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

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

Tactic:
[

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

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
⊢ (normAtPlace w) ((negAt s) x) = (normAtPlace w) x
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

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

Tactic:
[

Goal state after:
case pos
K : Type u_1
inst✝ : Field K
s : Set { w // w.IsReal }
x : mixedSpace K
w : { w // w.IsReal }
hw : w ∈ s
⊢ ((negAt s).symm x).1 w = ((negAt s) x).1 w
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

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

Tactic:
[

Goal state after:
case neg
K : Type u_1
inst✝ : Field K
s : Set { w // w.IsReal }
x : mixedSpace K
w : { w // w.IsReal }
hw : w ∉ s
⊢ ((negAt s).symm x).1 w = ((negAt s) x).1 w
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: case refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ
h : minkowskiBound K I ≤ volume (convexBodySum K B)
hB : 0 ≤ B
h1 : 0 < (↑(finrank ℚ K))⁻¹
h2 : 0 ≤ B / ↑(finrank ℚ K)
h_fund :
  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)
    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
a : K
ha : a ∈ ↑↑I
hx : (mixedEmbedding K) a ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
h_nz : ⟨(mixedEmbedding K) a, hx⟩ ≠ 0
h_mem : ↑⟨(mixedEmbedding K) a, hx⟩ ∈ convexBodySum K B
⊢ ↑|(Algebra.norm ℚ) a| ^ (∑ x, ↑x.mult)⁻¹ ≤ (∏ i, ↑i a ^ ↑i.mult) ^ (∑ i, ↑i.mult)⁻¹

Tactic:
[

Goal state after:
case refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ
h : minkowskiBound K I ≤ volume (convexBodySum K B)
hB : 0 ≤ B
h1 : 0 < (↑(finrank ℚ K))⁻¹
h2 : 0 ≤ B / ↑(finrank ℚ K)
h_fund :
  IsAddFundamentalDomain (↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup)
    (fundamentalDomain (fractionalIdealLatticeBasis K I)) volume
this : Countable ↥(span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
a : K
ha : a ∈ ↑↑I
hx : (mixedEmbedding K) a ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
h_nz : ⟨(mixedEmbedding K) a, hx⟩ ≠ 0
h_mem : ↑⟨(mixedEmbedding K) a, hx⟩ ∈ convexBodySum K B
⊢ ↑|(Algebra.norm ℚ) a| ^ (∑ x, ↑x.mult)⁻¹ ≤ (∏ i, ↑i a ^ ↑i.mult) ^ (∑ i, ↑i.mult)⁻¹
Module: Mathlib.RingTheory.Invariant.Galois

Goal state before:
pp: A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Field K
inst✝¹² : Field L
inst✝¹¹ : Algebra A K
inst✝¹⁰ : Algebra B L
inst✝⁹ : IsFractionRing A K
inst✝⁸ : IsFractionRing B L
inst✝⁷ : Algebra A B
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra A L
inst✝⁴ : IsScalarTower A K L
inst✝³ : IsScalarTower A B L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsIntegralClosure B A L
inst✝ : FiniteDimensional K L
h✝ : IsGalois K L
h : IntermediateField.fixedField ⊤ = ⊥
this : MulSemiringAction Gal(L/K) B := IsIntegralClosure.MulSemiringAction A K L B
b : B
k : K
hk : (algebraMap K L) k = (algebraMap B L) b
hb : IsIntegral A ((algebraMap K L) k)
⊢ ∃ a, (algebraMap A B) a = b

Tactic:
isIntegral_algebraMap_iff (FaithfulSMul.algebraMap_injective K L)

Goal state after:
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝¹⁵ : CommRing A
inst✝¹⁴ : CommRing B
inst✝¹³ : Field K
inst✝¹² : Field L
inst✝¹¹ : Algebra A K
inst✝¹⁰ : Algebra B L
inst✝⁹ : IsFractionRing A K
inst✝⁸ : IsFractionRing B L
inst✝⁷ : Algebra A B
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra A L
inst✝⁴ : IsScalarTower A K L
inst✝³ : IsScalarTower A B L
inst✝² : IsIntegrallyClosed A
inst✝¹ : IsIntegralClosure B A L
inst✝ : FiniteDimensional K L
h✝ : IsGalois K L
h : IntermediateField.fixedField ⊤ = ⊥
this : MulSemiringAction Gal(L/K) B := IsIntegralClosure.MulSemiringAction A K L B
b : B
k : K
hk : (algebraMap K L) k = (algebraMap B L) b
hb : IsIntegral A k
⊢ ∃ a, (algebraMap A B) a = b
Module: Mathlib.RingTheory.Invariant.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
G : Type u_3
inst✝¹⁶ : Finite G
inst✝¹⁵ : Group G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : Algebra.IsInvariant A B G
P : Ideal A
Q : Ideal B
inst✝¹² : Q.LiesOver P
inst✝¹¹ : P.IsPrime
inst✝¹⁰ : Q.IsPrime
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra (A ⧸ P) K
inst✝⁵ : IsFractionRing (A ⧸ P) K
inst✝⁴ : Algebra (B ⧸ Q) L
inst✝³ : IsFractionRing (B ⧸ Q) L
inst✝² : Algebra (A ⧸ P) L
inst✝¹ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝ : IsScalarTower (A ⧸ P) K L
this✝ : Algebra.IsIntegral A B
this : Algebra.IsAlgebraic K L
x y : B ⧸ Q
hy : y ∈ nonZeroDivisors (B ⧸ Q)
a : A
ha : (Quotient.mk P) a ≠ 0
b : B
h : (Quotient.mk P) a • x = y * (Quotient.mk Q) b
⊢ (Polynomial.map (algebraMap K L) (minpoly K ((algebraMap (B ⧸ Q) L) x / (algebraMap (B ⧸ Q) L) y))).Splits

Tactic:
[

Goal state after:
A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
G : Type u_3
inst✝¹⁶ : Finite G
inst✝¹⁵ : Group G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : Algebra.IsInvariant A B G
P : Ideal A
Q : Ideal B
inst✝¹² : Q.LiesOver P
inst✝¹¹ : P.IsPrime
inst✝¹⁰ : Q.IsPrime
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra (A ⧸ P) K
inst✝⁵ : IsFractionRing (A ⧸ P) K
inst✝⁴ : Algebra (B ⧸ Q) L
inst✝³ : IsFractionRing (B ⧸ Q) L
inst✝² : Algebra (A ⧸ P) L
inst✝¹ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝ : IsScalarTower (A ⧸ P) K L
this✝ : Algebra.IsIntegral A B
this : Algebra.IsAlgebraic K L
x y : B ⧸ Q
hy : y ∈ nonZeroDivisors (B ⧸ Q)
a : A
ha : (Quotient.mk P) a ≠ 0
b : B
h : (Quotient.mk P) a • x = y * (Quotient.mk Q) b
⊢ (Polynomial.map (algebraMap K L) (minpoly K ((algebraMap (B ⧸ Q) L) x / (algebraMap (B ⧸ Q) L) y))).Splits
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: case refine_1.inl
K : Type u_1
inst✝ : Field K
A : Set (mixedSpace K)
hA : ∀ (x : mixedSpace K), x ∈ A ↔ (fun w ↦ ‖x.1 w‖, x.2) ∈ A
s : Set { w // w.IsReal }
x : mixedSpace K
hx : x ∈ A
right✝ : x ∈ {x | ∀ (w : { w // w.IsReal }), 0 < x.1 w}
⊢ (negAt s) x ∈ A

Tactic:
[

Goal state after:
case refine_1.inl
K : Type u_1
inst✝ : Field K
A : Set (mixedSpace K)
hA : ∀ (x : mixedSpace K), x ∈ A ↔ (fun w ↦ ‖x.1 w‖, x.2) ∈ A
s : Set { w // w.IsReal }
x : mixedSpace K
hx : x ∈ A
right✝ : x ∈ {x | ∀ (w : { w // w.IsReal }), 0 < x.1 w}
⊢ (negAt s) x ∈ A
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: case e'_4
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : 1 < finrank ℚ K
a : ℕ → ℝ := fun n ↦ ↑n ^ (n * 2) / ((4 / π) ^ n * ↑n.factorial ^ 2)
n m : ℕ
hmn✝ : 2 ≤ m
h_m : 4 / 9 * (3 * π / 4) ^ m ≤ a m
this : 3 ≤ (1 + 1 / ↑m) ^ (2 * m)
⊢ a (m + 1) = a m * (1 + 1 / ↑m) ^ (2 * m) / (4 / π)

Tactic:
[

Goal state after:
case e'_4
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : 1 < finrank ℚ K
a : ℕ → ℝ := fun n ↦ ↑n ^ (n * 2) / ((4 / π) ^ n * ↑n.factorial ^ 2)
n m : ℕ
hmn✝ : 2 ≤ m
h_m : 4 / 9 * (3 * π / 4) ^ m ≤ a m
this : 3 ≤ (1 + 1 / ↑m) ^ (2 * m)
⊢ a (m + 1) = a m * (1 + 1 / ↑m) ^ (2 * m) / (4 / π)
Module: Mathlib.RingTheory.Invariant.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
G : Type u_3
inst✝¹⁶ : Finite G
inst✝¹⁵ : Group G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : Algebra.IsInvariant A B G
P : Ideal A
Q : Ideal B
inst✝¹² : Q.LiesOver P
inst✝¹¹ : P.IsPrime
inst✝¹⁰ : Q.IsPrime
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra (A ⧸ P) K
inst✝⁵ : IsFractionRing (A ⧸ P) K
inst✝⁴ : Algebra (B ⧸ Q) L
inst✝³ : IsFractionRing (B ⧸ Q) L
inst✝² : Algebra (A ⧸ P) L
inst✝¹ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝ : IsScalarTower (A ⧸ P) K L
this✝ : Algebra.IsIntegral A B
this : Algebra.IsAlgebraic K L
x y : B ⧸ Q
hy : y ∈ nonZeroDivisors (B ⧸ Q)
a : A
ha : (algebraMap A (A ⧸ P)) a ≠ 0
b : B
h : (algebraMap A (A ⧸ P)) a • x = y * (algebraMap B (B ⧸ Q)) b
val✝ : Fintype G
p : A[X]
hp : Polynomial.map (algebraMap A B) p = MulSemiringAction.charpoly G b
h_monic : p.Monic
h_eval : (aeval b) p = 0
q : A[X] := p.comp (C a * X)
d : L := (algebraMap (B ⧸ Q) L) x / (algebraMap (B ⧸ Q) L) y
⊢ (algebraMap K L).comp (algebraMap (A ⧸ P) K) = (algebraMap (B ⧸ Q) L).comp (algebraMap (A ⧸ P) (B ⧸ Q))

Tactic:
[

Goal state after:
A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
G : Type u_3
inst✝¹⁶ : Finite G
inst✝¹⁵ : Group G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : Algebra.IsInvariant A B G
P : Ideal A
Q : Ideal B
inst✝¹² : Q.LiesOver P
inst✝¹¹ : P.IsPrime
inst✝¹⁰ : Q.IsPrime
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra (A ⧸ P) K
inst✝⁵ : IsFractionRing (A ⧸ P) K
inst✝⁴ : Algebra (B ⧸ Q) L
inst✝³ : IsFractionRing (B ⧸ Q) L
inst✝² : Algebra (A ⧸ P) L
inst✝¹ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝ : IsScalarTower (A ⧸ P) K L
this✝ : Algebra.IsIntegral A B
this : Algebra.IsAlgebraic K L
x y : B ⧸ Q
hy : y ∈ nonZeroDivisors (B ⧸ Q)
a : A
ha : (algebraMap A (A ⧸ P)) a ≠ 0
b : B
h : (algebraMap A (A ⧸ P)) a • x = y * (algebraMap B (B ⧸ Q)) b
val✝ : Fintype G
p : A[X]
hp : Polynomial.map (algebraMap A B) p = MulSemiringAction.charpoly G b
h_monic : p.Monic
h_eval : (aeval b) p = 0
q : A[X] := p.comp (C a * X)
d : L := (algebraMap (B ⧸ Q) L) x / (algebraMap (B ⧸ Q) L) y
⊢ (algebraMap K L).comp (algebraMap (A ⧸ P) K) = (algebraMap (B ⧸ Q) L).comp (algebraMap (A ⧸ P) (B ⧸ Q))
Module: Mathlib.RingTheory.Invariant.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
G : Type u_3
inst✝¹⁶ : Finite G
inst✝¹⁵ : Group G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : Algebra.IsInvariant A B G
P : Ideal A
Q : Ideal B
inst✝¹² : Q.LiesOver P
inst✝¹¹ : P.IsPrime
inst✝¹⁰ : Q.IsPrime
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra (A ⧸ P) K
inst✝⁵ : IsFractionRing (A ⧸ P) K
inst✝⁴ : Algebra (B ⧸ Q) L
inst✝³ : IsFractionRing (B ⧸ Q) L
inst✝² : Algebra (A ⧸ P) L
inst✝¹ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝ : IsScalarTower (A ⧸ P) K L
this✝ : Algebra.IsIntegral A B
this : Algebra.IsAlgebraic K L
x y : B ⧸ Q
hy : y ∈ nonZeroDivisors (B ⧸ Q)
a : A
ha : (algebraMap A (A ⧸ P)) a ≠ 0
b : B
h : (algebraMap A (A ⧸ P)) a • x = y * (algebraMap B (B ⧸ Q)) b
val✝ : Fintype G
p : A[X]
hp : Polynomial.map (algebraMap A B) p = MulSemiringAction.charpoly G b
h_monic : p.Monic
h_eval : (aeval b) p = 0
q : A[X] := p.comp (C a * X)
d : L := (algebraMap (B ⧸ Q) L) x / (algebraMap (B ⧸ Q) L) y
comm₁ : (algebraMap K L).comp (algebraMap (A ⧸ P) K) = (algebraMap (B ⧸ Q) L).comp (algebraMap (A ⧸ P) (B ⧸ Q))
⊢ (algebraMap (A ⧸ P) (B ⧸ Q)).comp (algebraMap A (A ⧸ P)) = (algebraMap B (B ⧸ Q)).comp (algebraMap A B)

Tactic:
[

Goal state after:
A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
G : Type u_3
inst✝¹⁶ : Finite G
inst✝¹⁵ : Group G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : Algebra.IsInvariant A B G
P : Ideal A
Q : Ideal B
inst✝¹² : Q.LiesOver P
inst✝¹¹ : P.IsPrime
inst✝¹⁰ : Q.IsPrime
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra (A ⧸ P) K
inst✝⁵ : IsFractionRing (A ⧸ P) K
inst✝⁴ : Algebra (B ⧸ Q) L
inst✝³ : IsFractionRing (B ⧸ Q) L
inst✝² : Algebra (A ⧸ P) L
inst✝¹ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝ : IsScalarTower (A ⧸ P) K L
this✝ : Algebra.IsIntegral A B
this : Algebra.IsAlgebraic K L
x y : B ⧸ Q
hy : y ∈ nonZeroDivisors (B ⧸ Q)
a : A
ha : (algebraMap A (A ⧸ P)) a ≠ 0
b : B
h : (algebraMap A (A ⧸ P)) a • x = y * (algebraMap B (B ⧸ Q)) b
val✝ : Fintype G
p : A[X]
hp : Polynomial.map (algebraMap A B) p = MulSemiringAction.charpoly G b
h_monic : p.Monic
h_eval : (aeval b) p = 0
q : A[X] := p.comp (C a * X)
d : L := (algebraMap (B ⧸ Q) L) x / (algebraMap (B ⧸ Q) L) y
comm₁ : (algebraMap K L).comp (algebraMap (A ⧸ P) K) = (algebraMap (B ⧸ Q) L).comp (algebraMap (A ⧸ P) (B ⧸ Q))
⊢ (algebraMap (A ⧸ P) (B ⧸ Q)).comp (algebraMap A (A ⧸ P)) = (algebraMap B (B ⧸ Q)).comp (algebraMap A B)
Module: Mathlib.RingTheory.Invariant.Galois

Goal state before:
pp: A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
G : Type u_3
inst✝¹⁶ : Finite G
inst✝¹⁵ : Group G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : Algebra.IsInvariant A B G
P : Ideal A
Q : Ideal B
inst✝¹² : Q.LiesOver P
inst✝¹¹ : P.IsPrime
inst✝¹⁰ : Q.IsPrime
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra (A ⧸ P) K
inst✝⁵ : IsFractionRing (A ⧸ P) K
inst✝⁴ : Algebra (B ⧸ Q) L
inst✝³ : IsFractionRing (B ⧸ Q) L
inst✝² : Algebra (A ⧸ P) L
inst✝¹ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝ : IsScalarTower (A ⧸ P) K L
this✝ : Algebra.IsIntegral A B
this : Algebra.IsAlgebraic K L
x y : B ⧸ Q
hy : y ∈ nonZeroDivisors (B ⧸ Q)
a : A
ha : (algebraMap A (A ⧸ P)) a ≠ 0
b : B
h : (algebraMap A (A ⧸ P)) a • x = y * (algebraMap B (B ⧸ Q)) b
val✝ : Fintype G
p : A[X]
hp : Polynomial.map (algebraMap A B) p = MulSemiringAction.charpoly G b
h_monic : p.Monic
h_eval : (aeval b) p = 0
q : A[X] := p.comp (C a * X)
d : L := (algebraMap (B ⧸ Q) L) x / (algebraMap (B ⧸ Q) L) y
comm₁ : (algebraMap K L).comp (algebraMap (A ⧸ P) K) = (algebraMap (B ⧸ Q) L).comp (algebraMap (A ⧸ P) (B ⧸ Q))
comm₂ : (algebraMap (A ⧸ P) (B ⧸ Q)).comp (algebraMap A (A ⧸ P)) = (algebraMap B (B ⧸ Q)).comp (algebraMap A B)
⊢ (aeval d) (Polynomial.map (algebraMap (A ⧸ P) K) (Polynomial.map (algebraMap A (A ⧸ P)) q)) = 0

Tactic:
[

Goal state after:
A : Type u_1
B : Type u_2
inst✝¹⁹ : CommRing A
inst✝¹⁸ : CommRing B
inst✝¹⁷ : Algebra A B
G : Type u_3
inst✝¹⁶ : Finite G
inst✝¹⁵ : Group G
inst✝¹⁴ : MulSemiringAction G B
inst✝¹³ : Algebra.IsInvariant A B G
P : Ideal A
Q : Ideal B
inst✝¹² : Q.LiesOver P
inst✝¹¹ : P.IsPrime
inst✝¹⁰ : Q.IsPrime
K : Type u_4
L : Type u_5
inst✝⁹ : Field K
inst✝⁸ : Field L
inst✝⁷ : Algebra K L
inst✝⁶ : Algebra (A ⧸ P) K
inst✝⁵ : IsFractionRing (A ⧸ P) K
inst✝⁴ : Algebra (B ⧸ Q) L
inst✝³ : IsFractionRing (B ⧸ Q) L
inst✝² : Algebra (A ⧸ P) L
inst✝¹ : IsScalarTower (A ⧸ P) (B ⧸ Q) L
inst✝ : IsScalarTower (A ⧸ P) K L
this✝ : Algebra.IsIntegral A B
this : Algebra.IsAlgebraic K L
x y : B ⧸ Q
hy : y ∈ nonZeroDivisors (B ⧸ Q)
a : A
ha : (algebraMap A (A ⧸ P)) a ≠ 0
b : B
h : (algebraMap A (A ⧸ P)) a • x = y * (algebraMap B (B ⧸ Q)) b
val✝ : Fintype G
p : A[X]
hp : Polynomial.map (algebraMap A B) p = MulSemiringAction.charpoly G b
h_monic : p.Monic
h_eval : (aeval b) p = 0
q : A[X] := p.comp (C a * X)
d : L := (algebraMap (B ⧸ Q) L) x / (algebraMap (B ⧸ Q) L) y
comm₁ : (algebraMap K L).comp (algebraMap (A ⧸ P) K) = (algebraMap (B ⧸ Q) L).comp (algebraMap (A ⧸ P) (B ⧸ Q))
comm₂ : (algebraMap (A ⧸ P) (B ⧸ Q)).comp (algebraMap A (A ⧸ P)) = (algebraMap B (B ⧸ Q)).comp (algebraMap A B)
⊢ (aeval d) (Polynomial.map (algebraMap (A ⧸ P) K) (Polynomial.map (algebraMap A (A ⧸ P)) q)) = 0
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝ : Field K
A : Set (mixedSpace K)
hA : ∀ (x : mixedSpace K), x ∈ A ↔ (fun w ↦ ‖x.1 w‖, x.2) ∈ A
⊢ (⋃ s, ⇑(negAt s) '' plusPart A) ∪ A ∩ ⋃ w, {x | x.1 w = 0} = A

Tactic:
by
  ext x
  rw [Set.mem_union, Set.mem_inter_iff, Set.mem_iUnion, Set.mem_iUnion]
  refine ⟨?_, fun h ↦ ?_⟩
  · rintro (⟨s, ⟨x, ⟨hx, _⟩, rfl⟩⟩ | h)
    · simp_rw +singlePass [hA, negAt_apply_norm_isReal, negAt_apply_snd]
      rwa [← hA]
    · exact h.left
  · obtain hx | hx := exists_or_forall_not (fun w ↦ x.1 w = 0)
    · exact Or.inr ⟨h, hx⟩
    · refine Or.inl ⟨signSet x,
        (mem_negAt_plusPart_of_mem A hA h hx).mpr ⟨fun w hw ↦ ?_, fun w hw ↦ ?_⟩⟩
      · exact lt_of_le_of_ne hw (hx w)
      · exact lt_of_le_of_ne (lt_of_not_ge hw).le (Ne.symm (hx w))

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

Goal state before:
pp: case inl
K : Type u_1
inst✝ : Field K
x : mixedSpace K
w : InfinitePlace K
hw : w.IsReal
⊢ normAtAllPlaces (fun w ↦ ‖x.1 w‖, x.2) w = normAtAllPlaces x w

Tactic:
[

Goal state after:
case inl
K : Type u_1
inst✝ : Field K
x : mixedSpace K
w : InfinitePlace K
hw : w.IsReal
⊢ normAtAllPlaces (fun w ↦ ‖x.1 w‖, x.2) w = normAtAllPlaces x w
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: case inr
K : Type u_1
inst✝ : Field K
x : mixedSpace K
w : InfinitePlace K
hw : w.IsComplex
⊢ normAtAllPlaces (fun w ↦ ‖x.1 w‖, x.2) w = normAtAllPlaces x w

Tactic:
[

Goal state after:
case inr
K : Type u_1
inst✝ : Field K
x : mixedSpace K
w : InfinitePlace K
hw : w.IsComplex
⊢ normAtAllPlaces (fun w ↦ ‖x.1 w‖, x.2) w = normAtAllPlaces x w
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
N : ℕ
hK : |discr K| ≤ ↑N
⊢ boundOfDiscBdd N - 1 < boundOfDiscBdd N

Tactic:
[

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
N : ℕ
hK : |discr K| ≤ ↑N
⊢ boundOfDiscBdd N - 1 < boundOfDiscBdd N
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
N : ℕ
hK : |discr K| ≤ ↑N
a : 𝓞 K
h : ℚ⟮↑a⟯ = ⊤
⊢ (minpoly ℤ ↑a).natDegree ≤ rankOfDiscrBdd N

Tactic:
rw [Field.primitive_element_iff_minpoly_natDegree_eq,
    minpoly.isIntegrallyClosed_eq_field_fractions' ℚ a.isIntegral_coe,
    (minpoly.monic a.isIntegral_coe).natDegree_map] at h
  exact h.symm ▸ rank_le_rankOfDiscrBdd hK

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

Goal state before:
pp: 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}
hK₁ : {w | w.IsReal}.Nonempty
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
⊢ ∃
    x ∈
      ⋃ f,
        ⋃ (_ : f.natDegree ≤ D ∧ ∀ (i : ℕ), f.coeff i ∈ Set.Icc (-↑C) ↑C),
          ↑(Polynomial.map (algebraMap ℤ A) f).roots.toFinset,
    K = ℚ⟮x⟯

Tactic:
[

Goal state after:
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}
hK₁ : {w | w.IsReal}.Nonempty
hK₂ : |discr ↥↑⟨K, hK₀⟩| ≤ ↑N
⊢ ∃
    x ∈
      ⋃ f,
        ⋃ (_ : f.natDegree ≤ D ∧ ∀ (i : ℕ), f.coeff i ∈ Set.Icc (-↑C) ↑C),
          ↑(Polynomial.map (algebraMap ℤ A) f).roots.toFinset,
    K = ℚ⟮x⟯
Module: Mathlib.NumberTheory.NumberField.ClassNumber

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
J : ↥(Ideal (𝓞 K))⁰
hJ : ClassGroup.mk0 J = C⁻¹
a : 𝓞 K
ha : a ∈ ↑↑J
h_nz : (Algebra.linearMap (𝓞 K) K) a ≠ 0
h_nm :
  ↑|(Algebra.norm ℚ) ((Algebra.linearMap (𝓞 K) K) a)| ≤
    ↑(FractionalIdeal.absNorm ↑((FractionalIdeal.mk0 K) J)) * (4 / π) ^ nrComplexPlaces K * ↑(finrank ℚ K)! /
        ↑(finrank ℚ K) ^ finrank ℚ K *
      √|↑(discr K)|
⊢ ∃ I,
    ClassGroup.mk0 I = C ∧
      ↑(absNorm ↑I) ≤ (4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)

Tactic:
obtain ⟨I₀, hI⟩ := dvd_iff_le.mpr ((span_singleton_le_iff_mem J).mpr (by exact ha))

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
C : ClassGroup (𝓞 K)
J : ↥(Ideal (𝓞 K))⁰
hJ : ClassGroup.mk0 J = C⁻¹
a : 𝓞 K
ha : a ∈ ↑↑J
h_nz : (Algebra.linearMap (𝓞 K) K) a ≠ 0
h_nm :
  ↑|(Algebra.norm ℚ) ((Algebra.linearMap (𝓞 K) K) a)| ≤
    ↑(FractionalIdeal.absNorm ↑((FractionalIdeal.mk0 K) J)) * (4 / π) ^ nrComplexPlaces K * ↑(finrank ℚ K)! /
        ↑(finrank ℚ K) ^ finrank ℚ K *
      √|↑(discr K)|
I₀ : Ideal (𝓞 K)
hI : span {a} = ↑J * I₀
⊢ ∃ I,
    ClassGroup.mk0 I = C ∧
      ↑(absNorm ↑I) ≤ (4 / π) ^ nrComplexPlaces K * (↑(finrank ℚ K)! / ↑(finrank ℚ K) ^ finrank ℚ K * √|↑(discr K)|)