l3lab/ntp-mathlib huggingface dataset:
theorem AdjoinMonic.isIntegral (z : AdjoinMonic k) : IsIntegral k z := by
  let ⟨p, hp⟩ := Ideal.Quotient.mk_surjective z
  rw [← hp]
  induction p using MvPolynomial.induction_on generalizing z with
    | h_C => exact isIntegral_algebraMap
    

Proof state:
case h_add
k : Type u
inst✝ : Field k
p✝ q✝ : MvPolynomial (MonicIrreducible k) k
ha : ∀ (z : AdjoinMonic k), (Ideal.Quotient.mk (maxIdeal k)) p✝ = z → IsIntegral k ((Ideal.Quotient.mk (maxIdeal k)) p✝)
hb : ∀ (z : AdjoinMonic k), (Ideal.Quotient.mk (maxIdeal k)) q✝ = z → IsIntegral k ((Ideal.Quotient.mk (maxIdeal k)) q✝)
z : AdjoinMonic k
hp : (Ideal.Quotient.mk (maxIdeal k)) (p✝ + q✝) = z
⊢ IsIntegral k ((Ideal.Quotient.mk (maxIdeal k)) (p✝ + q✝))

Next tactic:
| h_add _ _ ha hb => exact (ha _ rfl).add (hb _ rfl)
theorem AdjoinMonic.isIntegral (z : AdjoinMonic k) : IsIntegral k z := by
  let ⟨p, hp⟩ := Ideal.Quotient.mk_surjective z
  rw [← hp]
  induction p using MvPolynomial.induction_on generalizing z with
    | h_C => exact isIntegral_algebraMap
    | h_add _ _ ha hb => 

Proof state:
case h_add
k : Type u
inst✝ : Field k
p✝ q✝ : MvPolynomial (MonicIrreducible k) k
ha : ∀ (z : AdjoinMonic k), (Ideal.Quotient.mk (maxIdeal k)) p✝ = z → IsIntegral k ((Ideal.Quotient.mk (maxIdeal k)) p✝)
hb : ∀ (z : AdjoinMonic k), (Ideal.Quotient.mk (maxIdeal k)) q✝ = z → IsIntegral k ((Ideal.Quotient.mk (maxIdeal k)) q✝)
z : AdjoinMonic k
hp : (Ideal.Quotient.mk (maxIdeal k)) (p✝ + q✝) = z
⊢ IsIntegral k ((Ideal.Quotient.mk (maxIdeal k)) (p✝ + q✝))

Next tactic:
exact (ha _ rfl).add (hb _ rfl)
theorem AdjoinMonic.isIntegral (z : AdjoinMonic k) : IsIntegral k z := by
  let ⟨p, hp⟩ := Ideal.Quotient.mk_surjective z
  rw [← hp]
  induction p using MvPolynomial.induction_on generalizing z with
    | h_C => exact isIntegral_algebraMap
    | h_add _ _ ha hb => exact (ha _ rfl).add (hb _ rfl)
    

Proof state:
case h_X
k : Type u
inst✝ : Field k
p : MvPolynomial (MonicIrreducible k) k
f : MonicIrreducible k
ih : ∀ (z : AdjoinMonic k), (Ideal.Quotient.mk (maxIdeal k)) p = z → IsIntegral k ((Ideal.Quotient.mk (maxIdeal k)) p)
z : AdjoinMonic k
hp : (Ideal.Quotient.mk (maxIdeal k)) (p * MvPolynomial.X f) = z
⊢ IsIntegral k ((Ideal.Quotient.mk (maxIdeal k)) (p * MvPolynomial.X f))

Next tactic:
| h_X p f ih =>
      · refine @IsIntegral.mul k _ _ _ _ _ (Ideal.Quotient.mk (maxIdeal k) _) (ih _ rfl) ?_
        refine ⟨f, f.2.1, ?_⟩
        erw [AdjoinMonic.algebraMap, ← hom_eval₂, Ideal.Quotient.eq_zero_iff_mem]
        exact le_maxIdeal k (Ideal.subset_span ⟨f, rfl⟩)
theorem AdjoinMonic.isIntegral (z : AdjoinMonic k) : IsIntegral k z := by
  let ⟨p, hp⟩ := Ideal.Quotient.mk_surjective z
  rw [← hp]
  induction p using MvPolynomial.induction_on generalizing z with
    | h_C => exact isIntegral_algebraMap
    | h_add _ _ ha hb => exact (ha _ rfl).add (hb _ rfl)
    | h_X p f ih =>
      · 

Proof state:
case h_X
k : Type u
inst✝ : Field k
p : MvPolynomial (MonicIrreducible k) k
f : MonicIrreducible k
ih : ∀ (z : AdjoinMonic k), (Ideal.Quotient.mk (maxIdeal k)) p = z → IsIntegral k ((Ideal.Quotient.mk (maxIdeal k)) p)
z : AdjoinMonic k
hp : (Ideal.Quotient.mk (maxIdeal k)) (p * MvPolynomial.X f) = z
⊢ IsIntegral k ((Ideal.Quotient.mk (maxIdeal k)) (p * MvPolynomial.X f))

Next tactic:
refine @IsIntegral.mul k _ _ _ _ _ (Ideal.Quotient.mk (maxIdeal k) _) (ih _ rfl) ?_
theorem AdjoinMonic.isIntegral (z : AdjoinMonic k) : IsIntegral k z := by
  let ⟨p, hp⟩ := Ideal.Quotient.mk_surjective z
  rw [← hp]
  induction p using MvPolynomial.induction_on generalizing z with
    | h_C => exact isIntegral_algebraMap
    | h_add _ _ ha hb => exact (ha _ rfl).add (hb _ rfl)
    | h_X p f ih =>
      · refine @IsIntegral.mul k _ _ _ _ _ (Ideal.Quotient.mk (maxIdeal k) _) (ih _ rfl) ?_
        

Proof state:
case h_X
k : Type u
inst✝ : Field k
p : MvPolynomial (MonicIrreducible k) k
f : MonicIrreducible k
ih : ∀ (z : AdjoinMonic k), (Ideal.Quotient.mk (maxIdeal k)) p = z → IsIntegral k ((Ideal.Quotient.mk (maxIdeal k)) p)
z : AdjoinMonic k
hp : (Ideal.Quotient.mk (maxIdeal k)) (p * MvPolynomial.X f) = z
⊢ IsIntegral k ((Ideal.Quotient.mk (maxIdeal k)) (MvPolynomial.X f))

Next tactic:
refine ⟨f, f.2.1, ?_⟩
theorem AdjoinMonic.isIntegral (z : AdjoinMonic k) : IsIntegral k z := by
  let ⟨p, hp⟩ := Ideal.Quotient.mk_surjective z
  rw [← hp]
  induction p using MvPolynomial.induction_on generalizing z with
    | h_C => exact isIntegral_algebraMap
    | h_add _ _ ha hb => exact (ha _ rfl).add (hb _ rfl)
    | h_X p f ih =>
      · refine @IsIntegral.mul k _ _ _ _ _ (Ideal.Quotient.mk (maxIdeal k) _) (ih _ rfl) ?_
        refine ⟨f, f.2.1, ?_⟩
        

Proof state:
case h_X
k : Type u
inst✝ : Field k
p : MvPolynomial (MonicIrreducible k) k
f : MonicIrreducible k
ih : ∀ (z : AdjoinMonic k), (Ideal.Quotient.mk (maxIdeal k)) p = z → IsIntegral k ((Ideal.Quotient.mk (maxIdeal k)) p)
z : AdjoinMonic k
hp : (Ideal.Quotient.mk (maxIdeal k)) (p * MvPolynomial.X f) = z
⊢ Polynomial.eval₂ (_root_.algebraMap k (MvPolynomial (MonicIrreducible k) k ⧸ maxIdeal k))
      ((Ideal.Quotient.mk (maxIdeal k)) (MvPolynomial.X f)) ↑f =
    0

Next tactic:
erw [AdjoinMonic.algebraMap, ← hom_eval₂, Ideal.Quotient.eq_zero_iff_mem]
theorem AdjoinMonic.isIntegral (z : AdjoinMonic k) : IsIntegral k z := by
  let ⟨p, hp⟩ := Ideal.Quotient.mk_surjective z
  rw [← hp]
  induction p using MvPolynomial.induction_on generalizing z with
    | h_C => exact isIntegral_algebraMap
    | h_add _ _ ha hb => exact (ha _ rfl).add (hb _ rfl)
    | h_X p f ih =>
      · refine @IsIntegral.mul k _ _ _ _ _ (Ideal.Quotient.mk (maxIdeal k) _) (ih _ rfl) ?_
        refine ⟨f, f.2.1, ?_⟩
        erw [AdjoinMonic.algebraMap, ← hom_eval₂, Ideal.Quotient.eq_zero_iff_mem]
        

Proof state:
case h_X
k : Type u
inst✝ : Field k
p : MvPolynomial (MonicIrreducible k) k
f : MonicIrreducible k
ih : ∀ (z : AdjoinMonic k), (Ideal.Quotient.mk (maxIdeal k)) p = z → IsIntegral k ((Ideal.Quotient.mk (maxIdeal k)) p)
z : AdjoinMonic k
hp : (Ideal.Quotient.mk (maxIdeal k)) (p * MvPolynomial.X f) = z
⊢ Polynomial.eval₂ MvPolynomial.C (MvPolynomial.X f) ↑f ∈ maxIdeal k

Next tactic:
exact le_maxIdeal k (Ideal.subset_span ⟨f, rfl⟩)
theorem AdjoinMonic.exists_root {f : k[X]} (hfm : f.Monic) (hfi : Irreducible f) :
    ∃ x : AdjoinMonic k, f.eval₂ (toAdjoinMonic k) x = 0 :=
  ⟨Ideal.Quotient.mk _ <| X (⟨f, hfm, hfi⟩ : MonicIrreducible k), by
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    

Proof state:
k : Type u
inst✝ : Field k
f : k[X]
hfm : Monic f
hfi : Irreducible f
⊢ Polynomial.eval₂ (toAdjoinMonic k)
      ((Ideal.Quotient.mk (maxIdeal k)) (MvPolynomial.X { val := f, property := (_ : Monic f ∧ Irreducible f) })) f =
    0

Next tactic:
erw [toAdjoinMonic, ← hom_eval₂, Ideal.Quotient.eq_zero_iff_mem]
theorem AdjoinMonic.exists_root {f : k[X]} (hfm : f.Monic) (hfi : Irreducible f) :
    ∃ x : AdjoinMonic k, f.eval₂ (toAdjoinMonic k) x = 0 :=
  ⟨Ideal.Quotient.mk _ <| X (⟨f, hfm, hfi⟩ : MonicIrreducible k), by
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [toAdjoinMonic, ← hom_eval₂, Ideal.Quotient.eq_zero_iff_mem]
    

Proof state:
k : Type u
inst✝ : Field k
f : k[X]
hfm : Monic f
hfi : Irreducible f
⊢ Polynomial.eval₂ MvPolynomial.C (MvPolynomial.X { val := f, property := (_ : Monic f ∧ Irreducible f) }) f ∈
    maxIdeal k

Next tactic:
exact le_maxIdeal k (Ideal.subset_span <| ⟨_, rfl⟩)
theorem toStepSucc.exists_root {n} {f : Polynomial (Step k n)} (hfm : f.Monic)
    (hfi : Irreducible f) : ∃ x : Step k (n + 1), f.eval₂ (toStepSucc k n) x = 0 := by
-- Porting note: original proof was `@AdjoinMonic.exists_root _ (Step.field k n) _ hfm hfi`,
-- but it timeouts.
  

Proof state:
k : Type u
inst✝ : Field k
n : ℕ
f : (Step k n)[X]
hfm : Monic f
hfi : Irreducible f
⊢ ∃ x, Polynomial.eval₂ (toStepSucc k n) x f = 0

Next tactic:
obtain ⟨x, hx⟩ := @AdjoinMonic.exists_root _ (Step.field k n) _ hfm hfi
theorem toStepSucc.exists_root {n} {f : Polynomial (Step k n)} (hfm : f.Monic)
    (hfi : Irreducible f) : ∃ x : Step k (n + 1), f.eval₂ (toStepSucc k n) x = 0 := by
-- Porting note: original proof was `@AdjoinMonic.exists_root _ (Step.field k n) _ hfm hfi`,
-- but it timeouts.
  obtain ⟨x, hx⟩ := @AdjoinMonic.exists_root _ (Step.field k n) _ hfm hfi
-- Porting note: using `hx` instead of `by apply hx` timeouts.
  

Proof state:
case intro
k : Type u
inst✝ : Field k
n : ℕ
f : (Step k n)[X]
hfm : Monic f
hfi : Irreducible f
x : AdjoinMonic (Step k n)
hx : Polynomial.eval₂ (toAdjoinMonic (Step k n)) x f = 0
⊢ ∃ x, Polynomial.eval₂ (toStepSucc k n) x f = 0

Next tactic:
exact ⟨x, by apply hx⟩
theorem toStepSucc.exists_root {n} {f : Polynomial (Step k n)} (hfm : f.Monic)
    (hfi : Irreducible f) : ∃ x : Step k (n + 1), f.eval₂ (toStepSucc k n) x = 0 := by
-- Porting note: original proof was `@AdjoinMonic.exists_root _ (Step.field k n) _ hfm hfi`,
-- but it timeouts.
  obtain ⟨x, hx⟩ := @AdjoinMonic.exists_root _ (Step.field k n) _ hfm hfi
-- Porting note: using `hx` instead of `by apply hx` timeouts.
  exact ⟨x, by 

Proof state:
k : Type u
inst✝ : Field k
n : ℕ
f : (Step k n)[X]
hfm : Monic f
hfi : Irreducible f
x : AdjoinMonic (Step k n)
hx : Polynomial.eval₂ (toAdjoinMonic (Step k n)) x f = 0
⊢ Polynomial.eval₂ (toStepSucc k n) x f = 0

Next tactic:
apply hx
private theorem toStepOfLE'.succ (m n : ℕ) (h : m ≤ n) :
    toStepOfLE' k m (Nat.succ n) (h.trans n.le_succ) =
    (toStepSucc k n) ∘ toStepOfLE' k m n h := by
  

Proof state:
k : Type u
inst✝ : Field k
m n : ℕ
h : m ≤ n
⊢ AlgebraicClosure.toStepOfLE' k m (Nat.succ n) (_ : m ≤ Nat.succ n) =
    ⇑(toStepSucc k n) ∘ AlgebraicClosure.toStepOfLE' k m n h

Next tactic:
ext x
private theorem toStepOfLE'.succ (m n : ℕ) (h : m ≤ n) :
    toStepOfLE' k m (Nat.succ n) (h.trans n.le_succ) =
    (toStepSucc k n) ∘ toStepOfLE' k m n h := by
  ext x
  

Proof state:
case h
k : Type u
inst✝ : Field k
m n : ℕ
h : m ≤ n
x : Step k m
⊢ AlgebraicClosure.toStepOfLE' k m (Nat.succ n) (_ : m ≤ Nat.succ n) x =
    (⇑(toStepSucc k n) ∘ AlgebraicClosure.toStepOfLE' k m n h) x

Next tactic:
convert Nat.leRecOn_succ h x
private theorem toStepOfLE'.succ (m n : ℕ) (h : m ≤ n) :
    toStepOfLE' k m (Nat.succ n) (h.trans n.le_succ) =
    (toStepSucc k n) ∘ toStepOfLE' k m n h := by
  ext x
  convert Nat.leRecOn_succ h x
  

Proof state:
case h.convert_1
k : Type u
inst✝ : Field k
m n : ℕ
h : m ≤ n
x : Step k m
⊢ m ≤ n + 1

Next tactic:
exact h.trans n.le_succ
/-- The canonical ring homomorphism to a step with a greater index. -/
def toStepOfLE (m n : ℕ) (h : m ≤ n) : Step k m →+* Step k n where
  toFun := toStepOfLE' k m n h
  map_one' := by
-- Porting note: original proof was `induction' h with n h ih; · exact Nat.leRecOn_self 1`
--                                   `rw [Nat.leRecOn_succ h, ih, RingHom.map_one]`
    

Proof state:
k : Type u
inst✝ : Field k
m n : ℕ
h : m ≤ n
⊢ AlgebraicClosure.toStepOfLE' k m n h 1 = 1

Next tactic:
induction' h with a h ih
/-- The canonical ring homomorphism to a step with a greater index. -/
def toStepOfLE (m n : ℕ) (h : m ≤ n) : Step k m →+* Step k n where
  toFun := toStepOfLE' k m n h
  map_one' := by
-- Porting note: original proof was `induction' h with n h ih; · exact Nat.leRecOn_self 1`
--                                   `rw [Nat.leRecOn_succ h, ih, RingHom.map_one]`
    induction' h with a h ih
    · 

Proof state:
case refl
k : Type u
inst✝ : Field k
m n : ℕ
⊢ AlgebraicClosure.toStepOfLE' k m m (_ : Nat.le m m) 1 = 1

Next tactic:
exact Nat.leRecOn_self 1
/-- The canonical ring homomorphism to a step with a greater index. -/
def toStepOfLE (m n : ℕ) (h : m ≤ n) : Step k m →+* Step k n where
  toFun := toStepOfLE' k m n h
  map_one' := by
-- Porting note: original proof was `induction' h with n h ih; · exact Nat.leRecOn_self 1`
--                                   `rw [Nat.leRecOn_succ h, ih, RingHom.map_one]`
    induction' h with a h ih
    · exact Nat.leRecOn_self 1
    · 

Proof state:
case step
k : Type u
inst✝ : Field k
m n a : ℕ
h : Nat.le m a
ih : AlgebraicClosure.toStepOfLE' k m a h 1 = 1
⊢ AlgebraicClosure.toStepOfLE' k m (Nat.succ a) (_ : Nat.le m (Nat.succ a)) 1 = 1

Next tactic:
rw [toStepOfLE'.succ k m a h]
/-- The canonical ring homomorphism to a step with a greater index. -/
def toStepOfLE (m n : ℕ) (h : m ≤ n) : Step k m →+* Step k n where
  toFun := toStepOfLE' k m n h
  map_one' := by
-- Porting note: original proof was `induction' h with n h ih; · exact Nat.leRecOn_self 1`
--                                   `rw [Nat.leRecOn_succ h, ih, RingHom.map_one]`
    induction' h with a h ih
    · exact Nat.leRecOn_self 1
    · rw [toStepOfLE'.succ k m a h]; 

Proof state:
case step
k : Type u
inst✝ : Field k
m n a : ℕ
h : Nat.le m a
ih : AlgebraicClosure.toStepOfLE' k m a h 1 = 1
⊢ (⇑(toStepSucc k a) ∘ AlgebraicClosure.toStepOfLE' k m a h) 1 = 1

Next tactic:
simp [ih]
/-- The canonical ring homomorphism to a step with a greater index. -/
def toStepOfLE (m n : ℕ) (h : m ≤ n) : Step k m →+* Step k n where
  toFun := toStepOfLE' k m n h
  map_one' := by
-- Porting note: original proof was `induction' h with n h ih; · exact Nat.leRecOn_self 1`
--                                   `rw [Nat.leRecOn_succ h, ih, RingHom.map_one]`
    induction' h with a h ih
    · exact Nat.leRecOn_self 1
    · rw [toStepOfLE'.succ k m a h]; simp [ih]
  map_mul' x y := by
-- Porting note: original proof was `induction' h with n h ih; · simp_rw [Nat.leRecOn_self]`
--                                   `simp_rw [Nat.leRecOn_succ h, ih, RingHom.map_mul]`
    

Proof state:
k : Type u
inst✝ : Field k
m n : ℕ
h : m ≤ n
x y : Step k m
⊢ OneHom.toFun
      { toFun := AlgebraicClosure.toStepOfLE' k m n h, map_one' := (_ : AlgebraicClosure.toStepOfLE' k m n h 1 = 1) }
      (x * y) =
    OneHom.toFun
        { toFun := AlgebraicClosure.toStepOfLE' k m n h, map_one' := (_ : AlgebraicClosure.toStepOfLE' k m n h 1 = 1) }
        x *
      OneHom.toFun
        { toFun := AlgebraicClosure.toStepOfLE' k m n h, map_one' := (_ : AlgebraicClosure.toStepOfLE' k m n h 1 = 1) }
        y

Next tactic:
induction' h with a h ih
/-- The canonical ring homomorphism to a step with a greater index. -/
def toStepOfLE (m n : ℕ) (h : m ≤ n) : Step k m →+* Step k n where
  toFun := toStepOfLE' k m n h
  map_one' := by
-- Porting note: original proof was `induction' h with n h ih; · exact Nat.leRecOn_self 1`
--                                   `rw [Nat.leRecOn_succ h, ih, RingHom.map_one]`
    induction' h with a h ih
    · exact Nat.leRecOn_self 1
    · rw [toStepOfLE'.succ k m a h]; simp [ih]
  map_mul' x y := by
-- Porting note: original proof was `induction' h with n h ih; · simp_rw [Nat.leRecOn_self]`
--                                   `simp_rw [Nat.leRecOn_succ h, ih, RingHom.map_mul]`
    induction' h with a h ih
    · 

Proof state:
case refl
k : Type u
inst✝ : Field k
m n : ℕ
x y : Step k m
⊢ OneHom.toFun
      { toFun := AlgebraicClosure.toStepOfLE' k m m (_ : Nat.le m m),
        map_one' := (_ : AlgebraicClosure.toStepOfLE' k m m (_ : Nat.le m m) 1 = 1) }
      (x * y) =
    OneHom.toFun
        { toFun := AlgebraicClosure.toStepOfLE' k m m (_ : Nat.le m m),
          map_one' := (_ : AlgebraicClosure.toStepOfLE' k m m (_ : Nat.le m m) 1 = 1) }
        x *
      OneHom.toFun
        { toFun := AlgebraicClosure.toStepOfLE' k m m (_ : Nat.le m m),
          map_one' := (_ : AlgebraicClosure.toStepOfLE' k m m (_ : Nat.le m m) 1 = 1) }
        y

Next tactic:
dsimp [toStepOfLE']
/-- The canonical ring homomorphism to a step with a greater index. -/
def toStepOfLE (m n : ℕ) (h : m ≤ n) : Step k m →+* Step k n where
  toFun := toStepOfLE' k m n h
  map_one' := by
-- Porting note: original proof was `induction' h with n h ih; · exact Nat.leRecOn_self 1`
--                                   `rw [Nat.leRecOn_succ h, ih, RingHom.map_one]`
    induction' h with a h ih
    · exact Nat.leRecOn_self 1
    · rw [toStepOfLE'.succ k m a h]; simp [ih]
  map_mul' x y := by
-- Porting note: original proof was `induction' h with n h ih; · simp_rw [Nat.leRecOn_self]`
--                                   `simp_rw [Nat.leRecOn_succ h, ih, RingHom.map_mul]`
    induction' h with a h ih
    · dsimp [toStepOfLE']; 

Proof state:
case refl
k : Type u
inst✝ : Field k
m n : ℕ
x y : Step k m
⊢ Nat.leRecOn (_ : Nat.le m m) (fun a => ⇑(toStepSucc k a)) (x * y) =
    Nat.leRecOn (_ : Nat.le m m) (fun a => ⇑(toStepSucc k a)) x *
      Nat.leRecOn (_ : Nat.le m m) (fun a => ⇑(toStepSucc k a)) y

Next tactic:
simp_rw [Nat.leRecOn_self]
/-- The canonical ring homomorphism to a step with a greater index. -/
def toStepOfLE (m n : ℕ) (h : m ≤ n) : Step k m →+* Step k n where
  toFun := toStepOfLE' k m n h
  map_one' := by
-- Porting note: original proof was `induction' h with n h ih; · exact Nat.leRecOn_self 1`
--                                   `rw [Nat.leRecOn_succ h, ih, RingHom.map_one]`
    induction' h with a h ih
    · exact Nat.leRecOn_self 1
    · rw [toStepOfLE'.succ k m a h]; simp [ih]
  map_mul' x y := by
-- Porting note: original proof was `induction' h with n h ih; · simp_rw [Nat.leRecOn_self]`
--                                   `simp_rw [Nat.leRecOn_succ h, ih, RingHom.map_mul]`
    induction' h with a h ih
    · dsimp [toStepOfLE']; simp_rw [Nat.leRecOn_self]
    · 

Proof state:
case step
k : Type u
inst✝ : Field k
m n : ℕ
x y : Step k m
a : ℕ
h : Nat.le m a
ih :
  OneHom.toFun
      { toFun := AlgebraicClosure.toStepOfLE' k m a h, map_one' := (_ : AlgebraicClosure.toStepOfLE' k m a h 1 = 1) }
      (x * y) =
    OneHom.toFun
        { toFun := AlgebraicClosure.toStepOfLE' k m a h, map_one' := (_ : AlgebraicClosure.toStepOfLE' k m a h 1 = 1) }
        x *
      OneHom.toFun
        { toFun := AlgebraicClosure.toStepOfLE' k m a h, map_one' := (_ : AlgebraicClosure.toStepOfLE' k m a h 1 = 1) }
        y
⊢ OneHom.toFun
      { toFun := AlgebraicClosure.toStepOfLE' k m (Nat.succ a) (_ : Nat.le m (Nat.succ a)),
        map_one' := (_ : AlgebraicClosure.toStepOfLE' k m (Nat.succ a) (_ : Nat.le m (Nat.succ a)) 1 = 1) }
      (x * y) =
    OneHom.toFun
        { toFun := AlgebraicClosure.toStepOfLE' k m (Nat.succ a) (_ : Nat.le m (Nat.succ a)),
          map_one' := (_ : AlgebraicClosure.toStepOfLE' k m (Nat.succ a) (_ : Nat.le m (Nat.succ a)) 1 = 1) }
        x *
      OneHom.toFun
        { toFun := AlgebraicClosure.toStepOfLE' k m (Nat.succ a) (_ : Nat.le m (Nat.succ a)),
          map_one' := (_ : AlgebraicClosure.toStepOfLE' k m (Nat.succ a) (_ : Nat.le m (Nat.succ a)) 1 = 1) }
        y

Next tactic:
simp_rw [toStepOfLE'.succ k m a h]
/-- The canonical ring homomorphism to a step with a greater index. -/
def toStepOfLE (m n : ℕ) (h : m ≤ n) : Step k m →+* Step k n where
  toFun := toStepOfLE' k m n h
  map_one' := by
-- Porting note: original proof was `induction' h with n h ih; · exact Nat.leRecOn_self 1`
--                                   `rw [Nat.leRecOn_succ h, ih, RingHom.map_one]`
    induction' h with a h ih
    · exact Nat.leRecOn_self 1
    · rw [toStepOfLE'.succ k m a h]; simp [ih]
  map_mul' x y := by
-- Porting note: original proof was `induction' h with n h ih; · simp_rw [Nat.leRecOn_self]`
--                                   `simp_rw [Nat.leRecOn_succ h, ih, RingHom.map_mul]`
    induction' h with a h ih
    · dsimp [toStepOfLE']; simp_rw [Nat.leRecOn_self]
    · simp_rw [toStepOfLE'.succ k m a h]; 

Proof state:
case step
k : Type u
inst✝ : Field k
m n : ℕ
x y : Step k m
a : ℕ
h : Nat.le m a
ih :
  OneHom.toFun
      { toFun := AlgebraicClosure.toStepOfLE' k m a h, map_one' := (_ : AlgebraicClosure.toStepOfLE' k m a h 1 = 1) }
      (x * y) =
    OneHom.toFun
        { toFun := AlgebraicClosure.toStepOfLE' k m a h, map_one' := (_ : AlgebraicClosure.toStepOfLE' k m a h 1 = 1) }
        x *
      OneHom.toFun
        { toFun := AlgebraicClosure.toStepOfLE' k m a h, map_one' := (_ : AlgebraicClosure.toStepOfLE' k m a h 1 = 1) }
        y
⊢ (⇑(toStepSucc k a) ∘ AlgebraicClosure.toStepOfLE' k m a h) (x * y) =
    (⇑(toStepSucc k a) ∘ AlgebraicClosure.toStepOfLE' k m a h) x *
      (⇑(toStepSucc k a) ∘ AlgebraicClosure.toStepOfLE' k m a h) y

Next tactic:
simp only at ih
/-- The canonical ring homomorphism to a step with a greater index. -/
def toStepOfLE (m n : ℕ) (h : m ≤ n) : Step k m →+* Step k n where
  toFun := toStepOfLE' k m n h
  map_one' := by
-- Porting note: original proof was `induction' h with n h ih; · exact Nat.leRecOn_self 1`
--                                   `rw [Nat.leRecOn_succ h, ih, RingHom.map_one]`
    induction' h with a h ih
    · exact Nat.leRecOn_self 1
    · rw [toStepOfLE'.succ k m a h]; simp [ih]
  map_mul' x y := by
-- Porting note: original proof was `induction' h with n h ih; · simp_rw [Nat.leRecOn_self]`
--                                   `simp_rw [Nat.leRecOn_succ h, ih, RingHom.map_mul]`
    induction' h with a h ih
    · dsimp [toStepOfLE']; simp_rw [Nat.leRecOn_self]
    · simp_rw [toStepOfLE'.succ k m a h]; simp only at ih; 

Proof state:
case step
k : Type u
inst✝ : Field k
m n : ℕ
x y : Step k m
a : ℕ
h : Nat.le m a
ih :
  AlgebraicClosure.toStepOfLE' k m a h (x * y) =
    AlgebraicClosure.toStepOfLE' k m a h x * AlgebraicClosure.toStepOfLE' k m a h y
⊢ (⇑(toStepSucc k a) ∘ AlgebraicClosure.toStepOfLE' k m a h) (x * y) =
    (⇑(toStepSucc k a) ∘ AlgebraicClosure.toStepOfLE' k m a h) x *
      (⇑(toStepSucc k a) ∘ AlgebraicClosure.toStepOfLE' k m a h) y

Next tactic:
simp [ih]
/-- The canonical ring homomorphism to a step with a greater index. -/
def toStepOfLE (m n : ℕ) (h : m ≤ n) : Step k m →+* Step k n where
  toFun := toStepOfLE' k m n h
  map_one' := by
-- Porting note: original proof was `induction' h with n h ih; · exact Nat.leRecOn_self 1`
--                                   `rw [Nat.leRecOn_succ h, ih, RingHom.map_one]`
    induction' h with a h ih
    · exact Nat.leRecOn_self 1
    · rw [toStepOfLE'.succ k m a h]; simp [ih]
  map_mul' x y := by
-- Porting note: original proof was `induction' h with n h ih; · simp_rw [Nat.leRecOn_self]`
--                                   `simp_rw [Nat.leRecOn_succ h, ih, RingHom.map_mul]`
    induction' h with a h ih
    · dsimp [toStepOfLE']; simp_rw [Nat.leRecOn_self]
    · simp_rw [toStepOfLE'.succ k m a h]; simp only at ih; simp [ih]
-- Porting note: original proof was `induction' h with n h ih; · exact Nat.leRecOn_self 0`
--                                   `rw [Nat.leRecOn_succ h, ih, RingHom.map_zero]`
  map_zero' := by
    

Proof state:
k : Type u
inst✝ : Field k
m n : ℕ
h : m ≤ n
⊢ OneHom.toFun
      (↑{
          toOneHom :=
            { toFun := AlgebraicClosure.toStepOfLE' k m n h,
              map_one' := (_ : AlgebraicClosure.toStepOfLE' k m n h 1 = 1) },
          map_mul' :=
            (_ :
              ∀ (x y : Step k m),
                OneHom.toFun
                    { toFun := AlgebraicClosure.toStepOfLE' k m n h,
                      map_one' := (_ : AlgebraicClosure.toStepOfLE' k m n h 1 = 1) }
                    (x * y) =
                  OneHom.toFun
                      { toFun := AlgebraicClosure.toStepOfLE' k m n h,
                        map_one' := (_ : AlgebraicClosure.toStepOfLE' k m n h 1 = 1) }
                      x *
                    OneHom.toFun
                      { toFun := AlgebraicClosure.toStepOfLE' k m n h,
                        map_one' := (_ : AlgebraicClosure.toStepOfLE' k m n h 1 = 1) }
                      y) })
      0 =
    0

Next tactic:
induction' h with a h ih