awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.Transcendental (goal)
Goal:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : Set ι
i : ι
h : i ∉ s
⊢ AlgebraicIndepOn R x (insert i s) ↔ AlgebraicIndepOn R x s ∧ Transcendental (↥(adjoin R (x '' s))) (x i)
have classical : AlgebraicIndependent.{u_1, u_3, v} (ι := Option { i : ι // i ∈ s }) R (A := A)
    ((fun (i_1 : (↑(insert i s) : Type u_1)) ↦ x (↑i_1 : ι)) ∘
      (⇑(subtypeInsertEquivOption (t := s) h).symm :
        (a : Option { i : ι // i ∈ s }) → { i_1 : ι // i_1 ∈ insert i s })) ↔
  (AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)) ∧
    Transcendental (↥(adjoin R (x '' s))) (x i)
Tactic: classical simp_rw [← algebraicIndependent_equiv (subtypeInsertEquivOption h).symm,
    AlgebraicIndepOn]
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.Transcendental (goal)
Goal:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : Set ι
i : ι
h : i ∉ s
⊢ AlgebraicIndependent.{u_1, u_3, v} (ι := Option { i : ι // i ∈ s }) R (A := A)
      ((fun (i_1 : (↑(insert i s) : Type u_1)) ↦ x (↑i_1 : ι)) ∘
        (⇑(subtypeInsertEquivOption (t := s) h).symm :
          (a : Option { i : ι // i ∈ s }) → { i_1 : ι // i_1 ∈ insert i s })) ↔
    (AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)) ∧
      Transcendental (↥(adjoin R (x '' s))) (x i)
have h₁ : Eq (α := Option { i : ι // i ∈ s } → A)
  ((fun (i_1 : (↑(insert i s) : Type u_1)) ↦ x (↑i_1 : ι)) ∘
    (⇑(subtypeInsertEquivOption (t := s) h).symm : (a : Option { i : ι // i ∈ s }) → { i_1 : ι // i_1 ∈ insert i s }))
  fun (o : Option (↑s : Type u_1)) ↦ o.elim (x i) fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)
Tactic: convert! option_iff (x := fun i : s ↦ x i) (a := x i) using 2
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.Transcendental (goal)
Goal:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : Set ι
i : ι
h : i ∉ s
h₁ :
  Eq (α := Option { i : ι // i ∈ s } → A)
    ((fun (i_1 : (↑(insert i s) : Type u_1)) ↦ x (↑i_1 : ι)) ∘
      (⇑(subtypeInsertEquivOption (t := s) h).symm : (a : Option { i : ι // i ∈ s }) → { i_1 : ι // i_1 ∈ insert i s }))
    fun (o : Option (↑s : Type u_1)) ↦ o.elim (x i) fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)
⊢ AlgebraicIndependent.{u_1, u_3, v} (ι := Option { i : ι // i ∈ s }) R (A := A)
      ((fun (i_1 : (↑(insert i s) : Type u_1)) ↦ x (↑i_1 : ι)) ∘
        (⇑(subtypeInsertEquivOption (t := s) h).symm :
          (a : Option { i : ι // i ∈ s }) → { i_1 : ι // i_1 ∈ insert i s })) ↔
    (AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)) ∧
      Transcendental (↥(adjoin R (x '' s))) (x i)
have h₂ : Transcendental (↥(adjoin R (x '' s))) (x i) ↔
  Transcendental (↥(adjoin R (range fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)))) (x i)
Tactic: convert! option_iff (x := fun i : s ↦ x i) (a := x i) using 2
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.Transcendental (goal)
Goal:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : Set ι
i : ι
h : i ∉ s
⊢ Eq (α := Option { i : ι // i ∈ s } → A)
    ((fun (i_1 : (↑(insert i s) : Type u_1)) ↦ x (↑i_1 : ι)) ∘
      (⇑(subtypeInsertEquivOption (t := s) h).symm : (a : Option { i : ι // i ∈ s }) → { i_1 : ι // i_1 ∈ insert i s }))
    fun (o : Option (↑s : Type u_1)) ↦ o.elim (x i) fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)
have h₁ : Function.comp (α := Option { i : ι // i ∈ s }) (fun (i_1 : (↑(insert i s) : Type u_1)) ↦ x (↑i_1 : ι))
    (⇑(subtypeInsertEquivOption (t := s) h).symm : (a : Option { i : ι // i ∈ s }) → { i_1 : ι // i_1 ∈ insert i s })
    none =
  Option.elim none (x i) fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)
Tactic: ext (_ | _)
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.Transcendental (goal)
Goal:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : Set ι
i : ι
h : i ∉ s
h₁ :
  Function.comp (α := Option { i : ι // i ∈ s }) (fun (i_1 : (↑(insert i s) : Type u_1)) ↦ x (↑i_1 : ι))
      (⇑(subtypeInsertEquivOption (t := s) h).symm : (a : Option { i : ι // i ∈ s }) → { i_1 : ι // i_1 ∈ insert i s })
      none =
    Option.elim none (x i) fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)
⊢ Eq (α := Option { i : ι // i ∈ s } → A)
    ((fun (i_1 : (↑(insert i s) : Type u_1)) ↦ x (↑i_1 : ι)) ∘
      (⇑(subtypeInsertEquivOption (t := s) h).symm : (a : Option { i : ι // i ∈ s }) → { i_1 : ι // i_1 ∈ insert i s }))
    fun (o : Option (↑s : Type u_1)) ↦ o.elim (x i) fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)
have h₂ : ∀ (val : { i : ι // i ∈ s }),
  ((fun (i_1 : (↑(insert i s) : Type u_1)) ↦ x (↑i_1 : ι)) ∘
        (⇑(subtypeInsertEquivOption (t := s) h).symm :
          (a : Option { i : ι // i ∈ s }) → { i_1 : ι // i_1 ∈ insert i s }))
      (some val) =
    (some val).elim (x i) fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)
Tactic: ext (_ | _)
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.Transcendental (goal)
Goal:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : Set ι
i : ι
h : i ∉ s
⊢ Transcendental (↥(adjoin R (x '' s))) (x i) ↔
    Transcendental (↥(adjoin R (range fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)))) (x i)
have h₁ : Transcendental (↥(adjoin R (range fun (x_1 : (↑s : Type u_1)) ↦ x (↑x_1 : ι)))) (x i) ↔
  Transcendental (↥(adjoin R (range fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)))) (x i)
Tactic: rw [Set.image_eq_range]
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.Transcendental (hypothesis)
Goal:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : Set ι
i : ι
hs : AlgebraicIndepOn R x s
hi : Transcendental (↥(adjoin R (x '' s))) (x i)
⊢ AlgebraicIndepOn R x (insert i s)
have a : Nontrivial R
Tactic: nontriviality R
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.Transcendental (hypothesis)
Goal:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : Set ι
i : ι
hs : AlgebraicIndepOn R x s
hi : Transcendental (↥(adjoin R (x '' s))) (x i)
h : Nontrivial R
⊢ AlgebraicIndepOn R x (insert i s)
have this : Nontrivial A
Tactic: have := hs.algebraMap_injective.nontrivial
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.Transcendental (goal)
Goal:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : Set ι
ind : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)
H :
  ∀ (t : Set ι),
    t.Finite →
      (AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑t : Type u_1)) ↦ x (↑i : ι)) →
        ∀ i ∉ s, i ∉ t → Transcendental (↥(adjoin R (x '' t))) (x i)
⊢ AlgebraicIndependent R x
have refine : ∀ (t : Set ι) (hfin : t.Finite), AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑t : Type u_1)) ↦ x (↑i : ι)
Tactic: refine algebraicIndependent_of_finite_type fun t hfin ↦ ?_
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.Transcendental (goal)
Goal:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : Set ι
ind : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)
H :
  ∀ (t : Set ι),
    t.Finite →
      (AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑t : Type u_1)) ↦ x (↑i : ι)) →
        ∀ i ∉ s, i ∉ t → Transcendental (↥(adjoin R (x '' t))) (x i)
t : Set ι
hfin : t.Finite
⊢ AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑t : Type u_1)) ↦ x (↑i : ι)
have suffices : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑(t ∩ s ∪ t \ s) : Type u_1)) ↦ x (↑i : ι)
Tactic: suffices AlgebraicIndependent R fun i : ↥(t ∩ s ∪ t \ s) ↦ x i from
    this.comp (Equiv.setCongr (t.inter_union_diff s).symm) (Equiv.injective _)
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.Transcendental (goal)
Goal:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : Set ι
ind : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)
H :
  ∀ (t : Set ι),
    t.Finite →
      (AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑t : Type u_1)) ↦ x (↑i : ι)) →
        ∀ i ∉ s, i ∉ t → Transcendental (↥(adjoin R (x '' t))) (x i)
t : Set ι
hfin : t.Finite
⊢ AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑(t ∩ s ∪ t \ s) : Type u_1)) ↦ x (↑i : ι)
have refine : ∀ {a : ι} {u : Set ι} (ha : a ∈ t \ s) (hu : u ⊆ t \ s) (ha' : a ∉ u)
  (h : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑(t ∩ s ∪ u) : Type u_1)) ↦ x (↑i : ι)),
  AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑(t ∩ s ∪ insert a u) : Type u_1)) ↦ x (↑i : ι)
Tactic: refine hfin.diff.induction_on_subset _ (ind.comp (inclusion <| by simp) (inclusion_injective _))
    fun {a u} ha hu ha' h ↦ ?_
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.Transcendental (hypothesis)
Goal:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : Set ι
ind : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)
H :
  ∀ (t : Set ι),
    t.Finite →
      (AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑t : Type u_1)) ↦ x (↑i : ι)) →
        ∀ i ∉ s, i ∉ t → Transcendental (↥(adjoin R (x '' t))) (x i)
t : Set ι
hfin : t.Finite
a : ι
u : Set ι
ha : a ∈ t \ s
hu : u ⊆ t \ s
ha' : a ∉ u
h : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑(t ∩ s ∪ u) : Type u_1)) ↦ x (↑i : ι)
⊢ AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑(t ∩ s ∪ insert a u) : Type u_1)) ↦ x (↑i : ι)
have this : a ∉ t ∩ s ∪ u
Tactic: have : a ∉ t ∩ s ∪ u := (·.elim (ha.2 ·.2) ha')
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.Transcendental (goal)
Goal:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : Set ι
ind : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)
H :
  ∀ (t : Set ι),
    t.Finite →
      (AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑t : Type u_1)) ↦ x (↑i : ι)) →
        ∀ i ∉ s, i ∉ t → Transcendental (↥(adjoin R (x '' t))) (x i)
t : Set ι
hfin : t.Finite
a : ι
u : Set ι
ha : a ∈ t \ s
hu : u ⊆ t \ s
ha' : a ∉ u
h : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑(t ∩ s ∪ u) : Type u_1)) ↦ x (↑i : ι)
this : a ∉ t ∩ s ∪ u
⊢ AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑(t ∩ s ∪ insert a u) : Type u_1)) ↦ x (↑i : ι)
have h₁ : ∀ (x_1 : (↑(t ∩ s ∪ insert a u) : Type u_1)),
  x (↑x_1 : ι) =
    Function.comp (β := { i : ι // i ∈ insert a (t ∩ s ∪ u) })
      ((fun (o : Option (↑(t ∩ s ∪ u) : Type u_1)) ↦ o.elim (x a) fun (i : (↑(t ∩ s ∪ u) : Type u_1)) ↦ x (↑i : ι)) ∘
        (⇑(subtypeInsertEquivOption (t := t ∩ s ∪ u) this) :
          (a : { i : ι // i ∈ insert a (t ∩ s ∪ u) }) → Option { i : ι // i ∈ t ∩ s ∪ u }))
      (⇑(Equiv.setCongr (s := t ∩ s ∪ insert a u) (t := insert a (t ∩ s ∪ u)) sorry) :
        (a_1 : (↑(t ∩ s ∪ insert a u) : Type u_1)) → (↑(insert a (t ∩ s ∪ u)) : Type u_1))
      x_1
Tactic: convert!
    (((image_eq_range .. ▸ h.option_iff_transcendental <| x a).2 <|
              H _ (hfin.subset (union_subset inter_subset_left <| hu.trans diff_subset)) h a ha.2
                this).comp
          _ (subtypeInsertEquivOption this).injective).comp
      (Equiv.setCongr union_insert) (Equiv.injective _) with
    x
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.Transcendental (goal)
Goal:
ι : Type u_1
R : Type u_3
A : Type v
x_1 : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : Set ι
ind : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑s : Type u_1)) ↦ x_1 (↑i : ι)
H :
  ∀ (t : Set ι),
    t.Finite →
      (AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑t : Type u_1)) ↦ x_1 (↑i : ι)) →
        ∀ i ∉ s, i ∉ t → Transcendental (↥(adjoin R (x_1 '' t))) (x_1 i)
t : Set ι
hfin : t.Finite
a : ι
u : Set ι
ha : a ∈ t \ s
hu : u ⊆ t \ s
ha' : a ∉ u
h : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑(t ∩ s ∪ u) : Type u_1)) ↦ x_1 (↑i : ι)
this : a ∉ t ∩ s ∪ u
x : (↑(t ∩ s ∪ insert a u) : Type u_1)
⊢ x_1 (↑x : ι) =
    Function.comp (β := { i : ι // i ∈ insert a (t ∩ s ∪ u) })
      ((fun (o : Option (↑(t ∩ s ∪ u) : Type u_1)) ↦
          o.elim (x_1 a) fun (i : (↑(t ∩ s ∪ u) : Type u_1)) ↦ x_1 (↑i : ι)) ∘
        (⇑(subtypeInsertEquivOption (t := t ∩ s ∪ u) this) :
          (a : { i : ι // i ∈ insert a (t ∩ s ∪ u) }) → Option { i : ι // i ∈ t ∩ s ∪ u }))
      (⇑(Equiv.setCongr (s := t ∩ s ∪ insert a u) (t := insert a (t ∩ s ∪ u)) ⋯) :
        (a_1 : (↑(t ∩ s ∪ insert a u) : Type u_1)) → (↑(insert a (t ∩ s ∪ u)) : Type u_1))
      x
have pos : ∀ (h : (↑x : ι) = a),
  x_1 (↑x : ι) =
    Function.comp (β := { i : ι // i ∈ insert a (t ∩ s ∪ u) })
      ((fun (o : Option (↑(t ∩ s ∪ u) : Type u_1)) ↦
          o.elim (x_1 a) fun (i : (↑(t ∩ s ∪ u) : Type u_1)) ↦ x_1 (↑i : ι)) ∘
        (⇑(subtypeInsertEquivOption (t := t ∩ s ∪ u) this) :
          (a : { i : ι // i ∈ insert a (t ∩ s ∪ u) }) → Option { i : ι // i ∈ t ∩ s ∪ u }))
      (⇑(Equiv.setCongr (s := t ∩ s ∪ insert a u) (t := insert a (t ∩ s ∪ u)) sorry) :
        (a_1 : (↑(t ∩ s ∪ insert a u) : Type u_1)) → (↑(insert a (t ∩ s ∪ u)) : Type u_1))
      x
Tactic: by_cases h : ↑x = a
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.Transcendental (goal)
Goal:
ι : Type u_1
R : Type u_3
A : Type v
x_1 : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : Set ι
ind : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑s : Type u_1)) ↦ x_1 (↑i : ι)
H :
  ∀ (t : Set ι),
    t.Finite →
      (AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑t : Type u_1)) ↦ x_1 (↑i : ι)) →
        ∀ i ∉ s, i ∉ t → Transcendental (↥(adjoin R (x_1 '' t))) (x_1 i)
t : Set ι
hfin : t.Finite
a : ι
u : Set ι
ha : a ∈ t \ s
hu : u ⊆ t \ s
ha' : a ∉ u
h : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑(t ∩ s ∪ u) : Type u_1)) ↦ x_1 (↑i : ι)
this : a ∉ t ∩ s ∪ u
x : (↑(t ∩ s ∪ insert a u) : Type u_1)
pos :
  ∀ (h : (↑x : ι) = a),
    x_1 (↑x : ι) =
      Function.comp (β := { i : ι // i ∈ insert a (t ∩ s ∪ u) })
        ((fun (o : Option (↑(t ∩ s ∪ u) : Type u_1)) ↦
            o.elim (x_1 a) fun (i : (↑(t ∩ s ∪ u) : Type u_1)) ↦ x_1 (↑i : ι)) ∘
          (⇑(subtypeInsertEquivOption (t := t ∩ s ∪ u) this) :
            (a : { i : ι // i ∈ insert a (t ∩ s ∪ u) }) → Option { i : ι // i ∈ t ∩ s ∪ u }))
        (⇑(Equiv.setCongr (s := t ∩ s ∪ insert a u) (t := insert a (t ∩ s ∪ u)) ⋯) :
          (a_1 : (↑(t ∩ s ∪ insert a u) : Type u_1)) → (↑(insert a (t ∩ s ∪ u)) : Type u_1))
        x
⊢ x_1 (↑x : ι) =
    Function.comp (β := { i : ι // i ∈ insert a (t ∩ s ∪ u) })
      ((fun (o : Option (↑(t ∩ s ∪ u) : Type u_1)) ↦
          o.elim (x_1 a) fun (i : (↑(t ∩ s ∪ u) : Type u_1)) ↦ x_1 (↑i : ι)) ∘
        (⇑(subtypeInsertEquivOption (t := t ∩ s ∪ u) this) :
          (a : { i : ι // i ∈ insert a (t ∩ s ∪ u) }) → Option { i : ι // i ∈ t ∩ s ∪ u }))
      (⇑(Equiv.setCongr (s := t ∩ s ∪ insert a u) (t := insert a (t ∩ s ∪ u)) ⋯) :
        (a_1 : (↑(t ∩ s ∪ insert a u) : Type u_1)) → (↑(insert a (t ∩ s ∪ u)) : Type u_1))
      x
have neg : ∀ (h : ¬(↑x : ι) = a),
  x_1 (↑x : ι) =
    Function.comp (β := { i : ι // i ∈ insert a (t ∩ s ∪ u) })
      ((fun (o : Option (↑(t ∩ s ∪ u) : Type u_1)) ↦
          o.elim (x_1 a) fun (i : (↑(t ∩ s ∪ u) : Type u_1)) ↦ x_1 (↑i : ι)) ∘
        (⇑(subtypeInsertEquivOption (t := t ∩ s ∪ u) this) :
          (a : { i : ι // i ∈ insert a (t ∩ s ∪ u) }) → Option { i : ι // i ∈ t ∩ s ∪ u }))
      (⇑(Equiv.setCongr (s := t ∩ s ∪ insert a u) (t := insert a (t ∩ s ∪ u)) sorry) :
        (a_1 : (↑(t ∩ s ∪ insert a u) : Type u_1)) → (↑(insert a (t ∩ s ∪ u)) : Type u_1))
      x
Tactic: by_cases h : ↑x = a
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.Transcendental (goal)
Goal:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : Set ι
ind : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)
H :
  ∀ (t : Set ι),
    t.Finite →
      (AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑t : Type u_1)) ↦ x (↑i : ι)) →
        ∀ i ∉ s, i ∉ t → Transcendental (↥(adjoin R (x '' t))) (x i)
⊢ AlgebraicIndependent R x
have refine : ∀ (t : Set ι) (hfin : t.Finite), AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑t : Type u_1)) ↦ x (↑i : ι)
Tactic: classical
  refine algebraicIndependent_of_finite_type fun t hfin ↦ ?_
  suffices AlgebraicIndependent R fun i : ↥(t ∩ s ∪ t \ s) ↦ x i from
    this.comp (Equiv.setCongr (t.inter_union_diff s).symm) (Equiv.injective _)
  refine hfin.diff.induction_on_subset _ (ind.comp (inclusion <| by simp) (inclusion_injective _))
    fun {a u} ha hu ha' h ↦ ?_
  have : a ∉ t ∩ s ∪ u := (·.elim (ha.2 ·.2) ha')
  convert!
    (((image_eq_range .. ▸ h.option_iff_transcendental <| x a).2 <|
              H _ (hfin.subset (union_subset inter_subset_left <| hu.trans diff_subset)) h a ha.2
                this).comp
          _ (subtypeInsertEquivOption this).injective).comp
      (Equiv.setCongr union_insert) (Equiv.injective _) with
    x
  by_cases h : ↑x = a <;> simp [h, Set.subtypeInsertEquivOption]
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.Transcendental (goal)
Goal:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : Set ι
ind : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)
H :
  ∀ (t : Set ι),
    t.Finite →
      (AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑t : Type u_1)) ↦ x (↑i : ι)) →
        ∀ i ∉ s, i ∉ t → Transcendental (↥(adjoin R (x '' t))) (x i)
refine :
  ∀ (t : Set ι) (hfin : t.Finite), AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑t : Type u_1)) ↦ x (↑i : ι)
⊢ AlgebraicIndependent R x
have refine₁ : ∀ (t : Set ι) (hfin : t.Finite),
  AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑(t ∩ s ∪ t \ s) : Type u_1)) ↦ x (↑i : ι)
Tactic: classical
  refine algebraicIndependent_of_finite_type fun t hfin ↦ ?_
  suffices AlgebraicIndependent R fun i : ↥(t ∩ s ∪ t \ s) ↦ x i from
    this.comp (Equiv.setCongr (t.inter_union_diff s).symm) (Equiv.injective _)
  refine hfin.diff.induction_on_subset _ (ind.comp (inclusion <| by simp) (inclusion_injective _))
    fun {a u} ha hu ha' h ↦ ?_
  have : a ∉ t ∩ s ∪ u := (·.elim (ha.2 ·.2) ha')
  convert!
    (((image_eq_range .. ▸ h.option_iff_transcendental <| x a).2 <|
              H _ (hfin.subset (union_subset inter_subset_left <| hu.trans diff_subset)) h a ha.2
                this).comp
          _ (subtypeInsertEquivOption this).injective).comp
      (Equiv.setCongr union_insert) (Equiv.injective _) with
    x
  by_cases h : ↑x = a <;> simp [h, Set.subtypeInsertEquivOption]
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.Transcendental (goal)
Goal:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : Set ι
ind : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)
H :
  ∀ (t : Set ι),
    t.Finite →
      (AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑t : Type u_1)) ↦ x (↑i : ι)) →
        ∀ i ∉ s, i ∉ t → Transcendental (↥(adjoin R (x '' t))) (x i)
refine :
  ∀ (t : Set ι) (hfin : t.Finite), AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑t : Type u_1)) ↦ x (↑i : ι)
refine₁ :
  ∀ (t : Set ι) (hfin : t.Finite),
    AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑(t ∩ s ∪ t \ s) : Type u_1)) ↦ x (↑i : ι)
⊢ AlgebraicIndependent R x
have refine₂ : ∀ (t : Set ι) (hfin : t.Finite) {a : ι} {u : Set ι} (ha : a ∈ t \ s) (hu : u ⊆ t \ s) (ha' : a ∉ u)
  (h : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑(t ∩ s ∪ u) : Type u_1)) ↦ x (↑i : ι)),
  AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑(t ∩ s ∪ insert a u) : Type u_1)) ↦ x (↑i : ι)
Tactic: classical
  refine algebraicIndependent_of_finite_type fun t hfin ↦ ?_
  suffices AlgebraicIndependent R fun i : ↥(t ∩ s ∪ t \ s) ↦ x i from
    this.comp (Equiv.setCongr (t.inter_union_diff s).symm) (Equiv.injective _)
  refine hfin.diff.induction_on_subset _ (ind.comp (inclusion <| by simp) (inclusion_injective _))
    fun {a u} ha hu ha' h ↦ ?_
  have : a ∉ t ∩ s ∪ u := (·.elim (ha.2 ·.2) ha')
  convert!
    (((image_eq_range .. ▸ h.option_iff_transcendental <| x a).2 <|
              H _ (hfin.subset (union_subset inter_subset_left <| hu.trans diff_subset)) h a ha.2
                this).comp
          _ (subtypeInsertEquivOption this).injective).comp
      (Equiv.setCongr union_insert) (Equiv.injective _) with
    x
  by_cases h : ↑x = a <;> simp [h, Set.subtypeInsertEquivOption]
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.Transcendental (goal)
Goal:
ι : Type u_1
R : Type u_3
A : Type v
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : Set ι
ind : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑s : Type u_1)) ↦ x (↑i : ι)
H :
  ∀ (t : Set ι),
    t.Finite →
      (AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑t : Type u_1)) ↦ x (↑i : ι)) →
        ∀ i ∉ s, i ∉ t → Transcendental (↥(adjoin R (x '' t))) (x i)
refine :
  ∀ (t : Set ι) (hfin : t.Finite), AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑t : Type u_1)) ↦ x (↑i : ι)
refine₁ :
  ∀ (t : Set ι) (hfin : t.Finite),
    AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑(t ∩ s ∪ t \ s) : Type u_1)) ↦ x (↑i : ι)
refine₂ :
  ∀ (t : Set ι) (hfin : t.Finite) {a : ι} {u : Set ι} (ha : a ∈ t \ s) (hu : u ⊆ t \ s) (ha' : a ∉ u)
    (h : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑(t ∩ s ∪ u) : Type u_1)) ↦ x (↑i : ι)),
    AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑(t ∩ s ∪ insert a u) : Type u_1)) ↦ x (↑i : ι)
⊢ AlgebraicIndependent R x
have refine₃ : ∀ (t : Set ι) (hfin : t.Finite) {a : ι} {u : Set ι} (ha : a ∈ t \ s) (hu : u ⊆ t \ s) (ha' : a ∉ u)
  (h : AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑(t ∩ s ∪ u) : Type u_1)) ↦ x (↑i : ι))
  (this : a ∉ t ∩ s ∪ u),
  AlgebraicIndependent.{u_1, u_3, v} R (A := A) fun (i : (↑(t ∩ s ∪ insert a u) : Type u_1)) ↦ x (↑i : ι)
Tactic: classical
  refine algebraicIndependent_of_finite_type fun t hfin ↦ ?_
  suffices AlgebraicIndependent R fun i : ↥(t ∩ s ∪ t \ s) ↦ x i from
    this.comp (Equiv.setCongr (t.inter_union_diff s).symm) (Equiv.injective _)
  refine hfin.diff.induction_on_subset _ (ind.comp (inclusion <| by simp) (inclusion_injective _))
    fun {a u} ha hu ha' h ↦ ?_
  have : a ∉ t ∩ s ∪ u := (·.elim (ha.2 ·.2) ha')
  convert!
    (((image_eq_range .. ▸ h.option_iff_transcendental <| x a).2 <|
              H _ (hfin.subset (union_subset inter_subset_left <| hu.trans diff_subset)) h a ha.2
                this).comp
          _ (subtypeInsertEquivOption this).injective).comp
      (Equiv.setCongr union_insert) (Equiv.injective _) with
    x
  by_cases h : ↑x = a <;> simp [h, Set.subtypeInsertEquivOption]