awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
K_1 : Type w
inst_3 : Field K_1
inst_4 : Algebra F K_1
K : IntermediateField F E
x : E
hx :
  Membership.mem (γ := IntermediateField F E)
    (IntermediateField.restrictScalars F (L' :=
      ↥(IntermediateField.restrictScalars F (L' := ↥K) (separableClosure (↥K) E)))
      (separableClosure (↥(IntermediateField.restrictScalars F (L' := ↥K) (separableClosure (↥K) E))) E))
    x
⊢ Membership.mem (γ := IntermediateField F E) (IntermediateField.restrictScalars F (L' := ↥K) (separableClosure (↥K) E))
    x
have refine_2 : ∀ (x : ↥(separableClosure (↥K) E))
  (hx :
    Membership.mem (α := E) (γ := IntermediateField F E)
      (IntermediateField.restrictScalars F (L' :=
        ↥(IntermediateField.restrictScalars F (L' := ↥K) (separableClosure (↥K) E)))
        (separableClosure (↥(IntermediateField.restrictScalars F (L' := ↥K) (separableClosure (↥K) E))) E))
      ((AlgHom.toRingHom (Algebra.ofId (↥(separableClosure (↥K) E)) E) : (a : ↥(separableClosure (↥K) E)) → E) x : E)),
  Membership.mem (α := E) (γ := IntermediateField F E)
    (IntermediateField.restrictScalars F (L' := ↥K) (separableClosure (↥K) E))
    ((AlgHom.toRingHom (Algebra.ofId (↥(separableClosure (↥K) E)) E) : (a : ↥(separableClosure (↥K) E)) → E) x : E)
Tactic: obtain ⟨x, rfl⟩ := (separableClosure.separableClosure_eq_bot K E).le hx
Cleared hypotheses: x, hx
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
K : Type w
inst_3 : Field K
inst_4 : Algebra F K
inst_5 : Algebra K E
inst_6 : IsScalarTower F K E
x : E
hx :
  Membership.mem (γ := IntermediateField F E)
    ((separableClosureOperator F E : (a : IntermediateField F E) → IntermediateField F E)
        (IntermediateField.restrictScalars F (separableClosure K E)) :
      IntermediateField F E)
    x
⊢ Membership.mem (γ := IntermediateField F E) (IntermediateField.restrictScalars F (separableClosure K E)) x
have obtain : ∀ (x : ↥(separableClosure K E))
  (hx :
    Membership.mem (α := E) (γ := IntermediateField F E)
      ((separableClosureOperator F E : (a : IntermediateField F E) → IntermediateField F E)
          (IntermediateField.restrictScalars F (separableClosure K E)) :
        IntermediateField F E)
      ((AlgHom.toRingHom (Algebra.ofId (↥(separableClosure K E)) E) : (a : ↥(separableClosure K E)) → E) x : E)),
  Membership.mem (α := E) (γ := IntermediateField F E) (IntermediateField.restrictScalars F (separableClosure K E))
    ((AlgHom.toRingHom (Algebra.ofId (↥(separableClosure K E)) E) : (a : ↥(separableClosure K E)) → E) x : E)
Tactic: obtain ⟨x, rfl⟩ := (separableClosure.separableClosure_eq_bot K E).le hx
Cleared hypotheses: x, hx
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
inst : Field F
⊢ sepDegree F F = 1
have rw : Module.rank F ↥(separableClosure F F) = 1
Tactic: rw [sepDegree, Subsingleton.elim (separableClosure F F) ⊥, IntermediateField.rank_bot]
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
inst : Field F
rw : Module.rank F ↥(separableClosure F F) = 1
⊢ sepDegree F F = 1
have rw₁ : Module.rank F ↥⊥ = 1
Tactic: rw [sepDegree, Subsingleton.elim (separableClosure F F) ⊥, IntermediateField.rank_bot]
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
inst : Field F
rw : Module.rank F ↥(separableClosure F F) = 1
rw₁ : Module.rank F ↥⊥ = 1
⊢ sepDegree F F = 1
have rw₂ : Eq.{u + 2} (α := Cardinal.{u}) 1 1
Tactic: rw [sepDegree, Subsingleton.elim (separableClosure F F) ⊥, IntermediateField.rank_bot]
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
inst : Field F
⊢ insepDegree F F = 1
have rw : Module.rank (↥(separableClosure F F)) F = 1
Tactic: rw [insepDegree, Subsingleton.elim (separableClosure F F) ⊤, IntermediateField.rank_top]
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
inst : Field F
rw : Module.rank (↥(separableClosure F F)) F = 1
⊢ insepDegree F F = 1
have rw₁ : Module.rank (↥⊤) F = 1
Tactic: rw [insepDegree, Subsingleton.elim (separableClosure F F) ⊤, IntermediateField.rank_top]
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
inst : Field F
rw : Module.rank (↥(separableClosure F F)) F = 1
rw₁ : Module.rank (↥⊤) F = 1
⊢ insepDegree F F = 1
have rw₂ : Eq.{u + 2} (α := Cardinal.{u}) 1 1
Tactic: rw [insepDegree, Subsingleton.elim (separableClosure F F) ⊤, IntermediateField.rank_top]
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
inst : Field F
⊢ finInsepDegree F F = 1
have rw : Eq (α := ℕ) ((Cardinal.toNat : (a : Cardinal.{u}) → ℕ) (insepDegree F F) : ℕ) (1 : ℕ)
Tactic: rw [finInsepDegree_def', insepDegree_self, Cardinal.one_toNat]
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
inst : Field F
rw : Eq (α := ℕ) ((Cardinal.toNat : (a : Cardinal.{u}) → ℕ) (insepDegree F F) : ℕ) (1 : ℕ)
⊢ finInsepDegree F F = 1
have rw₁ : Eq (α := ℕ) ((Cardinal.toNat : (a : Cardinal.{u}) → ℕ) 1 : ℕ) (1 : ℕ)
Tactic: rw [finInsepDegree_def', insepDegree_self, Cardinal.one_toNat]
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
inst : Field F
rw : Eq (α := ℕ) ((Cardinal.toNat : (a : Cardinal.{u}) → ℕ) (insepDegree F F) : ℕ) (1 : ℕ)
rw₁ : Eq (α := ℕ) ((Cardinal.toNat : (a : Cardinal.{u}) → ℕ) 1 : ℕ) (1 : ℕ)
⊢ finInsepDegree F F = 1
have rw₂ : Eq (α := ℕ) 1 1
Tactic: rw [finInsepDegree_def', insepDegree_self, Cardinal.one_toNat]
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
inst_3 : Algebra.EssFiniteType F E
d : (s : Finset E) → ℕ := fun (s : Finset E) ↦ finInsepDegree (↥(adjoin F (↑s : Set E))) E
⊢ Set.Nonempty.{v} (α := Finset E)
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
have have : ∀ (s : Finset E) (hs : Eq (α := IntermediateField F E) (adjoin F (↑s : Set E)) ⊤),
  Set.Nonempty.{v} (α := Finset E)
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
Tactic: have ⟨s, hs⟩ := IntermediateField.fg_top F E
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
inst_3 : Algebra.EssFiniteType F E
d : (s : Finset E) → ℕ := fun (s : Finset E) ↦ finInsepDegree (↥(adjoin F (↑s : Set E))) E
s : Finset E
hs : Eq (α := IntermediateField F E) (adjoin F (↑s : Set E)) ⊤
⊢ Algebra.IsAlgebraic (↥(Algebra.adjoin F (↑s : Set E))) E
have rw : Algebra.IsIntegral (↥⊤) E
Tactic: rw [← isAlgebraic_adjoin_iff_top, hs, Algebra.isAlgebraic_iff_isIntegral]
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (hypothesis)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
inst_3 : Algebra.EssFiniteType F E
d : (s : Finset E) → ℕ := fun (s : Finset E) ↦ finInsepDegree (↥(adjoin F (↑s : Set E))) E
s : Finset E
hs : Eq (α := IntermediateField F E) (adjoin F (↑s : Set E)) ⊤
⊢ Set.Nonempty.{v} (α := Finset E)
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
have this : Algebra.IsAlgebraic (↥(Algebra.adjoin F (↑s : Set E))) E
Tactic: have : Algebra.IsAlgebraic (Algebra.adjoin F (s : Set E)) E := by
      rw [← isAlgebraic_adjoin_iff_top, hs, Algebra.isAlgebraic_iff_isIntegral]
      refine Algebra.isIntegral_of_surjective topEquiv.surjective
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
inst_3 : Algebra.EssFiniteType F E
d : (s : Finset E) → ℕ := fun (s : Finset E) ↦ finInsepDegree (↥(adjoin F (↑s : Set E))) E
s : Finset E
hs : Eq (α := IntermediateField F E) (adjoin F (↑s : Set E)) ⊤
this : Algebra.IsAlgebraic (↥(Algebra.adjoin F (↑s : Set E))) E
⊢ Set.Nonempty.{v} (α := Finset E)
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
have have : ∀ (t : Set E) (hts : t ⊆ (↑s : Set E))
  (ht : IsTranscendenceBasis.{v, u, v} (ι := { x : E // x ∈ t }) F (Subtype.val (p := fun (x : E) ↦ x ∈ t))),
  Set.Nonempty.{v} (α := Finset E)
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
Tactic: have ⟨t, hts, ht⟩ := exists_isTranscendenceBasis_subset (R := F) (s : Set E)
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
inst_3 : Algebra.EssFiniteType F E
d : (s : Finset E) → ℕ := fun (s : Finset E) ↦ finInsepDegree (↥(adjoin F (↑s : Set E))) E
s : Finset E
hs : Eq (α := IntermediateField F E) (adjoin F (↑s : Set E)) ⊤
this : Algebra.IsAlgebraic (↥(Algebra.adjoin F (↑s : Set E))) E
t : Set E
hts : t ⊆ (↑s : Set E)
ht : IsTranscendenceBasis.{v, u, v} (ι := { x : E // x ∈ t }) F (Subtype.val (p := fun (x : E) ↦ x ∈ t))
⊢ Set.Nonempty.{v} (α := Finset E)
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
have lift : ∀ (t : Finset E) (hts : (↑t : Set E) ⊆ (↑s : Set E))
  (ht :
    IsTranscendenceBasis.{v, u, v} (ι := { x : E // x ∈ (↑t : Set E) }) F
      (Subtype.val (p := fun (x : E) ↦ x ∈ (↑t : Set E)))),
  Set.Nonempty.{v} (α := Finset E)
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
Tactic: lift t to Finset E using s.finite_toSet.subset hts
Cleared hypotheses: t, hts, ht
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (hypothesis)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
inst_3 : Algebra.EssFiniteType F E
d : (s : Finset E) → ℕ := fun (s : Finset E) ↦ finInsepDegree (↥(adjoin F (↑s : Set E))) E
⊢ ∃ (s : Finset E),
    MaximalFor (α := IntermediateField F E)
      (fun (t : Set E) ↦
        IsTranscendenceBasis.{v, u, v} (ι := { x : E // x ∈ t }) F (Subtype.val (p := fun (x : E) ↦ x ∈ t)))
      (fun (t : Set E) ↦ restrictScalars F (L' := ↥(adjoin F t)) (separableClosure (↥(adjoin F t)) E)) (↑s : Set E)
have Hexists : Set.Nonempty.{v} (α := Finset E)
  {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
Tactic: have Hexists : {s : Finset E | IsTranscendenceBasis F ((↑) : s → E)}.Nonempty := by
    have ⟨s, hs⟩ := IntermediateField.fg_top F E
    have : Algebra.IsAlgebraic (Algebra.adjoin F (s : Set E)) E := by
      rw [← isAlgebraic_adjoin_iff_top, hs, Algebra.isAlgebraic_iff_isIntegral]
      refine Algebra.isIntegral_of_surjective topEquiv.surjective
    have ⟨t, hts, ht⟩ := exists_isTranscendenceBasis_subset (R := F) (s : Set E)
    lift t to Finset E using s.finite_toSet.subset hts
    exact ⟨t, ht⟩
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (hypothesis)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
inst_3 : Algebra.EssFiniteType F E
d : (s : Finset E) → ℕ := fun (s : Finset E) ↦ finInsepDegree (↥(adjoin F (↑s : Set E))) E
Hexists :
  Set.Nonempty.{v} (α := Finset E)
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
s : Finset E :=
  Function.argminOn d
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))} Hexists
⊢ ∃ (s : Finset E),
    MaximalFor (α := IntermediateField F E)
      (fun (t : Set E) ↦
        IsTranscendenceBasis.{v, u, v} (ι := { x : E // x ∈ t }) F (Subtype.val (p := fun (x : E) ↦ x ∈ t)))
      (fun (t : Set E) ↦ restrictScalars F (L' := ↥(adjoin F t)) (separableClosure (↥(adjoin F t)) E)) (↑s : Set E)
have hs : Membership.mem (γ := Set (Finset E))
  {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
  (Function.argminOn d
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))} Hexists)
Tactic: have hs := d.argminOn_mem _ Hexists
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
inst_3 : Algebra.EssFiniteType F E
d : (s : Finset E) → ℕ := fun (s : Finset E) ↦ finInsepDegree (↥(adjoin F (↑s : Set E))) E
Hexists :
  Set.Nonempty.{v} (α := Finset E)
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
s : Finset E :=
  Function.argminOn d
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))} Hexists
hs :
  Membership.mem (γ := Set (Finset E))
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
    (Function.argminOn d
      {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))} Hexists)
⊢ ∃ (s : Finset E),
    MaximalFor (α := IntermediateField F E)
      (fun (t : Set E) ↦
        IsTranscendenceBasis.{v, u, v} (ι := { x : E // x ∈ t }) F (Subtype.val (p := fun (x : E) ↦ x ∈ t)))
      (fun (t : Set E) ↦ restrictScalars F (L' := ↥(adjoin F t)) (separableClosure (↥(adjoin F t)) E)) (↑s : Set E)
have refine : ∀ (t : Set E)
  (ht :
    (fun (t : Set E) ↦
        IsTranscendenceBasis.{v, u, v} (ι := { x : E // x ∈ t }) F (Subtype.val (p := fun (x : E) ↦ x ∈ t)))
      t)
  (H :
    LT.lt.{v} (α := IntermediateField F E)
      ((fun (t : Set E) ↦
          (restrictScalars F (L' := ↥(adjoin F t)) (separableClosure (↥(adjoin F t)) E) : IntermediateField F E))
        (↑s : Set E))
      ((fun (t : Set E) ↦
          (restrictScalars F (L' := ↥(adjoin F t)) (separableClosure (↥(adjoin F t)) E) : IntermediateField F E))
        t)),
  False
Tactic: refine ⟨s, hs, fun t ht ↦ not_lt_iff_le_imp_ge.mp fun H ↦ ?_⟩
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (hypothesis)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
inst_3 : Algebra.EssFiniteType F E
d : (s : Finset E) → ℕ := fun (s : Finset E) ↦ finInsepDegree (↥(adjoin F (↑s : Set E))) E
Hexists :
  Set.Nonempty.{v} (α := Finset E)
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
s : Finset E :=
  Function.argminOn d
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))} Hexists
hs :
  Membership.mem (γ := Set (Finset E))
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
    (Function.argminOn d
      {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))} Hexists)
t : Set E
ht :
  (fun (t : Set E) ↦
      IsTranscendenceBasis.{v, u, v} (ι := { x : E // x ∈ t }) F (Subtype.val (p := fun (x : E) ↦ x ∈ t)))
    t
H :
  LT.lt.{v} (α := IntermediateField F E)
    ((fun (t : Set E) ↦
        (restrictScalars F (L' := ↥(adjoin F t)) (separableClosure (↥(adjoin F t)) E) : IntermediateField F E))
      (↑s : Set E))
    ((fun (t : Set E) ↦
        (restrictScalars F (L' := ↥(adjoin F t)) (separableClosure (↥(adjoin F t)) E) : IntermediateField F E))
      t)
⊢ False
have this : t.Finite
Tactic: have : t.Finite := by
    simp [Set.Finite, ← Cardinal.mk_lt_aleph0_iff, ht.cardinalMk_eq hs, Cardinal.natCast_lt_aleph0]
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
inst_3 : Algebra.EssFiniteType F E
d : (s : Finset E) → ℕ := fun (s : Finset E) ↦ finInsepDegree (↥(adjoin F (↑s : Set E))) E
Hexists :
  Set.Nonempty.{v} (α := Finset E)
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
s : Finset E :=
  Function.argminOn d
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))} Hexists
hs :
  Membership.mem (γ := Set (Finset E))
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
    (Function.argminOn d
      {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))} Hexists)
t : Set E
ht :
  (fun (t : Set E) ↦
      IsTranscendenceBasis.{v, u, v} (ι := { x : E // x ∈ t }) F (Subtype.val (p := fun (x : E) ↦ x ∈ t)))
    t
H :
  LT.lt.{v} (α := IntermediateField F E)
    ((fun (t : Set E) ↦
        (restrictScalars F (L' := ↥(adjoin F t)) (separableClosure (↥(adjoin F t)) E) : IntermediateField F E))
      (↑s : Set E))
    ((fun (t : Set E) ↦
        (restrictScalars F (L' := ↥(adjoin F t)) (separableClosure (↥(adjoin F t)) E) : IntermediateField F E))
      t)
this : t.Finite
⊢ False
have lift : ∀ (t : Finset E)
  (ht :
    IsTranscendenceBasis.{v, u, v} (ι := { x : E // x ∈ (↑t : Set E) }) F
      (Subtype.val (p := fun (x : E) ↦ x ∈ (↑t : Set E))))
  (H :
    LT.lt.{v} (α := IntermediateField F E)
      ((fun (t : Set E) ↦
          (restrictScalars F (L' := ↥(adjoin F t)) (separableClosure (↥(adjoin F t)) E) : IntermediateField F E))
        (↑s : Set E))
      ((fun (t : Set E) ↦
          (restrictScalars F (L' := ↥(adjoin F t)) (separableClosure (↥(adjoin F t)) E) : IntermediateField F E))
        (↑t : Set E))),
  False
Tactic: lift t to Finset E using this
Cleared hypotheses: t, ht, H, this
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
inst_3 : Algebra.EssFiniteType F E
d : (s : Finset E) → ℕ := fun (s : Finset E) ↦ finInsepDegree (↥(adjoin F (↑s : Set E))) E
Hexists :
  Set.Nonempty.{v} (α := Finset E)
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
s : Finset E :=
  Function.argminOn d
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))} Hexists
hs :
  Membership.mem (γ := Set (Finset E))
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
    (Function.argminOn d
      {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))} Hexists)
t : Finset E
ht :
  IsTranscendenceBasis.{v, u, v} (ι := { x : E // x ∈ (↑t : Set E) }) F
    (Subtype.val (p := fun (x : E) ↦ x ∈ (↑t : Set E)))
H :
  LT.lt.{v} (α := IntermediateField F E)
    ((fun (t : Set E) ↦
        (restrictScalars F (L' := ↥(adjoin F t)) (separableClosure (↥(adjoin F t)) E) : IntermediateField F E))
      (↑s : Set E))
    ((fun (t : Set E) ↦
        (restrictScalars F (L' := ↥(adjoin F t)) (separableClosure (↥(adjoin F t)) E) : IntermediateField F E))
      (↑t : Set E))
⊢ Module.Finite (↥(adjoin F (↑s : Set E))) E
have inst : Algebra.EssFiniteType (↥(adjoin F (↑s : Set E))) E
Tactic: apply +allowSynthFailures Algebra.finite_of_essFiniteType_of_isAlgebraic
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
E : Type v
inst_1 : Field F
inst_2 : Field E
inst_3 : Algebra F E
inst_4 : Algebra.EssFiniteType F E
d : (s : Finset E) → ℕ := fun (s : Finset E) ↦ finInsepDegree (↥(adjoin F (↑s : Set E))) E
Hexists :
  Set.Nonempty.{v} (α := Finset E)
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
s : Finset E :=
  Function.argminOn d
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))} Hexists
hs :
  Membership.mem (γ := Set (Finset E))
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
    (Function.argminOn d
      {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))} Hexists)
t : Finset E
ht :
  IsTranscendenceBasis.{v, u, v} (ι := { x : E // x ∈ (↑t : Set E) }) F
    (Subtype.val (p := fun (x : E) ↦ x ∈ (↑t : Set E)))
H :
  LT.lt.{v} (α := IntermediateField F E)
    ((fun (t : Set E) ↦
        (restrictScalars F (L' := ↥(adjoin F t)) (separableClosure (↥(adjoin F t)) E) : IntermediateField F E))
      (↑s : Set E))
    ((fun (t : Set E) ↦
        (restrictScalars F (L' := ↥(adjoin F t)) (separableClosure (↥(adjoin F t)) E) : IntermediateField F E))
      (↑t : Set E))
inst : Algebra.EssFiniteType (↥(adjoin F (↑s : Set E))) E
⊢ Module.Finite (↥(adjoin F (↑s : Set E))) E
have inst₁ : Algebra.IsAlgebraic (↥(adjoin F (↑s : Set E))) E
Tactic: apply +allowSynthFailures Algebra.finite_of_essFiniteType_of_isAlgebraic
-- Lean 4 proof step from Mathlib.FieldTheory.SeparableClosure (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
inst_3 : Algebra.EssFiniteType F E
d : (s : Finset E) → ℕ := fun (s : Finset E) ↦ finInsepDegree (↥(adjoin F (↑s : Set E))) E
Hexists :
  Set.Nonempty.{v} (α := Finset E)
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
s : Finset E :=
  Function.argminOn d
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))} Hexists
hs :
  Membership.mem (γ := Set (Finset E))
    {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
    (Function.argminOn d
      {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))} Hexists)
t : Finset E
ht :
  IsTranscendenceBasis.{v, u, v} (ι := { x : E // x ∈ (↑t : Set E) }) F
    (Subtype.val (p := fun (x : E) ↦ x ∈ (↑t : Set E)))
H :
  LT.lt.{v} (α := IntermediateField F E)
    ((fun (t : Set E) ↦
        (restrictScalars F (L' := ↥(adjoin F t)) (separableClosure (↥(adjoin F t)) E) : IntermediateField F E))
      (↑s : Set E))
    ((fun (t : Set E) ↦
        (restrictScalars F (L' := ↥(adjoin F t)) (separableClosure (↥(adjoin F t)) E) : IntermediateField F E))
      (↑t : Set E))
⊢ Algebra.IsAlgebraic (↥(adjoin F (↑s : Set E))) E
have h : ∀ (x : E),
  (↑s : Set E) =
    Set.range (ι :=
      ↥(Function.argminOn d
          {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))} Hexists))
      (Subtype.val (p := fun (x : E) ↦
        x ∈
          Function.argminOn d
            {s : Finset E | IsTranscendenceBasis.{v, u, v} (ι := ↥s) F (Subtype.val (p := fun (x : E) ↦ x ∈ s))}
            Hexists))
Tactic: convert! hs.isAlgebraic_field