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

pp: K : Type u_1
inst✝¹ : Field K
f : InfinitePlace K → ℝ≥0
w₀ : { w // w.IsComplex }
inst✝ : NumberField K
B : ℝ≥0
⊢ NullMeasurableSet {x | |x.re| < 1 ∧ |x.im| < ↑B ^ 2} volume

Tactic:
]

Goal state after:
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

K : Type u_1
inst✝¹ : Field K
f : InfinitePlace K → ℝ≥0
w₀ : { w // w.IsComplex }
inst✝ : NumberField K
B : ℝ≥0
⊢ NullMeasurableSet {x | |x.re| < 1 ∧ |x.im| < ↑B ^ 2} volume
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
r : ℝ
h✝ : 0 ≤ r
x : K
⊢ ‖(canonicalEmbedding K) x‖ ≤ r ↔ ‖(canonicalEmbedding K) x‖ ≤ r

Tactic:
]

Goal state after:
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 (Set.Ioo (-1) 1 ×ˢ Set.Ioo (-↑B ^ 2) (↑B ^ 2)) = 4 * ↑B ^ 2

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
f : InfinitePlace K → ℝ≥0
w₀ : { w // w.IsComplex }
inst✝ : NumberField K
B : ℝ≥0
⊢ volume (Set.Ioo (-1) 1 ×ˢ Set.Ioo (-↑B ^ 2) (↑B ^ 2)) = 4 * ↑B ^ 2
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : Basis (K →+* ℂ) ℂ ((K →+* ℂ) → ℂ) := Pi.basisFun ℂ (K →+* ℂ)
e : (K →+* ℂ) ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := Fintype.equivOfCardEq ⋯
M : Matrix (K →+* ℂ) (K →+* ℂ) ℂ := B.toMatrix fun i ↦ (canonicalEmbedding K) ((integralBasis K) (e i))
this :
  (LinearIndependent ℂ fun i ↦ (canonicalEmbedding K) ((integralBasis K) (e i))) ∧
    Submodule.span ℂ (Set.range fun i ↦ (canonicalEmbedding K) ((integralBasis K) (e i))) = ⊤
⊢ Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℂ ((K →+* ℂ) → ℂ)

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : Basis (K →+* ℂ) ℂ ((K →+* ℂ) → ℂ) := Pi.basisFun ℂ (K →+* ℂ)
e : (K →+* ℂ) ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := Fintype.equivOfCardEq ⋯
M : Matrix (K →+* ℂ) (K →+* ℂ) ℂ := B.toMatrix fun i ↦ (canonicalEmbedding K) ((integralBasis K) (e i))
this :
  (LinearIndependent ℂ fun i ↦ (canonicalEmbedding K) ((integralBasis K) (e i))) ∧
    Submodule.span ℂ (Set.range fun i ↦ (canonicalEmbedding K) ((integralBasis K) (e i))) = ⊤
⊢ Basis (Free.ChooseBasisIndex ℤ (𝓞 K)) ℂ ((K →+* ℂ) → ℂ)
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : Basis (K →+* ℂ) ℂ ((K →+* ℂ) → ℂ) := Pi.basisFun ℂ (K →+* ℂ)
e : (K →+* ℂ) ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := Fintype.equivOfCardEq ⋯
M : Matrix (K →+* ℂ) (K →+* ℂ) ℂ := B.toMatrix fun i ↦ (canonicalEmbedding K) ((integralBasis K) (e i))
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ (fun i ↦ (integralBasis K) (e i)) (RingHom.equivRatAlgHom K ℂ)
⊢ N.transpose.det ≠ 0

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : Basis (K →+* ℂ) ℂ ((K →+* ℂ) → ℂ) := Pi.basisFun ℂ (K →+* ℂ)
e : (K →+* ℂ) ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := Fintype.equivOfCardEq ⋯
M : Matrix (K →+* ℂ) (K →+* ℂ) ℂ := B.toMatrix fun i ↦ (canonicalEmbedding K) ((integralBasis K) (e i))
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ (fun i ↦ (integralBasis K) (e i)) (RingHom.equivRatAlgHom K ℂ)
⊢ N.transpose.det ≠ 0
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : Basis (K →+* ℂ) ℂ ((K →+* ℂ) → ℂ) := Pi.basisFun ℂ (K →+* ℂ)
e : (K →+* ℂ) ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := Fintype.equivOfCardEq ⋯
M : Matrix (K →+* ℂ) (K →+* ℂ) ℂ := B.toMatrix fun i ↦ (canonicalEmbedding K) ((integralBasis K) (e i))
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ (fun i ↦ (integralBasis K) (e i)) (RingHom.equivRatAlgHom K ℂ)
⊢ N.det ^ 2 ≠ 0

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : Basis (K →+* ℂ) ℂ ((K →+* ℂ) → ℂ) := Pi.basisFun ℂ (K →+* ℂ)
e : (K →+* ℂ) ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := Fintype.equivOfCardEq ⋯
M : Matrix (K →+* ℂ) (K →+* ℂ) ℂ := B.toMatrix fun i ↦ (canonicalEmbedding K) ((integralBasis K) (e i))
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ (fun i ↦ (integralBasis K) (e i)) (RingHom.equivRatAlgHom K ℂ)
⊢ N.det ^ 2 ≠ 0
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
vol_box : ∀ (B : ℝ≥0), volume {x | |x.re| < 1 ∧ |x.im| < ↑B ^ 2} = 4 * ↑B ^ 2
⊢ (∏ x, ENNReal.ofReal (2 * ↑(f ↑x))) *
      ((∏ x ∈ Finset.univ.erase w₀,
          volume (if x = w₀ then {x_1 | |x_1.re| < 1 ∧ |x_1.im| < ↑(f ↑x) ^ 2} else ball 0 ↑(f ↑x))) *
        volume (if w₀ = w₀ then {x | |x.re| < 1 ∧ |x.im| < ↑(f ↑w₀) ^ 2} else ball 0 ↑(f ↑w₀))) =
    (∏ x, ENNReal.ofReal (2 * ↑(f ↑x))) *
      ((∏ x ∈ Finset.univ.erase w₀, ENNReal.ofReal ↑(f ↑x) ^ 2 * ↑pi) * (4 * ↑(f ↑w₀) ^ 2))

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
f : InfinitePlace K → ℝ≥0
w₀ : { w // w.IsComplex }
inst✝ : NumberField K
vol_box : ∀ (B : ℝ≥0), volume {x | |x.re| < 1 ∧ |x.im| < ↑B ^ 2} = 4 * ↑B ^ 2
⊢ (∏ x, ENNReal.ofReal (2 * ↑(f ↑x))) *
      ((∏ x ∈ Finset.univ.erase w₀,
          volume (if x = w₀ then {x_1 | |x_1.re| < 1 ∧ |x_1.im| < ↑(f ↑x) ^ 2} else ball 0 ↑(f ↑x))) *
        volume (if w₀ = w₀ then {x | |x.re| < 1 ∧ |x.im| < ↑(f ↑w₀) ^ 2} else ball 0 ↑(f ↑w₀))) =
    (∏ x, ENNReal.ofReal (2 * ↑(f ↑x))) *
      ((∏ x ∈ Finset.univ.erase w₀, ENNReal.ofReal ↑(f ↑x) ^ 2 * ↑pi) * (4 * ↑(f ↑w₀) ^ 2))
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : Basis (K →+* ℂ) ℂ ((K →+* ℂ) → ℂ) := Pi.basisFun ℂ (K →+* ℂ)
e : (K →+* ℂ) ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := Fintype.equivOfCardEq ⋯
M : Matrix (K →+* ℂ) (K →+* ℂ) ℂ := B.toMatrix fun i ↦ (canonicalEmbedding K) ((integralBasis K) (e i))
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ (fun i ↦ (integralBasis K) (e i)) (RingHom.equivRatAlgHom K ℂ)
⊢ N.det ^ 2 = (algebraMap ℚ ℂ) (Algebra.discr ℚ (⇑(integralBasis K) ∘ ⇑e.symm.symm))

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : Basis (K →+* ℂ) ℂ ((K →+* ℂ) → ℂ) := Pi.basisFun ℂ (K →+* ℂ)
e : (K →+* ℂ) ≃ Free.ChooseBasisIndex ℤ (𝓞 K) := Fintype.equivOfCardEq ⋯
M : Matrix (K →+* ℂ) (K →+* ℂ) ℂ := B.toMatrix fun i ↦ (canonicalEmbedding K) ((integralBasis K) (e i))
N : Matrix (K →+* ℂ) (K →+* ℂ) ℂ :=
  Algebra.embeddingsMatrixReindex ℚ ℂ (fun i ↦ (integralBasis K) (e i)) (RingHom.equivRatAlgHom K ℂ)
⊢ N.det ^ 2 = (algebraMap ℚ ℂ) (Algebra.discr ℚ (⇑(integralBasis K) ∘ ⇑e.symm.symm))
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
⊢ Set.range ⇑(latticeBasis K) = Set.range (⇑(canonicalEmbedding K).toIntAlgHom.toLinearMap ∘ ⇑(integralBasis K))

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (K →+* ℂ) → ℂ
⊢ Set.range ⇑(latticeBasis K) = Set.range (⇑(canonicalEmbedding K).toIntAlgHom.toLinearMap ∘ ⇑(integralBasis K))
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
⊢ x ∈ Submodule.span ℤ (⇑(canonicalEmbedding K).toIntAlgHom.toLinearMap '' Set.range ⇑(integralBasis K)) ↔
    x ∈ ((canonicalEmbedding K).comp (algebraMap (𝓞 K) K)).range

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (K →+* ℂ) → ℂ
⊢ x ∈ Submodule.span ℤ (⇑(canonicalEmbedding K).toIntAlgHom.toLinearMap '' Set.range ⇑(integralBasis K)) ↔
    x ∈ ((canonicalEmbedding K).comp (algebraMap (𝓞 K) K)).range
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
⊢ x ∈ ⇑(canonicalEmbedding K).toIntAlgHom.toLinearMap '' ↑(Submodule.span ℤ (Set.range ⇑(integralBasis K))) ↔
    x ∈ ((canonicalEmbedding K).comp (algebraMap (𝓞 K) K)).range

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (K →+* ℂ) → ℂ
⊢ x ∈ ⇑(canonicalEmbedding K).toIntAlgHom.toLinearMap '' ↑(Submodule.span ℤ (Set.range ⇑(integralBasis K))) ↔
    x ∈ ((canonicalEmbedding K).comp (algebraMap (𝓞 K) K)).range
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
⊢ (∃ x_1 ∈ ↑(Submodule.span ℤ (Set.range ⇑(integralBasis K))), (canonicalEmbedding K).toIntAlgHom.toLinearMap x_1 = x) ↔
    ∃ x_1 ∈ (algebraMap (𝓞 K) K).range, (canonicalEmbedding K) x_1 = x

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (K →+* ℂ) → ℂ
⊢ (∃ x_1 ∈ ↑(Submodule.span ℤ (Set.range ⇑(integralBasis K))), (canonicalEmbedding K).toIntAlgHom.toLinearMap x_1 = x) ↔
    ∃ x_1 ∈ (algebraMap (𝓞 K) K).range, (canonicalEmbedding K) x_1 = x
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
⊢ ∑ x_1, (canonicalEmbedding K) (((integralBasis K).repr x) x_1 • (integralBasis K) x_1) ∈
    Submodule.span ℚ (Set.range ⇑(latticeBasis K))

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
⊢ ∑ x_1, (canonicalEmbedding K) (((integralBasis K).repr x) x_1 • (integralBasis K) x_1) ∈
    Submodule.span ℚ (Set.range ⇑(latticeBasis 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 : Free.ChooseBasisIndex ℤ (𝓞 K)
x✝ : i ∈ Finset.univ
⊢ (latticeBasis K) i ∈ Set.range ⇑(latticeBasis K)

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i : Free.ChooseBasisIndex ℤ (𝓞 K)
x✝ : i ∈ Finset.univ
⊢ (latticeBasis K) i ∈ Set.range ⇑(latticeBasis 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 : Free.ChooseBasisIndex ℤ (𝓞 K)
⊢ ((Basis.restrictScalars ℚ (latticeBasis K)).repr ⟨(canonicalEmbedding K) x, ⋯⟩) i = ((integralBasis K).repr x) i

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : K
i : Free.ChooseBasisIndex ℤ (𝓞 K)
⊢ ((Basis.restrictScalars ℚ (latticeBasis K)).repr ⟨(canonicalEmbedding K) x, ⋯⟩) i = ((integralBasis K).repr x) i
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: case e_a.e_a
K : Type u_1
inst✝¹ : Field K
f : InfinitePlace K → ℝ≥0
w₀ : { w // w.IsComplex }
inst✝ : NumberField K
vol_box : ∀ (B : ℝ≥0), volume {x | |x.re| < 1 ∧ |x.im| < ↑B ^ 2} = 4 * ↑B ^ 2
w' : { w // w.IsComplex }
hw' : w' ∈ Finset.univ.erase w₀
⊢ ENNReal.ofReal ↑(f ↑w') ^ 2 * ↑pi = ENNReal.ofReal ↑(f ↑w') ^ 2 * ↑pi

Tactic:
]

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
F : Type u_3
inst✝ : Field F
w : InfinitePlace K
f : k →+* K
x : k
⊢ (place (w.embedding.comp f)) x = (place ⋯.choose) (f x)

Tactic:
]

Goal state after:
k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
F : Type u_3
inst✝ : Field F
w : InfinitePlace K
f : k →+* K
x : k
⊢ (place (w.embedding.comp f)) x = (place ⋯.choose) (f x)
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝¹ : Field k
K : Type u_2
inst✝ : Field K
f : k →+* K
w : InfinitePlace K
hφ : w.IsReal
⊢ ComplexEmbedding.IsReal (w.embedding.comp f)

Tactic:
]

Goal state after:
k : Type u_1
inst✝¹ : Field k
K : Type u_2
inst✝ : Field K
f : k →+* K
w : InfinitePlace K
hφ : w.IsReal
⊢ ComplexEmbedding.IsReal (w.embedding.comp f)
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝¹ : Field k
K : Type u_2
inst✝ : Field K
f : k →+* K
w : InfinitePlace K
hφ : ComplexEmbedding.IsReal w.embedding
⊢ ComplexEmbedding.IsReal (w.embedding.comp f)

Tactic:
]

Goal state after:
k : Type u_1
inst✝¹ : Field k
K : Type u_2
inst✝ : Field K
f : k →+* K
w : InfinitePlace K
hφ : ComplexEmbedding.IsReal w.embedding
⊢ ComplexEmbedding.IsReal (w.embedding.comp f)
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝¹ : Field k
K : Type u_2
inst✝ : Field K
f : k →+* K
w : InfinitePlace K
hf : ¬(w.comap f).IsReal
⊢ ¬w.IsReal

Tactic:
]

Goal state after:
k : Type u_1
inst✝¹ : Field k
K : Type u_2
inst✝ : Field K
f : k →+* K
w : InfinitePlace K
hf : ¬(w.comap f).IsReal
⊢ ¬w.IsReal
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)
⊢ (canonicalEmbedding K).toRatAlgHom ((integralBasis K) i) = (canonicalEmbedding K) ((integralBasis K) i)

Tactic:
]

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)
⊢ (canonicalEmbedding K).toRatAlgHom ((integralBasis K) i) = (canonicalEmbedding K) ((integralBasis K) i)
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝¹ : Field k
K : Type u_2
inst✝ : Field K
f : k ≃+* K
w : InfinitePlace K
⊢ ComplexEmbedding.IsReal w.embedding ↔ ComplexEmbedding.IsReal w.embedding

Tactic:
]

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

Goal state before:
pp: k : Type u_1
inst✝¹ : Field k
K : Type u_2
inst✝ : Field K
f : k →+* K
w : InfinitePlace K
h : (w.comap f).IsReal
⊢ (w.comap f).IsReal

Tactic:
]

Goal state after:
k : Type u_1
inst✝¹ : Field k
K : Type u_2
inst✝ : Field K
f : k →+* K
w : InfinitePlace K
h : (w.comap f).IsReal
⊢ (w.comap f).IsReal
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝¹ : Field k
K : Type u_2
inst✝ : Field K
f : k →+* K
w : InfinitePlace K
h : (w.comap f).IsReal
⊢ w.embedding.comp f = w.embedding.comp f

Tactic:
]

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

Goal state before:
pp: k : Type u_1
inst✝¹ : Field k
K : Type u_2
inst✝ : Field K
f : k →+* K
w : InfinitePlace K
⊢ (if (w.comap f).IsReal then 1 else 2) ≤ if w.IsReal then 1 else 2

Tactic:
]

Goal state after:
k : Type u_1
inst✝¹ : Field k
K : Type u_2
inst✝ : Field K
f : k →+* K
w : InfinitePlace K
⊢ (if (w.comap f).IsReal then 1 else 2) ≤ if w.IsReal then 1 else 2
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
σ : Gal(K/k)
w : InfinitePlace K
⊢ ¬w.IsReal ↔ ¬w.IsReal

Tactic:
]

Goal state after:
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
w w' : InfinitePlace K
h :
  w.embedding.comp (algebraMap k K) = w'.embedding.comp (algebraMap k K) ∨
    conjugate (w.embedding.comp (algebraMap k K)) = w'.embedding.comp (algebraMap k K)
⊢ ∃ σ, σ • w = w'

Tactic:
]

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
w w' : InfinitePlace K
h :
  w.embedding.comp (algebraMap k K) = w'.embedding.comp (algebraMap k K) ∨
    conjugate (w.embedding.comp (algebraMap k K)) = w'.embedding.comp (algebraMap k K)
⊢ ∃ σ, σ • w = w'
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: case h
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
w w' : InfinitePlace K
h : w.embedding.comp (algebraMap k K) = w'.embedding.comp (algebraMap k K)
σ : Gal(K/k)
hσ : w.embedding.comp ↑σ.symm = w'.embedding
⊢ mk w'.embedding = mk w'.embedding

Tactic:
]

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

Goal state before:
pp: case h
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
w w' : InfinitePlace K
h : conjugate (w.embedding.comp (algebraMap k K)) = w'.embedding.comp (algebraMap k K)
σ : Gal(K/k)
hσ : ((starRingEnd ℂ).comp w.embedding).comp ↑σ.symm = w'.embedding
⊢ w.embedding.comp ↑σ.symm = w'.embedding ∨ conjugate (w.embedding.comp ↑σ.symm) = w'.embedding

Tactic:
]

Goal state after:
case h
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
w w' : InfinitePlace K
h : conjugate (w.embedding.comp (algebraMap k K)) = w'.embedding.comp (algebraMap k K)
σ : Gal(K/k)
hσ : ((starRingEnd ℂ).comp w.embedding).comp ↑σ.symm = w'.embedding
⊢ w.embedding.comp ↑σ.symm = w'.embedding ∨ conjugate (w.embedding.comp ↑σ.symm) = w'.embedding
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ Fintype.card (K →+* ℂ) = Fintype.card (K →+* ℂ)

Tactic:
]

Goal state after:
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
w : InfinitePlace K
σ : Gal(K/k)
⊢ mk (w.embedding.comp (algebraMap k K)) = mk ((w.embedding.comp ↑σ.symm).comp (algebraMap k K))

Tactic:
]

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
w : InfinitePlace K
σ : Gal(K/k)
⊢ mk (w.embedding.comp (algebraMap k K)) = mk ((w.embedding.comp ↑σ.symm).comp (algebraMap k 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
⊢ w.mult ≤ (w.comap (algebraMap k K)).mult ↔ w.mult ≤ (w.comap (algebraMap k K)).mult

pp: k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra k K
w : InfinitePlace K
⊢ (w.comap (algebraMap k K)).mult ≤ w.mult

Tactic:
]

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
⊢ (w.comap (algebraMap k K)).mult ≤ w.mult
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝⁴ : Field k
K : Type u_2
inst✝³ : Field K
F : Type u_3
inst✝² : Field F
inst✝¹ : Algebra k K
inst✝ : Algebra k F
w : InfinitePlace F
h : IsUnramified k w
f : K →ₐ[k] F
⊢ (w.comap ↑f).mult ≤ w.mult

Tactic:
]

Goal state after:
k : Type u_1
inst✝⁴ : Field k
K : Type u_2
inst✝³ : Field K
F : Type u_3
inst✝² : Field F
inst✝¹ : Algebra k K
inst✝ : Algebra k F
w : InfinitePlace F
h : IsUnramified k w
f : K →ₐ[k] F
⊢ (w.comap ↑f).mult ≤ w.mult
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: k : Type u_1
inst✝⁶ : Field k
K : Type u_2
inst✝⁵ : Field K
F : Type u_3
inst✝⁴ : Field F
inst✝³ : Algebra k K
inst✝² : Algebra k F
inst✝¹ : Algebra K F
inst✝ : IsScalarTower k K F
w : InfinitePlace F
h : IsUnramified k w
⊢ ((w.comap (algebraMap K F)).comap (algebraMap k K)).mult ≤ (w.comap (algebraMap K F)).mult

Tactic:
]

Goal state after:
k : Type u_1
inst✝⁶ : Field k
K : Type u_2
inst✝⁵ : Field K
F : Type u_3
inst✝⁴ : Field F
inst✝³ : Algebra k K
inst✝² : Algebra k F
inst✝¹ : Algebra K F
inst✝ : IsScalarTower k K F
w : InfinitePlace F
h : IsUnramified k w
⊢ ((w.comap (algebraMap K F)).comap (algebraMap k K)).mult ≤ (w.comap (algebraMap K F)).mult
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
⊢ (¬(if (w.comap (algebraMap k K)).IsReal then 1 else 2) = if w.IsReal then 1 else 2) ↔
    ¬w.IsReal ∧ (w.comap (algebraMap k K)).IsReal

Tactic:
]

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
⊢ (¬(if (w.comap (algebraMap k K)).IsReal then 1 else 2) = if w.IsReal then 1 else 2) ↔
    ¬w.IsReal ∧ (w.comap (algebraMap k K)).IsReal
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
⊢ w.IsComplex ∧ (w.comap (algebraMap k K)).IsReal ↔ w.IsComplex ∧ (w.comap (algebraMap k K)).IsReal

Tactic:
]

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

Goal state before:
pp: case pos
k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra k K
w₁ w₂ : InfinitePlace K
h : ¬ComplexEmbedding.IsReal w₂.embedding ∧ (w₂.comap (algebraMap k K)).IsReal
h_eq : w₁ = w₂
⊢ w₁.embedding ≠ conjugate w₂.embedding

Tactic:
]

Goal state after:
case pos
k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra k K
w₁ w₂ : InfinitePlace K
h : ¬ComplexEmbedding.IsReal w₂.embedding ∧ (w₂.comap (algebraMap k K)).IsReal
h_eq : w₁ = w₂
⊢ w₁.embedding ≠ conjugate w₂.embedding
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
vol_box : ∀ (B : ℝ≥0), volume {x | |x.re| < 1 ∧ |x.im| < ↑B ^ 2} = 4 * ↑B ^ 2
⊢ (↑2 ^ nrRealPlaces K * ∏ x, ENNReal.ofReal ↑(f ↑x)) *
      ((∏ x ∈ Finset.univ.erase w₀, ENNReal.ofReal ↑(f ↑x) ^ 2) * ↑pi ^ (nrComplexPlaces K - 1) *
        (↑2 ^ 2 * ↑(f ↑w₀) ^ 2)) =
    (↑(2 ^ nrRealPlaces K * 2 ^ 2 * pi ^ (nrComplexPlaces K - 1)) * ∏ x, ENNReal.ofReal ↑(f ↑x)) *
      ((∏ x ∈ Finset.univ.erase w₀, ENNReal.ofReal ↑(f ↑x) ^ 2) * ↑(f ↑w₀) ^ 2)

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
f : InfinitePlace K → ℝ≥0
w₀ : { w // w.IsComplex }
inst✝ : NumberField K
vol_box : ∀ (B : ℝ≥0), volume {x | |x.re| < 1 ∧ |x.im| < ↑B ^ 2} = 4 * ↑B ^ 2
⊢ (↑2 ^ nrRealPlaces K * ∏ x, ENNReal.ofReal ↑(f ↑x)) *
      ((∏ x ∈ Finset.univ.erase w₀, ENNReal.ofReal ↑(f ↑x) ^ 2) * ↑pi ^ (nrComplexPlaces K - 1) *
        (↑2 ^ 2 * ↑(f ↑w₀) ^ 2)) =
    (↑(2 ^ nrRealPlaces K * 2 ^ 2 * pi ^ (nrComplexPlaces K - 1)) * ∏ x, ENNReal.ofReal ↑(f ↑x)) *
      ((∏ x ∈ Finset.univ.erase w₀, ENNReal.ofReal ↑(f ↑x) ^ 2) * ↑(f ↑w₀) ^ 2)
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: case neg
k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra k K
w₁ w₂ : InfinitePlace K
h : IsRamified k w₂
h_eq : w₁.embedding = conjugate w₂.embedding
⊢ w₂ = w₂

Tactic:
]

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 →+* ℂ) → ℂ
h_zero : (commMap K) x = 0
h_mem : x ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))
φ : K →+* ℂ
⊢ x φ = 0

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : (K →+* ℂ) → ℂ
h_zero : (commMap K) x = 0
h_mem : x ∈ Submodule.span ℝ (Set.range ⇑(canonicalEmbedding K))
φ : K →+* ℂ
⊢ x φ = 0