awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
A B : IntermediateField F E
pos : ∀ (hA : Algebra.IsAlgebraic.{u, v} F ↥A), Module.rank F ↥(A ⊔ B) ≤ Module.rank F ↥A * Module.rank F ↥B
⊢ Module.rank F ↥(A ⊔ B) ≤ Module.rank F ↥A * Module.rank F ↥B
have neg : ∀ (hA : ¬Algebra.IsAlgebraic.{u, v} F ↥A), Module.rank F ↥(A ⊔ B) ≤ Module.rank F ↥A * Module.rank F ↥B
Tactic: by_cases hA : Algebra.IsAlgebraic F A
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
A B : IntermediateField F E
hA : ¬Algebra.IsAlgebraic.{u, v} F ↥A
⊢ Module.rank F ↥(A ⊔ B) ≤ Module.rank F ↥A * Module.rank F ↥B
have pos : ∀ (hB : Algebra.IsAlgebraic.{u, v} F ↥B), Module.rank F ↥(A ⊔ B) ≤ Module.rank F ↥A * Module.rank F ↥B
Tactic: by_cases hB : Algebra.IsAlgebraic F B
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
A B : IntermediateField F E
hA : ¬Algebra.IsAlgebraic.{u, v} F ↥A
pos : ∀ (hB : Algebra.IsAlgebraic.{u, v} F ↥B), Module.rank F ↥(A ⊔ B) ≤ Module.rank F ↥A * Module.rank F ↥B
⊢ Module.rank F ↥(A ⊔ B) ≤ Module.rank F ↥A * Module.rank F ↥B
have neg : ∀ (hB : ¬Algebra.IsAlgebraic.{u, v} F ↥B), Module.rank F ↥(A ⊔ B) ≤ Module.rank F ↥A * Module.rank F ↥B
Tactic: by_cases hB : Algebra.IsAlgebraic F B
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality (hypothesis)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
A B : IntermediateField F E
hA : ¬Algebra.IsAlgebraic.{u, v} F ↥A
hB : ¬Algebra.IsAlgebraic.{u, v} F ↥B
⊢ Module.rank F ↥(A ⊔ B) ≤ Module.rank F ↥A * Module.rank F ↥B
have hA₁ : Algebra.Transcendental.{u, v} F ↥A
Tactic: rw [← Algebra.transcendental_iff_not_isAlgebraic] at hA hB
Cleared hypotheses: hA, hB
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality (hypothesis)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
A B : IntermediateField F E
hA : ¬Algebra.IsAlgebraic.{u, v} F ↥A
hB : ¬Algebra.IsAlgebraic.{u, v} F ↥B
hA₁ : Algebra.Transcendental.{u, v} F ↥A
⊢ Module.rank F ↥(A ⊔ B) ≤ Module.rank F ↥A * Module.rank F ↥B
have hB₁ : Algebra.Transcendental.{u, v} F ↥B
Tactic: rw [← Algebra.transcendental_iff_not_isAlgebraic] at hA hB
Cleared hypotheses: hA, hB
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality (hypothesis)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
A B : IntermediateField F E
hA : Algebra.Transcendental.{u, v} F ↥A
hB : Algebra.Transcendental.{u, v} F ↥B
⊢ Module.rank F ↥(A ⊔ B) ≤ Module.rank F ↥A * Module.rank F ↥B
have this : Algebra.Transcendental.{u, v} F ↥(A ⊔ B)
Tactic: haveI : Algebra.Transcendental F ↥(A ⊔ B) := .ringHom_of_comp_eq (RingHom.id F)
    (inclusion le_sup_left) Function.surjective_id (inclusion_injective _) rfl
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality (hypothesis)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
A B : IntermediateField F E
hA : Algebra.Transcendental.{u, v} F ↥A
hB : Algebra.Transcendental.{u, v} F ↥B
this : Algebra.Transcendental.{u, v} F ↥(A ⊔ B)
⊢ Module.rank F ↥(A ⊔ B) ≤ Module.rank F ↥A * Module.rank F ↥B
have this₁ : Infinite.{v + 1} ↥A
Tactic: haveI := Algebra.Transcendental.infinite F A
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality (hypothesis)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
A B : IntermediateField F E
hA : Algebra.Transcendental.{u, v} F ↥A
hB : Algebra.Transcendental.{u, v} F ↥B
this_1 : Algebra.Transcendental.{u, v} F ↥(A ⊔ B)
this : Infinite.{v + 1} ↥A
⊢ Module.rank F ↥(A ⊔ B) ≤ Module.rank F ↥A * Module.rank F ↥B
have this₁ : Infinite.{v + 1} ↥B
Tactic: haveI := Algebra.Transcendental.infinite F B
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
A B : IntermediateField F E
hA : Algebra.Transcendental.{u, v} F ↥A
hB : Algebra.Transcendental.{u, v} F ↥B
this_1 : Algebra.Transcendental.{u, v} F ↥(A ⊔ B)
this_2 : Infinite.{v + 1} ↥A
this : Infinite.{v + 1} ↥B
⊢ Module.rank F ↥(A ⊔ B) ≤ Module.rank F ↥A * Module.rank F ↥B
have neg : #↥(A ⊔ B) ≤ #↥A * #↥B
Tactic: simp_rw [Algebra.Transcendental.rank_eq_cardinalMk]
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
A B : IntermediateField F E
hA : Algebra.Transcendental.{u, v} F ↥A
hB : Algebra.Transcendental.{u, v} F ↥B
this_1 : Algebra.Transcendental.{u, v} F ↥(A ⊔ B)
this_2 : Infinite.{v + 1} ↥A
this : Infinite.{v + 1} ↥B
⊢ #↥(A ⊔ B) ≤ #↥A * #↥B
have neg : Cardinal.lift.{u, v} #↥(adjoin F ((↑A : Set E) ∪ (↑B : Set E))) ≤ Cardinal.lift.{u, v} (max #↥A #↥B)
Tactic: rw [sup_def, mul_mk_eq_max, ← Cardinal.lift_le.{u}]
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
A B : IntermediateField F E
hA : Algebra.Transcendental.{u, v} F ↥A
hB : Algebra.Transcendental.{u, v} F ↥B
this_1 : Algebra.Transcendental.{u, v} F ↥(A ⊔ B)
this_2 : Infinite.{v + 1} ↥A
this : Infinite.{v + 1} ↥B
⊢ Cardinal.lift.{u, v} #↥(adjoin F ((↑A : Set E) ∪ (↑B : Set E))) ≤ Cardinal.lift.{u, v} (max #↥A #↥B)
have neg : max (max (Cardinal.lift.{v, u} #F) (Cardinal.lift.{u, v} #(↑((↑A : Set E) ∪ (↑B : Set E)) : Type v))) ℵ₀ ≤
  Cardinal.lift.{u, v} (max #↥A #↥B)
Tactic: refine (lift_cardinalMk_adjoin_le _ _).trans ?_
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
A B : IntermediateField F E
hA : Algebra.Transcendental.{u, v} F ↥A
hB : Algebra.Transcendental.{u, v} F ↥B
this_1 : Algebra.Transcendental.{u, v} F ↥(A ⊔ B)
this_2 : Infinite.{v + 1} ↥A
this : Infinite.{v + 1} ↥B
⊢ max (max (Cardinal.lift.{v, u} #F) (Cardinal.lift.{u, v} #(↑((↑A : Set E) ∪ (↑B : Set E)) : Type v))) ℵ₀ ≤
    max (max (Cardinal.lift.{v, u} #F) (Cardinal.lift.{u, v} (max #↥A #↥B))) ℵ₀
have h₁ : Cardinal.lift.{u, v} #(↑((↑A : Set E) ∪ (↑B : Set E)) : Type v) ≤ Cardinal.lift.{u, v} (max #↥A #↥B)
Tactic: gcongr
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
A B : IntermediateField F E
hA : Algebra.Transcendental.{u, v} F ↥A
hB : Algebra.Transcendental.{u, v} F ↥B
this_1 : Algebra.Transcendental.{u, v} F ↥(A ⊔ B)
this_2 : Infinite.{v + 1} ↥A
this : Infinite.{v + 1} ↥B
⊢ Cardinal.lift.{u, v} #(↑((↑A : Set E) ∪ (↑B : Set E)) : Type v) ≤ Cardinal.lift.{u, v} (max #↥A #↥B)
have h₁ : #(↑((↑A : Set E) ∪ (↑B : Set E)) : Type v) ≤ max #↥A #↥B
Tactic: rw [Cardinal.lift_le]
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
A B : IntermediateField F E
hA : Algebra.Transcendental.{u, v} F ↥A
hB : Algebra.Transcendental.{u, v} F ↥B
this_1 : Algebra.Transcendental.{u, v} F ↥(A ⊔ B)
this_2 : Infinite.{v + 1} ↥A
this : Infinite.{v + 1} ↥B
⊢ max (max (Cardinal.lift.{v, u} #F) (Cardinal.lift.{u, v} #(↑((↑A : Set E) ∪ (↑B : Set E)) : Type v))) ℵ₀ ≤
    Cardinal.lift.{u, v} (max #↥A #↥B)
have _ : max (max (Cardinal.lift.{v, u} #F) (Cardinal.lift.{u, v} #(↑((↑A : Set E) ∪ (↑B : Set E)) : Type v))) ℵ₀ ≤
  max (max (Cardinal.lift.{v, u} #F) (Cardinal.lift.{u, v} (max #↥A #↥B))) ℵ₀
Tactic: calc
    _ ≤ Cardinal.lift.{v} #F ⊔ Cardinal.lift.{u} (#A ⊔ #B) ⊔ ℵ₀ := by
      gcongr
      rw [Cardinal.lift_le]
      exact (mk_union_le _ _).trans_eq (by simp)
    _ = _ := by
      simp [lift_mk_le_lift_mk_of_injective (algebraMap F A).injective]
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality (goal)
Goal:
F : Type u
E : Type v
inst : Field F
inst_1 : Field E
inst_2 : Algebra F E
A B : IntermediateField F E
hA : Algebra.Transcendental.{u, v} F ↥A
hB : Algebra.Transcendental.{u, v} F ↥B
this_1 : Algebra.Transcendental.{u, v} F ↥(A ⊔ B)
this_2 : Infinite.{v + 1} ↥A
this : Infinite.{v + 1} ↥B
_ :
  max (max (Cardinal.lift.{v, u} #F) (Cardinal.lift.{u, v} #(↑((↑A : Set E) ∪ (↑B : Set E)) : Type v))) ℵ₀ ≤
    max (max (Cardinal.lift.{v, u} #F) (Cardinal.lift.{u, v} (max #↥A #↥B))) ℵ₀
⊢ max (max (Cardinal.lift.{v, u} #F) (Cardinal.lift.{u, v} #(↑((↑A : Set E) ∪ (↑B : Set E)) : Type v))) ℵ₀ ≤
    Cardinal.lift.{u, v} (max #↥A #↥B)
have _₁ : max (max (Cardinal.lift.{v, u} #F) (Cardinal.lift.{u, v} (max #↥A #↥B))) ℵ₀ = Cardinal.lift.{u, v} (max #↥A #↥B)
Tactic: calc
    _ ≤ Cardinal.lift.{v} #F ⊔ Cardinal.lift.{u} (#A ⊔ #B) ⊔ ℵ₀ := by
      gcongr
      rw [Cardinal.lift_le]
      exact (mk_union_le _ _).trans_eq (by simp)
    _ = _ := by
      simp [lift_mk_le_lift_mk_of_injective (algebraMap F A).injective]
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (goal)
Goal:
R : Type u_1
A : Type w
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
⊢ trdeg R A =
    ⨆ (ι :
      { s : Set A //
        IsTranscendenceBasis.{w, u_1, w} (ι := { x : A // x ∈ s }) R (Subtype.val (p := fun (x : A) ↦ x ∈ s)) }),
      #(↑(↑ι : Set A) : Type w)
have refine : ∀ (s : { s : Set A // AlgebraicIndepOn R _root_.id s }),
  #(↑(↑s : Set A) : Type w) ≤
    ⨆ (ι :
      { s : Set A //
        IsTranscendenceBasis.{w, u_1, w} (ι := { x : A // x ∈ s }) R (Subtype.val (p := fun (x : A) ↦ x ∈ s)) }),
      #(↑(↑ι : Set A) : Type w)
Tactic: refine (ciSup_le' fun s ↦ ?_).antisymm
    (ciSup_le' fun s ↦ le_ciSup_of_le bddAbove_of_small ⟨s, s.2.1⟩ le_rfl)
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (goal)
Goal:
R : Type u_1
A : Type w
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
s : { s : Set A // AlgebraicIndepOn R _root_.id s }
⊢ #(↑(↑s : Set A) : Type w) ≤
    ⨆ (ι :
      { s : Set A //
        IsTranscendenceBasis.{w, u_1, w} (ι := { x : A // x ∈ s }) R (Subtype.val (p := fun (x : A) ↦ x ∈ s)) }),
      #(↑(↑ι : Set A) : Type w)
have choose : ∀ (t : Set A)
  (ht :
    (↑s : Set A) ⊆ t ∧
      IsTranscendenceBasis.{w, u_1, w} (ι := { x : A // x ∈ t }) R (Subtype.val (p := fun (x : A) ↦ x ∈ t))),
  #(↑(↑s : Set A) : Type w) ≤
    ⨆ (ι :
      { s : Set A //
        IsTranscendenceBasis.{w, u_1, w} (ι := { x : A // x ∈ s }) R (Subtype.val (p := fun (x : A) ↦ x ∈ s)) }),
      #(↑(↑ι : Set A) : Type w)
Tactic: choose t ht using exists_isTranscendenceBasis_superset s.2
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (goal)
Goal:
ι : Type u
R : Type u_1
A : Type w
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
i : AlgebraicIndependent R x
⊢ IsTranscendenceBasis R x ↔
    ∀ (κ : Type w) (w : κ → A), AlgebraicIndependent R w → ∀ (j : ι → κ), w ∘ j = x → Surjective j
have mp : IsTranscendenceBasis R x →
  ∀ (κ : Type w) (w : κ → A), AlgebraicIndependent R w → ∀ (j : ι → κ), w ∘ j = x → Surjective j
Tactic: fconstructor
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (goal)
Goal:
ι : Type u
R : Type u_1
A : Type w
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
i : AlgebraicIndependent R x
mp :
  IsTranscendenceBasis R x →
    ∀ (κ : Type w) (w : κ → A), AlgebraicIndependent R w → ∀ (j : ι → κ), w ∘ j = x → Surjective j
⊢ IsTranscendenceBasis R x ↔
    ∀ (κ : Type w) (w : κ → A), AlgebraicIndependent R w → ∀ (j : ι → κ), w ∘ j = x → Surjective j
have mpr : (∀ (κ : Type w) (w : κ → A), AlgebraicIndependent R w → ∀ (j : ι → κ), w ∘ j = x → Surjective j) →
  IsTranscendenceBasis R x
Tactic: fconstructor
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (hypothesis)
Goal:
ι : Type u
R : Type u_1
A : Type w
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
κ : Type w
w : κ → A
i' : AlgebraicIndependent R w
j : ι → κ
i : AlgebraicIndependent R (w ∘ j)
p : IsTranscendenceBasis R (w ∘ j)
⊢ Surjective j
have p₁ : range (w ∘ j) = range w
Tactic: have p := p.2 (range w) i'.coe_range (range_comp_subset_range _ _)
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (hypothesis)
Goal:
ι : Type u
R : Type u_1
A : Type w
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
κ : Type w
w : κ → A
i' : AlgebraicIndependent R w
j : ι → κ
i : AlgebraicIndependent R (w ∘ j)
p_1 : IsTranscendenceBasis R (w ∘ j)
p : range (w ∘ j) = range w
⊢ Surjective j
have p₁ : w '' range j = w '' univ
Tactic: rw [range_comp, ← @image_univ _ _ w] at p
Cleared hypotheses: p
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (goal)
Goal:
ι : Type u
R : Type u_1
A : Type w
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
i : AlgebraicIndependent R x
p : ∀ (κ : Type w) (w : κ → A), AlgebraicIndependent R w → ∀ (j : ι → κ), w ∘ j = x → Surjective j
⊢ IsTranscendenceBasis R x
have right : ∀ (s : Set A), AlgebraicIndepOn R _root_.id s → range x ⊆ s → range x = s
Tactic: use i
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (goal)
Goal:
ι : Type u
R : Type u_1
A : Type w
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
i : AlgebraicIndependent R x
p : ∀ (κ : Type w) (w : κ → A), AlgebraicIndependent R w → ∀ (j : ι → κ), w ∘ j = x → Surjective j
w : Set A
i' : AlgebraicIndepOn R _root_.id w
h : range x ⊆ w
⊢ (Function.comp (β := (↑w : Type w)) (Subtype.val (p := fun (x : A) ↦ x ∈ w)) fun (i : ι) ↦
      Subtype.mk (p := fun (x : A) ↦ x ∈ w) (x i) ⋯) =
    x
have h₁ : ∀ (x_1 : ι),
  Function.comp (β := (↑w : Type w)) (Subtype.val (p := fun (x : A) ↦ x ∈ w))
      (fun (i : ι) ↦ Subtype.mk (p := fun (x : A) ↦ x ∈ w) (x i) sorry) x_1 =
    x x_1
Tactic: ext
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (hypothesis)
Goal:
ι : Type u
R : Type u_1
A : Type w
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
i : AlgebraicIndependent R x
p : ∀ (κ : Type w) (w : κ → A), AlgebraicIndependent R w → ∀ (j : ι → κ), w ∘ j = x → Surjective j
w : Set A
i' : AlgebraicIndepOn R _root_.id w
h : range x ⊆ w
⊢ range x = w
have p₁ : Surjective.{u + 1, w + 1} (β := (↑w : Type w)) fun (i : ι) ↦ Subtype.mk (p := fun (x : A) ↦ x ∈ w) (x i) sorry
Tactic: specialize p w ((↑) : w → A) i' (fun i => ⟨x i, range_subset_iff.mp h i⟩) (by ext; simp)
Cleared hypotheses: p
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (hypothesis)
Goal:
ι : Type u
R : Type u_1
A : Type w
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
i : AlgebraicIndependent R x
w : Set A
i' : AlgebraicIndepOn R _root_.id w
h : range x ⊆ w
p : Surjective.{u + 1, w + 1} (β := (↑w : Type w)) fun (i : ι) ↦ Subtype.mk (p := fun (x : A) ↦ x ∈ w) (x i) ⋯
⊢ range x = w
have q : Eq.{w + 1} (α := Set A)
  ((fun (s : Set { x : A // x ∈ w }) ↦ Subtype.val (p := fun (x : A) ↦ x ∈ w) '' s)
    (range fun (i : ι) ↦ Subtype.mk (p := fun (x : A) ↦ x ∈ w) (x i) sorry))
  ((fun (s : Set { x : A // x ∈ w }) ↦ Subtype.val (p := fun (x : A) ↦ x ∈ w) '' s) univ)
Tactic: have q := congr_arg (fun s => ((↑) : w → A) '' s) p.range_eq
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (hypothesis)
Goal:
ι : Type u
R : Type u_1
A : Type w
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
i : AlgebraicIndependent R x
w : Set A
i' : AlgebraicIndepOn R _root_.id w
h : range x ⊆ w
p : Surjective.{u + 1, w + 1} (β := (↑w : Type w)) fun (i : ι) ↦ Subtype.mk (p := fun (x : A) ↦ x ∈ w) (x i) ⋯
q :
  Eq.{w + 1} (α := Set A)
    (Set.image (α := { x : A // x ∈ w }) (Subtype.val (p := fun (x : A) ↦ x ∈ w))
      (range fun (i : ι) ↦ Subtype.mk (p := fun (x : A) ↦ x ∈ w) (x i) ⋯))
    (Set.image (α := { x : A // x ∈ w }) (Subtype.val (p := fun (x : A) ↦ x ∈ w)) univ)
⊢ range x = w
have q₁ : Eq.{w + 1} (α := Set A) ((fun (x_1 : ι) ↦ (↑(Subtype.mk (p := fun (x : A) ↦ x ∈ w) (x x_1) sorry) : A)) '' univ)
  (Set.image (α := { x : A // x ∈ w }) (Subtype.val (p := fun (x : A) ↦ x ∈ w)) univ)
Tactic: rw [← image_univ, image_image] at q
Cleared hypotheses: q
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (goal)
Goal:
ι : Type u
R : Type u_1
A : Type w
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
hx : IsTranscendenceBasis R x
⊢ Algebra.IsAlgebraic (↥(adjoin R (range x))) A
have isAlgebraic₁ : ∀ (x_1 : A), IsAlgebraic (↥(adjoin R (range x))) x_1
Tactic: constructor
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (goal)
Goal:
ι : Type u
R : Type u_1
A : Type w
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
hx : IsTranscendenceBasis R x
a : A
⊢ IsAlgebraic (↥(adjoin R (range x))) a
have isAlgebraic₁ : ¬AlgebraicIndependent.{u, u_1, w} R (A := A) fun (o : Option ι) ↦ o.elim a x
Tactic: rw [← not_iff_comm.1 (hx.1.option_iff_transcendental _).symm]
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (hypothesis)
Goal:
ι : Type u
R : Type u_1
A : Type w
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
hx : IsTranscendenceBasis R x
a : A
ai : AlgebraicIndependent.{u, u_1, w} R (A := A) fun (o : Option ι) ↦ o.elim a x
⊢ False
have h₁ : range x ⊆ range fun (o : Option ι) ↦ o.elim a x
Tactic: have h₁ : range x ⊆ range fun o : Option ι => o.elim a x := by
    rintro x ⟨y, rfl⟩
    exact ⟨some y, rfl⟩
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (goal)
Goal:
ι : Type u
R : Type u_1
A : Type w
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
hx : IsTranscendenceBasis R x
a : A
ai : AlgebraicIndependent.{u, u_1, w} R (A := A) fun (o : Option ι) ↦ o.elim a x
h₁ : range x ⊆ range fun (o : Option ι) ↦ o.elim a x
h : range x = range fun (o : Option ι) ↦ o.elim a x
⊢ a ∈ range x
have rw : Membership.mem (γ := Set A) (range fun (o : Option ι) ↦ o.elim a x) a
Tactic: rw [h]
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (hypothesis)
Goal:
ι : Type u
R : Type u_1
A : Type w
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
hx : IsTranscendenceBasis R x
a : A
ai : AlgebraicIndependent.{u, u_1, w} R (A := A) fun (o : Option ι) ↦ o.elim a x
h₁ : range x ⊆ range fun (o : Option ι) ↦ o.elim a x
h : range x = range fun (o : Option ι) ↦ o.elim a x
⊢ False
have this : a ∈ range x
Tactic: have : a ∈ range x := by
      rw [h]
      exact ⟨none, rfl⟩
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (goal)
Goal:
ι : Type u
R : Type u_1
A : Type w
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
hx : IsTranscendenceBasis R x
a : A
ai : AlgebraicIndependent.{u, u_1, w} R (A := A) fun (o : Option ι) ↦ o.elim a x
h₁ : range x ⊆ range fun (o : Option ι) ↦ o.elim a x
h : range x = range fun (o : Option ι) ↦ o.elim a x
this : a ∈ range x
⊢ False
have rcases : ∀ (b : ι) (ai : AlgebraicIndependent.{u, u_1, w} R (A := A) fun (o : Option ι) ↦ o.elim (x b) x)
  (h₁ : range x ⊆ range fun (o : Option ι) ↦ o.elim (x b) x) (h : range x = range fun (o : Option ι) ↦ o.elim (x b) x),
  False
Tactic: rcases this with ⟨b, rfl⟩
Cleared hypotheses: a, ai, h₁, h, this
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (hypothesis)
Goal:
ι : Type u
R : Type u_1
A : Type w
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
hx : IsTranscendenceBasis R x
b : ι
ai : AlgebraicIndependent.{u, u_1, w} R (A := A) fun (o : Option ι) ↦ o.elim (x b) x
h₁ : range x ⊆ range fun (o : Option ι) ↦ o.elim (x b) x
h : range x = range fun (o : Option ι) ↦ o.elim (x b) x
⊢ False
have this : some b = none
Tactic: have : some b = none := ai.injective rfl
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (hypothesis)
Goal:
ι : Type u
R : Type u_1
A : Type w
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
hx : IsTranscendenceBasis R x
a : A
ai : AlgebraicIndependent.{u, u_1, w} R (A := A) fun (o : Option ι) ↦ o.elim a x
h₁ : range x ⊆ range fun (o : Option ι) ↦ o.elim a x
⊢ False
have h₂ : range x ≠ range fun (o : Option ι) ↦ o.elim a x
Tactic: have h₂ : range x ≠ range fun o : Option ι => o.elim a x := by
    intro h
    have : a ∈ range x := by
      rw [h]
      exact ⟨none, rfl⟩
    rcases this with ⟨b, rfl⟩
    have : some b = none := ai.injective rfl
    simpa
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (goal)
Goal:
ι : Type u
R : Type u_1
A : Type w
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
ind : AlgebraicIndependent R x
⊢ IsTranscendenceBasis R x ↔ Algebra.IsAlgebraic (↥(adjoin R (range x))) A
have refine : ∀ (alg : Algebra.IsAlgebraic (↥(adjoin R (range x))) A) (s : Set A) (ind_s : AlgebraicIndepOn R _root_.id s)
  (hxs : range x ⊆ s) (hxs' : ¬range x = s), False
Tactic: refine ⟨(·.isAlgebraic), fun alg ↦ ⟨ind, fun s ind_s hxs ↦ of_not_not fun hxs' ↦ ?_⟩⟩
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (hypothesis)
Goal:
ι : Type u
R : Type u_1
A : Type w
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
ind : AlgebraicIndependent R x
alg : Algebra.IsAlgebraic (↥(adjoin R (range x))) A
s : Set A
ind_s : AlgebraicIndepOn R _root_.id s
hxs : range x ⊆ s
hxs' : ¬range x = s
⊢ False
have this : ¬s ⊆ range x
Tactic: have : ¬ s ⊆ range x := (hxs' <| hxs.antisymm ·)
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (goal)
Goal:
ι : Type u
R : Type u_1
A : Type w
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
ind : AlgebraicIndependent R x
alg : Algebra.IsAlgebraic (↥(adjoin R (range x))) A
s : Set A
ind_s : AlgebraicIndepOn R _root_.id s
hxs : range x ⊆ s
hxs' : ¬range x = s
this : ¬s ⊆ range x
⊢ False
have have : ∀ (a : A) (has : a ∈ s) (hax : a ∉ range x), False
Tactic: have ⟨a, has, hax⟩ := not_subset.mp this
-- Lean 4 proof step from Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis (goal)
Goal:
ι : Type u
R : Type u_1
A : Type w
x : ι → A
inst : CommRing R
inst_1 : CommRing A
inst_2 : Algebra R A
inst_3 : Nontrivial R
ind : AlgebraicIndependent R x
alg : Algebra.IsAlgebraic (↥(adjoin R (range x))) A
s : Set A
ind_s : AlgebraicIndepOn R _root_.id s
hxs : range x ⊆ s
hxs' : ¬range x = s
this : ¬s ⊆ range x
a : A
has : a ∈ s
hax : a ∉ range x
⊢ range x = Subtype.val (p := fun (x : A) ↦ x ∈ s) '' range (Set.inclusion (s := range x) (t := s) hxs)
have rw : range x = range (Subtype.val (p := fun (x : A) ↦ x ∈ s) ∘ Set.inclusion (s := range x) (t := s) hxs)
Tactic: rw [← range_comp, val_comp_inclusion, Subtype.range_val]