Theorem: mellin_convergent_of_isBigO_scalar
File: Mathlib/Analysis/MellinTransform.lean

Proof steps:
state_before: E : Type u_1
inst✝ : NormedAddCommGroup E
a b : ℝ
f : ℝ → ℝ
s : ℝ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s
⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume
state_after: case intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
a b : ℝ
f : ℝ → ℝ
s : ℝ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s
c1 : ℝ
hc1 : 0 < c1
hc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume
⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume
tactic: obtain ⟨c1, hc1, hc1'⟩ := mellin_convergent_top_of_isBigO hfc.aestronglyMeasurable hf_top hs_top
state_before: case intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
a b : ℝ
f : ℝ → ℝ
s : ℝ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s
c1 : ℝ
hc1 : 0 < c1
hc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume
⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume
state_after: case intro.intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
a b : ℝ
f : ℝ → ℝ
s : ℝ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s
c1 : ℝ
hc1 : 0 < c1
hc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume
c2 : ℝ
hc2 : 0 < c2
hc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume
⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume
tactic: obtain ⟨c2, hc2, hc2'⟩ :=
    mellin_convergent_zero_of_isBigO hfc.aestronglyMeasurable hf_bot hs_bot
state_before: case intro.intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
a b : ℝ
f : ℝ → ℝ
s : ℝ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s
c1 : ℝ
hc1 : 0 < c1
hc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume
c2 : ℝ
hc2 : 0 < c2
hc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume
⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume
state_after: case intro.intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
a b : ℝ
f : ℝ → ℝ
s : ℝ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s
c1 : ℝ
hc1 : 0 < c1
hc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume
c2 : ℝ
hc2 : 0 < c2
hc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume
this : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1
⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume
tactic: have : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1 := by
    rw [union_assoc, Ioc_union_Ioi (le_max_right _ _),
      Ioc_union_Ioi ((min_le_left _ _).trans (le_max_right _ _)), min_eq_left (lt_min hc2 hc1).le]
state_before: case intro.intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
a b : ℝ
f : ℝ → ℝ
s : ℝ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s
c1 : ℝ
hc1 : 0 < c1
hc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume
c2 : ℝ
hc2 : 0 < c2
hc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume
this : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1
⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi 0) volume
state_after: case intro.intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
a b : ℝ
f : ℝ → ℝ
s : ℝ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s
c1 : ℝ
hc1 : 0 < c1
hc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume
c2 : ℝ
hc2 : 0 < c2
hc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume
this : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1
⊢ (IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume ∧
      IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc c2 c1) volume) ∧
    IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume
tactic: rw [this, integrableOn_union, integrableOn_union]
state_before: case intro.intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
a b : ℝ
f : ℝ → ℝ
s : ℝ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s
c1 : ℝ
hc1 : 0 < c1
hc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume
c2 : ℝ
hc2 : 0 < c2
hc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume
this : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1
⊢ (IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume ∧
      IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc c2 c1) volume) ∧
    IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume
state_after: case intro.intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
a b : ℝ
f : ℝ → ℝ
s : ℝ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s
c1 : ℝ
hc1 : 0 < c1
hc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume
c2 : ℝ
hc2 : 0 < c2
hc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume
this : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1
⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Icc c2 c1) volume
tactic: refine ⟨⟨hc2', integrableOn_Icc_iff_integrableOn_Ioc.mp ?_⟩, hc1'⟩
state_before: case intro.intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
a b : ℝ
f : ℝ → ℝ
s : ℝ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s
c1 : ℝ
hc1 : 0 < c1
hc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume
c2 : ℝ
hc2 : 0 < c2
hc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume
this : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1
⊢ IntegrableOn (fun t => t ^ (s - 1) * f t) (Icc c2 c1) volume
state_after: case intro.intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
a b : ℝ
f : ℝ → ℝ
s : ℝ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s
c1 : ℝ
hc1 : 0 < c1
hc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume
c2 : ℝ
hc2 : 0 < c2
hc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume
this : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1
⊢ ContinuousOn (fun t => t ^ (s - 1)) (Ioi 0)
tactic: refine
    (hfc.continuousOn_mul ?_ isOpen_Ioi).integrableOn_compact_subset
      (fun t ht => (hc2.trans_le ht.1 : 0 < t)) isCompact_Icc
state_before: case intro.intro.intro.intro
E : Type u_1
inst✝ : NormedAddCommGroup E
a b : ℝ
f : ℝ → ℝ
s : ℝ
hfc : LocallyIntegrableOn f (Ioi 0) volume
hf_top : f =O[atTop] fun x => x ^ (-a)
hs_top : s < a
hf_bot : f =O[𝓝[>] 0] fun x => x ^ (-b)
hs_bot : b < s
c1 : ℝ
hc1 : 0 < c1
hc1' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioi c1) volume
c2 : ℝ
hc2 : 0 < c2
hc2' : IntegrableOn (fun t => t ^ (s - 1) * f t) (Ioc 0 c2) volume
this : Ioi 0 = Ioc 0 c2 ∪ Ioc c2 c1 ∪ Ioi c1
⊢ ContinuousOn (fun t => t ^ (s - 1)) (Ioi 0)
state_after: no goals
tactic: exact ContinuousAt.continuousOn fun t ht => continuousAt_rpow_const _ _ <| Or.inl <| ne_of_gt ht