l3lab/ntp-mathlib huggingface dataset:
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    

Proof state:
case rel
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝ y✝ : R
hab : Rel r x✝ y✝
c d : R
hcd : Setoid.r c d
⊢ Setoid.r (x✝ + c) (y✝ + d)

Next tactic:
| rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      

Proof state:
case rel
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝ y✝ : R
hab : Rel r x✝ y✝
c d : R
hcd : Setoid.r c d
⊢ Setoid.r (x✝ + c) (y✝ + d)

Next tactic:
refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      

Proof state:
case rel
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝ y✝ : R
hab : Rel r x✝ y✝
c d : R
hcd : Setoid.r c d
⊢ EqvGen (Rel r) (y✝ + c) (y✝ + d)

Next tactic:
induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      

Proof state:
case rel.rel
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝¹ y✝¹ : R
hab : Rel r x✝¹ y✝¹
c d x✝ y✝ : R
hcd : Rel r x✝ y✝
⊢ EqvGen (Rel r) (y✝¹ + x✝) (y✝¹ + y✝)

Next tactic:
| rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => 

Proof state:
case rel.rel
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝¹ y✝¹ : R
hab : Rel r x✝¹ y✝¹
c d x✝ y✝ : R
hcd : Rel r x✝ y✝
⊢ EqvGen (Rel r) (y✝¹ + x✝) (y✝¹ + y✝)

Next tactic:
exact EqvGen.rel _ _ hcd.add_right
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      

Proof state:
case rel.refl
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝¹ y✝ : R
hab : Rel r x✝¹ y✝
c d x✝ : R
⊢ EqvGen (Rel r) (y✝ + x✝) (y✝ + x✝)

Next tactic:
| refl => exact EqvGen.refl _
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => 

Proof state:
case rel.refl
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝¹ y✝ : R
hab : Rel r x✝¹ y✝
c d x✝ : R
⊢ EqvGen (Rel r) (y✝ + x✝) (y✝ + x✝)

Next tactic:
exact EqvGen.refl _
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      

Proof state:
case rel.symm
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝¹ y✝¹ : R
hab : Rel r x✝¹ y✝¹
c d x✝ y✝ : R
a✝ : EqvGen (Rel r) x✝ y✝
h : EqvGen (Rel r) (y✝¹ + x✝) (y✝¹ + y✝)
⊢ EqvGen (Rel r) (y✝¹ + y✝) (y✝¹ + x✝)

Next tactic:
| symm _ _ _ h => exact h.symm _ _
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => 

Proof state:
case rel.symm
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝¹ y✝¹ : R
hab : Rel r x✝¹ y✝¹
c d x✝ y✝ : R
a✝ : EqvGen (Rel r) x✝ y✝
h : EqvGen (Rel r) (y✝¹ + x✝) (y✝¹ + y✝)
⊢ EqvGen (Rel r) (y✝¹ + y✝) (y✝¹ + x✝)

Next tactic:
exact h.symm _ _
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      

Proof state:
case rel.trans
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝¹ y✝¹ : R
hab : Rel r x✝¹ y✝¹
c d x✝ y✝ z✝ : R
a✝¹ : EqvGen (Rel r) x✝ y✝
a✝ : EqvGen (Rel r) y✝ z✝
h : EqvGen (Rel r) (y✝¹ + x✝) (y✝¹ + y✝)
h' : EqvGen (Rel r) (y✝¹ + y✝) (y✝¹ + z✝)
⊢ EqvGen (Rel r) (y✝¹ + x✝) (y✝¹ + z✝)

Next tactic:
| trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => 

Proof state:
case rel.trans
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝¹ y✝¹ : R
hab : Rel r x✝¹ y✝¹
c d x✝ y✝ z✝ : R
a✝¹ : EqvGen (Rel r) x✝ y✝
a✝ : EqvGen (Rel r) y✝ z✝
h : EqvGen (Rel r) (y✝¹ + x✝) (y✝¹ + y✝)
h' : EqvGen (Rel r) (y✝¹ + y✝) (y✝¹ + z✝)
⊢ EqvGen (Rel r) (y✝¹ + x✝) (y✝¹ + z✝)

Next tactic:
exact h.trans _ _ _ h'
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
    

Proof state:
case refl
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝ c d : R
hcd : Setoid.r c d
⊢ Setoid.r (x✝ + c) (x✝ + d)

Next tactic:
| refl => induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
    | refl => 

Proof state:
case refl
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝ c d : R
hcd : Setoid.r c d
⊢ Setoid.r (x✝ + c) (x✝ + d)

Next tactic:
induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
    | refl => induction hcd with
      

Proof state:
case refl.rel
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝¹ c d x✝ y✝ : R
hcd : Rel r x✝ y✝
⊢ Setoid.r (x✝¹ + x✝) (x✝¹ + y✝)

Next tactic:
| rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
    | refl => induction hcd with
      | rel _ _ hcd => 

Proof state:
case refl.rel
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝¹ c d x✝ y✝ : R
hcd : Rel r x✝ y✝
⊢ Setoid.r (x✝¹ + x✝) (x✝¹ + y✝)

Next tactic:
exact EqvGen.rel _ _ hcd.add_right
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
    | refl => induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      

Proof state:
case refl.refl
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝¹ c d x✝ : R
⊢ Setoid.r (x✝¹ + x✝) (x✝¹ + x✝)

Next tactic:
| refl => exact EqvGen.refl _
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
    | refl => induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => 

Proof state:
case refl.refl
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝¹ c d x✝ : R
⊢ Setoid.r (x✝¹ + x✝) (x✝¹ + x✝)

Next tactic:
exact EqvGen.refl _
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
    | refl => induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      

Proof state:
case refl.symm
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝¹ c d x✝ y✝ : R
a✝ : EqvGen (Rel r) x✝ y✝
h : Setoid.r (x✝¹ + x✝) (x✝¹ + y✝)
⊢ Setoid.r (x✝¹ + y✝) (x✝¹ + x✝)

Next tactic:
| symm _ _ _ h => exact h.symm _ _
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
    | refl => induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => 

Proof state:
case refl.symm
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝¹ c d x✝ y✝ : R
a✝ : EqvGen (Rel r) x✝ y✝
h : Setoid.r (x✝¹ + x✝) (x✝¹ + y✝)
⊢ Setoid.r (x✝¹ + y✝) (x✝¹ + x✝)

Next tactic:
exact h.symm _ _
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
    | refl => induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      

Proof state:
case refl.trans
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝¹ c d x✝ y✝ z✝ : R
a✝¹ : EqvGen (Rel r) x✝ y✝
a✝ : EqvGen (Rel r) y✝ z✝
h : Setoid.r (x✝¹ + x✝) (x✝¹ + y✝)
h' : Setoid.r (x✝¹ + y✝) (x✝¹ + z✝)
⊢ Setoid.r (x✝¹ + x✝) (x✝¹ + z✝)

Next tactic:
| trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
    | refl => induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => 

Proof state:
case refl.trans
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x✝¹ c d x✝ y✝ z✝ : R
a✝¹ : EqvGen (Rel r) x✝ y✝
a✝ : EqvGen (Rel r) y✝ z✝
h : Setoid.r (x✝¹ + x✝) (x✝¹ + y✝)
h' : Setoid.r (x✝¹ + y✝) (x✝¹ + z✝)
⊢ Setoid.r (x✝¹ + x✝) (x✝¹ + z✝)

Next tactic:
exact h.trans _ _ _ h'
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
    | refl => induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
    

Proof state:
case symm
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x y : R
a✝ : EqvGen (Rel r) x y
hxy : ∀ {c d : R}, Setoid.r c d → Setoid.r (x + c) (y + d)
c d : R
hcd : Setoid.r c d
⊢ Setoid.r (y + c) (x + d)

Next tactic:
| symm x y _ hxy => exact (hxy hcd.symm).symm
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
    | refl => induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
    | symm x y _ hxy => 

Proof state:
case symm
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x y : R
a✝ : EqvGen (Rel r) x y
hxy : ∀ {c d : R}, Setoid.r c d → Setoid.r (x + c) (y + d)
c d : R
hcd : Setoid.r c d
⊢ Setoid.r (y + c) (x + d)

Next tactic:
exact (hxy hcd.symm).symm
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
    | refl => induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
    | symm x y _ hxy => exact (hxy hcd.symm).symm
    

Proof state:
case trans
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x y z : R
a✝¹ : EqvGen (Rel r) x y
a✝ : EqvGen (Rel r) y z
h : ∀ {c d : R}, Setoid.r c d → Setoid.r (x + c) (y + d)
h' : ∀ {c d : R}, Setoid.r c d → Setoid.r (y + c) (z + d)
c d : R
hcd : Setoid.r c d
⊢ Setoid.r (x + c) (z + d)

Next tactic:
| trans x y z _ _ h h' => exact (h hcd).trans _ _ _ (h' <| EqvGen.refl _)
/-- `EqvGen (RingQuot.Rel r)` is a ring congruence. -/
def ringCon (r : R → R → Prop) : RingCon R where
  r := EqvGen (Rel r)
  iseqv := EqvGen.is_equivalence _
  add' {a b c d} hab hcd := by
    induction hab generalizing c d with
    | rel _ _ hab =>
      refine' (EqvGen.rel _ _ hab.add_left).trans _ _ _ _
      induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
    | refl => induction hcd with
      | rel _ _ hcd => exact EqvGen.rel _ _ hcd.add_right
      | refl => exact EqvGen.refl _
      | symm _ _ _ h => exact h.symm _ _
      | trans _ _ _ _ _ h h' => exact h.trans _ _ _ h'
    | symm x y _ hxy => exact (hxy hcd.symm).symm
    | trans x y z _ _ h h' => 

Proof state:
case trans
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
a b x y z : R
a✝¹ : EqvGen (Rel r) x y
a✝ : EqvGen (Rel r) y z
h : ∀ {c d : R}, Setoid.r c d → Setoid.r (x + c) (y + d)
h' : ∀ {c d : R}, Setoid.r c d → Setoid.r (y + c) (z + d)
c d : R
hcd : Setoid.r c d
⊢ Setoid.r (x + c) (z + d)

Next tactic:
exact (h hcd).trans _ _ _ (h' <| EqvGen.refl _)
theorem eqvGen_rel_eq (r : R → R → Prop) : EqvGen (Rel r) = RingConGen.Rel r := by
  

Proof state:
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
⊢ EqvGen (Rel r) = RingConGen.Rel r

Next tactic:
ext x₁ x₂
theorem eqvGen_rel_eq (r : R → R → Prop) : EqvGen (Rel r) = RingConGen.Rel r := by
  ext x₁ x₂
  

Proof state:
case h.h.a
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
x₁ x₂ : R
⊢ EqvGen (Rel r) x₁ x₂ ↔ RingConGen.Rel r x₁ x₂

Next tactic:
constructor
theorem eqvGen_rel_eq (r : R → R → Prop) : EqvGen (Rel r) = RingConGen.Rel r := by
  ext x₁ x₂
  constructor
  · 

Proof state:
case h.h.a.mp
R : Type uR
inst✝³ : Semiring R
S : Type uS
inst✝² : CommSemiring S
T : Type uT
A : Type uA
inst✝¹ : Semiring A
inst✝ : Algebra S A
r : R → R → Prop
x₁ x₂ : R
⊢ EqvGen (Rel r) x₁ x₂ → RingConGen.Rel r x₁ x₂

Next tactic:
intro h