ 2) =
cexp (↑x * I) - ∑ m in range 4, (↑x * I) ^ m / ↑(Nat.factorial m) +
(cexp (-↑x * I) - ∑ m in range 4, (-↑x * I) ^ m / ↑(Nat.factorial m)) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp only [sum_range_succ, neg_mul, pow_succ, pow_zero, mul_one, range_zero, sum_empty,
Nat.factorial, Nat.cast_one, ne_eq, one_ne_zero, not_false_eq_true, div_self,
zero_add, div_one, Nat.mul_one, Nat.cast_succ, Nat.cast_mul, Nat.cast_ofNat, mul_neg,
neg_neg] | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
... | Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ cexp (↑x * I) + cexp (-(↑x * I)) - (2 - ↑x * ↑x) =
cexp (↑x * I) -
(1 / (↑0 + 1) + ↑x * I / (↑0 + 1) + ↑x * I * (↑x * I) / (↑0 + 1 + 1) +
↑x * I * (↑x * I * (↑x * I)) / (↑0 + 1 + 1 + 1 + 1 + 1 + 1)) +
(cexp (-(↑x * I)) -
(1 / (↑0 + 1) + -(↑x * I) / (↑0 + 1) + ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | apply Complex.ext | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
... | Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
case a
x : ℝ
hx : abs' x ≤ 1
⊢ (cexp (↑x * I) + cexp (-(↑x * I)) - (2 - ↑x * ↑x)).re =
(cexp (↑x * I) -
(1 / (↑0 + 1) + ↑x * I / (↑0 + 1) + ↑x * I * (↑x * I) / (↑0 + 1 + 1) +
↑x * I * (↑x * I * (↑x * I)) / (↑0 + 1 + 1 + 1 + 1 + 1 + 1)) +
(cexp (-(↑x * I)) -
(1 / (↑0 + 1) + -(... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [div_eq_mul_inv, normSq] | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
... | Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
case a
x : ℝ
hx : abs' x ≤ 1
⊢ (cexp (↑x * I) + cexp (-(↑x * I)) - (2 - ↑x * ↑x)).im =
(cexp (↑x * I) -
(1 / (↑0 + 1) + ↑x * I / (↑0 + 1) + ↑x * I * (↑x * I) / (↑0 + 1 + 1) +
↑x * I * (↑x * I * (↑x * I)) / (↑0 + 1 + 1 + 1 + 1 + 1 + 1)) +
(cexp (-(↑x * I)) -
(1 / (↑0 + 1) + -(... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [div_eq_mul_inv, normSq] | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
... | Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
case a
x : ℝ
hx : abs' x ≤ 1
⊢ cos x + (cexp (-(↑x * I))).re - (2 - x * x) =
cos x - (1 + -(x * x * (1 + 1)⁻¹)) + ((cexp (-(↑x * I))).re - (1 + -(x * x * (1 + 1)⁻¹))) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ring_nf | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
... | Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
case a
x : ℝ
hx : abs' x ≤ 1
⊢ sin x + (cexp (-(↑x * I))).im =
sin x - (x + -(x * (x * x) * ((1 + 1 + 1 + 1 + 1 + 1) * ((1 + 1 + 1 + 1 + 1 + 1)⁻¹ * (1 + 1 + 1 + 1 + 1 + 1)⁻¹)))) +
((cexp (-(↑x * I))).im -
(-x + x * (x * x) * ((1 + 1 + 1 + 1 + 1 + 1) * ((1 + 1 + 1 + 1 + 1 + 1)⁻¹ * (1 + 1 + 1 + 1 + 1 + ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ring_nf | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
... | Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs
((cexp (↑x * I) - ∑ m in range 4, (↑x * I) ^ m / ↑(Nat.factorial m) +
(cexp (-↑x * I) - ∑ m in range 4, (-↑x * I) ^ m / ↑(Nat.factorial m))) /
2) ≤
Complex.abs ((cexp (↑x * I) - ∑ m in range 4, (↑x * I) ^ m / ↑(Nat.factorial m)) / 2) +
Complex.abs ((... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [add_div] | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
... | Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs
((cexp (↑x * I) - ∑ m in range 4, (↑x * I) ^ m / ↑(Nat.factorial m)) / 2 +
(cexp (-↑x * I) - ∑ m in range 4, (-↑x * I) ^ m / ↑(Nat.factorial m)) / 2) ≤
Complex.abs ((cexp (↑x * I) - ∑ m in range 4, (↑x * I) ^ m / ↑(Nat.factorial m)) / 2) +
Complex.abs ((cexp (... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | exact Complex.abs.add_le _ _ | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
... | Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs ((cexp (↑x * I) - ∑ m in range 4, (↑x * I) ^ m / ↑(Nat.factorial m)) / 2) +
Complex.abs ((cexp (-↑x * I) - ∑ m in range 4, (-↑x * I) ^ m / ↑(Nat.factorial m)) / 2) =
Complex.abs (cexp (↑x * I) - ∑ m in range 4, (↑x * I) ^ m / ↑(Nat.factorial m)) / 2 +
Complex.abs (cex... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [map_div₀] | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
... | Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs (cexp (↑x * I) - ∑ m in range 4, (↑x * I) ^ m / ↑(Nat.factorial m)) / 2 +
Complex.abs (cexp (-↑x * I) - ∑ m in range 4, (-↑x * I) ^ m / ↑(Nat.factorial m)) / 2 ≤
Complex.abs (↑x * I) ^ 4 * (↑(Nat.succ 4) * (↑(Nat.factorial 4) * ↑4)⁻¹) / 2 +
Complex.abs (-↑x * I) ^ 4 *... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | gcongr | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
... | Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
case h₁.h
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs (cexp (↑x * I) - ∑ m in range 4, (↑x * I) ^ m / ↑(Nat.factorial m)) ≤
Complex.abs (↑x * I) ^ 4 * (↑(Nat.succ 4) * (↑(Nat.factorial 4) * ↑4)⁻¹) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | exact Complex.exp_bound (by simpa) (by decide) | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
... | Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs (↑x * I) ≤ 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simpa | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
... | Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ 0 < 4 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | decide | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
... | Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
case h₂.h
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs (cexp (-↑x * I) - ∑ m in range 4, (-↑x * I) ^ m / ↑(Nat.factorial m)) ≤
Complex.abs (-↑x * I) ^ 4 * (↑(Nat.succ 4) * (↑(Nat.factorial 4) * ↑4)⁻¹) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | exact Complex.exp_bound (by simpa) (by decide) | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
... | Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs (-↑x * I) ≤ 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simpa | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
... | Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ 0 < 4 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | decide | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
... | Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs (↑x * I) ^ 4 * (↑(Nat.succ 4) * (↑(Nat.factorial 4) * ↑4)⁻¹) / 2 +
Complex.abs (-↑x * I) ^ 4 * (↑(Nat.succ 4) * (↑(Nat.factorial 4) * ↑4)⁻¹) / 2 ≤
abs' x ^ 4 * (5 / 96) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | norm_num [Nat.factorial] | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|cos x - (1 - x ^ 2 / 2)| = Complex.abs (Complex.cos x - (1 - (x : ℂ) ^ 2 / 2)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs ((Complex.exp (x * I) + Complex.exp (-x * I) - (2 - (x : ℂ) ^ 2)) / 2) := by
... | Mathlib.Data.Complex.Exponential.1834_0.1bixbwhfBeJKySp | theorem cos_bound {x : ℝ} (hx : |x| ≤ 1) : |cos x - (1 - x ^ 2 / 2)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ abs' (sin x - (x - x ^ 3 / 6)) = Complex.abs (Complex.sin ↑x - ↑(x - x ^ 3 / 6)) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [← abs_ofReal] | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
| Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs ↑(sin x - (x - x ^ 3 / 6)) = Complex.abs (Complex.sin ↑x - ↑(x - x ^ 3 / 6)) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
rw [← abs_ofReal]; | Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs (Complex.sin ↑x - ↑(x - x ^ 3 / 6)) =
Complex.abs (((cexp (-↑x * I) - cexp (↑x * I)) * I - ↑(2 * x - x ^ 3 / 3)) / 2) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [Complex.sin, sub_div, add_div, neg_div, mul_div_cancel_left _ (two_ne_zero' ℂ), div_div,
show (3 : ℂ) * 2 = 6 by norm_num] | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
(2 * x - x ^ 3 / 3 ... | Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ 3 * 2 = 6 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | norm_num | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
(2 * x - x ^ 3 / 3 ... | Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ (cexp (-↑x * I) - cexp (↑x * I)) * I - ↑(2 * x - x ^ 3 / 3) =
(cexp (-↑x * I) - ∑ m in range 4, (-↑x * I) ^ m / ↑(Nat.factorial m) -
(cexp (↑x * I) - ∑ m in range 4, (↑x * I) ^ m / ↑(Nat.factorial m))) *
I | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp only [sum_range_succ, neg_mul, pow_succ, pow_zero, mul_one, ofReal_sub, ofReal_mul,
ofReal_ofNat, ofReal_div, range_zero, sum_empty, Nat.factorial, Nat.cast_one, ne_eq,
one_ne_zero, not_false_eq_true, div_self, zero_add, div_one, mul_neg, neg_neg,
Nat.mul_one, Nat.cast_suc... | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
(2 * x - x ^ 3 / 3 ... | Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ (cexp (-(↑x * I)) - cexp (↑x * I)) * I - (2 * ↑x - ↑x * (↑x * ↑x) / 3) =
(cexp (-(↑x * I)) -
(1 / (↑0 + 1) + -(↑x * I) / (↑0 + 1) + ↑x * I * (↑x * I) / (↑0 + 1 + 1) +
-(↑x * I * (↑x * I * (↑x * I))) / (↑0 + 1 + 1 + 1 + 1 + 1 + 1)) -
(cexp (↑x * I) -
(1... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | apply Complex.ext | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
(2 * x - x ^ 3 / 3 ... | Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
case a
x : ℝ
hx : abs' x ≤ 1
⊢ ((cexp (-(↑x * I)) - cexp (↑x * I)) * I - (2 * ↑x - ↑x * (↑x * ↑x) / 3)).re =
((cexp (-(↑x * I)) -
(1 / (↑0 + 1) + -(↑x * I) / (↑0 + 1) + ↑x * I * (↑x * I) / (↑0 + 1 + 1) +
-(↑x * I * (↑x * I * (↑x * I))) / (↑0 + 1 + 1 + 1 + 1 + 1 + 1)) -
(cexp (↑x ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [div_eq_mul_inv, normSq] | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
(2 * x - x ^ 3 / 3 ... | Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
case a
x : ℝ
hx : abs' x ≤ 1
⊢ ((cexp (-(↑x * I)) - cexp (↑x * I)) * I - (2 * ↑x - ↑x * (↑x * ↑x) / 3)).im =
((cexp (-(↑x * I)) -
(1 / (↑0 + 1) + -(↑x * I) / (↑0 + 1) + ↑x * I * (↑x * I) / (↑0 + 1 + 1) +
-(↑x * I * (↑x * I * (↑x * I))) / (↑0 + 1 + 1 + 1 + 1 + 1 + 1)) -
(cexp (↑x ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [div_eq_mul_inv, normSq] | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
(2 * x - x ^ 3 / 3 ... | Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
case a
x : ℝ
hx : abs' x ≤ 1
⊢ sin x - (cexp (-(↑x * I))).im - (2 * x - x * (x * x) * 3⁻¹) =
sin x - (x + -(x * (x * x) * ((1 + 1 + 1 + 1 + 1 + 1) * ((1 + 1 + 1 + 1 + 1 + 1)⁻¹ * (1 + 1 + 1 + 1 + 1 + 1)⁻¹)))) -
((cexp (-(↑x * I))).im -
(-x + x * (x * x) * ((1 + 1 + 1 + 1 + 1 + 1) * ((1 + 1 + 1 + 1 + 1 ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | ring | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
(2 * x - x ^ 3 / 3 ... | Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs
((cexp (-↑x * I) - ∑ m in range 4, (-↑x * I) ^ m / ↑(Nat.factorial m) -
(cexp (↑x * I) - ∑ m in range 4, (↑x * I) ^ m / ↑(Nat.factorial m))) *
I /
2) ≤
Complex.abs ((cexp (-↑x * I) - ∑ m in range 4, (-↑x * I) ^ m / ↑(Nat.factorial m)) * I / 2) ... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [sub_mul, sub_eq_add_neg, add_div] | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
(2 * x - x ^ 3 / 3 ... | Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs
((cexp (-↑x * I) - ∑ m in range 4, (-↑x * I) ^ m / ↑(Nat.factorial m)) * I / 2 +
-((cexp (↑x * I) - ∑ m in range 4, (↑x * I) ^ m / ↑(Nat.factorial m)) * I) / 2) ≤
Complex.abs ((cexp (-↑x * I) - ∑ m in range 4, (-↑x * I) ^ m / ↑(Nat.factorial m)) * I / 2) +
Com... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | exact Complex.abs.add_le _ _ | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
(2 * x - x ^ 3 / 3 ... | Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs ((cexp (-↑x * I) - ∑ m in range 4, (-↑x * I) ^ m / ↑(Nat.factorial m)) * I / 2) +
Complex.abs (-((cexp (↑x * I) - ∑ m in range 4, (↑x * I) ^ m / ↑(Nat.factorial m)) * I) / 2) =
Complex.abs (cexp (↑x * I) - ∑ m in range 4, (↑x * I) ^ m / ↑(Nat.factorial m)) / 2 +
Compl... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [add_comm, map_div₀] | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
(2 * x - x ^ 3 / 3 ... | Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs (cexp (↑x * I) - ∑ m in range 4, (↑x * I) ^ m / ↑(Nat.factorial m)) / 2 +
Complex.abs (cexp (-↑x * I) - ∑ m in range 4, (-↑x * I) ^ m / ↑(Nat.factorial m)) / 2 ≤
Complex.abs (↑x * I) ^ 4 * (↑(Nat.succ 4) * (↑(Nat.factorial 4) * ↑4)⁻¹) / 2 +
Complex.abs (-↑x * I) ^ 4 *... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | gcongr | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
(2 * x - x ^ 3 / 3 ... | Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
case h₁.h
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs (cexp (↑x * I) - ∑ m in range 4, (↑x * I) ^ m / ↑(Nat.factorial m)) ≤
Complex.abs (↑x * I) ^ 4 * (↑(Nat.succ 4) * (↑(Nat.factorial 4) * ↑4)⁻¹) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | exact Complex.exp_bound (by simpa) (by decide) | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
(2 * x - x ^ 3 / 3 ... | Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs (↑x * I) ≤ 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simpa | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
(2 * x - x ^ 3 / 3 ... | Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ 0 < 4 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | decide | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
(2 * x - x ^ 3 / 3 ... | Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
case h₂.h
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs (cexp (-↑x * I) - ∑ m in range 4, (-↑x * I) ^ m / ↑(Nat.factorial m)) ≤
Complex.abs (-↑x * I) ^ 4 * (↑(Nat.succ 4) * (↑(Nat.factorial 4) * ↑4)⁻¹) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | exact Complex.exp_bound (by simpa) (by decide) | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
(2 * x - x ^ 3 / 3 ... | Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs (-↑x * I) ≤ 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simpa | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
(2 * x - x ^ 3 / 3 ... | Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ 0 < 4 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | decide | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
(2 * x - x ^ 3 / 3 ... | Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ Complex.abs (↑x * I) ^ 4 * (↑(Nat.succ 4) * (↑(Nat.factorial 4) * ↑4)⁻¹) / 2 +
Complex.abs (-↑x * I) ^ 4 * (↑(Nat.succ 4) * (↑(Nat.factorial 4) * ↑4)⁻¹) / 2 ≤
abs' x ^ 4 * (5 / 96) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | norm_num [Nat.factorial] | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) :=
calc
|sin x - (x - x ^ 3 / 6)| = Complex.abs (Complex.sin x - (x - x ^ 3 / 6 : ℝ)) := by
rw [← abs_ofReal]; simp
_ = Complex.abs (((Complex.exp (-x * I) - Complex.exp (x * I)) * I -
(2 * x - x ^ 3 / 3 ... | Mathlib.Data.Complex.Exponential.1866_0.1bixbwhfBeJKySp | theorem sin_bound {x : ℝ} (hx : |x| ≤ 1) : |sin x - (x - x ^ 3 / 6)| ≤ |x| ^ 4 * (5 / 96) | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ abs' x ^ 4 * (5 / 96) + x ^ 2 / 2 ≤ 1 * (5 / 96) + 1 / 2 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | gcongr | theorem cos_pos_of_le_one {x : ℝ} (hx : |x| ≤ 1) : 0 < cos x :=
calc 0 < 1 - x ^ 2 / 2 - |x| ^ 4 * (5 / 96) :=
sub_pos.2 <|
lt_sub_iff_add_lt.2
(calc
|x| ^ 4 * (5 / 96) + x ^ 2 / 2 ≤ 1 * (5 / 96) + 1 / 2 := by
| Mathlib.Data.Complex.Exponential.1898_0.1bixbwhfBeJKySp | theorem cos_pos_of_le_one {x : ℝ} (hx : |x| ≤ 1) : 0 < cos x | Mathlib_Data_Complex_Exponential |
case h₁.h
x : ℝ
hx : abs' x ≤ 1
⊢ abs' x ^ 4 ≤ 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | exact pow_le_one _ (abs_nonneg _) hx | theorem cos_pos_of_le_one {x : ℝ} (hx : |x| ≤ 1) : 0 < cos x :=
calc 0 < 1 - x ^ 2 / 2 - |x| ^ 4 * (5 / 96) :=
sub_pos.2 <|
lt_sub_iff_add_lt.2
(calc
|x| ^ 4 * (5 / 96) + x ^ 2 / 2 ≤ 1 * (5 / 96) + 1 / 2 := by
gcongr
· | Mathlib.Data.Complex.Exponential.1898_0.1bixbwhfBeJKySp | theorem cos_pos_of_le_one {x : ℝ} (hx : |x| ≤ 1) : 0 < cos x | Mathlib_Data_Complex_Exponential |
case h₂.h
x : ℝ
hx : abs' x ≤ 1
⊢ x ^ 2 ≤ 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rw [sq, ← abs_mul_self, abs_mul] | theorem cos_pos_of_le_one {x : ℝ} (hx : |x| ≤ 1) : 0 < cos x :=
calc 0 < 1 - x ^ 2 / 2 - |x| ^ 4 * (5 / 96) :=
sub_pos.2 <|
lt_sub_iff_add_lt.2
(calc
|x| ^ 4 * (5 / 96) + x ^ 2 / 2 ≤ 1 * (5 / 96) + 1 / 2 := by
gcongr
· exact pow_le_one _ (abs_non... | Mathlib.Data.Complex.Exponential.1898_0.1bixbwhfBeJKySp | theorem cos_pos_of_le_one {x : ℝ} (hx : |x| ≤ 1) : 0 < cos x | Mathlib_Data_Complex_Exponential |
case h₂.h
x : ℝ
hx : abs' x ≤ 1
⊢ abs' x * abs' x ≤ 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | exact mul_le_one hx (abs_nonneg _) hx | theorem cos_pos_of_le_one {x : ℝ} (hx : |x| ≤ 1) : 0 < cos x :=
calc 0 < 1 - x ^ 2 / 2 - |x| ^ 4 * (5 / 96) :=
sub_pos.2 <|
lt_sub_iff_add_lt.2
(calc
|x| ^ 4 * (5 / 96) + x ^ 2 / 2 ≤ 1 * (5 / 96) + 1 / 2 := by
gcongr
· exact pow_le_one _ (abs_non... | Mathlib.Data.Complex.Exponential.1898_0.1bixbwhfBeJKySp | theorem cos_pos_of_le_one {x : ℝ} (hx : |x| ≤ 1) : 0 < cos x | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
⊢ 1 * (5 / 96) + 1 / 2 < 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | norm_num | theorem cos_pos_of_le_one {x : ℝ} (hx : |x| ≤ 1) : 0 < cos x :=
calc 0 < 1 - x ^ 2 / 2 - |x| ^ 4 * (5 / 96) :=
sub_pos.2 <|
lt_sub_iff_add_lt.2
(calc
|x| ^ 4 * (5 / 96) + x ^ 2 / 2 ≤ 1 * (5 / 96) + 1 / 2 := by
gcongr
· exact pow_le_one _ (abs_non... | Mathlib.Data.Complex.Exponential.1898_0.1bixbwhfBeJKySp | theorem cos_pos_of_le_one {x : ℝ} (hx : |x| ≤ 1) : 0 < cos x | Mathlib_Data_Complex_Exponential |
x : ℝ
hx0 : 0 < x
hx : x ≤ 1
⊢ abs' x ^ 4 * (5 / 96) + x ^ 3 / 6 ≤ x * (5 / 96) + x / 6 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | gcongr | theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x :=
calc 0 < x - x ^ 3 / 6 - |x| ^ 4 * (5 / 96) :=
sub_pos.2 <| lt_sub_iff_add_lt.2
(calc
|x| ^ 4 * (5 / 96) + x ^ 3 / 6 ≤ x * (5 / 96) + x / 6 := by
| Mathlib.Data.Complex.Exponential.1912_0.1bixbwhfBeJKySp | theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x | Mathlib_Data_Complex_Exponential |
case h₁.h
x : ℝ
hx0 : 0 < x
hx : x ≤ 1
⊢ abs' x ^ 4 ≤ x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | calc
|x| ^ 4 ≤ |x| ^ 1 :=
pow_le_pow_of_le_one (abs_nonneg _)
(by rwa [_root_.abs_of_nonneg (le_of_lt hx0)]) (by decide)
_ = x := by simp [_root_.abs_of_nonneg (le_of_lt hx0)] | theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x :=
calc 0 < x - x ^ 3 / 6 - |x| ^ 4 * (5 / 96) :=
sub_pos.2 <| lt_sub_iff_add_lt.2
(calc
|x| ^ 4 * (5 / 96) + x ^ 3 / 6 ≤ x * (5 / 96) + x / 6 := by
gcongr
· | Mathlib.Data.Complex.Exponential.1912_0.1bixbwhfBeJKySp | theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x | Mathlib_Data_Complex_Exponential |
x : ℝ
hx0 : 0 < x
hx : x ≤ 1
⊢ abs' x ≤ 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rwa [_root_.abs_of_nonneg (le_of_lt hx0)] | theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x :=
calc 0 < x - x ^ 3 / 6 - |x| ^ 4 * (5 / 96) :=
sub_pos.2 <| lt_sub_iff_add_lt.2
(calc
|x| ^ 4 * (5 / 96) + x ^ 3 / 6 ≤ x * (5 / 96) + x / 6 := by
gcongr
· calc
... | Mathlib.Data.Complex.Exponential.1912_0.1bixbwhfBeJKySp | theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x | Mathlib_Data_Complex_Exponential |
x : ℝ
hx0 : 0 < x
hx : x ≤ 1
⊢ 1 ≤ 4 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | decide | theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x :=
calc 0 < x - x ^ 3 / 6 - |x| ^ 4 * (5 / 96) :=
sub_pos.2 <| lt_sub_iff_add_lt.2
(calc
|x| ^ 4 * (5 / 96) + x ^ 3 / 6 ≤ x * (5 / 96) + x / 6 := by
gcongr
· calc
... | Mathlib.Data.Complex.Exponential.1912_0.1bixbwhfBeJKySp | theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x | Mathlib_Data_Complex_Exponential |
x : ℝ
hx0 : 0 < x
hx : x ≤ 1
⊢ abs' x ^ 1 = x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [_root_.abs_of_nonneg (le_of_lt hx0)] | theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x :=
calc 0 < x - x ^ 3 / 6 - |x| ^ 4 * (5 / 96) :=
sub_pos.2 <| lt_sub_iff_add_lt.2
(calc
|x| ^ 4 * (5 / 96) + x ^ 3 / 6 ≤ x * (5 / 96) + x / 6 := by
gcongr
· calc
... | Mathlib.Data.Complex.Exponential.1912_0.1bixbwhfBeJKySp | theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x | Mathlib_Data_Complex_Exponential |
case h₂.h
x : ℝ
hx0 : 0 < x
hx : x ≤ 1
⊢ x ^ 3 ≤ x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | calc
x ^ 3 ≤ x ^ 1 := pow_le_pow_of_le_one (le_of_lt hx0) hx (by decide)
_ = x := pow_one _ | theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x :=
calc 0 < x - x ^ 3 / 6 - |x| ^ 4 * (5 / 96) :=
sub_pos.2 <| lt_sub_iff_add_lt.2
(calc
|x| ^ 4 * (5 / 96) + x ^ 3 / 6 ≤ x * (5 / 96) + x / 6 := by
gcongr
· calc
... | Mathlib.Data.Complex.Exponential.1912_0.1bixbwhfBeJKySp | theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x | Mathlib_Data_Complex_Exponential |
x : ℝ
hx0 : 0 < x
hx : x ≤ 1
⊢ 1 ≤ 3 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | decide | theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x :=
calc 0 < x - x ^ 3 / 6 - |x| ^ 4 * (5 / 96) :=
sub_pos.2 <| lt_sub_iff_add_lt.2
(calc
|x| ^ 4 * (5 / 96) + x ^ 3 / 6 ≤ x * (5 / 96) + x / 6 := by
gcongr
· calc
... | Mathlib.Data.Complex.Exponential.1912_0.1bixbwhfBeJKySp | theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x | Mathlib_Data_Complex_Exponential |
x : ℝ
hx0 : 0 < x
hx : x ≤ 1
⊢ x * (5 / 96) + x / 6 < x | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | linarith | theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x :=
calc 0 < x - x ^ 3 / 6 - |x| ^ 4 * (5 / 96) :=
sub_pos.2 <| lt_sub_iff_add_lt.2
(calc
|x| ^ 4 * (5 / 96) + x ^ 3 / 6 ≤ x * (5 / 96) + x / 6 := by
gcongr
· calc
... | Mathlib.Data.Complex.Exponential.1912_0.1bixbwhfBeJKySp | theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x | Mathlib_Data_Complex_Exponential |
x : ℝ
hx0 : 0 < x
hx : x ≤ 1
⊢ abs' x ≤ 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rwa [_root_.abs_of_nonneg (le_of_lt hx0)] | theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x :=
calc 0 < x - x ^ 3 / 6 - |x| ^ 4 * (5 / 96) :=
sub_pos.2 <| lt_sub_iff_add_lt.2
(calc
|x| ^ 4 * (5 / 96) + x ^ 3 / 6 ≤ x * (5 / 96) + x / 6 := by
gcongr
· calc
... | Mathlib.Data.Complex.Exponential.1912_0.1bixbwhfBeJKySp | theorem sin_pos_of_pos_of_le_one {x : ℝ} (hx0 : 0 < x) (hx : x ≤ 1) : 0 < sin x | Mathlib_Data_Complex_Exponential |
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