awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (hypothesis)
Goal:
k : Type u
inst : Field k
K_1 : Type u
inst_1 : Field K_1
L : Type v
M : Type w
inst_2 : Field L
inst_3 : Algebra K_1 L
inst_4 : Field M
inst_5 : Algebra K_1 M
inst_6 : IsAlgClosed M
inst_7 : Algebra.IsAlgebraic K_1 L
R : Type u
inst_8 : CommRing R
inst_9 : IsDomain R
S : Type v
inst_10 : CommRing S
inst_11 : IsDomain S
inst_12 : Algebra R S
inst_13 : Algebra R M
inst_14 : IsTorsionFree R S
inst_15 : IsTorsionFree R M
inst_16 : Algebra.IsAlgebraic R S
K : Type u_1
inst_17 : Field K
inst_18 : IsAlgClosed K
hfin : Fintype K
n : ℕ := Fintype.card K
f : K[X] := HPow.hPow (α := K[X]) X (n + 1) - 1
hfsep : f.Separable
⊢ (Fintype.card K).succ ≤ Fintype.card K
have hroot : n.succ = Fintype.card.{u_1} (↑(f.rootSet K) : Type u_1)
Tactic: have hroot : n.succ = Fintype.card (f.rootSet K) := by
    rw [card_rootSet_eq_natDegree hfsep (IsAlgClosed.splits_domain _)]
    unfold f
    rw [← C_1, natDegree_X_pow_sub_C]
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (goal)
Goal:
k : Type u
inst : Field k
K_1 : Type u
inst_1 : Field K_1
L : Type v
M : Type w
inst_2 : Field L
inst_3 : Algebra K_1 L
inst_4 : Field M
inst_5 : Algebra K_1 M
inst_6 : IsAlgClosed M
inst_7 : Algebra.IsAlgebraic K_1 L
R : Type u
inst_8 : CommRing R
inst_9 : IsDomain R
S : Type v
inst_10 : CommRing S
inst_11 : IsDomain S
inst_12 : Algebra R S
inst_13 : Algebra R M
inst_14 : IsTorsionFree R S
inst_15 : IsTorsionFree R M
inst_16 : Algebra.IsAlgebraic R S
K : Type u_1
inst_17 : Field K
inst_18 : IsAlgClosed K
hfin : Fintype K
n : ℕ := Fintype.card K
f : K[X] := HPow.hPow (α := K[X]) X (n + 1) - 1
hfsep : f.Separable
hroot : n.succ = Fintype.card.{u_1} (↑(f.rootSet K) : Type u_1)
⊢ (Fintype.card K).succ ≤ Fintype.card K
have h : Fintype.card.{u_1} (↑(f.rootSet K) : Type u_1) ≤ Fintype.card K
Tactic: rw [hroot]
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (hypothesis)
Goal:
k : Type u
inst : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst_1 : Field K
inst_2 : Field J
inst_3 : CommRing R
inst_4 : CommRing S
inst_5 : Field L
inst_6 : Field M
inst_7 : Algebra R M
inst_8 : IsTorsionFree R M
inst_9 : IsAlgClosure R M
inst_10 : Algebra K M
inst_11 : IsAlgClosure K M
inst_12 : Algebra S L
inst_13 : IsTorsionFree S L
inst_14 : IsAlgClosure S L
inst_15 : Algebra R S
inst_16 : Algebra R L
inst_17 : IsScalarTower R S L
inst_18 : Algebra K J
inst_19 : Algebra J L
inst_20 : IsAlgClosure J L
inst_21 : Algebra K L
inst_22 : IsScalarTower K J L
inst_23 : IsDomain R
inst_24 : IsDomain S
inst_25 : IsTorsionFree R S
inst_26 : Algebra.IsAlgebraic R L
⊢ L ≃ₐ[R] M
have this : IsTorsionFree R L
Tactic: have : IsTorsionFree R L := .trans_faithfulSMul R S L
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (hypothesis)
Goal:
k : Type u
inst : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst_1 : Field K
inst_2 : Field J
inst_3 : CommRing R
inst_4 : CommRing S
inst_5 : Field L
inst_6 : Field M
inst_7 : Algebra R M
inst_8 : IsTorsionFree R M
inst_9 : IsAlgClosure R M
inst_10 : Algebra K M
inst_11 : IsAlgClosure K M
inst_12 : Algebra S L
inst_13 : IsTorsionFree S L
inst_14 : IsAlgClosure S L
inst_15 : Algebra R S
inst_16 : Algebra R L
inst_17 : IsScalarTower R S L
inst_18 : Algebra K J
inst_19 : Algebra J L
inst_20 : IsAlgClosure J L
inst_21 : Algebra K L
inst_22 : IsScalarTower K J L
inst_23 : IsDomain R
inst_24 : IsDomain S
inst_25 : IsTorsionFree R S
inst_26 : Algebra.IsAlgebraic R L
this : IsTorsionFree R L
⊢ L ≃ₐ[R] M
have this₁ : IsAlgClosure R L
Tactic: have : IsAlgClosure R L :=
    { isAlgClosed := IsAlgClosure.isAlgClosed S
      isAlgebraic := ‹_› }
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (hypothesis)
Goal:
k : Type u
inst : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst_1 : Field K
inst_2 : Field J
inst_3 : CommRing R
inst_4 : CommRing S
inst_5 : Field L
inst_6 : Field M
inst_7 : Algebra R M
inst_8 : IsTorsionFree R M
inst_9 : IsAlgClosure R M
inst_10 : Algebra K M
inst_11 : IsAlgClosure K M
inst_12 : Algebra S L
inst_13 : IsTorsionFree S L
inst_14 : IsAlgClosure S L
inst_15 : IsDomain R
inst_16 : IsDomain S
hSR : S ≃+* R
this_1 : Algebra R S := hSR.symm.toRingHom.toAlgebra
this_2 : Algebra S R := hSR.toRingHom.toAlgebra
this : Algebra R L :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S L) (algebraMap R S))
⊢ { e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = RingHom.comp (algebraMap R M) hSR.toRingHom }
have this₁ : IsScalarTower R S L
Tactic: haveI : IsScalarTower R S L := .of_algebraMap_eq fun _ => rfl
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (hypothesis)
Goal:
k : Type u
inst : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst_1 : Field K
inst_2 : Field J
inst_3 : CommRing R
inst_4 : CommRing S
inst_5 : Field L
inst_6 : Field M
inst_7 : Algebra R M
inst_8 : IsTorsionFree R M
inst_9 : IsAlgClosure R M
inst_10 : Algebra K M
inst_11 : IsAlgClosure K M
inst_12 : Algebra S L
inst_13 : IsTorsionFree S L
inst_14 : IsAlgClosure S L
inst_15 : IsDomain R
inst_16 : IsDomain S
hSR : S ≃+* R
this_1 : Algebra R S := hSR.symm.toRingHom.toAlgebra
this_2 : Algebra S R := hSR.toRingHom.toAlgebra
this_3 : Algebra R L :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S L) (algebraMap R S))
this : IsScalarTower R S L
⊢ { e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = RingHom.comp (algebraMap R M) hSR.toRingHom }
have this₁ : IsScalarTower S R L
Tactic: haveI : IsScalarTower S R L := .of_algebraMap_eq (by simp [RingHom.algebraMap_toAlgebra])
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (hypothesis)
Goal:
k : Type u
inst : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst_1 : Field K
inst_2 : Field J
inst_3 : CommRing R
inst_4 : CommRing S
inst_5 : Field L
inst_6 : Field M
inst_7 : Algebra R M
inst_8 : IsTorsionFree R M
inst_9 : IsAlgClosure R M
inst_10 : Algebra K M
inst_11 : IsAlgClosure K M
inst_12 : Algebra S L
inst_13 : IsTorsionFree S L
inst_14 : IsAlgClosure S L
inst_15 : IsDomain R
inst_16 : IsDomain S
hSR : S ≃+* R
this_1 : Algebra R S := hSR.symm.toRingHom.toAlgebra
this_2 : Algebra S R := hSR.toRingHom.toAlgebra
this_3 : Algebra R L :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S L) (algebraMap R S))
this_4 : IsScalarTower R S L
this : IsScalarTower S R L
⊢ { e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = RingHom.comp (algebraMap R M) hSR.toRingHom }
have this₁ : FaithfulSMul R S
Tactic: have : FaithfulSMul R S := (faithfulSMul_iff_algebraMap_injective R S).mpr hSR.symm.injective
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (hypothesis)
Goal:
k : Type u
inst : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst_1 : Field K
inst_2 : Field J
inst_3 : CommRing R
inst_4 : CommRing S
inst_5 : Field L
inst_6 : Field M
inst_7 : Algebra R M
inst_8 : IsTorsionFree R M
inst_9 : IsAlgClosure R M
inst_10 : Algebra K M
inst_11 : IsAlgClosure K M
inst_12 : Algebra S L
inst_13 : IsTorsionFree S L
inst_14 : IsAlgClosure S L
inst_15 : IsDomain R
inst_16 : IsDomain S
hSR : S ≃+* R
this_1 : Algebra R S := hSR.symm.toRingHom.toAlgebra
this_2 : Algebra S R := hSR.toRingHom.toAlgebra
this_3 : Algebra R L :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S L) (algebraMap R S))
this_4 : IsScalarTower R S L
this_5 : IsScalarTower S R L
this : FaithfulSMul R S
⊢ { e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = RingHom.comp (algebraMap R M) hSR.toRingHom }
have this₁ : Algebra.IsAlgebraic R L
Tactic: have : Algebra.IsAlgebraic R L := (IsAlgClosure.isAlgebraic.extendScalars
    (show Function.Injective (algebraMap S R) from hSR.injective))
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (goal)
Goal:
k : Type u
inst : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst_1 : Field K
inst_2 : Field J
inst_3 : CommRing R
inst_4 : CommRing S
inst_5 : Field L
inst_6 : Field M
inst_7 : Algebra R M
inst_8 : IsTorsionFree R M
inst_9 : IsAlgClosure R M
inst_10 : Algebra K M
inst_11 : IsAlgClosure K M
inst_12 : Algebra S L
inst_13 : IsTorsionFree S L
inst_14 : IsAlgClosure S L
inst_15 : IsDomain R
inst_16 : IsDomain S
hSR : S ≃+* R
this_1 : Algebra R S := hSR.symm.toRingHom.toAlgebra
this_2 : Algebra S R := hSR.toRingHom.toAlgebra
this_3 : Algebra R L :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S L) (algebraMap R S))
this_4 : IsScalarTower R S L
this_5 : IsScalarTower S R L
this_6 : FaithfulSMul R S
this : Algebra.IsAlgebraic R L
⊢ { e : L ≃+* M // e.toRingHom.comp (algebraMap S L) = RingHom.comp (algebraMap R M) hSR.toRingHom }
have refine : Eq (α := S →+* M)
  (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
    (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (equivOfAlgebraic' R S L M))) (algebraMap S L))
  (RingHom.comp (algebraMap R M) hSR.toRingHom)
Tactic: refine ⟨equivOfAlgebraic' R S L M, ?_⟩
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (goal)
Goal:
k : Type u
inst : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst_1 : Field K
inst_2 : Field J
inst_3 : CommRing R
inst_4 : CommRing S
inst_5 : Field L
inst_6 : Field M
inst_7 : Algebra R M
inst_8 : IsTorsionFree R M
inst_9 : IsAlgClosure R M
inst_10 : Algebra K M
inst_11 : IsAlgClosure K M
inst_12 : Algebra S L
inst_13 : IsTorsionFree S L
inst_14 : IsAlgClosure S L
inst_15 : IsDomain R
inst_16 : IsDomain S
hSR : S ≃+* R
this_1 : Algebra R S := hSR.symm.toRingHom.toAlgebra
this_2 : Algebra S R := hSR.toRingHom.toAlgebra
this_3 : Algebra R L :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S L) (algebraMap R S))
this_4 : IsScalarTower R S L
this_5 : IsScalarTower S R L
this_6 : FaithfulSMul R S
this : Algebra.IsAlgebraic R L
⊢ Eq (α := S →+* M)
    (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
      (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (equivOfAlgebraic' R S L M))) (algebraMap S L))
    (RingHom.comp (algebraMap R M) hSR.toRingHom)
have a : ∀ (x : S),
  Eq.{w + 1} (α := M)
    ((@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
            (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (equivOfAlgebraic' R S L M))) (algebraMap S L) :
          (a : S) → M)
        x :
      M)
    ((RingHom.comp (algebraMap R M) hSR.toRingHom : (a : S) → M) x : M)
Tactic: ext x
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (goal)
Goal:
k : Type u
inst : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst_1 : Field K
inst_2 : Field J
inst_3 : CommRing R
inst_4 : CommRing S
inst_5 : Field L
inst_6 : Field M
inst_7 : Algebra R M
inst_8 : IsTorsionFree R M
inst_9 : IsAlgClosure R M
inst_10 : Algebra K M
inst_11 : IsAlgClosure K M
inst_12 : Algebra S L
inst_13 : IsTorsionFree S L
inst_14 : IsAlgClosure S L
inst_15 : IsDomain R
inst_16 : IsDomain S
hSR : S ≃+* R
this_1 : Algebra R S := hSR.symm.toRingHom.toAlgebra
this_2 : Algebra S R := hSR.toRingHom.toAlgebra
this_3 : Algebra R L :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S L) (algebraMap R S))
this_4 : IsScalarTower R S L
this_5 : IsScalarTower S R L
this_6 : FaithfulSMul R S
this : Algebra.IsAlgebraic R L
x : S
⊢ Eq.{w + 1} (α := M)
    ((@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _
            (RingEquiv.toRingHom (AlgEquiv.toRingEquiv (equivOfAlgebraic' R S L M))) (algebraMap S L) :
          (a : S) → M)
        x :
      M)
    ((RingHom.comp (algebraMap R M) hSR.toRingHom : (a : S) → M) x : M)
have a : Eq.{w + 1} (α := M) ((equivOfAlgebraic' R S L M : (a : L) → M) ((algebraMap S L : (a : S) → L) x : L) : M)
  ((algebraMap R M : (a : R) → M) ((hSR : (a : S) → R) x : R) : M)
Tactic: simp only [RingEquiv.toRingHom_eq_coe, Function.comp_apply, RingHom.coe_comp,
    AlgEquiv.coe_ringEquiv, RingEquiv.coe_toRingHom]
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (goal)
Goal:
k : Type u
inst : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst_1 : Field K
inst_2 : Field J
inst_3 : CommRing R
inst_4 : CommRing S
inst_5 : Field L
inst_6 : Field M
inst_7 : Algebra R M
inst_8 : IsTorsionFree R M
inst_9 : IsAlgClosure R M
inst_10 : Algebra K M
inst_11 : IsAlgClosure K M
inst_12 : Algebra S L
inst_13 : IsTorsionFree S L
inst_14 : IsAlgClosure S L
inst_15 : IsDomain R
inst_16 : IsDomain S
hSR : S ≃+* R
this_1 : Algebra R S := hSR.symm.toRingHom.toAlgebra
this_2 : Algebra S R := hSR.toRingHom.toAlgebra
this_3 : Algebra R L :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S L) (algebraMap R S))
this_4 : IsScalarTower R S L
this_5 : IsScalarTower S R L
this_6 : FaithfulSMul R S
this : Algebra.IsAlgebraic R L
x : S
| ((equivOfAlgebraic' R S L M : (a : L) → M) ((algebraMap S L : (a : S) → L) x : L) : M)
have rw : ((equivOfAlgebraic' R S L M : (a : L) → M)
    ((algebraMap S L : (a : S) → L) ((hSR.symm : (a : R) → S) ((hSR : (a : S) → R) x : R) : S) : L) :
  M)
Tactic: rw [← hSR.symm_apply_apply x]
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (goal)
Goal:
k : Type u
inst : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst_1 : Field K
inst_2 : Field J
inst_3 : CommRing R
inst_4 : CommRing S
inst_5 : Field L
inst_6 : Field M
inst_7 : Algebra R M
inst_8 : IsTorsionFree R M
inst_9 : IsAlgClosure R M
inst_10 : Algebra K M
inst_11 : IsAlgClosure K M
inst_12 : Algebra S L
inst_13 : IsTorsionFree S L
inst_14 : IsAlgClosure S L
inst_15 : IsDomain R
inst_16 : IsDomain S
hSR : S ≃+* R
this_1 : Algebra R S := hSR.symm.toRingHom.toAlgebra
this_2 : Algebra S R := hSR.toRingHom.toAlgebra
this_3 : Algebra R L :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S L) (algebraMap R S))
this_4 : IsScalarTower R S L
this_5 : IsScalarTower S R L
this_6 : FaithfulSMul R S
this : Algebra.IsAlgebraic R L
x : S
⊢ Eq.{w + 1} (α := M) ((equivOfAlgebraic' R S L M : (a : L) → M) ((algebraMap S L : (a : S) → L) x : L) : M)
    ((algebraMap R M : (a : R) → M) ((hSR : (a : S) → R) x : R) : M)
have a : Eq.{w + 1} (α := M)
  ((equivOfAlgebraic' R S L M : (a : L) → M)
      ((algebraMap S L : (a : S) → L) ((hSR.symm : (a : R) → S) ((hSR : (a : S) → R) x : R) : S) : L) :
    M)
  ((algebraMap R M : (a : R) → M) ((hSR : (a : S) → R) x : R) : M)
Tactic: conv_lhs => rw [← hSR.symm_apply_apply x]
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (goal)
Goal:
k : Type u
inst : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst_1 : Field K
inst_2 : Field J
inst_3 : CommRing R
inst_4 : CommRing S
inst_5 : Field L
inst_6 : Field M
inst_7 : Algebra R M
inst_8 : IsTorsionFree R M
inst_9 : IsAlgClosure R M
inst_10 : Algebra K M
inst_11 : IsAlgClosure K M
inst_12 : Algebra S L
inst_13 : IsTorsionFree S L
inst_14 : IsAlgClosure S L
inst_15 : IsDomain R
inst_16 : IsDomain S
hSR : S ≃+* R
this_1 : Algebra R S := hSR.symm.toRingHom.toAlgebra
this_2 : Algebra S R := hSR.toRingHom.toAlgebra
this_3 : Algebra R L :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S L) (algebraMap R S))
this_4 : IsScalarTower R S L
this_5 : IsScalarTower S R L
this_6 : FaithfulSMul R S
this : Algebra.IsAlgebraic R L
x : S
⊢ Eq.{w + 1} (α := M)
    ((equivOfAlgebraic' R S L M : (a : L) → M)
        ((algebraMap S L : (a : S) → L) ((hSR.symm : (a : R) → S) ((hSR : (a : S) → R) x : R) : S) : L) :
      M)
    ((algebraMap R M : (a : R) → M) ((hSR : (a : S) → R) x : R) : M)
have a : Eq.{w + 1} (α := M)
  ((equivOfAlgebraic' R S L M : (a : L) → M) ((algebraMap R L : (a : R) → L) ((hSR : (a : S) → R) x : R) : L) : M)
  ((algebraMap R M : (a : R) → M) ((hSR : (a : S) → R) x : R) : M)
Tactic: change equivOfAlgebraic' R S L M (algebraMap R L (hSR x)) = _
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (goal)
Goal:
k : Type u
inst : Field k
K : Type u_1
J : Type u_2
R : Type u
S : Type u_3
L : Type v
M : Type w
inst_1 : Field K
inst_2 : Field J
inst_3 : CommRing R
inst_4 : CommRing S
inst_5 : Field L
inst_6 : Field M
inst_7 : Algebra R M
inst_8 : IsTorsionFree R M
inst_9 : IsAlgClosure R M
inst_10 : Algebra K M
inst_11 : IsAlgClosure K M
inst_12 : Algebra S L
inst_13 : IsTorsionFree S L
inst_14 : IsAlgClosure S L
inst_15 : IsDomain R
inst_16 : IsDomain S
hSR : S ≃+* R
this_1 : Algebra R S := hSR.symm.toRingHom.toAlgebra
this_2 : Algebra S R := hSR.toRingHom.toAlgebra
this_3 : Algebra R L :=
  RingHom.toAlgebra (@RingHom.comp _ _ _ _ Semiring.toNonAssocSemiring _ (algebraMap S L) (algebraMap R S))
this_4 : IsScalarTower R S L
this_5 : IsScalarTower S R L
this_6 : FaithfulSMul R S
this : Algebra.IsAlgebraic R L
x : S
⊢ Eq.{w + 1} (α := M)
    ((equivOfAlgebraic' R S L M : (a : L) → M) ((algebraMap R L : (a : R) → L) ((hSR : (a : S) → R) x : R) : L) : M)
    ((algebraMap R M : (a : R) → M) ((hSR : (a : S) → R) x : R) : M)
have a : Eq.{w + 1} (α := M) ((algebraMap R M : (a : R) → M) ((hSR : (a : S) → R) x : R) : M)
  ((algebraMap R M : (a : R) → M) ((hSR : (a : S) → R) x : R) : M)
Tactic: rw [AlgEquiv.commutes]
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (goal)
Goal:
k : Type u
inst : Field k
F : Type u_1
K : Type u_2
A : Type u_3
inst_1 : Field F
inst_2 : Field K
inst_3 : Field A
inst_4 : Algebra F K
inst_5 : Algebra F A
inst_6 : Algebra.IsAlgebraic F K
L : IntermediateField F A
hL : ∀ (x : K), Splits.{u_3} (R := ↥L) (Polynomial.map (algebraMap F ↥L) (minpoly F x))
f : K →ₐ[F] A
x : K
⊢ Splits.{u_3} (R := ↥L) (Polynomial.map (algebraMap F ↥L) (minpoly.{u_1, u_3} F (B := A) ((f : (a : K) → A) x : A)))
have rw : Splits.{u_3} (R := ↥L) (Polynomial.map (algebraMap F ↥L) (minpoly F x))
Tactic: rw [minpoly.algHom_eq f f.injective]
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : IsAlgClosed K
inst_2 : CharZero K
f g : K[X]
hf0 : f ≠ 0
⊢ (∀ (x : K), f.IsRoot x → g.IsRoot x) ↔ f ∣ HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g
have refine : (∀ (x : K), f.IsRoot x → g.IsRoot x) → f ∣ HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g
Tactic: refine ⟨?_, isRoot_of_isRoot_of_dvd_derivative_mul hf0⟩
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : IsAlgClosed K
inst_2 : CharZero K
f g : K[X]
hf0 : f ≠ 0
⊢ (∀ (x : K), f.IsRoot x → g.IsRoot x) → f ∣ HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g
have pos : ∀ (hg0 : g = 0),
  (∀ (x : K), f.IsRoot x → g.IsRoot x) → f ∣ HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g
Tactic: by_cases hg0 : g = 0
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : IsAlgClosed K
inst_2 : CharZero K
f g : K[X]
hf0 : f ≠ 0
pos :
  ∀ (hg0 : g = 0),
    (∀ (x : K), f.IsRoot x → g.IsRoot x) → f ∣ HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g
⊢ (∀ (x : K), f.IsRoot x → g.IsRoot x) → f ∣ HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g
have neg : ∀ (hg0 : ¬g = 0),
  (∀ (x : K), f.IsRoot x → g.IsRoot x) → f ∣ HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g
Tactic: by_cases hg0 : g = 0
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : IsAlgClosed K
inst_2 : CharZero K
f g : K[X]
hf0 : f ≠ 0
hg0 : ¬g = 0
⊢ (∀ (x : K), f.IsRoot x → g.IsRoot x) → f ∣ HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g
have pos : ∀ (hdf0 : Eq (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) (0 : K[X])),
  (∀ (x : K), f.IsRoot x → g.IsRoot x) → f ∣ HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g
Tactic: by_cases hdf0 : derivative f = 0
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : IsAlgClosed K
inst_2 : CharZero K
f g : K[X]
hf0 : f ≠ 0
hg0 : ¬g = 0
pos :
  ∀ (hdf0 : Eq (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) (0 : K[X])),
    (∀ (x : K), f.IsRoot x → g.IsRoot x) → f ∣ HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g
⊢ (∀ (x : K), f.IsRoot x → g.IsRoot x) → f ∣ HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g
have neg : ∀ (hdf0 : ¬Eq (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) (0 : K[X])),
  (∀ (x : K), f.IsRoot x → g.IsRoot x) → f ∣ HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g
Tactic: by_cases hdf0 : derivative f = 0
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : IsAlgClosed K
inst_2 : CharZero K
f g : K[X]
hf0 : f ≠ 0
hg0 : ¬g = 0
hdf0 : Eq (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) (0 : K[X])
⊢ (∀ (x : K), f.IsRoot x → g.IsRoot x) → f ∣ HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g
have pos : (∀ (x : K), IsRoot ((C : (a : K) → K[X]) (f.coeff 0)) x → g.IsRoot x) →
  (C : (a : K) → K[X]) (f.coeff 0) ∣
    HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) ((C : (a : K) → K[X]) (f.coeff 0)) : K[X]) g
Tactic: rw [eq_C_of_derivative_eq_zero hdf0]
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (hypothesis)
Goal:
K : Type u_1
inst : Field K
inst_1 : IsAlgClosed K
inst_2 : CharZero K
f g : K[X]
hf0 : f ≠ 0
hg0 : ¬g = 0
hdf0 : ¬Eq (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) (0 : K[X])
⊢ (∀ (x : K), f.IsRoot x → g.IsRoot x) → f ∣ HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g
have hdg : HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g ≠ 0
Tactic: have hdg : f.derivative * g ≠ 0 := mul_ne_zero hdf0 hg0
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : IsAlgClosed K
inst_2 : CharZero K
f g : K[X]
hf0 : f ≠ 0
hg0 : ¬g = 0
hdf0 : ¬Eq (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) (0 : K[X])
hdg : HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g ≠ 0
⊢ (∀ (x : K), f.IsRoot x → g.IsRoot x) → f ∣ HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g
have neg : (∀ (x : K), f.IsRoot x → g.IsRoot x) →
  ∀ (a : K),
    Multiset.count a f.roots ≤
      Multiset.count a (HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g).roots
Tactic: rw [IsAlgClosed.dvd_iff_roots_le_roots hf0 hdg, Multiset.le_iff_count]
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : IsAlgClosed K
inst_2 : CharZero K
f g : K[X]
hf0 : f ≠ 0
hg0 : ¬g = 0
hdf0 : ¬Eq (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) (0 : K[X])
hdg : HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g ≠ 0
⊢ (∀ (x : K), f.IsRoot x → g.IsRoot x) → f ∣ HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g
have neg : (∀ (x : K), f.IsRoot x → g.IsRoot x) →
  ∀ (a : K),
    Multiset.count a f.roots ≤
      Multiset.count a (HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g).roots
Tactic: classical rw [IsAlgClosed.dvd_iff_roots_le_roots hf0 hdg, Multiset.le_iff_count]
-- Lean 4 proof step from Mathlib.FieldTheory.IsAlgClosed.Basic (goal)
Goal:
K : Type u_1
inst : Field K
inst_1 : IsAlgClosed K
inst_2 : CharZero K
f g : K[X]
hf0 : f ≠ 0
hg0 : ¬g = 0
hdf0 : ¬Eq (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) (0 : K[X])
hdg : HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g ≠ 0
⊢ (∀ (x : K), f.IsRoot x → g.IsRoot x) →
    ∀ (a : K),
      Multiset.count a f.roots ≤
        Multiset.count a (HMul.hMul (α := K[X]) ((derivative : (a : K[X]) → K[X]) f : K[X]) g).roots
have neg : (∀ (x : K), f.IsRoot x → g.IsRoot x) →
  ∀ (a : K), rootMultiplicity a f ≤ rootMultiplicity a ((derivative : (a : K[X]) → K[X]) f) + rootMultiplicity a g
Tactic: simp only [count_roots, rootMultiplicity_mul hdg]