mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
f : InfinitePlace K → ℝ≥0
w₀ : { w // w.IsComplex }
inst✝ : NumberField K
B : ℝ≥0
⊢ volume (⇑Complex.measurableEquivRealProd.symm ⁻¹' {x | |x.re| < 1 ∧ |x.im| < ↑B ^ 2}) = 4 * ↑B ^ 2

Tactic:
simp_rw [Set.preimage_ofPred_eq, Complex.measurableEquivRealProd_symm_apply]
      rw [show {a : ℝ × ℝ | |a.1| < 1 ∧ |a.2| < B ^ 2} =
        Set.Ioo (-1 : ℝ) (1 : ℝ) ×ˢ Set.Ioo (-(B : ℝ) ^ 2) ((B : ℝ) ^ 2) by
          ext; simp_rw [Set.mem_ofPred_eq, Set.mem_prod, Set.mem_Ioo, abs_lt]]
      simp_rw [volume_eq_prod, prod_prod, Real.volume_Ioo, sub_neg_eq_add, one_add_one_eq_two,
        ← two_mul, ofReal_mul zero_le_two, ofReal_pow (coe_nonneg B), ofReal_ofNat,
        ofReal_coe_nnreal, ← mul_assoc, show (2 : ℝ≥0∞) * 2 = 4 by norm_num]

Goal state after:
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i✝ : Free.ChooseBasisIndex ℤ (𝓞 K)
f : K →ₗ[ℚ] ↥(Submodule.span ℚ (Set.range ⇑(latticeBasis K))) :=
  LinearMap.codRestrict (Submodule.span ℚ (Set.range ⇑(latticeBasis K))) (canonicalEmbedding K).toRatAlgHom.toLinearMap
    ⋯
i : Free.ChooseBasisIndex ℤ (𝓞 K)
this : f ((integralBasis K) i) = (Basis.restrictScalars ℚ (latticeBasis K)) i
⊢ (↑(Basis.restrictScalars ℚ (latticeBasis K)).repr ∘ₗ f) ((integralBasis K) i) =
    ↑(integralBasis K).repr ((integralBasis K) i)

Tactic:
LinearMap.coe_comp,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i✝ : Free.ChooseBasisIndex ℤ (𝓞 K)
f : K →ₗ[ℚ] ↥(Submodule.span ℚ (Set.range ⇑(latticeBasis K))) :=
  LinearMap.codRestrict (Submodule.span ℚ (Set.range ⇑(latticeBasis K))) (canonicalEmbedding K).toRatAlgHom.toLinearMap
    ⋯
i : Free.ChooseBasisIndex ℤ (𝓞 K)
this : f ((integralBasis K) i) = (Basis.restrictScalars ℚ (latticeBasis K)) i
⊢ (⇑↑(Basis.restrictScalars ℚ (latticeBasis K)).repr ∘ ⇑f) ((integralBasis K) i) =
    ↑(integralBasis K).repr ((integralBasis K) i)
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (K →+* ℂ) → ℂ
h_mem : x ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding.latticeBasis K))
h_zero : (commMap K) x = 0
i : Free.ChooseBasisIndex ℤ (𝓞 K)
⊢ (canonicalEmbedding.latticeBasis K) i ∈ Set.range ⇑(canonicalEmbedding K)

Tactic:
exact ⟨integralBasis K i, (canonicalEmbedding.latticeBasis_apply K i).symm⟩

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra k K
φ : K →+* ℂ
h : IsMixed k φ
⊢ ¬ComplexEmbedding.IsReal (mk φ).embedding

Tactic:
by rcases embedding_mk_eq φ with (_ | _) <;> aesop

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra k K
w : InfinitePlace K
h : IsUnramified k w
hw : ComplexEmbedding.IsReal (conjugate (w.embedding.comp (algebraMap k K)))
⊢ ComplexEmbedding.IsReal (conjugate w.embedding)

Tactic:
IsSelfAdjoint.star_iff,

Goal state after:
k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra k K
w : InfinitePlace K
h : IsUnramified k w
hw : IsSelfAdjoint (w.embedding.comp (algebraMap k K))
⊢ ComplexEmbedding.IsReal (conjugate w.embedding)
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
⊢ convexBodySumFun x = 0 ↔ x = 0

Tactic:
← forall_normAtPlace_eq_zero_iff,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
⊢ convexBodySumFun x = 0 ↔ ∀ (w : InfinitePlace K), (normAtPlace w) x = 0
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝ : Field K
x : mixedSpace K
⊢ (∃ w, (normAtPlace w) x ≠ 0) ↔ ¬x = 0

Tactic:
← forall_normAtPlace_eq_zero_iff,

Goal state after:
K : Type u_1
inst✝ : Field K
x : mixedSpace K
⊢ (∃ w, (normAtPlace w) x ≠ 0) ↔ ¬∀ (w : InfinitePlace K), (normAtPlace w) x = 0
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: case inl
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : ℝ
hB✝ : B ≤ 0
hB : B < 0
x : mixedSpace K
⊢ x ∈ convexBodySum K B ↔ x ∈ ∅

Tactic:
refine ⟨fun hx => ?_, fun h => h.elim⟩

Goal state after:
case inl
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : ℝ
hB✝ : B ≤ 0
hB : B < 0
x : mixedSpace K
hx : x ∈ convexBodySum K B
⊢ x ∈ ∅
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ 2⁻¹ ^ nrComplexPlaces K * (↑(sqrt ‖discr K‖₊)).toReal = 2⁻¹ ^ nrComplexPlaces K * √|↑(discr K)|

Tactic:
ENNReal.coe_toReal,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ 2⁻¹ ^ nrComplexPlaces K * ↑(sqrt ‖discr K‖₊) = 2⁻¹ ^ nrComplexPlaces K * √|↑(discr K)|
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
⊢ (ENNReal.ofReal ↑(FractionalIdeal.absNorm ↑I)).toReal * (2⁻¹ ^ nrComplexPlaces K * (↑(sqrt ‖discr K‖₊)).toReal) =
    ↑(FractionalIdeal.absNorm ↑I) * 2⁻¹ ^ nrComplexPlaces K * √|↑(discr K)|

Tactic:
ENNReal.coe_toReal,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
⊢ (ENNReal.ofReal ↑(FractionalIdeal.absNorm ↑I)).toReal * (2⁻¹ ^ nrComplexPlaces K * ↑(sqrt ‖discr K‖₊)) =
    ↑(FractionalIdeal.absNorm ↑I) * 2⁻¹ ^ nrComplexPlaces K * √|↑(discr K)|
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i✝ : ChooseBasisIndex ℤ (𝓞 K)
f : K →ₗ[ℚ] ↥(Submodule.span ℚ (Set.range ⇑(latticeBasis K))) :=
  LinearMap.codRestrict (Submodule.span ℚ (Set.range ⇑(latticeBasis K))) (mixedEmbedding K).toRatAlgHom.toLinearMap ⋯
i : ChooseBasisIndex ℤ (𝓞 K)
this : f ((integralBasis K) i) = (Basis.restrictScalars ℚ (latticeBasis K)) i
⊢ (↑(Basis.restrictScalars ℚ (latticeBasis K)).repr ∘ₗ f) ((integralBasis K) i) =
    ↑(integralBasis K).repr ((integralBasis K) i)

Tactic:
LinearMap.coe_comp,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i✝ : ChooseBasisIndex ℤ (𝓞 K)
f : K →ₗ[ℚ] ↥(Submodule.span ℚ (Set.range ⇑(latticeBasis K))) :=
  LinearMap.codRestrict (Submodule.span ℚ (Set.range ⇑(latticeBasis K))) (mixedEmbedding K).toRatAlgHom.toLinearMap ⋯
i : ChooseBasisIndex ℤ (𝓞 K)
this : f ((integralBasis K) i) = (Basis.restrictScalars ℚ (latticeBasis K)) i
⊢ (⇑↑(Basis.restrictScalars ℚ (latticeBasis K)).repr ∘ ⇑f) ((integralBasis K) i) =
    ↑(integralBasis K).repr ((integralBasis K) i)
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: K : Type u_4
L : Type u_5
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
v : InfinitePlace K
inst✝¹ : NumberField K
inst✝ : NumberField L
⊢ (unramifiedPlacesOver L v).ncard + 2 * (ramifiedPlacesOver L v).ncard = finrank K L

Tactic:
let : Algebra K ℂ := v.embedding.toAlgebra
  rw [← AlgHom.card K L ℂ, ramifiedPlacesOver_ncard, unramifiedPlacesOver_ncard,
    ← Set.ncard_union_eq (disjoint_unmixedEmbeddingsOver_mixedEmbeddingsOver L v.embedding),
    union_unmixedEmbeddingsOver_mixedEmbeddingsOver, Set.ncard_eq_toFinset_card]
  apply (card_nbij AlgHom.toRingHom (fun σ _ ↦ by simpa using ⟨by aesop⟩)
    AlgHom.coe_ringHom_injective.injOn (fun ψ hψ ↦ ?_)).symm
  simp only [Set.Finite.toFinset_ofPred, coe_filter, mem_univ, true_and, Set.mem_ofPred_eq] at hψ
  exact ⟨⟨ψ, fun _ ↦ by simp [RingHom.algebraMap_toAlgebra, ← hψ.over]⟩, by simp⟩

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ := (minkowskiBound K I * ↑(convexBodySumFactor K)⁻¹).toReal ^ (1 / ↑(finrank ℚ K))
h_le : minkowskiBound K I ≤ volume (convexBodySum K B)
x✝ : K
⊢ ↑(FractionalIdeal.absNorm ↑I) * 2 ^ (2 * ↑(nrComplexPlaces K)) * √‖discr K‖ * ↑(finrank ℚ K).factorial *
      π⁻¹ ^ nrComplexPlaces K =
    ↑(FractionalIdeal.absNorm ↑I) * (4 / π) ^ nrComplexPlaces K * ↑(finrank ℚ K).factorial * √|↑(discr K)|

Tactic:
rw [Int.norm_eq_abs, zpow_mul, show (2 : ℝ) ^ (2 : ℤ) = 4 by norm_cast, div_pow,
        inv_eq_one_div, div_pow, one_pow, zpow_natCast]
      ring

Goal state after:
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: case inr
K : Type u_1
inst✝ : Field K
s : Set { w // w.IsReal }
x : mixedSpace K
w : InfinitePlace K
hw : w.IsComplex
⊢ (normAtPlace w) ((negAt s) x) = (normAtPlace w) x

Tactic:
normAtPlace_apply_of_isComplex hw,

Goal state after:
case inr
K : Type u_1
inst✝ : Field K
s : Set { w // w.IsReal }
x : mixedSpace K
w : InfinitePlace K
hw : w.IsComplex
⊢ ‖((negAt s) x).2 ⟨w, hw⟩‖ = ‖x.2 ⟨w, hw⟩‖
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: case inr
K : Type u_1
inst✝ : Field K
x : realSpace K
w : InfinitePlace K
hx : 0 ≤ x w
hw : w.IsComplex
⊢ (normAtPlace w) (fun w ↦ x ↑w, fun w ↦ ↑(x ↑w)) = x w

Tactic:
normAtPlace_apply_of_isComplex hw,

Goal state after:
case inr
K : Type u_1
inst✝ : Field K
x : realSpace K
w : InfinitePlace K
hx : 0 ≤ x w
hw : w.IsComplex
⊢ ‖(fun w ↦ x ↑w, fun w ↦ ↑(x ↑w)).2 ⟨w, hw⟩‖ = x w
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : Nat.succ 1 ≤ finrank ℚ K
⊢ 4 / 9 * (3 * π / 4) ^ 2 ≤ 4 / 9 * (3 * π / 4) ^ finrank ℚ K

Tactic:
by
      gcongr
      linarith [Real.pi_gt_three]

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Cyclotomic.Embeddings

Goal state before:
pp: case neg
n : ℕ
inst✝² : NeZero n
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
h : IsCyclotomicExtension {n} ℚ K
this : NumberField K
hn : ¬2 < n
h1 : ¬1 < n
⊢ φ n = 1

Tactic:
convert! totient_one

Goal state after:
n : ℕ
inst✝² : NeZero n
K : Type u
inst✝¹ : Field K
inst✝ : CharZero K
h : IsCyclotomicExtension {n} ℚ K
this : NumberField K
hn : ¬2 < n
h1 : ¬1 < n
⊢ n = 1
Module: Mathlib.RingTheory.Ideal.Quotient.Index

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : CommRing R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
I : Ideal R
N : Submodule R M
inst✝ : Finite (R ⧸ I)
s : Finset M
hs : span R ↑s = N
val✝ : Fintype (R ⧸ I)
e : (↥N ⧸ comap N.subtype (I • N)) ≃ₗ[R] (R ⧸ I) ⊗[R] ↥N :=
  (comap N.subtype (I • N)).quotEquivOfEq (I • ⊤) ⋯ ≪≫ₗ (quotTensorEquivQuotSMul (↥N) I).symm
H : (Finsupp.linearCombination R Subtype.val).range = N
c : ↥s →₀ R
⊢ (Finsupp.linearCombination R Subtype.val) c ∈ N

Tactic:
rw [← H, LinearMap.mem_range]; exact exists_apply_eq_apply _ _

Goal state after:
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: case refine_2.refine_2
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈max (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsComplex}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
this✝¹ : CharZero ↥K
this✝ : NumberField ↥K
w₀ : InfinitePlace ↥↑⟨K, hK₀⟩
hw₀ : w₀ ∈ {w | w.IsComplex}
this : minkowskiBound (↥K) 1 < ↑(convexBodyLT'Factor ↥K) * ↑(boundOfDiscBdd N)
x : 𝓞 ↥K
hx₁ : ℚ⟮↑x⟯ = ⊤
hx₂ : ∀ (w : InfinitePlace ↥K), w ↑x < √(1 + ↑(boundOfDiscBdd N) ^ 2)
hx : IsIntegral ℤ ((algebraMap (𝓞 ↥K) ↥K) x)
i : ℕ
⊢ max (√(1 + ↑(boundOfDiscBdd N) ^ 2)) 1 ^ finrank ℚ ↥K * ↑((finrank ℚ ↥K).choose (finrank ℚ ↥K / 2)) ≤
    max (√(1 + ↑(B ^ 2))) 1 ^ D * ↑↑(D.choose (D / 2))

Tactic:
NNReal.coe_pow

Goal state after:
case refine_2.refine_2
A : Type u_2
inst✝¹ : Field A
inst✝ : CharZero A
N : ℕ
D : ℕ := rankOfDiscrBdd N
B : ℝ≥0 := boundOfDiscBdd N
C : ℕ := ⌈max (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2))⌉₊
x✝¹ : { F // FiniteDimensional ℚ ↥F }
K : IntermediateField ℚ A
hK₀ : FiniteDimensional ℚ ↥K
x✝ : ⟨K, hK₀⟩ ∈ {K | {w | w.IsComplex}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
this✝¹ : CharZero ↥K
this✝ : NumberField ↥K
w₀ : InfinitePlace ↥↑⟨K, hK₀⟩
hw₀ : w₀ ∈ {w | w.IsComplex}
this : minkowskiBound (↥K) 1 < ↑(convexBodyLT'Factor ↥K) * ↑(boundOfDiscBdd N)
x : 𝓞 ↥K
hx₁ : ℚ⟮↑x⟯ = ⊤
hx₂ : ∀ (w : InfinitePlace ↥K), w ↑x < √(1 + ↑(boundOfDiscBdd N) ^ 2)
hx : IsIntegral ℤ ((algebraMap (𝓞 ↥K) ↥K) x)
i : ℕ
⊢ max (√(1 + ↑(boundOfDiscBdd N) ^ 2)) 1 ^ finrank ℚ ↥K * ↑((finrank ℚ ↥K).choose (finrank ℚ ↥K / 2)) ≤
    max (√(1 + ↑B ^ 2)) 1 ^ D * ↑↑(D.choose (D / 2))
Module: Mathlib.RingTheory.LocalRing.Quotient

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsLocalRing R
inst✝² : Module.Finite R S
inst✝¹ : Free R S
ι : Type u_4
inst✝ : Fintype ι
b : Basis ι R S
x : S
i : ι
⊢ (Ideal.Quotient.mk pS) x =
    (Finsupp.linearCombination (R ⧸ p) ⇑(basisQuotient b))
      ((Finsupp.linearEquivFunOnFinite (R ⧸ p) (R ⧸ p) ι).symm (⇑(Ideal.Quotient.mk p) ∘ ⇑(b.repr x)))

Tactic:
Finsupp.linearCombination_eq_fintype_linearCombination_apply (R ⧸ p),

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : IsLocalRing R
inst✝² : Module.Finite R S
inst✝¹ : Free R S
ι : Type u_4
inst✝ : Fintype ι
b : Basis ι R S
x : S
i : ι
⊢ (Ideal.Quotient.mk pS) x =
    (Fintype.linearCombination (R ⧸ p) ⇑(basisQuotient b)) (⇑(Ideal.Quotient.mk p) ∘ ⇑(b.repr x))
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: case refine_2
A : Type u_1
inst✝¹¹ : CommRing A
B : Type u_2
inst✝¹⁰ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁹ : CommRing K
inst✝⁸ : Algebra A K
inst✝⁷ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝⁶ : CommRing L
inst✝⁵ : Algebra B L
inst✝⁴ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I : FractionalIdeal M K
C : Type u_5
W : Type u_6
inst✝³ : CommRing C
inst✝² : CommRing W
inst✝¹ : Algebra C W
P : Submonoid C
inst✝ : IsLocalization P W
g : B →+* C
hg : N ≤ Submonoid.comap g P
x : K
hx : x ∈ ↑I
⊢ (IsLocalization.map W (g.comp f) ⋯) x ∈ ↑(span C (⇑(IsLocalization.map W g hg) '' ↑(extended L hf I)))

Tactic:
refine Submodule.subset_span ⟨IsLocalization.map L f hf x, ?_, ?_⟩

Goal state after:
case refine_2.refine_1
A : Type u_1
inst✝¹¹ : CommRing A
B : Type u_2
inst✝¹⁰ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁹ : CommRing K
inst✝⁸ : Algebra A K
inst✝⁷ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝⁶ : CommRing L
inst✝⁵ : Algebra B L
inst✝⁴ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I : FractionalIdeal M K
C : Type u_5
W : Type u_6
inst✝³ : CommRing C
inst✝² : CommRing W
inst✝¹ : Algebra C W
P : Submonoid C
inst✝ : IsLocalization P W
g : B →+* C
hg : N ≤ Submonoid.comap g P
x : K
hx : x ∈ ↑I
⊢ (IsLocalization.map L f hf) x ∈ ↑(extended L hf I)

case refine_2.refine_2
A : Type u_1
inst✝¹¹ : CommRing A
B : Type u_2
inst✝¹⁰ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁹ : CommRing K
inst✝⁸ : Algebra A K
inst✝⁷ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝⁶ : CommRing L
inst✝⁵ : Algebra B L
inst✝⁴ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
I : FractionalIdeal M K
C : Type u_5
W : Type u_6
inst✝³ : CommRing C
inst✝² : CommRing W
inst✝¹ : Algebra C W
P : Submonoid C
inst✝ : IsLocalization P W
g : B →+* C
hg : N ≤ Submonoid.comap g P
x : K
hx : x ∈ ↑I
⊢ (IsLocalization.map W g hg) ((IsLocalization.map L f hf) x) = (IsLocalization.map W (g.comp f) ⋯) x
Module: Mathlib.RingTheory.Ideal.Int

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
inst✝² : Algebra.IsIntegral ℤ R
P : Ideal R
inst✝¹ : P.IsPrime
inst✝ : NeZero P
⊢ Prime (absNorm (under ℤ P))

Tactic:
Nat.prime_iff_prime_int,

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDomain R
inst✝² : Algebra.IsIntegral ℤ R
P : Ideal R
inst✝¹ : P.IsPrime
inst✝ : NeZero P
⊢ _root_.Prime ↑(absNorm (under ℤ P))
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: R : Type u_1
inst✝⁸ : CommRing R
inst✝⁷ : IsDomain R
S : Type u_3
inst✝⁶ : CommRing S
inst✝⁵ : IsDomain S
inst✝⁴ : IsIntegrallyClosed R
inst✝³ : IsIntegrallyClosed S
inst✝² : Algebra R S
inst✝¹ : Module.Finite R S
inst✝ : IsTorsionFree R S
x : R
hx : x ∈ ⇑(Algebra.intNorm R S) '' ↑⊥
⊢ x = 0

Tactic:
by simpa using hx

Goal state after:
Module: Mathlib.RingTheory.Ideal.Norm.RelNorm

Goal state before:
pp: case pos
R : Type u_1
inst✝¹³ : CommRing R
inst✝¹² : IsDomain R
S : Type u_3
inst✝¹¹ : CommRing S
inst✝¹⁰ : IsDomain S
inst✝⁹ : IsIntegrallyClosed R
inst✝⁸ : IsIntegrallyClosed S
inst✝⁷ : Algebra R S
inst✝⁶ : Module.Finite R S
inst✝⁵ : IsTorsionFree R S
inst✝⁴ : IsDedekindDomain R
inst✝³ : IsDedekindDomain S
P : Ideal S
p : Ideal R
hPp : P.LiesOver p
inst✝² : p.IsMaximal
inst✝¹ : P.IsPrime
inst✝ : IsGalois (FractionRing R) (FractionRing S)
this✝¹ : P.IsMaximal
G : Type u_3 := Gal(FractionRing S/FractionRing R)
this✝ : MulSemiringAction Gal(FractionRing S/FractionRing R) S :=
  IsIntegralClosure.MulSemiringAction R (FractionRing R) (FractionRing S) S
this : IsGaloisGroup G R S
hp : p = ⊥
h : P.inertiaDeg R ≠ 0
hP : P = ⊥
⊢ ⊥.inertiaDeg R ≠ 0

Tactic:
rwa [hP] at h

Goal state after:
Module: Mathlib.RingTheory.Prime

Goal state before:
pp: case insert
R : Type u_1
inst✝² : CommMonoidWithZero R
inst✝¹ : IsCancelMulZero R
α : Type u_2
inst✝ : DecidableEq α
p : α → R
i : α
s : Finset α
his : i ∉ s
ih :
  ∀ {x y a : R},
    (∀ i ∈ s, Prime (p i)) →
      x * y = a * ∏ i ∈ s, p i →
        ∃ t u b c, t ∪ u = s ∧ Disjoint t u ∧ a = b * c ∧ x = b * ∏ i ∈ t, p i ∧ y = c * ∏ i ∈ u, p i
a : R
hp : ∀ i_1 ∈ insert i s, Prime (p i_1)
hpi : Prime (p i)
t u : Finset α
b c : R
htus : t ∪ u = s
htu : Disjoint t u
hbc : a * p i = b * c
hx : (b * ∏ i ∈ t, p i) * (c * ∏ i ∈ u, p i) = a * p i * ∏ x ∈ s, p x
hit : i ∉ t
hiu : i ∉ u
⊢ ∃ t_1 u_1 b_1 c_1,
    t_1 ∪ u_1 = insert i s ∧
      Disjoint t_1 u_1 ∧
        a = b_1 * c_1 ∧ b * ∏ i ∈ t, p i = b_1 * ∏ i ∈ t_1, p i ∧ c * ∏ i ∈ u, p i = c_1 * ∏ i ∈ u_1, p i

Tactic:
obtain ⟨d, rfl⟩ | ⟨d, rfl⟩ : p i ∣ b ∨ p i ∣ c := hpi.dvd_or_dvd ⟨a, by rw [← hbc, mul_comm]⟩

Goal state after:
case insert.inl
R : Type u_1
inst✝² : CommMonoidWithZero R
inst✝¹ : IsCancelMulZero R
α : Type u_2
inst✝ : DecidableEq α
p : α → R
i : α
s : Finset α
his : i ∉ s
ih :
  ∀ {x y a : R},
    (∀ i ∈ s, Prime (p i)) →
      x * y = a * ∏ i ∈ s, p i →
        ∃ t u b c, t ∪ u = s ∧ Disjoint t u ∧ a = b * c ∧ x = b * ∏ i ∈ t, p i ∧ y = c * ∏ i ∈ u, p i
a : R
hp : ∀ i_1 ∈ insert i s, Prime (p i_1)
hpi : Prime (p i)
t u : Finset α
c : R
htus : t ∪ u = s
htu : Disjoint t u
hit : i ∉ t
hiu : i ∉ u
d : R
hbc : a * p i = p i * d * c
hx : (p i * d * ∏ i ∈ t, p i) * (c * ∏ i ∈ u, p i) = a * p i * ∏ x ∈ s, p x
⊢ ∃ t_1 u_1 b c_1,
    t_1 ∪ u_1 = insert i s ∧
      Disjoint t_1 u_1 ∧
        a = b * c_1 ∧ p i * d * ∏ i ∈ t, p i = b * ∏ i ∈ t_1, p i ∧ c * ∏ i ∈ u, p i = c_1 * ∏ i ∈ u_1, p i

case insert.inr
R : Type u_1
inst✝² : CommMonoidWithZero R
inst✝¹ : IsCancelMulZero R
α : Type u_2
inst✝ : DecidableEq α
p : α → R
i : α
s : Finset α
his : i ∉ s
ih :
  ∀ {x y a : R},
    (∀ i ∈ s, Prime (p i)) →
      x * y = a * ∏ i ∈ s, p i →
        ∃ t u b c, t ∪ u = s ∧ Disjoint t u ∧ a = b * c ∧ x = b * ∏ i ∈ t, p i ∧ y = c * ∏ i ∈ u, p i
a : R
hp : ∀ i_1 ∈ insert i s, Prime (p i_1)
hpi : Prime (p i)
t u : Finset α
b : R
htus : t ∪ u = s
htu : Disjoint t u
hit : i ∉ t
hiu : i ∉ u
d : R
hbc : a * p i = b * (p i * d)
hx : (b * ∏ i ∈ t, p i) * (p i * d * ∏ i ∈ u, p i) = a * p i * ∏ x ∈ s, p x
⊢ ∃ t_1 u_1 b_1 c,
    t_1 ∪ u_1 = insert i s ∧
      Disjoint t_1 u_1 ∧
        a = b_1 * c ∧ b * ∏ i ∈ t, p i = b_1 * ∏ i ∈ t_1, p i ∧ p i * d * ∏ i ∈ u, p i = c * ∏ i ∈ u_1, p i
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: R : Type u
K : Type v
L : Type z
p : R
inst✝⁹ : CommRing R
inst✝⁸ : Field K
inst✝⁷ : Field L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : Algebra R K
inst✝³ : IsScalarTower R K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝ : Module.Finite K L
P : R[X] := minpoly R B.gen
hei : P.IsEisensteinAt (R ∙ p)
hndiv : ¬p ^ 2 ∣ P.coeff 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
this : Module.IsTorsionFree R L
Q₁ : R[X]
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
hQ : (aeval B.gen) Q = p • z
hQzero : ¬Q = 0
j : ℕ
hj : j + 1 ∈ range (Q.natDegree + 1)
H : j + 1 + 1 ≤ P.natDegree
Hj : Q.natDegree + 1 = j + 1 + (Q.natDegree - j)
g : ℕ → R
hg : ∀ m ≤ j, m ∈ range (Q.natDegree + 1) → Q.coeff m = p * g m
k : ℕ
hk : k ∈ range (j + 1)
⊢ Q.coeff k • B.gen ^ k = (algebraMap R L) p * g k • B.gen ^ k

Tactic:
hg k (mem_range_succ_iff.1 hk)
        (mem_range_succ_iff.2
          (le_trans (mem_range_succ_iff.1 hk) (succ_le_iff.1 (mem_range_succ_iff.1 hj)).le)),

Goal state after:
R : Type u
K : Type v
L : Type z
p : R
inst✝⁹ : CommRing R
inst✝⁸ : Field K
inst✝⁷ : Field L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra R L
inst✝⁴ : Algebra R K
inst✝³ : IsScalarTower R K L
inst✝² : IsDomain R
inst✝¹ : IsFractionRing R K
inst✝ : IsIntegrallyClosed R
B : PowerBasis K L
hp : _root_.Prime p
hBint : IsIntegral R B.gen
z : L
hzint : IsIntegral R z
this✝ : Module.Finite K L
P : R[X] := minpoly R B.gen
hei : P.IsEisensteinAt (R ∙ p)
hndiv : ¬p ^ 2 ∣ P.coeff 0
hP : P = minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
this : Module.IsTorsionFree R L
Q₁ : R[X]
Q : R[X] := Q₁ %ₘ P
hQ₁ : Q = Q₁ %ₘ P
hQ : (aeval B.gen) Q = p • z
hQzero : ¬Q = 0
j : ℕ
hj : j + 1 ∈ range (Q.natDegree + 1)
H : j + 1 + 1 ≤ P.natDegree
Hj : Q.natDegree + 1 = j + 1 + (Q.natDegree - j)
g : ℕ → R
hg : ∀ m ≤ j, m ∈ range (Q.natDegree + 1) → Q.coeff m = p * g m
k : ℕ
hk : k ∈ range (j + 1)
⊢ (p * g k) • B.gen ^ k = (algebraMap R L) p * g k • B.gen ^ k
Module: Mathlib.RingTheory.DedekindDomain.Different

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁷ : CommRing A
inst✝¹⁶ : Field K
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Field L
inst✝¹³ : Algebra A K
inst✝¹² : Algebra B L
inst✝¹¹ : Algebra A B
inst✝¹⁰ : Algebra K L
inst✝⁹ : Algebra A L
inst✝⁸ : IsScalarTower A K L
inst✝⁷ : IsScalarTower A B L
inst✝⁶ : IsFractionRing A K
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : FiniteDimensional K L
inst✝³ : IsIntegralClosure B A L
inst✝² : Algebra.IsSeparable K L
I : Submodule B L
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b : Basis ι K L
hb : ∀ (i : ι), IsIntegral A (b i)
a x : L
ha : a ∈ I
hx : x ∈ Iᵛ
hinv : IsUnit (traceMatrix K ⇑b).det
H : (traceMatrix K ⇑b).cramer (traceMatrix K ⇑b *ᵥ b.equivFun (x * a)) = (traceMatrix K ⇑b).det • b.equivFun (x * a)
this : Function.Injective (traceMatrix K ⇑b).mulVec
⊢ IsIntegral A (discr K ⇑b • a * x)

Tactic:
traceMatrix_of_basis_mulVec

Goal state after:
A : Type u_1
K : Type u_2
L : Type u
B : Type u_3
inst✝¹⁷ : CommRing A
inst✝¹⁶ : Field K
inst✝¹⁵ : CommRing B
inst✝¹⁴ : Field L
inst✝¹³ : Algebra A K
inst✝¹² : Algebra B L
inst✝¹¹ : Algebra A B
inst✝¹⁰ : Algebra K L
inst✝⁹ : Algebra A L
inst✝⁸ : IsScalarTower A K L
inst✝⁷ : IsScalarTower A B L
inst✝⁶ : IsFractionRing A K
inst✝⁵ : IsIntegrallyClosed A
inst✝⁴ : FiniteDimensional K L
inst✝³ : IsIntegralClosure B A L
inst✝² : Algebra.IsSeparable K L
I : Submodule B L
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : Fintype ι
b : Basis ι K L
hb : ∀ (i : ι), IsIntegral A (b i)
a x : L
ha : a ∈ I
hx : x ∈ Iᵛ
hinv : IsUnit (traceMatrix K ⇑b).det
H : ((traceMatrix K ⇑b).cramer fun i ↦ (Algebra.trace K L) (x * a * b i)) = (traceMatrix K ⇑b).det • b.equivFun (x * a)
this : Function.Injective (traceMatrix K ⇑b).mulVec
⊢ IsIntegral A (discr K ⇑b • a * x)