Module: Mathlib.Topology.Homotopy.Lifting

Goal state before:
pp: case succ.refine_1
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)) ∩ frontier {a | a ≤ t n}
⊢ Γ s = q.invFun (γ s, (↑q (Γ (t n))).2)

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)) ∩ closure {a | a ≤ t n}
⊢ x✝ ∈ Set.Icc 0 (t n)

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
⊢ ContinuousOn (fun s ↦ q.invFun (γ s, (↑q (Γ (t n))).2)) (Set.Icc 0 (t (n + 1)) ∩ closure {a | ¬a ≤ t n})

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 ∘ fun s ↦ if s ≤ t n then Γ s else q.invFun (γ s, (↑q (Γ (t n))).2)) s = γ s

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
⊢ (fun s ↦ if s ≤ t n then Γ s else q.invFun (γ s, (↑q (Γ (t n))).2)) 0 = e

Tactic:
· cases frontier_Iic_subset _ hs.2
    rw [← pΓtn]
    refine (q.symm_apply_mk_proj ?_).symm
    rw [q.mem_source, pΓtn]
    exact t_sub ⟨le_rfl, t_mono n.le_succ⟩

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)) ∩ closure {a | a ≤ t n}
⊢ x✝ ∈ Set.Icc 0 (t n)

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
⊢ ContinuousOn (fun s ↦ q.invFun (γ s, (↑q (Γ (t n))).2)) (Set.Icc 0 (t (n + 1)) ∩ closure {a | ¬a ≤ t n})

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 ∘ fun s ↦ if s ≤ t n then Γ s else q.invFun (γ s, (↑q (Γ (t n))).2)) s = γ s

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
⊢ (fun s ↦ if s ≤ t n then Γ s else q.invFun (γ s, (↑q (Γ (t n))).2)) 0 = e