Theorem: ENNReal.limsup_const_mul_of_ne_top
File: Mathlib/Order/Filter/ENNReal.lean

tactic: by_cases ha_zero : a = 0
state_before: α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
⊢ limsup (fun x => a * u x) f = a * limsup u f

tactic: let g := fun x : ℝ≥0∞ => a * x
state_before: case neg
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
⊢ limsup (fun x => a * u x) f = a * limsup u f

tactic: have hg_bij : Function.Bijective g :=
  Function.bijective_iff_has_inverse.mpr
    ⟨fun x => a⁻¹ * x,
      ⟨fun x => by simp [← mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top], fun x => by
        simp [← mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]⟩⟩
state_before: case neg
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
⊢ limsup (fun x => a * u x) f = a * limsup u f

tactic: have hg_mono : StrictMono g :=
  Monotone.strictMono_of_injective (fun _ _ _ => by rwa [mul_le_mul_left ha_zero ha_top]) hg_bij.1
state_before: case neg
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
hg_bij : Function.Bijective g
⊢ limsup (fun x => a * u x) f = a * limsup u f

tactic: let g_iso := StrictMono.orderIsoOfSurjective g hg_mono hg_bij.2
state_before: case neg
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
hg_bij : Function.Bijective g
hg_mono : StrictMono g
⊢ limsup (fun x => a * u x) f = a * limsup u f

tactic: refine' (OrderIso.limsup_apply g_iso _ _ _ _).symm
state_before: case neg
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
hg_bij : Function.Bijective g
hg_mono : StrictMono g
g_iso : ℝ≥0∞ ≃o ℝ≥0∞ := StrictMono.orderIsoOfSurjective g hg_mono (_ : Function.Surjective g)
⊢ limsup (fun x => a * u x) f = a * limsup u f

tactic: all_goals isBoundedDefault
state_before: case neg.refine'_1
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
hg_bij : Function.Bijective g
hg_mono : StrictMono g
g_iso : ℝ≥0∞ ≃o ℝ≥0∞ := StrictMono.orderIsoOfSurjective g hg_mono (_ : Function.Surjective g)
⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) f fun x => u x

case neg.refine'_2
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
hg_bij : Function.Bijective g
hg_mono : StrictMono g
g_iso : ℝ≥0∞ ≃o ℝ≥0∞ := StrictMono.orderIsoOfSurjective g hg_mono (_ : Function.Surjective g)
⊢ IsCoboundedUnder (fun x x_1 => x ≤ x_1) f fun x => u x

case neg.refine'_3
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
hg_bij : Function.Bijective g
hg_mono : StrictMono g
g_iso : ℝ≥0∞ ≃o ℝ≥0∞ := StrictMono.orderIsoOfSurjective g hg_mono (_ : Function.Surjective g)
⊢ IsBoundedUnder (fun x x_1 => x ≤ x_1) f fun x => ↑g_iso (u x)

case neg.refine'_4
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
hg_bij : Function.Bijective g
hg_mono : StrictMono g
g_iso : ℝ≥0∞ ≃o ℝ≥0∞ := StrictMono.orderIsoOfSurjective g hg_mono (_ : Function.Surjective g)
⊢ IsCoboundedUnder (fun x x_1 => x ≤ x_1) f fun x => ↑g_iso (u x)

tactic: simp_rw [ha_zero, zero_mul, ← ENNReal.bot_eq_zero]
state_before: case pos
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : a = 0
⊢ limsup (fun x => a * u x) f = a * limsup u f

tactic: exact limsup_const_bot
state_before: case pos
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : a = 0
⊢ limsup (fun x => ⊥) f = ⊥

tactic: simp [← mul_assoc, ENNReal.inv_mul_cancel ha_zero ha_top]
state_before: α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
x : ℝ≥0∞
⊢ (fun x => a⁻¹ * x) (g x) = x

tactic: simp [← mul_assoc, ENNReal.mul_inv_cancel ha_zero ha_top]
state_before: α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
x : ℝ≥0∞
⊢ g ((fun x => a⁻¹ * x) x) = x

tactic: rwa [mul_le_mul_left ha_zero ha_top]
state_before: α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
hg_bij : Function.Bijective g
x✝² x✝¹ : ℝ≥0∞
x✝ : x✝² ≤ x✝¹
⊢ g x✝² ≤ g x✝¹

tactic: isBoundedDefault
state_before: case neg.refine'_4
α : Type u_1
f : Filter α
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a ≠ ⊤
ha_zero : ¬a = 0
g : ℝ≥0∞ → ℝ≥0∞ := fun x => a * x
hg_bij : Function.Bijective g
hg_mono : StrictMono g
g_iso : ℝ≥0∞ ≃o ℝ≥0∞ := StrictMono.orderIsoOfSurjective g hg_mono (_ : Function.Surjective g)
⊢ IsCoboundedUnder (fun x x_1 => x ≤ x_1) f fun x => ↑g_iso (u x)