l3lab/ntp-mathlib huggingface dataset:
theorem hcomp_half (F : Homotopy p₀ q₀) (G : Homotopy p₁ q₁) (t : I) :
    F.hcomp G (t, ⟨1 / 2, by norm_num, by 

Proof state:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p₀ q₀ : Path x₀ x₁
p₁ q₁ : Path x₁ x₂
F : Homotopy p₀ q₀
G : Homotopy p₁ q₁
t : ↑I
⊢ 1 / 2 ≤ 1

Next tactic:
norm_num
theorem hcomp_half (F : Homotopy p₀ q₀) (G : Homotopy p₁ q₁) (t : I) :
    F.hcomp G (t, ⟨1 / 2, by norm_num, by norm_num⟩) = x₁ :=
  show ite _ _ _ = _ by 

Proof state:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p₀ q₀ : Path x₀ x₁
p₁ q₁ : Path x₁ x₂
F : Homotopy p₀ q₀
G : Homotopy p₁ q₁
t : ↑I
⊢ (if ↑(t, { val := 1 / 2, property := (_ : 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1) }).2 ≤ 1 / 2 then
      extend (eval F (t, { val := 1 / 2, property := (_ : 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1) }).1)
        (2 * ↑(t, { val := 1 / 2, property := (_ : 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1) }).2)
    else
      extend (eval G (t, { val := 1 / 2, property := (_ : 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1) }).1)
        (2 * ↑(t, { val := 1 / 2, property := (_ : 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1) }).2 - 1)) =
    x₁

Next tactic:
norm_num
/--
Suppose `p` is a path, then we have a homotopy from `p` to `p.reparam f` by the convexity of `I`.
-/
def reparam (p : Path x₀ x₁) (f : I → I) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
    Homotopy p (p.reparam f hf hf₀ hf₁) where
  toFun x := p ⟨σ x.1 * x.2 + x.1 * f x.2,
    show (σ x.1 : ℝ) • (x.2 : ℝ) + (x.1 : ℝ) • (f x.2 : ℝ) ∈ I from
      convex_Icc _ _ x.2.2 (f x.2).2 (by 

Proof state:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
f : ↑I → ↑I
hf : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
x : ↑I × ↑I
⊢ 0 ≤ ↑(σ x.1)

Next tactic:
unit_interval
/--
Suppose `p` is a path, then we have a homotopy from `p` to `p.reparam f` by the convexity of `I`.
-/
def reparam (p : Path x₀ x₁) (f : I → I) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
    Homotopy p (p.reparam f hf hf₀ hf₁) where
  toFun x := p ⟨σ x.1 * x.2 + x.1 * f x.2,
    show (σ x.1 : ℝ) • (x.2 : ℝ) + (x.1 : ℝ) • (f x.2 : ℝ) ∈ I from
      convex_Icc _ _ x.2.2 (f x.2).2 (by unit_interval) (by 

Proof state:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
f : ↑I → ↑I
hf : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
x : ↑I × ↑I
⊢ 0 ≤ ↑x.1

Next tactic:
unit_interval
/--
Suppose `p` is a path, then we have a homotopy from `p` to `p.reparam f` by the convexity of `I`.
-/
def reparam (p : Path x₀ x₁) (f : I → I) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
    Homotopy p (p.reparam f hf hf₀ hf₁) where
  toFun x := p ⟨σ x.1 * x.2 + x.1 * f x.2,
    show (σ x.1 : ℝ) • (x.2 : ℝ) + (x.1 : ℝ) • (f x.2 : ℝ) ∈ I from
      convex_Icc _ _ x.2.2 (f x.2).2 (by unit_interval) (by unit_interval) (by 

Proof state:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
f : ↑I → ↑I
hf : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
x : ↑I × ↑I
⊢ ↑(σ x.1) + ↑x.1 = 1

Next tactic:
simp
/--
Suppose `p` is a path, then we have a homotopy from `p` to `p.reparam f` by the convexity of `I`.
-/
def reparam (p : Path x₀ x₁) (f : I → I) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
    Homotopy p (p.reparam f hf hf₀ hf₁) where
  toFun x := p ⟨σ x.1 * x.2 + x.1 * f x.2,
    show (σ x.1 : ℝ) • (x.2 : ℝ) + (x.1 : ℝ) • (f x.2 : ℝ) ∈ I from
      convex_Icc _ _ x.2.2 (f x.2).2 (by unit_interval) (by unit_interval) (by simp)⟩
  map_zero_left x := by norm_num
  map_one_left x := by norm_num
  prop' t x hx := by
    cases' hx with hx hx
    · rw [hx]
      simp [hf₀]
    · rw [Set.mem_singleton_iff] at hx
      rw [hx]
      simp [hf₁]
  continuous_toFun := by
    -- Porting note: was `continuity` in auto-param
    

Proof state:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
f : ↑I → ↑I
hf : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
⊢ Continuous fun x =>
    p { val := ↑(σ x.1) * ↑x.2 + ↑x.1 * ↑(f x.2), property := (_ : ↑(σ x.1) • ↑x.2 + ↑x.1 • ↑(f x.2) ∈ I) }

Next tactic:
refine continuous_const.path_eval ?_
/--
Suppose `p` is a path, then we have a homotopy from `p` to `p.reparam f` by the convexity of `I`.
-/
def reparam (p : Path x₀ x₁) (f : I → I) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
    Homotopy p (p.reparam f hf hf₀ hf₁) where
  toFun x := p ⟨σ x.1 * x.2 + x.1 * f x.2,
    show (σ x.1 : ℝ) • (x.2 : ℝ) + (x.1 : ℝ) • (f x.2 : ℝ) ∈ I from
      convex_Icc _ _ x.2.2 (f x.2).2 (by unit_interval) (by unit_interval) (by simp)⟩
  map_zero_left x := by norm_num
  map_one_left x := by norm_num
  prop' t x hx := by
    cases' hx with hx hx
    · rw [hx]
      simp [hf₀]
    · rw [Set.mem_singleton_iff] at hx
      rw [hx]
      simp [hf₁]
  continuous_toFun := by
    -- Porting note: was `continuity` in auto-param
    refine continuous_const.path_eval ?_
    

Proof state:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
f : ↑I → ↑I
hf : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
⊢ Continuous fun x =>
    { val := ↑(σ x.1) * ↑x.2 + ↑x.1 * ↑(f x.2), property := (_ : ↑(σ x.1) • ↑x.2 + ↑x.1 • ↑(f x.2) ∈ I) }

Next tactic:
apply Continuous.subtype_mk
/--
Suppose `p` is a path, then we have a homotopy from `p` to `p.reparam f` by the convexity of `I`.
-/
def reparam (p : Path x₀ x₁) (f : I → I) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
    Homotopy p (p.reparam f hf hf₀ hf₁) where
  toFun x := p ⟨σ x.1 * x.2 + x.1 * f x.2,
    show (σ x.1 : ℝ) • (x.2 : ℝ) + (x.1 : ℝ) • (f x.2 : ℝ) ∈ I from
      convex_Icc _ _ x.2.2 (f x.2).2 (by unit_interval) (by unit_interval) (by simp)⟩
  map_zero_left x := by norm_num
  map_one_left x := by norm_num
  prop' t x hx := by
    cases' hx with hx hx
    · rw [hx]
      simp [hf₀]
    · rw [Set.mem_singleton_iff] at hx
      rw [hx]
      simp [hf₁]
  continuous_toFun := by
    -- Porting note: was `continuity` in auto-param
    refine continuous_const.path_eval ?_
    apply Continuous.subtype_mk
    

Proof state:
case h
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
f : ↑I → ↑I
hf : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
⊢ Continuous fun x => ↑(σ x.1) * ↑x.2 + ↑x.1 * ↑(f x.2)

Next tactic:
apply Continuous.add
/--
Suppose `p` is a path, then we have a homotopy from `p` to `p.reparam f` by the convexity of `I`.
-/
def reparam (p : Path x₀ x₁) (f : I → I) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
    Homotopy p (p.reparam f hf hf₀ hf₁) where
  toFun x := p ⟨σ x.1 * x.2 + x.1 * f x.2,
    show (σ x.1 : ℝ) • (x.2 : ℝ) + (x.1 : ℝ) • (f x.2 : ℝ) ∈ I from
      convex_Icc _ _ x.2.2 (f x.2).2 (by unit_interval) (by unit_interval) (by simp)⟩
  map_zero_left x := by norm_num
  map_one_left x := by norm_num
  prop' t x hx := by
    cases' hx with hx hx
    · rw [hx]
      simp [hf₀]
    · rw [Set.mem_singleton_iff] at hx
      rw [hx]
      simp [hf₁]
  continuous_toFun := by
    -- Porting note: was `continuity` in auto-param
    refine continuous_const.path_eval ?_
    apply Continuous.subtype_mk
    apply Continuous.add <;> 

Proof state:
case h.hf
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
f : ↑I → ↑I
hf : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
⊢ Continuous fun x => ↑(σ x.1) * ↑x.2

Next tactic:
apply Continuous.mul
/--
Suppose `p` is a path, then we have a homotopy from `p` to `p.reparam f` by the convexity of `I`.
-/
def reparam (p : Path x₀ x₁) (f : I → I) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
    Homotopy p (p.reparam f hf hf₀ hf₁) where
  toFun x := p ⟨σ x.1 * x.2 + x.1 * f x.2,
    show (σ x.1 : ℝ) • (x.2 : ℝ) + (x.1 : ℝ) • (f x.2 : ℝ) ∈ I from
      convex_Icc _ _ x.2.2 (f x.2).2 (by unit_interval) (by unit_interval) (by simp)⟩
  map_zero_left x := by norm_num
  map_one_left x := by norm_num
  prop' t x hx := by
    cases' hx with hx hx
    · rw [hx]
      simp [hf₀]
    · rw [Set.mem_singleton_iff] at hx
      rw [hx]
      simp [hf₁]
  continuous_toFun := by
    -- Porting note: was `continuity` in auto-param
    refine continuous_const.path_eval ?_
    apply Continuous.subtype_mk
    apply Continuous.add <;> 

Proof state:
case h.hg
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
f : ↑I → ↑I
hf : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
⊢ Continuous fun x => ↑x.1 * ↑(f x.2)

Next tactic:
apply Continuous.mul
/--
Suppose `p` is a path, then we have a homotopy from `p` to `p.reparam f` by the convexity of `I`.
-/
def reparam (p : Path x₀ x₁) (f : I → I) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
    Homotopy p (p.reparam f hf hf₀ hf₁) where
  toFun x := p ⟨σ x.1 * x.2 + x.1 * f x.2,
    show (σ x.1 : ℝ) • (x.2 : ℝ) + (x.1 : ℝ) • (f x.2 : ℝ) ∈ I from
      convex_Icc _ _ x.2.2 (f x.2).2 (by unit_interval) (by unit_interval) (by simp)⟩
  map_zero_left x := by norm_num
  map_one_left x := by norm_num
  prop' t x hx := by
    cases' hx with hx hx
    · rw [hx]
      simp [hf₀]
    · rw [Set.mem_singleton_iff] at hx
      rw [hx]
      simp [hf₁]
  continuous_toFun := by
    -- Porting note: was `continuity` in auto-param
    refine continuous_const.path_eval ?_
    apply Continuous.subtype_mk
    apply Continuous.add <;> apply Continuous.mul
    · 

Proof state:
case h.hf.hf
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
f : ↑I → ↑I
hf : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
⊢ Continuous fun x => ↑(σ x.1)

Next tactic:
exact continuous_induced_dom.comp (unitInterval.continuous_symm.comp continuous_fst)
/--
Suppose `p` is a path, then we have a homotopy from `p` to `p.reparam f` by the convexity of `I`.
-/
def reparam (p : Path x₀ x₁) (f : I → I) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
    Homotopy p (p.reparam f hf hf₀ hf₁) where
  toFun x := p ⟨σ x.1 * x.2 + x.1 * f x.2,
    show (σ x.1 : ℝ) • (x.2 : ℝ) + (x.1 : ℝ) • (f x.2 : ℝ) ∈ I from
      convex_Icc _ _ x.2.2 (f x.2).2 (by unit_interval) (by unit_interval) (by simp)⟩
  map_zero_left x := by norm_num
  map_one_left x := by norm_num
  prop' t x hx := by
    cases' hx with hx hx
    · rw [hx]
      simp [hf₀]
    · rw [Set.mem_singleton_iff] at hx
      rw [hx]
      simp [hf₁]
  continuous_toFun := by
    -- Porting note: was `continuity` in auto-param
    refine continuous_const.path_eval ?_
    apply Continuous.subtype_mk
    apply Continuous.add <;> apply Continuous.mul
    · exact continuous_induced_dom.comp (unitInterval.continuous_symm.comp continuous_fst)
    · 

Proof state:
case h.hf.hg
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
f : ↑I → ↑I
hf : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
⊢ Continuous fun x => ↑x.2

Next tactic:
continuity
/--
Suppose `p` is a path, then we have a homotopy from `p` to `p.reparam f` by the convexity of `I`.
-/
def reparam (p : Path x₀ x₁) (f : I → I) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
    Homotopy p (p.reparam f hf hf₀ hf₁) where
  toFun x := p ⟨σ x.1 * x.2 + x.1 * f x.2,
    show (σ x.1 : ℝ) • (x.2 : ℝ) + (x.1 : ℝ) • (f x.2 : ℝ) ∈ I from
      convex_Icc _ _ x.2.2 (f x.2).2 (by unit_interval) (by unit_interval) (by simp)⟩
  map_zero_left x := by norm_num
  map_one_left x := by norm_num
  prop' t x hx := by
    cases' hx with hx hx
    · rw [hx]
      simp [hf₀]
    · rw [Set.mem_singleton_iff] at hx
      rw [hx]
      simp [hf₁]
  continuous_toFun := by
    -- Porting note: was `continuity` in auto-param
    refine continuous_const.path_eval ?_
    apply Continuous.subtype_mk
    apply Continuous.add <;> apply Continuous.mul
    · exact continuous_induced_dom.comp (unitInterval.continuous_symm.comp continuous_fst)
    · continuity
    · 

Proof state:
case h.hg.hf
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
f : ↑I → ↑I
hf : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
⊢ Continuous fun x => ↑x.1

Next tactic:
continuity
/--
Suppose `p` is a path, then we have a homotopy from `p` to `p.reparam f` by the convexity of `I`.
-/
def reparam (p : Path x₀ x₁) (f : I → I) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
    Homotopy p (p.reparam f hf hf₀ hf₁) where
  toFun x := p ⟨σ x.1 * x.2 + x.1 * f x.2,
    show (σ x.1 : ℝ) • (x.2 : ℝ) + (x.1 : ℝ) • (f x.2 : ℝ) ∈ I from
      convex_Icc _ _ x.2.2 (f x.2).2 (by unit_interval) (by unit_interval) (by simp)⟩
  map_zero_left x := by norm_num
  map_one_left x := by norm_num
  prop' t x hx := by
    cases' hx with hx hx
    · rw [hx]
      simp [hf₀]
    · rw [Set.mem_singleton_iff] at hx
      rw [hx]
      simp [hf₁]
  continuous_toFun := by
    -- Porting note: was `continuity` in auto-param
    refine continuous_const.path_eval ?_
    apply Continuous.subtype_mk
    apply Continuous.add <;> apply Continuous.mul
    · exact continuous_induced_dom.comp (unitInterval.continuous_symm.comp continuous_fst)
    · continuity
    · continuity
    · 

Proof state:
case h.hg.hg
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
f : ↑I → ↑I
hf : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
⊢ Continuous fun x => ↑(f x.2)

Next tactic:
continuity
/--
Suppose `p` is a path, then we have a homotopy from `p` to `p.reparam f` by the convexity of `I`.
-/
def reparam (p : Path x₀ x₁) (f : I → I) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
    Homotopy p (p.reparam f hf hf₀ hf₁) where
  toFun x := p ⟨σ x.1 * x.2 + x.1 * f x.2,
    show (σ x.1 : ℝ) • (x.2 : ℝ) + (x.1 : ℝ) • (f x.2 : ℝ) ∈ I from
      convex_Icc _ _ x.2.2 (f x.2).2 (by unit_interval) (by unit_interval) (by simp)⟩
  map_zero_left x := by 

Proof state:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
f : ↑I → ↑I
hf : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
x : ↑I
⊢ ContinuousMap.toFun
      (ContinuousMap.mk fun x =>
        p { val := ↑(σ x.1) * ↑x.2 + ↑x.1 * ↑(f x.2), property := (_ : ↑(σ x.1) • ↑x.2 + ↑x.1 • ↑(f x.2) ∈ I) })
      (0, x) =
    p.toContinuousMap x

Next tactic:
norm_num
/--
Suppose `p` is a path, then we have a homotopy from `p` to `p.reparam f` by the convexity of `I`.
-/
def reparam (p : Path x₀ x₁) (f : I → I) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
    Homotopy p (p.reparam f hf hf₀ hf₁) where
  toFun x := p ⟨σ x.1 * x.2 + x.1 * f x.2,
    show (σ x.1 : ℝ) • (x.2 : ℝ) + (x.1 : ℝ) • (f x.2 : ℝ) ∈ I from
      convex_Icc _ _ x.2.2 (f x.2).2 (by unit_interval) (by unit_interval) (by simp)⟩
  map_zero_left x := by norm_num
  map_one_left x := by 

Proof state:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
f : ↑I → ↑I
hf : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
x : ↑I
⊢ ContinuousMap.toFun
      (ContinuousMap.mk fun x =>
        p { val := ↑(σ x.1) * ↑x.2 + ↑x.1 * ↑(f x.2), property := (_ : ↑(σ x.1) • ↑x.2 + ↑x.1 • ↑(f x.2) ∈ I) })
      (1, x) =
    (Path.reparam p f hf hf₀ hf₁).toContinuousMap x

Next tactic:
norm_num
/--
Suppose `p` is a path, then we have a homotopy from `p` to `p.reparam f` by the convexity of `I`.
-/
def reparam (p : Path x₀ x₁) (f : I → I) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
    Homotopy p (p.reparam f hf hf₀ hf₁) where
  toFun x := p ⟨σ x.1 * x.2 + x.1 * f x.2,
    show (σ x.1 : ℝ) • (x.2 : ℝ) + (x.1 : ℝ) • (f x.2 : ℝ) ∈ I from
      convex_Icc _ _ x.2.2 (f x.2).2 (by unit_interval) (by unit_interval) (by simp)⟩
  map_zero_left x := by norm_num
  map_one_left x := by norm_num
  prop' t x hx := by
    

Proof state:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
f : ↑I → ↑I
hf : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
t x : ↑I
hx : x ∈ {0, 1}
⊢ (ContinuousMap.mk fun x =>
        ContinuousMap.toFun
          {
              toContinuousMap :=
                ContinuousMap.mk fun x =>
                  p
                    { val := ↑(σ x.1) * ↑x.2 + ↑x.1 * ↑(f x.2),
                      property := (_ : ↑(σ x.1) • ↑x.2 + ↑x.1 • ↑(f x.2) ∈ I) },
              map_zero_left :=
                (_ :
                  ∀ (x : ↑I),
                    p
                        { val := ↑(σ (0, x).1) * ↑(0, x).2 + ↑(0, x).1 * ↑(f (0, x).2),
                          property := (_ : ↑(σ (0, x).1) • ↑(0, x).2 + ↑(0, x).1 • ↑(f (0, x).2) ∈ I) } =
                      p x),
              map_one_left :=
                (_ :
                  ∀ (x : ↑I),
                    p
                        { val := ↑(σ (1, x).1) * ↑(1, x).2 + ↑(1, x).1 * ↑(f (1, x).2),
                          property := (_ : ↑(σ (1, x).1) • ↑(1, x).2 + ↑(1, x).1 • ↑(f (1, x).2) ∈ I) } =
                      p (f x)) }.toContinuousMap
          (t, x))
      x =
    p.toContinuousMap x

Next tactic:
cases' hx with hx hx
/--
Suppose `p` is a path, then we have a homotopy from `p` to `p.reparam f` by the convexity of `I`.
-/
def reparam (p : Path x₀ x₁) (f : I → I) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
    Homotopy p (p.reparam f hf hf₀ hf₁) where
  toFun x := p ⟨σ x.1 * x.2 + x.1 * f x.2,
    show (σ x.1 : ℝ) • (x.2 : ℝ) + (x.1 : ℝ) • (f x.2 : ℝ) ∈ I from
      convex_Icc _ _ x.2.2 (f x.2).2 (by unit_interval) (by unit_interval) (by simp)⟩
  map_zero_left x := by norm_num
  map_one_left x := by norm_num
  prop' t x hx := by
    cases' hx with hx hx
    · 

Proof state:
case inl
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
f : ↑I → ↑I
hf : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
t x : ↑I
hx : x = 0
⊢ (ContinuousMap.mk fun x =>
        ContinuousMap.toFun
          {
              toContinuousMap :=
                ContinuousMap.mk fun x =>
                  p
                    { val := ↑(σ x.1) * ↑x.2 + ↑x.1 * ↑(f x.2),
                      property := (_ : ↑(σ x.1) • ↑x.2 + ↑x.1 • ↑(f x.2) ∈ I) },
              map_zero_left :=
                (_ :
                  ∀ (x : ↑I),
                    p
                        { val := ↑(σ (0, x).1) * ↑(0, x).2 + ↑(0, x).1 * ↑(f (0, x).2),
                          property := (_ : ↑(σ (0, x).1) • ↑(0, x).2 + ↑(0, x).1 • ↑(f (0, x).2) ∈ I) } =
                      p x),
              map_one_left :=
                (_ :
                  ∀ (x : ↑I),
                    p
                        { val := ↑(σ (1, x).1) * ↑(1, x).2 + ↑(1, x).1 * ↑(f (1, x).2),
                          property := (_ : ↑(σ (1, x).1) • ↑(1, x).2 + ↑(1, x).1 • ↑(f (1, x).2) ∈ I) } =
                      p (f x)) }.toContinuousMap
          (t, x))
      x =
    p.toContinuousMap x

Next tactic:
rw [hx]
/--
Suppose `p` is a path, then we have a homotopy from `p` to `p.reparam f` by the convexity of `I`.
-/
def reparam (p : Path x₀ x₁) (f : I → I) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
    Homotopy p (p.reparam f hf hf₀ hf₁) where
  toFun x := p ⟨σ x.1 * x.2 + x.1 * f x.2,
    show (σ x.1 : ℝ) • (x.2 : ℝ) + (x.1 : ℝ) • (f x.2 : ℝ) ∈ I from
      convex_Icc _ _ x.2.2 (f x.2).2 (by unit_interval) (by unit_interval) (by simp)⟩
  map_zero_left x := by norm_num
  map_one_left x := by norm_num
  prop' t x hx := by
    cases' hx with hx hx
    · rw [hx]
      

Proof state:
case inl
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
f : ↑I → ↑I
hf : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
t x : ↑I
hx : x = 0
⊢ (ContinuousMap.mk fun x =>
        ContinuousMap.toFun
          {
              toContinuousMap :=
                ContinuousMap.mk fun x =>
                  p
                    { val := ↑(σ x.1) * ↑x.2 + ↑x.1 * ↑(f x.2),
                      property := (_ : ↑(σ x.1) • ↑x.2 + ↑x.1 • ↑(f x.2) ∈ I) },
              map_zero_left :=
                (_ :
                  ∀ (x : ↑I),
                    p
                        { val := ↑(σ (0, x).1) * ↑(0, x).2 + ↑(0, x).1 * ↑(f (0, x).2),
                          property := (_ : ↑(σ (0, x).1) • ↑(0, x).2 + ↑(0, x).1 • ↑(f (0, x).2) ∈ I) } =
                      p x),
              map_one_left :=
                (_ :
                  ∀ (x : ↑I),
                    p
                        { val := ↑(σ (1, x).1) * ↑(1, x).2 + ↑(1, x).1 * ↑(f (1, x).2),
                          property := (_ : ↑(σ (1, x).1) • ↑(1, x).2 + ↑(1, x).1 • ↑(f (1, x).2) ∈ I) } =
                      p (f x)) }.toContinuousMap
          (t, x))
      0 =
    p.toContinuousMap 0

Next tactic:
simp [hf₀]
/--
Suppose `p` is a path, then we have a homotopy from `p` to `p.reparam f` by the convexity of `I`.
-/
def reparam (p : Path x₀ x₁) (f : I → I) (hf : Continuous f) (hf₀ : f 0 = 0) (hf₁ : f 1 = 1) :
    Homotopy p (p.reparam f hf hf₀ hf₁) where
  toFun x := p ⟨σ x.1 * x.2 + x.1 * f x.2,
    show (σ x.1 : ℝ) • (x.2 : ℝ) + (x.1 : ℝ) • (f x.2 : ℝ) ∈ I from
      convex_Icc _ _ x.2.2 (f x.2).2 (by unit_interval) (by unit_interval) (by simp)⟩
  map_zero_left x := by norm_num
  map_one_left x := by norm_num
  prop' t x hx := by
    cases' hx with hx hx
    · rw [hx]
      simp [hf₀]
    · 

Proof state:
case inr
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
f : ↑I → ↑I
hf : Continuous f
hf₀ : f 0 = 0
hf₁ : f 1 = 1
t x : ↑I
hx : x ∈ {1}
⊢ (ContinuousMap.mk fun x =>
        ContinuousMap.toFun
          {
              toContinuousMap :=
                ContinuousMap.mk fun x =>
                  p
                    { val := ↑(σ x.1) * ↑x.2 + ↑x.1 * ↑(f x.2),
                      property := (_ : ↑(σ x.1) • ↑x.2 + ↑x.1 • ↑(f x.2) ∈ I) },
              map_zero_left :=
                (_ :
                  ∀ (x : ↑I),
                    p
                        { val := ↑(σ (0, x).1) * ↑(0, x).2 + ↑(0, x).1 * ↑(f (0, x).2),
                          property := (_ : ↑(σ (0, x).1) • ↑(0, x).2 + ↑(0, x).1 • ↑(f (0, x).2) ∈ I) } =
                      p x),
              map_one_left :=
                (_ :
                  ∀ (x : ↑I),
                    p
                        { val := ↑(σ (1, x).1) * ↑(1, x).2 + ↑(1, x).1 * ↑(f (1, x).2),
                          property := (_ : ↑(σ (1, x).1) • ↑(1, x).2 + ↑(1, x).1 • ↑(f (1, x).2) ∈ I) } =
                      p (f x)) }.toContinuousMap
          (t, x))
      x =
    p.toContinuousMap x

Next tactic:
rw [Set.mem_singleton_iff] at hx