mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Invariant.Basic

Goal state before:
pp: case refine_2
A : Type u_1
B : Type u_2
k : Type u_3
inst✝¹³ : CommRing A
inst✝¹² : CommRing B
inst✝¹¹ : Algebra A B
G : Type u_4
inst✝¹⁰ : Finite G
inst✝⁹ : Group G
inst✝⁸ : MulSemiringAction G B
inst✝⁷ : Algebra.IsInvariant A B G
P : Ideal A
Q : Ideal B
inst✝⁶ : Q.LiesOver P
inst✝⁵ : CommRing k
inst✝⁴ : Algebra (A ⧸ P) k
inst✝³ : Algebra (B ⧸ Q) k
inst✝² : IsScalarTower (A ⧸ P) (B ⧸ Q) k
inst✝¹ : IsDomain k
inst✝ : FaithfulSMul (B ⧸ Q) k
σ : k ≃ₐ[A ⧸ P] k
f : B ⧸ Q →ₐ[A ⧸ P] k := IsScalarTower.toAlgHom (A ⧸ P) (B ⧸ Q) k
hf : Function.Injective ⇑f
τ₁ : B ⧸ Q →ₐ[A ⧸ P] B ⧸ Q
h₁ : ∀ (x : B ⧸ Q), (algebraMap (B ⧸ Q) k) (τ₁ x) = ↑σ ((algebraMap (B ⧸ Q) k) x)
τ₂ : B ⧸ Q →ₐ[A ⧸ P] B ⧸ Q
h₂ : ∀ (x : B ⧸ Q), (algebraMap (B ⧸ Q) k) (τ₂ x) = ↑σ.symm ((algebraMap (B ⧸ Q) k) x)
⊢ Function.RightInverse (⇑τ₂) (↑↑τ₁.toRingHom).toFun

Tactic:
·

Goal state after:
case refine_2
A : Type u_1
B : Type u_2
k : Type u_3
inst✝¹³ : CommRing A
inst✝¹² : CommRing B
inst✝¹¹ : Algebra A B
G : Type u_4
inst✝¹⁰ : Finite G
inst✝⁹ : Group G
inst✝⁸ : MulSemiringAction G B
inst✝⁷ : Algebra.IsInvariant A B G
P : Ideal A
Q : Ideal B
inst✝⁶ : Q.LiesOver P
inst✝⁵ : CommRing k
inst✝⁴ : Algebra (A ⧸ P) k
inst✝³ : Algebra (B ⧸ Q) k
inst✝² : IsScalarTower (A ⧸ P) (B ⧸ Q) k
inst✝¹ : IsDomain k
inst✝ : FaithfulSMul (B ⧸ Q) k
σ : k ≃ₐ[A ⧸ P] k
f : B ⧸ Q →ₐ[A ⧸ P] k := IsScalarTower.toAlgHom (A ⧸ P) (B ⧸ Q) k
hf : Function.Injective ⇑f
τ₁ : B ⧸ Q →ₐ[A ⧸ P] B ⧸ Q
h₁ : ∀ (x : B ⧸ Q), (algebraMap (B ⧸ Q) k) (τ₁ x) = ↑σ ((algebraMap (B ⧸ Q) k) x)
τ₂ : B ⧸ Q →ₐ[A ⧸ P] B ⧸ Q
h₂ : ∀ (x : B ⧸ Q), (algebraMap (B ⧸ Q) k) (τ₂ x) = ↑σ.symm ((algebraMap (B ⧸ Q) k) x)
⊢ Function.RightInverse (⇑τ₂) (↑↑τ₁.toRingHom).toFun
Module: Mathlib.LinearAlgebra.FreeModule.Finite.Quotient

Goal state before:
pp: case refine_1
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Free ℤ M
inst✝ : Module.Finite ℤ M
N : Submodule ℤ M
h : Finite (M ⧸ N)
x : M
⊢ ((LinearMap.lsmul ℤ M) ↑(Nat.card (M ⧸ N))) x ∈ N

pp: case refine_2
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Free ℤ M
inst✝ : Module.Finite ℤ M
N : Submodule ℤ M
h : Finite (M ⧸ N)
⊢ Function.Injective ⇑(LinearMap.codRestrict N ((LinearMap.lsmul ℤ M) ↑(Nat.card (M ⧸ N))) ⋯)

Tactic:
·

Goal state after:
case refine_1
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Free ℤ M
inst✝ : Module.Finite ℤ M
N : Submodule ℤ M
h : Finite (M ⧸ N)
x : M
⊢ ((LinearMap.lsmul ℤ M) ↑(Nat.card (M ⧸ N))) x ∈ N
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: case mp
F : Type u_3
inst✝¹ : Field F
inst✝ : Fintype F
hF : ringChar F ≠ 2
a : F
ha : a ≠ 0
⊢ (∃ r, a = ↑r * ↑r) → ∃ r, a = r * r

pp: case mpr
F : Type u_3
inst✝¹ : Field F
inst✝ : Fintype F
hF : ringChar F ≠ 2
a : F
ha : a ≠ 0
⊢ (∃ r, a = r * r) → ∃ r, a = ↑r * ↑r

Tactic:
·

Goal state after:
case mp
F : Type u_3
inst✝¹ : Field F
inst✝ : Fintype F
hF : ringChar F ≠ 2
a : F
ha : a ≠ 0
⊢ (∃ r, a = ↑r * ↑r) → ∃ r, a = r * r
Module: Mathlib.LinearAlgebra.FreeModule.Finite.Quotient

Goal state before:
pp: case refine_2
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Free ℤ M
inst✝ : Module.Finite ℤ M
N : Submodule ℤ M
h : Finite (M ⧸ N)
⊢ Function.Injective ⇑(LinearMap.codRestrict N ((LinearMap.lsmul ℤ M) ↑(Nat.card (M ⧸ N))) ⋯)

Tactic:
·

Goal state after:
case refine_2
M : Type u_3
inst✝² : AddCommGroup M
inst✝¹ : Free ℤ M
inst✝ : Module.Finite ℤ M
N : Submodule ℤ M
h : Finite (M ⧸ N)
⊢ Function.Injective ⇑(LinearMap.codRestrict N ((LinearMap.lsmul ℤ M) ↑(Nat.card (M ⧸ N))) ⋯)
Module: Mathlib.FieldTheory.Finite.Basic

Goal state before:
pp: case mpr
F : Type u_3
inst✝¹ : Field F
inst✝ : Fintype F
hF : ringChar F ≠ 2
a : F
ha : a ≠ 0
⊢ (∃ r, a = r * r) → ∃ r, a = ↑r * ↑r

Tactic:
·

Goal state after:
case mpr
F : Type u_3
inst✝¹ : Field F
inst✝ : Fintype F
hF : ringChar F ≠ 2
a : F
ha : a ≠ 0
⊢ (∃ r, a = r * r) → ∃ r, a = ↑r * ↑r
Module: Mathlib.LinearAlgebra.FreeModule.Finite.CardQuotient

Goal state before:
pp: case calc_1
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Free ℤ M
inst✝² : Module.Finite ℤ M
N : Submodule ℤ M
E : Type u_2
inst✝¹ : EquivLike E M ↥N
inst✝ : AddEquivClass E M ↥N
e : E
b : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := Free.chooseBasis ℤ M
h : finrank ℤ ↥N = finrank ℤ M
a : Free.ChooseBasisIndex ℤ M → ℤ := smithNormalFormCoeffs b h
b' : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := smithNormalFormTopBasis b h
ab : Basis (Free.ChooseBasisIndex ℤ M) ℤ ↥N := smithNormalFormBotBasis b h
ab_eq :
  ∀ (i : Free.ChooseBasisIndex ℤ M),
    ↑((smithNormalFormBotBasis b h) i) = smithNormalFormCoeffs b h i • (smithNormalFormTopBasis b h) i
e' : M ≃ₗ[ℤ] ↥N := b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))
f : M →ₗ[ℤ] M := N.subtype ∘ₗ ↑e'
f_apply : ∀ (x : M), f x = ↑((b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))) x) := fun x ↦ rfl
ha : ∀ (i : Free.ChooseBasisIndex ℤ M), f (b' i) = a i • b' i
⊢ ((LinearMap.toMatrix b' b') f).det.natAbs = (Matrix.diagonal a).det.natAbs

pp: case calc_2
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Free ℤ M
inst✝² : Module.Finite ℤ M
N : Submodule ℤ M
E : Type u_2
inst✝¹ : EquivLike E M ↥N
inst✝ : AddEquivClass E M ↥N
e : E
b : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := Free.chooseBasis ℤ M
h : finrank ℤ ↥N = finrank ℤ M
a : Free.ChooseBasisIndex ℤ M → ℤ := smithNormalFormCoeffs b h
b' : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := smithNormalFormTopBasis b h
ab : Basis (Free.ChooseBasisIndex ℤ M) ℤ ↥N := smithNormalFormBotBasis b h
ab_eq :
  ∀ (i : Free.ChooseBasisIndex ℤ M),
    ↑((smithNormalFormBotBasis b h) i) = smithNormalFormCoeffs b h i • (smithNormalFormTopBasis b h) i
e' : M ≃ₗ[ℤ] ↥N := b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))
f : M →ₗ[ℤ] M := N.subtype ∘ₗ ↑e'
f_apply : ∀ (x : M), f x = ↑((b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))) x) := fun x ↦ rfl
ha : ∀ (i : Free.ChooseBasisIndex ℤ M), f (b' i) = a i • b' i
⊢ ∏ i, (a i).natAbs = Nat.card (M ⧸ N)

Tactic:
·

Goal state after:
case calc_1
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Free ℤ M
inst✝² : Module.Finite ℤ M
N : Submodule ℤ M
E : Type u_2
inst✝¹ : EquivLike E M ↥N
inst✝ : AddEquivClass E M ↥N
e : E
b : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := Free.chooseBasis ℤ M
h : finrank ℤ ↥N = finrank ℤ M
a : Free.ChooseBasisIndex ℤ M → ℤ := smithNormalFormCoeffs b h
b' : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := smithNormalFormTopBasis b h
ab : Basis (Free.ChooseBasisIndex ℤ M) ℤ ↥N := smithNormalFormBotBasis b h
ab_eq :
  ∀ (i : Free.ChooseBasisIndex ℤ M),
    ↑((smithNormalFormBotBasis b h) i) = smithNormalFormCoeffs b h i • (smithNormalFormTopBasis b h) i
e' : M ≃ₗ[ℤ] ↥N := b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))
f : M →ₗ[ℤ] M := N.subtype ∘ₗ ↑e'
f_apply : ∀ (x : M), f x = ↑((b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))) x) := fun x ↦ rfl
ha : ∀ (i : Free.ChooseBasisIndex ℤ M), f (b' i) = a i • b' i
⊢ ((LinearMap.toMatrix b' b') f).det.natAbs = (Matrix.diagonal a).det.natAbs
Module: Mathlib.LinearAlgebra.FreeModule.Finite.CardQuotient

Goal state before:
pp: case pos
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Free ℤ M
inst✝² : Module.Finite ℤ M
N : Submodule ℤ M
E : Type u_2
inst✝¹ : EquivLike E M ↥N
inst✝ : AddEquivClass E M ↥N
e : E
b : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := Free.chooseBasis ℤ M
h✝ : finrank ℤ ↥N = finrank ℤ M
a : Free.ChooseBasisIndex ℤ M → ℤ := smithNormalFormCoeffs b h✝
b' : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := smithNormalFormTopBasis b h✝
ab : Basis (Free.ChooseBasisIndex ℤ M) ℤ ↥N := smithNormalFormBotBasis b h✝
ab_eq :
  ∀ (i : Free.ChooseBasisIndex ℤ M),
    ↑((smithNormalFormBotBasis b h✝) i) = smithNormalFormCoeffs b h✝ i • (smithNormalFormTopBasis b h✝) i
e' : M ≃ₗ[ℤ] ↥N := b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))
f : M →ₗ[ℤ] M := N.subtype ∘ₗ ↑e'
f_apply : ∀ (x : M), f x = ↑((b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))) x) := fun x ↦ rfl
ha : ∀ (i : Free.ChooseBasisIndex ℤ M), f (b' i) = a i • b' i
i j : Free.ChooseBasisIndex ℤ M
h : i = j
⊢ (Finsupp.single j (a j)) i = Matrix.diagonal a i j

pp: case neg
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Free ℤ M
inst✝² : Module.Finite ℤ M
N : Submodule ℤ M
E : Type u_2
inst✝¹ : EquivLike E M ↥N
inst✝ : AddEquivClass E M ↥N
e : E
b : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := Free.chooseBasis ℤ M
h✝ : finrank ℤ ↥N = finrank ℤ M
a : Free.ChooseBasisIndex ℤ M → ℤ := smithNormalFormCoeffs b h✝
b' : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := smithNormalFormTopBasis b h✝
ab : Basis (Free.ChooseBasisIndex ℤ M) ℤ ↥N := smithNormalFormBotBasis b h✝
ab_eq :
  ∀ (i : Free.ChooseBasisIndex ℤ M),
    ↑((smithNormalFormBotBasis b h✝) i) = smithNormalFormCoeffs b h✝ i • (smithNormalFormTopBasis b h✝) i
e' : M ≃ₗ[ℤ] ↥N := b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))
f : M →ₗ[ℤ] M := N.subtype ∘ₗ ↑e'
f_apply : ∀ (x : M), f x = ↑((b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))) x) := fun x ↦ rfl
ha : ∀ (i : Free.ChooseBasisIndex ℤ M), f (b' i) = a i • b' i
i j : Free.ChooseBasisIndex ℤ M
h : ¬i = j
⊢ (Finsupp.single j (a j)) i = Matrix.diagonal a i j

Tactic:
·

Goal state after:
case pos
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Free ℤ M
inst✝² : Module.Finite ℤ M
N : Submodule ℤ M
E : Type u_2
inst✝¹ : EquivLike E M ↥N
inst✝ : AddEquivClass E M ↥N
e : E
b : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := Free.chooseBasis ℤ M
h✝ : finrank ℤ ↥N = finrank ℤ M
a : Free.ChooseBasisIndex ℤ M → ℤ := smithNormalFormCoeffs b h✝
b' : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := smithNormalFormTopBasis b h✝
ab : Basis (Free.ChooseBasisIndex ℤ M) ℤ ↥N := smithNormalFormBotBasis b h✝
ab_eq :
  ∀ (i : Free.ChooseBasisIndex ℤ M),
    ↑((smithNormalFormBotBasis b h✝) i) = smithNormalFormCoeffs b h✝ i • (smithNormalFormTopBasis b h✝) i
e' : M ≃ₗ[ℤ] ↥N := b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))
f : M →ₗ[ℤ] M := N.subtype ∘ₗ ↑e'
f_apply : ∀ (x : M), f x = ↑((b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))) x) := fun x ↦ rfl
ha : ∀ (i : Free.ChooseBasisIndex ℤ M), f (b' i) = a i • b' i
i j : Free.ChooseBasisIndex ℤ M
h : i = j
⊢ (Finsupp.single j (a j)) i = Matrix.diagonal a i j
Module: Mathlib.LinearAlgebra.FreeModule.Finite.CardQuotient

Goal state before:
pp: case neg
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Free ℤ M
inst✝² : Module.Finite ℤ M
N : Submodule ℤ M
E : Type u_2
inst✝¹ : EquivLike E M ↥N
inst✝ : AddEquivClass E M ↥N
e : E
b : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := Free.chooseBasis ℤ M
h✝ : finrank ℤ ↥N = finrank ℤ M
a : Free.ChooseBasisIndex ℤ M → ℤ := smithNormalFormCoeffs b h✝
b' : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := smithNormalFormTopBasis b h✝
ab : Basis (Free.ChooseBasisIndex ℤ M) ℤ ↥N := smithNormalFormBotBasis b h✝
ab_eq :
  ∀ (i : Free.ChooseBasisIndex ℤ M),
    ↑((smithNormalFormBotBasis b h✝) i) = smithNormalFormCoeffs b h✝ i • (smithNormalFormTopBasis b h✝) i
e' : M ≃ₗ[ℤ] ↥N := b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))
f : M →ₗ[ℤ] M := N.subtype ∘ₗ ↑e'
f_apply : ∀ (x : M), f x = ↑((b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))) x) := fun x ↦ rfl
ha : ∀ (i : Free.ChooseBasisIndex ℤ M), f (b' i) = a i • b' i
i j : Free.ChooseBasisIndex ℤ M
h : ¬i = j
⊢ (Finsupp.single j (a j)) i = Matrix.diagonal a i j

Tactic:
·

Goal state after:
case neg
M : Type u_1
inst✝⁴ : AddCommGroup M
inst✝³ : Free ℤ M
inst✝² : Module.Finite ℤ M
N : Submodule ℤ M
E : Type u_2
inst✝¹ : EquivLike E M ↥N
inst✝ : AddEquivClass E M ↥N
e : E
b : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := Free.chooseBasis ℤ M
h✝ : finrank ℤ ↥N = finrank ℤ M
a : Free.ChooseBasisIndex ℤ M → ℤ := smithNormalFormCoeffs b h✝
b' : Basis (Free.ChooseBasisIndex ℤ M) ℤ M := smithNormalFormTopBasis b h✝
ab : Basis (Free.ChooseBasisIndex ℤ M) ℤ ↥N := smithNormalFormBotBasis b h✝
ab_eq :
  ∀ (i : Free.ChooseBasisIndex ℤ M),
    ↑((smithNormalFormBotBasis b h✝) i) = smithNormalFormCoeffs b h✝ i • (smithNormalFormTopBasis b h✝) i
e' : M ≃ₗ[ℤ] ↥N := b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))
f : M →ₗ[ℤ] M := N.subtype ∘ₗ ↑e'
f_apply : ∀ (x : M), f x = ↑((b'.equiv ab (Equiv.refl (Free.ChooseBasisIndex ℤ M))) x) := fun x ↦ rfl
ha : ∀ (i : Free.ChooseBasisIndex ℤ M), f (b' i) = a i • b' i
i j : Free.ChooseBasisIndex ℤ M
h : ¬i = j
⊢ (Finsupp.single j (a j)) i = Matrix.diagonal a i j
Module: Mathlib.RingTheory.Norm.Basic

Goal state before:
pp: case mpr
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : Free R S
inst✝ : Module.Finite R S
x : S
⊢ x = 0 → (norm R) x = 0

pp: case mp
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : Free R S
inst✝ : Module.Finite R S
x : S
⊢ (norm R) x = 0 → x = 0

Tactic:
·

Goal state after:
case mpr
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : Free R S
inst✝ : Module.Finite R S
x : S
⊢ x = 0 → (norm R) x = 0
Module: Mathlib.RingTheory.Norm.Basic

Goal state before:
pp: case mp
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : Free R S
inst✝ : Module.Finite R S
x : S
⊢ (norm R) x = 0 → x = 0

Tactic:
·

Goal state after:
case mp
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : Free R S
inst✝ : Module.Finite R S
x : S
⊢ (norm R) x = 0 → x = 0
Module: Mathlib.RingTheory.Norm.Basic

Goal state before:
pp: case mp
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : Free R S
inst✝ : Module.Finite R S
x : S
b : Basis (Free.ChooseBasisIndex R S) R S := Free.chooseBasis R S
decEq : DecidableEq (Free.ChooseBasisIndex R S) := Classical.decEq (Free.ChooseBasisIndex R S)
v : Free.ChooseBasisIndex R S → R
v_ne : v ≠ 0
hv : (leftMulMatrix b) x *ᵥ v = 0
⊢ x = 0

Tactic:
← b.equivFun.apply_symm_apply v,

Goal state after:
case mp
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : Free R S
inst✝ : Module.Finite R S
x : S
b : Basis (Free.ChooseBasisIndex R S) R S := Free.chooseBasis R S
decEq : DecidableEq (Free.ChooseBasisIndex R S) := Classical.decEq (Free.ChooseBasisIndex R S)
v : Free.ChooseBasisIndex R S → R
v_ne : v ≠ 0
hv : (leftMulMatrix b) x *ᵥ b.equivFun (b.equivFun.symm v) = 0
⊢ x = 0
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative

Goal state before:
pp: case pos
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, Associated x y → P x → P y
ha0 : a = 0
⊢ P a

pp: case neg
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, Associated x y → P x → P y
ha0 : ¬a = 0
⊢ P a

Tactic:
·

Goal state after:
case pos
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)
this : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, Associated x y → P x → P y
ha0 : a = 0
⊢ P a
Module: Mathlib.RingTheory.Norm.Basic

Goal state before:
pp: case mp.refine_1
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : Free R S
inst✝ : Module.Finite R S
x : S
b : Basis (Free.ChooseBasisIndex R S) R S := Free.chooseBasis R S
decEq : DecidableEq (Free.ChooseBasisIndex R S) := Classical.decEq (Free.ChooseBasisIndex R S)
v : Free.ChooseBasisIndex R S → R
v_ne : v ≠ 0
hv : ⇑(b.repr (x * ∑ i, v i • b i)) = 0
i : Free.ChooseBasisIndex R S
⊢ (b.repr (x * ∑ i, v i • b i)) i = (b.repr 0) i

pp: case mp.refine_2
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : Free R S
inst✝ : Module.Finite R S
x : S
b : Basis (Free.ChooseBasisIndex R S) R S := Free.chooseBasis R S
decEq : DecidableEq (Free.ChooseBasisIndex R S) := Classical.decEq (Free.ChooseBasisIndex R S)
v : Free.ChooseBasisIndex R S → R
v_ne : v ≠ 0
hv : ⇑(b.repr (x * ∑ i, v i • b i)) = 0
⊢ ∑ i, v i • b i ≠ 0

Tactic:
·

Goal state after:
case mp.refine_1
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : Free R S
inst✝ : Module.Finite R S
x : S
b : Basis (Free.ChooseBasisIndex R S) R S := Free.chooseBasis R S
decEq : DecidableEq (Free.ChooseBasisIndex R S) := Classical.decEq (Free.ChooseBasisIndex R S)
v : Free.ChooseBasisIndex R S → R
v_ne : v ≠ 0
hv : ⇑(b.repr (x * ∑ i, v i • b i)) = 0
i : Free.ChooseBasisIndex R S
⊢ (b.repr (x * ∑ i, v i • b i)) i = (b.repr 0) i
Module: Mathlib.RingTheory.Norm.Basic

Goal state before:
pp: case mp.refine_2
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : Free R S
inst✝ : Module.Finite R S
x : S
b : Basis (Free.ChooseBasisIndex R S) R S := Free.chooseBasis R S
decEq : DecidableEq (Free.ChooseBasisIndex R S) := Classical.decEq (Free.ChooseBasisIndex R S)
v : Free.ChooseBasisIndex R S → R
v_ne : v ≠ 0
hv : ⇑(b.repr (x * ∑ i, v i • b i)) = 0
⊢ ∑ i, v i • b i ≠ 0

Tactic:
·

Goal state after:
case mp.refine_2
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : Ring S
inst✝⁴ : Algebra R S
inst✝³ : IsDomain R
inst✝² : IsDomain S
inst✝¹ : Free R S
inst✝ : Module.Finite R S
x : S
b : Basis (Free.ChooseBasisIndex R S) R S := Free.chooseBasis R S
decEq : DecidableEq (Free.ChooseBasisIndex R S) := Classical.decEq (Free.ChooseBasisIndex R S)
v : Free.ChooseBasisIndex R S → R
v_ne : v ≠ 0
hv : ⇑(b.repr (x * ∑ i, v i • b i)) = 0
⊢ ∑ i, v i • b i ≠ 0
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative

Goal state before:
pp: case neg.refine_1
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)
this✝¹ : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, Associated x y → P x → P y
ha0 : ¬a = 0
this✝ : Nontrivial α
this : StrongNormalizationMonoid α := UniqueFactorizationMonoid.strongNormalizationMonoid
⊢ ∀ p ∈ normalizedFactors a, Prime p

pp: case neg.refine_2
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)
this✝¹ : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, Associated x y → P x → P y
ha0 : ¬a = 0
this✝ : Nontrivial α
this : StrongNormalizationMonoid α := UniqueFactorizationMonoid.strongNormalizationMonoid
⊢ ∀ p ∈ normalizedFactors a, ∀ q ∈ normalizedFactors a, p ∣ q → p = q

Tactic:
·

Goal state after:
case neg.refine_1
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)
this✝¹ : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, Associated x y → P x → P y
ha0 : ¬a = 0
this✝ : Nontrivial α
this : StrongNormalizationMonoid α := UniqueFactorizationMonoid.strongNormalizationMonoid
⊢ ∀ p ∈ normalizedFactors a, Prime p
Module: Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative

Goal state before:
pp: case neg.refine_2
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)
this✝¹ : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, Associated x y → P x → P y
ha0 : ¬a = 0
this✝ : Nontrivial α
this : StrongNormalizationMonoid α := UniqueFactorizationMonoid.strongNormalizationMonoid
⊢ ∀ p ∈ normalizedFactors a, ∀ q ∈ normalizedFactors a, p ∣ q → p = q

Tactic:
·

Goal state after:
case neg.refine_2
α : Type u_1
inst✝¹ : CommMonoidWithZero α
inst✝ : UniqueFactorizationMonoid α
P : α → Prop
a : α
h0 : P 0
h1 : ∀ {x : α}, IsUnit x → P x
hpr : ∀ {p : α} (i : ℕ), Prime p → P (p ^ i)
hcp : ∀ {x y : α}, IsRelPrime x y → P x → P y → P (x * y)
this✝¹ : DecidableEq α := Classical.decEq α
P_of_associated : ∀ {x y : α}, Associated x y → P x → P y
ha0 : ¬a = 0
this✝ : Nontrivial α
this : StrongNormalizationMonoid α := UniqueFactorizationMonoid.strongNormalizationMonoid
⊢ ∀ p ∈ normalizedFactors a, ∀ q ∈ normalizedFactors a, p ∣ q → p = q
Module: Mathlib.RingTheory.Norm.Basic

Goal state before:
pp: case pos
K : Type u_4
inst✝³ : Field K
L : Type u_7
inst✝² : DivisionRing L
inst✝¹ : Algebra K L
inst✝ : Module.Finite K L
x : L
hx : x = 0
⊢ (norm K) x⁻¹ = ((norm K) x)⁻¹

pp: case neg
K : Type u_4
inst✝³ : Field K
L : Type u_7
inst✝² : DivisionRing L
inst✝¹ : Algebra K L
inst✝ : Module.Finite K L
x : L
hx : ¬x = 0
⊢ (norm K) x⁻¹ = ((norm K) x)⁻¹

Tactic:
·

Goal state after:
case pos
K : Type u_4
inst✝³ : Field K
L : Type u_7
inst✝² : DivisionRing L
inst✝¹ : Algebra K L
inst✝ : Module.Finite K L
x : L
hx : x = 0
⊢ (norm K) x⁻¹ = ((norm K) x)⁻¹
Module: Mathlib.RingTheory.Norm.Basic

Goal state before:
pp: case h.refine_1
K : Type u_4
L : Type u_5
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
x : L
s : Finset ↥K⟮x⟯
b : Basis (↥s) K ↥K⟮x⟯
⊢ (Subalgebra.toSubmodule K⟮x⟯.toSubalgebra).FG

pp: case h.refine_2
K : Type u_4
L : Type u_5
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
x : L
s : Finset ↥K⟮x⟯
b : Basis (↥s) K ↥K⟮x⟯
⊢ x ∈ K⟮x⟯.toSubalgebra

Tactic:
·

Goal state after:
case h.refine_1
K : Type u_4
L : Type u_5
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
x : L
s : Finset ↥K⟮x⟯
b : Basis (↥s) K ↥K⟮x⟯
⊢ (Subalgebra.toSubmodule K⟮x⟯.toSubalgebra).FG
Module: Mathlib.RingTheory.Norm.Basic

Goal state before:
pp: case h.refine_2
K : Type u_4
L : Type u_5
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
x : L
s : Finset ↥K⟮x⟯
b : Basis (↥s) K ↥K⟮x⟯
⊢ x ∈ K⟮x⟯.toSubalgebra

Tactic:
·

Goal state after:
case h.refine_2
K : Type u_4
L : Type u_5
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
x : L
s : Finset ↥K⟮x⟯
b : Basis (↥s) K ↥K⟮x⟯
⊢ x ∈ K⟮x⟯.toSubalgebra
Module: Mathlib.RingTheory.Norm.Basic

Goal state before:
pp: case h.refine_2
K : Type u_4
L : Type u_5
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
x : L
s : Finset ↥K⟮x⟯
b : Basis (↥s) K ↥K⟮x⟯
⊢ x ∈ K⟮x⟯.toSubalgebra

Tactic:
exact IntermediateField.subset_adjoin K _ (Set.mem_singleton x)

Goal state after:
Module: Mathlib.RingTheory.Norm.Basic

Goal state before:
pp: case neg
K : Type u_4
L : Type u_5
F : Type u_6
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Field F
inst✝¹ : Algebra K L
inst✝ : Algebra K F
x : L
hf : (Polynomial.map (algebraMap K F) (minpoly K x)).Splits
injKxL : Function.Injective ⇑(algebraMap (↥K⟮x⟯) L)
hx : ¬IsIntegral K x
⊢ (algebraMap K F) ((norm K) (AdjoinSimple.gen K x)) = ((minpoly K x).aroots F).prod

pp: case pos
K : Type u_4
L : Type u_5
F : Type u_6
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Field F
inst✝¹ : Algebra K L
inst✝ : Algebra K F
x : L
hf : (Polynomial.map (algebraMap K F) (minpoly K x)).Splits
injKxL : Function.Injective ⇑(algebraMap (↥K⟮x⟯) L)
hx : IsIntegral K x
⊢ (algebraMap K F) ((norm K) (AdjoinSimple.gen K x)) = ((minpoly K x).aroots F).prod

Tactic:
·

Goal state after:
case neg
K : Type u_4
L : Type u_5
F : Type u_6
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Field F
inst✝¹ : Algebra K L
inst✝ : Algebra K F
x : L
hf : (Polynomial.map (algebraMap K F) (minpoly K x)).Splits
injKxL : Function.Injective ⇑(algebraMap (↥K⟮x⟯) L)
hx : ¬IsIntegral K x
⊢ (algebraMap K F) ((norm K) (AdjoinSimple.gen K x)) = ((minpoly K x).aroots F).prod