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 |
|---|---|---|---|---|---|---|
α : Type u_1
inst✝⁴ : LinearOrderedField α
inst✝³ : Archimedean α
β : Type u_2
inst✝² : Ring β
inst✝¹ : Nontrivial β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
⊢ IsCauSeq abs' fun m => ∑ n in range m, abv (x ^ n) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp only [abv_pow abv, geom_sum_eq hx1'] | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m :=
have hx1' : abv x ≠ 1 := fun h => by simp [h, lt_irrefl] at hx1
isCauSeq_series_of_abv_isCauSeq
(by
| Mathlib.Data.Complex.Exponential.129_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝⁴ : LinearOrderedField α
inst✝³ : Archimedean α
β : Type u_2
inst✝² : Ring β
inst✝¹ : Nontrivial β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
⊢ IsCauSeq abs' fun m => (abv x ^ m - 1) / (abv 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... | conv in _ / _ => rw [← neg_div_neg_eq, neg_sub, neg_sub] | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m :=
have hx1' : abv x ≠ 1 := fun h => by simp [h, lt_irrefl] at hx1
isCauSeq_series_of_abv_isCauSeq
(by
simp only [abv_pow abv, geo... | Mathlib.Data.Complex.Exponential.129_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝⁴ : LinearOrderedField α
inst✝³ : Archimedean α
β : Type u_2
inst✝² : Ring β
inst✝¹ : Nontrivial β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
m : ℕ
| (abv x ^ m - 1) / (abv 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... | rw [← neg_div_neg_eq, neg_sub, neg_sub] | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m :=
have hx1' : abv x ≠ 1 := fun h => by simp [h, lt_irrefl] at hx1
isCauSeq_series_of_abv_isCauSeq
(by
simp only [abv_pow abv, geo... | Mathlib.Data.Complex.Exponential.129_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝⁴ : LinearOrderedField α
inst✝³ : Archimedean α
β : Type u_2
inst✝² : Ring β
inst✝¹ : Nontrivial β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
m : ℕ
| (abv x ^ m - 1) / (abv 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... | rw [← neg_div_neg_eq, neg_sub, neg_sub] | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m :=
have hx1' : abv x ≠ 1 := fun h => by simp [h, lt_irrefl] at hx1
isCauSeq_series_of_abv_isCauSeq
(by
simp only [abv_pow abv, geo... | Mathlib.Data.Complex.Exponential.129_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝⁴ : LinearOrderedField α
inst✝³ : Archimedean α
β : Type u_2
inst✝² : Ring β
inst✝¹ : Nontrivial β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
m : ℕ
| (abv x ^ m - 1) / (abv 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... | rw [← neg_div_neg_eq, neg_sub, neg_sub] | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m :=
have hx1' : abv x ≠ 1 := fun h => by simp [h, lt_irrefl] at hx1
isCauSeq_series_of_abv_isCauSeq
(by
simp only [abv_pow abv, geo... | Mathlib.Data.Complex.Exponential.129_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝⁴ : LinearOrderedField α
inst✝³ : Archimedean α
β : Type u_2
inst✝² : Ring β
inst✝¹ : Nontrivial β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
⊢ IsCauSeq abs' fun m => (1 - abv x ^ m) / (1 - abv x) | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have : 0 < 1 - abv x := sub_pos.2 hx1 | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m :=
have hx1' : abv x ≠ 1 := fun h => by simp [h, lt_irrefl] at hx1
isCauSeq_series_of_abv_isCauSeq
(by
simp only [abv_pow abv, geo... | Mathlib.Data.Complex.Exponential.129_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝⁴ : LinearOrderedField α
inst✝³ : Archimedean α
β : Type u_2
inst✝² : Ring β
inst✝¹ : Nontrivial β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
⊢ IsCauSeq abs' fun m => (1 - abv x ^ m) / (1 - abv 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... | refine' @isCauSeq_of_mono_bounded _ _ _ _ ((1 : α) / (1 - abv x)) 0 _ _ | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m :=
have hx1' : abv x ≠ 1 := fun h => by simp [h, lt_irrefl] at hx1
isCauSeq_series_of_abv_isCauSeq
(by
simp only [abv_pow abv, geo... | Mathlib.Data.Complex.Exponential.129_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m | Mathlib_Data_Complex_Exponential |
case refine'_1
α : Type u_1
inst✝⁴ : LinearOrderedField α
inst✝³ : Archimedean α
β : Type u_2
inst✝² : Ring β
inst✝¹ : Nontrivial β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
⊢ ∀ n ≥ 0, abs' ((1 - abv x ^ n) / (1 - abv x)) ≤ 1 / (1 - abv 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... | intro n _ | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m :=
have hx1' : abv x ≠ 1 := fun h => by simp [h, lt_irrefl] at hx1
isCauSeq_series_of_abv_isCauSeq
(by
simp only [abv_pow abv, geo... | Mathlib.Data.Complex.Exponential.129_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m | Mathlib_Data_Complex_Exponential |
case refine'_1
α : Type u_1
inst✝⁴ : LinearOrderedField α
inst✝³ : Archimedean α
β : Type u_2
inst✝² : Ring β
inst✝¹ : Nontrivial β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
n : ℕ
a✝ : n ≥ 0
⊢ abs' ((1 - abv x ^ n) / (1 - abv x)) ≤ 1 / (1 - abv 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 [abs_of_nonneg] | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m :=
have hx1' : abv x ≠ 1 := fun h => by simp [h, lt_irrefl] at hx1
isCauSeq_series_of_abv_isCauSeq
(by
simp only [abv_pow abv, geo... | Mathlib.Data.Complex.Exponential.129_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m | Mathlib_Data_Complex_Exponential |
case refine'_1
α : Type u_1
inst✝⁴ : LinearOrderedField α
inst✝³ : Archimedean α
β : Type u_2
inst✝² : Ring β
inst✝¹ : Nontrivial β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
n : ℕ
a✝ : n ≥ 0
⊢ (1 - abv x ^ n) / (1 - abv x) ≤ 1 / (1 - abv x)
case refine'_1
α : Ty... | /-
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 isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m :=
have hx1' : abv x ≠ 1 := fun h => by simp [h, lt_irrefl] at hx1
isCauSeq_series_of_abv_isCauSeq
(by
simp only [abv_pow abv, geo... | Mathlib.Data.Complex.Exponential.129_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m | Mathlib_Data_Complex_Exponential |
case refine'_1.h
α : Type u_1
inst✝⁴ : LinearOrderedField α
inst✝³ : Archimedean α
β : Type u_2
inst✝² : Ring β
inst✝¹ : Nontrivial β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
n : ℕ
a✝ : n ≥ 0
⊢ 1 - abv x ^ 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 sub_le_self _ (abv_pow abv x n ▸ abv_nonneg _ _) | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m :=
have hx1' : abv x ≠ 1 := fun h => by simp [h, lt_irrefl] at hx1
isCauSeq_series_of_abv_isCauSeq
(by
simp only [abv_pow abv, geo... | Mathlib.Data.Complex.Exponential.129_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m | Mathlib_Data_Complex_Exponential |
case refine'_1
α : Type u_1
inst✝⁴ : LinearOrderedField α
inst✝³ : Archimedean α
β : Type u_2
inst✝² : Ring β
inst✝¹ : Nontrivial β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
n : ℕ
a✝ : n ≥ 0
⊢ 0 ≤ (1 - abv x ^ n) / (1 - abv 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... | refine' div_nonneg (sub_nonneg.2 _) (sub_nonneg.2 <| le_of_lt hx1) | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m :=
have hx1' : abv x ≠ 1 := fun h => by simp [h, lt_irrefl] at hx1
isCauSeq_series_of_abv_isCauSeq
(by
simp only [abv_pow abv, geo... | Mathlib.Data.Complex.Exponential.129_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m | Mathlib_Data_Complex_Exponential |
case refine'_1
α : Type u_1
inst✝⁴ : LinearOrderedField α
inst✝³ : Archimedean α
β : Type u_2
inst✝² : Ring β
inst✝¹ : Nontrivial β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
n : ℕ
a✝ : n ≥ 0
⊢ abv x ^ 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 _ (by positivity) hx1.le | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m :=
have hx1' : abv x ≠ 1 := fun h => by simp [h, lt_irrefl] at hx1
isCauSeq_series_of_abv_isCauSeq
(by
simp only [abv_pow abv, geo... | Mathlib.Data.Complex.Exponential.129_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝⁴ : LinearOrderedField α
inst✝³ : Archimedean α
β : Type u_2
inst✝² : Ring β
inst✝¹ : Nontrivial β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
n : ℕ
a✝ : n ≥ 0
⊢ 0 ≤ abv 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... | positivity | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m :=
have hx1' : abv x ≠ 1 := fun h => by simp [h, lt_irrefl] at hx1
isCauSeq_series_of_abv_isCauSeq
(by
simp only [abv_pow abv, geo... | Mathlib.Data.Complex.Exponential.129_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m | Mathlib_Data_Complex_Exponential |
case refine'_2
α : Type u_1
inst✝⁴ : LinearOrderedField α
inst✝³ : Archimedean α
β : Type u_2
inst✝² : Ring β
inst✝¹ : Nontrivial β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
⊢ ∀ n ≥ 0, (1 - abv x ^ n) / (1 - abv x) ≤ (1 - abv x ^ Nat.succ n) / (1 - abv 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... | intro n _ | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m :=
have hx1' : abv x ≠ 1 := fun h => by simp [h, lt_irrefl] at hx1
isCauSeq_series_of_abv_isCauSeq
(by
simp only [abv_pow abv, geo... | Mathlib.Data.Complex.Exponential.129_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m | Mathlib_Data_Complex_Exponential |
case refine'_2
α : Type u_1
inst✝⁴ : LinearOrderedField α
inst✝³ : Archimedean α
β : Type u_2
inst✝² : Ring β
inst✝¹ : Nontrivial β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
n : ℕ
a✝ : n ≥ 0
⊢ (1 - abv x ^ n) / (1 - abv x) ≤ (1 - abv x ^ Nat.succ n) / (1 - abv 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 [← one_mul (abv x ^ n), pow_succ] | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m :=
have hx1' : abv x ≠ 1 := fun h => by simp [h, lt_irrefl] at hx1
isCauSeq_series_of_abv_isCauSeq
(by
simp only [abv_pow abv, geo... | Mathlib.Data.Complex.Exponential.129_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m | Mathlib_Data_Complex_Exponential |
case refine'_2
α : Type u_1
inst✝⁴ : LinearOrderedField α
inst✝³ : Archimedean α
β : Type u_2
inst✝² : Ring β
inst✝¹ : Nontrivial β
abv : β → α
inst✝ : IsAbsoluteValue abv
x : β
hx1 : abv x < 1
hx1' : abv x ≠ 1
this : 0 < 1 - abv x
n : ℕ
a✝ : n ≥ 0
⊢ (1 - 1 * abv x ^ n) / (1 - abv x) ≤ (1 - abv x * abv x ^ n) / (1 - ab... | /-
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 isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m :=
have hx1' : abv x ≠ 1 := fun h => by simp [h, lt_irrefl] at hx1
isCauSeq_series_of_abv_isCauSeq
(by
simp only [abv_pow abv, geo... | Mathlib.Data.Complex.Exponential.129_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series {β : Type*} [Ring β] [Nontrivial β] {abv : β → α} [IsAbsoluteValue abv]
(x : β) (hx1 : abv x < 1) : IsCauSeq abv fun n => ∑ m in range n, x ^ m | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝¹ : LinearOrderedField α
inst✝ : Archimedean α
a x : α
hx1 : abs' x < 1
⊢ IsCauSeq abs' fun m => ∑ n in range m, a * x ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | have : IsCauSeq abs fun m => a * ∑ n in range m, (x ^ n) :=
(CauSeq.const abs a *
(show CauSeq α abs from ⟨fun n => ∑ m in range n, x ^ m, isCauSeq_geo_series x hx1⟩)).2 | theorem isCauSeq_geo_series_const (a : α) {x : α} (hx1 : |x| < 1) :
IsCauSeq abs fun m => ∑ n in range m, (a * x ^ n) := by
| Mathlib.Data.Complex.Exponential.149_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series_const (a : α) {x : α} (hx1 : |x| < 1) :
IsCauSeq abs fun m => ∑ n in range m, (a * x ^ n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝¹ : LinearOrderedField α
inst✝ : Archimedean α
a x : α
hx1 : abs' x < 1
this : IsCauSeq abs' fun m => a * ∑ n in range m, x ^ n
⊢ IsCauSeq abs' fun m => ∑ n in range m, a * x ^ n | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simpa [mul_sum] using this | theorem isCauSeq_geo_series_const (a : α) {x : α} (hx1 : |x| < 1) :
IsCauSeq abs fun m => ∑ n in range m, (a * x ^ n) := by
have : IsCauSeq abs fun m => a * ∑ n in range m, (x ^ n) :=
(CauSeq.const abs a *
(show CauSeq α abs from ⟨fun n => ∑ m in range n, x ^ m, isCauSeq_geo_series x hx1⟩)).2
| Mathlib.Data.Complex.Exponential.149_0.1bixbwhfBeJKySp | theorem isCauSeq_geo_series_const (a : α) {x : α} (hx1 : |x| < 1) :
IsCauSeq abs fun m => ∑ n in range m, (a * x ^ n) | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
n : ℕ
r : α
hr0 : 0 ≤ r
hr1 : r < 1
h : ∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)
⊢ IsCauSeq abv fun m => ∑ n in range m, f 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 har1 : |r| < 1 := by rwa [abs_of_nonneg hr0] | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
| Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
n : ℕ
r : α
hr0 : 0 ≤ r
hr1 : r < 1
h : ∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)
⊢ abs' r < 1 | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | rwa [abs_of_nonneg hr0] | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
n : ℕ
r : α
hr0 : 0 ≤ r
hr1 : r < 1
h : ∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)
har1 : abs' r < 1
⊢ IsCauSeq abv fun m => ∑ n in range m, f 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... | refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_geo_series_const (abv (f n.succ) * r⁻¹ ^ n.succ) har1) | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
| Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
n : ℕ
r : α
hr0 : 0 ≤ r
hr1 : r < 1
h : ∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)
har1 : abs' r < 1
⊢ ∀ (m : ℕ), Nat.succ n ≤ m → abv (f m) ≤ abv (f (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... | intro m hmn | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
n : ℕ
r : α
hr0 : 0 ≤ r
hr1 : r < 1
h : ∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)
har1 : abs' r < 1
m : ℕ
hmn : Nat.succ n ≤ m
⊢ abv (f m) ≤ abv (f (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... | cases' Classical.em (r = 0) with r_zero r_ne_zero | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
case inl
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
n : ℕ
r : α
hr0 : 0 ≤ r
hr1 : r < 1
h : ∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)
har1 : abs' r < 1
m : ℕ
hmn : Nat.succ n ≤ m
r_zero : r = 0
⊢ abv (f 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... | have m_pos := lt_of_lt_of_le (Nat.succ_pos n) hmn | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
case inl
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
n : ℕ
r : α
hr0 : 0 ≤ r
hr1 : r < 1
h : ∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)
har1 : abs' r < 1
m : ℕ
hmn : Nat.succ n ≤ m
r_zero : r = 0
m_pos : 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... | have := h m.pred (Nat.le_of_succ_le_succ (by rwa [Nat.succ_pred_eq_of_pos m_pos])) | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
n : ℕ
r : α
hr0 : 0 ≤ r
hr1 : r < 1
h : ∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)
har1 : abs' r < 1
m : ℕ
hmn : Nat.succ n ≤ m
r_zero : r = 0
m_pos : 0 < m
⊢ Nat... | /-
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 [Nat.succ_pred_eq_of_pos m_pos] | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
case inl
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
n : ℕ
r : α
hr0 : 0 ≤ r
hr1 : r < 1
h : ∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)
har1 : abs' r < 1
m : ℕ
hmn : Nat.succ n ≤ m
r_zero : r = 0
m_pos : 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... | simpa [r_zero, Nat.succ_pred_eq_of_pos m_pos, pow_succ] using this | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
case inr
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
n : ℕ
r : α
hr0 : 0 ≤ r
hr1 : r < 1
h : ∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)
har1 : abs' r < 1
m : ℕ
hmn : Nat.succ n ≤ m
r_ne_zero : ¬r = 0
⊢ abv ... | /-
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... | generalize hk : m - n.succ = k | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
case inr
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
n : ℕ
r : α
hr0 : 0 ≤ r
hr1 : r < 1
h : ∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)
har1 : abs' r < 1
m : ℕ
hmn : Nat.succ n ≤ m
r_ne_zero : ¬r = 0
k : 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 r_pos : 0 < r := lt_of_le_of_ne hr0 (Ne.symm r_ne_zero) | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
case inr
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
n : ℕ
r : α
hr0 : 0 ≤ r
hr1 : r < 1
h : ∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)
har1 : abs' r < 1
m : ℕ
hmn : Nat.succ n ≤ m
r_ne_zero : ¬r = 0
k : 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... | replace hk : m = k + n.succ := (tsub_eq_iff_eq_add_of_le hmn).1 hk | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
case inr
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
n : ℕ
r : α
hr0 : 0 ≤ r
hr1 : r < 1
h : ∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)
har1 : abs' r < 1
m : ℕ
hmn : Nat.succ n ≤ m
r_ne_zero : ¬r = 0
k : 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... | induction' k with k ih generalizing m n | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
case inr.zero
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
r : α
hr0 : 0 ≤ r
hr1 : r < 1
har1 : abs' r < 1
r_ne_zero : ¬r = 0
r_pos : 0 < r
n : ℕ
h : ∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)
m : ℕ
hmn : Na... | /-
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 [hk, Nat.zero_add, mul_right_comm, inv_pow _ _, ← div_eq_mul_inv, mul_div_cancel] | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
case inr.zero.h
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
r : α
hr0 : 0 ≤ r
hr1 : r < 1
har1 : abs' r < 1
r_ne_zero : ¬r = 0
r_pos : 0 < r
n : ℕ
h : ∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f m)
m : ℕ
hmn : ... | /-
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 (ne_of_lt (pow_pos r_pos _)).symm | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
case inr.succ
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
r : α
hr0 : 0 ≤ r
hr1 : r < 1
har1 : abs' r < 1
r_ne_zero : ¬r = 0
r_pos : 0 < r
k : ℕ
ih :
∀ (n : ℕ),
(∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (... | /-
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 kn : k + n.succ ≥ n.succ := by
rw [← zero_add n.succ]; exact add_le_add (Nat.zero_le _) (by simp) | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
r : α
hr0 : 0 ≤ r
hr1 : r < 1
har1 : abs' r < 1
r_ne_zero : ¬r = 0
r_pos : 0 < r
k : ℕ
ih :
∀ (n : ℕ),
(∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f 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 [← zero_add n.succ] | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
r : α
hr0 : 0 ≤ r
hr1 : r < 1
har1 : abs' r < 1
r_ne_zero : ¬r = 0
r_pos : 0 < r
k : ℕ
ih :
∀ (n : ℕ),
(∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f 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... | exact add_le_add (Nat.zero_le _) (by simp) | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
r : α
hr0 : 0 ≤ r
hr1 : r < 1
har1 : abs' r < 1
r_ne_zero : ¬r = 0
r_pos : 0 < r
k : ℕ
ih :
∀ (n : ℕ),
(∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f 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 | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
case inr.succ
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
r : α
hr0 : 0 ≤ r
hr1 : r < 1
har1 : abs' r < 1
r_ne_zero : ¬r = 0
r_pos : 0 < r
k : ℕ
ih :
∀ (n : ℕ),
(∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (... | /-
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... | erw [hk, Nat.succ_add, pow_succ' r, ← mul_assoc] | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
case inr.succ
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
r : α
hr0 : 0 ≤ r
hr1 : r < 1
har1 : abs' r < 1
r_ne_zero : ¬r = 0
r_pos : 0 < r
k : ℕ
ih :
∀ (n : ℕ),
(∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (... | /-
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
le_trans (by rw [mul_comm] <;> exact h _ (Nat.le_of_succ_le kn))
(mul_le_mul_of_nonneg_right ?_ hr0) | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
r : α
hr0 : 0 ≤ r
hr1 : r < 1
har1 : abs' r < 1
r_ne_zero : ¬r = 0
r_pos : 0 < r
k : ℕ
ih :
∀ (n : ℕ),
(∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f 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_comm] | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
r : α
hr0 : 0 ≤ r
hr1 : r < 1
har1 : abs' r < 1
r_ne_zero : ¬r = 0
r_pos : 0 < r
k : ℕ
ih :
∀ (n : ℕ),
(∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f 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... | exact h _ (Nat.le_of_succ_le kn) | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
case inr.succ
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
r : α
hr0 : 0 ≤ r
hr1 : r < 1
har1 : abs' r < 1
r_ne_zero : ¬r = 0
r_pos : 0 < r
k : ℕ
ih :
∀ (n : ℕ),
(∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (... | /-
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 ih _ h _ (by simp) rfl | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
α : Type u_1
inst✝³ : LinearOrderedField α
inst✝² : Archimedean α
β : Type u_2
inst✝¹ : Ring β
abv : β → α
inst✝ : IsAbsoluteValue abv
f : ℕ → β
r : α
hr0 : 0 ≤ r
hr1 : r < 1
har1 : abs' r < 1
r_ne_zero : ¬r = 0
r_pos : 0 < r
k : ℕ
ih :
∀ (n : ℕ),
(∀ (m : ℕ), n ≤ m → abv (f (Nat.succ m)) ≤ r * abv (f 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 | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n := by
have har1 : |r| < 1 := by rwa [abs_of_nonneg hr0]
refine'
isCauSeq_series_of_abv_le_of_isCauSeq n.succ _
(isCauSeq_g... | Mathlib.Data.Complex.Exponential.159_0.1bixbwhfBeJKySp | theorem series_ratio_test {f : ℕ → β} (n : ℕ) (r : α) (hr0 : 0 ≤ r) (hr1 : r < 1)
(h : ∀ m, n ≤ m → abv (f m.succ) ≤ r * abv (f m)) :
IsCauSeq abv fun m => ∑ n in range m, f n | Mathlib_Data_Complex_Exponential |
α✝ : Type u_1
inst✝⁴ : LinearOrderedField α✝
inst✝³ : Archimedean α✝
β : Type u_2
inst✝² : Ring β
abv : β → α✝
inst✝¹ : IsAbsoluteValue abv
α : Type u_3
inst✝ : AddCommMonoid α
n : ℕ
f : ℕ → ℕ → α
⊢ ∑ m in range n, ∑ k in range (m + 1), f k (m - k) = ∑ m in range n, ∑ k in range (n - m), f m 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... | rw [sum_sigma', sum_sigma'] | theorem sum_range_diag_flip {α : Type*} [AddCommMonoid α] (n : ℕ) (f : ℕ → ℕ → α) :
(∑ m in range n, ∑ k in range (m + 1), f k (m - k)) =
∑ m in range n, ∑ k in range (n - m), f m k := by
| Mathlib.Data.Complex.Exponential.186_0.1bixbwhfBeJKySp | theorem sum_range_diag_flip {α : Type*} [AddCommMonoid α] (n : ℕ) (f : ℕ → ℕ → α) :
(∑ m in range n, ∑ k in range (m + 1), f k (m - k)) =
∑ m in range n, ∑ k in range (n - m), f m k | Mathlib_Data_Complex_Exponential |
α✝ : Type u_1
inst✝⁴ : LinearOrderedField α✝
inst✝³ : Archimedean α✝
β : Type u_2
inst✝² : Ring β
abv : β → α✝
inst✝¹ : IsAbsoluteValue abv
α : Type u_3
inst✝ : AddCommMonoid α
n : ℕ
f : ℕ → ℕ → α
⊢ ∑ x in Finset.sigma (range n) fun m => range (m + 1), f x.snd (x.fst - x.snd) =
∑ x in Finset.sigma (range n) fun 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... | exact
sum_bij (fun a _ => ⟨a.2, a.1 - a.2⟩)
(fun a ha =>
have h₁ : a.1 < n := mem_range.1 (mem_sigma.1 ha).1
have h₂ : a.2 < Nat.succ a.1 := mem_range.1 (mem_sigma.1 ha).2
mem_sigma.2
⟨mem_range.2 (lt_of_lt_of_le h₂ h₁),
mem_range.2 ((tsub_lt_tsub_iff_right (Nat.l... | theorem sum_range_diag_flip {α : Type*} [AddCommMonoid α] (n : ℕ) (f : ℕ → ℕ → α) :
(∑ m in range n, ∑ k in range (m + 1), f k (m - k)) =
∑ m in range n, ∑ k in range (n - m), f m k := by
rw [sum_sigma', sum_sigma']
| Mathlib.Data.Complex.Exponential.186_0.1bixbwhfBeJKySp | theorem sum_range_diag_flip {α : Type*} [AddCommMonoid α] (n : ℕ) (f : ℕ → ℕ → α) :
(∑ m in range n, ∑ k in range (m + 1), f k (m - k)) =
∑ m in range n, ∑ k in range (n - m), f m k | Mathlib_Data_Complex_Exponential |
α✝ : Type u_1
inst✝⁴ : LinearOrderedField α✝
inst✝³ : Archimedean α✝
β : Type u_2
inst✝² : Ring β
abv : β → α✝
inst✝¹ : IsAbsoluteValue abv
α : Type u_3
inst✝ : AddCommMonoid α
n : ℕ
f : ℕ → ℕ → α
x✝¹ x✝ : (_ : ℕ) × ℕ
a₁ a₂ : ℕ
ha✝ : { fst := a₁, snd := a₂ } ∈ Finset.sigma (range n) fun m => range (m + 1)
b₁ b₂ : ℕ
hb✝... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simpa using h | theorem sum_range_diag_flip {α : Type*} [AddCommMonoid α] (n : ℕ) (f : ℕ → ℕ → α) :
(∑ m in range n, ∑ k in range (m + 1), f k (m - k)) =
∑ m in range n, ∑ k in range (n - m), f m k := by
rw [sum_sigma', sum_sigma']
exact
sum_bij (fun a _ => ⟨a.2, a.1 - a.2⟩)
(fun a ha =>
have h₁ : a.1 <... | Mathlib.Data.Complex.Exponential.186_0.1bixbwhfBeJKySp | theorem sum_range_diag_flip {α : Type*} [AddCommMonoid α] (n : ℕ) (f : ℕ → ℕ → α) :
(∑ m in range n, ∑ k in range (m + 1), f k (m - k)) =
∑ m in range n, ∑ k in range (n - m), f m k | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Semiring β
inst✝ : IsAbsoluteValue abv
γ : Type u_3
f : γ → β
s : Finset γ
this : DecidableEq γ
⊢ abv (∑ k in ∅, f k) ≤ ∑ k in ∅, abv (f 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... | simp [abv_zero abv] | theorem abv_sum_le_sum_abv {γ : Type*} (f : γ → β) (s : Finset γ) :
abv (∑ k in s, f k) ≤ ∑ k in s, abv (f k) :=
haveI := Classical.decEq γ
Finset.induction_on s (by | Mathlib.Data.Complex.Exponential.227_0.1bixbwhfBeJKySp | theorem abv_sum_le_sum_abv {γ : Type*} (f : γ → β) (s : Finset γ) :
abv (∑ k in s, f k) ≤ ∑ k in s, abv (f k) | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Semiring β
inst✝ : IsAbsoluteValue abv
γ : Type u_3
f : γ → β
s✝ : Finset γ
this : DecidableEq γ
a : γ
s : Finset γ
has : a ∉ s
ih : abv (∑ k in s, f k) ≤ ∑ k in s, abv (f k)
⊢ abv (∑ k in insert a s, f k) ≤ ∑ k in insert a s, abv (f 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... | rw [sum_insert has, sum_insert has] | theorem abv_sum_le_sum_abv {γ : Type*} (f : γ → β) (s : Finset γ) :
abv (∑ k in s, f k) ≤ ∑ k in s, abv (f k) :=
haveI := Classical.decEq γ
Finset.induction_on s (by simp [abv_zero abv]) fun a s has ih => by
| Mathlib.Data.Complex.Exponential.227_0.1bixbwhfBeJKySp | theorem abv_sum_le_sum_abv {γ : Type*} (f : γ → β) (s : Finset γ) :
abv (∑ k in s, f k) ≤ ∑ k in s, abv (f k) | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Semiring β
inst✝ : IsAbsoluteValue abv
γ : Type u_3
f : γ → β
s✝ : Finset γ
this : DecidableEq γ
a : γ
s : Finset γ
has : a ∉ s
ih : abv (∑ k in s, f k) ≤ ∑ k in s, abv (f k)
⊢ abv (f a + ∑ x in s, f x) ≤ abv (f a) + ∑ x in s, abv (f 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 le_trans (abv_add abv _ _) (add_le_add_left ih _) | theorem abv_sum_le_sum_abv {γ : Type*} (f : γ → β) (s : Finset γ) :
abv (∑ k in s, f k) ≤ ∑ k in s, abv (f k) :=
haveI := Classical.decEq γ
Finset.induction_on s (by simp [abv_zero abv]) fun a s has ih => by
rw [sum_insert has, sum_insert has]; | Mathlib.Data.Complex.Exponential.227_0.1bixbwhfBeJKySp | theorem abv_sum_le_sum_abv {γ : Type*} (f : γ → β) (s : Finset γ) :
abv (∑ k in s, f k) ≤ ∑ k in s, abv (f k) | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | norm_num | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | norm_num | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 :
(∑ m in range K, ∑ k in range (m + 1), a k * b (m - k)) =
∑ m in range K, ∑ n in range (K - m), a m * b n :=
by simpa using sum_range_diag_flip K fun m n => a m * b n | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simpa using sum_range_diag_flip K fun m n => a m * b n | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 :
(fun i => ∑ k in range (K - i), a i * b k) = fun i => a i * ∑ k in range (K - i), b k := by
simp [Finset.mul_sum] | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 [Finset.mul_sum] | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 :
(∑ i in range K, a i * ∑ k in range (K - i), b k) =
(∑ i in range K, a i * ((∑ k in range (K - i), b k) - ∑ k in range K, b k)) +
∑ i in range K, a i * ∑ k in range K, b k :=
by rw [← sum_add_distrib]; simp [(mul_add _ _ _).symm] | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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_add_distrib] | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simp [(mul_add _ _ _).symm] | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 two_mul_two : (4 : α) = 2 * 2 := by norm_num | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | norm_num | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 hQ0 : Q ≠ 0 := fun h => by simp [h, lt_irrefl] at hQε0 | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 [h, lt_irrefl] at hQε0 | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 h2Q0 : 2 * Q ≠ 0 := mul_ne_zero two_ne_zero hQ0 | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 hε : ε / (2 * P) * P + ε / (4 * Q) * (2 * Q) = ε := by
rw [← div_div, div_mul_cancel _ (Ne.symm (ne_of_lt hP0)), two_mul_two, mul_assoc, ← div_div,
div_mul_cancel _ h2Q0, add_halves] | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 [← div_div, div_mul_cancel _ (Ne.symm (ne_of_lt hP0)), two_mul_two, mul_assoc, ← div_div,
div_mul_cancel _ h2Q0, add_halves] | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 hNMK : max N M + 1 < K :=
lt_of_lt_of_le (by rw [two_mul]; exact lt_add_of_pos_left _ (Nat.succ_pos _)) hK | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 [two_mul] | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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_add_of_pos_left _ (Nat.succ_pos _) | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 hKN : N < K :=
calc
N ≤ max N M := le_max_left _ _
_ < max N M + 1 := (Nat.lt_succ_self _)
_ < K := hNMK | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 hsumlesum :
(∑ i in range (max N M + 1),
abv (a i) * abv ((∑ k in range (K - i), b k) - ∑ k in range K, b k)) ≤
∑ i in range (max N M + 1), abv (a i) * (ε / (2 * P)) | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
case hsumlesum
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | gcongr with m hmJ | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
case hsumlesum.h.h
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range ... | /-
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 le_of_lt
(hN (K - m)
(le_tsub_of_add_le_left
(le_trans
(by
rw [two_mul]
gcongr
· exact le_of_lt (mem_range.1 hmJ)
· exact le_trans (le_max_left _ _) (le_of_lt (lt_add_one _))... | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 [two_mul] | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
case h₁
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, pro... | /-
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 le_of_lt (mem_range.1 hmJ) | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
case h₂
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, pro... | /-
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 le_trans (le_max_left _ _) (le_of_lt (lt_add_one _)) | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 hsumltP : (∑ n in range (max N M + 1), abv (a n)) < P :=
calc
(∑ n in range (max N M + 1), abv (a n)) = |∑ n in range (max N M + 1), abv (a n)| :=
Eq.symm (abs_of_nonneg (sum_nonneg fun x _ => abv_nonneg abv (a x)))
_ < P := hP (max N M + 1) | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 [h₁, h₂, h₃, sum_mul, ← sub_sub, sub_right_comm, sub_self, zero_sub, abv_neg abv] | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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' lt_of_le_of_lt (abv_sum_le_sum_abv _ _) _ | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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... | suffices
(∑ i in range (max N M + 1),
abv (a i) * abv ((∑ k in range (K - i), b k) - ∑ k in range K, b k)) +
((∑ i in range K, abv (a i) * abv ((∑ k in range (K - i), b k) - ∑ k in range K, b k)) -
∑ i in range (max N M + 1),
abv (a i) * abv ((∑ k in range (K - i), ... | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 [hε] at this | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#ali... | simpa [abv_mul abv] using this | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
case h₁
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, pro... | /-
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_of_le_of_lt hsumlesum
(by rw [← sum_mul, mul_comm]; gcongr) | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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_mul, mul_comm] | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
case h₂
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, pro... | /-
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 (le_of_lt hNMK)] | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
case h₂
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, pro... | /-
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
(∑ i in (range K).filter fun k => max N M + 1 ≤ k,
abv (a i) * abv ((∑ k in range (K - i), b k) - ∑ k in range K, b k)) ≤
∑ i in (range K).filter fun k => max N M + 1 ≤ k, abv (a i) * (2 * Q) := by
gcongr
rw [sub_eq_add_neg]
refine' le_trans (abv_add _ _ _)... | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
case h.h
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, pr... | /-
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_eq_add_neg] | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
case h.h
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, pr... | /-
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' le_trans (abv_add _ _ _) _ | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
case h.h
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, pr... | /-
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 [two_mul, abv_neg abv] | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
case h.h
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, pr... | /-
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 cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
case h.h.h₁
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b 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... | exact le_of_lt (hQ _) | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
case h.h.h₂
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b 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... | exact le_of_lt (hQ _) | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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_mul, ← sum_range_sub_sum_range (le_of_lt hNMK)] | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 := lt_of_le_of_lt (abv_nonneg _ _) (hQ 0) | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, property :=... | /-
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 cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
case bc
α : Type u_1
β : Type u_2
inst✝² : LinearOrderedField α
abv : β → α
inst✝¹ : Ring β
inst✝ : IsAbsoluteValue abv
a b : ℕ → β
ha : IsCauSeq abs' fun m => ∑ n in range m, abv (a n)
hb : IsCauSeq abv fun m => ∑ n in range m, b n
ε : α
ε0 : 0 < ε
Q : α
hQ : ∀ (i : ℕ), abv (↑{ val := fun m => ∑ n in range m, b n, pro... | /-
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'
lt_of_le_of_lt (le_abs_self _)
(hM _ (le_trans (Nat.le_succ_of_le (le_max_right _ _)) (le_of_lt hNMK)) _
(Nat.le_succ_of_le (le_max_right _ _))) | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib.Data.Complex.Exponential.240_0.1bixbwhfBeJKySp | theorem cauchy_product {a b : ℕ → β} (ha : IsCauSeq abs fun m => ∑ n in range m, abv (a n))
(hb : IsCauSeq abv fun m => ∑ n in range m, b n) (ε : α) (ε0 : 0 < ε) :
∃ i : ℕ,
∀ j ≥ i,
abv
(((∑ k in range j, a k) * ∑ k in range j, b k) -
∑ n in range j, ∑ m in range (n + 1),... | Mathlib_Data_Complex_Exponential |
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