awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (hypothesis)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ (σ : Lifts F E K), (↑S : Set E) ⊆ (↑σ.carrier : Set E)
⊢ Nonempty (E →ₐ[F] K)
have this : IsExtendible.{u_1, u_2, u_3} (F := F) (E := E) (K := K) ⊥
Tactic: have : (⊥ : Lifts F E K).IsExtendible := fun S ↦ have ⟨σ, hσ⟩ := h S; ⟨σ, bot_le, hσ⟩
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (goal)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ (σ : Lifts F E K), (↑S : Set E) ⊆ (↑σ.carrier : Set E)
this : IsExtendible.{u_1, u_2, u_3} (F := F) (E := E) (K := K) ⊥
⊢ Nonempty (E →ₐ[F] K)
have have : ∀ (ϕ : Lifts F E K)
  (hϕ :
    Maximal (fun (x : Lifts F E K) ↦ Membership.mem (γ := Set (Lifts F E K)) {ϕ : Lifts F E K | ϕ.IsExtendible} x) ϕ),
  Nonempty (E →ₐ[F] K)
Tactic: have ⟨ϕ, hϕ⟩ := zorn_le₀ {ϕ : Lifts F E K | ϕ.IsExtendible}
    fun c hext hc ↦ (isEmpty_or_nonempty c).elim
      (fun _ ↦ ⟨⊥, this, fun ϕ hϕ ↦ isEmptyElim (⟨ϕ, hϕ⟩ : c)⟩)
      fun _ ↦ ⟨_, union_isExtendible c hc hext, le_union c hc⟩
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (goal)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ (σ : Lifts F E K), (↑S : Set E) ⊆ (↑σ.carrier : Set E)
this : IsExtendible.{u_1, u_2, u_3} (F := F) (E := E) (K := K) ⊥
ϕ : Lifts F E K
hϕ : Maximal (fun (x : Lifts F E K) ↦ Membership.mem (γ := Set (Lifts F E K)) {ϕ : Lifts F E K | ϕ.IsExtendible} x) ϕ
⊢ Nonempty (E →ₐ[F] K)
have suffices : ϕ.carrier = ⊤
Tactic: suffices ϕ.carrier = ⊤ from ⟨ϕ.emb.comp <| ((equivOfEq this).trans topEquiv).symm⟩
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (goal)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ (σ : Lifts F E K), (↑S : Set E) ⊆ (↑σ.carrier : Set E)
this : IsExtendible.{u_1, u_2, u_3} (F := F) (E := E) (K := K) ⊥
ϕ : Lifts F E K
hϕ : Maximal (fun (x : Lifts F E K) ↦ Membership.mem (γ := Set (Lifts F E K)) {ϕ : Lifts F E K | ϕ.IsExtendible} x) ϕ
⊢ ϕ.carrier = ⊤
have by_contra! : ∀ (this : ϕ.carrier ≠ ⊤), False
Tactic: by_contra!
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (goal)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ (σ : Lifts F E K), (↑S : Set E) ⊆ (↑σ.carrier : Set E)
this_1 : IsExtendible.{u_1, u_2, u_3} (F := F) (E := E) (K := K) ⊥
ϕ : Lifts F E K
hϕ : Maximal (fun (x : Lifts F E K) ↦ Membership.mem (γ := Set (Lifts F E K)) {ϕ : Lifts F E K | ϕ.IsExtendible} x) ϕ
this : ϕ.carrier ≠ ⊤
⊢ False
have obtain : ∀ (α : E) (hα : α ∉ ϕ.carrier), False
Tactic: obtain ⟨α, -, hα⟩ := SetLike.exists_of_lt this.lt_top
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (hypothesis)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ (σ : Lifts F E K), (↑S : Set E) ⊆ (↑σ.carrier : Set E)
this_1 : IsExtendible.{u_1, u_2, u_3} (F := F) (E := E) (K := K) ⊥
ϕ : Lifts F E K
hϕ : Maximal (fun (x : Lifts F E K) ↦ Membership.mem (γ := Set (Lifts F E K)) {ϕ : Lifts F E K | ϕ.IsExtendible} x) ϕ
this : ϕ.carrier ≠ ⊤
α : E
hα : α ∉ ϕ.carrier
x : Algebra.{u_2, u_3} (↥ϕ.carrier) K := ϕ.emb.toAlgebra
Λ : Type (max u_2 u_3) := AlgHom.{u_2, u_2, u_3} (↥ϕ.carrier) (↥(↥ϕ.carrier)⟮α⟯) K
⊢ False
have this₁ : FiniteDimensional.{u_2, u_2} ↥ϕ.carrier ↥(↥ϕ.carrier)⟮α⟯
Tactic: have := finiteDimensional_adjoin (S := {α}) fun _ _ ↦ ((alg.tower_top ϕ.carrier).isIntegral).1 _
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (hypothesis)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ (σ : Lifts F E K), (↑S : Set E) ⊆ (↑σ.carrier : Set E)
this_1 : IsExtendible.{u_1, u_2, u_3} (F := F) (E := E) (K := K) ⊥
ϕ : Lifts F E K
hϕ : Maximal (fun (x : Lifts F E K) ↦ Membership.mem (γ := Set (Lifts F E K)) {ϕ : Lifts F E K | ϕ.IsExtendible} x) ϕ
this_2 : ϕ.carrier ≠ ⊤
α : E
hα : α ∉ ϕ.carrier
x : Algebra.{u_2, u_3} (↥ϕ.carrier) K := ϕ.emb.toAlgebra
Λ : Type (max u_2 u_3) := AlgHom.{u_2, u_2, u_3} (↥ϕ.carrier) (↥(↥ϕ.carrier)⟮α⟯) K
this : FiniteDimensional.{u_2, u_2} ↥ϕ.carrier ↥(↥ϕ.carrier)⟮α⟯
L : (σ : Λ) → Lifts F E K :=
  fun (σ : Λ) ↦ { carrier := restrictScalars F (L' := ↥ϕ.carrier) (↥ϕ.carrier)⟮α⟯, emb := AlgHom.restrictScalars F σ }
⊢ False
have hL : ∀ (σ : Λ), ϕ < L σ
Tactic: have hL (σ : Λ) : ϕ < L σ := lt_iff.mpr
    ⟨by simpa only [L, restrictScalars_adjoin_eq_sup, left_lt_sup, adjoin_simple_le_iff],
      AlgHom.coe_ringHom_injective σ.comp_algebraMap⟩
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (hypothesis)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ (σ : Lifts F E K), (↑S : Set E) ⊆ (↑σ.carrier : Set E)
this_1 : IsExtendible.{u_1, u_2, u_3} (F := F) (E := E) (K := K) ⊥
ϕ : Lifts F E K
hϕ : Maximal (fun (x : Lifts F E K) ↦ Membership.mem (γ := Set (Lifts F E K)) {ϕ : Lifts F E K | ϕ.IsExtendible} x) ϕ
this_2 : ϕ.carrier ≠ ⊤
α : E
hα : α ∉ ϕ.carrier
x : Algebra.{u_2, u_3} (↥ϕ.carrier) K := ϕ.emb.toAlgebra
Λ : Type (max u_2 u_3) := AlgHom.{u_2, u_2, u_3} (↥ϕ.carrier) (↥(↥ϕ.carrier)⟮α⟯) K
this : FiniteDimensional.{u_2, u_2} ↥ϕ.carrier ↥(↥ϕ.carrier)⟮α⟯
L : (σ : Λ) → Lifts F E K :=
  fun (σ : Λ) ↦ { carrier := restrictScalars F (L' := ↥ϕ.carrier) (↥ϕ.carrier)⟮α⟯, emb := AlgHom.restrictScalars F σ }
hL : ∀ (σ : Λ), ϕ < L σ
⊢ False
have ϕ_ext : ϕ.IsExtendible
Tactic: have ⟨(ϕ_ext : ϕ.IsExtendible), ϕ_max⟩ := maximal_iff_forall_gt.mp hϕ
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (hypothesis)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ (σ : Lifts F E K), (↑S : Set E) ⊆ (↑σ.carrier : Set E)
this_1 : IsExtendible.{u_1, u_2, u_3} (F := F) (E := E) (K := K) ⊥
ϕ : Lifts F E K
hϕ : Maximal (fun (x : Lifts F E K) ↦ Membership.mem (γ := Set (Lifts F E K)) {ϕ : Lifts F E K | ϕ.IsExtendible} x) ϕ
this_2 : ϕ.carrier ≠ ⊤
α : E
hα : α ∉ ϕ.carrier
x : Algebra.{u_2, u_3} (↥ϕ.carrier) K := ϕ.emb.toAlgebra
Λ : Type (max u_2 u_3) := AlgHom.{u_2, u_2, u_3} (↥ϕ.carrier) (↥(↥ϕ.carrier)⟮α⟯) K
this : FiniteDimensional.{u_2, u_2} ↥ϕ.carrier ↥(↥ϕ.carrier)⟮α⟯
L : (σ : Λ) → Lifts F E K :=
  fun (σ : Λ) ↦ { carrier := restrictScalars F (L' := ↥ϕ.carrier) (↥ϕ.carrier)⟮α⟯, emb := AlgHom.restrictScalars F σ }
hL : ∀ (σ : Λ), ϕ < L σ
ϕ_ext : ϕ.IsExtendible
⊢ False
have ϕ_max : ∀ ⦃y : Lifts F E K⦄, ϕ < y → y ∉ {ϕ : Lifts F E K | ϕ.IsExtendible}
Tactic: have ⟨(ϕ_ext : ϕ.IsExtendible), ϕ_max⟩ := maximal_iff_forall_gt.mp hϕ
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (hypothesis)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ (σ : Lifts F E K), (↑S : Set E) ⊆ (↑σ.carrier : Set E)
this_1 : IsExtendible.{u_1, u_2, u_3} (F := F) (E := E) (K := K) ⊥
ϕ : Lifts F E K
hϕ : Maximal (fun (x : Lifts F E K) ↦ Membership.mem (γ := Set (Lifts F E K)) {ϕ : Lifts F E K | ϕ.IsExtendible} x) ϕ
this_2 : ϕ.carrier ≠ ⊤
α : E
hα : α ∉ ϕ.carrier
x : Algebra.{u_2, u_3} (↥ϕ.carrier) K := ϕ.emb.toAlgebra
Λ : Type (max u_2 u_3) := AlgHom.{u_2, u_2, u_3} (↥ϕ.carrier) (↥(↥ϕ.carrier)⟮α⟯) K
this : FiniteDimensional.{u_2, u_2} ↥ϕ.carrier ↥(↥ϕ.carrier)⟮α⟯
L : (σ : Λ) → Lifts F E K :=
  fun (σ : Λ) ↦ { carrier := restrictScalars F (L' := ↥ϕ.carrier) (↥ϕ.carrier)⟮α⟯, emb := AlgHom.restrictScalars F σ }
hL : ∀ (σ : Λ), ϕ < L σ
ϕ_ext : ϕ.IsExtendible
ϕ_max : ∀ ⦃y : Lifts F E K⦄, ϕ < y → y ∉ {ϕ : Lifts F E K | ϕ.IsExtendible}
⊢ False
have ϕ_max₁ : ∀ ⦃y : Lifts F E K⦄, ϕ < y → ¬∀ (S : Finset E), ∃ τ ≥ y, (↑S : Set E) ⊆ (↑τ.carrier : Set E)
Tactic: simp_rw [Set.mem_setOf, IsExtendible] at ϕ_max
Cleared hypotheses: ϕ_max
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (hypothesis)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ (σ : Lifts F E K), (↑S : Set E) ⊆ (↑σ.carrier : Set E)
this_1 : IsExtendible.{u_1, u_2, u_3} (F := F) (E := E) (K := K) ⊥
ϕ : Lifts F E K
hϕ : Maximal (fun (x : Lifts F E K) ↦ Membership.mem (γ := Set (Lifts F E K)) {ϕ : Lifts F E K | ϕ.IsExtendible} x) ϕ
this_2 : ϕ.carrier ≠ ⊤
α : E
hα : α ∉ ϕ.carrier
x : Algebra.{u_2, u_3} (↥ϕ.carrier) K := ϕ.emb.toAlgebra
Λ : Type (max u_2 u_3) := AlgHom.{u_2, u_2, u_3} (↥ϕ.carrier) (↥(↥ϕ.carrier)⟮α⟯) K
this : FiniteDimensional.{u_2, u_2} ↥ϕ.carrier ↥(↥ϕ.carrier)⟮α⟯
L : (σ : Λ) → Lifts F E K :=
  fun (σ : Λ) ↦ { carrier := restrictScalars F (L' := ↥ϕ.carrier) (↥ϕ.carrier)⟮α⟯, emb := AlgHom.restrictScalars F σ }
hL : ∀ (σ : Λ), ϕ < L σ
ϕ_ext : ϕ.IsExtendible
ϕ_max : ∀ ⦃y : Lifts F E K⦄, ϕ < y → ¬∀ (S : Finset E), ∃ τ ≥ y, (↑S : Set E) ⊆ (↑τ.carrier : Set E)
⊢ False
have ϕ_max₁ : ∀ ⦃y : Lifts F E K⦄, ϕ < y → ∃ (S : Finset E), ∀ (τ : Lifts F E K) (_ : τ ≥ y), ¬(↑S : Set E) ⊆ (↑τ.carrier : Set E)
Tactic: push Not at ϕ_max
Cleared hypotheses: ϕ_max
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (goal)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ (σ : Lifts F E K), (↑S : Set E) ⊆ (↑σ.carrier : Set E)
this_1 : IsExtendible.{u_1, u_2, u_3} (F := F) (E := E) (K := K) ⊥
ϕ : Lifts F E K
hϕ : Maximal (fun (x : Lifts F E K) ↦ Membership.mem (γ := Set (Lifts F E K)) {ϕ : Lifts F E K | ϕ.IsExtendible} x) ϕ
this_2 : ϕ.carrier ≠ ⊤
α : E
hα : α ∉ ϕ.carrier
x : Algebra.{u_2, u_3} (↥ϕ.carrier) K := ϕ.emb.toAlgebra
Λ : Type (max u_2 u_3) := AlgHom.{u_2, u_2, u_3} (↥ϕ.carrier) (↥(↥ϕ.carrier)⟮α⟯) K
this : FiniteDimensional.{u_2, u_2} ↥ϕ.carrier ↥(↥ϕ.carrier)⟮α⟯
L : (σ : Λ) → Lifts F E K :=
  fun (σ : Λ) ↦ { carrier := restrictScalars F (L' := ↥ϕ.carrier) (↥ϕ.carrier)⟮α⟯, emb := AlgHom.restrictScalars F σ }
hL : ∀ (σ : Λ), ϕ < L σ
ϕ_ext : ϕ.IsExtendible
ϕ_max :
  ∀ ⦃y : Lifts F E K⦄, ϕ < y → ∃ (S : Finset E), ∀ (τ : Lifts F E K) (_ : τ ≥ y), ¬(↑S : Set E) ⊆ (↑τ.carrier : Set E)
⊢ False
have choose : ∀ (S : (σ : Λ) → Finset E) (hS : ∀ (σ : Λ) (τ : Lifts F E K) (_ : τ ≥ L σ), ¬(↑(S σ) : Set E) ⊆ (↑τ.carrier : Set E)),
  False
Tactic: choose S hS using fun σ : Λ ↦ ϕ_max (hL σ)
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (goal)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ (σ : Lifts F E K), (↑S : Set E) ⊆ (↑σ.carrier : Set E)
this_1 : IsExtendible.{u_1, u_2, u_3} (F := F) (E := E) (K := K) ⊥
ϕ : Lifts F E K
hϕ : Maximal (fun (x : Lifts F E K) ↦ Membership.mem (γ := Set (Lifts F E K)) {ϕ : Lifts F E K | ϕ.IsExtendible} x) ϕ
this_2 : ϕ.carrier ≠ ⊤
α : E
hα : α ∉ ϕ.carrier
x : Algebra.{u_2, u_3} (↥ϕ.carrier) K := ϕ.emb.toAlgebra
Λ : Type (max u_2 u_3) := AlgHom.{u_2, u_2, u_3} (↥ϕ.carrier) (↥(↥ϕ.carrier)⟮α⟯) K
this : FiniteDimensional.{u_2, u_2} ↥ϕ.carrier ↥(↥ϕ.carrier)⟮α⟯
L : (σ : Λ) → Lifts F E K :=
  fun (σ : Λ) ↦ { carrier := restrictScalars F (L' := ↥ϕ.carrier) (↥ϕ.carrier)⟮α⟯, emb := AlgHom.restrictScalars F σ }
hL : ∀ (σ : Λ), ϕ < L σ
ϕ_ext : ϕ.IsExtendible
ϕ_max :
  ∀ ⦃y : Lifts F E K⦄, ϕ < y → ∃ (S : Finset E), ∀ (τ : Lifts F E K) (_ : τ ≥ y), ¬(↑S : Set E) ⊆ (↑τ.carrier : Set E)
S : (σ : Λ) → Finset E
hS : ∀ (σ : Λ) (τ : Lifts F E K) (_ : τ ≥ L σ), ¬(↑(S σ) : Set E) ⊆ (↑τ.carrier : Set E)
⊢ False
have have : ∀ (θ : Lifts F E K) (hθϕ : θ ≥ ϕ) (hθ : (↑({α} ∪ Finset.biUnion Finset.univ S) : Set E) ⊆ (↑θ.carrier : Set E)), False
Tactic: have ⟨θ, hθϕ, hθ⟩ := ϕ_ext ({α} ∪ Finset.univ.biUnion S)
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (hypothesis)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ (σ : Lifts F E K), (↑S : Set E) ⊆ (↑σ.carrier : Set E)
this_1 : IsExtendible.{u_1, u_2, u_3} (F := F) (E := E) (K := K) ⊥
ϕ : Lifts F E K
hϕ : Maximal (fun (x : Lifts F E K) ↦ Membership.mem (γ := Set (Lifts F E K)) {ϕ : Lifts F E K | ϕ.IsExtendible} x) ϕ
this_2 : ϕ.carrier ≠ ⊤
α : E
hα : α ∉ ϕ.carrier
x : Algebra.{u_2, u_3} (↥ϕ.carrier) K := ϕ.emb.toAlgebra
Λ : Type (max u_2 u_3) := AlgHom.{u_2, u_2, u_3} (↥ϕ.carrier) (↥(↥ϕ.carrier)⟮α⟯) K
this : FiniteDimensional.{u_2, u_2} ↥ϕ.carrier ↥(↥ϕ.carrier)⟮α⟯
L : (σ : Λ) → Lifts F E K :=
  fun (σ : Λ) ↦ { carrier := restrictScalars F (L' := ↥ϕ.carrier) (↥ϕ.carrier)⟮α⟯, emb := AlgHom.restrictScalars F σ }
hL : ∀ (σ : Λ), ϕ < L σ
ϕ_ext : ϕ.IsExtendible
ϕ_max :
  ∀ ⦃y : Lifts F E K⦄, ϕ < y → ∃ (S : Finset E), ∀ (τ : Lifts F E K) (_ : τ ≥ y), ¬(↑S : Set E) ⊆ (↑τ.carrier : Set E)
S : (σ : Λ) → Finset E
hS : ∀ (σ : Λ) (τ : Lifts F E K) (_ : τ ≥ L σ), ¬(↑(S σ) : Set E) ⊆ (↑τ.carrier : Set E)
θ : Lifts F E K
hθϕ : θ ≥ ϕ
hθ : (↑({α} ∪ Finset.biUnion Finset.univ S) : Set E) ⊆ (↑θ.carrier : Set E)
⊢ False
have hθ₁ : α ∈ (↑θ.carrier : Set E) ∧ ∀ (i : Λ), (↑(S i) : Set E) ⊆ (↑θ.carrier : Set E)
Tactic: simp_rw [Finset.coe_union, Set.union_subset_iff, Finset.coe_singleton, Set.singleton_subset_iff,
    Finset.coe_biUnion, Finset.coe_univ, Set.mem_univ, Set.iUnion_true, Set.iUnion_subset_iff] at hθ
Cleared hypotheses: hθ
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (goal)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ (σ : Lifts F E K), (↑S : Set E) ⊆ (↑σ.carrier : Set E)
this_1 : IsExtendible.{u_1, u_2, u_3} (F := F) (E := E) (K := K) ⊥
ϕ : Lifts F E K
hϕ : Maximal (fun (x : Lifts F E K) ↦ Membership.mem (γ := Set (Lifts F E K)) {ϕ : Lifts F E K | ϕ.IsExtendible} x) ϕ
this_2 : ϕ.carrier ≠ ⊤
α : E
hα : α ∉ ϕ.carrier
x : Algebra.{u_2, u_3} (↥ϕ.carrier) K := ϕ.emb.toAlgebra
Λ : Type (max u_2 u_3) := AlgHom.{u_2, u_2, u_3} (↥ϕ.carrier) (↥(↥ϕ.carrier)⟮α⟯) K
this : FiniteDimensional.{u_2, u_2} ↥ϕ.carrier ↥(↥ϕ.carrier)⟮α⟯
L : (σ : Λ) → Lifts F E K :=
  fun (σ : Λ) ↦ { carrier := restrictScalars F (L' := ↥ϕ.carrier) (↥ϕ.carrier)⟮α⟯, emb := AlgHom.restrictScalars F σ }
hL : ∀ (σ : Λ), ϕ < L σ
ϕ_ext : ϕ.IsExtendible
ϕ_max :
  ∀ ⦃y : Lifts F E K⦄, ϕ < y → ∃ (S : Finset E), ∀ (τ : Lifts F E K) (_ : τ ≥ y), ¬(↑S : Set E) ⊆ (↑τ.carrier : Set E)
S : (σ : Λ) → Finset E
hS : ∀ (σ : Λ) (τ : Lifts F E K) (_ : τ ≥ L σ), ¬(↑(S σ) : Set E) ⊆ (↑τ.carrier : Set E)
θ : Lifts F E K
hθϕ : θ ≥ ϕ
hθ : α ∈ (↑θ.carrier : Set E) ∧ ∀ (i : Λ), (↑(S i) : Set E) ⊆ (↑θ.carrier : Set E)
⊢ restrictScalars F (L' := ↥ϕ.carrier) (↥ϕ.carrier)⟮α⟯ ≤ θ.carrier
have rw : ϕ.carrier ≤ θ.carrier ∧ α ∈ θ.carrier
Tactic: rw [restrictScalars_adjoin_eq_sup, sup_le_iff, adjoin_simple_le_iff]
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (hypothesis)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ (σ : Lifts F E K), (↑S : Set E) ⊆ (↑σ.carrier : Set E)
this_1 : IsExtendible.{u_1, u_2, u_3} (F := F) (E := E) (K := K) ⊥
ϕ : Lifts F E K
hϕ : Maximal (fun (x : Lifts F E K) ↦ Membership.mem (γ := Set (Lifts F E K)) {ϕ : Lifts F E K | ϕ.IsExtendible} x) ϕ
this_2 : ϕ.carrier ≠ ⊤
α : E
hα : α ∉ ϕ.carrier
x : Algebra.{u_2, u_3} (↥ϕ.carrier) K := ϕ.emb.toAlgebra
Λ : Type (max u_2 u_3) := AlgHom.{u_2, u_2, u_3} (↥ϕ.carrier) (↥(↥ϕ.carrier)⟮α⟯) K
this : FiniteDimensional.{u_2, u_2} ↥ϕ.carrier ↥(↥ϕ.carrier)⟮α⟯
L : (σ : Λ) → Lifts F E K :=
  fun (σ : Λ) ↦ { carrier := restrictScalars F (L' := ↥ϕ.carrier) (↥ϕ.carrier)⟮α⟯, emb := AlgHom.restrictScalars F σ }
hL : ∀ (σ : Λ), ϕ < L σ
ϕ_ext : ϕ.IsExtendible
ϕ_max :
  ∀ ⦃y : Lifts F E K⦄, ϕ < y → ∃ (S : Finset E), ∀ (τ : Lifts F E K) (_ : τ ≥ y), ¬(↑S : Set E) ⊆ (↑τ.carrier : Set E)
S : (σ : Λ) → Finset E
hS : ∀ (σ : Λ) (τ : Lifts F E K) (_ : τ ≥ L σ), ¬(↑(S σ) : Set E) ⊆ (↑τ.carrier : Set E)
θ : Lifts F E K
hθϕ : θ ≥ ϕ
hθ : α ∈ (↑θ.carrier : Set E) ∧ ∀ (i : Λ), (↑(S i) : Set E) ⊆ (↑θ.carrier : Set E)
⊢ False
have this₁ : restrictScalars F (L' := ↥ϕ.carrier) (↥ϕ.carrier)⟮α⟯ ≤ θ.carrier
Tactic: have : ϕ.carrier⟮α⟯.restrictScalars F ≤ θ.carrier := by
    rw [restrictScalars_adjoin_eq_sup, sup_le_iff, adjoin_simple_le_iff]; exact ⟨hθϕ.1, hθ.1⟩
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (goal)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ (σ : Lifts F E K), (↑S : Set E) ⊆ (↑σ.carrier : Set E)
this_1 : IsExtendible.{u_1, u_2, u_3} (F := F) (E := E) (K := K) ⊥
ϕ : Lifts F E K
hϕ : Maximal (fun (x : Lifts F E K) ↦ Membership.mem (γ := Set (Lifts F E K)) {ϕ : Lifts F E K | ϕ.IsExtendible} x) ϕ
this_2 : ϕ.carrier ≠ ⊤
α : E
hα : α ∉ ϕ.carrier
x : Algebra.{u_2, u_3} (↥ϕ.carrier) K := ϕ.emb.toAlgebra
Λ : Type (max u_2 u_3) := AlgHom.{u_2, u_2, u_3} (↥ϕ.carrier) (↥(↥ϕ.carrier)⟮α⟯) K
this : FiniteDimensional.{u_2, u_2} ↥ϕ.carrier ↥(↥ϕ.carrier)⟮α⟯
L : (σ : Λ) → Lifts F E K :=
  fun (σ : Λ) ↦ { carrier := restrictScalars F (L' := ↥ϕ.carrier) (↥ϕ.carrier)⟮α⟯, emb := AlgHom.restrictScalars F σ }
hL : ∀ (σ : Λ), ϕ < L σ
ϕ_ext : ϕ.IsExtendible
ϕ_max :
  ∀ ⦃y : Lifts F E K⦄, ϕ < y → ∃ (S : Finset E), ∀ (τ : Lifts F E K) (_ : τ ≥ y), ¬(↑S : Set E) ⊆ (↑τ.carrier : Set E)
S : (σ : Λ) → Finset E
hS : ∀ (σ : Λ) (τ : Lifts F E K) (_ : τ ≥ L σ), ¬(↑(S σ) : Set E) ⊆ (↑τ.carrier : Set E)
⊢ False
have have : ∀ (θ : Lifts F E K) (hθϕ : θ ≥ ϕ) (hθ : (↑({α} ∪ Finset.biUnion Finset.univ S) : Set E) ⊆ (↑θ.carrier : Set E)), False
Tactic: classical
  have ⟨θ, hθϕ, hθ⟩ := ϕ_ext ({α} ∪ Finset.univ.biUnion S)
  simp_rw [Finset.coe_union, Set.union_subset_iff, Finset.coe_singleton, Set.singleton_subset_iff,
    Finset.coe_biUnion, Finset.coe_univ, Set.mem_univ, Set.iUnion_true, Set.iUnion_subset_iff] at hθ
  have : ϕ.carrier⟮α⟯.restrictScalars F ≤ θ.carrier := by
    rw [restrictScalars_adjoin_eq_sup, sup_le_iff, adjoin_simple_le_iff]; exact ⟨hθϕ.1, hθ.1⟩
  exact hS ⟨(θ.emb.comp <| inclusion this).toRingHom, hθϕ.2⟩ θ ⟨this, fun _ ↦ rfl⟩ (hθ.2 _)
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (goal)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ (σ : Lifts F E K), (↑S : Set E) ⊆ (↑σ.carrier : Set E)
this_1 : IsExtendible.{u_1, u_2, u_3} (F := F) (E := E) (K := K) ⊥
ϕ : Lifts F E K
hϕ : Maximal (fun (x : Lifts F E K) ↦ Membership.mem (γ := Set (Lifts F E K)) {ϕ : Lifts F E K | ϕ.IsExtendible} x) ϕ
this_2 : ϕ.carrier ≠ ⊤
α : E
hα : α ∉ ϕ.carrier
x : Algebra.{u_2, u_3} (↥ϕ.carrier) K := ϕ.emb.toAlgebra
Λ : Type (max u_2 u_3) := AlgHom.{u_2, u_2, u_3} (↥ϕ.carrier) (↥(↥ϕ.carrier)⟮α⟯) K
this : FiniteDimensional.{u_2, u_2} ↥ϕ.carrier ↥(↥ϕ.carrier)⟮α⟯
L : (σ : Λ) → Lifts F E K :=
  fun (σ : Λ) ↦ { carrier := restrictScalars F (L' := ↥ϕ.carrier) (↥ϕ.carrier)⟮α⟯, emb := AlgHom.restrictScalars F σ }
hL : ∀ (σ : Λ), ϕ < L σ
ϕ_ext : ϕ.IsExtendible
ϕ_max :
  ∀ ⦃y : Lifts F E K⦄, ϕ < y → ∃ (S : Finset E), ∀ (τ : Lifts F E K) (_ : τ ≥ y), ¬(↑S : Set E) ⊆ (↑τ.carrier : Set E)
S : (σ : Λ) → Finset E
hS : ∀ (σ : Λ) (τ : Lifts F E K) (_ : τ ≥ L σ), ¬(↑(S σ) : Set E) ⊆ (↑τ.carrier : Set E)
have have₁ have₂ :
  ∀ (θ : Lifts F E K) (hθϕ : θ ≥ ϕ) (hθ : (↑({α} ∪ Finset.biUnion Finset.univ S) : Set E) ⊆ (↑θ.carrier : Set E)), False
⊢ False
have have₃ : ∀ (θ : Lifts F E K) (hθϕ : θ ≥ ϕ) (hθ : α ∈ (↑θ.carrier : Set E) ∧ ∀ (i : Λ), (↑(S i) : Set E) ⊆ (↑θ.carrier : Set E)),
  False
Tactic: classical
  have ⟨θ, hθϕ, hθ⟩ := ϕ_ext ({α} ∪ Finset.univ.biUnion S)
  simp_rw [Finset.coe_union, Set.union_subset_iff, Finset.coe_singleton, Set.singleton_subset_iff,
    Finset.coe_biUnion, Finset.coe_univ, Set.mem_univ, Set.iUnion_true, Set.iUnion_subset_iff] at hθ
  have : ϕ.carrier⟮α⟯.restrictScalars F ≤ θ.carrier := by
    rw [restrictScalars_adjoin_eq_sup, sup_le_iff, adjoin_simple_le_iff]; exact ⟨hθϕ.1, hθ.1⟩
  exact hS ⟨(θ.emb.comp <| inclusion this).toRingHom, hθϕ.2⟩ θ ⟨this, fun _ ↦ rfl⟩ (hθ.2 _)
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (goal)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
alg : Algebra.IsAlgebraic F E
h : ∀ (S : Finset E), ∃ (σ : Lifts F E K), (↑S : Set E) ⊆ (↑σ.carrier : Set E)
this_1 : IsExtendible.{u_1, u_2, u_3} (F := F) (E := E) (K := K) ⊥
ϕ : Lifts F E K
hϕ : Maximal (fun (x : Lifts F E K) ↦ Membership.mem (γ := Set (Lifts F E K)) {ϕ : Lifts F E K | ϕ.IsExtendible} x) ϕ
this_2 : ϕ.carrier ≠ ⊤
α : E
hα : α ∉ ϕ.carrier
x : Algebra.{u_2, u_3} (↥ϕ.carrier) K := ϕ.emb.toAlgebra
Λ : Type (max u_2 u_3) := AlgHom.{u_2, u_2, u_3} (↥ϕ.carrier) (↥(↥ϕ.carrier)⟮α⟯) K
this : FiniteDimensional.{u_2, u_2} ↥ϕ.carrier ↥(↥ϕ.carrier)⟮α⟯
L : (σ : Λ) → Lifts F E K :=
  fun (σ : Λ) ↦ { carrier := restrictScalars F (L' := ↥ϕ.carrier) (↥ϕ.carrier)⟮α⟯, emb := AlgHom.restrictScalars F σ }
hL : ∀ (σ : Λ), ϕ < L σ
ϕ_ext : ϕ.IsExtendible
ϕ_max :
  ∀ ⦃y : Lifts F E K⦄, ϕ < y → ∃ (S : Finset E), ∀ (τ : Lifts F E K) (_ : τ ≥ y), ¬(↑S : Set E) ⊆ (↑τ.carrier : Set E)
S : (σ : Λ) → Finset E
hS : ∀ (σ : Λ) (τ : Lifts F E K) (_ : τ ≥ L σ), ¬(↑(S σ) : Set E) ⊆ (↑τ.carrier : Set E)
have have₁ have₂ :
  ∀ (θ : Lifts F E K) (hθϕ : θ ≥ ϕ) (hθ : (↑({α} ∪ Finset.biUnion Finset.univ S) : Set E) ⊆ (↑θ.carrier : Set E)), False
have₃ :
  ∀ (θ : Lifts F E K) (hθϕ : θ ≥ ϕ)
    (hθ : α ∈ (↑θ.carrier : Set E) ∧ ∀ (i : Λ), (↑(S i) : Set E) ⊆ (↑θ.carrier : Set E)), False
⊢ False
have have₄ : ∀ (θ : Lifts F E K) (hθϕ : θ ≥ ϕ) (hθ : α ∈ (↑θ.carrier : Set E) ∧ ∀ (i : Λ), (↑(S i) : Set E) ⊆ (↑θ.carrier : Set E))
  (this : restrictScalars F (L' := ↥ϕ.carrier) (↥ϕ.carrier)⟮α⟯ ≤ θ.carrier), False
Tactic: classical
  have ⟨θ, hθϕ, hθ⟩ := ϕ_ext ({α} ∪ Finset.univ.biUnion S)
  simp_rw [Finset.coe_union, Set.union_subset_iff, Finset.coe_singleton, Set.singleton_subset_iff,
    Finset.coe_biUnion, Finset.coe_univ, Set.mem_univ, Set.iUnion_true, Set.iUnion_subset_iff] at hθ
  have : ϕ.carrier⟮α⟯.restrictScalars F ≤ θ.carrier := by
    rw [restrictScalars_adjoin_eq_sup, sup_le_iff, adjoin_simple_le_iff]; exact ⟨hθϕ.1, hθ.1⟩
  exact hS ⟨(θ.emb.comp <| inclusion this).toRingHom, hθϕ.2⟩ θ ⟨this, fun _ ↦ rfl⟩ (hθ.2 _)
-- Lean 4 proof step from Mathlib.FieldTheory.Extension (goal)
Goal:
F : Type u_1
E : Type u_2
K : Type u_3
inst : Field F
inst_1 : Field E
inst_2 : Field K
inst_3 : Algebra F E
inst_4 : Algebra F K
x : Lifts F E K
s : E
h1 : IsIntegral (↥x.carrier) s
h2 : (Polynomial.map x.emb.toRingHom (minpoly (↥x.carrier) s)).Splits
I2 : degree (R := ↥x.carrier) (minpoly (↥x.carrier) s) ≠ 0
this_1 : Algebra.{u_2, u_3} (↥x.carrier) K := x.emb.toAlgebra
carrier : IntermediateField F E := restrictScalars F (L' := ↥x.carrier) (↥x.carrier)⟮s⟯
this : Algebra ↥x.carrier ↥carrier := Subalgebra.algebra (toSubalgebra (↥x.carrier)⟮s⟯)
⊢ rootOfSplits h2 ⋯ ∈ aroots (T := ↥x.carrier) (minpoly (↥x.carrier) s) K
have rw : Eq.{u_3 + 1} (α := K) ((aeval (rootOfSplits h2 sorry) : (a : (↥x.carrier)[X]) → K) (minpoly (↥x.carrier) s) : K) (0 : K)
Tactic: rw [mem_aroots, and_iff_right (minpoly.ne_zero h1)]