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 |
|---|---|---|---|---|---|---|
case hx
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S x y : ℝ
hy : x + R ≤ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
⊢ 0 ≤ x + R | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | positivity | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case hx
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S x y : ℝ
hy : x + R ≤ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
⊢ 0 ≤ y | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | positivity | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S x y : ℝ
hy : x + R ≤ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
this : y ^ (-b) ≤ (x + R) ^ (-b)
⊢ y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | refine' this.trans _ | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S x y : ℝ
hy : x + R ≤ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
this : y ^ (-b) ≤ (x + R) ^ (-b)
⊢ (x + R) ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | rw [← mul_rpow, rpow_neg, rpow_neg] | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S x y : ℝ
hy : x + R ≤ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
this : y ^ (-b) ≤ (x + R) ^ (-b)
⊢ ((x + R) ^ b)⁻¹ ≤ ((1 / 2 * x) ^ b)⁻¹ | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | gcongr | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.h.h₁
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S x y : ℝ
hy : x + R ≤ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
this : y ^ (-b) ≤ (x + R) ^ (-b)
⊢ 1 / 2 * x ≤ x + R | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | linarith | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.hx
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S x y : ℝ
hy : x + R ≤ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
this : y ^ (-b) ≤ (x + R) ^ (-b)
⊢ 0 ≤ 1 / 2 * x
case intro.hx
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ,... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | all_goals positivity | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.hx
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S x y : ℝ
hy : x + R ≤ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
this : y ^ (-b) ≤ (x + R) ^ (-b)
⊢ 0 ≤ 1 / 2 * x | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | positivity | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.hx
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S x y : ℝ
hy : x + R ≤ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
this : y ^ (-b) ≤ (x + R) ^ (-b)
⊢ 0 ≤ x + R | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | positivity | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.hx
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S x y : ℝ
hy : x + R ≤ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
this : y ^ (-b) ≤ (x + R) ^ (-b)
⊢ 0 ≤ 1 / 2 | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | positivity | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.hy
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S x y : ℝ
hy : x + R ≤ y
hx1 : 0 < x
hx2 : -2 * R < x
hxR : 0 < x + R
hy' : 0 < y
this : y ^ (-b) ≤ (x + R) ^ (-b)
⊢ 0 ≤ x | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | positivity | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
⊢ (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atTop] fun x => |x| ^ (-b) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | obtain ⟨c, hc, hc'⟩ := hf.exists_pos | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
hc' : IsBigOWith c atTop ⇑f fun x => |x| ^ (-b)
⊢ (fun x => ‖Continuo... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | simp only [IsBigO, IsBigOWith, eventually_atTop] at hc' ⊢ | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
hc' : ∃ a, ∀ b_1 ≥ a, ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖
⊢ ∃ c a, ∀ b_1 ≥ a... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | obtain ⟨d, hd⟩ := hc' | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
d : ℝ
hd : ∀ b_1 ≥ d, ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖
⊢ ∃ c a, ∀ b... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | refine' ⟨c * (1 / 2) ^ (-b), ⟨max (1 + max 0 (-2 * R)) (d - R), fun x hx => _⟩⟩ | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
d : ℝ
hd : ∀ b_1 ≥ d, ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖
x : ℝ
hx : x... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | rw [ge_iff_le, max_le_iff] at hx | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
d : ℝ
hd : ∀ b_1 ≥ d, ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖
x : ℝ
hx : 1... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | have hx' : max 0 (-2 * R) < x := by linarith | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
d : ℝ
hd : ∀ b_1 ≥ d, ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖
x : ℝ
hx : 1 + max 0 (-2 * R) ≤ x ∧... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | linarith | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
d : ℝ
hd : ∀ b_1 ≥ d, ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖
x : ℝ
hx : 1... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | rw [max_lt_iff] at hx' | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
d : ℝ
hd : ∀ b_1 ≥ d, ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖
x : ℝ
hx : 1... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | rw [norm_norm,
ContinuousMap.norm_le _
(mul_nonneg (mul_nonneg hc.le <| rpow_nonneg_of_nonneg one_half_pos.le _) (norm_nonneg _))] | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
d : ℝ
hd : ∀ b_1 ≥ d, ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖
x : ℝ
hx : 1... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | refine' fun y => (hd y.1 (by linarith [hx.1, y.2.1])).trans _ | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
d : ℝ
hd : ∀ b_1 ≥ d, ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖
x : ℝ
hx : 1 + max 0 (-2 * R) ≤ x ∧... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | linarith [hx.1, y.2.1] | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
d : ℝ
hd : ∀ b_1 ≥ d, ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖
x : ℝ
hx : 1... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | have A : ∀ x : ℝ, 0 ≤ |x| ^ (-b) := fun x => by positivity | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
d : ℝ
hd : ∀ b_1 ≥ d, ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖
x✝ : ℝ
hx : 1 + max 0 (-2 * R) ≤ x✝... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | positivity | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
d : ℝ
hd : ∀ b_1 ≥ d, ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖
x : ℝ
hx : 1... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | rw [mul_assoc, mul_le_mul_left hc, norm_of_nonneg (A _), norm_of_nonneg (A _)] | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
d : ℝ
hd : ∀ b_1 ≥ d, ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖
x : ℝ
hx : 1... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | convert claim x (by linarith only [hx.1]) y.1 y.2.1 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
d : ℝ
hd : ∀ b_1 ≥ d, ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖
x : ℝ
hx : 1 + max 0 (-2 * R) ≤ x ∧... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | linarith only [hx.1] | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case h.e'_3.h.e'_5
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
d : ℝ
hd : ∀ b_1 ≥ d, ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖
x : ℝ
hx : 1 + m... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | apply abs_of_nonneg | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case h.e'_3.h.e'_5.h
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
d : ℝ
hd : ∀ b_1 ≥ d, ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖
x : ℝ
hx : 1 +... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | linarith [y.2.1] | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
case h.e'_4.h.e'_6.h.e'_5
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atTop] fun x => |x| ^ (-b)
R S : ℝ
claim : ∀ (x : ℝ), max 0 (-2 * R) < x → ∀ (y : ℝ), x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b)
c : ℝ
hc : c > 0
d : ℝ
hd : ∀ b_1 ≥ d, ‖f b_1‖ ≤ c * ‖|b_1| ^ (-b)‖
x : ℝ
hx ... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | exact abs_of_pos hx'.1 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib.Analysis.Fourier.PoissonSummation.131_0.1MbUAOzT9Ye0D34 | /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc... | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
⊢ (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | have h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ^ (-b) := by
convert hf.comp_tendsto tendsto_neg_atTop_atBot using 1
ext1 x; simp only [Function.comp_apply, abs_neg] | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
| Mathlib.Analysis.Fourier.PoissonSummation.178_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
⊢ ⇑(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | convert hf.comp_tendsto tendsto_neg_atTop_atBot using 1 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
have h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ... | Mathlib.Analysis.Fourier.PoissonSummation.178_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case h.e'_8
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
⊢ (fun x => |x| ^ (-b)) = (fun x => |x| ^ (-b)) ∘ Neg.neg | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | ext1 x | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
have h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ... | Mathlib.Analysis.Fourier.PoissonSummation.178_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case h.e'_8.h
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S x : ℝ
⊢ |x| ^ (-b) = ((fun x => |x| ^ (-b)) ∘ Neg.neg) x | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | simp only [Function.comp_apply, abs_neg] | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
have h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ... | Mathlib.Analysis.Fourier.PoissonSummation.178_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)
⊢ (fun x => ‖ContinuousMap.restrict (Icc (x + R) (x + S)) f‖) =O[atBot] fun x => |x| ^ (-b) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | have h2 := (isBigO_norm_Icc_restrict_atTop hb h1 (-S) (-R)).comp_tendsto tendsto_neg_atBot_atTop | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
have h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ... | Mathlib.Analysis.Fourier.PoissonSummation.178_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)
h2 :
((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a =... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | have : (fun x : ℝ => |x| ^ (-b)) ∘ Neg.neg = fun x : ℝ => |x| ^ (-b) := by
ext1 x; simp only [Function.comp_apply, abs_neg] | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
have h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ... | Mathlib.Analysis.Fourier.PoissonSummation.178_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)
h2 :
((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a =... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | ext1 x | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
have h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ... | Mathlib.Analysis.Fourier.PoissonSummation.178_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case h
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)
h2 :
((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk ... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | simp only [Function.comp_apply, abs_neg] | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
have h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ... | Mathlib.Analysis.Fourier.PoissonSummation.178_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)
h2 :
((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a =... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | rw [this] at h2 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
have h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ... | Mathlib.Analysis.Fourier.PoissonSummation.178_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)
h2 :
((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a =... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | refine' (isBigO_of_le _ fun x => _).trans h2 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
have h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ... | Mathlib.Analysis.Fourier.PoissonSummation.178_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)
h2 :
((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a =... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | rw [norm_norm, Function.comp_apply, norm_norm, ContinuousMap.norm_le _ (norm_nonneg _)] | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
have h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ... | Mathlib.Analysis.Fourier.PoissonSummation.178_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)
h2 :
((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a =... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | rintro ⟨x, hx⟩ | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
have h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ... | Mathlib.Analysis.Fourier.PoissonSummation.178_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case mk
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)
h2 :
((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | rw [ContinuousMap.restrict_apply_mk] | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
have h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ... | Mathlib.Analysis.Fourier.PoissonSummation.178_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case mk
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)
h2 :
((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | refine' (le_of_eq _).trans (ContinuousMap.norm_coe_le_norm _ ⟨-x, _⟩) | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
have h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ... | Mathlib.Analysis.Fourier.PoissonSummation.178_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case mk.refine'_1
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)
h2 :
((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (Contin... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | rw [ContinuousMap.restrict_apply_mk, ContinuousMap.comp_apply, ContinuousMap.coe_mk,
ContinuousMap.coe_mk, neg_neg] | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
have h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ... | Mathlib.Analysis.Fourier.PoissonSummation.178_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case mk.refine'_2
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)
h2 :
((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (Contin... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | exact ⟨by linarith [hx.2], by linarith [hx.1]⟩ | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
have h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ... | Mathlib.Analysis.Fourier.PoissonSummation.178_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)
h2 :
((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a =... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | linarith [hx.2] | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
have h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ... | Mathlib.Analysis.Fourier.PoissonSummation.178_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[atBot] fun x => |x| ^ (-b)
R S : ℝ
h1 : ⇑(ContinuousMap.comp f (ContinuousMap.mk fun a => -a)) =O[atTop] fun x => |x| ^ (-b)
h2 :
((fun x => ‖ContinuousMap.restrict (Icc (x + -S) (x + -R)) (ContinuousMap.comp f (ContinuousMap.mk fun a =... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | linarith [hx.1] | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
have h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ... | Mathlib.Analysis.Fourier.PoissonSummation.178_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
(hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[cocompact ℝ] fun x => |x| ^ (-b)
K : Compacts ℝ
⊢ (fun x => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight x))‖) =O[cocompact ℝ] fun x =>
|x| ^ (-b) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_closedBall 0 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) := by
| Mathlib.Analysis.Fourier.PoissonSummation.200_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case intro
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[cocompact ℝ] fun x => |x| ^ (-b)
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Metric.closedBall 0 r
⊢ (fun x => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight x))‖) =O[cocompact ℝ] fun x =>
|x| ^ (-b) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | rw [closedBall_eq_Icc, zero_add, zero_sub] at hr | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) := by
obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_clo... | Mathlib.Analysis.Fourier.PoissonSummation.200_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case intro
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[cocompact ℝ] fun x => |x| ^ (-b)
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
⊢ (fun x => ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight x))‖) =O[cocompact ℝ] fun x =>
|x| ^ (-b) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | have :
∀ x : ℝ,
‖(f.comp (ContinuousMap.addRight x)).restrict K‖ ≤ ‖f.restrict (Icc (x - r) (x + r))‖ := by
intro x
rw [ContinuousMap.norm_le _ (norm_nonneg _)]
rintro ⟨y, hy⟩
refine' (le_of_eq _).trans (ContinuousMap.norm_coe_le_norm _ ⟨y + x, _⟩)
· simp_rw [ContinuousMap.restrict_apply, ... | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) := by
obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_clo... | Mathlib.Analysis.Fourier.PoissonSummation.200_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[cocompact ℝ] fun x => |x| ^ (-b)
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
⊢ ∀ (x : ℝ),
‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight x))‖ ≤
‖ContinuousMap.restrict (Icc (x - r) (x + r)) f‖ | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | intro x | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) := by
obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_clo... | Mathlib.Analysis.Fourier.PoissonSummation.200_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[cocompact ℝ] fun x => |x| ^ (-b)
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
x : ℝ
⊢ ‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight x))‖ ≤
‖ContinuousMap.restrict (Icc (x - r) (x + r)) f‖ | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | rw [ContinuousMap.norm_le _ (norm_nonneg _)] | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) := by
obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_clo... | Mathlib.Analysis.Fourier.PoissonSummation.200_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[cocompact ℝ] fun x => |x| ^ (-b)
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
x : ℝ
⊢ ∀ (x_1 : ↑↑K),
‖(ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight x))) x_1‖ ≤
‖ContinuousMap.restrict (Icc (x - r) (x + ... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | rintro ⟨y, hy⟩ | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) := by
obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_clo... | Mathlib.Analysis.Fourier.PoissonSummation.200_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case mk
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[cocompact ℝ] fun x => |x| ^ (-b)
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
x y : ℝ
hy : y ∈ ↑K
⊢ ‖(ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight x))) { val := y, property := hy }‖ ≤
‖ContinuousMap.r... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | refine' (le_of_eq _).trans (ContinuousMap.norm_coe_le_norm _ ⟨y + x, _⟩) | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) := by
obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_clo... | Mathlib.Analysis.Fourier.PoissonSummation.200_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case mk.refine'_1
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[cocompact ℝ] fun x => |x| ^ (-b)
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
x y : ℝ
hy : y ∈ ↑K
⊢ ‖(ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight x))) { val := y, property := hy }‖ =
‖(Cont... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | simp_rw [ContinuousMap.restrict_apply, ContinuousMap.comp_apply, ContinuousMap.coe_addRight] | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) := by
obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_clo... | Mathlib.Analysis.Fourier.PoissonSummation.200_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case mk.refine'_2
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[cocompact ℝ] fun x => |x| ^ (-b)
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
x y : ℝ
hy : y ∈ ↑K
⊢ y + x ∈ Icc (x - r) (x + r) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | exact ⟨by linarith [(hr hy).1], by linarith [(hr hy).2]⟩ | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) := by
obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_clo... | Mathlib.Analysis.Fourier.PoissonSummation.200_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[cocompact ℝ] fun x => |x| ^ (-b)
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
x y : ℝ
hy : y ∈ ↑K
⊢ x - r ≤ y + x | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | linarith [(hr hy).1] | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) := by
obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_clo... | Mathlib.Analysis.Fourier.PoissonSummation.200_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[cocompact ℝ] fun x => |x| ^ (-b)
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
x y : ℝ
hy : y ∈ ↑K
⊢ y + x ≤ x + r | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | linarith [(hr hy).2] | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) := by
obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_clo... | Mathlib.Analysis.Fourier.PoissonSummation.200_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case intro
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
hf : ⇑f =O[cocompact ℝ] fun x => |x| ^ (-b)
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
this :
∀ (x : ℝ),
‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight x))‖ ≤
‖ContinuousMap.restrict (Icc (x - r) (x ... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | simp_rw [cocompact_eq, isBigO_sup] at hf ⊢ | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) := by
obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_clo... | Mathlib.Analysis.Fourier.PoissonSummation.200_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case intro
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
this :
∀ (x : ℝ),
‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight x))‖ ≤
‖ContinuousMap.restrict (Icc (x - r) (x + r)) f‖
hf : (⇑f =O[atBot] fun x => |x| ^ (... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | constructor | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) := by
obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_clo... | Mathlib.Analysis.Fourier.PoissonSummation.200_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.left
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
this :
∀ (x : ℝ),
‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight x))‖ ≤
‖ContinuousMap.restrict (Icc (x - r) (x + r)) f‖
hf : (⇑f =O[atBot] fun x => |x... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | refine' (isBigO_of_le atBot _).trans (isBigO_norm_Icc_restrict_atBot hb hf.1 (-r) r) | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) := by
obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_clo... | Mathlib.Analysis.Fourier.PoissonSummation.200_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.left
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
this :
∀ (x : ℝ),
‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight x))‖ ≤
‖ContinuousMap.restrict (Icc (x - r) (x + r)) f‖
hf : (⇑f =O[atBot] fun x => |x... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | simp_rw [norm_norm] | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) := by
obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_clo... | Mathlib.Analysis.Fourier.PoissonSummation.200_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.left
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
this :
∀ (x : ℝ),
‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight x))‖ ≤
‖ContinuousMap.restrict (Icc (x - r) (x + r)) f‖
hf : (⇑f =O[atBot] fun x => |x... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | exact this | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) := by
obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_clo... | Mathlib.Analysis.Fourier.PoissonSummation.200_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.right
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
this :
∀ (x : ℝ),
‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight x))‖ ≤
‖ContinuousMap.restrict (Icc (x - r) (x + r)) f‖
hf : (⇑f =O[atBot] fun x => |... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | refine' (isBigO_of_le atTop _).trans (isBigO_norm_Icc_restrict_atTop hb hf.2 (-r) r) | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) := by
obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_clo... | Mathlib.Analysis.Fourier.PoissonSummation.200_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.right
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
this :
∀ (x : ℝ),
‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight x))‖ ≤
‖ContinuousMap.restrict (Icc (x - r) (x + r)) f‖
hf : (⇑f =O[atBot] fun x => |... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | simp_rw [norm_norm] | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) := by
obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_clo... | Mathlib.Analysis.Fourier.PoissonSummation.200_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
case intro.right
E : Type u_1
inst✝ : NormedAddCommGroup E
f : C(ℝ, E)
b : ℝ
hb : 0 < b
K : Compacts ℝ
r : ℝ
hr : ↑K ⊆ Icc (-r) r
this :
∀ (x : ℝ),
‖ContinuousMap.restrict (↑K) (ContinuousMap.comp f (ContinuousMap.addRight x))‖ ≤
‖ContinuousMap.restrict (Icc (x - r) (x + r)) f‖
hf : (⇑f =O[atBot] fun x => |... | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | exact this | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) := by
obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_clo... | Mathlib.Analysis.Fourier.PoissonSummation.200_0.1MbUAOzT9Ye0D34 | theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
(hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
|x| ^ (-b) | Mathlib_Analysis_Fourier_PoissonSummation |
f g : SchwartzMap ℝ ℂ
hfg : 𝓕 ⇑f = ⇑g
⊢ ∑' (n : ℤ), f ↑n = ∑' (n : ℤ), g ↑n | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | simp_rw [← hfg] | /-- **Poisson's summation formula** for Schwartz functions. -/
theorem SchwartzMap.tsum_eq_tsum_fourierIntegral (f g : SchwartzMap ℝ ℂ) (hfg : 𝓕 f = g) :
∑' n : ℤ, f n = (∑' n : ℤ, g n) := by
-- We know that Schwartz functions are `O(‖x ^ (-b)‖)` for *every* `b`; for this argument we take
-- `b = 2` and work w... | Mathlib.Analysis.Fourier.PoissonSummation.253_0.1MbUAOzT9Ye0D34 | /-- **Poisson's summation formula** for Schwartz functions. -/
theorem SchwartzMap.tsum_eq_tsum_fourierIntegral (f g : SchwartzMap ℝ ℂ) (hfg : 𝓕 f = g) :
∑' n : ℤ, f n = (∑' n : ℤ, g n) | Mathlib_Analysis_Fourier_PoissonSummation |
f g : SchwartzMap ℝ ℂ
hfg : 𝓕 ⇑f = ⇑g
⊢ ∑' (n : ℤ), f ↑n = ∑' (n : ℤ), 𝓕 ⇑f ↑n | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | rw [Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay f.continuous one_lt_two
(f.isBigO_cocompact_rpow (-2))] | /-- **Poisson's summation formula** for Schwartz functions. -/
theorem SchwartzMap.tsum_eq_tsum_fourierIntegral (f g : SchwartzMap ℝ ℂ) (hfg : 𝓕 f = g) :
∑' n : ℤ, f n = (∑' n : ℤ, g n) := by
-- We know that Schwartz functions are `O(‖x ^ (-b)‖)` for *every* `b`; for this argument we take
-- `b = 2` and work w... | Mathlib.Analysis.Fourier.PoissonSummation.253_0.1MbUAOzT9Ye0D34 | /-- **Poisson's summation formula** for Schwartz functions. -/
theorem SchwartzMap.tsum_eq_tsum_fourierIntegral (f g : SchwartzMap ℝ ℂ) (hfg : 𝓕 f = g) :
∑' n : ℤ, f n = (∑' n : ℤ, g n) | Mathlib_Analysis_Fourier_PoissonSummation |
f g : SchwartzMap ℝ ℂ
hfg : 𝓕 ⇑f = ⇑g
⊢ 𝓕 ⇑f =O[cocompact ℝ] fun x => |x| ^ (-2) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | rw [hfg] | /-- **Poisson's summation formula** for Schwartz functions. -/
theorem SchwartzMap.tsum_eq_tsum_fourierIntegral (f g : SchwartzMap ℝ ℂ) (hfg : 𝓕 f = g) :
∑' n : ℤ, f n = (∑' n : ℤ, g n) := by
-- We know that Schwartz functions are `O(‖x ^ (-b)‖)` for *every* `b`; for this argument we take
-- `b = 2` and work w... | Mathlib.Analysis.Fourier.PoissonSummation.253_0.1MbUAOzT9Ye0D34 | /-- **Poisson's summation formula** for Schwartz functions. -/
theorem SchwartzMap.tsum_eq_tsum_fourierIntegral (f g : SchwartzMap ℝ ℂ) (hfg : 𝓕 f = g) :
∑' n : ℤ, f n = (∑' n : ℤ, g n) | Mathlib_Analysis_Fourier_PoissonSummation |
f g : SchwartzMap ℝ ℂ
hfg : 𝓕 ⇑f = ⇑g
⊢ ⇑g =O[cocompact ℝ] fun x => |x| ^ (-2) | /-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Distribution.Schwar... | exact g.isBigO_cocompact_rpow (-2) | /-- **Poisson's summation formula** for Schwartz functions. -/
theorem SchwartzMap.tsum_eq_tsum_fourierIntegral (f g : SchwartzMap ℝ ℂ) (hfg : 𝓕 f = g) :
∑' n : ℤ, f n = (∑' n : ℤ, g n) := by
-- We know that Schwartz functions are `O(‖x ^ (-b)‖)` for *every* `b`; for this argument we take
-- `b = 2` and work w... | Mathlib.Analysis.Fourier.PoissonSummation.253_0.1MbUAOzT9Ye0D34 | /-- **Poisson's summation formula** for Schwartz functions. -/
theorem SchwartzMap.tsum_eq_tsum_fourierIntegral (f g : SchwartzMap ℝ ℂ) (hfg : 𝓕 f = g) :
∑' n : ℤ, f n = (∑' n : ℤ, g n) | Mathlib_Analysis_Fourier_PoissonSummation |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | change Icc a b ⊆ { x | f x ≤ B x } | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | set s := { x | f x ≤ B x } ∩ Icc a b | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | have : IsClosed s := by
simp only [inter_comm]
exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | simp only [inter_comm] | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le' | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | apply this.Icc_subset_of_forall_exists_gt ha | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
case intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : Continuo... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | cases' hxB.lt_or_eq with hxB hxB | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
case intro.inl
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : Cont... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | refine' nonempty_of_mem (inter_mem _ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩)) | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
case intro.inl
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : Cont... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x :=
A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB) | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
case intro.inl
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : Cont... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
case intro.inl
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : Cont... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | exact this.mono fun y => le_of_lt | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
case intro.inr
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : Cont... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩ | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
case intro.inr.intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
hf' : ∀ x ∈ Ico a b, ∀ (r : ℝ), f' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r
B B' : ℝ → ℝ
ha : f a ≤ B... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | specialize hf' x xab r hfr | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
case intro.inr.intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (I... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z :=
(hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici
(Ioi_mem_nhds hrB) | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
case intro.inr.intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (I... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | exact (hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
case intro.inr.intro.intro.intro.intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWi... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | refine' ⟨z, _, hz⟩ | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
case intro.inr.intro.intro.intro.intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWi... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | have := (hfz.trans hzB).le | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
case intro.inr.intro.intro.intro.intro.intro
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f f' : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWi... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB,
sub_le_sub_iff_right] at this | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.84_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b,... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by
apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound
· rwa [sub_self, mul_zero, add_zero]
· exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const))
· intro x hx
exact (hB' x hx).add ... | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.149_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ Ico a b,... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.149_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
case ha
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ ... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | rwa [sub_self, mul_zero, add_zero] | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.149_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
case hB
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈ ... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const)) | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.149_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
case hB'
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | intro x hx | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.149_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
case hB'
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x ∈... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.149_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
case bound
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | intro x _ _ | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.149_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
case bound
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : ℝ → ℝ
a b : ℝ
hf : ContinuousOn f (Icc a b)
B B' : ℝ → ℝ
ha : f a ≤ B a
hB : ContinuousOn B (Icc a b)
hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x
bound : ∀ x... | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import... | rw [mul_one] | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib.Analysis.Calculus.MeanValue.149_0.ReDurB0qNQAwk9I | /-- General fencing theorem for continuous functions with an estimate on the derivative.
Let `f` and `B` be continuous functions on `[a, b]` such that
* `f a ≤ B a`;
* `B` has right derivative `B'` at every point of `[a, b)`;
* for each `x ∈ [a, b)` the right-side limit inferior of `(f z - f x) / (z - x)`
is bounded... | Mathlib_Analysis_Calculus_MeanValue |
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