mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: case inr.hi.hab
k : Type u_1
inst✝ : Field k
p q r : ℕ
u v w : k
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
d : ℕ
a b c : k[X]
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
eq_d : a.natDegree = d + 1
ad : derivative a = 0
bd : derivative b = 0
cd : derivative c = 0
ca : k[X]
ca_nz : ca ≠ 0
cb : k[X]
cb_nz : cb ≠ 0
cc : k[X]
cc_nz : cc ≠ 0
ch : ℕ := ⋯
chn0 : ch ≠ 0
eq_a : a = (expand k ch) ca
eq_deg_a : a.natDegree = ca.natDegree * ch
eq_b : b = (expand k ch) cb
eq_deg_b : b.natDegree = cb.natDegree * ch
eq_c : c = (expand k ch) cc
eq_deg_c : c.natDegree = cc.natDegree * ch
hnca : ¬ca.natDegree = 0
⊢ IsCoprime ((expand k ch) ca) ((expand k ch) cb)

Tactic:
← eq_a,

Goal state after:
case inr.hi.hab
k : Type u_1
inst✝ : Field k
p q r : ℕ
u v w : k
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
d : ℕ
a b c : k[X]
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
eq_d : a.natDegree = d + 1
ad : derivative a = 0
bd : derivative b = 0
cd : derivative c = 0
ca : k[X]
ca_nz : ca ≠ 0
cb : k[X]
cb_nz : cb ≠ 0
cc : k[X]
cc_nz : cc ≠ 0
ch : ℕ := ringChar k
chn0 : ch ≠ 0
eq_a : a = (expand k ch) ca
eq_deg_a : a.natDegree = ca.natDegree * ch
eq_b : b = (expand k ch) cb
eq_deg_b : b.natDegree = cb.natDegree * ch
eq_c : c = (expand k ch) cc
eq_deg_c : c.natDegree = cc.natDegree * ch
hnca : ¬ca.natDegree = 0
⊢ IsCoprime a ((expand k ch) cb)
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: k : Type u_1
inst✝ : Field k
n : ℕ
hn : 3 ≤ n
chn : ↑n ≠ 0
a b c a' b' : k[X]
d : k[X] := gcd a b
hb : d ≠ 0 ∧ b' ≠ 0
eq_a : a = a' * d
eq_b : b = b' * d
hd : d ≠ 0
c' : k[X]
heq : a' ^ n + b' ^ n = c' ^ n
hc : d ≠ 0 ∧ c' ≠ 0
eq_c : c = c' * d
ca' : k
ha : d ≠ 0 ∧ ca' ≠ 0
ha' : C ca' = a'
cb' : k
hb' : C cb' = b'
cc' : k
hc' : C cc' = c'
⊢ IsUnit (C ca') ∧ IsUnit (C cb') ∧ IsUnit (C cc')

Tactic:
rw [← hb', C_ne_zero] at hb

Goal state after:
k : Type u_1
inst✝ : Field k
n : ℕ
hn : 3 ≤ n
chn : ↑n ≠ 0
a b c a' b' : k[X]
d : k[X] := gcd a b
eq_a : a = a' * d
eq_b : b = b' * d
hd : d ≠ 0
c' : k[X]
heq : a' ^ n + b' ^ n = c' ^ n
hc : d ≠ 0 ∧ c' ≠ 0
eq_c : c = c' * d
ca' : k
ha : d ≠ 0 ∧ ca' ≠ 0
ha' : C ca' = a'
cb' : k
hb : d ≠ 0 ∧ cb' ≠ 0
hb' : C cb' = b'
cc' : k
hc' : C cc' = c'
⊢ IsUnit (C ca') ∧ IsUnit (C cb') ∧ IsUnit (C cc')
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: case inr
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
hw : w.IsComplex
⊢ Nontrivial (mixedSpace K)

Tactic:
have : Nonempty {w : InfinitePlace K // IsComplex w} := ⟨⟨w, hw⟩⟩

Goal state after:
case inr
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
hw : w.IsComplex
this : Nonempty { w // w.IsComplex }
⊢ Nontrivial (mixedSpace K)
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 (w.embedding.comp (algebraMap k K))
⊢ ComplexEmbedding.IsReal w.embedding

Tactic:
rw [← isReal_mk_iff, ← comap_mk, mk_embedding] at hw

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 : (w.comap (algebraMap k K)).IsReal
⊢ ComplexEmbedding.IsReal w.embedding
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ 0 ≤ ↑(discr K) ↔ Even (nrComplexPlaces K)

Tactic:
discr_eq_basisMatrix_det_sq,

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ 0 ≤ (basisMatrix K).det ^ 2 ↔ Even (nrComplexPlaces K)
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
φ : K →+* ℂ
H : ∀ (σ : Gal(K/k)), ComplexEmbedding.IsConj φ σ → σ = 1
hφ : (mk φ).IsComplex ∧ ((mk φ).comap (algebraMap k K)).IsReal
⊢ False

Tactic:
rw [comap_mk, isReal_mk_iff, ← not_isReal_iff_isComplex, isReal_mk_iff,
    ← ComplexEmbedding.isConj_one_iff (k := k)] at hφ

Goal state after:
k : Type u_1
inst✝³ : Field k
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra k K
inst✝ : IsGalois k K
φ : K →+* ℂ
H : ∀ (σ : Gal(K/k)), ComplexEmbedding.IsConj φ σ → σ = 1
hφ : ¬ComplexEmbedding.IsConj φ 1 ∧ ComplexEmbedding.IsReal (φ.comp (algebraMap k K))
⊢ False
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: case refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : ℝ
x : mixedSpace K
hx : x ∈ convexBodySum K B
y : mixedSpace K
hy : y ∈ convexBodySum K B
⊢ ‖y‖ ≤ B

Tactic:
exact le_trans (norm_le_convexBodySumFun y) hy

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

Goal state before:
pp: case inl
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (K →+* ℂ) → ℂ
hx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)
w : { w // w.IsReal }
⊢ ↑(x (↑w).embedding).re =
    ∑ x_1, 1 w x_1 * x (↑x_1).embedding +
      ∑ x_1, (0 w (x_1, 0) * x (↑x_1).embedding + 0 w (x_1, 1) * x (ComplexEmbedding.conjugate (↑x_1).embedding))

Tactic:
one_apply,

Goal state after:
case inl
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (K →+* ℂ) → ℂ
hx : ∀ (φ : K →+* ℂ), (starRingEnd ℂ) (x φ) = x (ComplexEmbedding.conjugate φ)
w : { w // w.IsReal }
⊢ ↑(x (↑w).embedding).re =
    ∑ x_1, (if w = x_1 then 1 else 0) * x (↑x_1).embedding +
      ∑ x_1, (0 w (x_1, 0) * x (↑x_1).embedding + 0 w (x_1, 1) * x (ComplexEmbedding.conjugate (↑x_1).embedding))
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) *
      (2⁻¹ ^ nrComplexPlaces K *
        (↑(sqrt ‖discr K‖₊) *
          (2 ^ finrank ℚ K * (↑(2 ^ nrRealPlaces K * (pi / 2) ^ nrComplexPlaces K) / ↑↑(finrank ℚ K).factorial)⁻¹))) =
    ↑(FractionalIdeal.absNorm ↑I) *
      (2⁻¹ ^ nrComplexPlaces K *
        (↑(sqrt ‖discr K‖₊) *
          (2 ^ finrank ℚ K * (2 ^ nrRealPlaces K * (π / 2) ^ nrComplexPlaces K / ↑(finrank ℚ K).factorial)⁻¹)))

Tactic:
NNReal.coe_mul,

Goal state after:
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) *
      (2⁻¹ ^ nrComplexPlaces K *
        (↑(sqrt ‖discr K‖₊) *
          (2 ^ finrank ℚ K *
            (↑(2 ^ nrRealPlaces K) * ↑((pi / 2) ^ nrComplexPlaces K) / ↑↑(finrank ℚ K).factorial)⁻¹))) =
    ↑(FractionalIdeal.absNorm ↑I) *
      (2⁻¹ ^ nrComplexPlaces K *
        (↑(sqrt ‖discr K‖₊) *
          (2 ^ finrank ℚ K * (2 ^ nrRealPlaces K * (π / 2) ^ nrComplexPlaces K / ↑(finrank ℚ K).factorial)⁻¹)))
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: case fst
K : Type u_1
inst✝ : Field K
s : Set { w // w.IsReal }
x : mixedSpace K
w : { w // w.IsReal }
⊢ ((negAt s).symm x).1 w = ((negAt s) x).1 w

Tactic:
by_cases hw : w ∈ s
    · simp_rw [negAt_apply_isReal_and_mem _ hw, negAt, prodCongr_symm,
        prodCongr_apply, piCongrRight_symm_apply, if_pos hw, symm_neg,
        ContinuousLinearEquiv.neg_apply]
    · simp_rw [negAt_apply_isReal_and_notMem _ hw, negAt, prodCongr_symm,
        prodCongr_apply, piCongrRight_symm_apply, if_neg hw, refl_symm,
        refl_apply]

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
A : Set (mixedSpace K)
inst✝ : NumberField K
hm : MeasurableSet A
⊢ plusPart A = A ∩ ⋂ w, {x | 0 < x.1 w}

pp: K : Type u_1
inst✝¹ : Field K
A : Set (mixedSpace K)
inst✝ : NumberField K
hm : MeasurableSet A
⊢ MeasurableSet (A ∩ ⋂ w, {x | 0 < x.1 w})

Tactic:
· ext; simp

Goal state after:
K : Type u_1
inst✝¹ : Field K
A : Set (mixedSpace K)
inst✝ : NumberField K
hm : MeasurableSet A
⊢ MeasurableSet (A ∩ ⋂ w, {x | 0 < x.1 w})
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:
rw [normAtPlace_apply_of_isComplex hw, Complex.norm_of_nonneg hx]

Goal state after:
Module: Mathlib.NumberTheory.RamificationInertia.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝²⁰ : CommRing A
inst✝¹⁹ : CommRing B
inst✝¹⁸ : Algebra A B
p : Ideal A
P : Ideal B
hPp : P.IsPrime
hp : P.LiesOver p
G : Type u_3
inst✝¹⁷ : Group G
inst✝¹⁶ : Finite G
inst✝¹⁵ : MulSemiringAction G B
inst✝¹⁴ : IsGaloisGroup G A B
C : Type u_4
inst✝¹³ : CommRing C
inst✝¹² : Algebra A C
inst✝¹¹ : Algebra B C
inst✝¹⁰ : Nonempty ↑(P.primesOver C)
inst✝⁹ : IsScalarTower A B C
GAC : Type u_5
inst✝⁸ : Group GAC
inst✝⁷ : Finite GAC
inst✝⁶ : MulSemiringAction GAC C
inst✝⁵ : IsGaloisGroup GAC A C
GBC : Type u_6
inst✝⁴ : Group GBC
inst✝³ : Finite GBC
inst✝² : MulSemiringAction GBC C
inst✝¹ : IsGaloisGroup GBC B C
inst✝ : Flat B C
Q : Ideal C
left✝ : Q.IsPrime
hQ : Q.LiesOver P
this : Q.LiesOver p
⊢ p.ramificationIdxIn B * P.ramificationIdxIn C = p.ramificationIdxIn C

Tactic:
rw [ramificationIdxIn_eq_ramificationIdx p P G, ramificationIdxIn_eq_ramificationIdx p Q GAC,
    ramificationIdxIn_eq_ramificationIdx P Q GBC, ← ramificationIdx_tower P Q]

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

Goal state before:
pp: case pos
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
k : ℕ
hk : φ n = k + k
key : 0 + 2 * nrComplexPlaces K = Module.finrank ℚ K
⊢ nrComplexPlaces K = φ n / 2

Tactic:
zero_add,

Goal state after:
case pos
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
k : ℕ
hk : φ n = k + k
key : 2 * nrComplexPlaces K = Module.finrank ℚ K
⊢ nrComplexPlaces K = φ n / 2
Module: Mathlib.NumberTheory.RamificationInertia.Galois

Goal state before:
pp: R : Type u_1
K : Type u_2
L : Type u_3
S : Type u_4
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
inst✝¹² : Field K
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra R K
inst✝⁹ : IsFractionRing R K
inst✝⁸ : Algebra S L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : IsScalarTower R S L
inst✝⁴ : IsScalarTower R K L
inst✝³ : IsIntegralClosure S R L
inst✝² : FiniteDimensional K L
inst✝¹ : IsDedekindDomain R
inst✝ : IsGalois K L
p : Ideal R
P₁ P₂ : Ideal S
hP₁ : P₁ ∈ p.primesOver S
hP₂ : P₂ ∈ p.primesOver S
this✝⁵ : IsDomain S
this✝⁴ : IsDedekindDomain S
this✝³ : Module.Finite R S
this✝² : P₁.IsPrime
this✝¹ : P₁.LiesOver p
this✝ : P₂.IsPrime
this : P₂.LiesOver p
⊢ ∃ σ, comap ((galRestrict R K L S) σ) P₁ = P₂

Tactic:
have : IsFractionRing S L := IsIntegralClosure.isFractionRing_of_finite_extension R K L S

Goal state after:
R : Type u_1
K : Type u_2
L : Type u_3
S : Type u_4
inst✝¹⁵ : CommRing R
inst✝¹⁴ : CommRing S
inst✝¹³ : Algebra R S
inst✝¹² : Field K
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra R K
inst✝⁹ : IsFractionRing R K
inst✝⁸ : Algebra S L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra R L
inst✝⁵ : IsScalarTower R S L
inst✝⁴ : IsScalarTower R K L
inst✝³ : IsIntegralClosure S R L
inst✝² : FiniteDimensional K L
inst✝¹ : IsDedekindDomain R
inst✝ : IsGalois K L
p : Ideal R
P₁ P₂ : Ideal S
hP₁ : P₁ ∈ p.primesOver S
hP₂ : P₂ ∈ p.primesOver S
this✝⁶ : IsDomain S
this✝⁵ : IsDedekindDomain S
this✝⁴ : Module.Finite R S
this✝³ : P₁.IsPrime
this✝² : P₁.LiesOver p
this✝¹ : P₂.IsPrime
this✝ : P₂.LiesOver p
this : IsFractionRing S L
⊢ ∃ σ, comap ((galRestrict R K L S) σ) P₁ = P₂
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 B 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.IsReal}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
this✝¹ : CharZero ↥K
this✝ : NumberField ↥K
w₀ : InfinitePlace ↥↑⟨K, hK₀⟩
hw₀ : w₀ ∈ {w | w.IsReal}
this : minkowskiBound (↥K) 1 < ↑(convexBodyLTFactor ↥K) * ↑B
x : 𝓞 ↥K
hx₁ : ℚ⟮↑x⟯ = ⊤
hx₂ : ∀ (w : InfinitePlace ↥K), w ↑x < ↑(max B 1)
hx : IsIntegral ℤ ((algebraMap (𝓞 ↥K) ↥K) x)
i : ℕ
⊢ max (↑B) 1 ^ finrank ℚ ↥K * ↑((finrank ℚ ↥K).choose (finrank ℚ ↥K / 2)) ≤ ↑(max B 1 ^ D * ↑(D.choose (D / 2)))

Tactic:
NNReal.coe_mul,

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 B 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.IsReal}.Nonempty ∧ |discr ↥↑K| ≤ ↑N}
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
this✝¹ : CharZero ↥K
this✝ : NumberField ↥K
w₀ : InfinitePlace ↥↑⟨K, hK₀⟩
hw₀ : w₀ ∈ {w | w.IsReal}
this : minkowskiBound (↥K) 1 < ↑(convexBodyLTFactor ↥K) * ↑B
x : 𝓞 ↥K
hx₁ : ℚ⟮↑x⟯ = ⊤
hx₂ : ∀ (w : InfinitePlace ↥K), w ↑x < ↑(max B 1)
hx : IsIntegral ℤ ((algebraMap (𝓞 ↥K) ↥K) x)
i : ℕ
⊢ max (↑B) 1 ^ finrank ℚ ↥K * ↑((finrank ℚ ↥K).choose (finrank ℚ ↥K / 2)) ≤ ↑(max B 1 ^ D) * ↑↑(D.choose (D / 2))
Module: Mathlib.RingTheory.LocalRing.Quotient

Goal state before:
pp: case hNN
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsLocalRing R
inst✝ : Module.Finite R S
s : Set S
H :
  Submodule.restrictScalars R (span (R ⧸ p) (⇑(Ideal.Quotient.mk pS) '' s)) =
    map (↑(IsScalarTower.toAlgHom R S (S ⧸ pS))) (span R s)
hs : span (R ⧸ p) (⇑(Ideal.Quotient.mk pS) '' s) = ⊤
⊢ ⊤ ≤ span R s ⊔ p • ⊤

Tactic:
Ideal.smul_top_eq_map

Goal state after:
case hNN
R : Type u_1
S : Type u_2
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : IsLocalRing R
inst✝ : Module.Finite R S
s : Set S
H :
  Submodule.restrictScalars R (span (R ⧸ p) (⇑(Ideal.Quotient.mk pS) '' s)) =
    map (↑(IsScalarTower.toAlgHom R S (S ⧸ pS))) (span R s)
hs : span (R ⧸ p) (⇑(Ideal.Quotient.mk pS) '' s) = ⊤
⊢ ⊤ ≤ span R s ⊔ Submodule.restrictScalars R pS
Module: Mathlib.RingTheory.Trace.Quotient

Goal state before:
pp: R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : Free R S
inst✝¹ : Module.Finite R S
inst✝ : IsLocalRing R
x : S
ι : Type u_2 := Free.ChooseBasisIndex R S
b : Basis ι R S := Free.chooseBasis R S
⊢ ((leftMulMatrix (basisQuotient b)) ((Ideal.Quotient.mk pS) x)).trace =
    (Ideal.Quotient.mk p) ((leftMulMatrix b) x).trace

Tactic:
AddMonoidHom.map_trace

Goal state after:
R : Type u_1
S : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : Free R S
inst✝¹ : Module.Finite R S
inst✝ : IsLocalRing R
x : S
ι : Type u_2 := Free.ChooseBasisIndex R S
b : Basis ι R S := Free.chooseBasis R S
⊢ ((leftMulMatrix (basisQuotient b)) ((Ideal.Quotient.mk pS) x)).trace =
    (((leftMulMatrix b) x).map ⇑(Ideal.Quotient.mk p)).trace
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 (sqrt (1 + B ^ 2)) 1 ^ D * ↑(D.choose (D / 2)))

Tactic:
NNReal.coe_mul,

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 (sqrt (1 + B ^ 2)) 1 ^ D) * ↑↑(D.choose (D / 2))
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: case refine_1
A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
x : K
x✝ : L
hy : x✝ ∈ span B (⇑(IsLocalization.map L f hf) '' ↑(spanSingleton M x))
a : A
hw : a • x ∈ ↑(spanSingleton M x)
⊢ (IsLocalization.map L f hf) ((algebraMap A K) a) * (IsLocalization.map L f hf) x ∈ B ∙ (IsLocalization.map L f hf) x

Tactic:
IsLocalization.map_eq,

Goal state after:
case refine_1
A : Type u_1
inst✝⁷ : CommRing A
B : Type u_2
inst✝⁶ : CommRing B
f : A →+* B
K : Type u_3
M : Submonoid A
inst✝⁵ : CommRing K
inst✝⁴ : Algebra A K
inst✝³ : IsLocalization M K
L : Type u_4
N : Submonoid B
inst✝² : CommRing L
inst✝¹ : Algebra B L
inst✝ : IsLocalization N L
hf : M ≤ Submonoid.comap f N
x : K
x✝ : L
hy : x✝ ∈ span B (⇑(IsLocalization.map L f hf) '' ↑(spanSingleton M x))
a : A
hw : a • x ∈ ↑(spanSingleton M x)
⊢ (algebraMap B L) (f a) * (IsLocalization.map L f hf) x ∈ B ∙ (IsLocalization.map L f hf) x
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: case pos
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝¹⁸ : CommRing A
inst✝¹⁷ : IsDomain A
inst✝¹⁶ : CommRing B
inst✝¹⁵ : IsDomain B
inst✝¹⁴ : Algebra A B
inst✝¹³ : IsTorsionFree A B
inst✝¹² : Field K
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra A K
inst✝⁹ : Algebra B L
inst✝⁸ : IsFractionRing A K
inst✝⁷ : IsFractionRing B L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra A L
inst✝⁴ : IsScalarTower A B L
inst✝³ : IsScalarTower A K L
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsDedekindDomain A
inst✝ : IsDedekindDomain B
I J : FractionalIdeal A⁰ K
h : 0 = J
hI : I = 0
⊢ I = J

Tactic:
← hI

Goal state after:
case pos
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝¹⁸ : CommRing A
inst✝¹⁷ : IsDomain A
inst✝¹⁶ : CommRing B
inst✝¹⁵ : IsDomain B
inst✝¹⁴ : Algebra A B
inst✝¹³ : IsTorsionFree A B
inst✝¹² : Field K
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra A K
inst✝⁹ : Algebra B L
inst✝⁸ : IsFractionRing A K
inst✝⁷ : IsFractionRing B L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra A L
inst✝⁴ : IsScalarTower A B L
inst✝³ : IsScalarTower A K L
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsDedekindDomain A
inst✝ : IsDedekindDomain B
I J : FractionalIdeal A⁰ K
h : I = J
hI : I = 0
⊢ I = J
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: case refine_1
p : ℕ
hp : Fact (Nat.Prime p)
⊢ ((cyclotomic p ℤ).comp (X + C 1)).Monic

Tactic:
refine (cyclotomic.monic p ℤ).comp (monic_X_add_C 1) fun h => ?_

Goal state after:
case refine_1
p : ℕ
hp : Fact (Nat.Prime p)
h : (X + C 1).natDegree = 0
⊢ False
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: case refine_3
p : ℕ
hp : Fact (Nat.Prime p)
⊢ eval (0 + 1) (∑ i ∈ range p, X ^ i) ∉ (ℤ ∙ ↑p) ^ 2

Tactic:
zero_add,

Goal state after:
case refine_3
p : ℕ
hp : Fact (Nat.Prime p)
⊢ eval 1 (∑ i ∈ range p, X ^ i) ∉ (ℤ ∙ ↑p) ^ 2
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: case refine_3
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ eval (0 + 1) (∑ i ∈ range p, (X ^ p ^ n) ^ i) ∉ (ℤ ∙ ↑p) ^ 2

Tactic:
zero_add,

Goal state after:
case refine_3
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ eval 1 (∑ i ∈ range p, (X ^ p ^ n) ^ i) ∉ (ℤ ∙ ↑p) ^ 2
Module: Mathlib.RingTheory.Prime

Goal state before:
pp: α : Type u_1
inst✝ : CommRing α
p : α
hp : Prime p
⊢ Prime (-p)

Tactic:
obtain ⟨h1, h2, h3⟩ := hp

Goal state after:
α : Type u_1
inst✝ : CommRing α
p : α
h1 : p ≠ 0
h2 : ¬IsUnit p
h3 : ∀ (a b : α), p ∣ a * b → p ∣ a ∨ p ∣ b
⊢ Prime (-p)
Module: Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral

Goal state before:
pp: case h.calc_3
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
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : (minpoly R B.gen).IsEisensteinAt (R ∙ p)
this : Module.Finite K L := finite B
P : R[X] := minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
finrank_K_L : Module.finrank K L = B.dim
deg_K_P : (minpoly K B.gen).natDegree = B.dim
deg_R_P : P.natDegree = B.dim
f : ℕ → L
hf : ∀ (i : ℕ), B.dim ≤ i → f i ∈ R[B.gen] ∧ (algebraMap R L) p * f i = B.gen ^ i
aux : ∀ i ∈ (range (Q.natDegree + 1)).erase 0, B.dim ≤ i + n
hintsum : IsIntegral R (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))
r : R
hr :
  (algebraMap R K) r = (Algebra.norm K) (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))
⊢ (algebraMap R K) p ^ Module.finrank K L *
      (Algebra.norm K) (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n)) =
    (algebraMap R K) p ^ n.succ * (algebraMap R K) r

Tactic:
finrank_K_L,

Goal state after:
case h.calc_3
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
Q : R[X]
hQ : (aeval B.gen) Q = p • z
hzint : IsIntegral R z
hei : (minpoly R B.gen).IsEisensteinAt (R ∙ p)
this : Module.Finite K L := finite B
P : R[X] := minpoly R B.gen
n : ℕ
hn : B.dim = n.succ
finrank_K_L : Module.finrank K L = B.dim
deg_K_P : (minpoly K B.gen).natDegree = B.dim
deg_R_P : P.natDegree = B.dim
f : ℕ → L
hf : ∀ (i : ℕ), B.dim ≤ i → f i ∈ R[B.gen] ∧ (algebraMap R L) p * f i = B.gen ^ i
aux : ∀ i ∈ (range (Q.natDegree + 1)).erase 0, B.dim ≤ i + n
hintsum : IsIntegral R (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))
r : R
hr :
  (algebraMap R K) r = (Algebra.norm K) (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n))
⊢ (algebraMap R K) p ^ B.dim *
      (Algebra.norm K) (z * B.gen ^ n - ∑ x ∈ (range (Q.natDegree + 1)).erase 0, Q.coeff x • f (x + n)) =
    (algebraMap R K) p ^ n.succ * (algebraMap R K) r