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✝³ y x : ℝ
hx : 0 ≤ x
n j : ℕ
hj : j ≥ n
x✝² : ℕ
x✝¹ : x✝² ∈ range j
x✝ : x✝² ∉ range n
⊢ 0 ≤ x ^ x✝² / ↑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... | positivity | theorem sum_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : ∑ i in range n, x ^ i / i ! ≤ exp x :=
calc
∑ i in range n, x ^ i / i ! ≤ lim (⟨_, isCauSeq_re (exp' x)⟩ : CauSeq ℝ Abs.abs) := by
refine' le_lim (CauSeq.le_of_exists ⟨n, fun j hj => _⟩)
simp only [exp', const_apply, re_sum]
norm_cast
... | Mathlib.Data.Complex.Exponential.1473_0.1bixbwhfBeJKySp | theorem sum_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : ∑ i in range n, x ^ i / i ! ≤ exp x | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
hx : 0 ≤ x
n : ℕ
⊢ CauSeq.lim { val := fun n => (↑(exp' ↑x) n).re, property := (_ : IsCauSeq abs' fun n => (↑(exp' ↑x) n).re) } = 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... | rw [exp, Complex.exp, ← cauSeqRe, lim_re] | theorem sum_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : ∑ i in range n, x ^ i / i ! ≤ exp x :=
calc
∑ i in range n, x ^ i / i ! ≤ lim (⟨_, isCauSeq_re (exp' x)⟩ : CauSeq ℝ Abs.abs) := by
refine' le_lim (CauSeq.le_of_exists ⟨n, fun j hj => _⟩)
simp only [exp', const_apply, re_sum]
norm_cast
... | Mathlib.Data.Complex.Exponential.1473_0.1bixbwhfBeJKySp | theorem sum_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) (n : ℕ) : ∑ i in range n, x ^ i / i ! ≤ exp x | Mathlib_Data_Complex_Exponential |
x y : ℝ
hx : 0 ≤ x
n k : ℕ
x✝ : k ∈ range (n + 1)
⊢ 0 ≤ (fun k => x ^ k / ↑k !) k | /-
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... | positivity | lemma pow_div_factorial_le_exp (hx : 0 ≤ x) (n : ℕ) : x ^ n / n ! ≤ exp x :=
calc
x ^ n / n ! ≤ ∑ k in range (n + 1), x ^ k / k ! :=
single_le_sum (f := fun k ↦ x ^ k / k !) (fun k _ ↦ by | Mathlib.Data.Complex.Exponential.1484_0.1bixbwhfBeJKySp | lemma pow_div_factorial_le_exp (hx : 0 ≤ x) (n : ℕ) : x ^ n / n ! ≤ exp x | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
hx : 0 ≤ x
⊢ 1 + x + x ^ 2 / 2 = ∑ i in range 3, x ^ i / ↑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, range_one, sum_singleton, _root_.pow_zero, factorial, cast_one,
ne_eq, one_ne_zero, not_false_eq_true, div_self, pow_one, mul_one, div_one, Nat.mul_one,
cast_succ, add_right_inj] | theorem quadratic_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) : 1 + x + x ^ 2 / 2 ≤ exp x :=
calc
1 + x + x ^ 2 / 2 = ∑ i in range 3, x ^ i / i ! := by
| Mathlib.Data.Complex.Exponential.1490_0.1bixbwhfBeJKySp | theorem quadratic_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) : 1 + x + x ^ 2 / 2 ≤ exp x | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
hx : 0 ≤ x
⊢ 1 + x + x ^ 2 / 2 = 1 / (↑0 + 1) + x / (↑0 + 1) + x ^ 2 / (↑0 + 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 quadratic_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) : 1 + x + x ^ 2 / 2 ≤ exp x :=
calc
1 + x + x ^ 2 / 2 = ∑ i in range 3, x ^ i / i ! := by
simp only [sum_range_succ, range_one, sum_singleton, _root_.pow_zero, factorial, cast_one,
ne_eq, one_ne_zero, not_false_eq_true, div_self, pow_one, m... | Mathlib.Data.Complex.Exponential.1490_0.1bixbwhfBeJKySp | theorem quadratic_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) : 1 + x + x ^ 2 / 2 ≤ exp x | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
hx : 0 < x
⊢ x + 1 < 1 + x + 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... | nlinarith | private theorem add_one_lt_exp_of_pos {x : ℝ} (hx : 0 < x) : x + 1 < exp x :=
(by | Mathlib.Data.Complex.Exponential.1500_0.1bixbwhfBeJKySp | private theorem add_one_lt_exp_of_pos {x : ℝ} (hx : 0 < x) : x + 1 < exp x | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
hx : 0 ≤ x
⊢ x + 1 ≤ 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... | rcases eq_or_lt_of_le hx with (rfl | h) | private theorem add_one_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) : x + 1 ≤ exp x := by
| Mathlib.Data.Complex.Exponential.1503_0.1bixbwhfBeJKySp | private theorem add_one_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) : x + 1 ≤ exp x | Mathlib_Data_Complex_Exponential |
case inl
x y : ℝ
hx : 0 ≤ 0
⊢ 0 + 1 ≤ rexp 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... | simp | private theorem add_one_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) : x + 1 ≤ exp x := by
rcases eq_or_lt_of_le hx with (rfl | h)
· | Mathlib.Data.Complex.Exponential.1503_0.1bixbwhfBeJKySp | private theorem add_one_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) : x + 1 ≤ exp x | Mathlib_Data_Complex_Exponential |
case inr
x✝ y x : ℝ
hx : 0 ≤ x
h : 0 < x
⊢ x + 1 ≤ 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... | exact (add_one_lt_exp_of_pos h).le | private theorem add_one_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) : x + 1 ≤ exp x := by
rcases eq_or_lt_of_le hx with (rfl | h)
· simp
| Mathlib.Data.Complex.Exponential.1503_0.1bixbwhfBeJKySp | private theorem add_one_le_exp_of_nonneg {x : ℝ} (hx : 0 ≤ x) : x + 1 ≤ exp x | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
hx : 0 ≤ x
⊢ 1 ≤ 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... | linarith [add_one_le_exp_of_nonneg hx] | theorem one_le_exp {x : ℝ} (hx : 0 ≤ x) : 1 ≤ exp x := by | Mathlib.Data.Complex.Exponential.1508_0.1bixbwhfBeJKySp | theorem one_le_exp {x : ℝ} (hx : 0 ≤ x) : 1 ≤ exp x | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
h : x ≤ 0
⊢ 0 < 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... | rw [← neg_neg x, Real.exp_neg] | theorem exp_pos (x : ℝ) : 0 < exp x :=
(le_total 0 x).elim (lt_of_lt_of_le zero_lt_one ∘ one_le_exp) fun h => by
| Mathlib.Data.Complex.Exponential.1511_0.1bixbwhfBeJKySp | theorem exp_pos (x : ℝ) : 0 < exp x | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
h : x ≤ 0
⊢ 0 < (rexp (-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 inv_pos.2 (lt_of_lt_of_le zero_lt_one (one_le_exp (neg_nonneg.2 h))) | theorem exp_pos (x : ℝ) : 0 < exp x :=
(le_total 0 x).elim (lt_of_lt_of_le zero_lt_one ∘ one_le_exp) fun h => by
rw [← neg_neg x, Real.exp_neg]
| Mathlib.Data.Complex.Exponential.1511_0.1bixbwhfBeJKySp | theorem exp_pos (x : ℝ) : 0 < exp x | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
⊢ rexp (abs' x) ≤ rexp x + 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... | cases le_total x 0 | lemma exp_abs_le (x : ℝ) : exp |x| ≤ exp x + exp (-x) := by
| Mathlib.Data.Complex.Exponential.1524_0.1bixbwhfBeJKySp | lemma exp_abs_le (x : ℝ) : exp |x| ≤ exp x + exp (-x) | Mathlib_Data_Complex_Exponential |
case inl
x✝ y x : ℝ
h✝ : x ≤ 0
⊢ rexp (abs' x) ≤ rexp x + 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... | simp [abs_of_nonpos, _root_.abs_of_nonneg, exp_nonneg, *] | lemma exp_abs_le (x : ℝ) : exp |x| ≤ exp x + exp (-x) := by
cases le_total x 0 <;> | Mathlib.Data.Complex.Exponential.1524_0.1bixbwhfBeJKySp | lemma exp_abs_le (x : ℝ) : exp |x| ≤ exp x + exp (-x) | Mathlib_Data_Complex_Exponential |
case inr
x✝ y x : ℝ
h✝ : 0 ≤ x
⊢ rexp (abs' x) ≤ rexp x + 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... | simp [abs_of_nonpos, _root_.abs_of_nonneg, exp_nonneg, *] | lemma exp_abs_le (x : ℝ) : exp |x| ≤ exp x + exp (-x) := by
cases le_total x 0 <;> | Mathlib.Data.Complex.Exponential.1524_0.1bixbwhfBeJKySp | lemma exp_abs_le (x : ℝ) : exp |x| ≤ exp x + exp (-x) | Mathlib_Data_Complex_Exponential |
x✝ y✝ x y : ℝ
h : x < y
⊢ rexp x < rexp y | /-
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_add_cancel y x, Real.exp_add] | @[mono]
theorem exp_strictMono : StrictMono exp := fun x y h => by
| Mathlib.Data.Complex.Exponential.1527_0.1bixbwhfBeJKySp | @[mono]
theorem exp_strictMono : StrictMono exp | Mathlib_Data_Complex_Exponential |
x✝ y✝ x y : ℝ
h : x < y
⊢ rexp x < rexp (y - x) * 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... | exact (lt_mul_iff_one_lt_left (exp_pos _)).2
(lt_of_lt_of_le (by linarith) (add_one_le_exp_of_nonneg (by linarith))) | @[mono]
theorem exp_strictMono : StrictMono exp := fun x y h => by
rw [← sub_add_cancel y x, Real.exp_add]
| Mathlib.Data.Complex.Exponential.1527_0.1bixbwhfBeJKySp | @[mono]
theorem exp_strictMono : StrictMono exp | Mathlib_Data_Complex_Exponential |
x✝ y✝ x y : ℝ
h : x < y
⊢ 1 < y - 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... | linarith | @[mono]
theorem exp_strictMono : StrictMono exp := fun x y h => by
rw [← sub_add_cancel y x, Real.exp_add]
exact (lt_mul_iff_one_lt_left (exp_pos _)).2
(lt_of_lt_of_le (by | Mathlib.Data.Complex.Exponential.1527_0.1bixbwhfBeJKySp | @[mono]
theorem exp_strictMono : StrictMono exp | Mathlib_Data_Complex_Exponential |
x✝ y✝ x y : ℝ
h : x < y
⊢ 0 ≤ y - 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 | @[mono]
theorem exp_strictMono : StrictMono exp := fun x y h => by
rw [← sub_add_cancel y x, Real.exp_add]
exact (lt_mul_iff_one_lt_left (exp_pos _)).2
(lt_of_lt_of_le (by linarith) (add_one_le_exp_of_nonneg (by | Mathlib.Data.Complex.Exponential.1527_0.1bixbwhfBeJKySp | @[mono]
theorem exp_strictMono : StrictMono exp | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
⊢ 1 < rexp x ↔ 0 < 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 [← exp_zero, exp_lt_exp] | @[simp]
theorem one_lt_exp_iff {x : ℝ} : 1 < exp x ↔ 0 < x := by | Mathlib.Data.Complex.Exponential.1569_0.1bixbwhfBeJKySp | @[simp]
theorem one_lt_exp_iff {x : ℝ} : 1 < exp x ↔ 0 < x | Mathlib_Data_Complex_Exponential |
x✝ y x : ℝ
⊢ rexp x < 1 ↔ x < 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... | rw [← exp_zero, exp_lt_exp] | @[simp]
theorem exp_lt_one_iff {x : ℝ} : exp x < 1 ↔ x < 0 := by | Mathlib.Data.Complex.Exponential.1573_0.1bixbwhfBeJKySp | @[simp]
theorem exp_lt_one_iff {x : ℝ} : exp x < 1 ↔ x < 0 | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
m : ℕ
hm : m ∈ filter (fun k => n ≤ k) (range j)
⊢ n ≤ 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 at hm | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
m : ℕ
hm : m < j ∧ n ≤ m
⊢ n ≤ 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... | tauto | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
m : ℕ
hm : m ∈ filter (fun k => n ≤ k) (range j)
⊢ m < j | /-
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 at hm | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
m : ℕ
hm : m < j ∧ n ≤ m
⊢ m < j | /-
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... | tauto | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
m : ℕ
hm : m ∈ filter (fun k => n ≤ k) (range j)
⊢ 1 / ↑(Nat.factorial m) = 1 / ↑(Nat.factorial ((fun m x => m - n) m hm + 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 [tsub_add_cancel_of_le] | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
m : ℕ
hm : m ∈ filter (fun k => n ≤ k) (range j)
⊢ n ≤ 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 at * | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
m : ℕ
hm : m < j ∧ n ≤ m
⊢ n ≤ 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... | tauto | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
a₁ a₂ : ℕ
ha₁ : a₁ ∈ filter (fun k => n ≤ k) (range j)
ha₂ : a₂ ∈ filter (fun k => n ≤ k) (range j)
h : (fun m x => m - n) a₁ ha₁ = (fun m x => m - n) a₂ ha₂
⊢ a₁ = 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... | rwa [tsub_eq_iff_eq_add_of_le, tsub_add_eq_add_tsub, eq_comm, tsub_eq_iff_eq_add_of_le,
add_left_inj, eq_comm] at h | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
a₁ a₂ : ℕ
ha₁ : a₁ ∈ filter (fun k => n ≤ k) (range j)
ha₂ : a₂ ∈ filter (fun k => n ≤ k) (range j)
h : a₂ + n - n = a₁
⊢ n ≤ a₂ + 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 at * | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
a₁ a₂ : ℕ
ha₁ : a₁ ∈ filter (fun k => n ≤ k) (range j)
ha₂ : a₂ ∈ filter (fun k => n ≤ k) (range j)
h : a₁ = (fun m x => m - n) a₂ ha₂ + n
⊢ n ≤ 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 at * | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
a₁ a₂ : ℕ
ha₁ : a₁ ∈ filter (fun k => n ≤ k) (range j)
ha₂ : a₂ ∈ filter (fun k => n ≤ k) (range j)
h : (fun m x => m - n) a₁ ha₁ = (fun m x => m - n) a₂ ha₂
⊢ n ≤ a₁ | /-
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 at * | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
a₁ a₂ : ℕ
ha₂ : a₂ ∈ filter (fun k => n ≤ k) (range j)
h : a₁ = a₂ - n + n
ha₁ : a₁ < j ∧ n ≤ a₁
⊢ n ≤ 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... | aesop | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
a₁ a₂ : ℕ
ha₁ : a₁ ∈ filter (fun k => n ≤ k) (range j)
ha₂ : a₂ ∈ filter (fun k => n ≤ k) (range j)
h : a₁ - n = a₂ - n
⊢ n ≤ a₁ | /-
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... | aesop | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
b : ℕ
hb : b ∈ range (j - n)
⊢ b = (fun m x => m - n) (b + n) (_ : b + n ∈ filter (fun k => n ≤ k) (range j)) | /-
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... | dsimp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
b : ℕ
hb : b ∈ range (j - n)
⊢ b = b + 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 [add_tsub_cancel_right] | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
⊢ ∑ m in range (j - n), 1 / ↑(Nat.factorial (m + n)) ≤ ∑ m in range (j - n), (↑(Nat.factorial n) * ↑(Nat.succ n) ^ m)⁻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_rw [one_div] | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
⊢ ∑ x in range (j - n), (↑(Nat.factorial (x + n)))⁻¹ ≤ ∑ m in range (j - n), (↑(Nat.factorial n) * ↑(Nat.succ n) ^ m)⁻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... | gcongr | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
case h.h
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
i✝ : ℕ
a✝ : i✝ ∈ range (j - n)
⊢ ↑(Nat.factorial n) * ↑(Nat.succ n) ^ i✝ ≤ ↑(Nat.factorial (i✝ + 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 [← Nat.cast_pow, ← Nat.cast_mul, Nat.cast_le, add_comm] | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
case h.h
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
i✝ : ℕ
a✝ : i✝ ∈ range (j - n)
⊢ Nat.factorial n * Nat.succ n ^ i✝ ≤ Nat.factorial (n + 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... | exact Nat.factorial_mul_pow_le_factorial | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
⊢ ∑ m in range (j - n), (↑(Nat.factorial n) * ↑(Nat.succ n) ^ m)⁻¹ =
(↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻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 [mul_inv, mul_sum.symm, sum_mul.symm, mul_comm, inv_pow] | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =
(↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - 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... | have h₁ : (n.succ : α) ≠ 1 :=
@Nat.cast_one α _ ▸ mt Nat.cast_inj.1 (mt Nat.succ.inj (pos_iff_ne_zero.1 hn)) | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
h₁ : ↑(Nat.succ n) ≠ 1
⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =
(↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - 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... | have h₂ : (n.succ : α) ≠ 0 := by positivity | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
h₁ : ↑(Nat.succ n) ≠ 1
⊢ ↑(Nat.succ n) ≠ 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... | positivity | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
h₁ : ↑(Nat.succ n) ≠ 1
h₂ : ↑(Nat.succ n) ≠ 0
⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =
(↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - 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... | have h₃ : (n.factorial * n : α) ≠ 0 := by positivity | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
h₁ : ↑(Nat.succ n) ≠ 1
h₂ : ↑(Nat.succ n) ≠ 0
⊢ ↑(Nat.factorial n) * ↑n ≠ 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... | positivity | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
h₁ : ↑(Nat.succ n) ≠ 1
h₂ : ↑(Nat.succ n) ≠ 0
h₃ : ↑(Nat.factorial n) * ↑n ≠ 0
⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =
(↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - 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... | have h₄ : (n.succ - 1 : α) = n := by simp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
h₁ : ↑(Nat.succ n) ≠ 1
h₂ : ↑(Nat.succ n) ≠ 0
h₃ : ↑(Nat.factorial n) * ↑n ≠ 0
⊢ ↑(Nat.succ n) - 1 = ↑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 | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
h₁ : ↑(Nat.succ n) ≠ 1
h₂ : ↑(Nat.succ n) ≠ 0
h₃ : ↑(Nat.factorial n) * ↑n ≠ 0
h₄ : ↑(Nat.succ n) - 1 = ↑n
⊢ (↑(Nat.factorial n))⁻¹ * ∑ m in range (j - n), (↑(Nat.succ n))⁻¹ ^ m =
(↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.fact... | /-
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 [geom_sum_inv h₁ h₂, eq_div_iff_mul_eq h₃, mul_comm _ (n.factorial * n : α),
← mul_assoc (n.factorial⁻¹ : α), ← mul_inv_rev, h₄, ← mul_assoc (n.factorial * n : α),
mul_comm (n : α) n.factorial, mul_inv_cancel h₃, one_mul, mul_comm] | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
⊢ (↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n)) / (↑(Nat.factorial n) * ↑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... | gcongr | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
case h
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
⊢ ↑(Nat.succ n) - ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - n) ≤ ↑(Nat.succ 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... | apply sub_le_self | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
case h.a
α : Type u_1
inst✝ : LinearOrderedField α
n j : ℕ
hn : 0 < n
⊢ 0 ≤ ↑(Nat.succ n) * (↑(Nat.succ n))⁻¹ ^ (j - 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... | positivity | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) :=
calc
(∑ m in filter (fun k => n ≤ k) (range j), (1 / m.factorial : α)) =
∑ m in range (j - n), (1 / ((m + n).... | Mathlib.Data.Complex.Exponential.1600_0.1bixbwhfBeJKySp | theorem sum_div_factorial_le {α : Type*} [LinearOrderedField α] (n j : ℕ) (hn : 0 < n) :
(∑ m in filter (fun k => n ≤ k) (range j),
(1 / m.factorial : α)) ≤ n.succ / (n.factorial * n) | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 1
n : ℕ
hn : 0 < n
⊢ abs (cexp x - ∑ m in range n, x ^ m / ↑(Nat.factorial m)) ≤ abs x ^ n * (↑(Nat.succ n) * (↑(Nat.factorial n) * ↑n)⁻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 [← lim_const (abv := Complex.abs) (∑ m in range n, _), exp, sub_eq_add_neg,
← lim_neg, lim_add, ← lim_abs] | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) := by
| Mathlib.Data.Complex.Exponential.1638_0.1bixbwhfBeJKySp | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 1
n : ℕ
hn : 0 < n
⊢ CauSeq.lim (cauSeqAbs (exp' x + -const (⇑abs) (∑ m in range n, x ^ m / ↑(Nat.factorial m)))) ≤
abs x ^ n * (↑(Nat.succ n) * (↑(Nat.factorial n) * ↑n)⁻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... | refine' lim_le (CauSeq.le_of_exists ⟨n, fun j hj => _⟩) | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _), exp, sub_eq_add_neg,
← lim_neg, lim_add, ← lim_abs]
| Mathlib.Data.Complex.Exponential.1638_0.1bixbwhfBeJKySp | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 1
n : ℕ
hn : 0 < n
j : ℕ
hj : j ≥ n
⊢ ↑(cauSeqAbs (exp' x + -const (⇑abs) (∑ m in range n, x ^ m / ↑(Nat.factorial m)))) j ≤
↑(const abs' (abs x ^ n * (↑(Nat.succ n) * (↑(Nat.factorial n) * ↑n)⁻¹))) j | /-
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_rw [← sub_eq_add_neg] | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _), exp, sub_eq_add_neg,
← lim_neg, lim_add, ← lim_abs]
refine' lim_le ... | Mathlib.Data.Complex.Exponential.1638_0.1bixbwhfBeJKySp | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 1
n : ℕ
hn : 0 < n
j : ℕ
hj : j ≥ n
⊢ ↑(cauSeqAbs (exp' x - const (⇑abs) (∑ m in range n, x ^ m / ↑(Nat.factorial m)))) j ≤
↑(const abs' (abs x ^ n * (↑(Nat.succ n) * (↑(Nat.factorial n) * ↑n)⁻¹))) j | /-
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... | show
abs ((∑ m in range j, x ^ m / m.factorial) - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _), exp, sub_eq_add_neg,
← lim_neg, lim_add, ← lim_abs]
refine' lim_le ... | Mathlib.Data.Complex.Exponential.1638_0.1bixbwhfBeJKySp | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 1
n : ℕ
hn : 0 < n
j : ℕ
hj : j ≥ n
⊢ abs (∑ m in range j, x ^ m / ↑(Nat.factorial m) - ∑ m in range n, x ^ m / ↑(Nat.factorial m)) ≤
abs x ^ n * (↑(Nat.succ n) * (↑(Nat.factorial n) * ↑n)⁻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 [sum_range_sub_sum_range hj] | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _), exp, sub_eq_add_neg,
← lim_neg, lim_add, ← lim_abs]
refine' lim_le ... | Mathlib.Data.Complex.Exponential.1638_0.1bixbwhfBeJKySp | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 1
n : ℕ
hn : 0 < n
j : ℕ
hj : j ≥ n
⊢ abs (∑ k in filter (fun k => n ≤ k) (range j), x ^ k / ↑(Nat.factorial k)) ≤
abs x ^ n * (↑(Nat.succ n) * (↑(Nat.factorial n) * ↑n)⁻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... | calc
abs (∑ m in (range j).filter fun k => n ≤ k, (x ^ m / m.factorial : ℂ)) =
abs (∑ m in (range j).filter fun k => n ≤ k,
(x ^ n * (x ^ (m - n) / m.factorial) : ℂ)) := by
refine' congr_arg abs (sum_congr rfl fun m hm => _)
rw [mem_filter, mem_range] at hm
rw [← mul_div_assoc, ← pow... | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _), exp, sub_eq_add_neg,
← lim_neg, lim_add, ← lim_abs]
refine' lim_le ... | Mathlib.Data.Complex.Exponential.1638_0.1bixbwhfBeJKySp | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 1
n : ℕ
hn : 0 < n
j : ℕ
hj : j ≥ n
⊢ abs (∑ m in filter (fun k => n ≤ k) (range j), x ^ m / ↑(Nat.factorial m)) =
abs (∑ m in filter (fun k => n ≤ k) (range j), x ^ n * (x ^ (m - n) / ↑(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... | refine' congr_arg abs (sum_congr rfl fun m hm => _) | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _), exp, sub_eq_add_neg,
← lim_neg, lim_add, ← lim_abs]
refine' lim_le ... | Mathlib.Data.Complex.Exponential.1638_0.1bixbwhfBeJKySp | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 1
n : ℕ
hn : 0 < n
j : ℕ
hj : j ≥ n
m : ℕ
hm : m ∈ filter (fun k => n ≤ k) (range j)
⊢ x ^ m / ↑(Nat.factorial m) = x ^ n * (x ^ (m - n) / ↑(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... | rw [mem_filter, mem_range] at hm | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _), exp, sub_eq_add_neg,
← lim_neg, lim_add, ← lim_abs]
refine' lim_le ... | Mathlib.Data.Complex.Exponential.1638_0.1bixbwhfBeJKySp | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 1
n : ℕ
hn : 0 < n
j : ℕ
hj : j ≥ n
m : ℕ
hm : m < j ∧ n ≤ m
⊢ x ^ m / ↑(Nat.factorial m) = x ^ n * (x ^ (m - n) / ↑(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... | rw [← mul_div_assoc, ← pow_add, add_tsub_cancel_of_le hm.2] | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _), exp, sub_eq_add_neg,
← lim_neg, lim_add, ← lim_abs]
refine' lim_le ... | Mathlib.Data.Complex.Exponential.1638_0.1bixbwhfBeJKySp | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 1
n : ℕ
hn : 0 < n
j : ℕ
hj : j ≥ n
⊢ ∑ m in filter (fun k => n ≤ k) (range j), abs (x ^ n * (x ^ (m - n) / ↑(Nat.factorial m))) ≤
∑ m in filter (fun k => n ≤ k) (range j), abs x ^ n * (1 / ↑(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_rw [map_mul, map_pow, map_div₀, abs_natCast] | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _), exp, sub_eq_add_neg,
← lim_neg, lim_add, ← lim_abs]
refine' lim_le ... | Mathlib.Data.Complex.Exponential.1638_0.1bixbwhfBeJKySp | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 1
n : ℕ
hn : 0 < n
j : ℕ
hj : j ≥ n
⊢ ∑ x_1 in filter (fun k => n ≤ k) (range j), abs x ^ n * (abs (x ^ (x_1 - n)) / ↑(Nat.factorial x_1)) ≤
∑ m in filter (fun k => n ≤ k) (range j), abs x ^ n * (1 / ↑(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... | gcongr | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _), exp, sub_eq_add_neg,
← lim_neg, lim_add, ← lim_abs]
refine' lim_le ... | Mathlib.Data.Complex.Exponential.1638_0.1bixbwhfBeJKySp | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) | Mathlib_Data_Complex_Exponential |
case h.h.h
x : ℂ
hx : abs x ≤ 1
n : ℕ
hn : 0 < n
j : ℕ
hj : j ≥ n
i✝ : ℕ
a✝ : i✝ ∈ filter (fun k => n ≤ k) (range j)
⊢ abs (x ^ (i✝ - n)) ≤ 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 [abv_pow abs] | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _), exp, sub_eq_add_neg,
← lim_neg, lim_add, ← lim_abs]
refine' lim_le ... | Mathlib.Data.Complex.Exponential.1638_0.1bixbwhfBeJKySp | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) | Mathlib_Data_Complex_Exponential |
case h.h.h
x : ℂ
hx : abs x ≤ 1
n : ℕ
hn : 0 < n
j : ℕ
hj : j ≥ n
i✝ : ℕ
a✝ : i✝ ∈ filter (fun k => n ≤ k) (range j)
⊢ abs x ^ (i✝ - n) ≤ 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 exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _), exp, sub_eq_add_neg,
← lim_neg, lim_add, ← lim_abs]
refine' lim_le ... | Mathlib.Data.Complex.Exponential.1638_0.1bixbwhfBeJKySp | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 1
n : ℕ
hn : 0 < n
j : ℕ
hj : j ≥ n
⊢ ∑ m in filter (fun k => n ≤ k) (range j), abs x ^ n * (1 / ↑(Nat.factorial m)) =
abs x ^ n * ∑ m in filter (fun k => n ≤ k) (range j), 1 / ↑(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 [abs_mul, abv_pow abs, abs_div, mul_sum.symm] | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _), exp, sub_eq_add_neg,
← lim_neg, lim_add, ← lim_abs]
refine' lim_le ... | Mathlib.Data.Complex.Exponential.1638_0.1bixbwhfBeJKySp | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 1
n : ℕ
hn : 0 < n
j : ℕ
hj : j ≥ n
⊢ abs x ^ n * ∑ m in filter (fun k => n ≤ k) (range j), 1 / ↑(Nat.factorial m) ≤
abs x ^ n * (↑(Nat.succ n) * (↑(Nat.factorial n) * ↑n)⁻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... | gcongr | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _), exp, sub_eq_add_neg,
← lim_neg, lim_add, ← lim_abs]
refine' lim_le ... | Mathlib.Data.Complex.Exponential.1638_0.1bixbwhfBeJKySp | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) | Mathlib_Data_Complex_Exponential |
case h
x : ℂ
hx : abs x ≤ 1
n : ℕ
hn : 0 < n
j : ℕ
hj : j ≥ n
⊢ ∑ m in filter (fun k => n ≤ k) (range j), 1 / ↑(Nat.factorial m) ≤ ↑(Nat.succ n) * (↑(Nat.factorial n) * ↑n)⁻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 sum_div_factorial_le _ _ hn | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _), exp, sub_eq_add_neg,
← lim_neg, lim_add, ← lim_abs]
refine' lim_le ... | Mathlib.Data.Complex.Exponential.1638_0.1bixbwhfBeJKySp | theorem exp_bound {x : ℂ} (hx : abs x ≤ 1) {n : ℕ} (hn : 0 < n) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n * ((n.succ : ℝ) * (n.factorial * n : ℝ)⁻¹) | Mathlib_Data_Complex_Exponential |
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
⊢ abs (cexp x - ∑ m in range n, x ^ m / ↑(Nat.factorial m)) ≤ abs x ^ n / ↑(Nat.factorial n) * 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 [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs] | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
| Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
⊢ CauSeq.lim (cauSeqAbs (exp' x + -const (⇑abs) (∑ m in range n, x ^ m / ↑(Nat.factorial m)))) ≤
abs x ^ n / ↑(Nat.factorial n) * 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... | refine' lim_le (CauSeq.le_of_exists ⟨n, fun j hj => _⟩) | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
| Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj : j ≥ n
⊢ ↑(cauSeqAbs (exp' x + -const (⇑abs) (∑ m in range n, x ^ m / ↑(Nat.factorial m)))) j ≤
↑(const abs' (abs x ^ n / ↑(Nat.factorial n) * 2)) j | /-
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_rw [← sub_eq_add_neg] | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj : j ≥ n
⊢ ↑(cauSeqAbs (exp' x - const (⇑abs) (∑ m in range n, x ^ m / ↑(Nat.factorial m)))) j ≤
↑(const abs' (abs x ^ n / ↑(Nat.factorial n) * 2)) j | /-
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... | show abs ((∑ m in range j, x ^ m / m.factorial) - ∑ m in range n, x ^ m / m.factorial) ≤
abs x ^ n / n.factorial * 2 | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj : j ≥ n
⊢ abs (∑ m in range j, x ^ m / ↑(Nat.factorial m) - ∑ m in range n, x ^ m / ↑(Nat.factorial m)) ≤
abs x ^ n / ↑(Nat.factorial n) * 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... | let k := j - n | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj : j ≥ n
k : ℕ := j - n
⊢ abs (∑ m in range j, x ^ m / ↑(Nat.factorial m) - ∑ m in range n, x ^ m / ↑(Nat.factorial m)) ≤
abs x ^ n / ↑(Nat.factorial n) * 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 hj : j = n + k := (add_tsub_cancel_of_le hj).symm | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj✝ : j ≥ n
k : ℕ := j - n
hj : j = n + k
⊢ abs (∑ m in range j, x ^ m / ↑(Nat.factorial m) - ∑ m in range n, x ^ m / ↑(Nat.factorial m)) ≤
abs x ^ n / ↑(Nat.factorial n) * 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 [hj, sum_range_add_sub_sum_range] | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj✝ : j ≥ n
k : ℕ := j - n
hj : j = n + k
⊢ abs (∑ k in range k, x ^ (n + k) / ↑(Nat.factorial (n + k))) ≤ abs x ^ n / ↑(Nat.factorial n) * 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... | calc
abs (∑ i : ℕ in range k, x ^ (n + i) / ((n + i).factorial : ℂ)) ≤
∑ i : ℕ in range k, abs (x ^ (n + i) / ((n + i).factorial : ℂ)) :=
abv_sum_le_sum_abv _ _
_ ≤ ∑ i : ℕ in range k, abs x ^ (n + i) / (n + i).factorial := by
simp [Complex.abs_natCast, map_div₀, abv_pow abs]
_ ≤ ∑ i : ℕ... | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj✝ : j ≥ n
k : ℕ := j - n
hj : j = n + k
⊢ ∑ i in range k, abs (x ^ (n + i) / ↑(Nat.factorial (n + i))) ≤
∑ i in range k, abs x ^ (n + i) / ↑(Nat.factorial (n + 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 [Complex.abs_natCast, map_div₀, abv_pow abs] | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
case calc_1
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj✝ : j ≥ n
k : ℕ := j - n
hj : j = n + k
⊢ ∑ i in range k, abs x ^ (n + i) / ↑(Nat.factorial (n + i)) ≤
∑ i in range k, abs x ^ (n + i) / (↑(Nat.factorial n) * ↑(Nat.succ n) ^ 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... | gcongr | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
case calc_1.h.h
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj✝ : j ≥ n
k : ℕ := j - n
hj : j = n + k
i✝ : ℕ
a✝ : i✝ ∈ range k
⊢ ↑(Nat.factorial n) * ↑(Nat.succ n) ^ i✝ ≤ ↑(Nat.factorial (n + 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... | exact mod_cast Nat.factorial_mul_pow_le_factorial | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
case calc_2
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj✝ : j ≥ n
k : ℕ := j - n
hj : j = n + k
⊢ ∑ i in range k, abs x ^ (n + i) / (↑(Nat.factorial n) * ↑(Nat.succ n) ^ i) =
∑ i in range k, abs x ^ n / ↑(Nat.factorial n) * (abs x ^ i / ↑(Nat.succ n) ^ 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... | refine' Finset.sum_congr rfl fun _ _ => _ | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
case calc_2
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj✝ : j ≥ n
k : ℕ := j - n
hj : j = n + k
x✝¹ : ℕ
x✝ : x✝¹ ∈ range k
⊢ abs x ^ (n + x✝¹) / (↑(Nat.factorial n) * ↑(Nat.succ n) ^ x✝¹) =
abs x ^ n / ↑(Nat.factorial n) * (abs x ^ x✝¹ / ↑(Nat.succ n) ^ 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... | simp only [pow_add, div_eq_inv_mul, mul_inv, mul_left_comm, mul_assoc] | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
case calc_3
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj✝ : j ≥ n
k : ℕ := j - n
hj : j = n + k
⊢ ∑ i in range k, abs x ^ n / ↑(Nat.factorial n) * (abs x ^ i / ↑(Nat.succ n) ^ i) ≤ abs x ^ n / ↑(Nat.factorial n) * 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 [← mul_sum] | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
case calc_3
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj✝ : j ≥ n
k : ℕ := j - n
hj : j = n + k
⊢ abs x ^ n / ↑(Nat.factorial n) * ∑ x_1 in range k, abs x ^ x_1 / ↑(Nat.succ n) ^ x_1 ≤
abs x ^ n / ↑(Nat.factorial n) * 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 exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
case calc_3.h
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj✝ : j ≥ n
k : ℕ := j - n
hj : j = n + k
⊢ ∑ x_1 in range k, abs x ^ x_1 / ↑(Nat.succ n) ^ x_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... | simp_rw [← div_pow] | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
case calc_3.h
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj✝ : j ≥ n
k : ℕ := j - n
hj : j = n + k
⊢ ∑ x_1 in range (j - n), (abs x / ↑(Nat.succ n)) ^ x_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... | rw [geom_sum_eq, div_le_iff_of_neg] | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
case calc_3.h
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj✝ : j ≥ n
k : ℕ := j - n
hj : j = n + k
⊢ 2 * (abs x / ↑(Nat.succ n) - 1) ≤ (abs x / ↑(Nat.succ n)) ^ (j - n) - 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... | trans (-1 : ℝ) | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj✝ : j ≥ n
k : ℕ := j - n
hj : j = n + k
⊢ 2 * (abs x / ↑(Nat.succ n) - 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... | linarith | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj✝ : j ≥ n
k : ℕ := j - n
hj : j = n + k
⊢ -1 ≤ (abs x / ↑(Nat.succ n)) ^ (j - n) - 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 only [neg_le_sub_iff_le_add, div_pow, Nat.cast_succ, le_add_iff_nonneg_left] | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj✝ : j ≥ n
k : ℕ := j - n
hj : j = n + k
⊢ 0 ≤ abs x ^ (j - n) / (↑n + 1) ^ (j - 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... | positivity | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
case calc_3.h
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj✝ : j ≥ n
k : ℕ := j - n
hj : j = n + k
⊢ abs x / ↑(Nat.succ n) - 1 < 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_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
case calc_3.h.h
x : ℂ
n : ℕ
hx : abs x / ↑(Nat.succ n) ≤ 1 / 2
j : ℕ
hj✝ : j ≥ n
k : ℕ := j - n
hj : j = n + k
⊢ abs x / ↑(Nat.succ n) ≠ 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... | linarith | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 := by
rw [← lim_const (abv := Complex.abs) (∑ m in range n, _),
exp, sub_eq_add_neg, ← lim_neg, lim_add, ← lim_abs]
refine' lim_le (CauSeq.le_of_exists ⟨n, fun j... | Mathlib.Data.Complex.Exponential.1670_0.1bixbwhfBeJKySp | theorem exp_bound' {x : ℂ} {n : ℕ} (hx : abs x / n.succ ≤ 1 / 2) :
abs (exp x - ∑ m in range n, x ^ m / m.factorial) ≤ abs x ^ n / n.factorial * 2 | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 1
⊢ abs (cexp x - 1) = abs (cexp x - ∑ m in range 1, 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 [sum_range_succ] | theorem abs_exp_sub_one_le {x : ℂ} (hx : abs x ≤ 1) : abs (exp x - 1) ≤ 2 * abs x :=
calc
abs (exp x - 1) = abs (exp x - ∑ m in range 1, x ^ m / m.factorial) := by | Mathlib.Data.Complex.Exponential.1706_0.1bixbwhfBeJKySp | theorem abs_exp_sub_one_le {x : ℂ} (hx : abs x ≤ 1) : abs (exp x - 1) ≤ 2 * abs x | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 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... | decide | theorem abs_exp_sub_one_le {x : ℂ} (hx : abs x ≤ 1) : abs (exp x - 1) ≤ 2 * abs x :=
calc
abs (exp x - 1) = abs (exp x - ∑ m in range 1, x ^ m / m.factorial) := by simp [sum_range_succ]
_ ≤ abs x ^ 1 * ((Nat.succ 1 : ℝ) * ((Nat.factorial 1) * (1 : ℕ) : ℝ)⁻¹) :=
(exp_bound hx (by | Mathlib.Data.Complex.Exponential.1706_0.1bixbwhfBeJKySp | theorem abs_exp_sub_one_le {x : ℂ} (hx : abs x ≤ 1) : abs (exp x - 1) ≤ 2 * abs x | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 1
⊢ abs x ^ 1 * (↑(Nat.succ 1) * (↑(Nat.factorial 1) * ↑1)⁻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... | simp [two_mul, mul_two, mul_add, mul_comm, add_mul, Nat.factorial] | theorem abs_exp_sub_one_le {x : ℂ} (hx : abs x ≤ 1) : abs (exp x - 1) ≤ 2 * abs x :=
calc
abs (exp x - 1) = abs (exp x - ∑ m in range 1, x ^ m / m.factorial) := by simp [sum_range_succ]
_ ≤ abs x ^ 1 * ((Nat.succ 1 : ℝ) * ((Nat.factorial 1) * (1 : ℕ) : ℝ)⁻¹) :=
(exp_bound hx (by decide))
_ = 2 * abs... | Mathlib.Data.Complex.Exponential.1706_0.1bixbwhfBeJKySp | theorem abs_exp_sub_one_le {x : ℂ} (hx : abs x ≤ 1) : abs (exp x - 1) ≤ 2 * abs x | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 1
⊢ abs (cexp x - 1 - x) = abs (cexp x - ∑ m in range 2, 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 [sub_eq_add_neg, sum_range_succ_comm, add_assoc, Nat.factorial] | theorem abs_exp_sub_one_sub_id_le {x : ℂ} (hx : abs x ≤ 1) : abs (exp x - 1 - x) ≤ abs x ^ 2 :=
calc
abs (exp x - 1 - x) = abs (exp x - ∑ m in range 2, x ^ m / m.factorial) := by
| Mathlib.Data.Complex.Exponential.1714_0.1bixbwhfBeJKySp | theorem abs_exp_sub_one_sub_id_le {x : ℂ} (hx : abs x ≤ 1) : abs (exp x - 1 - x) ≤ abs x ^ 2 | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 1
⊢ 0 < 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... | decide | theorem abs_exp_sub_one_sub_id_le {x : ℂ} (hx : abs x ≤ 1) : abs (exp x - 1 - x) ≤ abs x ^ 2 :=
calc
abs (exp x - 1 - x) = abs (exp x - ∑ m in range 2, x ^ m / m.factorial) := by
simp [sub_eq_add_neg, sum_range_succ_comm, add_assoc, Nat.factorial]
_ ≤ abs x ^ 2 * ((Nat.succ 2 : ℝ) * (Nat.factorial 2 * (... | Mathlib.Data.Complex.Exponential.1714_0.1bixbwhfBeJKySp | theorem abs_exp_sub_one_sub_id_le {x : ℂ} (hx : abs x ≤ 1) : abs (exp x - 1 - x) ≤ abs x ^ 2 | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 1
⊢ abs x ^ 2 * (↑(Nat.succ 2) * (↑(Nat.factorial 2) * ↑2)⁻¹) ≤ abs 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... | gcongr | theorem abs_exp_sub_one_sub_id_le {x : ℂ} (hx : abs x ≤ 1) : abs (exp x - 1 - x) ≤ abs x ^ 2 :=
calc
abs (exp x - 1 - x) = abs (exp x - ∑ m in range 2, x ^ m / m.factorial) := by
simp [sub_eq_add_neg, sum_range_succ_comm, add_assoc, Nat.factorial]
_ ≤ abs x ^ 2 * ((Nat.succ 2 : ℝ) * (Nat.factorial 2 * (... | Mathlib.Data.Complex.Exponential.1714_0.1bixbwhfBeJKySp | theorem abs_exp_sub_one_sub_id_le {x : ℂ} (hx : abs x ≤ 1) : abs (exp x - 1 - x) ≤ abs x ^ 2 | Mathlib_Data_Complex_Exponential |
case h
x : ℂ
hx : abs x ≤ 1
⊢ ↑(Nat.succ 2) * (↑(Nat.factorial 2) * ↑2)⁻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... | norm_num [Nat.factorial] | theorem abs_exp_sub_one_sub_id_le {x : ℂ} (hx : abs x ≤ 1) : abs (exp x - 1 - x) ≤ abs x ^ 2 :=
calc
abs (exp x - 1 - x) = abs (exp x - ∑ m in range 2, x ^ m / m.factorial) := by
simp [sub_eq_add_neg, sum_range_succ_comm, add_assoc, Nat.factorial]
_ ≤ abs x ^ 2 * ((Nat.succ 2 : ℝ) * (Nat.factorial 2 * (... | Mathlib.Data.Complex.Exponential.1714_0.1bixbwhfBeJKySp | theorem abs_exp_sub_one_sub_id_le {x : ℂ} (hx : abs x ≤ 1) : abs (exp x - 1 - x) ≤ abs x ^ 2 | Mathlib_Data_Complex_Exponential |
x : ℂ
hx : abs x ≤ 1
⊢ abs x ^ 2 * 1 = 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 [mul_one] | theorem abs_exp_sub_one_sub_id_le {x : ℂ} (hx : abs x ≤ 1) : abs (exp x - 1 - x) ≤ abs x ^ 2 :=
calc
abs (exp x - 1 - x) = abs (exp x - ∑ m in range 2, x ^ m / m.factorial) := by
simp [sub_eq_add_neg, sum_range_succ_comm, add_assoc, Nat.factorial]
_ ≤ abs x ^ 2 * ((Nat.succ 2 : ℝ) * (Nat.factorial 2 * (... | Mathlib.Data.Complex.Exponential.1714_0.1bixbwhfBeJKySp | theorem abs_exp_sub_one_sub_id_le {x : ℂ} (hx : abs x ≤ 1) : abs (exp x - 1 - x) ≤ abs x ^ 2 | Mathlib_Data_Complex_Exponential |
x : ℝ
hx : abs' x ≤ 1
n : ℕ
hn : 0 < n
⊢ 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... | have hxc : Complex.abs x ≤ 1 := mod_cast hx | 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
| 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 |
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