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 |
|---|---|---|---|---|---|---|
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
X✝ Y✝ Z✝ : T
h₁ : X✝ ⟶ Y✝
h₂ : Y✝ ⟶ Z✝
⊢ { obj := fun t => mk (fun i => (X C i).obj t) fun i j => (d C i j).app t,
map := fun {X Y} h ... | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb8... | ext i | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
h... | Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
case h
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
X✝ Y✝ Z✝ : T
h₁ : X✝ ⟶ Y✝
h₂ : Y✝ ⟶ Z✝
i : ι
⊢ Hom.f
({ obj := fun t => mk (fun i => (X C i).obj t) fun i j => (d C i j).app t,
... | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb8... | dsimp | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
h... | Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
case h
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.78, u_2} T
C : HomologicalComplex (T ⥤ V) c
X✝ Y✝ Z✝ : T
h₁ : X✝ ⟶ Y✝
h₂ : Y✝ ⟶ Z✝
i : ι
⊢ (X C i).map (h₁ ≫ h₂) = (X C i).map h₁ ≫ (X C i).map h₂ | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb8... | rw [Functor.map_comp] | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t :=
{ X := fun i => (C.X i).obj t
d := fun i j => (C.d i j).app t
d_comp_d' := fun i j k _ _ => by
h... | Mathlib.Algebra.Homology.Functor.34_0.DFL0Z3rvWsk3pdU | /-- A complex of functors gives a functor to complexes. -/
@[simps obj map]
def asFunctor {T : Type*} [Category T] (C : HomologicalComplex (T ⥤ V) c) :
T ⥤ HomologicalComplex V c where
obj t | Mathlib_Algebra_Homology_Functor |
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.4461, u_2} T
X✝ Y✝ : HomologicalComplex (T ⥤ V) c
f : X✝ ⟶ Y✝
t t' : T
g : t ⟶ t'
⊢ ((fun C => asFunctor C) X✝).map g ≫ (fun t => Hom.mk fun i => (Hom.f f i).app t) t' =
(fun t => Hom.... | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb8... | ext i | /-- The functorial version of `HomologicalComplex.asFunctor`. -/
@[simps]
def complexOfFunctorsToFunctorToComplex {T : Type*} [Category T] :
HomologicalComplex (T ⥤ V) c ⥤ T ⥤ HomologicalComplex V c where
obj C := C.asFunctor
map f :=
{ app := fun t =>
{ f := fun i => (f.f i).app t
comm' :... | Mathlib.Algebra.Homology.Functor.63_0.DFL0Z3rvWsk3pdU | /-- The functorial version of `HomologicalComplex.asFunctor`. -/
@[simps]
def complexOfFunctorsToFunctorToComplex {T : Type*} [Category T] :
HomologicalComplex (T ⥤ V) c ⥤ T ⥤ HomologicalComplex V c where
obj C | Mathlib_Algebra_Homology_Functor |
case h
V : Type u
inst✝² : Category.{v, u} V
inst✝¹ : HasZeroMorphisms V
ι : Type u_1
c : ComplexShape ι
T : Type u_2
inst✝ : Category.{?u.4461, u_2} T
X✝ Y✝ : HomologicalComplex (T ⥤ V) c
f : X✝ ⟶ Y✝
t t' : T
g : t ⟶ t'
i : ι
⊢ Hom.f (((fun C => asFunctor C) X✝).map g ≫ (fun t => Hom.mk fun i => (Hom.f f i).app t) t')... | /-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Homology.HomologicalComplex
#align_import algebra.homology.functor from "leanprover-community/mathlib"@"8e25bb6c1645bb80670e13848b79a54aa45cb8... | exact (f.f i).naturality g | /-- The functorial version of `HomologicalComplex.asFunctor`. -/
@[simps]
def complexOfFunctorsToFunctorToComplex {T : Type*} [Category T] :
HomologicalComplex (T ⥤ V) c ⥤ T ⥤ HomologicalComplex V c where
obj C := C.asFunctor
map f :=
{ app := fun t =>
{ f := fun i => (f.f i).app t
comm' :... | Mathlib.Algebra.Homology.Functor.63_0.DFL0Z3rvWsk3pdU | /-- The functorial version of `HomologicalComplex.asFunctor`. -/
@[simps]
def complexOfFunctorsToFunctorToComplex {T : Type*} [Category T] :
HomologicalComplex (T ⥤ V) c ⥤ T ⥤ HomologicalComplex V c where
obj C | Mathlib_Algebra_Homology_Functor |
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
⊢ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | by_cases ha_zero : a = 0 | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u := by
| Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case pos
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : a = 0
⊢ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | simp_rw [ha_zero, zero_mul, ← ENNReal.bot_eq_zero] | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
· | Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case pos
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : a = 0
⊢ limsup (fun x => ⊥) f = ⊥ | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | exact limsup_const_bot | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
· simp_rw [ha_zero, zero_mul, ← ENNReal.bot_eq_zero]
| Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
⊢ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | let g := fun x : ℝ≥0∞ => a * x | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
· simp_rw [ha_zero, zero_mul, ← ENNReal.bot_eq_zero]
exact limsup_const_bot
| Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
⊢ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | have hg_bij : Function.Bijective g :=
Function.bijective_iff_has_inverse.mpr
⟨fun x => a⁻¹ * x,
⟨fun x => by simp [← mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
simp [← mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]⟩⟩ | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
· simp_rw [ha_zero, zero_mul, ← ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ℝ≥0∞ => a * x
| Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
x : ℝ≥0∞
⊢ (fun x => a⁻¹ * x) (g x) = x | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | simp [← mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top] | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
· simp_rw [ha_zero, zero_mul, ← ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ℝ≥0∞ => a * x
have hg_bij : Function.Bijective g :=... | Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
x : ℝ≥0∞
⊢ g ((fun x => a⁻¹ * x) x) = x | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | simp [← mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top] | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
· simp_rw [ha_zero, zero_mul, ← ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ℝ≥0∞ => a * x
have hg_bij : Function.Bijective g :=... | Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
hg_bij : Function.Bijective g
⊢ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | have hg_mono : StrictMono g :=
Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1 | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
· simp_rw [ha_zero, zero_mul, ← ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ℝ≥0∞ => a * x
have hg_bij : Function.Bijective g :=... | Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
hg_bij : Function.Bijective g
x✝² x✝¹ : ℝ≥0∞
x✝ : x✝² ≤ x✝¹
⊢ g x✝² ≤ g x✝¹ | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | rwa [mul_le_mul_left ha_zero ha_top] | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
· simp_rw [ha_zero, zero_mul, ← ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ℝ≥0∞ => a * x
have hg_bij : Function.Bijective g :=... | Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
hg_bij : Function.Bijective g
hg_mono : StrictMono g
⊢ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2 | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
· simp_rw [ha_zero, zero_mul, ← ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ℝ≥0∞ => a * x
have hg_bij : Function.Bijective g :=... | Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
hg_bij : Function.Bijective g
hg_mono : StrictMono g
g_iso : ℝ≥0∞ ≃o ℝ≥0∞ := StrictMono.orderIsoOfSurjective g hg_mono (_ : Function.Surjective g)
⊢ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | exact (OrderIso.limsup_apply g_iso).symm | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u := by
by_cases ha_zero : a = 0
· simp_rw [ha_zero, zero_mul, ← ENNReal.bot_eq_zero]
exact limsup_const_bot
let g := fun x : ℝ≥0∞ => a * x
have hg_bij : Function.Bijective g :=... | Mathlib.Order.Filter.ENNReal.33_0.m2m9iHbAOFxzXph | theorem limsup_const_mul_of_ne_top {u : α → ℝ≥0∞} {a : ℝ≥0∞} (ha_top : a ≠ ⊤) :
(f.limsup fun x : α => a * u x) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
⊢ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | by_cases ha_top : a ≠ ⊤ | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u := by
| Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case pos
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
⊢ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | exact limsup_const_mul_of_ne_top ha_top | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u := by
by_cases ha_top : a ≠ ⊤
· | Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : ¬a ≠ ⊤
⊢ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | push_neg at ha_top | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u := by
by_cases ha_top : a ≠ ⊤
· exact limsup_const_mul_of_ne_top ha_top
| Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ⊤
⊢ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | by_cases hu : u =ᶠ[f] 0 | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u := by
by_cases ha_top : a ≠ ⊤
· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
| Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case pos
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ⊤
hu : u =ᶠ[f] 0
⊢ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | have hau : (a * u ·) =ᶠ[f] 0 := hu.mono fun x hx => by simp [hx] | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u := by
by_cases ha_top : a ≠ ⊤
· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =ᶠ[f] 0
· | Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ⊤
hu : u =ᶠ[f] 0
x : α
hx : u x = OfNat.ofNat 0 x
⊢ (fun x => a * u x) x = OfNat.ofNat 0 x | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | simp [hx] | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u := by
by_cases ha_top : a ≠ ⊤
· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =ᶠ[f] 0
· have hau : (a * u ·) =ᶠ[f] 0 := hu.mono fun x hx => by | Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case pos
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ⊤
hu : u =ᶠ[f] 0
hau : (fun x => a * u x) =ᶠ[f] 0
⊢ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | simp only [limsup_congr hu, limsup_congr hau, Pi.zero_apply, ← ENNReal.bot_eq_zero,
limsup_const_bot] | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u := by
by_cases ha_top : a ≠ ⊤
· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =ᶠ[f] 0
· have hau : (a * u ·) =ᶠ[f] 0 := hu.mono fun x hx => by simp [hx]
| Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case pos
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ⊤
hu : u =ᶠ[f] 0
hau : (fun x => a * u x) =ᶠ[f] 0
⊢ ⊥ = a * ⊥ | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | simp | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u := by
by_cases ha_top : a ≠ ⊤
· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =ᶠ[f] 0
· have hau : (a * u ·) =ᶠ[f] 0 := hu.mono fun x hx => by simp [hx]
simp o... | Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ⊤
hu : ¬u =ᶠ[f] 0
⊢ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | have hu_mul : ∃ᶠ x : α in f, ⊤ ≤ ite (u x = 0) (0 : ℝ≥0∞) ⊤ := by
rw [EventuallyEq, not_eventually] at hu
refine' hu.mono fun x hx => _
rw [Pi.zero_apply] at hx
simp [hx] | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u := by
by_cases ha_top : a ≠ ⊤
· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =ᶠ[f] 0
· have hau : (a * u ·) =ᶠ[f] 0 := hu.mono fun x hx => by simp [hx]
simp o... | Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ⊤
hu : ¬u =ᶠ[f] 0
⊢ ∃ᶠ (x : α) in f, ⊤ ≤ if u x = 0 then 0 else ⊤ | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | rw [EventuallyEq, not_eventually] at hu | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u := by
by_cases ha_top : a ≠ ⊤
· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =ᶠ[f] 0
· have hau : (a * u ·) =ᶠ[f] 0 := hu.mono fun x hx => by simp [hx]
simp o... | Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ⊤
hu : ∃ᶠ (x : α) in f, ¬u x = OfNat.ofNat 0 x
⊢ ∃ᶠ (x : α) in f, ⊤ ≤ if u x = 0 then 0 else ⊤ | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | refine' hu.mono fun x hx => _ | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u := by
by_cases ha_top : a ≠ ⊤
· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =ᶠ[f] 0
· have hau : (a * u ·) =ᶠ[f] 0 := hu.mono fun x hx => by simp [hx]
simp o... | Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ⊤
hu : ∃ᶠ (x : α) in f, ¬u x = OfNat.ofNat 0 x
x : α
hx : ¬u x = OfNat.ofNat 0 x
⊢ ⊤ ≤ if u x = 0 then 0 else ⊤ | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | rw [Pi.zero_apply] at hx | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u := by
by_cases ha_top : a ≠ ⊤
· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =ᶠ[f] 0
· have hau : (a * u ·) =ᶠ[f] 0 := hu.mono fun x hx => by simp [hx]
simp o... | Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ⊤
hu : ∃ᶠ (x : α) in f, ¬u x = OfNat.ofNat 0 x
x : α
hx : ¬u x = 0
⊢ ⊤ ≤ if u x = 0 then 0 else ⊤ | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | simp [hx] | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u := by
by_cases ha_top : a ≠ ⊤
· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =ᶠ[f] 0
· have hau : (a * u ·) =ᶠ[f] 0 := hu.mono fun x hx => by simp [hx]
simp o... | Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ⊤
hu : ¬u =ᶠ[f] 0
hu_mul : ∃ᶠ (x : α) in f, ⊤ ≤ if u x = 0 then 0 else ⊤
⊢ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | have h_top_le : (f.limsup fun x : α => ite (u x = 0) (0 : ℝ≥0∞) ⊤) = ⊤ :=
eq_top_iff.mpr (le_limsup_of_frequently_le hu_mul) | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u := by
by_cases ha_top : a ≠ ⊤
· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =ᶠ[f] 0
· have hau : (a * u ·) =ᶠ[f] 0 := hu.mono fun x hx => by simp [hx]
simp o... | Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ⊤
hu : ¬u =ᶠ[f] 0
hu_mul : ∃ᶠ (x : α) in f, ⊤ ≤ if u x = 0 then 0 else ⊤
h_top_le : limsup (fun x => if u x = 0 then 0 else ⊤) f = ⊤
⊢ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | have hfu : f.limsup u ≠ 0 := mt limsup_eq_zero_iff.1 hu | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u := by
by_cases ha_top : a ≠ ⊤
· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =ᶠ[f] 0
· have hau : (a * u ·) =ᶠ[f] 0 := hu.mono fun x hx => by simp [hx]
simp o... | Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
case neg
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ⊤
hu : ¬u =ᶠ[f] 0
hu_mul : ∃ᶠ (x : α) in f, ⊤ ≤ if u x = 0 then 0 else ⊤
h_top_le : limsup (fun x => if u x = 0 then 0 else ⊤) f = ⊤
hfu : limsup u f ≠ 0
⊢ limsup (fun x => a * u x) f = a * limsup u f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | simp only [ha_top, top_mul', h_top_le, hfu, ite_false] | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u := by
by_cases ha_top : a ≠ ⊤
· exact limsup_const_mul_of_ne_top ha_top
push_neg at ha_top
by_cases hu : u =ᶠ[f] 0
· have hau : (a * u ·) =ᶠ[f] 0 := hu.mono fun x hx => by simp [hx]
simp o... | Mathlib.Order.Filter.ENNReal.50_0.m2m9iHbAOFxzXph | theorem limsup_const_mul [CountableInterFilter f] {u : α → ℝ≥0∞} {a : ℝ≥0∞} :
f.limsup (a * u ·) = a * f.limsup u | Mathlib_Order_Filter_ENNReal |
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u v : α → ℝ≥0∞
⊢ limsup (u * v) f ≤ limsup (fun x => limsup u f * v x) f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | refine limsup_le_limsup ?_ | theorem limsup_mul_le [CountableInterFilter f] (u v : α → ℝ≥0∞) :
f.limsup (u * v) ≤ f.limsup u * f.limsup v :=
calc
f.limsup (u * v) ≤ f.limsup fun x => f.limsup u * v x := by
| Mathlib.Order.Filter.ENNReal.71_0.m2m9iHbAOFxzXph | theorem limsup_mul_le [CountableInterFilter f] (u v : α → ℝ≥0∞) :
f.limsup (u * v) ≤ f.limsup u * f.limsup v | Mathlib_Order_Filter_ENNReal |
α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u v : α → ℝ≥0∞
⊢ u * v ≤ᶠ[f] fun x => limsup u f * v x | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | filter_upwards [@eventually_le_limsup _ f _ u] with x hx using mul_le_mul' hx le_rfl | theorem limsup_mul_le [CountableInterFilter f] (u v : α → ℝ≥0∞) :
f.limsup (u * v) ≤ f.limsup u * f.limsup v :=
calc
f.limsup (u * v) ≤ f.limsup fun x => f.limsup u * v x := by
refine limsup_le_limsup ?_
| Mathlib.Order.Filter.ENNReal.71_0.m2m9iHbAOFxzXph | theorem limsup_mul_le [CountableInterFilter f] (u v : α → ℝ≥0∞) :
f.limsup (u * v) ≤ f.limsup u * f.limsup v | Mathlib_Order_Filter_ENNReal |
α : Type u_1
f✝ : Filter α
β : Type u_2
inst✝¹ : Countable β
f : Filter α
inst✝ : CountableInterFilter f
g : Filter β
u : α → β → ℝ≥0∞
⊢ ∀ᶠ (a : α) in f, ∀ (b : β), u a b ≤ limsup (fun a' => u a' b) f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | rw [eventually_countable_forall] | theorem limsup_liminf_le_liminf_limsup {β} [Countable β] {f : Filter α} [CountableInterFilter f]
{g : Filter β} (u : α → β → ℝ≥0∞) :
(f.limsup fun a : α => g.liminf fun b : β => u a b) ≤
g.liminf fun b => f.limsup fun a => u a b :=
have h1 : ∀ᶠ a in f, ∀ b, u a b ≤ f.limsup fun a' => u a' b := by
| Mathlib.Order.Filter.ENNReal.86_0.m2m9iHbAOFxzXph | theorem limsup_liminf_le_liminf_limsup {β} [Countable β] {f : Filter α} [CountableInterFilter f]
{g : Filter β} (u : α → β → ℝ≥0∞) :
(f.limsup fun a : α => g.liminf fun b : β => u a b) ≤
g.liminf fun b => f.limsup fun a => u a b | Mathlib_Order_Filter_ENNReal |
α : Type u_1
f✝ : Filter α
β : Type u_2
inst✝¹ : Countable β
f : Filter α
inst✝ : CountableInterFilter f
g : Filter β
u : α → β → ℝ≥0∞
⊢ ∀ (i : β), ∀ᶠ (x : α) in f, u x i ≤ limsup (fun a' => u a' i) f | /-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | exact fun b => ENNReal.eventually_le_limsup fun a => u a b | theorem limsup_liminf_le_liminf_limsup {β} [Countable β] {f : Filter α} [CountableInterFilter f]
{g : Filter β} (u : α → β → ℝ≥0∞) :
(f.limsup fun a : α => g.liminf fun b : β => u a b) ≤
g.liminf fun b => f.limsup fun a => u a b :=
have h1 : ∀ᶠ a in f, ∀ b, u a b ≤ f.limsup fun a' => u a' b := by
rw... | Mathlib.Order.Filter.ENNReal.86_0.m2m9iHbAOFxzXph | theorem limsup_liminf_le_liminf_limsup {β} [Countable β] {f : Filter α} [CountableInterFilter f]
{g : Filter β} (u : α → β → ℝ≥0∞) :
(f.limsup fun a : α => g.liminf fun b : β => u a b) ≤
g.liminf fun b => f.limsup fun a => u a b | Mathlib_Order_Filter_ENNReal |
u : ℂ
hu : 0 < u.re
v : ℂ
⊢ IntervalIntegrable (fun x => ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)) volume 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | apply IntervalIntegrable.mul_continuousOn | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) := by
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hf
u : ℂ
hu : 0 < u.re
v : ℂ
⊢ IntervalIntegrable (fun x => ↑x ^ (u - 1)) volume 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' intervalIntegral.intervalIntegrable_cpow' _ | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continu... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hf
u : ℂ
hu : 0 < u.re
v : ℂ
⊢ -1 < (u - 1).re | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rwa [sub_re, one_re, ← zero_sub, sub_lt_sub_iff_right] | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continu... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hg
u : ℂ
hu : 0 < u.re
v : ℂ
⊢ ContinuousOn (fun x => (1 - ↑x) ^ (v - 1)) (uIcc 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | apply ContinuousAt.continuousOn | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continu... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hg.hcont
u : ℂ
hu : 0 < u.re
v : ℂ
⊢ ∀ x ∈ uIcc 0 (1 / 2), ContinuousAt (fun x => (1 - ↑x) ^ (v - 1)) 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | intro x hx | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continu... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hg.hcont
u : ℂ
hu : 0 < u.re
v : ℂ
x : ℝ
hx : x ∈ uIcc 0 (1 / 2)
⊢ ContinuousAt (fun x => (1 - ↑x) ^ (v - 1)) 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [uIcc_of_le (by positivity : (0 : ℝ) ≤ 1 / 2)] at hx | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continu... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u : ℂ
hu : 0 < u.re
v : ℂ
x : ℝ
hx : x ∈ uIcc 0 (1 / 2)
⊢ 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | positivity | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continu... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hg.hcont
u : ℂ
hu : 0 < u.re
v : ℂ
x : ℝ
hx : x ∈ Icc 0 (1 / 2)
⊢ ContinuousAt (fun x => (1 - ↑x) ^ (v - 1)) 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | apply ContinuousAt.cpow | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continu... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hg.hcont.hf
u : ℂ
hu : 0 < u.re
v : ℂ
x : ℝ
hx : x ∈ Icc 0 (1 / 2)
⊢ ContinuousAt (fun x => 1 - ↑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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | exact (continuous_const.sub continuous_ofReal).continuousAt | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continu... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hg.hcont.hg
u : ℂ
hu : 0 < u.re
v : ℂ
x : ℝ
hx : x ∈ Icc 0 (1 / 2)
⊢ ContinuousAt (fun x => v - 1) 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | exact continuousAt_const | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continu... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hg.hcont.h0
u : ℂ
hu : 0 < u.re
v : ℂ
x : ℝ
hx : x ∈ Icc 0 (1 / 2)
⊢ 0 < (1 - ↑x).re ∨ (1 - ↑x).im ≠ 0 | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [sub_re, one_re, ofReal_re, sub_pos] | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continu... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case hg.hcont.h0
u : ℂ
hu : 0 < u.re
v : ℂ
x : ℝ
hx : x ∈ Icc 0 (1 / 2)
⊢ x < 1 ∨ (1 - ↑x).im ≠ 0 | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | exact Or.inl (hx.2.trans_lt (by norm_num : (1 / 2 : ℝ) < 1)) | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continu... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u : ℂ
hu : 0 < u.re
v : ℂ
x : ℝ
hx : x ∈ Icc 0 (1 / 2)
⊢ 1 / 2 < 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | norm_num | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) := by
apply IntervalIntegrable.mul_continu... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.61_0.in2QiCFW52coQT2 | /-- Auxiliary lemma for `betaIntegral_convergent`, showing convergence at the left endpoint. -/
theorem betaIntegral_convergent_left {u : ℂ} (hu : 0 < re u) (v : ℂ) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 (1 / 2) | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
⊢ IntervalIntegrable (fun x => ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)) volume 0 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' (betaIntegral_convergent_left hu v).trans _ | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 := by
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
⊢ IntervalIntegrable (fun x => ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)) volume (1 / 2) 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [IntervalIntegrable.iff_comp_neg] | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
⊢ IntervalIntegrable (fun x => ↑(-x) ^ (u - 1) * (1 - ↑(-x)) ^ (v - 1)) volume (-(1 / 2)) (-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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_3
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
⊢ (fun x => ↑(-x) ^ (u - 1) * (1 - ↑(-x)) ^ (v - 1)) = fun x => ↑(x + 1) ^ (v - 1) * (1 - ↑(x + 1)) ^ (u - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | ext1 x | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_3.h
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
x : ℝ
⊢ ↑(-x) ^ (u - 1) * (1 - ↑(-x)) ^ (v - 1) = ↑(x + 1) ^ (v - 1) * (1 - ↑(x + 1)) ^ (u - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | conv_lhs => rw [mul_comm] | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
x : ℝ
| ↑(-x) ^ (u - 1) * (1 - ↑(-x)) ^ (v - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [mul_comm] | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
x : ℝ
| ↑(-x) ^ (u - 1) * (1 - ↑(-x)) ^ (v - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [mul_comm] | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
x : ℝ
| ↑(-x) ^ (u - 1) * (1 - ↑(-x)) ^ (v - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [mul_comm] | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_3.h
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
x : ℝ
⊢ (1 - ↑(-x)) ^ (v - 1) * ↑(-x) ^ (u - 1) = ↑(x + 1) ^ (v - 1) * (1 - ↑(x + 1)) ^ (u - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | congr 2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_3.h.e_a.e_a
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
x : ℝ
⊢ 1 - ↑(-x) = ↑(x + 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | push_cast | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_3.h.e_a.e_a
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
x : ℝ
⊢ 1 - -↑x = ↑x + 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | ring | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_3.h.e_a.e_a
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
x : ℝ
⊢ ↑(-x) = 1 - ↑(x + 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | push_cast | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_3.h.e_a.e_a
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
x : ℝ
⊢ -↑x = 1 - (↑x + 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | ring | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_5
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
⊢ -(1 / 2) = 1 / 2 - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | norm_num | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h.e'_6
u v : ℂ
hu : 0 < u.re
hv : 0 < v.re
⊢ -1 = 0 - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | norm_num | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 := by
refine' (betaIntegral_convergent_left hu v).trans _
rw... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.78_0.in2QiCFW52coQT2 | /-- The Beta integral is convergent for all `u, v` of positive real part. -/
theorem betaIntegral_convergent {u v : ℂ} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1) : ℝ → ℂ) volume 0 1 | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : ℂ
⊢ betaIntegral v u = betaIntegral u v | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [betaIntegral, betaIntegral] | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v := by
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : ℂ
⊢ ∫ (x : ℝ) in 0 ..1, ↑x ^ (v - 1) * (1 - ↑x) ^ (u - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : ℝ => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1)) neg_one_lt_zero.ne 1 | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : ℂ
this :
∫ (x : ℝ) in 0 ..1, ↑(-1 * x + 1) ^ (u - 1) * (1 - ↑(-1 * x + 1)) ^ (v - 1) =
(-1)⁻¹ • ∫ (x : ℝ) in -1 * 0 + 1 ..-1 * 1 + 1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
⊢ ∫ (x : ℝ) in 0 ..1, ↑x ^ (v - 1) * (1 - ↑x) ^ (u - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [inv_neg, inv_one, neg_one_smul, ← intervalIntegral.integral_symm] at this | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : ℝ => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1)) neg_one_lt_zero.ne 1
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : ℂ
this :
∫ (x : ℝ) in 0 ..1, ↑(-1 * x + 1) ^ (u - 1) * (1 - ↑(-1 * x + 1)) ^ (v - 1) =
∫ (x : ℝ) in -1 * 1 + 1 ..-1 * 0 + 1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
⊢ ∫ (x : ℝ) in 0 ..1, ↑x ^ (v - 1) * (1 - ↑x) ^ (u - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | simp? at this says
simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : ℝ => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, ← inter... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : ℂ
this :
∫ (x : ℝ) in 0 ..1, ↑(-1 * x + 1) ^ (u - 1) * (1 - ↑(-1 * x + 1)) ^ (v - 1) =
∫ (x : ℝ) in -1 * 1 + 1 ..-1 * 0 + 1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
⊢ ∫ (x : ℝ) in 0 ..1, ↑x ^ (v - 1) * (1 - ↑x) ^ (u - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | simp only [neg_mul, one_mul, ofReal_add, ofReal_neg, ofReal_one, sub_add_cancel'', neg_neg,
mul_one, add_left_neg, mul_zero, zero_add] at this | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : ℝ => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, ← inter... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : ℂ
this : ∫ (x : ℝ) in 0 ..1, (-↑x + 1) ^ (u - 1) * ↑x ^ (v - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
⊢ ∫ (x : ℝ) in 0 ..1, ↑x ^ (v - 1) * (1 - ↑x) ^ (u - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | conv_lhs at this => arg 1; intro x; rw [add_comm, ← sub_eq_add_neg, mul_comm] | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : ℝ => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, ← inter... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : ℂ
this : ∫ (x : ℝ) in 0 ..1, (-↑x + 1) ^ (u - 1) * ↑x ^ (v - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
| ∫ (x : ℝ) in 0 ..1, (-↑x + 1) ^ (u - 1) * ↑x ^ (v - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | arg 1; intro x; rw [add_comm, ← sub_eq_add_neg, mul_comm] | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : ℝ => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, ← inter... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : ℂ
this : ∫ (x : ℝ) in 0 ..1, (-↑x + 1) ^ (u - 1) * ↑x ^ (v - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
| ∫ (x : ℝ) in 0 ..1, (-↑x + 1) ^ (u - 1) * ↑x ^ (v - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | arg 1; intro x; rw [add_comm, ← sub_eq_add_neg, mul_comm] | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : ℝ => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, ← inter... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : ℂ
this : ∫ (x : ℝ) in 0 ..1, (-↑x + 1) ^ (u - 1) * ↑x ^ (v - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
| ∫ (x : ℝ) in 0 ..1, (-↑x + 1) ^ (u - 1) * ↑x ^ (v - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | arg 1 | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : ℝ => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, ← inter... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : ℂ
this : ∫ (x : ℝ) in 0 ..1, (-↑x + 1) ^ (u - 1) * ↑x ^ (v - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
| fun x => (-↑x + 1) ^ (u - 1) * ↑x ^ (v - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | intro x | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : ℝ => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, ← inter... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case h
u v : ℂ
this : ∫ (x : ℝ) in 0 ..1, (-↑x + 1) ^ (u - 1) * ↑x ^ (v - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
x : ℝ
| (-↑x + 1) ^ (u - 1) * ↑x ^ (v - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [add_comm, ← sub_eq_add_neg, mul_comm] | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : ℝ => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, ← inter... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u v : ℂ
this : ∫ (x : ℝ) in 0 ..1, ↑x ^ (v - 1) * (1 - ↑x) ^ (u - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 1)
⊢ ∫ (x : ℝ) in 0 ..1, ↑x ^ (v - 1) * (1 - ↑x) ^ (u - 1) = ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) * (1 - ↑x) ^ (v - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | exact this | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v := by
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : ℝ => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, ← inter... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.92_0.in2QiCFW52coQT2 | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u : ℂ
hu : 0 < u.re
⊢ betaIntegral u 1 = 1 / u | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | simp_rw [betaIntegral, sub_self, cpow_zero, mul_one] | theorem betaIntegral_eval_one_right {u : ℂ} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.104_0.in2QiCFW52coQT2 | theorem betaIntegral_eval_one_right {u : ℂ} (hu : 0 < re u) : betaIntegral u 1 = 1 / u | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u : ℂ
hu : 0 < u.re
⊢ ∫ (x : ℝ) in 0 ..1, ↑x ^ (u - 1) = 1 / u | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [integral_cpow (Or.inl _)] | theorem betaIntegral_eval_one_right {u : ℂ} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.104_0.in2QiCFW52coQT2 | theorem betaIntegral_eval_one_right {u : ℂ} (hu : 0 < re u) : betaIntegral u 1 = 1 / u | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u : ℂ
hu : 0 < u.re
⊢ (↑1 ^ (u - 1 + 1) - ↑0 ^ (u - 1 + 1)) / (u - 1 + 1) = 1 / u | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel] | theorem betaIntegral_eval_one_right {u : ℂ} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
· | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.104_0.in2QiCFW52coQT2 | theorem betaIntegral_eval_one_right {u : ℂ} (hu : 0 < re u) : betaIntegral u 1 = 1 / u | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u : ℂ
hu : 0 < u.re
⊢ u - 1 + 1 ≠ 0 | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [sub_add_cancel] | theorem betaIntegral_eval_one_right {u : ℂ} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.104_0.in2QiCFW52coQT2 | theorem betaIntegral_eval_one_right {u : ℂ} (hu : 0 < re u) : betaIntegral u 1 = 1 / u | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u : ℂ
hu : 0 < u.re
⊢ u ≠ 0 | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | contrapose! hu | theorem betaIntegral_eval_one_right {u : ℂ} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.104_0.in2QiCFW52coQT2 | theorem betaIntegral_eval_one_right {u : ℂ} (hu : 0 < re u) : betaIntegral u 1 = 1 / u | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u : ℂ
hu : u = 0
⊢ u.re ≤ 0 | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [hu, zero_re] | theorem betaIntegral_eval_one_right {u : ℂ} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.104_0.in2QiCFW52coQT2 | theorem betaIntegral_eval_one_right {u : ℂ} (hu : 0 < re u) : betaIntegral u 1 = 1 / u | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
u : ℂ
hu : 0 < u.re
⊢ -1 < (u - 1).re | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rwa [sub_re, one_re, ← sub_pos, sub_neg_eq_add, sub_add_cancel] | theorem betaIntegral_eval_one_right {u : ℂ} (hu : 0 < re u) : betaIntegral u 1 = 1 / u := by
simp_rw [betaIntegral, sub_self, cpow_zero, mul_one]
rw [integral_cpow (Or.inl _)]
· rw [ofReal_zero, ofReal_one, one_cpow, zero_cpow, sub_zero, sub_add_cancel]
rw [sub_add_cancel]
contrapose! hu; rw [hu, zero_re]... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.104_0.in2QiCFW52coQT2 | theorem betaIntegral_eval_one_right {u : ℂ} (hu : 0 < re u) : betaIntegral u 1 = 1 / u | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : ℂ
a : ℝ
ha : 0 < a
⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s + t - 1) * betaIntegral s t | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne' | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t := by
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s + t - 1) * betaIntegral s t | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [betaIntegral] | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =
↑a ^ (s + t - 1) * ∫ (x : ℝ) in 0 ..1, ↑x ^ (s - 1) * (1 - ↑x) ^ (t - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc] | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
| Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
⊢ ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc] | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
⊢ s + t - 1 = 1 + (s - 1) + (t - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | abel | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
⊢ s + t - 1 = 1 + (s - 1) + (t - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | abel | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =
↑a ^ (s + t - 1) * ∫ (x : ℝ) in 0 ..1, ↑x ^ (s - 1) * (1 - ↑x) ^ (t - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [A, mul_assoc, ← intervalIntegral.integral_const_mul, ← real_smul, ← zero_div a, ←
div_self ha.ne', ← intervalIntegral.integral_comp_div _ ha.ne', zero_div] | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
⊢ ∫ (x : ℝ) in 0 ..a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) =
∫ (x : ℝ) in 0 ..a, ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | simp_rw [intervalIntegral.integral_of_le ha.le] | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
⊢ ∫ (x : ℝ) in Ioc 0 a, ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) ∂volume =
∫ (x : ℝ) in Ioc 0 a, ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 1)) ∂volume | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | refine' set_integral_congr measurableSet_Ioc fun x hx => _ | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
x : ℝ
hx : x ∈ Ioc 0 a
⊢ ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s - 1) * ↑a ^ (t - 1) * (↑(x / a) ^ (s - 1) * (1 - ↑(x / a)) ^ (t - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [mul_mul_mul_comm] | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
x : ℝ
hx : x ∈ Ioc 0 a
⊢ ↑x ^ (s - 1) * (↑a - ↑x) ^ (t - 1) = ↑a ^ (s - 1) * ↑(x / a) ^ (s - 1) * (↑a ^ (t - 1) * (1 - ↑(x / a)) ^ (t - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | congr 1 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case e_a
s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
x : ℝ
hx : x ∈ Ioc 0 a
⊢ ↑x ^ (s - 1) = ↑a ^ (s - 1) * ↑(x / a) ^ (s - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [← mul_cpow_ofReal_nonneg ha.le (div_pos hx.1 ha).le, ofReal_div, mul_div_cancel' _ ha'] | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case e_a
s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
x : ℝ
hx : x ∈ Ioc 0 a
⊢ (↑a - ↑x) ^ (t - 1) = ↑a ^ (t - 1) * (1 - ↑(x / a)) ^ (t - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [(by norm_cast : (1 : ℂ) - ↑(x / a) = ↑(1 - x / a)), ←
mul_cpow_ofReal_nonneg ha.le (sub_nonneg.mpr <| (div_le_one ha).mpr hx.2)] | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
x : ℝ
hx : x ∈ Ioc 0 a
⊢ 1 - ↑(x / a) = ↑(1 - x / 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | norm_cast | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case e_a
s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
x : ℝ
hx : x ∈ Ioc 0 a
⊢ (↑a - ↑x) ^ (t - 1) = (↑a * ↑(1 - x / a)) ^ (t - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | push_cast | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
case e_a
s t : ℂ
a : ℝ
ha : 0 < a
ha' : ↑a ≠ 0
A : ↑a ^ (s + t - 1) = ↑a * (↑a ^ (s - 1) * ↑a ^ (t - 1))
x : ℝ
hx : x ∈ Ioc 0 a
⊢ (↑a - ↑x) ^ (t - 1) = (↑a * (1 - ↑x / ↑a)) ^ (t - 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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | rw [mul_sub, mul_one, mul_div_cancel' _ ha'] | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t := by
have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.113_0.in2QiCFW52coQT2 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
s t : ℂ
hs : 0 < s.re
ht : 0 < t.re
⊢ Gamma s * Gamma t = Gamma (s + t) * betaIntegral s t | /-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | have conv_int := integral_posConvolution
(GammaIntegral_convergent hs) (GammaIntegral_convergent ht) (ContinuousLinearMap.mul ℝ ℂ) | /-- Relation between Beta integral and Gamma function. -/
theorem Gamma_mul_Gamma_eq_betaIntegral {s t : ℂ} (hs : 0 < re s) (ht : 0 < re t) :
Gamma s * Gamma t = Gamma (s + t) * betaIntegral s t := by
-- Note that we haven't proved (yet) that the Gamma function has no zeroes, so we can't formulate
-- this as a... | Mathlib.Analysis.SpecialFunctions.Gamma.Beta.134_0.in2QiCFW52coQT2 | /-- Relation between Beta integral and Gamma function. -/
theorem Gamma_mul_Gamma_eq_betaIntegral {s t : ℂ} (hs : 0 < re s) (ht : 0 < re t) :
Gamma s * Gamma t = Gamma (s + t) * betaIntegral s t | Mathlib_Analysis_SpecialFunctions_Gamma_Beta |
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