mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
z : ↑U
hz : z ∈ {u | meromorphicOrderAt f ↑u = ⊤}
x✝ : Set ↑U
| x✝ ⊆ {u | meromorphicOrderAt f ↑u = ⊤} ∧ IsOpen[instTopologicalSpaceSubtype] x✝ ∧ z ∈ x✝

Tactic:
;

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
z : ↑U
hz : z ∈ {u | meromorphicOrderAt f ↑u = ⊤}
x✝ : Set ↑U
| x✝ ⊆ {u | meromorphicOrderAt f ↑u = ⊤} ∧ IsOpen[instTopologicalSpaceSubtype] x✝ ∧ z ∈ x✝
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
z : ↑U
hz : z ∈ {u | meromorphicOrderAt f ↑u = ⊤}
x✝ : Set ↑U
| x✝ ⊆ {u | meromorphicOrderAt f ↑u = ⊤}

Tactic:
;

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
z : ↑U
hz : z ∈ {u | meromorphicOrderAt f ↑u = ⊤}
x✝ : Set ↑U
| x✝ ⊆ {u | meromorphicOrderAt f ↑u = ⊤}
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
z : ↑U
hz : z ∈ {u | meromorphicOrderAt f ↑u = ⊤}
x✝ : Set ↑U
| {u | meromorphicOrderAt f ↑u = ⊤}

Tactic:
;

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
z : ↑U
hz : z ∈ {u | meromorphicOrderAt f ↑u = ⊤}
x✝ : Set ↑U
| {u | meromorphicOrderAt f ↑u = ⊤}
Module: Mathlib.Analysis.Meromorphic.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
z : ↑U
hz : z ∈ {u | meromorphicOrderAt f ↑u = ⊤}
x✝ : Set ↑U
| fun u ↦ meromorphicOrderAt f ↑u = ⊤

Tactic:
;

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
U : Set 𝕜
hf : MeromorphicOn f U
z : ↑U
hz : z ∈ {u | meromorphicOrderAt f ↑u = ⊤}
x✝ : Set ↑U
| fun u ↦ meromorphicOrderAt f ↑u = ⊤
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case succ.refine_2
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
γ : C(↑I, X)
e : E
γ_0 : γ 0 = p e
U : X → Set X := fun x ↦ ⋯.choose
mem_base : ∀ (x : X), x ∈ ⋯.choose
U_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose
h✝ : ∀ (x : X), IsOpen[inst✝¹] (p ⁻¹' ⋯.choose)
H : (x : X) → ↑(p ⁻¹' ⋯.choose) ≃ₜ ↑⋯.choose × ↑(p ⁻¹' {x})
x✝¹ : ∀ (x : X) (x_1 : ↑(p ⁻¹' ⋯.choose)), ↑((H x) x_1).1 = p ↑x_1
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub✝ : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
Γ : ↑I → E
cont : ContinuousOn Γ (Set.Icc 0 (t n))
eqOn : Set.EqOn (p ∘ Γ) (⇑γ) (Set.Icc 0 (t n))
Γ_0 : Γ 0 = e
x : X
t_sub : Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
pΓtn : p (Γ (t n)) = γ (t n)
this : Nonempty ↑(p ⁻¹' {x})
q : Bundle.Trivialization (↑(p ⁻¹' {x})) p := ⋯.toTrivialization
x✝ : ↑I
h : x✝ ∈ Set.Icc 0 (t (n + 1)) ∩ {b | b ≤ t n}
⊢ x✝ ∈ Set.Icc 0 (t n)

Tactic:
;

Goal state after:
case succ.refine_2
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
γ : C(↑I, X)
e : E
γ_0 : γ 0 = p e
U : X → Set X := fun x ↦ ⋯.choose
mem_base : ∀ (x : X), x ∈ ⋯.choose
U_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose
h✝ : ∀ (x : X), IsOpen[inst✝¹] (p ⁻¹' ⋯.choose)
H : (x : X) → ↑(p ⁻¹' ⋯.choose) ≃ₜ ↑⋯.choose × ↑(p ⁻¹' {x})
x✝¹ : ∀ (x : X) (x_1 : ↑(p ⁻¹' ⋯.choose)), ↑((H x) x_1).1 = p ↑x_1
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub✝ : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
Γ : ↑I → E
cont : ContinuousOn Γ (Set.Icc 0 (t n))
eqOn : Set.EqOn (p ∘ Γ) (⇑γ) (Set.Icc 0 (t n))
Γ_0 : Γ 0 = e
x : X
t_sub : Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
pΓtn : p (Γ (t n)) = γ (t n)
this : Nonempty ↑(p ⁻¹' {x})
q : Bundle.Trivialization (↑(p ⁻¹' {x})) p := ⋯.toTrivialization
x✝ : ↑I
h : x✝ ∈ Set.Icc 0 (t (n + 1)) ∩ {b | b ≤ t n}
⊢ x✝ ∈ Set.Icc 0 (t n)
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case succ.refine_3
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
γ : C(↑I, X)
e : E
γ_0 : γ 0 = p e
U : X → Set X := fun x ↦ ⋯.choose
mem_base : ∀ (x : X), x ∈ ⋯.choose
U_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose
h✝ : ∀ (x : X), IsOpen[inst✝¹] (p ⁻¹' ⋯.choose)
H : (x : X) → ↑(p ⁻¹' ⋯.choose) ≃ₜ ↑⋯.choose × ↑(p ⁻¹' {x})
x✝ : ∀ (x : X) (x_1 : ↑(p ⁻¹' ⋯.choose)), ↑((H x) x_1).1 = p ↑x_1
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub✝ : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
Γ : ↑I → E
cont : ContinuousOn Γ (Set.Icc 0 (t n))
eqOn : Set.EqOn (p ∘ Γ) (⇑γ) (Set.Icc 0 (t n))
Γ_0 : Γ 0 = e
x : X
t_sub : Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
pΓtn : p (Γ (t n)) = γ (t n)
this : Nonempty ↑(p ⁻¹' {x})
q : Bundle.Trivialization (↑(p ⁻¹' {x})) p := ⋯.toTrivialization
⊢ Set.MapsTo ((fun x_1 ↦ (x_1, (↑q (Γ (t n))).2)) ∘ γ.toFun) (Set.Icc 0 (t (n + 1)) ∩ closure {a | t n < a})
    (q.baseSet ×ˢ Set.univ)

Tactic:
;

Goal state after:
case succ.refine_3
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
γ : C(↑I, X)
e : E
γ_0 : γ 0 = p e
U : X → Set X := fun x ↦ ⋯.choose
mem_base : ∀ (x : X), x ∈ ⋯.choose
U_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose
h✝ : ∀ (x : X), IsOpen[inst✝¹] (p ⁻¹' ⋯.choose)
H : (x : X) → ↑(p ⁻¹' ⋯.choose) ≃ₜ ↑⋯.choose × ↑(p ⁻¹' {x})
x✝ : ∀ (x : X) (x_1 : ↑(p ⁻¹' ⋯.choose)), ↑((H x) x_1).1 = p ↑x_1
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub✝ : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
Γ : ↑I → E
cont : ContinuousOn Γ (Set.Icc 0 (t n))
eqOn : Set.EqOn (p ∘ Γ) (⇑γ) (Set.Icc 0 (t n))
Γ_0 : Γ 0 = e
x : X
t_sub : Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
pΓtn : p (Γ (t n)) = γ (t n)
this : Nonempty ↑(p ⁻¹' {x})
q : Bundle.Trivialization (↑(p ⁻¹' {x})) p := ⋯.toTrivialization
⊢ Set.MapsTo ((fun x_1 ↦ (x_1, (↑q (Γ (t n))).2)) ∘ γ.toFun) (Set.Icc 0 (t (n + 1)) ∩ closure {a | t n < a})
    (q.baseSet ×ˢ Set.univ)
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 1 = ?m.47

Tactic:
;

Goal state after:
⊢ 1 = ?m.47
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 2 = ?m.57

Tactic:
;

Goal state after:
⊢ 2 = ?m.57
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 3 = ?m.67

Tactic:
;

Goal state after:
⊢ 3 = ?m.67
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 4 = ?m.77

Tactic:
;

Goal state after:
⊢ 4 = ?m.77
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 5 = ?m.87

Tactic:
;

Goal state after:
⊢ 5 = ?m.87
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 6 = ?m.97

Tactic:
;

Goal state after:
⊢ 6 = ?m.97
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 7 = ?m.107

Tactic:
;

Goal state after:
⊢ 7 = ?m.107
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 8 = ?m.117

Tactic:
;

Goal state after:
⊢ 8 = ?m.117
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 9 = ?m.127

Tactic:
;

Goal state after:
⊢ 9 = ?m.127
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 10 = ?m.137

Tactic:
;

Goal state after:
⊢ 10 = ?m.137
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case succ.refine_4
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
γ : C(↑I, X)
e : E
γ_0 : γ 0 = p e
U : X → Set X := fun x ↦ ⋯.choose
mem_base : ∀ (x : X), x ∈ ⋯.choose
U_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose
h✝ : ∀ (x : X), IsOpen[inst✝¹] (p ⁻¹' ⋯.choose)
H : (x : X) → ↑(p ⁻¹' ⋯.choose) ≃ₜ ↑⋯.choose × ↑(p ⁻¹' {x})
x✝ : ∀ (x : X) (x_1 : ↑(p ⁻¹' ⋯.choose)), ↑((H x) x_1).1 = p ↑x_1
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub✝ : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
Γ : ↑I → E
cont : ContinuousOn Γ (Set.Icc 0 (t n))
eqOn : Set.EqOn (p ∘ Γ) (⇑γ) (Set.Icc 0 (t n))
Γ_0 : Γ 0 = e
x : X
t_sub : Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
pΓtn : p (Γ (t n)) = γ (t n)
this : Nonempty ↑(p ⁻¹' {x})
q : Bundle.Trivialization (↑(p ⁻¹' {x})) p := ⋯.toTrivialization
s : ↑I
hs : s ∈ Set.Icc 0 (t (n + 1))
⊢ p (if s ≤ t n then Γ s else q.invFun (γ s, (↑q (Γ (t n))).2)) = γ s

Tactic:
;

Goal state after:
case succ.refine_4
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
γ : C(↑I, X)
e : E
γ_0 : γ 0 = p e
U : X → Set X := fun x ↦ ⋯.choose
mem_base : ∀ (x : X), x ∈ ⋯.choose
U_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose
h✝ : ∀ (x : X), IsOpen[inst✝¹] (p ⁻¹' ⋯.choose)
H : (x : X) → ↑(p ⁻¹' ⋯.choose) ≃ₜ ↑⋯.choose × ↑(p ⁻¹' {x})
x✝ : ∀ (x : X) (x_1 : ↑(p ⁻¹' ⋯.choose)), ↑((H x) x_1).1 = p ↑x_1
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub✝ : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
Γ : ↑I → E
cont : ContinuousOn Γ (Set.Icc 0 (t n))
eqOn : Set.EqOn (p ∘ Γ) (⇑γ) (Set.Icc 0 (t n))
Γ_0 : Γ 0 = e
x : X
t_sub : Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
pΓtn : p (Γ (t n)) = γ (t n)
this : Nonempty ↑(p ⁻¹' {x})
q : Bundle.Trivialization (↑(p ⁻¹' {x})) p := ⋯.toTrivialization
s : ↑I
hs : s ∈ Set.Icc 0 (t (n + 1))
⊢ p (if s ≤ t n then Γ s else q.invFun (γ s, (↑q (Γ (t n))).2)) = γ s
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 11 = ?m.147

Tactic:
;

Goal state after:
⊢ 11 = ?m.147
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 12 = ?m.157

Tactic:
;

Goal state after:
⊢ 12 = ?m.157
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 13 = ?m.167

Tactic:
;

Goal state after:
⊢ 13 = ?m.167
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 14 = ?m.178

Tactic:
;

Goal state after:
⊢ 14 = ?m.178
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case succ.refine_5
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
γ : C(↑I, X)
e : E
γ_0 : γ 0 = p e
U : X → Set X := fun x ↦ ⋯.choose
mem_base : ∀ (x : X), x ∈ ⋯.choose
U_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose
h✝ : ∀ (x : X), IsOpen[inst✝¹] (p ⁻¹' ⋯.choose)
H : (x : X) → ↑(p ⁻¹' ⋯.choose) ≃ₜ ↑⋯.choose × ↑(p ⁻¹' {x})
x✝ : ∀ (x : X) (x_1 : ↑(p ⁻¹' ⋯.choose)), ↑((H x) x_1).1 = p ↑x_1
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub✝ : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
Γ : ↑I → E
cont : ContinuousOn Γ (Set.Icc 0 (t n))
eqOn : Set.EqOn (p ∘ Γ) (⇑γ) (Set.Icc 0 (t n))
Γ_0 : Γ 0 = e
x : X
t_sub : Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
pΓtn : p (Γ (t n)) = γ (t n)
this : Nonempty ↑(p ⁻¹' {x})
q : Bundle.Trivialization (↑(p ⁻¹' {x})) p := ⋯.toTrivialization
⊢ (if 0 ≤ t n then Γ 0 else q.invFun (γ 0, (↑q (Γ (t n))).2)) = e

Tactic:
;

Goal state after:
case succ.refine_5
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
γ : C(↑I, X)
e : E
γ_0 : γ 0 = p e
U : X → Set X := fun x ↦ ⋯.choose
mem_base : ∀ (x : X), x ∈ ⋯.choose
U_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose
h✝ : ∀ (x : X), IsOpen[inst✝¹] (p ⁻¹' ⋯.choose)
H : (x : X) → ↑(p ⁻¹' ⋯.choose) ≃ₜ ↑⋯.choose × ↑(p ⁻¹' {x})
x✝ : ∀ (x : X) (x_1 : ↑(p ⁻¹' ⋯.choose)), ↑((H x) x_1).1 = p ↑x_1
t : ℕ → ↑I
t_0 : t 0 = 0
t_mono : Monotone t
t_sub✝ : ∀ (n : ℕ), ∃ i, Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
n_max : ℕ
h_max : ∀ m ≥ n_max, t m = 1
n : ℕ
Γ : ↑I → E
cont : ContinuousOn Γ (Set.Icc 0 (t n))
eqOn : Set.EqOn (p ∘ Γ) (⇑γ) (Set.Icc 0 (t n))
Γ_0 : Γ 0 = e
x : X
t_sub : Set.Icc (t n) (t (n + 1)) ⊆ ⇑γ ⁻¹' ⋯.choose
pΓtn : p (Γ (t n)) = γ (t n)
this : Nonempty ↑(p ⁻¹' {x})
q : Bundle.Trivialization (↑(p ⁻¹' {x})) p := ⋯.toTrivialization
⊢ (if 0 ≤ t n then Γ 0 else q.invFun (γ 0, (↑q (Γ (t n))).2)) = e
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 14 = ?m.177

Tactic:
;

Goal state after:
⊢ 14 = ?m.177
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 15 = ?m.187

Tactic:
;

Goal state after:
⊢ 15 = ?m.187
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 16 = ?m.197

Tactic:
;

Goal state after:
⊢ 16 = ?m.197
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 17 = ?m.207

Tactic:
;

Goal state after:
⊢ 17 = ?m.207
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 18 = ?m.217

Tactic:
;

Goal state after:
⊢ 18 = ?m.217
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
x y z : X
e : E
hpe : x = p e
γ : Path x y
γ' : Path y z
x✝ : ↑I
⊢ p
      (if h : ↑x✝ ≤ 1 / 2 then { toContinuousMap := cov.liftPath (↑γ) e ⋯, source' := ⋯, target' := ⋯ } ⟨2 * ↑x✝, ⋯⟩
      else
        { toContinuousMap := cov.liftPath (↑γ') ((cov.liftPath (↑γ) e ⋯) 1) ⋯, source' := ⋯, target' := ⋯ }
          ⟨2 * ↑x✝ - 1, ⋯⟩) =
    if h : ↑x✝ ≤ 1 / 2 then γ ⟨2 * ↑x✝, ⋯⟩ else γ' ⟨2 * ↑x✝ - 1, ⋯⟩

Tactic:
;

Goal state after:
E : Type u_1
X : Type u_2
inst✝¹ : TopologicalSpace E
inst✝ : TopologicalSpace X
p : E → X
cov : IsCoveringMap p
x y z : X
e : E
hpe : x = p e
γ : Path x y
γ' : Path y z
x✝ : ↑I
⊢ p
      (if h : ↑x✝ ≤ 1 / 2 then { toContinuousMap := cov.liftPath (↑γ) e ⋯, source' := ⋯, target' := ⋯ } ⟨2 * ↑x✝, ⋯⟩
      else
        { toContinuousMap := cov.liftPath (↑γ') ((cov.liftPath (↑γ) e ⋯) 1) ⋯, source' := ⋯, target' := ⋯ }
          ⟨2 * ↑x✝ - 1, ⋯⟩) =
    if h : ↑x✝ ≤ 1 / 2 then γ ⟨2 * ↑x✝, ⋯⟩ else γ' ⟨2 * ↑x✝ - 1, ⋯⟩
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 19 = ?m.227

Tactic:
;

Goal state after:
⊢ 19 = ?m.227
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 20 = ?m.237

Tactic:
;

Goal state after:
⊢ 20 = ?m.237
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 21 = ?m.247

Tactic:
;

Goal state after:
⊢ 21 = ?m.247
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 22 = ?m.258

Tactic:
;

Goal state after:
⊢ 22 = ?m.258
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
t : ↑I
a : A
⊢ (p ∘ fun ta ↦ (cov.liftPath (H.comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const (↑I) ta.2))) (f ta.2) ⋯) ta.1)
      (t, a) =
    H (t, a)

Tactic:
;

Goal state after:
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
t : ↑I
a : A
⊢ (p ∘ fun ta ↦ (cov.liftPath (H.comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const (↑I) ta.2))) (f ta.2) ⋯) ta.1)
      (t, a) =
    H (t, a)
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
a : A
⊢ (cov.liftPath (H.comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const ↑I (0, a).2))) (f (0, a).2) ⋯) (0, a).1 = f a

Tactic:
;

Goal state after:
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
a : A
⊢ (cov.liftPath (H.comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const ↑I (0, a).2))) (f (0, a).2) ⋯) (0, a).1 = f a
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: ⊢ 0 < 2⁻¹

Tactic:
;

Goal state after:
⊢ 0 < 2⁻¹
Module: Mathlib.Analysis.Complex.ExponentialBounds

Goal state before:
pp: t : |2⁻¹| = 2⁻¹
⊢ 2⁻¹ < 1

Tactic:
;

Goal state after:
t : |2⁻¹| = 2⁻¹
⊢ 2⁻¹ < 1
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
a : A
⊢ Continuous[_, inst✝²] fun x ↦
    (cov.liftPath (H.comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const (↑I) a))) (f a) ⋯) x

Tactic:
;

Goal state after:
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
a : A
⊢ Continuous[_, inst✝²] fun x ↦
    (cov.liftPath (H.comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const (↑I) a))) (f a) ⋯) x
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case refine_1
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
⊢ (∀ (a : A), Continuous[_, inst✝²] fun x ↦ (cov.liftHomotopy H f H_0) (x, a)) ∧
    p ∘ ⇑(cov.liftHomotopy H f H_0) = ⇑H ∧ ∀ (a : A), (cov.liftHomotopy H f H_0) (0, a) = f a

Tactic:
;

Goal state after:
case refine_1
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
⊢ (∀ (a : A), Continuous[_, inst✝²] fun x ↦ (cov.liftHomotopy H f H_0) (x, a)) ∧
    p ∘ ⇑(cov.liftHomotopy H f H_0) = ⇑H ∧ ∀ (a : A), (cov.liftHomotopy H f H_0) (0, a) = f a
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case refine_1
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
a : A
⊢ Continuous[_, inst✝²] fun x ↦
    (cov.liftPath (H.comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const (↑I) a))) (f a) ⋯) x

Tactic:
;

Goal state after:
case refine_1
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
a : A
⊢ Continuous[_, inst✝²] fun x ↦
    (cov.liftPath (H.comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const (↑I) a))) (f a) ⋯) x
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
H' : ↑I × A → E
x✝¹ : (∀ (a : A), Continuous[_, inst✝²] fun x ↦ H' (x, a)) ∧ p ∘ H' = ⇑H ∧ ∀ (a : A), H' (0, a) = f a
H'_cont : ∀ (a : A), Continuous[_, inst✝²] fun x ↦ H' (x, a)
H'_lifts : p ∘ H' = ⇑H
H'_0 : ∀ (a : A), H' (0, a) = f a
x✝ : ↑I × A
t✝ : ↑I
a✝ : A
t : ℝ
a : t ∈ I
⊢ (p ∘ fun x ↦ H' (x, a✝)) ⟨t, a⟩ = (H.comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const ↑I (t✝, a✝).2))) ⟨t, a⟩

Tactic:
;

Goal state after:
E : Type u_1
X : Type u_2
A : Type u_3
inst✝² : TopologicalSpace E
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
H' : ↑I × A → E
x✝¹ : (∀ (a : A), Continuous[_, inst✝²] fun x ↦ H' (x, a)) ∧ p ∘ H' = ⇑H ∧ ∀ (a : A), H' (0, a) = f a
H'_cont : ∀ (a : A), Continuous[_, inst✝²] fun x ↦ H' (x, a)
H'_lifts : p ∘ H' = ⇑H
H'_0 : ∀ (a : A), H' (0, a) = f a
x✝ : ↑I × A
t✝ : ↑I
a✝ : A
t : ℝ
a : t ∈ I
⊢ (p ∘ fun x ↦ H' (x, a✝)) ⟨t, a⟩ = (H.comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const ↑I (t✝, a✝).2))) ⟨t, a⟩
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
f g : ℂ → E
l : Filter ℂ
u : ℂ → ℝ
hBf : ∃ c < a, ∃ B, f =O[l] fun z ↦ expR (B * expR (c * |u z|))
hBg : ∃ c < a, ∃ B, g =O[l] fun z ↦ expR (B * expR (c * |u z|))
c₁✝ c₂✝ B₁✝ B₂✝ : ℝ
hc : c₁✝ ≤ c₂✝
hB₀ : 0 ≤ B₂✝
hB : B₁✝ ≤ B₂✝
z : ℂ
⊢ expR (B₁✝ * expR (c₁✝ * |u z|)) ≤ expR (B₂✝ * expR (c₂✝ * |u z|))

Tactic:
;

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
f g : ℂ → E
l : Filter ℂ
u : ℂ → ℝ
hBf : ∃ c < a, ∃ B, f =O[l] fun z ↦ expR (B * expR (c * |u z|))
hBg : ∃ c < a, ∃ B, g =O[l] fun z ↦ expR (B * expR (c * |u z|))
c₁✝ c₂✝ B₁✝ B₂✝ : ℝ
hc : c₁✝ ≤ c₂✝
hB₀ : 0 ≤ B₂✝
hB : B₁✝ ≤ B₂✝
z : ℂ
⊢ expR (B₁✝ * expR (c₁✝ * |u z|)) ≤ expR (B₂✝ * expR (c₂✝ * |u z|))
Module: Mathlib.Analysis.Complex.PhragmenLindelof

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
f g : ℂ → E
l : Filter ℂ
u : ℂ → ℝ
hBg : ∃ c < a, ∃ B, g =O[l] fun z ↦ expR (B * expR (c * |u z|))
this :
  ∀ {c₁ c₂ B₁ B₂ : ℝ},
    c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * |u z|))‖ ≤ ‖expR (B₂ * expR (c₂ * |u z|))‖
cf : ℝ
hcf : cf < a
Bf : ℝ
hOf : f =O[l] fun z ↦ expR (Bf * expR (cf * |u z|))
⊢ ∃ c < a, ∃ B, (f - g) =O[l] fun z ↦ expR (B * expR (c * |u z|))

Tactic:
;

Goal state after:
E : Type u_1
inst✝ : NormedAddCommGroup E
a : ℝ
f g : ℂ → E
l : Filter ℂ
u : ℂ → ℝ
hBg : ∃ c < a, ∃ B, g =O[l] fun z ↦ expR (B * expR (c * |u z|))
this :
  ∀ {c₁ c₂ B₁ B₂ : ℝ},
    c₁ ≤ c₂ → 0 ≤ B₂ → B₁ ≤ B₂ → ∀ (z : ℂ), ‖expR (B₁ * expR (c₁ * |u z|))‖ ≤ ‖expR (B₂ * expR (c₂ * |u z|))‖
cf : ℝ
hcf : cf < a
Bf : ℝ
hOf : f =O[l] fun z ↦ expR (Bf * expR (cf * |u z|))
⊢ ∃ c < a, ∃ B, (f - g) =O[l] fun z ↦ expR (B * expR (c * |u z|))
Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: E : Type u_1
X : Type u_2
A : Type u_3
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
f₀ f₁ : C(A, X)
S : Set A
F : f₀.HomotopyRel f₁ S
inst✝ : PreconnectedSpace A
f₀' f₁' : C(A, E)
he : ∃ a ∈ S, f₀' a = f₁' a
h₀ : p ∘ ⇑f₀' = ⇑f₀
h₁ : p ∘ ⇑f₁' = ⇑f₁
F_0 : ∀ (a : A), F (0, a) = p (f₀' a)
t✝ : ↑I
a : A
ha : a ∈ S
t t' : ↑I
⊢ p ((cov.liftPath ((↑F).comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const ↑I (t✝, a).2))) (f₀' (t✝, a).2) ⋯) t) =
    p
      ((cov.liftPath ((↑F).comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const ↑I (t✝, a).2))) (f₀' (t✝, a).2) ⋯)
        t')

Tactic:
;

Goal state after:
E : Type u_1
X : Type u_2
A : Type u_3
inst✝³ : TopologicalSpace E
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace A
p : E → X
cov : IsCoveringMap p
H : C(↑I × A, X)
f : C(A, E)
H_0 : ∀ (a : A), H (0, a) = p (f a)
f₀ f₁ : C(A, X)
S : Set A
F : f₀.HomotopyRel f₁ S
inst✝ : PreconnectedSpace A
f₀' f₁' : C(A, E)
he : ∃ a ∈ S, f₀' a = f₁' a
h₀ : p ∘ ⇑f₀' = ⇑f₀
h₁ : p ∘ ⇑f₁' = ⇑f₁
F_0 : ∀ (a : A), F (0, a) = p (f₀' a)
t✝ : ↑I
a : A
ha : a ∈ S
t t' : ↑I
⊢ p ((cov.liftPath ((↑F).comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const ↑I (t✝, a).2))) (f₀' (t✝, a).2) ⋯) t) =
    p
      ((cov.liftPath ((↑F).comp ((ContinuousMap.id ↑I).prodMk (ContinuousMap.const ↑I (t✝, a).2))) (f₀' (t✝, a).2) ⋯)
        t')