mathlib-initiative/mathlib-tactics huggingface dataset:
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, (mixedEmbedding 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, (mixedEmbedding K) (((integralBasis K).repr x) x_1 • (integralBasis K) x_1) ∈
    Submodule.span ℚ (Set.range ⇑(latticeBasis K))
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
⊢ minkowskiBound K 1 <
    ↑(convexBodyLT'Factor K) *
      ↑((if w₀ ∈ Finset.univ.erase w₀ then NNReal.sqrt B ^ w₀.mult else 1) *
          if True then NNReal.sqrt B ^ w₀.mult else 1)

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
⊢ minkowskiBound K 1 <
    ↑(convexBodyLT'Factor K) *
      ↑((if w₀ ∈ Finset.univ.erase w₀ then NNReal.sqrt B ^ w₀.mult else 1) *
          if True then NNReal.sqrt B ^ w₀.mult else 1)
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)
x✝ : i ∈ 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 : ChooseBasisIndex ℤ (𝓞 K)
x✝ : i ∈ univ
⊢ (latticeBasis K) i ∈ Set.range ⇑(latticeBasis K)
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

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

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
i : ChooseBasisIndex ℤ (𝓞 K)
⊢ ((Basis.restrictScalars ℚ (latticeBasis K)).repr ⟨(mixedEmbedding 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 : ChooseBasisIndex ℤ (𝓞 K)
⊢ ((Basis.restrictScalars ℚ (latticeBasis K)).repr ⟨(mixedEmbedding K) x, ⋯⟩) i = ((integralBasis K).repr x) i
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:
]

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: 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)
⊢ (mixedEmbedding K).toRatAlgHom ((integralBasis K) i) = (mixedEmbedding K) ((integralBasis K) i)

Tactic:
]

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)
⊢ (mixedEmbedding K).toRatAlgHom ((integralBasis K) i) = (mixedEmbedding K) ((integralBasis K) i)
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: case pos
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w ↦ if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ : |((↑⟨w₀, hw₀⟩).embedding ↑a).re| < 1 ∧ |((↑⟨w₀, hw₀⟩).embedding ↑a).im| < ↑(NNReal.sqrt B) ^ 2
w : InfinitePlace K
h_eq : w = w₀
⊢ w ↑a < √(1 + ↑B ^ 2)

Tactic:
]

Goal state after:
case pos
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w ↦ if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ : |((↑⟨w₀, hw₀⟩).embedding ↑a).re| < 1 ∧ |((↑⟨w₀, hw₀⟩).embedding ↑a).im| < ↑(NNReal.sqrt B) ^ 2
w : InfinitePlace K
h_eq : w = w₀
⊢ w ↑a < √(1 + ↑B ^ 2)
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: K : Type u_4
L : Type u_5
inst✝² : Field K
inst✝¹ : Field L
inst✝ : Algebra K L
v : InfinitePlace K
⊢ (unmixedEmbeddingsOver L v.embedding).ncard = (unmixedEmbeddingsOver L v.embedding).ncard

Tactic:
]

Goal state after:
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
this : Algebra K ℂ := v.embedding.toAlgebra
⊢ #⋯.toFinset = Fintype.card (L →ₐ[K] ℂ)

Tactic:
]

Goal state after:
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
this : Algebra K ℂ := v.embedding.toAlgebra
⊢ #⋯.toFinset = Fintype.card (L →ₐ[K] ℂ)
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I
this :
  (latticeBasis K).det (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e) =
    (algebraMap ℚ ℝ) ((integralBasis K).det (⇑(basisOfFractionalIdeal K I) ∘ ⇑e))
⊢ ↑(FractionalIdeal.absNorm ↑I) = ↑(FractionalIdeal.absNorm ↑I)

Tactic:
]

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

Goal state before:
pp: case pos
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w ↦ if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ : |(w₀.embedding ↑a).re| < 1 ∧ |(w₀.embedding ↑a).im| < ↑(NNReal.sqrt B) ^ 2
w : InfinitePlace K
h_eq : w = w₀
⊢ (w₀.embedding ↑a).re ^ 2 + (w₀.embedding ↑a).im ^ 2 < 1 + ↑B ^ 2

Tactic:
]

Goal state after:
case pos
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w ↦ if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ : |(w₀.embedding ↑a).re| < 1 ∧ |(w₀.embedding ↑a).im| < ↑(NNReal.sqrt B) ^ 2
w : InfinitePlace K
h_eq : w = w₀
⊢ (w₀.embedding ↑a).re ^ 2 + (w₀.embedding ↑a).im ^ 2 < 1 + ↑B ^ 2
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I
⊢ ((latticeBasis K).toMatrix (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e)).det =
    ((algebraMap ℚ ℝ).mapMatrix ((integralBasis K).toMatrix (⇑(basisOfFractionalIdeal K I) ∘ ⇑e))).det

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I
⊢ ((latticeBasis K).toMatrix (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e)).det =
    ((algebraMap ℚ ℝ).mapMatrix ((integralBasis K).toMatrix (⇑(basisOfFractionalIdeal K I) ∘ ⇑e))).det
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: case pos.refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w ↦ if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ : |(w₀.embedding ↑a).re| < 1 ∧ |(w₀.embedding ↑a).im| < ↑(NNReal.sqrt B) ^ 2
w : InfinitePlace K
h_eq : w = w₀
⊢ |(w₀.embedding ↑a).re| < 1

Tactic:
]

Goal state after:
case pos.refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w ↦ if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ : |(w₀.embedding ↑a).re| < 1 ∧ |(w₀.embedding ↑a).im| < ↑(NNReal.sqrt B) ^ 2
w : InfinitePlace K
h_eq : w = w₀
⊢ |(w₀.embedding ↑a).re| < 1
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
⊢ finrank ℤ (𝓞 K) = finrank ℤ (𝓞 K)

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
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I := Fintype.equivOfCardEq ⋯
this :
  LinearIndependent ℝ (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e) ∧
    Submodule.span ℝ (Set.range (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e)) = ⊤
⊢ Basis (ChooseBasisIndex ℤ (𝓞 K)) ℝ (mixedSpace K)

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I := Fintype.equivOfCardEq ⋯
this :
  LinearIndependent ℝ (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e) ∧
    Submodule.span ℝ (Set.range (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e)) = ⊤
⊢ Basis (ChooseBasisIndex ℤ (𝓞 K)) ℝ (mixedSpace K)
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: case pos.refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w ↦ if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ : |(w₀.embedding ↑a).re| < 1 ∧ |(w₀.embedding ↑a).im| < ↑(NNReal.sqrt B) ^ 2
w : InfinitePlace K
h_eq : w = w₀
⊢ |(w₀.embedding ↑a).im| < ↑(NNReal.sqrt B ^ 2)

Tactic:
]

Goal state after:
case pos.refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w ↦ if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ : |(w₀.embedding ↑a).re| < 1 ∧ |(w₀.embedding ↑a).im| < ↑(NNReal.sqrt B) ^ 2
w : InfinitePlace K
h_eq : w = w₀
⊢ |(w₀.embedding ↑a).im| < ↑(NNReal.sqrt B ^ 2)
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I := Fintype.equivOfCardEq ⋯
this :
  LinearIndependent ℝ (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e) ∧
    Submodule.span ℝ (Set.range (⇑(mixedEmbedding K) ∘ ⇑(basisOfFractionalIdeal K I) ∘ ⇑e)) = ⊤
⊢ ⊤ ≤ ⊤

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
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I := Fintype.equivOfCardEq ⋯
⊢ ¬↑I = 0

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
e : ChooseBasisIndex ℤ (𝓞 K) ≃ ChooseBasisIndex ℤ ↥↑↑I := Fintype.equivOfCardEq ⋯
⊢ ¬↑I = 0
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
x : mixedSpace K
⊢ Set.range ⇑(fractionalIdealLatticeBasis K I) =
    Set.range (⇑(mixedEmbedding K).toIntAlgHom.toLinearMap ∘ ⇑(basisOfFractionalIdeal K I))

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
x : mixedSpace K
⊢ Set.range ⇑(fractionalIdealLatticeBasis K I) =
    Set.range (⇑(mixedEmbedding K).toIntAlgHom.toLinearMap ∘ ⇑(basisOfFractionalIdeal K I))
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
x : mixedSpace K
⊢ x ∈ Submodule.span ℤ (⇑(mixedEmbedding K).toIntAlgHom.toLinearMap '' Set.range ⇑(basisOfFractionalIdeal K I)) ↔
    x ∈ ⇑(mixedEmbedding K) '' ↑↑I

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
x : mixedSpace K
⊢ x ∈ Submodule.span ℤ (⇑(mixedEmbedding K).toIntAlgHom.toLinearMap '' Set.range ⇑(basisOfFractionalIdeal K I)) ↔
    x ∈ ⇑(mixedEmbedding K) '' ↑↑I
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
x : mixedSpace K
⊢ x ∈ ⇑(mixedEmbedding K).toIntAlgHom.toLinearMap '' ↑(Submodule.span ℤ (Set.range ⇑(basisOfFractionalIdeal K I))) ↔
    x ∈ ⇑(mixedEmbedding K) '' ↑↑I

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
x : mixedSpace K
⊢ x ∈ ⇑(mixedEmbedding K).toIntAlgHom.toLinearMap '' ↑(Submodule.span ℤ (Set.range ⇑(basisOfFractionalIdeal K I))) ↔
    x ∈ ⇑(mixedEmbedding K) '' ↑↑I
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
x : mixedSpace K
⊢ x ∈ ⇑(mixedEmbedding K).toIntAlgHom.toLinearMap '' ↑↑I ↔ x ∈ ⇑(mixedEmbedding K) '' ↑↑I

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
x : mixedSpace K
⊢ x ∈ ⇑(mixedEmbedding K).toIntAlgHom.toLinearMap '' ↑↑I ↔ x ∈ ⇑(mixedEmbedding K) '' ↑↑I
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: case e_a.e_a.e_a.e_a.e_a
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
⊢ ↑(nrRealPlaces K) + 2 * ↑(nrComplexPlaces K) - ↑(nrComplexPlaces K) - ↑(nrRealPlaces K) + ↑(nrComplexPlaces K) =
    2 * ↑(nrComplexPlaces K)

Tactic:
]

Goal state after:
case e_a.e_a.e_a.e_a.e_a
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
⊢ ↑(nrRealPlaces K) + 2 * ↑(nrComplexPlaces K) - ↑(nrComplexPlaces K) - ↑(nrRealPlaces K) + ↑(nrComplexPlaces K) =
    2 * ↑(nrComplexPlaces K)
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w ↦ if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ :
  |((↑⟨w₀, hw₀⟩).embedding ↑a).re| < 1 ∧
    |((↑⟨w₀, hw₀⟩).embedding ↑a).im| < ↑(if ↑⟨w₀, hw₀⟩ = w₀ then NNReal.sqrt B else 1) ^ 2
w : InfinitePlace K
h_eq : ¬w = w₀
⊢ 1 ≤ 1 + ↑B ^ 2

Tactic:
]

Goal state after:
case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w₀ : InfinitePlace K
hw₀ : w₀.IsComplex
B : ℝ≥0
hB : minkowskiBound K 1 < ↑(convexBodyLT'Factor K) * ↑B
this : minkowskiBound K 1 < volume (convexBodyLT' K (fun w ↦ if w = w₀ then NNReal.sqrt B else 1) ⟨w₀, hw₀⟩)
a : 𝓞 K
h_nz : a ≠ 0
h_le : ∀ (w : InfinitePlace K), w ≠ ↑⟨w₀, hw₀⟩ → w ↑a < ↑(if w = w₀ then NNReal.sqrt B else 1)
h_le₀ :
  |((↑⟨w₀, hw₀⟩).embedding ↑a).re| < 1 ∧
    |((↑⟨w₀, hw₀⟩).embedding ↑a).im| < ↑(if ↑⟨w₀, hw₀⟩ = w₀ then NNReal.sqrt B else 1) ^ 2
w : InfinitePlace K
h_eq : ¬w = w₀
⊢ 1 ≤ 1 + ↑B ^ 2
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
⊢ DiscreteTopology ↥(Submodule.span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I)))

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
⊢ DiscreteTopology ↥(Submodule.span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I)))
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
⊢ MeasureTheory.IsAddFundamentalDomain (↥(Submodule.span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))))
    (ZSpan.fundamentalDomain (fractionalIdealLatticeBasis K I)) MeasureTheory.volume

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
⊢ MeasureTheory.IsAddFundamentalDomain (↥(Submodule.span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))))
    (ZSpan.fundamentalDomain (fractionalIdealLatticeBasis K I)) MeasureTheory.volume
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ finrank ℚ K = finrank ℚ K

Tactic:
]

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

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ
h : B < 0
⊢ 0 < minkowskiBound K I

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
I : (FractionalIdeal (𝓞 K)⁰ K)ˣ
B : ℝ
h : B < 0
⊢ 0 < minkowskiBound K I
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ 1 = 1

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
⊢ DiscreteTopology ↥(ZLattice.comap ℝ (mixedEmbedding.integerLattice K) ↑↑(toMixed K))

Tactic:
]

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
⊢ DiscreteTopology ↥(ZLattice.comap ℝ (mixedEmbedding.integerLattice K) ↑↑(toMixed 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
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) * 4 ^ nrComplexPlaces K * √|↑(discr K)| * ↑(finrank ℚ K).factorial *
      (1 / π ^ nrComplexPlaces K) =
    ↑(FractionalIdeal.absNorm ↑I) * (4 ^ nrComplexPlaces K / π ^ nrComplexPlaces K) * ↑(finrank ℚ K).factorial *
      √|↑(discr K)|

Tactic:
]

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) * 4 ^ nrComplexPlaces K * √|↑(discr K)| * ↑(finrank ℚ K).factorial *
      (1 / π ^ nrComplexPlaces K) =
    ↑(FractionalIdeal.absNorm ↑I) * (4 ^ nrComplexPlaces K / π ^ nrComplexPlaces K) * ↑(finrank ℚ K).factorial *
      √|↑(discr K)|
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: 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
x : mixedSpace K
hx✝ : x ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
hx : x ∈ ⇑(mixedEmbedding K) '' ↑↑I
h_nz : ⟨x, hx✝⟩ ≠ 0
h_mem : ↑⟨x, hx✝⟩ ∈ convexBodySum K B
⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ↑|(Algebra.norm ℚ) a| ≤ (B / ↑(finrank ℚ K)) ^ finrank ℚ K

Tactic:
]

Goal state after:
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
x : mixedSpace K
hx✝ : x ∈ (span ℤ (Set.range ⇑(fractionalIdealLatticeBasis K I))).toAddSubgroup
hx : x ∈ ⇑(mixedEmbedding K) '' ↑↑I
h_nz : ⟨x, hx✝⟩ ≠ 0
h_mem : ↑⟨x, hx✝⟩ ∈ convexBodySum K B
⊢ ∃ a ∈ ↑I, a ≠ 0 ∧ ↑|(Algebra.norm ℚ) a| ≤ (B / ↑(finrank ℚ K)) ^ finrank ℚ K