internlm/Lean-Github huggingface dataset:
Theorem: MellinOfSmooth1a
Proof state before:
case h.mk
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
x y : ℝ
⊢ (x, y) ∈ S ↔ (x, y) ∈ Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
Tactic:
constructor
Proof state after:
case h.mk.mp
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
x y : ℝ
⊢ (x, y) ∈ S → (x, y) ∈ Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}

case h.mk.mpr
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
x y : ℝ
⊢ (x, y) ∈ Tx ×ˢ Ty ∩ {(x, y) | x ≤ y} → (x, y) ∈ S
Theorem: MellinOfSmooth1a
Proof state before:
case h.mk.mp
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
x y : ℝ
⊢ (x, y) ∈ S → (x, y) ∈ Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
Tactic:
exact fun h ↦ ⟨⟨⟨h.1, le_trans h.2.1 h.2.2.2⟩, ⟨h.2.2.1, h.2.2.2⟩⟩, h.2.1⟩
Proof state after:
no goals
Theorem: MellinOfSmooth1a
Proof state before:
case h.mk.mpr
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
x y : ℝ
⊢ (x, y) ∈ Tx ×ˢ Ty ∩ {(x, y) | x ≤ y} → (x, y) ∈ S
Tactic:
exact fun h ↦  ⟨h.1.1.1, ⟨h.2, h.1.2.1, h.1.2.2⟩⟩
Proof state after:
no goals
Theorem: MellinOfSmooth1a
Proof state before:
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
z : ℝ × ℝ
hz : z ∈ S
⊢ 0 < 2 ^ (-ε)
Tactic:
apply rpow_pos_of_pos
Proof state after:
case hx
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
z : ℝ × ℝ
hz : z ∈ S
⊢ 0 < 2
Theorem: MellinOfSmooth1a
Proof state before:
case hx
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
z : ℝ × ℝ
hz : z ∈ S
⊢ 0 < 2
Tactic:
norm_num
Proof state after:
no goals
Theorem: MellinOfSmooth1a
Proof state before:
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
SsubI : S ⊆ Ioi 0 ×ˢ Ioi 0
⊢ S ⊆ Tx ×ˢ Ty
Tactic:
simp_rw [Seq, inter_subset_left]
Proof state after:
no goals
Theorem: MellinOfSmooth1a
Proof state before:
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
SsubI : S ⊆ Ioi 0 ×ˢ Ioi 0
SsubT : S ⊆ Tx ×ˢ Ty
⊢ MeasurableSet S
Tactic:
rw [Seq]
Proof state after:
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
SsubI : S ⊆ Ioi 0 ×ˢ Ioi 0
SsubT : S ⊆ Tx ×ˢ Ty
⊢ MeasurableSet (Tx ×ˢ Ty ∩ {(x, y) | x ≤ y})
Theorem: MellinOfSmooth1a
Proof state before:
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
SsubI : S ⊆ Ioi 0 ×ˢ Ioi 0
SsubT : S ⊆ Tx ×ˢ Ty
⊢ MeasurableSet (Tx ×ˢ Ty ∩ {(x, y) | x ≤ y})
Tactic:
apply MeasurableSet.inter ?_ <| measurableSet_le measurable_fst measurable_snd
Proof state after:
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
SsubI : S ⊆ Ioi 0 ×ˢ Ioi 0
SsubT : S ⊆ Tx ×ˢ Ty
⊢ MeasurableSet (Tx ×ˢ Ty)
Theorem: MellinOfSmooth1a
Proof state before:
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
SsubI : S ⊆ Ioi 0 ×ˢ Ioi 0
SsubT : S ⊆ Tx ×ˢ Ty
⊢ MeasurableSet (Tx ×ˢ Ty)
Tactic:
simp [measurableSet_prod, Tx, Ty]
Proof state after:
no goals
Theorem: MellinOfSmooth1a
Proof state before:
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
SsubI : S ⊆ Ioi 0 ×ˢ Ioi 0
SsubT : S ⊆ Tx ×ˢ Ty
Smeas : MeasurableSet S
⊢ IntegrableOn F (Ioi 0 ×ˢ Ioi 0) volume
Tactic:
apply IntegrableOn.congr_fun (f := F') ?_ ?_ (by simp [measurableSet_prod])
Proof state after:
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
SsubI : S ⊆ Ioi 0 ×ˢ Ioi 0
SsubT : S ⊆ Tx ×ˢ Ty
Smeas : MeasurableSet S
⊢ IntegrableOn F' (Ioi 0 ×ˢ Ioi 0) volume

𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
SsubI : S ⊆ Ioi 0 ×ˢ Ioi 0
SsubT : S ⊆ Tx ×ˢ Ty
Smeas : MeasurableSet S
⊢ EqOn F' F (Ioi 0 ×ˢ Ioi 0)
Theorem: MellinOfSmooth1a
Proof state before:
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
SsubI : S ⊆ Ioi 0 ×ˢ Ioi 0
SsubT : S ⊆ Tx ×ˢ Ty
Smeas : MeasurableSet S
⊢ IntegrableOn F' (Ioi 0 ×ˢ Ioi 0) volume

𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
SsubI : S ⊆ Ioi 0 ×ˢ Ioi 0
SsubT : S ⊆ Tx ×ˢ Ty
Smeas : MeasurableSet S
⊢ EqOn F' F (Ioi 0 ×ˢ Ioi 0)
Tactic:
swap
Proof state after:
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
SsubI : S ⊆ Ioi 0 ×ˢ Ioi 0
SsubT : S ⊆ Tx ×ˢ Ty
Smeas : MeasurableSet S
⊢ EqOn F' F (Ioi 0 ×ˢ Ioi 0)

𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
SsubI : S ⊆ Ioi 0 ×ˢ Ioi 0
SsubT : S ⊆ Tx ×ˢ Ty
Smeas : MeasurableSet S
⊢ IntegrableOn F' (Ioi 0 ×ˢ Ioi 0) volume
Theorem: MellinOfSmooth1a
Proof state before:
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
SsubI : S ⊆ Ioi 0 ×ˢ Ioi 0
SsubT : S ⊆ Tx ×ˢ Ty
Smeas : MeasurableSet S
⊢ MeasurableSet (Ioi 0 ×ˢ Ioi 0)
Tactic:
simp [measurableSet_prod]
Proof state after:
no goals
Theorem: MellinOfSmooth1a
Proof state before:
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
SsubI : S ⊆ Ioi 0 ×ˢ Ioi 0
SsubT : S ⊆ Tx ×ˢ Ty
Smeas : MeasurableSet S
⊢ EqOn F' F (Ioi 0 ×ˢ Ioi 0)
Tactic:
simp only [F, F', f, g, mul_ite, mul_one, mul_zero, Function.uncurry_apply_pair]
Proof state after:
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
SsubI : S ⊆ Ioi 0 ×ˢ Ioi 0
SsubT : S ⊆ Tx ×ˢ Ty
Smeas : MeasurableSet S
⊢ EqOn (S.piecewise (fun x => ↑(DeltaSpike Ψ ε x.2) / ↑x.2 * ↑x.1 ^ (s - 1)) fun x => 0)
    (Function.uncurry fun x y => (if 0 < x / y ∧ x / y ≤ 1 then ↑(DeltaSpike Ψ ε y) / ↑y else 0) * ↑x ^ (s - 1))
    (Ioi 0 ×ˢ Ioi 0)
Theorem: MellinOfSmooth1a
Proof state before:
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
SsubI : S ⊆ Ioi 0 ×ˢ Ioi 0
SsubT : S ⊆ Tx ×ˢ Ty
Smeas : MeasurableSet S
⊢ EqOn (S.piecewise (fun x => ↑(DeltaSpike Ψ ε x.2) / ↑x.2 * ↑x.1 ^ (s - 1)) fun x => 0)
    (Function.uncurry fun x y => (if 0 < x / y ∧ x / y ≤ 1 then ↑(DeltaSpike Ψ ε y) / ↑y else 0) * ↑x ^ (s - 1))
    (Ioi 0 ×ˢ Ioi 0)
Tactic:
intro ⟨x, y⟩ hz
Proof state after:
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
SsubI : S ⊆ Ioi 0 ×ˢ Ioi 0
SsubT : S ⊆ Tx ×ˢ Ty
Smeas : MeasurableSet S
x y : ℝ
hz : (x, y) ∈ Ioi 0 ×ˢ Ioi 0
⊢ S.piecewise (fun x => ↑(DeltaSpike Ψ ε x.2) / ↑x.2 * ↑x.1 ^ (s - 1)) (fun x => 0) (x, y) =
    Function.uncurry (fun x y => (if 0 < x / y ∧ x / y ≤ 1 then ↑(DeltaSpike Ψ ε y) / ↑y else 0) * ↑x ^ (s - 1)) (x, y)
Theorem: MellinOfSmooth1a
Proof state before:
case pos
𝕂 : Type u_1
inst✝ : _root_.RCLike 𝕂
Ψ : ℝ → ℝ
diffΨ : ContDiff ℝ 1 Ψ
suppΨ : Function.support Ψ ⊆ Icc (1 / 2) 2
ε : ℝ
εpos : 0 < ε
s : ℂ
hs : 0 < s.re
f' : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x)
f : ℝ → ℂ := fun x => ↑(DeltaSpike Ψ ε x) / ↑x
g : ℝ → ℂ := fun x => if 0 < x ∧ x ≤ 1 then 1 else 0
F : ℝ × ℝ → ℂ := Function.uncurry fun x y => f y * g (x / y) * ↑x ^ (s - 1)
S : Set (ℝ × ℝ) := {(x, y) | 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε}
F' : ℝ × ℝ → ℂ :=
  S.piecewise
    (fun x =>
      match x with
      | (x, y) => f y * ↑x ^ (s - 1))
    fun x => 0
Tx : Set ℝ := Ioc 0 (2 ^ ε)
Ty : Set ℝ := Icc (2 ^ (-ε)) (2 ^ ε)
Seq : S = Tx ×ˢ Ty ∩ {(x, y) | x ≤ y}
SsubI : S ⊆ Ioi 0 ×ˢ Ioi 0
SsubT : S ⊆ Tx ×ˢ Ty
Smeas : MeasurableSet S
x y : ℝ
hS : 0 < x ∧ x ≤ y ∧ 2 ^ (-ε) ≤ y ∧ y ≤ 2 ^ ε
hz : 0 < x ∧ 0 < y
⊢ (if True then ↑(DeltaSpike Ψ ε y) / ↑y * ↑x ^ (s - 1) else 0) =
    Function.uncurry (fun x y => (if 0 < x / y ∧ x / y ≤ 1 then ↑(DeltaSpike Ψ ε y) / ↑y else 0) * ↑x ^ (s - 1)) (x, y)
Tactic:
simp [hS, div_pos hz.1 hz.2, (div_le_one hz.2).mpr hS.2.1]
Proof state after:
no goals