l3lab/ntp-mathlib huggingface dataset:
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · 

Proof state:
case pos
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ↑x ≤ 1 / 2
h₂ : 2 * ↑x ≤ 1 / 2
h₃ : ¬↑x ≤ 1 / 4
h₄ : ¬↑x + 1 / 4 ≤ 1 / 2
h₅ : 2 * (↑x + 1 / 4) - 1 ≤ 1 / 2
⊢ p { val := 2 * (2 * ↑x), property := (_ : 2 * ↑{ val := 2 * ↑x, property := (_ : 2 * ↑x ∈ I) } ∈ I) } =
    q { val := 2 * (2 * (↑x + 1 / 4) - 1), property := (_ : (fun x => x ∈ I) (2 * (2 * (↑x + 1 / 4) - 1))) }

Next tactic:
exfalso
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    

Proof state:
case pos.h
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ↑x ≤ 1 / 2
h₂ : 2 * ↑x ≤ 1 / 2
h₃ : ¬↑x ≤ 1 / 4
h₄ : ¬↑x + 1 / 4 ≤ 1 / 2
h₅ : 2 * (↑x + 1 / 4) - 1 ≤ 1 / 2
⊢ False

Next tactic:
linarith
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · 

Proof state:
case neg
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ↑x ≤ 1 / 2
h₂ : 2 * ↑x ≤ 1 / 2
h₃ : ¬↑x ≤ 1 / 4
h₄ : ¬↑x + 1 / 4 ≤ 1 / 2
h₅ : ¬2 * (↑x + 1 / 4) - 1 ≤ 1 / 2
⊢ p { val := 2 * (2 * ↑x), property := (_ : 2 * ↑{ val := 2 * ↑x, property := (_ : 2 * ↑x ∈ I) } ∈ I) } =
    r { val := 2 * (2 * (↑x + 1 / 4) - 1) - 1, property := (_ : (fun x => x ∈ I) (2 * (2 * (↑x + 1 / 4) - 1) - 1)) }

Next tactic:
exfalso
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    

Proof state:
case neg.h
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ↑x ≤ 1 / 2
h₂ : 2 * ↑x ≤ 1 / 2
h₃ : ¬↑x ≤ 1 / 4
h₄ : ¬↑x + 1 / 4 ≤ 1 / 2
h₅ : ¬2 * (↑x + 1 / 4) - 1 ≤ 1 / 2
⊢ False

Next tactic:
linarith
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · 

Proof state:
case pos
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ↑x ≤ 1 / 2
h₂ : ¬2 * ↑x ≤ 1 / 2
h✝¹ : ↑x ≤ 1 / 4
h✝ : 2 * (2 * ↑x) - 1 ≤ 1 / 2
⊢ q { val := 2 * (2 * ↑x) - 1, property := (_ : 2 * ↑{ val := 2 * ↑x, property := (_ : 2 * ↑x ∈ I) } - 1 ∈ I) } =
    q { val := 2 * (2 * (2 * ↑x) - 1), property := (_ : (fun x => x ∈ I) (2 * (2 * (2 * ↑x) - 1))) }

Next tactic:
exfalso
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    

Proof state:
case pos.h
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ↑x ≤ 1 / 2
h₂ : ¬2 * ↑x ≤ 1 / 2
h✝¹ : ↑x ≤ 1 / 4
h✝ : 2 * (2 * ↑x) - 1 ≤ 1 / 2
⊢ False

Next tactic:
linarith
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · 

Proof state:
case neg
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ↑x ≤ 1 / 2
h₂ : ¬2 * ↑x ≤ 1 / 2
h✝¹ : ↑x ≤ 1 / 4
h✝ : ¬2 * (2 * ↑x) - 1 ≤ 1 / 2
⊢ q { val := 2 * (2 * ↑x) - 1, property := (_ : 2 * ↑{ val := 2 * ↑x, property := (_ : 2 * ↑x ∈ I) } - 1 ∈ I) } =
    r { val := 2 * (2 * (2 * ↑x) - 1) - 1, property := (_ : (fun x => x ∈ I) (2 * (2 * (2 * ↑x) - 1) - 1)) }

Next tactic:
exfalso
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    

Proof state:
case neg.h
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ↑x ≤ 1 / 2
h₂ : ¬2 * ↑x ≤ 1 / 2
h✝¹ : ↑x ≤ 1 / 4
h✝ : ¬2 * (2 * ↑x) - 1 ≤ 1 / 2
⊢ False

Next tactic:
linarith
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · 

Proof state:
case pos
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ↑x ≤ 1 / 2
h₂ : ¬2 * ↑x ≤ 1 / 2
h✝¹ : ¬↑x ≤ 1 / 4
h✝ : ↑x + 1 / 4 ≤ 1 / 2
⊢ q { val := 2 * (2 * ↑x) - 1, property := (_ : 2 * ↑{ val := 2 * ↑x, property := (_ : 2 * ↑x ∈ I) } - 1 ∈ I) } =
    p { val := 2 * (↑x + 1 / 4), property := (_ : (fun x => x ∈ I) (2 * (↑x + 1 / 4))) }

Next tactic:
exfalso
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    

Proof state:
case pos.h
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ↑x ≤ 1 / 2
h₂ : ¬2 * ↑x ≤ 1 / 2
h✝¹ : ¬↑x ≤ 1 / 4
h✝ : ↑x + 1 / 4 ≤ 1 / 2
⊢ False

Next tactic:
linarith
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · 

Proof state:
case pos
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ↑x ≤ 1 / 2
h₂ : ¬2 * ↑x ≤ 1 / 2
h✝² : ¬↑x ≤ 1 / 4
h✝¹ : ¬↑x + 1 / 4 ≤ 1 / 2
h✝ : 2 * (↑x + 1 / 4) - 1 ≤ 1 / 2
⊢ q { val := 2 * (2 * ↑x) - 1, property := (_ : 2 * ↑{ val := 2 * ↑x, property := (_ : 2 * ↑x ∈ I) } - 1 ∈ I) } =
    q { val := 2 * (2 * (↑x + 1 / 4) - 1), property := (_ : (fun x => x ∈ I) (2 * (2 * (↑x + 1 / 4) - 1))) }

Next tactic:
have h : 2 * (2 * (x : ℝ)) - 1 = 2 * (2 * (↑x + 1 / 4) - 1) := by linarith
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · have h : 2 * (2 * (x : ℝ)) - 1 = 2 * (2 * (↑x + 1 / 4) - 1) := by 

Proof state:
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ↑x ≤ 1 / 2
h₂ : ¬2 * ↑x ≤ 1 / 2
h✝² : ¬↑x ≤ 1 / 4
h✝¹ : ¬↑x + 1 / 4 ≤ 1 / 2
h✝ : 2 * (↑x + 1 / 4) - 1 ≤ 1 / 2
⊢ 2 * (2 * ↑x) - 1 = 2 * (2 * (↑x + 1 / 4) - 1)

Next tactic:
linarith
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · have h : 2 * (2 * (x : ℝ)) - 1 = 2 * (2 * (↑x + 1 / 4) - 1) := by linarith
    

Proof state:
case pos
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ↑x ≤ 1 / 2
h₂ : ¬2 * ↑x ≤ 1 / 2
h✝² : ¬↑x ≤ 1 / 4
h✝¹ : ¬↑x + 1 / 4 ≤ 1 / 2
h✝ : 2 * (↑x + 1 / 4) - 1 ≤ 1 / 2
h : 2 * (2 * ↑x) - 1 = 2 * (2 * (↑x + 1 / 4) - 1)
⊢ q { val := 2 * (2 * ↑x) - 1, property := (_ : 2 * ↑{ val := 2 * ↑x, property := (_ : 2 * ↑x ∈ I) } - 1 ∈ I) } =
    q { val := 2 * (2 * (↑x + 1 / 4) - 1), property := (_ : (fun x => x ∈ I) (2 * (2 * (↑x + 1 / 4) - 1))) }

Next tactic:
simp [h₂, h₁, h, dif_neg (show ¬False from id), dif_pos True.intro, if_false, if_true]
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · have h : 2 * (2 * (x : ℝ)) - 1 = 2 * (2 * (↑x + 1 / 4) - 1) := by linarith
    simp [h₂, h₁, h, dif_neg (show ¬False from id), dif_pos True.intro, if_false, if_true]
  · 

Proof state:
case neg
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ↑x ≤ 1 / 2
h₂ : ¬2 * ↑x ≤ 1 / 2
h✝² : ¬↑x ≤ 1 / 4
h✝¹ : ¬↑x + 1 / 4 ≤ 1 / 2
h✝ : ¬2 * (↑x + 1 / 4) - 1 ≤ 1 / 2
⊢ q { val := 2 * (2 * ↑x) - 1, property := (_ : 2 * ↑{ val := 2 * ↑x, property := (_ : 2 * ↑x ∈ I) } - 1 ∈ I) } =
    r { val := 2 * (2 * (↑x + 1 / 4) - 1) - 1, property := (_ : (fun x => x ∈ I) (2 * (2 * (↑x + 1 / 4) - 1) - 1)) }

Next tactic:
exfalso
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · have h : 2 * (2 * (x : ℝ)) - 1 = 2 * (2 * (↑x + 1 / 4) - 1) := by linarith
    simp [h₂, h₁, h, dif_neg (show ¬False from id), dif_pos True.intro, if_false, if_true]
  · exfalso
    

Proof state:
case neg.h
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ↑x ≤ 1 / 2
h₂ : ¬2 * ↑x ≤ 1 / 2
h✝² : ¬↑x ≤ 1 / 4
h✝¹ : ¬↑x + 1 / 4 ≤ 1 / 2
h✝ : ¬2 * (↑x + 1 / 4) - 1 ≤ 1 / 2
⊢ False

Next tactic:
linarith
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · have h : 2 * (2 * (x : ℝ)) - 1 = 2 * (2 * (↑x + 1 / 4) - 1) := by linarith
    simp [h₂, h₁, h, dif_neg (show ¬False from id), dif_pos True.intro, if_false, if_true]
  · exfalso
    linarith
  · 

Proof state:
case pos
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ¬↑x ≤ 1 / 2
h✝¹ : ↑x ≤ 1 / 4
h✝ : 2 * ↑x ≤ 1 / 2
⊢ r { val := 2 * ↑x - 1, property := (_ : 2 * ↑x - 1 ∈ I) } =
    p { val := 2 * (2 * ↑x), property := (_ : (fun x => x ∈ I) (2 * (2 * ↑x))) }

Next tactic:
exfalso
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · have h : 2 * (2 * (x : ℝ)) - 1 = 2 * (2 * (↑x + 1 / 4) - 1) := by linarith
    simp [h₂, h₁, h, dif_neg (show ¬False from id), dif_pos True.intro, if_false, if_true]
  · exfalso
    linarith
  · exfalso
    

Proof state:
case pos.h
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ¬↑x ≤ 1 / 2
h✝¹ : ↑x ≤ 1 / 4
h✝ : 2 * ↑x ≤ 1 / 2
⊢ False

Next tactic:
linarith
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · have h : 2 * (2 * (x : ℝ)) - 1 = 2 * (2 * (↑x + 1 / 4) - 1) := by linarith
    simp [h₂, h₁, h, dif_neg (show ¬False from id), dif_pos True.intro, if_false, if_true]
  · exfalso
    linarith
  · exfalso
    linarith
  · 

Proof state:
case pos
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ¬↑x ≤ 1 / 2
h✝² : ↑x ≤ 1 / 4
h✝¹ : ¬2 * ↑x ≤ 1 / 2
h✝ : 2 * (2 * ↑x) - 1 ≤ 1 / 2
⊢ r { val := 2 * ↑x - 1, property := (_ : 2 * ↑x - 1 ∈ I) } =
    q { val := 2 * (2 * (2 * ↑x) - 1), property := (_ : (fun x => x ∈ I) (2 * (2 * (2 * ↑x) - 1))) }

Next tactic:
exfalso
theorem trans_assoc_reparam {x₀ x₁ x₂ x₃ : X} (p : Path x₀ x₁) (q : Path x₁ x₂) (r : Path x₂ x₃) :
    (p.trans q).trans r =
      (p.trans (q.trans r)).reparam
        (fun t => ⟨transAssocReparamAux t, transAssocReparamAux_mem_I t⟩) (by continuity)
        (Subtype.ext transAssocReparamAux_zero) (Subtype.ext transAssocReparamAux_one) := by
  ext x
  simp only [transAssocReparamAux, Path.trans_apply, mul_inv_cancel_left₀, not_le,
    Function.comp_apply, Ne.def, not_false_iff, bit0_eq_zero, one_ne_zero, mul_ite, Subtype.coe_mk,
    Path.coe_reparam]
  -- TODO: why does split_ifs not reduce the ifs??????
  split_ifs with h₁ h₂ h₃ h₄ h₅
  · rfl
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    linarith
  · have h : 2 * (2 * (x : ℝ)) - 1 = 2 * (2 * (↑x + 1 / 4) - 1) := by linarith
    simp [h₂, h₁, h, dif_neg (show ¬False from id), dif_pos True.intro, if_false, if_true]
  · exfalso
    linarith
  · exfalso
    linarith
  · exfalso
    

Proof state:
case pos.h
X : Type u
Y : Type v
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x₀✝ x₁✝ x₀ x₁ x₂ x₃ : X
p : Path x₀ x₁
q : Path x₁ x₂
r : Path x₂ x₃
x : ↑I
h₁ : ¬↑x ≤ 1 / 2
h✝² : ↑x ≤ 1 / 4
h✝¹ : ¬2 * ↑x ≤ 1 / 2
h✝ : 2 * (2 * ↑x) - 1 ≤ 1 / 2
⊢ False

Next tactic:
linarith