l3lab/ntp-mathlib huggingface dataset:
/-- The `Subfield` generated by a set. -/
def closure (s : Set K) : Subfield K where
  carrier := {z : K | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}
  zero_mem' := ⟨0, Subring.zero_mem _, 1, Subring.one_mem _, div_one _⟩
  one_mem' := ⟨1, Subring.one_mem _, 1, Subring.one_mem _, div_one _⟩
  neg_mem' {x} := by
    rintro ⟨y, hy, z, hz, x_eq⟩
    exact ⟨-y, Subring.neg_mem _ hy, z, hz, x_eq ▸ neg_div _ _⟩
  inv_mem' x := by rintro ⟨y, hy, z, hz, x_eq⟩; exact ⟨z, hz, y, hy, x_eq ▸ (inv_div _ _).symm ⟩
  add_mem' x_mem y_mem := by
    obtain ⟨nx, hnx, dx, hdx, rfl⟩ := id x_mem
    obtain ⟨ny, hny, dy, hdy, rfl⟩ := id y_mem
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set K
nx : K
hnx : nx ∈ Subring.closure s
dx : K
hdx : dx ∈ Subring.closure s
x_mem :
  nx / dx ∈
    {
          toSubsemigroup :=
            { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
              mul_mem' :=
                (_ :
                  ∀ {a b : K},
                    a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                      b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                        a * b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
          one_mem' := (_ : ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier
ny : K
hny : ny ∈ Subring.closure s
dy : K
hdy : dy ∈ Subring.closure s
y_mem :
  ny / dy ∈
    {
          toSubsemigroup :=
            { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
              mul_mem' :=
                (_ :
                  ∀ {a b : K},
                    a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                      b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                        a * b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
          one_mem' := (_ : ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier
⊢ nx / dx + ny / dy ∈
    {
          toSubsemigroup :=
            { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
              mul_mem' :=
                (_ :
                  ∀ {a b : K},
                    a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                      b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                        a * b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
          one_mem' := (_ : ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier

Next tactic:
by_cases hx0 : dx = 0
/-- The `Subfield` generated by a set. -/
def closure (s : Set K) : Subfield K where
  carrier := {z : K | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}
  zero_mem' := ⟨0, Subring.zero_mem _, 1, Subring.one_mem _, div_one _⟩
  one_mem' := ⟨1, Subring.one_mem _, 1, Subring.one_mem _, div_one _⟩
  neg_mem' {x} := by
    rintro ⟨y, hy, z, hz, x_eq⟩
    exact ⟨-y, Subring.neg_mem _ hy, z, hz, x_eq ▸ neg_div _ _⟩
  inv_mem' x := by rintro ⟨y, hy, z, hz, x_eq⟩; exact ⟨z, hz, y, hy, x_eq ▸ (inv_div _ _).symm ⟩
  add_mem' x_mem y_mem := by
    obtain ⟨nx, hnx, dx, hdx, rfl⟩ := id x_mem
    obtain ⟨ny, hny, dy, hdy, rfl⟩ := id y_mem
    by_cases hx0 : dx = 0; · 

Proof state:
case pos
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set K
nx : K
hnx : nx ∈ Subring.closure s
dx : K
hdx : dx ∈ Subring.closure s
x_mem :
  nx / dx ∈
    {
          toSubsemigroup :=
            { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
              mul_mem' :=
                (_ :
                  ∀ {a b : K},
                    a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                      b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                        a * b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
          one_mem' := (_ : ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier
ny : K
hny : ny ∈ Subring.closure s
dy : K
hdy : dy ∈ Subring.closure s
y_mem :
  ny / dy ∈
    {
          toSubsemigroup :=
            { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
              mul_mem' :=
                (_ :
                  ∀ {a b : K},
                    a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                      b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                        a * b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
          one_mem' := (_ : ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier
hx0 : dx = 0
⊢ nx / dx + ny / dy ∈
    {
          toSubsemigroup :=
            { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
              mul_mem' :=
                (_ :
                  ∀ {a b : K},
                    a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                      b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                        a * b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
          one_mem' := (_ : ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier

Next tactic:
rwa [hx0, div_zero, zero_add]
/-- The `Subfield` generated by a set. -/
def closure (s : Set K) : Subfield K where
  carrier := {z : K | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}
  zero_mem' := ⟨0, Subring.zero_mem _, 1, Subring.one_mem _, div_one _⟩
  one_mem' := ⟨1, Subring.one_mem _, 1, Subring.one_mem _, div_one _⟩
  neg_mem' {x} := by
    rintro ⟨y, hy, z, hz, x_eq⟩
    exact ⟨-y, Subring.neg_mem _ hy, z, hz, x_eq ▸ neg_div _ _⟩
  inv_mem' x := by rintro ⟨y, hy, z, hz, x_eq⟩; exact ⟨z, hz, y, hy, x_eq ▸ (inv_div _ _).symm ⟩
  add_mem' x_mem y_mem := by
    obtain ⟨nx, hnx, dx, hdx, rfl⟩ := id x_mem
    obtain ⟨ny, hny, dy, hdy, rfl⟩ := id y_mem
    by_cases hx0 : dx = 0; · rwa [hx0, div_zero, zero_add]
    

Proof state:
case neg
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set K
nx : K
hnx : nx ∈ Subring.closure s
dx : K
hdx : dx ∈ Subring.closure s
x_mem :
  nx / dx ∈
    {
          toSubsemigroup :=
            { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
              mul_mem' :=
                (_ :
                  ∀ {a b : K},
                    a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                      b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                        a * b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
          one_mem' := (_ : ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier
ny : K
hny : ny ∈ Subring.closure s
dy : K
hdy : dy ∈ Subring.closure s
y_mem :
  ny / dy ∈
    {
          toSubsemigroup :=
            { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
              mul_mem' :=
                (_ :
                  ∀ {a b : K},
                    a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                      b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                        a * b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
          one_mem' := (_ : ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier
hx0 : ¬dx = 0
⊢ nx / dx + ny / dy ∈
    {
          toSubsemigroup :=
            { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
              mul_mem' :=
                (_ :
                  ∀ {a b : K},
                    a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                      b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                        a * b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
          one_mem' := (_ : ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier

Next tactic:
by_cases hy0 : dy = 0
/-- The `Subfield` generated by a set. -/
def closure (s : Set K) : Subfield K where
  carrier := {z : K | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}
  zero_mem' := ⟨0, Subring.zero_mem _, 1, Subring.one_mem _, div_one _⟩
  one_mem' := ⟨1, Subring.one_mem _, 1, Subring.one_mem _, div_one _⟩
  neg_mem' {x} := by
    rintro ⟨y, hy, z, hz, x_eq⟩
    exact ⟨-y, Subring.neg_mem _ hy, z, hz, x_eq ▸ neg_div _ _⟩
  inv_mem' x := by rintro ⟨y, hy, z, hz, x_eq⟩; exact ⟨z, hz, y, hy, x_eq ▸ (inv_div _ _).symm ⟩
  add_mem' x_mem y_mem := by
    obtain ⟨nx, hnx, dx, hdx, rfl⟩ := id x_mem
    obtain ⟨ny, hny, dy, hdy, rfl⟩ := id y_mem
    by_cases hx0 : dx = 0; · rwa [hx0, div_zero, zero_add]
    by_cases hy0 : dy = 0; · 

Proof state:
case pos
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set K
nx : K
hnx : nx ∈ Subring.closure s
dx : K
hdx : dx ∈ Subring.closure s
x_mem :
  nx / dx ∈
    {
          toSubsemigroup :=
            { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
              mul_mem' :=
                (_ :
                  ∀ {a b : K},
                    a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                      b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                        a * b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
          one_mem' := (_ : ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier
ny : K
hny : ny ∈ Subring.closure s
dy : K
hdy : dy ∈ Subring.closure s
y_mem :
  ny / dy ∈
    {
          toSubsemigroup :=
            { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
              mul_mem' :=
                (_ :
                  ∀ {a b : K},
                    a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                      b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                        a * b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
          one_mem' := (_ : ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier
hx0 : ¬dx = 0
hy0 : dy = 0
⊢ nx / dx + ny / dy ∈
    {
          toSubsemigroup :=
            { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
              mul_mem' :=
                (_ :
                  ∀ {a b : K},
                    a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                      b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                        a * b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
          one_mem' := (_ : ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier

Next tactic:
rwa [hy0, div_zero, add_zero]
/-- The `Subfield` generated by a set. -/
def closure (s : Set K) : Subfield K where
  carrier := {z : K | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}
  zero_mem' := ⟨0, Subring.zero_mem _, 1, Subring.one_mem _, div_one _⟩
  one_mem' := ⟨1, Subring.one_mem _, 1, Subring.one_mem _, div_one _⟩
  neg_mem' {x} := by
    rintro ⟨y, hy, z, hz, x_eq⟩
    exact ⟨-y, Subring.neg_mem _ hy, z, hz, x_eq ▸ neg_div _ _⟩
  inv_mem' x := by rintro ⟨y, hy, z, hz, x_eq⟩; exact ⟨z, hz, y, hy, x_eq ▸ (inv_div _ _).symm ⟩
  add_mem' x_mem y_mem := by
    obtain ⟨nx, hnx, dx, hdx, rfl⟩ := id x_mem
    obtain ⟨ny, hny, dy, hdy, rfl⟩ := id y_mem
    by_cases hx0 : dx = 0; · rwa [hx0, div_zero, zero_add]
    by_cases hy0 : dy = 0; · rwa [hy0, div_zero, add_zero]
    

Proof state:
case neg
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set K
nx : K
hnx : nx ∈ Subring.closure s
dx : K
hdx : dx ∈ Subring.closure s
x_mem :
  nx / dx ∈
    {
          toSubsemigroup :=
            { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
              mul_mem' :=
                (_ :
                  ∀ {a b : K},
                    a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                      b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                        a * b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
          one_mem' := (_ : ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier
ny : K
hny : ny ∈ Subring.closure s
dy : K
hdy : dy ∈ Subring.closure s
y_mem :
  ny / dy ∈
    {
          toSubsemigroup :=
            { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
              mul_mem' :=
                (_ :
                  ∀ {a b : K},
                    a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                      b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                        a * b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
          one_mem' := (_ : ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier
hx0 : ¬dx = 0
hy0 : ¬dy = 0
⊢ nx / dx + ny / dy ∈
    {
          toSubsemigroup :=
            { carrier := {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z},
              mul_mem' :=
                (_ :
                  ∀ {a b : K},
                    a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                      b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →
                        a * b ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}) },
          one_mem' := (_ : ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = 1) }.toSubsemigroup.carrier

Next tactic:
exact
      ⟨nx * dy + dx * ny, Subring.add_mem _ (Subring.mul_mem _ hnx hdy) (Subring.mul_mem _ hdx hny),
        dx * dy, Subring.mul_mem _ hdx hdy, (div_add_div nx ny hx0 hy0).symm⟩
theorem mem_closure_iff {s : Set K} {x} :
    x ∈ closure s ↔ ∃ y ∈ Subring.closure s, ∃ z ∈ Subring.closure s, y / z = x := by
  

Proof state:
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set K
x : K
⊢ x ∈ closure s ↔ ∃ y ∈ Subring.closure s, ∃ z ∈ Subring.closure s, y / z = x

Next tactic:
change x ∈ (closure s).carrier ↔ ∃ y ∈ Subring.closure s, ∃ z ∈ Subring.closure s, y / z = x
theorem mem_closure_iff {s : Set K} {x} :
    x ∈ closure s ↔ ∃ y ∈ Subring.closure s, ∃ z ∈ Subring.closure s, y / z = x := by
  change x ∈ (closure s).carrier ↔ ∃ y ∈ Subring.closure s, ∃ z ∈ Subring.closure s, y / z = x
  

Proof state:
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set K
x : K
⊢ x ∈ (closure s).toSubring.toSubsemiring.toSubmonoid.toSubsemigroup.carrier ↔
    ∃ y ∈ Subring.closure s, ∃ z ∈ Subring.closure s, y / z = x

Next tactic:
simp only [closure, exists_prop, Set.mem_setOf_eq]
/-- An induction principle for closure membership. If `p` holds for `1`, and all elements
of `s`, and is preserved under addition, negation, and multiplication, then `p` holds for all
elements of the closure of `s`. -/
@[elab_as_elim]
theorem closure_induction {s : Set K} {p : K → Prop} {x} (h : x ∈ closure s) (Hs : ∀ x ∈ s, p x)
    (H1 : p 1) (Hadd : ∀ x y, p x → p y → p (x + y)) (Hneg : ∀ x, p x → p (-x))
    (Hinv : ∀ x, p x → p x⁻¹) (Hmul : ∀ x y, p x → p y → p (x * y)) : p x := by
    

Proof state:
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set K
p : K → Prop
x : K
h : x ∈ closure s
Hs : ∀ x ∈ s, p x
H1 : p 1
Hadd : ∀ (x y : K), p x → p y → p (x + y)
Hneg : ∀ (x : K), p x → p (-x)
Hinv : ∀ (x : K), p x → p x⁻¹
Hmul : ∀ (x y : K), p x → p y → p (x * y)
⊢ p x

Next tactic:
letI : Subfield K :=
      ⟨⟨⟨⟨⟨p, by intro _ _; exact Hmul _ _⟩, H1⟩,
        by intro _ _; exact Hadd _ _, @add_neg_self K _ 1 ▸ Hadd _ _ H1 (Hneg _ H1)⟩,
          by intro _; exact Hneg _⟩, Hinv⟩
/-- An induction principle for closure membership. If `p` holds for `1`, and all elements
of `s`, and is preserved under addition, negation, and multiplication, then `p` holds for all
elements of the closure of `s`. -/
@[elab_as_elim]
theorem closure_induction {s : Set K} {p : K → Prop} {x} (h : x ∈ closure s) (Hs : ∀ x ∈ s, p x)
    (H1 : p 1) (Hadd : ∀ x y, p x → p y → p (x + y)) (Hneg : ∀ x, p x → p (-x))
    (Hinv : ∀ x, p x → p x⁻¹) (Hmul : ∀ x y, p x → p y → p (x * y)) : p x := by
    letI : Subfield K :=
      ⟨⟨⟨⟨⟨p, by 

Proof state:
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set K
p : K → Prop
x : K
h : x ∈ closure s
Hs : ∀ x ∈ s, p x
H1 : p 1
Hadd : ∀ (x y : K), p x → p y → p (x + y)
Hneg : ∀ (x : K), p x → p (-x)
Hinv : ∀ (x : K), p x → p x⁻¹
Hmul : ∀ (x y : K), p x → p y → p (x * y)
⊢ ∀ {a b : K}, a ∈ p → b ∈ p → a * b ∈ p

Next tactic:
intro _ _
/-- An induction principle for closure membership. If `p` holds for `1`, and all elements
of `s`, and is preserved under addition, negation, and multiplication, then `p` holds for all
elements of the closure of `s`. -/
@[elab_as_elim]
theorem closure_induction {s : Set K} {p : K → Prop} {x} (h : x ∈ closure s) (Hs : ∀ x ∈ s, p x)
    (H1 : p 1) (Hadd : ∀ x y, p x → p y → p (x + y)) (Hneg : ∀ x, p x → p (-x))
    (Hinv : ∀ x, p x → p x⁻¹) (Hmul : ∀ x y, p x → p y → p (x * y)) : p x := by
    letI : Subfield K :=
      ⟨⟨⟨⟨⟨p, by intro _ _; 

Proof state:
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set K
p : K → Prop
x : K
h : x ∈ closure s
Hs : ∀ x ∈ s, p x
H1 : p 1
Hadd : ∀ (x y : K), p x → p y → p (x + y)
Hneg : ∀ (x : K), p x → p (-x)
Hinv : ∀ (x : K), p x → p x⁻¹
Hmul : ∀ (x y : K), p x → p y → p (x * y)
a✝ b✝ : K
⊢ a✝ ∈ p → b✝ ∈ p → a✝ * b✝ ∈ p

Next tactic:
exact Hmul _ _
/-- An induction principle for closure membership. If `p` holds for `1`, and all elements
of `s`, and is preserved under addition, negation, and multiplication, then `p` holds for all
elements of the closure of `s`. -/
@[elab_as_elim]
theorem closure_induction {s : Set K} {p : K → Prop} {x} (h : x ∈ closure s) (Hs : ∀ x ∈ s, p x)
    (H1 : p 1) (Hadd : ∀ x y, p x → p y → p (x + y)) (Hneg : ∀ x, p x → p (-x))
    (Hinv : ∀ x, p x → p x⁻¹) (Hmul : ∀ x y, p x → p y → p (x * y)) : p x := by
    letI : Subfield K :=
      ⟨⟨⟨⟨⟨p, by intro _ _; exact Hmul _ _⟩, H1⟩,
        by 

Proof state:
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set K
p : K → Prop
x : K
h : x ∈ closure s
Hs : ∀ x ∈ s, p x
H1 : p 1
Hadd : ∀ (x y : K), p x → p y → p (x + y)
Hneg : ∀ (x : K), p x → p (-x)
Hinv : ∀ (x : K), p x → p x⁻¹
Hmul : ∀ (x y : K), p x → p y → p (x * y)
⊢ ∀ {a b : K},
    a ∈
        { toSubsemigroup := { carrier := p, mul_mem' := (_ : ∀ {a b : K}, p a → p b → p (a * b)) },
              one_mem' := H1 }.toSubsemigroup.carrier →
      b ∈
          { toSubsemigroup := { carrier := p, mul_mem' := (_ : ∀ {a b : K}, p a → p b → p (a * b)) },
                one_mem' := H1 }.toSubsemigroup.carrier →
        a + b ∈
          { toSubsemigroup := { carrier := p, mul_mem' := (_ : ∀ {a b : K}, p a → p b → p (a * b)) },
                one_mem' := H1 }.toSubsemigroup.carrier

Next tactic:
intro _ _
/-- An induction principle for closure membership. If `p` holds for `1`, and all elements
of `s`, and is preserved under addition, negation, and multiplication, then `p` holds for all
elements of the closure of `s`. -/
@[elab_as_elim]
theorem closure_induction {s : Set K} {p : K → Prop} {x} (h : x ∈ closure s) (Hs : ∀ x ∈ s, p x)
    (H1 : p 1) (Hadd : ∀ x y, p x → p y → p (x + y)) (Hneg : ∀ x, p x → p (-x))
    (Hinv : ∀ x, p x → p x⁻¹) (Hmul : ∀ x y, p x → p y → p (x * y)) : p x := by
    letI : Subfield K :=
      ⟨⟨⟨⟨⟨p, by intro _ _; exact Hmul _ _⟩, H1⟩,
        by intro _ _; 

Proof state:
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set K
p : K → Prop
x : K
h : x ∈ closure s
Hs : ∀ x ∈ s, p x
H1 : p 1
Hadd : ∀ (x y : K), p x → p y → p (x + y)
Hneg : ∀ (x : K), p x → p (-x)
Hinv : ∀ (x : K), p x → p x⁻¹
Hmul : ∀ (x y : K), p x → p y → p (x * y)
a✝ b✝ : K
⊢ a✝ ∈
      { toSubsemigroup := { carrier := p, mul_mem' := (_ : ∀ {a b : K}, p a → p b → p (a * b)) },
            one_mem' := H1 }.toSubsemigroup.carrier →
    b✝ ∈
        { toSubsemigroup := { carrier := p, mul_mem' := (_ : ∀ {a b : K}, p a → p b → p (a * b)) },
              one_mem' := H1 }.toSubsemigroup.carrier →
      a✝ + b✝ ∈
        { toSubsemigroup := { carrier := p, mul_mem' := (_ : ∀ {a b : K}, p a → p b → p (a * b)) },
              one_mem' := H1 }.toSubsemigroup.carrier

Next tactic:
exact Hadd _ _
/-- An induction principle for closure membership. If `p` holds for `1`, and all elements
of `s`, and is preserved under addition, negation, and multiplication, then `p` holds for all
elements of the closure of `s`. -/
@[elab_as_elim]
theorem closure_induction {s : Set K} {p : K → Prop} {x} (h : x ∈ closure s) (Hs : ∀ x ∈ s, p x)
    (H1 : p 1) (Hadd : ∀ x y, p x → p y → p (x + y)) (Hneg : ∀ x, p x → p (-x))
    (Hinv : ∀ x, p x → p x⁻¹) (Hmul : ∀ x y, p x → p y → p (x * y)) : p x := by
    letI : Subfield K :=
      ⟨⟨⟨⟨⟨p, by intro _ _; exact Hmul _ _⟩, H1⟩,
        by intro _ _; exact Hadd _ _, @add_neg_self K _ 1 ▸ Hadd _ _ H1 (Hneg _ H1)⟩,
          by 

Proof state:
K : Type u
L : Type v
M : Type w
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Field M
s : Set K
p : K → Prop
x : K
h : x ∈ closure s
Hs : ∀ x ∈ s, p x
H1 : p 1
Hadd : ∀ (x y : K), p x → p y → p (x + y)
Hneg : ∀ (x : K), p x → p (-x)
Hinv : ∀ (x : K), p x → p x⁻¹
Hmul : ∀ (x y : K), p x → p y → p (x * y)
⊢ ∀ {x : K},
    x ∈
        {
                toSubmonoid :=
                  { toSubsemigroup := { carrier := p, mul_mem' := (_ : ∀ {a b : K}, p a → p b → p (a * b)) },
                    one_mem' := H1 },
                add_mem' := (_ : ∀ {a b : K}, p a → p b → p (a + b)),
                zero_mem' := (_ : p 0) }.toSubmonoid.toSubsemigroup.carrier →
      -x ∈
        {
                toSubmonoid :=
                  { toSubsemigroup := { carrier := p, mul_mem' := (_ : ∀ {a b : K}, p a → p b → p (a * b)) },
                    one_mem' := H1 },
                add_mem' := (_ : ∀ {a b : K}, p a → p b → p (a + b)),
                zero_mem' := (_ : p 0) }.toSubmonoid.toSubsemigroup.carrier

Next tactic:
intro _