Theorem: PhragmenLindelof.eqOn_right_half_plane_of_superexponential_decay
File: Mathlib/Analysis/Complex/PhragmenLindelof.lean

Proof steps:
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f g✝ : ℂ → E
z : ℂ
g : ℂ → E
hfd : DiffContOnCl ℂ f {z | 0 < z.re}
hgd : DiffContOnCl ℂ g {z | 0 < z.re}
hfexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)
hgexp : ∃ c < 2, ∃ B, g =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)
hre : SuperpolynomialDecay atTop expR fun x => ‖f ↑x - g ↑x‖
hfim : ∃ C, ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hgim : ∃ C, ∀ (x : ℝ), ‖g (↑x * I)‖ ≤ C
⊢ EqOn f g {z | 0 ≤ z.re}
state_after: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f g✝ : ℂ → E
z : ℂ
g : ℂ → E
hfd : DiffContOnCl ℂ f {z | 0 < z.re}
hgd : DiffContOnCl ℂ g {z | 0 < z.re}
hfexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)
hgexp : ∃ c < 2, ∃ B, g =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)
hre : SuperpolynomialDecay atTop expR fun x => ‖f ↑x - g ↑x‖
hfim : ∃ C, ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hgim : ∃ C, ∀ (x : ℝ), ‖g (↑x * I)‖ ≤ C
⊢ EqOn (f - g) 0 {z | 0 ≤ z.re}
tactic: suffices EqOn (f - g) 0 {z : ℂ | 0 ≤ z.re} by
    simpa only [EqOn, Pi.sub_apply, Pi.zero_apply, sub_eq_zero] using this
state_before: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f g✝ : ℂ → E
z : ℂ
g : ℂ → E
hfd : DiffContOnCl ℂ f {z | 0 < z.re}
hgd : DiffContOnCl ℂ g {z | 0 < z.re}
hfexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)
hgexp : ∃ c < 2, ∃ B, g =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)
hre : SuperpolynomialDecay atTop expR fun x => ‖f ↑x - g ↑x‖
hfim : ∃ C, ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hgim : ∃ C, ∀ (x : ℝ), ‖g (↑x * I)‖ ≤ C
⊢ EqOn (f - g) 0 {z | 0 ≤ z.re}
state_after: case refine_1
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f g✝ : ℂ → E
z : ℂ
g : ℂ → E
hfd : DiffContOnCl ℂ f {z | 0 < z.re}
hgd : DiffContOnCl ℂ g {z | 0 < z.re}
hfexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)
hgexp : ∃ c < 2, ∃ B, g =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)
hre : SuperpolynomialDecay atTop expR fun x => ‖f ↑x - g ↑x‖
hfim : ∃ C, ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hgim : ∃ C, ∀ (x : ℝ), ‖g (↑x * I)‖ ≤ C
⊢ ∃ c < 2, ∃ B, (f - g) =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)

case refine_2
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
a b C : ℝ
f g✝ : ℂ → E
z : ℂ
g : ℂ → E
hfd : DiffContOnCl ℂ f {z | 0 < z.re}
hgd : DiffContOnCl ℂ g {z | 0 < z.re}
hfexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)
hgexp : ∃ c < 2, ∃ B, g =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)
hre : SuperpolynomialDecay atTop expR fun x => ‖f ↑x - g ↑x‖
hfim : ∃ C, ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C
hgim : ∃ C, ∀ (x : ℝ), ‖g (↑x * I)‖ ≤ C
⊢ ∃ C, ∀ (x : ℝ), ‖(f - g) (↑x * I)‖ ≤ C
tactic: refine eq_zero_on_right_half_plane_of_superexponential_decay (hfd.sub hgd) ?_ hre ?_