awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
inst_3 : Module.Free R S
inst_4 : AddCommGroup A
inst_5 : Module R A
inst_6 : Module S A
inst_7 : IsScalarTower R S A
inst_8 : Module.Free S A
f : A →ₗ[S] A
h : Nontrivial R
h_1 : Nontrivial A
this_1 : Nontrivial S
ιS : Type u_2
bS : Module.Basis ιS R S
ιA : Type u_3
bA : Module.Basis ιA S A
this_2 : Nonempty ιS
this : Nonempty ιA
val : Fintype ιS
val_1 : Infinite ιA
inl :
  Eq.{u_1 + 1} (α := R)
    ((LinearMap.det : (a : A →ₗ[R] A) → R)
        ((↑R :
            {S : Type u_2} →
              {M M₂ : Type u_3} →
                [inst : Semiring R] →
                  [inst_9 : Semiring S] →
                    [inst_10 : AddCommMonoid M] →
                      [inst_11 : AddCommMonoid M₂] →
                        [inst_12 : Module R M] →
                          [inst_13 : Module R M₂] →
                            [inst_14 : Module S M] →
                              [inst_15 : Module S M₂] → [CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
          f) :
      R)
    ((Algebra.norm R : (a : S) → R) 1 : R)
⊢ Eq.{u_1 + 1} (α := R)
    ((LinearMap.det : (a : A →ₗ[R] A) → R)
        ((↑R :
            {S : Type u_2} →
              {M M₂ : Type u_3} →
                [inst : Semiring R] →
                  [inst_9 : Semiring S] →
                    [inst_10 : AddCommMonoid M] →
                      [inst_11 : AddCommMonoid M₂] →
                        [inst_12 : Module R M] →
                          [inst_13 : Module R M₂] →
                            [inst_14 : Module S M] →
                              [inst_15 : Module S M₂] → [CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
          f) :
      R)
    ((Algebra.norm R : (a : S) → R) ((LinearMap.det : (a : A →ₗ[S] A) → S) f : S) : R)
have inl₁ : Eq.{u_1 + 1} (α := R)
  ((LinearMap.det : (a : A →ₗ[R] A) → R)
      ((↑R :
          {S : Type u_2} →
            {M M₂ : Type u_3} →
              [inst : Semiring R] →
                [inst_9 : Semiring S] →
                  [inst_10 : AddCommMonoid M] →
                    [inst_11 : AddCommMonoid M₂] →
                      [inst_12 : Module R M] →
                        [inst_13 : Module R M₂] →
                          [inst_14 : Module S M] →
                            [inst_15 : Module S M₂] → [CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
        f) :
    R)
  (1 : R)
Tactic: rw [det_eq_one_of_not_module_finite (Module.not_finite_of_infinite_basis bA), map_one,
      det_eq_one_of_not_module_finite (Module.not_finite_of_infinite_basis (bS.smulTower bA))]
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
inst_3 : Module.Free R S
inst_4 : AddCommGroup A
inst_5 : Module R A
inst_6 : Module S A
inst_7 : IsScalarTower R S A
inst_8 : Module.Free S A
f : A →ₗ[S] A
h : Nontrivial R
h_1 : Nontrivial A
this_1 : Nontrivial S
ιS : Type u_2
bS : Module.Basis ιS R S
ιA : Type u_3
bA : Module.Basis ιA S A
this_2 : Nonempty ιS
this : Nonempty ιA
val : Fintype ιS
val_1 : Infinite ιA
inl :
  Eq.{u_1 + 1} (α := R)
    ((LinearMap.det : (a : A →ₗ[R] A) → R)
        ((↑R :
            {S : Type u_2} →
              {M M₂ : Type u_3} →
                [inst : Semiring R] →
                  [inst_9 : Semiring S] →
                    [inst_10 : AddCommMonoid M] →
                      [inst_11 : AddCommMonoid M₂] →
                        [inst_12 : Module R M] →
                          [inst_13 : Module R M₂] →
                            [inst_14 : Module S M] →
                              [inst_15 : Module S M₂] → [CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
          f) :
      R)
    ((Algebra.norm R : (a : S) → R) 1 : R)
inl₁ :
  Eq.{u_1 + 1} (α := R)
    ((LinearMap.det : (a : A →ₗ[R] A) → R)
        ((↑R :
            {S : Type u_2} →
              {M M₂ : Type u_3} →
                [inst : Semiring R] →
                  [inst_9 : Semiring S] →
                    [inst_10 : AddCommMonoid M] →
                      [inst_11 : AddCommMonoid M₂] →
                        [inst_12 : Module R M] →
                          [inst_13 : Module R M₂] →
                            [inst_14 : Module S M] →
                              [inst_15 : Module S M₂] → [CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
          f) :
      R)
    (1 : R)
⊢ Eq.{u_1 + 1} (α := R)
    ((LinearMap.det : (a : A →ₗ[R] A) → R)
        ((↑R :
            {S : Type u_2} →
              {M M₂ : Type u_3} →
                [inst : Semiring R] →
                  [inst_9 : Semiring S] →
                    [inst_10 : AddCommMonoid M] →
                      [inst_11 : AddCommMonoid M₂] →
                        [inst_12 : Module R M] →
                          [inst_13 : Module R M₂] →
                            [inst_14 : Module S M] →
                              [inst_15 : Module S M₂] → [CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
          f) :
      R)
    ((Algebra.norm R : (a : S) → R) ((LinearMap.det : (a : A →ₗ[S] A) → S) f : S) : R)
have inl₂ : Eq.{u_1 + 1} (α := R) 1 1
Tactic: rw [det_eq_one_of_not_module_finite (Module.not_finite_of_infinite_basis bA), map_one,
      det_eq_one_of_not_module_finite (Module.not_finite_of_infinite_basis (bS.smulTower bA))]
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
inst_3 : Module.Free R S
inst_4 : AddCommGroup A
inst_5 : Module R A
inst_6 : Module S A
inst_7 : IsScalarTower R S A
inst_8 : Module.Free S A
f : A →ₗ[S] A
h : Nontrivial R
h_1 : Nontrivial A
this_1 : Nontrivial S
ιS : Type u_2
bS : Module.Basis ιS R S
ιA : Type u_3
bA : Module.Basis ιA S A
this_2 : Nonempty ιS
this : Nonempty ιA
val : Fintype ιS
val_1 : Fintype ιA
⊢ Eq.{u_1 + 1} (α := R)
    ((LinearMap.det : (a : A →ₗ[R] A) → R)
        ((↑R :
            {S : Type u_2} →
              {M M₂ : Type u_3} →
                [inst : Semiring R] →
                  [inst_9 : Semiring S] →
                    [inst_10 : AddCommMonoid M] →
                      [inst_11 : AddCommMonoid M₂] →
                        [inst_12 : Module R M] →
                          [inst_13 : Module R M₂] →
                            [inst_14 : Module S M] →
                              [inst_15 : Module S M₂] → [CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
          f) :
      R)
    ((Algebra.norm R : (a : S) → R) ((LinearMap.det : (a : A →ₗ[S] A) → S) f : S) : R)
have inl : Eq.{u_1 + 1} (α := R)
  ((LinearMap.det : (a : A →ₗ[R] A) → R)
      ((↑R :
          {S : Type u_2} →
            {M M₂ : Type u_3} →
              [inst : Semiring R] →
                [inst_9 : Semiring S] →
                  [inst_10 : AddCommMonoid M] →
                    [inst_11 : AddCommMonoid M₂] →
                      [inst_12 : Module R M] →
                        [inst_13 : Module R M₂] →
                          [inst_14 : Module S M] →
                            [inst_15 : Module S M₂] → [CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
        f) :
    R)
  (det.{u_1, u_2} (n := ιS)
      ((Algebra.leftMulMatrix bS : (a : S) → Matrix ιS ιS R) ((LinearMap.det : (a : A →ₗ[S] A) → S) f)) :
    R)
Tactic: rw [Algebra.norm_eq_matrix_det bS, ← AlgHom.coe_toRingHom, ← det_toMatrix bA, det_det,
    ← det_toMatrix (bS.smulTower' bA), restrictScalars_toMatrix, RingHom.coe_coe]
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
inst_3 : Module.Free R S
inst_4 : AddCommGroup A
inst_5 : Module R A
inst_6 : Module S A
inst_7 : IsScalarTower R S A
inst_8 : Module.Free S A
f : A →ₗ[S] A
h : Nontrivial R
h_1 : Nontrivial A
this_1 : Nontrivial S
ιS : Type u_2
bS : Module.Basis ιS R S
ιA : Type u_3
bA : Module.Basis ιA S A
this_2 : Nonempty ιS
this : Nonempty ιA
val : Fintype ιS
val_1 : Fintype ιA
inl :
  Eq.{u_1 + 1} (α := R)
    ((LinearMap.det : (a : A →ₗ[R] A) → R)
        ((↑R :
            {S : Type u_2} →
              {M M₂ : Type u_3} →
                [inst : Semiring R] →
                  [inst_9 : Semiring S] →
                    [inst_10 : AddCommMonoid M] →
                      [inst_11 : AddCommMonoid M₂] →
                        [inst_12 : Module R M] →
                          [inst_13 : Module R M₂] →
                            [inst_14 : Module S M] →
                              [inst_15 : Module S M₂] → [CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
          f) :
      R)
    (det.{u_1, u_2} (n := ιS)
        ((Algebra.leftMulMatrix bS : (a : S) → Matrix ιS ιS R) ((LinearMap.det : (a : A →ₗ[S] A) → S) f)) :
      R)
⊢ Eq.{u_1 + 1} (α := R)
    ((LinearMap.det : (a : A →ₗ[R] A) → R)
        ((↑R :
            {S : Type u_2} →
              {M M₂ : Type u_3} →
                [inst : Semiring R] →
                  [inst_9 : Semiring S] →
                    [inst_10 : AddCommMonoid M] →
                      [inst_11 : AddCommMonoid M₂] →
                        [inst_12 : Module R M] →
                          [inst_13 : Module R M₂] →
                            [inst_14 : Module S M] →
                              [inst_15 : Module S M₂] → [CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
          f) :
      R)
    ((Algebra.norm R : (a : S) → R) ((LinearMap.det : (a : A →ₗ[S] A) → S) f : S) : R)
have inl₁ : Eq.{u_1 + 1} (α := R)
  ((LinearMap.det : (a : A →ₗ[R] A) → R)
      ((↑R :
          {S : Type u_2} →
            {M M₂ : Type u_3} →
              [inst : Semiring R] →
                [inst_9 : Semiring S] →
                  [inst_10 : AddCommMonoid M] →
                    [inst_11 : AddCommMonoid M₂] →
                      [inst_12 : Module R M] →
                        [inst_13 : Module R M₂] →
                          [inst_14 : Module S M] →
                            [inst_15 : Module S M₂] → [CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
        f) :
    R)
  (det.{u_1, u_2} (n := ιS)
      (((↑(Algebra.leftMulMatrix bS) : S →+* Matrix ιS ιS R) : (a : S) → Matrix ιS ιS R)
        ((LinearMap.det : (a : A →ₗ[S] A) → S) f)) :
    R)
Tactic: rw [Algebra.norm_eq_matrix_det bS, ← AlgHom.coe_toRingHom, ← det_toMatrix bA, det_det,
    ← det_toMatrix (bS.smulTower' bA), restrictScalars_toMatrix, RingHom.coe_coe]
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
R : Type u_1
S : Type u_2
A : Type u_3
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
inst_3 : Module.Free R S
inst_4 : AddCommGroup A
inst_5 : Module R A
inst_6 : Module S A
inst_7 : IsScalarTower R S A
inst_8 : Module.Free S A
f : A →ₗ[S] A
⊢ Eq.{u_1 + 1} (α := R)
    ((LinearMap.det : (a : A →ₗ[R] A) → R)
        ((↑R :
            {S : Type u_2} →
              {M M₂ : Type u_3} →
                [inst : Semiring R] →
                  [inst_9 : Semiring S] →
                    [inst_10 : AddCommMonoid M] →
                      [inst_11 : AddCommMonoid M₂] →
                        [inst_12 : Module R M] →
                          [inst_13 : Module R M₂] →
                            [inst_14 : Module S M] →
                              [inst_15 : Module S M₂] → [CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
          f) :
      R)
    ((Algebra.norm R : (a : S) → R) ((LinearMap.det : (a : A →ₗ[S] A) → S) f : S) : R)
have nontriviality : Subsingleton R →
  Eq.{u_1 + 1} (α := R)
    ((LinearMap.det : (a : A →ₗ[R] A) → R)
        ((↑R :
            {S : Type u_2} →
              {M M₂ : Type u_3} →
                [inst : Semiring R] →
                  [inst_9 : Semiring S] →
                    [inst_10 : AddCommMonoid M] →
                      [inst_11 : AddCommMonoid M₂] →
                        [inst_12 : Module R M] →
                          [inst_13 : Module R M₂] →
                            [inst_14 : Module S M] →
                              [inst_15 : Module S M₂] → [CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
          f) :
      R)
    ((Algebra.norm R : (a : S) → R) ((LinearMap.det : (a : A →ₗ[S] A) → S) f : S) : R)
Tactic: classical
  nontriviality R
  nontriviality A
  have := Module.nontrivial S A
  let ⟨ιS, bS⟩ := Module.Free.exists_basis (R := R) (M := S)
  let ⟨ιA, bA⟩ := Module.Free.exists_basis (R := S) (M := A)
  have := bS.index_nonempty
  have := bA.index_nonempty
  cases fintypeOrInfinite ιS; swap
  · rw [Algebra.norm_eq_one_of_not_module_finite (Module.not_finite_of_infinite_basis bS),
      det_eq_one_of_not_module_finite (Module.not_finite_of_infinite_basis (bS.smulTower bA))]
  cases fintypeOrInfinite ιA; swap
  · rw [det_eq_one_of_not_module_finite (Module.not_finite_of_infinite_basis bA), map_one,
      det_eq_one_of_not_module_finite (Module.not_finite_of_infinite_basis (bS.smulTower bA))]
  rw [Algebra.norm_eq_matrix_det bS, ← AlgHom.coe_toRingHom, ← det_toMatrix bA, det_det,
    ← det_toMatrix (bS.smulTower' bA), restrictScalars_toMatrix, RingHom.coe_coe]
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
R : Type u_1
S : Type u_2
inst : CommRing R
inst_1 : CommRing S
inst_2 : Algebra R S
inst_3 : Module.Free R S
A : Type u_6
inst_4 : Ring A
inst_5 : Algebra R A
inst_6 : Algebra S A
inst_7 : IsScalarTower R S A
inst_8 : Module.Free S A
a : A
⊢ Eq.{u_1 + 1} (α := R) ((norm R : (a : S) → R) ((norm S : (a : A) → S) a : S) : R) ((norm R : (a : A) → R) a : R)
have rw : Eq.{u_1 + 1} (α := R)
  (((LinearMap.det.{u_6, u_1} (M := A) (A := R) : MonoidHom.{u_6, u_1} (A →ₗ[R] A) R) : (a : A →ₗ[R] A) → R)
      ((↑R :
          {S : Type u_2} →
            {M M₂ : Type u_6} →
              [inst : Semiring R] →
                [inst_9 : Semiring S] →
                  [inst_10 : AddCommMonoid M] →
                    [inst_11 : AddCommMonoid M₂] →
                      [inst_12 : Module R M] →
                        [inst_13 : Module R M₂] →
                          [inst_14 : Module S M] →
                            [inst_15 : Module S M₂] →
                              [LinearMap.CompatibleSMul M M₂ R S] → (fₗ : M →ₗ[S] M₂) → M →ₗ[R] M₂)
        (S := S) ((lmul S A : (a : A) → Module.End S A) a)) :
    R)
  (((LinearMap.det.{u_6, u_1} (M := A) (A := R) : MonoidHom.{u_6, u_1} (A →ₗ[R] A) R) : (a : A →ₗ[R] A) → R)
      ((lmul R A : (a : A) → Module.End R A) a : Module.End R A) :
    R)
Tactic: rw [norm_apply S, norm_apply R a, ← LinearMap.det_restrictScalars]
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
R : Type u_1
inst : CommRing R
L : Type u_6
K : Type u_7
inst_1 : Field K
inst_2 : Field L
inst_3 : Algebra K L
inst_4 : Algebra R L
inst_5 : Algebra R K
inst_6 : IsScalarTower R K L
x : L
hx : IsIntegral R x
⊢ IsIntegral.{u_1, u_7} R (A := K) ((norm K : (a : L) → K) x : K)
have pos : ∀ (h : FiniteDimensional K L), IsIntegral.{u_1, u_7} R (A := K) ((norm K : (a : L) → K) x : K)
Tactic: by_cases h : FiniteDimensional K L
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
R : Type u_1
inst : CommRing R
L : Type u_6
K : Type u_7
inst_1 : Field K
inst_2 : Field L
inst_3 : Algebra K L
inst_4 : Algebra R L
inst_5 : Algebra R K
inst_6 : IsScalarTower R K L
x : L
hx : IsIntegral R x
pos : ∀ (h : FiniteDimensional K L), IsIntegral.{u_1, u_7} R (A := K) ((norm K : (a : L) → K) x : K)
⊢ IsIntegral.{u_1, u_7} R (A := K) ((norm K : (a : L) → K) x : K)
have neg : ∀ (h : ¬FiniteDimensional K L), IsIntegral.{u_1, u_7} R (A := K) ((norm K : (a : L) → K) x : K)
Tactic: by_cases h : FiniteDimensional K L
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
R : Type u_1
inst : CommRing R
L : Type u_6
K : Type u_7
inst_1 : Field K
inst_2 : Field L
inst_3 : Algebra K L
inst_4 : Algebra R L
inst_5 : Algebra R K
inst_6 : IsScalarTower R K L
x : L
hx : IsIntegral R x
h : FiniteDimensional K L
F : IntermediateField K L := K⟮x⟯
⊢ IsIntegral.{u_1, u_7} R (A := K) ((norm K : (a : L) → K) x : K)
have pos : IsIntegral R
  (HPow.hPow (α := K) ((norm K : (a : ↥F) → K) (gen K x) : K) (Fintype.card.{u_6} (Free.ChooseBasisIndex (↥F) L)))
Tactic: rw [← norm_norm (S := F), ← coe_gen K x, ← IntermediateField.algebraMap_apply,
    norm_algebraMap_of_basis (Module.Free.chooseBasis F L) (gen K x), map_pow]
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
R : Type u_1
inst : CommRing R
L : Type u_6
K : Type u_7
inst_1 : Field K
inst_2 : Field L
inst_3 : Algebra K L
inst_4 : Algebra R L
inst_5 : Algebra R K
inst_6 : IsScalarTower R K L
x : L
hx : IsIntegral R x
h : FiniteDimensional K L
F : IntermediateField K L := K⟮x⟯
⊢ IsIntegral R
    (HPow.hPow (α := K) ((norm K : (a : ↥F) → K) (gen K x) : K) (Fintype.card.{u_6} (Free.ChooseBasisIndex (↥F) L)))
have pos : IsIntegral.{u_1, u_7} R (A := K) ((norm K : (a : ↥F) → K) (gen K x) : K)
Tactic: apply IsIntegral.pow
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
R : Type u_1
inst : CommRing R
L : Type u_6
K : Type u_7
inst_1 : Field K
inst_2 : Field L
inst_3 : Algebra K L
inst_4 : Algebra R L
inst_5 : Algebra R K
inst_6 : IsScalarTower R K L
x : L
hx : IsIntegral R x
h : FiniteDimensional K L
F : IntermediateField K L := K⟮x⟯
⊢ IsIntegral.{u_1, u_7} R (A := K) ((norm K : (a : ↥F) → K) (gen K x) : K)
have pos : IsIntegral.{u_1, u_6} R (A := AlgebraicClosure ↥F) (Multiset.prod (aroots (minpoly K x) (AlgebraicClosure ↥F)))
Tactic: rw [← isIntegral_algebraMap_iff (algebraMap K (AlgebraicClosure F)).injective,
    norm_gen_eq_prod_roots _ (IsAlgClosed.splits _)]
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
R : Type u_1
inst : CommRing R
L : Type u_6
K : Type u_7
inst_1 : Field K
inst_2 : Field L
inst_3 : Algebra K L
inst_4 : Algebra R L
inst_5 : Algebra R K
inst_6 : IsScalarTower R K L
x : L
hx : IsIntegral R x
h : FiniteDimensional K L
F : IntermediateField K L := K⟮x⟯
⊢ IsIntegral.{u_1, u_6} R (A := AlgebraicClosure ↥F) (Multiset.prod (aroots (minpoly K x) (AlgebraicClosure ↥F)))
have pos : ∀ (y : AlgebraicClosure ↥F)
  (hy : Membership.mem (γ := Multiset (AlgebraicClosure ↥F)) (aroots (minpoly K x) (AlgebraicClosure ↥F)) y),
  eval₂ (algebraMap R (AlgebraicClosure ↥F)) y (minpoly R x) = 0
Tactic: refine IsIntegral.multiset_prod (fun y hy ↦ ⟨minpoly R x, minpoly.monic hx, ?_⟩)
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
R : Type u_1
inst : CommRing R
L : Type u_6
K : Type u_7
inst_1 : Field K
inst_2 : Field L
inst_3 : Algebra K L
inst_4 : Algebra R L
inst_5 : Algebra R K
inst_6 : IsScalarTower R K L
x : L
hx : IsIntegral R x
h : FiniteDimensional K L
F : IntermediateField K L := K⟮x⟯
y : AlgebraicClosure ↥F
hy : Membership.mem (γ := Multiset (AlgebraicClosure ↥F)) (aroots (minpoly K x) (AlgebraicClosure ↥F)) y
⊢ eval₂ (algebraMap R (AlgebraicClosure ↥F)) y (minpoly R x) = 0
have pos : Eq.{u_6 + 1} (α := AlgebraicClosure ↥F)
  ((aeval y : (a : K[X]) → AlgebraicClosure ↥F) (Polynomial.map (algebraMap R K) (minpoly R x)) : AlgebraicClosure ↥F)
  (0 : AlgebraicClosure ↥F)
Tactic: suffices (aeval y) ((minpoly R x).map (algebraMap R K)) = 0 by simpa
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
R : Type u_1
inst : CommRing R
L : Type u_6
K : Type u_7
inst_1 : Field K
inst_2 : Field L
inst_3 : Algebra K L
inst_4 : Algebra R L
inst_5 : Algebra R K
inst_6 : IsScalarTower R K L
x : L
hx : IsIntegral R x
h : FiniteDimensional K L
F : IntermediateField K L := K⟮x⟯
y : AlgebraicClosure ↥F
hy : Membership.mem (γ := Multiset (AlgebraicClosure ↥F)) (aroots (minpoly K x) (AlgebraicClosure ↥F)) y
⊢ Eq.{u_6 + 1} (α := AlgebraicClosure ↥F)
    ((aeval y : (a : K[X]) → AlgebraicClosure ↥F) (Polynomial.map (algebraMap R K) (minpoly R x)) : AlgebraicClosure ↥F)
    (0 : AlgebraicClosure ↥F)
have pos : ∀ (P : K[X]) (hP : Polynomial.map (algebraMap R K) (minpoly R x) = minpoly K x * P),
  Eq.{u_6 + 1} (α := AlgebraicClosure ↥F)
    ((aeval y : (a : K[X]) → AlgebraicClosure ↥F) (Polynomial.map (algebraMap R K) (minpoly R x)) : AlgebraicClosure ↥F)
    (0 : AlgebraicClosure ↥F)
Tactic: obtain ⟨P, hP⟩ := minpoly.dvd K x (show aeval x ((minpoly R x).map (algebraMap R K)) = 0 by simp)
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
L : Type u_6
K : Type u_7
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
x : L
⊢ (norm K : (a : L) → K) x = HPow.hPow (α := K) ((norm K : (a : ↥K⟮x⟯) → K) (gen K x) : K) (finrank (↥K⟮x⟯) L)
have pos : ∀ (h : FiniteDimensional K L),
  (norm K : (a : L) → K) x = HPow.hPow (α := K) ((norm K : (a : ↥K⟮x⟯) → K) (gen K x) : K) (finrank (↥K⟮x⟯) L)
Tactic: by_cases h : FiniteDimensional K L
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
L : Type u_6
K : Type u_7
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
x : L
pos :
  ∀ (h : FiniteDimensional K L),
    (norm K : (a : L) → K) x = HPow.hPow (α := K) ((norm K : (a : ↥K⟮x⟯) → K) (gen K x) : K) (finrank (↥K⟮x⟯) L)
⊢ (norm K : (a : L) → K) x = HPow.hPow (α := K) ((norm K : (a : ↥K⟮x⟯) → K) (gen K x) : K) (finrank (↥K⟮x⟯) L)
have neg : ∀ (h : ¬FiniteDimensional K L),
  (norm K : (a : L) → K) x = HPow.hPow (α := K) ((norm K : (a : ↥K⟮x⟯) → K) (gen K x) : K) (finrank (↥K⟮x⟯) L)
Tactic: by_cases h : FiniteDimensional K L
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
L : Type u_6
K : Type u_7
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
x : L
h : ¬FiniteDimensional K L
⊢ (norm K : (a : L) → K) x = HPow.hPow (α := K) ((norm K : (a : ↥K⟮x⟯) → K) (gen K x) : K) (finrank (↥K⟮x⟯) L)
have neg : 1 = HPow.hPow (α := K) ((norm K : (a : ↥K⟮x⟯) → K) (gen K x) : K) (finrank (↥K⟮x⟯) L)
Tactic: rw [norm_eq_one_of_not_module_finite h]
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
L : Type u_6
K : Type u_7
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
x : L
h : ¬FiniteDimensional K L
⊢ 1 = HPow.hPow (α := K) ((norm K : (a : ↥K⟮x⟯) → K) (gen K x) : K) (finrank (↥K⟮x⟯) L)
have pos : ∀ (hx : IsIntegral K x), 1 = HPow.hPow (α := K) ((norm K : (a : ↥K⟮x⟯) → K) (gen K x) : K) (finrank (↥K⟮x⟯) L)
Tactic: by_cases hx : IsIntegral K x
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
L : Type u_6
K : Type u_7
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
x : L
h : ¬FiniteDimensional K L
pos : ∀ (hx : IsIntegral K x), 1 = HPow.hPow (α := K) ((norm K : (a : ↥K⟮x⟯) → K) (gen K x) : K) (finrank (↥K⟮x⟯) L)
⊢ 1 = HPow.hPow (α := K) ((norm K : (a : ↥K⟮x⟯) → K) (gen K x) : K) (finrank (↥K⟮x⟯) L)
have neg : ∀ (hx : ¬IsIntegral K x), 1 = HPow.hPow (α := K) ((norm K : (a : ↥K⟮x⟯) → K) (gen K x) : K) (finrank (↥K⟮x⟯) L)
Tactic: by_cases hx : IsIntegral K x
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (hypothesis)
Goal:
L : Type u_6
K : Type u_7
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
x : L
h : ¬FiniteDimensional K L
hx : IsIntegral K x
H : FiniteDimensional (↥K⟮x⟯) L
⊢ FiniteDimensional K L
have this : FiniteDimensional.{u_7, u_6} K ↥K⟮x⟯
Tactic: have : FiniteDimensional K K⟮x⟯ := adjoin.finiteDimensional hx
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (hypothesis)
Goal:
L : Type u_6
K : Type u_7
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
x : L
h : ¬FiniteDimensional K L
hx : IsIntegral K x
⊢ 1 = HPow.hPow (α := K) ((norm K : (a : ↥K⟮x⟯) → K) (gen K x) : K) (finrank (↥K⟮x⟯) L)
have h₁ : ¬FiniteDimensional (↥K⟮x⟯) L
Tactic: have h₁ : ¬ FiniteDimensional K⟮x⟯ L := fun H ↦ h <| by
        have : FiniteDimensional K K⟮x⟯ := adjoin.finiteDimensional hx
        exact Finite.trans K⟮x⟯ L
-- Lean 4 proof step from Mathlib.RingTheory.Norm.Transitivity (goal)
Goal:
L : Type u_6
K : Type u_7
inst : Field K
inst_1 : Field L
inst_2 : Algebra K L
x : L
h : ¬FiniteDimensional K L
hx : ¬IsIntegral K x
⊢ 1 = HPow.hPow (α := K) ((norm K : (a : ↥K⟮x⟯) → K) (gen K x) : K) (finrank (↥K⟮x⟯) L)
have neg : 1 = HPow.hPow (α := K) 1 (finrank (↥K⟮x⟯) L)
Tactic: rw [norm_eq_one_of_not_module_finite]