l3lab/ntp-mathlib huggingface dataset:
/-- A monoidal functor induces a monoidal functor between
the categories of `G`-actions within those categories. -/
@[simps]
def mapAction : MonoidalFunctor (Action V G) (Action W G) where
  toFunctor := F.toFunctor.mapAction G
  ε :=
    { hom := F.ε
      comm := fun g => by
        dsimp [FunctorCategoryEquivalence.inverse, Functor.mapAction]
        rw [Category.id_comp, F.map_id, Category.comp_id] }
  μ := fun X Y =>
    { hom := F.μ X.V Y.V
      comm := fun g => F.toLaxMonoidalFunctor.μ_natural (X.ρ g) (Y.ρ g) }
  -- using `dsimp` before `simp` speeds these up
  μ_natural {X Y X' Y'} f g := by ext; dsimp; simp
  associativity X Y Z := by ext; dsimp; simp
  left_unitality X := by ext; 

Proof state:
case h
V : Type (u + 1)
inst✝³ : LargeCategory V
G✝ : MonCat
W : Type (u + 1)
inst✝² : LargeCategory W
inst✝¹ : MonoidalCategory V
inst✝ : MonoidalCategory W
F : MonoidalFunctor V W
G : MonCat
X : Action V G
⊢ (MonoidalCategory.leftUnitor ((Functor.mapAction F.toFunctor G).obj X)).hom.hom =
    (MonoidalCategory.tensorHom (Hom.mk F.ε) (𝟙 ((Functor.mapAction F.toFunctor G).obj X)) ≫
        (fun X Y => Hom.mk (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor X.V Y.V)) (𝟙_ (Action V G)) X ≫
          (Functor.mapAction F.toFunctor G).map (MonoidalCategory.leftUnitor X).hom).hom

Next tactic:
dsimp
/-- A monoidal functor induces a monoidal functor between
the categories of `G`-actions within those categories. -/
@[simps]
def mapAction : MonoidalFunctor (Action V G) (Action W G) where
  toFunctor := F.toFunctor.mapAction G
  ε :=
    { hom := F.ε
      comm := fun g => by
        dsimp [FunctorCategoryEquivalence.inverse, Functor.mapAction]
        rw [Category.id_comp, F.map_id, Category.comp_id] }
  μ := fun X Y =>
    { hom := F.μ X.V Y.V
      comm := fun g => F.toLaxMonoidalFunctor.μ_natural (X.ρ g) (Y.ρ g) }
  -- using `dsimp` before `simp` speeds these up
  μ_natural {X Y X' Y'} f g := by ext; dsimp; simp
  associativity X Y Z := by ext; dsimp; simp
  left_unitality X := by ext; dsimp; 

Proof state:
case h
V : Type (u + 1)
inst✝³ : LargeCategory V
G✝ : MonCat
W : Type (u + 1)
inst✝² : LargeCategory W
inst✝¹ : MonoidalCategory V
inst✝ : MonoidalCategory W
F : MonoidalFunctor V W
G : MonCat
X : Action V G
⊢ (MonoidalCategory.tensorHom (𝟙 (𝟙_ W)) (𝟙 (F.obj X.V)) ≫ (MonoidalCategory.leftUnitor (F.obj X.V)).hom) ≫
      𝟙 (F.obj X.V) =
    MonoidalCategory.tensorHom F.ε (𝟙 (F.obj X.V)) ≫
      LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor (𝟙_ V) X.V ≫
        F.map ((MonoidalCategory.tensorHom (𝟙 (𝟙_ V)) (𝟙 X.V) ≫ (MonoidalCategory.leftUnitor X.V).hom) ≫ 𝟙 X.V)

Next tactic:
simp
/-- A monoidal functor induces a monoidal functor between
the categories of `G`-actions within those categories. -/
@[simps]
def mapAction : MonoidalFunctor (Action V G) (Action W G) where
  toFunctor := F.toFunctor.mapAction G
  ε :=
    { hom := F.ε
      comm := fun g => by
        dsimp [FunctorCategoryEquivalence.inverse, Functor.mapAction]
        rw [Category.id_comp, F.map_id, Category.comp_id] }
  μ := fun X Y =>
    { hom := F.μ X.V Y.V
      comm := fun g => F.toLaxMonoidalFunctor.μ_natural (X.ρ g) (Y.ρ g) }
  -- using `dsimp` before `simp` speeds these up
  μ_natural {X Y X' Y'} f g := by ext; dsimp; simp
  associativity X Y Z := by ext; dsimp; simp
  left_unitality X := by ext; dsimp; simp
  right_unitality X := by
    

Proof state:
V : Type (u + 1)
inst✝³ : LargeCategory V
G✝ : MonCat
W : Type (u + 1)
inst✝² : LargeCategory W
inst✝¹ : MonoidalCategory V
inst✝ : MonoidalCategory W
F : MonoidalFunctor V W
G : MonCat
X : Action V G
⊢ (MonoidalCategory.rightUnitor ((Functor.mapAction F.toFunctor G).obj X)).hom =
    MonoidalCategory.tensorHom (𝟙 ((Functor.mapAction F.toFunctor G).obj X)) (Hom.mk F.ε) ≫
      (fun X Y => Hom.mk (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor X.V Y.V)) X (𝟙_ (Action V G)) ≫
        (Functor.mapAction F.toFunctor G).map (MonoidalCategory.rightUnitor X).hom

Next tactic:
ext
/-- A monoidal functor induces a monoidal functor between
the categories of `G`-actions within those categories. -/
@[simps]
def mapAction : MonoidalFunctor (Action V G) (Action W G) where
  toFunctor := F.toFunctor.mapAction G
  ε :=
    { hom := F.ε
      comm := fun g => by
        dsimp [FunctorCategoryEquivalence.inverse, Functor.mapAction]
        rw [Category.id_comp, F.map_id, Category.comp_id] }
  μ := fun X Y =>
    { hom := F.μ X.V Y.V
      comm := fun g => F.toLaxMonoidalFunctor.μ_natural (X.ρ g) (Y.ρ g) }
  -- using `dsimp` before `simp` speeds these up
  μ_natural {X Y X' Y'} f g := by ext; dsimp; simp
  associativity X Y Z := by ext; dsimp; simp
  left_unitality X := by ext; dsimp; simp
  right_unitality X := by
    ext
    

Proof state:
case h
V : Type (u + 1)
inst✝³ : LargeCategory V
G✝ : MonCat
W : Type (u + 1)
inst✝² : LargeCategory W
inst✝¹ : MonoidalCategory V
inst✝ : MonoidalCategory W
F : MonoidalFunctor V W
G : MonCat
X : Action V G
⊢ (MonoidalCategory.rightUnitor ((Functor.mapAction F.toFunctor G).obj X)).hom.hom =
    (MonoidalCategory.tensorHom (𝟙 ((Functor.mapAction F.toFunctor G).obj X)) (Hom.mk F.ε) ≫
        (fun X Y => Hom.mk (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor X.V Y.V)) X (𝟙_ (Action V G)) ≫
          (Functor.mapAction F.toFunctor G).map (MonoidalCategory.rightUnitor X).hom).hom

Next tactic:
dsimp
/-- A monoidal functor induces a monoidal functor between
the categories of `G`-actions within those categories. -/
@[simps]
def mapAction : MonoidalFunctor (Action V G) (Action W G) where
  toFunctor := F.toFunctor.mapAction G
  ε :=
    { hom := F.ε
      comm := fun g => by
        dsimp [FunctorCategoryEquivalence.inverse, Functor.mapAction]
        rw [Category.id_comp, F.map_id, Category.comp_id] }
  μ := fun X Y =>
    { hom := F.μ X.V Y.V
      comm := fun g => F.toLaxMonoidalFunctor.μ_natural (X.ρ g) (Y.ρ g) }
  -- using `dsimp` before `simp` speeds these up
  μ_natural {X Y X' Y'} f g := by ext; dsimp; simp
  associativity X Y Z := by ext; dsimp; simp
  left_unitality X := by ext; dsimp; simp
  right_unitality X := by
    ext
    dsimp
    

Proof state:
case h
V : Type (u + 1)
inst✝³ : LargeCategory V
G✝ : MonCat
W : Type (u + 1)
inst✝² : LargeCategory W
inst✝¹ : MonoidalCategory V
inst✝ : MonoidalCategory W
F : MonoidalFunctor V W
G : MonCat
X : Action V G
⊢ (MonoidalCategory.tensorHom (𝟙 (F.obj X.V)) (𝟙 (𝟙_ W)) ≫ (MonoidalCategory.rightUnitor (F.obj X.V)).hom) ≫
      𝟙 (F.obj X.V) =
    MonoidalCategory.tensorHom (𝟙 (F.obj X.V)) F.ε ≫
      LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor X.V (𝟙_ V) ≫
        F.map ((MonoidalCategory.tensorHom (𝟙 X.V) (𝟙 (𝟙_ V)) ≫ (MonoidalCategory.rightUnitor X.V).hom) ≫ 𝟙 X.V)

Next tactic:
simp only [MonoidalCategory.rightUnitor_conjugation,
      LaxMonoidalFunctor.right_unitality, Category.id_comp, Category.assoc,
      LaxMonoidalFunctor.right_unitality_inv_assoc, Category.comp_id, Iso.hom_inv_id]
/-- A monoidal functor induces a monoidal functor between
the categories of `G`-actions within those categories. -/
@[simps]
def mapAction : MonoidalFunctor (Action V G) (Action W G) where
  toFunctor := F.toFunctor.mapAction G
  ε :=
    { hom := F.ε
      comm := fun g => by
        dsimp [FunctorCategoryEquivalence.inverse, Functor.mapAction]
        rw [Category.id_comp, F.map_id, Category.comp_id] }
  μ := fun X Y =>
    { hom := F.μ X.V Y.V
      comm := fun g => F.toLaxMonoidalFunctor.μ_natural (X.ρ g) (Y.ρ g) }
  -- using `dsimp` before `simp` speeds these up
  μ_natural {X Y X' Y'} f g := by ext; dsimp; simp
  associativity X Y Z := by ext; dsimp; simp
  left_unitality X := by ext; dsimp; simp
  right_unitality X := by
    ext
    dsimp
    simp only [MonoidalCategory.rightUnitor_conjugation,
      LaxMonoidalFunctor.right_unitality, Category.id_comp, Category.assoc,
      LaxMonoidalFunctor.right_unitality_inv_assoc, Category.comp_id, Iso.hom_inv_id]
    

Proof state:
case h
V : Type (u + 1)
inst✝³ : LargeCategory V
G✝ : MonCat
W : Type (u + 1)
inst✝² : LargeCategory W
inst✝¹ : MonoidalCategory V
inst✝ : MonoidalCategory W
F : MonoidalFunctor V W
G : MonCat
X : Action V G
⊢ MonoidalCategory.tensorHom (𝟙 (F.obj X.V)) F.ε ≫
      LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor X.V (𝟙_ V) ≫
        F.map (MonoidalCategory.rightUnitor X.V).hom ≫
          F.map (MonoidalCategory.rightUnitor X.V).inv ≫ F.map (MonoidalCategory.rightUnitor X.V).hom =
    MonoidalCategory.tensorHom (𝟙 (F.obj X.V)) F.ε ≫
      LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor X.V (𝟙_ V) ≫ F.map (MonoidalCategory.rightUnitor X.V).hom

Next tactic:
rw [← F.map_comp, Iso.inv_hom_id, F.map_id, Category.comp_id]
@[simp]
theorem mapAction_ε_inv_hom : (inv (F.mapAction G).ε).hom = inv F.ε := by
  

Proof state:
V : Type (u + 1)
inst✝³ : LargeCategory V
G✝ : MonCat
W : Type (u + 1)
inst✝² : LargeCategory W
inst✝¹ : MonoidalCategory V
inst✝ : MonoidalCategory W
F : MonoidalFunctor V W
G : MonCat
⊢ (inv (mapAction F G).toLaxMonoidalFunctor.ε).hom = inv F.ε

Next tactic:
rw [← cancel_mono F.ε, IsIso.inv_hom_id, ← F.mapAction_toLaxMonoidalFunctor_ε_hom G,
    ← Action.comp_hom, IsIso.inv_hom_id, Action.id_hom]
@[simp]
theorem mapAction_μ_inv_hom (X Y : Action V G) :
    (inv ((F.mapAction G).μ X Y)).hom = inv (F.μ X.V Y.V) := by
  

Proof state:
V : Type (u + 1)
inst✝³ : LargeCategory V
G✝ : MonCat
W : Type (u + 1)
inst✝² : LargeCategory W
inst✝¹ : MonoidalCategory V
inst✝ : MonoidalCategory W
F : MonoidalFunctor V W
G : MonCat
X Y : Action V G
⊢ (inv (LaxMonoidalFunctor.μ (mapAction F G).toLaxMonoidalFunctor X Y)).hom =
    inv (LaxMonoidalFunctor.μ F.toLaxMonoidalFunctor X.V Y.V)

Next tactic:
rw [← cancel_mono (F.μ X.V Y.V), IsIso.inv_hom_id, ← F.mapAction_toLaxMonoidalFunctor_μ_hom G,
    ← Action.comp_hom, IsIso.inv_hom_id, Action.id_hom]
@[simp]
theorem stirlingSeq_zero : stirlingSeq 0 = 0 := by
  

Proof state:
⊢ stirlingSeq 0 = 0

Next tactic:
rw [stirlingSeq, cast_zero, mul_zero, Real.sqrt_zero, zero_mul,
    div_zero]
@[simp]
theorem stirlingSeq_one : stirlingSeq 1 = exp 1 / Real.sqrt 2 := by
  

Proof state:
⊢ stirlingSeq 1 = rexp 1 / Real.sqrt 2

Next tactic:
rw [stirlingSeq, pow_one, factorial_one, cast_one, mul_one, mul_one_div, one_div_div]
theorem log_stirlingSeq_formula (n : ℕ) :
    log (stirlingSeq n) = Real.log n ! - 1 / 2 * Real.log (2 * n) - n * log (n / exp 1) := by
  

Proof state:
n : ℕ
⊢ Real.log (stirlingSeq n) = Real.log ↑n ! - 1 / 2 * Real.log (2 * ↑n) - ↑n * Real.log (↑n / rexp 1)

Next tactic:
cases n
theorem log_stirlingSeq_formula (n : ℕ) :
    log (stirlingSeq n) = Real.log n ! - 1 / 2 * Real.log (2 * n) - n * log (n / exp 1) := by
  cases n
  · 

Proof state:
case zero
⊢ Real.log (stirlingSeq zero) = Real.log ↑zero ! - 1 / 2 * Real.log (2 * ↑zero) - ↑zero * Real.log (↑zero / rexp 1)

Next tactic:
simp
theorem log_stirlingSeq_formula (n : ℕ) :
    log (stirlingSeq n) = Real.log n ! - 1 / 2 * Real.log (2 * n) - n * log (n / exp 1) := by
  cases n
  · simp
  · 

Proof state:
case succ
n✝ : ℕ
⊢ Real.log (stirlingSeq (succ n✝)) =
    Real.log ↑(succ n✝)! - 1 / 2 * Real.log (2 * ↑(succ n✝)) - ↑(succ n✝) * Real.log (↑(succ n✝) / rexp 1)

Next tactic:
rw [stirlingSeq, log_div, log_mul, sqrt_eq_rpow, log_rpow, Real.log_pow, tsub_tsub]
theorem log_stirlingSeq_formula (n : ℕ) :
    log (stirlingSeq n) = Real.log n ! - 1 / 2 * Real.log (2 * n) - n * log (n / exp 1) := by
  cases n
  · simp
  · rw [stirlingSeq, log_div, log_mul, sqrt_eq_rpow, log_rpow, Real.log_pow, tsub_tsub]
      <;> 

Proof state:
case succ.hx
n✝ : ℕ
⊢ 0 < 2 * ↑(succ n✝)

Next tactic:
positivity
theorem log_stirlingSeq_formula (n : ℕ) :
    log (stirlingSeq n) = Real.log n ! - 1 / 2 * Real.log (2 * n) - n * log (n / exp 1) := by
  cases n
  · simp
  · rw [stirlingSeq, log_div, log_mul, sqrt_eq_rpow, log_rpow, Real.log_pow, tsub_tsub]
      <;> 

Proof state:
case succ.hx
n✝ : ℕ
⊢ Real.sqrt (2 * ↑(succ n✝)) ≠ 0

Next tactic:
positivity
theorem log_stirlingSeq_formula (n : ℕ) :
    log (stirlingSeq n) = Real.log n ! - 1 / 2 * Real.log (2 * n) - n * log (n / exp 1) := by
  cases n
  · simp
  · rw [stirlingSeq, log_div, log_mul, sqrt_eq_rpow, log_rpow, Real.log_pow, tsub_tsub]
      <;> 

Proof state:
case succ.hy
n✝ : ℕ
⊢ (↑(succ n✝) / rexp 1) ^ succ n✝ ≠ 0

Next tactic:
positivity
theorem log_stirlingSeq_formula (n : ℕ) :
    log (stirlingSeq n) = Real.log n ! - 1 / 2 * Real.log (2 * n) - n * log (n / exp 1) := by
  cases n
  · simp
  · rw [stirlingSeq, log_div, log_mul, sqrt_eq_rpow, log_rpow, Real.log_pow, tsub_tsub]
      <;> 

Proof state:
case succ.hx
n✝ : ℕ
⊢ ↑(succ n✝)! ≠ 0

Next tactic:
positivity
theorem log_stirlingSeq_formula (n : ℕ) :
    log (stirlingSeq n) = Real.log n ! - 1 / 2 * Real.log (2 * n) - n * log (n / exp 1) := by
  cases n
  · simp
  · rw [stirlingSeq, log_div, log_mul, sqrt_eq_rpow, log_rpow, Real.log_pow, tsub_tsub]
      <;> 

Proof state:
case succ.hy
n✝ : ℕ
⊢ Real.sqrt (2 * ↑(succ n✝)) * (↑(succ n✝) / rexp 1) ^ succ n✝ ≠ 0

Next tactic:
positivity
/-- The sequence `log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))` has the series expansion
   `∑ 1 / (2 * (k + 1) + 1) * (1 / 2 * (m + 1) + 1)^(2 * (k + 1))`
-/
theorem log_stirlingSeq_diff_hasSum (m : ℕ) :
    HasSum (fun k : ℕ => ↑1 / (↑2 * ↑(k + 1) + ↑1) * (((1:ℝ)/(↑2 * ↑(m + 1) + ↑1)) ^ 2) ^ ↑(k + 1))
      (log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))) := by
  

Proof state:
m : ℕ
⊢ HasSum (fun k => 1 / (2 * ↑(k + 1) + 1) * ((1 / (2 * ↑(m + 1) + 1)) ^ 2) ^ (k + 1))
    (Real.log (stirlingSeq (m + 1)) - Real.log (stirlingSeq (m + 2)))

Next tactic:
change HasSum
    ((fun b : ℕ => (1:ℝ) / ((2:ℝ) * b + (1:ℝ)) * (((1:ℝ) / (2 * (m + 1 :) + 1)) ^ 2) ^ b) ∘ succ) _
/-- The sequence `log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))` has the series expansion
   `∑ 1 / (2 * (k + 1) + 1) * (1 / 2 * (m + 1) + 1)^(2 * (k + 1))`
-/
theorem log_stirlingSeq_diff_hasSum (m : ℕ) :
    HasSum (fun k : ℕ => ↑1 / (↑2 * ↑(k + 1) + ↑1) * (((1:ℝ)/(↑2 * ↑(m + 1) + ↑1)) ^ 2) ^ ↑(k + 1))
      (log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))) := by
  change HasSum
    ((fun b : ℕ => (1:ℝ) / ((2:ℝ) * b + (1:ℝ)) * (((1:ℝ) / (2 * (m + 1 :) + 1)) ^ 2) ^ b) ∘ succ) _
  

Proof state:
m : ℕ
⊢ HasSum ((fun b => 1 / (2 * ↑b + 1) * ((1 / (2 * ↑(m + 1) + 1)) ^ 2) ^ b) ∘ succ)
    (Real.log (stirlingSeq (m + 1)) - Real.log (stirlingSeq (m + 2)))

Next tactic:
refine' (hasSum_nat_add_iff (a := _ - _) -- Porting note: must give implicit arguments
    (f := (fun b : ℕ => ↑1 / (↑2 * b + ↑1) * (((1:ℝ) / (2 * ↑(m + 1) + 1)) ^ 2) ^ b)) 1).mpr _
/-- The sequence `log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))` has the series expansion
   `∑ 1 / (2 * (k + 1) + 1) * (1 / 2 * (m + 1) + 1)^(2 * (k + 1))`
-/
theorem log_stirlingSeq_diff_hasSum (m : ℕ) :
    HasSum (fun k : ℕ => ↑1 / (↑2 * ↑(k + 1) + ↑1) * (((1:ℝ)/(↑2 * ↑(m + 1) + ↑1)) ^ 2) ^ ↑(k + 1))
      (log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))) := by
  change HasSum
    ((fun b : ℕ => (1:ℝ) / ((2:ℝ) * b + (1:ℝ)) * (((1:ℝ) / (2 * (m + 1 :) + 1)) ^ 2) ^ b) ∘ succ) _
  refine' (hasSum_nat_add_iff (a := _ - _) -- Porting note: must give implicit arguments
    (f := (fun b : ℕ => ↑1 / (↑2 * b + ↑1) * (((1:ℝ) / (2 * ↑(m + 1) + 1)) ^ 2) ^ b)) 1).mpr _
  

Proof state:
m : ℕ
⊢ HasSum (fun b => 1 / (2 * ↑b + 1) * ((1 / (2 * ↑(m + 1) + 1)) ^ 2) ^ b)
    (Real.log (stirlingSeq (m + 1)) - Real.log (stirlingSeq (m + 2)) +
      ∑ i in range 1, 1 / (2 * ↑i + 1) * ((1 / (2 * ↑(m + 1) + 1)) ^ 2) ^ i)

Next tactic:
convert (hasSum_log_one_add_inv <|
    cast_pos.mpr (succ_pos m)).mul_left ((↑(m + 1) : ℝ) + 1 / 2) using 1
/-- The sequence `log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))` has the series expansion
   `∑ 1 / (2 * (k + 1) + 1) * (1 / 2 * (m + 1) + 1)^(2 * (k + 1))`
-/
theorem log_stirlingSeq_diff_hasSum (m : ℕ) :
    HasSum (fun k : ℕ => ↑1 / (↑2 * ↑(k + 1) + ↑1) * (((1:ℝ)/(↑2 * ↑(m + 1) + ↑1)) ^ 2) ^ ↑(k + 1))
      (log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))) := by
  change HasSum
    ((fun b : ℕ => (1:ℝ) / ((2:ℝ) * b + (1:ℝ)) * (((1:ℝ) / (2 * (m + 1 :) + 1)) ^ 2) ^ b) ∘ succ) _
  refine' (hasSum_nat_add_iff (a := _ - _) -- Porting note: must give implicit arguments
    (f := (fun b : ℕ => ↑1 / (↑2 * b + ↑1) * (((1:ℝ) / (2 * ↑(m + 1) + 1)) ^ 2) ^ b)) 1).mpr _
  convert (hasSum_log_one_add_inv <|
    cast_pos.mpr (succ_pos m)).mul_left ((↑(m + 1) : ℝ) + 1 / 2) using 1
  · 

Proof state:
case h.e'_5
m : ℕ
⊢ (fun b => 1 / (2 * ↑b + 1) * ((1 / (2 * ↑(m + 1) + 1)) ^ 2) ^ b) = fun i =>
    (↑(m + 1) + 1 / 2) * (2 * (1 / (2 * ↑i + 1)) * (1 / (2 * ↑(succ m) + 1)) ^ (2 * i + 1))

Next tactic:
ext k
/-- The sequence `log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))` has the series expansion
   `∑ 1 / (2 * (k + 1) + 1) * (1 / 2 * (m + 1) + 1)^(2 * (k + 1))`
-/
theorem log_stirlingSeq_diff_hasSum (m : ℕ) :
    HasSum (fun k : ℕ => ↑1 / (↑2 * ↑(k + 1) + ↑1) * (((1:ℝ)/(↑2 * ↑(m + 1) + ↑1)) ^ 2) ^ ↑(k + 1))
      (log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))) := by
  change HasSum
    ((fun b : ℕ => (1:ℝ) / ((2:ℝ) * b + (1:ℝ)) * (((1:ℝ) / (2 * (m + 1 :) + 1)) ^ 2) ^ b) ∘ succ) _
  refine' (hasSum_nat_add_iff (a := _ - _) -- Porting note: must give implicit arguments
    (f := (fun b : ℕ => ↑1 / (↑2 * b + ↑1) * (((1:ℝ) / (2 * ↑(m + 1) + 1)) ^ 2) ^ b)) 1).mpr _
  convert (hasSum_log_one_add_inv <|
    cast_pos.mpr (succ_pos m)).mul_left ((↑(m + 1) : ℝ) + 1 / 2) using 1
  · ext k
    

Proof state:
case h.e'_5.h
m k : ℕ
⊢ 1 / (2 * ↑k + 1) * ((1 / (2 * ↑(m + 1) + 1)) ^ 2) ^ k =
    (↑(m + 1) + 1 / 2) * (2 * (1 / (2 * ↑k + 1)) * (1 / (2 * ↑(succ m) + 1)) ^ (2 * k + 1))

Next tactic:
rw [← pow_mul, pow_add]
/-- The sequence `log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))` has the series expansion
   `∑ 1 / (2 * (k + 1) + 1) * (1 / 2 * (m + 1) + 1)^(2 * (k + 1))`
-/
theorem log_stirlingSeq_diff_hasSum (m : ℕ) :
    HasSum (fun k : ℕ => ↑1 / (↑2 * ↑(k + 1) + ↑1) * (((1:ℝ)/(↑2 * ↑(m + 1) + ↑1)) ^ 2) ^ ↑(k + 1))
      (log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))) := by
  change HasSum
    ((fun b : ℕ => (1:ℝ) / ((2:ℝ) * b + (1:ℝ)) * (((1:ℝ) / (2 * (m + 1 :) + 1)) ^ 2) ^ b) ∘ succ) _
  refine' (hasSum_nat_add_iff (a := _ - _) -- Porting note: must give implicit arguments
    (f := (fun b : ℕ => ↑1 / (↑2 * b + ↑1) * (((1:ℝ) / (2 * ↑(m + 1) + 1)) ^ 2) ^ b)) 1).mpr _
  convert (hasSum_log_one_add_inv <|
    cast_pos.mpr (succ_pos m)).mul_left ((↑(m + 1) : ℝ) + 1 / 2) using 1
  · ext k
    rw [← pow_mul, pow_add]
    

Proof state:
case h.e'_5.h
m k : ℕ
⊢ 1 / (2 * ↑k + 1) * (1 / (2 * ↑(m + 1) + 1)) ^ (2 * k) =
    (↑(m + 1) + 1 / 2) *
      (2 * (1 / (2 * ↑k + 1)) * ((1 / (2 * ↑(succ m) + 1)) ^ (2 * k) * (1 / (2 * ↑(succ m) + 1)) ^ 1))

Next tactic:
push_cast
/-- The sequence `log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))` has the series expansion
   `∑ 1 / (2 * (k + 1) + 1) * (1 / 2 * (m + 1) + 1)^(2 * (k + 1))`
-/
theorem log_stirlingSeq_diff_hasSum (m : ℕ) :
    HasSum (fun k : ℕ => ↑1 / (↑2 * ↑(k + 1) + ↑1) * (((1:ℝ)/(↑2 * ↑(m + 1) + ↑1)) ^ 2) ^ ↑(k + 1))
      (log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))) := by
  change HasSum
    ((fun b : ℕ => (1:ℝ) / ((2:ℝ) * b + (1:ℝ)) * (((1:ℝ) / (2 * (m + 1 :) + 1)) ^ 2) ^ b) ∘ succ) _
  refine' (hasSum_nat_add_iff (a := _ - _) -- Porting note: must give implicit arguments
    (f := (fun b : ℕ => ↑1 / (↑2 * b + ↑1) * (((1:ℝ) / (2 * ↑(m + 1) + 1)) ^ 2) ^ b)) 1).mpr _
  convert (hasSum_log_one_add_inv <|
    cast_pos.mpr (succ_pos m)).mul_left ((↑(m + 1) : ℝ) + 1 / 2) using 1
  · ext k
    rw [← pow_mul, pow_add]
    push_cast
    

Proof state:
case h.e'_5.h
m k : ℕ
⊢ 1 / (2 * ↑k + 1) * (1 / (2 * (↑m + 1) + 1)) ^ (2 * k) =
    (↑m + 1 + 1 / 2) * (2 * (1 / (2 * ↑k + 1)) * ((1 / (2 * (↑m + 1) + 1)) ^ (2 * k) * (1 / (2 * (↑m + 1) + 1)) ^ 1))

Next tactic:
field_simp
/-- The sequence `log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))` has the series expansion
   `∑ 1 / (2 * (k + 1) + 1) * (1 / 2 * (m + 1) + 1)^(2 * (k + 1))`
-/
theorem log_stirlingSeq_diff_hasSum (m : ℕ) :
    HasSum (fun k : ℕ => ↑1 / (↑2 * ↑(k + 1) + ↑1) * (((1:ℝ)/(↑2 * ↑(m + 1) + ↑1)) ^ 2) ^ ↑(k + 1))
      (log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))) := by
  change HasSum
    ((fun b : ℕ => (1:ℝ) / ((2:ℝ) * b + (1:ℝ)) * (((1:ℝ) / (2 * (m + 1 :) + 1)) ^ 2) ^ b) ∘ succ) _
  refine' (hasSum_nat_add_iff (a := _ - _) -- Porting note: must give implicit arguments
    (f := (fun b : ℕ => ↑1 / (↑2 * b + ↑1) * (((1:ℝ) / (2 * ↑(m + 1) + 1)) ^ 2) ^ b)) 1).mpr _
  convert (hasSum_log_one_add_inv <|
    cast_pos.mpr (succ_pos m)).mul_left ((↑(m + 1) : ℝ) + 1 / 2) using 1
  · ext k
    rw [← pow_mul, pow_add]
    push_cast
    field_simp
    

Proof state:
case h.e'_5.h
m k : ℕ
⊢ (2 * ↑k + 1) * ((2 * (↑m + 1) + 1) ^ (2 * k) * (2 * (↑m + 1) + 1)) =
    ((↑m + 1) * 2 + 1) * ((2 * ↑k + 1) * (2 * (↑m + 1) + 1) ^ (2 * k))

Next tactic:
ring
/-- The sequence `log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))` has the series expansion
   `∑ 1 / (2 * (k + 1) + 1) * (1 / 2 * (m + 1) + 1)^(2 * (k + 1))`
-/
theorem log_stirlingSeq_diff_hasSum (m : ℕ) :
    HasSum (fun k : ℕ => ↑1 / (↑2 * ↑(k + 1) + ↑1) * (((1:ℝ)/(↑2 * ↑(m + 1) + ↑1)) ^ 2) ^ ↑(k + 1))
      (log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))) := by
  change HasSum
    ((fun b : ℕ => (1:ℝ) / ((2:ℝ) * b + (1:ℝ)) * (((1:ℝ) / (2 * (m + 1 :) + 1)) ^ 2) ^ b) ∘ succ) _
  refine' (hasSum_nat_add_iff (a := _ - _) -- Porting note: must give implicit arguments
    (f := (fun b : ℕ => ↑1 / (↑2 * b + ↑1) * (((1:ℝ) / (2 * ↑(m + 1) + 1)) ^ 2) ^ b)) 1).mpr _
  convert (hasSum_log_one_add_inv <|
    cast_pos.mpr (succ_pos m)).mul_left ((↑(m + 1) : ℝ) + 1 / 2) using 1
  · ext k
    rw [← pow_mul, pow_add]
    push_cast
    field_simp
    ring
  · 

Proof state:
case h.e'_6
m : ℕ
⊢ Real.log (stirlingSeq (m + 1)) - Real.log (stirlingSeq (m + 2)) +
      ∑ i in range 1, 1 / (2 * ↑i + 1) * ((1 / (2 * ↑(m + 1) + 1)) ^ 2) ^ i =
    (↑(m + 1) + 1 / 2) * Real.log (1 + (↑(succ m))⁻¹)

Next tactic:
have h : ∀ (x : ℝ) (_ : x ≠ 0), 1 + x⁻¹ = (x + 1) / x := by
      intro x hx; rw [_root_.add_div, div_self hx, inv_eq_one_div]
/-- The sequence `log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))` has the series expansion
   `∑ 1 / (2 * (k + 1) + 1) * (1 / 2 * (m + 1) + 1)^(2 * (k + 1))`
-/
theorem log_stirlingSeq_diff_hasSum (m : ℕ) :
    HasSum (fun k : ℕ => ↑1 / (↑2 * ↑(k + 1) + ↑1) * (((1:ℝ)/(↑2 * ↑(m + 1) + ↑1)) ^ 2) ^ ↑(k + 1))
      (log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))) := by
  change HasSum
    ((fun b : ℕ => (1:ℝ) / ((2:ℝ) * b + (1:ℝ)) * (((1:ℝ) / (2 * (m + 1 :) + 1)) ^ 2) ^ b) ∘ succ) _
  refine' (hasSum_nat_add_iff (a := _ - _) -- Porting note: must give implicit arguments
    (f := (fun b : ℕ => ↑1 / (↑2 * b + ↑1) * (((1:ℝ) / (2 * ↑(m + 1) + 1)) ^ 2) ^ b)) 1).mpr _
  convert (hasSum_log_one_add_inv <|
    cast_pos.mpr (succ_pos m)).mul_left ((↑(m + 1) : ℝ) + 1 / 2) using 1
  · ext k
    rw [← pow_mul, pow_add]
    push_cast
    field_simp
    ring
  · have h : ∀ (x : ℝ) (_ : x ≠ 0), 1 + x⁻¹ = (x + 1) / x := by
      

Proof state:
m : ℕ
⊢ ∀ (x : ℝ), x ≠ 0 → 1 + x⁻¹ = (x + 1) / x

Next tactic:
intro x hx
/-- The sequence `log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))` has the series expansion
   `∑ 1 / (2 * (k + 1) + 1) * (1 / 2 * (m + 1) + 1)^(2 * (k + 1))`
-/
theorem log_stirlingSeq_diff_hasSum (m : ℕ) :
    HasSum (fun k : ℕ => ↑1 / (↑2 * ↑(k + 1) + ↑1) * (((1:ℝ)/(↑2 * ↑(m + 1) + ↑1)) ^ 2) ^ ↑(k + 1))
      (log (stirlingSeq (m + 1)) - log (stirlingSeq (m + 2))) := by
  change HasSum
    ((fun b : ℕ => (1:ℝ) / ((2:ℝ) * b + (1:ℝ)) * (((1:ℝ) / (2 * (m + 1 :) + 1)) ^ 2) ^ b) ∘ succ) _
  refine' (hasSum_nat_add_iff (a := _ - _) -- Porting note: must give implicit arguments
    (f := (fun b : ℕ => ↑1 / (↑2 * b + ↑1) * (((1:ℝ) / (2 * ↑(m + 1) + 1)) ^ 2) ^ b)) 1).mpr _
  convert (hasSum_log_one_add_inv <|
    cast_pos.mpr (succ_pos m)).mul_left ((↑(m + 1) : ℝ) + 1 / 2) using 1
  · ext k
    rw [← pow_mul, pow_add]
    push_cast
    field_simp
    ring
  · have h : ∀ (x : ℝ) (_ : x ≠ 0), 1 + x⁻¹ = (x + 1) / x := by
      intro x hx; 

Proof state:
m : ℕ
x : ℝ
hx : x ≠ 0
⊢ 1 + x⁻¹ = (x + 1) / x

Next tactic:
rw [_root_.add_div, div_self hx, inv_eq_one_div]