Theorem: DirichletCharacter.not_LSeriesSummable_at_one
File: Mathlib/NumberTheory/LSeries/Dirichlet.lean

tactic: refine fun h ↦ (Real.not_summable_indicator_one_div_natCast hN 1) ?_
state_before: N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
⊢ ¬LSeriesSummable (fun n => χ ↑n) 1
state_after: N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
h : LSeriesSummable (fun n => χ ↑n) 1
⊢ Summable ({n | ↑n = 1}.indicator fun n => 1 / ↑n)

tactic: refine h.norm.of_nonneg_of_le (fun m ↦ Set.indicator_apply_nonneg (fun _ ↦ by positivity))
  (fun n ↦ ?_)
state_before: N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
h : LSeriesSummable (fun n => χ ↑n) 1
⊢ Summable ({n | ↑n = 1}.indicator fun n => 1 / ↑n)
state_after: N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
h : LSeriesSummable (fun n => χ ↑n) 1
n : ℕ
⊢ {n | ↑n = 1}.indicator (fun n => 1 / ↑n) n ≤ ‖term (fun n => χ ↑n) 1 n‖

tactic: rw [norm_term_eq, one_re, Real.rpow_one, Set.indicator]
state_before: N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
h : LSeriesSummable (fun n => χ ↑n) 1
n : ℕ
⊢ {n | ↑n = 1}.indicator (fun n => 1 / ↑n) n ≤ ‖term (fun n => χ ↑n) 1 n‖
state_after: N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
h : LSeriesSummable (fun n => χ ↑n) 1
n : ℕ
⊢ (if n ∈ {n | ↑n = 1} then 1 / ↑n else 0) ≤ if n = 0 then 0 else ‖χ ↑n‖ / ↑n

tactic: split_ifs with h₁ h₂
state_before: N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
h : LSeriesSummable (fun n => χ ↑n) 1
n : ℕ
⊢ (if n ∈ {n | ↑n = 1} then 1 / ↑n else 0) ≤ if n = 0 then 0 else ‖χ ↑n‖ / ↑n
state_after: case pos
N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
h : LSeriesSummable (fun n => χ ↑n) 1
n : ℕ
h₁ : n ∈ {n | ↑n = 1}
h₂ : n = 0
⊢ 1 / ↑n ≤ 0

case neg
N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
h : LSeriesSummable (fun n => χ ↑n) 1
n : ℕ
h₁ : n ∈ {n | ↑n = 1}
h₂ : ¬n = 0
⊢ 1 / ↑n ≤ ‖χ ↑n‖ / ↑n

case pos
N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
h : LSeriesSummable (fun n => χ ↑n) 1
n : ℕ
h₁ : n ∉ {n | ↑n = 1}
h✝ : n = 0
⊢ 0 ≤ 0

case neg
N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
h : LSeriesSummable (fun n => χ ↑n) 1
n : ℕ
h₁ : n ∉ {n | ↑n = 1}
h✝ : ¬n = 0
⊢ 0 ≤ ‖χ ↑n‖ / ↑n

tactic: all_goals positivity
state_before: case pos
N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
h : LSeriesSummable (fun n => χ ↑n) 1
n : ℕ
h₁ : n ∉ {n | ↑n = 1}
h✝ : n = 0
⊢ 0 ≤ 0

case neg
N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
h : LSeriesSummable (fun n => χ ↑n) 1
n : ℕ
h₁ : n ∉ {n | ↑n = 1}
h✝ : ¬n = 0
⊢ 0 ≤ ‖χ ↑n‖ / ↑n
state_after: no goals

tactic: positivity
state_before: N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
h : LSeriesSummable (fun n => χ ↑n) 1
m : ℕ
x✝ : m ∈ {n | ↑n = 1}
⊢ 0 ≤ 1 / ↑m
state_after: no goals

tactic: rw [h₂, cast_zero, div_zero]
state_before: case pos
N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
h : LSeriesSummable (fun n => χ ↑n) 1
n : ℕ
h₁ : n ∈ {n | ↑n = 1}
h₂ : n = 0
⊢ 1 / ↑n ≤ 0
state_after: no goals

tactic: rw [h₁, χ.map_one, norm_one]
state_before: case neg
N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
h : LSeriesSummable (fun n => χ ↑n) 1
n : ℕ
h₁ : n ∈ {n | ↑n = 1}
h₂ : ¬n = 0
⊢ 1 / ↑n ≤ ‖χ ↑n‖ / ↑n
state_after: no goals

tactic: positivity
state_before: case neg
N : ℕ
hN : N ≠ 0
χ : DirichletCharacter ℂ N
h : LSeriesSummable (fun n => χ ↑n) 1
n : ℕ
h₁ : n ∉ {n | ↑n = 1}
h✝ : ¬n = 0
⊢ 0 ≤ ‖χ ↑n‖ / ↑n
state_after: no goals