l3lab/ntp-mathlib huggingface dataset:
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  

Proof state:
case tail
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b c d : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a c
b✝ c✝ : α
a✝¹ : ReflTransGen r a b✝
a✝ : r b✝ c✝
a_ih✝ : Join (ReflTransGen r) b✝ c
⊢ Join (ReflTransGen r) c✝ c

Next tactic:
case tail d e _ hde ih =>
  rcases ih with ⟨b, hdb, hcb⟩
  have : ∃ a, ReflTransGen r e a ∧ ReflGen r b a := by
    clear hcb
    induction hdb
    case refl => exact ⟨e, refl, ReflGen.single hde⟩
    case tail f b _ hfb ih =>
    rcases ih with ⟨a, hea, hfa⟩
    cases' hfa with _ hfa
    · exact ⟨b, hea.tail hfb, ReflGen.refl⟩
    · rcases h _ _ _ hfb hfa with ⟨c, hbc, hac⟩
      exact ⟨c, hea.trans hac, hbc⟩
  rcases this with ⟨a, hea, hba⟩
  cases' hba with _ hba
  · exact ⟨b, hea, hcb⟩
  · exact ⟨a, hea, hcb.tail hba⟩
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b c d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a c
d e : α
a✝ : ReflTransGen r a d
hde : r d e
ih : Join (ReflTransGen r) d c
⊢ Join (ReflTransGen r) e c

Next tactic:
case tail d e _ hde ih =>
  rcases ih with ⟨b, hdb, hcb⟩
  have : ∃ a, ReflTransGen r e a ∧ ReflGen r b a := by
    clear hcb
    induction hdb
    case refl => exact ⟨e, refl, ReflGen.single hde⟩
    case tail f b _ hfb ih =>
    rcases ih with ⟨a, hea, hfa⟩
    cases' hfa with _ hfa
    · exact ⟨b, hea.tail hfb, ReflGen.refl⟩
    · rcases h _ _ _ hfb hfa with ⟨c, hbc, hac⟩
      exact ⟨c, hea.trans hac, hbc⟩
  rcases this with ⟨a, hea, hba⟩
  cases' hba with _ hba
  · exact ⟨b, hea, hcb⟩
  · exact ⟨a, hea, hcb.tail hba⟩
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  case tail d e _ hde ih =>
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b c d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a c
d e : α
a✝ : ReflTransGen r a d
hde : r d e
ih : Join (ReflTransGen r) d c
⊢ Join (ReflTransGen r) e c

Next tactic:
rcases ih with ⟨b, hdb, hcb⟩
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  case tail d e _ hde ih =>
  rcases ih with ⟨b, hdb, hcb⟩
  

Proof state:
case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b✝ c d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a c
d e : α
a✝ : ReflTransGen r a d
hde : r d e
b : α
hdb : ReflTransGen r d b
hcb : ReflTransGen r c b
⊢ Join (ReflTransGen r) e c

Next tactic:
have : ∃ a, ReflTransGen r e a ∧ ReflGen r b a := by
    clear hcb
    induction hdb
    case refl => exact ⟨e, refl, ReflGen.single hde⟩
    case tail f b _ hfb ih =>
    rcases ih with ⟨a, hea, hfa⟩
    cases' hfa with _ hfa
    · exact ⟨b, hea.tail hfb, ReflGen.refl⟩
    · rcases h _ _ _ hfb hfa with ⟨c, hbc, hac⟩
      exact ⟨c, hea.trans hac, hbc⟩
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  case tail d e _ hde ih =>
  rcases ih with ⟨b, hdb, hcb⟩
  have : ∃ a, ReflTransGen r e a ∧ ReflGen r b a := by
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b✝ c d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a c
d e : α
a✝ : ReflTransGen r a d
hde : r d e
b : α
hdb : ReflTransGen r d b
hcb : ReflTransGen r c b
⊢ ∃ a, ReflTransGen r e a ∧ ReflGen r b a

Next tactic:
clear hcb
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  case tail d e _ hde ih =>
  rcases ih with ⟨b, hdb, hcb⟩
  have : ∃ a, ReflTransGen r e a ∧ ReflGen r b a := by
    clear hcb
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b✝ c d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a c
d e : α
a✝ : ReflTransGen r a d
hde : r d e
b : α
hdb : ReflTransGen r d b
⊢ ∃ a, ReflTransGen r e a ∧ ReflGen r b a

Next tactic:
induction hdb
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  case tail d e _ hde ih =>
  rcases ih with ⟨b, hdb, hcb⟩
  have : ∃ a, ReflTransGen r e a ∧ ReflGen r b a := by
    clear hcb
    induction hdb
    

Proof state:
case refl
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b✝ c d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a c
d e : α
a✝ : ReflTransGen r a d
hde : r d e
b : α
⊢ ∃ a, ReflTransGen r e a ∧ ReflGen r d a
case tail
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b✝¹ c d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a c
d e : α
a✝² : ReflTransGen r a d
hde : r d e
b b✝ c✝ : α
a✝¹ : ReflTransGen r d b✝
a✝ : r b✝ c✝
a_ih✝ : ∃ a, ReflTransGen r e a ∧ ReflGen r b✝ a
⊢ ∃ a, ReflTransGen r e a ∧ ReflGen r c✝ a

Next tactic:
case refl => exact ⟨e, refl, ReflGen.single hde⟩
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  case tail d e _ hde ih =>
  rcases ih with ⟨b, hdb, hcb⟩
  have : ∃ a, ReflTransGen r e a ∧ ReflGen r b a := by
    clear hcb
    induction hdb
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b✝ c d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a c
d e : α
a✝ : ReflTransGen r a d
hde : r d e
b : α
⊢ ∃ a, ReflTransGen r e a ∧ ReflGen r d a

Next tactic:
case refl => exact ⟨e, refl, ReflGen.single hde⟩
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  case tail d e _ hde ih =>
  rcases ih with ⟨b, hdb, hcb⟩
  have : ∃ a, ReflTransGen r e a ∧ ReflGen r b a := by
    clear hcb
    induction hdb
    case refl => 

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b✝ c d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a c
d e : α
a✝ : ReflTransGen r a d
hde : r d e
b : α
⊢ ∃ a, ReflTransGen r e a ∧ ReflGen r d a

Next tactic:
exact ⟨e, refl, ReflGen.single hde⟩
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  case tail d e _ hde ih =>
  rcases ih with ⟨b, hdb, hcb⟩
  have : ∃ a, ReflTransGen r e a ∧ ReflGen r b a := by
    clear hcb
    induction hdb
    case refl => exact ⟨e, refl, ReflGen.single hde⟩
    

Proof state:
case tail
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b✝¹ c d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a c
d e : α
a✝² : ReflTransGen r a d
hde : r d e
b b✝ c✝ : α
a✝¹ : ReflTransGen r d b✝
a✝ : r b✝ c✝
a_ih✝ : ∃ a, ReflTransGen r e a ∧ ReflGen r b✝ a
⊢ ∃ a, ReflTransGen r e a ∧ ReflGen r c✝ a

Next tactic:
case tail f b _ hfb ih =>
    rcases ih with ⟨a, hea, hfa⟩
    cases' hfa with _ hfa
    · exact ⟨b, hea.tail hfb, ReflGen.refl⟩
    · rcases h _ _ _ hfb hfa with ⟨c, hbc, hac⟩
      exact ⟨c, hea.trans hac, hbc⟩
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  case tail d e _ hde ih =>
  rcases ih with ⟨b, hdb, hcb⟩
  have : ∃ a, ReflTransGen r e a ∧ ReflGen r b a := by
    clear hcb
    induction hdb
    case refl => exact ⟨e, refl, ReflGen.single hde⟩
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b✝¹ c d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a c
d e : α
a✝¹ : ReflTransGen r a d
hde : r d e
b✝ f b : α
a✝ : ReflTransGen r d f
hfb : r f b
ih : ∃ a, ReflTransGen r e a ∧ ReflGen r f a
⊢ ∃ a, ReflTransGen r e a ∧ ReflGen r b a

Next tactic:
case tail f b _ hfb ih =>
    rcases ih with ⟨a, hea, hfa⟩
    cases' hfa with _ hfa
    · exact ⟨b, hea.tail hfb, ReflGen.refl⟩
    · rcases h _ _ _ hfb hfa with ⟨c, hbc, hac⟩
      exact ⟨c, hea.trans hac, hbc⟩
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  case tail d e _ hde ih =>
  rcases ih with ⟨b, hdb, hcb⟩
  have : ∃ a, ReflTransGen r e a ∧ ReflGen r b a := by
    clear hcb
    induction hdb
    case refl => exact ⟨e, refl, ReflGen.single hde⟩
    case tail f b _ hfb ih =>
    

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b✝¹ c d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a c
d e : α
a✝¹ : ReflTransGen r a d
hde : r d e
b✝ f b : α
a✝ : ReflTransGen r d f
hfb : r f b
ih : ∃ a, ReflTransGen r e a ∧ ReflGen r f a
⊢ ∃ a, ReflTransGen r e a ∧ ReflGen r b a

Next tactic:
rcases ih with ⟨a, hea, hfa⟩
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  case tail d e _ hde ih =>
  rcases ih with ⟨b, hdb, hcb⟩
  have : ∃ a, ReflTransGen r e a ∧ ReflGen r b a := by
    clear hcb
    induction hdb
    case refl => exact ⟨e, refl, ReflGen.single hde⟩
    case tail f b _ hfb ih =>
    rcases ih with ⟨a, hea, hfa⟩
    

Proof state:
case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a✝² b✝¹ c d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a✝² c
d e : α
a✝¹ : ReflTransGen r a✝² d
hde : r d e
b✝ f b : α
a✝ : ReflTransGen r d f
hfb : r f b
a : α
hea : ReflTransGen r e a
hfa : ReflGen r f a
⊢ ∃ a, ReflTransGen r e a ∧ ReflGen r b a

Next tactic:
cases' hfa with _ hfa
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  case tail d e _ hde ih =>
  rcases ih with ⟨b, hdb, hcb⟩
  have : ∃ a, ReflTransGen r e a ∧ ReflGen r b a := by
    clear hcb
    induction hdb
    case refl => exact ⟨e, refl, ReflGen.single hde⟩
    case tail f b _ hfb ih =>
    rcases ih with ⟨a, hea, hfa⟩
    cases' hfa with _ hfa
    · 

Proof state:
case intro.intro.refl
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b✝¹ c d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a c
d e : α
a✝¹ : ReflTransGen r a d
hde : r d e
b✝ f b : α
a✝ : ReflTransGen r d f
hfb : r f b
hea : ReflTransGen r e f
⊢ ∃ a, ReflTransGen r e a ∧ ReflGen r b a

Next tactic:
exact ⟨b, hea.tail hfb, ReflGen.refl⟩
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  case tail d e _ hde ih =>
  rcases ih with ⟨b, hdb, hcb⟩
  have : ∃ a, ReflTransGen r e a ∧ ReflGen r b a := by
    clear hcb
    induction hdb
    case refl => exact ⟨e, refl, ReflGen.single hde⟩
    case tail f b _ hfb ih =>
    rcases ih with ⟨a, hea, hfa⟩
    cases' hfa with _ hfa
    · exact ⟨b, hea.tail hfb, ReflGen.refl⟩
    · 

Proof state:
case intro.intro.single
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a✝² b✝¹ c d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a✝² c
d e : α
a✝¹ : ReflTransGen r a✝² d
hde : r d e
b✝ f b : α
a✝ : ReflTransGen r d f
hfb : r f b
a : α
hea : ReflTransGen r e a
hfa : r f a
⊢ ∃ a, ReflTransGen r e a ∧ ReflGen r b a

Next tactic:
rcases h _ _ _ hfb hfa with ⟨c, hbc, hac⟩
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  case tail d e _ hde ih =>
  rcases ih with ⟨b, hdb, hcb⟩
  have : ∃ a, ReflTransGen r e a ∧ ReflGen r b a := by
    clear hcb
    induction hdb
    case refl => exact ⟨e, refl, ReflGen.single hde⟩
    case tail f b _ hfb ih =>
    rcases ih with ⟨a, hea, hfa⟩
    cases' hfa with _ hfa
    · exact ⟨b, hea.tail hfb, ReflGen.refl⟩
    · rcases h _ _ _ hfb hfa with ⟨c, hbc, hac⟩
      

Proof state:
case intro.intro.single.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a✝² b✝¹ c✝ d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac✝ : ReflTransGen r a✝² c✝
d e : α
a✝¹ : ReflTransGen r a✝² d
hde : r d e
b✝ f b : α
a✝ : ReflTransGen r d f
hfb : r f b
a : α
hea : ReflTransGen r e a
hfa : r f a
c : α
hbc : ReflGen r b c
hac : ReflTransGen r a c
⊢ ∃ a, ReflTransGen r e a ∧ ReflGen r b a

Next tactic:
exact ⟨c, hea.trans hac, hbc⟩
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  case tail d e _ hde ih =>
  rcases ih with ⟨b, hdb, hcb⟩
  have : ∃ a, ReflTransGen r e a ∧ ReflGen r b a := by
    clear hcb
    induction hdb
    case refl => exact ⟨e, refl, ReflGen.single hde⟩
    case tail f b _ hfb ih =>
    rcases ih with ⟨a, hea, hfa⟩
    cases' hfa with _ hfa
    · exact ⟨b, hea.tail hfb, ReflGen.refl⟩
    · rcases h _ _ _ hfb hfa with ⟨c, hbc, hac⟩
      exact ⟨c, hea.trans hac, hbc⟩
  

Proof state:
case intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b✝ c d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a c
d e : α
a✝ : ReflTransGen r a d
hde : r d e
b : α
hdb : ReflTransGen r d b
hcb : ReflTransGen r c b
this : ∃ a, ReflTransGen r e a ∧ ReflGen r b a
⊢ Join (ReflTransGen r) e c

Next tactic:
rcases this with ⟨a, hea, hba⟩
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  case tail d e _ hde ih =>
  rcases ih with ⟨b, hdb, hcb⟩
  have : ∃ a, ReflTransGen r e a ∧ ReflGen r b a := by
    clear hcb
    induction hdb
    case refl => exact ⟨e, refl, ReflGen.single hde⟩
    case tail f b _ hfb ih =>
    rcases ih with ⟨a, hea, hfa⟩
    cases' hfa with _ hfa
    · exact ⟨b, hea.tail hfb, ReflGen.refl⟩
    · rcases h _ _ _ hfb hfa with ⟨c, hbc, hac⟩
      exact ⟨c, hea.trans hac, hbc⟩
  rcases this with ⟨a, hea, hba⟩
  

Proof state:
case intro.intro.intro.intro
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a✝¹ b✝ c d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a✝¹ c
d e : α
a✝ : ReflTransGen r a✝¹ d
hde : r d e
b : α
hdb : ReflTransGen r d b
hcb : ReflTransGen r c b
a : α
hea : ReflTransGen r e a
hba : ReflGen r b a
⊢ Join (ReflTransGen r) e c

Next tactic:
cases' hba with _ hba
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  case tail d e _ hde ih =>
  rcases ih with ⟨b, hdb, hcb⟩
  have : ∃ a, ReflTransGen r e a ∧ ReflGen r b a := by
    clear hcb
    induction hdb
    case refl => exact ⟨e, refl, ReflGen.single hde⟩
    case tail f b _ hfb ih =>
    rcases ih with ⟨a, hea, hfa⟩
    cases' hfa with _ hfa
    · exact ⟨b, hea.tail hfb, ReflGen.refl⟩
    · rcases h _ _ _ hfb hfa with ⟨c, hbc, hac⟩
      exact ⟨c, hea.trans hac, hbc⟩
  rcases this with ⟨a, hea, hba⟩
  cases' hba with _ hba
  · 

Proof state:
case intro.intro.intro.intro.refl
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b✝ c d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a c
d e : α
a✝ : ReflTransGen r a d
hde : r d e
b : α
hdb : ReflTransGen r d b
hcb : ReflTransGen r c b
hea : ReflTransGen r e b
⊢ Join (ReflTransGen r) e c

Next tactic:
exact ⟨b, hea, hcb⟩
/-- A sufficient condition for the Church-Rosser property. -/
theorem church_rosser (h : ∀ a b c, r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d)
    (hab : ReflTransGen r a b) (hac : ReflTransGen r a c) : Join (ReflTransGen r) b c := by
  induction hab
  case refl => exact ⟨c, hac, refl⟩
  case tail d e _ hde ih =>
  rcases ih with ⟨b, hdb, hcb⟩
  have : ∃ a, ReflTransGen r e a ∧ ReflGen r b a := by
    clear hcb
    induction hdb
    case refl => exact ⟨e, refl, ReflGen.single hde⟩
    case tail f b _ hfb ih =>
    rcases ih with ⟨a, hea, hfa⟩
    cases' hfa with _ hfa
    · exact ⟨b, hea.tail hfb, ReflGen.refl⟩
    · rcases h _ _ _ hfb hfa with ⟨c, hbc, hac⟩
      exact ⟨c, hea.trans hac, hbc⟩
  rcases this with ⟨a, hea, hba⟩
  cases' hba with _ hba
  · exact ⟨b, hea, hcb⟩
  · 

Proof state:
case intro.intro.intro.intro.single
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a✝¹ b✝ c d✝ : α
h : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d
hac : ReflTransGen r a✝¹ c
d e : α
a✝ : ReflTransGen r a✝¹ d
hde : r d e
b : α
hdb : ReflTransGen r d b
hcb : ReflTransGen r c b
a : α
hea : ReflTransGen r e a
hba : r b a
⊢ Join (ReflTransGen r) e c

Next tactic:
exact ⟨a, hea, hcb.tail hba⟩
theorem reflTransGen_of_transitive_reflexive {r' : α → α → Prop} (hr : Reflexive r)
    (ht : Transitive r) (h : ∀ a b, r' a b → r a b) (h' : ReflTransGen r' a b) : r a b := by
  

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b c d : α
r' : α → α → Prop
hr : Reflexive r
ht : Transitive r
h : ∀ (a b : α), r' a b → r a b
h' : ReflTransGen r' a b
⊢ r a b

Next tactic:
induction' h' with b c _ hbc ih
theorem reflTransGen_of_transitive_reflexive {r' : α → α → Prop} (hr : Reflexive r)
    (ht : Transitive r) (h : ∀ a b, r' a b → r a b) (h' : ReflTransGen r' a b) : r a b := by
  induction' h' with b c _ hbc ih
  · 

Proof state:
case refl
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b c d : α
r' : α → α → Prop
hr : Reflexive r
ht : Transitive r
h : ∀ (a b : α), r' a b → r a b
⊢ r a a

Next tactic:
exact hr _
theorem reflTransGen_of_transitive_reflexive {r' : α → α → Prop} (hr : Reflexive r)
    (ht : Transitive r) (h : ∀ a b, r' a b → r a b) (h' : ReflTransGen r' a b) : r a b := by
  induction' h' with b c _ hbc ih
  · exact hr _
  · 

Proof state:
case tail
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b✝ c✝ d : α
r' : α → α → Prop
hr : Reflexive r
ht : Transitive r
h : ∀ (a b : α), r' a b → r a b
b c : α
a✝ : ReflTransGen r' a b
hbc : r' b c
ih : r a b
⊢ r a c

Next tactic:
exact ht ih (h _ _ hbc)
theorem Equivalence.eqvGen_iff (h : Equivalence r) : EqvGen r a b ↔ r a b :=
  Iff.intro
    (by
      

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b : α
h : Equivalence r
⊢ EqvGen r a b → r a b

Next tactic:
intro h
theorem Equivalence.eqvGen_iff (h : Equivalence r) : EqvGen r a b ↔ r a b :=
  Iff.intro
    (by
      intro h
      

Proof state:
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b : α
h✝ : Equivalence r
h : EqvGen r a b
⊢ r a b

Next tactic:
induction h with
      | rel => assumption
      | refl => exact h.1 _
      | symm => apply h.symm; assumption
      | trans _ _ _ _ _ hab hbc => exact h.trans hab hbc
theorem Equivalence.eqvGen_iff (h : Equivalence r) : EqvGen r a b ↔ r a b :=
  Iff.intro
    (by
      intro h
      induction h with
      

Proof state:
case rel
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b : α
h : Equivalence r
x✝ y✝ : α
a✝ : r x✝ y✝
⊢ r x✝ y✝

Next tactic:
| rel => assumption
theorem Equivalence.eqvGen_iff (h : Equivalence r) : EqvGen r a b ↔ r a b :=
  Iff.intro
    (by
      intro h
      induction h with
      | rel => 

Proof state:
case rel
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ζ : Type u_6
r : α → α → Prop
a b : α
h : Equivalence r
x✝ y✝ : α
a✝ : r x✝ y✝
⊢ r x✝ y✝

Next tactic:
assumption